{"Bibliographic":{"Title":"Hydrologic optics. Volume VI: Surfaces","Authors":"","Publication date":"1976","Publisher":""},"Administrative":{"Date created":"08-17-2023","Language":"English","Rights":"CC 0","Size":"0000721150"},"Pages":["HYDROLOGIC OPTICS\nVolume VI. Surfaces\nR.W. PREISENDORFER\nU.S. DEPARTMENT OF COMMERCE\nNATIONAL OCEANIC & ATMOSPHERIC ADMINISTRATION\nENVIRONMENTAL RESEARCH LABORATORIES\nHONOLULU, HAWAII\n1976","GB\n665\nP645\nV.6\nOF COMMUNITY\nHYDROLOGIC OPTICS\nWITH STATES OF Avenue\nVolume VI. Surfaces\nR.W. Preisendorfer\nJoint Tsunami Research Effort\nHonolulu, Hawaii\n1976\nATMOSPHERIC SCIENCES\nLIBRARY\nJUN 1 1977\nN.O.A.A.\nU. S. Dept. of Commerce\nU.S. DEPARTMENT OF COMMERCE\nNational Oceanic and Atmospheric Administration\nEnvironmental Research Laboratories\nPacific Marine Environmental Laboratory\n77\n1297","ii\nFrom a planetary point of view the ocean's\nsingle most important aspect is neither its\ndepth nor its volume but its surface: the\nlargest and most homogeneous environment on\nthe planet.\nFerren MacIntyre\nThe Top Millimeter of the Ocean\nScientific American, May 1974\np. 62-77\nWe are now descending toward our watery\nworld with its star directly behind us.\nFrom this great height the disk of the\nplanet is still featureless and its clouds\nsuffuse it with a marbled blue-white glow.\nDirectly below us on the ocean surface, as\nseen through the scattered clouds, is a\ndazzling reflection of the planet's sun.\nI marvel at the enormous extent of this\nreflected pattern and its marked oval\nshape running in a northeast-southwes1 -\ndirection. This means strong trade winds\nover the ocean and the neighboring coast-\nline. As we descend, the crew's spirit\nvisibly rises. It's good to be home again.\nKarl Ostheimer\nThe Ages of Arkadia","CONTENTS\niii\nVOLUME VI\nChapter 12\nOptical Properties of the Air-Water Surface\n12.0\nIntroduction\n1\n12.1\nReflectance and Transmittance Properties of the\nStatic Surface\n3\nThe Geometric Law of Reflection\n3\nThe Geometric Law of Refraction\n5\nThe Fresnel Laws for Reflectance\n10\nThe Fresnel Laws for Transmittance\n15\nExample 1: Reflectance Under Uniform\nRadiance Distributions\n16\nExample 2: Reflectance Under Cardicidal\nRadiance Distributions\n21\nExample 3: Reflectance Under Zonal Radiance\nDistributions\n28\n12.2\nRadiative Transfer and the Static Surface\n34\nIrradiance Interaction Between the Surface\nand the Hydrosol\n35\nThe Threefold Irradiance Interaction: Aerosol,\nAir-Water Surface, and Hydrosol\n38\nThe Threefold Radiance Interaction: for the\nStatic Surface\n39\nContrast Transmittance Formulas for the Static\nSurface\n41\nContrast Transmittance Formulas for Extended\nPaths Across the Static Air-Water Surface\n44\n12.3\nElementary Hydrodynamics of the Air-Water\nSurface\n46\nThe Fluid Transfer Process\n46\nPhysics of the Fluid Transfer Process\n47\nGeneral Equations of Motion of a Fluid\n49\nSpecial Equations of Motion for the Air and\nWater Masses\n51\nSurface Kinematic Condition\n55\nSurface Pressure Condition\n57\nSinusoidal Wave Forms\n60\nLinearized Equations of Motion\n62\nClassical Wave Model\n63\nKelvin-Helmholtz Model\n67","CONTENTS\niv\nElementary Hydrodynamics of the Air-Water\n12.3\nSurface--Continued\n70\nKelvin-Helmholtz Instability\n71\nCapillary and Gravity Waves\n72\nEnergy of Surface Waves\n76\nSuperposition of Waves\nSpectrum of the Air-Water Surface\n78\nHarmonic Analyses of the Dynamic Air-Water\n12.4\n82\nSurface\nThe Roots of Harmonic Analysis\n83\nHarmonic Synthesis vs. Harmonic Analysis\n84\nIntegrals vs. Series in Harmonic Analysis\n86\nFourier Series Representations of the\n87\nAir-Water Surface\nHydrodynamic Basis for Harmonic Analysis\n91\nof Air-Water Surfaces\nThe Periodogram Basis of the Energy Spectrum\n94\nFourier Integral Representations of the\nAir-Water Surface. Case 1: The Surface\n101\nis Aperiodic\nFourier Integral Representations of the\nAir-Water Surface. Case 2: The Surface\nis Periodic or Random\n109\nA Working Representation of the Dynamic\nAir-Water Surface and its Directional\n115\nEnergy Spectrum\nGeometrical Applications of the Directional\n120\nEnergy Spectrum\n12.5\nWave Slope Data\n132\nThe Logarithmic Wind Profile Model\n132\nVisual Observations on Wave Slopes\n133\nHulburt's Observations of Wave Slopes\n136\nDuntley's Immersed-Wire Measurements of\n138\nWave Slopes\nIntuitive Picture of the Gaussian Slope\nDistribution\n142\nThe Wave-Slope Wind-Speed Law (Duntley)\n145\nCox and Munk's Photographic Analysis of\n145\nthe Glitter Pattern\nThe Wave-Slope Wind-Speed Law (Cox and Munk)\n149\nSchooley's Flash Photography Measurements\nof Wave Slopes\n151\nWave Generation and Decay Data\n152\n12.6\nGeneration of Waves: Shallow Depths, Small\n153\nFetches\nGeneration of Waves : Deep Depths, Large\n157\nFetches\nDecay of Waves\n161\n12.7\nWave Spectrum Data\n166\nWave Spectra by Aerial Stereo Photography\n166\nWave Spectra by Floating-Buoy Motion\n173\nWave Spectra from Submarine Echo Recordings\n180","CONTENTS\nV\n12.8\nEmpirical Wave Spectra Models\n181\nThe Neumann Spectrum\n181\nDerivation of the Neumann Spectrum\n183\nThree Laws Derived from the Neumann Spectrum\n186\nAlternate Forms of the One-Dimensional Spectrum\n189\nGeneral Properties of Gamma Type Spectra\n190\nWind Speed, Wavelength, and Wave Energy\n193\n12.9\nTheoretical Wave Spectra Models\n194\nThe Wave Elevation Distribution\n194\nThe Wave Slope Distribution\n197\nThe Wavelength Distribution\n202\nThe Bretschneider Spectrum\n204\nThe Wave Height Distribution\n205\nModels of Wind-Generated Spectra\n205\nSpectral Transport Theory\n208\n12.10\nInstantaneous Radiance Field Over a Dynamic\nAir-Water Surface\n210\nThe Geometrical Setting\n211\nThe Integral Equation for the Instantaneous\nSurface Radiance N+ (S)\n212\n12.11\nTime-Averaged Radiance Field Over a Dynamic\nAir -Water - Surface\n216\nDirect and Indirect Radiance Averages\n216\nThe Stationarity Condition\n219\nThe Independence Condition\n220\nThe Weighting Functions\n221\nThe Time-Averaged Integral Equation for N+ (S)\n222\nStructure of the Weighting Functions\n224\nThe Instantaneous and Time-Averaged Equations\nfor N+ (S)\n234\n12.12\nInstantaneous and Time-Averaged Radiance\nFields Within a Natural Hydrosol\n237\nTwo Types of Time-Averaged Radiance Fields\n238\nEquations of Transfer for Time-Averaged\nRadiance Fields\n239\nConnection Between Fixed Depth and Cosurface\nTime-Averaged Radiances\n243\n12.13\nSynthesis of Time-Averaged Radiance Fields\n246\nComparison with the Static Case\n250\n12.14\nObservations on the Theory of Time-Averaged\nRadiance Fields for Dynamic Air-Water\nSurfaces\n250\nA Hierarchy of Approximate Theories\n251\nIllustrations of Some Classical Partial\nTheories\n253\nConcluding Observations\n260","CONTENTS\nvi\nSimulation of the Reflectance of the Air-Water\n12.15\n260\nSurface by Mechanical Devices\n261\nThe Central Idea of the Sea State Simulator\n261\nErgodic Hypothesis\n263\nThe Discrete Case\n266\nThe Continuous Case\nSome General Observations on the Ergodic Cup\n268\nDevice\n269\nSea Simulator Devices Beyond the Ergodic Cup\n269\nBibliographic Notes for Chapter 12\n12.16\nChapter 13\nOperational Formulations of Concepts for\nExperimental Procedures\n273\nIntroduction\n13.0\nOperational Definitions of the Principal\n13.1\n273\nRadiometric Concepts\n274\nRadiant Flux\n274\nIrradiance\n278\nSpherical Irradiance and Scalar Irradiance\n281\nRadiance\n284\nOperational Definition of Beam Transmittance\n13.2\n285\nGeneral Two-Path Method\n286\nGeneral One-Path Method\nOperational Definitions of Path Radiances\n13.3\n287\nand Path Functions\n287\nOperational Formulation of Path Radiance\n288\nOperational Formulation of Path Function\nOperational Definition of Volume Attenuation\n13.4\n290\nFunction\n292\nCanonical Equation Method\nA General Theory of Perturbed Light Fields,\n13.5\nwith Applications to Forward Scattering\n293\nEffects in Beam Transmittance Measurements\n293\nIntroduction\nGeneral Representation of a Perturbed Light\n295\nField\nLinearized Representation of Slightly\n297\nPerturbed Light Fields\n299\nApplication to Bright-Target Technique\n300\nApplication to Dark-Target Technique\nAn Outline of Possible Experimental Procedures\n302\nof a in Perturbed Light Fields\n305\nOrder of Magnitude Estimates\n307\nSummary and Conclusions","CONTENTS\nvii\n13.6\nOperational Definition of Volume Scattering\nFunction\n308\no-Recovery Procedures\n312\nDetermining the Volume Scattering Matrix\nin the Polarized Case\n314\n13.7\nDirect Measurement of the Volume Total\nScattering Function\n316\nThe General Method\n317\nObservations\n318\nTwo Special Methods\n319\nCylindrical Medium\n319\nSpherical Medium\n320\n13.8\nOperational Definition of Volume Absorption\nFunction\n321\nProcedures for Stratified Light Fields\n322\nProcedures for Deep Media\n323\nGeneral Global Method\n323\nFurther Procedures for General Media\n323\n13.9\nOperational Procedures for Apparent Optical\nProperties\n324\nThe Fundamental Irradiance Quartet\n325\nDiscussion of the Reflectance Functions\n327\nDiscussion of the Distribution Functions\n329\nDiscussion of the K-Functions\n330\n13.10\nTheory of Measurement of Local and Global\nR and T Properties\n331\nExample 1: R and T Factors in Homogeneous\nPolarity-Free Settings\n332\nExample 2: Homogeneous Media with Polarity\n333\nExample 3: Forward and Backward Scattering\nFunctions\n335\nExample 4 : R and T Operators for Radiance\n337\nGeneral Observations on Inverse Problems in\nHydrologic Optics\n339\n13.11\nOn the Consistency of the Operational\nFormulations\n339\nOn the Relative Consistency of the Unpolarized\nand Polarized Theories of Radiative Transfer\n341\n13.12\nBibliographic Notes for Chapter 13\n345\nBibliography for Volumes I-VI\n347\nIndex to Volumes I-VI\n369","","ix\nPREFACE\nThe central problem addressed in this volume is the\nprediction of the reflected radiance distribution from a ran-\ndom sea surface, and the transmitted radiance distribution\nentering the body of the sea below the random surface. The\nmain results are presented in equations (18) and (44) of Sec.\n12.11. These equations describe the predicted radiance in\nterms of the wind-generated statistical parameters of the sea\nsurface, such as mean square wave height, wave slope, and sky\nradiance distribution. 1 With possible future oceanographic\nand meteorologic applications of this theory in mind, I have\nappended in Sec. 12.14 a hierarchy of approximate versions of\nthe exact theory ranging from the exact time-averaged radi-\nance theory down to the simple model of contrast transmit-\ntance of a random sea surface which I solved many years ago\n(Preface, Vol. I) as one of my first research problems in the\nsubject of hydrologic optics.\nThe inverse problem to that considered in this volume\nis\nof considerable interest to the ongoing research problem\nof global weather and climate prediction: given the observed\nglitter pattern on the oceans and seas of the earth (as ob-\nserved by satellite), it is required to find the surface wind\nspeeds over those hydrosols. The theory developed below pre-\nsents all the elements needed to solve this problem via pat\n-\ntern recognition methods: We can now compute reflected radi-\nance distributions under a wide variety of wind speed, fetch\nand duration conditions, and thereby can select the appro-\npriate wind speed for the observed glitter pattern. Correc-\ntions for intervening path radiance can be made so that the\ninherent ocean surface radiance can be estimated. Knowledge\nof the wind speed, duration, and fetch are thus forthcoming\nand in turn help in the determination of evaporation rates\nand advective ocean currents, both important parts of the\nglobal weather prediction problem.\nAnother application of the present theory is to the\nheating rate of the oceans and seas by transmitted visible\nradiant energy past the random air-water surface into the\nbody of the hydrosol where it undergoes absorption and\nhence transformation into thermal energy. Knowledge of the\noptical state of the sea, given by the solution of the in-\nverse problem above, permits a local estimate (via (44) of\nSec. 12.11) of the average irradiance transmittance of sun\nand sky light into the sea with proper allowance for wave\nslope and wave height effects (which are critical in sub-\npolar regions), so that thermal conversion estimates can\nbe made. For the latter estimates we must in turn have\nreliable ways of documenting the oceanic volume absorption\nfunction over the visible spectrum. Several ways of achieving\nthese measurements are given in Chapter 13, which closes the\npresent work. Such knowledge of the thermal component of the","PREFACE\nX\nenergy budget in the sea, along with an understanding of the\nvertical and horizontal mixing processes in the upper layer\nof the ocean will be instrumental in predicting short period\nclimatic fluctuations.\nThe observations above indicate the important role of\nradiative transfer calculations in the problem of weather\nand climate prediction. The theory presented in the six\nvolumes of this work on hydrologic optics, while ostensibly\ndirected toward oceanographic matters, also applies to meteo-\nrologic problems of radiative transfer. In particular the\nwork applies to the difficult problem of predicting radiative\ntransfer in the visible wavelengths (prior to absorption)\nthrough vertically and horizontally structured clouds, haze,\nand various aerosols. The solution procedures of Vol. IV,\nin particular those of Sec. 7:11, should help cast some light\non this problem which appears to be of central concern in\n2,3\nboth short- and long-range weather prediction efforts.\nIn the future, satellites will play an important role\nas observation platforms from which we can obtain instantane-\nous optical surveys of the physical properties of the lower\natmosphere and upper ocean dynamics needed in furthering our\nunderstanding of the combined atmosphere-ocean prediction\nproblem. Because any part of the ocean is masked over the\nvisible spectrum approximately fifty percent of the time by\nclouds, it is important that we endow our electromagnetic\nprobes with the ability to scan over longer electromagnetic\nwavelengths. Infra red spectral examination will give infor-\nmation of the temperature structure of the atmosphere. New\ntechniques of solving the associated inverse problems are\n5\nIn addition, the use of land-and space-based\npromising.\nmicrowave scanners of the ocean surface itself may eventually\npermit direct observation of surface advection currents.\nIn order to achieve a sufficiently broad theoretical\nframework for the preceding microwave observations, we must\nincrease the scope of the classical equation of transfer from\nits present form (which includes thermal radiation) to one\nthat also describes the radiance of a partially coherent field\n6,7,8,9\nof electromagnetic waves in a scattering-absorbing medium.\nThe reader who comes upon these six volumes of Hydro-\nlogic Optics should, after having perused the above remarks,\nview this work not as a record of closed and completed re-\nsearch in radiative transfer theory, but simply as one step\nin a long sequence of steps in our quest for understanding the\ndynamics of the world we live in. I dedicate this work to all\nscientists who may find it of help in their individual quests\nand studies, and in particular to the memory of my mentor during\nmy early years at Scripps Institution of Oceanography, Profes-\nsor Carl Eckart.\nI acknowledge my gratitude, for years of counsel and con-\ntractual support (Bureau of Ships, Office of Naval Research)\nduring the formation and writing of this work (1952-69) to\nProfessor S. Q. Duntley of the Visibility Laboratory, Scripps\nInstitution of Oceanography. My thanks also go to Mr. John E.\nTyler of the Visibility Laboratory for his contractual support\n(National Science Foundation) and for providing the inspira-\ntion, based on real world data, leading to many parts of the","PREFACE\nxi\ntheory presented in this work. These volumes in their pres-\nent form, and in their wide distribution, would not have been\npossible but for the generous support and services provided\nby the Late Dr. Gaylord Miller, director of the Honolulu-based\nJoint Tsunami Research Effort of the Pacific Marine Environ-\nmental Laboratory, Environmental Research Laboratories, NOAA. I\nam also grateful that Dr. Miller, over the past six years, had\nprovided an excellent research atmosphere in which I could ex-\ntend the interaction principles from radiative transfer theory\ninto the field of long ocean surface wave propagation.\nThe final manuscript was typed by Ms Louise F. Lembeck.\nR.W.P.\nHonolulu, Hawaii\nDecember 1976\nA brief exposition of the salient ideas of the present\napproach to the statistical radiance problem is given in:\nPreisendorfer, R. W. \"General Theory of Radiative Transfer\nAcross the Random Atmosphere-Ocean Interface,\" J. Quant.\nSpectrosc. Radiat. Transfer, 11, 723 (1971).\nThe references to the published chapters of Hydrologic Optics\nin the above paper are now emended to the present volumes I\nand VI, published by the Environmental Research Laboratories\nof NOAA (1976).\n2 The role of modern radiative transfer theory in the\natmosphere is outlined in;\n\"Problems of Atmospheric Radiation in GARP,\" \" GARP Publications\nSeries No. 5, July 1970, World Meteorological Organization,\nCase postale No. 1, CH-1211 Geneva 20, Switzerland.\n3 A more general overview of the role of modern radiative\ntransfer theory in the problem of weather and climate predic-\ntion is given in:\n\"The Physical Basis of Climate and Climate Modelling,\" GARP\nPublications Series No. 16, April 1975 (address as in 2).\nIn both GARP publications the need is stressed for nu-\nmerical methods of radiative transfer which appear to be\navailable now in the present work on Hydrologic Optics.\n4 In particular the work of:\nChahine, M. T. , \"An Analytical Transformation for Remote\nSensing of Clear-Column Atmospheric Temperature Profiles,\"\nJ. Atm. Sci., 32, 1946 (1975),\nand its references lead to useful approaches to the inverse\ntemperature problem.\n5 A new approach to the inverse temperature problem is\nbased on invariant imbedding methods wherein the integral\nequation of the first kind governing the temperature function\nis solved after being reduced to an initial value problem:\nKagiwada, H. H., Kalaba, R. E., \"New Methods for Atmospheric\nTemperature Inversion\" Rand Corporation Report R-1810-DOC,\nOctober 1975 (prepared for the Department of Commerce).","xii\nPREFACE\n6 See closing paragraph of Bibliographic Notes for Chap-\nter 13, below.\n7A possible way of building a bridge from electromag-\nnetic theory to radiative transfer theory, in the case of in-\ncoherent electromagnetic fields, is sketched in Chapter XIV\nof [251] (see references at end of this volume)\n8\nNo comprehensive theory of the scattering of microwaves\nby the random sea surface seems to be available at present.\nThe work in:\nBeckmann, P., and Spizzichino, A., The Scattering of Electro-\nmagnetic Waves from Rough Surfaces, Macmillan, N. Y. (1963)\nprovides an excellent introduction to the problem. However,\nthe statistical treatment of the problem, the adopted model\nof the random sea surface, and the form of the adopted elec-\ntromagnetic equations are all too rudimentary to provide the\nappropriate framework for the presently envisioned problems.\nWe have in mind a sufficiently detailed model of both the sea\nand the electromagnetic field that will reproduce not only\nthe equations of Sec. 12.11 below (for visible wavelengths)\nbut also the (as yet) underived corresponding equations for\nthe microwave context. A beginning in this direction may be\nbased on the (still) excellent treatment of the electromag-\nnetic field in Chapters 3, 4, and 5 of:\nSilver, S. (Ed.), Microwave Antenna Theory and Design (origi-\nnally Volume 12 in the 1949 Massachusetts Institute of Tech-\nnology Radiation Laboratory Series) Dover Publications, N.Y.\n(1965)\nalong with the instantaneous sea surface model given in (71)\nof Sec. 12.4 below.\nIt is possible to derive an integrodifferential equa-\ntion of transfer for the specific intensity of any field\ngoverned by a linear space-time wave operator in a random\nsetting. Such a derivation was carried out, for example,\nfor sound fields in a random sea, in my lectures on :\nGeophysical Random Processes, Oceanography course 220, Scripps\nInstitution of Oceanography, spring 1967 (Chapter 9).\nThe reduced wave equation for the acoustic field is\nquite similar in form to that for the E-components of the\nelectromagnetic field. Using the approach outlined in notes\n7 and 8, above and the techniques explored in these lectures,\nthe requisite generalized transfer equations for microwaves\nmay be forthcoming, which in turn would provide a tool for\nanalyzing and remotely probing the random sea surface. In\nthis connection, the techniques in §63 of the following ref-\nerence may also be helpful:\nTatarskii, V. I., The effects of the Turbulent Atmosphere on\nWave Propagation, U.S. Department of Commerce, National Tech-\nnical Information Service, Springfield, Va. 22151 (1971).\n10\nFor some references to these works, see the Preface to\nVolume IV, and also Sec. 8.10 of Volume V of Hydrologic Optics.\nSeveral further studies , conducted at the Joint Tsunami Research\nEffort, Honolulu, on the transport theory of linear hydrody-\nnamics are being prepared for publication.","CHAPTER 12\nOPTICAL PROPERTIES OF THE AIR-WATER SURFACE\n12.0 Introduction\nThe study of the penetration of light into, and the reflection\nof light from natural hydrosols at some point must consider\nthe geometric and physical properties of the air-water sur-\nface. In this chapter we shall study the salient geometric\nand radiometric features of the air-water surface in both its\nstatic and dynamic states. We shall be guided in these stud-\nies, especially as regards the selection of auxiliary material\noutside the discipline of radiative transfer, by fixing at the\noutset two main goals for the present chapter. These are the\ncomplete description of the transmission and reflection prop-\nerties of the air-water surface, in both its static and dynamic\nstates, taking into account the optical interactions of the\nsurface with the atmosphere above and the hydrosphere below.\nIn order to achieve these goals, and still keep the discussion\nessentially self contained, we shall draw on the fields of\nphysical optics, hydrodynamics, and harmonic analysis, in ad-\ndition to the principles of radiative transfer theory.\nOur first goal, then, is the description of the general\nradiative transfer process across a static air-water surface.\nWere there never a breath of air, nor tidal, seismic, or other\nagents to disturb the air-water surfaces on the natural hy -\ndrosols of the earth, our task in the present chapter need\nnot extend past sections 12.1 and 12.20 below. In those sec-\ntions we describe and solve completely the main optical prob-\nlems of the static air-water surface as they arise in the\nstudy of radiative transfer across such surfaces. For all\nthat is currently needed in such a case is knowledge of three\nlaws of physical optics and the interaction principle of ra-\ndiative transfer. The three laws from physical optics are\nthe geometric reflection and refraction laws and Fresnel's\nreflectance formula.\nSuch an ideal static state of the air-water surface is\nrarely found in nature and consequently, in our quest for the\nsecond main goal, we must face a complex, dynamic air-water\nfilm which during each second and at each point reflects\nlight from many different portions of the sky and transmits\nlight from many different portions of the underwater domain\nto the eye of the beholder. Ordinarily, when one looks at a\nwind-blown sea with its underriding phalanxes of swells and\ngravity waves and with its crinkly skin of capillary waves,\none absorbs the sensation as an unanalyzed whole and uncriti-\ncally accepts the external reality of the dynamic surface in","VOL. VI\nAIR-WATER SURFACE PROPERTIES\n2\nall its complexity. However, under the direction of aroused\nscientific curiosity or under the necessity of solving some\nproblem in which the successful description of the dynamic\nsurface plays an essential role, the analytical and critical\nfaculties of the observer come into play and soon a concep-\ntual webwork begins to join the previously loose-knit visual\nsensations together. For example, in the radiative transfer\ncontext, the dynamic surface could be conceptually frozen in-\nto a state of immobility, as one would stop a high speed\nmovie film of the surface, and the mental image then could\nbe scrutinized for patterns and possible orderings of data.\nEach visualized brilliant highlight of reflected sunlight\nbecomes a static tiny patch of light with a geometrically\nprecise relation between four further constructs envisioned\nby the observer: the position of the sun, the observer, and\nthe position and orientation of the reflecting facet of the\nair-water surface. Furthermore it is possible, and this must\nbe taken into account in any complete theory of radiative\ntransfer across a dynamic air-water surface, to envision and\ndescribe in detail the radiometric interaction of the complex\ncurved surface with itself: Virtually infinite numbers of\nmultiple interreflections within the frozen concavities of\nthe surface are possible before the resultant flux is once\nagain moving free in the atmosphere above the surface or\ntowards the observer. The conceptual webwork in which the\ndynamic air-water surface is being envisioned thickens as\nfurther knowledge is introduced concerning the orderly and\nlawful motions the seemingly chaotic jumble of waves must in\nreality obey: So that when the conceptually frozen motion\nof the air-water surface is allowed to take its natural course\ninto the next frame of the film, its movement is inexorably\nprescribed by the laws of hydrodynamics. Once again, when\nthe various parts of light field of this new configuration\nare examined, they are found to obey the same general type of\nradiative equations as in the preceding frame, and so the\nconceived order increases, and conceptual chaos decreases.\nHowever, conceptual order on both the radiative transfer and\nhydrodynamic levels notwithstanding, the growing number of\nframes in the conceptual film as second succeeds second makes\nit impossible to succinctly and completely describe numeri-\ncally the radiometric relations from frame to frame. For\nthis reason the concepts of harmonic analysis and statistics\nare called into service to succinctly summarize the averaged\nor statistical features of the radiative and hydrodynamic\nprocesses extant in the dynamic air-water surface.\nThe requisite amount of hydrodynamic theory for con-\nstructing the present optical theory of the air-water sur-\nface is given in section 12.3, and those parts of harmonic\nanalysis required for the present task are developed in\nsection 12.4. In order to give depth to the present statis-\ntical theory and to prepare for useful applications of the\ntheory to air-water radiative transfer phenomena, there fol-\nlows in sections 12.5 to 12.8 a review of some recent experi-\nmental studies of wave generation and decay and of certain\nstatistical properties of wind-generated seas in equilibrium\nwith the generating wind. Section 12.9 contains an attempt\nto understand the reviewed empirical data from a unifying\nstatistical-theoretical vantage point. Then in sections 12.10","STATIC PROPERTIES\nSEC. 12.1\n3\nto 12.14 a statistical theory of radiative transfer across\na dynamic air-water surface is developed and applied to some\nillustrative examples. The chapter concludes with a brief\nstudy of possible devices which may be used in the laboratory\nto simulate the optical properties of randomly moving surfaces\nof natural hydrosols.\n12.1 Reflectance and Transmittance Properties of the Static\nSurface\nWe begin the discussions of the reflectance and trans-\nmittance properties of the static surface with the simplest\nand most useful of the laws of geometric optics for our pres-\nent purposes, namely:\nThe Geometric Law of Reflection\nFigure 12.1 (a) depicts a portion Y of an optical medium\nnear a plane boundary interface S which separates Y into\nparts X' and X in which the indices of refraction are\nrespectively n' and n. The surface S is a mathematical\nsurface, i.e., one which has no thickness and serves merely\nto separate X' from X. A narrow beam of radiant flux in\nX' is incident along a direction E' at point X on the\ninterface S. If n (= k) is the unit normal to the surface\nS and is directed, as shown in (a) of Fig. 12. 1, from X to\nX' , then the part of the incident beam that is reflected at\nX back into X' is directed along & where 5',5 and n\nare related by the law of reflection:\nE - E'\n(1)\n=\nn\n& - is\nwhere \" I 5- E 1 I 11 denotes the magnitude of the difference\nE - E' of the two unit vectors 5' and E. From (1) we\nfind (since E + E' is perpendicular to 5 - 5') on dotting\n(5 + E') into each side:\nE . n = - & . n\n(2)\n,\nwhich shows that the angles between the reflection direction\nE and n, and between the incident direction E' and n\nare equal. Suppose we write:\n\" o' \"\nfor\narc cos ( - E'.n\nand\n\" 0 \"\nfor arc cos (5.n)\nthen (2) implies that:\n0 = O'\n(3)\nIn addition, (1) summarizes the fact that 5',5 and n lie\nin a common plane, the plane of incidence. The significance","4\nAIR-WATER SURFACE PROPERTIES\nVOL. VI\nY\n8\n01\nD1\nk=(n)\nD'\nX'\n(a)\nS\nX\nx\nY\n0'\n>\nD'\nk=(n)\nX'\n(b)\nS\nX\nBy\nG\nn (=k)\nn\n(c)\nS\nn''\n0\nPlane of\n0\nIncidence\nng\nFIG. 12.1 Geometry for reflection and refraction laws\nfor surface S. .","SEC. 12.1\nSTATIC PROPERTIES\n5\nof the law of reflection for our present purposes is that\nknowledge of any two of the three unit vectors E' , 5, and n\nis sufficient to determine the third. Thus e.g., knowing is\nand E , we can find n. An important application of this\nwill occur later in Sec. 12.10 when we are devising ways of\ninferring the instantaneous orientation n of a wave facet's\nnormal on the dynamic surface of a natural hydrosol, having\nmeasured E' and E.\nThe Geometric Law of Refraction\nFig. 12.1(b) depicts the refraction of a ray incident\nalong is at X on the interface S between X' and X, , and\nrefracted along E. The directions E,5' , n are related\nby the law of refraction:\n(no - n' E') X n = 0\n(4)\nwhere n' and n are again the indices of refraction of X'\nand X, , respectively. This law summarizes two important facts:\nfirst, the refraction direction & along with E and n lie\nin a common plane, which is the plane of incidence (cf. Fig.\n12.2(a)). Second, since:\n& X n = sin 0\n,\nwhere O' and 0 are defined above, (4) implies:\nn' sin O' = n sin 0\n(5)\nwhich is the scalar version of (4) (and the most common rep-\nresentation of the law of refraction) known as Snell's Law.\nA simple graphical interpretation of (5) is shown in (c) of\nFig. 12.1. The salient geometric fact to observe in connec-\ntion with (5) is that if n > n' then 0 < 0' . Table 1\nis a tabulation of angle pairs O' , 0 related by (5), cor-\nresponding to the relative index of refraction m = 4/3\nwhere we have written:\n\"m\" for n/n'\n(6)\nand where n' is the index of refraction of the incident\nmedium and n that of the refracting medium. The most com-\nmon pair of media to which Table 1 is applied is the air-\nwater pair, for which n' = 1, n = 4/3.","VOL. VI\nAIR-WATER SURFACE PROPERTIES\n6\nTABLE 1\nSnell's Law (m = 4/3)\nO'\n0\n0\n0'\n0\n14°00'\n10°27'\n5°15'\n7°00'\n0°00'\n0°00'\n35'\n10'\n22'\n10\n08'\n10\n42'\n20\n20\n30'\n15'\n20\n49\n30\n37'\n30\n23'\n30\n57\n40'\n45'\n40\n40\n30'\n11°04'\n50'\n52\n38\n50\n50\n15°00'\n12'\n6°00'\n8°00\n1°00\n45'\n19\n10\n07'\n53'\n10\n10\n26\n20\n15'\n1°00'\n20\n20\n34\n30\n22\n30\n08'\n30\n41\n40\n40\n29\n15'\n40\n48\n50\n50\n37\n23'\n50\n56\n16°00'\n9°00'\n44'\n2°00'\n30'\n12°03'\n10\n52'\n10\n38\n10\n11\n20\n20\n59\n45'\n20\n18\n7°07'\n30\n30'\n30'\n53'\n25\n40\n14'\n2°00'\n40\n40'\n50\n33'\n22'\n50\n50\n08'\n17°00'\n40\n10°00'\n29\n3°00'\n15'\n47\n10'\n37\n23'\n10'\n10\n55\n20'\n44\n20\n30'\n20\n13°02'\n30'\n51\n38\n30\n30\n40'\n09\n59\n45'\n40'\n40\n17\n8°06\n50'\n50'\n52'\n50\n18°00'\n13°24'\n14\n3°00'\n11°00'\n4°00'\n31\n21\n10'\n10\n07\n10\n20'\n39\n29\n15\n20\n20\n46\n36\n30'\n22'\n30'\n30\n40'\n53\n44\n30'\n40\n40'\n14°01'\n50'\n51\n37\n50\n50'\n19°00'\n08'\n8°58'\n12°00'\n4°00'\n45'\n9°06'\n10'\n15\n10\n52\n10\n20'\n23\n13'\n4°00'\n20\n20\n30\n30'\n07\n30'\n21\n30'\n37\n40'\n15'\n40\n28'\n40\n50\n45\n36\n22'\n50\n50'\n20°00'\n52\n13°00'\n431\n4°30'\n6°00'\n10'\n59\n10'\n50'\n37'\n10\n15°06\n20\n58'\n45\n20\n20\n14\n10°05\n30\n30\n52'\n30\n40\n21\n13\n5°00'\n40\n40\n28\n50\n50'\n20\n50\n07","SEC. 12.1\nSTATIC PROPERTIES\n7\nTABLE 1\nSnell's Law (m = 4/3) - Continued.\n0\n0\n0\n0'\n0\n21°00'\n15°36'\n28°00'\n20°37'\n35°00'\n25°29'\n10\n43'\n10'\n44\n10'\n36\n20\n50'\n20\n51\n20\n42\n30\n57'\n30\n58\n30'\n49\n40'\n16°05'\n21°05'\n40\n40'\n56\n50\n12'\n50\n12'\n50'\n26°03'\n22°00'\n19'\n29°00'\n19\n36°00'\n09\n10\n26'\n10\n26\n10'\n16\n20\n34'\n20\n33'\n20'\n23\n30\n41\n30\n40\n30\n30\n40\n48'\n40\n47\n40\n36\n50\n55\n50\n54\n50'\n43\n23°00'\n17° 02'\n30°00'\n22°01\n37°00'\n50\n10\n10'\n10\n08\n10'\n57\n20-\n17\n20\n15\n20'\n27°03'\n30\n24\n30\n22\n30'\n10\n40\n31\n40'\n29\n40+\n17\n50\n38\n50'\n36\n50\n23\n24°00'\n46\n31°00'\n43\n38°00\n30\n10\n53\n10\n50\n10\n37\n20'\n18°00\n20\n57\n20\n43\n30\n07\n30\n23°04\n30'\n50\n40\n14\n40\n11\n40'\n57\n50\n22\n50\n18\n50'\n28°03'\n25°00\n29'\n32°00'\n25\n39°00\n10\n10\n36\n10'\n32\n10\n16\n20\n43\n20\n39'\n20\n23\n30'\n50'\n30\n46\n30\n30\n40'\n57\n40\n53\n40\n36\n50'\n19°05'\n50'\n24°00'\n50\n43\n26°00'\n12\n33°00'\n07'\n40°00'\n49\n10\n19\n10\n13'\n10\n56\n20\n26\n20\n20\n20\n29°02'\n30\n33\n30\n27'\n30\n09\n40'\n40\n40\n34\n40\n16\n50\n47\n50\n41\n50\n22\n27°00'\n54\n34°00\n48\n41°00\n29\n10\n20°02\n10\n55\n10\n35'\n20\n09\n20'\n25°01'\n20\n41\n30\n16\n30\n08'\n30\n48\n40\n23\n40\n15'\n40\n54\n50\n30\n50\n22\n50'\n30°01'","VOL. VI\nAIR-WATER SURFACE PROPERTIES\n8\nTABLE 1\nSnell's Law (m = 4/3)--Continued.\n0'\n0\n0'\n0\ne'\n0\n56°00'\n38°27\n34°28'\n49°00'\n42°00'\n30°07'\n32\n34'\n10'\n10\n14'\n10'\n20\n37\n40\n20\n20'\n20\n43\n30\n46'\n30'\n27'\n30'\n48'\n40\n52'\n40\n33'\n40\n53'\n50\n58'\n50\n39'\n50'\n59'\n35°04\n57°00'\n50°00'\n43°00'\n46'\n39°04'\n10'\n10'\n10\n52'\n10\n16\n20\n09'\n20'\n59'\n20\n30\n14'\n31°05'\n30\n22'\n30'\n27'\n40\n19'\n40\n40'\n11'\n50\n25'\n33\n18\n50\n50\n58°00'\n30'\n51°00\n39'\n44°00\n24'\n10\n35'\n45'\n10\n30\n10\n40'\n51'\n20\n20\n37\n20\n45'\n30\n57'\n30\n43\n30\n50'\n36°02\n40\n40'\n49\n40\n55'\n08\n50\n55\n50\n50\n59°00'\n40°00'\n52°00'\n14'\n45°00'\n32° 02'\n19'\n10'\n05'\n10\n08'\n10\n20\n10'\n20\n25'\n14\n20\n30\n15'\n30\n31'\n20'\n30'\n40\n20'\n40'\n37'\n27\n40'\n25'\n50\n33\n50\n42'\n50\n60°00'\n30\n53°00'\n48'\n46°00'\n39'\n35'\n10'\n53\n10\n45\n10'\n40'\n20'\n59\n20\n20'\n51\n37°05'\n30\n45\n58\n30'\n30'\n40\n50'\n10'\n33°04'\n40'\n40'\n55'\n50'\n16'\n50\n10'\n50'\n61°00\n41°00\n54°00'\n21\n47°00'\n16\n27\n10\n04\n22\n10'\n10'\n32'\n20\n09\n20'\n20'\n28\n30\n14\n30'\n38\n30'\n34\n431\n40\n19\n40'\n40'\n40\n50\n23'\n46\n50'\n49\n50'\n62°00\n28\n55°00'\n54\n48°00'\n33°52'\n07\n38°00\n10\n58'\n10'\n10'\n20\n11\n34°04'\n20\n05\n20'\n15\n34°04'\n30'\n11\n30\n30'\n40\n19\n16\n16'\n40'\n40'\n50\n23\n21\n22\n50'\n50'","SEC. 12.1\nSTATIC PROPERTIES\n9\nTABLE 1\nSnell's Law (m = 4/3) -Continued.\n0'\n0\n0\n0\n0'\n0\n63°00'\n41°56'\n70°00'\n44°49'\n77°00'\n46°57'\n10\n42°01'\n10\n52\n10'\n47°00'\n20\n05'\n20\n56\n20'\n02\n30\n10'\n30\n59\n30'\n04\n40\n14\n40\n45°03'\n40'\n07\n50\n19\n50\n06\n50'\n09\n64°00'\n23'\n71°00'\n10\n78°00'\n11\n10\n27'\n10'\n13\n10'\n14\n20'\n32\n20\n17\n20'\n16\n30'\n36\n30\n20\n30'\n18\n40'\n41\n40'\n24\n40'\n20\n50\n45\n50\n27\n50'\n23\n65°00'\n49\n72°00'\n30'\n79°00\n25\n10'\n54\n10'\n33\n10\n27\n20'\n58\n20'\n37\n20\n29\n30'\n43°02'\n30\n40'\n30\n31\n40\n06\n40\n43'\n40\n33\n50\n11\n50\n46\n50\n35\n66°00'\n15'\n73°00'\n50'\n80°00\n37\n10\n19'\n10\n53'\n10\n39\n20\n23'\n20\n56'\n20\n41\n30\n27'\n30'\n59'\n30\n42\n40'\n31.\n40'\n46°02'\n40\n44\n50\n35\n50\n05'\n50\n46\n67°00\n40'\n74°00\n08'\n81°00'\n48\n10'\n44\n10\n11\n10\n50\n20'\n48'\n20\n14\n20\n51\n30'\n52'\n30\n17\n30\n53\n40'\n56\n40\n20'\n40\n55\n50'\n44°00'\n50\n23\n50\n56\n68°00'\n04\n75°00\n25'\n82°00\n58\n10'\n07\n10\n28'\n10\n59\n20'\n11\n20\n31\n20\n48°01\n30\n15'\n30\n34\n30\n02\n40\n19'\n40\n36\n40\n04\n50\n23'\n50\n39\n50\n05\n69°00\n27'\n76°00'\n42'\n83°00\n07\n10\n30\n10\n44'\n10\n08\n20\n34\n20\n47\n20\n09'\n30\n38\n30\n50\n30\n10'\n40\n41\n40\n52\n40\n12\n50\n45\n50\n55'\n50\n13","VOL. VI\nAIR-WATER SURFACE PROPERTIES\n10\nTABLE 1\nSnell's Law (m = 4/3)--Continued.\n0 1\n0 ,\n0\n0\n0\n0\n84°00'\n48°14'\n86°00'\n48°26'\n88°00'\n48°33'\n10'\n33'\n10'\n15'\n10'\n27'\n20'\n16'\n20'\n27'\n20'\n34'\n30'\n18'\n30'\n28'\n30'\n34'\n40'\n19'\n40'\n29'\n40'\n34'\n50'\n20'\n50'\n29'\n50'\n35'\n85°00'\n21'\n00'\n30'\n89°00'\n35'\n35'\n10'\n22'\n10'\n31'\n10'\n20'\n23'\n20'\n31'\n20'\n35'\n30'\n23'\n30'\n32'\n30'\n35'\n40'\n24'\n40'\n32'\n40'\n35'\n50'\n25'\n50'\n33'\n50'\n35'\n90°00'\n35'\nThe Fresnel Laws for Reflectance\nThe laws of reflection (1) and refraction (4) may be\nderived from Maxwell's equations for electromagnetic waves\nin dielectric media in a very simple manner (see, e.g.,\n[292]). The derivations automatically yield not only (1)\nand (4) but the amount of radiant flux reflected back into\nX' (as in Fig. 12.1(a)) and refracted into X (as in Fig.\n12.1(b)). We consider now the laws, originally derived by\nFresnel from Maxwell's equations, which govern the amount\nof reflected radiant flux.\nIn explaining the basis of Fresnel laws, it is neces-\nsary to revert momentarily from the radiometric picture of\nlight to the electromagnetic picture of light (re Sec. 2.2).\nFigure 12. 2 gives a perspective view of the situation in\nFig. 12.1. Recall that the three vectors 5', E, n (= k)\nlie in a common plane, the plane of incidence. Along the\nincident ray direction E' moves an electric vector E'\nwhich, by the transverse nature of electromagnetic waves,\noscillates in a plane normal to E' A small circular patch\nof this plane, for three orientations, is depicted in Fig.\n12.2. Now, in analyzing what happens to E' as it strikes S\nat X, it is found convenient and possible (because of the\nlinearity of the Maxwellian theory) to resolve E' into the\nequivalent sum of two components whose magnitudes are\nand and which are, respectively, perpendicular.\nand parallel to the plane of incidence, and which still lie\nin the plane of E'. If it is known how is reflected\nand refracted at X, and similarly for E11, then the behav-\nior of E' at X is completely determined. It can be shown\n(see, e.g., [292]) that the magnitude of the reflected per-\npendicular component is related to the magnitude E at","11\nSTATIC PROPERTIES\nSEC. 12.1\nincident\nreflected ray\nplane of incidence\nE\nE11\nray\n(=n)\nE\n57\n0\nE'll\nX\nX'\nS\nX\n0\nE\nE\nII\nrefracted ray\nFIG. 12.2 Direction space conventions for general\nreflectance calculations.\nby:\nX\nsin\n(0' - 0)\nE1 =\n-\nsin (e' + 0)\nFurther, the reflected parallel component E11 is related to\nE at X by:\nE11 = tan tan\nwhere 0 is the angle of the refracted ray and where 0\nand 0' are related by Snell's law (5).\nWe can use the preceding relations along with the clas-\nsical results (4) and (5) to predict the connection between\nincident and reflected radiance as would be observed using\nradiance meters at X. Thus if N(x,5') is the incident ra-\ndiance at X and N(x,5) the reflected radiance, we first\nnote that'\n*To within a fixed factor, involving the dielectric con-\nstant of X for a given wavelength. As long as one works in\nan arbitrary, but fixed homogeneous medium, the connections\n(7) are adequate, since instruments can be appropriately\ncalibrated.","AIR-WATER SURFACE PROPERTIES\nVOL. VI\n12\nN(x,5') =\nand\n(7)\nN(x,5) |E(x)|2 4\nhere E'(x) 2 is the mean square amplitude of the incident\nelectric vector E', where the average is taken over some\nsuitable interval T of time (say on the order of a hundredth\nof a second). Similar definitions hold for N(x,5). For a\nderivation of the general relations in (7), , along with a dis-\ncussion of the conditions under which they generally can be\nused, the reader is referred to Sec. 124 of Ref. [251]. It\nsuffices to observe here that the main conditions of validity\nof (7) hold in virtually all natural radiometric environments\nlighted by the sun or by most man-made artificial sources in\neither the atmosphere or the hydrosphere. These main condi-\ntions are made explicit below.\nNext we observe that for the case of steady unpolarized\nlight, E(x,t) arrives at X at time t with steady sinusoidal\nfrequency, but with random orientation, over the interval T\n(= 10-2 sec.). Let E' (x,t) be the magnitude of E(x,t). If\nE'(x) is the maximum value attained by E' (x,t), then E' (x,t)\n= =E'(x) cos wt, and:\nE1(x,t)E(x,t)siny(t)\n(8)\nE(x,t)E(x,t)cosu(t)\n(9)\nwhere 4(t) is the angle E(x,t) makes with the plane of in-\ncidence at time t (Fig. 12.2). The reflected vector E(x,t)\nis then given at each instant by:\nwhere ell and e are unit vectors perpendicular and parallel\nto the plane of incidence and such that In view\nof (8) (9) we have:\nanB(xt)\nE(x,t)\n+\n=\n-\n[\nsin\n(0'-0)\n(0'-0)\ntan\n=\n+\nsin\n(0'+0)\ntan\n(10)\nIt follows that, since 4(t) oscillates randomly over all\nvalues in the interval 0 4(t) < 2 TT , and with great fre-\nquency during T (about 6 x 10 14 cycles per second) the mean\nsquare value E(x) of E(x,t) over T is:\n(11","SEC. 12.1\nSTATIC PROPERTIES\n13\nThe conditions on the randomness of 4(t) and on the great\nfrequency w, which lead to (11), are also those that enter\ninto the derivation of (7), so that (7) and (11) combine to\nyield:\n=\n(12)\nwhere we write:\n\"r(n,5',5)\"\n\"r(5',5)\"\n\"r(0')\"\nfor\nor\n(13a)\nThe number r (n,5',5), (or r (5',5), if the unit outward normal\nn to the surface is understood) is the Fresnel reflectance of\nthe surface S for unpolarized electromagnetic fields. Equa-\ntion (12) is the most important and frequently used form of\nthe radiance reflectance law. Tabulations of r(e) are given\nin Table 2, and are adapted from [183]. The relative index\nof refraction m is not explicitly shown in the notation. If\nit is needed explicitly, we could write \"r(m,E',5)\" for\nr (5',5), or for r(0'), as convenience indicates.\nTo convert from radiance to degrees, use the relation,1 radian\n= 57.296 degrees 112 57.30 degrees. (Reference [183] also tab-\nulates reflectances for linearly polarized light so that, to-\ngether with (14) below, reflectances for arbitrary incident\norientations of the E-vector are determinable.)\nThe radiance reflectance law (12) can be supplemented\nby the law of reflection for linearly polarized radiance,\nwith fixed orientation of the E vector at angle 4, as in\nFig. 12. 2. The result is:\nN(x,5) =\n(13)\nwhere we write:\n\"r(E',5;4)\"\n\"r(0';\"))\"\nfor\nor\n+\n(14)\n.\nIn general, both the incident radiant flux and the\nreflected radiant flux at an interface S between two media\nX' and X of different indices of refraction will be partial-\nly polarized, so that (11) and (13) are ideal special cases.\nBy using the operational definitions of polarized radiance\ngiven in Sec. 2.10, the detailed empirical study of reflected,\nrefracted and scattered polarized radiance fields is possible\nin natural optical media. However, for many practical pur-\nposes formulas (11) and (13) serve adequately (separately or\njointly) to give quantitative estimates of the reflected ra-\ndiance at interfaces S. Observe that r (5', (5;4) in (14) may","VOL. VI\nAIR-WATER SURFACE PROPERTIES\n14\nTABLE 2\nFresnel Reflection m = 4/3\n(Superscripts refer to the number of decimal zeros before the\ntabulated entry Thus 11 20408 163\" stands for 0. 02408 163.\nThis holds for all entires down the table until the next\nsuperscript at: 0 10834 505.)\n0\n0\nr\nr\n(radians)\n(radians)\n28708\n037\n0.80\n20408\n163\n0.00\n29875\n821\n0.82\n20408\n165\n0.02\n31192\n993\n0.84\n20408\n191\n0.04\n32677\n961\n0.86\n20408\n306\n0.06\n34351\n455\n0.88\n20408\n616\n0.08\n36236\n826\n0.90\n20409\n273\n0.10\n38360\n401\n0.92\n20410\n474\n0.12\n40751\n873\n0.94\n20412\n465\n0.14\n43444\n754\n0.96\n20415\n541\n0.16\n46476\n892\n0.98\n20420\n053\n0.18\n49891\n050\n1.00\n20426\n410\n0.20\n576\n1.02\n53735\n20435\n082\n0.22\n58065\n163\n1.04\n20446\n607\n0.24\n62941\n719\n1.06\n20461\n597\n0.26\n68435\n355\n1.08\n20480\n744\n0.28\n74625\n517\n1.10\n20504\n828\n0.30\n81602\n284\n1.12\n20534\n725\n0.32\n89467\n841\n1.14\n20571\n419\n0.34\n98338\n183\n1.16\n20616\n011\n0.36\n0\n10834\n505\n1.18\n20669\n735\n0.38\n11963\n817\n20733\n967\n1.20\n0.40\n13238\n780\n20810\n245\n1.22\n0.42\n14678\n768\n20900\n287\n1.24\n0.44\n16305\n844\n21006\n009\n1.26\n0.46\n18145\n151\n21129\n552\n1.28\n0.48\n20225\n368\n21273\n303\n1.30\n0.50\n22579\n233\n21439\n928\n1.32\n0.52\n25244\n160\n21632\n403\n1.34\n0.54\n28262\n953\n21854\n051\n1.36\n0.56\n31684\n645\n22108\n585\n1.38\n0.58\n35565\n483\n22400\n153\n1.40\n0.60\n39970\n095\n22733\n394\n1.42\n0.62\n44972\n871\n23113\n495\n1.44\n0.63\n50659\n613\n23546\n261\n1.46\n0.66\n57129\n513\n24038\n191\n1.48\n0.68\n64497\n533\n24596\n565\n1.500\n0.70\n66495\n071\n25229\n540\n1.505\n0.72\n68559\n275\n25946\n265\n1.510\n0.74\n70692\n527\n26757\n005\n1.515\n0.76\n72897\n299\n27673\n281\n1.520\n0.78","SEC. 12.1\nSTATIC PROPERTIES\n15\nTABLE 2\nFresnel Reflection m = 4/3--Continued.\n0\n0\nr\n(radians)\nr\n(radians)\n1.525\n75176\n165\n1.550\n87779\n435\n1.530\n77531\n800\n1.555\n90563\n010\n1.535\n79966\n989\n1.560\n93441\n860\n1.540\n82484\n628\n1.565\n96419\n529\n1.545\n85087\n730\n1.570\n99499\n715\n2\n1.00000\n000\nalso be written as:\nr(5',5;4) r(5',5;T/2)sin26 (15)\nFor brevity we usually write for r(E',5;T/2) and\n\"r\"(0)\" for r(5',5;0). These values are tabulated in [183].\nThe Fresnel Laws for Transmittance\nHaving found the quantitative law for reflection of\nradiance (12) or (13)) we can deduce with relative ease the\nassociated law for the transmission of radiance across the\ninterface S (Fig. 12.1(b)). Suppose the radiant flux con-\ntent of a beam incident at X via a solid angle D' is P(S',D')\nwhere S' is a small plane surface normal to 5' at X. In\nterms of radiance this is:\nP(S',D') = N(S',D')A(S')R(D')\nNow the flux comprising the reflected radiance leaves S at\nX in a set of directions D1 such that:\n=\n,\nwhich is a simple consequence of (1). Furthermore, the pro-\njection of S' along 5' down onto S defines on S a patch of\nsurface S\" which, when subsequently projected on a plane per-\npendicular to E clearly defines another patch of surface S1,\nsuch that A(S) = A(S). It then follows from (12) that the\nconnection between the incident and reflected radiant flux\nP(S,,D,) at X is:\nP(S,,D)P(S),D')r(s,)\n(16)\nSince no absorption of radiant flux takes place at x, it is\nnow clear that an amount P (S 2,D2), where","VOL. VI\nAIR-WATER SURFACE PROPERTIES\n16\n(17)\n= -\nis transmitted through the interface S along the various\nrefraction directions within the refracted direction set D2.\nHere S 2 is the projection of S' on a plane normal to E, in\nthe manner that S was defined. It is this amount of flux\nthat now goes on to comprise the transmitted radiance in X.\nIt follows that the n2-1aw for radiance ((14) of Sec. 2.6)\nnow takes the form:\nN(x,5)\nN(x,5')\n(18)\nt(5',5)\n=\nn2\nwhere we have written:\n\"t(5',5)\" for 1 .-r(5',5)\n(19)\nA complete derivation of (18) can be based on the discussion\nfollowing (4) of Sec. 2.6. Equation (18) also holds for the\npolarized case, in which case we would use r (E' in (17).\nExample 1: Reflectance Under Uniform\nRadiance Distributions\nAs an illustration of the use of the Fresnel reflec-\ntance law (12), we shall develop an exact formula for the\nreflectance of an interface between two media of relative\nindex of refraction m > 1, as irradiated by unpolarized ra-\ndiant flux from the side of index of refraction 1. To point\nup the fact that irradiation is incident in this direction\nwe call the associated reflectance the external reflectance.\nIf the flux was incident from the side with index m, then\nthe reflectance would be internal reflectance. Figure 12.3\n(a) depicts the point X on a surface S irradiated by radiant\nflux streaming onto X over the hemisphere E-(x) = E(k'),\nwhere k' is the unit inward normal to S at X. Then by (8)\nof Sec. 2.5, the irradiance on S at X is\n(20)\nwhere E(5) in (8) of Sec. 2.5 is now specifically of the form\nE- (x), as introduced in Sec. 3.3, which is customarily used\nfor work with actual surfaces. Now according to (2), the ra-\ndiance along direction E' is reflected along direction E,\nwhere:\n(21)\nE'.K' = E . K\nand where k is the unit outward normal to S at X. Using this\nand (12), we have:","SEC. 12.1\nSTATIC PROPERTIES\n17\n(a)\n(x)\nS\n0\n8'\n(x)\nk\nS\n((x)\n'\nE_(x)\n(b)\nFIG. 12.3 Direction-space conventions for general\nreflectance calculations.\nN(x,5) E e k =\n(22)\nThis equation relates the irradiance on S induced at X by\nN(x,5'), to the resultant radiant emittance of S at X asso-\nciated with the reflected flux. Hence the total radiant\nemittance associated with H(x,k') is:\nW(x,k) = / E_(x)\nN(x,\")r(,)E\n(23)\nwhere W(x,k) is defined in (22) of Sec. 2.4. Then, by (19)\nof Sec. 3.3, we have as the external reflectance r- (x) of\nS at X for irradiance:\nr_(x) = W(x,k)/H(x,k')\n(24)\nCombining this with (20) and (23),\nr (x) E_(x)\n(25)\n=\nI\nE_(x)\nIn the present example, we require N(x,5') to be independent\nof E'; so that (25) reduces to:","VOL. VI\nAIR-WATER SURFACE PROPERTIES\n18\n(26)\nr\nEquation (26) can be readied for evaluation by intro-\nducing an appropriate coordinate system. Such a coordinate\nsystem is depicted in (b) of Fig. 12.3, which serves as a\ntransition diagram between the standard orientations of Figs.\n12.1, 12.2, and the general situation depicted in Fig. 12.3\n(a). Thus, in the framework of Fig. 12.3(b), (26) becomes:\nwith O' sin sin de' O' do' de'\nO' do'\n(27)\no'\nThe transition from the solid angle measure So to the 0',0'\nrepresentation of ds (5') is given in (9) of Sec. 2.5, and\nr(0') is given in (12). The fixed radiance value over E_ (x)\nhas been cancelled from the integrals. The denominator of\n(27) is clearly IT, that is:\n2\n2\n(28)\n0' O' de' do'= TT\nsin =\nand the numerator is reducible to:\nde'\n0'\nsin\n0\ncos\n,\n(29)\nso that:\nr sin 0' de'\ncos O'\nin which all reference to X has been dropped. Evaluating\nthe integral in (29) for relative index of refraction m(> 1),\nwe have:\nr = = 1/ + -\n+ [8m4 1n m\n+ 1n [(m-1) ( (m+1) ]\n(30)","SEC. 12.1\nSTATIC PROPERTIES\n19\nThis representation of r- was first worked out by Walsh\n[310] and applied in his studies of reflectances of polished\nglass surfaces. For glass with m = 1.5, it follows from (30)\nthat r- = 0.092. In the present studies, the relative index\nof refraction m = 4/3 = 1.33 for water is of central inter-\nest and for this, the associated r_ , as given by (30), is\n0.066. Thus under a uniformly overcast sky, approximately\n6.6 percent of the incident radiant flux on a static air-\nwater surface is reflected from the surface. A correspond-\ning exact algebraic formula for the internal reflectance r+\napparently has never been worked out. Numerical integrations\nby Judd [131] indicate that r+ = 0.596 for m = 1.5 and r+\n= 0.472 for m = 4/3. These values are listed, along with\nothers, in Table 3.\nTABLE 3\nReflectance of unpolarized light at a plane\nboundary between two media as a function of\ntheir relative index of refraction, m.\nReflectance for Completely\nReflectance for\nDiffuse Incidence\nPerpendicular\nm\nIncidence\nExternal\nInternal\nReflection\nReflection\n1.00\n0.00000\n0.0000\n0.000\n1.01\n0.00002\n0.0028\n0.022\n1.02\n0.00010\n0.0055\n0.044\n1.03\n0.00022\n0.0082\n0.064\n1.04\n0.00038\n0.0108\n0.084\n1.05\n0.00059\n0.0134\n0.103\n1.06\n0.00085\n0.0158\n0.122\n1.07\n0.00114\n0.0183\n0.140\n1.08\n0.00148\n0.0206\n0.158\n1.09\n0.00185\n0.0230\n0.175\n1.10\n0.00227\n0.0252\n0.192\n1.11\n0.00272\n0.0274\n0.208\n1.12\n0.00320\n0.0294\n0.224\n1.13\n0.00372\n0.0314\n0.240\n1.14\n0.00428\n0.0334\n0.254\n1.15\n0.00487\n0.0353\n0.269\n1.16\n0.00549\n0.0371\n0.283\n1.17\n0.00614\n0.0389\n0.296\n1.18\n0.00682\n0.0407\n0.309\n1.19\n0.00753\n0.0425\n0.322\n1.20\n0.00826\n0.0443\n0.335\n1.21\n0.00903\n0.0461\n0.347\n1.22\n0.00982\n0.0478\n0.359\n1.23\n0.1064\n0.0496\n0.371\n1.24\n0.01148\n0.0513\n0.382\n1.25\n0.01235\n0.0530\n0.393","AIR-WATER SURFACE PROPERTIES\nVOL. VI\n20\nTABLE 3\nReflectance of unpolarized light at a plane\nboundary between two media as a function of\ntheir relative index of refraction, m. - -Continued. -\nReflectance for Completely\nReflectance for\nDiffuse Incidence\nPerpendicular\nm\nInternal\nIncidence\nExternal\nReflection\nReflection\n0.0546\n0.404\n1.26\n0.012]2\n0.404\n1.27\n0.01415\n0.0563\n1.28\n0.01508\n0.0579\n0.424\n0.0596\n0.434\n1.29\n0.01604\n1.30\n0.01701\n0.0612\n0.444\n0.0628\n0.454\n1.31\n0.01801\n0.0644\n0.463\n1.32\n0.01902\n0.472\n1.33\n0.02006\n0.0660\n1.34\n0.02111\n0.0676\n0.480\n0.0692\n0.489\n1.35\n0.02218\n1.36\n0.02327\n0.0707\n0.497\n1.37\n0.02437\n0.0723\n0.505\n0.513\n1.38\n0.02549\n0.0738\n1.39\n0.02663\n0.0754\n0.520\n1.40\n0.02778\n0.0769\n0.528\n1.41\n0.02894\n0.0784\n0.536\n1.42\n0.03012\n0.0800\n0.543\n1.43\n0.03131\n0.0815\n0.550\n0.557\n1.44\n0.03252\n0.0830\n1.45\n0.0337\n0.0845\n0.564\n1.46\n0.03497\n0.0860\n0.571\n1.47\n0.03621\n0.0875\n0.577\n1.48\n0.03746\n0.0890\n0.584\n0.590\n1.49\n0.03873\n0.0904\n1.50\n0.4000\n0.0919\n0.596\n1.51\n0.04129\n0.0934\n0.602\n1.52\n0.04258\n0.0948\n0.608\n1.53\n0.04389\n0.0963\n0.614\n0.619\n1.54\n0.04520\n0.0977\n1.55\n0.04652\n0.0992\n0.624\n1.56\n0.04785\n0.1006\n0.630\n1.57\n0.04919\n0.1020\n0.635\n1.58\n0.05054\n0.1035\n0.640\n0.645\n1.59\n0.05189\n0.1049\n1.60\n0.05325\n0.1063\n0.650","SEC. 12.1\nSTATIC PROPERTIES\n21\nExample 2. Reflectance Under Cardioidal\nRadiance Distributions\nWhat would be the reflectance of the sea surface if it\nwere absolutely calm and exposed to a heavily overcast sky?\nThis is the problem we pose and solve in this example. Now,\nthe actual form of the radiance distribution under a heavily\novercast sky is not uniform as that discussed in Example 1,\nbut more nearly of a cardioidal form:\n(31)\nwhere Eo is any fixed horizontal direction, i. e , so . k = 0 ,\nand where k is a unit vector directed toward the zenith.\n(Recall that 5' is the direction of flow of the photons com-\nprising N(x,E').) Thus if in Fig. 12.3(b) k is directed\ntoward the zenith and E' is any downward radiance, then (31)\ngives the (unpolarized) radiance N(x,5'). Equation (31) is\nan empirical law, found by Moon and Spencer [186]. Further\nempirical confirmation of (31) was made by Hopkinson [112].\nEquation (31) is of the same general family as that in (14)\nof Sec. 6.6. That is, (31) is closely related to the solu-\ntions of the classical diffusion theory for plane-parallel\nmedia. In the case of (14) of Sec. 6.6, which holds for prac-\ntical situations such as the present one, the radiance repre-\nsented there is at a relatively great depth in a plane-parallel\nmedium (assuming Fick's law for photons holds in that medium).\nA closely related form to (31) was predicted theoretically by\nSchwarzschild [282] and later by Chandrasekhar [43]. Fig 12.4\n1.0\nISOTROPIC\n8\nRAYLEIGH\n.6\nCARDIOID\n4\n2\no\n.10\n.20\n30\n.40\n.50\n.60\n.70\n.80\n.90\n1.00\nu = COS 0\nFIG. 12.4 Emergent radiance distributions for an atmos-\nsphere with no appreciable absorption. For the cases of Ray-\nleigh and isotropic scattering functions, as compared with an\nemergent cardioidal radiance distribution.","VOL. VI\nAIR-WATER SURFACE PROPERTIES\n22\ncompares the empirical cardioidal radiance law (31) with two\ntheoretical radiance distributions based on isotropic scatter-\ning and Rayleigh scattering forms for O. These computed dis-\ntributions are partly based on those on page 135 in [43]. In\nthe present example we shall find an exact representation of\nr_ (x) for the case of a cardioidal radiance distribution of\nthe form (31) and for the same setting as in Example 1. We\nshall in fact use a general form of (31) in which 2 is replaced\nby an arbitrary real number n. In this way we shall general-\nize (30), which is the case for n = 0.\nThus, using the coordinate frame of Fig. 12.3 (b), the\ngeneral form of (31) may be written:\nN(x,0',0') = N(x,m/2,0') (1 + n cos 0')\n(32)\n.\nFurthermore (25), with N(x,5') given by (32) , now becomes:\n0']r(0') cos O' sin o' do' do'\n(m,n)\nr\n=\n0'] cos e' sin 0' de' do'\n(33)\nwhere we have written \"r_ (m,n)\" for r_ (x) to point up the de-\npendence of the reflectance on the two parameters m >1; (the\nrelative index of refraction) and n (the shape index of the\nradiance distribution). Clearly r_(m,0) is the r_ of (30),\nso that r_ (m,n), when evaluated, will be a proper generaliza-\ntion of Walsh's formula for external reflectance. Once again,\nirradiation is from the side with index of refraction 1. We\nnow outline the manner in which the exact form of r (m,n) may\nbe obtained.\nWe begin by making a preliminary simplification of (33)\nby performing the integrations over the azimuth angles ''\nIT/2\n]r(0') cos o' sin O' O'\n(34)\nr\n0']\nde'\nO'\nsin\n0'\ncos\nThe denominator of this fraction, which we shall denote by\n\"H' (n)' \" , is easily evaluated:\nH' (n) = 1\n(35)\nThe numerator of r- (m,n), which we shall designate by\n(m,n)\" , is relatively difficult to evaluate because of\nthe presence of the factor r (0') in the integrand. It is\nfound that by a suitable pair of transformations of variables,\ndone in tandem, the relatively complex numerator W' (m,n) of\nr. - (m,n) may be systematically disassembled into manageable","SEC. 12.1\nSTATIC PROPERTIES\n23\npieces. Thus, , first we write \"0'\" for the difference 0'-0 -\noccurring in the representation (13a) of r(0'), and using\ntrigonometry with Snell's law (5), we eventually arrive at\nthe following representation of Fresnel's reflectance law:\n(') = [2m2/(m2-1)] (cos ''-a)2 + (sec ''-a) 2 2] 2\n(36)\nin which we have now written \"r_(o')\" for r. (0') and:\n\"a\" for (m2 + 1) / 21\n(37)\nand where 0 < 0' < arc cos (1/m). The term COS ''' now plays\nthe prominent role in r(o'), and we may thus simplify W' (m,n)\nby writing \"X\" for cos ' and \"s\" for 1/m, so that with this\nsecond transformation of variables, we eventually obtain:\nN'(m,n) =\ncs)/1s2x1\nX [(sx-1) (x-s)/(x-a)2) dx\n(38)\nClearly W'(m,n) can be written in the form:\nN'(m,n) = A(m) + nB(m)\n(39)\nwhere we write:\n\"A(m)\" for\n(40)\nand\n\"B(m)\" for dx\n(41)\nHence:\nr (m,n) = 6[A(m) + nB(m)]/(3 + 2n)\n(42)\nThe classical expression (30) found by Walsh is readily\nforthcoming from (42) by setting n = 0 :","AIR-WATER SURFACE PROPERTIES\nVOL. VI\n24\n(43)\nr_(m,0) = 2 A(m)\nThus the task of evaluating A(m) has already been done. It\nremains to find B(m). Some algebraic experimentation on the\nintegrated form of B(m) suggests that a natural representa-\ntion of B(m) is the following:\nB(m) =\n(44)\nwhere we have written:\n(\"B11\" for\n- (1/7) (p7-a7/d)\n(45)\n\"B12\" for (x/3)\n(46)\n(47)\n\"B13\" for\n\"B14\" for - m(p-q1/2)\n(48)\n(\"Bei\" for (1/3)\n(49)\n(50)\n\"Boo\"\" for\n\"B23\" for\nand\n\"A\" for (51)\n\"B24\" for (-m/r)(p-mq1/8) 1n A\n(52)\nL\n(53)\nfor\n\"B32\" for\n(54)","SEC. 12.1\nSTATIC PROPERTIES\n25\n\"B33\" for\n+ A\n(55)\n\"Bas\" for\n- (20m4/r7/8)\n(56)\n,\nand, finally, where we have written:\n\"p\" for m-1\n(57)\n\"q\" for m²-1\n(58)\n\"r\" for m ² +1\n(59)\nEquation (44), when used in (42) along with A(m), as given by\n(30) (recall (43)), yields an exact expression for r_ (m,n).\nTable 4 lists some values of r (m,n) as computed from the\nexact formula for (42), for the indicated ranges of m and n.\nThe help of Mr. James Bates, Mrs. Alma Schaules, Mrs. Mar-\ngaret Rethwish, Mrs. Margaret Church, and Mrs. Dolores Rein-\nbold is acknowledged in performing and checking the calcula-\ntions, at various stages of the work, leading to Table 4.\nFigure 12.5 summarizes the information of Table 4 in a\nway that reveals the m-dependence of r_(,n) as essentially\na linear function of m for 1.2 < m < 1.9 and with 1/2 S. Sup-\npose Ts, t is the transformation which assigns to each point\nX at time S the unique point (s) at which a fluid packet is\nsubsequently located at time t. Thus,\n(1)\nx(t) = x(s) T s,t\nWhatever the analytical form this transformation has, it is\nclear that it must satisfy the following simple properties:\n(2)\nTr, S T S, t = r, t\n(3)\nTt,t = I\nThe first of these properties is the semigroup property for\nthe family {Ts, t: s < t} of such fluid transformations, and\nis quite analogous to the semigroup properties encountered\nseveral times earlier in this work, especially in Chapters\n3, 7, and 8, within the radiative transfer context. A par-\nticularly deep analogy exists between the set of transformations","SEC. 12.3\nELEMENTARY HYDRODYNAMICS\n47\n(at time s)\n+\nX\n(at time t)\nThe trajectory\ny\nof a fluid packet\nPs,t(x)\nFIG. 12.9 The general trajectory of a fluid packet\nconsidered in deriving the equations of hydrodynamic theory.\n{T. t s < t } as defined above, and the transfer process for\nradiative transfer theory (see Chapter III of Ref. [216]), and\nit is on this latter analogy that an invariant imbedding (or\ngeneral group- - theoretic) approach to hydrodynamics may possi-\nbly be built. The second of the preceding relations, namely\n(3), asserts the identity property of the family of transfor-\nmations {Ts S S, then the requisite location X (t) of that\nfluid packet is obtainable by a simple integration:\n(\" v(x(t'),t') dt\nx(t)\nx(s)\n=\n+\ndt'\n(7)\nFor this reason it is customary in classical hydrodynamics\nto consider an equation of motion solved if the velocity\nfield V can be determined for a given fluid. The velocity\nfunction V also plays an important role in the further re-\nduction of the equations of motion to differential form.\nThus the acceleration (x(t), t) may be rendered into a form\nwhich is a total derivative of V along Ps. t (x) . That is,\nfrom (5) we see that a(x(t),t) is the time derivative of the\ncomposite function of velocity V and position X. This de-\nrivative is evaluated using the concepts of vector analysis.\nThe result is:\na(x(t),t) = Dv(x(t),t) Dt\n(8)\nwhere we have written:\n\"D\"\na\nfor\n+\nV\n(9)\nV\n.\nDt\nat\nand where \"V\" denotes the gradient operator. (See (5) and\n(6) of Sec. 3.15 for an earlier use of D/Dt.) This operator\nknown as the mobile-derivative, or Lagrangian derivative\noperator, is used quite generally to find the rate of change\nof some function describing a small aggregate of things mov-\ning along some path in space, whether the aggregate be com-\nprised of photons or fluid particles or other substances.\nThus if f is a general function which assigns a quantity\nf(x(t),t) to each time t and associated position X (t) of the\npacket along the path Ps,t(x), then:","VOL. VI\nAIR-WATER SURFACE PROPERTIES\n50\nf (x(t),t) - f(x(s),s)\nDf(x(t),t) Dt = lim\nt S\n= af at +\n(10)\nV\nV\n.\nIn particular, if f is the mass function for the fluid packet\nwe see from (6) that:\nDM(x(t),t) =\n(11)\nDt\n.\nEquation (11) is the equation of continuity for the mass of\nthe fluid packet. Now when the fluid packet is quite small,\nas we intend it to be in these discussions, then the mass\nM(x, (t),t) is expressible as:\nM(x,(t),t) = p(x(t),t)V(x(t),t)\n(12)\nwhere p and V are respectively the mass density and volume\nfunctions for the fluid packet. Now it is easy to verify by\n(10) that the mobile derivative operator acting on a product\nof functions works exactly analogously to the ordinary deriv-\native operator. Hence from (11) and (12) :\nDt DM = = DV + V Dp Dt = 0 ;\nwhence:\n1 Dp . 1 Dv\n(13)\np Dt + = 0\n.\nIn this way we have split the variation of M into two parts:\na purely geometric part involving the volume V of the packet\nand an ideal physical part involving the density of the pack-\net. It is easy to see that the purely geometric part of (13)\nis represented by:\n1\nDV\n(14)\nV Dt ,\nwhich is quite plausible intuitively, and incidentally an\nexcellent way of picturing the geometric significance of the\ndivergence of the velocity field.\nWith these preliminaries established we can cast the\nbasic equations of motion (6) into the more familiar dif-\nferential (i. , local) form:","SEC. 12.3\nELEMENTARY HYDRODYNAMICS\n51\nDv = dv\n+\nV.V.V =\n(15)\nDt at\n1 DM = ap\n+\nVp + pV . V = 0\nV\n(16)\nV Dt at\nwhere we have written:\n\"f\"\nfor F/M\n,\ni. , f is the force per unit mass on the fluid packet.\nAn outstanding feature of the equations of motion,\nnamely the nonlinearity of the hydrodynamic equations, is\nevident in (15) wherein the velocity function V is multi-\nplied by its derivatives in the term V Vv. It is this\nparticular nonlinearity which has launched the search for\ncountless linear and simplified nonlinear hydrodynamic\nmodels of fluids. For, by suitably choosing the components\nof f, the behavior of p, and the behavior of V itself, vari-\nous simplifications of (15) and (16) can be effected which\nlead to tractable equations of motion. We shall now adopt\nthose assumptions which lead us to the hydrodynamic models\nof the air-sea surface that are of interest in the optical\nstudies of the present chapter.\nSpecial Equations of Motion for\nthe Air and Water Masses\nThe model we shall adopt for the equations of motion\ndescribing the air-water surface and the masses it bounds,\nrests on two main sets of assumptions: one set is about the\nfluids on each side of the air-water surface, and the other\nset is about the form of the surface itself and the forces\nin its immediate neighborhood. We first consider the move-\nments of a packet X in either the air or water mass. We\nshall limit the forces on the packet X to consist only of\ngravity and normal surface pressures arising from contact\nwith other packets. (Thus viscosity forces are assumed neg-\nligible along with tidal and coriolis forces. ) The force of\ngravity on a unit mass of matter at or near the surface of\nthe earth is of magnitude g, the gravitational acceleration\nconstant (980 cm/sec2 12 32 ft/sec2) and of direction - k in\na terrestrially based coordinate frame (Fig. 12.10). If\n(y) is the unit inward normal to the packet X at a point y\non its surface S and p(y) is the associated (scalar) normal\npressure then the net force on the small packet is:\nMf = I p(y)n(y)dA(y) - k / p(x)g dV(x)\n= - / X - k /xp(x) g d V(x) X\n(17)","VOL. VI\nAIR-WATER SURFACE PROPERTIES\n52\ntowards a local description\nof the fluid motion\ndA(y)\ny\n$\nn(y)\np(y)\nx(t)\nx\ngravity force\nFIG. 12.10 Considering the forces on a fluid packet in\nthe derivation of its equations of motion.\nwhere the transition from the surface integral to the volume\nintegral 'is by means of Gauss' theorem. From (15) we have:\nM Dv Dt = Mf = - +\nX\nwhich over the small volume of X may be written as:\nVp dV 0\n=\n.\nSince X is arbitrary, it follows that (15) can be cast into\nthe form:\nDv Dt = -\n(18)\n,\nContinuing to work on this version of (15), we observe that:\n(19)\nV(g z) = g k\n.\nIn other words there is a function X , namely that whose\nvalue at point (x,y, z) is X (x,y,y,z) = g z, and whose gradient\nis gk. This allows us to write (18) as:","SEC. 12.3\nELEMENTARY HYDRODYNAMICS\n53\nDt Dv = Vp\n(20)\nwhere Z is now the z-coordinate of the fluid packet. It is\ntempting to remove the gradient operator outside the brackets\nin (20). Thus (20) supplies a strong motivation for requir-\ning p to be some nonzero constant independent of location at\nall times; in other words for requiring the fluid to be in-\ncompressible. Therefore we shall assume:\nap = 0\nVp=0\nand\n(21)\nat\nand see where it leads us. This assumption permits, first\nof all, (20) to be written as :\nDt Dv = - V [82+2 p\n(22)\nand secondly allows (16) to be simplified to:\nV . v = 0\n(23)\n.\nHence the equations of motion of the fluid are now of the\nform (22), (23) as a result of assumptions (17) and (21).\nWe next concentrate on the term V Vv in Dv/Dt. This\nis a nonlinear term and a traditional mathematical trouble\nspot for classical hydrodynamics. Its presence is eased out\nof the present picture by noting that it is the vectorial\ncounterpart to the simpler scalar case v(dv/dx). This lat-\nter term may be written as (1/2) dv2/dx. A search for a vec-\ntorial counterpart to this scalar situation uncovers the\nidentity:\n7v2\nVv\n2v\n+\nV\n(24)\nV\nX\nX\no\nV\n.\nwhere we have written\n|v|\n\" V 11\nfor\ni.e. , V is the magnitude of the velocity vector V. Using\nthis identity (24) in (22) we have:\nat dv (1/2 7v2 2v X V x v) = [8z + p\n(25)\n.\nIt seems that we have traded one complication (namely V. Vv)\nfor another (namely V x V X v). However a further simplifi-\ncation follows if we observe that the term V x V describes\nthe local rotation of the fluid motion, that is, by Stokes\ntheorem:","VOL. VI\nAIR-WATER SURFACE PROPERTIES\n54\nS ( v) n dA = So v t d ds\nwhere t is a unit tangent to any closed curve C bounding a\nsurface S in the fluid, and n is a unit outward normal to S.\nThe line integral evidentally describes the circulation of\nthe fluid around the curve C. Now in the present fluids (air\nwater) and for our present purposes, this circulation turns\nout to be a relatively unimportant motion of the fluid as com-\npared to its translatory motion. This observation permits us\nto make one more assumption, namely:\n(26)\n7 x v = 0\nin addition to (17) and (21). As a result (25) becomes:\nat av + V + p\n(27)\n= 0 . .\nBy a theorem in vector analysis, we now can assert that, by\nvirtue of (26), there exists a scalar valued function de-\nfined in the domain of the fluid (either air or water) such\nthat:\n(28)\nv=\nWe call 0 the velocity potential for the appropriate fluid.\nThe minus sign in (28) is conventional, though it can be\njustified using simple physical interpretations of o. Using\n(28), equation (27) can now be written as:\n:-[- do at + 2 1 v2 + g Z + P p = 0\nV\n.\nThis means that all three spatial derivatives of the bracketed\nquantity are zero, so that, at most, the bracketed quantity\ncan be an arbitrary function of time, say C. Hence:\n- do at + Z\n(29)\nFurther, the continuity equation (23) now becomes (using (28)):\n. (VO) = 0\n,\nthat is:\n2 = 0\n(30)","SEC. 12.3\nELEMENTARY HYDRODYNAMICS\n55\nIn this way we finally arrive at the required forms of the\nequations of motion for the air-water masses. Thus, we have\nrecast Newton's law of motion (6a) in the guise of (29) (the\npressure equation) and the law of conservation of mass (6b) in\nthe guise of (30) (the potential equation) under the following\nassumptions (repeated here for reference) :\n(i)\nV\n0\nap/dt = 0\n(incompressible fluid)\np\n=\n,\n(ii)\nV\n0\n(irrotational motion)\n=\n(iii) f consists only of gravitational and scalar\npressure forces.\nOur main task henceforce will be to solve, with the help of\n(29), the potential equation (30), subject to suitable bound-\nary conditions, so that V can be determined and hence also\nthe form of the fluid transfer operators T. t, via (7) and\n(1). The principal boundary conditions required for this\ntask are those for the air-water surface, to which we now turn.\nSurface Kinematic Condition\nThe first of the principal boundary conditions to which\nthe equations of motion (29) and (30) are to be subjected will\nnow be considered. This condition ties together the movement\nof the air-water film with the motion of the bodies of air and\nwater on either side of it.\nSuppose that 5 is the function which assigns to each\npair of spatial variables X and y and each time t the eleva-\ntion (x,y,t of the air-water surface above (or below) point\n(x,y) in some datum plane at time t. This datum plane may be\na mean sea surface, an average bottom surface, or some other\nhypothetical surface. The function 5 is the function of prin-\ncipal interest in the study of the air-water surface. It is\none of the principal problems of hydrodynamics to describe 5\nas a function of x, y, and t, given appropriate mathematical\nconstraints based on geophysical conditions. Indeed, for the\nremainder of this chapter we shall be concerned with ways and\nmeans of describing the spatial and temporal behavior of 5 in\norder that the problem of radiative transfer at the dynamic\nair-water surface can be solved on various levels of detail.\nNow suppose that at time S the center of a small water\npacket is located at point (x',y's (x',y', x)) ( (=x(s)). = Thus,\nfor all practical purposes the center of this particular pack-\net describes the location of the air-water surface at instant\nS. (See Fig. 12.11.) Suppose that at a little later time t,\nthe packet still comprises part of the air-water boundary but\nthat its center's location is now at (x,y,5 (x,y, (=x(t)).\nThen, on the one hand, since the packet journeys with the air-\nwater film over the time interval (s,t):","VOL. VI\nAIR-WATER SURFACE PROPERTIES\n56\nsurface kinematic condition\n5(x,y,t)\ny\n5(x',y',s)\n(x,y)\n(x',y')\nX\nFIG. 12.11 Following the motion of a fluid packet at\nthe air-water surface yields an important boundary condition\n(the surface kinematic condition) for the hydrodynamic equa-\ntions governing water waves.\n=\n=\nS\n= lims+t\nlisttxx\n(31)\n=\nt-s\nOn the other hand, since the packet is part of the whole\nfluid mass, we have:\n(32)\nby (4).","SEC. 12.3\nELEMENTARY HYDRODYNAMICS\n57\nBy equating the z-components of each of the vectors in\n(31) and (32), we have the desired condition on 5. Thus by\n(10), (30), and (32) , we can write:\nW = Dr\n(33)\nDt\nwhere \"W\" denotes the z-component V z of V, the other two\ncomponents being denoted by \"U\" and \"V\" Equation (33)\nis the desired connection. An alternate form of (33) may be\nobtained as follows. By (28), W may be represented as:\nW do dz\n= -\n(34)\nUsing (34) in the following expanded version of (33) :\ndo\nas\nas\nas\nU\n=\n-\n+\nV\n+\ndz at\ndx\ndy\n,\n.\nand using (28) once again to replace U and V by -derivatives -\nwe finally arrive at:\ndo as\ndo as do as\n+\n=\n-\n(35)\ndx dx dy dy az at\nwhich is called the surface kinematic condition. It serves\nto tie together the surface elevation function 5 with the\nvelocity potential 0 at the surface.\nSurface Pressure Condition\nThe second and last of the conditions required for the\nair-water surface in the present study concerns an analytic con-\nnection between the pressures within the air and within the\nwater media in the immediate neighborhood of the air-water\nsurface.\nTo see the nature of this connection first imagine the\nair-water surface to be flat calm and the air and water\nmasses to be at rest. Then each small patch of air-water\nsurface is in static equilibrium, so that all forces on it\nadd up to zero. The forces on the patch are those of the\npressing downward of air on its upper side, the resistive\npushing upward of the water below, and tensive forces acting\nwithin the plane of the film and arising from the molecular\nforces of the fluids on either side of the film.\nIf the surface tension forces were to be disturbed, as\nmay be accomplished, for example, by placing a chemical wet-\nting agent in the water, then there is an abrupt tearing mo-\ntion of the air-water film, very much like the tearing motion","VOL. VI\nAIR-WATER SURFACE PROPERTIES\n58\nof a rubber balloon that has been punctured. (This experi-\nment with water surfaces can. be performed using droplets of\ncertain household detergents or toothpastes. By this means,\nthe tension can actually be measured. It turns out that the\nsurface tension force is on the order of 74 dynes per centi-\nmeter at room temperature.) Such an experiment serves to\npoint up vividly the important role that surface tension\nplays in the configuration of the air-water surface. More-\nover, if one now blows gently on a clean air-water surface,\nthe breath of air pushes a roundish concave dimple in the\nwater and as long as the gentle air stream is maintained the\ndimple will persist and the small patch of curved surface\nwill remain in equilibrium with the three principal forces:\nair pressure, water pressure, and surface tension. At each\ninstant these forces are adding up to zero. The same phenom-\nenon occurs when one blows up a toy balloon: there is, in\nthe resultant configuration, a well-defined relation between\nthe air pressures inside and outside the balloon, the curva-\nture of the balloon, and the tensile forces within the bal-\nloon's surface. Figure 12.12(a) depicts the common essence\nof these two situations. A small rectangular patch of sur-\nface S is in equilibrium with pressures Pa and Pw (force per\nunit area, respectively, induced say by air and water masses)\nand surface tension T (force per unit boundary length) act-\ning over it. For simplicity, the surface is assumed for the\nmoment to take the shape of a circular cylinder of radius R\nand that the dimensions of the patch of surface are 2R0 by a\n(measured into the plane of the figure) Hence the area of\nthe patch is 2R0a, and the net downward force on the patch\ninduced by the pressures is:\n(Pa - Pw) 2Rea\nThis force is exactly balanced by the upward component of end the\ntension acting over the two edges of the patch indicated a\non in the figure. This upward component is -2T a0. Here T\nis force per unit length so that T a is total force acting in\nthe tangent planes to the surface on each side of the patch.\n0 is the approximant to tan 0, the actual number required to\nfind the upward component of the tension. Since the patch is\nin equilibrium we have:\n- 2T a0 = (Pa - Pw) 2ROa a\n,\n1\nwhence:\n= (Pw - Pa)R\nor alternately:\nT\n(36)\n1\nPw-Pa=R -","SEC. 12.3\nELEMENTARY HYDRODYNAMICS\n59\nForces and distances\nin downward direction\nare positive.\n0\n0\nR\ne.\n(a)\ne\n0\nT\nT1\nPa\nPw\nS\nR1\nR2\n(b)\nS\nFIG. 12.12 The surface pressure condition introduces\nsurface tension forces into the equations of motion for water\nwaves.\nwhich is the required surface pressure condition.\nObserve how the assumption that 0 is small is built into\nthe derivation. Our use of (36) below will remain within the\ndomain of this approximation.\nThe reader may now readily show, using the same principles,\nthat if the patch were not of cylindrical form but of double\ncurvature with principal radii of curvature R1 and R2 (b) of\nFig. 12.12) then\nPw - Pa = T 1 ( R 1 1 + R 1 2\n)\n(37)\nwhich reduces to (36) when R2, 2' say, is infinite. Further\ngeneralizations may be made, such as having T 1 depend on di-\nrection of extension within the surface, but such generality","VOL. VI\nAIR-WATER SURFACE PROPERTIES\n60\nwill not be needed here; indeed, (36) will suffice for our\npurposes.\nSinusoidal Wave Forms\nOur present object is to see if the equations of motion\n(29) and (30) along with the surface conditions (35) and (36)\nyield solutions of the surface function 5 which are recogniz-\nable as the periodic waves and ripples we see on the surfaces\nof ponds, lakes, seas and other natural hydrosols. That is, ,\nwe are looking for functions 5 such that:\n(38)\n5(x,y,t) = a cos(kx - ot + E)\nClearly the graph of 5 as given in (38) is sinusoidal\nand of amplitude a. The whole sinusoidal wave form moves to\nthe right, as in Fig. 12.13, with a speed such that the argu-\nment of cos in (38) is constant. In particular, if the con-\nstant value of the argument of cos is zero (so that we move\nwith a crest of the wave) then\nkx - ot = 0\nk\ndirection of motion of wave\nZ\n(amplitude) a\n2a (height)\nz(x,y,t)=zo+5(x,y,t)=\nZo\nzo+a cos(kx-ot+e)\nX\nFIG. 12. 13 A small-amplitude sinusoidal water-surface\nwave as predicted by the linearized equations of hydrodynamics\n(the contour lines are normal to the plane of the diagram).","SEC. 12. 3\nELEMENTARY HYDRODYNAMICS\n61\nimplies\nX = olk o t\nso that the speed C of the wave form is o/k. We shall write\n\"C\" for o/k .\n(39)\nThus the elevation function 5 in (38) constitutes a model of\na long-crested (i.e. , essentially cylindrical) sinusoidal\nwave, of amplitude a and phase speed (celerity) c, and\nphase E. Observe in particular that the wave surface, as given\nin (38), is cylindrical with the cylinder generators perpen-\ndicular to the plane of Fig. 12.13. We shall need to con-\nsider only such cylindrical (one-dimensional) waves in order\nto develop a workable model of dynamic air-water surfaces.\nQuite complex seas can be synthesized by suitably superimpos- -\ning wave forms of the type (38). Thus in general a one-\ndimensional sinusoidal wave train moving at an angle 0 with\nrespect to the x-axis (Fig. 12.14) may be represented as:\n5(x,y,t) = a cos (k . r\n- ot + E)\n(40)\nZ\ny\nX\nFIG. 12.14 The small-amplitude sinusoidal water wave\nof Fig. 12.13 now traveling in a general direction over the\nxy direction.","62\nAIR-WATER SURFACE PROPERTIES\nVOL. VI\nwhere we have written:\n\"k\" for (u,v)\n\"r\"\nfor\n(x,y)\n.\nand so\nk r r = ux + vy = kx cos 0 + ky sin 0\nwhere we write\n|\n\"k\"\nfor\nk\nThe length k of the vector k is the wave number of the wave\nform in (40). The wavelength l and the celerity of the wave\nform are clearly:\nl = 2/k , C = o/k\n.\nFor the most part we can simplify the exposition by set-\nting V = 0 in (40), so that the waves progress parallel to the\nx-axis. Henceforth this simplification will be in force.\nThere will be no essential loss in generality by adopting this\nsimplification.\nLinearized Equations of Motion\nWe now turn to the actual details of the search for sinu-\nsoidal solutions of the form (38) of the equations of motion\n(29) and (30), subject to the conditions (35) and (36). Some\npreliminary experimentation with these equations shows that\nthe desired solutions are forthcoming if we assume that the\nsinusoidal wave forms have small slope ds/dx (Fig. 12.13) and\nfurthermore that the speed V of the fluid packets (air and\nwater) are small so that V2 may be set to zero in (29). Hence\nwe in effect must make the assumptions:\n(iv) V 2 <<1\n(v) (ds/dx) <<1\n(vi) product terms negligible in (35)\n(vii) E (t) = 0\nwhere the numbering continues the list of assumptions begun\nbelow (30). Assumption (vii) makes use of the arbitrariness\nof the function C in (29). As a result we have from (29),\n(iv) and (vii)\n- at do + gz + = = 0\n(41)","SEC. 12.3\nELEMENTARY HYDRODYNAMICS\n63\nand from (30) and (38) (i. e. , wave forms independent of y):\n(42)\nand from (35) and (vi):\ndo = as\n(43)\n-\nFinally, the curvature 1/R of the sinusoidal wave form is\nsuch that, by (v), 1/R is essentially 225/2x2, so that from\n(36) :\n(44)\nEquations (41) and (42), are the linearized equations of\nfluid motion and (43) and (44) the associated linearized sur-\nface conditions. Equations (41) and (42) describe in either\nthe air domain or the water domain. To distinguish between\nthese domains when using o (or other concepts) we shall append\nto (or other symbols) a subscript \"a\" or \"w\", as\nthe case may be.\nClassical Wave Model\nWe now show how (41) - (44) yield sinusoidal wave forms\nin a case of extreme simplicity and of surprisingly wide\napplicability. We assume that the waves are so small and\nmildly curved that 225/ax2 is negligible so that, by (44)\nPa = Pw at the surface. Since the pressures play no further\nrole in the fluid flows, we may reset our pressure scales so\nthat Pa = Pw = 0, so that (41) becomes:\n5=10\n(45)\nat the surface. This shows how to find 5 once 0 is deter-\nmined for either the air or water mass in the neighborhood of\nthe surface. From (43) and (45) we see that:\n=\n-\nIn other words, we have:","64\nAIR-WATER SURFACE PROPERTIES\nVOL. VI\nas\n0\n(46)\n=\nat the air-water boundary. It is interesting to note in pass-\ning that this equation has the general form of a diffusion\nequation with respect to depth z, At the boundary we choose\n0 (x,z,t) to be a sinusoidal wave form of the kind:\n(x,0,t) = be _(i(kx - ot + E)\n(47)\nwhere b is an arbitrary constant and k, and o are parameters\nto be suitably determined. Using this form of 0 in (46) we\nreduce (46) to:\n(48)\nfor Z at the air-water surface.\nTaking the hint from (47) let us assume that\n= @(z)d(x,o,t)\n(49)\nin other words, that at every depth z, (x,z,t t) may be\nwritten as the product of the values of function $ of Z and\n0 (x,0,t). In this way we can set into motion a standard sep-\naration of variables technique in the solution of the partial\ndifferential equation (42). Using (49) in (42) we arrive at:\nday\n(50)\nWe have now used all four equations (41) through (44) to\narrive at (50).\nThe general solution of (50) is of the form\n+ d e-kz\n(51)\n.\nThe constants d+ may be evaluated for each hydrosol of con-\nstant depth h by noting that\n(0) = 1\n(52)\nas required by (49), and that:\n20 = 0\n(53)\naz","SEC. 12.3\nELEMENTARY HYDRODYNAMICS\n65\nat depth h. Here we are using the fact that there is no ver-\ntical motion of the fluid at the lower boundary, and translat-\ning it into mathematical form via (28). By (49), condition\n(53) may be expressed as:\n(54)\n,\nat depth h. From (51) and (52) we have:\n(55)\n=\n,\nand from (54) we have:\nekh - d e-kh = 0 ,\n(56)\nwhence:\n(57)\n-kh\n,\nand\n(58)\nWe may then express p(z) as\n(z) = cosh kh cosh k(z-h)\n(59)\nIn the case of infinitely deep media, we have for every fixed\ndepth Z :\n(60)\ncosh(kh)\n,\nso that in such media:\n(z) = e-kz\n(61)\nIn this way we arrive at the following representations for\nthe velocity potential for finitely deep hydrosols :\n$(x,z,t) = b coshk(z-h)i(k-ot) cosh (k\n(62)\n(finite depth)\nand\n(x,z,t) = b e - kz + i(kx - ot + E ) -\n(63)\n(infinite depth)","AIR-WATER SURFACE PROPERTIES\nVOL. VI\n66\nfor infinitely deep hydrosols.\nHaving found 0, we can now deduce from it a multitude\nof physical results. For example, using 0, as given in (62),\nin (28), we can find the velocity of the water packets at any\ndepth z, and any time t. Using (7) the trajectories of the\npackets are determinable in detail (they turn out to be ellip-\nses which in deep media decrease exponentially with size as\ndepth increases). However, the most important result for the\npresent studies is the determination of the celerity C of the\nwave form as given in (39). According to (39), to find C, we\nmust know o, and this in turn is characterized by (48). Using\n(62) in (48) and setting Z = 0, we have:\n1/2\ntan (kh)\nh\n(64)\nFor very deep waters (h = 00), we have:\nC = vg/k = o/k\n(65)\nOceanographers and other geophysicists occasionally prefer\nto have C in terms of the wave length 1. Since, as a perusal\nof (40) has shown,\nl = 2/k\n(66)\n(64) may be written:\n[\n1/2\n2nh\ngl\n(67)\ntanh\nC =\n2TT\nl\nand (65) becomes:\ng\no\n(68)\nC =\nwhich is the classical equation for the celerity C of deep\nwater gravity waves in terms of their wavelengths 1. Equa-\ntion (66), incidentally, shows the reciprocal relation between\nthe wave number k and wavelength 1 associated with any sinu-\nsoidal wave.\nFrom (45) and (62) we can now obtain an explicit formula\nfor the form of the air-water surface. Retaining only real\nparts (which is tantamount to using only the real part of 0\nthroughout this discussion) we have (to within some arbitrary\nphase angle E of cos)\n(x,y,y,t) = a cos(kx-ot + E)\n(69)\nwhere:\na = - b o/g","SEC. 12.3\nELEMENTARY HYDRODYNAMICS\n67\nKelvin-Hlemholtz Model\nWe finally arrive at the heart of the present discussion\nof hydrodynamics with a derivation of the Kelvin-Helmholtz\nmodel for waves on the air-water surface. The Kelvin-Helmholtz\ntheory builds on the classical wave model just completed by\nassuming two further physical features of the air-water masses:\nFirst, the air and water masses are no longer at rest, but\nrather set into motion with speeds Ua and Uw, respectively,\nalong the X axis. This simulates the movement of wind over a\nhydrosol which itself may be drifting along at some speed Uw.\nSecond, the surface tension forces are allowed to act within\nthe air-water boundary, so that capillary waves-- or ripples\nas they were called by Kelvin--can be explicitly incorporated\nin the theory of surface waves. The first of these additional\nfeatures was studied about a hundred years ago (1868) by Helm-\nholtz. The second additional feature was combined with Helm-\nholtz's hypothesis several years later (1871) by Kelvin.\nBoth men used their respective theories to carry out some of\nthe first studies of the phenomenon of wind-generated waves.\nEach model, as crude as it was, showed that there were crit-\nical wind speeds, relative to the speed of the hydrosol, at\nwhich waves of either gravity or capillary type would begin\nto grow exponentially in amplitude. Below these critical\nwind speeds, the wave forms are stable and the celerity C is\na well-defined number depending on the speeds Ua and Uw, the\ndensities Pa, PW' and surface tension T\nIn accordnace with the introductory remarks above we\nimagine the air and water masses set into translatory motion\nalong the X axis with speeds Ua and Uw, respectively, and that\na sinusoidal wave motion is superimposed on these motions at\nthe interface. Hence we assume that the velocity potentials\noa and OW for these media are of the form:\nUax + a\n(70)\nU W X + 0 W 1\n(71)\nThe\npotentials\nand are to be viewed as small oscilla-\ntory perturbations of Uax a and U X and such that da and\nand their associated motions are subject to the conditions\n(iv) through (vii) leading to the classical linearized equa-\ntions (41) through (44). Hence all our results for the clas-\nsical wave model are applicable to those components of the\nmotions generated by Qa and D.W. In particular for the pres-\nent model we shall assume infinitely deep media, so that\nand OW are given by (63) as follows:\ne +kz+i(kx-ot)\n(72)\n*For references to these models, see pp. 22, 374, 459\nof [149].","VOL. VI\nAIR-WATER SURFACE PROPERTIES\n68\n(73)\nwhere, as agreed, distance is measured positive downward into\nthe water. Furthermore since a translated sinusoid is still\nsinusoidal, the form of the air-water surface will be given\nby:\n==aei(kz-ot)\n(74)\nwhich is the complex form of (69). To return to the physi-\ncally meaningful setting we shall need only take the real\nparts of all complex expressions, as usual.\nHaving fixed the form of the velocity potential func-\ntions and the desired sinusoidal form of the air-water sur-\nface, it remains to see what conditions are imposed on the\ncelerity C of the required sinusoidal wave form by the ad-\nditional wind speed and surface tension conditions. Toward\nthis end, we return to the surface kinematic condition (35)\nand note its present forms for the air and water masses:\n(75)\n(76)\nat\nwhich follow by recalling that V = 0 for either fluid, i.e.,\nthat Va = Vw = 0, by hypothesis. Furthermore, the pressure\nequation (29) for each fluid now takes the form (let ( (t) = 0):\n(77)\n(78)\ng z\nwhich follows by recalling that, for either fluid,\nV2 = U2 + W2\nand that:\nU\n=\nW = =\nFinally, the surface pressure condition (36) now becomes\n(79)","SEC. 12.3\nELEMENTARY HYDRODYNAMICS\n69\nEquations (75) - (79) tie together the parameters k, o so that\nthe celerity C is rigidly determined for the present physi-\ncal situation. Using the representations of 5, Pa, and OW\nin (70) - (74), the requisite C may be found as follows. First,\nequations (75) and (76) become:\n(80)\na(c-up)\n(81)\nafter substituting (70)-(74) and simplifying.\nEquation (77) becomes:\n(82)\nafter using the assumptions that U and its derivatives\nare small. In a similar way, from (78):\n(83)\nWe now connect (82) and (83) by means of the surface pressure\ncondition:\nPw e8\n(84)\nUsing the current forms for ow, 5 in (84) and simplifying,\nthe result, with the aid of (80), (81), and recalling (39),\nis:\n(85)\n=\n-\nThis is basically the required condition on the celerity C.\nWe can solve for C explicitly and easily by noting that if\nUa = Uw = 0, we = = determine the celerity C o for the stationary\ncase. Setting the values of Ua, Uw in (85) to zero, we find:\n(86)\nWith this, (85) becomes:","70\nAIR-WATER SURFACE PROPERTIES\nVOL. VI\n-\nSolving this quadratic for C, and simplifying, we have:\n(87)\nwhich is the requisite expression for the celerity C of the\nair-water wave form.\nKelvin-Helmholtz Instability\nThe first thing one usually does when a quadratic\nequation is solved is to look under the radical sign to see\nwhen the radicand takes on positive, zero, or negative values.\nIn the latter case, one would then expect the roots to be\ncomplex numbers and usually some interesting physical insight\nis forthcoming in the associated physical phenomenon (cf.,\ne.g., (13)-(16) of Sec. 8.5). In the present case we note\nthat the celerity C in the Kelvin-Helmholtz model, as given\nin (87), becomes complex when\nPaPw\n(88)\nthat is, when\n(89)\nThis indicates that there exists a relative speed | U - Uwl\nbetween the air and water masses at which instabilities in\nthe wave forms may occur and grow. In short, if the wind\nblows fast enough over the water surface, waves of any\ngiven amplitude and length will build up in time. This may\nbe seen by rewriting (74) as:\nt = a e ik ( 2 - ct)\nwith C = a + iB, where B>0. Then:","SEC. 12.3\nELEMENTARY HYDRODYNAMICS\n71\nThe first factor (in brackets) is a complex number with\nfinite magnitude |a|. The second factor is an exponential\nwith positive exponent and is the one that attests to the\ninstability of the wave forms, which has been called the\nKelvin-Helmholtz instability.\nIn this way, a relatively simple model of the sea\nsurface is developed with the property that it predicts the\ngrowth of wind generated waves. A11 the terms on the right\nof (89) are computable for the air-water case, and it turns\nout that when | Ua - Uw > 6.6 m/sec for waves of l = 1.7 cm,\ninstabilities, according to this model, should occur. This\npredicted speed is somewhat higher than the observed wave\ngenerating wind speeds and lower than others predicted by\nother theories. We shall briefly reconsider this matter in\nSec. 12.9 wherein some modern theories of wind generated\nwaves are surveyed.\nCapillary and Gravity Waves\nAnother dividend of the Kelvin-Helmholtz model of\nthe dynamic air-water surface is the formula it yields for\nthe celerity of the surface waves in otherwise still air\nand water. Thus, by (87) and (86) if Ua = Uw = 0, then:\n1/2\n1\n(90)\nC\n=\no\nThis equation for C may be checked by noting that\nif we set T1 = 0 (no surface tension), and assume the hydro-\nsol (Pw = 1) is bounded by a vacuum (Pa = 0) then the equa-\ntion for C in (90) reduces to that in (65). On the other\nhand, if we could arrange hydrodynamic studies in gravity-\nfree space (as we soon will be able to do) so that g = 0, we\nwould be able to observe wave forms on the air-water surface\nwhich are sustained by surface tension forces alone, and\nwith celerity given by:\n2T\n1\n(91)\nT\nk\n=\n=\n.\n1\nThis shows that the surface tension waves increase in celerity\nwith decreasing wavelength. This is completely inverse to the\ngravity wave relation as given in (68). It follows that when\nboth gravity and tensile forces are present over an air-water\nsurface, the celerity of a given wave form is the result of\nthe combination of these two causes, and that if we could\nvary k (or 1) in (90) from small to large values we would\nfind a relatively complicated dependence of C on 1. We can\nstudy this relation best be rewriting (90) in the form):\n2TT 1 a + at l 2 TT g Pw+Pa Pw-Pa\n(92)","VOL. VI\nAIR-WATER SURFACE PROPERTIES\n72\nFunctions of this form increase without bound as A 0 or\nand have a minimum for some finite value Am of l which\nA\nmay be found by standard calculus techniques. It turns out\nthat for the air-water case:\n^m=1.7cm =\n(93)\nand that the associated minimum celerity cm is:\ncm = 23 cm/se\n(94)\nWaves with smaller or greater wavelengths than 1.7 cm\ntravel with speeds greater than 23 cm/sec. These speeds may\ngenerally be computed from the following rearrangement of\n(92)\nl\n(95)\n+\nAm\nAn approximate useful form of (95) is forthcoming if\nwe note that 1m = 3 cm, for then (95) may be reduced to:\nc = 12.5/2+\n(96)\nwhich yields C in centimeters per second when l is in\ncentimeters.\nThe minimum 1 given in (93) is both mathematically so\nwell defined and physically meaningful that it has been used\nto define the difference between capillary waves and gravity\nwaves. Thus if a surface wave in a natural hydrosol has\nwavelength greater than Am, it is called a gravity wave;\nif its wavelength is equal to or smaller than Am, it is called\na capillary wave (or ripple). By means of (92) or (95) one\ncan see that there will always be some tensive effects in a\ngravity wave and some gravity effects in a ripple, but as l\ndeparts from Am on either side of Am, one term in (92) or\n(95) will soon begin to markedly dominate.\nEnergy of Surface Waves\nWe now take up the matter of the energy of waves asso-\nciated with the surface of a natural hydrosol, for the pur-\npose of laying the groundwork for the concept of the power\nspectrum of the dynamic air-water surface.\nImagine a flat calm air-water surface with motionless\nair and water masses above and below the surface. Each water\nand air packet is motionless with respect to the terrestrial\nreference frame. Hence the total kinetic energy of the air-\nwater system is zero. The potential energy of the system\nrelative to the terrestrial reference frame is some finite\nnumber which we may take as a fiducial point and effectively\nset to zero. Now the system is set into motion, say in the\nframework of the Kelvin-Helmholtz model considered above.","SEC. 12.3\nELEMENTARY HYDRODYNAMICS\n73\nAs a result, each air packet or water packet is set into\nmotion with an oscillatory motion superimposed on a transla-\ntory motion. Thus each fluid packet has a well-defined speed\nfrom which its kinetic energy is computable at each instant.\nFurthermore, the change of position of each fluid packet with-\nin the earth's gravitational field changes the packet's poten-\ntial energy, which may now be reckoned relative to the zero\nfiducial potential energy fixed above. Finally, the tensile\nforces within the air-water surface, being brought into play\nby the wave motion, act exactly analogously to the forces of\na thin deformed rubbersheet between the air and water masses.\nThus there is potential energy built up in the air-water sur-\nface as work is done to stretch it into the shape of the pass-\ning wave form. As a result of all this motion and change of\nposition and surface deformation, there is a continual inter-\nchange between the potential and kinetic energies of the mov-\ning air-water system. As our studies of the Kelvin-Helmholtz\nmodel have shown, the surface motion and configuration com-\npletely characterizes the motion and configuration throughout\nthe entire air-water mass. Hence we may associate the kinetic\nand potential energies of the system--which in truth arises\nfrom the activity of the entire medium--solely with the waves\non the air-water interface, and conveniently speak of the\nenergy of the entire system simply as the energy of the waves.\nLet us now consider a sample calculation of the energy\nof the waves on a natural hydrosol. Figure 12.14 is a setting\nwhich is of sufficient generality in which to perform the\ncomputation. A sinusoidal wave train is moving over the xy\nplane in the direction of the unit vector W. Suppose we\nslice through the medium along the direction of w with two\nparallel vertical planes one unit distance apart and remove\nthe slice for examination. A portion of the excised slice,\none wavelength long, is shown in Fig. 12.15. As noted earlier,\nthe apparent progressive motion of the wave is induced by the\noscillations. of fluid packets in relatively small elliptical\norbits in a frame of reference locked to the main body of the\nfluid. For simplicity we shall assume Ua = Uw = 0, and con-\ncentrate only on the kinetic and potential energy changes\nbrought about by the water packets moving in these local el-\nliptical orbits.\nThe kinetic energy of a small water packet of unit\nvolume mass is (1/2) pv2 where V is its orbital speed. The\ntotal kinetic energy in the volume of the water slice of\nFig. 12.15 is:\nTOTAL (x, z) dx dz 2\ndx dz\ndx\np\nwhere the last equality is the result of an application of\nGreen's theorem to the excised volume, and where we once\nagain assume small wave slopes so that do/dz will simultate\na the normal derivative of at the air-water surface.\nUsing (48) we have:","VOL. VI\nAIR-WATER SURFACE PROPERTIES\n74\nZ\nFIG. 12.15 A vertical slice of the hydrosol for study\nof the kinetic and potential energy of a progressing water\nwave.\nand working with the real part of 0 as given in (63), this\nbecomes:\nfor E = 0 and Z = 0. Recall that the connection between b\nand the amplitude a of the surface wave is:\na = - bo/g ,\nas given in (69) Hence:\nadidas a2 g cos2 2 (kx-ot) -\nIntegrating this, as required:\ncos (kx-ot) dx\nwhere, by (66), k = 2/; so that we have at last:","SEC. 12.3\nELEMENTARY HYDRODYNAMICS\n75\ndx dz 1/ p a superscript(2) (97)\nwhich shows that the average kinetic energy of gravity waves\nper unit wavelength over the slice in Fig. 12.15 varies di-\nrectly as the square of the amplitude of the wave, p and g\nbeing fixed constants. Recalling that the slice is of a\nunit thickness, (97) shows that, alternately, (1/4) pa g is\nthe average area density of kinetic energy at each point in\nthe horizontal plane over which the wave train in Fig. 12.14\nis moving.\nIt is now easy to show, without the necessity of further\ndetailed calculation and using only energy conservation con-\nsiderations, that the the potential energy of the displaced\nwater mass slice of Fig. 12.15 is precisely (1/4)pa2g (see,\ne.g., Art. 174 in Ref. [149]).\nFinally, the potential energy of stretching of the sur- -\nface from a straight line into the sinusoidal curve, in Fig.\n12.15, is obtained by simply applying the formula: work equals\nforce times distance. Here T1 is the force, and the distance\nis the difference between the wavelength of the wave and its\nactual length considered as a curve:\nT\ndx\n(kx-ot)\ndx\n.\nTotal energy E(k) of the wave per unit horizontal area is E\nthen: *\ni.e.,\n((k) = K(k)a2\n(98)\n*Oceanographers frequently use wave height H instead of\namplitude a, where H is measured from crest to trough, so that\na = H/2.","VOL. VI\nAIR-WATER SURFACE PROPERTIES\n76\nwhere we have written:\n\"K(k)\" for , 2\nThus, all other factors being fixed, the total energy density\nE per unit horizontal area, associated with a sinusoidal wave\ntrain of amplitude a, varies directly as a . Observe that\nthe energy density E is independent of l (or k) for the pure-\nly gravity component of the energy, but that it depends on k2\n(or 1/X2) for the capillary component of the energy. Thus\nas a wave is imagined to shrink down to capillary size, we\nsee that capillary waves could in principle store a sizable\nfraction of the energy of motion on steady state-wind-blown\nsurfaces, especially freshly wind-blown surfaces over which\nthe gravity waves have not yet built up. However, under\nfully risen seas driven by strong winds, the gravity waves\ntake the lion share of the total energy. This matter will\nbe discussed quantitatively in (27) of Sec. 12.8 when enough\ntheoretical machinery will have been constructed and enough\nempirical knowledge will have been gained. Observe finally,\nthat the energy density E of a wave train is independent of\nthe direction of travel k of that wave train.\nSuperposition of Waves\nAs one of the final topics in the present development\nof hydrodynamics for hydrologic optics we observe an extreme-\nly important property of the classic and Helmholtz wave models\nstudied above. This is the readily verified property that\nthe sum of two velocity potentials P1,P2 associated with two\nwave trains 51 1,52, each train being governed by the linearized\nequations (41) - -(44), is again a solution of the set (41) - (44).\nAt this point the reader should verify that, by a simple rota-\ntion of axes, (41) - (44) are transformed to forms which hold\nfor (40). This means that, in view of (45), the linearized\nwave models can be generalized to describe air-water surfaces\nwhere functions 5 are linear combinations of arbitrary finite\nnumbers of one-dimensional wave trains of the kind pictured\nin Fig. 12.14. As a result, the dynamic air-water surfaces\nof many types of wind-blown hydrosols can be arbitrarily\nclosely represented by linear combinations of the kind:\np\n5(x,y,t) = { a cos n + E n )\n(99)\nn=o\nwhere p is an integer and where kn (=(un,vn)) is the vector\nwave number defined for (40) The minus sign before o is\nchosen so that the associated wave component with wave number\nK travels in the direction of kn. Alternatively, we may\nwrite (99) as :\np\n(x,y,y,t) a cos n\nn=o","SEC. 12.3\nELEMENTARY HYDRODYNAMICS\n77\nThe nth wave train again travels in the direction of Kn' and\nhas celerity cn, where:\nC n = o n /k n\nand where:\n2\n2\nk n =\nn + V n\n.\nIn mathematical discussions of the dynamic air-water surface,\nfor analytical convenience, all finite stops are pulled out\nin (99) by setting p equal to 80, so that (99) becomes a\nFourier series representation of the function 5. In con-\ntrast to this, when p < 8 , (99) is the finite Fourier series\n(or Fourier polynomial) representation of 5. A convenient\ngraphical means of picturing the wave train components of the\nFourier representation of an air-water surface, as in (99),\nis to plot the vector wave numbers kn on the uv plane, as in\nFig. 12.16 (b) .\nThe vector in Fig. 12.16(b) represents the wave train\ndepicted in (a) of that figure. Thus the direction of k\ngives the direction of travel of the train, and its magnitude\nk (= k) contains the means of computing the wavelength of\nthe train (1 = 2/k). This vector characterization of a wave\ntrain and other superpositions can be carried out in some de-\ntail using analogies with force vectors in mechanics. For\nexample Fig. 12. 16 (c) shows the vector wave-number means of\nfinding the resultant of two wave trains. For if\nV\ny\nV\nA\nki\nk2\nX\nu\nu\n(c)\n(a)\n(b)\nFIG. 12.16 Depicting a sinusoidal wave by means of\nthe k-vector.","78\nAIR-WATER SURFACE PROPERTIES\nVOL. VI\na cos\nand\na cos\nare two wave trains, their sum is representable as:\n2a cos (Ak - Aot) cos (K-)\nwhere\n2\nand\nso\n.\n2\nFrom wave-vector diagrams one can, after some practice, tell\nat a glance the general appearance of a sea. Figure 12.17\nillustrates some representative \"seas.\" In Fig. 12.17(a) we\nhave wave vectors clustered about a single direction and of\nrelatively small magnitude. Since small k indicates large\nwavelength, (a) depicts a heavy swell configuration with a\nwell-defined direction. Figure 12.17(b) is also a highly\ndirectional sea but of relatively smaller wavelengths. Figure\n12.17 (c) and (d) indicate jumbled seas with criss-crossing\nwave trains, with slightly larger wavelengths on the average\nin case (c) than in case (d). We shall return to the discus-\nsion of the Fourier representation of the air-water surface\nin Sec. 12.4.\nSpectrum of the Air-Water Surface\nThe present discussion of the hydrodynamics of the air-\nwater surface concludes with one of the more significant con-\ncepts to be added to the repertory of oceanographics in the\npast decades, namely the concept of the spectrum of the sea\nsurface, or air-water surfaces for general hydrosols. As we\nshall see, this concept also plays an important role in the\nstudy of radiative transfer across dynamic air-water surfaces.\nThe basis for the concept lies in the Fourier series repre-\nsentation of the air-water surface and this, in turn, rests\non the classical wave model developed earlier in this section.","SEC. 12.3\nELEMENTARY HYDRODYNAMICS\n79\nV\nV\n+\n(a)\n(b)\nu\nu\nV\nV\n+\n+\n(c)\n(d)\n+\nu\nu\n+\n+\n+\n+\nFIG. 12.17 Depicting the appearance of a sea surface\nusing the k-vectors of its component sinusoidal waves.\nA preliminary description of the spectrum of an air-\nwater surface would be achieved by simply pointing to the set\nof all coefficients an in the finite Fourier representation\n(99) and saying that the set of all the a comprises the\nspectrum of the elevation function 5. Indeed, if 5 is\nexactly represented by (99), then the spectrum of 5 is the\nrange of a function A which assigns to each kn (= (Un,Vn))\nthe number\nn , n = 1,\n(100)\np.\nThe spectrum A associated with a finite Fourier series repre-\nsentation of 5 is described as nondense, discrete because of\nthe finite number of separate wave numbers kn involved in the\nrepresentation. The energy spectrum of 5 in (98) is the set\nof all numbers (1/2) am, n = 1, p. The factor\n\"1/2\"\nis\nincluded for formal reasons which will become clear later in\nthis discussion and in (35) of Sec. 12.4. However, the rea-\nson for squaring the rests in (98) of Sec. 12.3.\nIn the case of the air-water surface for natural hydro-\nsols it is usually found that a great range of wave numbers\nis associated with the analyzed surface function. These num-\nbers are closely packed together and found to occur virtual-\nly everywhere on extensive reaches within the wave number\ndiagrams of the kind displayed in Fig. 12.18. We shall say\nthat the spectrum of 5 is dense, discrete over a region R\nof the uv plane if every neighborhood W (k) of every point","VOL. VI\nAIR-WATER SURFACE PROPERTIES\n80\nW\nV\nu\nAu\nFIG. 12.18 The k-vectors of a sea surface with a\ndense discrete spectrum.\nin a has points kj k in it for which the associated\nai in the Fourier Series are not zero.\nA dense discrete spectrum function is still of the form\n(100) but now it has a countably infinite number of ai in its\nrange (i.e., set of values). Since the k's are densely packed\nover various parts of the uv plane in the case of a dense\ndiscrete spectrum, it is possible in principle to define a\nspectral density function, much in the way we defined irradi-\nance in Sec. 2.4 as a flux density function. Indeed, the\nsame general properties of additivity and continuity can be\nused to make rigorous the heuristic discussion on which we\nnow embark. To see how such a definition will go in outline,\nconsider an arbitrary region W in the uv plane for an air-\nwater function 5. In this region there is a set of k's whose\nindices run over some set J(W) of integers. We next select\nthe square a? of all the coefficients ai in the Fourier ser-\nies representation of 5 whose indices i are in J(W) and\nform their sum:\nziefewsi\n(*)\niej(W)\nNow let k (= (u,v)) be a point in and for simplicity W\ncould be a rectangular region of sides Au, Av, so that its\n\"area\" A(W) is AuAv (Fig. 12.18). At any rate we can form\nthe quotient:\n1\nA(W)","SEC. 12.3\nELEMENTARY HYDRODYNAMICS\n81\nWe shall assume that this quotient has a limit as W+{k}\n(here the additivity and continuity properties of with\nrespect to W would enter) and we shall write:\n\"E(k)\" or \"E(u,v)\" for lim\nW+ (k) A(w)\n(101)\nThis function E which assigns to each k the number E(k) is\ncalled the spectral (energy) density * function (or the\nenergy spectrum). The connection between the spectral den-\nsity function E given in (98) is clearly:\n(102)\nfor a small region W about point k of area A(W) (= AuAv).\nThus to find the total energy density E per unit horizontal\narea over the sea surface** we perform either the sum:\nor the integration:\n5.5\n8\n(u,v)k(k) du dv\nwhere:\n=\n.\nThe former (infinite sum) operation is useful when the a\nare known directly, the integral when E is known from prior\nanalysis, or, when it would be expedient to use the calculus.\nFinally, according to (101), the arbitrary form of W,\nand the preceding equality, we deduce that:\nis\n8\n(u,v)\ndu\ndv\n(103)\n.\n- 00 - 00\n*Occasionally \"intensity\" is used instead of \"density.\"\nThe logical basis for the summation of the a to obtain\ntotal energy rests in the derivation of (98) now for the case\nof O being a finite linear combination of orthogonal sinusoids.\nThen the passage to the infinite limit may be made.","VOL. VI\nAIR-WATER SURFACE PROPERTIES\n82\nWe have now taken the exposition of classical hydrodynamics\nas far as we need to in the present work. To proceed any\nfurther along the present path would take us into the domain\nof harmonic analysis of air-water surfaces. This study is\nreserved for the following section. For the present it suf-\nfices to note that we have laid the groundwork for an intui-\ntive understanding of the spectral density function asso-\nciated with a dynamic air-water surface. The spectrum of\nthe elevation and spectral density function play the role of\ncentral unifying concepts in the several important problems\nconcerned with the dynamic air-water surfaces. Mathematical-\nly, the spectrum is equivalent to knowledge of the coeffi-\ncients of the Fourier series for the elevation function 5.\nPhysically, the spectral density function has manifold appli-\ncations. On the one hand it has been used in one of its\nearliest applications to explain microseisms generated by the\ndynamic air-water surface [167]. On the other hand it is use-\nful in describing the reflectance properties of the sea sur-\nface with respect to irradiation by radar, sound, and light\n[25], [85], [56]. In the present work we shall show that the\nspectral density function is closely connected with the solu-\ntion of radiative transfer problems at the air-water boundary\nof natural hydrosols (Sec. 12.9) But before we relate it to\nradiative transfer problems it will be of considerable help\nto have a battery of associated harmonic analysis concepts at\nhand which will facilitate the formulation and solution of\nthese problems and which will further the general discussions\nof recent experimental and theoretical studies of the physical\nand geometric properties of the dynamic air-water surface. To\nthis task we now turn.\n12.4 Harmonic Analysis of the Dynamic Air-Water Surface\nWe shall devote some attention in this section to the\ntopics in harmonic analysis required for our present studies\nof radiative transfer across the dynamic air-water surface.\nThe battery of concepts of harmonic analysis, as they are\napplied to the air-water surface, are relatively new, having\nbeen intensively applied during the past decade by increasing\nnumbers of workers in mathematical and experimental oceanog-\nraphy. A survey of the history of the subject is out of\nplace in this work, but it can be begun by consulting the\nreferences, [320], [307], [191], and others listed during the\ndiscussion below.\nOur primary aim in the discussion below is to prepare\nthe ground for answering some of the initial basic questions\nraised by researchers entering this domain of ideas for the\nfirst time. The most frequently occurring questions are:\nWhat are the sources of the ideas of harmonic analysis?\nWhat is the difference between harmonic analysis and synthe-\nsis? Sometimes one sees Fourier integral representations of\nan analyzed function, and other times a Fourier series repre-\nsentation. Is there some way of deciding between these two\nmodes of representation for a given context? Is there some\nspecial justification for choosing the tools of harmonic\nanalysis for use in describing the sea surface and the dynam-\nic surfaces of natural hydrosols in general? Even if such\nharmonic analyses of the dynamic air-water surfaces can be\nmade, why is the energy spectrum singled out for so much","SEC. 12.4\nHARMONIC ANALYSIS\n83\nattention? What are some of the things one can do with the\nnotions of harmonic analysis in pursuing the studies of\nhydrologic optics?\nThe Roots of Harmonic Analysis\nModern harmonic analysis may be said to have begun\nwith the seminal paper by Wiener [320] in 1930 which was\nconcerned with the rigorous mathematical foundation of the\nmethod of periodogram analysis. The immediate mathematical\nbasis of Wiener's work rested on that of Plancherel [207] a\nmathematician of the early twentieth century. Periodogram\nanalysis was begun by Schuster [278], [277] in 1897 in his\nstudies of geophysical optical and magnetic phenomena.\nSchuster's concern was with the detection of \"hidden\" perio-\ndicities in these physical processes as they evolve in space\nand time. One of the principal optical conundrums of that\nday, to which Schuster addressed himself and his periodogram\nanalysis method, was the nature of the composition of white\nlight. The physicist Gouy [100] used Fourier's series (which\nin turn date back to the first decade of the 1800's in Four-\nier's studies of heat [93]] to represent white and other com-\nposite flows of light. Implicit in Gouy's analysis was the\nbelief that white light did in fact consist of a composition\nof distinct individual flows of colored light. However, in\nthose early days, the sophisticated way of viewing \"white\"\nand other \"mixtures\" of light, namely as superimposed quan-\ntized electromagnetic fields, had not yet been evolved, so\nthat for a time the natural philosophy of light as envisioned\nby Gouy and others was partially beset by Schuster's conclu-\nsion that when \"white\" light was analyzed by a diffraction\ngrating, the monochromatic components were manufactured on\nthe spot by the special geometric arrangement of matter in\nthe grating. This is only a partial statement of the present\nview, and it turns out that Gouy and Schuster both saw only\none facet of reality. In the present view, light may be con-\nceived as having in reality a composition synthesized of pre-\ndominantly monochromatic wave trains of light which manifest\neither particulate or wave structure (and hence color) which\nthen may be observed or not depending on what mode of obser-\nvation is used. Thus the present view is that the actual\nobservation can either select or manufacture the appropriate\nradiometric component from the composite field, depending on\nthe state of the observed field and the state of the observ-\ning instrument. (See, e.g., [150], [151].)\nThe physical basis for the harmonic analysis of light,\nas begun by Schuster at the turn of the century, currently\nrests in Maxwell's equations (in classical or quantized form)\nwhose solutions may take the form of sinusoidal functions\n(plane waves) and which in turn, because of the linearity of\nMaxwell's equations, may be synthesized as linear combinations\nof those sinusoids. Most of the modern field theories, while\ndiffering from Maxwell's equations in form and content, never-\ntheless share these two essential features (eigenfunctions and\nlinearity) with their logical ancestors, so that these impor-\ntant physical bases for harmonic analysis still stand today.\nAs we have seen in (99) of Sec. 12.3, and as we shall see later","VOL. VI\nAIR-WATER SURFACE PROPERTIES\n84\nin more detail, the physical basis for the harmonic analysis\nof the air-water surface rests in the linearized equations\nof hydrodynamics.\nHarmonic analysis, then, originated principally in the\nwork of Fourier, Plancherel, Wiener, Schuster, and then fanned\nout into the work of a host of physicists and mathematicians\nsince Wiener. For a historical sketch of Fourier analysis,\nsee [38]. Some modern mathematical references on harmonic\nanalysis are [170], [269], [162]. Working references on har-\nmonic analysis are [313], [21], [321], [38], [24], [294], and\n[191]. Two especially useful reference texts on harmonic\nanalysis are [152] and [29]. The powerful concept of distri-\nbutions promises to become a working tool in the harmonic\nanalysis of the physically more realistic functions (nonsum-\nmable over noncompact domains). The studies [89], [159], [31],\n[327] make a beginning in this direction, and serve as supple-\nments to Wiener's pioneering studies in [320].\nHarmonic Synthesis vs. Harmonic Analysis\nThe term \"harmonic synthesis\" denotes the addition (or\nsuperposition) of a finite or infinite number of sinusoidal\nfunctions to form a new function. An example of a harmonic\nsynthesis is given in (99) of Sec. 12.3 In that case the\nelevation function 5 describing the dynamic air-water surface\nwas the result of the synthesis. The term \"synthesis\" thus\ndenotes a building-up or construction of an object from sim-\npler components. \"Analysis,\" on the other hand denotes the\nbreaking-down or taking apart of an object into simpler com-\nponents. \"Harmonic analysis\" therefore denotes the analysis\nof an object into its harmonic or sinusoidal components.\nBeyond these simple definitions lies an interesting and\nrelatively deep distinction between \"analysis\" and \"synthesis.\"\nof these two ideas, that of synthesis of an object appears to\nbe a relatively straightforward process: given certain compo-\nnents, we can put them together in a prescribed manner and end\nup with a composite, synthesized end-object. Thus, a synthe-\nsized object presents no mystery about what went into its\nstructure. On the other hand, when confronted with a given\nobject, (say white light, or the sea surface) which comes to\nus from nature as an unanalyzed apparently indivisible whole,\nthere seems to be an element of artificiality and arbitrari-\nness in the subsequent analysis of the given primitive object\ninto the preselected components (say sinusoidal functions)\nThat is, while conceptual synthesis of objects seems to be\nstraightforward, conceptual analysis of objects on the other hand,\nraises the question of the reality of the analytical compo-\nnents in the original object.\nSome thought shows that in the case of the dynamic air-\nwater surface, just as in the historic case of the analysis\nof white light, the recording or observing of various harmonic-\nor sinusoidal--components that is, the observed presence of\nwaves of a given wavelength, depends jointly on the mode of\norigin of the waves and their mode of observation. Thus if\none drops a pebble into an otherwise still air-water surface,","SEC. 12.4\nHARMONIC ANALYSIS\n85\nan instant later there is a set of small circular ripples\nspreading out from the point of entry of the pebble. If we\nchoose to analyze this dynamic surface by means of the plane\nwave components cos (k . r ot + E) introduced in Sec. 12.3,\nit is clear on the one hand that there is not a single plane\nwave in sight in the system of circular ripples so that a\nvisual analysis into plane wave components is impossible.\nOn the other hand, it is a simple matter (and a stroke of\ngenius) to mathematically analyze this disturbance into a\nFourier series of plane wave components. This analysis can\nbe accomplished with arbitrarily great precision in practice,\nand exactly in principle, and therein lies the importance of\nFourier analysis: It is a convenient tool with simple analy-\ntic properties. Here, then, is an instance where the observ-\ning instrument (the mathematical theory or one of its hardware\nrealizations) can manufacture plane-wave harmonic components\nand endow the analysis of the object (the circular ripples)\nwith elements not inherent in the object. However for many\npractical and theoretical purposes of conducting scientific\nwork this forced and willful analysis is satisfactory.\nIn this way those readers coming on the notions of har-\nmonic analysis for the first time can be prepared to view har-\nmonic analysis as a useful powerful tool which, in its resul-\ntant representations, may or may not use components inherent\nin the original object. The importance and worth of the anal-\nysis therefore rests ultimately not in the conceptual objects\nit resolves physical data into (for these, as we have just\nseen, may have no natural counterparts) but whether the anal-\nysis can be uniquely reversed by a synthesis which faithfully\nyields all the features of the original physical process se-\nlected for representation.\nAnother simple example from natural everyday activity\nmay serve to emphasize the preceding viewpoint of harmonic\nanalysis. Imagine that an observer is standing on a hillside\noverlooking a quiet sunlit meadow in the center of which is a\nsmall pond. He spies a hawk suddenly swoop straight down out\nof the sun and alight on a small furry creature several yards\nto the right of the pond's edge. Notice now how the sun and\nthe pond played central roles in helping the reader to visual-\nize the scene. They serve as primitive coordinates relative\nto the observer in locating the preceding activity in the\nspace above and on the meadow. This descriptive activity is\nformalized in theoretical discussions by replacing the hawk's\ninstantaneous position by a vector V. Once a coordinate frame\nhas been selected, V can then be analyzed into components\nalong the x,y and Z axes in the customary manner. Thus:\nv.i)i+(v.j)j+(v.k)k\n(1)\nV\n=\nis the representation of V relative to the components (i,j,k).\nThe main point now being made is that the choice of the compo-\nnents (i,j,k) is completely arbitrary, and that neither these\ncomponents nor any others are inherent in the structure of V\nand in the original pastoral scene. Yet it is manifest that\nfor all practical purposes this kind of analysis is useful\nand occasionally indispensable in painting faithful symbolic\npictures of reality. The preceding analysis of V is of the\nsimplest kind of geometric analysis. Yet, as we shall see,","VOL. VI\nAIR-WATER SURFACE PROPERTIES\n86\nit shares the algebraic heart of the concept of harmonic\nanalysis with the most awesomely complex examples ever, or\nyet to be manufactured. For, the dot products V i, V j,\nV . . k are a form of analysis of V and as such are analogous\n(or correspond) to taking the Fourier transform of V, or\nFourier integration over a finite interval of the function\nV. The adding up of the components (v i)i, (v j)j, and\n(v k)k is a form of synthesis and is analogous to the sum\nof a Fourier series, or the inverse of the Fourier transform\nof V.\nIntegrals vs. Series in Harmonic Analysis\nThe question of whether one uses Fourier integrals or\nFourier series to represent a given function f is often re-\nsolved by deciding on the domain of f. If the domain of f\nis\n(closed and) bounded, then the representation of f may use\nFourier series. If the domain of f is (not closed or) not\nbounded, then the representation of f may use Fourier inte-\ngrals. Here are some examples of closed, bounded domains:\n[0,2], [-2, 2n], [a,b], where a,b are finite numbers. Two\ndimensional closed, bounded sets are: [0,2]] x [0,2], [a,b]\nx [c,d], where a,b,c, and d are finite numbers, etc. In each\nof these examples the endpoints of the intervals are included\nand the sides of the rectangles are included in the domain,\nand this is what makes the intervals closed. We have included\nthe condition \"closed\" in parentheses above, since this condi-\ntion is of secondary importance in geophysical practice.\nExamples of unbounded domains are: [-00,00], [0,00]\n[a,wo] a finite. Further: X [-00,00], [a,00] X [b, oo] are\nexamples of two-dimensional unbounded regions. These domains\nare closed or open depending on whether the endpoints or sides\n(as the case may be) are all included or all omitted, respec-\ntively. The mathematical basis for the preceding criteria may\nbe found in [269] or [170].\nIt is clear that if f is defined over some finite plane\nrectangle D which represents an extensive portion of the sea,\nthen the whole infinite plane containing D may be partitioned\ninto a checkerboard using copies of D over each copy of which\nthe structure of f is repeated. It is clear also that, while\nf can be so extended to an infinite domain, its essential do-\nmain is finite and consequently its representation may be\neffected by Fourier series.\nFourier series representations of the dynamic air-water\nsurface may then suffice for all conceivable practical situa-\ntions in hydrologic optics. Fourier integrals are not excluded,\nhowever, but their principal role, except in the simplest cases,\nwill be that of a tool to be used in theoretical studies of\nthe air-water surface prior to or instead of numerical studies.\nIn the present exposition, therefore, we may and shall use\nFourier series or integral representations as the need for\neach type of tool arises. The practical criteria for the use\nof one or the other mode of representation are thus clear: to\nstudy a finite (albeit large) region of the air-water surface,\none can use Fourier series; to study an infinite region (i.e.,\nE ) of the air-water surface where most of the energy of the\n2","SEC. 12.4\nHARMONIC ANALYSIS\n87\ndisturbed surface is contained in a finite region (e.g., a\nstorm center) outside of which the energy of the surface can\nbe made arbitrarily small, one can use Fourier integrals;\nand finally to study the surface in an arbitrary steady state\nover an infinite region one uses the Fourier-Lebesgue-Stieltjes\nrepresentation, to be described below. The Fourier series\nrepresentation thus is adapted to handle what have been cus-\ntomarily called the periodic functions (because they may be\nprescribed essentially over a finite region and the remainder\nof their extent is obtained by mere replication); the Fourier\nintegral represents what are called transient functions or\naperiodic transient functions; and finally the Fourier-\nLebes gue-Stieltjes representation is relatively recent and\napplies to what are called random functions or aperiodic sta-\ntionary functions (mathematically they are simply bounded\nmeasurable functions). Interesting parallels of the present\nrepresentation problems with those arising in communication\nengineering may be studied in [152].\nFourier Series Representations\nof the Air-Water Surface\nA formal representation of the dynamic air-water sur-\nface by means of Fourier series may be obtained as follows:\nLet f be a real or complex valued function defined on the\nclosed interva-l [a,a+2p] of the real line whose real or\nimaginary parts describe the surface along a given directed\nline. Here a and p(>0) are arbitrary finite real numbers\nfixed throughout this discussion. It was originally shown in\nessence by Fourier (Art. 171 of [93] et seq.) that the value\n(x) of f at each X in [a,a+2p] may be represented by trigo-\nnometric series of the form:\n8\nnux\nnTX\n(\n{\n)\na\nb\ncos\n+\nsin\n(\n)\nn\np\nn\np\nn=1\nn=1\n(2)\nwhere the sign \"n\" means that the trigonometric series on\nthe right was manufactured by constructing the coefficients\nan and bn from f in the following manner. To be specific,\nlet f be real valued. (If f is complex valued, we work sepa-\nrately with its real and imaginary parts. ) We write:\n\"a\n11\nfor\nf(x) cos ( p dx\n(3)\nn\na+2p\n1\nnTx\n\"bn\"\nfor\nf(x) sin dx\n(4)\nwhere n = 0, 1, 2,\nThe series (2) is the Fourier series\nassociated with f. Practical working conditions on f which\nwill ensure that its associated Fourier series converges to f\nin well-defined ways, are the well-known Dirichlet conditions","VOL. VI\nAIR-WATER SURFACE PROPERTIES\n88\nwhich may be found for example in [38], and which certainly\ncover many of the hydrodynamic situations encountered in\nhydrologic optics.\nIt is possible, by adopting complex variable concepts ,\nto collapse (2) and the formulas (3) and (4) into very com-\npact forms without any essential loss of physical signifi-\ncance. The advantage gained in doing so rests principally\nin expediting subsequent formal manipulations of the series\nand in uncovering the essential algebraic ideas behind Fourier\n(and, generally, harmonic) analysis. When the manipulations\nare over and numerical work is to begin, the following steps\ncan be reversed. To begin, write:\nnux\n\" y \" for p\nand recall from complex analysis that:\neiy = cos y t i sin y =\n(5)\n,\nwhich is Euler's formula. Next, from (3), (4), and (5) :\ne\ne -iy dx\n.\nLet us now write:\n\"Cn\" for\n\"C-n\" for\n\"co\" for ao/2\nThen:\n+ f(x) dx an\ncos y =\ncn - p a+2p f(x) sin y bn\nFrom these two equations and (2) we see that:","SEC. 12.4\nHARMONIC ANALYSIS\n89\nsin y\ni sin y ] + C-, y sin y ]\n.\nbe\nHence\n(6)\ne\nand\ne dx\n(7)\nfor n = 0, +1, +2,\nTo summarize the preceding procedure, we may say that\nif f is a real (or complex) valued function of one variable\ndefined over [a,a+2p], then its associated Fourier series, in\ncomplex form, is obtained by the construction summarized in\n(6), , in which Cn is found from f according to the operation\nin (7).\nIt is particularly important to note that the series in\n(6) , while ostensibly complex valued, is real valued if f is,\nand this may be readily seen by the working out of a particu-\nlar example or retracing the steps from (7) to (2).\nCommon occurrences of the parameter a are 0 and - IT,\nwhile that of p is most often TT. Thus, for a = 0, P = IT,\n(6)\nand (7) become:\n\"\n-inx\ndx\ne\nWe now show how functions of several real variables may\nbe methodically analyzed into their Fourier series representa-\ntions by repeated applications of (6) and (7) . We consider\nthe elevation function 5 which represents the dynamic air- -\nwater surface over the closed spatial rectangle [a,a+2p]\nX [b, b+2q] and over a time interval [c, c+2r] where a,b,c and\np, q,r are all arbitrary finite real numbers. We begin by\nfixing the time t in [c,c+2r] and the y coordinate in [b,b+2q]","VOL. VI\nAIR-WATER SURFACE PROPERTIES\n90\nwith the result that 5(.,y,t) is a function of a single vari-\nable X over the interval [a,a+2p]. Then we have at once from\n(6) and (7):\n(8)\n(9)\ndx\nfor every y in [b,b+2q] and t in [c,c+2r] and m = 0, , +1, =2,\nKeeping t fixed in [c,c+2r], now let y vary in [b,b+2q]\nso that we can apply (6) and (7) to the Fourier analysis of\nthe function Cm(.,t):\n(10)\n(11)\n.\nCombining (8) and (10), we have:\n5(x,y,t) 2\n(12)\n=\nwhere:\nxya dy\n(x,y,t) dy\n(13)\nWe can apply (6) and (7) once again, now to the function\nBmn ()) and find:\n(14)\ndt\n(15)","SEC. 12.4\nHARMONIC ANALYSIS\n91\nUsing (14) in (12) and (13) in (15) and reducing the results,\nwe finally arrive at:\n5\n2 m=-00 n=-00 mns\n(16)\nS=-00\nb+2q C+2r -in r dx dy dt\n(17)\nwhich is the desired Fourier series representation of 5 over\nthe space-time block [a,a+2p] x [b,b+2q] X [c,c+2r].\nHydrodynamic Basis for Harmonic\nAnalysis of Air-Water Surfaces\nThe Fourier series representation (16) of a general air-\nwater surface over a space-time block was derived mechanically\nand on a strictly mathematical level. We now return to that\nrepresentation and introduce some appropriate concepts from\nhydrodynamic theory. One immediate result will be a formal\nsimplification of the representation of 5 (x,y, t) to one which\nuses only a double sum over the spatial index m and n. This\nsimplification is made possible by introducing the equations\nof motion of the air-water surface into the analysis and let-\nting them carry some of the burden of describing 5 over the\ntime interval [c,c+2r]. .\nTo begin, set C = 0 and let the initial displacement\nconfiguration of the air-water surface be given by 5(x,y,0)\nfor X and y in [a,a+2p], [b, ,b+2q], and let initial vertical\ndisplacement speeds be given by 25(x,y,0)/at, for X and y\nover the same intervals. Suppose further that the subsequent\ndisplacements during the time interval [0,2r] are described\nwithin the rectangle [a,a+2p] X [b,b+2q] by 5. Then a Fourier\nseries representation of 5 may be obtained as follows. First,\nlet t = 0, so that:\n8\n5(x,y,0)\n2\n(18)\nn=-00\nwhere\n(x,y,0,) e -in ny q\ndx\ndy\n(19)\n.\nFurthermore, let:\n25(x,y,0) at 2 ny\n(20)\nn=-00","VOL. VI\nAIR-WATER SURFACE PROPERTIES\n92\nwhere:\nq\ndy\n.\n(21)\nNow from (16) we see that at t = 0, on comparison with (18) :\nand by differentiating (16) with respect to t, setting t = 0,\nand comparing it with (21) :\nA\nmns\nThis merely serves to suggest that in the case of (18) and\n(20) we should also expect a special condition linking Amn\nand Bmn . Some experimentation shows that the requisite condi-\ntion is:\n(22)\nwhere we have written:\n\"Omn\" for Km (mm h)\n(23)\nand where in turn we write:\n(24)\n\"Kmn\" \" for\nThe basis for the condition (22) rests in (64) of Sec. 12.3,\nsince for each pair (m, n) of integers, we have an associated\nplane wave component of the dynamic surface whose parameter\nO mn is governed by g and the depth h of the medium. More\ngenerally, we could write:\n(25)\n\"Omn\" for\nusing (90) of Sec. 12.3. In any event, by imposing the condi-\ntion (22), we let the natural motions of the waves take up the\ntask of describing 5(x,y,t) for t in the range 0 t r having","HARMONIC ANALYSIS\n93\nSEC. 12.4\ninitially prescribed E(x,y,0), and ag(x,y,0)/at. That is, by\n(19), (21), and (22),\ndy\n(26)\nwhich fixes the values of Amn given the initial conditions of\nthe wave motion. Then write:\n\"z(x,y,t)\" for\nIt follows that:\nz(x,y,0)\nand that:\nat\nWe now appeal to the theorem in hydrodynamics which states\nthat a fluid undergoing irrotational motion (cf., , assumption\n(ii) below (30) in Sec. 12.3) has its motion uniquely deter-\nmined for all t > 0, given its initial displacement and ini-\ntial displacement speed (see, e.g., Art. 57 of [149] or Sec.\n3.77 of [181]). Hence, by the uniqueness of the subsequent\nflows:\n5(x,y,t)=z(x,y,t) =\nfor t in [0,r]. That is:\n(27)\nm= - 00 n= - 00\nfor (x,y,t,) in the given space-time - block.\nThe preceding representation of 5 (x,y t) can be returned\nto real notation by retracing the steps from (2) to (7). The\nresult may be cast into the form:\n5(x,y,t)\n+ TT","94\nAIR-WATER SURFACE PROPERTIES\nVOL. VI\nand where, for m > 1, n > 1,\n-\nThis form may be further reduced by noting that sin 0 = cos.\n(90-0) which will help change over the sines and leave only\ncosines. Finally, by renaming a as \"C, \" and a mn (after the\ngeneral manner in (10), (11) of Sec. 6.3), as \"C2j\" and bmn\nas \"C2j-1\" where\n2j\n(m+n-1)\n(m+n-2) + 2n,\n=\nform>1, n 2\n.\n2j -1 = (m+n-1) . (m+n-2) + (2n-1), for m 1, >.1, ,\nThen\n5(x,y,t)\n(28)\nwhere for each pair (m,n) we have written:\nEl / a\n\"uj\"\nfor\nFI ,\n\"vj\"\nfor\n\"oj\"\nfor\nmn\n\"kj\" for\nand where Ei is an appropriate phase (0 to 90°). In this way\nwe can close our analysis of 5,having finally returned to the\nappropriate general version of (99) of Sec. 12. 3.\nThe Periodogram Basis of the Energy\nSpectrum\nIn his search for hidden periodicities in natural phe-\nnomena, Schuster [277], [278] hit upon a simple but remarkably\neffective analytical scheme for uncovering the periodicities\nwhich existed in the data of interest. His method (cf. [280])\neventually was developed by Wiener [320], into the method of\nautocorrelation (or autocovariance) analysis, a powerful tool\nof harmonic analysis. To see the early form of the method as\nevolved by Schuster, suppose a function f, defined over some\nrelatively long time interval, [c,c+2T] is to be examined for\nperiodic components. It is tempting to try to reconstruct\nSchuster's line of reasoning as he sought a means of discover-\ning hidden periodicities. Perhaps it went, in essence, like\nthis (cf. [278]) : Suppose we write:","SEC. 12.4\nHARMONIC ANALYSIS\n95\nC+2T\n\"A(T)\"\nfor\nf(x) cos ux dx\nand\nC+2T\nf(x) sin ux dx\n\"Bu(T)\"\nfor\n.\nNow if f were periodic, and in fact equal to a COS ux, and\n2T = 2/u, then\nC+2T\ncos ux dx\n.\nHence, under such a form of f, Au(T) would eventually increase\nlinearly with time T. Stating this in another more precise\nway,\n= f(x) cos\ndx\nux\n.\n= a\nThat is, the process of finding the limit of Au(T)/T as T+00, ,\nwould yield the amplitude a of the periodic phenomenon repre- -\nsented by f, if f were exactly of the form a cos ux. Still\nfurther, if f were of the form a cos VX with u # V, then it\nis clear that\ncos VX cos ux dx\n=\n= 0\nIn a similar way these observations may be shown to hold for\nthe case of Bu(T). This suggests that the operations :\n] cos ux\ndx\nand\nthe\n[\n]\nsin ux dx\napplied to a function f defined over the time domain would\nextract from f the amplitude of any sinusoidal component of\nfrequency u that f may have, and discard all other components\nof different frequency. Thus if we write:","96\nAIR-WATER SURFACE PROPERTIES\nVOL. VI\nA u (T)\n\"A\"\"\nfor\nlim\nT\n-00\nand\nB (T)\nu\n\"BU\"\nfor\nlim\nT\n-00\nthen either of the numbers:\n| A | + Bu |\nor:\nwould serve as a quantitative measure of the presence of\nsinusoidal components of f of frequency u. (Simply adding\nAu and Bu might cause cancellation to zero, and thus belie\nthe presence of a component.) The squaring operation is by\nfar the more tractable of the two in analytic work, and there-\nfore is chosen as the requisite measure. The function which\nassigns to each \"u\" the value A + B2 (or some constant factor\ntimes this) is the periodogram of f. It is essentially the\nenergy spectrum of f, as we shall see below.\nWhen Wiener studied Schuster's periodogram method, he\ncontributed further to the evidence of the efficacy of the\nmethod. Suppose that f is data from some physical process\nrecorded over the time interval [-T,T]. To emphasize the\nfact that f is known only over the interval [-T,T] and that\nit is set to zero for t > T we write \"fT\" for this f.\n(Hence in the preceding analysis, we would choose C = -\nT.)\nThe autocorre lation (autocovariance) function OT of f was\nthen defined by Wiener (following G. I. Taylor [295]) by\nwriting:\n\"qp(x)\"\nfor 2T\ndt\n(29)\nThe bar over \"f\" in (29) denotes the complex conjugate of f.\nIf the data are real valued, then fT = fT. Observe that the\ninfinite limits of integration mask an essentially finite in-\ntegration range (namely [-T,T]) for each value X (because\nof our definition of f). However by being able to put in\nthese infinite limits we open the door to the formal Fourier\ntransform techniques of Sec. 7.14 and are able in particular\nto gain some deep insight into (x). To prepare for the\napplication of the Fourier transform, suppose that f has the\nFourier series development:","SEC. 12.4\nHARMONIC ANALYSIS\n97\n(30)\nwhere:\ndx\n(31)\nHere we have used (6), (7), and have set a = - T, p = T,\nand have written:\n\"un\" for THE nT\n(32)\nBecause of the finite domain over which f is not zero, C\nin (31) is, , to within a multiplicative factor, the Fourier\ntransform of fT. That is, using the notation of (10) of\nSec. 7.14 (with kernel ((x,w) = e-iwx)\ndx\ndx\n= 2T Cn =\n(33)\nObserve next that we can express PT as a convolution (Sec.\n7.14):\nWe are now ready to apply the Fourier transform operator of\nto PT. Converting into a convolution operation on fTR\nand ET, and using the convolution theorem (6) of Sec. 7.14,\nwe have:","VOL. VI\nAIR-WATER SURFACE PROPERTIES\n98\n(34)\nThe fact that we can here use the convolution representation\nfor PT, in the argument of the operator F, rests on a proof\nsimilar to that of (5) of Sec. 7.14. The notation \"fT R\"\nmeans the \"reverse of fT\", i.e., (fT R) (x) = fT(-x). Com-\nparing (33) and (34), we see from (33) that F applied to\nany function fT yields the value cn (in the spectrum of f)\nmultiplied by 2T, the length of the interval over which fT\nis generally not zero. Hence, from (34), it is clear that\nthe corresponding value (for the integer n) in the spectrum\nof PT is (1/2) Cn/2, (the interval over which T is not zero\nis [-2T,2T]). Therefore the Fourier series representation of\nPT is:\n(35)\nIn this way we have demonstrated that: the spectrum of the\nautocorrelation function OT of a function f is the energy\nspectrum of f (as defined below (100) of Sec. 12.3). This\nfact, which can be shown to hold for quite general functions\nf, is known as the Wiener-Khintchine theorem. Its earliest\nforms may be found in [320], [135].\nThe relation between the periodogram of f and its energy\nspectrum is readily seen by referring to (6) and recalling\nthat:\nThen:\nand + = 4c\n=410n12\n(36)\nHence Wiener's choice of the autocorrelation function (29)\nto be the basis for the generalization of Schuster's period-\nogram is an appropriate one.\nIt is possible to find out some essential information\nabout the periodic components of a function f by a direct\nexamination of its autocorrelation function prior to an har-\nmonic analysis of f. This is one of the important intuitive","SEC. 12.4\nHARMONIC ANALYSIS\n99\nfeatures of the autocorrelation function. Some examples will\nmake this clear.\nFirst, we select a sinusoidal f and compute its autocor-\nrelation function. Let f be a cos ux over [-T,T], where\n2T = 2n/u, and zero elsewhere on the real line; and then let\nT+00:\nH\n( (x) = lim\n2T\ncos u(x+t) cos ut dt\nT\nS\n= lim\n[cos ux cos ut - sin ux sin ut ] cos ut dt\n2T\nT\nT\na 2 cos ux\n/\n2\n= lim\ncos\ndt\nut\n=\ncos\nux\n2T\n.\n-T\n(37)\nOn the basis of this, it becomes plausible that the autocor-\nrelation function of an extensive periodic function is again\nperiodic and with the same period as the given function.\nNote how, in this simple example, the Wiener-Khintchine theo-\nrem is illustrated. As a simple exercise, the reader may now\nevaluate PT for finite T and note in detail the manner in\nwhich T approaches 0 computed above.\nSecond, we select a function which is very definitely\naperiodic. A simple example is that of the real valued func-\ntion f depicted in Fig. 12. 19, such that:\nf(x)\nif\nT\n=\na\n<\nf(x)\n0\nof\nT\n=\nX\n>\n.\nThen\n(x+t)\nf(t)\ndt\n(2T\n)\nfor\nX\n2T\n(38)\n<\n2T\n-\n,\nand\n= 0 ,\nfor\nX\n2T\n>\n(39)\n.\nThis example shows that for the given f, (x) is a de-\ncreasing function and that for large values of x, becomes\nzero for the aperiodic instance at hand. Actually this is\nquite representative of aperiodic functions in general which\nextend to + 00 and -00: PT(x) is sharply decreasing in value\ntoward zero as x| increases from 0. A rule of thumb is","VOL. VI\nAIR-WATER SURFACE PROPERTIES\n100\nFig. 12. 19\nf(t)\nf(x+t)\na\n+T\n-T\nFig. 12.20\na2\nT\ncosux\n=\n2\na2\n(a)\n2\nX\n2T\n(b)\nX\n+2T\n-2T\nFIG. 12.19 Calculating the autocorrelation function\nof a box-shaped function.\nFIG. 12.20 Part (a) depicts the autocorrelation func-\ntion of a sunusoid with shift X. Part (b) depicts the auto-\ncorrelation function of Fig. 12.19.\nthat: the more sharply decreasing T with increasing\nthe more aperiodic is the f from which is formed. Graphs\nof for the two examples are shown in (a) and (b) of Fig.\n12.20. A simple graphical interpretation of the calculation\nof PT in the case of the present aperiodic function is shown\nin Fig. 12.19. Observe how the graph of f is shifted X units\nto the left to represent the values f (x+t) when X is positive.\nFurther examples of autocorrelation functions may be found in\nthe paper by Rice [313] and the book by Lee [152].\nOnce a visual analysis of has been made on which\ntraces of periodicities of f are evident, quantitative esti-\nmates of the components giving rise to these periodicities\nare available by simply taking the Fourier transform of PT,","SEC. 12.4\nHARMONIC ANALYSIS\n101\nand this, by (34), yields 1/2 cn | 2 , and hence the energy\nspec-\ntrum of f.\nFourier Integral Representations of the Air-Water\nSurface. Case 1: The Surface is Aperiodic\nThe concept of the energy spectrum which, as we have\njust seen, is the logical outgrowth of the periodogram method\nof analysis devised for optical problems by Schuster, also\nlends itself to the difficult problem of analytically de-\nscribing the dynamic air-water surface. The mathematical\nsetting in which the concept attains its full powers of de-\nscription in the present task is that of two dimensional\nFourier integral theory, which we shall now briefly outline.\nWe shall consider two cases: First, the case where the sea\nsurface is aperiodic and whose amplitudes die away rapidly\noutside some finite (e. g. circular) region. * Second, we\nconsider the cases of periodic and random seas.\nThe main goal at hand in Case 1 is to find a way of\ngenerally describing (i.e., in one fell swoop) the contribu-\ntions to 5(x,y,1 t) by wave components of an aperiodic sea sur-\nface whose amplitudes belong to either discrete or continuous\nspectra, or both. Visual observations of the dynamic sea sur-\nface result in plots of ocean wave numbers which fall into two\nmain classes as indicated schematically in Fig. 12.21: The\nfirst class is that of wave components whose wave numbers k\nare part of a discrete set. The second class consists of\nthose wave numbers which are part of a continuous set. In\nother words, in a given sea it may be possible that all the\npoints from a region C in the uv plane have nonzero wave com-\nponents in that sea in addition to the finite (or infinite)\nnumber of points k in a discrete set D. Such a sea surface\ncan be represented as follows: Let the amplitude associated\nwith the wave number ki (= (Ui,Vi)) in D be A(ki) (=A(u,vi)) =\nand that with k (=(u,v)) in C be A(k) (=A(u,v)). A(k) dV(k)\nis then interpreted as the common amplitude of the wave com-\nponents cos (k x - ot+e) in the region of area dV(k) about\nin the uv plane. Hence our work leading to (28) and our\nk\nintuitive understanding of A (k) leads us to write:\nn\n5(x,t) = A(k) cos\ni=1\nA(k)\n(k\nx - ot + E) dV(k)\n(40)\ncos\n*This is also called the transient case in communication\nengineering. An interesting parallel between the present cases\nin hydrodynamic theory and their communication theory counter-\nparts may be generated by comparing the ensuing development\nwith that in [152].","VOL. VI\nAIR-WATER SURFACE PROPERTIES\n102\nV\n(u,v)\ndiscrete\nset D\ncontinuous set C\n= (Ui,vi)\nu\nFIG. 12.21 A given sea surface may have a spectrum\nwhich is partly discrete and partly continuous.\nwhere we have written \"x\" for (x,y), where o and E are gen-\nerally functions of k, and where n is finite or infinite.\nExamples of continuous spectra will be considered in Sec. 12.5.\nFor the present we consider theoretical methods by which, at\nleast in principle, the raw data of (x, t), can be processed\nso as to yield estimates of A(ki) and A (k) where kj is in D\nand k is in C, and hence yield the representation (40). We\nshall concentrate at first on the aperiodic case, as repre-\nsented by the continuous amplitude spectrum A(k), as that is\nthe more difficult of the two spectra to handle analytically. *\nThis will take us to (49) below. Then we shall go on to the\ngeneral combined spectrum problem summarized in (51).\nWe now consider, for convenience, two copies of Euclidean\ntwo space E2 one which we shall call the 'x - plane' and the\nother which we shall call the 'k - plane' The X - plane will\nbe the setting for the representations of the air-water sur-\nface and k - plane the settings for the wave numbers (and\nhence the amplitude and energy spectra of 5). If R is any\nmeasurable subset of , then:\nf(x) dV(x)\nR\n*Note that the purely aperiodic sea surface whose ampli-\ntudes die away at great distances necessarily has no finite\ndiscrete spectrum (since individual sine functions never die\naway). The latter spectra are considered in case 2.","SEC. 12.4\nHARMONIC ANALYSIS\n103\nis the integral of a measurable function f over R and which\nis an alternate notation to:\n.\nWe shall need the concepts of Lebesgue and Stieltjes integra-\ntion in order to carry out the present exposition, and any one\nof several available standard texts on advanced calculus (or\n[320]) may be consulted for terminology. However, in keeping\nwith the general mathematical level of this work our principal\nemphasis in this discussion will be on the physical meanings\nof the terms and equations, and not on the formalism.\nRecall that a function f on E2 is square integrable\n(briefly: \"in 2(E)\") if:\nexists and is finite, and where Sr(y) is a circular domain in\nE2 of center y and radius r. For example, if for some time t\nand radius ro, 5 (x, t) = 0 for /x/ > ro, and\nexists and is finite, then mathematically 5(.,t) is in L2 (E2);\nbut more interestingly, the mean square height and hence total\nenergy of the air-water surface at time t is finite. Hurri-\ncanes and other turbulent sea areas outside of which the sea\nis relatively calm may be represented by such functions.\nIf 5 (.,s) and 5(.,t) are two \"seas\" at different times,\nS < t, a measure of the difference of the energies is given\nby:\nMore generally, the square of the \"distance\" between two\nfunctions f and fa in L2(E2) is:\n.\nFurthermore if, as time goes on,\nlim 15(x,s) 5(x,t) 2 dV(x) = 0,\n,","VOL. VI\nAIR-WATER SURFACE PROPERTIES\n104\nor more generally, if fa is a member of a general family {fa}\nof functions, and\n0\n=\nthen we will write:\n5(x,t) = 1.i.m.s+t 5(x,s)\nor more generally:\n(41)\ng(x) = 1.1.m.a+h fa(x)\nand say that 5(x,t) (or g(x)) is the limit in the mean of\n5 (x,s) (or fa(x)). Recall (or verify) from integration theory\nthat if (41) holds, then:\nA dV(x) = a+b dV(x)\n(42)\nfor every measurable set AC\nNow if at some time t, the air-water surface is given\nby 5 (., t) which is in then let us write: *\n11 for 1.i.m. (43)\n\"A(k,t)'\nwhere x = (x,y) and k = (u,v), and so:\nx . k = xu + yv\nThen by Plancherel's theorem (cf., e.g., [321]) A(,t) is in\nand\n5(x,t) 1.i.m. A(k,t)ei(xk)dv(k)\n(44)\n=\n*The manner of placement of the factors 1/2m before (43)\nand (44) is to some extent arbitrary, and may be done in any\nway so long as the product of the factors before the integrals\nis 1/ (4t22). Thus we can, e.g., omit 1/2m from before (43)\nand place 1/(42) before (44), etc.","SEC. 12.4\nHARMONIC ANALYSIS\n105\nand furthermore:\nSE A(k,t) = dV(x) , (45)\nso that the energy content of the air-water surface is given\nby means of A(,t), according to (45) (cf., (98) of Sec.\n12.3; also see (29) and (35) in which X = 0). Relation (45)\nis known as a Parseval identity.\nNow, from a computational point of view the operation\nin (41) is a bete noire because \"l.i.m.' denotes an opera-\ntion realizable only in the mathematical ivory tower and not\nthe numerical laboratory. Whenever Lebesgue integration\ntheory (or measure theory in general) yields up an object g\nwhich is in the form of a limit in the mean, and that object,\nsuch as A(k,t) in (43) above, is of physical interest, it is\npossible to compute the value A(k,t) almost everywhere by\nmeans of the general formula:\ng(x)\ndV(x')\n=\n(46)\nFor example, replacing the general function g by Ak(,t):\nA(k,t) =\ndV(k')\nlime\n(k+k',t)\n=\ndV(k')\n(47)\nfor almost every K. That is, for all k in E2, except perhaps\nfor a subset Z of E2 of at most zero area, (47) will yield a\nusable value A(k,t) for X outside of Z. Hence whenever we\nreplace \"l.i.m.\" by \"lim\" in the above manner we can visually\nclear the way, at least in principle, for a transition from\nabstract set theory to practical numerical work. The opera-\ntion (46) or (47) is clearly an averaging process of the func-\ntion involved, thereby removing complicated wiggles and spikes\nfrom the function in question over the set Sr (x) or Sr(k).\nAs an illustration of the preceding remark, let us re-\nconsider (43). According to (47) we require an initial ex-\npression for A(k+k', t), and this is given by (43) in the\nform:","VOL. VI\nAIR-WATER SURFACE PROPERTIES\n106\nthat is:\nA(k+k',t) = 1.i.m. S+00\nTherefore for almost every k:\nA(k+k',t) d(k')\nA(k,t)\n=\n=\ne-i(xk)dv(x)\n.\n(48)\nThe equality above involving the ordinary limit operation is\nobtained with the help of (42). Equation (48) is the required\noperational rule for computing A(k,t), and the result holds\nalmost everywhere in In a similar manner the calculable\nrepresentation of 5 (x,t) takes the form:\n5(x,t)\n.\n=\nei(x*k)dv(k)\n(49)\nThe Fourier integral synthesis of s(x,t) in (49) holds\nwhen the indefinite integral of the amplitude function A(,t)\nis continuous, so that the preceding theory is adequate to\ncope with purely continuous spectra, with no Dirac-delta\nspikes representing isolated wave components.\nNow when the integral of the spectrum function A(.,t) is\nnot continuous, so that the limit operation in (48) may not\nnecessarily be possible, a different tactic must be used in\nfinding A(,t). Evidently, if trouble arises when we go to\nthe limit r+0 in (48), it may be profitable to delay this\noperation until a more judicious stage in the analysis of 5.\nThus for each real number s>0, and time t let us write:\n(k')\n\"A(k,t;s)\" for 1.i.m.\n-i(xk)dv(x) . (50)","SEC. 12.4\nHARMONIC ANALYSIS\n107\nIn this way we are in effect defining a measure, the amplitude\nmeasure, which assigns (for each given k and t) to the circu-\nlar subset Ss (k) the number A(k,t;s), such that the square of\nthe absolute magnitude is the energy of all wave components\nin\nthe region Ss (k). In communication theory, this has a one-\ndimensional analog known as the integrated Fourier transform\nor integrated spectrum of 5. At this point our heuristic dis-\ncussion leading to (101) of Sec. 12.3 may be helpful. Clearly\nSs (k) in (50) could be replaced by any arbitrarily shaped sub-\nset of the k - plane. However, simple circular neighborhoods\nwill suffice in the present discussion. In order to define\nA(k,t;s) as we have done above we implicitly required 5(.,t)\nto be in L2 (B2), or less stringently, we require 5(.,t) to be\nintegrable over regions of finite area, and:\n5(.,t)\n1 x x\nto be in L2 (E2). . In. this way 5(.,t) can describe a broad\nrange of wild and tumbled seas (still no whitecaps or breakers,\nunfortunately, since 5 is to be single valued) and essentially\nall we require is that 5(.,t) be bounded and measurable. The\nassociated A(k,t;s) may now vary discontinuously with respect\nto S, and k.\nIn the special form of. (50) we now recognize, by virtue\nof the general forms of the definitions (43) and (44), that it\nis possible to use Plancherel's theorem to conclude:\n(k')\n(x,t)\n=\nA(k,t;s)ei(xk) dV(k)\n= 1.i.m.\nFor practical work, we can remove \"l.i.m.\" and replace it by\n\"lim\" by integrating over a small circular region Sq(y) : of\ncenter Y, radius q. This we do, and so arrive at:\n(k')\ndV(x) =\nIsa(v)\ndV(k)\nDividing each side by the area of Ss(0) and rewriting the inner\nintegral on the right side of the preceding equation, we have:","VOL. VI\nAIR-WATER SURFACE PROPERTIES\n108\ndV(x) =\ndV(k)\nWe observe that:\n1\n=\n.\nHence, since 5(.,t) is integrable over Sq(y),\nei(yek)A(k,t;s)dV(k)\nThe right-hand side is the penultimate step in forming a\nStieltjes integral with the amplitude measure A(k,t;s) as the\nStieltjes weighting function with respect to the geometric\narea measure on the k- plane.\nDividing each side of the preceding equation by the\ncommon area q of Sq(0) and Sq(y), and letting q - 0, we\nfinally arrive at\n5(y,t)\nei(y)v(k)\n(51)\nwhich holds for almost every y in Comparing this repre-\nsentation with (49), it is clear that we have reached our main\ngoal in Case 1 of representing 5 when 5 is composed of wave\ntrains having both discrete and continuous spectra. Observe\nhow (51) reduces to (49) if A(k,t;.) is absolutely continuous\nwith respect to area measure in the spectral domain. Then","SEC. 12.4\nHARMONIC ANALYSIS\n109\nfor almost every k. In practice we can attain the complex\nversion of (40) using the two-dimensional complex versions of\nthe results of Sec. 21 in [321]. In practice, however, it is\ncustomary to use only (49) for the purely aperiodic case.\nPeriodic seas or single samples of random seas are handled\nsomewhat differently as we shall now see. This approach con-\ncentrates not on 5 but on its autocovariance-- or autocorrela-\ntion function.\nFourier Integral Representations of the Air-Water\nSurface. Case 2: The Surface is Periodic\nor Random\nIn the present case the Fourier transform of 5 over all\nof E generally does not exist. We accordingly introduce for\n2\nthe purposes of Case 2, the two-dimensional time-dependent\nautocorrelation function of 5 by writing:\n/st(0)\n1\nfor\n:(x+y,t)5(y,t)dV(y)\nlim\n(52)\nr->0\n2\nThis function plays an important part in both the theoretical\nand numerical studies of the dynamic air-water surface. It\nmay be used when the surface 5 has a periodic structure over\nE. or when 5 is simply a bounded measurable function (i.e., 5\nis continuing or random). * It is easy to deduce the following\ntwo properties of o: For every t,\n(xxt) (-x,t)\n(53)\nand\n((0,t)\n(54)\nThe\nfirst property establishes the fact that is an even\nfunction of its spatial variables whenever 5 is real valued\n(as it is assumed to be throughout this chapter) the second\nestablishes the fact that its magnitude is a nonincreasing\nfunction of x as x increases. This shows that (,t)\nis a bounded measurable function and satisfies the same gen-\neral conditions that 5(.,t) does. Indeed, the present analy-\nsis rests its case on the mathematically (and physically) ob-\nserved fact that, while 5 may have no Fourier transform, its\nautocovariance often does, in the sense defined in the dis-\ncussion following (50), and we assume this to be true in what\nfollows.\nUsing (52) we can then construct an amplitude measure\nfor (,t), in the manner we did for 5(,t) in (50), and then\ngo on to find the representation of (,t) analogous to (51).\n*As in Case 1, the parallel between the present case and\nthe corresponding situation in communication theory should be\nstudied by the serious reader. An overview of the present\ncases is given in Chapter 9 of [152].","VOL. VI\n110\nAIR-WATER SURFACE PROPERTIES\nHowever, our present interest in $(.,t) derives from its\nability to yield the energy spectrum of 5, without having\ndirect knowledge of the spectrum (i.e., Fourier coefficients)\nof 5. Furthermore, we are at present interested primarily\nin continuous spectra of individual samples of random func-\ntions 5, since this is the nontrivial part of the present\nrepresentation problem. (For periodic 5 we would work in\nthe Fourier series context considered earlier - - with finite or\ninfinite numbers of terms. ) Finally, our task is to obtain a\ncalculable representation of the amplitude measure of $(.,1 t).\nAccordingly, in analogy to (50), let us now assume at the\noutset that 0 has the same regularity properties* as 5(.,t),\nand let us write:\n(x.k')\ndV(k')\n\"E(k,t;s)\"\nfor\n1.i.m.\nr-00\n-(x.k)\ndV(x)\n(55)\nE(k,t, ) is the energy measure function. To convert this\n1.i.m. to something we can work with numerically, we can fol-\nlow the procedure given in (46), or if further theoretical\nwork is necessary we can use the theorem in Lebesgue measure\ntheory which states (cf. e.g., [320]) that: if\nf(x) = 1.i.m. ,\nthen there is a subsequence\n(fnk X } of\nsuch that the subsequence converges to f almost everywhere.\nWe can construct such a sequence as follows: Write\n(56)\n\"In(x,t)\"\nfor each positive integer n and where, in turn, we have\nwritten:\nE\n*We shall increase the regularity 0 and its transform\n(defined below) as we proceed. This increase is dictated by\nthe rather limited mathematical apparatus available to us in\nthis work and by physical considerations applied to 0 and E.\nThe Bochner theory of positive-definite functions (cf., , e.g.,\n[320]) is an ideal vehicle for the theory of the correlation\nfunction.","SEC. 12.4\nHARMONIC ANALYSIS\n111\n(x,t) if x is in Sn(0)\n\"5n(x,t)\" for\n0\nif\nis not in Sn(0)\nx\n(57)\nIt follows that:\n$(x,t) =\n(58)\nIf \"En(k,t;s)\" denotes the amplitude measure for on(x,t),\nusing the general definition (55), then by the aforementioned\ntheorem,\na\ndV(x)\ndV(k')\n.\n(59)\nFurthermore: there is a subsequence of { On such that\nE(k,t;s)=inn(k,t;s) =\nalmost everywhere on the k-plane. By the Lebesgue bounded\nconvergence theorem, we can move this limit operation inside\nEn(k,t;s), and use (58) to obtain:\ndV(k')\n.\n(60)\nWriting\n\"E(k,t)\" for ims+o\n(61)\nwe call E(k,t) the unresolved spectral energy density function\nor unresolved energy spectrum of 5. We are now implicitly\nassuming that E(k,t;) is absolutely continuous with respect\nto area measure, and that its derivative is absolutely inte-\ngrable.\nWe find from (60) :","VOL. VI\n112\nAIR-WATER SURFACE PROPERTIES\n(62)\nalmost everywhere on the k-plane. In this way we arrive at\nthe important representation of the energy spectrum of 5 in\nterms of the Fourier transform of the autocorrelation func-\ntion 0 of 5. Observe now E(k, t) is obtained without Fourier-\nanalyzing 5 directly. In the case now at hand the Fourier\ntransform of a random (i.e., bounded) function 5 generally\ndoes not exist over However, by our present assumptions\non E(k,t), we see that E(.,t) and (,t) can form a Fourier\npair. Thus almost everywhere on the x-plane:\n(xxt)\n(63)\n=\nThe unresolved energy spectrum E(k,t) is an even func-\ntion of K. This may be seen by first using (53) to deduce:\ni.e.\n,\n(x,t) COS(XK)dV(x)\n(62a)\nwhere + is the right half of the x-plane (i.e., all x = (x,y)\nsuch that X > 0). The evenness of E(.,t) now follows from\nthat of the cosine function. (In general: the parity (even-\nness, oddness) of a function and its Fourier transform agree.)\nThis property of E .,t) has a physical interpretation\nwhich is important to bring out at this time: E(k, t) is a\nmeasure of the combined energy of wave trains moving in both\ndirections k and -k (see (77) below). This interpretation\ncan be made plausible by studying the one-dimensional case\nas it is given in (30) - (35), and by pairing (35) with (63).\nThe complex coefficients cn in (30) contain directional in-\nformation in their arguments (phases). This information is\nunfortunately \"folded together\" in going over into cnl 2 in\n(35). It is for this reason that we have called E(k,t) the\n\"unresolved\" energy spectrum.\nWe may directly observe this phenomenon of folding to-\ngether of directional information in E(k,t) and also relate\nE(k,t) to A(k,t) by repeating for the present setting the\nderivation of (34). Thus, analogously to (34), we have:","SEC. 12.4\nHARMONICANALYSIS\n113\n(64)\nNext, from (50) for the case of 5n, we have:\nA\n=\ndV(x)\ndV(k')\n=\nThe integration over Ss (k) smooths the 1.i.m. functions and\nserves to make the amplitudes accessible to numerical compu-\ntation and physically meaningful, as we have already noted\nseveral times above. Thus :\ndV(x)\ndV(k')\nis a working formula for An (k,t), where we write:\n(65)\nand which may be compared to (61). Hence for continuous\nspectra:\ndV(x)\n(66)\nalmost everywhere on the k-plane. Using (66) in (64), and\nusing (62) applied to En and on:\n(67)\nfor every n > 1. Equation (67) is another working formula\nfor the connections between En and An, and is the analog of\n(34) which holds in the Fourier series case. Going to the\nlimit, (67) is idealized by:\nE(k,t)\n(68)\n=","VOL. VI\nAIR-WATER SURFACE PROPERTIES\n114\nWe continue the present discussion with some pertinent obser-\nvations on the time dependence of the various functions 5, 0,\nE, and A discussed above, and their relations to the unre-\nsolved energy spectrum. In our studies of the Fourier series\nanalysis of the sea surface, we found that introducing the\ncondition (22), derived from hydrodynamics, into the Fourier\nrepresentation simplified the analytic form of the amplitude\nfunctions. This may be seen by comparing (16) with (27) A\nsimilar tactic may be employed in the present case. Thus we\nmay assume:\n-iot\n(69)\nfor all t, where o is given by (23), or (25), in continuous\nform. This assumption, therefore, adopts the linearized\nmodel of the air-water surface discussed in Sec. 12.3. As a\nresult, the present version of (44), on recalling (66), be-\ncomes :\nA n i(xk-ot) dV(k)\n(70)\nwhich holds almost everywhere on the x-plane. It follows\nfrom (68) and (69) that E(k,t) = E(k,0) for all t, so that\nthe energy content of the ir-water surface is constant dur-\ning the time the linearized conditions hold. Thus also\n(x,t) = (x,0), by (63). These observations show that the\nlinearized model is adequate only for steady state seas whose\nwaves do not (or no longer) interact with the wind or with\nother waves, and in which no energy dissipation processes\noccur.\nThe application of the linearized theory of the air-\nwater surface, therefore, is practically limited both in\nhydrodynamical and harmonic analyses to steady state and rela-\ntively calm seas in a small region. The most the associated\nharmonic analysis will yield is the unresolved energy spec-\ntrum in the form (62). This is quite adequate for many prac-\ntical investigations (cf. [191]] and is retained as a useful\ntool for this reason. However, if the energy spectrum is to\nbe further resolved, more information must be drawn from\neither theory or nature herself. In the former case, for\nexample, we are led into the domain of nonlinear wave theory.,\na subject under continuing development (cf., e.g., [191]).\nIn the further appeal to nature, we must Fourier-analyze data\ntaken over E2 at more than one instant in time. Thus, we can\nbegin anew and apply Plancherel's and Wiener's theories over\nthe space-time domain E3 (= E2 X E1) for the air-water surface.\nHere E2 is the x-plane and E1 is the time domain (i.e., (-00,00))\nThis procedure would yield the exact basis for data-reduction\nschemes yielding energy spectra for the air-water surface\nwithout invoking any specific assumptions about the hydrody-\nnamical behavior of the natural hydrosols. Such a procedure\nwould result in completely analogous Fourier integral counter-\nparts to (16) and (17).","SEC. 12.4\nHARMONIC ANALYSIS\n115\nThere are several middle courses that can be taken\nbetween the preceding two extreme extensions of the analytic\nprocedures for the energy spectrum, and we now turn to de-\nscribe such courses.\nA Working Representation of the Dynamic Air-Water\nSurface and its Directional Energy Spectrum\nOne procedure leading to a middle-road air-water surface\nrepresentation theory of the kind discussed above would be to\npostulate, on the basis of our preceding work in hydrodynamic\ntheory (cf. (28)) and harmonic analysis, (cf. (44), (51)) that\nthe air-water surface has a representation of the form:\n5(x,t)\n=\ncos\n(71)\n(k.x-ot+e)\nR\ndA(k)\n=\nThis is the spectral representation of the dynamic air-water\nsurface which may, if required, be considered as a real sta-\ntionary stochastic process* in the x-plane (cf. Chapter XI\n[66]]. It is useful in studying periodic or random seas (cf.\nSec. 5.5 of [101]). Here the integral is of the Fourier-\nStieltjes type, where E and o are given functions of u,v,\nthe coordinates of the k-plane. In particular, o is an even\nfunction of u,v and can be defined by means of linearized\nhydrodynamic theory (cf. (64) or (90) of Sec. 12.3 and (23)\nabove). The amplitude measure used above is of the form\nA(k,s), i.e., the steady state version of A(k,t;s) discussed\nin detail following definition (50). Because of the Stieltjes\nform of (71) we can represent 5(x,t) either in series or in-\ntegral form or both. If A(k,s) is assumed continuous with\nrespect to the area measure V() of the k-plane, then we can\nwrite:\n\"da(k)\"\nA(k,s)\n\" a(k)\"\nfor\nlims+0\n(72)\nor\ndV(k)\n2\nTS\nwhich defines the amplitude spectrum a. In order to expound\nthe essential ideas of the present representation, we shall\nassume that the amplitude measure A(k,t;.) is indeed continuous\n*In this context, we would view 5(x,t) as but one sample\nof a large set {5a (x,t) : aeA} of surface representations Here\na is an index drawn from a set A of indices, which may be fi-\nnite or infinite, depending on the type of stochastic process\nadopted. Concurrently, there would be associated with 5a(x,t)\na sample Aa(k) of the amplitude measure.","AIR-WATER SURFACE PROPERTIES\nVOL. VI\n116\nwith respect to V(), and we shall work with one sample of\nthe surface at a time. In this way we avoid the complica-\ntions of Stieltjes integration and stochastic processes,\nwhich one may better tackle once the main issues below have\nbeen clarified. Equation (71) may then be written:\n5(x,t) = cos x.x-ot+E)\ndV(k)\n(73)\n.\nIn the discrete case for A(k,s), we could write out the\nrepresentation of 5 as in series form. The autocorrelation\nfunction o is now to be defined by an averaging process over\nboth the k-plane and the time domain (in distinction to (52))\nas follows. We write:\nT\n\"o(x,t)\" for lim TT 2 -T\n5(y,s)dV(y) ds\n(74)\nThus (74) allows consideration of the case where 5 could be\neither periodic or random (cf. Case 2 above) . Using (73) in\nthis definition, and writing*\nplane\n1\na(k')8(k-k')av(k')\n\"E(k)\"\nfor\n2\n=12a,(k)\n(75)\nwe find:\n(x,t) = S E(k) cos (kx-ot) dV(k)\n(76)\n*The connection of E(k) with existing treatments in the\nharmonic analysis of the air-water surface may be made by\nnoting that E (u,v) (k)) is basically the E (u, v) of Longuet-\nHiggins in [166]. The rather singular mode of definition of\nE(k) in (75) is prompted by dimensional considerations and our\ndecision to bypass stochastic theory in the present exposition.\n(See the discussion after (101), below.) The ac-delta func-\ntion in (75) is dimensionless, as a study of the substitution\nof (73) in (74) will show. The subscript \"1\" in (75) is a\nmomentary reminder that a is multiplied by a unit-valued\nfunction of dimension L-1, i.e., , inverse length.","SEC. 12.4\nHARMONIC ANALYSIS\n117\nE(k) is the resolved energy spectrum or directional energy\nspectrum of 5. The reader will find it instructive to de-\nrive in detail the form of (76) when the spectrum is dis-\ncrete and finite. To begin, suppose (71) is such that:\n5(x,t) = j=1\nwhere the aj are real.\nIt follows that:\n=\nso that:\n5 (x+y,t+s) 5(x,t) = [a] j=1 .\n.\nl=1\nThen perform the integration and limit operations in (74) for\nthe full autocorrelation over space and time:\n(x+y,t+s)5(x,t)dv(x)dt;\nand observe that:\n{\n0 if k # 0\ndV(x)\n= s(k)\n=\n1 if k = 0\nalong with:\nif 0 # 0\n0\n= = (8(0)\n=\n-T\n1\nif 0=0\nThe bracketed statements are for discrete variables k, the\nDirac-delta for continuous k (or o, respectively). The req-\nuisite averages are facilitated by means of the general analy-\ntic fact that: If a,B are two complex valued functions, such\nthat\n(Qa)(RB) =","VOL. VI\nAIR-WATER SURFACE PROPERTIES\n118\nthen\nHere \"s\" stands for either a general integral, sum, or linear\naveraging operation. The result is that, for the present dis-\ncrete case:\n(y,s)\naji\n(kj*y-ojs)\ncos\nj=1\nThis equation is the discrete version of (76). The deriva-\ntion of (76) itself is fundamentally similar to that just out-\n-\nlined.\nThe relationship between the unresolved energy spectrum\nE(k) of the linearized theory (we shall omit \"t\" since E(k,t)\n= E(k,0) for all t) and the resolved energy spectrum of the\npresent theory may now be deduced. This relation is important\nto know since we cannot invert (76) to find E(k) directly from\n0 as in the case of (62) and (63). This is due to the term\not in (76). From (63) and the evenness of E(k):\n$(x,0) = dV(k)\n/\nE(k) cos (xk) dV(k)\n.\nSetting t = 0 in (76) we have:\nE(k) cos (kx) dV(k)\n$(x,0) =\nS\n[E(k) E(-k)] cos (kx) dV(k)\n=\nwhere is the right half of the k-plane` i.e., all (u,v)\nsuch that u > 0. Hence when the present model is applied to\nthe linearized theory, discussed above, which yields E(k), we\nhave (since the two preceding representations of (x, 0) hold for\nall X, and since E (k) is essentially unique) :","SEC. 12.4\nHARMONIC ANALYSIS\n119\n(77)\n+\nActually, on the basis of the meanings of E and E, the rela- -\ntion (76) holds quite generally, and may be used not only in\nlinearized theory but also to find E(k) when estimates of the\nkind E(k) only are available from real data. Thus suppose\nthat at the same time (t = 0) the temporal derivative of\n(x,0) is estimable. This may be done, for example, by tak-\ning two stereo photos of the air-water surface close together\nin time (cf. [45], [191]). Then from (76)\n(x,0) = E(k) sin (kx) dV(k)\n(E (k) - E(-k)) sin (kx) dV(k)\n=\n= $ 2000 (E(k) - E(-k)) sin (kV(k)\nwhere we have used the fact that the sine is an odd function,\nand o is an even function of k (i.e. , of (u,v)). It follows\nthat ' (x, 0) and (1/2) (k) (E(k) - E(-k)) are two-dimensional\nFourier (sine) transform pairs. Hence:\n1 ( o k k [E(k) - E(-k)] sin (kx) dV(x)\n.\nLet us write:\n\"D(k)\" for 1 (E(k) - E(-k)\n(78)\n.\nThen we have:\n(x,0) sin (kx) dV(x) .\n(79)\nWe can now state that if ' (x,0) and are estimable\nfrom real data, then in principle we can find E(k) according\nto the rule:\nD(k) + (E(k) = E(k)\n(80)","VOL. VI\nAIR-WATER SURFACE PROPERTIES\n120\nHere D(k) is the unresolved difference spectrum and E(k) the\nunresolved (sum) spectrum defined earlier in (61). The time\nvariable t has been suppressed; all three spectra in (80) per-\ntain to a common instant in time. Equations (62a) (with t = 0),\n(73), (79), and (80) together solve the main air-water surface\nrepresentation problem stated at the outset of this discussion.\nFor by (62a) with t = 0, we obtain the unresolved (sum) energy\nspectrum E(k), and from (79) we obtain the unresolved differ-\nence spectrum D(k). The directional energy spectrum E(k) then\nfollows from (80), and the amplitude spectrum a(k) for the\nair-water surface is obtained from (75). The representation\nof the air-water surface function 5 then follows from (73).\nNumerical procedures for realizing the preceding theo-\nretical relations are developed in the one-dimensional case\nin [24] and in the two-dimensional case in [45].\nGeometrical Applications of the\nDirectional Energy Spectrum\nIt will be shown next how the directional energy spec-\ntrum E, and several of its important variants to be intro-\nduced below, may be used to determine four important geo-\nmetric properties of the air-water surface. These properties\nare instrumental in the solution of radiative transfer prob-\nlems associated with the dynamic air-water surface. The req-\nuisite properties leading to the solution are: the mean\nsquare elevation of the surface above mean sea level, the\nmean square slopes of the surface in each of two given per-\npendicular directions in the mean sea level surface, and the\nmean of the product of these two slopes. We now discuss, in\nturn, each of these properties. The derivations will hold\nfor either periodic 5 or random 5, where 5 is real valued on\nFirst, the mean square elevation of the air-water sur-\nface is defined by writing\nT\n1\n1\n\"I2\"\n(x,t)\ndV(x)\ndt\nfor\nlim\nT+00\n2T\n2\nnr\n-T Sr(0)\n(81)\nObserve that if the air-water surface is statistically sta-\ntionary (i.e., if the integral of 52 over E2 is independent\nof t) then the additional averaging integration over t is\nunnecessary. It is at once clear from (74) and (76) that\n(82)\n= (0,0) = E(k)\ndV(k)\n.\nIf the spectrum of 5 is discrete, then 52 is a (finite or\ninfinite) sum of squares of component amplitudes of 5, as the\nexample following (76) illustrates.","SEC. 12.4\nHARMONIC ANALYSIS\n121\nSecond, the mean square slope of the air-water surface,\nas it is sectioned by planes parallel to the xz-plane, is\ndefined by writing:\nfor dV(x) dt.\n(83)\nFor example, if 5 is given by (73) with a continuous spectrum\nand we write out x in the form \"(x,y)\", when necessary for\nclarity, then:\nas(x,y,t) = u(k) sin (x*k-ot+e) dV(k)\n.\n(84)\nUsing this in (83) along with (75), we have:\nE(k)\ndV(k)\n(85)\nwhich is the general form of the representation using the\ndirectional energy spectrum.\nThird, defining the mean square slopes in planes par-\nallel to the yz-plane by writing:\nfor\ndV(x)\ndt.\n(86)\nIt follows that:\nv2\nE(k)\ndV(k)\n(87)\nwhere again we have used the rectangular coordinate repre-\nsentations x = (x,y) and k = (u,v).\nEquations (85) and (87) use special weighted integrals\nof E (k) , the general form of which is defined by writing:","VOL. VI\nAIR-WATER SURFACE PROPERTIES\n122\n1.\nE(k) uPv9 dv(k)\n(88)\n\"mpq\"\nfor\nIt turns out that the preceding three mean square quantities\nare representable as :\n(89)\n(90)\n(91)\nWe call mpq the pqth moment of E. It is found that, in\nnatural hydrosols M01' M10, and the mean of 5 are for most\ncases zero. However, M11, i.e., the mean of (25/2x) (25/dy)\nis often nonzero. Clearly:\n(92)\ndy\nwhich is the fourth and final geometric feature of the air-\nwater surface needed for our hydrologic optics studies. The\nspecific integral representation of M11 is:\n(93)\ndV(k)\nuv\nWe will now free ourselves from the restriction of\ndescribing the energy spectrum E in one particular coori-\nnate system over the k-plane. The first result will be a\npractical rule for expressing the mean square slope of the\nair-water surface along any direction in terms of the three\nbasic means mo2' M20, M11. Let the coordinate frame in the\nk-plane be rotated an angle radians as shown in Fig. 12.22.\nA point k with coordinates (u,1 v) will now have coordinates\n(u',v') in the new frame, where:\n= u cos + V sin 0\n(94)\nv' = - u sin + V cos\nand reciprocally :\nu=u' cos - V' sin\n(95)\nv=u' sin 0 + v' cos 0","SEC. 12.4\nHARMONIC ANALYSIS\n123\nold\nnew\nvsind\nk'=(u',v')\nV\n=(u,v)\nvicos\nV\nu'\nu'sin\nu\nv'sin\nu\nu'cos\nFIG. 12.22 Diagram for constructing the equations for\na rotation transformation of angle 0 about the origin.\nNow, the mean square slope of the air-water surface in\nany direction 0 may be obtained by passing a vertical plane\nthrough the u' 1 axis of a coordinate frame rotated 0 radians\nand then computing M20 in the new frame. To emphasize the\nrelative orientation of this frame, we shall denote M20 in\nthe rotated frame by \"m2()\". The \"2\" denotes the mean\nsquare aspect of the moment. We compute m () as follows:\nStrictly by the definitions, and (85), if É' is the energy\nspectrum with respect to variables measured in the rotated\nframe, then:\nE'(k)(u') 2 dV(k')\nm 2() =\nE' (u' v') (u') 2 du' dv'\n(96)","VOL. VI\nAIR-WATER SURFACE PROPERTIES\n124\nwhere we have used the representation k' = (u',v'), , and the\nfact that the element of area in the new system is du' dv'\nin contrast to du dv (= dV (k)) in the old system. To perform\nthis conversion in a systematic way, observe that the con-\nversion of variables from one frame to another is governed\nby the trans formation (94). The change of area elements then\nrequires the Jacobian of (94) which in this case is:\ndu'\ncos sin\na(u',v')\n=\n=\ndu,v)\nav'\nav\n- sin cos\ndu\nav\n= o = 1\nThen since:\ndu' dv'= du\ndv\n= du dv\n,\nwe should have:\n=E(u,v)\n,\nso that (96) becomes\nm2() = E(u,v) (u cos 0 + V sin 0)2 2 du dv\nu2\ndV(k)\n= cos E(k)\n+ 2 sin o SB E(k)\nuvdv(k)\ncos 0\nv2\ndV(k)\n+\nThe desired result then follows, and takes the form:\n.(97)\n= + cos 0 sin 0 + m","SEC. 12.4\nHARMONIC ANALYSIS\n125\nTherefore, the mean square slope m (0) along an arbitrary\ndirection in some frame of reference is calculable if in\nthat frame the three numbers m20, mo2, and M11 are known.\nIt should be noted that this connection is quite general;\nhowever to effect it, the representation 5 of the surface is\nto have integrable second derivatives.\nAs a final topic in the discussion of various geometric\napplications of the directional energy spectrum, we consider\ncertain types of experimentally determined energy spectra\nwhich are concerned only with the frequencies (or period) of\nthe water waves and not with their directional properties.\nThis suggests that we establish a polar coordinate system in\nthe k-plane, as depicted in Fig. 12.23. The transformation\nfrom the uv coordinates to the k, coordinates is:\nu(k,o) = u = k cos 0\n,\n(98)\nv(k,o) = V = k sin 0\n,\nwhere:\nk = (u,v)\n(99)\nk = (u2 2 + v2)\nk\nV\n&\nu\nFIG. 12.23 Changing from cartesian to polar spectrum\ncoordinates.","AIR-WATER SURFACE PROPERTIES\nVOL. VI\n126\nLet us write\n\"S(k,o)\" for E(u(k,o), v(k,))\n(100)\nThe Jacobian for (98) is:\ndu\ndu\n- k sin\ncos\nak\ndo\na(u,v)\n=\n=\na(k,o)\nav\nav\nsin o\nk cos o\nak\ndo\n= k cos2 0 + k sin = k\nTherefore we have the connection\nS(k) = S(k,o) = E (u,v) k E(k) k\n(101)\nwhen (u,v) and (k,o) are but two representations of one and\nthe same point k in the k-plane, with their coordinates re-\nlated by (98). The function S is the polar form of the di-\nrectional energy spectrum E.\nSome words about the dimensions of the energy spectra\nE and S are in order, as it will become increasingly impor-\ntant to keep track of the dimensions of the various derivates\nof E and S below, preparatory to the study of experimental\ndata in the following section. The basic point of departure\nin the dimensional analysis can be (74). A dimensional anal-\nysis of the autocorrelation function 0 shows that it has di-\nmension L2. The measure V in (76) in the two dimensional\ncase is an \"area\" measure in the k-plane. The dimension of\nk is L-1 so that the dimension of V in this case is L-2.\nSince the cosine function is dimensionless, it follows from\n(76) that the dimension X of E is the solution of the equa-\ntion\nthat is, dim (E(k)) = L4. An alternate analysis may be based\non (71) and (75) by noting via (73) that dim (A (k)) = L, and\ndim (a(k)) = L3. Then since dim (s (k-k' ) = 1, the result\nfollows from (75). According to (101), the dimension of S(k)\nis then L4L-1 = L3.\nReturning to the main line of discussion we next write:\nfan\n\"Ik\"\nfor\nS(k,o)\ndo\n(102)\nTk is the spatial frequency spectrum associated with the wave\nnumbers k. From this definition and (101) we see that dim(Tk)\n= L3. Furthermore,\n(103)","SEC. 12.4\nHARMONIC ANALYSIS\n127\nwhich may be established by observing that:\ndu dv = S(k)\ndodk\n1.10 dodk = It dk dk\nIt is also occasionally convenient to have available a\nfrequency spectrum associated with the temporal frequency o\noccurring in the theories of this and the preceding section.\nTo lay the ground work for the introduction of the temporal\nfrequency, we must decide on a workable connection between\nand k. From (64) of Sec. 12.3 we have, on the basis of\no\nthe linear theory of hydrodynamics:\ntanh (kh)\n1/2\n(104)\nwhich in infinitely deep media (h = 00) becomes:\no = = gk\n(105)\nWhen surface tension effects are to be taken into account in\nspectrum studies, we could use (90) of Sec. 12.3:\nwhich we could cast into\no2 = Bk + yk 3\n(106)\nwhere we have written\ng(Pw-Pa)\n\"B\"\nfor\n,\nPa+Pw\nand:\n\"y\"\nfor\nWe shall work, for illustrative purposes, with (105);\nhenceforth, unless specificially stated otherwise, U and k\nare to be functionally related by (105). In many practical","AIR-WATER SURFACE PROPERTIES\nVOL. VI\n128\ncomputations, we may set PW = 1, Pa = 0; so that B = g,\nY = T1 (cgs system) .\nThe definition of the directional spectrum based on\ntemporal frequency O and direction is now obtained by\nwriting\n(107)\n\"F(0,0)\" for\nFrom (98) and (105) we have the transformation from the uv\ncoordinate frame to the 00 coordinate frame in the k-plane\no k c os\nQ\n.\nHence\ndu\n20\n(u,v)\n=\n=\n2(0,0)\nFrom this and (107) we have:\nF(k) = F(0,0) = (108)\nFurthermore from (101) and (108) we have:\nso that:\n20 S(k,4) = F(a,4)\n(109)\nFrom this relation the dimension of F(k) is readily found:\ndim =\n-\nCorresponding to Tk, in (102), we can write:","SEC. 12.4\nHARMONIC ANALYSIS\n129\n\"To\" for do\n(110)\nfor the (temporal) frequency spectrum and observe that:\n(111)\nsince\nE(k) du dv = F(k) do\ndo\ndo\ndo\nTo do\nFrom (110) we have dim(Tg) = L ST.\nOceanographers often find it convenient to work with\ntemporal periods T rather than temporal frequencies O.\nSince:\n(112)\n,\nwe can write:\n\"TT\" for Total\n(113)\nwhere the derivative do/dt is now a one-dimensional version\nof the Jacobian. T is the (temporal) period spectrum.\n(114)\nso that dim (T) = dim (o2) @ dim (To) = T-2.L2 T; that is,\ndim =\nFurthermore, directly from (111) and (113) :\n(115)\n=\nThe importance of the various frequency and period\nspectra, S (k,0), Tk, To, and TT for hydrologic optics rests\nin the means (!) they make available for converting data on\nwave spectra of the surfaces of natural hydrosols into useful","VOL. VI\nAIR-WATER SURFACE PROPERTIES\n130\ninformation about the mean square elevations and slopes of\nthose surfaces. As an illustration, consider S(k,) and\nTk. From (89) and (103), we have:\n(116)\nFurthermore from (85) and (90):\n= u2 dV(k) = E(u,v) u2 du\ndv\nS(k,o) u2 (k,) do dk\n=\nTherefore by (98):\nS(k,) k2 cos2 2 do dk\n(117)\n.\nIn a similar manner we find:\nS(k,) k2 sin2 do dk\n(118)\n....... (k,) k2 sin cos do dk\n(119)\n.\nAs a simple instructive exercise, the reader should now find,\nin a similar manner, the preceding four moments in terms of\nF(0,0). (See, e.g., 42)-(46) of Sec. 12.9.)\nA considerable simplification in computations of the\nmoments Moo, mo2' M20 can be effected whenever the direction-\nal spectrum S 1S isotropic, i.e.,\n(120)\nfor every o, 0 < 0 < 2 TT . Under the isotropy condition on\nS, (117)- - (119) become:","SEC. 12.4\nHARMONIC ANALYSIS\n131\n(121)\n(122)\nmay\n(123)\nFrom these relations and (97) we find:\n(124)\nfor\n2TT, under the isotropy condition (120).\n0,\nevery\nIn general, isotropy or not, we have, from (117) and (118):\n(125)\nIn this way we see that from knowledge of S(k,o), and\nTk the four important moments moo' and\ndeterminable. From these moments and distri-\nm.\nare\nbutions governing 5, 25/ax, and as/dy, which we shall study\nin Sec. 12.9, we can deduce in turn the basic functional\nrelations governing radiative transfer phenomena at the air-\nwater surface (Sec. 12.12). In this connection, (121) - (125)\nand their following variants will be found helpful (Sec.\n12.8):\nFirst, we have quite generally:\n(126)\nwhich follows from (109) and (110). This, together with\n(114), provides connections among the three versions of the\none-dimensional spectrum. Furthermore, under the isotropy\ncondition:\nF(0,4)\n(127)\nwe obtain from (121) or (122);\n(128)\n(129)\n=","VOL. VI\nAIR-WATER SURFACE PROPERTIES\n132\nalong with:\n(130)\nm = 0\n.\n11\n12.5 Wave Slope Data\nIn this and the following four sections we shall pre-\nsent some experimental data on the dynamic air-water surface\nwith respect to those properties which play essential roles\nin the study of radiative transfer across the surface. The\nprincipal problem in such radiative transfer studies is the\nprediction of the instantaneous and time-averaged reflec-\ntance and transmittance of the air-water surface under wide\nranges of meteorologic and hydrologic conditions. The se-\nlection of the type of experimental data sampled below is\ngoverned by the observation that the theory of Sec. 12.11,\nwhich attempts to resolve this principal problem, requires\nknowledge of the statistical distributions of wave slopes\nand elevations. It turns out that these wave distributions\nare basically gaussian in structure for all natural hydro-\nsols under wide conditions, so that they are completely de-\ntermined by their mean square quantities, i.e., mean square\nslopes and elevations. Our studies of Sec. 12.4 showed that\nthese mean square quantities in turn are all uniformly de-\nrivable from the directional energy spectrum (in any of sev-\neral alternate forms) Experimental investigations show\nfurther that these wave spectra obey (both in the transient\nand steady states) remarkably regular laws which exhibit\ntheir dependence on the speed, fetch, and duration of the\nwinds generating and sustaining the configuration of the\ndynamic air-water surface. Therefore in engineering calcu-\nlations leading to estimates of the reflectance and trans-\nmittance of the dynamic air-water surface, knowledge of\nwind speeds, fetch, and wind duration is essential for an\nestimate of the associated wave spectrum. The preceding\nconsiderations fairly well dictate the selection of the\nfollowing five topics for discussion: wind profile data,\nwave slope data, wave height data, wave spectrum data, and\nauxiliary data derivable from wave spectrum data.\nThe data selected below and in the following four sec-\ntions is intended as a representative selection and, as such,\ndoes not exhaust the present fund of experimental knowledge\nin these selected areas (references to sources are given at\nappropriate places in the discussions) The main purpose of\nthe selection is to illustrate the principal kinds of data\nneeded in the optical studies of the dynamic air-water sur-\nface.\nThe Logarithmic Wind Profile Model\nThe use of the results of the experimental studies of\nwind-generated wave slopes described in this chapter requires\naccurate knowledge of the wind speeds at or just above the\nair-water boundary. Since it is not always possible to mea-\nsure the wind speed at those particular heights, it is neces-\nsary to have some rule which relates the wind speed at some","SEC. 12.5\nWAVE SLOPE DATA\n133\narbitrary altitude above the wind-roughened surface to that\nat or just above the surface.\nExperimental studies show that the wind speed varies\nvery nearly logarithmically with altitude above dynamic air-\nwater surfaces. At the surface the mean air speed is rela-\ntively small as the air drags and burbles over the ruffled\nwater surface. As altitude increases, the air drag of the\ncupped and capped water surface falls off, the air turbu-\nlence burbles are less pronounced, and a transition to smooth\nlaminar flow sets in. The air speed Ua (z) at altitude Z may\nbe represented generally by the following law:\n(1)\nwhere a, b, C are constants phraseable in thermodynamic and\nhydrodynamic concepts. The work of Lake [148] suggests cer-\ntain values of the constants b and C which should be valid\nunder wide ranges of conditions. These are:\nb = 6.0 cm\n(2)\nC = 0.18 cm\nHence if z,z+ are two altitudes above the mean level of the\nair-water surface, (1) with (2) yields:\nz-6\nz-6\nUa(2)\nIn\nlog\n0.18\n0.18\n10\n(3)\nUa(2\")\n=\nZ'-6\nZ'-6\nIn\n0.18\n0.18\nwhere Z is in centimeters.\nAs an example of (3), , let Z = 41 feet and z' = 8 inches.\nConverting to centimeters, Z = 1230 cm and z' = 20 cm (1\ninch = 2.5 cm), so that Ua (z)/U(z') = 2.0. Hence the wind\nspeed at 41 feet above the mean air-water level should be\nabout twice that at 8 inches, according to the logarithmic\nwind profile model.\nThe logarithmic wind profile model is only one of\nseveral models currently in use. For an alternative alge-\nbraic model of comparable simplicity, the reader may refer\nto [286].\nVisual Observations on Wave Slopes\nWhen one looks out over a wind-ruffled sea or lake on\na clear, sunny day, the surface sparkles with myriads of\npoints of light. Each sparkle represents an image of the\nsun reflected in a small patch of the air-water surface,\ntipped just right so that a ray from the sun is deviated,\naccording to the law of reflection (1) of Sec. 12.1, direct-\nly into the beholder's eye. If one could face a low sun\nwhen the sea is dead calm, the sun's image would be a still,\nbright spot on the glassy surface, which is just as far below","VOL. VI\nAIR-WATER SURFACE PROPERTIES\n134\nthe observer's horizon, angularly, as the sun is above the\nhorizon. The moment the surface is disturbed, say by a\nbreeze, a rash of images breaks out over the surface. As\nthe breeze continues steadily, the sea is kept agitated at\na constant rate and a steady-state - glitter pattern is even-\ntually established over the water surface. For a gentle\nbreeze, this glitter pattern is narrow, forming a luminous\npath from the observer out to the sun (which in Russian is\ngiven the romantic description of the \"road to happiness\") ;\nfor a brisker breeze the steady glitter pattern is broader,\nand on some rare bright windy days the entire sea is aglow\nbefore the observer. Ordinary visual observations of this\nkind establish in the mind of a perceptive observer certain\ncausal relations one comes to expect between the position\nof the sun, the strength of a wind, and the shape and extent\nof the steady glitter pattern the wind generates. Such an\nobserver, if of a turn of mind that is analytic, would re-\nflect on the geometric relations between his position, the\nglitter's position, and the sun's position and attempt to\nfind a succinct symbolic expression connecting these posi-\ntions, with the ultimate goal in mind of making quantitative\nand precise what he already knows intuitively and from di-\nrect observation. Figure 12.24 depicts the essential geo-\nmetric elements for the analysis of visual observations of\nsun glitter patterns. The observer at 0 receives along di-\nrection E, reflected flux from a wave facet at F which has\nredirected flux from the sun arriving along direction &\nBy knowing & and E' one can readily determine the orienta-\ntion of the normal n to the reflecting facet, using (1) of\nSec. 12.1.\nThus, using the angles defined in Fig. 12.24, we may\nrepresent E and E as:\n& = i cos w - j sin w k cos d\n(4)\n(5)\nE' = i sin 0\ncos 0\nFrom these we find:\n& - E' = i (cos w - sin 0) - j sin w + k (cos d - cos 0)\n(6)\nso that\n5-5'12 = (cos w - sin 0) 2 + sin2 2 w + (cos d - cos 0) 2\n(7)\n= 2 (1 - cos w sin 0) + cos d (cos d - 2 cos 0).\nLet the vector n have components n1, n2, n3 along the\nx, y, and Z axes, respectively; then by (1) of Sec. 12.1:\n(cos\nsin\n0)\nw\n-\n(8)\nn 1 =\nA(w,e,d)\nsin w\n(9)\nn\nA(w,e,d)\n(cos d - cos 0)\n(10)\nn\n=\nA(w,0, d)","SEC. 12.5\nWAVE SLOPE DATA\n135\n0\nplane of sun through O\nof\nplane\nthrough\nd\nT\nE'\n(observer) O\nE\nn\n(reflecting\nF\nfacet)\nk\nglitter boundary\nangular half width (w)\nFIG. 12.24 Scheme for deriving the sun glitter pattern\nequations.\nwhere we have written:\n\"A(w,0,d)\" for [2(1 - cos w sin 0) + cos d (cos d - - 2 cos 0) 1/2\n(11)\nso that \"A(w,e,d)\" is simply another name for the distance\n15-5'\nEquations (8) - (10) permit the determination of n\nknowing the angular half-width W and angular depth d of a\nreflecting wave facet, as measured by a sun-based reference\nframe at the observer, in which the rays of the sun are par-\nallel to the XZ plane and streaming down at an angle 0, as\nshown in Fig. 12.24. Of the three components of n, the most\nimportant for our present purposes is n3. This component\ngives a measure of the tilt of the facet's normal from the\nvertical. The number n3 can be used to give a measure of\nthe roughness of the sea surface as follows. If one observes\nthe angular half width w of a glitter pattern (the largest w\nfor facets in the pattern), and the associated angle of de-\npression d to the boundary of the pattern (shown dashed in\nFig. 12.24), one can use (10) to compute the tilt of a wave\nfacet at the boundary of the pattern. For example, if one\nis standing on shore or on a ship deck, d is essentially 90°.\nIf, for a particular observation, w = 30° and 0 = 120°, then:\n1/2\nA (30, 120, 90) = = 2 1\n,","AIR-WATER SURFACE PROPERTIES\nVOL. VI\n136\nso that:\nn 3 = /1/2 1/2 = /1/2\nSince n3 is the cosine of the angle T of tilt, the requisite\nangle is 45° If 0 and d are kept the same but w is now 60°\nthen\n(60, 120, 90) = [2(1-14)] 1/2 = 4-3 1/2\n= 1.06\nA\nand:\n1/2\n= 0.47\nn 3 = 1.06\n,\nfrom which we find that the angle T of tilt of the normal\nfrom the vertical is about 62°. Thus for a given sun alti-\ntude the greater the half width w of the glitter pattern,\nthe greater the maximum tilt T of the wave facet of the water\nsurface and hence the \"rougher\" that surface is. The tilt\nT is given via (10) by:\ncos d - cos 0\n(12)\narc cos T =\nA(w,0,d)\nHulburt's Observations of Wave Slopes\nIn 1933, thoughts such as those described above were\nbeing considered by Hulbert in his studies of the polariza-\ntion of reflected light from the sea [113]. In particular\nhe published a graphical means of finding the tilt T for\nthe case d = 90°. This celebrated graph (cf. [182]] is re-\nproduced, with slight modifications, in Fig. 12.25. To\nfind the maximum tilt of wave facets in a sea with a glitter\npattern of half width w and sun angle a(= 0 - 90) above the\nhorizon, first find the point (w, in the grid of the figure.\nThe T-curve going through (w,a) determines the requisite tilt\nT. The first of the two examples worked out above may il-\nlustrate the use of the graph. This graph may be consider-\nably extended, as required, using (8) - (10) , to completely de-\nscribe the wave-facet geometry behind the sparkle pattern of\na wind-ruffled sea.\nOne of the more important findings by Hulburt for our\npresent purposes is his quantitative explanation of why the\nrim of a wind blown sea on a clear day is so sharp - that is,\nwhy it has such great contrast against the horizon sky. If\nthe sea were calm, Fresnel's reflectance formula (12) or (14)\nof Sec. 12.1 would show that the reflectance approaches unity\nfor grazing lines of sight as one views the sea surface near\nthe horizon. Hence the reflected radiance from the sea just\nbelow the horizon rim should, in this case, nearly match that","SEC. 12.5\nWAVE SLOPE DATA\n137\n609\n30°\n35°\n50°\n25°\n40°\n40°\n20°\n30°\n45°\n15°\n50°\n55°\n20°\n109\n60°\n65°\n70°\n10°\nw\no\n5°\n10°\n15°\n.20°\n25°\n30°\nFIG. 12.25 Graph for determining maximum wave slope\noccurring in a glitter pattern of angular half-width\nwith a sun angle (= 0 - 90°) above the horizon.\nof the sky just above, so that the edge of the sea would\nblend imperceptibly into the sky, even on the clearest days.\nHowever, experience and intuition tell us that when the sur-\nface of the sea at or near the horizon is ruffled, we should\nsee reflected light not from the horizon, but from points\nhigher in the celestial dome above the reflecting facets.\nThe sky is generally darker blue the nearer the dome. This\nfact, coupled with the smaller Fresnel reflectance for the\ntwo angles now involved, contribute to the qualitative ex-\nplanation of the dark horizon rim. Hulburt's contribution\nlies in determining quantitatively that portion of the sky\nsending radiance down to the observed facet. It turns out\nthat [113] : \"the light of the rim of a breezy sea comes\nmainly\nfrom the region of the sky 25° to 35° above the hori-\nzon and hence the reflecting facets of the sea which are\nvisible to the observer are tilted up on the average about\n15° from the horizontal. \"\nThe experiments leading to this conclusion were con-\nducted in the Atlantic between Chesapeake Bay and Long Island\nSound. The wind range associated with this conclusion was\n3-18 knots (1. .5-9.2 m/sec). It is in this sense that the\nterm \"breezy sea\" in the preceding conclusion is to be inter-\npreted. Hulburt goes on to apply this conclusion to explana-\ntions of several interesting visual phenomena associated with\nwind-blown natural hydrosols: looking across a tree-rimmed\nlake whose surface is ruffled, one can see reflected blue sky\nvirtually up to the distant lake shore, rather than reflected","VOL. VI\nAIR-WATER SURFACE PROPERTIES\n138\ngreen from the tree leaves; similarly, dark or light colored\nsmall ships on a breezy sea have essentially no observable\nreflections in the sea; the blueness of roughened seas is\naccentuated by the waves picking the deeper near-zenith blue\nof the sky to reflect; reflections of extensive clouds in\nroughened water appear to be displaced nearer the observer\nthan the simple law of reflection (1) of Sec. 12.4 would pre-\ndict; and so on (cf. [182]).\nHulburt concludes his paper with the thought-provoking\nremarks: \"If all the facets had the same T the sun path\nwould consist of two straight lines of sparkles* stretching\nout from the observer at an angle 2w to each other, the bear-\ning of the sun being half way. If, as is actually the case,\nthe region in between the two lines is filled with sparkles,\nthere must be facets of a less T than the T for the edge of\nthe path, with a distribution of I such as to give the ob-\nserved sun path. Thus observations of the distribution of\nlight across the path for various altitudes of the sun would\nyield the distribution of T among the sea facets, and hence,\nwith sufficient mathematical ingenuity, the profile of the\nwavelets of the sea. This may be regarded as a difficult\nway to find out the shape of the wavelets.\"\nThis remark (in which we have inserted our own angle\nnotation) contains the germ of a fruitful approach to the\nquantitative study of the distribution of water wave slopes.\nThe closing sentence, no more than a subjective interjection,\nmay be cast aside, leaving a positive suggestion in which we\nsee the possibility of photographing the sea and in principle\nbeing able to count in the photograph the number of wave\nfacets in a given area having a given tilt. The execution\nof this intricate task, which in all its manifold details\nindeed requires mathematical ingenuity, was successfully com-\npleted twenty years later in 1954 by Cox and Munk [56]. We\nshall subsequently turn to an exposition of the main results\nof their study.\nDuntley's Immersed-Wire Measurements\nof Wave Slopes\nIn November of 1949 in Lake Winnipesaukee, New Hamp-\nshore, Duntley [82], [73] performed his first experiments to\ndetermine the statistical properties of the slopes of the\nair-water surface. The purpose of the experiments was to\ntest the hypothesis that the slopes of a steady wind-blown\nsurface at a given point were distributed in time according\nto a gaussian law. The existence of such a regular law for\nthe distribution of slopes would be of great importance in\nfurthering the study of the transmission of radiant flux\npast the dynamic surface. Applications to surface and sub-\nmarine visibility, absorbed solar insolation in the sea, and\nmarine biology would, in any event, ultimately be concerned\nwith such findings.\n*A minor point, but this is not quite correct. See, e.g.,\nFig. 12.30, and the explanation of the grid overlay. The\nmain idea of the quotation is, however, correct and was deemed\nworth preserving.","SEC. 12.5\nWAVE SLOPE DATA\n139\nRECORDING\nAMPLIFIER\nDETECTOR\nOSCILLOGRAPH\nSUBTRACTOR\nAMPLIFIER\n2000\nCYCLE\nELECTRONIC\nGENERATOR\nAMPLIFIER\nDETECTOR\nCOMPUTER\nAIR\nWATER\nFIG. 12.26 Schematic diagram of Duntley's immersed\nwire technique of measuring water-wave slopes.\nAccordingly an instrument was designed and built to\nsense and record simultaneously the two principal slopes\n5x(= 25/2x) and Sy (= 25/dy) of the waves (cf. notation of\nSec. 12.3) at a fixed location in the water and over a suit-\nable period of time. The instrument, called the sea state\nmeter, is schematically depicted in Fig. 12.26. Two pairs\nof vertical, parallel stainless steel wires spaced about 1-\ninch apart were mounted and oriented so that the planes of\neach pair were mutually orthogonal. Both pairs were powered\nby the same 2000-cycle alternator and their impedance con-\ntinuously recorded. The principle of the method was that\nthe electrical impedance of a pair of immersed wires was\naltered measurably and in a reproducible manner when their\nrelative submerged lengths changed. The difference in sub-\nmerged length is directly proportional to the slope of a.\npassing wave. Therefore the instrument could monitor the\ninstantaneous slopes of passing waves. Provision was also\nmade for the simultaneous recording of wave amplitudes. A\ntypical sample of amplitude and slope records is shown in\nFig. 12.27.\nAfter numerous records of wave slopes were studied, it\nwas concluded that the frequency of occurrence of a given\nslope could be closely approximated by a gaussian distribu-\ntion, and that the hypothesis leading to the experiment was\naccordingly verified in its essential aspects. The visuali-\nzation of this result may be helped by means of Fig. 12.28,\nwhich represents the first data obtained by the sea state\nmeter. The up-down wind recording is depicted. The output\nof the sea state meter was also recorded in terms of the","VOL. VI\nAIR-WATER SURFACE PROPERTIES\n140\n13 DECEMBER 1949\nWIND VELOCITY: 5 KNOTS\n7 P.M.\nAMPLITUDE\no\n-3\"\n11\n6\n7\n2\n3\n4\n5\no\nTIME IN SECONDS\n(WITH THE WIND)\nSLOPE\n+20°\n-20°\n(ACROSS THE WIND)\nSLOPE\n+20°.\n-20°\nFIG. 12.27 A sample of Duntley's wave-amplitude and\nwave-slope measurements (cf. Fig. 12.26).\nnumber no of times the wave slopes assumed the value tan o.\nIf during an experimental run a large number n o of crossings\nfor a given wire pair were recorded for tan 0 = 0, i.e., for\nhorizontal slopes, then the ratio no/no would be a measure\nof the relative number of times the wave slope for that given\nwire pair was of `magnitude tan o. It was found that the ratio\nno/no depended on 0 in the following manner:\n/\n(13)\nwhere 02 is the observed mean square slope. (Note that h2 in\nFig. 12.28 is such that h2 = 1/202.)\nFurther experiments showed that the mean square slopes\ncould differ for each pair of wire sensors, indicating that\nthe capillary waves and their gravity wave supports had a\npredominant directional flow. To distinguish between the\ntwo slope distributions, let us agree to use a wind-based\nreference frame whose xz-plane is parallel to the wind direc-\ntion. Then let us write:\n\"ou\" for (mgo)1/2 1/2\n(14)\nand","141\nWAVE SLOPE DATA\nSEC. 12.5\nCOUNTS\n200\n100\nWIND\n18 KNOTS\nZ\nZ\n-\n-30°\n-20°\n-10°\no\n10°\n20°\n30°\nSLOPE (Z = tan )\n300\n300\nh 2 tan 2\nCROSS-WIND\n/no=\n200\n1000\n200\nS=\nh2\na\n100\nh2=19.2\nh2=19.2\n70\nS=52.2\nS=52.2\n40\n100\n.00\n.04\n.08\n.00\n.02\n.04\ntan 2\n2\ntan\nFIG. 12.28 Duntley's experimental verification of the\ngaussian wave slope law (cf. Figs. 12. 26, 12.27).\n\"oc\" for (m 02 02) 1/2 1/2\n(15)\nwhere m20 and m 02 are the mean square slopes defined in gen-\neral in (88) of Sec. 12.4. The \"u\" and \"C\" denote the up-\ndown wind and crosswind directions, respectively. The first\nset of experiments with the wires spaces at 1 inch (2.5 cm)\napart yielded 02/02 = 1.9. In a later experiment the wire\nspacing was decreased to 0.9 cm, i.e , about one-third inch\nwith the resulting ratio 02/02 = 1.6. This decrease is be-\nlieved to reflect the increased resolving power of the wires.\nHowever, reduction of the spacing much further below 0.9 cm\nwould cause a pronounced meniscus between the wires which\nwould affect their resolving power. Other surface tension\neffects such as the wires contributing their own capillary\nwaves to the melee (the \"fish-line\" problem, Art. 272, [149])\nshow the inherent limitations of the immersed wire method.","142\nAIR-WATER SURFACE PROPERTIES\nVOL. VI\nHowever, this method was of sufficient sensitivity to clearly\nverify the gaussian wave slope hypothesis and hence to lay\nthe empirical foundations for subsequent statistical studies\nof the dynamic air-water surface.\nIntuitive Picture of the Gaussian\nSlope Distribution\nSince the gaussian wave slope distribution will play an\nessential role in the remainder of this chapter, we shall\npause here to emphasize the physical and geometrical meaning\nof the distribution. To simplify the exposition, we shall\nassume that and °c are equal to that, for any orientation\nof the immersed wires in the sea state meter, the n, count\nobeys (13). Part (a) of Fig. 12.29 depicts an instantaneous\ncross section of the water surface at point P by a vertical\nplane through the unit normal to the water surface. The angle\nof tilt of the normal from the vertical is 0. Now as the\nangle 0 varies from 0 to 90°, its slope tan 0 varies from 0\nto 80. In part (b) of Fig. 12.29, a two-dimensional domain\nwith rectangular coordinate system is displayed, along each\naxis of which we have measured out equal intervals of tan 0,\nstarting at the origin and going out to 00. This is the slope\ndomain of the water wave slopes. Imagine that the sea state\nmeter is turned on and run for a relatively long time T, long\nenough so that hundreds of wave slope recordings can take\nplace for each wire pair. We next record the total amount\nof time that wave slopes are in the interval (tan +\nT\n+ (1/2) d (tan 0)), (tan - (1/2) d (tan ) where d(tan ) is\na small increment of slope. In these ,recordings the normal\nn to the wave slopes may have any orientation about the ver-\ntical; it is the tangent of the tilt angle and not its azi-\nmuthal location which is essential at the moment. However,\nfor visualization purposes the two-dimensional plot in (b)\nof Fig. 12.29 is designed to account for the various azimuth-\nal orientations the wave normals may take, and these orienta-\ntions may be easily obtained from the recorded data. Thus if\nthe wire pairs record slopes tan Pu and tan Oc for the up-down\nand crosswind directions, then the actual tilt angle is re-\nlated to\nby:\nC\ntan20 tan 2 tan\n(A derivation of this is given in (16) of Sec. 12.9. The\nazimuth a of the wave normal relative to the upwind direction\nis given by:\ntan a = tan tan de/du 25/2c\nWith these preliminaries established, we now compute the\nfraction To of time that the wave slopes tan are in the\ncircular ring of radius tan and thickness d(tan ) in the\nslope space, as shown in (b) of Fig. 12.29. With the numbers","SEC. 12.5\nWAVE SLOPE DATA\n143\n0\nP\n(a)\n(b) = ce 2 I tan2 o2\nupwind\ncrosswind\ntan\nd(tan\nFIG. 12.29 Toward a visualization of the empirical\ngaussian wave-slope law.\nTo/T as ordinates we can construct a surface of revolution-- - -\na bell -shaped - surface such that the volume determined by the\ncylindrical shell of radius tan and thickness d (tan ) is\n2 tan o (T) d (tan )\nFrom the manner of defining To/T we see that, except for a\nconstant factor, To/T is given by (13). Hence the volume of\nthe cylindrical shell is:\n-1/2 tan20\no2\n2 C tan 0 e\nd(tan o\nand this \"volume\" is to be interpreted as the fraction d\nof time the slope of the water surface is in that region\nof","AIR-WATER SURFACE PROPERTIES\nVOL. VI\n144\nof the infinite slope domain defined by the ring of radius\ntan 0 and thickness d (tan ). We shall select the constant\nC so that the total time the slope is in the slope domain is\nunity. Writing ad hoc:\n\"E\"\nfor\ntan ,\nwe have:\nso that\n== d = 2 C\n= 2 C o2\nwhence\nC - 2710\n(16)\nFinally, performing a similar integration from 0 = 0 to some\narbitrary 0, we find that the fractional time to in which the\nwave slopes occur in the interval (0, tan 0) at a given point\non the sea surface is:\n(1.2)\nto\n(17)\n=\nIn this way a simple intuitive picture of the gaussian prop-\nerty of water wave slopes may be obtained. The reader should\nnote that (17) may also be interpreted as a probability in\neither a spatial or temporal context. In a spatial context,\n(17) states that the probability that wave slopes occur in\nthe range (0, tan 0) over a given region of sea surface is\nto. The temporal interpretation is similar (cf. (29)).","SEC. 12.5\nWAVE SLOPE DATA\n145\nThe Wave Slope Wind-Speed Law (Duntley)\nAnother important property of wind generated waves un-\ncovered by Duntley's immersed-wire method of measuring wave\nslopes is the remarkable linear dependence of and of on\nwind speed [73]. In the initial experiments with the wires\nspaced at 2.5 cm, data were taken with wind speeds from 3.5\nto 40 knots (1.8-20 m/sec) and the resultant linear relations\nwere:\n= (0.0053+0.0022)U a\n(18)\ne = (0.0028 + 0.0015)U,\n(19)\n,\na\nwhere ou,oc are as defined in (14) and (15) The air speed\nU was measured with an integrating anemometer located 8\ninches above the wave crests, and the units of Ua are in\nknots (1 knot = 0. 515 m/sec).\nLater experiments at the same location using a wire\nspace of 0.9 cm yielded:\n0°= (0.0052 + 0.0011)U,\n(20)\n= (0.0032 + 0.0014) U\n(21)\nFurther discussion, which will aid in the application of\nthese results, is given below in the exposition of Cox and\nMunk's experimental study of the wave-slope wind-speed law.\nSee also (11) of Sec. 12.8.\nCox and Munk's Photographic Analysis\nof the Glitter Pattern\nIn 1954 Cox and Munk [56] published the results of an\nextensive analysis of photographs of the sun's glitter pat-\ntern on the surface of the sea. The data were taken in the\nregion of the Pacific around Hawaii during September of 1951,\nthe object being to learn something about the distribution\nof water wave slopes generated by various wind speeds. Two\nimportant quantitative results were forthcoming from the\nanalysis: first, that the statistical distribution of wave\nslopes over a steady state glitter pattern could described\nby a Gram-Charlier distribution which reduces in most natu-\nral settings to a gaussian distribution; and second, that\nthe mean square of the wave slopes over the same region was\nlinearly proportional to the mean of the generating wind\nspeed at the air-water surface. These observations confirmed\nsimilar findings by Duntley [73] arrived at earlier by com-\npletely different methods, as we have seen above.\nThe central task of Cox and Munk in the glitter pat-\ntern study resolved itself into two main phases: (1) to","AIR-WATER SURFACE PROPERTIES\nVOL. VI\n146\nidentify a glitter point on the sea surface, by means of geo-\nmetrically controlled measurements on carefully made photo-\ngraphs, and to determine the slope of the sea surface at that\npoint; (2) to interpret the average brightness of the sea sur-\nface over a small region around a point in terms of the rela-\ntive number of waves in that region having given slopes. By\nconsidering many such regions, a statistical distribution of\nslopes could be determined optically. The pertinent geomet-\nric details leading to the completion of these two phases\nand interpretations of results are given in [55]. However,\nit is of interest to briefly illustrate the manner in which\nthe photographic data were initially processed for the first\nphase of the analysis and this is shown in Fig. 30 (A), (B)\nwhich was kindly made available by Cox and Munk. The photo-\ngraphs were taken in late afternoon (A) or near sunset (B)\nin late summer of 1951, looking due west, from an altitude\nof 2000 feet in the vicinity of Maui an island in the Hawai-\nian group. This pair of photographs shows the effect of the\nsun's elevation on the glitter pattern. (In other photo-\ngraphs available in Part I of [55], or [57] the effects of\nwind speed are shown.) In (A) of Fig. 12.30 the sun is at\n50° above the horizon and the camera axis is tilted 50° below\nthe horizon. In (B) of Fig. 12.30, the sun is at 30° eleva-\ntion and the camera tilted 30° below the horizon. In each\ncase the wind speed was about 9 knots at 41 feet above mean\nsea level. The superimposed grids on the photographs form\nthe heart of the method and are the realization of the req-\nuisite \"mathematical ingenuity\" which Hulburt (see above)\nhad hoped some day would be applied to the glitter pattern\nanalysis. The radial curves connect those reflecting facets\non the water whose lines of steepest slope (or whose normals)\nhave the same angle a measured from the vertical plane of the\nsun; the closed curves connect those reflecting facets on the\nwater surface whose lines of steepest slope (or whose normals)\nhave a common angle B from the vertical direction. (The\nprincipal analytic equations leading to these grids are of\nthe form (8) - (10), plus some auxiliary equations showing how\nthe camera images are related to their real counterparts.)\nHence the regions enclosed by the oval shaped curves are\nanalogous to the circular rings discussed in Fig. 12.29.\nWe shall extract the main results of Cox and Munk's in-\nvestigation for summary here.\nLet 5x(= 25/2x) and by (= 25/dy) be the slopes of the\nair-water surface along the X and y axes which are now ori-\nented so that the X axis is parallel to the mean wind direc-\ntion and the y axis is perpendicular to that direction. To\nemphasize that this particular orientation is now adopted,\nwe shall again use (89) - (91) of Sec. 12.3 to write:\n(mgo)1/2\n(22)\n\"ou\"\nfor\n20\nand\n(23)\n\"Oc\" C\nfor\n02","SEC. 12.5\nWAVE SLOPE DATA\n147\nFIG. 12.30 A sample of the sun-glitter photographs\ntaken by Cox and Munk, and the grid overlay which\nidentifies points on the water at which the normal\nto the reflecting surface facet has direction angles\na and B (see text).","VOL. VI\nAIR-WATER SURFACE PROPERTIES\n148\nHence ou is the root mean square slope of the sea surface in\nthe upwind direction, and °c is the root mean square slope of\nthe sea surface in the crosswind direction. Furthermore,\nwriting:\n(24)\n5x/o2\nfor\n\"E\"\nu\n(25)\n5y/oc ,\n\"n\"\nfor\nCox and Munk's experimental findings showed that 5x and by\nwere on the average zero over the photographed region, and\nthat the probability p(5x25y) of finding the sea surface at\nan arbitrary point with slope components (5x25y) is given by:\n(26)\ne\nwhere we have written:\n(27)\n\"G(E,n)\" for 1 +\nand where the H1 are Hermite polynomials and the cij are con-\nstants for the given sea region and wind conditions: The\ndistribution (26) is known as a Gram-Charlier distribution\n[59]. This reduces to the gaussian distribution:\ne -(1/2)(E+n)\n(28)\nwhen Cij = 0 for all i,j. The first few coefficients cij of\nthe distribution describe the skewness and peakedness of the\ndistribution. However, for most practical settings (28) may\nbe used as the basic law governing the distribution of slopes.\nAt this point it is of interest to observe the dual\nphysical conditions under which the gaussian slope distribu-\ntion law was obtained; first, by Duntley and then by Cox and\nMunk: Duntley's sea state meter measurements, as we have\nseen, were made at a fixed point in the air-water surface\nwith the averages taken over time; Cox and Munk's glitter\nphotographs were made at a fixed point in time with the aver-\nages taken over the air-water surface. The resultant slope\ndistribution in each case was described by a gaussian law.\nThis lends some credence to the physical applications of the\nso-called ergodic hypothesis which states that these two\naverages (and related analytical averages) can be inter-\nchanged one for the other in many optical and physical dis-\ncussions of the dynamic air-water surface. As an illustra-\ntion of this, we return to (17) and re-interpret it as a","SEC. 12.5\nWAVE SLOPE DATA\n149\nstatement about the spatial rather than a temporal property\nof the wave slopes. Toward this end, we shall set\nin (28) and note that 52+n2 = (52 + 52)/02, in which\nis invariant with respect to the rotation of the xy axes see\n(16) of Sec. 12.9). This magnitude we shall again denote by\n\"tan20\". Hence (28) becomes:\nSince the probability here is derived from a frequency inter-\npretation of the occurrence of wave slopes over the sea sur-\nface at a given instant, it follows that the fractional area\na of sea surface which has wave slopes lying in the range\n(0, tan 0) at a given instant in time is:\n(1-0-14)\n(29)\nso that (cf. (17)).\nThe Wave-Slope Wind-Speed Law (Cox and Munk)\nIn addition to deducing the Gram-Charlier representa-\ntion of the wave slopes of the air-water surface, Cox and\nMunk used their photographic analyses [56] to corroborate\nDuntley's finding that the mean square slope 02 of a steady\nwind generated water surface varied linearly with the wind\nspeed Ua.\nThe relations obtained for a natural hydrosol (clean\nsurface) were:\n===0.000 + 0.00316) Ua 0 0.004 , r = 0.945\n(30)\n= , r==0.956 (31)\n+ 0.004 , r = 0.986 . (32)\nThe associated relations were also measured for an\nartificially induced slick surface: a mixture of oil was\npumped on the water, consisting of 40% used crank case, 40%\ndiesel oil, and 20% fish oil. With 200 gallons of this mix-\nture, a coherent slick 2000 by 2000 feet could be laid in 25\nminutes for winds not exceeding 20 miles per hour. The re-\nsultant relations were:\n= 0.005 - (0.00078) , r = 0.70\n(33)\n(0.00084)U +0.002 , r=0.78\n(34)\n0.004 ,\n0.77\n(35)\nr","AIR-WATER SURFACE PROPERTIES\nVOL. VI\n150\n.04\n9°50'\n.02\n5°43'\n12°38'\n.04\n9°50'\n.02\n5°43'\n0°\n15°48'\n.08\n14°47'\n13°46'\n06\n12°38'\n6.04\n9°50'\n.02\n0°\n10\n15\no\n5\nWm/sec\nFIG. 12.31 Mean square slope components for upwind\n(OR) and crosswind (o2) directions and their sum, as func-\ntions of wind speed at 41 feet above sea level. Open circles\ndenote clean sea surface readings; solid circles denote slick\nsurface readings. Solid and dashed lines are respectively,\nregression lines for clean and slick surfaces.\nIn (30) ) - (35) , - , the air speed U a is in units of meters per\nsecond and as measured at a height 41 feet above sea level.\nThe form of presentation above is the usual manner of dis-\nplaying the result of a least-square fit of a straight line","SEC. 12.5\nWAVE SLOPE DATA\n151\nto measured data. The slope of each regression line is in\nparentheses, the standard deviation of the slope data follows\nthe sign \"+\", with correlation coefficients given by r. The\nnumbers before the parentheses represent the contributions to\nmean square slopes from nonlocal disturbances, i.e., swells\nand other waves not generated by the local wind. Hence the\nworking parts of the six preceding formulas are obtained by\nomitting these nonlocal disturbance terms and standard terms.\nFigure 12.31, adapted from [57], summarizes these results\ngraphically.\nIt is instructive to compare the two versions of the\nwave-slope wind-speed law obtained by Duntley and by Cox and\nMunk. For example, converting (20) and (21) to meters per\nsecond, wind-speed units and referring wind speeds to 41 feet\nabove sea level (cf. (3)), we have:\n0.515 m/sec\n2 @ 41 feet\n= (0.0052 + 0.0011) U a\nknot\n@ 8 inches\nout = (0.0050 + 0.0011) U\n(Duntley)\na\nwhere Ua is now in meters per second and measured at 41 feet.\nThe change-of-scale factors are shown in parentheses. These\nmultiply U: and their inverses multiply 0.0052 + 0011. By\nan odd fluke the two conversion factors essentially cancel.\nSimilarly, we have:\n= (0.0031 + 0.0014)U\n(Duntley)\n(31a)\na\nwhere Ua is now in meters per second and referred to 41 feet\nabove sea level. Comparing Duntley's our with that of Cox\nand Munk in (30), we see that the rate of increase of our\nwith U a varies from 0.0061 U to 0.0039 Ua, with average\n0.00500 U for Duntley, as compared with the average 0.00316\nUa for Cox and Munk, for the case of a clean surface. The\ndifferences here are believed to be attributable to two main\nfactors: Duntley's measurements were taken in the vicinity\nof outcropping boulders near the lake shoreline, about 25\nfeet from the sea state meter. The reflected waves from the\nsupports of the sea state meter and shoreline would tend to\nincrease the observed o 2 and of. Secondly, the immersed\nwires of the sea state meter themselves may have generated\nadditional capillary waves (the fish line problem [149]] and\n2\nthereby have raised the o estimates.\nSchooley's Flash Photography Measurements\nWave Slopes\nIn the fall of 1952, subsequent to the work of Duntley\nand Cox and Munk, Schooley obtained the results of a flash\nphotography technique of measuring wave slope distributions\nin a dynamic air-water surface [274]. The technique consists\nin taking photographs with a flash camera directed vertically\ndownward toward the moving water surface. Schooley used a\nbridge over the Anacostia River, Washington, D.C. as a camera","VOL. VI\nAIR-WATER SURFACE PROPERTIES\n152\nHdk\nF\nC\n20°\n10°\nW\nFIG. 12.32 Camera geometry for Schooley's flash\nphotography measurements of wave slopes.\nsupport for his experiment. The resulting photographs reveal\nflash reflection patterns that can readily be converted into\nwater wave slope data. The associated geometry is vastly\nsimpler than in the case of Cox and Munk. However, the pres-\nence of the bridge and river shoreline, as in the similar\ncase of the presence of Duntley's sea state meter supports\nand nearby shoreline, contribute a measure of unwanted hydro-\ndynamic effects. Nevertheless, the simple geometry of the\nexperiment, depicted schematically in Fig. 12.32, leads with\na minimum of analytical effort to results not too greatly\ndifferent from the general type of results of Cox and Munk,\nand Duntley. Thus, for example, the standard deviation of\nthe crosswind wave slopes was found to be approximately 2.5,\n5, and 7.8 degrees for surface wind speeds of 5, 10, and 20\nknots, respectively. Under the same conditions the up-down\nwind standard deviations were found to be 4. 2, 7.5, and 10\ndegrees. The differences between Schooley's results and Cox\nand Munk's results is that Schooley's results, wind speed\nfor wind speed, are seen to yield too low mean square slopes.\nThis perhaps could be attributed, as suggested by Cox and\nMunk, to the short wind fetch distances on the river not per-\nmitting sufficient build up of wave slopes; and also imper-\nfect resolution of the highlighted areas on Schooley's photo-\ngraphs. (For an indirect route to the wave-slope wind-speed\nlaw from a rather unexpected direction, see (12) of Sec. 12.8.)\n12.6 Wave Generation and Decay Data\nIn this section we shall review those phases of the\ngrowth and decay of wind-generated waves pertinent to our","SEC. 12.6\nGENERATION AND DECAY DATA\n153\npresent optical studies of the air-water surface. Most of\nthe properties presented below are relatively precise quanti-\ntative empirical facts about our ordinary intuitive under-\nstanding of wave phenomena. These data have been gathered\nover the years by various workers and are beginning to form\na coherent picture of the generation and decay of wind-gener-\nated waves on the air-water surface. For workers in the field\nof hydrologic optics, some knowledge of the basic facts of\nthe generation and decay of water waves will be helpful in\npredicting the optical state of the sea in a region, having\ngiven the wind histories, fetch lengths, and some data on\ninitial wave heights and wave speeds.\nGeneration of Waves: Shallow Depths, Small Fetches\nThe generation of waves is replete with intricate\neffects. We shall describe these features on two levels of\ndetail: first for shallow. depth and small fetches, which\nwill allow detailed phenomena to be observed. Then the wave\ngeneration over oceanic depths and distances will be discussed.\nWhen the air above a still air-water surface is slowly\nset into smooth horizontal motion, there is no perceptible\ndisturbance in the smooth surface. If this smoothness (lami-\nnar type) of flow is maintained as the air speed increases\nsomewhat, the surface remains unperturbed. However, if the\nsurface has any sort of nonplanar irregularity, produced per-\nhaps by the dropping of a bit of foreign matter on the sur-\nface, or if the flow of the air is but the slightest nonlami-\nnar, then the irregularity may grow, under the prodding of\nthe air flow into a wave of increasing length and height.\nOur studies of the Kelvin-Helmholtz instability ((89) of Sec.\n12.3) showed that an air speed of 6.6 m/sec is sufficient to\ncause such irregularities to develop into growing wavelike\ndisturbances in the surface. Actually, in nature purely\nlaminar air flows are rare: every breeze, no matter how\nsmall its speed over the water surface, has small irregular\nvertical components which cause pressure dimples in the\nwater. The passing air can swirl into and push these dimples,\nthereby yielding up some of its energy to the water and which,\nin turn, promptly cashes it in for additional wave amplitude\nand wavelength. And so waves begin to grow.\nIn 1951 Roll [266] made some detailed observations on\nthe growth of waves in the ponds of the tidal flats at Neu-\nwerk off the Frisian coast in the Netherlands. Wave growths\nwere observed over fetches of 70 m starting on the windward\nside of the ponds. Despite the shallowness of the ponds, the\ndepth of the water usually exceeded half a wavelength, and\nthe smallness of the ponds still permitted steady state con-\nditions to be attained shortly after changes of wind direc-\ntion. In Fig. 12.33 (adapted from [307]] a set of curves are\ngiven, each curve being associated with a given fetch. The\ncurves show the wavelength of the wind-generated waves as a\nfunction of wind speed Ua These graphs under careful study\ntell a remarkable story. For example, if the wind were blow-\ning at 200 cm/sec, small transitional capillary waves of 1.7\ncm wavelength would be formed first. This wavelength, as we","VOL.\nVI\nAIR-WATER SURFACE PROPERTIES\n154\n20\n30\n20.5\n3.2\n40m\n2.7\n40\n5\n2.30\n1.0\n1.7\n1.2\n0.7\n1.2\n0.9\n1.2\n0.8\n35.5\n1.2\n5.6\n1.7,322.23.2.7\n11.5\n30\n8\n500\n600\n400\n$20\nu\nScott Russell\n15.5/\n/10.5\n40\nJeffreys\n5\nStanton\n20\n35\nNeuwerk\n1.0\n9.5\n5.5\n0.8\n2\n40\n5.5\n5.6\n10\n0.6\n30\n.\n0.5\n0.9\n20\n0.5\n40\n0.3\n0,5\n0.7\n0.4\n0.2\n0.20\n20\n0.3\n1600\n1000\n1200\n1400\n200\n400\n600\n800\nWind velocity U a in cm sec.\nFIG. 12.33 Wavelength l of wind generated waves as a\nfunction of air speed Ua (at 35 cm above the surface)\nand fetch F, from data by Roll taken in 1951.\nhave seen ((93) of Sec. 12.3), lies on the borderland between\ncapillary (or surface tension waves) and the larger gravity\nwaves. The graphs show that as the waves progress and the\nfetch increases, these transitional waves suddenly split\napart at about 20 m fetch, aided no doubt by the nonlaminar\nflow of the air. As the fetch increases farther, say to\n30 m, there are now two sets of waves, one of wavelength\nabout 1 cm, the other of wavelength about 7 cm. Generally\nspeaking, as the fetch increases still farther, the big waves\nget bigger and the smaller waves get smaller. The steepness\nof the waves at the time of splitting may approach the ratio\n1 to 7 (height to length) but generally the gravity waves'\nslopes diminish with fetch. The graphs also show that the\ngreater the wind speed, the smaller the fetch at which the\nsplit takes place. This splitting is signalled in the graph\nby a vertical tangent to the particular fetch curve.\nIn exact radiative transfer calculations both wave\nheight and wave slope data are important. Thus the labora-\ntory experiment of Cox [54] documenting the growth with fetch\nof the mean square wave slope serves to complement the find-\nings of Roll on wavelengths and wave heights. Figure 12.35\ndepicts mean square slope as a function of wind speed for\nthree different fetches in a laboratory water tunnel shown\nin Fig. 12.34. Figure 12.36 gives some information on the\ngrowth of mean square slopes with fetch for five different\nwind speeds.","SEC. 12.6\nGENERATION AND DECAY DATA\n155\n4\nI\n3\n2\nII\n7\n10\n9\nB\n6\n0000\n5\nFIG. 12. 34 The laboratory wind and water tunnel, as de-\nvised by Cox, for measurements of wind generated wave slopes.\nA cup anemometer (1) measures the wind speed in the entrance\nnozzle (2) as the wind sweeps over the water (dotted) and is\ndrawn through the tunnel by a fan (3) The wind speed is\ncontrolled by a damper (4). Observations of the waves are\npossible through a set of viewing ports all along the ceiling\nof the tunnel. The inset shows the detail of a viewing port.\nLight sources (5) send light through a diffusing plate (6)\nand on through a plate-glass window (7) and past a neutral\nfilter wedge (8) A telescope tube (9) focusses an image of\nthe water surface on a pinhole in front of a photocell (10)\nwhich leads to a recorder (not shown). Counterflowing waves\ngenerated at the end of the tunnel were dampened by a gravel\nbeach (11). The dimensions of the tunnel are 14 cm (depth),\n26.3 cm (breadth), 6.) 1 m (length).\nThe phenomenon of a minimum-speed wave, of 1.7 cm wave-\nlength, splitting into a distinct pair of capillary and gravity\nwaves as observed by Roll, was also observed by Schooley [276].\nHowever, the more interesting contribution by Schooley in\n[276] was the experimental verification of the detailed geo-\nmetric structure of the wind-generated capillary waves. Re-\ncall from our hydrodynamic studies in Sec. 12.3 that the\nlinearized theory was able to predict precisely the celerity-\nwavelength relationship for capillary waves ( (95) of Sec. 12.3).\nHowever, the true shape of the capillary waves of finite\namplitude was beyond the ken of the linearized theory. The\nexact shape of capillary waves (as in the case of gravity\nwaves) of finite amplitude, such as those seen being generated\nby the wind, is a difficult problem in nonlinear differential\nequation theory which has only recently been solved by Crapper","AIR-WATER SURFACE PROPERTIES\nVOL. VI\n156\n0.08\nF = 3.20 m\n.08\n0.04\nF = 2.14m\n04\n9.\n0.08\n-O\nF = 1.47m\n.04\n12\n8\n4\no\nWIND SPEED (m sec-1)\nFIG. 12.35 Upwind mean square slope our as measured in\nthe wind tunnel of Fig. 12.34, for the indicated fetches F.\nU= 3.5m sec\n.04\n3.25\n3.0\n.02\n2.75\n2.5\no\n2\n3\n4\nFETCH (m)\nFIG. 12.36 Upwind mean square slope o 2 as a function\nof fetch for various windspeeds as derived from the graphs\nof Fig. 12.35","SEC. 12.6\nGENERATION AND DECAY DATA\n157\na/x = 0.73\n0.53\n0.34\n0.20\n0,10\n0.05\nFIG. 12.37 Theoretical capillary wave profiles for\nvarious amplitude/wavelength ratios (a/ ) , as pre-\ndicted via Crapper's analysis.\n[60]. Schooley verified the theory by reproducing capillary\nwave shapes of the predicted kind. A sample of shapes is\nshown in Fig. 12.37. According to the linearized hydrodynam-\nic theory, solitary gravity waves customarily occur in sinu-\nsoidal form with very small (infinitesimal) heights in the\nair-water surface and have relatively gently sloped configu-\nrations. For gravity waves of finite height, the configura-\ntions are approximately trochoidal with configurations not\nexceeding 1 to 7, depth to length; and if they have vertical\ncrest-cusps-- - these will occur with tangents not exceeding 30°\nfrom the horizontal (cf. Arts. 250-252, [149]). The shape of\nfinite capillary waves, on the other hand, seem turned inside\nout relative to the trochoidal gravity wave shapes; that is,\nin capillaries the cusp is a wedge of air pointing down into\nthe water; whereas in finite gravity waves, the cusp is a\nwedge of water pointing up into the air.\nGeneration of Waves: Deep Depths, Large Fetches\nExtensive observations over the years of the relations\nbetween wind speed, wind duration, fetch, wave height, and\nwave speed of oceanic waves and waves of other extensive nat-\nural hydrosols, have resulted in a comprehensive, orderly\npicture of the empirical and theoretical interconnections\namong arbitrary pairs of these parameters. One of the first\nsystematic modern compilations was made by Sverdrup and Munk\n[293], which was later extended by Bretschneider [32] with","1000\n0.1\n103\n102\n10\nfeet\nfeet\n8.0\n6.0\n4.0\n2.0\n80\n60\n40\n06\n008\n006\n10.0\n1.0","SEC. 12.6\nGENERATION AND DECAY DATA\n159\nadditional data. Appropriate references for these data are\ngiven in [32]. These data are redrawn and briefly summarized\nin Fig. 12.38. For a related laboratory study, see [129]. A\nuseful related reference on wave generation, decay, spectra,\nand statistics is [205].\nIn using Fig. 12.38, it is important to note that, in\nthe present discussion and that on wave decay below, \"wave\nheight 11 means \"significant wave height,\" a concept evolved\nby oceanographers in an attempt to attain a usable measure\nof the height of sea waves when all are visually different\none from the other and no mathematically clearcut definition\nof \"wave\" or \"wave height\" is available. The concept of\nsignificant wave height may be defined as follows: record\nall the wave heights going by a fixed point during a test\nperiod Tp (recall that a wave height is the vertical distance\nfrom crest to trough and is twice the amplitude) and rearrange\nthe recorded heights in order of decreasing height. Say\nthe\nheights are H1 H2\nHN so that H1 > H2 >\n> HN.\nDivide the group into three equal parts (suppose for this\npurpose, N = 3M) ; take the first part H\nHM,\nand\nfind the average: = (1/M) (H + H2 +\n2\nHM).\n(A\n+\npractical method of determining M, and also a clarification\nof what is meant by \"wave\" is given below.)\nThe resultant average is the significant wave height\nfor the wave record taken during the test period Tp. When\nseas are steady, that is in equilibrium, the significant\nwave height is usually independent of T. for large Tp.\nHence, in a word, the significant wave height of a sample is\nthe average of the heights of the highest one-third of the\nwaves in that sample. The significant wave heights in Fig.\n12.38 are steady state (i.e., equilibrium) heights, in the\nsense explained in the example below.\nA significant wave period can be associated with the\nsignificant wave height as follows. As is well known, waves\noften travel in closely packed visually observable groups\nwhich are the superimposed results of aggregates of component\nwaves. The average period T of these groups (each thought of\nas a single \"wave\") observed over a test period Tp is called\nthe significant wave period. If we form the quotient /T,\nwe obtain and estimate of the number of such wave groups ob-\nserved during Tp. Then (1/3) (T, is the number M (to the\nnearest integer) of waves used in the computation of the\nsignificant wave height above. The significant wave period\nT and associated wavelength L are related by: T = /2TL/g\nin the notation of Fig. 12.38. The significant wave period\nused in the Fetch graph is the steady state period.\nIn seas whose wave spectra are fairly narrow, in the\nsense that wave heights are confined to a narrow wave height\nor frequency range, it may be shown [164] that:\n(significant wave height) = 1.600 (average wave height)\n= 1.600 (H )\n(1)\nand that:","AIR-WATER SURFACE PROPERTIES\nVOL. VI\n160\n(significant wave height) = 1.416 (root mean square height)\n(2)\ni.e.,\n(H1/3'\n=1.416(H)\nIn the paper [164], a comparison with experimental data was\nmade which shows that the relationships (1) and (2) are\nquite accurate. In this way we can use Fig. 12.38 and (2)\nto estimate the mean square elevation 52, used in the theory\nof Sec. 12.11, under various hydrologic conditions. The con-\nnection between the mean square elevation 52 and the mean\nsquare height H2 is:\n(H)2 = 45 ²\n(3)\nWe shall use \"H1/3\" when it is necessary to be quite specific\nabout significant wave height; otherwise, for brevity, the\nbar and \"1/3\" will be dropped from \"H1/3\",\nAs an example of the use of Fig. 12.38, suppose we\nrequired the root mean square elevation of the wind generated\nwaves in an oceanic region for which the following wind and\nfetch information is known: The waves in the region were\ngenerated by 20 mile per hour surface winds over a fetch F\nof about 20 miles during a five hour period, t. These esti-\nmates are of course order-of-magnitude estimates at best in\npresent-day meteorologic technology and will be treated\naccordingly. Hence, converting to the units used by the\nFetch Graph, F = 20 x (5 X 103) = 105 feet. t = 5 x 3600 = 1.8\nx 104 sec. U = 20 x 1.47 = 29.4 feet/sec. We shall use the\ngraph of gH/U ², since we know F, and U, and of course g = 32.16\nfeet/sec2. But what role does t play? It is known from ex-\ntensive observations and theoretical analysis that for a\ngiven fetch length and wind speed, there is a time t beyond\nwhich the generated waves reach an equilibrium steady state\nin height and period. This limiting time is the time it\nwould take the energy front associated with the significant\nwaves to propagate, at the continuously changing group veloc-\nity, from the beginning to the end of the fetch [32]. This\nempirical connection between t, U and F is included in Fig.\n12.38. To see if the given time of 5 hours is a steady state\ntime, we compute gF/U2\ngF/U* = 32.16 x 10\" = 3.72 x 10\n3\nWith this value as abscissa, we find the corresponding ordi-\nnate on the tU/F vs. gF/U2 curve to be\ntU/F = 4\nso that\nt = 4F/U = 1.36 x 104 seconds","SEC. 12.6\nGENERATION AND DECAY DATA\n161\nwhich is the time for the wind-generated waves to reach a\nsteady state over the given fetch and under the given wind\nspeed. Since our given generating time is 1.8 x 104 sec, it\nappears that we may use the gH/U2 vs. gF/U2 curve to estimate\nH. We have already reckoned gF/U2; we find that the corre-\nsponding ordinate is:\ngH/U2 = 0.15\nso that\nH = 0.15 = 4.03 feet\nwhich is the steady state significant wave height at the end\nof the 20-mile fetch generated by the 20 mile per hour winds.\nBy (3) the associated root mean square elevation is\n= 1.42 feet.\nIf it turns out that a given wave generating wind has a dura-\ntion less then the steady state wind, then clearly Fig. 12.38\ncan only give an upper bound to 52.\nDecay of Waves\nWhen the wave-generating ocean winds decrease or die\naway, their wave progeny may continue for thousands of miles,\ndissipating and decreasing in their energies and heights con-\ntinuously as they travel along into relatively still water.\nInitially, the greater waves leave behind their smaller com-\npanions by sheer speed ((68) of Sec. 12.3), the net result\nbeing a spreading out in space of the original collection of\nwaves. (This is analogous to the dispersion of light waves\nin material media.) ) Empirical studies have shown that the\nheight and period of decaying water wave groups, at any stage\nin their decay-run, depend principally on three factors:\nthe length of the fetch over which they were generated, and\nthe height and period with which they began their decay - run.\nIn order to describe the observed dependences on these\nparameters, we shall write:\n\"TF\"\nfor\nsignificant wave period at end\nof fetch F\n\"HF\"\nfor\nsignificant wave height at end\nof fetch F\n\"TD\"\nfor\nsignificant wave period at end\nof decay distance D\n\"HD\"\nfor\nsignificant wave height at end\nof decay distance D.","VOL. VI\nAIR-WATER SURFACE PROPERTIES\n162\n16\nI\nII\n14\nIII\nIV\n2\n10\n8\n4\no\n20\nDECAY CURVES FOR TF =10 sec.\nHF\nU\nTF\nF\ntmin\nCase\nn.mil.\nknots\nfeet\nhrs.\nNOTE: AT D=0,\nsec\n16\nHF =HD\n100\n40\n9.5\n20.5\nI\n10\n29\n17.5\n16.1\nII\n10\n200\n32\n12.8\n10\n400\n23.5\nIII\n12\n56\n10.1\n10\n800\n20.8\nIV\nI\n8\nIV\n4\nIV\nI\no\n3000\n2000\n1000\n600\n200\nDECAY DISTANCE IN NAUTICAL MILES\nFIG. 12. 39 (a) An example of how wave heights decrease\nand wave periods increase with decay distance. This set of\ngraphs for Tf = 10 sec and the indicated cases (after Bret-\nschneider) .\nFigure 12.39 (a) and Fig. 12.39(b), adapted from [32], , gives an\nexample of these four quantities as a function of the decay dis-\ntance D. A11 graphs in (a) refer to a TF of 10 sec. Observe","GENERATION AND DECAY DATA\nSEC. 12.6\n163\n18\nV\nVI\n16\nVII\nVIII\n14\nNOTE: AT D=0, TF= To\n12\n10\n8\nVIII\n4\nVII VI\nV\no\n20\nDECAY CURVES FOR HF =20 feet\nHF\nF\nU\nTF\ntmin\nCase\n16\nfeet\nn.mil.\nknots\nhrs.\nsec.\nV\n20\n100\n39\n10.5\n9.9\nVI\n20\n200\n33\n16.5\n10.75\n12\n20\n400\n31\n28.5\n11.9\nVIII\n20\n800\n30\n47.0\n13.4\n8\nVII\nVII\nVI\n4\nV\nO\n200 600 1000\n2000\n3000\nDECAY DISTANCE IN NAUTICAL MILES\nFIG. wave 12. 39 (b) An example of how wave heights\nand for periods increase with decay distance. This decrease set\nBretschneider). graphs Hf = 20 feet and the indicated cases (after\nof\nshorter that for the fixed decay distances and fetch periods\nfor fetch the greater the period increase. TF, the\nmore, the same fetch period, the shorter the fetch Further- the","AIR-WATER SURFACE PROPERTIES\nVOL. VI\n164\nswifter the fall-off - of the relative height HD/HF with D. To\nsee this intuitively, note that larger wind speeds are needed\nto generate a given wave period in shorter fetches; the re-\nsultant waves are higher and steeper and consequently decay\nmore quickly (the nonlinear wave-dissipating - mechanisms are\nmore effective in waves of finite amplitude)\nA complementary situation to that in Fig. 12.39 (a) is\ndepicted in Fig. 12.39 (b) in which the common datum is now\nan HF of 20 feet.\nThe general shape of the curves is such that, on theo-\nretical grounds, the energy content of the waves should ul-\ntimately decrease exponentially, a fact which holds for both\nlarge and small fetches and hydrosol depths. However, ini-\ntial rates of decay are governed by the geometry of ine ini-\ntial waves, and their initial fetch, and are generally not\nof a simple exponential type. (In analogy to the asymptotic\nradiance theorem of Chapter 10, there may be a precisely\nphrasable and valid counterpart in ocean wave propagation\ntheory, so that in the limit, ocean waves decay at a rate\nindependent of their origin and dependent only on the inher-\nent hydrodynamic and geometric structure of the hydrosol.)\nD\n102\n3\n4\nH,\n10\n10\n.01\n0.1\n1.0\n10\nD\n100\n10\nD\nDECAY GRAPH FOR\nI\nF\nHEIGHT DECREASE\nD\nD\nvs\nHF HD\nWITH D AS A PARAMETER\nF\nD in Nautical Miles\n#\n\"\n#\nF\nHD in Feet\n\"\nHF\n.01\n1.0\n10\n0.1\nD\n.01\nHF\nFIG. 12.40 (a) Relationships between fetch, wave period\nat end of fetch, decay, and wave period at end of decay\n(after Bretschneider).","SEC. 12.6\nGENERATION AND DECAY DATA\n165\nD\nT22\n.001\n.01\n.I\n1.0\n10\n100\n100\n10\nD\n1.0\nDECAY GRAPH FOR\nF\nPERIOD INCREASE\nD\nD\nvs\nTo2 TF 2\n.I\nWITH AS A PARAMETER\nD in Nautical Miles\n18\n\"\n#\nF\nTD in Seconds\n\"\nHF\n.01\n.001\n.01\n.I\n1.0\nD\nT2\nFIG. 12. 40 (b) Relationships between fetch, wave height\nat end of fetch, decay, and wave period at end of decay (after\nBretschneider)\nFigures 12.40 (a) and 12.40(b) depict the general empir-\nical relations governing the decay of waves, as compiled by\nBretschneider [32]. Figure 12.40(a) depicts D/HF vs. D/HD\nwith D/F as parameter. Hence this figure may be used to\nfind HD knowing the fetch F, decay distance D and Hp. Fig-\nure 12.40 (b) on the other hand may be used to find TD given\nTF and D, exactly analogously to the example given above for\nH. For example, to find HD when HF = 5 feet, F = 20 nautical\nmiles, and D = 20 nautical miles; form the ratios D/F = 1,\nD/HF = 4. Starting at ordinate D/F = 1 on the left hand scale\nof Fig. 12.40 (a), move to the right until the abscissa D/H = 4\nis reached. This point lies between the D/HD curve 6 and the\nD/HD curve 8. Let us say that it lies on the D/HD curve 7.\nThen\nin\nH\nso that\nHD = D/7 = 20/7 = 2.85 feet","VOL. VI\nAIR-WATER SURFACE PROPERTIES\n166\n12.7 Wave Spectrum Data\nThe energy spectrum of the dynamic air-water surface is\na powerful modern analytical tool which may be used to describe\nthe intricate geometric and physical features of the air-water\nsurface. In particular, our study of the energy spectrum in\nSec. 12.4 showed how it may be used as a source of informa-\ntion on the mean square slopes and heights of the air-water\nsurface, quantitites which, as will be shown in Sec. 12.11,\nare of importance in the description of radiative transfer\nphenomena at the surface. In this section we shall review\nsome current means of obtaining experimental data on wave\nenergy spectra of the dynamic air-water surface. We shall\nreview three such procedures based on three fundamentally\ndifferent principles.\nThe first procedure to be described uses aerial stereo\nphotographs of the sea surface taken from flying aircraft;\nthe second records the motions of small floating buoys with\nsensitive positional and orientational recorders inside; the\nthird uses sonic echo transducers directed upward toward the\nwaves from the deck of a hovering submerged submarine. Each\nof these three procedures has one principal goal: the deter-\nmination of the one-dimensional spectrum Tk (or its equiva-\nlents To,TT) as defined in (102) of Sec. 12.4, or hopefully,\nwith enough probes and resolution, the full two-dimensional\nspectrum (k) (or its equivalents S,F) as defined in (75) of\nSec. 12.4. In regard to the matter of resolution, radiative\ntransferists perusing the following exposition will observe\nthat in each case the resolution does not quite extend into\nthe capillary region of the wave spectra. This region is of\nparticular importance to optical calculations. However, each\nmethod is capable in principle of having its resolution in-\ncreased, and it is for this reason that the reviews are pre-\nsented in some detail, to supplement the descriptions of the\nelectrical and optical methods of Duntley, and Cox and Munk\ngiven in Sec. 12.5.\nWave Spectra by Aerial Stereo Photography\nImagine that the dynamic air-water surface may be in-\nstantaneously frozen in its motion and suppose that a survey\nteam is dispatched to make a contour map of the surface much\nin the manner that a contour map of hilly and mountainous\ncountry is made by survey parties with theodolites and level\nrods. Such a contour map of the frozen air-water surface\nalong with the main analytical relations of harmonic analysis\ndeveloped in Sec. 12.4, may be used to construct a detailed\npicture of the unresolved energy spectrum E(k, t) of the sur-\nface, as defined in (61) of Sec. 12.4. This seemingly mirac-\nulous event of freezing the water surface was in effect ac-\ncomplished (optically) in November of 1954 when two camera-\nladen airplanes flew over a certain stretch of the North\nAtlantic Ocean and, as schematically depicted in Fig. 12.41,\ntook a set of simultaneous pairs of photographs of that\nregion of ocean. Each pair of photos allowed the same patch\nof surface to be viewed from two different vantage points,\nthereby producing a set of stereo photographic pairs of the\nregion; that is, photographs were obtained which, when viewed","167\nWAVE SPECTRUM DATA\nSEC. 12.7\n2000\n74°\n742\n3000'\nSHIP\nBUOY\n500\n2520\nFIG. 12.41 Schematic arrangements of the Stereo Wave\nObservation Project aircraft carrying the synchronized\ncameras. Their common field of view is shown. Stereo\nphotographs of the air-water surface in the common field\nwere the basis of an extensive computation program leading\nto a spectrum of the observed sea surface.\nthrough special optical equipment, would permit the hills\nand hollows of the dynamic surface to be directly perceived\nand measured. Within view of both cameras was the Woods Hole\nresearch vessel, Atlantis, and also a floating buoy. The\nAtlantis and the buoy were to help provide a photogrammatic\nbasis for the processing of the stereo pairs and the buoy,\nin addition, was recording the motion of the sea directly\nvia its own response to the passing waves.\nMonths of preparation went into that flight in November\n1954 to provide in a matter of seconds, a detailed stereo\nphotographic record of nearly five million square feet of\nocean surface; months of work were yet ahead to numerically\nprocess the data to obtain the desired results. This mammoth\nproject was designated as the Stereo Wave Observation Project\n(\"SWOP\"), and was carried out under the auspices of the New\nYork University College of Engineering Research Division, by\nthe departments of Meteorology, Oceanography, and the Engineer-\ning Statistics Group, and with the help of various local naval\ninstallations and the Woods Hole Oceanographic Institution.\nA full report of the history of the project, its conception,\ndevelopment, and its results are contained in [45], [52]. Our\nmain purpose here is to review only a few of the manifold re-\nsults of the project, namely those that touch on the computed\nform of the two-dimensional energy spectrum F(0,0) the one-\ndimensional spectrum To, and the wave-slope wind-speed law.","168\nAIR-WATER SURFACE PROPERTIES\nVOL. VI\nTo prepare the photographs for numerical processing,\neach photo of the best pairs of photos was reduced to 5400\nnumbers on a rectangular grid. The wave pole (the floating\nbuoy) records were converted to a discrete time series of\nabout 1800 numbers each. The purpose of the wave pole rec-\nords was to provide an independent check on the information\nderived from the 10,800 stereo grid points. The guiding\nprinciple behind the ensuing computations is represented by\nthe formulas:\nE(k)\n(x,0) cos (kx) dV(x)\n(1)\nE(k) = 12 E(k) + E(-k))\n(2)\n(3)\n(x,0) = 1.a\n(kx)\ndV(k)\n(4)\ncos\nwhich follow from (61)-(80) of Sec. 12.4 with phase E = 0.\nFurthermore the autocorrelation integral over (cf., (74)\nof Sec. 12.4) is assumed independent of t, reflecting the\nassumed stationary nature of the physical processes taking\nplace over the ocean surface at the time of the photographs.\nThe theoretical procedure used in the SWOP data reduc-\ntion was first to generate a contour map of the air-water\nsurface under study. A sample contour sheet is shown in\nFig. 12.42. The locations of the ship and buoy may be dis-\ncerned near the center of the map. The contour interval is\n3 feet, with vertical distances accurate to two feet (about\n0.2 mm on the map), and horizontal distances accurate to 2\nfeet. Next, the autocorrelation function $(x,0) was computed\nby discretizing the contour map data and using a discrete ver-\nsion of (74) of Sec. 12.4:\n5(x+y,s) (y,s) dV(y) . (5)\n(x,0) = lim\nTT\nOnce (x,0) in (5) was found, the discrete version of\n(1) was computed to find E(k). Equations (2) and (3) show\nthe connection between the unresolved energy spectrum E(k),\nthe resolved energy spectrum E(k), and the amplitude spectrum\na(k) of the air-water surface. It is clear from (2) that by\nmeans of a single stereo photograph pair, one can find only\nthe average of E(k) and E(-k), unless it is happily found\nthat there is a certain direction, say ko, such that E(k') = 0\nfor all k' whenever k' ko 0. In other words it may happen\nthat all the waves on the sea are traveling within 90° of a\nfixed direction ko, which is usually the mean wind direction\nin an open sea. If this is the case (and the assumption is","SEC. 12.7\nWAVE SPECTRUM DATA\n169\nPROJECT SWOP\nis\n.30.\nBUOY\nScale 1:3,000\nContour values given in tenths of a millimeter\no\n200\n400\nft\nSpot heights given in hundredths of a millimeter\nCONTOUR INTERVAL 0.30 mm\n(10 mm = 9.8425 ft.)\nFIG. 12.42 The contour map resulting from the stereo\nphotographs of Fig. 12.41. The ship and buoy are visible\nnear the center of the map.","AIR-WATER SURFACE PROPERTIES\nVOL. VI\n170\n.040\n035\n11\n020\n.030\n17\n.015\n.025\n.045\n.020\n010\n020\n.040\n18\n12\n.015\n.005\n.035\n.010\n.045\no\n.015\n.010\n.030\nns\n19\n.040\n.005\n.005\n.025\n.010\n13\n035\no\n1.005\n.015\no\n20\n.030\n.015\no\nloid\n.005\n.025\nOIO\n,010\n21\no\n.020\n.030\n010f\n.005\n00\n14\n22\n.025\n.015\no\n.010\no\n.020\n.010\n.005\n23\n.030\n.015\n0\n.005\n.005\nF.010\n.025\nof\no\n15\ntoos\n.020\n24\n.005\no\n.015\n0\n.025\n.005\n.010\n25\n:020\no\n.005\n16\n005\n+.015\n26\no\n.010\no\n.005\n.005\n27\no\no\n-50\nO\n50\n50\n-50\no\nFIG. 12.43 Seventeen histograms of the directional spec-\ntrum F(0,0) of air-water surface as derived from the data\nSWOP experiment (Fig. 12.42) Each histogram is a plot of\nF(0,0) as a function of 0 for a given o (see text).\noccasionally not unreasonable) then E(k) = E (k)/2 for all k\nwhose directions lie within 90° of ko, and E(k) = 0 elsewhere.\nThis assumption was adopted by the SWOP analysis.\nFigure 12.43 shows the computed energy spectrum in the\nform of histograms, as they emerged from the computations\nusing a discrete version of (1). . For computation purposes,\nthe temporal frequency form F(o, ) of the energy spectrum\nwas adopted (cf. (107) of Sec. 12.4). Hence the ordinates\nof each histogram in Fig. 12.43 are values of F(0,0). The\nabscissas take values over angles 0 varying + 90° from the","SEC. 12.7\nWAVE SPECTRUM DATA\n171\nmean wind direction ko. Each histogram is distinguished from\nanother by its temporal frequency o, which took one of seven-\nteen discrete values o = 2nj/96, j = 11, 12,\n26, 27.\nThe analysts of the SWOP report were able to obtain,\nafter a considerable amount of subjective curve fitting, an\nindealized energy spectrum from the preceding data in the\nform:\n(6)\nwhere we have written:\n(%))\n\"f(o,0)\"\nfor\n2\ncos\n]\n4 °\n(7)\ncos\nfor in the range [ - IT/2, /2], and where:\nc, = 3.05 x 104 cm2 sec-5\nHere Ua is the wind speed in centimeters per second observed\nduring the experiment, namely 18.7 knots at an anemometer\nheight of 15 feet. (A11 SWOP units are in the c.g.s. system\nunless noted otherwise.) This determination of (6), espe-\ncially as regards the two powers of o and Ua occurring in\nthe equation, is a remarkable feat considering that all this\ncomes from only one stereo pair. As we shall see later this\nenergy spectrum yields a cone-dimensional Neumann type of\nspectrum T obtained by completely different means in differ-\nent times [189]. Indeed, by integrating (6) over all (and\nrecalling F(0,0) = 0 for 101 > 90°) we have from (110) of\nSec. 12.4:\n(8)\nwhich is the gestalt of the Neumann spectrum [189] in tempo-\nral frequency form (see (8) of Sec. 12.8). In Fig. 12.44\nthe theoretical Neumann (temporal frequency) spectrum To is\ncompared with the stereo spectrum obtained by integrating the\nF(0,0) in Fig. 12.43 over for each o = 2j/96. The spec-\ntrum To was also estimated by means of the bobbing buoy, and\nthis version of To is denoted by \"wave pole spectrum\" in the\nfigure. The general visual agreement among the three spec-\ntra, arising from such diverse means of development, is quite\n*Readers who will study the SWOPTreport [45] in conjunc-\ntion with the present work, should note that our u,v pair\nwith SWOP's a,B, respectively. Furthermore our 0,0 pair with\nU,O; our E(u,v) with A*(a,B)]2/2; our F(0,0) with [A(u,0)]2/2.","AIR-WATER SURFACE PROPERTIES\nVOL. VI\n172\nTo\nt2sec\ncm sec\n0.70\n600\n0.60\nSTEREO SPECTRUM\n500\n0.50\nTHEORETICAL NEUMANN SPECTRUM To\nWAVE POLE SPECTRUM\n400\n0.40\n300\n0.30\n200\n0.20\n100\n0.10\n25\n30\n35\n40\n45\n50\nj= o\n5\n10\n15\n20\n1.25\n1.50\n1.75\n2.00\n2.25\n2.50\n2.75\n3.00\n3.25\n025\n050\n0.75\n1.00\no=\n2\nT= 96 48 3224 16 12\n8\n6\n4\n3\nFIG. 12.44 By integrating each histogram in Fig. 12.43\nover 0, plots of the temporal energy spectrum T o could be\nmade and compared with independent estimates of To, as given\nby Neumann and by the wave pole (buoy) readings of the SWOP\nexperiment.\ngood. It must be noted, however, that a comparison such as\nthis, based on only one stereo pair, and an ad hoc fit at\nthat, is hardly conclusive evidence for the Neumann spectrum.\nFurther study has shown [33] that the Bretschneider spectrum\n(cf. Sec. 12.9) fits the SWOP data just as well.\nAs a final matter in the survey of the SWOP results,\nwe cite the version of the wave-slope - nd-speed law evolved\nby the stereo project. It was found that:\n==0.99 X 10 3 U a\n(9)\n,\n0.60\n10\nU\n(10)\nX\n,\nwhere Ua is in meters per second and presumably measured at\n15 feet above sea level. Once again 02 depends linearly on\nU. The difference between these estimates and those of Cox\nand Munk ( (40) - (42) of Sec. 12.5) or those of Duntley ( (18) -\n(21) of Sec. 12.5) is moderate, involving factors of 3 to 6,\nconsidering that only one wind speed was studied by SWOP.\nIn view of these findings, the stereo method of deter-\nmining energy spectra seems on the whole marginally promis-\ning and certainly deserves to be further studied to see if it\nit feasible to increase the resolving power of the stereo\npairs and to place more of the burden of the routine calcula-\ntions on modern general - purpose computers with the goal in\nmind of a fully automated spectrum program. For on the one","173\nWAVE SPECTRUM DATA\nSEC. 12.7\nhand oceanographers will require as many accurate energy\nspectra gathered under as many diverse conditions as possible\nin the near future in order to evolve a comprehensive trans-\nport theory of the time dependent energy spectrum E(k,t) ; and\nthe radiative transferists on the other hand will require\nstill greater resolution of the capillary component of the\nenergy spectrum, for the purpose of predicting radiant flux\nactivity in natural waters, as affected by the air-water sur-\nface. Whether more comes of the stereo approach to wave\nenergy spectra or not, the stereo wave observation project\ndescribed above will stand as one of the more imaginative\nand prodigious projects of modern day optical oceanography.\nWave Spectra by Floating-Buoy Motion\nThe second of the two methods of determining wave spec-\ntra rests on the fact that, to a first approximation, the\nlocus in space a small floating object on the air-water sur-\nface executes an orbital motion very nearly that of the\npackets of fluid at the surface (see observations following\n(63) of Sec. 12.3). If the object is small, light, and flat,\nnot only will it take up the orbital motion of the surface\nwater but also the orientations of the surface. If the flat\nfloating object is a buoy which contains a recording acceler-\nometer, then the twice-integrated acceleration vector will\ngive 5 as a function of x,y and, especially, as a function\nof time t (cf. (7) : (8) of Sec. 12.3). A pair of gyroscopes\ninside the floating buoy can give 25/ax and as/dy as functions\nof time, provided the buoy is kept oriented by a sea anchor\nas depicted in Fig. 12.45. Hence the basic ingredients for\na first order statistical description of the dynamic air-\nwater surface would be available by this means.\nThe preceding scheme for recording 5, 25/ax, and aslay\nwas used by Longuet-Higgins, Cartwright, and Smith [169] to\nobtain a working estimate of the energy spectrum F(0,0) (cf.\n(107) of Sec. 12.4) in some experiments in 1955 at the National\nInstitution of Oceanography, England. The method was origi-\nnally suggested by N. F. Barber [9] and was developed by\nLonguett-Higgins [163]. We shall now give a resumé of the\nmethod, as developed in [169].\nThe preceding description forms the physical part of\nthe floating buoy method. It yet remains to translate the\n5 , 25/ax, and as/dy data into a mathematical expression for\nF(0,0). The desired end product is a five term finite Four-\nier series expansion of F(0,0) defined by writing:\n1\n(0,0)\"\nfor\n(a\nb\nsin\n0)\n0\na\n+\nCOS\n+\n1\n1\n+ (a cos 2 o + 2 sin 20) (11)\nwhere F1 (0,0) is the finite approximant to F(0,0), and where\nao, a, b1, and b2 are the required coefficients depending\non the temporal frequency O. It was found that the first five\ncoefficients of the Fourier expansion of F(0,0) were adequate","VOL. VI\n174\nAIR-WATER SURFACE PROPERTIES\nDIRECTION\nOF WIND\nP\nY\nX\nJ\nD\nFIG. 12.45 Illustrating the floating buoy which carries\nrecording linear and angular accelerometers to yield gravity\nwave-slope and wave elevation data for use in determining the\ndirectional spectrum of the dynamic air-water surrace. A\ndrogue bucket D and float arrangement of a pontoon P and line\njunction J keep the X and y axes of the float aligned parallel\nand perpendicular to the wind direction, respectively.\nfor the experiment under discussion. However, it is possible\nto obtain more terms of the Fourier expansion if the higher\nderivatives of 5 could be measured. A note was made in [169]\nto the effect that such a program was in progress.\nNow, in order to obtain the five coefficients an and bn,\nrecall from (3), (4), , and (7) of Sec. 12.4 that, on setting\np = IT, a = 0, we have in general:\npe\n-ino\nb\nF(0,0)\ndo\n(12)\nan\n=\nn\nTT\nThat is, the an and bn are integrals of products of cosines or\nsines with F(o, ) . Our studies of the moments m pq in (116)- -\n(119) of Sec. 12.4 suggest that the an and bn are therefore\nclosely related to the various moments mpq when the spectrum\nF(0,0) (rather than S(k,o)) is used (cf. (88) of Sec. 12.4).\nThe first few of these moments use only 5 and 25/ax, as/dy.\nTo see the exact connections, we start with (71) of Sec. 12.4\n(with E = 0) for a continuous spectrum:","SEC. 12.7\nWAVE SPECTRUM DATA\n175\n5(x,t) = i(kx-ot) dA(k)\na(k) cos (k.x-ot) dV(k)\n=\na(k) cos (x cos 0 + y sin ) - ot] do do\n(13)\nwhere k = 02/g, k = (u(o,0), v(o,0)), , and where we have used\nthe transformations following (107) of Sec. 12.4. From (13),\nby differentiation:\n25(x,t) = cos 0 _i(k-t) dA(k)\n(14)\nand once again from (13)\n25(x,t) dy = k sin 0 _i(kx-ot) dA(k) .\n(15)\nWith these preliminaries established we now continue\nwith the main part of the description of how the coefficients\nao, a1, b1, a 2' b2 are obtained in practice. The records of\n5, as/ax, 25/ are time records obtained over a period of\ntime at essentially one point in space. Therefore the req-\nuisite statistical information contained in their records is\nunlocked by performing appropriate autocorrelation and cross\ncorrelation operations on these functions. Thus, for brev-\nity, let us write \"E1\", \"E2\", \"E3\" for 5, 25/ax, and as/dy,\nrespectively. Then writing:\n\"Pij(t)\" for lim dt (16)\nwe can obtain the spectrum of Pij(t), i,j = 1, 2, 3, in the\nusual manner by writing:\n\"Dij(a)\" for li -iot dt (17).\nWhenever i = j (i.e., when we take the autocorrelation func-\ntion of 5i) Dij (o) is real, since Pij is an even function.\nThus if we explicitly represent Dij(o) as:","VOL. VI\nAIR-WATER SURFACE PROPERTIES\n176\n(18)\nDij(o)Cij()iij()\n,\nthen Qii (o) = 0. The functions Cij for i j j are called the\nco-spectra of 5 and Ej; and Qij are called the quadrature\nspectra of 5 and Ej. It is now easy to show, using the\nrepresentations (13)-(15) of 51, E2, 53 in (17), that:\n(19)\n= cos2 F(0,0) do\n(20)\n= pet sin2 F(0,0) do\n(21)\n= sin cos F(0,0) do\n(22)\nso = k cos F(0,0) do\n(23)\nQ15 (4) k sin F(0,0) do\n(24)\nThese relations give the requisite working link between the\ncomputables Cij, Qij and the desirables ao, a1, b1, a2, b2;\nfor by placing F, (o, as given by (11) into (12), and sort-\ning out the results using (19)-(24) - (for F1 (0,0)) and by the\nuniqueness of the desired Fourier coefficients we eventually\narrive at:\nao - 1 c,\n(25)\n(26)\n(27)\nOne final point in the computations of the coefficients\nin (11) is to be noted. The spectrum F(0,0) is never nega-\ntive, by definition. It is possible, however, for its finite\napproximant F,(0,0) to be negative. Indeed, since","SEC. 12.7\nWAVE SPECTRUM DATA\n177\ndo'\n(28)\nwhere we have written\nfor 1 + 2 cos (0'-0) + 2 cos 2(0'-0) ;\n(29)\nit is clear that W1 and hence F1 can be negative for some 0.\nTo avoid negative values of F (0,0), a new approximant of\nF(0,0) is defined by writing:\n\"F3(0,0)\" for cos 0 + b1 sin 0)\ncos 2 + b2 sin 20). (30)\nThe theory leading to the coefficients a a1, b, in\n(30) is still that summarized in (25)-(27) . Once these five\nnumbers have been found, F3 (0,0) is constructed as shown, the\nresult.being a nonnegative function of and representable as:\n(31)\nwhere we write:\n\"W3('-)\" for 1 + 1 3 cos (0'-0) cos 2(0'-0)\n.\n(32)\nIt is F3 (o, ) (and not F1 (0,0)) that is the chosen finite\napproximant to F(0,0) in [169]. This completes the exposi-\ntion of one form of theory of obtaining the energy spectrum\nF(0,0) from floating-buoy motions.\nAn example of the results of the study [169] will now\nbe given. Figure 12.46 shows histograms of the surface ele-\nvation 5 of the sea, and the angles corresponding to the\nslopes as/ax, as/ay. The gaussian forms of the distributions\nare evident, at least on a visual level, attesting to the\nlinearity of the waves (i.e., the slopes were still small\nenough so that a 112 tan a, for either pitch or roll angle a).\nThe recorded results are for an anemometer wind speed of 23\nknots, with a fetch of over 300 miles. The root mean square\nelevation was 2.6 feet.\nFigure 12.47 gives a plot of computed T, (= C11 (o))\nfor the same recorded run (number 5) as that in Fig. 12.46.\nThe dashed line is the form of Phillips' theoretical limit-\ning spectrum for an equilibrium sea (cf. [198]).","VOL. VI\nAIR-WATER SURFACE PROPERTIES\n178\nPERCENTAGE\nOF ORDINATES\n10\n(a) SURFACE ELEVATION\n(5)\n6\n4\n2\n5\n10 FT\n-10\n-5\no\nI2F\n(b) ANGLE OF PITCH\n8\nas\ndx\n4\nupwind\n2\n.3 RADIANS\n-3\n-2\n-1\nO\nI\n(c) )ANGLE OF ROLL\nas\ndy\n4\ncrosswind\n-3\n-2\n:1\no\n.I\n2\n3 RADIANS\nFIG. 12.46 Statistical distribution of vertical and\nangular displacements of the recording buoy of Fig. 12.45.\n(For an anemometer wind speed of 23 knots with a fetch of\nover 300 miles )\nFigure 12.48 depicts F (0,0) as computed from record\nNo. 5. Hence Figs. 12.46-12.48 give a complete spectral\ndocumentation and geometric documentation of the sea surface\nunder the given conditions, which may be used to predict the\naverage reflectance and transmittance of the air-water sur-\nface, under the same conditions, using the theory of Sec.\n12.11.\nThe co-spectra and quadrature spectra played essential\nroles in the present method. For a general study of the\nempirical estimations of such spectra, see [99].","SEC. 12.7\nWAVE SPECTRUM DATA\n179\n100 ft2 sec\n10\n1.0\nTo\n(= C(o))\nO.I\n.OI\n0.1\n1.0\n10\no (rad/sec)\nFIG. 12.4 47 A plot of To vs. o for the same data run of Fig. 12.46.\np) ft2 sec\na = 2.4 0.1\no = 0.4 1.0\no\no\no = 2.6\nQ.I\no = 0.6 10.0\no\no\no = 2.8 0.01\no = 0.8\n1.0\no\no\no = 3.0 0.01\no = 1.0\n1.0\no\no\no = 3.2 QOI\no = 1.2\n1.0\no\no\no = 3.4 0.01\n= 1.4\n1.0\no\no\no = 3.6 0.01\no = 1.6\n0.1\no\no\no = 3.8 0.01\no = 1.8\n0.1\no\no\no = 4.0 0.01\no = 2.0 0.1\no\no\n-180°\n0°\n180°\n= 2.2 0.1\no\nFIG. 12.48 Directional spectrum F3(0,0) for the same data\nrun of Fig. 12.46. Figures 12.45-12.48 are adapted from a\nstudy by Longuet-Higgins, - Cartwright, and Smith (Record No. 5).","VOL. VI\nAIR-WATER SURFACE PROPERTIES\n180\nWave Spectra from Submarine Echo Recordings\nWe close our survey of some experimental means of deter-\nmining the energy spectrum of the dynamic air-water surface\nby noting that submarines have been used to good effect as\nsubmerged hovering platforms on which echo sounders have been\nmounted, directed upward at the sea surface, and the relative\nwave heights measured by continually dividing the echo travel\ntime by the speed of sound in water. For example, the sub-\nmarine U.S.S. Red Fin in the fall of 1960, 200 miles east of\nCape Hatteras, North Carolina, obtained for the U.S. Hydro-\ngraphic Office, records of echo soundings of the surface at\nkeel depths of 80 to 100 feet, with the submarine drifting at\nabout 1 knot and sending a sonic beam of 3° spread up toward\nthe surface. Further details are given by de Leonibus [64].\nSince only one echo sounder was used, the data are equiva-\nlent to taking wave height readings from an anchored pole in\nthe sea, so that only To or TT, or any of the other one-\ndimensional spectra can be determined. A sample of the data\nin the form of the temporal frequency spectrum T (cf. (110)\nof Sec. 12.4) is given in Fig. 12.49.\n9.6\n8 8\nKEEL\nREL\nDATE\nSHIP'S\nDEPTH\nHEAD\nNOV\n1/3\n(ft)\n(°T)\nRUN\n1960\nTIME\nE(ft2)\nSPEED\n8.0\n2150\n8-3-5\n16\n2220\n4.11\n5.7\nI kn\n80\n030\n7.2\n2225\n015\n4.56\n6.0\nI kn.\n80\n8-3-6\n16\n2255\n2302\n6.2\nI kn\n80\n060\n8-3-7\n16\n2332\n4.85\n6.4\n90. Per cent Confidence Intervals\n.78 P((f) < Pq(f) 1.27 P (f)\n5.6\nTo\nE = mean square wave elevation\n4.8\nsec\n11/2\n= significant wave height\nHi/ = 2.83\n(cf (2) of sec 12.6)\n4.0\n3.2\n2.4\n1.6\n0.8\n2w\n2w\n2w\n2w\n2w2w\n2w 2w 2w\n2w\n2w\n2W\n2W\n2w2w\n2TT\n2TH\n2W\n2T\n2w\n2W\n2w\n2w\n2w\n2w\n2w\n2T\n2TT\n2w\n2w\n3029\n2.8\n2.7\n2.62.5\n24\n232.2\n6.5\n6055\n5.1\n4.8\n4.5\n4.2\n34\n3.3 3.1\n4.0\n3.8\n3.6\n24\n18\n14.4\n12\n10.3\n90\n8.072\n2\nrad/sec\no :\nFIG. 12.49 Wave energy spectrum T Q as determined by\nde Leonibus using sonar equipment mounted on a submerged\nhovering submarine.","SEC. 12.8\nEMPIRICAL MODELS\n181\n12.8 Empirical Wave Spectra Models\nWe conclude our review of experimental data on the dy-\nnamic air-water surface with a discussion of certain empiri-\ncal models of wind-wave spectra. These models and the ideas\nthey represent are perhaps the single most important out-\ngrowth of the current developments in the theory of ocean\nwave analysis and forecasting. The first definitive wave\nspectrum model to be found is that of Neumann [189] which in\nturn was inspired by the pioneering work of Pierson [199],\nand Pierson and Marks [202] in using modern generalized har-\nmonic analysis on ocean wave records. In this section we\nshall briefly review the genesis of the Neumann spectrum,\ndiscus's its pertinence to the problems of hydrologic optics,\nreview some alternate spectra that were developed subsequent\nto the Neumann spectrum and, finally, give an overview of\nthe properties of one-dimensional spectra common to all cur-\nrent models. This overview should be helpful in eventual\napplied hydrologic optics studies of wind-generated seas.\nThe Neumann Spectrum\nThe Neumann spectrum for gravity waves of period T is\nof the form:\n()\n{\nTHE\n4\nT\ngt\n- 2\n(1)\nexp\nU\n4 11 2\n2\na\nwhere\nC = 8.27x10 4 sec\n,\nand where T. is defined and related to the two-dimensional\n(directional) spectra in (113) of Sec. 12.4, g is the accel-\neration of gravity (980 cm sec-2), and Ua is wind speed in\ncentimeters per second and is of \"anemometer height,' which\nvaries from 10 to 20 feet above mean sea level. The period\nT (= 2n/o) is given in units of seconds.\nFigure 12.50 depicts the characteristic shape of the\nNeumann spectrum for several wind speeds. The ordinate corre-\nsponding to the rationalized frequency f = 1/T (i.e., 2f = o)\nis interpretable as being proportional to the amount of total\nwave energy per unit frequency at frequency f, over a unit\nhorizontal area of surface (cf. (11) of Sec. 12.3) contributed\nby all waves passing a point on the sea surface in all possi-\nble directions. The dynamic air-water surface is understood\nto be in equilibrium with the wind over deep water and at the\nend of a long fetch.\nThe total area under the curves in Fig. 12.50 increases\nswiftly with wind speed, in fact according to the fifth power\nof Ua, as we shall see. Furthermore the maximum points on\ncurves systematically move to lower frequencies {longer periods - -\ni.e., bigger waves) with increasing Ua. This decrease, too,","VOL. VI\nAIR-WATER SURFACE PROPERTIES\n182\nWave spectra for fully\narisen sea at a wind speed\nof 20, 30, and 40 Knots,\nrespectively.\n40 Knots\n30 Knots\n20 Knots\n0.20\n0.15\n0.10\n0.05\no\nf = 1/4\nFIG. 12.50 Showing the way the wave energy spectrum in-\ncreases rapidly with increasing wind speed and how the peaks\ndescend to lower frequencies with increasing wind speed (after\nNeumann)\nis predictable and follows a simple law such that the product\nof the wind speed Ua and the abscissa of the maximum is a con-\nstant. In this way, radiative transferists and others who are\nacquainted with classical heat radiation theory, will see a\nstriking resemblance between the Neumann spectrum and the\nradiant emittance curve of a Planckian radiator. While no\nconnections between these two types of spectra have yet been\nmade except on an obvious intuitive statistical level, there\nis without a doubt a strong underlying analogy between the\nspectra of these two apparently dissimilar phenomena (i.e.,\nlight wave transport, ocean wave transport) and perhaps in\nthe future some unifying theory, underlying both phenomena\nand yielding each spectrum as a special case, will be evolved.","183\nEMPIRICAL MODELS\nSEC. 12.8\nDerivation of the Neumann Spectrum\nA brief outline of the derivation of the Neumann spec-\ntrum, as it was originally made in [189], will be instructive\nfor radiative transferists because it will illustrate the\nphysical level to which the spectrum's descriptions apply.\nMoreover, an insight into the problem of extending the Neu-\nmann spectrum into the capillary region may possibly be\nforthcoming by doing so.\nThe Neumann spectrum is the result of observing the\nstatistical connection between two concepts long used in\nempirical wave generation studies, namely the wave steepness\nand wave age of the dynamic air-water surface.\nThe wave steepness of an idealized sinusoidal wave\nform of height H and length l is H/X (recall that H = 2a\nwhere a is the amplitude of a wave). For visual observa-\ntions of the sea surface, it is customary to replace l by\nthe squared period T2, and study the alternate measure of\nwave steepness H/t2 rather than H/. This alternate measure\nis suggested by the ideal connection:\n2\n(2)\n(x = ct)\n,\nbetween wavelength l and wave period T for gravity waves in\ndeep water. This follows from (65) and (66) of Sec. 12.3\nand the connection T = 2T//O. The real waves at sea of course\ndo not follow idealized wave forms, but are instead viewable\nas the superposition of many such forms. Instead of the wave\nperiod T, Neumann used the apparent wave period T which is\nthe time between successive crests on a wave record or in a\nvisual observation at sea. Furthermore the apparent wave\nheight (also denoted by \"H\" below) was also adopted and is\ndefined as the average of the wave heights between two suc-\ncessive crests. See Fig. 12.51. From now on \"H\" denotes\napparent wave height. It can be shown [204], [189], that a\nversion of (2) can be defined for the apparent wave descrip-\ntion. The result is:\nTHE\n(3)\nwhere 1 is now an apparent wavelength. Equation (3) may be\nthought of as an average of (2) over the wavelength pattern\ndepicted in Fig. 12.51.\nThe wave age of an apparent wave in the air-water sur-\nface is the ratio of the wave speed (celerity) C to the wind\nspeed Ua, namely c/U. The closer the ratio is to 0, the\n\"younger\" the sea. A ripe old wave age is near 1 and is\nattained under long-fetch, steady-wind conditions. (In Fig.\n12.38, wave age is denoted by \"Co/U\".)\nIn the study of real seas, the ideal steepness parameter\nH/t2 is replaced by H/t2, and the ideal wave age c/U is often\nreplaced by the apparent wave age T/U, as suggested by the","AIR-WATER SURFACE PROPERTIES\nVOL. VI\n184\nT1\n71\nT1\nH.\"\nFIG. 12.51 Visual determination of apparent wave\nheights and apparent wave periods.\nrelation C = (g/2) T, in its averaged form: c = (g/2TT) T.\nExamples of wave steepness and wave age as used by Sverdrup\nand Munk, et al. may be seen in Fig. 12.38 (there, however,\nwave height is significant wave height).\nThe results of the visual observations of the sea\ntaken in 1948 by Neumann in Long Branch, New Jersey, are\nplotted in Fig. 12.52. Apparent wave steepness H/2 (mea-\nsured on a logarithmic scale) in its relation to the square\nof apparent wave age (T/U) 2 scatters widely over the figure.\nIt is reasonable to expect this scatter, since for a given\nT/U in a steady wind, one expects all manners of steepness\nfrom very small amounts and up to a certain limit. The mean\nline of upper limits is marked by the sloping solid line in\nFig. 12.52. Since the vertical scale is logarithmic the\nempirical relation Neumann encountered is necessarily of the\nform:\nH = 0.219 exp { - 2.438 (/u)).\n(4)\nNeumann notes the interesting numerical closeness of 2.438\nand (g/2)2, , in which g = 9.8 m/sec2. Using this and (2), ,\nNeumann wrote (4) as:\nH = (constant) . gt2 . exp {\n(5)\nwhere Ua is in meters per second, H in meters.","SEC. 12.8\nEMPIRICAL MODELS\n185\nexp {\nH\nObservations\nLong Branch Wave Records\n6.\nx\nMay 3, 1948\nMay 5, 1948\n0.2\nOctober 6, 1948\nX\nX\nOctober 7, 1948\n0.1\nH\nX\nX\nT2\nx\n0.05\nx\nX\nx\nX\nO\n0.2\n0.4\n0.6\n0.8\n1.0\n1/2\nFIG. 12.52 The central result of Neumann's classic wave\nspectrum experiment: The semilog relation between the square\nof apparent wave age (T/U) 2 and apparent wave steepness H/2,\nleading in turn to the Neumann spectrum.\nWe now recall ((98) of Sec. 12.3, with T1 1 = 0) that the\nenergy E(T) of a single gravity wave of height H(T) and period\nt is:\n(T)\n(6)\nThis formula gives total energy (kinetic and potential) (per\nunit area of horizontal reference surface) of the dynamic sea.\nNeumann showed that H in (5) is an energy density spectrum in\nthe sense that H2 dt is proportional to the energy E(T) dt of\nwaves per unit area on the sea with periods in the range T +\nT + (1/2) dt, and in this way arrived at the continuous ver-\nsion of (6):","VOL. VI\nAIR-WATER SURFACE PROPERTIES\n186\n(7)\ndT\nwith H(T) given by (5). Here, then, E(T1,T2) is the energy\nof the waves per unit horizontal area of sea surface on a\nfully arisen sea, and with periods in the interval [T1,T]].\nHence, by squaring each side of (5) we arrive at (1), after\nidentifying (with Neumann) it with T for the purposes of the\npresent derivation. Using (112) and (114) of Sec. 12.4 we\ncan cast (1) into temporal frequency form:\n(8)\nwhere C is the (constant) 2 of (5) and is given as in (1) Ua\nis in centimeters per second, g in centimeters per second\nsquared, and o is in radians per second (i.e., o = 2T/T,\nwhere T is the period). In arriving at (1), we used the re-\nlations in (98) and (103) of Sec. 12.3, along with (115) of\nSec. 12.4, which yield the connection\n/\nfor every pair\n(8a)\nfor every T. The constant C in (1) was determined by Neumann\n[189], and is closely compatible with C1 in (6) of Sec. 12.7.\nThe value C1 was calculated via an alternate method by Pier-\nson [199], and was subsequently used in the SWOP, i.e., in\n[45]. See also (39) in [57]. The connection that necessar-\nily exists between them is c, = (2ng)2C.\nThree Laws Derived from the Neumann Spectrum\nWe now consider three laws that may be derived from\nthe Neumann spectrum (8), laws which are of help in gaining\ndeeper insight into the structure of the graphs in Fig. 12.50\nand also in providing cross-checks on the empirical spectrum\nitself by providing alternate theoretical and empirical ap-\nproaches to the spectrum.\nThe first law is the temporal-frequency wind-speed dis-\nplacement law, which relates the maximum frequency Omax with\nthe wind speed U. We find max by obtaining dTo/do via (8)\nby setting","SEC. 12.8\nEMPIRICAL MODELS\n187\ndT\n0\n,\nand by solving for the necessary value of O. The result is\nmax = 12/38\nor\nmax Ua = 223 g\n(9)\nHence the frequency at which the highest density of wave\nenergy occurs depends inversely on the generating wind speed:\nthe greater the wind speed U, the smaller the maximum fre-\nquency max --or the greater the period - - or the longer the\nwave will be. The resemblance between (9) and the Wien dis-\nplacement law derived from the Planck law of radiant emit- -\ntance of an ideal blackbody is quite interesting, especially\nfrom the point of view of what concepts in the thermal radia-\ntion context pair with what in the oceanic wave context.\nThe second law to be derived is the mean-elevation wind-\nspeed law, and this also has an interesting connection with\nclassical thermodynamics. The tefan-Boltzmann law of heat\nradiation, which states that the radiant emittance of a Planck-\nian radiator over the whole spectrum is proportional to T4,\nhas the following analog in the ocean wave case. From (115)\nof Sec. 12.4 and (8) we have:\n= do exp { 2 do\nthat is:\n(10)\n=","AIR-WATER SURFACE PROPERTIES\nVOL. VI\n188\nThus the mean square elevation of the dynamic air-water\nsurface - - as far as the gravity waves are concerned--increases\nas the fifth power of the wind speed U\nThe coefficient of Ua works out to be\nso that\nwhen Ua is in meters per second.\nThe third law we shall derive from the Neumann spectrum\nis the wave-slope wind-speed law and is a special case of the\nwave-slope wind-speed law discovered by Duntley and corrobo-\nrated by Cox and Munk (cf. (18) - (21), and (30) - (35) of Sec.\n12.5). Assuming the directional spectrum F(0,0) to be iso-\ntropic, as in (127) of Sec. 12.4, we find from (128) of Sec.\n12.4 and (8) above:\n{- 2\nso that:\n(11)\n=\nThe essential fact to observe here is that the mean\nsquare slopes increase linearly with wind speed U. It is\nfor this reason that the Neumann spectrum, although derived\nfrom observations on gravity waves, seems to extend meaning-\nfully down to the capillary-wave level. This fact was first\nnoted by Pierson and also by Cox and Munk [57], and may be\nrendered in the following form using (11) and the value of\nC in (1):\n(12)\nwhere oc and out are the cross and upwind components of the\nmean square slope of the surface, and U is in meters per\nsecond. (Compare (12) with (18) - (21) and (30) - (35) of Sec.\n12.5.)","SEC. 12.8\nEMPIRICAL MODELS\n189\nAlternate Forms of the One-Dimensional Spectrum\nAfter the pioneering efforts of Neumann, several vari-\nants of (1) were suggested by various writers, the principal\ndifferences between the new proposed forms and (1) being in\nthe powers to which T was to be raised both in the algebraic\nfactor and in the exponential factor. The great complexity\nof the dynamic air-water surface and the difficulty with\nwhich representative seas could be observed has left the\nquestion of the appropriate form of TT only partially re-\nsolved to the present time (1966). However, some recent\nwork by Pierson and Moskowitz [203] indicates that with im-\nproved data and observing conditions, along with some power -\nful dimensional analysis arguments, * the form of the spectrum\nbest suited to describe the gravity wave portion of the spec-\ntrum is that advocated by Bretschneider [34]:\ne -D\n(13)\nwhere C and D are dimensionless parameters as suggested by\nPierson and Moskowitz [203]:\nC = 8.10 x 10\nD = 0.74\nand where the cgs system of units is used. Still further\nvariations of the basic theme are given by:\n(14)\nsuggested by Roll and Fischer [267]; and also:\n(15)\nsuggested in [203]. From these forms we see that the ques-\ntion of the power 0-5 outside the exponential seems to be\nresolved but that the power of o in the exponential is yet\nat issue. The work in [203] suggests that (13) is superior\nto (14) and (15), and even (8), in describing gravity wave\nspectra. In view of the multiplicity of these spectral forms,\nthe time seems ripe for a deeper study in both observation\ntechniques of ocean wave spectra and their theoretical basis\nin nonlinear hydrodynamic theory.\n*The power of dimensional analysis in studying mathemat-\nical models of fluid flow is also illustrated, e.g., in [22].","AIR-WATER SURFACE PROPERTIES\nVOL. VI\n190\nGeneral Properties of Gamma-Type Spectra\nThe Neumann spectrum (1), or (8), the Bretschneider\nspectrum (13), the Roll-Fischer spectrum (14) and some others\nof current interest are all special cases of the general type\nof spectrum:\n(16)\nwhich we shall call the gamma-type one-dimensional spectrum,\nfor the reason that it has the gestalt of one of the equiva-\nlent forms of the integrands used to define the gamma func-\ntion\nof classical analysis. * There are four parameters de-\nscribing a gamma-type spectrum: p and q are positive real\nnumbers--usually integers--which define the form or geometric\nshape of To, while a and b are respectively ordinate and ab-\nscissa scale factors with dim a = L2T1+P dim b = L T-? The\nNeumann spectrum (8) is the special case of (16) in which\np = 6, q = 2, and where a = (1/8) C g2 (2 TT) 3, , b = 2 g.\nWe shall now study some general properties of gamma-type\nspectra, properties which perhaps can help choose between\nvarious future special cases of (16) for particular hydrologic\noptics applications. We begin with the observation that the\ndisplacement law (9) holds for any To of the form (16). In-\ndeed, we find on setting\nd\n0\ndo\nthat necessarily:\nmax Ua a = b (9) 1/9\n(17)\nwhich is the displacement law for the temporal frequency o max\nand wind speed Ua. Hence in any of the gamma-type models pro-\nposed so far, we may expect an exact inverse relation between\nwind speed Ua and the maximum of the frequency wave spectrum.\nWe take up next the matter of deriving the general mo-\nment \"so of To, using the definition of mpg in (88) of Sec.\n12. 4, and the assumption of isotropy of s(k,o) or F(0,0) (120)\nof Sec. 12.4).\n*Alternately, generalizations of T given in (1) may\ntake the form of X2 distributions with finite numbers of\ndegrees of freedom.","SEC. 12.8\nEMPIRICAL MODELS\n191\nWe begin with:\nus\ndV(k)\n= do dk\nks coss do\ndk\nIt is clear that:\nfor odd S. Accordingly, we shall henceforth assume that\nS = 2r for some nonnegative integer r. Then:\ndk\n(18)\n.\nWe may convert this into temporal frequency form; the result\nis\ndo\n(19)\nevaluate in closed form when To is a spectrum\nWe\ncan\nof the gamma type (10) The resultant form is:\n(20)\nT\nOn setting r = 0 and then r = 1 in (20) we obtain the two\nmoments of greatest current interest in hydrologic optics,\nnamely moo, the mean square elevation of the air-water sur- -\nface; and 220 the mean square slope of the surface. Thus by\n(89)-(91) - of Sec. 12.4:\n(21)\nand:","VOL. VI\nAIR-WATER SURFACE PROPERTIES\n192\n5-p\na\n(22)\nT\n=\nqg\nFrom this we see that the Neumann spectrum (the case p = 6)\nis the only one of the proposed spectra for gravity waves\nwhich reproduces the observed linear dependence of M20\non\nwind speed U for capillary-type waves. Interestingly\nenough, the Bretschneider spectrum (13), for which p = 5,\npredicts that m20 is independent of wind speed (!), and is\ntherefore a spectrum clearly inapplicable to the capillary\nwave domain. On the other hand m for the Bretschneider\n00\nspectrum increases as the fourth power of U, which has some\nexperimental corroboration.\nIf the Neumann spectrum does well in the capillary re-\ngion as regards the wave-slope wind-speed law, then we may\ninquire as to how well it does with respect to the higher\nmoments of the spectrum. For example, it is known that the\nfourth order moments m '40' m describe the gaussian curva-\nture of an isotropic sea surface [57], and that the mean\ngaussian curvature of the surface is approximated by 1/R2\nwhere R is the radius of an osculating sphere in contact\nwith the surface. Now Schooley [275], has observed from\nlaboratory wind wave experiments that the radius R of curva-\nture of the wind blown surface (comprised mostly of short\ngravity waves and long capillary waves, i.e., waves in the\ntransitional range) can be given approximately by:\n= 0.046\nU\n(23)\n,\na\nwhere the surface wind speed is in knots and R is in centi-\nmeters. The range of observed linearity was between 8 knots\n(where waves began to be formed) and 20 knots over a fetch\nof 26 inches.\nTurning now to the fourth moment m 40' obtained by set-\nting r = 2 in (20), we see that:\n9-p\n[ ]]\n(2)\n3a\n(24)\nT\n.\n4\nIt follows that for p = 6, the Neumann spectrum case, the\nexponent of b/U is 3 so at once it is evident that 40 be-\ncomes infinite for U = 0. Moreover m 40 decreases as\nthe\ninverse third power of wind speed. These observations show\nthat on this level the Neumann spectrum at last.meets its\nlimit of validity in the capillary region, since, according\nto Schooley's observations cited above, gaussian curvature\nshould increase as the square of the wind speed U. Any gamma-\ntype model which has p = 11 would exhibit this feature observed\nby Schooley. However, then m for this model (cf. (22))\n20\nwould not have the observed linear behavior with wind speed.\nThese observations reinforce our earlier conclusion that\ngamma-type one-dimensional spectra are only first approxima-\ntions to a more complex analytical spectral formula as yet to","SEC. 12.8\nEMPIRICAL MODLES\n193\nbe determined over the whole frequency spectrum, and which\ncompletely describes the structure of the dynamic air-water\nsurface. It must also be borne in mind that empirical laws\nsuch as the wave-slope wind-speed law and the curvature law\n(23) against which the gamma spectra have been pitted are de-\ntermined only for limited wind speed ranges and wave length\nranges. Thus what may be a linear law in one range of wind\nspeed could very well blossom into a polynomial law in wind\nspeeds over a greater range; and perhaps ultimately a ration-\nal function in wind speeds (i.e., a ratio of two polynomials)\nis forthcoming which (e.g.) eventually goes to zero with in-\ncreasing wind speeds. (The sea is blown smooth in the limit.)\nWind Speed, Wavelength, and Wave Energy\nWe close this discussion of models of ocean wave spec-\ntra by combining the Neumann spectrum with the Kelvin-\nHelmholtz wave theory, the purpose being to predict the rela-\ntive energy dependence of such waves on wind speed or wave\nlength. Thus, from (98) of Sec. 12.3 we have (using wave\nheight H)\nE(k)=\nk2\nfor the energy of a sinusoidal wave with wave number k and\nheight H, and which is propagated jointly by surface tension\nand gravity forces. If we have a continuum of waves in which\n(1/8) H2 (T) is replaced by TT (as in (7)) or (1/8) H2 (k) is\nreplaced by Tk, then by the same kind of reasoning that led\nto (8a) :\ndk\n(25)\nis the energy content per unit horizontal surface area of\nthe set of waves with wave numbers in the interval [k,.\nIn particular, on recalling (116) and (121) of Sec. 12.4,\nthe total energy for Kelvin-Helmholtz waves, under an iso-\ntropic direction spectrum, is:\nE = + T m\n(26)\n1\n20\nIf the Neumann spectrum is used for the waves, then from (10)\nand (11), we can write (26) as :\nE=C,U a + C 2 U5 a 5\n(27)\n,\nwhere we have written:","VOL. VI\nAIR-WATER SURFACE PROPERTIES\n194\n3\nT\nC\nV\nIT\n1\n(28)\n\"C\"\n11\nfor\n4g\nand:\nfor 32 3 p C 2 3 TT 2\n(29)\n\"C_\"\ng\nIt turns out that, on using T1 = 74 dynes/cm,\nC = 6.1 x 10-4 dyne cm 2 sec\n(30)\nC = 3.2 x 10-8 gm cm 5\n3\nsec3\nIn the cgs system, so that Ua is in centimeters per second\nand E in ergs per centimeter squared.\nFrom (27) we can see that for small winds, i.e.,\nUa < < 1 (in whatever units) the linear power of Ua will be\nrelatively dominant, so that capillary waves will contribute\nmost of the energy E. For greater winds, i.e. , U > > 1,\nthe fifth power of U takes dominance, and gravity waves\nwill hold the lion's share of energy. In this way we have\nmade quantitative the intuition we already possessed about\nthe energy E in our discussions of (98) of Sec. 12.3.\n12.9 Theoretical Wave Spectra Models\nIn this section we shall review some of the empirical\ndata and models discussed in Secs. 12.5-12.8 from a relative-\nly theoretical vantage point with the purpose in mind of\n\"rationalizing\" the empirical results and embedding them in\na plausible conceptual framework. In particular, we shall\nshow how one can view the gaussian frequency distributions\nof wave elevations and slopes as the result of the steady\nconfluence of great numbers of independent simple wave events.\nSome of the wave spectra of Sec. 12.8 can also be viewed in\nthis way. However, a matter which cannot yet be so simply\nviewed is the generation process of the wave spectra. The\nmathematical description of this process is still incomplete.\nSome current approaches to this description problem will be\noutlined in the closing paragraphs of this section.\nThe Wave Elevation Distribution\nExperimental evidence for the gaussian form of the dis-\ntribution of wave crest elevations about a mean sea level\nunder open sea, steady state wind conditions is now largely\nestablished (cf., e.g., Fig. 12.46). We can construct a sim-\nple mathematical basis for this empirical fact as follows.\nImagine a uniformly graduated vertical pole fixed into the\nsea bottom and visualize the wave profile moving up and down\nalong the pole as the waves go by (Fig. 12.53). Suppose that","SEC. 12.9\nTHEORETICAL MODELS\n195\n!z\nis\nFIG. 12.53 Setting for a theoretical derivation of the\nwave-elevation distribution and its close relatives.\nan arbitrary large number Z1 , Z2,\nZn, of depth readings\n,\nalong the pole are taken, and let Q (z) be the\nrelative\nnum-\nber of times the depth occurred in the small Az interval\nabout depth Zi then is an empirically determined proba-\nbility function. For a large number n of readings spaced\nreasonably far apart in time, our intuition tells us that a\nhistogram of 1 (zi) versus Zi would cluster around some mean\ndepth, say a, and fall off on either side of depth a. Hence\n01\nattains a maximum value at a. Even though we do not know\nthe form of $1 , at least that much should be granted, and\ngranting this is the first crucial step in the derivation.\nThe next thing we do is compute the mean elevation a\nfrom the sample = 1, 2,\n+\n...\n+\n(1)\nThen we agree to refer all elevations to a, so that if Si is\nany observed elevation, we write:\nfor\nSuperscript(2)-i - a\nand\n\"(5)\" for\n(2)\nNow using 0, we compute the probability P n (5, 5","VOL. VI\nAIR-WATER SURFACE PROPERTIES\n196\n5n of elevations\nthat any arbitrary given set 51, 52,\nabout the mean level a will occur. The second crucial step\nin the present derivation is the assumption that these n\nelevations are statistically independent, so that\n... $(5) .\n(3)\nNow for our present purposes, Pn can be thought of as a func-\ntion of a single variable, namely the mean depth a, with the\nZ being arbitrary and fixed throughout the discussion. Since\nthe samples 21, Z2? Zn are depths drawn from a population\nwhose mean is (estimated by) the depth a, it is at least in-\ntuitively clear that, since has a maximum at 5 = 0,Pn(5,,\n, 5n) should have a maximum when the variable a takes\non the true values of the population mean of Z - , Z\n(this is a maximum likelihood argument; see for example [59]].\nHence we set\n(4)\n.\nUsing the representation (3) of Pn, the condition (4) yields\nds, dist\n(5)\n+\nwhere the prime denotes differentiation of 0 with respect to\nits argument 5. From (2) we have\ndei\nn; so that (5) can be reduced to\nfor each i = 1,2,...,\n(7)\n+\nBy (1) and (2):\n(8)\n+ + 50)\nfor any constant . Subtracting (8) from (7), term by term:\n115, ) + +\nSince our choice of the 2i (and hence the 5i) was arbitrary,\nwe have:","SEC. 12.9\nTHEORETICAL MODELS\n197\n'(5j)\n(10)\n9(5j)\nfor every i, which is a simple differential equation in 0,\nthe general solution of which is:\n$(5) = 2/2\n(11)\nThe requirement that have a maximum at 5 = 0 requires\nbe negative. Therefore, H = - 1/m for some positive number\nm so that (11) can be written:\n$(5) = Ae -52/2m\n(12)\nIt turns out that A = 1/22 mm under the requirement that:\npo\n00\n$(5)ds 1\n(13)\n.\n- 00\nBy a straightforward computation, it is clear that m is sim-\nply the mean square deviation m of the elevation from the\nmean elevation discussed all during this chapter (cf. (89) of\nSec. 12.3). In this way, assuming only the independence of\nwave elevations in an arbitrary sequence of recordings, and\nthat the mean wave elevation occurs most often of all eleva-\ntions, we arrive at the gaussian frequency distribution of\nwave elevations. of course there are several fine points on\nthe differentiability of the empirical function o and the\ncontinuity of its variables 5 that would be more fully dis-\ncussed in a rigorous derivation. However, even then, the\nessential two properties leading to (12) will be those enun-\nciated above.\nThe Wave Slope Distribution\nExperimental evidence that the form of the distribu-\ntion of wave slopes about the zero mean slope is, to a first\napproximation, a gaussian distribution, may now be considered\nwell established, principally through the work of Duntley\n(cf. (13) of Sec. 12.5) and Cox and Munk (cf. (26) of Sec.\n12.5). Our present aim is to deduce this observed slope\ndistribution from a small set of physically plausible assump-\ntions; in this way we may explain its observed form.\nIn order to lend a semblance of concreteness to the\nderivation, we shall adopt the Duntley sea state meter (Sec.\n12.5) as the instrument yielding the various hypothetical\nreadings used in the derivation. Thus imagine that the\nslope readings are processed and reduced to pairs (51,ng),\n(52'n2)\n(5n,nn) where i is the ith up-down wind slope","AIR-WATER SURFACE PROPERTIES\nVOL. VI\n198\nreading and ni the corresponding cross-wind slope reading\ntaken at the same time, and normalized using (24) and (25)\nof Sec. 12. 5. We use the wind reference frame for concrete-\nness. There is no essential modification of the derivation\nif a terrestrial or sun-based reference frame is adopted.\nIf these pairs of readings are plotted in a En coordinate\nsystem, a scattering of points, such as those schematically\nappearing in part (a) of Fig. 12.54, is obtained. By using\nE and n, the scatter diagram is found to be basically sym-\nmetric about the origin and decreases in density in all di-\nrections away from the origin. We shall adopt this property\nas an initial assumption. If we had used a different refer-\nence frame obtained by rotating the En frame through an angle\n0, the inherent circular form of the distribution, of course,\nwould not change, only the coordinates of its individual\nunit area\n(E,\n0\n(a)\nwind\ny\np\n0\n(b)\nX\nFIG. 12.54 Setting for a theoretical derivation\nof the wave-slope distribution.","THEORETICAL MODELS\n199\nSEC. 12.9\npoints would change. Furthermore, the distance of (5,n)\nfrom the origin would not change during a rotation of coor-\ndinates. This is clear geometrically from Fig. 12.54, but\nactually can be verified from simple transformation equa-\ntions. Since this fact plays an important role in the deri-\nvation, we will now indicate its proof, since not all physi-\ncal quantities are invariant under such changes of coordinate\nsystems. For example, if we had adopted the concept of the\ninclination of the surface; that is, angles rather than tan-\ngents of angles, then the present derivation would not go\nthrough as naturally; in this way we see the judiciousness\nof the choice of wave slopes rather than wave inclincations,\nby the original experimenters, as the appropriate variables\nwhich are distributed in a gaussian manner.\nA rotation of coordinates of the sea surface through\nan angle 0 is depicted in (b) of Fig. 12.54. The new coor-\ndinates (x',y') of point P relative to the x'y' frame are\nrelated to the coordinates of P in the xy frame by (the geo-\nmetrical details in Fig. 12.22 may be used here also):\nx' = X cos 0 + y sin 0\n(14)\ny'\n= - X cos 0 + y cos 0\n,\nand conversely:\nx = x' cos 0 - y' sin 0\n(15)\ny = x' sin 0 + y' cos 0\n.\nIf the orientation of the air-water surface is given by\n(5x,5y) in the original frame, then in the new frame we have,\nby the chain rule for partial differentiation:\n5x' = + by 5x cos 0 + by sin 0\ndx dy sin 0 + by cos 0\n.\nFrom these representations, we see that\n.\nThis result is a purely geometric property of slopes. Simi-\nlarly, by now going over into the (5,n) coordinate system,\nin which the distribution is centrally symmetric about the\norigin, it follows from (12) or (15) that:\n(16)\n=\n,\nas was to be shown. The stablishment of (16) constitutes\nthe first step of the derivation.\nThe second step of the derivation adopts the second\nand final assumption; namely, that of the statistical","AIR-WATER SURFACE PROPERTIES\nVOL. VI\n200\nindependence of the wave slope components E and n. The anal-\nytic form this assumption takes is obtained as follows. Sup-\npose that $(5,n) gives the fractional number of crosses per\nunit area (in slope space) at (5,n) in the scatter diagram\n(a) of Fig. 12.54. Experimental evidence indicates that\nthere is some frequency function such that:\n(17)\n(E,n)=o(5)(n)\n,\nand by the circular symmetry of the scatter diagram (a) of\nFig. 12.54, this is independent of 0. In other words, if\nwe took vertical slices (i.e., with & fixed) of the scatter\ndiagram, as shown in (a) of Fig. 12.54, and the relative\nfrequency of crosses in that slice is found as a function of\nn, then the functional form of that frequency function is to\nbe independent of 5. In a similar way we could consider hori-\nzontal slices of constant n. This property of would, then,\nbe independent of the orientation 0 of the coordinate frame\nrelative to the wind-based frame. As noted above, we have\nessentially erased the assymetry of the wind in the slope\ndistribution by adopting the normalized coordinates 5x/ou)\nand n (= 5y/oc).\nIt follows from (16) and (17) that 0 obeys the function-\nal equation:\n(18)\n=\n.\nThis may be seen by representing (5,n) in two successive\nways by rotating the reference frame between two fixed ori-\nentations so that the E axis goes through the point (5,n) .\nFor the general prerotation case, we have:\n(E,n) $(5)0(n)\n(19)\nfor the post rotation case we have\n(20)\nwhere 5' = (E2+n2) 1/2 n'==.\nBy the circular symmetry of the scatter diagram in (a)\nof Fig. 12.54,\n$(5,n) = $(5',n')\nwhenever E,n and 5',n' are such that E2 + n 2 = (5') 2 + (n')\nThis last condition is by (16) now the case, by virtue of\nrotation transformations of the kind (14) or (15). Equation\n(18) now follows from (19) and (20) and the preceding equation.\nThe determination of the functional relation (18) for con-\nstitutes the third step of the present derivation\nThe fourth and final step of the derivation involves\nonly the mechanics of calculus. The basic assumptions and\nthe essential physical ideas (16) and (17) behind the wave\nslope distribution gave rise to (18), and now by differentia-\ntion of (18) with respect to E and to n, we have:","SEC. 12.9\nTHEORETICAL MODELS\n201\n(21)\n(22)\nwhence\nno'(E)(n)E(E)' (n)\nor\n(23)\nES(E)\nSince E and n are independent slope variables, this equality\nfor arbitrary E and n requires each side to be of some arbi-\ntrary fixed value, say l which we can choose to be -1 by the\nmaximum property of the scatter diagram (a) of Fig. 12.54.\nThen:\n$'(E) = - En(E)\n(24)\nand:\n$'(n) = - nd(n)\nIn either case, we find:\n=\n(25)\n$(n) = b'e-(1/8)n2\nwhere a',b' are arbitrary constants. From (17):\n$(E,n) = a'b' e-(1/a)(ER+nR)\nand by definition of :\nwe find:\nso that:\n(26)","VOL. VI\nAIR-WATER SURFACE PROPERTIES\n202\nIt may be well to summarize the basic assumptions leading\nto (26). (i) the scatter diagram in (a) of Fig. 12.54 can\nbe normalized and symmetrized by adopting the coordinates\nE,N as in (24) and (25) of Sec. 12.5, and the density of\npoints is a maximum at the origin. (ii) The wave slope com-\nponents E,n are independent (cf. (17)). The rest of the\nderivation, granting calculus operations possible, follows\nautomatically. Therefore except for the symmetry assumption\nthe basic assumptions leading to (26) are identical in kind\nto those leading to (12).\nObserve, that if ou = oc, then since 52 + 52 = tan 2 0,\nfor in the context of Fig. 12.29, we obtain from (26) the\nprobability distribution:\n(tan\") 20 2\n(27)\np(tan\ne\ncorresponding to (13) of Sec. 12.5. Furthermore, (26), as\nit stands, is the gaussian part of Cox and Munk's probability\ndistribution for p(5x,5y) given in (26) of Sec. 12.5.\nObserve, finally, that the functional equation (18)\nwhich governs is the mathematical heart of the present\nderivation: all the physical assumptions were chosen so as\nto lead to (18) and all the essential mathematical deductions\nled from (18). Mathematical readers will note that the func-\ntional equation (18) is to the wave slope distribution, as\nthe semigroup property is to the beam transmittance function\n(cf. (6) of Sec. 3.10 and also (3) of Sec. 3.11). Further-\nmore, the basic mathematical arguments leading to (12) and\n(26) are applicable to a wide range of physical phenomena\nbeyond the present context (cf., e.g., [158]).\nAn alternate derivation of the slope distribution\np(5x,5y) which starts with the assumption that the sea sur-\nface is a sum of sine curves, is given for the one-dimensional\ncase (5y = 0) by Cox and Munk in part III of [55]. This der-\nvation leads to a one-dimensional Gram-Charlier representation\nof the form (26) of Sec. 12.5 (with n = 0) by considering a\nlarge finite number of superimposed sine waves. An interest-\ning result of the derivation is some direct evidence for the\nexistence of continuous wave spectra for ocean waves.\nThe Wavelength Distribution\nWe now return to the setting of Fig. 12.53 and re-\nexamine the waves passing the wave pole. We shall use the\nnotion of apparent wavelengths depicted in Fig. 12.51, and\nadduce a simple argument leading to a probability distribu-\ntion of wavelengths and then go on to deduce a gamma type\nof wave spectrum (16) of Sec. 12.8) for the temporal fre-\nquency O. We shall initially assume that the waves are all\ntraveling back and forth in a narrow band of directions\nabout a given direction (shown by the arrows in Fig. 12.53).","SEC. 12.9\nTHEORETICAL MODELS\n203\nBy repeating the argument carried out in the case of\nwave slopes and leading to (26), or repeating the argument\nleading to (12), we can show that the relative number nx/no\nof occurrence of waves of wavelength l passing the wave pole\nin any direction in the band is:\n6.\nwhere we have assumed our = = m (now the mean square wave-\nlength) and where no is the number of waves of average wave-\nlength passing the pole. This representation is deliberately\nchosen to be similar to (13) of Sec. 12.5 so as to point up\nthe basic similarity between the present derivation and that\nof (17) of Sec. 12.5. Indeed, we may use Fig. 12.29 and (a)\nof Fig. 12.54 to represent the distribution of wavelengths of\nwaves traveling in various directions past the wave pole. The\ndistances from the origin to the crosses in (a) of Fig. 12.54\nare now values of 1. From this mode of representation, we\nsee that the relative number dng of waves of wavelength l\noccurring in the sector of the ring of radius l and thick-\nness dl is:\n22\nd X = e 2m d\n,\nwhere we choose C so that the total relative number of waves\nover the whole X-plane is 1. Hence, analogously to (17) of\nSec. 12.5, we have:\n(28)\nThe density distribution dñ/dx as a function of l is\nof particular interest at present. Writing \"TA\" for dn/dx we\nhave:\n(29)\nAs it stands, T1 is simple a Rayleigh distribution in the\nwavelength parameter 1. This Rayleigh distribution occurs\noften in statistical physical situations where some parameter\n(such as 1) varies randomly over the interval (0,00) rather\nthan over (-00,00). Indeed, a Rayleigh distribution may be\nthought of as the distribucion of the lengths (i.e., , scalar\ncomponents) of vectors (i.e., directional entities) which are\nsuitably randomly distributed over some domain. It is now\nclear that we can relax the condition of a narrow band of di-\nrections and redo the derivation with = 2 TT, if desired.\nWe will arrive once again at (29).","VOL. VI\nAIR-WATER SURFACE PROPERTIES\n204\nThe Bretschneider Spectrum\nWe can convert (29) into a gamma type spectrum of the\nform given in (16) of Sec. 12.8, as follows. We assume that\nthere is some constant, say B, such that:\n=Bt2\n(30)\nfor apparent wavelengths and periods. For example, if we\nhave the purely periodic sinusoidal waves of the classical\nwave theory, then, according to (2) of Sec. 12. 8, B = g/2TT.\nIf the waves are distributed according to the Neumann spec-\ntrum, then by (3) of Sec. 12.8, B = (2/3)(g/2tt). In general\nthen, B depends on the distribution in question. From (30)\nwe have:\n(31)\n=\nso that:\ndi = - 8TT28-3 do\n(32)\nWriting:\n\"To\" for Ta\n(33)\nwe arrive at:\ndo , (34)\nthat is:\n(35)\nwhere we have written:\n(36)\n\"a\"\n(37)\n\"b\"\nfor\nHence in the context of (16) of Sec. 12.8, p = 5 and q = 4,\nand we have deduced the form of Bretschneider's spectrum (13)\nof Sec. 12.8. (Note that To in (35) must be renormalized to\nbe used for the evaluations discussed in Sec. 12.4.) In addi-\ntion to the empirical evidence for To referred to in the dis-\ncussions of (13) of Sec. 12.8, we now have some theoretical\nevidence in its favor in the form of (35). But (35) must","SEC. 12.9\nTHEORETICAL MODELS\n205\nstill be viewed in the light of the critique of (21) and (22)\nof Sec. 12.8 and also as a consequence of the assumptions\nthat led to it.\nThe Wave Height Distribution\nIn a manner exactly analogous to that just completed,\nwe can show (by replacing \"X\" everywhere by \"H\") that, corre-\nsponding to (29), , we have:\n2\nH\nTHE\n-\n(38)\ne\nTH is the probability distribution for wave height H(= 2X\n(amplitude)) so that TH and T 1 are both Rayleigh distribu-\ntions. The quantity m is now the root mean square wave\nheight. A derivation of (38) was first given by Longuet-\nHiggins, and various consequences drawn from it are available\nin [164]. In view of (29), these consequences for H ( or\namplitude a) can then be applied, mutatis mutandis, to the\nwavelengths l.\nA study of the derivation of (38) and the properties\nof vector wave numbers in Fig. 12.16 seems to give the key\nto the answer to the question of why wave numbers will not\nwork in the derivation of a formula like (29) the overall\nresultant of two wave forms is governed not by k = (1/2) .\n(K1 +k2) but by Ak = (1/2) (K1-k2). (See the discussion of\n(99) of Sec. 12.3.) However, Rayleigh distributions are\nderived from superposition formulas of the type (1/n) (k +\n+kn). This leaves the reciprocal l of k/2m as the only re-\nmaining possible simple parameter associated with wavelengths.\nThe fact that the empirically verified formula (35) follows\nfrom (29) seems to bear out that l (rather than k) is the\nappropriate variable to use in the deduction of the gaussian\ntype distribution (27).\nModels of Wind-Generated Spectra\nWe turn now to review some relatively recent and ad-\nvanced theories of wind-generated waves. These theories go\ndeeper into the physical wave generation processes than those\nreviewed above, by incorporating the hydrodynamic equations\ndirectly into the statistical treatment and by using harmonic\nanalysis techniques of representing the dynamic air-water\nsurface and the wind blowing over it. Despite the formidable\nanalytical tools needed for a full unfolding of the descrip-\ntion of the wave generation process, it is still possible to\nlist and describe in simple terms the principal physical\neffects of the wind which have been considered pertinent to\nthe generation process. These are as follows:","VOL. VI\nAIR-WATER SURFACE PROPERTIES\n206\n(i) The Kelvin-Helmholtz Model. In this model the air\nflows smoothly with horizontal streamlines over the flat water\nsurface at speed Ua with the body of water possibly in its own\nsmooth translatory motion of speed Uw. Then, as noted in (89)\nof Sec. 12. 3, when Ua-Uwl is great enough, the tiny irregu-\nlarities usually found on the real air-water surface, as pre-\ndicted, grow exponentially with time. The active physical\nmechanism in this model is not immediately clear since the\ninstability is simply the manifestation of the two roots of a\nquadratic equation becoming complex as |Ua-Uw becomes suffi-\nciently large. However, Bernoulli effects, which are permis-\nsible in the simple physical setting of the Kelvin-Helmholtz\nmodel, may be the pertinent operative effects in this model.\n(ii) The Sheltering Model. When one places his hand out\nthe window of a moving vehicle and holds it against the force\nof the wind, one notes that the normal force on the palm in-\ncreases systematically as the palm is turned full into the\nwind. If one replaces the palm with a plane surface and lets\nthe stream of air of speed Ua push against it at an angle 0\nfrom the plane, then the pressure normal to the surface is\nknown (Art. 77, [149]] to vary as:\nsin 0\n(4) + sin 0\nwith 0, and as:\nPaUR\nwith wind speed Ua and air density Pa. If the plane surface\nis instead a facet of water surface, then the normal wind\nforce on the facet is very nearly of the form:\n(39)\nPa a a\nwhere 25/2x now approximates tan 0, which for small 0 in turn\napproximates sin 0 (and hence is proportional to sin e/[(4/)\n+ sin e]). The belief that the term (39) constituted the\nprincipal wave generating effect of the wind was called the\nsheltering hypothesis by Jeffreys [121] and was used as the\nstarting point for his sheltering model. What is \"sheltered\"\nin this theory are the leeward sides of waves on a wind-blown\nsea. The normal force (39) acts on the windward side of the\nwaves and because the leeward side has only an ineffective\nturbulent pressure on it, (39) acts as a driving force to\npush into and thereby feed kinetic energy to the waves. The\nhorizontal driving force is spa (25/ax)2, and if Ua is suffi-\nciently great, instability will arise as in the Kelvin-Helmholtz\nmodel. The sheltering coefficient S forms a serious obstacle\nto the use of the sheltering model as it must be determined\nempirically, a determination which to this day (1966) is quite\nbeyond available techniques.\n(iii) The Laminar Flow Model. Wuest [324] in 1949 and\nLock [161] in 1954 took into account the viscosity of the air","207\nTHEORETICAL MODELS\nSEC. 12.9\nwhich allows a boundary layer to be formed. The wind can\nthereby impart a forward dragging force on the water surface.\nIf a small, vertical, oscillatory motion in the lamina of\nair is introduced, say by the irregularities in the air-water\nsurface, and if the lamina speed is great enough, then insta-\nbilities will arise as in the Kelvin-Helmholtz model and waves\nwill be generated.\n(iv) The Stochastic Pressure Model. In addition to the\nhorizontal push and viscous drag of wind on already formed\nwaves, the sudden downward gusts and sudden partial vacuums\nover an otherwise still water surface can effectively act to\nset the water mass into motion. In 1954 Eckart [86] combined\nthe equations of hydrodynamics with the techniques of harmonic\nanalysis to form a stochastic model of wind pressures acting\nin a storm area over a part of the sea. The model goes far\ntoward the description of waves by this pressure mechanism,\nbut on comparison with observed wind pressures, it turns out\nthat the theory requires about ten times as much wind pres-\nsure as observed in the actual storms to generate the actual\nobserved wave heights. Despite this inability of the sto-\nchastic wind pressure model to account fully for the observed\nwind-generated waves, the pioneering use of harmonic analysis\ntechniques set the stage for the construction of later models.\n(v) The Stochastic Resonance Model. A direct outgrowth\nof Eckart's model is that proposed in 1957 by Phillips [197].\nThe central feature of this model is the spatial Fourier anal-\nysis of air pressures over the air-water surface with an eye\ntoward the selective resonance that one or more of these com-\nponents may set up in the free water surface. One practical\nfeature of this model is the derivation of the form of the\ntime-dependent resolved directional energy spectrum E(k,t)\n(cf. (75) of Sec. 12.4 see also (61) and (77) of Sec. 12.4).\n(See also [195], [196].)\n(vi) The Shear Flow Model. This model, proposed in\n1957 by Miles [179], builds on both the Kelvin-Helmholtz\ninstability model and the Jeffreys sheltering model and goes\nbeyond the latter in being able to calculate an equivalent\nof the sheltering coefficient. The actual mechanism of ener-\ngy transfer from the air to the water is through an inviscid\nshear flow in the air layer above the water; the transfer\nproceeds at a rate proportional to the wind-profile curvature,\nd2Ua(y)/dy2 at the elevation where Ua = C (c is the celerity\nof the water waves).\nThis model appears to yield some results in quantita-\ntive agreement with experimental data but is not wholly con-\nfirmed at present (1971).\nThe preceding six models of wave-generating wind-proc-\nesses are all but parts of a currently incomplete and vastly\ncomplex puzzle as to how wind generates water waves in air-\nwater surfaces. The final answer is certain to combine each\nof the preceding models as an integral part with varying","AIR-WATER SURFACE PROPERTIES\nVOL. VI\n208\nweights of importance, with perhaps a mechanism or two so\nfar neglected. Such a synthesis has yet to be made. *\nSpectral Transport Theory\nThe six models of wave generation processes reviewed\nabove form but a small part of the entire problem of de-\nscribing the evolution of waves on the air-water surface.\nOnce the waves have been generated by either wind or other\nagents, their interactions one with the other, and their de-\ncay in time and space must be taken into account. Thus,\nquite analogously to photon (radiative) transport processes,\nwe can formulate the general spectral transport process in\nterms of a general nonlinear integrodifferential equation\nfor the directional energy spectrum E (x, t, k) : If D/Dt is\nthe Lagrangian derivative operator (5) of Sec. 3.15), then:\nDE(x,t,k)\na(x,t,k)E(x,t,k)\n=\n-\nDt\nE°(x,t,k)\nk')\ndV\n(k')\n+\n+\n(40)\nwhere in the integrand we have written\n\"E' 11\nfor\nE(x,t,k')\nand\n\"E\"\nfor\nE(x,t,k)\nand where P is the interaction functional describing the net\nrate of increase of E(x,t,k) as a result of interactions of\nwaves described by E(x,t,k'), k # k'. The first term on the\nright is a decay term and the last term E°(x,t,k) describes\nthe source term for E, and may be found by a suitable synthe-\nsis of any or all of the wind generation models discussed\nabove. The central problem in current spectral transport\ntheory is the exact delineation of the form of the interaction\nfunctional @. The basic similarity of (40) to the equation\nof transfer, or the nonlinear Boltzmann equation of gas dy-\nnamics, or still more generally, its kinship with the Boltz-\nmann equation of general stochastic theory [48] should lead,\nwith the possible help of the nonlinear hydrodynamic equa-\ntions (15), (16) of Sec. 12.3, and turbulence theory [10] to\nsome specific representations of P. Once this is done, solu-\ntion procedures of (40) of various kinds can be studied rang-\ning from iterative and perturbation procedures to closed forms\n*For the status of this problem at the time of publication,\nsee T. P. Barnett, and K. E. Kenyon, \"Recent Advances in the\nStudy of Wind Waves, 11 Rep. Prog. Phys. , 38, 667 (1976).","SEC. 12.9\nTHEORETICAL MODELS\n209\nin the simpler cases of P. At present very little is known\nabout the form of P. Some approximate perturbation tech-\nniques by Hasselmann [105] have yielded low order approxima-\ntions to P, but otherwise our knowledge of Pis sparse. See\nalso [106], [297], and [268].\nAs an indication of the use of E(x,t,k) once (40) has\nbeen solved, we recall, e.g., the definition of F(0,0) in\n(107) of Sec. 12.4. If we then write:\n(x,t,o,0)\" for\n(41)\nwhere now k = (u,v), and also write:\n\"To(x,t)\"\nfor ,\n(42)\nwe would then have the general space-variable, time-depen-\ndent version of To in (110) of Sec. 12.4, and we would be\nable to have an overview of the Neumann, Bretschneider and\nother gamma-type spectra (16) of Sec. 12.8 and thereby be\nable to see how they fit into the general scheme of things.\nFurthermore, corresponding to (116)- - (119) of Sec. 12.4, we\nwould, for example, have for each point x and time t on the\nair-water surface:\n(mean square wave\n(43)\nelevation)\n$(x,t,o,0)o+c) do do\n(44)\n(mean square wave slope\nalong X axis)\ndo do\n(45)\n(mean square wave slope\nalong y axis)\nF(x,t,o,) sin cos do do\n(46)\nThese moments, in turn, would supply the basic input data\nneeded in the theory of the calculation of the average reflec-\ntion and transmission properties of the dynamic air-water sur-\nface with respect to incident radiant flux. We shall now turn\nto the construction of such a theory.","VOL. VI\nAIR-WATER SURFACE PROPERTIES\n210\n12.10 Instantaneous Radiance Field over a Dynamic Air-Water\nSurface\nWe go on now to the final optical problem of the\npresent chapter: We shall show how a general solution may\nbe found to the problem of the time-averaged radiance dis-\ntribution of wind-blown, air-water surfaces. This will be\ndone by suitably combining the interaction principle of\nChapter 3 with parts of the hydrodynamic and harmonic analy-\nses above. The problem will be solved in four main stages.\nThe first stage will be accomplished in the present section\nwherein we shall in imagination stop the complex randomly\nmoving surface and take an instantaneous mathematical snap-\nshot, so to speak, of the light field as it plays over the\nhills and in the hollows of the frozen surface. This infor-\nmation of the instantaneous light field will be made suffi-\nciently general so that it will hold over the entire surface\nand for any instant in a continuum of instants. Then, in\nSec. 12.11, we shall perform certain averaging operations on\nthe instantaneous radiance fields over a sufficiently long\ntime interval to obtain the requisite time-average radiance\nfields over the surface. In Sec. 12.12 the theory of the\ntime-averaged light field within the hydrosol will be devel-\noped, and finally, in Sec. 12.13 the results will be synthe-\nsized into a complete mathematical solution of the problem\nof time-averaged light fields in natural hydrosols. Applica-\ntions and concluding observations are made in Sec. 12.14.\nBecause of the necessity to work explicitly in both\naerosols and hydrosols in the next few sections, it is also\nnecessary to take into account the change in radiance when\ngoing from one medium to another with differing indices of\nrefraction. In order to avoid cumbersome appearances of the\nsquare n2 of the index of refraction in the various formula-\ntions, we shall make the standing convention for Secs. 12.10-\n12.14 inclusive: n°-convention: all radiances N appearing\nin the equations of those sections will be understood to be\ndivided by n2, where n is the index of refraction of the\nmedium (air or water) at the point at which the radiance is\nconsidered. The basis for this convention rests in the\nradiance invariance law (4) of Sec. 2.6. *\n*It may be well to repeat once again the convention used\nthroughout this work, concerning unpolarized flux (cf. Sec.\n1.1). The significance of this convention for the discus-\nsions in Sec. 12.10-12.14 inclusive is that the equations are\nmaterially simplified without loss of essential generality.\nTo develop the corresponding theory for polarized flux, one\nmerely replaces N by its observable vector counterpart N (cf.\nSec. 2.10) and the Fresnel reflectance by its matrix counter-\npart (cf. [292]) and the volume scattering function by its\nown corresponding counterpart (cf. Sec. 13.6). In this way,\nproducts of the form Nr, No are replaced throughout by the\nvectors Nr,No. Since the theory is linear, it follows that\nthis is the only change needed to elevate the scalar theory\nto its vector (polarized) level.","SEC. 12.10\nINSTANTANEOUS RADIANCE FIELD\n211\nThe Geometrical Setting\nFigure 12.55 depicts a portion of a dynamic air-water\nsurface S at time t. The surface S is of arbitrary geometric\nstructure, such as is found in natural wind-blown settings,\nand not necessarily of a simple sinusoidal structure, nor\neven a superposition of sinusoidal surfaces. Only relatively\nrecently have realistic analytical representations of the dy-\nnamic air-water surface, along the lines of (71) of Sec. 12.4,\nbeen pressed into use in the study of hydrodynamic problems.\nAn example of a parametric representation of the Lagrangian\nformulation (1) of Sec. 12.3) of hydrosol motions may be\nfound in [200]. Therefore with the knowledge that various\nrealistic analytical models of the random water surface S\nexist for possible computation use, we can proceed with the\nmain task of describing the interreflection process of the\nlight field in the vicinity of S.\nWe establish a local reference frame at each point X\nof S, as shown in Fig. 12.55. To S at X we assign a unit\noutward normal n(x), and this in turn induces a partition of\nthe unit sphere of directions into (x,t) and E. (x,t) respec-\ntively, the outward and inward hemispheres of directions to\nat X at time t. E+(x,t) consists of all directions in E1)\nS\nmaking an angle less than 90° with n (x). The set E. (x,t) con-\nsists of all other directions in E. The main reference frame\nfor the hydrosol is established relative to a mean horizontal\nn\nE+(x,t)\n(x,t)\nx\n+\n(mean surface)\nS\nS\nD(S x 1)\nD°(x) I)\nFIG. 12.55 Direction conventions at an arbitrary point\nX on the dynamic air-water surface at a given instant t.","VOL.\nVI\nAIR-WATER SURFACE PROPERTIES\n212\nplane surface S, shown in Fig. 12.55. The unit outward nor-\nmal to S is k and distances are measured from S positive\ndownward (in the direction of -k) as usual.\nThe Integral Equation for the Instantaneous\nSurface Radiance N (S)\nThe general equations governing the radiance distribu-\ntions at the points of surface S have been derived via the\ninteraction principle in Example 4 of Sec. 3.5. In particu-\nlar, the requisite equations are given in (14) - (17) of Sec.\n3.5. That set of four equations may now be used to obtain\nthe two basic surface radiance distributions N+ and N+ in\nthe present problem. Before proceeding, however, it may be\nhelpful to the reader, in order to achieve complete under-\nstanding of what follows, to study Example 4 of Sec. 3.5,\nand an appropriate amount of the material prerequisite to it.\nThe particular forms of (14) - (17) of Sec. 3.5 most\nappropriate for the present setting may be found with the\nhelp of Fig. 12.56. This figure shows schematically that,\ndespite the awesome complexity of the air-water surface,\nits \"outside\" never sees its \"inside,\" as, for example, is\nthe case in (e) or (f) of Fig. 3.16. This fact allows a\nsimplification in the structure of the set (14)-(17) of Sec.\n3.5. For it is quite clear that the auxiliary functional\nequations (16) and (17) of Sec. 3.5 now become:\nn\n(x,t)\nII\nx\nSS\nmean surface\nNC(S)\n(x,t)\nN+(S)\nNF(S)\nN:(S)\nx\nFIG. 12.56 The four radiances associated with an arbi-\ntrary point x on the dynamic air-water surface at a given\ninstant t.","SEC. 12.10\nINSTANTANEOUS RADIANCE FIELD\n213\n(1)\n=\nN](S) = N](s)\n(2)\nOf the two equations (14) and (15) of Sec. 3.5, the one we\nshall single out for illustration is that describing (S),\nthat is, the radiance distribution of the air-water surface\nas seen from above the water. The solution procedure for\nN°(S) can be inferred from that for N+ (S) which will be\ngiven in detail in the course of the present and following\nsections. The representation of N++ (S) is the more immedi-\nately useful of the two and fortunately can be discussed\nfor the most part independently of specific knowledge of the\nradiance distributions in the hydrosol and aerosol bounding\nthe surface. However, in the final analysis-- as in the static\ncase of Sec. 12.2--all three media: aerosol, boundary, and\nhydrosol, actively participate in the solution procedure and\nboth N+ (S) and N+ (S) must be determined. Accordingly, the\ntime-averaged form for N+ (S) is summarized in (44) of Sec.\n12.11.\nIn accordance with the preceding remarks, we consider\n(14) of Sec. 3.5 in which (7 (S) have been replaced by their\nrepresentations given in (1) and (2):\nN+(s) = A+ + [N+(s) S)t((S)]r_(s)\n[N+(s) x_(S)t_(S)]t+(s)\n(3)\n+\nwhere we have written:\n\"A+(S)\" for N°(S)r°(S) +\n(4)\nA further simplification may now be made in the structure of\n(3) which in no way will incur any loss of generality of the\nresults. We shall assume that the aerosol above S is per-\nfectly transparent and that the hydrosol below S is perfect- -\nly opaque. In other words. we shall white convexity S from\nabove and black convexity S from below (cf. Sec. 3.8 and\nExample 6 of Sec. 3.9) and thereby isolate it for study pend-\ning the description of its subsequent interactions with the\nmedia above and below it via the generalized principles of\ninvariance. This will be done in Sec. 12.12 after suitably\naveraging all the equations involved. With these convexifi-\ncations invoked, (3) becomes:\n(5)\n+\n.\nEquation (5) clearly determines N+(s) under the present con-\nditions since it may be formally solved to yield:","AIR-WATER SURFACE PROPERTIES\nVOL. VI\n214\n(6)\n= -\nin terms of the incident external radiances N°(S) from the\nsky and No (S) from the water body, and the reflectances and\ntransmittances of the air-water surface. For the present,\nthen, No (S) will be assumed given.\nWe can convert (5) from operator form to its specific\nfunctional form (as illustrated in Example 4 of Sec. 3.5)\npreparatory to the averaging process. The result is (dropping\nsuper and subscripts \"+\", for brevity):\nN(x,E,t) =\n0°(x,t)\n(x,t)\n/\nD(S,x,t)\n(7)\nwhere & is in E+(x,t), and where D(S,x,t) is the time-depen-\ndent version of D(S,x) defined in Example 4 of Sec. 3.5.\nD°(x,t) is the subset of E_ (x,t) over which radiant flux may\narrive at X at time t from the sky (see Fig. 12.55). There-\nfore the first integral on the right in (7) gives the contri-\nbution to N(x,E,t) by reflected skylight. The second term\ngives the contribution to N(x,E,t) by transmitted hydrosol\nlight. The final term is the interreflection term where, be-\ncause of the local concavities of S, S may feed light to it-\nself. It is clear that the natural movement of the water\nsurface is so slow relative to the speed of light that the\ninterreflection process over each instantaneous configura-\ntion of the water surface virtually attains its steady state\nbefore the configuration can sensibly change: thousands of\nterms in the natural solution can be formed over a gravity\nwave of 1 meterwavelength before that gravity wave can change\nits linear proportions by 1 percent. Therefore (7) gives an\ninstantaneous - - or as is sometimes described, a quasi-steady\nstate--description of the light field over S as it is lit by\nthe sky, the sea, and itself.\nThe exact interconnection among the domains of integra-\ntion of the integrals in (7) will be needed subsequently, and\nis as follows. First, by Fig. 12.55 it is clear that for\nevery X on S and every time t,\n(1)\n(8)\n.","SEC. 12.10\nINSTANTANEOUS RADIANCE FIELD\n215\nThe inward hemisphere E. (x,t) in turn has the decomposi-\ntion:\n0°(x,t)UD(S,x,t) = = E_(x,t)\n(9)\n.\nCombining (8) and (9) we have:\nE(x,t)uD(x,t)UD(S,x,t) = (1)\n(10)\n.\nNext we may introduce the characteristic functions X\nof a set E is such that x(x,E) = 1 whenever X is in E, and\nx(x,E) = 0 whenever X is not in E. It follows from this def-\ninition and (10) that for every direction 5 in E, we have:\nX(E,E,(x,t)) + (E,D°(x,t)) t X (E,D(S,x,t)) = 1. (11)\nThe introduction of the characteristic function serves the\npurpose of allowing the domains of integration in (7) to be\nuniformly replaced by E and the integrands to be prepared\nfor the ensuing averaging process:\nN(x,E,t) = -\nD(S,\nt)\nr\nE E E ( x, t), X E S\n(12)\nThis is the required integral equation governing the surface\nradiance of the air-water surface S, at an arbitrary point X\nof S, at time t and for all directions 5 in E+(x,t). The in-\ncident radiance N° consists of two parts: the part N°(E)\nfrom the sky and the part N° (x,5) from the body of the hydro-\nsol below. When these two radiances are given or assumed\nknown, (12) completely determines N(x,E,t). We note in pass-\ning a certain asymmetry in the decomposition of (1) above.\nThus we have partitioned E. into Do and D as in (9), but have\nnot made the analogous partition of E+. This is a result of\nour decision to treat the aerosol above S and the hydrosol\nbelow S in two distinct ways using the technique of convexi-\nfying the surface S. The consequences of this decision are\nsummarized in the discussion of (20) of Sec. 12.12, below.","VOL. VI\nAIR-WATER SURFACE PROPERTIES\n216\n12. 11 Time-Averaged Radiance Field Over a Dynamic Air-Water\nSurface\nIn order to obtain the time averaged radiance distribu-\ntion over a dynamic air-water surface it appears that we need\nonly average each side of equation (12) of Sec. 12.10 over a\nsufficiently long time interval. While this operation indeed\nseems the natural one to perform and outwardly seems straight\nforward, in order to achieve a useful averaged version of (12)\nof Sec. 12.10, some care must be taken in first of all estab-\nlishing a meaningful definition of a time averaged radiance\ndistribution over E. Once this is done there is the addition-\nal task of assuming enough regularity properties of the aver-\naged radiance, reflectance, transmittance and characteristic\nfunctions appearing in (12), to allow the averaging process\nto culminate in the workable static description of the dynamic\nsurface. The latter feature is of course the prime virtue of\nthe time-averaged radiance field. We shall devote most of the\nefforts in this section to achieve this end. But throughout\nthe discussion of all these subsidiary details the reader\nshould retain the single idea which prompts all the action:\nthat the light field over an arbitrarily curved air-water\nsurface is given for every instant by (12) of Sec. 12. 10 and\nthat by adding together such representations for each time in\nan interval of times, an averaged time-independent version of\n(12) of Sec. 12.10 can be obtained.\nDirect and Indirect Radiance Averages\nA study of (12) of Sec. 12.10 preparatory to a time\naveraging operation reveals that the radiance values N(x,E,t)\nare to be taken at points X of the moving surface while the\ndirection argument & is held fixed. A natural method of aver-\naging such a radiance function is suggested when one imagines\na fixed observation point somewhere off in the atmosphere\nfrom which one can view the water surface along a rigidly\nfixed line of sight. Figure 12.57 depicts the essential geo-\nmetric arrangements for such an averaging process for three\ndifferent instants in time. Let a point X on the mean water\nsurface S be fixed and let a straight line directed along a\ngiven & in E+ pass through X. Then as time proceeds, the\npoint of intersection of the line of sight with S is a moving\npoint x(t) on S defined by the farthest intersection of S\nwith the line of sight from X (or equivalently, the nearest\npoint of intersection to the observer). We shall write:\nT\nI\n1\n\"N+(X,5)\"\nN(x(t),5,t)dt\n(1)\nfor\nlim\nT\nT-00\nwhenever & is in tri and call Ñ (x, E) the upward time-averaged\nradiance of S at X along the direction & in E+. In this\nmanner we can assign to each X of S a certain radiance in any\ndirection E of E+.\nOn the other hand, as can be seen from Fig. 12.58, down-\nward radiances can also be observed to flow along the line\nthrough X and in directions & in E.. In this case we use the","SEC. 12.11\nTIME AVERAGED RADIANCE FIELD\n217\nFig. 12.57\nobserver\n((x)\nS\n(a)\nS\n(x)\n(b)\nx\nx(t)\nk\n(x)\nExternal\nInternal Point\n+\nPoint\nS\n(c)\nk\n+(x)\nS\n(a)\n(x)\nk\nExternal\n(t)\nPoint\nS\n(b)\nInternal\nPoint\nS\n+\nx(t)\n(c)\nFig. 12.58\nFIG. 12.57 Geometric construction convention for finding\ninstantaneous position X (t) of observed point of dynamic air-\nwater surface (downward line of sight)\nFIG.\n12.58 Further cases of the constructions in Fig. 12.57\n(upward line of sight).","VOL. VI\nAIR-WATER SURFACE PROPERTIES\n218\nfirst external point of intersection x(t) of S (again, this is\nthe external point nearest the observer) with the line of sight\nextended back in the direction - E. Observe, however, that this\npoint may not be always be defined because the surface S may\nnever rise above the back part of the line through x (i.e.,\nas one moves along the direction - E, from x, S is never en-\ncountered). For this reason, the radiances N(x(t),5,t) with\nE in E_ must be appropriately weighted by ((E,D(S,x,t)) prior\nto averaging in order to account for the occasional absence of\na point of S contributing to N(x, (t) E,t). Therefore we shall\nwrite:\n\"N_(X,5)\"\nN(x,(t),s,t)x(E,D(S,X,t))dt\nfor\nlim\nT\n(2)\nwhenever & is in E_ and call Ñ_ (x, E) the downward time-averaged\nradiance of S at X along the direction 5 in E.. In this way,\nvia (1) and (2), we can assign a time-averaged radiance to X\nin S for every direction E in E.\nDefinitions (1) and (2) give a precise meaning to the\ntime-averaged radiance for the radiances N(x,5,t) occurring\non the left side of (12) of Sec. 12.10. We shall now give\nthe associated definitions of time averages of N (x-rm5', t)\noccurring under the integral sign on the right side of (12)\nof Sec. 12.10. It is at once clear that we must at least fol-\nlow suit after (1) and (2) and construct definitions separate-\nly for the upward and downward directed radiances, so that we\nshall be able in principle to consistently relate the averages\nof N under the integral sign to those of N not under the inte-\ngral sign. But we must go further and take cognizance of the\ndisplaced (or retarded) argument x(t)-rm' of N(x(t)-rm5', is t).\nFigure 12.59 shows by means of four typical cases what is en-\ntailed in taking this retardation of argument into account.\n(There are eight basic types of cases in all. The four depict-\ned in Fig. 12.59 will suggest the remaining cases to the read-\ner. For example, the companion case to case (a) is that which\nextends the path at X (t) downward and to the right, analogous-\nly to case (c). In each case the point x(t) is first con-\nstructed from the point X on S after the manner described above\nfor (1) and (2). Then, once is is chosen, a new point x' (t)\non S and distinct from (t) is found by going from X (t) in\nthe direction -E' until the first intersection x' (t) with S,\nsuch that E' is in E+(x'(t)). The point x' (t) so found is\nx(t)-xms' as required in N(x(t)-rm5,5,t). In this way, to each\nX on S and E' in E+ we can assign a radiance Ñ' (x, E') , where\nwe have written:\nST\n1\n\"N'(X,5')\"\nN(x(t)-r_5',5',t)dt\n(3)\nfor\nlim\nT\nand for E' in E_ we write:","SEC. 12.11\nTIME AVERAGED RADIANCE FIELD\n219\n+\nS\nS\nx'(t)\n(b)\n+\nS\nx(t)\nS\n(d)\nFIG. 12.59 Geometric construction convention for finding\nthe secondary point x' (t) on dynamic air-water surface.\n\"N'(X,5')\" for lim\n(4)\nThe Stationarity Condition\nIn order to obtain an equation which determines a physi-\ncally meaningful time -averaged - radiance distribution associ-\nated with the mean surface S we shall be encouraged by the\nstatistical stationarity property, frequently encountered over\nair-water surfaces (recall the discussion after (28) of Sec.\n12.5), to postulate that:","AIR-WATER SURFACE PROPERTIES\nVOL. VI\n220\n(5)\nfor every E in E+, and x in S. In view of the definitions\n(1) - (4) and condition (5) above it is clear that the time-\naveraged radiance distribution to be determined by (12) of\nSec. 12.10 compresses a complex time dependent radiance func-\ntion over a highly convoluted random surface S into a steady\nstate radiance distribution associated with the plane mean\nsurface §. On some reflection of the matter, condition (5)\nrelating the four averages introduced above appears to be\nbut one condition of several that may connect various types\nof averages of radiances of the moving air-water surface.\nThe present condition relates directly observable radiances\n(1), (2)) with those that are observable only indirectly,\nthat is from within the concavities of the surface ((3) , (4)).\nThe natural random movements of the surface tend to make\ncondition (5) occur very nearly if not exactly, so that con-\ndition (5), which is admittedly introduced as a mathematical\nconvenience, is in the last analysis apparently closely met\nin the natural setting. It should be emphasized that equa-\ntion (12) of Sec. 12.10 could have been averaged mechanical-\nly by applying the operator\nit dt\n(6)\nto each side. However, by introducing the averages (2) and\n(4) in addition to (1) and (3) and relating them via (5) , we\nare using our intuition to add to the mathematical form of\n(12) of Sec. 12.10 enough detail about the radiometric struc-\nture of the sea surface to lead toward a realistic time-\naveraged radiance distribution. This distribution is governed\nby an integral equation whose specific form will be determined\nbelow.\nThe Independence Condition\nWe begin the reduction of (12) of Sec. 12.10 by choos-\ning any E in E and writing:\nN(x,E,t = + (x;5',5) ds (5')\nN(x-r_E\",E\",t)x(E\" D(S,x,t))r (x;5';5)ds(E')\n(7)\nwhere \"A+(x,5,t)\" denotes the integral involving N° in (12)\nof Sec. 12.10. The second integral in (12) of Sec. 12.10 has\nbeen partitioned over (1) into E_ and E+.","TIME AVERAGED RADIANCE FIELD\n221\nSEC. 12.11\nNext, choosing 5 in E+ and applying the averaging oper-\nator (6) we have,\n(,5) = A+(X,5) + D(S,x,t)).\nr_(x;5';5)dd(E\")\nr_(x;5';E)da(E')\n(8)\nwhere we have written:\n\"Ã+(X,5)\" for into (x,5,t) dt .\n(9)\nNow the dynamic air-water surface at a point is generally in\ncompletely random motion* so that functions such as N, X, t+,\nand r- occurring in (8) and (9) are very likely to be pair- -\nwise statistically independent over every time interval in\nthe sense that:\n(10)\nwhere f and g are any pairs of time-dependent functions made\nup of N, X, r-, t_ or their products for arbitrary values of\ntheir arguments other than time. At any rate we shall adopt\ncondition (10) (the independence condition) and add it to\n(5) in the list of regularity properties leading to the req-\nuisite integral equation below.\nThe Weighting Functions\nThe effect of adopting (10) is to permit the operator\n(6) to be applied individually to each function in the inte-\n-\ngrands of the integrals in (8). Recalling that \"x\" in (8)\ndenotes the point X (t) on S determined according to the pro-\ncedure depicted in Figs. 12.57-12.59, let us write:\n*If the motion were periodic and of prescribed form\nthen an alternative approach is clearly indicated. By far\nthe more realistic case is that of random motion.","VOL. VI\n222\nAIR-WATER SURFACE PROPERTIES\n\"Q°(X,5)\"\nfor\n(11)\n\"Q(X,E)\"\nfor\ndt (12)\n\"Q+(x,5)\"\nfor\n(13)\n\"R_(X;E';E)\"\nfor\ndt\n(14)\n\"T+(X'E';E)\"\nfor\ndt. . (15)\nWe shall refer to the preceding five functions as the\nweighting functions for N. It will be occasionally useful\n(in view of (9) of Sec. 12.10) to combine (11) and (12) by\nwriting\n\"Q_(X,5)\" for Q°(X,5)+Q(X,5)\n(13a)\n.\nThe Time-Averaged Integral Equation for N+(s)\nWith (10) in effect and adopting (11)-(15), equation\n(18) becomes:\nN2) = +\nds(E)\n+\n+\n(16)\nfor 5 in E+. Returning to (8), choosing E in E., multiplying\neach side of (8) by X(E,D(S,x,t)) and then applying the opera-\ntor (2) with the independence condition (10) in effect, we\nhave:","TIME AVERAGED RADIANCE FIELD\nSEC. 12.11\n223\nN(X,5) = (X;5';E) +\nda(s)\n1s. N(X,E')Q(X;E')R, (X;5';5) do(E') }ace.or\n(17)\nfor E in E. . By adopting the function S(X,5) such that:\nif if EEE_ EEE+\n(16) and (17) may be written as a single integral equation\nfor N(x,E):\nN(X,5) = S(X,5)\nwhere\nA+(X,5) = (X;5';E) +\ndo(E')\nEEE, X E S\n(18)\nThis is the requisite integral equation governing the time-\naveraged surface radiance N(X,5) of the mean air-water sur-\nface S at point KX with time-averaged incident sky radiance\ndistribution No(x,) as the basic input. The term A+(X,5)\nis a convenient contraction of two terms, one of which is the\nintegral of N°Q°R_ over E, and the other the integral of\nNoQ+T+ over E, as may be seen by comparing (18) with (16) and\n(17). In the first integral, No is the time-averaged - sky\nradiance defined over E. In the second integral N° is time-\naveraged upward hydrosol radiance defined over E. In sum,","AIR-WATER SURFACE PROPERTIES\nVOL. VI\n224\n(18) is the time-averaged version of (12) of Sec. 12.10,\nwhich in turn is the expanded version of (5) of Sec. 12.10.\nWhen necessary for clarity, we may write \"N(X, E)' in (18)\n\"N+ (x, E)\". (Recall the convention of dropping the \"++\"\nas\nsign in going over to (7) of Sec. 12.10.)\nStructure of the Weighting Functions\nIn order to solve equation (18) for the time-averaged\nradiance N(X,5), the structures of the weighting functions\noccurring in (18) must be known in detail and some particu-\nlar model of the air-water surface must be adopted, if nu-\nmerical estimates of N are desired. We shall now consider\nthese matters, and begin with R_ (x,5';5), as defined in (14).\nIn Fig. 12.60(a), the incident and reflected directions\nE' and & are related to the unit outward normal n(x,t) to the\nsurface at X by means of (1) of Sec. 12.1. By that relation,\nwe can express & in terms of E' and n(x,t) as follows:\n-2n(x,t)]n(x,t)\n(19)\n.\nThis leads to a representation of -(x;5';5) of the form:\nr (x;5';5) = ]) (20)\nwhere r(5',5) is the Fresnel reflectance function given in\n(12) or (13) of Sec. 12.1, and S is the Dirac-delta function\ndefined on E3, and has dimension of steradian Thus, un-\nless n(x,t) E' and & are related at X exactly as in (19),\nr_(x;5';5) will be zero, in accordance with the requirements\nof specular reflection. From (20) :\n(\nn(x,t)])dt\n(21)\nIn (21), as time varies, so will n(x,t) and at occa-\nsional instants the fixed directions E' and E will be re-\nlated as in (19). At such instants the argument of S is the\nzero vector and the integration will pick up a contribution\ntoward the average value of r_ (x;5';5). This observation\nfollows by virtue of the analogous property of S that:\nis the number of zeros of f in the interval (0,T) over which\nf can be written as a succession of monotonic functions (cf.,\ne.g., [95]). Hence, in general:","SEC. 12.11\nTIME AVERAGED RADIANCE FIELD\n225\nk\n(x,t)\nReflection Case\nE'\n(a)\nE\nn(x, t)\nRefraction Case\n(b)\nFIG. 12.60 Showing how the incident direction (E') and\nreflected (or refracted) direction (5) are related to the\nwave normal n(x,t) at point X on the dynamic air-water sur-\nface at an instant t. A11 three lie in a common plane.\n08(f(t))\nis the average density of zeros per unit time of f over the\ntime interval (0,00). For example, if f(t) = sin wt, then\n(nt/w) + E\nflow\nw cos wt | (sin wt)dt = n + 1\nwhere E is any positive number such that E < IT/W. Hence if\nT = n w / w , we have:","VOL. VI\nAIR-WATER SURFACE PROPERTIES\n226\nIw cos wt 18 =\nTherefore, for every E, 0 < E < IT/W:\nwt / (sin wt)dt = lim T Itte cos wt .\nS (sin wt)dt = =\nFrom this we see that the average density of zeros per unit\ntime of sin wt on the interval (0, ,00) is w/ = 2f (where W\n= 2mf, and f is the number of cycles per second). The great-\ner w, the greater the density of zeros of sin wt.\nWith the preceding example in mind (which is a sugges-\ntive analogy only) let us assume once again that the air-water\nsurface is statistically stationary so that time averages at\na point X on S are independent of location X. In particular,\nlet us write:\n\"p(5',5)\" for dt . .\n(22)\nThus p(5',5) may be viewed as the fraction of a unit of time\nthat the wave normal at X assumes the correct orientation\nwithin a unit solid angle which is directed along n) relative\nto the pair 5', E, so that a reflection can take place from\nE' to & at X. It follows from (14) that:\n(23)\n=\nThe probabilistic interpretation of p ( 5 ' , 5) is straightforward:\nfor each given E' E, p ( 5 ', 5) is the probability (based on\ntemporal frequencies) that the wave normal's orientation is\nn(x,t) per unit solid angle where n(x,t) is determined from\nE' and E by means of the equation:\nn(x,t) - 5.81\nAs an example of the general frequency distribution\np(5',5), we reconsider (28) of Sec. 12.5, where 5x and by\nare the slope parameters which are fixed once 5',5 are chosen\n(cf. (8)-(10) of Sec. 12.5). In the case oc = 'u' p(5',5)\ndepends only on the angle between the normal n and the","SEC. 12.11\nTIME AVERAGED RADIANCE FIELD\n227\nvertical k, where nok = cos . Hence if is and E are such\nthat > 90°, then p(s), 5) = 0 for the case of the gaussian\nwave slope distribution. In the case °c = uu (=0) we have\nthe simple representation:\ntan2\nif < 90°\n2\np(5',5) =\n(24)\n0\nif > 90°\nwhere\nk\n(25)\n= arc cos\nand the normalization constant (1/22) is for the slope\ndomain (cf. (16) of Sec. 12.5). The directions 5',5 in (25)\ndetermine the unit normal n to S for a reflection operation\nin conjunction with (23). The representation of p(5',5) in\n(24) is for a gaussian wave slope distribution of the air-\nwater surface. If a particular Neumann spectrum ((1) of Sec.\n12.8) is known to apply to a given region of the sea, or some\nother given natural hydrosol, then we know from (12) or more\ngenerally (22) of Sec. 12.8 how to estimate o as a function\nof wind speed Ua, since o2 varies linearly with Ua. In par-\nticular, if U 0, then o = 0 and p(E', 5) = s (E-EY/|5-E'\nk) for all is and E. As wind speed U builds up (24) yields\n-\nthe associated distributions with Ua as a parameter.\nIt should be noted in passing that the limit in (22)\ncan in some models of S be dependent on X. The presence of\nswells and other spatially or temporally periodic phenomena\non S can in principle be included in (5' E) and in the theory\nbelow. However, for simplicity of exposition we shall limit\nthe discussion to statistically stationary surfaces S.\nIn a similar way we can represent '+(X;E';E), defined\nin (15), as :\nT(X;5;) p(E,)t(E',) =\n(26)\nwhere t(5',5) is the Fresnel transmittance function defined\nin (19) of Sec. 12.1 and where 5',5 and the normal n to the\nsurface are related by means of Sec. 12. 1 (cf. (b) of Fig.\n12.60). Thus, for a given pair 5',5, p(5',5) and t(5',5)\nare evaluated analogously to the case of R. (x;5 5), but now\nof course 5' and E are to be related through a refraction\nrather than a reflection operation. Once n is found from\nthe refraction pair 5',5, the value of p(5',5) for the gauss-\nian model, for example, is given by (24). To find n from E'","AIR-WATER SURFACE PROPERTIES\nVOL. VI\n228\nand & observe that by (4) of Sec. 12.1 1, n is the unit vector\nwhich may be associated with n's' n g in the following way:\nObserve that if we set n = n', we formally obtain the reflec-\ntion case from this formula for n. This relation is made\nquite clear by studying Fig. 12.1 (c).\nWe turn next to the study of Qo(XXE) and Q(X,5), de-\nfined in (11) and (12). Observe first that, by (10) of Sec.\n12.10, and (13a) we have:\nQ+(X,5) + Q_(X,E) = 1\ni.e.,\n2+(5) + Q°(X,5) + Q(X,5) = 1\n(27)\nwhere we have written (cf. (13)):\n\"Q+(x,5)\" for lim dt . (28)\nIt is clear that Q,Qo and Qt are so interrelated that knowl-\nedge of any two of them is sufficient to determine the third.\nIt turns out that, for our present exposition, it is conveni-\nent to determine Q and Q+ so that the average Q° which occurs\nin (18) and (45) below, is determined indirectly through (27)\nand knowledge of Q and Q+. Care must be taken in the deter-\nmination of Q and Q+ by adopting compatible statistical models\nfor each so that Q° turns out to be nonnegative.\nThe representation of Q+ is readily obtained. From its\ndefinition above, Q+ (x, E) is evidently the probability that\na given direction E is in the outer hemisphere E+ (x,t) at X\non the surface S at any time t, i.e., Q+(X,5) is the proba-\nbility that E n 0 where n is the unit outer normal to S at\nX. This dot product characterization n° E>0 permits still\nanother way of viewing Q+ (x, E) as the probability that n is\nin E+ (5), i.e., in the directional hemisphere determined by\nE. This characterization of Q+ (x, E) leads directly to:\nds(n)\n(29)\n=\nin which:\n(30)\nIn the gaussian model of the air-water surface, it follows","SEC. 12.11\nTIME AVERAGED RADIANCE FIELD\n229\nthat Q+ (X,5) depends on the mean square slopes o2, of, and\nhence on the wind speed Ua, after the manner discussed in\nSec. 12.5 or Sec. 12.8.\nFor example, if the gaussian model is adopted, and we\nassume that the slope distribution is isotropic: °U = °c =\n= o (m1/2 = m1/2), then it has been shown by Keith MacAdam (in\na private communication) that the integral (29) has the\nvalue:\n(29a)\nwhere we write:\ne-us/2\n1\n\"E(t)\"\nfor\ndu\ne\n-t\nand where 4 at present is the angle (between 0° and 90° in\nmagnitude) that E makes with the horizontal plane §, and\nsuch that 4 > 0 if & lies below S. Hence, in particular,\nQ+ (x,k) = 1, Q+ (x,-k) = 0, and Q+(x,5) = 1/2 for & in S.\nSince, by (27),\n(X,5) + Q_(X,5) = 1\nwe have at once that:\n- ( E (tan y/o)]\n(29b)\n.\nWe come finally to the determination of Q(X,E), as de-\nfined in (12). It is simpler to consider Q(x,5) as obtained\nby means of a space average rather than a time average. For\nthis purpose we shall adopt the ergodic hypothesis for S\nwhich in the case of Q(x,E) takes the form:\n(31)\nwhere the integration is taken at a fixed time t over any\nstraight path on the mean surface S, and where X (i.e.,\nx(t)) is on S and determined for each X after the manner\nexplained in Figs. 12.57 and 12.58 above. We can now inter-\npret Q(X,5) as the fraction of a unit length along the path\nPwhich a straight line of sight from x(t) on S, and extend-\ning along -E, meets S. When phrased in this way, the inter-\npretation of Q(X,5) is reminiscent of the ordinate-crossing\nproperties of random functions (cf., e.g., [168], [87]).\nOne such property, which is pertinent to the present problem,\nmay be stated as follows: If S is a surface with a gaussian\ndistribution of elevations (cf., e.g., (12) of Sec. 12.9),\nthen the number of times 2(5) per unit length a horizontal\nstraight path P a distance 5 above S crosses S, is given by","VOL. VI\nAIR-WATER SURFACE PROPERTIES\n230\n52\n2 (0) e - 2m 00\n(32)\nwhere m2 (0) is given by (97) of Sec. 12.4 and moo is the root\nmean square elevation of S above § (cf., (89) of Sec. 12.4).\nExcept for the value n (0) the s-dependence of n (5) is intui-\ntively obvious. A derivation of (32) may be found, e.g., in\n[166]. However, for our present purposes, (12) of Sec. 12.9\nwill do just as well. We may use (12) of Sec. 12.9 for our\npresent problem as follows: the relative probability q (5)\n(i.e., without the property of normalization over all possi-\nbilities) that a horizontal path @ a distance 5 above S\nmeets S is:\n2\n5\n2m\n(33)\ne 00\n.\nThis relative probability may be used to obtain a rough but\nworkable estimate of Q(X,5) by computing the probability of\nintersection with S of an inclined path of direction &\nthrough a point X on S, where & is in E. In particular,\nit\nwill yield the conditional probability of an intersection,\ngiven that E is in E. (x,t). As shown in (a) of Fig. 12.61,\nthe inclined path at each elevation & above § has a relative\nprobability q (5) per unit length of meeting S, were it to\ncontinue horizontally. However, the path is continually\nchanging its elevation so that the probability that it meets\nthe surface S at least once is a sum of probabilities of the\nform:\n(jox tan 41) 2\n-\n2m\n8\n00\n1\nAX tan 141\n(34)\nj=1\n00\non the assumption that the elevations of S above § are inde-\npendent from one discrete station to the next along the x-\naxis as shown in (b) of Fig. 12.61. The factor m172 is\nplaced into (34) so as to make the relative probability di-\nmensionless. Along the x-axis (which is to lie in S), we\nhave marked out unit distances AX of arbitrarily small but\nfixed extent. The path through X is inclined at an angle 4\nwith the x-axis where 4 is positive above, and negative be-\nlow §. At j units out from x, the relative probability that\nthe surface is in the vertical slot beginning at jox tan\nand of height AX tan 14 is given by\n(jx tan 141) 2\n2m\n11/10\n00\nAX tan 141","SEC. 12.11\nTIME AVERAGED RADIANCE FIELD\n231\nS\n5\n(a)\nZ\nE\ntan\nAX\n(b)\n2\nX (in S\nQ.\nQ+\n(c)\n1/2\nE\nE\n-90\n+90\n-90\n90\n+\nQ°\nFIG. 12.61 Parts (a) and (b) depict the geometrical rela-\ntions needed for the derivation of the form of the function\nq- . Part (c) shows the general qualitative behavior of the\nQ- - functions with respect to angle 4.","VOL. VI\nAIR-WATER SURFACE PROPERTIES\n232\nThe relative probability that the given path intersects S at\nleast once is then the sum (34) given above. Actually, since\nthe surface is statistically stationary, the horizontal dis-\ntances jAx along the x-axis in (b) of Fig. 12.61 are only\nschematic. The whole diagram may be collapsed like an accor-\ndion onto the z-axis and the sum (34) is imagined to take\nplace in a vertical strip of arbitrary unit width Ax and pro-\nceeding vertically in steps of magnitude Ax tan 141, begin-\nning with Ax tan 141. Writing:\n\"AS\" for AX tan /4/\n\"5j\" for jox tan 14\nthe sum (34) reduces to:\nj=1\nand by requiring AX = m 1/2 where l is at present an arbi-\ntrary fixed positive dimensionless parameter (to be discussed\nbelow), the sum can go over into its continuous version:\n$ 5 e-y2/2 dy\ntan 141\nand which, when normalized, becomes the required conditional\nprobability 9- (x,5):\n2-y2/2 dy\n(35)\ntan 14\nThus we have found q- (X,E), the conditional probability of\nat least one intersection of the path along & with the sea\nsurface S, on the condition that E is in E_(x,t). Our deri-\nvation has assumed that this conditional probability is inde-\npendent of the position of E in E_ (x,t). Therefore:\nQ(X,)=q_(x,)q_(x,)\n(35a)\nwhich with (29b) determines Q(X,5). From this and (13a) we\nfind at once that:\n(X,5) = Q°(X,E) Q(X,E)\n=Q°(X,5) + 9. (X,E)Q_(\n,","SEC. 12.11\nTIME AVERAGED RADIANCE FIELD\n233\nso that for a gaussian sea surface model:\n=\n(35b)\nwhich with (29b) determines Q°(X,5). From the intuitive\nmeanings of the Q- functions, we require the following limit-\ning values, which will be helpful in checking specific sta-\ntistical models (of which the gaussian is but a single in-\nstance)\nQ-function\nE in §\nE = k\n===\nQ+(X,5) =\n1\n1/2\n0\nQ_(X,5) =\n0\n1/2\n1\nQ°(X,5) =\n0\n0\n1\nQ(X,5) =\n0\n1/2\n0\n(X,5) =\n1.\n1\n0\nThe general expected shapes of the Q-curves are sche-\nmatically depicted in (c) of Fig. 12.61. Thus Q- is expected\nto be a monotonic decreasing function of 4, while Q is to in-.\ncrease in the range (-90°,0) and is to coincide with Q- in\nthe interval (0,90°). Q+ on the other hand is to be a mono-\ntonic increasing function of 4 and such that Q+ + Q- = 1.\nThese properties hold not only for the illustrative example\nof the gaussian sea presented above, but also for general sea\nsurface statistics.\nSome final comments are needed on the presence of the\narbitrary dimensionless parameter A in the lower limit of\n(35). The determination of l can rest in the following ob-\nservations. First observe that the choice of AX in the pre-\nceding derivation is not fixed by any physical conditions\nassociated with the simple model (32). A more detailed model\nwould give, for example, the joint probability distribution\nfor wave elevations above or below two neighboring points on\nS. Second, observe that this distribution would then provide\na horizontal distance between two points of § above and below\neach of which the ir-water surface would oscillate indepen-\ndently. Suppose this distance were Hoo Then we could write\n\"X\" for Hor /mood , and so l would be intimately tied to the\nstatistics of the sea surface. In particular, we choose AX\nso that it is l times m 1/2 00 or simply Hoo For such a AX the\nindependence condition needed in the preceding derivation\nwould be satisfied. The l chosen should be the smallest of","VOL. VI\nAIR-WATER SURFACE PROPERTIES\n234\nsuch ratios. It might be helpful in practice to keep in mind\nthat Poo should decrease and mood should increase with increase\nof the generating wind speed (recall, e.g., (10) of Sec. 12.10).\nObserve finally that (35) is a special case of:\n(36)\n9_ tan q(y)\ndy\nwhere & .k = sin 141, and where we write:\n\"90\" for (aly) dy\n9(5) is the relative probability that S is at elevation 5\nabove S; and (36) is derived under the same assumptions\nadopted for (35).\nThis completes the discussion of the structure of the\nweighting functions R_, T+, Q, Q°, Q+, and Q-. For a sta-\ntistically stationary air-water surface whose Fresnel reflec-\ntance is independent of location over the mean plane S, these\nweighting functions and the time averaged radiance distribu-\ntion N are independent of location X in S; hence \"X\" may be\ndropped from the notation when these conditions prevail.\nEquation (18) may now be solved in principle for air-water\nsurfaces with known statistical properties.\nThe Instantaneous and Time-Averaged\nEquations for N+ (S)\nHaving developed the theory of the time-averaged radi-\nance N+(S), starting with (3) of Sec. 12.10 and culminating\nin (18) above, we complete the general theory of time-aver- -\naged radiance fields for the dynamic air-water surface S by\noutlining the derivation of the time-averaged equations for\nN (S), the radiance output of S into the body of the hydro-\nsol (cf. Fig. 12.56).\nStarting with (15) of Sec. 3.5 we have:\nN*(S) = A_(S) + = N_(S)t_(S) +\n(37)\nwhere we have written\n(38)\n\"A_(S)\" for +\nHere No is the downward incident radiance distribution on S\nfrom the sky and No is the upward incident radiance distribu-\ntion on S from the hydrosol. By invoking the black-convexi-\nfication hypothesis on the hydrosol side of S, (37) reduces\nto\nN°(S) = A_(S) + N_(S)t_(S)\n(39)\n.","SEC. 12.11\nTIME AVERAGED RADIANCE FIELD\n235\nHence in view of (1), (6) of Sec. 12.10 N+(S) is completely\n8\ndetermined once No and N- are known. N_ is assumed given.\nNo is determinable in averaged form in the manner to be shown\nin (10) of Sec. 12.13.\nWe can convert (39) into explicit numerical form in ex-\nactly the manner that (5) of Sec. 12.10 was converted into\n(12) of Sec. 12.10. The result is:\n=\nE E E ( x, t), X E S\n(40)\nThis is the requisite representation of the instantane-\nous radiance N(x,5,t) with E in E-(x,t), at an arbitrary\npoint X of S at time t. The radiances N(x-rm5',5',t) are\nthose governed by the integral equation (12) of Sec. 12.10.\nEquation (40) with (12) of Sec. 12.10 completes the descrip-\ntion of the radiance distribution over E = E+(x,t) UE_ (x,t)\nat an arbitrary point X of S at time t. The input radiance\ndistribution N°(x,., t) is derived from knowledge of the sky\nradiance over D°(x,t) and of the hydrosol radiance over\nE+(x,t). It will be shown in Sec. 12.12 how to find the hy-\ndrosol radiance given N(x, , , t) over E-(x,t). . Then in Sec.\n12.12 a solution will be synthesized from the appropriate\npieces of the analysis.\nContinuing with the preparations for the present aver-\naging process, let us write, analogously to (14) and (15) :\n\"T_(x;5';5)\" for (41)\nfor lim . (42)\nFinally, we write:\n\"N*(X,E)\"\nfor\n(43)","VOL. VI\nAIR-WATER SURFACE PROPERTIES\n236\nwhenever 5 is in E. We currently do not need two cases for\nN+(x,E), as in (1) and (2) for N+(x,E), because of the black-\nconvexification condition on the hydrosol side of S. We are\nnow retaining superscripts and subscripts on \"N\" to keep\nN+ (x,5) and conceptually distinct. The latter, of\ncourse, is governed by (18). Finally, (t) in (43) is ob-\ntained as in Figs. 12.57 and 12.58 by imagining the hydrosol\nand aerosol to be reversed in position relative to S(i.e.,\n\"reflected\" in S). Hence 41)-(43) are well defined. Next,\nwe apply the averaging operation (43) to each side of (40),\nand assuming the same stationary and independence conditions\non the various quantities in (40), just as was assumed in\nobtaining (18), we arrive at:\n(,5) = (x,5) + ds(s)\nwhere\nA_(x,5) = (X;E';5) +\n.dr((E)\nE E E , X E S\n(44)\nThis is the requisite equation governing the time-\naveraged surface radiance it(s) of the mean air-water sur-\nface S at point x with the time-averaged incident sky radiance\nNo(X,E) as the basic input. The term A_ (x,5) is a convenient\ncontraction of two terms, one of which is the integral of\nN°Q°T_ over E, and the other the integral of over E,\nas may be seen by comparing (44) with (40). In the first in-\ntegral N° is the time-averaged sky radiance over E. In the\nsecond integral No is the time-averaged upward hydrosol radi-\nance defined over E. In sum (44) is the time-averaged ver-\nsion of (39), with the radiances (,) as given in (18).\nEquation (44) is the companion formula to (18) and together\nthey specify N++ (S), (S) given No and No as defined, e.g.,\nin (38).\nUnder the same conditions leading to (26) we can deduce\nthat:\n(45)\n=\nwhere p(5',5) is evaluated, e.g., as in (24), but with the\nunit normal determined by requiring E',5 to be a Fresnel\ntransmittance pair of directions. In addition,\n(46)\n(X;E';E) = 5',5)r(E',5)\nwhich is the companion to (23). Observe that the is 1 in (46)","SEC. 12.12\nEQUATIONS OF TRANSFER\n237\nis to be incident from the hydrosol side of S, whereas E1 in\n(23) is incident from the aerosol side of S. A similar re-\nversal of location of 5' holds for (26) and (45) This in-\nformation is to be kept in mind when the Fresnel reflectance\n'(5', ,5) and the Fresnel transmittance (5',5) (as given in\nSec. 12.1) are used for computations of (,), N+(X,5) us-\ning (18) and (45). Furthermore p(s), 5) in (45) is computed\nfrom knowledge of the orientation of the outward unit normal\nn to S given the transmittance pair 5',5. The equation\nwhich determines n from E' and & in the Fresnel transmittance\ncase is:\nn'E'-nE\nwhich follows from (4) of Sec. 12.1. On the other hand\np(5',5) in (46) is evaluated when n is obtained from the re-\nflectance pair E' E, by means of:\nE-E'\nn\n=\n15-5'\nwhich\nis\n(1) of Sec. 12.1. In the case of statistically sta-\ntionary air-water surfaces \"X\" may be dropped from the nota-\ntion in (44) .\n12.12 Instantaneous and Time-Averaged Radiance Fields Within\na Natural Hydrosol\nThe integral equation and integral representation for\nthe time-averaged radiance distribution over the dynamic air-\nwater surface, as given in (18) and (44) of Sec. 12.11, will\nhere be supplemented with a description of the time-averaged\nradiance field below the mean surface S. In this way the\ntheory of the time-averaged light field within natural hydro-\nsols is made completely self-contained and can be reduced to\na steady state plane-parallel medium problem. For, as a study\nof the derivation of (18) and (44) of Sec. 12.11 would show,\nthe derivations began with the assumption that the initial\nsky radiance distribution N° was given over S, along with the\nwhite and black convexifications of the surface S so that the\nmultiple interreflection process over S could be isolated and\nstudied by itself. In reality, however, the incident radi-\nances N° (x,E,t) on S from below (i.e., 5 in E+ (x,t)) are in-\ntimately related to the fully self-interreflected radiances\nover S. Hence in a very definite sense, the input radiances\nN° (x,E,t) with E in E+ (x, t) leading to N(x,5) are dependent\non the answer N(X,5) sought. It is precisely at this point\nthat the power of the principles of invariance or, more gen-\nerally, the interaction principle, becomes manifest. For by\nblack convexifying the lower surface of S we could defer this\ncomplication of interactions between S and the body X of the\nnatural hydrosol until the present stage of analysis. In the\nstage now to be considered, we imagine the dynamic air-water\nsurface S peeled off the hydrosol leaving only the body X of\nthe hydrosol. The instantaneous output of S (after full in-\nterreflections) will now be the instantaneous input to X at","238\nAIR-WATER SURFACE PROPERTIES\nVOL. VI\nevery point X on S at time t. And, conversely, the instan-\ntaneous output of X will be the instantaneous input to S.\nThe solution of the time-averaged light field problem\nas outlined above will be developed in the following three\nstages: First we shall introduce two kinds of time-averaged\nradiance fields. The equation of transfer will be derived\nfor each of these fields in the second stage. Each field\nperforms a certain descriptive task and when these fields\nare linked together in the third stage, they will form a\nbridge between the theory and practice of measuring time-\naveraged light fields in the sea.\nTwo Types of Time-Averaged Radiance Fields\nThe measurement of time-varying light fields in lakes,\noceans, and other natural hydrosols is most conveniently\naccomplished at an arbitrary fixed depth Z below (or above)\nthe mean surface S. On the other hand, it was found conven-\nient in Sec. 12.11 to describe the time varying light field\nover the dynamic air-water surface S at points fixed on and\nmoving with S. Hence the time-averaged radiances N(x,5) in\n(18) and (44) of Sec. 12.11 are averages obtained by moving\nwith the surface in a certain perscribed way (Figs . 12.57-\n12.59). The empirical and theoretical descriptions of time-\naveraged light fields therefore adopt two distinct types of\naverages. The empirical averages may be defined formally by\nwriting:\n1\n\"N(z,E)\"\nfor\nlim\nN(z,E,t)dt\n(1)\nT\nin which Z and E are fixed ( < Z < 00). Further for the\ntheoretical averages, we write:\n1th\n1\n\"N(5',5)\"\nN(5'+5(t),5,t)\nfor\nlim\ndt\n(2)\nT\nin which 5' and & are fixed (5' > 0). For 5' = 0, N(5',5)\ntakes the form (2) of Sec. 12.11 for E in E. The geometric\nsetting for (1) and (2) is given in Fig. 12.62. (t) is the\nelevation of S above S. (t) and all other distances are mea-\nsured positive downward. For the purposes of the following\ndiscussion, we say that N is a fixed depth average, and N a\ncosurface average. Thus (18) of Sec. 12.11 is an integral\nequation for the cosurface average N(0,5), EEE, by virtue of\nthe stationarity condition. For simplicity of notation the\nhorizontal X and y coordinates have been suppressed in the\nvarious functions above, and these variables are also fixed\nin (1) and (2). The point X on S has the representation\n(x,y,0). The X and y coordinates are eventually averaged\nput in statistically stationary media, the media most common-\nly adopted for discussion in practice, and which we shall as-\nsume throughout this discussion. It is to be noted that the\ncosurface average for depths below § is simply a mathematical","239\nSEC. 12.12\nEQUATIONS OF TRANSFER\nNo(E)\n+\nS\n5(t)\nX\n5'\nNg(E)\n(t)\ndistances positive\ndownward from S\nFIG. 12.62 Setting for the definition of fixed-depth and\ncosurface time averages (vertical path case).\ndevice for facilitating the present averaging computations.\nAnother possibility would be to solve the hydrodynamic equa-\ntions (29) and (30) of Sec. 12.3 subject to suitable boundary\nconditions, and to make explicit use of the solutions.\nEquations of Transfer for Time-Averaged\nRadiance Fields\nWe begin with the derivation of the equation of trans-\nfer for fixed depth averaged radiance fields, and to see the\nessential idea of the derivation, the vertical downward case\n(i.e., , us = -k) will be considered first. Thus for the mo-\nment, E, as in Fig. 12.62, will be directed vertically down-\nward. At any time t, the radiance N(z,E,t) consists of two\nparts (cf., (2) of Sec. 3.15) namely the residual radiance\ntransmitted over the path through X from S to depth z, and\nthe path radiance built up over the same path. However, be-\ncause the depth Z has been fixed, it is possible for a wave\nto pass which is so deep that a point at depth Z would lie\ncompletely exposed to the atmosphere. In such an event the\nradiance reading is vi(5), and is either the surface radi-\nance of S or the zenith radiance of the sky as seen from\nabove the surface. When N° (5) is the surface radiance of S,\nit is given by (12) of Sec. 12.10. See inset of Fig. 12.62.\nTo anticipate this case we must devise a means of describing\nuniformly the situations when the depth Z is either exposed\nto the atmosphere or not. Thus let G be a function of a real\nvariable with the property:","AIR-WATER SURFACE PROPERTIES\nVOL. VI\n240\n0\nif\nx < 0\nG(x) =\n(3)\n1\nif\nX > 0\nThen from two applications of (2) of Sec. 3.15 for the cases\nZ - 5(t) > 0 and Z - 5(t) < 0, we have:\nN(z,E,t) = N°(5(t),5,t)T z-5(t) (t)(5(t),)G(z-(t))\n+\n12 $(n,5,t) (n,E)G(n - 5(t)) dn (4)\n+\nThe initial radiance N°(5(t),5,t) is the instantaneous radi-\nance of the lower part of S in the direction E at time t, and\nas given in (40) of Sec. 12.11.\nThe case of the inclined downward path (5 in E_) now\nfollows readily by measuring all distances parallel to the\nline along the direction E and going through a fixed point X\non S, as shown in Fig. 12.63. This figure is labeled analo-\ngously to Fig. 12.61. To emphasize that the distances for\nk\nE.\nS\nS\nn\nX\n5'\nZ\n0\nFIG. 12.63 Setting for the definition of fixed-depth and\ncosurface time averages (slant path case).","SEC. 12.12\nEQUATIONS OF TRANSFER\n241\nbeam transmittance are measured parallel to E, \"z\" is re-\nplaced by \"r\" and \"s(t)\" by \"5' (t)\", as shown. Equation (4)\nmay be converted to the slant path case by introducing these\nnotational changes. In a similar manner the case for & in\nE+ can be constructed. However for our present purpose it\nis unnecessary to go into these cases in detail since they\nall are neatly expressible in the associated integrodifferen-\ntial equation for N(x,E,t):\ndN(z,E,t) dr a(z)N(z,E,t) + N+(z,E,t)]\n(5)\n-\nwhere Z is any depth or altitude, - 00 Z < is any direc-\ntion in E and s(t) is the elevation of S with respect to S\n(with \"x\", \"y\" omitted throughout this discussion for brevity).\nThe last term is the net contribution to N(z,E,t) by the radi-\nance of the surface S; and this contribution is confined to\nthe \"paper thin\" thickness of S, hence the presence Dirac-\ndelta function s(z - 5(t)). It may be verified that, * for a\ngiven path Qr (x, E) with endpoints in or out of the medium X,\nthe appropriate integral of (5) is (4) or its slant path\ncounterpart as the case may be. It should be noted that (5)\nis a quasi-steady state form of the equation of transfer, and\nnot a pure time dependent form as explained in the discussion\nof (7) of Sec. 12.10. The fixed-depth average form of (5) can\nnow be taken, on application of definitions (1), and (2), and\nthe invocation of the independence condition (10) of Sec. 12.11\nfor the four functions N°, N, G, and 8; the result is:\ndN(2,5)T(z)-a(z)N(z,) dr + N+(2,5)] + 3(2)[N°(0,5) - N°(5)]\n(6)\nwhere we have written:\n\"G(z)\"\nG(z - 5 (t))\nfor\nlim\ndt\n(7)\n,\nand\n*Mainly what is required here is knowledge of the iden-\ntity d(f(z)G(z))/dz = (df (z)/dz)G(z) + f (0)8(z), where G is\ngiven in (3) . , S is the Dirac- delta function, and f is any\ndepth dependent function. Recall also the derivation of (3)\nof Sec. 3.15, and note that [G(z)] = G(z); [1 - G(z) ]G(z) =\n2\n0;\nand in general G(y)G (z) = G(y) for y < 2. (The latter prop-\nerty yields the preceding two.)","VOL. VI\nAIR-WATER SURFACE PROPERTIES\n242\ndt\n(8)\n\"(z)\"\nfor\n,\nN°(2,5) is the fixed-depth time-average of N * (z,E,t) and is\neasily seen to be related to N(z,5) in the expected way:\n=\n(9)\n=\n.\nThe averaged radiance (0,5) is, by (2) (in which 5' = 0),\nthe cosurface time-averaged radiance and therefore by the\nstationarity assumption is governed by (44) of Sec. 12.11\n(in which X = (x,y,0)). In the case of an air-water surface\nS whose elevations 5(t) are governed by a gaussian distribu-\ntion (cf. (12) of Sec. 12.9) we have: (z) = (2) and:\n(10)\nEquation (6) is the requisite equation of transfer for the\nfixed-depth time-averaged radiance function N. Observe that\n(6) is a steady state equation whose radiance function is de-\nfined for every Z in the depth range [-00,00] and that (6) has\nthe gestalt of an equation of transfer applied to an infinite\nplane-parallel medium with continuous source Nn (z) = T(2)\n[N (0,5) - N (5) ], < Z and in which the volume atten-\nuation function is Ga and the volume scattering function is\nGo. It follows that the theory of Chapter 7, and in particu-\nlar that of Sec. 7.13, is directly applicable to the present\nsetting and can in principle determine N ( 2, E) for all 2 in\n[-00,00] and all & in E, given the \"internal source\" Nn(2)\n(with EEE) and the sky radiance N°(E) (with E E E_) as a\nboundary condition.\nThe cosurface time-averaged radiance field has an equa-\ntion of transfer obtained by applying (2) directly to the\nequation of transfer for N(5'+5(t),5,t):\ndN(5',5) = a(5')NC5',5) + N+(5',5)\n(11)\ndr\nwhere the connection:\n(12)\n=\nis readily established using the independence condition (10)\nof Sec. 12.11. Equation (11) is vastly simpler to work with\nthan (6) and indeed the entire theory of radiative transfer","SEC. 12.12\nEQUATIONS OF TRANSFER\n243\non stratified source-free plane-parallel media is available\nfor the solution of the problem of determining NC5', 5) for\n5' in [0,00] and all E in E. The boundary radiance of (11)\nis N(0,5) as governed by (44) of Sec. 12.11. Therefore by\nvirtue of (18) and (44) of Sec. 12.11, and (11) above, the\nproblem of the time-averaged radiance field over a dynamic\nair-water surface has been reduced to the case of the static\nair-water surface considered in Sec. 12. and all results\nof that section may now be transferred, mutatis mutandis, to\nthe present context for the cosurface radiance field N in a\nmedium whose upper boundary is S.\nConnection Between Fixed Depth and\nCosurface Time-Averaged Radiances\nBy observing on the one hand the relative ease with\nwhich the mathematical problem of the time-average radiance\nfield is solved by adopting the cosurface average N and its\ngoverning equation (11), and observing on the other hand the\nfact that the experimentally determined average light field\nN is more conveniently expressed in terms of fixed depth aver-\nages, we are led to seek a connection between the two averages\nN and N.\nConsider a vertical path as in (a) of Fig. 12.64 and the\ndifferential of path radiance N*(z,E,t) at fixed depth Z at\ntime t as generated by the path function value N, (5' + 5 (t) ,5,t)\nover the differential path length d5' with fixed cosurface\ndepth 5':\ndN*(z,E,t) = d5' . (13)\nThe presence of G(y(t)), (cf. (3)) takes into account the pos\nsibility that depth Z may lie above the water surface at time\nt. Applying the time averaging operator to each side of (13),\nand using (10) of Sec. 12.11, we have, by (1) and (2):\ndN*(z,e) = N*(5',)(z-5')as'\n(14)\nwhere we have written:\n\"0(z-5')\" for\n(15)\nThe difference Z-5' arises because of the defined form of y (t)\nand the fact that under random conditions:\n= Z-3'\nso that the time average (15), under random conditions on S,\nwould depend only on the difference Z-5' From (14) we have\non integrating over all cosurface depths 5':","VOL. VI\nAIR-WATER SURFACE PROPERTIES\n244\nk\n(b)\n(a)\n+\nS\nS\nZ\n5'\nobservation\nd5'\npoint\ny(t)=-z+(5'+5(+))\nZ\nd5'\ny(t)=z-(5'+5(+))\nobservation point\ndepths measured positive\ndownward from S or S\nFIG. 12.64 Deriving the connection between fixed-depth and\ncosurface time averages.\nN*(2,5) =\n(16)\nwhich is the desired connection between the two time- eraged\nradiances N* and N*.\nThe connection between the time-averaged residua radi-\nances is obtained by reconsidering the general form of the\nargument leading to (4), but now within the setting (a of Fig.\n12.64: The instantaneous connection for downward radia ce is:\nN° (2,E,t) = No(5(t),,t)G(z-(t))T (2-5(t)(5(t),5)\n(17)","EQUATIONS OF TRANSFER\nSEC. 12.12\n245\nAveraging this, results in:\n= + N°(5)[1-G(z)]\n(18)\n.\nIn this way we come to the requisite connection (for & = - k):\nN(z,5) = N°(2,5) + N*(2,5)\n=\n(19)\nIn a similar manner the upward radiance connections (i.e.,\n& = k) can be made (see (b) of Fig. 12.64). The requisite\nform is:\nN(z,5) = N°(0,E)G(-z) +\n(20)\nThe asymmetry between (19) and (20) arises from two sources:\nfirst there is the usual asymmetry indigenous to deep plane-\nparallel media (such as those now being considered) where\nthere is incident external downward radiance, but no such up-\nward radiance. This accounts for the term N°(E) [1-G(z)] in\n(19) and its absence in (20) Secondly, we agreed at the out-\nset of the present study (in Sec. 12.10) to white convexify\nthe upper surface of S. This is tantamount to assuming that\nthe atmosphere is relatively transparent compared to the hy-\ndrosol, so that Tr = 1 in air. This assumption is quite\nrealistic for distances r on the order of ten moderate grav-\nity wave lengths. Under such conditions (z) = G(z), as may\nbe verified from (15). This accounts for the G(-z) rather\nthan $ (-z) in the residual radiance term in (20). The minus\nsign and the use of 5'-Z rather than Z-5' in the integral term\nof (20) reflects the change of direction in going from (19) to (20).\nWe note in passing that a completely symmetric formu-\nlation of the present problem is possible if both the upper\nand lower surfaces of S are black convexified at the outset\nand the aerosol and the hydrosol are both treated exactly\nalike--namely - as two contiguous general scattering-absorbing\nmedia with an interface. (See Example 7 of Sec. 3.9. ) We\nshall leave such a formulation to interested students of the\nsubject. The derivations above are readily extended to cover\nsuch a symmetric formulation. Such a formulation is useful\nwhen a dense layer of fog rests on a dynamic air-water sur-\nface and the radiative transfer interaction is required in an\ninstantaneous or averaged form. We have chosen the present\nroute to (19) and (20) because of expository reasons and be-\ncause they represent the more frequently occurring situation.","VOL. VI\nAIR-WATER SURFACE PROPERTIES\n246\nIn the event that the air-water surface is to be viewed\nfrom a great height (aircraft and satellite heights) then one\nmay use N(0,5), with E in E+, as determined in the manner\nshown in Sec. 12.13, as the time-averaged inherent radiance\nof the surface and then go on to compute its apparent radi-\nance in the usual manner for a given path of sight.\nEquations (19) and (20) are particularly designed to\ngive the exact time-averaged radiance field in the immediate\nvicinity of the air-water surface, in particular over the\ndepth region with 2m above and below S, and generally at\ndepths in the hydrosol where bright moving beams of refracted\nsunlight are still observable.\nOne may generalize (19) and (20) from the vertical up-\ndown case to the case where & is arbitrary nonhorizontal\npaths by merely replacing y (t) in Iv(t): and G(y(t)) by - (sec\n0)\n(t) in (19) and (20) to account for the slant path attenu-\nation (see Fig. 12.63). A11 other terms are unaffected by\nvirtue of the assumed statistically stationary character of\nthe motions of the hydrosol X and its boundary S. Hence (19)\napplies to all directions & in E. and (20) to E in E+. Fur-\nthermore, by invoking the ergodic hypothesis we may represent\n(2-3') for a path directed along E, by:\n(sgn(r) (z-s')) = (\"G(I)T\n(21)\n(5',5)dt'\nwhere we have written:\n\"r\" for - sec 0 (z-s')\nand where & k = cos 0. \"sgn (r)\" means the algebraic sign\nof r, namely + or -. The beam transmittance Tr(5),5) is\nreadily evaluated in homogeneous media. In fact!\n(22)\nand the time-average G(r) may be evaluated by means of (14)\nfor seas with gaussian elevation distributions. Hence in\nsuch practical settings $ (z-5') is known. This completes\nthe establishment of the connections between fixed-depth and\ncosurface averages of the radiance distributions beneath a\ndynamic air-water surface.\nAs a check on the connections (19) and (20) we can let\nthe wind speed U go to zero, so that m 'oo' and o go to\nzero. Consequently $ (z-5) goes to Tz-5', (z) for Z > 0\ngoes to 1, and G(-z) goes to zero for every positive z; and\nso (19) and (20) reduce to the integral forms of the equa-\ntion of transfer for the static case (re: Sec. 12.2).\n12.13 Synthesis of Time-Averaged Radiance Fields\nThe complete description of the time-averaged light\nfield in a natural hydrosol with a wind-blown air-water sur-\nface will now be attained by gathering together the various\npieces of the description fashioned in the preceding three\nsections.","SEC. 12.13\nSYNTHESIS OF RADIANCE FIELDS\n247\nThe synthesis to be given is facilitated by casting\nequations (18) and (44) of Sec. 12.11 into operator form.\nToward this end, and with (18) of Sec. 12.1 11 in mind, let us\nwrite:\n\"R°(S)\" for (5)\n(1)\n\"T+(s)\" for S (5)\n(2)\nwhere we have dropped reference to the point XX in the mean\nsurface §. For example, \"S(E)\" is therefore a contracted\nname for S(x,5), defined in (18) of Sec. 12.11. Furthermore,\nwriting:\n.\n\"R (s) for S (5) m (3)\nwe can then write (18) of Sec. 12.11 as:\n(S) =NOR_(S) + + (S)R_(S)\n(4)\nwhere \"N+(S)\" denotes the function N governed by (18) of\nSec.\n12.11, with the subscript \"++\" as a reminder that N is the\naveraged response of S over E+(x,t). The surface radiance\nsuperscript is now dropped as being understood. N° is the\ntime-averaged sky radiance distribution (defined over E-)\nand N+ (X) is the time-averaged response of the body X of the\naveraged hydrosol whose boundary is S. Thus N+(X) is the\ntime-averaged radiance function N° over E+ (x,t) occurring in\n(12) of Sec. 12.10 and in (40) of Sec. 12.11. (Recall the\nn2 -convention stated at the outset of Sec. 12.10.)\nIn a similar manner (44) of Sec. 12.11 can be cast into\nthe form (again all radiances being surface type radiances,\nthe superscript \"++\" may now be dropped)\nN°T°(S) +\n(5)\nwhere we have written:\n\"R+(S)\nfor\n(6)","VOL. VI\nAIR-WATER SURFACE PROPERTIES\n248\n(7)\n\"T°(S)\"\nfor\n\"T_(§)\" for Is\n(8)\n.\nThe functions R+, T+ + are defined in (23), (26), (45), and\n(46) of Sec. 12.11.\nBy virtue of the conclusions following (11) of Sec.\n12.12 we have:\n(9)\nwhere R_ (X) is the time-averaged reflectance operator for\nthe medium X. This operator equation is obtained by apply-\ning the steady state theory of radiative transfer in plane-\nparallel media, with appropriate modifications, to the time-\naveraged equation of transfer (11) of Sec. 12.12, defined in\nthe medium with upper boundary §. The salient modification\nrelative to the static case is that R_ (X) is an integral\noperator with a representation of the form:\n(9a)\nR =\nwith E in E. In other words the integration is over all of\n(1) rather than just E_ (as is the case for R(a,b) in the\nstatic theory of Chapters 3 and 7). Furthermore, the re-\nsponse of X can be along every E in E. The visualization of\nthis new situation is quite easy, after the derivations of\nSecs. 12.10-12.12 have been thoroughly assimilated. Other-\nwise, the theory of R_(X) (and the other three time-averaged\noperators R+(X), (X)) is exactly analogous to the static\nplane-parallel case and is essentially as developed in Chap-\nters 3 and 7. The further exploration of this aspect of the\ntime-averaged theory will be left to future students of the\ndiscipline of radiative transfer.\nWe continue the present synthesis by showing that the\nthree equations (4), , (5), and (9) may be formally solved to\nyield the three radiance fields (s) and N+ (X). . By (4) we\nhave:\n(10)\n+\nwhere we have written:","SEC. 12.13\nSYNTHESIS OF RADIANCE FIELDS\n249\n\"R_(s) for R°(S)[I - R (S)]\n(11)\nfor T°(S)[I - R_(S)]`\n(12)\nFurthermore, (5) and (9) may be combined to yield:\n(_s) = N°T°(S) + N (S)T_(S) + N_(S)R_(X)R (S)\nwhich in turn yields the following formal solution for (s):\nin terms of N+(S) and No:\nN_(s) N°T(s,X) + ()J_(s,X)\n(13)\nwhere we have written\nfor -\n(14)\n\"J_(s,X)\" for -\n(15)\nThe operator (15) acts like a transmittance operator for\nN+ (S) because N+(S) is the averaged surface radiance of S\nleaving S and descending down onto S to be transmitted\nthrough S and hence to contribute to N+ (S). With the help\nof (9), we can write (10) so that, like (13), it involves\nonly the inputs No and N (S) to S:\n+ N_(S)Q_(s,X)\n(16)\nwhere we have written\n\"R_(S,X)\" for R (X)(\n(17)\nWe have reached the penultimate step in the formal\nsolution of the time-averaged surface radiances N+(S), N-(S).\nEquations (13) and (16) are a system of operator equations in\nthe requisite unknowns N+(S) with given R and T operators and\ngiven input radiance No associated with the sky radiance dis-\ntribution. These equations can be solved formally on the\noperator level in the manner illustrated repeatedly through-\nout Chapters 3 and 7. The results are:\nN_(S) = NOT =\n(18)\n(19)","VOL. VI\nAIR-WATER SURFACE PROPERTIES\n250\nwhere we have written\n\"J\nDo(s,8)\n(20)\nfor\nRo(s) + T Q (s,X)\n(21)\n\"\nfor\nEquations (18) and (19) completely solve, in principle,\nthe problem of the time-averaged radiance distribution of\nthe dynamic air-water surface irradiated by skylight No\n.\n(Recall the n°-convention stated at the outset of Sec. 12.10.)\nThe operator J is a general time-averaged complete trans-\nmittance of S, and R is a general time-averaged complete re-\nflectance of S, where the inner structure of R- and J. is\ncompletely determinable by retracing the thread of reasoning\nbeginning with (20) and (21) and working back to Sec. 12.10.\nComparison with the Static Case\nIn closing, it is of inerest to compare (18) and (19)\nwith the representation of the radiance of the static air-\nwater case given in (13) and (14) of Sec. 12.2. (At this\npoint the reader should recall the n2-convention for radi-\nances stated at the outset of Sec. 12.10.) The operator\nT(1,0,2,) in (16) of Sec. 12.2 is a special case of T OC-\ncurring in (18). Furthermore, the operator@(-1,0,2, in (17)\nof Sec. 12.2 is a special case of R- occurring in (19). This\noperator, when applied to No, yields N+(S). The similarity\nin the structure of these operators with their static counter-\nparts is quite striking and is traceable, of course, to the\ninteraction principle underlying all algebraic descriptions\nof radiative transfer phenomena. The reader will find it\ninstructive to show that J. and R- reduce exactly to (16)\nand (17), respectively, of Sec. 12.2 as the dynamic air-water\nsurface S continuously approaches S. This may be done for\nexample by adopting the Neumann spectrum model for S (Sec.\n12.8), letting Ua 0, (i.e., letting the equilibrium wind\nspeed go to zero) and using the gaussian representations of\nthe weighting functions Q, Q°, Q+, Q-. Or it may be done by\nsimple intuitive considerations on the necessary properties\nthe Q-functions must have for any reasonable model of the\nair-water surface, as the air-water surface continuously ap-\npraches the static plane form.\n12.14 Observations on the Theory of Time-Averaged Radiance\nFields for Dynamic Air-Water Surfaces\nThe theory of time-averaged radiance fields developed\nin the preceding four sections contains a great variety of\nspecial cases of practical interest in applied hydrologic\noptics. It is the purpose of this section to classify and\ndiscuss the main set of these special cases and to indicate\ntheir use in practice.","SEC. 12.14\nOBSERVATIONS ON THE THEORY\n251\nA Hierarchy of Approximate Theories\nStarting with the basic equations (12) of Sec. 12.10\nand (40) of Sec. 12.11, we may construct a sequence of ap-\nproximate theories of the time-averaged light fields for the\ndynamic air-water surface of natural hydrosols. As one pro-\nceeds along this sequence, the descriptive powers of the\nmodels decrease and the numerical tractability increases.\nThe main sequence of approximations is as follows:\n(i) The Exact Time-Averaged Theory. Starting with\n(12) of Sec. 12.10 and (40) of Sec. 12.11 with N° given along\nwith the prescribed motion of the air-water surface S, the\nradiance N(x,E,t) at point X on S in the direction E for every\nt in some interval (0,T) of time may be computed. The result-\nant set of radiances may then be averaged to obtain a numeri-\ncal estimate of N(X,5) which is, in principle, exact in the\nsense that no special assumptions are used to obtain the\naverages.\n(ii) The Statistical Time-Averaged Theory. The theory\nsummarized in (18) and (44) of Sec. 12.11 and in the equations\nof Sec. 12.1 is based on certain observed or theoretical sta-\ntistical regularity properties of the dynamic air-water sur-\nface such as the stationarity condition (5) of Sec. 12.11, the\nindependence condition (10) of Sec. 12.11, and the ergodic\nhypothesis (31) of Sec. 12.11 (which allows space averages to\nbe interchanged with time averages). This theory contains\nthe effects of the radiometric interaction of S with itself\nand with the body of the hydrosol in addition to the hiding\neffects of finite wave slopes and finite wave heights from\nthe sky radiance distribution.\n(iii) The Wave-Slope, Wave-Height Time-Averaged Theory.\nThe statistical time-averaged theory in (ii) is made some-\nwhat more tractable but less descriptive by dropping the\nself-interaction term in (18) of Sec. 12.11. The result is:\nN(X,5) = S(X,5)A,(X,5)\n(1)\nfor all E in E, and which is a straightforward integral rep-\nresentation (and no longer an integral equation) of N(X,5)\nwhich may be evaluated once Ñ, Q°, Q+, R_, T+ are prescribed.\nIt is clear that the effects of finite wave heights and fi-\nnite wave slopes are still included in this approximation.\nFrom (19) of Sec. 12.13, the upward time-averaged radiance\nof S is given in the statistical theory by\n(2)\nIn the present wave-slope, wave-height mode, R. takes\nthe form:\na = Ro(S) + °S [I - R (X)R (S)] R_(X)I(S)\n(3)","AIR-WATER SURFACE PROPERTIES\nVOL. VI\n252\nThat is, the effect of dropping the self-interaction\nterm is to require:\nR (S) = 0\n(4)\nT (§) = 0\nwhence, in particular\na_(S) = R°(S)\nTo(s) = T°(S)\n(5)\nT(S,X)\n(iv) The Wave-Slope Time-Averaged Theory. The wave-\nslope, wave-height time-averaged theory of (iii) is further\nsimplified by neglecting effects of the wave heights of the\ndynamic ir-water surface. Thus the immediate effect is to\nrequire q (x,5) = 0, whence by (35a) of Sec. 12.11 Q(X,5) = 0,\nand so that by (27) of Sec. 12.11 we now have:\nQ°(X,5) + 2+(x,5) = 1\nFurthermore, by the definition of S(X,5) in (18) of\nSec. 12.11, and (1) (,5) = 0 for & in E. Otherwise, the\ngeneral form of A+(X,5) is the same. The resultant theory is\nof great importance to optical studies of the dynamic air-\nwater surface since the distribution of wave slopes rather\nthan wave heights is more often the dominant physical deter-\nminant of the time-averaged radiance of air-water surfaces.\nFor example, as a fresh wind blows over an otherwise calm\nwater surface on a clear day, a fine dark patina quickly\noverspreads the windblown surface in which the higher parts\nof the dark sky are mirrored. On the other hand, when the\nobserver-sun geometry permits, a bright glitter pattern\nsprings into existence as the wind produces a great range of\nwave slopes of capillary waves. In such transient settings\nthe overriding importance of the wave slope state of the sea\nis clear.\n(v) Partial Time-Averaged Theories. Occasionally it\nis of interest to study the reflected or transmitted radi-\nance from a dynamic air-water surface S which is due only to\nthe light of the sun or only to the light of the sky or only\nto the radiance incident on the underside of S from the hy-\ndrosol proper and in which self-interaction, the wave-height\nand wave-slope effects may or may not be included. Equations\n(5) and (9) of Sec. 12.13 summarize these component radiances\nin clear intuitive form. For example, (S) in (5) of Sec.\n12.13 describes the inherent radiance of the lower side of\nthe time-averaged surface § as being composed of transmitted\ntime-averaged sky light No, self-interaction light N+(S),\nand reflected time-averaged hydrosol light, N+(X), and is\ntherefore, of the same general form as the static radiance of\ncurved surfaces considered in Example 1 of Sec. 3.5. Similar-\nly, N+ (s) in (10) of Sec. 12.13, namely the upward time-aver-\naged radiance of §, is seen to be composed of reflected sky","SEC. 12.14\nOBSERVATIONS ON THE THEORY\n253\nlight and transmitted hydrosol light. Any one of the five\nterms on the right of (5) or (10) of Sec. 12.13 may be iso-\nlated for the study of N+ (S) in a partial time-averaged\ntheory.\nIllustrations of Some Classical Partial Theories\nSome of the features of the theory of time-averaged\nlight fields as developed above can be illustrated by select-\ning for examination each of the four terms in (18) and (19)\nof Sec. 12.13 By doing so we shall also deduce some of the\nclassical instances of the theory of time-averaged reflected\nand transmitted radiance at the surface of natural hydrosols.\nWe begin with the component of the time-averaged upward\nradiance N+ (S) of S contributed by reflected sky and sunlight.\nThus from (19) of Sec. 12.13 we consider NoR (S) For sim-\nplicity of exposition,we shall omit the self-interactions ef-\nfect of S so that a. (S) = R_(S), which follows from (5).\nHence from (1) of Sec. 12. 13 and the present selection of\neffects:\nNCE)\n=\n,\nwith E in E+ and where we have dropped reference to the\npoint X in S, the uniformity of optical properties over all\nof S being assumed.\nThe factor Q°(5') in the integrand of (6), as defined\nin (11) of Sec. 12.11, takes into account the shielding\neffect of the wave heights and wave slopes against the light\nof the sky and sun. 2°(5') may be evaluated using, e.g.,\n(27), (29), and (35) of Sec. 12.11. Even when wave heights\nare considered negligible (i.e., q-(x,5) = 0), so that Q(5)\n= 0, we see from (29b) and (35b) of Sec. 12.11 and (6) that\nwave slope shielding may be in effect. By means of (23) of\nSec. 12.11, we may represent N(5) in greater detail as follows:\nÑ(E)\n(7)\n=\nwhere (5' 5) is the value of the Fresnel reflectance func-\ntion for incident and reflected directions E'E , respectively,\nin E. and E+. The function p(5', 5) is the probability that\nthe wave normal to S is such that the incident and reflected\ndirections E' and E occur for Fresnel reflection. If the\nsea surface is statistically gaussian, then either (24) of\nSec. 12.11 may be used or its anisotropic generalization (28)\nof Sec. 12.5, after an appropriate change of variables in (7),\nif desired, from direction space E to slope space 2 (i.e.,\nthe two-dimensional Euclidean plane).","AIR-WATER SURFACE PROPERTIES\nVOL. VI\n254\nAs a specific illustration of (7), suppose that the\nsun is of uniform radiance N° and positioned such that the\nradiance of its center is directed along 50, and that its\ndirection set is E0 CE. Then since SEE) is quite small,\nwe may approximate the contribution to N(E) by sunlight, by:\nÑ(5) =\n(8)\n.\nA plot of Ñ(5) as a function of E will, therefore, yield a\nradiometric map of the sun's glitter pattern over E+. If\np(50,5) is given by (24) of Sec. 12.11 or (28) of Sec. 12.5,\nthen the glitter pattern is seen to be a function of the\nroot mean slope o, and, owing to the general presence of\nQ°(5)), to the root mean elevation 00 It is, in fact, quite\npossible to use (8) to form a systematic tabulation of the\nvalues:\n(9)\nIII\nas a function of Ej, 5 and the statistical sea surface param-\neters o and moo. Then, when the radiance distribution of the\nsky hemisphere E- is known over a partition {E1,\nEnd\nof E. with central direction Ej, j = 1, .... n, for each Ej,\nwe can compute:\n(10)\nwhich is a discrete version of (7). A different kind of ap-\nproximate formula for computing reflected sunlight from a\nroughened water surface is given by Cox and Munk in [56].\nEquation (10) also may be used to estimate the time-averaged\nradiance of an air-water surface which has an extraordinary\nground swell under the usual statistically stationary jumbled\nsurface. This would be accomplished by literally tilting the\ncoordinate axes used throughout the preceding derivations so\nthat the previously horizontal reference frame now lies in\nthe rising or descending part of the swell. Computations then\nproceed as usual in this new frame of reference. With some\ncare, the additional shielding effect due to the swell's geom-\netry can be included to supplement the Q° factor shielding\neffect.\nAs another example of a classical representation of\n(s) we consider the upward transmitted component N+ (X).\n(S) in (10) of Sec. 12.13. Once again, for simplicity, the\nself-interaction effect is omitted so that T+(S) = T+(S) and\nwe have for the case at hand:\n(11)\n=\nwhere E is in E and where N° (5') is the radiance of the hy-\ndrosol incident on the lower side of § in the direction 5'.","OBSERVATIONS ON THE THEORY\nSEC. 12.14\n255\nE=1\nPt\nPt\nD\nb\nS\n41.4°\nI\nN\nNb\nD\nFIG. 12.65 Setting for the classical apparent contrast\nreduction formula by time-averaged refraction at the air-\nwater surface.\nWe have momentarily set aside the n°-convention, stated at\nthe outset in Sec. 12.10, so as to avoid an accidental mis-\napplication of (11) by readers using only this discussion\nas a reference source (n for air is taken as n = 1) .\nEquation (11) can be unfolded layer by layer for par-\nticular applications just as we did with (6). Thus, by (26)\nof Sec. 12.11, we can write:\nN(E) = [ \"E]]\n(12)\nwhere E is in E.","256\nAIR-WATER SURFACE PROPERTIES\nVOL. VI\nAs a particular illustration of (12) let us estimate\nthe time-averaged - radiance in the vertical direction k for\nthe case where N°(5') is of the form:\nNt\nE'\nis\nin\nst\nN° (5') =\nif\nNb\nE'\nis\nin\nIf\n0\nif E' is in E\nwhere Et and Eb are the sets of directions which partition\nE+, shown in Fig. 12.65. The set of directions Et is sub-\ntended by a circular disk D of uniform apparent time-averaged\nradiance Nt and Eb is the set of directions of its background.\nAs a result of this choice of direction k, and form of No,\n(12) becomes:\n(5')\n(13)\n.\nNow the presence of the Fresnel transmittance factor in (13)\nis such as to limit the range of integration of E' in E+ to\na solid angle Ef of half angle 41.4° within which t (5', k) > 0,\nand outside of which t (5', k) = 0 owing to total internal re-\nflection. Hence the range of integration may be limited to\nEtnE+ and , when working toward actual numerical esti-\nmates of N(k). To see the general order of magnitude of N(k),\nlet us set:\nQ+(5')t(E',k) =\nfor all E' over E+, so that (13) becomes :\n(E',k)dd(E')\n.\n(14)\nThe integrals in this expression have very simple inter-\npretations: the integral over Et, for example, is the proba-\nbility that the normals to the wave slopes at point P in Fig.\n12.65 are tilted at less than the angle It from k, the angle\nfor which the refracted rays from the rim of the submerged\ndisk D emerge from the hydrosol along the direction k. (See\ninsert, Fig. 12.65.) Using the gaussian model of the sea\nsurface (24) of Sec. 12.11, we have at once from (17) of Sec.\n12.5","SEC. 12.14\nOBSERVATIONS ON THE THEORY\n257\ntan2t\n-\n1st\n202\n=1-e\n(15)\nand\ntan2ot\nSep\n(16)\nWith these representations, (14) may take the form:\ntan2ot\ntan &t\n(17)\nThis approximate representation of Ñ(k) is of historic inter-\nest, being essentially the result of the first calculation of\na time-averaged radiance viewed through a moving air-water\nsurface (cf. [82]) Equation (17) may be rewritten to use\nthe observed time-averaged radiances Nt, Nb seen just above\nthe surface S. For this purpose we use (18) of Sec. 12.1 but\nretain the present assumption that t (5',5) = 1, so that (17)\nbecomes:\ntan ot\ntan\not\nÑ(k) = + e 202\n(18)\nOne of the first calculations of contrast reduction by\ntime varying refraction was based on (18). Thus, as in [82],\nlet us define the time-averaged apparent contrast of the ap-\nparent radiance Nt of the center of the submerged disk with\nrespect to the background radiance Nb by writing:\nÑ(k)\n\"C\"\nfor\n(19)\n.\nThis is an extension of the standard notion of contrast, de-\nfined in Sec. 9.5, to the time-averaged case. The apparent\ncontrast C for the static case is, by definition of Sec. 9.5,\nsimply:","VOL. VI\nAIR-WATER SURFACE PROPERTIES\n258\n= -\n(20)\nHere we are implicitly assuming that the time-averaged radi-\nances Nt and Nb are the same as would be measured in the\nstatic case, i.e., , we are assuming that Nt = t/n2, Nb = /n2.\nIn order words, we are hypothesizing for the purposes of the\npresent discussion that the movements of the surface S do not\nchange the average values of the apparent radiances over the\ndisk D and over its background. With this assumption in mind\nwe can obtain a simple and useful connection between C and C\nusing (18)\ntan2 ot\nC [1 - - 202 2\n(21)\n=\nFrom this representation of C, despite the multitude of\nassumptions on which it has been based, we gain valuable quan-\ntitative information about the effect of the movement of an\nair-water surface on the apparent contrast of submerged objects\nviewed through that surface. Equation (21) is particularly\nuseful when the submerged target is angularly small and the\nwater surface is freshly crinkled with capillary waves. Then\nwe can readily see how C varies with Pt and o: the larger\nQt, for a given o, the larger C is, meaning the less the dim-\ninution of contrast by wave action. Thus small objects tend\nto be readily blended into their background by the water sur-\nface disturbances. On the other hand for a given target size,\nC goes down as o increases. For example, if we use the rela-\ntion\no2=YU a\nfor some constant (cf. (28) and (29) of Sec. 12. 5 and (11)\nof Sec. 12.8), (21) can be written as :\na\nC=C\n(22)\nwhich for small targets and fresh breezes over an otherwise\ncalm surface can be reduced to the rule of thumb :\n2\nC C ( tan 2YU\n(23)\n=","SEC. 12.14\nOBSERVATIONS ON THE THEORY\n259\nThis equation allows one to readily see the direct variation\nbetween pt and C, and the inverse variation between Ua and C.\nAs a final illustration of some partial representations\nof the time-averaged radiance N+(S), we consider (19) of Sec.\n12.13. Thus we are to consider (compare with (2)):\nOnce again let us omit self-interaction effects over\nS\nso that (4) holds. Furthermore we concentrate on the trans-\nmitted upward-flux term in the preceding relation, thus we\nconsider the second term of\n= + - (_)(I°(S)\n(24)\nThe interpretation of the second term of (24) is straightfor-\nward: given the incident radiance distribution No from sun\nand sky, operate on No with To (S) as given in (7). The re-\nsult is a time-averaged radiance distribution transmitted\ndownward across S. The latter distribution is operated on\nby the interreflection operator:\n[I -\nwhich may be approximated arbitrarily closely by computing\na sufficient number of terms of its representing infinite\noperator series (in practice two terms beyond the identity I\nshould be sufficient)\n[I -\n(25)\nwhere we write:\n\"Tj+1(s,x)\" for\n(26)\nand:\nT'(S,X)\" or \"T(S,X)\" for R (X)R (S)\n(27)\n.\nThe result is a time-averaged downward radiance distribution\nto be operated on by R_ (X)T(S). The reader may deepen his\nunderstanding of (24) by referring back to a related but\nmore readily visualized example in terms of irradiance. Thus,\nthe exact irradiance counterpart to (19) of Sec. 12.13 is\n(2) of Sec. 12.2 and, in particular, the interacting counter-\npart to (24) is the second term (2) of Sec. 12.2. This shows\nthat the algebraic structures of both the irradiance and\nradiance cases (in either the static or dynamic context) are\nidentical. The practical difference between these cases is","VOL. VI\nAIR-WATER SURFACE PROPERTIES\n260\nthat in the irradiance case one works with numbers; in the\nradiance case one works with functions and, in addition to\nextensive numerical details, the order of operations on func-\ntions in this case must be scrupulously observed.\nConcluding Observations\nOne of the two main goals of this chapter has been the\npair of equations (18) and (19) of Sec. 12.13. They describe\nthe time-averaged radiance distributions +(S), (S) directed\ninto the aerosol and hydrosol, respectively, above and below\nthe dynamic air-water surface. The main purpose of these\nequations is to form a solid mathematical and physical founda-\ntion for practical techniques of describing and predicting\nthe average radiance distributions of wind-blown seas, lakes,\nand other natural hydrosols. The preceding discussion of the\nhierarchies of approximate theories leading from (18) and (19)\nof Sec. 12.13 shows that this purpose has been adequately\nfulfilled. Thus we were able to deduce the two classical for-\nmulas for the time-averaged apparent radiance and apparent\ncontrasts of submerged objects as in (18) and (21). However,\n(18) and (19) of Sec. 12.13 allow us to go beyond these clas-\nsical formulas and establish (7) and, more completely, (24)\nwhich can take into account the shielding effects of wave\nslopes and wave elevations from sky - and sunlight without the\ncomplications of radiometric self-interactions of the dynamic\nair-water surface.\nThe complete description of the averaged light field,\nwith self-interactions, shieldings, and hydrosol light-fields,\nall in concert, as given in (18) and (19) of Sec. 12.13, has\nnot yet been subject to numerical applications. Before this\ncan be done, further study of the operator R_ (X) in (9) of\nSec. 12.13 must be made. To a first approximation we may use\nthe R(0,00) operators (with kernels R(0, 0; E'; E) ) in Sec. 7.6\ninstead of R- (X) However, interested students after careful\nstudy will see what more can be done here.\nThe theory of the optical properties of the air-water\nsurface has thus been brought to a level of completion sum-\nmarized by (18) and (19) of Sec. 12.13.\n12.15 Simulation of the Reflectance of the Air-Water Surface\nby Mechanical Devices\nWe conclude this chapter on optical properties of the\nair-water surface with descriptions of some mechanical devices\nwhich may be used to predict, with relatively little advanced\nmathematical computations, the average sun and sky radiance\nreflected from wind-blown water surfaces. The theoretical\ndiscussions throughout this chapter have demonstrated that a\nvirtually complete theory of radiative transfer across ran-\ndomly moving water surfaces can be formulated so that the\nreflected and transmitted radiances can, in principle, be\ncomputed in great detail. However, it is not always possible\nor economical to solve daily problems of applied hydrologic\noptics with such relatively powerful batteries of tools. In\nthe first and main part of this section we shall round out","SIMULATION BY DEVICES\n261\nSEC. 12.15\nour theoretical studies by describing in detail a practical\ndevice which predicts the reflected radiance distribution and\nwhich is based on the theory of reflected radiance from a sta-\ntistically uniform sea surface lighted by sky and sun. The\nmotivation for such a device is the need for a simple labora-\ntory means of predicting the apparent contrast of floating\nobjects with respect to the background sea, as seen by ob-\nservers aloft. The radiance of the sea forming the background\ndepends, as we have seen, in part on the radiance distribution\nof the sky above the observed point, and the spectrum of the\nwaves in the neighborhood of the point. Measurements of the\nreflected radiance distribution and spectra from aircraft are\npossible but occasionally inconvenient. The sea surface simu-\nlator described below was devised to permit laboratory mea-\nsurements of reflected radiance distributions under widely\nsimulated conditions of sky lighting and sea state.\nThe Central Idea of the Sea State Simulator\nThe geometric interpretation of the equation:\n1 tan 2 0\n2\nas = (1 - e\nof\nholds the key to the idea of the sea state simulator. As we\nsaw in the derivation of this equation (cf. (29) of Sec. 12.5)\na is the fractional horizontal area of a random sea surface\nO\nover which the wave normals are tilted at angles ''' in the\nrange (0, 0). -If, therefore, a fixed surface of finite later-\nal extent could be constructed so that the normals to it obey\nthe preceding geometric condition on then we would have a\nstationary surface whose slope properties, at least statis-\ntically, are identical with those of a random moving sea sur-\nface whose mean square slope is O2. More genrally, it seems\npossible to be able to construct a surface S such that the\nfractional horizontal area covered by points of S at which\nthe slope components are in the intervals (5x + (1/2)A5x),\n(sy + (1/2)Asy), , is given by:\n2\n([(s)\n1\nAS\ne\n2\nu°\nX\ny\nC\nThe construction of §S will now be described.\nErgodic Hypothesis\nFor the present discussion the sea surface is repre-\nsented by a real-valued elevation function 5 defined on the\nEuclidean plane E2 such that at each time t, and at each\npoint (x,y) of E2, 5 (x,y, t) is the associated elevation (now\nmeasured positive upward) of the sea surface with respect to","VOL. VI\nAIR-WATER SURFACE PROPERTIES\n262\nthe datum plane The slope components of the sea surface\nare then represented by the functions 5x = 25/ax, by = as/dy\ndefined on E2 for each time t. Let G be the Heavisíde func-\ntion:\nG(x) = if x 20 x < 0\nand let f be some given real-valued function on Then\nfor each point (5,n) in write:\nFurther, for each t and subset E of E2' let \"E\" denote the\narea of E, and write:\n\"RE(5x\"ty;t')\"\nfor\ndn'\n5 is said to be ergodic over ECE, during [0,T], T < 80, with\nrespect to f, if for each point (E,n) E E, each C,\nand t E [0,T],\n(1)\n5 is normally ergodic if 5 is ergodic and\nand\nQT(5xp5yi5,n) = Rg(5x+5yit) = _ exp exp\n(2)\nWe come now to the main definition of the present discussion.\nThe graph of a function 5 forms an ergodic cup (cap) on a sub-\nset E of E2 if and only if (i) E is compact, (ii) 5 is inde-\npendent of t and normally ergodic on E, and (iii) 5 is contin-\nuous and concave (convex) upward on E.","SEC. 12.15\nSIMULATION BY DEVICES\n263\nThe ergodic cups we shall consider will be concave (or\nconvex) stationary surfaces of arbitrarily small lateral and\nvertical extent which summarize the salient statistical prop-\nerties of the slopes of randomly moving irregular surfaces\nof large or possibly infinite lateral extent. In this discus-\nsion we will assume that the sea surface may be described by\na normally ergodic function on E2, 2' with parameters ou,oc.\nThe discussion will be concerned principally with the descrip-\ntion of the parametric equations of the simplest associated\nergodic cup (in principle there is an infinite variety of\ncups for each pair ou,o c). . The simulation problem is there-\nfore considered here only in terms of its statistical com-\nponents. Thus the question of whether an ergodic cup repro-\nduces all the optical properties of the original surface\n(i.e., those in addition to the mean square slope) is beyond\nthe scope of the following discussion.\nOne may construct assemblies of ergodic cups (or caps)\nsuitably arranged in (say) finite rectangle arrays. The\narray may then be placed in an environment with real or simu-\nlated lighting conditions and the resulting reflected radi-\nance pattern photometered. To be specific in our construc-\ntions we shall discuss only ergodic cups. However, every\nresult obtained is immediately applicable to ergodic caps.\nThe Discrete Case\nThe following problem is fundamental in the determina-\ntion of an ergodic cup:\nGiven: - An ellipse e in E2, of area A, with semimajor\nand semiminor axes a and b, respectively; and a set of m\npositive numbers kj, j = 1,\nm.\n(Fig.12.66)\nRequired: To find m ellipses e 1 ,\ne\n,\nm\nwith the following properties:\n(a) ej cej+1, = 1, , m, where em+1=e such that\nall ellipses are concentric, and have parallel major axes.\n(b) A11 ellipses ej have eccentricity [1 -\n(c) If Aj+1 is the area of the elliptical annulus be-\ntween e and ej+1, j = 1, m then Aj+1/Aj = kj. The so-\nlution to this problem is straightforward and is obtained as\nfollows:\nFrom (c):\nAj=kk 1 2 kj - 1 A 1\nwhere A is the area of e 1 By (a) and (c)\n:\nA = A1 (1+k 1 +k k 2 + +k 1 k 2 k m ) E\nAK\n.\nm\n1\nSo that:\nk\nk\nj-1\n1\nAj A K m","VOL. VI\nAIR-WATER SURFACE PROPERTIES\n264\nThe area of ej is then:\nA1 + A R +\n+\nwhich, along with (b), determines the semimajor and semimajor\naxes aj, bj, of ej by means of equations:\naj/bj=a/b\nHence:\n(3)\n(4)\nClearly aj < aj+1, bj < bj+1, j = 1, m, so that the\nremaining requirements of (a) complete the solution.\nThe fundamental problem solved above may be used to\nfind a solution of the following discrete form of the ergodic\ncup problem which, in turn, yields the continuous solution of\nthe main problem by an appropriate limit argument. Starting\nwith:\nexp\nand assuming up choose m+2 nonnegative numbers Cj (n),\nj = 0, 1, m+1, such that 0 < Cj (n) 0, . then H(x,E(E)) is the usual form of irradiance\nused in practice. We can denote H (x,E(E)) more briefly by\n\"H(x,5)\". In practical work S may be selected as a relatively\nsmall collecting surface, say S(x), fitted over the photoelec-\ntric element of the radiant flux meter, and we write:","SEC. 13.1\nPRINCIPAL RADIOMETRIC CONCEPTS\n275\n\"H(x,E)\"\nfor\nP(S(x),E(E))/A(S(x))\n(2)\nThe irradiance meter can be calibrated to read H(x,5) direct-\nly. Precautions must be taken, however, to have S(x) collect\nradiant flux in accordance with the cosine law which holds for\nirradiance (re: (15) of Sec. 2.4, and (8) of Sec. 2.8), other-\nwise theoretical calculations of H(X,5) from independent radi-\nance measurements (as in (8), and (17) of Sec. 2.5) may not\nagree with direct measurements of H(x,5) using irradiance\nmeters.\nThe difference between H(x,5) as measured by an irra-\ndiance meter and as computed from a radiance distribution is\nreadily estimable if the meter is calibrated in the laboratory\nso as to learn beforehand its response characteristic to arbi-\ntrary incident beams of flux. Thus, let D(E') be a relative-\nly narrow bundle of directions whose magnitude is fixed and\non the order of a thirtieth of a steradian. Then a plot of\n(x,D(E')) can be made versus E' as E' varies from the inward\nnormal E of the collector surface to directions normal to E,\nand during this variation the flux content of the beam is\nheld fixed in such a way that its radiance is unity. Thus\nwrite:\nH(x,D(E'))\n\"f(E,5')\"\nfor\n(3)\n(E'.E)R(D(E'))\nIf the collector is a cosine collector with given X and E,\nthen f(5',5) will be constant and of unit value with respect\nto E', and independent of the shape of D(E') as long as\nSD(E() is kept small, as agreed above.\nThe quantities f(5,5') may be used as follows to pre-\ndict the departure of irradiance readings from their values\ncomputed using knowledge of N (x,5'). Suppose that the radi-\nance distribution N(x, ) is given at x, and that the irradi-\nance meter has an inward unit normal E. Then, according to\n(8) of Sec. 2.5 the actual irradiance is:\nH(x,E) =\nAccording to (26) of Sec. 2.5 this representation may be\nrendered into an approximating summation formula of the type:\n(4)\nwhere N is the radiance related to H(x,Di) by the usual oper-\national connection (re: (2) of Sec. 2.5):","OPERATIONAL CONCEPTS\nVOL. VI\n276\nand X(5,5:) is defined in (20) of Sec. 2.5. On the other hand\nH(x,5) may be extimated by using the irradiance meter. Thus,\nby (3):\n(5)\nH(x,D(E')) = N(x,5')\n.\nUsing the partition solid angles Di employed in (4), the equa-\ntion (5) may be written:\n=\nHence:\nn\n(6)\ni=1\nis the anticipated reading of H(x,5) using the irradiance\nmeter. Comparing (4) and (6) we see that the total difference\nAH(x,5) between (4) and (6) resides in the following sum of\nweighted differences:\n(7)\n-\nSummarizing, we may say that the departure of an irradi-\nance meter from its ideal cosine-collecting characteristics is\nmeasured by the values f(E,E') defined in (3) ; and that the\ncorresponding total error AH(x,5) between the readings of the\nirradiance meter irradiated by a radiance distribution N(x, )\nand the actual irradiances calculated from N(x,.) may be esti-\nmated by means by (7). Alternate means of estimating AH(x,E)\ncan be based on slight variants of f, as defined in (3). Spe-\ncific examples of such error estimates have been studied by\nTyler [299].\nIt is useful in practice to have a specific partition of\nthe hemisphere (E) through which the radiant flux is collect-\ned prior to an irradiance calculation of the general type in-\ndicated in (4). Figure 13.1 depicts one such partition used\nin actual computations. To be specific, we let 5 be k, the\nvertical direction in terrestrial coordinate systems (re Sec.\n2.4) and consider the computation of the upward irradiance\nH(x,k) at a point X in the medium. The hemisphere (1) (E) is now\nthe upward hemisphere of directions E+ and this hemisphere is\npartitioned into a grid of spherical rectangles by equally\nspaced longitude and latitude circles as follows. We divide\nE+ into m+1 zones by means of latitude circles. There is, to\nbegin with, a zone in the form of a spherical cap about the\ndirection k. The half-angle width of the cap is A0/2 where\nis a fixed increment defined by the relation:\nm = zzo\n(8)","PRINCIPAL RADIOMETRIC CONCEPTS\nSEC.\n13.1\n277\n3No + N1\nCAP\n4\nNo\nN1\n100\nA\nNili = 2,3, 2)\nGENERAL\nZONE\nNm - I\nEQUATORIAL\nZONE\n@\n3Nm+ Nm-\n4\n108\nNm\nFIG. 13.1 A useful decompositon of direction space E for\nuse in computing irradiances.\nIn other words, by choosing m, we fix so; or, conversely,\nsuitably fixing determines an integral m, whichever best\nsuits the needs of the computations. Once the magnitudes of m\nand have been decided on we can fix the location of the\nmain m-1 zones of E+ by writing\"\n\"0,\" for ine\n(9)\nfor i = 1, , 2,\n, m-1. The angles i are the central lati-\ntudes of the m-1 zones depicted in Fig. 13.1. The cap and the\nhalf zone of width A0/2 (the equatorial zone) whose lower bound-\nary is the equator of E, make up the remaining two zones of E+ .\nNext of divide E+ into n+1 lunes by means of n+1 equally\nspaced longitude semicircles of angular width AO radians, such\nthat the relation:","VOL. VI\nOPERATIONAL CONCEPTS\n278\nholds. The longitude of the center of the jth lune is jao,\nwhich we shall denote by \"oj\" As a result of these divi-\nsions, E+ is partitioned into (m+1) (n+1) regions each of\nwhich can be indexed by a pair (i,j) of integers, i = 0, 1,\nn such that the parts of the\nm, and j = 0, 1,\nspherical cap are indexed by (0,j) and the regions of the\nand\nfirst zone below the cap by (1,j) j = 0, 1,\nn,\nthose in the ith zone below the cap by (i,j), j = 0, 1,\n,\nn. The coordinates of the centers of the main (n+1) (m-1) re-\ngions (i.e., all regions except those on the cap and the\nm-1, j = 0,\nequatorial zone) are (01,0j), where i = 1,\n..., n. The weighted solid angle content of the (m+1) (n+1)\nregions of the present partition of E+ are given below:\n(10)\nm-2\nsin 0 cos 0 for i =\nsin(4) cos(4).\n= [sin cos +\n(11)\nA000\n8\n(4) cos (4) AO\n(12)\n.\nHere \"AR! , i = 0, , m denotes the solid angle content of\nthe ith of the m+1 regions (in an arbitrary lune) weighted by\nthe cosine of the angle between the central direction of the\nregion and k. Let us write \"Nij\" for the radiance N(x,0i,4j),\nwhere (01,0j) is the direction of the center of the jth region\nin the ith zone for i = 1, m-1, n = 0, 1, n, and let\nthe radiances be assigned to the cap regions and the equatori-\nal regions as shown in Fig. 13.1. Then (4) reduces to:\n(13)\ni=o j=o\nAn example of the use of (13) is displayed in Ref. [306].\nFurther practical means of computing irradiance are discussed\nin Examples 14 and 15 of Sec. 2.11.\nSpherical Irradiance and Scalar Irradiance\nSpherical irradiance and scalar irradiance are both mea-\nsures of the radiant energy per unit volume at a point in an\noptical medium. The definitions and connections among h4tt, h\nand u are given in detail in Sec. 2.7, and may be summarized\nas:\n(14)\nv(x)u(x) = h (x) If 4h4tt(x)\n,\nwhere v(x) is the speed of light at x, u(x) the radiance den-\nsity, h(x) the scalar irradiance, and h4tt(x) the spherical","279\nSEC. 13.1\nPRINCIPAL RADIOMETRIC CONCEPTS\nirradiance at X. As demonstrated in Sec. 2.7, it is h4tt which\nis operationally meaningful while u(x) and h(x) are constructs\nderived from the theory of geometrical radiometry (radiative\ntransfer theory in a vacuum) and related to h4tt (x) by theoret-\nical arguments. The quantity h4tt (x) may be given the follow-\ning operational definition. Let Sr(x) be a spherical collect-\ning surface of center X and radius r. Suppose that about\neach point y of the boundary surface of Sr(x) the surface is\na cosine collector. Let P(Sr(x)) be the radiant flux recorded\nas incident on (x) in some radiometric environment. Then we\nwrite:\nP(S(x))\n(15)\n\"h4(x)\"\nfor\n4ur2\nWhen r is sufficiently small, the radiance distributions\nover the set of points that would be occupied by Sr(x) are in-\ndependent of location over that set, and also h 4 TT (x) is sensi-\nbly independent of r. Hence in this sense \"r.\" need not enter\ninto the notation for the spherical irradiance.\nIn constructing a realization of Sr (x) (determining Sr\nwith actual instruments) it is impossible to obtain a collect-\ning surface about each point of which the material exhibits co-\nsine collecting properties, zero reflectance and unit transmit-\ntance, the properties needed in order that the measurements not\ndisturb the light field at X. To study an important effect of\nthe departure of the surface of Sr (x) from this ideal, let us\nsuppose that at each point y of the surface of Sr (x) behaves\nin accordance with f(E(y) ,5') as given in (3) when the experi-\nment giving rise to (3) is now repeated for Sr(x). While we\nmay not now have f(E(y) E') independent of E, where 5(y) is\nthe inward unit normal to (x) at some given point y on its\nsurface, we should at least be able to manufacture* a surface\nso that f (E(y),5') is independent of y. Assuming this done,\nwe repeat the general arguments leading to (5) of Sec. 2.7.\nThus the radiant flux recorded by the spherical collector,\nwhen radiance is incident on Sr(x) over small solid angle Si\nabout i (see Fig. 2.17), is given by:\nwhere Sr(5j) is that part of (x) such that (y) 0,\ni.e., ST(51) is the illuminated hemisphere of Sr(x).\" - Let us\nwrite:\n(16)\n\"CT\"\nfor\no\n(5)\n*If the y-dependence is not eliminable, then the analy-\nsis leading to (17) can be extended quite readily by using\nbounds on the y-variability of f (5 (y) 5') Hence (17) will\nbe generalized to bracketing inequalities around h4tt (x).","OPERATIONAL CONCEPTS\nVOL. VI\n280\nIt is clear that, by our hypothesis about f, Cr is independent\nof . Geometrically, Cr is the effective cross section area\nof the sphere presented to the beam of light in Fig. 2.17. In\nthe case of Sr(x) being an ideal cosine collector, we have\nf(5,5') = 1 for all E' in (5), so that Cr = 2. For real\ncollectors usually\nWe now sum the terms over all i to obtain:\nwhich we recognize as Crh(x), the total radiant flux recorded\nby Sr (x). The recorded spherical irradiance is then, by (15),\ngiven as:\nhaw(x)=\n(17)\nwhich is a generalization of (14) . Thus if a spherical col-\nlector is used to measure h4tt (x) the connection with h (x) is\nstill the simple linear type of connection given in (14) from\nwhich the constants may be divided out in practice. That is,\nas the spherical collector moves down into a lake, say, we\nhave by (17), the following useful relations:\n(18)\n=\nfor every pair of depths y,z in the medium. The last equality\nfollows if the index of refraction is constant with respect\nto depth. The linearity summarized in (17) is the most impor-\ntant feature required of a spherical irradiance meter or for\nthat matter, any other meter used in applied radiative trans -\nfer theory. Thus while it is generally too much to ask that\na spherical collector Sr (x) measure exactly h (x) or u(x), we\ncertainly can obtain these quantities to within known,\nfixed numerical factors using the general connection between\nh4m (x) and h (x) given in (17).\nThe preceding analysis was based on the agreement to\nuse a spherical collector. Actually the theory of Example 15\nSec. 2.11 allows any of a wide class of collectors to measure\nnot only h (x) but N(x, ) as well. Thus, for h (x) one may use,\nby the argument leading to (72) of Sec. 2. 11, the relation\n(17a)\nif one already has a flat plate irradiance collector. This\nmay also be extended to practical measurement conditions anal-\nogous to (17).","SEC. 13.1\nPRINCIPAL RADIOMETRIC CONCEPTS\n281\nRadiance\nWe conclude the present discussion on radiometric con-\ncepts with some practical observations on the operational\ndefinition of radiance. Following the discussion of Sec. 2.5,\nwe fit a radiant flux meter with a cylindrical tube so that\neach point on the collecting surface S(x) of the radiant flux\nmeter is exposed to radiant flux incident in a small set D(E)\nof directions about the unit inward normal to S(x). If the\nradiant flux reading of this assembly is (S(x), , D(E)) then\nwe agree to write:\n\"N(x,E)\" for (5))/A(S(x))r(D(E))\n(19)\nWith S(x) and D(E) understood, the definitional identity for\nradiance is simply:\nN = P/As = H/S\n,\nwhere,\nH = P/A\nThus we see that the measurement of N requires a knowledge of\nthe amount P of radiant flux collected, the area on which it\nis collected, and the solid angle So within which it is col-\nlected. Since the latter two parameters can be instrument\nconstants it is evident that the technique for measuring rela-\ntive radiance requires:\n1) a collecting area of fixed size,\n2) a limited solid angle of fixed size, and\n3) an electrooptical coupling system whose characteris-\ntics are invariant or known under wide ranges of\nchange of the light field levels to be measured.\nAn important feature of radiance distribution measure-\nments is the detail or resolution with which they are made.\nThis detail is governed by the size of the solid angle of ac-\nceptance of the optical system. If the solid angle is small\nthe resolution will be high but the flux collected will be\nsmall. Alternatively a large solid angle will collect a large\namount of flux but will not accurately report the directional\nstructure of the light field.\nA natural lower limit for the resolving power of a radi-\nance photometer might be set by the angular subtense of the\nsun's disc which is 34 min 4 sec of arc (average) in air, or\nabout 24 min from an underwater station (underwater solid\nangle equals 0.00527m). This level of resolving power would\nmake it possible to describe the underwater light field in the\nvicinity of the sun with great detail if the air-water inter-\nface remained perfectly flat. In a more practical situation,\nhowever, the air-water interface would be rippled and the","VOL. VI\nOPERATIONAL CONCEPTS\n282\nimage of the sun would be replaced by a glitter pattern whose\nsize would depend on the recent wind history at the surface\nand to some extent on the angle of observation. Under these\nconditions an angular subtense equal to half the angular size\nof the glitter pattern might give adequate resolving power.\nSome information on the size of glitter patterns has been\ngiven by Cox and Munk [57].\nAt great depths in optically deep water the glitter pat-\ntern will not be observable. The selection of resolving power\nof the radiance tube then depends only on the accuracy with\nwhich one desires to determine the average relative values of\nradiance at any point along the radiance distributon curve.\nAs we have seen, the restriction of the solid angle of\nacceptance can be accomplished by means of a radiance tube or\nby an appropriate optical system. The radiance tube was sug-\ngested in 1936 by Gershun in his paper on the Light Field\n[98] and is frequently referred to as a \"Gershun tube\" for\nthis reason. It consists of an internally baffled and black-\nened tube constructed so that internal reflections from the\nwalls of the tube cannot reach the detector. The angle of\nacceptance of a radiance tube is defined by a simple geometry\nas shown in Fig. 13.2. Note that the angle Y1 is the solid\nangle within which every point of the area A collects flux.\nThe angle Y2 establishes the penumbral limits of the device.\nRays within the shadowed area will be seen by only a portion\nof the detector area.\nThe theory of the baffle tube is illustrated in Fig.\n13.3. For effective baffling it is necessary that the baffles\nbe closer together at the forward end of the tube. At suffi-\nciently large entry angles the rays are not specularly reflect-\ned from the black paint, and subsequent multiple reflections\nfrom wall to wall of the tube quickly reduce their contribu-\ntion to the collected flux. To quantitatively describe the\nrequisite baffling, observe that baffles A- -deep, spaced B--\napart at locations (1) and (2) in Fig. 13.3 will eliminate\nthe rays between the angles tan 1 (D/L) and tan 1 (2A/B). Now,\ntan (2A/B) = tan 1 [D/(L-(B/2))] and:\nA\nY2\nY1\nFIG. 13.2 Placement of baffles in a radiance tube.","SEC. 13.1\nPRINCIPAL RADIOMETRIC CONCEPTS\n283\nL\na\nD\nb\na\nA\na\ne\nB\nC\n2\n2\nI\n2\n3\nFIG. 13.3 Theory of radiance tube baffles.\ntan 1 2A = tan 1 ( DG) D\n1\n(2A/C) >tan - 1 (2A/B), , and all rays parallel to ray\nHence tan -\nef across the aperture D will be stopped by baffles (1) or\n(2). Next we observe that:\ntan a = B/2 A\ntan b = C/2\nand since b > a, , we have:\nA > A\nC/2\nB/2\nso that:\nC < B\nor\nwhich shows how the baffles, in order to be effective, should\nbegin to crowd closer together as they near the opening of the\ntube.\nSome experiments with early models of radiance meters\nbrought out a rather unexpected disadvantage of the radiance\ntube for underwater measurements when the radiance meter's\ntube is of great length. To obtain the radiance distribution\naccurately at a point underwater with a meter which has a","OPERATIONAL CONCEPTS\nVOL. VI\n284\nrelatively long tube, the radiance meter should not be ro-\ntated around its detector end, but rather it should be ro-\ntated about the forward end of the radiance tube. If, as is\nusual, the radiance meter is rotated around an axis at the\ndetector end of the radiance tube, then the radiance measured\nin each direction will be effectively for a different depth,\nnamely the depth of the forward end of the tube. Since the\nvariation of radiance with depth in relatively turbid media\nis quite marked over short distances and since it is nonlin-\near and different in every direction, correction of the data\nto a single depth under such circumstances would be very\ndifficult. The use of a carefully designed optical system\nfor restricting the solid angle of acceptance makes it pos-\nsible to obtain the same resolution with a shorter tube,\nthereby alleviating this difficulty. In this connection, ap-\npropriate internal baffling and black finish are just as im-\nportant for an optical system as for the baffle tube. In any\ncase, for very accurate results in water having relatively\nlarge volume attenuation function a, measurements with an\noptical system may possibly have to be corrected for signifi-\ncant depth changes at the end of the measuring tube, as the\ntube scans its environment.\nIn closing it may be observed that radical new radiance\ndistribution measurement techniques may be possible by adopt-\ning some of the observations of Examples 14 and 15, Sec. 2.11,\nbuilt around novel collection devices and Legendre polynomial\nanalysis and synthesis. Another interesting possibility is\nthe use of Fourier analysis techniques on the radiance dis-\ntribution N(x, ) ) over the compact set (1) of the directions E.\nIn particular, the sampling theory [29] coupled with resolu-\ntion requirements may obviate the need for scanning each\npoint of E, but only a discrete set of appropriately placed\npoints in E. This of course holds also for the Legendre poly-\nnomial analysis, and indeed, for any orthogonal family of\nfunctions defined on E.\nFinally, it may be noted that the time-consuming sweep\nover all directions in order to determine radiance distribu-\ntions may be altogether eliminated by employing a \"fisheye\"\nlens which maps each ray in a fixed incoming 2 solid angle\ninto a unique point in a small circular area at the base of\nthe lens. This lens would allow an instantaneous photorecord-\ning of a hemispherical region of radiance values, which may\nbe analyzed by a specially programmed computer. Such a de-\nvice has been studied by John Tyler and Raymond Smith at\nScripps Institution of Oceanography.\n13.2 Operational Definition of Beam Transmittance\nWe consider next some possible experimental means of\ndetermining the beam transmittance of a general path of sight\nin a natural hydrosol. The development in Sec. 3.10 of the\nconcept of beam transmittance starting from the interaction\nprinciple exhibits the physical foundations and basic meaning\nof the beam transmittance concept, and we now show how we can\nbuild on that development in several ways, so that beam trans-\nmittances can be obtained using standard radiance measuring\nequipment.","SEC. 13.2\nBEAM TRANSMITTANCE\n285\nGeneral Two-Path - Method\nLet PT r (x1 , E) and Qx(x2,5) be two parallel paths of\nlength r as depicted in Fig. 13.4. We now show how measure-\nments of the radiances at the extremities of these paths can\nlead to a determination of Tr(x,5), the beam transmittance of\nthe path Q (x,5). We need only arrange matters so that all\nthree paths are in a regular neighborhood of paths (Def. 2,\nSec. 9.5), i.e., so that:\n(1)\n= =\nand\n(2)\nSuch paths are often encountered in real optical media, so\nthat what follows is of more than academic interest. By means\nof (5) of Sec. 3.13, the apparent radiances at the terminal\npoints of the paths Pr(x,,5), Pr(x2,5) are expressible as:\n(3)\n+\nN(Y2,5) + N*(Y2,5)\n(4)\n.\ny,\ny=x+r8\nPr(xpE)\n2\nPr(x,E)\nPr(x2,E)\nE\n&\nX1\nX\nE\nx2\nFIG. 13.4 The two-path method for beam transmittance.","286\nOPERATIONAL CONCEPTS\nVOL. VI\nWriting:\n\"AN(y,5)\" for N(y,,5) - N(Y2,5)\n(5)\n\"AN(x,E)\" for\n(6)\nand subtracting (4) from (3), we have:\n(7)\nNow for theoretical or practical purposes it is possible to\nuse either the natural radiances occurring at X1, X2, Y1, Y2\nfor computations with (7) or to place artificial sources at\nthe points X1, X2. In either case the method proceeds by mea-\nsuring the four radiances under such conditions, and then us-\ning (7) to determine Tr(x,5). In either case therefore, (7)\nwill yield up numerical determinations of Tr(x,5) so long as\n(1) and (2) hold. Equation (7) may thus provide an operation-\nal definition of Tr(x,5).\nGeneral One-Path Method\nThe preceding development from (1) to (7) may be rein-\nterpreted so that there is only one path, namely Pr(x,5), over\nwhich at time t there is an artificial or natural radiance\n1\ndistribution such that (3) holds and a small time later at T2,\n(4) holds. For example, a light beam along the path Pr(x,5)\nmay change radiance arbitrarily or may blink periodically so\nthat in any case it has two distinct radiances N(x,5,t,) and\nN(x,E,to) with corresponding observable radiances N(y,E,t,),\nN(y,E,tz). If (1) and (2) hold in the present case, i.e., if:\n(8)\n= =\n(9)\nthen:\nAN(y,E,t)\n(10)\n,\nwhere now we have written:\n\"AN(y,E,t)\" for N(y,E,t)- N(y,,t2)\n(11)\n\"AN(x,E,t)\" for .\n(12)\nThus by either spatial modulation of radiances, as in (7), or\ntemporal modulation of radiances, as in (10), the beam trans-\nmittance over the corresponding regular neighborhood of paths\nmay be operationally defined.","SEC. 13.3\nPATH RADIANCES\n287\n13.3 Operational Definitions of Path Radiances and Path\nFunctions\nThe path radiance and path function were conceptually\nformulated by means of the interaction principle in Sec. 3.12,\nand so are firmly grounded in the basic principles of radia-\ntive transfer theory. We now turn those conceptual formula-\ntions into useful operational definitions of the path radiance\nand path function.\nOperational Formulation of Path Radiance\nThere are two general ways in which we may obtain the\npath radiance N°(z, E) of a path Pr(x,5) in an optical medium,\nas depicted, e.g., in Fig. 13.5. Each method is based on the\nfundamental relation:\nN(z,5) = N(x,5)Ty(x,5) + N*(2,5)\n(1)\nwhich is an instance of (5) of Sec. 3.13.\nSuppose that it is possible to set:\nN(x,5) = 0\n,\nthen (1) yields N. (z,E) immediately. This is the dark target\nmethod of determining N*(2,5). Workable equivalents to the\nX\nFIG. 13.5 A path Pr r (x,5) with initial point x, internal\npoint y, and end point Z.","288\nOPERATIONAL CONCEPTS\nVOL. VI\ncondition N(x, 5) = 0 may be obtained by actually viewing a\nsmall black light trap placed at X.\nThe alternative method of finding N* (z,5) is to first\ndetermine the beam transmittance of P r (x,), as in Sec. 13.2.\nThen, (1) allows the rigorous computation of N°(2,5) in the\nform:\n(2)\n-\nassuming all three quantities on the right in (2) are measur-\nable. The two radiances on the right in (2) may be as they\nare found in nature, or as generated by artificial sources.\nOperational Formulation of Path Function\nAn operational formulation of N* (x,5) can be based on\n(15) of Sec. 3.12. For suppose Pr (x,5) is a path so short\nthat the radiance distributions over it are independent of\nlocation and that for all practical purposes Tr(x,5) = 1.\nThen, very nearly:\nNa(X,5)=\n(3)\nin which N°(x,E) may be determined by means of the preceding\ndark target method or by means of (2). In general, (15) of\nSec. 3.12 may be written rigorously as :\nN(2,5)=N(x,5)ro(r)\n(4)\nwhere o(r) /r 0 as r 0 (cf., (2) of Sec. 3.12). From this,\n(3) follows for small r.\nAn alternative means of finding N* (z,5) under special\nlighting conditions and in homogeneous media is given by (1)\nof Sec. 4.3. For, then:\n(5)\na path Pr (x,5) (as in Fig. 3.33) which lies in a homo-\nover\ngeneous, uniformly lighted stretch of optical medium. Know-\ning 5) and a, then yields N+ (x,5). Formulation (5) casts\nlight on what is meant in (3) by \"small r.\" We shall con-\nsider r \"small\" when ar < 1, i.e., when r < 1/a = La' where L\nis the attenuation length of the medium.\nActual measurement procedures leading to N* (x, 5) can be\nbased on a device of the kind depicted in Fig. 13.6. The de-\nvice consists of a radiance meter directed into a black tar-\nget. The ambient radiance distribution N(x, at a general","SEC. 13.3\nPATH RADIANCES\n289\nN(x',E')\nradiant flux\nsensor\nX\nFIG. 13.6 Dark target arrangement for determining path\nfunction value N*(x,5).\npoint X is scattered into the direction E by the material com-\nprising the cylinder around the short path segment Pr r (x, 5)\n(shown shaded) between the radiance meter and the black tar-\nget. Conditions can usually be arranged so that the assump-\ntions leading to (3) are satisfied with reasonable accuracy.\nAn illustration of a path function meter of the kind described\nhere may be found in Ref. [80].\nFurther, by combining (2) and (3) we have: very nearly:\nN(z,E) -N(x,5)Ty(x,5)\nN*(x,5)\n(5)\n,\nr\nthe setting for which is depicted in Fig. 3.33. We shall\nadopt (3) as the operational definition of V*(x,5).\nStill further, if the volume scattering function is\nknown, then it follows at once from (8) of Sec. 3.14 that:\n(6)\nfor each j, = 1,\nn over some suitable partition {E,,\n..., En of E. Hence by measuring the functions N(x,) and\no(x;*;*), N*(x,.) can be obtained by direct computation.\nComputations of this kind were explored in Ref. [213].\nFinally, the K-method of determining the path function\nmay be used in virtually all real media. The K-method, as","PATH RADIANCES\nVOL. VI\n290\ndeveloped in Ref. [219], is based on the canonical form of\nthe equation of transfer. To illustrate the method, we con-\nsider an arbitrarily stratified plane-parallel medium. The\nassociated canonical form of the equation of transfer is\ngiven by (21) of Sec. 4.5, and is repeated here:\nN*(2,5)\nN(2,5) = a(2)+k(2,5) COS 0\nSolving this for N*(2,E):\nN+(2,5)=[a(z) + K(z,5) cos e]N(z,E)\n(7)\nHence if all quantities on the right side of (7) are known,\nor determinable, then N* (z,5) is determinable. Examples of\nthe use of (7) using real data are given in Ref. [219].\n13.4 Operational Definition of Volume Attenuation Function\nThe volume attenuation functon, as developed in Sec.\n3.11 from first principles may be obtained in several alter-\nnate empirical procedures by performing measuring operations\nin natural waters which contain either natural or artificial\nsources of radiant energy. In this section we shall outline\nsome of the more fundamental of these procedures.\nWe consider first the most direct operational means of\ndefining a. From (1) of Sec. 3.11 we have:\n+ 0(r)\n(1)\nwhere 0(r) is a quantity which goes to zero with r, and is on\nthe order of magnitude of the quantity ar. Thus, if beam\ntransmittance measurements are available, we have, very nearly:\n(2)\nfor relatively small r, that is for r on the order of 1/2 to\n1 meter in most oceanic waters and for wavelengths in the\nvicinity of 500 mu. of course for more turbid waters, with\nrespect to the given wavelength, r must be chosen correspond-\ningly smaller in order for the term 0 (r) to be negligible.\nIn virtually all natural optical media a will be independent\nof E, and we henceforth drop \"E\" from the notation.\nAn alternate operational definition of a, one that is\nquite general and which we shall adopt here, is that based\non (2) or (5) of Sec. 3. 11. Thus:\ndT (x,5)\nr\n(z) = - Tr(x,5)\n(3)\ndr","SEC. 13.4\nVOLUME ATTENUATION FUNCTION\n291\nin which we have written:\n\"z\" for X + r E\n(4)\nThe physical situation associated with (3) is depicted\nin Fig. 13.5. The important point to observe in (3) is that\nif the path Q (x,5) is varied in X by holding X, fixed and\nvarying r (which is the intended meaning of the derivative\nin (3)), then the result of the indicated operations on Tr(x,5)\nis precisely a (z), the value of a at terminal point Z of\nPr(x,5).\nAs a simple application of (3), consider an experiment-\nal arrangement such as that depicted in Fig. 13.4, and as\nsummarized in (7) of Sec. 13.2. Then by using the results of\nthis arrangement, (3) becomes :\n= d ln [ANGX.52\n(5)\nwhere now y = X + rE, as shown in Fig. 13.4.\nFurther applications of (3) are possible. For example,\nthe results of the one-path method of determining Tr(x,5), as\nin (10) of Sec. 13.2, may be used in (3). Also suggested by\n(3) is the possibility of self luminous moving probes in the\nsea, that is probes with self contained light sources and\nradiance pickups. For further details in this direction, the\nreader is referred to [238], which contains a general theory\nfor such advanced techniques.\nAn alternate method of measuring a to those considered\nabove makes use of a frequently occurring regularity of the\nnatural lighting conditions in homogeneous media, and is\ncalled the dark target technique (cf. Sec. 3.3). For sup-\npose the conditions are just right in a medium so that the\nclassical canonical equation for apparent radiance (2) of\nSec. 4.4 holds:\n(6)\na\nwhere us is a horizontal direction (so that cos 0 = 0). Next,\nplace a black target of some kind at a distance r from the\nobservation point, so that No (z,5) = 0 in (6). Then under\nthese conditions :\n(7)\n.\nHere we observe that N x (z,E)/a is the equilibrium radiance\nNg(2,5) and which in this case is simply the measurable hori-\nzóntal radiance N(z,5) at depth Z in direction E. Letting\n00, we have:\nr\nlim","VOL. VI\nOPERATIONAL CONCEPTS\n292\nan observable quantity, namely Nq (z,E) (cf., (4) of Sec. 4.4). .\nHence (7) becomes:\nwg\n(8)\nin which N(z,E), Na (z,5) and r are in principle measurable.\nHence if Na (z,5) is measured for Z and E and then a small\nblack target placed at a distance r on the same path from the\nobservation point, (8) yields a means for obtaining a. Of\ncourse, care must be taken so that the N, (z,E) is not perturbed\ntoo greatly over the path segment between the target and ob -\nservation point.\nObserve that if one sets r = 80, then (8) becomes:\nN = Nq = N\n(9)\nwhich determines a once N and N* are determined.\nA complementary procedure (the bright target technique).\nto that just described arranges matters so that N* is effec-\ntively zero, with the result that (6) implies:\n(10)\n.\nFinally, we observe that a may be determined from the\nrelation:\n(11)\na=a+s\nwhen the volume absorption and total scattering functions are\nknown. Relation (11) follows from the definition in (4) of\nSec. 4.2. Ways of independently measuring a and S will be\nconsidered in Sec. 13.7 and Sec. 13.8.\nCanonical Equation Method\nUsing (21) of Sec. 4.5, rearranged in the following\nform:\n(12)\n- K(z,5) cos 0\n,\nwe find that a(z) is determinable via direct light measure-\nments of N(z,5) and, say, a dark target technique for Nx (z,5)\nas in Sec. 13.3.","SEC. 13.5\nPERTURBED LIGHT FIELDS\n293\n13.5\nA General Theory of Perturbed Light Fields, with Appli-\ncations to Forward Scattering Effects in Beam Trans-\nmittance Measurements\nA close study of the preceding three sections would sug-\ngest that the problem of the measurement of the optical prop-\nerties of a given medium is complicated by the fact that the\nact of measurement perturbs the distribution of radiant flux\nin the immediate vicinity of the measuring apparatus. Conse-\nquently, the numbers derived from a measurement process may\nnot faithfully reflect the inherent optical properties of the\nmedium under study, but rather contain along with the informa-\ntion sought the effect of the presence of the measuring appa-\nratus. In the present section we shall develop a general\nformulation of the equation of transfer for a perturbed radi-\nance field-in an arbitrary optical medium. The resultant\ntheory is then applied to the problem of the measurement of\nthe volume attenuation function and the beam transmittance in\nnatural waters. The treatment is sufficiently general to\nhold in any natural optical medium, in particular the various\nnatural hydrosols of the earth and the atmosphere.\nThe theory we shall develop below leads to several new\nmeasuring techniques for the volume attenuation function a\nwhich take into account the perturbation effect on the light\nfield of a standard measuring apparatus used for the deter-\nmination of a. In addition, the theory provides a means of\nconsistently estimating the relatively elusive forward scat-\ntering value o of the volume scattering function O. Final-\nly two criteria are given for estimating the order of magni-\ntude of the forward scattering effects encountered in beam\ntransmittance measurements.\nIntroduction\nIt is a cardinal axiom of experimental physics that the\nact of observing a given phenomenon necessarily disturbs the\nphenomenon under observation. It follows that the \"true\"\nnature of the observed is obscured by some such disturbance\ngenerated by the observer. This axiom holds in particular in\nthe field of experimental radiative transfer. An important\nillustration of this is afforded by the operational procedures,\nsuch as those discussed in Sec. 13.4, for the determination\nof the volume attenuation function a. By way of introduction\nto the present methods of circumventing the effects of these\nperturbations, the procedures for finding a will be briefly\nreviewed. The general theory of a perturbed light field is\nthen formulated and applied to the case of the determination\nof a, which results in the perturbed light field counterparts\nto the classical procedures. The theory developed below\nyields five distinct approaches to the problem of the deter-\nmination of a, each of which may be transformed into an oper-\national procedure.\nThe procedures for determining a discussed in Sec. 13.4\nassume that the light field is unperturbed or perturbed in an\ninessential manner as the probing for the relevant informa-\ntion goes on. By ignoring such perturbations an investigator","OPERATIONAL CONCEPTS\nVOL. VI\n294\nis rewarded with analytical formulations of ohm-law simplic-\nity, as we saw, e.g. , in (5) , (8), and (9) of Sec. 13.4. The\nprice for this is paid by having the resulting prediction\ncurve for a more often than not pass unconcernedly through an\narray of nonconformist data points. The two main techniques\nnow in use may be classified as the bright-target and dark-\ntarget techniques.\nFigure 13.7 depicts the essential geometrical elements\nof each technique. In the bright-target technique, T is a\nself-luminous target viewed by a radiance meter G at a dis-\ntance r. It is assumed that the target is angularly small-\nin fact, of zero solid angular subtense - - when viewed at each\npoint of the path P. r between T and G. In addition, the effect\nof the ambient light field is removed by either a direct shield-\ning of P from its surrounds or by taking the difference of the\nG-readings found by turning T on and then off. This then is\nthe bright-target method, of Sec. 13.4, resulting in (10) of\nSec. 13.4. The assumption is made that this on - off procedure\ndoes not perturb the ambient light field. Finally, G is as-\nsumed to be an ideal collector: any flux entering G and not on\nP is not recorded. A11 these assumptions combine to reduce\nthe general equation of transfer:\nEp(r)\n(r)\nEg(r')\nT\nG\nb\na\np\nr\nl\nr'\nr\nFIG. 13.7 Illustrating the geometry of the perturbed light\nfield. T is a generalized target, G is the Gershun tube of\na\nradiance meter. Both T and G induce a perturbation of the\nradiance distribution about a point p on the path Pr. In the\ngeneral case, T need not be on the axis of G. The subsets of\nthe direction space E about p which are occupied by the direc-\ntions of the target, the Gershun tube, and the perturbation\nare indicated in the figure.","SEC. 13.5\nPERTURBED LIGHT FIELDS\n295\ndN/dr = - aN + 1 N*\n(1)\nto the particularly simple form:\ndN/dr = - aN\nfor the radiance along\nPr.\nThus if No o and N are the inherent\nand apparent radiances of T along Qx, then the operation,\n(2)\n(which follows from the preceding approximate form of the equa-\ntion of transfer) on the measurable quantities r, Nr, No is\ntaken to yield the required value of a.\nThe dark-target approach, on the other hand, assigns\nzero inherent radiance to T. The same assumptions adopted\nabove remain in force. In addition, Pr is chosen so that N*\nis constant along Or. As we saw in (8) of Sec. 13.4, the so-\nlution of (1) leads to the following operation,\n(3)\non the measurable quantities Ng = N*/a, Nr, and r, and is\ntaken to yield the required value of a.\nGeneral Representation of a Perturbed\nLight Field\nIn actuality, the placing of a target T of inherent radi-\nance N o in the light field perturbs the light field. Further,\nthe target, being a material object, occupies a finite volume\nof space so that it fills a finite subregion E+ (r') of direc-\ntion space (1) as viewed at each distance r' on Pr. From this\nvantage point the presence of T causes a perturbation sensibly\nextending over a subset, say Ep (r') of E. Finally the tube of\nthe radiance meter, being a material object of finite dimen-\nsions, will also contribute its share to the perturbation and,\nin addition, will record flux entering G which is not strictly\non Pr; the collection of such directions which carry accept-\nable flux will be denoted in general by \"Eg(r')\".\nWith these observations in mind, the general equation of\ntransfer is replaced by the following form for the new context:\ndN'(r',5) dr = x(r')N'(r' ,5) + N'\n0(r';5;5')dd(E')\nN(r',s)o(r';)(E')\n+\nE-Ep(r')\nN'","OPERATIONAL CONCEPTS\nVOL. VI\n296\nHere, and in the sequel, the symbols \"N' and \"N\" will de-\nnote the perturbed and unperturbed radiance functions, re-\nspectively. Thus the preceding equation governs the rate of\nchange of N', as it is actually measured by the real radiance\nmeter in the perturbed light field. The equation may be re-\nwritten more compactly as:\ndN'(r',E)\n-a'(r')N'(r',E)\n+\n=\ndr\n(r',5')\n+\nNx(r',5)\n(4)\nwhere in the case of an instrumental perturbation we write:\n\"a\" (r')\"\nfor\na (r') -\n,\nand where, as usual:\nN*(r',5)\n,E)o(r';)(E')\n=\nThis is the general equation of transfer for a perturbed\nlight field. Its domain of applicability is quite wide. The\nnotion of radiance meter is here intended to cover all types\nof radiance detectors, including such organic detectors as\nhuman eyes. Since all material radiance detectors occupy fi-\nnite regions in space, they always give rise to a perturbed a,\nnamely the a' of (4). In view of this fact one can raise the\nquestion: Is it meaningful to talk about a \"true a\" in prac-\ntical contexts? Any operational procedure designed to deter-\nmine the a of a medium must necessarily be made through the\nintermediation of a physical recording apparatus. It is,\ntherefore, meaningful to talk or think only about the a' for\nthat instrument, or collection of a' values relative to a\ngiven collection of radiance detectors. The \"true a\" is there-\nfore a constitutive rather than an operational concept, a\nuseful fiction with which one may create the theory of attenua-\ntion and about which one may conveniently cluster for refer-\nence the operationally obtainable a' values.\nAn examination of (4) indicates that the problem of the\ndetermination of a is intimately connected with the determina-\ntion of the forward scattering values o (r';E;E) (which will\nhenceforth be denoted by (r')\" or \"100\") of the volume scat-\ntering function o. Here again the physical limitations of the\nrelevant instruments, in this case the o-meters (cf., Sec. 13.6),","SEC. 13.5\nPERTURBED LIGHT FIELDS\n297\nprevent an exact determination of o o (r') Even if an instru-\nment could, by some clever ruse, be forced to look directly\ndown the one-dimensional path to the primary source, what\nprinciple will allow the separation of the so-called forward\nscattered flux from the unscattered, transmitted flux? This\nraises the question: Is such an attempted separation meaning-\nful in practice in which steady state fluxes are measured? The\nanswer, clearly, is that it is not. For a discussion of this\nmatter, see Sec. 18 of Ref. [251] and the summary below. But\nyet, even with strict experimental justification absent, there\nappears to be some unavoidable compulsion to conceptually de-\ncompose the forward flux into scattered and unscattered com-\nponents. The motivation for such a procedure is apparently\nan esthetic requirement: one in which the gap in the experi-\nmental definition of the o function for the singular forward\ndirection be closed by the inclusion of the value 'o(r').\nThe desirability of obtaining 00 clearly stems from the\nfact that it is an important constitutive construct that is,\none which helps bring order and completeness into the classi-\nfication and theoretical study of turbid media. It is with\nthis in mind that the study of oo is carried out in conjunc-\ntion with the study of operational determinations of a.\nFrom the preceding observations, it is seen that even\non a phenomenological (or macroscopic) level, the study of\nlight is beset by limitations on the exact experimental deter-\nminations of the three basic notions: N, a, and O. The seem-\ning indeterminacy of N may be sidestepped in principle by de-\nfining N operationally as the apparent limit of a sequence\n{Nn} of radiance functions given by a sequence of radiance\nmeters which approaches as a limit the ideal (sg = 0) radiance\nmeter. The corresponding operational values of a and °0, how-\never, are determined only after more elaborate procedures are\nspecified; their operational definitions are subject, in the\nsense explained above, to some quite fundamental difficulties.\nSome relatively simple ways in which these difficulties can\nbe overcome on a practical level will now be considered.\nLinearized Representation of Slightly\nPerturbed Light Fields\nAs an illustration of the use of the general represen-\ntation (4) of a perturbed light field, we reconsider the prob-\n1em of experimentally determining the volume attenuation func-\ntion a. The discussions which follow apply to arbitrary opti-\ncal media, e.g., natural hydrosols or aerosols. The equipments\nused in the usual procedures have been designed so as to mini-\nmize, within reasonable limits, the induced perturbations.\nNevertheless, small but detectable perturbations are encoun-\ntered. We shall assume that such perturbations may be repre-\nsented by certain linearization conditions imposed on the gen-\neral structure of (4). Towards this end we postulate three\nconditions for a linearly perturbed light field:\n(i) The functions and o(r';.;5) are constant\nover Eg (r') and Et (r'), respectively, for each r 0 Kr' 1 (k is introduced to\ninsure the fulfillment of the condition that the target must\nat least fill the field of the radiance tube). Then the\ncondition b-(a2/2r) = 0 requires that the range of the tar-\nget be r = 2l2/bk2.\nIn illustration, let b/l = 1/30, l = 0.30 meter.\nChoose two k-values, e.g., k, = 12, k2 = 2. These numbers\nnow fix a and r for each choice of k:\n= 9.00 meters\n= 0.432 meters\nr = 4.50 meters\n==0.300 meters.\nNow returning to (14) in which b- (a2/2r) = 0, we have for two\nsuch experimental arrangements:","304\nOPERATIONAL CONCEPTS\nVOL.\nVI\nF(a;a,b,r,oo)\nNq -N'(r)\nNq\n(a)\n(a)\na\nA(a)\nB(a)\n(b)\n(a)\na\nFIG. 13.10 Two numerical procedures for determining\nPart (a) : Experiment 2; Part (b) : Experiment 4 (see text).\n-ar\n(r,)q(a\nr\n-ar\nwhere\nThis set of equations may be arranged to read:","SEC. 13.5\nPERTURBED LIGHT FIELDS\n305\n-ar III B(a)\nEach side represents a computable function of a (the ai and\nri being known and fixed during the experiment). Hence both\nA and B can be graphed over a certain domain of a values. If\nthe curves are graphed over the same set of axes (Fig. 13.10\n(b)) the point of intersection of the graphs defines the re-\nquired a. The graph of Fig. 13.10 (b) merely represents the\nidea of this solution procedure; it need not represent an\nactual set of A and B graphs.\nExperiment 5. Equations (8) and (14) are the basis for\nthis experiment. Equation (8) is solved for ° 0 and the re-\nsultant expression for ° o is substituted in (14). After this\nis done, the remaining procedure is in principle covered by\nthe analytical steps outlined for Experiment 2.\nOrder of Magnitude Estimates\nA given bright or dark-target arrangement can be given\na quick preliminary analysis by means of equations which ap-\nproximate (8) and (14). The appropriate equation for (8) is\ngiven by (9). Turning to (14) for the purpose of obtaining\na useful approximation, we see that a lower bound on the o\neffect may be obtained by setting Set (r') = Set (r) so that (14)\nreduces to:\n= exp\n.\n(16)\nThus if (16) predicts a measurable deviation from the unper-\nturbed radiance N o (1-e-ar), the actual - effect produces an\neven greater deviation (an illustration is given below).\nUn-\nfortunately, a correspondingly good upper bound is not found\nin such a simple way, so that no general bracketing expres-\nsion can be simply given for N' (r) in the dark-target case.\nEach problem is best handled separately using (14), or its\nvariant (15).\nFrom the preceding analyses it is evident that the a-\nmeter technique (i.e., the bright-target technique) appears\nto be the more simple to handle analytically. From (9a) one\ncan estimate the percent difference between a and a':\n2\n= 100 X a'-a = -\n(17)\n,\nthe minus sign denoting that a' is always less than a.\nTo gain a rough idea of the order of magnitude of the\nforward scattering effect in a typical hydrosol and aerosol,","VOL. VI\nOPERATIONAL CONCEPTS\n306\nwe choose for the hydrosol: oo = 1.92/meter steradian;\na = 0.402/meter; * and for the aerosol, Oo = 9 x 10-4/ meter-\nsteradian, a = 32 X 10-5/meter. ** The corresponding values\nof A for a set of a-meters (characterized by their a/r ratios)\nare given below.\n- A (in percent)\na/r\nHydrosol\nAerosol\n0.075\n1/200\n0.044\n0.176\n0.300\n1/100\n1/50\n0.704\n1.20\n4.80\n2.82\n1/25\n19.2\n1.12.5\n11.3\nWe conclude with an example of the use of (16) for the\ncase of the present hydrosol. Using the set-up suggested in\nExperiment 4, and observing that the path lengths were chosen\nso that b-(a2/2r) = 0, we have:\nN' (r1) = -\n9 meters.\nN' 0.080) = (0.92) (1 e-are)\n= 4.5 meters.\nIt appears that a definitely measurable perturbation of\nthe light field would be induced in the present case. Thus,\nsome radical procedure, such as that outlined in Experiment 4,\nshould be followed in order to obtain an accurate estimate\nof a.\n*The a and oo are associated with a wavelength of about\n478 mu; a and oo are based on measurements taken by J. E. Tyler\nin Lake Pend Oreille and are representative of moderately\nclear lake and nearshore ocean water (Sec. 1.6). Depending\non the medium, the ratio oo/a may range over several orders\nof magnitude. °0 was estimated using the method of -recovery\n(Sec. 13.6).\nBased on Waldram's ([177] p. 48) data for industrial\nhaze, o/a for clear air is on the order of a seventh of that\nfor industrial haze. The associated wavelength is 570 mu; 00\nwas estimated by extrapolation, a highly perilous operation.","SEC. 13.5\nPERTURBED LIGHT FIELDS\n307\nSummary and Conclusions\nThe general equation of transfer (4) for an arbitrarily\nperturbed light field is formulated. From this is deduced\nthe linearized equation or transfer (5) which is applicable\nto the study of slightly perturbed light fields such as those\ninduced during the measurement of the volume attenuation\nfunction (or beam transmittance) by means of the bright- or\ndark-target techniques. The general solutions (6) and (7)\nof the linearized equation lead to analytical expressions\nwhich may be used to estimate the true value of a when either\nthe bright-target approach (8) or the dark-target approach\n(14) is used. The general solution of the linearized equa-\ntion yields in particular five possible experimental proce-\ndures leading to an estimate of a. Finally, two methods are\nbased on (16) and (17) for estimating the order of magnitude\nof the forward scattering effects encountered in beam trans-\nmittance measurements.\nThere remains still another possibility for determining\na and o for a given optical medium, and that is by using\nradical radiometric techniques based on electromagnetic\ntheory (see, e.g., Sec. 126 of Ref. [251]) Admittedly, this\nchanges the main rule of the radiative transfer game (\"to\nsolve radiative transfer problems using only geometrical radi-\nometry and the interaction principle\") However if the solu-\ntion of the problem of determining the structure of the graph\nof o(x;5;.) for 5' near E, continues to elude the most inci-\nsive transfer techniques, then we must not be too proud to\ncall for help from other fields of mathematical physics.\nElectromagnetic theory and quantum theory are right next\ndoor, so to speak, and surely could cast some light on the\nqualitative and quantitative nature of forward- and near-\nforward scattering in natural optical media. Our current\ninability to meaningfully measure or meaningfully predict\nwhat happens for scattering angles less than 1/2°, indicates\none of the more serious shortcomings of an otherwise complete\nand elegant theory of radiative transfer. The reader who\nwishes to see the nature of the o -problem is asked to com-\nplete the graph of o in Fig. 1.72 in a logically defensible\nway.\nIn answering this challenge, the reader is cautioned\nnot to use such shortcuts to o (at least not without justi-\nfication) as may be suggested by the simple superposition of\nsingly scattered radiant flux components from each member of\na dense aggregate of particles comprising the scattering\nvolume. (See, e.g., [308].) It is to be specifically noted\nthat the volume scattering function, as it is correctly used\nin radiative transfer theory, is itself viewable as the so-\nZution of a complex multiple scattering problem defined with-\nin the experimenter's irradiated scattering volume (see Sec. 5\nof [251], and the closing remarks in Sec. 13.12, below).","308\nOPERATIONAL CONCEPTS\nVOL. VI\n13.6 Operational Definition of Volume Scattering Function\nThe operational definition of the volume scattering\nfunction, to which we now devote our attention, is second in\nimportance only to that for beam transmittances (or equiva-\nlently the volume attenuation function) As we found in\nChapter 9, the two concepts a and o form a fundamental set\nfor radiative transfer theory, from which all other optical\nproperties may be deduced. It is therefore important that\nthe operational definition for o be given at least as much\ncare in formulation and experimental realization as that\ngiven to a (or Tr) Thus we start with the basic relation\n(8) of Sec. 3.14 (or (4) of Sec. 3.17) which relates the\nobservable radiances N* and N at a point (and as measured by\nradiance meters without polarizers) with the values of o at\nthat point:\nN*(x,5)\n(1)\n=\nWe shall base our operational definition of o on this\nrelation in order that the practical numerical uses of 0, so\nfound, will be consistent with the theoretical uses of O.\nIn order to isolate the values o (x;5'; 5) of o(x;*;.)\nfor distinct pairs 5',5 of directions, we choose N(x,) so\nthat it is zero over all of E except a small conical subset\nE' about E' of solid angle magnitude S(E) Over E', N(x,.) is\nto be uniform of magnitude N(x,5'). Then (1) becomes.\nN*(x,E)\n=\n= + o(S((E))\n(2)\nwhere o(S(E')) is a quantity such that 0(Q(E'))/8(E) goes to\nzero with SL(E) Matters can usually be arranged in either\nnatural or artificial light fields, so that (2) holds. It\nfollows from (2) that, very nearly:\n(3)\n(i.e., , to within\nEquation (3) suggests the experimental arrangement de-\npicted in Fig. 13.11. An element of volume of the optical\nmedium about point X is irradiated in the direction E' by a\nsource S so that, at x, the irradiating flux is of radiance\nmagnitude N(x,5') and arrives in a solid angle of magnitude\nS.( ) . The incident radiant flux is scattered in all direc-\ntions by the element of volume; some of the scattered flux\nbeing directed in particular along the direction E. At\na\ndistance r' from X a radiance meter G records the path radi-\nances NE(x)E) generated throughout the element of volume,\nwhere l is the length of the element of volume along the di-\nrection E. Matters are so arranged that the beam transmittance","SEC. 13.6\nVOLUME SCATTERING FUNCTION\n309\nr\nl(E)\nSI(E')\nA\nd\nHR\nX\nS\n0 = arc cos E1.E\nA(8)\nSr\nN°(x,E)\nr'\nG\n1\nFIG. 13.11 Arrangement for measuring volume scattering\nfunction.\nTr' (x, 5) is essentially 1, and that the path of sight from G\nto X is essentially dark so that no further flux is added to\nthat comprising No(x,5). From (2) of Sec. 3.12, we may ap-\nproximate N. (x, 5) by:\n()\n&(E)\n,\nso that (3) above yields:\nN*(x,5)\n(x;5';5) = N(x,E')2(E)Q(E\")\n(4)\nwhich we adopt as the operational definition of o(x;5';5).\nMost natural optical media are isotropic, so that the\nvalues o (x;5';5) depend only on 5' . E (= arc cos 0, in Fig.\n13.11). Further, since the o measuring experiment takes\nplace at a fixed location X with irradiation fixed along E',\nwe can usually drop references to X and E' in discussions of\nexperimental results. Thus let us write ad hoc:\n\"o(0)\"\nfor\no(x;5';5)\n\"NT\"\nfor N(x,5')","OPERATIONAL CONCEPTS\nVOL. VI\n310\nN*(x,5)\n\"N*(0)\"\nfor\n\"e(0)\"\nfor\ne(E')\nand:\n\"SiT\"\nfor\nS.(E)\n.\nWe now may inquire about the directional distribution\nof the scattered flux. First we rewrite (4) as :\n(5)\nWhen this operation on the observable quantities Nr , Sr, N°(0)\nand l (0) is examined in detail, we uncover the following set\nof experimental facts:\n(0) is found to be independent of the amount of\n(i)\nirradiation\n(ii) o(0) is independent of the magnitude of l (0).\n(iii) If 0, r, r', , d, and No o are all held fixed and G\nis swung around the beam, o(0) remains fixed.\n(iv) 0(0) is independent of the absolute orientation\nof S and G about X (medium is isotropic)\nThese four experimental findings form the empirical basis for\nthe conclusion that o (0) is an inherent optical property of\nthe medium. (The theoretical basis for this property of o\nis developed in Example 1 of Sec. 3.17.) Clearly, on the\nbasis of (i) , o (0) does not depend on the absolute amount of\nirradiation on the element of volume of the medium. Further-\nmore, on the basis of (ii), the relative amount of flux ob-\nserved to be scattered at a given angle 0 by a small irradi-\nated volume does not depend on the length of the path of\nsight through that small volume. Finally, according to (iii)\nand (iv), o (0) does not depend on the spatial orientation of\nthe plane formed by the direction & irradiating beam and the\ndirection E of flow of the scattered flux. The function o,\nwhich thus depends only on 0 at point X is called the volume\nscattering function at X. Its operational definition is\ngiven by (4) or (5) or succinctly by the equivalent form:\n0(8) =\n(6)\nwhere we have written:\nN ( O )\n\"Nx(0) \"\nfor\n(7)\ne(0)\n,\nN* (0) is a quantity independent of the length l (0) (fact (ii)).\nNr and Sr refer to the radiance and solid angle subtense (at\nthe point x) of the irradiating source. The dimensions of o","SEC. 13.6\nVOLUME SCATTERING FUNCTION\n311\nare: per unit length per unit solid angle. Both the unit of\nlength and solid angle are clearly in the direction E of ob-\nservation of the irradiated volume.\nAn alternate form of (4) may be obtained by observing\nthat, by the definition of surface radiance:\nN°(9)\n(8)\n,\nwhere A(0) is the area of the projection of the element of\nvolume on a plane perpendicular to & (Fig. 13.11), , and SIT\nis the solid angle subtense of the radiance meter's collect-\ning surface at X. Using (8) in (4) we have:\n(9)\nWe next observe that (to within second order .effects)\nand that:\nwhere Hr is irradiance produced by N. over Str, V(0) is the\nvolume of the irradiated element, and J() is the intensity\nof the radiant flux scattered in the direction 5 by the ele-\nment of volume (cf. Sec. 2.9 for a detailed study of the con-\ncept of radiant intensity) With these observations, (6)\nbecomes:\n(8)\n(10)\nSome experimental arrangements may favor (10) over (4) or (5);\nbut actually, to measure (0), one must in essence measure\nNl (0), so that (4) (or (6)) is in the last analysis opera-\ntionally more basic than (10). It is of interest to note\nthat the alternate form (10) can be anticipated by a purely\ndimensional analysis of the path function (cf. note (h) fol-\nlowing Table 3 of Sec. 2.12).","VOL. VI\nOPERATIONAL CONCEPTS\n312\no-Recovery Procedures\nWe consider next the problem of determining the volume\nscattering function at a point X in a medium when it is not\npossible or not convenient to control the light field about\nX in the very special way which led the associated theory\nfrom (1) to (4). Thus, suppose that we know both N* *(x,) and\nN(x,) at point X. Is it possible to recover the information\nabout o(x;;)? We are led by the query to rewrite (1) as :\n(11)\nwhere the n direction sets E partition (1) (see, e.g., Fig.\n13.1). If the partition is fine enough so that N*(x,5i),\nwith Ei in Ei, is representative of all the values No(x,5)\nwith E in Ei, and similarly with the values N(x,5j) and\no(x;5i;Ej), then writing:\n(12)\n\"N*j\"\nfor\nN*(x,5j)\n(13)\n\"N\"\nfor\nN(x,5j)\n(14)\no'ij for\n(11) is approximated by:\n(15)\nN* = No'\nwhere N x and N are n component vectors defined by their n\ncomponents in (12) and (13). Furthermore, o' is an nxn\nmatrix whose entries are defined in (14).\nEquation (15) may be used as a basis for an operational\ndetermination of O. For, apparently what we must do in gen-\neral is to find n linearly independent vectors, Ni, each of\nthe kind appearing in front of o' in (15). If N*i are the\ncorresponding path function values to these n radiance vec-\ntors, then we can make an invertible nxn matrix n whose\nrows are the Ni. As a result, we have from (15) :\no' = n-1 n\n(16)\nwhere N* is the nxn matrix whose rows are the Nxi. In order\nto obtain the n Ni, one may, for example, select n different\nlocations in a part of the medium over which o is known not\nto change. The n linearly independent vectors are also very\nreadily obtained by means of artificial lighting arrangements\nintroduced into the natural light field if necessary.","SEC. 13.6\nVOLUME SCATTERING FUNCTION\n313\nAn alternate means of finding o using natural light\nfields makes explicit use of the generally available prop- -\nerty of o thato(x;5;)depends only on 5' . E. For then\n(1) can be rewritten over again in the form (11), , but now\neach (5') is a spherical zone on (1) symmetric about E', 1 , as\nin Fig. 13.12. If we write, ad hoc\n\"oj\"\nfor .0(x;5';5j)\n(17)\nand\nI\n\"hi(E')\"\nfor\nN(x,5)dd(E)\n(18)\n(E')\n\"N*(5')\" for N*(x,5')\n(19)\nthen (11) may be approximated by:\n(20)\nNow suppose n different orientations 5- of E' 1 are chosen and\nlet h be the matrix made up of rows, the ith of which is:\ni\nFIG. 13.12 General zone of direction space used in sigma- -\nrecovery technique.","VOL. VI\nOPERATIONAL CONCEPTS\n314\n(51)\nand arrange matters, if possible, so that h is invertible.\nThen the system of equation (20) takes the form:\n\"+=ho\"\nwhere N* is the vector whose ith component is N* (51), and o'!\nis the n component vector whose ith component is i From\nthis we have:\n0\" = h ' N*\n(21)\n=\nDetailed calculations based on (21) may be found in\nRef. [214], along with a survey of practical schemes for in-\nverting h.\nDetermining the Volume Scattering\nMatrix in the Polarized Case\nWhen the radiance meter is fitted with analyzers for\npolarized radiance, as described in Sec. 2. 10, then a whole\nnew order of reality is opened to the empirical exploration\nof polarized natural light fields. A11 unpolarized or pola-\nrized radiative transfer phenomena previously viewed with an\nordinary radiance tube now spring from a one-dimensional sim-\nplicity to a rich four-dimensional complexity. In the case\nof the volume scattering function, the simple scalar opera-\ntional definition (4) is raised to a matricial operational\ndefinition in a manner now to be explained.\nIrradiate an element of volume about point X by radi-\nant flux such that the associated incident polarized radi-\nances are in turn four linearly independent vectors:\n1N(x,5')\n(22)\n3N(x,5')\nNNx\nThese radiance vectors are local observable vectors, as de-\nfined in Sec. 2.10. The geometric aspects of the scattering\nsituation in the present case are described once again by\nFig. 13.11. Let the path radiances generated by the irradi-\nated volume be:\n(23)\n3\n(Ng(x,5)","SEC. 13.6\nVOLUME SCATTERING FUNCTION\n315\nrespectively. Thus iN(x,5') generates iN (x,5), i = 1, 2,\n3, 4, where l is the length of the path of sight through the\nirradiated scattering volume.\nLet N(x,5') be the matrix formed by using the vectors\n(22) as rows; and let No(x) be the matrix whose rows are\nthe vectors (23). Then, emulating (4) as closely as possible,\nwe write:\n[N(x,5')]]' Ng(x,5)\n\"o(x;E';E)\"\nfor\n(24)\ne(E)S(E)\n.\nThus, by measuring the four linearly independent vectors\n(22) and their associated vectors (23) and by performing the\nindicated operations in (24), a 4 x 4 matrix o (x;5'; 5) is ob- -\ntained for each choice of x, E', and E. This matrix is the\nLocal Observable Volume Scattering Matrix.\nIt is not within the scope of the present work to pur-\nsue further this matter of the polarized version of the vol-\nume scattering function. A discussion of the operational\ndefinition of o and of how it enters into the theory both on\nfoundational and practical levels, may be found in Chapter 12\nof Ref. [251].\nHowever, before leaving this matter it is important to\nobserve that there exists at least one set of linearly inde-\npendent polarized radiance vectors that experimenters may use\nfor their irradiation vectors. Thus consider the first three\nof the linearly polarized observable radiance vectors listed\nin Sec. 2.10, along with the right circularly polarized radi-\nance vector. That is, consider:\n(vertical linear)\nN°=\n(0,2N,N,N)\n(horizontal linear)\nN = 1 2\n(N,N,2N,N)\n(45° linear)\n(N,N,N,0)\n(right circular)\nwhere N = 0. The determinant A of these four vectors is:","VOL. VI\nOPERATIONAL CONCEPTS\n316\n2 0 1 1\n2 0 1 1\n2\n1\n1\n=N\n0\nA = N 2\n0\n2\n1\n1\n=\n2\n1\n1\n2\n1\n1\n1\n2\n1\n0 0 1 1\n1\n1\n1\n0\n2\n0\n0\n0\n2 1 1\n0 2 1 1\n= N 2\n1 2 1\n= N\n1 1 2 1\n0 1 -1\n0 0 -1 1\n2 0 0\n= 2N(-2+1) = -2N # 0\n(25)\n1 2 1\n= N\n0 -1 -1\nHence the four given vectors above are linearly independent.\n13.7 Direct Measurement of the Volume Total Scattering\nFunction\nOur purpose in this section is to outline a method of\ndirect measurement of the volume total scattering function S.\nWe will be concerned principally with the derivation of the\nformula behind the method and a few words about the kinds of\nequipment that may be needed for the realization of the\nmethod. The general method outlined below will be applicable\nto any scattering medium, but it appears to be most useful in\nthe experimental study of natural hydrosols. A few addition-\nal preliminary observations will help orient the reader and\nplace the present discussion in the appropriate perspective.\nThe two fundamental inherent optical properties of scat-\ntering-absorbing media are the volume scattering function o,\nand the volume attenuation function a. The pair (a,o) is\nfundamental in the sense of Def. 3 of Sec. 9.1, i.e., that\nfrom these two, all other attenuation functions, either in-\nherent or apparent, may effectively be found in accordance\nwith certain well-defined rules of computation. However, the\npair (a,0), while being fundamental in the sense just explained,\nis by no means unique. Thus the pair (a,o) is also fundament-\nal, where a is the volume absorption function. The connect-\ning link between these two fundamental pairs of optical func-\ntions is provided by the notion of the volume total scatter-\ning function S, defined by writing:\nS\nods\n\"s\"\nfor\n(cf., (3) of Sec. 4.2). For then, by definition (i.e., (4)\nof Sec. 4.2) we have a = a + S, so that the pair (a,a) is\nknown once (a,o) is, and conversely.\nBy means of the work in Chapter 3, and also that of\nSecs. 13.4-13.6, the theory and practice of the direct mea-\nsurement of a and o, is now well established. As was shown,\nthe measurement of a is accomplished by various beam trans- -\nmittance procedures all of which, at their core, spring from","SEC. 13.7\nTOTAL SCATTERING FUNCTION\n317\na single analytical relation, the interaction principle of\nChapter 3. The direct measurement of o is accomplished by\nspecially designed instruments known as o-meters which mechan-\nically mimic the general definition of O. Even the volume ab- -\nsorption function a, once an elusive quantity which could be\ndetermined only indirectly, now has a direct, simply realiz-\nable method of determination either in the field or in the\nlaboratory as we shall see in Sec. 13.8. We now round out\nthis list by discussing a direct method of determining S, and\nthereby supplementing the usual indirect methods which ob -\ntain S from an integration of o over E, or from a and a by\nmeans of the formula: S = a - a.\nThe General Method\nThe general direct method of determining S makes use of\nthe close connection between the volume scattering function o\nand the path function N* at some point X in an arbitrary op-\ntical medium X (cf., (8) of Sec. 3.14)\nN*(x,5)\n=\n(1)\n.\nRecall that N* (x,5) is a radiance per unit length at in\nthe direction E generated by scattered radiances N(x,5') at\nX arriving along the directions E' (see (7) of Sec. 13.6).\nHere E is as usual the collection of all units vectors (the\nunit sphere) in E3 The practice of measuring N* (x, 5) is\nrelatively highly developed both in the atmosphere and the\nhydrosphere, and we shall make use of this fact in develop-\ning the general method of measuring S.\nRecall that the value S (x) of the volume total scatter-\ning function at X is given by:\n(x) =\n(2)\n,\nin any isotropic medium.\nThen returning to (1) and in-\ntegrating each side over (1) we have:\n= o(x;5';E) (5') ] ds(E)\n(3)\nThe order of integration may generally be reversed on\nthe right hand side of (3), the result being:\n(4)\nThe inner integral is the value S (x) of S at X. Since\nS (x) is independent of E', it may be placed before the outer\nintegral sign of (4), thus:","VOL. VI\nOPERATIONAL CONCEPTS\n318\ns(x)+ N(x,5')dd(E')\n(5)\n.\nThe integral term in (5) is the value h (x) of the scalar ir-\nradiance function h at X. As a result, the right side of (3)\nbecomes:\ns(x)h(x)\nIn analogy to h(x), , we define (as in (23) of Sec. 9.3) the\nleft side of (3) by writing\n1\nN+(x,E)ds(E)\n(6)\n\"h*(x)\"\nfor\nCombining these results, we have the desired basic formula\nfor the general direct method of determining S (x) :\ns (x) = held\n(7)\nObservations\nObserve first of all that equation (7) is quite general.\nNo assumption has been made about the angular distribution of\nthe radiance about point X. Furthermore, o is quite arbitrary\nin the angular structure at X. The only assumption made, and\na quite reasonable one at that, concerns the isotropy of X at\npoint X, i.e., that o is invariant under a rotation of\nthe local coordinate frame about X; in other words, that\no (x;5';5) = o (x;51;51) whenever E . us = 51 o E 1 (re: (8) of\nSec. 7.12).\nObserve next the structural similarity between (7) and\nthe classical formula:\na = N* N\n(8)\nfor the determination of the volume attenuation function a\n(see (9) of Sec. 13.4). In order to obtain the simple form\n(8), quite a few assumptions about the medium and light field\nmust be made. This is not, however, the case for S.\nFurther, it is interesting to observe that (8) is asso-\nciated with summations of N over all points along a line of\nfixed direction, while (7) is associated with summations of\nN over all directions of lines through a fixed point. In\nthis sense, the concepts a and S may be classed as dual con-\ncepts with respect to the phase space X X E (cf., , Fig. 3.34). .","SEC. 13.7\nTOTAL SCATTERING FUNCTION\n319\nFinally, we note that the measurement of S (x) can be\naccomplished by rotating a short-path radiance meter about x,\nwhich thereby determines N, (x, E) for each 5. The result is\nthen integrated either automatically or manually over E to\nobtain h* (x). The theory and practice of measurement proce-\ndures for h(x) are well known (cf., (17) of Sec. 13.1).\nTwo Special Methods\nIf one has control over the lighting conditions and the\nhomogeneity of the medium, as may be possible in the labora-\ntory, the basic relation (1) and the general formula (7) yield\na number of particularly simple methods for the determination\nof X. We now briefly consider two such methods.\nCylindrical Medium\nSuppose a narrow circular cylindrical tube of length r\nis filled uniformly with the scattering material under study.\nThe inner walls of the tube are lighted so that at all points\nalong the axis of the tube, the radiance distribution is angu-\nlarly uniform of magnitude N. Thus, under the assumptions of\nan angularly uniform N and a homogeneous medium, (1) reduces\nto:\nN* = N s\n(9)\n.\nIf an observation point is at one end of the tube so that a\nline of sight may be directed along the axis of symmetry to\nthe other end, which has zero inherent radiance, one would\nexpect to observe an apparent radiance Nr of magnitude:\nr1-ear\n(10)\nSuppose the tube is constructed so that r may be varied.\nThen if ar is small, (10) yields:\nNT = N s r\n(11)\nHence by plotting Nr vs. r, one may look for the region of\nlinearity of Nr. The slope of the plot in this region is\nsimply Ns. If N is known, then S is determinable.\nBy lengthening r, (10) indicates that eventually the\nreadings Nr must level out, the plateau being of magnitude:\n(12)\na\nKnowing Nr and N, we may then estimate the ratio p = s/a,\nwhich is the well-known scattering-attenuation ratio or","OPERATIONAL CONCEPTS\nVOL. VI\n320\nalbedo for single scattering, a quantity which plays an impor-\ntant role in both the theory and application of radiative\ntransfer.\nIt is clear that if experiments leading to both (11)\nand (12) have been made, then a and S are both determinable\nby this simple scheme, whence follows the volume absorption\ncoefficient also: a = a - S.\nSpherical Medium\nSuppose an integrating sphere of internal radius r is\nfilled uniformly with the scattering material under study.\nSuppose further that the inherent radiance distribution of\nthe inner surface is uniform and of equal magnitude at each\npoint. The sphere is fitted with a small viewing port which\nallows a radiance tube an unrestricted view along a diametral\nline from the port, through the center, to a small circular\nregion of inherent radiance zero on the far portion of the\ninner surface. Let the observed radiance of the circular\nregion be N, and suppose a scalar irradiance probe records\nthe amount h at the center of the spherical cavity. Then,\nbecause of the symmetry of the light field at the center of\nthe cavity, except perhaps for the tiny dark patch and obser-\nvation port, we would expect N, to be essentially independent\nof direction and (for not too dense a medium so that the black\npatch is clearly visible) very nearly of magnitude N/2r (cf.\n(3) of Sec. 13.3). It follows that h* = 2N/r and that, by\nmeans of (7):\nS = 2TN hr\n(13)\nIf, on the other hand, the medium is made optically dense, so\nthat the black patch is not visible, then the observed N would\nbe very nearly Nx/a, where N x is the value of the path func-\ntion at the center of the sphere. Therefore in this case:\nS = 4Na h\n(14)\nso that under the present circumstances, unless a is already\nknown, one may determine only:\ns/a = 4 NN\n(15)\nSeveral variants of the above methods are immediately\nrealizable. For example, a set of three spheres of diameters\none, two, and three units, say, are constructed and, using\noptically rare media of equal density, plot the quantity 2N/h.","SEC. 13.8\nVOLUME ABSORPTION FUNCTION\n321\nThen, according to (13), if this varies linearly with r, the\nconditions of (13) are satisfied and the slope of the line is\nnone other than the required value of S.\n13.8 Operational Definition of Volume Absorption Function\nThe volume absorption function a (x), by its very nature,\nmust initially enter the domain of radiative transfer theory\nin an indirect manner, that is, by means of the defintion\nwhich characterizes a (x) as the difference between a (x) and\nS (x), as was done in (4) of Sec. 4.2. In this section we\nshall deduce from this definition alternate operational pro-\ncedures which lead to determinations of a (x) by means of\ndirect measurements of radiometric quantities.\nThe formula which serves as the basis for the present\noperational definitions is (15) of Sec. 8.8. Thus, solving\nfor a (x) we have:\n(1)\nThis operational definition of (a)x inverts the usual\nway of looking at the divergence relation (15) of Sec. 8.8.\nIn (1) we imagine the relation to have its roots in the oper-\nations of the real radiometric world. By performing the in-\ndicated operations on the right side in (1), a(x) is thereby\ndetermined virtually independently of (x) and S (x).\nFig. 13.13 depicts an operational realization of equa-\ntion (1). Suppose a janus plate (i.e., an instrument which\nmeasures net irradiance H(x,5), as depicted in Fig. 2.21) is\noriented so that it measures H(x,i), H(x,j), H(x,k) at a\npoint X in an optical medium. Suppose further that the\njanus plate is moved back and forth along the i-axis a dis-\ntance r about each side of x, as shown in Fig. 13.13. Then:\nH(x+ri)- H(x - ri)\n-\n2r\n,\napproximates to:\naH(x,i)\ndr\n,\nfor suitably chosen r. The remaining two derivatives occur-\nring in the representation of the divergence V . H can be ap-\nproximated similarly. If the scalar irradiance (x) at X is\ndetermined, then (1) leads to a (x) Equation (1) also sug-\ngests several alternate modes of measurement of a (x) in which\nspherically or cylindrically symmetric light fields, artifi-\ncially induced, can lead to appropriate forms of V H using\nspherical and cylindrical coordinate systems. These special\ninstances of (1) are best left for adaptation, by interested\nresearchers, to individual cases (cf. , e.g., (43) of Sec. 9.2).","VOL. VI\nOPERATIONAL CONCEPTS\n322\nZ\n(\n2r\nH(x,i)\nFIG. 13.13 Required motions of a janus-plate probe about\na point X which can lead to an estimate of the divergence of\nthe irradiance vector.\nProcedures for Stratified Light Fields\nIn natural light fields, which are usually found to be\nstratified, (1) reduces to:\ndH(z,k)\n(2)\nwhich shows how to find a(z) at depth Z in terms of the ver-\ntical derivative of net irradiance H(z,k).\nWhen a natural stratified light field decays at a regu-\nlar (the asymptotic) rate k with depth Z in an homogenous\nmedium, then (2) suggests the following integrated form:\na = h(y)[1-e-k(2-y) ] k[H(z,k) - H(y,k)\n(3)\nfor any two depths y,zy < z in the medium. By measuring\nH(z,k) and H(y,k), , along with h(y) and k, it follows that\na(=a(z) for all depths z) is determinable using (3).","SEC. 13.8\nVOLUME ABSORPTION FUNCTION\n323\nProcedures for Deep Media\nIf the optical medium is known to be optically infinite-\nly deep, then, in (3) we let Z = 80, so that, since H(00,k) = 0, ,\nwe have:\nH(y,-k) = ah(y)\n(4)\nwhich is the basis for the formula:\nH(y,-k)[1- R(y,-)]\n(5)\nfor every depth y; and was considered earlier in Sec. 10.8,\nalong with alternate useful estimates of a.\nGeneral Global Method\nFormulas (3) and (5) are based on special cases of the\ngeneral formula:\na P(S,-)/vU(X)\n(6)\nwhich holds for any homogeneous subset X of an optical medium\nin which the speed of light is V and whose radiant energy con-\ntent is steady, with magnitude U(X). Thus by measuring U(X)\nand the net inward flux P(S,-) across the boundary S of X, a\nis determinable (cf., (33) of Sec. 8.8 for a derivation and\ndiscussion. See also (8) of Sec. 10.8). Equation (6) should\nsuggest to experimenters several powerful means of determin-\ning the volume absorption coefficient a in both laboratory\nand natural optical media, (Compare (5) of Sec. 5.13 for an\nearlier application of the principle behind (6) )\nFurther Procedures for General Media\nWe close this discussion of the methods of experimental\ndetermination of a (x) by observing that in stratified light\nfields, (25) of Sec. 9.2 may be solved for a(z) so that: we\nobtain the following exact formula for a (z):\n(7)\nThus a (z) may be obtained at any depth knowing the D, R, and\nK functions for irradiance. Equation (7) is quite general,\nso that the medium may be arbitrarily stratified, of arbi-\ntrary depth, and in which the angular dependence of N(z,.)\nis arbitrary at each a.","VOL. VI\nOPERATIONAL CONCEPTS\n324\nFinally, , we note that if we have a spherically symmetric\nfield about a submerged light source, the estimation of a (s)\nat radial distance S from the source is given by (44) of Sec.\n9.2.\n13.9 Operational Procedures for Apparent Optical Properties\nThe operational definitions of the apparent optical\nproperties of stratified natural hydrosols are given in de-\ntail in Sec. 9. 2, so that our present discussion may be lim-\nited to a brief summary of their definitions with particular\nattention to various features of the general depth behavior\nof the properties observed in natural waters. These features\nshould be helpful in devising experimental procedures for the\nmeasurement of the apparent optical properties.\nThe principal apparent optical properties for strati-\nfied plane-parallel media are given in the following list.\n(1)\n=\nH(z,F)\n(2)\nthe\n=\ndH(z,+)\n1\n(3)\ndz\ndh(z,\n1\n(4)\n(5)\nThe preceding group of properties falls into two divi-\nsions: the first consists of the distribution function pair\n(1). The second division is the main group (2) - (5) of appar-\nent optical properties and consists of the seven concepts\nshown. Closely associated with these concepts and lying\nhalfway between them and the inherent optical properties a, o\nare the hybrid optical properties:\n(6)\n(z,+)=a(z)D(z,) =\n(7)\n(z, ) = s(z)D(z,=)\n(8)\na (z,-) = a(z)D(z,+)\nalong with:\nf(z, and b(z,\n(9)\n,\nas given in Table 4 of Sec. 9.6, or (7) and (8) of Sec. 8.3.","SEC. 13.9\nAPPARENT OPTICAL PROPERTIES\n325\nThe Fundamental Irradiance Quartet\nThe principal family of apparent optical properties of\nstratified natural waters consists at present of a set of\nseven quantities (2)-(5) - whose numerical values depend on the\nangular structure of the light field as well as on the physi-\ncal composition of the water.\nThe first observation we can make which is pertinent\nto the determination of the seven apparent optical properties\nis that they can be obtained from four basic irradiance mea-\nsurements. These measurements take the form of two pairs of\nirradiance quantities: one pair consists of ordinary irradi-\nances, the other of scalar irradiances. In each of these\npairs, one member is assigned to upwelling flux, the other to\ndownwelling flux in the medium. The reason that there are\nprecisely four such quantities stems from our conceptual de-\ncomposition of the flow of radiant energy in any natural hy-\ndrosol (stratified or not) into two streams : an upward flow-\ning stream and a downward flowing stream across each horizont-\nal plane in the medium (cf., the two-flow theory of irradiance\nin Chapter 8).\nThe four basic irradiances are:\nH(z,+)\nH(z,-)\nh(z,+)\n(z,-)\n(10)\nH(z,+) and H(z,-) are the upwelling (+) and downwelling (-)\nirradiances, respectively (Sec. 2.4). They are induced by\nthe upwelling and downwelling flux streams at depth Z. These\nquantities may be obtained from field radiance measurements,\nor they may be measured by flat plate collectors exposed to\nthe appropriate hemispheres. See Fig. 13.14. In like manner,\nh(z,+) and h(z,-) are the upwelling (+) and downwelling (-)\nscalar irradiances, and refer to upwelling and downwelling\nflux, respectively, at depth Z (Sec. 2.7). They may be ob-\ntained by simple numerical procedures from field radiance\nmeasurements. Alternatively, appropriately shielded spheri-\ncal collectors may be used to measure these quantities. A\npossible experimental arrangement is shown in Fig. 13.15.\nSee also Fig. 2.18. Observe that the collectors are complete\nspheres in each case. The sphere that measures h (z-), for\nexample, should be shielded from the upwelling flux by some\ndevice which at the same time impedes as little as possible\nthe interchange of flux across the horizontal plane at depth\nZ. In analogy to our discussions of the relation between h\nand II we can show that the downwelling spherical irradi-\nance (z,-), measured by the shielded sphere shown sche-\nmatically in Fig. 13.15(a), is related to h ( z - ) by:\n(11)\n=\n.","VOL. VI\nOPERATIONAL CONCEPTS\n326\nIncident Flux at Upper Boundary\ndH(z,+)\na(z)= h(z) I dz\n-\nH(z,+) = H (z,+) - H(z,-)\nZ\n/\nh4t((z)\nH(z,+)\n+\n1\nH(z,-)\nFIG. . 13. 14 Schematic arrangements for measuring up (+)\nand downward (-) irradiance and scalar irradiance at depth Z.\nThese measurements lead to exact calculations of the volume\nabsorption function values a(z). .\nh47 477 ( z, 9 -)\nZ\nshield\nshield\nh 4TT (z,+)\n(b)\n(a)\nFIG. 13.15 Schematic arrangements for measuring up (+) and\ndownward (-) spherical irradiances at depth Z.","APPARENT OPTICAL PROPERTIES\n327\nSEC. 13.9\nSimilarly, the upwelling spherical irradiance h (z,+), mea-\nsured by the other shielded sphere shown schematically in\nFig. 13.15(b), is related to h(z,+) by:\n(12)\n.\nThese relations are simply definitional identities based on\nthe discussion of Sec. 2.7. The demonstration of the connec-\ntion between h4tt and the spherical irradiances defined above,\nassuming ideal shielding is straightforward. The result is:\n= (z,-) + han (z, +)\n(13)\nFurthermore, as in (9) of Sec. 2.7:\nh ( z) = h(z,-) + h(z,+)\n(14)\nThe practical connection between h4t 4 TT (z) and h(z) is\ndiscussed in Sec. 13.1; in particular, see (17) of Sec. 13.1.\nThis connection holds also between h4tt(z,=) and h (z, )\nQb-\nserve that the connection factor Cr in (17) of Sec. 13.1 also\nholds for the hemispherical irradiance connections.\nDiscussion of the Reflectance Functions\nThe physical interpretation of R(z,-) in (2) is straight-\nforward: It represents the ratio of upwelling irradiance at\ndepth Z to the downwelling irradiance at depth Z, WO that R(z,-)\nmay be thought of as the reflectance, with respect to the down\nwelling flux, of a hypothetical plane surface at depth Z in the\nmedium. For completeness, we have included in (2) the reflec-\ntance R(z,+) for the upwelling stream. However, this is simply\nthe numerical reciprocal of R(z,-). In actuality, R(z,-) de-\npends on the scattering properties of the entire medium above\nand below level z. It will also depend in part on the reflec-\ntance properties of the upper and lower boundaries of the me-\ndium if these are within sight of the flux collectors. R(z,-)\nis not an inherent property of the medium (as defined in Sec.\n9.3) for experiments and theory show in general that for a\ngiven medium and a given depth in that medium, the value\nR(z,-) changes with the external lighting conditions. Some\ntheoretical relations helpful in studying these variations\nare given in Sec. 9.4.\nRelations (2) are completely general: They apply to\nany medium, be it deep or shallow, irradiated by the sun in\na clear sky or by any type of overcast. Because of this gen-\nerality, very little can be said about exactly how the values\nor R(z,-) should depend on depth. No simple statements beyond\nthose made in Chapters 9, 10, and 11 can be made which assert\nthat R(z,-) should always increase with depth, or that it\nshould always decrease with depth, or that it should go through\nmaxima or minima at certain depths, and so on.\nDespite this unwillingness of R(z,-) to have its charac-\nteristics typed generally and in very fine detail, there are","328\nOPERATIONAL CONCEPTS\nVOL. VI\ncertain gross characteristics as we have seen in Chapter 9,\n10, and 11, which make it an indispensable tool in engineer-\ning calculations: in optically deep homogeneous hydrosols,\nit is an observational fact that R(z,-) varies very little\nwith depth. Near the surface of these media, it shows rela-\ntively high variability with depth which depends on the state\nof the surface and incident lighting patterns, but soon set-\ntles down and approaches a constant value independent of depth.\nR(z,-) thereby takes on the status of an apparent optical\nproperty of the medium. Furthermore, in media that have no\nself-luminous organisms, R(z,-) behaves as any respectable\nreflectance should: It is never greater than 1. In fact, in\nmost clear natural hydrosols the values of R(z,-) are usually\nfound to be somewhere in the neighborhood of 0.02, give or\ntake 0.01, for midspectrum wavelengths (around 550 mu). For\nturbid, nearshore or inland waters, R(z,-) can rise to 0.08\nor more. In media containing self-luminous organisms distrib-\nuted throughout some layer it is quite possible, however, for\nthe values of R(z,-) to approach 1 as this layer is approached,\nand even become greater than 1 just before it enters the lay-\ner. Some examples of R(z,-) are given in Table 1.\nWhile the problem of the find detail of the depth de-\npendence of R(z,-) is mainly of academic interest, we note\nthat there is no dearth of theoretical approaches to this in-\nteresting problem. One model of the light field which is\nparticularly useful in the study of this problem is the two-D\ntheory of Chapter 8. This model is relatively simple to use\nand is still sufficiently detailed to supply a multitude of\nTABLE 1\nExamples of the values of D(z, K(z, a (z), R(z,\nm\nD(z,-)\nD (z, +)\n(z, )\nK(z,+)\na (z)\nR(z,-)\n(meters)\n4.24\n1.247\n2.704\n0.0215\n-\n-\n7.33\n0.129\n0.126\n-\n-\n-\n10.42\n1.288\n2.727\n0.153\n0.150\n0.115\n0.0184\n13.50\n0.178\n0.174\n-\n-\n-\n-\n16.58\n1.291\n2.778\n0.174\n0.172\n0.118\n0.0204\n22.77\n0.171\n0.170\n-\n-\n-\n-\n28.96\n1.313\n2.781\n0.169\n0.169\n0.117\n0.0227\n35.13\n0.167\n0.167\n-\n-\n-\n-\n41.30\n1.315\n2.757\n0.165\n0.165\n0.117\n0.0235\n47.50\n0.162\n0.163\n-\n-\n-\n-\n53.71\n1.307\n2.763\n0.158\n0.158\n0.112\n0.0234\n59.90\n0.154\n0.154\n-\n-\n-\n-\nExplanation of Table 1: Depths and units are in terms\nof meters. Data are associated with a wavelength of 480 mu\nand were derived from radiance information summarized in Ref.\n[298]. The optical medium (Lake Pend Oreille, Idaho) was\nfound to be essentially homogeneous; the volume attenuation\ncoefficient being a = 0.402 per meter. The sky was clear\nand sunny with the sun at about 40° from the zenith. The\nvalues a (z) were obtained by means of (2) of Sec. 13.8.","SEC. 13.9\nAPPARENT OPTICAL PROPERTIES\n329\nexamples of the depth dependence of R(z,-) : It supplies\ncases in which R(z,-) can increase or decrease over pre-\nselected depth ranges. (See Sec. 10.4.) In all cases, how-\never, the model states that there is some value Roo which\nR(z,-) approaches asymptotically with depth in optically in-\nfinitely deep media. This asymptotic value depends in a cal-\nculable way on both the inherent optical properties of the\nmedium and on the limiting lighting conditins. Further dis-\ncussion on the behavior or R(z,-) at great depths are made in\nChapter 10.\nDiscussion of the Distribution Functions\nA particularly simple means of characterizing the depth\ndependence of the shape of radiance distributions, without\nresorting to an actual measurement of the radiance over all\ndirections at each depth, is given by the distribution func-\ntions:\nD(z,+) = h(z,+)\nIt is readily seen from the definitions of h and H that if\nthe shape of the radiance distribution changes with depth,\nthen D(z,-) and D(z, +) will change with depth; and conversely,\nif the values of the distribution functions vary with depth,\nthe radiance distributions must be changing shape with depth\n(cf. Sec. 8.5). It is clear from the definitions that D(z,-)\ngives an index of the shape of the radiance distribution in\nthe upper hemisphere (i.e., for the downwelling flux), and\nD(z,+) does a similar job of characterizing the shape of the\nradiance distribution in the lower hemisphere (i.e., for the\nupwelling flux).\nNumerical analysis of detailed experimental studies of\nthe light field by Tyler in Lake Pend Oreille show that both\nD(z,+) and D(z,-) exhibit relatively little change with depth\n(cf. Ref. [306]) Furthermore, this independence of depth is\nfound whether the external lighting conditions are sunny or\novercast. Under either of these conditions, and for blue-\ngreen light, the values D(z,-) hovered very closely in the\nneighborhood of 1.3, while the values D(z,+) clustered around\n2.7. Examples of D(z,-) and D(z, +) are given in Table 1. It\nappears at present that these values should be typical of the\nvalues that one may find in many natural hydrosols and for\nthe blue-green portion of the spectrum.\nof course, as in the case of R(z,-), the quantities\nD(z,-) and D(z,+) will obstinately refuse to have any sweep-\ning generalizations made about the fine structure of their\ndepth dependence. However, as in the case of R(z,-), , simple\ntheoretical tools exist which can be directed toward such\nproblems if the need ever arises to discuss depth dependence\nin detail. These are given in Chapters 9 and 10. Further-\nmore, the ultimate depth dependence of D (z, ) and D(z,+) in","VOL. VI\nOPERATIONAL CONCEPTS\n330\ndeep media is quite regular and predictable, as was shown in\nChapter 10, as a consequence of the proof of the asumptotic\nradiance hypothesis.\nThe observed constancy of the distribution functions\nwith depth has important practical consequences. In homo-\ngeneous media exhibiting this type of behavior a few well-\nselected measurements of the inherent optical properties to-\ngether with radiance distributions near the surface would\nsuffice as the basis for an estimate of the quantity and\nquality of the light field for all depths in the medium.\nSuch estimates could be made by means of the two-D model of\nChapter 8' or the simple radiance model built around (2) of\nSec. 4.4.\nIn addition to characterizing the depth dependence of\nthe angular structure of radiance distributions, as explained\nin Sec. 8. 5, D(z,- and D(z,+) play indispensable roles in\nthe equations of applied radiative transfer theory, particu-\nlarly in those equations which link the inherent and apparent\noptical properties of a medium. These roles are illustrated\nas a matter of course in the discussions throughout Chapters\n8, 9, and 10.\nDiscussion of the K-Functions\nThe reflectance function, as we have seen, gives a run-\nning account of the relative magnitudes of the irradiance of\neach stream of radiant flux. The quantities which character-\nize the individual depth dependence of the upwelling and down-\nwelling irradiances and of the scalar irradiance of the streams\nat each depth are called the K-functions. The motivations\nfor the operational definitions of these functions are sup-\nplied by both theoretical and experimental precedent extend-\ning back over at least fifty years of applied radiative trans-\nfer theory, and we shall now devote some discussion to the\nconcepts centering around their definitions.\nThe theoretical motivation for the K-functions for ir-\nradiance and scalar irradiance stems from an attempt to in-\ncrease the usefulness of the Schuster equations for the two-\nflow analysis of the light field. The detailed developement\nof this approach and its practical applications have been\nthoroughly explored in Chapter 9 (see Sec. 9.2).\nThe experimental motivation for the K-functions rests\nin early empirical relations of the kind:\nO e-Kz\n(15)\nwhich simultaneously were to characterize the depth depen-\ndence of I z and define its logarithmic depth-rate of decay\nK. In the above relation, the quantity I took many forms:\nin some studies it was downwelling irradiance, in others it\nwas a scalar irradiance-like quantity; in still others, its\nexact nature was not quite clear. In every case, however,\nit was intended to be some measurable \"intensity\" of radia-\ntion. Nevertheless, there was no universal agreement as to\nwhat \"intensity\" meant and what radiometric quantity it","SEC. 13.10\nLOCAL AND GLOBAL PROPERTIES\n331\nshould represent. As a result, there was no agreement as to\nwhat was really measured. A plot of I on semilog paper with\ndepth as abscissa yielded -K as the slope of a curve which\noften appeared visually as a straight line. K could thus be\ndefined operationally, via the equation:\n(16)\nIt suffices to observe here that these early theoreti-\ncal and experimental approaches to characterize a K-like op-\ntical property of natural hydrosols were inadequate to the\nsubsequent needs for precision and completeness in modern\nhydrologic optics. In current basic research I z is explicitly\nreplaced by any of the three precisely defined irradiances\nH(z,-), H(z,+) and h(z). Furthermore, it has become neces-\nsary to distinguish not only between the magnitudes 1(z,-)\nH(z,+), and h(z), but also their logarithmic rate of change\nwith depth. Careful measurements (see, e.g., Table 1) show\nthat their logarithmic rates of change are generally differ-\nent, and the difference far exceeds the range of experimental\nerror. In general, semilog plots of H(z,-), H(z, +0, and (z)\nalso exhibit noticeable departures from linearity, especially\nin near-surface regions. This fact, of course, is part of\nthe folklore of the study of hydrologic optics which has been\nextant for many years, but this nonlinearity has been con-\nsidered more of an annoyance than a source of enlightening\ninformation. In particular this nonlinearity made it impos-\nsible to rigorously define a single unambiguous fixed number\nK, of the kind appearing in (16) which otherwise could be\nused to help classify the optical properties of the medium.\nThe current views in hydrologic optics are such that\nthe departures from linearity by semilog plots of H(z,-),\nH(z, and (z), and even h(z,+) are a source of extremely\nuseful insight into the intricate structure of real light\nfields in natural hydrosols. Far from being ignored, these\ndepartures from linearity should be welcomed as harbingers of\nnew and deeper understanding. The logarithmic slopes of the\nH(z,=), h(z,=), and h (z) plots are defined in general as in\n(3)-(5). Useful interrelations among these magnitudes which\ncan help guide the reduction of empirical data and further\nunderstanding of radiative processes in natural hydrosols\nare developed throughout Chapter 9 and 10.\n13.10 Theory of Measurement of Local and Global R and T\nProperties\nIn this section we invert the usual way of looking at\nthe principles of invariance, and more generally the interac-\ntion principle, and show that, by so doing, we encounter new\nways in which to measure the inherent and apparent properties\nof optical media. We have already used to advantage this\npoint of view throughout all the preceding sections to find\noperational definitions of such local properties as a, o, a,\nS, the K functions, R(z, and (z, The results of Sec.\n7.8 are also pertinent to the present discussion. Now we wish\nto show how these procedures can be extended to the complete","VOL. VI\nOPERATIONAL CONCEPTS\n332\nset of R and T factors (or operators) for plane-parallel\nmedia. As a consequence we shall be able to solve many prob-\nlems of the second class in radiative transfer theory (re:\nSec. 2 of Ref. [251]), i.e., problems which require the de-\ntermination of the apparent and inherent optical properties\n(either local or global) of an optical medium, given the\nradiometric field throughout the medium or on its boundaries\n(or both). This problem is also referred to more descriptive-\nly as the inverse problem of radiative transfer theory. We\nnow go on to consider some examples of inverse problems in-\nvolving local and global optical properties.\nExample 1: R and T Factors in Homogeneous\nPolarity-Free Settings\nTo show the basic point of view taken in the formula-\ntion of inverse problems consider a plane-parallel medium\nX(a,b) in which we can measure the upward and downward irradi-\nances H(y,+) at depths y in X(a,b). Can we make enough mea-\nsurements, and of the right kind so as to be able to compute\nthe reflectance and transmittance R(x, z) and T(x,z) of an\ninternal submedium X(x,z) ? An examination of the principles\nof invariance for general irradiance settings (1), and (2)\nof Sec. 8.1):\nH(y,+) H(z,+)T(z,y) + H(y,-)R(y,z)\n(1)\nH(y,-) = H (x,-)T(x,y) + H(y,+)R(y,x)\n(2)\nshows that we should first measure the upward and downward\nirradiances at levels X and Z in X(x,z). The relations (1)\nand (2) reduce in this case to:\n(3)\nH(x,+) = H(z,+)T(2,x) + H(x,-)R(x,z)\nH(z,-) = H(x,-)T(x,z) + H(z,+)R(z,x)\n(4)\nin which x < Z. Next if it is possible to invoke the symmetry\nconditions :\n(5)\nR(x,z) = R(z,x) =\nT(x,z) = T(z,x) = T(2-x)\n(6)\nthen (3) and (4) clearly permit the determination of R(|z-x|)\nand T(2-x|). This is tantamount to adopting a one-D theory\nfor irradiance (Sec. 8.6). Thus, assuming that (5) and (6)\nhold, (3) and (4) imply:\n(7)\n;\n(x)-H(\n(8)\nH°(x,-) - H(z,x+)","SEC. 13.10\nLOCAL AND GLOBAL PROPERTIES\n333\nObserve that in deep natural media, by letting Z 8 and\nholding X fixed, (7) implies:\n(9)\n=\nand\n= T(x,z)\n(10)\n.\nFurthermore, by letting Z - X in (7) we have:\n(11)\n= =\nwhere \"b (x)\" denotes the common value of b(x,-) and (x,+),\nwhich exists by (5) above and (3) and (4) of Sec. 8.2 (see\nalso (15) of Sec. 1.4 and (11) and (12) of Sec. 8.3). Fur-\nther, by letting Z+X in (8) we have:\n= = a(x)D - f(x) (12)\nwhere \"f(x)\" denotes the common value of f (x, -) and f(x,+)\nimplied by (6) and (3) and (4) of Sec. 8.2 (see also (16) of\nSec. 1.4 and (11) and (12) of Sec. 8.3). Further, D is the\nfixed value of the distribution functions. Equations (11)\nand (12) show that we can estimate f (x) and b(x) by measuring\nappropriate irradiances in situ along with a. However, some\nattention should first be given to the restrictive assumptions\nabout the shape of the upward and downward radiance distribu-\ntions. This we shall do in the next example.\nExample 2: Homogeneous Media with Polarity\nAs a second example of the inverse problem in hydrologic\noptics we reconsider the problem of determining the R and T\nfactors in homogeneous media. We postulate for the present\ndiscussion that the symmetry conditions (5) and (6) do not\nhold. This therefore simulates the empirical setting of the\ntwo-D theory of Sec. 8.5, which as we saw is a quite impor-\ntant setting in hydrologic optics.\nWe begin by observing that for each submedium X(x,z)\nthere are generally four R and T factors to be determined.\nIn the present case, therefore, the principles of invariance\n(3) and (4) may best be used in the form:\n(H(x,+),H(z,-) = (H(z,+),H(x,-))M(x,z) -\n(13)\nwhere M(x,z) is the operator defined in Sec. 7.4, now adapted\nto the irradiance context.\nIt is clear that we cannot reach our goal of finding\nthe four factors R(x,z), T(x,z), R(z,x), T(z,x) by measuring\nonly the incident and response irradiances at levels X and z,\nfor (13) represents only two equations. We must therefore\nfind two more equations which have as unknowns the same four","VOL. VI\nOPERATIONAL CONCEPTS\n334\nfactors in M(x,z). Now one way in which this may be done is\nto measure the irradiances at the boundary of another slab\nX(x', 2') for which z'-x' = z-x, i.e., , which has the same\nthickness as X(x,z). Then, since the medium is homogeneous,\nM(x, z) = M(|z-x|) = = M(x', 2') (see (19) and (20)\nof Sec. 8.7). Let us follow this lead to see where it takes\nthe discussion. The matricial statement of the principles of\ninvariance for X(x',z') is:\n(H(x'+),H(z',-)) = (H(z),+),H(x',-))M(x,z) .\n(14)\nNow let us write:\nH(z,+)\nH(x,-)\n(15)\n\"H2\"\nfor\nH(z',+) H(x',-)\nand:\nH(x,+)\nH(z,-)\n(16)\n\"Hx\"\nfor\n.\nH(x',+) H(z',-)\nBy means of these definitions (13) and (14) may be\nrepresented as:\n(17)\nIt follows that if Hz has an inverse, we may determine M(x,z)\nby means of the realtion:\n(18)\nWhat is the physical requirement placed on Hz in order that\nH 2 exist? First of all, we require that the two vectors:\n(H(z,+),H(x,-))\nand\n(H(z',+),H(x',-))\nnot be linear combinations of one another. That is, there\nshould exist no nonzero real number C such that:\nc(H(2,+),H(x,-)) + (H(z',+),H(x',-)) = 0\nIn other words, we must have no C such that:\nH(z,+) + H(z',+) = 0\nH(x,-)+H(x',-) 0\nAn equivalent way of stating this is that:\n(19)\n.","LOCAL AND GLOBAL PROPERTIES\n335\nSEC. 13.10\nThus, whenever (19) holds, we may use (18) to solve for the R\nand T factors for X(x,z).\nHow likely is it that the condition (19) holds? A pe-\nrusal of (22) and (23) of Sec. 8.6 shows that we must stay\naway from one-D settings in optically deep media. On the\nother hand, the results of Example 1 of Sec. 8.7 show that\ntwo-D settings in deep or shallow media will often give rise\nto the condition (19). In these latter settings the theory\nmay take over nicely. Thus (18) may be used to exactly com-\nplement the one-D and two-D theories in the task of determin-\ning the reflectance and transmittance factors for irradiance.\nThe net result of Examples 1 and 2 is to provide the\nexperimenter with two complementary means of determining the\nfour R and T factors for submedia X(x,z) in a given homogene-\nous plane-parallel medium X(a,b). The associated local opti-\ncal properties are then found using (3) and (4) of Sec. 8.2\nand their companions leading to (7) of Sec. 8.2. (Note also\n(11) and (12) of Sec. 8.3.) Observe that the R and T factors\nand their local counterparts discussed in the preceding two\nexamples are apparent optical properties and that it was pos-\nsible to determine these factors only after establishing a\none-D or two-D assumption.\nExample 3: Forward and Backward\nScattering Functions\nWe consider next the inverse problem which requires\ndetermination of the forward and backward scattering functions\nf(z, b (z, +), when the irradiance flows in a stratified\nplane-parallel medium are given.\nIt is at once clear from (9) and (10) of Sec. 8.3 how\none can go about finding f (z,-) and b(z,=). We can measure\nH(z, over a small depth range about various depths Z in\norder to obtain all radiometric terms in (9) and (10) along\nwith their derivatives. However, it is also clear that there\nare more unknowns than equations which govern them. There is\nonly one possible way out of this difficulty: the measurements\nmust be taken at a minimum of two depths, say X and z, and we\nrequire in addition two small miracles to take place simultan-\neously, namely:\na(x,-) = f(z,-) - a(z,-)\n(=T(-))\n(i)\nb(x,-) = b(z,-)\n(=p(-))\nand\nH(x,+)\nH(z,+)\ndet\n# 0\n(ii)\nH(x,-)\nH(z,-)\nFor, suppose (i) and (ii) hold. Then, by (i) above and (10)\nof Sec. 8. 3 we have:","OPERATIONAL CONCEPTS\nVOL. VI\n336\nH(x,+)p(+) + H(x,-)t(-) = dH(x,-)\nH(z,+)p(+) + H(z,-)t(-) = dH(2,-)\nThis may be written:\n[\nH(x,+)\nH(z,+)\n= (dH(x)-) dx\n)\ndH (z,-)\n(p(+),t(-))\ndz\nH(x,-)\nH(z,-)\nBy (ii) we then have:\n- 1\n)\nH(x,+)\nH(z,+)\n(p(+),t(-)) = dH(xx-) dx\ndH(z,-)\ndz\n,\nH(x,-)\nH(z,-)\n(20)\nIn this way p(+) and T (-) are determinable. A similar equa-\ntion holds for p(-) and T (+), using (9) of Sec. 8.3. Since\na(z,+) = a(z)D(=) and a is determinable by beam transmittance,\nand since D(+) are generally known (Sec. 8.5), b(+) and f(-)\nthen follow from (11) and (12) of Sec. 8.3.\nIt should be noted that conditions (i) and (ii) above\nincorporate rather stringent requirements on the irradiance\nfield. Before using the present method, condition (ii),\nwhich is the more critical of the two, must be verified. It\nis clear from the asymptotic radiance theorem that in deep\nmedia and for depths far from the surface, condition (ii) is\nnot likely to hold. As in Example 2, we must work near the\nsurface of deep natural hydrosols, or in shallow hydrosols,\npreferably those whose lower boundaries are visible or at\nleast whose presence is detectable by not having exactly\nfixed exponential decrease of irradiance with depth through-\nout the medium. For example, (20) may be used whenever\nK(y,+) # K(y,- -) over the depth range X y\n< Z.\nAlternate, less fundamental approaches to finding f and\nb may be based on the one-D and two-D models of Chapter 8.\nFor example, in the one-D model, we have the general relation\nk = [aD(aD + b) 11/2. Measurements of D, a and k will yield\nestimates of the backward scattering coefficientb. The for-\nward scattering coefficient then follows from the relation\nS = f + b. Figures 1.41-1.45 may facilitate such estimates.\nFurthermore, limit calculations based on (3) and (4) of Sec.\n8.2 using measured values T (xx R(z,x), are potential means\nof finding b and f.","SEC. 13.10\nLOCAL AND GLOBAL PROPERTIES\n337\nExample 4 : R and T Operators\nfor Radiance\nIn this example we consider the problem of determining\nthe R and T operators for a submedium X(x,z) of an optical\nmedium X (a,b), given sufficient radiance measurements. The\nprinciples of invariance for X(x,z) are:\nN+(2)T(2,x) + N_(x)R(x,z)\n= N.(z)R(z,x) + N_(x)T(x,z)\nUsing the operator M(x,z) of Sec. 7.4, now once again in the\nradiance context, these principles may be written:\nN+(x),N_(z)) = (N+ (z),N_(x))M(x,z)\n(21)\n.\nClearly (21) by itself is not sufficient to determine\nM(x,z) For it is not generally possible to predict the re-\nsponse of X(x,z) to every radiance distribution on the basis\nof only one irradiation as indicated in (21). Therefore we\nmust irradiate X(x,z) in a sufficient number of ways so as to\nextract the essential form of M(x,z). One practical way of\ndoing this is to switch from (21) to its matrix approximant.\nThis tactic was employed repeatedly in our earlier studies\nand so need not be elaborated here. See, for example, Secs.\n7.7 and 7.8 for the details of the transition from (21) to\nmatrix form.\nThus, suppose E+ and E. are partitioned into n and m\npieces, respectively, as in (1) and (2) of Sec. 7.7. These\npartitions, in turn, induce decompositions of the N+ (z) and\nN+(x) appearing in (21). Thus, N+ (z) and N+ (x) go over into\nn-component vectors and N (x) and N_ (z) go over into n-com-\nponent vectors. As a result we can write:\n\"N 2\"\nfor\n(N+(2),N_(x)]\nand\n(22)\n\"Nx\" for [N+(x),N_(z)]\nand thus N z and N X are m+n component vectors. Furthermore\nthe component R and T operators of M(x,z) go over into matrices\ndimensioned as follows:\n[\nT(z,x)\nR(z,x)\nn\n(23)\nR(x,z)\nT(x,z)\nm\nn\nm\nThus T(z,x) becomes an nxn matrix T(z,x) and R(x,z) becomes\nan m x n matrix R(x,z), and so on. Let us denote the resultant\nmatrix approximant to M(x,z) as \"M(x,z)\".","VOL. VI\nOPERATIONAL CONCEPTS\n338\nNm+n\nbe m+n linearly independent\nNow let N1, N2\n,\nvectors of the kind defined in (22) . Write:\nN1\nZ\n2\nN\nZ\n(24)\n.\nfor\n\"N2\"\nm+n\nN\nZ\nand:\n1\nN\nX\n2\nN'\nX\n(25)\n\"Nx\"\n11\nfor\n.\nm+n\nN\nX\nwhere NJ is the response vector to N as governed by (21).\nIt follows that (21) may be written down m+n times, once for\neach pair The resultant system of m+n equations may\nbe given the compact form:\n(26)\nNx = N M(x,z)\nSince N z is invertible, we can in principle determine M(x,z):\nM(x,z) = N 2 N X\n(27)\nThe importance of working with inherent optical prop-\nerties becomes manifest in the present example. That is to\nsay, when we must change the irradiation pattern on X(x,z)\nin order to obtain more conditions we require the interaction\noperators to be invariant under the change of irradiation\npattern. This requires that the same operator M(x,2) appear\nin (23) and (26). Since the R and T operators for radiance\nare inherent optical properties, this change of irradiation\npattern is permissible. Otherwise, if the operators were\napparent optical properties, changing the lighting conditions\nwould serve no useful purpose in the solution of the inverse","SEC. 13.11\nCONSISTENCY QUESTIONS\n339\nproblem, for with each change in the lighting conditions four\nmore new operators appear and so there will always be two\nmore operators than radiometric equations available for solu-\ntion.\nGeneral Observations on Inverse Problems\nin Hydrologic Optics\nObserve that the inverse problem of determining the\nlocal optical properties p(y) and (y) is readily solved in\nhydrologic optics by using the results of Examples 2 or 3 and\nthe theory of Sec. 7.3. Hence the inverse problem in hydro-\nlogic optics is completely solvable by appropriately using\nthe general concepts assembled in Chapter 7.\nOne important reason why the inverse problem presents\nno novel difficulties in hydrodologic optics (in principle)\nrests on the fact that the optical medium whose optical prop-\nerties are sought is directly accessible to experimental prob-\ning. This fact was used throughout the preceding examples.\nIn branches of radiative transfer other than geophysical\noptics, such as the current fields of astrophysical optics or\nplanetary optics, the problem of determining, say, a and o\nthroughout a stellar or planetary atmosphere is much more dif-\nficult when the atmosphere cannot be directly probed inter-\nnally. Indeed, under such a condition, unless some specific\nlaws governing at least the internal depth behavior of o and\na are available, or some equivalent information is available\nor even good guesses possible, then the general (inverse) prob-\n1ems of the second class, are insoluble.\n13.11 On the Consistency of the Operational Formulations\nWe conclude this chapter with a check on the consis-\ntency of the operational definitions of the main optical\nproperties introduced throughout the chapter. The method we\nshall employ is that which attempts to assemble all the vari-\nous operationally defined pieces into a structure which,\nhopefully, will be recognizable as one of the forms of trans-\nport equations - - either for radiance, irradiance, or some other\nappropriate radiometric quantity.\nTo see how the method proceeds, we select for illustra-\ntion those concepts which should fall together into the form\nof the equation of transfer for unpolarized radiance fields.\nThus consider a regular neighborhood C(A,B) of paths, as\nshown in Fig. 13.16. The common beam transmittance Tr (x,E)\nfor the members of C(A,B) is given by (7) of Sec. 13.2 as:\nN(Y,,5) - N(Yax5)\n(1)\nThe common path radiance N°(y,5) for the members of\n(A,B) is given in (2) of Sec. 13.3 in the form:\nN*(y,5) = N(y,5) - N(x,E)T-\n(2)","VOL. VI\nOPERATIONAL CONCEPTS\n340\nE\ny1\nB\ny\nPr(x1,E)\nY/2\n(x,E)\npr(x2, E)\nA\nx1\nr\n+\n+\nx\n+\nC(A,B)\nx2\nFIG. 13.16 A regular neighborhood C(A,B) of paths in a\ngeneral optical medium used in the consistency check of the\noperational formulations of the concepts of unpolarized radi-\native transfer theory.\nIn particular, for paths Pr(x,,5), Pr(x ,5) in C(A,B), (2)\nimplies:\nHence (1) and (2) are mutually consistent. Therefore, we\ncan proceed with (2) in which Tr(x,5) is defined as in (1),\nand write (2) as:\n(3)\nWe next introduce into (3) the operational definition of a\nby means of (1) of Sec. 13.4:\nT_(x,5) = 1 - ra(x,5) + ro(r)\nThe result is:\nN(y,)[1- ra(x,)]N(x,) + N*(y,5) + 0,(r)\n(4)\nwhere 01 (r) is a quantity such that 01 (r)/r goes to zero with\nr. Equation (4) can be rearranged so as to read:","SEC. 13.11\nCONSISTENCY QUESTIONS\n341\nN(y,5) - N(x,5) x(x,E)N(x,E) N*(y,5) + r (r) .\n= - +\n(5)\nUsing (4) of Sec. 13.3, (5) can be written as:\nN(y,5) - N(x,5) r = a(x,E)N(x,E) + N* (x, + (9,6)\n(6)\nwhere o, 2 (r) /r goes to zero with r. Now letting r 0, (6)\nleads to\nIN(x,5)\n= - a(x,5)N(x,5) + (*(x,5)\n(7)\ndr\nor simply\ndN\ndr = - aN + N*\nwhich is the desired equation of transfer for N with volume\nattenuation function a and path function N*. By introducing\no into (7) via (1) of Sec. 13.6 or any of the operational\nformulations in Sec. 13.6 built up from that equation, we\nsee that the operational definitions of Tr(x,5) N°(x,E),\nN* (x,5), a (x,5) and o (x;E';E) are indeed mutually consistent.\nOn-the Relative Consistency of the\nUnpolarized and Polarized\nTheories of Radiative\nTransfer\nThere remains the question of whether the equation of\ntransfer (7) is consistent with respect to the finer-grained\ntheory of the polarized light field, and whether it forms a\nfaithful picture of the radiometric features of an optical\nmedium. The problem, essentially, is this: Suppose one mea-\nsures radiance distributions in the sea or air using a radi-\nance meter without a polarizer attached (cf. Fig. 2.25 and\nSec. 2.10). Suppose the measurements determine the a, and o\nof the medium under study, as defined operationally in the\npresent section. Suppose further that we then place this a\nand o into (7), compute the radiance field throughout the\nmedium and then compare the computed field with the measured\nfield. The question now is: Can these two radiance fields\nbe equal, in principle? Notice that we qualify the question\nas one of a matter of principle. Surely, measurement and\ncalculation techniques, even in the relatively advanced tech-\nnology of today, cannot culminate in an exact corroboration\nof the two radiance fields. Therefore, what we are primarily\nafter here is the resolution of a subtler matter concerned\nwith the physical foundation, and the internal mathematical\nconsistency of the polarized theory. The question therefore\nsplits into two parts. The first question is: Does the\npolarized theory contain the unpolarized theory as a special\ncase, and if so, does the special case agree with that which","OPERATIONAL CONCEPTS\nVOL. VI\n342\nwe have been using all along in this work? (The fact that we\ndare ask this question at this late hour, and the fact that\nthis work exists publicly, means that the answer to the pre-\nceding question is happily in the affirmative. However, there\nis a surprise ending to the story which, like the denouement\nof any interesting mystery, should not be glimpsed premature-\nly.)\nTo answer the first question, as stated above, we need\nonly return to (7) of Sec. 3.15, the equation of transfer for\npolarized radiance, and write it out in component form, using\nthe notation of Sec. 2. 10:\nd(N)\n(8)\n=\nj\n1,\n2,\n3,\n4\n=\nwhere Pij is the entry in the ith row and jth column of the\nstandard observable volume scattering matrix p, and N is the\njth component of the radiance vector N, j = 1, 2, 3, 4. p is\nrelated to o in (24) of Sec. 13.6, in the manner shown in\nSec. 112 of [251]. According to (8) of Sec. 2.10, the ob-\nserved radiance N without a polarizer attached to the radi-\nance tube is given in terms of the polarized radiance vector's\ncomponents as:\n(9)\n.\nThis suggests that we consider (8) for the cases j = 1 and\n= 2, and add the associated equations together, thus:\n= - +\nds\n(10)\n+\n=\n-\n.\nWe now specifically adopt the assumption that: the radiance\nfield in a given optical medium is unpolarized. It follows\nfrom the list of observable radiance vectors in Sec. 2.10\nthat the unpolarized radiance vector N has the form:\nN = 1 (N,N,N,N)\n(11)\n,\ni.e., , all N are equal, to a common function (1/2)N, where\nN would be measured by the radiance tube without a polarizer.\nUsing this form of N in (10) we deduce that\ndN\ndn\n(12)\nN\n.","SEC. 13.11\nCONSISTENCY QUESTIONS\n343\nWriting\n\"0\" for 2 1 +\n(13)\nwe see that the mathematical structure of (12) is identical*\nto that of (7). Hence the theory of the unpolarized light\nfield is a logical consequence of the theory of the polarized\nlight field, and so they are relatively consistent in a mathe-\nmatical sense: An added dividend to the present inquiry is\nthe exact form of the volume scattering function o in terms\nof the components Pii of p under the present assumption. (See\n(24) of Sec. 13.6 and the remarks following (7) of Sec. 3.15.)\nNote carefully that all we have shown is that the form\nof (7) is correct in unpolarized light fields. Its content\n(i.e., the magnitudes of a and o as determined by the opera-\ntions of this chapter), while mutually consistent, as shown\nabove, need not agree, e.g., with the o of (13). We shall\nlook into this matter in a moment.\nIt is one thing to postulate that the world behaves in\na certain way and another to experimentally determine whether\nor not it does indeed behave that way. It is an experimental\nfact (look up into the sunlit sky with polaroid sunglasses)\nthat the process of scattering of light, on either the phe-\nnomenological level (such as that inhabited by a radiative\ntransfer theory) or the microscopic electromagnetic level,\nalters the state of polarization of the light. Thus unpo-\nlarized light becomes polarized after being scattered (i.e.,\ningoing (1/2) (N,N,N,N) emerges as polarized flux after en-\ncountering a scattering volume) and polarized light subtly\nalters its state to another state after another scattering.\nThe net effect of a multitude of such scatterings in an ex-\ntensive optical medium on the radiance distribution within it\nis to reach an asymptotic state indigenous to the phase func-\ntion of the medium (see the discussion of (19) of Sec. 4.6,\nand the consequences of the asymptotic radiance theorem in\nSec. 10.7). The second of the two main questions in the pres-\nent discussion hinges on these physical facts, and may now be\nphrased as follows: Is the basic equation of transfer for the\nradiance N = 1N + 2N, strictly of the form (7) in a polarized\nlight field?\nTo answer this second question, we turn again to (10)\nand observe immediately that it cannot generally be placed\ninto the form (7). Hence the answer to the second question\nis negative. Just where and by how much do the two equations\ndiffer? They differ in the path function term. Thus by defi-\nnition (9) we can write (7) as:\n*What could have gone wrong here, for example, would have\nbeen the appearance of an unexpected source term on the right\nside of (12) arising from contributions to N by N and N; or\nthe inability to factor the integrand into an N-type and a o-\ntype factor.","VOL. VI\nOPERATIONAL CONCEPTS\n344\n= - a(,N+2N) + (1N+2N)odr\n(14)\nwhere o is determined as in (4) of Sec. 13.6. Subtracting\nthe right-hand side of (14) from that of (10) term by term,\nwe are led to write, ad hoc:\n\"N*\"\nfor\n(15)\n+ +\nThe function N+ has the dimensions of the path function, name-\nly radiance per unit length. Clearly the function values\nN*(x,5) are zero for each X and & over a given optical medium\nif and only-if (7) and (14) are equivalent equations of trans-\nfer over that medium under all lighting conditions. From (15)\nwe see at once that such an equivalence holds if and only if\nP11 + P12 = P22 + P21 = o\nP31+P32=0\n(16)\nP41 + P42 = 0\nIn order words, if (16) holds in a certain optical medium,\nthen the equation of transfer (7) is an exact model of the\nlight field (i.e., 1 N + 2N) in that medium even though the\nlight field in that medium is polarized. On the other hand,\nif (16) does not hold, then (7) cannot in principle describe\nthe polarized radiance field in such a medium under all light-\ning conditions. Equation (16), therefore, provides a practi-\ncal test for the theoretical applicability of (7) to natural\noptical media with polarized light fields. The set (16) can\nalso be written in terms of the components ij of o, if de-\nsired. A convenient measure of the values N*,N* would be via\nthe corresponding equilibrium radiances Nq = N*/a, Nq = N*/a.\nWe can write the exact equation of transfer (10) for\nN(= ,N + 2N) in source-free steady media, in a form which is\ndirectly tied to the preceding equivalence criterion:\ndN\nN*\naN\nN*\n+\n=\ndr\n(17)\n1\n2\nHence if the conditions (16) do not hold, then this fact is\nmanifested in (17) by the appearance of a nonzero increment\nN of the path function N* (a spurious source term) which\narises from the scattering contribution of the four components\nof the polarized light field. We have observed above that\nthis in reality will always be the case. In such a case, then,\nit is the magnitude of N* which critically gauges the departure","SEC. 13.12\nBIBLIOGRAPHIC NOTES\n345\nof the classical equation of transfer (7) from its exact\ncounterpart (17) within polarized light fields. It follows\nthat the classical equation of transfer in real light fields\nexhibiting polarization, is only an approximate equation. To\nthe author's knowledge, no systematic test of the conditions\n(16) nor any estimate of the magnitude of N* in (17) has been\nmade at the time of publication [1976]. An important question\nfor the classical unpolarized theory of radiative transfer\nnow rests in the practical applicability of (7) to the study\nof light fields in natural hydrosols: while (2.) generates a\nmathematically self-consistent theory of unpolarized light\nfields, how well (in a quantitative sense) does it describe\nN (= N + 2N) within actual (polarized) light fields found in\nnature?\nAs anticipated above, the denouement of this problem\nstill stands at this late date in the history of the theory,\nand awaits a definitive answer from those who are the only\nones who can definitively answer it: the experimenters.\nTheoretical reasoning, such as that above, can be carried\nonly so far. There eventually comes a time in the construc-\ntion of any physical theory when all the theorizing must mo-\nmentarily stop, and the court of last appeal be faced: Nature\nherself.\n13.12 Bibliographic Notes for Chapter 13\nThe development of the operational formulation for beam\ntransmittance in Sec. 13.2 is based on the work in Ref. [238],\nThe operational formulations of path function and path radi-\nance, as given in Sec. 13.3, are patterned in part on the\ntheory of Sec. 3.12. The K-method of determining the path\nfunction was developed in Ref. [219]. Path function calcula-\ntions using the integral method (6) of Sec. 13.3 were explored\nin Ref. [214]. The operational definition of volume attenua-\ntion function in Sec. 13.4 is based in part on Sec. 3.11 and\nthe work of Ref. [238]. Interesting and useful parallels of\nthe methods of Sec. 13.4 as developed in meteorologic optics\nmay be found in Ref. [231]. Further parallels with a measure-\nments in meteorologic optics may be found in Ref. [177] and\nRef. [80]. The theory of Sec. 13.5 is based directly on Ref.\n[217]. The developments of the volume scattering function in\n(11) - (25) of Sec. 13.6 are for the most part new, with basic\npoints of view developed in Sec. 18 of Ref. [251]. Some ru-\ndimentary forms of o-definitions may be found in Ref. [177]\nand which are recognizable as early forms of (4) of Sec. 13.6.\nThe discussion of the general empirical properties of o, as\ngiven from (5) to (10) of Sec. 13.6 is based in part on ob-\nservations in Ref. [229]. The theory of the volume total\nscattering function in Sec. 13.7 is drawn from Ref. [241],\nwhile that of Sec. 13.8 is based on Ref. [241] and Ref. [220].\nAn experimental device based on this theory is described in\nRef. [299], Alternate means of measuring S are discussed in\nSec. 9.4.4 of Ref. [177]. The remarks in Sec. 13.9 are a con-\nsolidation of those on apparent optical properties found in\nRef. [305].\nThe discussions of Sec. 13.10 are based in part on Ref.\n[243] and only begin to explore the inverse problem in radia-\ntive transfer theory. The inverse problems (problems of the\nsecond class) of radiative transfer are classified in Ref.","VOL. VI\n346\nOPERATIONAL CONCEPTS\n[251], and are studied in the meteorologic context in Ref.\n[231] and Ref. [238], and in the hydrologic context in Ref.\n[247].\nFor further study of the questions raised in the clos-\ning discussion of Sec. 13.11, see [308]. This reference\nwill serve only to familiarize the reader with the problem in\nthe context of single scattering theory. Chandrasekhar's\nwork [43] gives an example of the use of the single scatter-\ning phase function for Rayleigh scattering in computing a\nmultiply-scattered polarized light field. The reference [53]\ncan serve to generate some tentative numerical answers using\nits tabulated solutions of polarized light fields. The prob-\n1em, however, of determining the experimenter's operationally\nfound o (in the form (4) or (24) of Sec. 13.6) in terms of\nthe electromagnetic properties of an irradiated aggregate of\ndisjoint and arbitrarily disposed microscopic scattering\nvolumes, has yet to be solved in sufficient generality and\ndetail. Such a result would contribute materially toward the\nefficient use of the equivalence criterion (16) of Sec. 13.11\nfor unpolarized and polarized radiative transfer theory, as\napplied to real optical media (see the closing remarks of Sec.\n13.5).\nFurther questions on the nature of the equation of\ntransfer, beyond these considered in Sec. 13.11, can now be\nraised, and offer interesting prospects for generalizing the\nattendant theory: what is the appropriate form of the equa-\ntion of transfer for partially coherent unpolarized radiance\nfields? For partially coherent polarized radiance fields?\nIs the equation of transfer for incoherent polarized light\nfields mutually consistent with the equation for partially\ncoherent polarized light fields - in the same general sense\nstudied above for (7) and (17) of Sec. 13.11? A11 these\nquestions and their close relatives can perhaps be studied\nwith precision and depth within the context of problems I,\nII, III, and IV of Sec. 141 in [251].","BIBLIOGRAPHY\n347\nBIBLIOGRAPHY FOR VOLUMES I-VI\n1. Ambarzumian, V. A., \"Diffuse reflection of light by a\nfoggy medium\" Compt. rend. (Doklady) Acad. Sci. U.R.S.S.\n38, 229 (1943).\n2.\nAmbarzumian, V. A., \"On the problem of the diffuse re-\nflection of light,\" J. Phys. Acad. Sci. U.S.S.R. 8 ,\n65 (1944).\n3. American Institute of Physics Handbook (McGraw-Hill, New\nYork, 1957).\n4. American Standards Association, \"Nomenclature, for radi-\nometry and photometry (258.1.1-1953),\" J. Opt. Soc.\nAm. 43, 809 (1953).\n5. Armed Forces NRC Vision Committee, Minutes and Proceed-\nings. 23rd Meeting, pp. 123-126 (March 1949).\n6. Atkins, W.R.G., and Poole, H. H., \"The angular scatter\n-\ning of blue, green and red light by sea water,\" Sci.\nProc. Roy. Dublin Soc. 26, 313 (1954).\n7. Austin, R. 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VI\nSURFACES\n348\n14. Bellman, R. , and Kalaba, R., \"On the principle of in-\nvariant imbedding and diffuse reflection from cylin-\ndrical regions,\" Proc. Nat1. Acad. Sci. 43, 514\n(1957).\n15. Bellman, R. E., Kalaba, R. E., and Prestrud, M. C.,\nInvariant Imbedding and Radiative Transfer on Slabs\nof Finite Thickness (American Elsevier Pub. Co., New\nYork, 1963).\n16. Bellman, R. E. Kagiwada, H. H., Kalaba, R. E., and\nPrestrud, M. C., Invariant Imbedding and Time-Depen- -\ndent Transport Processes (American Elsevier Pub. Co.,\nNew York, 1964).\n17. Bellman, R. E., Kalaba, R. E., and Wing, G. M. \"Invari-\nant imbedding and neutron transport theory,\" J. Math.\nand Mech. I, 149(1958) ; II, 741 (1958) III, 249 (1959)\n18. Benford, F., , \"Reflection and transmission by parallel\nplates,\" J. Opt. Soc. Am. 7, 1017 (1923).\n19. Benford, F., \"Transmission and reflection by a diffus-\ning medium,\" Gen. Elec. Rev. 49, 46 5(1946).\n20. 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J. , Models of Random Seas Based on the\nLagrangian Equations of Motion (N.Y.U. Coll. of Eng.,\nRes. Div. April 1961).\n201. Pierson, W. J. , Some New Unsolved Problems in Connec-\ntion with Random Processes of Interest in Geophysics\n(N.Y.U. Coll. of Eng., Res. Div., January 1962).\n202. Pierson, W. J., and Marks, W. \"The power spectrum anal-\nysis of ocean-wave records, Trans. Am. Geophys. Union\n33, 834 (1952).\n203. Pierson, W. J., and Moskowitz, L., A Proposed Spectral\nForm for Fully Developed Wind Seas Based on the Simi-\nlarity Theory of S. A. Kitaigorodskii (N.Y.U. College\nof Engineering and Science, October 1963).\n204. Pierson, W. J., and Neumann, G., \"A detailed comparison\nof theoretical wave spectra and wave forecasting\nmethods,\" \" Deutsch Hydrographische Zeitschrift Band\n10, 73 and 134 (1957).\n205. Pierson, W. J. , Neumann, G., and James, R. W., Practi-\ncal Methods for Observing and Forecasting Ocean Waves\nby Means of Wave Spectra and Statistics (N.Y.U. 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W., \"Apparent radiance of submerged\nobjects,\" J. Opt. Soc. Am. 45, 404 (1955).\n213. Preisendorfer, R. W., Calculation of the Path Function.\nTheory and Numerical Example (Contract NObs-50274,\nIndex Number 714-100, Visibility Laboratory, Univer-\nsity of California, San Diego, March 1956), (manu-\nscript completed August 1954).","BIBLIOGRAPHY\n361\n214. Preisendorfer, R. W., A New Method for the Determina-\ntion of the Volume Scattering Function in Environ-\nmental Optics (Contract NObs-50274, Index Number\n714-100, Visibility Laboratory, University of Cali-\nfornia, San Diego, March 1956), (manuscript completed\n1954).\n215. Preisendorfer, R. W. \"Exact reflectance under a cardi-\noidal luminance distribution,\" Quart. J. Roy. Met.\nSoc. 83, 540 (1957).\n216. Preisendorfer, R. W., , \"A mathematical foundation for\nradiative transfer theory,\" J. Math. and Mech. 6,\n685 (1957).\n217. Preisendorfer, R. W., A General Theory of Perturbed\nLight Fields, with Applications to Forward Scattering\nEffects in Beam Transmittance Measurements (Scripps\nInst. of Ocean. Ref. 58-37, University of California,\nSan Diego, 1958).\n218.\nPreisendorfer, R. W. , The Planetary Hydrosphere Problem\n(Scripps Inst. of Ocean. Ref. 58-40, University of\nCalifornia, San Diego, 1957).\n219. Preisendorfer, R. W. The K-Method of Determining the\nPath Function (Scripps Inst. of Ocean. Ref. 58-39,\nUniversity of California, San Diego, 1958).\n220. Preisendorfer, R. W. The Divergence of the Light Field\nin Optical Media (Scripps Inst. of Ocean. Ref. 58-41,\nUniversity of California, San Diego, 1957).\n221. Preisendorfer, R. W., Unified Irradiance Equations\n(Scripps Inst. of Ocean. Ref. 58-43, University of\nCalifornia, San Diego, 1957).\n222.\nPreisendorfer, R. W., Directly Observable Quantities\nfor Light Fields in Natural Hydrosols (Scripps Inst.\nof Ocean. Ref. 58-46, University of California, San\nDiego, 1958).\n223. Preisendorfer, R. W., Canonical Forms of the Equation\nof Transfer (Scripps Inst. of Ocean. Ref. 58-47,\nUniversity of California, San Diego, 1958).\n224. Preisendorfer, R. W., A Proof of the Asymptotic Radi-\nance Hypothesis (Scripps Inst. of Ocean. Ref. 58-57,\nUniversity of California, San Diego, 1958).\n225. Preisendorfer, R. W., On the Existence of Characteris-\ntic Diffuse Light in Natural Waters (Scripps Inst.\nof Ocean. Ref. 58-59, University of California, San\nDiego, 1958).\n226. Preisendorfer, R. W., Some Practical Consequences of\nthe Asymptotic Radiance Hypothesis (Scripps Inst. of\nOcean. Ref. 58-60, University of California, San Diego,\n1958).","VOL. VI\nSURFACES\n362\n227. Preisendorfer, R. W. Photic Field Theory for Natural\nHydrosols (Scripps Inst. of Ocean. Ref. 58-66, Uni-\nversity of California, San Diego, 1958).\n228. Preisendorfer, R. W., and Harris, J. L., The Contrast\nTransmittance Distribution as a Possible Tool in Vis-\nibility Calculations (Scripps Inst. of Oceanography\nRef. 58-68, University of California, San Diego, 1958).\n229. Preisendorfer, R. W., and Tyler, J. E., The Measurement\nof Light in Natural Waters: Radiometric Concepts and\nOptical Properties (Scripps Inst. of Ocean. Ref. 58-\n69, University of California, San Diego, 1958).\n230. Preisendorfer, R. W., and Richardson, W. H., Simple\nFormulas for the Volume Absorption Coefficient in\nAsymptotic Light Fields (Scripps Inst. of Ocean. Ref.\n58-79, University of California, San Diego, 1958).\n231. Preisendorfer, R. W., Theory of Attenuation Measurements\nin Planetary Atmospheres (Scripps Inst. of Ocean. Ref.\n58-81, University of California, San Diego, 1958).\n232. Preisendorfer, R. W., A Sea Surface Simulator, I. Theo-\nretical Analysis: Simulation of Reflectance Properties\nof the Sea Surface by Arrays of Ergodic Caps (Report\n1-1, Task 1, Contract NObs-72039, Visibility Labora-\ntory, University of California, San Diego, June 1958).\n233. Preisendorfer, R. W., \"Invariant imbedding relation for\nthe principles of invariance,\" Proc. Nat. Acad. Sci.\n44, 320 (1958).\n234. Preisendorfer, R. W., \"Functional relations for the R\nand T operators on plane-parallel media,\" Proc. Nat.\nAcad. Sci. 44, 323 (1958).\n235. Preisendorfer, R. W., \"Time-dependent principles of in-\nvariance,\" Proc. Nat. Acad. Sci. 44, 328 (1958).\n236. Preisendorfer, R. W. , Temporal Metric Spaces in Radia-\ntive Transfer Theory (five papers: I Temporal Semi -\nmetrics; II Epoch Times and Characteristic Functions;\nIII Characteristic Spheroids and Ellipsoids; IV Tem-\nporal Diameters; V Local Temporal Diameters), (Scripps\nInst. of Ocean. Refs. : I 59-2; II 59-7; III 59-10;\nIV 59-17; V 59-18, University of California, San\nDiego, 1959).\n237. Preisendorfer, R. W., A Study of Light Storage Phenomena\nin Scattering Media (Scripps Inst. of Ocean. Ref.\n59-12, University of California, San Diego, 1959).\n238. Preisendorfer, R. W., On Ground-Based Measurements of\nthe Optical Properties of the Atmosphere Aloft\n(Scripps Inst. of Ocean. Ref. 59-19, University of\nCalifornia, San Diego, 1959).","BIBLIOGRAPHY\n363\n239. Preisendorfer, R. W. Radiance Bounds (Scripps Inst. of\nOcean. Ref. 59-20, University of California, San Diego,\n1959).\n240. Preisendorfer, R. W. The Universal Radiative Transport\nEquation (Scripps Inst. of Ocean. Ref. 59-21, Univer-\nsity of California, San Diego, 1959).\n241. Preisendorfer, R. W., On the Direct Measurement of the\nTotal Scattering Function (Scripps Inst. of Ocean.\nRef. 59-41, University of California, San Diego,\n1959)\n242. Preisendorfer, R. W. The Covariation of the Diffuse\nAttenuation and Distribution Functions in Plane-\nParallel Media (Scripps Inst. of Ocean. Ref. 59-52,\nUniversity of California, San Diego, 1959).\n243. Preisendorfer, R. W. Principles of Invariance for Di-\nrectly Observable Irradiances in Plane-Parallet Media\n(Scripps Inst. of Ocean. Ref. 59-73, University of\nCalifornia, San Diego, 1959).\nPreisendorfer, R. W. \"Theoretical proof of the exis-\n244.\ntence of characteristic diffuse light in natural\nwaters,\" J. Mar. Res. 18, 1 (1959).\n245. Preisendorfer, R. W., On the Structure of the Light\nField at Shallow Depths in Deep Homogeneous Hydrosols\n(Report 3-5, Task 3, Contract NObs-72039, Visibility\nLaboratory, University of California, San Diego,\nMarch 1959).\n246. Preisendorfer, R. W., , General Analytical Representa-\ntions of the Observable Reflectance Function (Report\n5-1, Task 5, Contract NObs-72039, Visibility Labora-\ntory, University of California, San Diego, November\n1959).\n247. Preisendorfer, R. W. \"Application of radiative trans-\nfer theory to light measurements in the sea,\" Sympo-\nsium on Radiant Energy in the Sea, International Union\nof Geodesy and Geophysics, Helsinki Meeting, August\n1960 (L'Institute Geographique National Monograph No.\n10, Paris, 1961).\n248. Preisendorfer, R. W., \"Generalized invariant imbedding\nrelation,\" Proc. Nat. Acad. Sci. 47, 591 (1961).\n249. Preisendorfer, R. W. Spatial Semi-groups in Neutron\nTransport Theory (General Atomic Report GA-2057, John\nJay Hopkins Laboratory of Pure and Applied Sciece,\nSan Diego, California, 1961).\n250. Preisendorfer, R. W., \"A model for radiance distribu-\ntions,\" in Physical Aspects of Light in the Sea Univ.\nof Hawaii Press, 1964), J. E. Tyler, ed.","VOL. VI\nSURFACES\n364\n251. Preisendorfer, R. W., Radiative Transfer on Discrete\nSpaces (Pergamon Press, New York, 1965).\n252. Redheffer, R., \"Remarks on the basis of network theory,\"\nJ. Math. and Phys. 28, 237 (1950).\n253. Redheffer, R., \"Novel uses of functional equations,\" J.\nRat. Mech. and Anal. 3, 271 (1954).\n254. Redheffer, R., \"On solutions of Riccati's equation as\nfunctions of the initial values,\" J. Rat. Mech. and\nAnal. 5, 835 (1956).\n255. Redheffer, R. , \"The Riccati equation: initial values\nand inequalities,\" Math. Ann. 133, 235 (1957).\n256. Redheffer, R., \"Inequalities for a matrix Riccati equa-\ntion,\" J. Math. and Mech. 8, 349 (1959).\n257. Redheffer, R., \"Supplementary note on matrix Riccati\nequations,\" J. Math. and Mech. 9, 745 (1960).\n258. Redheffer, R., \"The Myciolski-Pazkowski diffusion prob- -\n1em,\" J. Rat. Mech. and Anal. 9, 607 (1960).\n259. Redheffer, R., \"On the relation of transmission-line\ntheory to scattering and transfer,\" J. Math. and\nPhys. 41, 1 (1962).\n260. Redmond, P. M. , Light Refraction by a Free Ocean Sur-\nface (AIAA paper No. 65-238, Am. Inst. of Aeronautics\nand Astronautics, New York, March 1965).\n261. Reid, W. T., \"Solutions of a Riccati matrix differential\nequation as functions of initial values,\" J. Math.\nand Mech. 8, 221 (1959).\n262. Reid, W. T., \"Properties of solutions of a Riccati ma-\ntrix differential equation,\" J. Math. and Mech. 9,\n749 (1960).\n263. Richardson, W. H., Determination of the Non-zero Asymp-\ntote of an Exponential Decay Function (Scripps Inst.\nof Ocean. Ref. 58-36, University of California, San\nDiego, 1958).\n264. Rickart, C. E., General Theory of Banach Algebras (D.\nVan Nostrand, New York, 1960).\n265. Riley, G. A., \"Theory of food-chain relations in the\nocean,\" \" in The Sea (Interscience Pub., , New York, 1963),\nM. N. Hill, Ed., vol. II, Chapt. 20.\n266. Roll, H. U., \"Neue messungen zur entstehung von wasser-\nwellen durch wind,\" Ann. Met. Hamburg 4, 269 (1951).","BIBLIOGRAPHY\n365\n267. Roll, H. U., and Fischer, G., \"Eine kritische bemerkung\nzum Neumann-spektrum des seeganges,\" Deutsche Hydrogr.\nZeits., Band 9, Heft 9 (1956)\n268. Rosenblatt, H., \"A random model of the sea surface gen-\nerated by a hurricane,\" J. Math. and Mech. 6, 235\n(1957).\n269. Rudin, W., Fourier Analysis on Groups (Interscience\nPub., New York, 1962).\n270. Ryde, J. W., and Cooper, B. S., \"The scattering of\nlight by turbid media,\" Proc. Roy. Soc. London 131A,\n451 (1931).\n271. Sasaki, T., , Okami, N., Oshiba, G., and Watanabe, S.,\n\"Angular distribution of light in deep water,\" Rec-\nords of Ocean. Works in Japan 5, 1 (1960)\n272. Schenck, H. , \"On the focusing of sunlight by ocean waves,\"\nJ. Opt. Soc. Am. 47, 653 (1957).\n273. Schmidt, H. W., \"Über reflexion und absorption von B-\nstrahlen,\" Ann. der Physik 23, (series 4), 671 (1907).\n274. Schooley, A. H., \"A simple optical method of measuring\nthe statistical distribution of water surface slopes,\"\nJ. Opt. Soc. Am. 44, 37 (1954).\n275. Schooley, A. H., \"Curvature distributions of wind-created\nwater waves,\" Trans. Am. Geophys. Un. 36, 273 (1955).\n276. Schooley, A. H., \"Profiles of wind-created water waves\nin the capillary-gravity transition region,\" J. Mar.\nRes. 16, 100 (1958).\n277. Schuster, A., \"On hidden periodicities,\" Terrestrial\nMagnetism 3, 3 (1897).\n278. Schuster, A., \"The periodogram of magnetic declination,\"\nCamb. Phil. Trans. 18, 108 (1899).\n279. Schuster, A., \"Radiation through a foggy atmosphere,\"\nAstrophys. J. 21, 1 (1905).\n280. Schuster, A., \"The periodogram and its optical analogy,\"\nProc. Roy. Soc. 77, 136 (1906)\n281. Schwarzschild, K., \"Ueber das gleichgewicht das sonnen-\natmosphare, Gesell. Wiss. Gottingen, Nachr. Math.- -\nphys. Klasse, p. 41, (1906).\n282. Schwarzschild, K., \"Über diffusion und absorption in der\nsonnenatmosphare, Sitzungsberichte der Königlich\nPreussischen Akad. der Wiss., p. 1183 (Jan.-June, 1914).","VOL. VI\nSURFACES\n366\n283. Secchi, P. A., Relazione della Esperienze Fatte a Bordo\ndella Pontificia Pirocorvetta L'Immacolata Concezione\nper Determinare la Trasparenza del Mare (circa 1866)\n[Reports on Experiments Made on Board the Papal Steam\nSloop L' Immacolata Concezione to Determine the Trans-\nparency of the Sea] (Translation available, Dept. of\nthe Navy, Office of Chief of Naval Operations, O.N.I.\nTrans. No. A-655, Op-923 M4B, 21 Dec. 1955).\n284. Sekera, Z., Radiative Transfer in a Planetary Atmosphere\n(Rand Corp. Report R-413-PR, June 1963).\n285. Silberstein, L., \"The transparency of turbid media,\"\nPhil. Mag. 4, 1291 (1927).\n286. Singer, I. A., and Raynor, G. S., \"Variation of wind\nprofile with meteorological parameters, 6th Midwest-\nern Conference on Fluid Mechanics (Univ. of Texas,\n1959), p. 98.\n287. Sliepcevich, C. M., Churchill, S. W., Clark, G. C. and\nChiao-min Chu, Attenuation of Thermal Radiation by a\nDispersion of oil Particles (Army Chem. Corps Con-\ntract No. DA18-108-CML-4695, AFSWP-749 ERI-2089-2-F,\nEng. Res. Inst., University of Michigan, Ann Arbor,\n1954).\n288. Sobolev, V., A Treatise on Radiative Transfer (Van Nos-\ntrand, New York, 1963).\n289. Sommerfeld, A., Partial Differential Equations in Phy: -\nics (Academic Press, New York, 1949)\n290. Spilhaus, A. F., Observations of Light Scattering in\nSea Water (Ph. D. Thesis, Dept. of Geology and Geo-\nphysics, M.I.T., February 1965, prepared under ONR\nContract Nonr 1841 (74), NR 083-157).\n291. Stokes, G. G., \"On the intensity of the light reflected\nfrom or transmitted through a pile of plates,\" Mathe-\nmatical and Physical Papers of Sir George Stokes (Cam-\nbridge Univ. Press, 1904), vol. IV, p. 145.\n292. Stratton, J. A., Electromagnetic Theory (McGraw-Hill,\nNew York, 1941).\n293. Sverdrup, H. U., and Munk, W. H., Wind, Sea, and Swell;\nTheory of Relationships for Forecasting (Hydrographic\nOffice Publication 601, 1947).\n294. Symposium on Applications of Autocorrelation Analysis to\nPhysical Problems, Woods Hole, Mass., 13-14 June 1949\n(O.N.R., Depart. of the Navy).\n295. Taylor, G. I., \"Diffusion by continuous movements,\" Proc.\nLondon Math. Soc. 20 (Ser. 2), 196 (1920).\n296. Thekaekara, M. P., \"The solar constant and spectral dis-\ntribution of solar radiant flux,\" Solar Energy 9, 7\n(1965).","BIBLIOGRAPHY\n367\n297. Tick, L. J., \"A nonlinear random model of gravity waves,\"\nJ. Math. and Mech. 8, 643 (1959).\n298. Tyler, J. E., \"Radiance distribution as a function of\ndepth in an underwater environment,\" Bull. Scripps\nInst. Ocean. 7, 362 (1960).\n299. Tyler, J. E., An Instrument for the Measurement of the\nVolume Absorption Coefficient of Horizontally Strati-\nfied Water (Report No. 5-4, Task 5, Contract NObs-\n72039, Bureau of Ships Project NS714-100, Visibility\nLaboratory, University of California, San Diego, Feb-\nruary 1960).\n300. Tyler, J. E., \"Scattering properties of distilled and\nnatural waters,\" Limnology and Oceanography 6, 451\n(1961).\n301. Tyler, J. E. \"Estimation of percent polarization in\ndeep oceanic water,\" J. Mar. Res. 21, 102 (1963).\n302. Tyler, J. E., \"Colour of the ocean,\" Nature 202, 1262\n(1964).\n303. Tyler, J. E., ed., Physical Aspects of Light in the Sea,\nA Symposium at the Tenth Pacific Science Congress,\nHonolulu, Hawaii, August 1961 (Univ. of Hawaii Press,\nHonolulu, Hawaii, 1964).\n304. Tyler, J. E., and Shaules, A., \"Irradiance on a flat\nobject underwater, Applied Optics 3, 105 (1964).\n305. Tyler, J. E., and Preisendorfer, R. W., \"Light,\" in The\nSea (Interscience Pub., New York, 1962), M. N. Hill,\ned., vol. I, Chapt. 8.\n306. Tyler, J. E., , Richardson, W. H., and Holmes, R. W.,\n\"Method for obtaining the optical properties of large\nbodies of water,\" J. Geophys. Res. 64, 667 (1959).\n307. Ursell, F., \"Wave generation by wind,\" in Surveys in\nMechanics (Camb. Monographs on Mech. and Applied Math.,\n1956), G. K. Batchelor, R. M. Davies, eds.\n308. van de Hulst, H. C., Light Scattering by Small Particles\n(John Wiley and Sons, New York, 1957).\n309. Volterra, V., Theory of Functionals and of Integral and\nIntegrodifferential Equations (Dover Pub., Inc., New\nYork, 1959).\n310. Walsh, J. W. T., \"The reflection factor of a polished\nglass surface for diffused light,\" Dept. Sci. Ind.\nRes. (Brit.), Illumination Research Tech. Pap. 2, 10\n(1926).\n311. Walsh, J. W. T., Photometry (Dover Pub., Inc., New York,\n1965).","SURFACES\nVOL. VI\n368\n312. Wang, Alan Ping-I, Scattering Processes (Ph. D. Math.\nThesis, University of California, Los Angles, 1966).\n313. Wax, N. , ed., , Selected Papers on Noise and Stochastic\nProcesses (Dover Pub., Inc., New York, 1954).\n314. Weinberg, A.M., and Wigner, E. P., The Physical Theory\nof Neutron Chain Reactors (Univ. of Chicago Press,\n1958).\n315. Whitney, L. V., \"The angular distribution of character-\nistic diffuse light in natural waters,\" J. Mar. Res.\n122 (1941).\n316. Whitney, L. V., \"A general law of diminution of light\nintensity in natural waters and the percent of dif-\nfuse light at different depths,\" J. Opt. Soc. Am. 31,\n714 (1941).\n317. Whitney, W. M. , Contrast Reduction by Time-Varying Re. -\nfraction (Contract NObs-50274, Index Number NS741-100,\nVisibility Laboratory, University of California, San\nDiego, March 1956).\n318. Whittaker, E. T., and Watson, G. N. , A Course of Modern\nAnalysis (Cambridge Univ. Press, 1952), 4th ed.\n319. Wick, G. C., \"Uber ebene diffusionsprobleme,\" Z. f.\nPhys, 121, 702 (1943).\n320. Wiener, N. \"Generalized harmonic analysis,\" Acta Math.\n55, 117 (1930).\n321. Wiener, N. , The Fourier Integral and Certain of Its\nApplications (Cambridge Univ. Press, 1933; reprinted\nby Dover Pub., Inc., , New York).\n322. Wilczynski, E. J., \"An application of group theory to\nhydrodynamics, Trans. Am. Math. Soc. 23, 339 (1912).\n323. Wilf, H. S. , \"Numerical integration of the transport\nequation with no angular truncation,\" J. Math. Phys.\n1, 225 (1960).\n324. Wuest, W. \"Beitrag zur entstehung von wasserwellen\ndurch wind,\" Z. Angew. Math. Mech. 29, 239 (1949).\n325. Yabushita, S., \"Tschebyscheff polynomial approximation\nmethod of the neutron transport equation,\" J. Math.\nPhys. 2, 543 (1961).\n326. Yosida, K., Functional Analysis (Academic Press, New\nYork, 1965).\n327. Zemanian, A. H., Distribution Theory and Transform Anal-\nysis (McGraw-Hill, New York, 1965)","INDEX TO VOLUMES I-VI\n369\na (alpha), I : 60\nalgebraic properties of R,T\na (ay), I : 55, 58, 60\nIV : 27\nabsolutely continuous\ninvariant imbedding rela-\nmeasure, II: 375\ntion, IV: 35\nAC property, II: 381\nalgebraic spherical harmonics,\nIV: 12\nIII: 141\nabsorbed flux, I : 55, 58\naltitude (in a reference\nabsorbed radiant energy,\nframe), II: 19\nIII: 94\nAmbarzumian's principle, II:\nabsorption (of a finitely\n228. IV: 34\ndeep slab of water, Ay)\namplitude spectrum VI: 115,\nI : 70\nsee spectrum\nmeasurements, I : 103\nanalytic properties (invari-\nlength, I : 110\nant imbedding relation),\nabsorption function (for\nIV: 68\nirradiance), V: 12, 97\nanemometer height, VI: 181\nsimple formula for in\nangle, polar, azimuthal, II:\nasymptotic light field,\n24\nV: 255\napparent optical properties,\nabsorption time constant\nI : 118\nIII: 82\ndefined, listed, I: 135,\nabstract spherical harmonic\nV : 106, 178, 181\nmethod, III: 143\ntransport equations for,\nAC property, II: 381, IV: 32\nV : 271\nacceleration (for a fluid\noperational definitions re-\npacket), VI: 48\nviewed, VI: 324\nadaptation level (for visi-\napparent radiance, I : 60\nbility), I : 160\nII: 362\naddition theorem for spheri-\napparent radiance\ncal harmonics, III: 148,\ncanonical representation,\n153\nIII: 16\naerial stereo photography\napparent-radiance equation,\n(for spectra), VI: 166\nII: 361\nair-water surface\napparent wave period, height\nelementary hydrodynamics\nVI: 183\nfor, VI: 46\nastrophysical optics, defined,\nFourier series, integral\nI : 1\nrepresentations, VI: 87,\nasymmetries of the Y-operator,\n101, 109\nIV: 179\nworking representation,\nasymptotic form of light\nVI: 115\nfields, V : 95\nintegral equation for\ncriterion for asymptoticity,\ninstantaneous surface\nV: 254\nradiance, VI: 215, 234\nasymptotic properties (of R,T)\nweighting functions, VI: 221\nIV: 33, IV: 104\ntime-averaged equation for\nor radiance, IV: 127\nsurface radiance, VI: 223,\nof polarized radiance,\n234\nIV: 128\nsynthesis of results, VI:\nasymptotic radiance hypothesis,\n246\nI : 41, V: 212\nalbedo, II: 216\nalgebraic proof, IV: 127\nalbedo for single scattering\nmain mathematical proof\n(see scattering-attenuation\n(using canonic equations for\nratio)\nK), V : 213-227\nalgebra\nintegral equation for limit\nof reflectance and transmit-\nradiance distribution,\ntance operators, II: 230\nV : 228, 237, 245\nnormal, II: 243\ntheoretical and experimental\nBanach, V: 244\nexample, V: 229, 230","SURFACES\nVOL. VI\n370\nasymptotic radiance hypothesis\nBoltzmann equation (for water\n--Cont'd.\nwaves), VI: 208\nsimple proof (using exponen-\nBouguer's work, III: 1\ntiality of h(z)), V : 230\nboundaries, I : 55\npractical consequences, V :\nreflecting, II: 340\n238\nboundary effects, V: 71, 46\nK characterization of\nbounds, on radiance, III: 47\nhypothesis, V: 242\nBretschneider's empirical\ncritique of Whitney's\nwave observations, VI: 158\n\"general law\" V : 248\nwave spectrum, VI: 204\nheuristic proof (using\nBright target technique (for\ndifferential equation for\na), VI: 292\nK), V: 253\nfor a , VI: 299\natmospheric radiative trans-\nbrightness (monochromatic) of\nfer, gross features, I : 27\nradiant flux, I : 10\nattenuated radiant energy,\n\"brightness\" is an untech-\nIII: 94\nnical term for the precise\nattenuating functions\nconcepts of radiance or lu-\ndepth dependence, V: 25\nminance (as the case may be)\nattenuation functions (for\n(irradiance), V: 11\ncandela, I: 20, II: 163, 179\ndepth dependence, V: 25\ncanonical equations\nfor radiance, V: 264\nsense of the term, III: 1\nfor irradiance, V: 265\nclassical, III: 91\nfor scalar irradiance,\nexperimental verification,\nV: 270\nIII: 13\nfor reflectance, V:\ngeneral media, III: 15\n149, 279\nstratified media, III: 18\nfor K-function, V : 275\npolarized radiance, III: 21\nfor other concepts, V:\nabstract versions, III: 24\n281\ncanonical forms of transport\nattenuation length, I: 90\nequation for K functions,\nIII: 99, 196\nV: 273, VI: 292\nattenuation time constant,\ncanonical representation\nIII: 76\nof apparent radiance, III,\nscattering, III: 10\n16\nautocorrelation, VI: 96\nof abstract functions, III,\n27\nback scattered flux, I: 55\nof irradiance fields, V : 98\nbackward scattering func-\nof a (attenuation function),\ntions, V: 11, 141\nVI: 292\nexperimental determination,\ncapillary waves, VI: 71\nVI: 335\ncardioidal radiance distribu-\nBanach algebra, II: 244\ntion, VI: 21\nBeam transmittance, I: 120\ncarnivores (in food chain),\nBeam transmittance function,\nI : 199\nII: 344\ncatalog of K-figurations\nvarious properties, II: 348\n(shallow depth theory),\noperational definition, VI:\nV: 201\n284\nnondegenerate\nBeebe, L., I: 143, 153\ndegenerate\nbete noir (in applied math),\nfirst, second kind\nVI: 105, 263\nforbidden, V: 202\nbiological sources, underwater\ncategorical synthesis method,\nlight field, I : 53\nIV: 164\nblondel, I: 21, II: 179\ncauchy sequence, III: 130\nBochner theory (of positive-\ncausality conditions, (for\ndefinite functions) VI:\nR,T), IV: 23\n110,","INDEX\n371\ncharacteristic\ncollimated flux\nrepresentation of N(y),\nscattering functions for,\nIV: 122\nI : 83\ncharacteristic ellipsoid\nproduced by sources, I : 114\nspheroid, III: 66, 68\ncolor\ncharacteristic equation\ncomponents, I: 146\nfor second order differ-\npurity, I : 149\nential equation, V : 39\ndominant wavelength, I : 149\nfor K function, V : 123, 252\ncolorimetric radiative trans-\nchromaticity (color), I : 146\nfer, I : 142\ncomponents, I: 148\ncommutativity (of R,T,P, T), ,\nplane, I : 147\nIV: 34\ndiagram, I : 149\ncomplete (general) solution\ncoordinates, I : 149\nof irradiance equations,\nClairaut's equation (for in-\nV: 42\nverse nth power irradi-\ncomplete metric space, III:\nance law), II: 126\n131\nclassical diffusion theory,\ncomplete (Planckian) radiator,\nIII: 134\nII: 162\nbasic diffusing equation,\ncomplete reflectance (for\nIII: 175\nirradiance), I: 79, V: 4,\nradiance distribution in,\n4, 79, 62, 64\nIII: 181\ncomplete transmittance (for\napproaches via higher order\nirradiance), I : 79, V : 4,\napproximations, III: 183\n62, 64, 79\nhierarchy of processes, III:\ncompleteness property of\n184\nspherical harmonics, III:\nplane-parallel solutions,\n142, 153\nIII: 193\ncone (in space time), III: 53\nspherical (point) solutions,\nconsistency\nIII: 200\ncheck for inherent optical\ndiscrete (extended) solu-\nproperties, I : 124\ntions, III: 203\ncheck for operational for-\ncontinuous (extended) solu-\nmulations, VI: 339\ntions, III: 206\nof unpolarized radiative\nprimary sources, III: 207\ntransfer theory, VI: 341\nfor higher order scattered\nexperimental check for un-\nscalar irradiance, III: 213\npolarized radiative trans-\ntime dependent, III: 214\nfer in a polarized setting,\nclassical models for irradi-\nVI: 345\nance fields, V : 19\nconstitutive definitions, II:\ncollimated-diffuse light\n8\nfields, V: 19\nconstructive extension of\nisotropic scattering models,\nM(x,y,z), IV: 50\nV: 23\ncontinuous sources in diffu-\nconnections with diffusion\nsion theory, III: 206\ntheory, V : 24\ncontraction mapping, III: 129\nclassical spherical harmonic\ncontraction mapping, princi-\nmethod, plane-parallel\nple of, III: 131\nmedia, III: 158\ncontraction property (of beam\nclassical theory, inadequacies,\ntransmittance), II: 348,\nV : 115\nIII: 129\nclassification of natural\ncontrast\nhydrosols, I : 138\napparent inherent, I : 44,\ncoherent (partially) radiance\nV : 165\nfield (problem), VI: 346\ntransmittance law, I : 89,\ncolligation of component 4-\n90, 99\noperators, IV: 176","VOL. VI\nINDEX\n372\ncontrast--Cont'd.\ndecomposed (light field) - -\nmultiplicative (semigroup)\nCont'd.\nproperty, I: 95\nequation of transfer, IV:\ncontrast reduction\n9, 14\nsubsurface, by scattering\ndefinitions: constitutive,\nand absorption, I : 44\noperational, II: 8\nby refractive effects,\ndepth (in a reference\nI: 48\nframe), II: 19\ncontrast transmittance, V : 162\nderivative property (of\nproperties, V : 168\nintegral transform), IV:\nin canonical equation for\n192\nradiance, V: 170\ndichroic material (and\nalternate representations,\npolarized light), II: 84\nV: 171\ndifferential equations\nas an apparent optical\nfor m(x,y), IV: 69\nproperty, V: 172\nfor M(x,y,z), IV: 71\neffect of shadows on, V: 174\nfor M(v,x;u,w), IV: 76\nat static air-water\nfor M(x,y), IV: 79\nsurface, VI: 41, 44\nfor Y(s,y), IV: 79\nat dynamic air-water\nfor R(y,b), IV: 80\nsurface, VI: 258\nfor dual operators, IV: 173\ncontravariation of K and D,\ndiffraction, limits on\nV: 144\nradiometry, II: 16\nconventions (used in this\ndiffuse absorption coeffi-\nwork)\ncient, V: 111\nnature of radiant flux, I : 6\ndiffuse (decomposed) irradi-\nunpolarized-flux, I : 7\nance, V: 14\nfrequency density (footnote)\ntransmittance for, V: 17\nconvexification, white, black\ndiffuse radiometric func-\nII: 316\ntions, III: 36\nconvolution theorem (of inte-\nstored energy, III: 123\ngral transform), IV: 192\ndiffusion constant (D), I : 64\napplied to energy spectrum,\nin terms of K, I : 111\nVI: 98\nin terms of K, III: 194\ncoordinate systems, terres-\ndiffusion equation\ntrial, II: 19\nscalar, III: 174\ncosine collector, II: 7\nwave, III: 184\ncosine law, for irradiance,\ntensor, etc., III: 184\nI : 13, II: 26\nfor h, I : 64\nfor radiant emittance, I: 14\ndiffusion function (D), III:\ncosine (mean value ), III: 180\n174, 180, 181\ncosurface time average, III:\ndiffusion length, I: 135\n238\nIII: 196\ncovariation of K and D, V :\ndiffusion model, I : 61\n128, 136, 140\nfor point sources, I : 110\nrule of thumb, V : 145\nempirical examples, I : 112\nCox and Munk's wave slope\ndiffusion processes\nobservations, VI: 145\na short list, III: 184\ndiffusion theory, three\ndark target technique (for N*),\napproaches, III: 172\nVI: 289\ndiffusion theory, connections\nfor a, VI: 291\nwith, V: 24\nfor a, 00, VI: 300\nsee also exact, below\ndecomposed (diffuce) irradi-\ndimensionless forms of radi-\nance, V: 14\nant energy fields, III: 97\ndecomposed (light field),\nDirac matrices, V: 8\nI : 63\ndirection, defined, II: 19\nradiometric functions,\nupward, downward, II: 21\nIII: 36","INDEX\n373\ndirection, defined--Cont'd.\nEckart's wave generation\nunit inward, II: 25\nmodel, VI: 207\nfor reflectance and\nelectric circuit analogy\ntransmittance, II: 212\n(with an optical medium)\ndirectly observable\nIII: 77, 123\nradiant energy, equation\nelectromagnetic view of light\nof transfer, III: 81\n(vs. phenomenological),\noptical properties, V:\nII: 13\n109, 178\nelsewhere (in space-time),\ndirectly transmitted radi-\nIII: 53\nance, II: 347\nempirical basis (of module\ndiscrete sources in dif-\nequations), IV: 106\nfusion theory, III: 202\nempirical wave spectrum\ndiscrete-space radiative\nmodels, VI: 181\ntransfer, V: 51\nenergy conservation princi-\ndisplacement law (in ocean\nple (for radiometry),\nwave theory), VI: 186\nII: 199\nanalogous to Wien's law\nenergy of surface waves,\ndistribution factor, I : 55\nVI: 72\ndistribution function,\nand wind speed, VI: 193\norigin, V: 10\nenergy spectrum, VI: 79\nfor diffuse irradiance,\nsee spectrum\nV: 15\nequation of continuity, VI:\nrepresentative values,\n50 et seq.\nV: 26\nequation of (fluid) notion,\nrepresentation via radi-\nIV: 48 et seq.\nance distribution, V: 27\nlinearized, VI: 62\ndepth dependence, V: 26\nequation of transfer, I: 60\nin canonical equation for\nfor radiance, II: 368\nirradiance, V: 99\ntime dependent and polar-\nexperimental, V: 115\nized form, II: 371\ncontravariation (with K-\nn-ary radiance, III: 36\nfunction), V: 144\nunscattered radiance, III:\ncovariation (with K-\n37\nfunction), V: 128, 136, 140\ndiffuse radiance, III: 37\nphysical and geometrical\npath function, III: 38\nfeatures, V: 128\nnatural solution, III: 43,\nasymptotic limits, V: 246\n127\nsummary discussion of prop-\nfor optical ringing, III:\nerties, VI: 329\n56\ndivergence law for vector ir-\nsolved symbolically, III :\nradiance, I : 44, 64\n65\ndominant wavelength, I: 149\nresidual radiant energy,\ndual operators\nIII: 76\nfirst appearance, IV: 156\nn-ary radiant energy, III:\nsecond appearance, IV: 158\n80\ndefinitive list, IV: 160\ndirectly observable radiant\nfor invariant imbedding\nenergy, III: 81\nrelation, IV: 166\ndimensionless (for radiant\nintegral representation,\nenergy), III: 97\nIV: 167\nscalar irradiance (diffusion\ndifferential equations for,\nequation), III: 175\nIV: 173\nlimitations (in group method)\nDuntley disks, I : 96\nIV: 130\nDuntley's wave slope obser-\nas local form of principles\nvations, IV: 138\nof invariance, IV: 4\nscattered radiance, III: 209\nE (epilson) function, III: 222\nscattered scalar irradiance,\nIII: 210, 213","VOL. VI\nINDEX\n374\nFick's Law (of diffusion), I :\nequation of transfer--Cont'd.\nfor time-averaged radiance\n64, III: 174\nfield interpretations of ra-\nfields, VI: 239\nfor perturbed light field,\ndiant flux, I : 12\nfine structure of light field\nVI: 293\nin a polarized light field,\nhypotheses, V : 199\nspecial relations, V : 208\nVI: 344\nequilibrium radiance, I : 85,\nfinite recurrence property,\nIII: 147, 154\nIII: 6\nequilibrium radiance function,\nfinitely deep hydrosols,\nreflectance and transmit-\nV : 151\nfor radiance, V: 151\ntance, I : 68\nfirst order scattered radiance\nfor irradiance, V: 13, 265\nfor scalar irradiance, V:\nequation for, III: 41\nfirst standard solution of\n270\nfor reflectance, V: 149, 279\ntwo-D model, V : 31\nfixed depth time average, III:\nfor K-functions, V: 275\nfor other concepts, V : 281\n238\nequilibrium-seeking theorem\nfloating buoy motion (for\nfor R, V : 152\nspectra), VI: 173\nfluid transfer process, VI:\nfor N,H, h, V: 270\nand universal radiative\n47\ntransport equation, V : 279\nflux (see radiant flux or\nequilibrium solutions (food\nluminous flux)\nflux density (radiant), I : 10\nchain), I : 203\nfood chain problem (in the\nequivalence classes of func-\ntions, III: 128\nsea), I : 196\nequivalence theorem for\nfoot candle, I : 20\nforward scattering functions,\nR(z, V : 159\nergodic cups and caps, VI:\nV : 11, 141\nexperimental determination,\n263\nergodic hypothesis, VI: 148\nVI: 335\nforward scattering media, V:\nand water wave slopes,\nVI: 261\n140\nexact diffusion theory\nFourier integral representa-\nbasic equation, III: 134,\ntion of air-water surface,\n190, 192, IV: 218\nVI: 101\ninfinite medium with point\nFourier series representation\nof air-water surface, VI:\nsource, III: 219\ninfinite medium with arbi-\n87\nFourier transform of exact\ntrary sources, III: 225\nscalar irradiance, III: 226\ndiffusion equation, III:\nsemi-infinite medium with\n192\nboundary point source, with\nframes (of reference, e.g.,\ninternal point source, III:\nterrestrial), II: 24\nfrequency density convention\n228, 233\non the Elliott functional\n(in this work), I : 7\nrelations, III: 236, IV: 153\nFresnel's laws for reflectance\nexponential law of change (gen-\nVI: 10\nfunctional relations for fc, fo\neral), IV: 197\ndifferential form, I : 201\nin exact diffusion theory,\nexponential property of diffu-\nIII: 236\nsion field (plane-parallel\nfundamental optical property,\ncase) III: 194\nV : 107, 178\nexponential representation of\nfuture (in space time), III:\nm(y),N(y), IV: 117\n53\nof a(y), IV: 117\nextension of operators, IV: 50 gamma-type wave spectra laws,\nVI: 190","INDEX\n375\nGelbstoff, I : 133\nHulburt's wave slope observa-\ngeophysical optics, defined\ntions, VI: 136\nI: 1\nhydrodynamic basis of harmonic\nGEOVAC (geophysical optics\nanalysis of air-water sur-\nvariable automatic compu-\nface, VI: 91\nter), I: 208\nhydrodynamics and radiative\nGershun tube, VI: 282\ntransfer, VI: 46\nglitter patterns\nhydrologic optics, defined,\non air-water surface,\nI : 1\nI : 32\nfuture problems, I: 205\nphotographed, VI: 147\nhydrologic range, I: 90\ncomputed, VI: 254\nglobal approximations for\nilluminance, I: 19\nradiance, III: 117\nmeasured at earth's surface,\nhigher order, III: 119\nI: 25, II: 166\nglobal concepts, IV: 148\nindependence condition, VI:\nglobal optical property,\n220\nV : 107, 180\nindex of refraction (relative)\nmeasurement of R,T,\nVI: 5\nVI: 331\ninelastic scatter, III: 5\nglobal y-operators\ninequality for K,a, III: 195\nintegral representation,\namong optical properties,\nIV: 162\nV: 114, 119\ngravity waves, VI: 71\ninfinitesimal generator, IV:\ngroup property (of integral\n112\ntransfer kernel), IV:\nof 2, IV: 116\n191\ninherent optical properties,\ngroup structure of natural\nI : 118,\nlight fields, II: 307\ndefined, listed, I: 119,\ngroups (method of), IV: 114\nV: 106, 178, 180\ninherent radiance, I : 60,\nHaidinger's brush (in pola-\nII: 363\nrized light), II: 84\ninner structure of natural\nharmonic analysis,\nlight fields, IV: 141\nof dynamic air-water\nintegral equation for scalar\nsurface, VI: 82\nirradiance, III: 189\nroots of, VI: 83\nintegral representations of\nvs. synthesis, VI: 84\nlocal Y-operators, IV:\nintegrals vs. series, VI:\n154\n86\nof global y-operators,\nhydrodynamic basis, VI:\nIV: 162\n91\nof dual operators, IV: 167\nHasselmann's wave spectrum\nintegral transform techniques\ntheory, VI: 209\nIV: 188\nheirarchy of time-averaged\ngroup property, IV: 191\nsea surface radiance\nderivative property, IV: 191\nmodels, VI: 250\nconvulution theorem, IV: 191\nhemisphere (E+)\nfor time-dependent radia-\nherbivore (in food chain)\ntive transfer, IV: 194\nI : 199\nintegrating sphere, II: 262\nherschel (luminance unit),\nintensity (radiant), I : 10,\nI : 21, II: 172\nII: 70\nheterochromatic radiative\nfield, I : 12\ntransfer, IV: 197\nsurface, I: 12\nhomogeneity (of o), I : 82\nluminous, II: 166\nhomogeneous (opticl medium)\ninteraction functional (for\nIV: 144\nwave spectrum), VI: 208\nrestricted, IV: 145","376\nINDEX\nVOL. VI\ninteraction method\ninteraction principle - Cont'd\na first synthesis, II: 222\nas a means and as an end,\nin other fields, II: 223\nII: 391\n(in footnote)\nfor time-averaged sea radi-\nsummary, II: 388\nance, VI: 246\nand quantum theory,\ninterdependence (plan) of\nII: 390\nchapters on this work,\ninteraction operators\nI : 5\nintegral struture, II,\ninterfaces, reflecting, V: 340\n372\ninternal-source problem,\nkernel, II: 380\nIV: 152\nsee also operators\nnew methods, V : 102\ninteracting media (invariant\ninternal sources and irradi-\nimbedding operators for)\nance fields, V : 37, 55,81\nV: 76\ninterreflection calculations\nair-water surface and hy-\nterminable and nontermin-\ndrosol, VI: 35\nable, II: 248\nthreefold irradiance in-\ninvariant imbedding relation,\nteraction (air, water,\nI : 71, 80\nair-water surface), VI:\non plane parallel media,\n38\nII: 297\nthreefold radiance inter-\nhistorical notes, II: 299\naction, VI: 39\ngeneralized form, II: 301\ntime-averaged radiance\nfor one-parameter media,\nfields (air-water),\nII: 327\nVI: 246\nin general media, II: 339\ninteraction principle, I : 4\ntime dependent, IV: 22\nphysical basis, II: 189\nalgebraic properties, IV: 35\nbasic statement, II: 205\nanalytic properties, IV: 68\nplace in radiative trans-\nfor deep hydrosols, IV: 104\nfer theory, II: 208\ndual form, IV: 166\nlevels of interpretation,\nfor irradiance fields, V :\nII: 208\n2, 61\nAmbarzumian's principle,\nin two-D model, V : 36\nII: 228, IV: 34\nincluding boundary effects,\napplications to plane sur-\nV: 50\nfaces, II: 217\nfor interacting media, V:\napplications to curved\n76\nsurfaces, II: 258\ninverse problems in hydrologic\napplications to plane-\noptics, VI: 339\nparallel media, II: 285\ninvertibility of operators,\napplications to general\nIV: 39\nspaces, II: 322\napplication to N from H,\non one-parameter media\nII: 145\nwith sources, II: 330\nirradiance, I : 12\nas basis for beam trans-\nscalar, I: 15, 106, II: 54\nmittance and volume\nhemispherical scalar, I : 16\nattenuation function,\nvector, I : 15\nII: 344\nnet, I : 16, 61, II: 26\nas basis for path func-\nupwelling (upward), I: 16,\ntion, path radiance, II:\n55, 58, 106\n351\ndownwelling (downward), I :\nas basis for volume\n16, 55, 58, 106\nscattering function, V:\nmeasured at earth's surface\n364\nI : 24\nas basis for equation of\nreflectance of air-water\ntransfer, II: 368\nsurface, I : 30","INDEX\n377\nirradiance--Cont'd\nisotropic scatter, IV: 83, 86\nreflectance in deep water,\nisotropic scattering (in vec-\nI : 67\ntor light field model),\ninvariant imbedding\nV: 94\nrelation, I : 71\nmodels, V: 23\ndefined, II: 14, 171,\nisotropy (of o) , I : 82\nIII: 274\niterated operators, II: 236\nmeaning, II: 16\ntypical orders of\njanus plate, II: 68\nmagnitude, II: 17\nJordan canonical form IV: 122\nin terrestrial frames,\njoule, II: 172\nII: 24\nupward, downward, II: 24\nK (kappa) K-function or dif-\ncosine law for, II: 26, 66\nfuse attenuation function\n(from radiance), II: 35, 131,\nfor diffusion model),\n138\nI : 65, III: 194\nradiance from II: 41\nKo (dimensionless form)\nspherical, II: 56\nIII: 271\nvector, and mechanical\nk (little kay), I : 58\nanalogy, II: 62\ninterchangeable with K (big\njanus plate (for net),\nkay), I : 83\nII: 68\nK-function\n-distance law (for\ngeneral, III: 15\nspheres), II: 103\ntwo-D model, V: 31, 39\n-distance law (for\none-D model, V: 53, 56\ncircular disks, II: 105\nin canonical equation, V :\n-distance law (for general\n100\nsurfaces), II: 106\ntheoretical form, V: 111\nvia line integrals, II: 109\ndiffuse absorption coeffi-\nvia surface integrals,\ncient, V: 111\nII: 115\nexperimental, V: 115\nlaws of the form 1/rn,\nsignificance of sign, V: 120\nII: 120\ncharacteristic equation for,\ndistributions, equivalence\nV: 123\nof with radiance distribu-\nconnections among (irradi-\ntions, II: 143\nance K-function), V: 123\ncomputation for parallel\ngeneral forms, V : 125\nplanes, II: 217\nfor radiance, V: 125\non plane-parallel media,\nintegral representations,\nII: 286\nV: 126\nvector, via spherical\nin spherical coordinates,\nharmonics, II: 177\nV: 127\nscalar, via higher order\ncontravariation (with dis-\nscattering, III: 213\ntribution function), V:\n(see also scalar irradiance,\n144\nvector irradiance)\ncovariation (with distribu-\ninteraction equations at\ntion function), V: 128,\nair-water surface, VI: 38\n136, 140\nirradiance distributions,\nphysical and geometrical\nunderwater, I : 42\nfeatures, V: 128\nirradiance (two-flow) equa-\nabsorptionlike character\ntions, V: 6\n(for irradiance), V: 138\nirradiance quartet, I : 135,\nconfigurations for shallow\nV: 115, VI: 325\ndepths, V : 201\nisomorphism between T2,G2\nasymptotic limits, V: 246\nIV: 44\ncanonical form of transport\nisotropic (optical medium),\nequations for, V : 273\nIV: 147,\ndiscussion of experimental\nprocess, IV: 147\nhistory, VI: 330","VOL. VI\nINDEX\n378\nlight storage phenomena in\nKa, III: 188\nnatural optical media,\nKK, III: 214\nIII: 121\nKg, III: 221\nlinear functional, II: 373\nKelvin-Helmholtz model,\npositive, II: 375\nVI: 67, 206\ninstability, VI: 70\nline of flux, II: 8\nlocal optical property, V:\nkernel\ninteraction, II: 380\n107, 179, 180\nof integral transform\noperational determination,\nVI: 333, 335\nIV: 191\nlocal principles of invari-\nKoschmieder's equation,\nance, IV: 18\nIII: 5\nlocal Y-operators, IV: 154\nlocal reflectance, IV: 3,\n1.i.m., VI: 104\nlagrangian (a mobile or sub-\nV: 7\nlocal residual (reduced)\nstantial) derivative,\ntransmittance operators,\nI: 371, VI: 49\nIV: 11\nlambert, I : 20\nlocal transmittance, IV: 3,\ntransmitter or reflector,\nV : 7\nII: 262\nlogarithmic wind profile,\nlaminar flow model (of wave\ngeneration) VI: 206\nVI: 132\nlogical descendents of Y,\nlaw of reflection (at water\nsurface), VI: 3\nIV: 171\nlumen, I : 19, II: 161, 179\nlaw of refraction (at water\nluminance, I : 19,\nsurface), VI: 5\ndistribution, relative,\nlight\nthis term is used throughout\nII: 156\nthe present work as an in-\ndistribution, general, II:\nformal correspondent to any\n163, 166\none of the defined concepts\ntypical magnitudes, II: 164\nof geometrical radiometry\npath, II: 179\nluminosity of a wavelength,\nand photometry. The mean-\ning intended for the term\nII: 153\nlight will be implicit in\nluminosity function,\neach context of its use.\nstandard, II: 151, 157, 159,\nThus light field may, e.g.,\n160\ncorrespond informally to\nfor individuals, II: 153\nphotopic, I : 145, II: 155,\nradiant energy, radiant\nflux, radiance distribution\n158\nirradiance function, Zumi-\nscotopic, II: 155\nrelative (of radiance), ,\nnous energy, luminous flux,\nluminance distribution,\nII: 157\ngeneralized, II: 184\nilluminance function, et-\nluminous emittance, II: 166,\ncetera.\n179\nscalar, II: 168, 179\nlight field\ndecay with depth, I : 37, 66\nvector, II: 179\nluminous energy, I : 19, II:\npolarization, underwater,\n167\nI : 50\nbiological sources, I : 53\nluminous energy density, II:\nartificial, I: 109\n167\nluminous flux, II: 166\ndecomposed, I: 63\ntime dependent, III: 49,\nluminous intensity, II: 166\nluxoid (via inverse nth power\nIV: 17\ninner structure, IV: 141\nfor irradiance), II: 130\nfine structure, V: 199,208\nperturbed, VI: 293\nmpq' VI: 120","INDEX\n379\nmu (millimicron)\nn-ary radiometric concepts,\n(= nanometer, nm), II: 192\nIII: 31\nmanhole (optical), I: 34\nradiance, III: 33\nmany-D models, V: 57\nscalar irradiance, III: 34\nmean square\nradiant energy, III: 34, 83\nwave elevation, VI: 120\ngeneral, III: 35\nwave slopes, VI: 121\ncanonical equation for\nmeasure, Riemann, Lebesque,\nradiance, III: 38\nStieltjes, II: 12\nnatural closed forms for\nillustration of, II: 373\nradiant energy, III: 86\nbasic theorems, II: 375\ntime-dependent properties,\ninteraction, II: 380\nIII: 89\nmedia\ndimensionless forms, III:\nplane-parallel, cylindrical,\n97\netcetera, see table of\nn2- convention, VI: 210\ncontents\nnanometer (=10-9m) = milli-\nmelanoidines (Gelbstoff),\nmicron (mu), II: 192\nI : 133\nnatural hydrosols\nmeter, radiance, II, 30\nclassified, I : 138\nfor polarized radiance,\ncharacterization (for\nII: 85\nvisibility), I : 195\nmethod of modules, IV: 103\nnatural illumination, I : 156\nsemigroups, IV: 108\nnatural (mode of) solution,\ngroups, IV: 114, 129, 135,\nII: 203, 247\n141\nfor radiance, III: 42\nfor irradiance fields,\ntruncated, III: 45, 69\nV: 80\ntime-dependent, III: 58\nmetric\nsymbolic integration,\nsupremum, III: 129\nIII: 65\nradio-, III: 128\nfor directly observable\nmetric space, III: 127\nradiant energy, III: 82\ncomplete, III: 131\ntime-dependent properties,\nMiles' wave generation model,\nIII: 90,\nIII: 207\ndimensionless form, III: 97\nmillimicron (mu), = 10-9 m\noperator-theoretic basis,\n= nanometer, II: 192\nIII: 127\nmobile (or substantial or\nfor scalar irradiance, III:\nlagrangian) derivative\n191\nI : 371, VI: 49\nNeumann spectrum, VI: 181\nmobius strip, II: 271\nnomographs for underwater\nmodes of classification of\nvisibility, I : 154\nnatural hydrosols, I : 140\nnorm-contracting property,\nmodules (method of), IV: 103\nof C operator, II: 147\nequations, IV: 106\nof R,T operators, II: 236\nmonotonicity condition on\nof operators, II: 322\nradiance distribution,\nnormal space (0 < p < 1),\nV : 28\nIII: 103\nmoon\nnormal operator algebra, II\nradiance of, II: 98\n243\nradiant intensity of, II: 99\nnorth-based reference frame,\nluminance of, II: 164\nII: 19\nmultiplicative (semigroup)\nnutrient (in food chain), I\nproperty\n199\nof contrast transmittance,\nI : 93\none-D (two-flow irradiance)\nof beam transmittance,\nmodel, I : 56\nI : 120, II: 348\none-D models for irradiance\nfields, V: 51","VOL. VI\nINDEX\n380\noptical medium--Cont'd\none-D models for irradiance\nwith internal sources,\nfields--Cont'd.\nfor undecomposed fields,\nIV: 152\ndefinition (formal), V : 108\nV: 52\ninternal sources, V: 55\noptical properties\ninherent, apparent, I: 118\nfor decomposed fields,\nII: 349\nV: 56\nconnections with observable\nlocal global, V: 107\nfundamental, V : 107\nfields, V: 160\ngeneral definition, V: 109\none-path method (for beam trans-\ndirectly observable, V: 109\nmittance), VI: 286\nontogeny (family roots) of two\nclassification, V: 178\nin asymptotic light fields,\nflow equations, V: 13\noperational definitions of the\nV : 238\ndensities, I : 10, II: 8\noptical reverberation case,\nprincipal radiometric con-\nIII: 86\noptical ringing problem, one-\ncepts, VI: 273\noperator-theoretic basis for\ndimensional, III: 49\nthree-dimensional, III: 66\nnatural solution, III: 127\noptical volume, III: 220\noperators\nalgebra, II: 230\nparseval identity, VI: 105\niterated, II: 236\npartially coherent polarized\nalgebra and radiative\nradiance fields (theory\ntransfer, II: 241\nformulation problem),\nalgebra, normal, II: 243\nVI: 346\nII: 319\ninteraction, structure of,\npartition relations, IV: 28\npast (in space-time), III: 53\nII: 372\nvolume transpectral, II: 386\npath function (radiant), I :\npath function, II: 383\n60, II: 172\npath radiance, II: 384\n(luminous) II: 179\ninteraction for general\nderivation, II: 351\nconnection with path\nspaces, II: 314, 378\ninteraction for surfaces,\nradiance, II: 354\nintegral representation,\nII: 377\nmiscellaneous examples,\nII: 367\noperator, II: 383\nII: 387\nR (path function), III: 32\nequation of transfer, III:\nT (path radiance), III: 32\n38\noperational definition,\nS (radiance), III: 33\ntime dependent, III: 68\nIII: 287\npath luminance, II: 179\ncontraction, III: 129\nU (scalar irradiance), III:\npath radiance, I: 63, 172\nderivation, II: 351\n129\nconnection with path func-\nV (= TU), III: 188\ntion, II: 354\noptical length, III: 270,1 146\noperator, II: 384\noptical medium\nfirst-order form, III: 11\ntransparent, III: 3\noperational definition,\nabsorbing, III: 3\nfundamental, III: 5\nVI: 287\nelectric circuit analogy,\nPauli matrices, II: 8\nperfectly diffusing (surface)\nIII: 77\nas a metric space, III: 132\nI : 21\nlight storage, III: 121\nperiodogram (basis of energy\none parameter, IV: 142\nspectrum), VI: 94\nhomogeneity, isotropy and\nperturbed light fields (gen-\nrelated properties, IV:\neral theory), VI: 293\nexperimental procedures,\n143 et seq.\nVI: 302","INDEX\n381\nphase density, of radiant\npolarized radiance\nflux, I: 10\ncanonical representation,\nphase function, IV: 146\nIII: 19\npolarized, VI: 343\nasymptotic properties,\nphase space density, II: 9\nIV: 128\nphenomenological, view of\nin a coherent electro-\nlight (vs. electromagnetic),\nmagnetic field, VI: 346\nII: 13\nPoynting vector field, II: 9\nphotoelectric effects (photo-\nprey-predator equations, I :\nelectric cells, photoemis-\n198\nsive, photoconductive,\nprimary radiance, equation\nphotovoltaic), II: 3\nfor, III: 41\nphotometry, geometrical, I : 18\nprimary scattered flux as\nII: 2, 165\nsource flux, III: 207\ngeneralized, II: 183\nprimary scattered irradiance,\nnonlinear, II: 185\nV: 43\nphoton,\nprinciples of invariance\nas viewed in this work,\nfor irradiance, I : 73, 79\nI : 7\non plane parallel media,\nas an aid to visualization,\nII: 294\nII: 10\non spherical, cylindrical,\nentering an eye from a star,\ntoroidal media, II: 325\nII: 18\non general media, II: 336\nphotopic luminosity curve,\nlocal form (equation of\nI: 18, 145\ntransfer), IV: 4\nphytoplankton (in food chain)\ndiffuse form, IV: 14, 15\nI : 199\ntime-dependent form local\nplanckian (complete) radiator,\nIV: 18\nII: 162, VI: 182\ntime-dependent form global,\nPlanck's quantum of action,\nIV: 23\nII: 9\nfor complete operators,\nplane of incidence (in reflec-\nIV: 48\ntion and transmission at\nglobal (for irradiance)\nwater surface), VI: 3, 5\nV : 2\nplane of scattering, II: 91\nlocal (for irradiance), V: 7\nplane-parallel medium, I : 55\nfor diffuse irradiance, V: 18\nsee also table of contents\nproblems of hydrologic optics,\npoint source\nI : 2, 205\noperational definition,\nY-operator (for internal\nII: 75\nsources) (basic definitions\ncriterion for, II: 105\nin Example 3, Sec. 3.9)\nin classical diffusion\nconnection with an opera-\ntheory, III: 198\ntor, IV: 53\nin exact diffusion theory,\ndifferential equation,\nIII: 219\nIV: 79\npolar (optical medium), IV:\npurely absorbing medium,\n150\nIII: 3\npolarity (of R,T), IV: 26,\nV : 34\nquantum, I : 7\npolarity theorem, IV: 150\nquantum mechanics\npolarization, defined, I : 50\nformal similarity with, V : 8\nunderwater properties, I: 52\nquantum-terminable calcula-\nconvention, II: 194 (footnote)\ntions, II: 254\nscattering function measured\nquantum theory and interac-\nVI: 314\ntion method, II: 370\nproblem of coherent electro-\nquasi-irrotational light\nmagnetic fields VI: 346\nfield, V: 88\npolarized equation of trans-\nquasi-steady state (food\nfer, II: 371, VI: 342\nchain), I : 202","INDEX\nVOL. VI\n382\nR-infinity (Roo), I : 67\nradiance--Cont'd.\ncorrection in visibility,\nin absorbing media, III: 3\nI : 172\nequilibrium, III: 6\nformulas, V: 113\nmaximum (Natural waters),\nradiance, I : 10\nIII: 12\nfield, I : 12\ntransmittance, III: 14\nsurface, I : 12\npolarized, III: 19\ninherent, I : 60\nresidual (reduced, unscat-\napparent, I : 60\ntered), III: 31\nequilibrium, I: 85\nn-ary (primary, secondary,\n-difference law, I : 92\netc.), III: 33\nempirical definition,\nnatural solution for, III: 42\nII: 30, 171, VI: 281\nbounds, III: 47\nmeter, II: 30, 85\nglobal approximations, III:\ntheoretical, II: 32\n117\nvia photon density, II: 33\ndistribution in diffusion\ntypical values, II: 33, 34,\ntheory, III: 119, 181, 197,\n97\n201\ndistributions (on E), II: 34\ndirectly transmitted, I : 347\nfunction (on X x E), II: 34\nresidual, I : 120, II: 347\nirradiance from, II: 35\nunattenuated, II: 347\n(from irradiance), II: 41\ncharacteristic representa-\nfield vs. surface, II: 44\ntion, IV: 125, 126\ninvariance property, II: 46\npolarized (asymptotic), IV:\nradiance-invariance law,\n128\nII: 46\nequilibrium, V : 151\noperational meaning of sur-\ntransmittance, V : 164\nface radiance, II: 49\nmultiplicity of representa-\nn2-1aw, I : 18, 87, II: 51\ntions, V : 177\npolarized, II: 83\ninteraction at air-water\nstd. Stokes vector, II: 88\nsurface, VI: 39\nstd. observable vector, II:\ntime-averaged interactions\n88\nat surface, VI: 253\npolarization composition\npartially coherent, VI: 346\ntheorem, II: 89\nradiance distribution\nlocal observable vector,\nbehavior with depth, I : 39\nII: 91\nasymptotic hypothesis, I : 41\nradiant flux content of\nby submerged point source,\npolarized, II: 94\nI : 113\ndistributions, elliptical,\nat air-water surface, VI: 40\nII: 131\nintegral equation for air-\ndistributions, polynomial,\nwater surface (instantane-\nII: 139\nous), VI: 215\ndistributions, equivalence\nintegral equation for air-\nof with irradiance distri-\nwater surface (time aver-\nbutions, II: 143\naged), VI: 223\npath, I : 63, II: 172\ninstantaneous and time\nD'-additivity (surfaces),\naveraged within hydrosol,\nII: 195\nVI: 237\nD'-continuity (surfaces),\npartial time averaged\nII: 195\ntheories, VI: 253\nof parallel planes, II: 244\nradiance model (classical)\nD'additivity (slabs), II: 282\nI : 58\nD'-continuity (slabs), II:\nneo-classical (air-water\n282\nsurface), VI: 253\non plane-parallel media,\nradiant density, I : 16\nII: 290\nII: 54, 172\nin transparent media, III: 2\nradiant emittance, I : 12, 28, 171","INDEX\n383\nradiant emittance--Cont'd.\nradiant flux--Cont'd.\nempirical, II: 29\nF-continuity, II: 11\nscalar, II: 61\nmonochromatic (or spectral),\nvector, II: 171\nII: 11\nradiant energy\nfinite vs. countable addi-\nIn this total work either\ntivity, II: 11\nradiant energy or radiant\nS-additivity, II: 12\nflux may be the undefined,\nS-continuity, II: 12\nprimitive concepts, taken\nD-addivity, II: 13\nas given by nature and ax-\nD-continuity, II: 13\niomatized by radiometrists\nnet inward, III: 76\nas their primary physical\nsource, III: 76\nnotions. In other fields,\nnet n-ary, III: 80\nsuch as electromagnetics,\nradiant intensity\nthey can be made to rest\nfield and surface, I: 10,\non one step lower: on the\n12\nconstructs (E,D,B,H) of\nempirical definition, II:\nthe electromagnetic field.\n70, 171\nThese steps into physical\nfield vs. specific, II: 72\nprimitivity descend even\ntheoretical, II: 73\nlower. But this nether\npoint sources and, II: 74\nregion is of no concern to\ncosine laws for, II: 77, 80\nus in this work. See foot-\nvector, II: 81\nnote, p. 32, Vol. II for\narea-law for general sur-\nthe local choice of @ or U\nfaces, II: 119\nas the more primitive no-\nradiative process defined,\ntion.\nII: 190\nradiant energy, II: 54, 172\nradiative transfer analogies,\nover space, II: 60\nIII: 77, 133\nover time, II: 61\nradiative transfer theory,\nn-ary, III: 34, 83\ndefined, I: 1\nresidual representation,\nbasic constructs, I : 4\nIII: 79\natmospheric features, I : 27\nequation of transfer for\nacross air-water surface,\nn-ary, III: 81\nI : 28\nnatural closed form repre-\ncolorimetric, I : 142\nsentation, III: 86\nas based on the interaction\noptical reverberation case,\nprinciple, II: 188\nIII: 86\nand operator algebra\nstandard growth and decay\non a metric space, III: 132\ncase, III: 87\nheterochromatic, IV: 197\ntime dependent properties,\nmultidimensional, IV: 198\nIII: 89\nacross static air-water\nscattered, absorbed, attenu-\nsurface, VI: 34\nated, III: 93\nand hydrodynamics, VI: 46\nstored, III: 123\nradiator, planckian or com-\ntime dependent (check),\nplete, II: 162\nIII: 216\nradiometric (as a metric),\nradiant flux, defined, I : 7\nIII: 128\nmonochromatic brightness of,\nradiometric concepts, opera-\nI : 10\ntional definitions, I : 11\nfield and surface interpre-\nradiometric functions\ntations, I: 12\ngeneral n-ary, III: 35\noperational definition, II:\ndiffuse, decomposed, III: 36\n2, 7, 171, IV: 274\nradiometric norm, II: 146, 232\ncalculations, II: 117\nradiometric-photometric trans-\nmeaning, II: 8\nition operator, II: 166, 169\nF-additivity, II: 10\nradiometrically adequate col-\nlector, II: 151","VOL. VI\nINDEX\n384\nradiometry, geometrical, I : 7, reflectance--Cont'd\nregularity properties,\nII: 2\ntransition to photometry,\nIV: 32\nasymptotic properties,\nII: 166\nmathematical basis of,\nIV: 33\nanalysis of differential\nIII: 169\nRadon-Nikodym\nequation for, IV: 80\ntheorem II: 185\npartition relations, IV:\nderivative, II: 376\n28\nrandom surface, realization\nconnections with invariant\nof, VI: 115\nimbedding operator, IV:\nrationalized units (polemic\n82\nsolution procedures, IV:\non), I : 21\nRayleigh scattering, I: 132,\n97, 83\ndual (complete), IV: 160\nVI: 21\nRayleigh wavelength distribu-\nfor irradiance (undecom-\ntion (for sea waves), VI:\nposed), V: 3\nin two-D model, V: 33, 35,\n203\nreciprocal (optical medium),\n45\nIV: 151\ncomplete (two-D), V : 62\nreciprocity theoren, IV: 151\ncomplete (one-D), V: 64\nreduced (residual irradiance)\nRoo formulas, V: 113\nV : 15\nexperimental, V: 115\nreference frame, terrestrial,\nconnections with attenuat-\nII: 20\ning functions, V: 118\nstratified, II: 24\nanalytic representation,\nlocal vs. standard (in pola-\nV : 146\nrized context), II: 91\nequilibrium-seeking prop-\nreflectance\nerty, V : 150\nfor irradiance at air-water\nintegral representations\nsurface, I: 30\nof R(z,-), V : 156, 196\nfor infinitely deep homoge-\nequivalence theorem for\nneous water, I : 67\nR(z,-), V : 159\nfor finitely deep homogene-\nequilibrium and attenua-\nous water, I : 68\ntion functions, V : 149\ncomplete (for irradiance),\nasymptotic limits, V: 246\nof static water surface\nI : 79\nempirical, for surfaces, II:\n(Fresnel), VI: 3, 13\n194\ninternal and external,\noperators, for surfaces,\nVI: 16\nII: 210\nunder cardioidal radiance\ntheoretical, for surfaces,\ndistribution, VI: 21\nII: 213\nr- (m,n), VI: 26\nlambert, II: 215\nunder zonal radiance dis-\nalgebra of operators, II:\ntributions, VI: 28\ndiscussion of operational\n230\noperators for plane-paralel\nproperties, VI: 327\nmedia, II: 279\noperational determination,\nsemigroup properties,\nVI: 337\nrefraction, subsurface, I :\nII: 300\nin diffusion theory, III:\n33\n198, 202\nregular neighborhoods of\nlocal, IV: 3\npaths, V: 166\ndifferential equations, IV:\nregularity properties (of\n6, 8, 25, 26, V : 65, 79,\nR,T), IV: 32\n123, 148\nrelative error in radiance\ncausality condition, IV: 23\ncomputations, III: 48\nalgebraic properties, IV: 27\nrelative time, III: 99","INDEX\n385\nresidual radiance, I : 20, 63\nscattered radiant energy,\nII: 347, III: 31\nIII: 94\nphenomenological interpre-\nscattered scalar irradiance\ntation, II: 349\nequation of transfer, III:\ntransfer equation, III: 74\n210\nradiant energy, III: 79\nscattering\nresidual (reduced) irradiance\nplane of, II: 91\nV: 15\nstandard function, II: 318\nresidual (reduced) transmit-\nscattering-attenuation ratio,\ntance, IV: 11\nIII: 10, IV: 86, 148\nrestricted homogeneity, IV:\nscattering functions (for\n145\nirradiance),\nreverberation, optical, III:\nforward, backward, V: 11,\n49\n141\nreversible (optical medium),\ntotal, V : 12\nIV: 148\nfor decomposed irradiance,\nRiccati equation in food\nV: 16\nchain, I : 203; for R, IV: 6\nforward scattering media,\nV: 140\no (sigma), I: 122\nscattering-order decomposi-\nS (ess), I : 58\ntion, III: 30\nSWOP (stereo wave observation\nSchooley's wave slope obser-\nproject), VI: 167\nvations, VI: 151\nscalar diffusion equation,\nSchuster, A., I. 57\nIII: 174\nsea state meter, VI: 139,\nhigher order form, III: 213\n260\nscalar illuminance, II: 167\nsea state simulator, VI: 261\nscalar irradiance, II: 54,\nsea surface radiance distri-\nIII: 11\nbution (time-averaged\nn-ary, III: 34, 217\ntheories), VI: 250\nequation of transfer (diffu-\nexact time-averaged theory,\nsion equation), III: 175\nVI: 251\nintegral equation, III: 189\nstatistical time-averaged\nscattered (equation formula),\ntheory, VI: 251\nIII: 210, 218\nwave-slope, wave height\nhigher order (equation formu-\ntime-averaged theory,\n1a), III: 213\nVI: 251\nintegral form, III: 214\nwave slope, time-averaged\ntime-dependent n-ary (dif-\ntheory, VI: 252\nfusion equation), III: 216\npartial time-averaged\nexact diffusion theory,\ntheories, VI: 252\nIII: 226\nSecchi disk, I: 96\noperational definition, VI:\nin visibility calculations\n278\nI : 169\nscalar luminous emittance,\nsecond standard solution of\nII: 168\ntwo-D model, V: 34\nscatter processes (Rayleigh,\n\"Seeliger's formula,\" III:\nCompton, etc.), II: 191\n132\ninelastic or transpectral,\nsemigroup (multiplicative)\nIII: 5\nproperties\nsingle, III: 10\nof reflected and trans-\nscatter time constant, III:\nmitted radiance flux,\n81\nII: 300\nscattered flux, I: 58,\nof 2 (a,b), II: 309\nfor collimated flux, I : 83\nof 3 (a,b), II: 298, 301\nforward, backward, I : 124\nof r (a,b), II: 303, 312\nhigher order, III: 211\nconnections among T3,T4,\nscattered radiance, equation\nII: 313\nof transfer, III: 209","VOL. VI\nINDEX\n386\nsemigroup properties-- Cont'd.\nspectral transport theory,\nof beacm transmittance,\nVI: 208\nI : 120, II: 348\nspectrum\nof T, IV: 56\nof air-water surface,\nmethod of, IV: 108\nVI: 78\ninfinitesimal generator,\nenergy, VI: 79\nIV: 112\ndensity, VI: 81\nfor reflectance and trans-\nperiodigram basis, VI: 94\nmittance, I : 95, V: 169\nintegrated, VI: 107\nin fluid dynamics, VI: 46,\nunresolved, VI: 111\n270\namplitude, VI: 115\nseparable (optical medium),\nresolved, VI: 117\nIV: 145\ndifference, VI: 119\nshallow depth theory (of ir-\ngeometrical application,\nradiance field)\nVI: 120\nexperimental basis, V: 187\nS(k,), VI: 126\nformulation, V: 193\nTk, VI: 126\ncomparison with experiment,\nF(0,0), VI: 128\nV: 197\nTT, VI: 129\nshear flow model (of wave\nTo, VI: 129\ngeneration), VI: 207\ndata on waves, III: 166\nsheltering model (of wave\nco-and quadration, VI: 176\ngeneration), VI: 206\nempirical models, VI: 181\nsighting range (interpreta-\nNeumann, VI: 181\ntion) I : 193\nthe remarkable trio of laws\nsignificant wave height (and\nanalogous to thermodynamic-\nperiod), VI: 159\nradiative laws, VI: 186\nsimple model for radiance,\nalternate forms of one-\nI : 61\ndimensional laws, VI: 189\nfor polarized light fields,\ngamma type, VI: 190\nIII: 21\nand wind speed, VI: 193\neventual exactness, V : 249\ntheoretical models, VI:\nsinusoidal wave forms, VI: 60\n194\nSnell's law, VI: 5\nwave elevation distribution,\nsolar (orradiance) constant,\nVI: 194\nI: 22\nwave slope distribution,\n(illuminance), I: 22\nVI: 197\nsolid angle, II: 37\nwave length distribution,\nsubtense of surfaces, II:\nVI: 202\n112\nwave height distribution,\nS-additivity property,\nVI: 205\nII: 114\nwind generated, models of\nS-continuity property,\nVI: 205\nII: 114\ntransport theory, VI: 208\nand the foundations of\nspectrum locus, I : 149\nEuclid's optics, II: 115\nspherical harmonic method\nsource term (for scalar\nIII: 134\nirradiance), I: 64\nbases, III: 135\nsources\nmotivating argument, sum-\nin one parameter media,\nmarized, III: 140\nII: 330\nalgebraic setting, III:\nof light fields, IV: 152\n141\nspace light (= path radiance),\ncompleteness property, III :\nII: 363\n142, 153\nspace-time diagrams, III:\nabstract method, III: 143\n51 et seq.\nfinite abstract forms, III:\nspecific intensity (see\n147, 149\nradiance, surface), II: 44\nclassical method, general\nspecific radiance, II: 44\nmedia, III: 149","INDEX\n387\nspherical harmonic method\nsurface interpretation of\nradiant flux, I : 12\n--Cont'd.\nsurface kinematic condition,\nfinite recurrence prop-\nerty, III: 154\nVI: 55\ngeneral differential\nsurface pressure condition,\nequations, III: 157\nVI: 57\nclassical method, plane-\nsurface waves, energy, VI:\nparallel media, III: 158\n72\ntruncated solution pro-\nsurfaces,\nsolid angle subtense of,\ncedure, III: 163\nspherical irradiance, II: 56\nII: 112\nVI: 278\ngeneral two-sided, II: 267\nspherical medium (and scat-\ngeneral one-sided, II: 271\ntering function), VI: 320\nsymbolic integration (term\nspherical (point source)\nby term for natural solu-\ndiffusion field, III: 200\ntion), III: 65\nstandard ellipsoid (for radi-\nsymmetric (optical medium),\nance distribution), V: 250\nIV: 149\nstandard growth and decay\ncase (for n-ary radiant\ntalbot: II: 179\nenergy), IV: 87\ntelegrapher's equation, III:\nstar product, IV: 45\n185\nphysical interpretation,\ntensor diffusion equation,\nIV: 46\nIII: 184\nfor invariant imbedding\nthermocline phenomena, I : 36\noperators, IV: 54\ntime-averaged radiance fields,\nfor , IV: 139\nVI: 246\nstatic surface (and radi-\nhierarchy of, VI: 251\native transfer), VI:\ntime constant\n34, 41, et seq.\nattenuation, III: 76\nsteradian, II: 38\nscattering, III: 81\nstochastic pressure model\nabsorption, III: 82\n(of wave generation)\ndimensionless forms:\nVI: 207\nIII: 100\nstochastic process, air\nfor n-ary radiant energy,\nwater surface, VI: 115\nIII: 109\nStokes vector (for radi-\ngeneral discussion, III:\nance), II: 88\n114\nstorage capacity (of an\ntime dependent equation of\noptical medium), III: 123\ntransfer, II: 371, IV: 18\nstratified media, I : 62\nlight field, III: 49\ncanonical equation for\nn-ary radiant energy\nradiance, III: 18\nfields, III: 89\ntime-transformed principles\noperators, III: 68\nof invariance, IV: 200\ninvariant imbedding rela-\nsubmarine echoes (for spec-\ntion, IV: 19\nprinciples of invariance,\ntra) VI: 180\nsubmarine light field, gen-\nIII: 23\neral representation, V : 93\ndifferential equations for\nsubsurface refractive\nR,T, IV: 25,26\nphenomena, I : 33\nintegral transfer method,\nIV: 194\nsun\nradiance of, II: 97\ntotal scattering functions,\nradiant intensity of,\nV : 12\ntransmission line equations,\nII: 98\nluminance of, II: 164\nformal similarity with, V: 8\nsun-based reference frame,\ntransmittance for irradiance\nII: 19\nat air-water surface\nsupremum metric, III: 129\n(t = 1 - r), I : 30","INDEX\nVOL. VI\n388\ntransport (transfer) equations\ntransmittance--Cont'd.\nfor finitely deep homoge-\nresidual, radiant energy,\nneous water, I : 68\nIII: 76\ncomplete (for irradiance)\nn-ary radiant energy,\nIII: 80\nI : 79\nbeam, I : 120\ndirectly observable radi-\ncontrast, I : 93\nant energy, III: 81\nempirical, for surfaces,\nfor ocean wave spectra,\nII: 194\nVI: 208\noperators, for surfaces,\ntristimulus functions, I: 144\nII: 210\ntrue absorption, III: 5\ntheoretical, for surfaces,\ntruncation error estimates,\nII: 213\nII: 250\nlambert, II: 262\ntruncated natural solution,\nalgebra of operators,\nIII: 45\nII: 230\ntime dependent, III: 69\noperators for plane-\ntruncated spherical harmonic\nparallel media, II: 279\nmethod, III: 163\nsemigroup properties,\ntwo-D models for irradiance\nII: 300\nfields, V: 25\nradiance, III: 14, 17\nfor undecomposed fields,\nlocal, IV: 3\nV: 30\ndifferential equations\nfirst standard solution,\n(undecomposed) IV: 2, 8\nV: 31\n(decomposed), IV: 16\nsecond standard solution,\nlocal residual, IV: 11\nV: 34\nlocal diffuse, IV: 12\nfor internal sources, V: 37\nglobal diffuse, IV: 12\nfor decomposed fields,\ncausality condition, IV:\nV: 43\ntime dependent differ-\ninadequacies, V: 115\nential equation, IV: 25, 26\neventual exactness, V: 247\nalgebraic properties,\ntwo-flow equations (for irra-\nIV: 27\ndiance), V : 6\npartition relation, IV: 29\nundecomposed form, V: 8\nregularity properties, IV: 32\ndecomposed form, V: 14\nasymptotic properties,\nequilibrium form, V: 13\nIV: 33\nontogeny, V : 13\nsemigroup property,\nfor reduced irradiance,\nIV: 56, 106\nV: 17\nsolution procedures,\ntwo-D (undecomposed) model,\nIV: 97, 83\nV: 30\ndual (complete), IV: 160\nstandard solutions, V: 31-\nfor irradiance (undecom-\n34\nposed), V : 3\ncomplete (general) solu-\nfor reduced and diffuse\ntion, V: 42\nirradiance, V : 17\nfor decomposed irradiance,\nin two-D model, V: 33, 35,\nV: 43\n45\nboundary conditions\ncomplete (two-D), V : 62\n(effects), V: 46\ncomplete (one-D), V: 64\none-D (undecomposed) model,\ndifferential equations, V:\nV: 52\n65, 79\nmany-D models, V: 57\nfor radiance, V : 164\nexact vs. two-D, V: 115\nof static water surface\nasymptotic behavior, V: 247\n(fresnel), VI: 3, 15\ntwo-flow (irradiance) model,\noperational determination\nI : 55, 56, 57\n(radiance), VI: 337\ntwo-path method (for beam\ntranspectral scatter, III: 5\ntransmittance), VI: 285","INDEX\n389\nunattenuated radiance, II: 347\nvolume backward scattering\nunified atmosphere-hydrosphere\nfunctions, I : 124\nproblem, II: 343\nvolume forward scattering\nunit source condition, III: 220\nfunctions, I : 124\nuniversal radiative transport\nvolume scattering function,\nequation, V : 263\nI : 122, II: 364\nfor radiometric concepts,\no-recovery procedures,\nV: 263\nVI: 312\nfor apparent optical prop-\npolarized, VI: 315\nerties, V: 271\nvolume total scattering\nand equilibrium principle,\nfunction, I: 60, III: 4\nV: 279\noperational definition,\nstandard cases, V : 281\nI : 123, VI: 316\nadditional cases, V : 281\nvolume transpectral scatter-\nunpolarized-flux convention\ning operator, II: 386\n(in this work), I : 7\nunscattered radiance, III : 31\nWalsh's formula (for\nequation of transfer, III: 37\nreflectance), VI: 18\nwater clarity (visualiza-\nvector analogy with color,\ntion), I : 194\nI : 146\nwatt, II: 171\nvector irradiance, III: 62\nwave age, VI: 183\nvia spherical harmonics,\nwave diffusion equation,\nIII: 177\nIII: 185\nin classical diffusion\nwave generation and decay,\ntheory, III: 198, 201, 207\nVI: 152\nscattered form, III: 210\nwind speed connection,\nn-ary, III: 217\nVI: 193\nmodel for, in hydrosols,\nwave height distribution,\nV : 87\nVI: 205\nvelocity (of a fluid packet)\nwave slope data, VI: 132\nVI: 48\nHulbert's observations,\nvelocity potential, VI: 54\nVI: 136\nviews of light (phenomeno-\nDuntley's observations,\nlogical vs. electromagnetic)\nVI: 138\nII: 13\nintuitive gaussian picture,\nvisibility underwater, I : 154\nVI: 142\neffect of depth and water\nwave slope-wind speeds law,\nclarity, I : 157\nVI: 145\nuse of nomographs, I: 163\nCox and Munk's observations,\nalong inclined paths of\nVI: 145\nsight, I : 165\nSchooley's observations,\nhorizontal paths of sight,\nVI: 151, 155\nI : 170\nwave slope distribution\nvolterra prey-predator equa -\n(theoretical), VI: 197\ntions, I : 198\nwave period (apparent), VI:\nvolume absorption function,\n183\nI : 60, III: 4\nwave-slope, wind-speed law\nmeasurement, I : 103\nderived, VI: 188\noperation of definition,\nwave spectrum data\nI : 124, VI: 321\naerial stereo photography,\nvolume attenuation function,\nVI: 166\nI : 60\nfloating-buoy motion, VI:\noperational definition,\n173\nI : 119, VI: 290\nsubmarine echoes, VI: 180\nempirical, I : 120, II: 349\nwave spectrum models (theo-\nfor undecomposed light\nretical), VI: 194\nfield, IV: 114\nwave steepness, VI: 183","VOL. VI\nINDEX\n390\nWhitney's \"general law\" of\nwavelength distribution\nlight field with depth,\n(theoretical),\nVI: 202\nV: 248\nwavelength, dominant,\nWiener-Khintchine theorem,\nI : 149\nVI: 98\nwaves, gravity and\nWien's displacement law\ncapillary, VI: 71\nin oceanography,\nsuperposition of,\nVI: 186\nVI: 76\nwind generated spectra,\nweighting functions\nmodels of, VI: 205\n(for statistics of\nwind profile (logarthmic),\nair-water surface),\nVI: 132\nVI : 221\nwindow (spectral), I : 134\nwhite light, I : 149,\nworld region, III: 52\nVI: 83\nU.S. GOVERNMENT PRINTING OFFICE: 1977 -778-782/45 REGION NO. 8"]}