{"Bibliographic":{"Title":"Hydrologic optics. Volume II: Foundations","Authors":"","Publication date":"1976","Publisher":""},"Administrative":{"Date created":"08-17-2023","Language":"English","Rights":"CC 0","Size":"0000771679"},"Pages":["HYDROLOGIC OPTICS\nVolume II. Foundations\nR.W. PREISENDORFER\nU.S. DEPARTMENT OF COMMERCE\nNATIONAL OCEANIC & ATMOSPHERIC ADMINISTRATION\nENVIRONMENTAL RESEARCH LABORATORIES\nHONOLULU, HAWAII\n1976","GB\n665\nP645\nV.2\nOF COMMUNICATION\nHYDROLOGIC OPTICS\n.\n-\nwith\nSTATES OF\nVolume II. Foundations\nR.W Preisendorfer\nJoint Tsunami Research Effort\nHonolulu, Hawaii\n1976\nATMOSPHERIC SCIENCES\nLIBRARY\nAUG 30 1976\nN.O.A.A.\nU. S. Dept. of Commerce\nU.S. DEPARTMENT OF COMMERCE\nNational Oceanic and Atmospheric\nAdministration\nEnvironmental Research Laboratories\nPacific Marine Environmental Laboratory\n16 3418","ii\nNature Seen Through Half-Shut Eyes\nA man who takes a magnifying glass into a\npicture gallery and examines the canvases at a\ndistance of 3 inches may acquire much interesting\ninformation about the texture of paint, but he\ndoes not see the pictures. It is better to stand\naway. If trivial details still intrude, it is\nbetter to half-shut the eyes. As a final step,\nit is well to shut the eyes completely and think\nabout what has been seen.\nJ.L. Synge\nScience, 5 October 1962","CONTENTS\niii\nVolume II\nChapter 2\nRadiometric and Photometric Concepts\n2.0\nIntroduction\n1\n2.1\nRadiant Flux\n2\nBasic Photoelectric Effects\n3\nOperational Definition of Radiant Flux\n5\n2.2\nThe Meaning of 'Radiant Flux'\n8\n2.3\nFundamental Geometric Properties of Radiant Flux\n10\n2.4\nIrradiance and Radiant Emittance\n14\nDefinition of Irradiance\n14\nThe Meaning of 'Irradiance'\n16\nTerrestrial Coordinate Systems\n19\nRepresentation of Irradiance in Terrestrial\nFrames\n24\nThe Cosine Law for Irradiance\n26\nRadiant Emittance\n28\n2.5\nRadiance\n30\nRadiance Distributions\n34\nIrradiance from Radiance\n35\nRadiance from Irradiance\n41\nField Radiance VS Surface Radiance\n44\n2.6\nAn Invariance Property of Radiance\n46\nThe Radiance-Invariance Law\n46\nThe Operational Meaning of Surface Radiance\n49\nThe n2-Law for Radiance\n51\n2.7\nScalar Irradiance, Radiant Energy, and Related\nConcepts\n54\nRadiant Density\n54\nScalar Irradiance\n55\nSpherical Irradiance\n56\nHemispherical Irradiance\n58\nRadiant Energy over Space\n60\nRadiant Energy over Time\n61\nScalar Radiant Emittance\n61\n2.8\nVector Irradiance\n62\nA Mechanical Analogy\n62\nGeneral Definition of Vector Irradiance\n65\nThe General Cosine Law for Irradiance\n66","CONTENTS\niv\n70\nRadiant Intensity\n2.9\nOperational Definition of Empirical Radiant\n70\nIntensity\n72\nField Intensity VS Surface Intensity\n73\nTheoretical Radiant Intensity\n74\nRadiant Intensity and Point Sources\n77\nCosine Law for Radiant Intensity\nGeneralized Cosine Law for Radiant Intensity\n80\n83\nPolarized Radiance\n2.10\nOperational Definition of Polarized Radiance\n85\nThe Standard Stokes and Standard Observable\n88\nVectors\n90\nAnalytic Link Between S and N\n91\nStandard and Local Reference Frames\nRadiant Flux Content of Polarized Radiance\n94\nExamples Illustrating the Radiometric Concepts\n95\n2.11\n96\nRadiance of the Sun and Moon\n1.\nRadiant Intensity of the Sun and Moon\n98\n2.\nRadiant Flux Incident on Portions of the\n3.\n101\nEarth\nIrradiance Distance - Law for Spheres\n103\n4.\nIrradiance Distance - Law for Circular Disks;\n5.\n105\nCriterion for a Point Source\nIrradiance Distance - Law for General Surfaces\n106\n6.\n109\nIrradiance via Line Integrals\n7.\n112\nSolid Angle Subtense of Surfaces\n8.\n115\nIrradiance via Surface Integrals\n9.\n117\nRadiant Flux Calculations\n10.\nIntensity Area-Law for General Surfaces\n119\n11.\n12. On the Possibility of Inverse nth Power\n120\nIrradiance Laws\nIrradiance from Elliptical Radiance\n13.\n131\nDistributions\n14. Irradiance from Polynomial Radiance\n138\nDistributions\nOn the Formal Equivalence of Radiance and\n15.\n143\nIrradiance Distributions\n151\nTransition from Radiometry to Photometry\n2.12\n152\nThe Individual Luminosity Functions\n157\nThe Standard Luminosity Functions\n161\nPhotometric Bedrock: the Lumen\n163\nLuminance Distributions\n165\nTransition to Geometrical Photometry\nGeneral Properties of the Radiometric-Photometric\n169\nTransition Operator\nThe Mathematical Basis for Geometrical Photometry\n169\nSummary and Examples (Tables of Radiometric and\n170\nPhotometric Concepts)\n183\nGeneralized Photometries\n2.13\nLinear Photometries\n184\n185\nNonlinear Photometries\n187\nBibliographic Notes for Chapter 2\n2.14","CONTENTS\nV\nChapter 3\nThe Interaction Principle\n3.0\nIntroduction\n188\nThe Physical Basis of the Linearity of the\nInteraction Principle\n189\nPlan of the Chapter\n193\n3.1\nA Preliminary Example\n194\nEmpirical Reflectances and Transmittances for\nSurfaces\n194\nThe Problem\n196\nThe Present Instance of the Interaction Principle 197\nSolution of the Problem\n198\nDiscussion of Solution\n199\nRelated Problems and Their Solutions\n200\nAn Alternate Form of the Principle\n201\nThe Natural Mode of Solution\n203\n3.2\nThe Interaction Principle\n205\nDiscussion of the Interaction Principle\n206\nThe Place of the Interaction Principle in\nRadiative Transfer Theory\n208\nThe Levels of Interpretation of the Interaction\nPrinciple\n208\n3.3\nReflectance and Transmittance Operators for\nSurfaces\n210\nGeometrical Conventions\n210\nThe Empirical Reflectances and Transmittances\n212\nThe Theoretical Reflectances and Transmittances\n213\nVariations of the Basic Theme\n215\n3.4\nApplications to Plane Surfaces\n217\nExample 1: Irradiances on Two Infinite Parallel\nPlanes\n217\nExample 2 : Irradiances on Two Infinite Parallel\nPlanes, Reexamined. A First Synthesis of the\nInteraction Method\n220\nExample 3: Irradiances on Finitely Many\nInfinite Parallel Planes\n223\nExample 4 : Irradiances on Infinitely Many\nInfinite Parallel Planes\n227\nExample 5 : The Algebra of Reflectance and\nTransmittance Operators for Planes. Radiometric\nNorm. Iterated Operators. Operator Algebras\nand Radiative Transfer.\n230\nExample 6: Radiances of Infinite Parallel Planes 244\nExample 7 : Terminable and Non Terminable Inter-\nreflection Calculations. A Terminable Calcula-\ntion. Truncation Error Estimates. Quantum-\nterminable Calculations.\n248\nExample 8: Two Interacting Finite Plane Surfaces 254","vi\nCONTENTS\nApplications to Curved Surfaces\n258\n3.5\nExample 1: Open Concave Surfaces\n258\nExample 2: Closed Concave Surfaces; the\n262\nIntegrating Sphere\nExample 3: Open and Closed Convex Surfaces\n266\nExample 4 : General Two-Sided Surfaces\n267\nExample 5 : General One-Sided Surfaces\n271\nReflectance and Transmittance Operators for\n3.6\nPlane-Parallel Media\n279\nGeometrical Conventions\n279\nThe Empirical Reflectances and Transmittances\n280\nThe Theoretical Reflectances and Transmittances\n282\nVariations of the Basic Theme\n284\nApplications to Plane-Parallel Media\n285\n3.7\nExample 1: Irradiances on Plane-Paralle1 Media\n286\nExample 2: Radiances in Plane-Parallel Media\n290\nExample 3: The Classical Principles of\n294\nInvariance\nExample 4 : The Invariant Imbedding Relation\n297\nExample 5 : Semigroup Properties of Transmitted\nand Reflected Radiant Flux\n300\nExample 6: The Generalized Invariant Imbedding\n301\nRelation\nExample 7 : Group-Theoretic Structure of Natural\nLight Fields. Group Theory, Radiative Transfer\nand Quantum Theory.\n307\nInteraction Operators for General Spaces\n314\n3.8\nGeometrical Conventions\n314\nThe Empirical Scattering Functions\n317\nThe Theoretical Scattering Functions\n318\nVariations of the Basic Theme\n322\n3.9\nApplications to General Spaces\n322\nExample 1 : Principles of Invariance on\nSpherical, Cylindrical, Toroidal Media\n325\nExample 2: Invariant Imbedding Relation for\nOne-Parameter Media\n327\nExample 3: One-Parameter Media with Internal\n330\nSources\nExample 4 : Principles of Invariance for General\nMedia\n336\nExample 5 : Invariant Imbedding Relation in\nGeneral Media\n339\nExample 6: Reflecting Boundaries and Interfaces\n340\nExample 7 : The Unified Atmosphere-Hydrosphere\n343\nProblem\nExample 8 : Several Interacting Separate Media\n344\nDerivation of the Beam Transmittance Function\n344\n3.10\nDerivation of the Volume Attenuation Function\n349\n3.11","vii\nCONTENTS\n351\n3.12\nDerivation of Path Radiance and Path Function\n351\nThe Path Radiance\n352\nThe Path Function\nThe Connection Between Path Function and Path\nRadiance\n354\n3.13\nDerivation of Apparent-Radiance Equation\n361\n3.14\nDerivation of the Volume Scattering Function\n364\nRegularity Properties of o\n366\nThe Integral Representation of the Path Function\n367\n3.15\nThe Equation of Transfer for Radiance\n368\nSteady State Equation of Transfer\n370\nTime Dependent and Polarized Equations of\nTransfer\n371\n3.16\nOn the Integral Structure of the Interaction\nOperators\n372\nThe Mathematical Prerequisites\n373\nInteraction Operators for Surfaces\n377\nInteraction Operators for General Media\n378\nInteraction Measures and Kernels\n380\n3.17\nFurther Examples of the Interaction Method\n383\nExample 1: The Path Function Operator\n383\nExample 2 : The Path Radiance Operator\n384\nExample 3: The Volume Transpectral Scattering\nOperator\n386\nMiscellaneous Examples\n387\n3.18\nSummary of the Interaction Method\n388\nSummary of the Interaction Method\n388\nRemarks on the Stages of the Interaction Method\n389\nThe Interaction Method and Quantum Theory\n390\nThe Interaction Principle as a Means and as an\nEnd. Conclusion\n391\n3.19\nBibliographic Notes for Chapter 3\n392","","ix\nPREFACE\nMy first encounter with radiometry and photometry was as\na student reading Sears' Optics. * The lucid exposition in\nSears' book, ably illuminated by the lectures of Prof. S. Q.\nDuntley, awakened my interest in the subject. Soon afterward\nthe geometer in me took over as I sought the foundations of\nthe subject. Following graduation from Massachusetts Institute\nof Technology in 1952, and during my first years as a mathe-\nmatics graduate student at Scripps Institution of Oceanography,\nI had the opportunity to develop my awakened interest in radi-\nometry, and to find the foundations of this subject in measure\ntheory. Eventually I found that radiometry, a beautiful union\nof Euclid's geometry and the axiomatized notion of radiant\nflux, is the ground on which radiative transfer theory could be\nbuilt. For a few heady years I had the leisure to explore this\nfoundation (see, e. g., [210], [211], [216]) much as the ancient\ngeometers explored the world of euclidean geometry and the\nbeginnings of mechanics. For one need not know much about the\nphysical world beyond what his senses reveal in order to be\nqualified to pursue radiometry, radiative transfer, and their\napplications to problems of visibility and radiant energy flow\nin the sea. In this sense radiometry and probability theory\nare very much alike. While radiometry is the marriage of\ngeometry and radiant flux, probability theory is the union\nof geometry and chance. In both disciplines, the mathematical\nvehicle for the physical concept is the notion of a measure.\nIndeed, the parallel between radiative transfer and that\nbranch of probability known as 'Markov chains' is exact, as I\nshowed sometime later in Chapter XIII of my monograph [251] on\nradiative transfer theory.\nThis volume, then, is the product of a labor of love,\nwherein very deep geometric predilections took over my first\nyears of scientific research, years in which for better or\nworse I half-shut my eyes to the multifarious richness of the\nreal world, and tried by thought alone to order my visual ex-\nperiences in a suitable mathematical frame. I was successful\nin that effort. For if one carves out of the chaos of his\nexperience a small enough piece, he can examine it and under-\nstand it, and eventually make its secrets part of himself\nonce and for all. But the price of this victory is quite dear:\nthe remaining portions of the world sweep by and onward while\none remains anchored to a spot, examining a few grains of\nearth for order and meaning.\n*Sears, F. W. Optics, Addison Wesley, Cambridge, Mass.\n(1949), 3rd ed.","PREFACE\nx\nThe interested student of radiative transfer theory\nmay take the following as a base on which to rest his own\nwork. He will then be spared the necessity for remaining\noverly long at a relatively isolated point in the conceptual\nlandscape of radiative transfer theory. If he can then see\nfarther and clearer because of this work, my efforts will\nhave done some good. If he feels that the final answer has\nyet to be found in the quest for the foundations, then I\nwish him good luck and a full measure of joy in pursuing\nthat quest.\nThe final manuscript was typed by Ms. Judy Marshall.\nR.W.P.\nHonolulu, Hawaii\nJanuary 1974","CHAPTER 2\nRADIOMETRIC AND PHOTOMETRIC CONCEPTS\n2.0 Introduction\nHaving completed our introductory survey of hydrologic\noptics, we now embark on a theoretical reconstruction of all\nthat we saw and did: we shall now start from scratch.\nIn this chapter we shall develop the concepts of ra-\ndiometry and photometry needed in the study of radiative\ntransfer. The mode of approach to these concepts is governed\nby the particular outlook of radiative transfer theory as it\nis applied to hydrologic optics. As we have defined it (Sec.\n1.0) hydrologic optics is the study of radiative transfer in\ngeneral hydrosols; and radiative transfer, in turn, employs\na phenomenological viewpoint of light. Therefore, the ap-\nproach to radiometry and photometry in hydrologic optics\ntakes place on a phenomenological level. In other words, the\nconcepts of radiometry and photometry which play a major role\nin hydrologic optics are those which are defined for natural\nlight fields in which operations have been performed on a\nmacroscopic level and with instruments which in certain key\ntypes of response are very similar, qualitatively, to the\nhuman eye. Thus, like the human eye, the sizes of the special\ninstruments used in radiative transfer measurements are large\ncompared to the sizes of the wave and particle structures of\nlight and matter. Therefore these instruments do not osten-\nsibly sense and record any of those features of light directly\ncharacterizable by its wave or particle structure, such as\ndiffraction and interference features. However, the radio-\nmetric instruments used are designed to extend and amplify\nunder precise quantitative control certain selected capabil-\nities inherent in the human eye, foremost of which is the\nability to sense and record the various brightnesses of mono-\nchromatic light in all directions about a given point in the\nobserver's locale. This, after all is said and done, is the\nprincipal goal of classical photometry. The complete route\nto this goal is necessarily through an intricate maze of\npsychometric, radiometric, electromagnetic and quantum con-\nstructs. However, the paths we shall take to photometry in\nthis chapter can fortunately bypass most of the usual detours\nalong the route, detours which study the manifold conceptual\nand experimental aspects of the subject. The topics we se-\nlect for discussion are mainly theoretical and on this level\nconstitute the minimal number just sufficient to allow the\nlogical establishment of those photometric concepts and var-\nious radiometric models of natural light fields used in","VOL. II\nRADIOMETRY AND PHOTOMETRY\n2\nradiative transfer measurements. We shall stand away from\nelectromagnetic complexity, and half-shut our eyes as we re-\nconstruct radiometry. In the next chapter we shall shut our\neyes completely and think about what we have seen in Vol. I.\nThe outline of this chapter is as follows. We begin\nin Sec. 2.1 with the operational definition of radiant flux.\nIt is always good practice to give as many means of visuali-\nzation of a newly defined concept as mutual consistency will\nallow. For this reason, and also to pave the way for a more\nversatile presentation of the concepts of hydrologic optics\nthan that of Chapter 1, we develop in Sec. 2.2 the three main\nways to conceptually view the notion of radiant flux. The\nprincipal properties of radiant flux, as they are used in geo-\nmetrical radiometry, are developed in Sec. 2.3. Then, in\nclose succession, the principal derived concepts of radiometry\nare developed: radiance and various forms of irradiance, a-\nlong with theorems governing and examples illustrating their\nsalient properties. Throughout our development we shall em-\nphasize the geometrical aspects of radiometry rather than\ntheir physical aspects. The latter aspects, to the degree\nthat we shall need to study them in this work, are reserved\nfor discussion in Sec. 2.1. However, some notice must also\nbe taken of the physical aspects of radiometry in preparing\nto construct the bridge between radiometric and photometric\nconcepts. Therefore, in Sec. 2.12, we pause to develop those\nconcepts of photometry which facilitate the operational defi-\nnition of the notion of luminous flux the photometric coun-\nterpart to radiant flux. With the radiometric discussions as\nmodel, the various derived photometric concepts are then\na\nreadily attained. The chapter closes with some remarks on\ngeneralized photometric concepts.\nOur present viewpoint of geometrical radiometry and\nphotometry may then be summarized in the following definitions\nof these disciplines, which we adopt: Radiometry is the sci-\nence of the measurement of radiant energy. Geometrical Radi-\nometry is the union of euclidean geometry and Radiometry: it\nmeasures and describes the flow of radiant energy of given\nfrequency through volumes, across surfaces, along lines, and\nat points in space. With this in mind we can go on to say\nthat: Geometrical Photometry measures the visual, erythemal,\nphotoelectric, or photographic response, by given receptors,\nto the quantities of geometrical radiometry, with respect to\ndifferent frequencies of radiant energy.\n2.1 Radiant Flux\nWe now take up the details of an operational defini-\ntion of radiant flux. The heart of the definition we shall\nadopt consists of the postulation of some physical device\nwhich can sense and record in quantitative detail the pres-\nence of light--or radiant energy in general- in a neighbor-\nhood of a point in space. There are several devices available\nfor such a purpose. Of those currently available, the photo-\nelectric devices are most satisfactory from the point of view\nof sensitivity and quantitative precision. We pause briefly\nto survey this class of devices.","SEC. 2.1\nRADIANT FLUX\n3\nBasic Photoelectric Effects\nThe class of light-measuring devices known collectively\nas photoelectric cells consists of three broad sets, each set\nbeing characterized by a distinctive mode of interaction of\nlight with matter and the particular form of electrical re-\nsponse arising from that interaction. These responses are de-\nnoted by the terms photoemissive, photoconductive, and\nphotovoltaic. A comparison of the characteristic features\nof these phenomena is readily made by means of Fig. 2.1.\nPart (a) of the figure depicts the electrical essence\nof a photoemissive cell (or phototube) Light, indicated by\nthe arrow, is incident on a negatively charged electrode.\nThe impact of the incident light dislodges electrons from the\nsurface of the electrode and these are drawn across the gap\nto the relatively positively charged electrode within the el-\nement. The seat of electromotive force is supplied by a bat-\ntery or other means and so continuously replenishes the sup-\nply of electrons on the negative electrode. The net result\nof the incident light is a small but measurable current of\nelectrons flowing through a current meter, as shown in the\nfigure. The swarm of electrons, liberated at the electrode\nby the incident light, streams across the gap between the\nelectrodes and thereby completes the circuit. If there is no\nincident light on the electrode, then under normal conditions,\nthere are no electrons liberated from the electrode to com-\nplete the circuit, and there is consequently no current reg-\nistered by the meter. Generally, the greater the amount of\nlight incident on the receiving electrode, the correspondingly\ngreater is the resultant current in the circuit. By a careful\ncalibration, the meter can be made to read directly the rate\nof incidence of radiant energy on the receiving electrode.\nThe photoemissive effect just described is the most recently\ndiscovered of the three effects. It was discovered in crude\nform in 1887 by Heinrich Hertz as a by-product of his classical\nresearches on electromagnetism. Under subsequent refinements,\nover the years, it has become the principal effect used in\nphotoelectric devices. The theory of the photoemissive effect\nwas not evolved until about eighteen years after its discovery.\nThe theory of the photoemissive effect itself forms a major\nepoch in the history of physics, for its completion eventually\nrequired the concept of the photon as introduced by Einstein\nin 1905.\nA photoconductive cell is schematically depicted in\npart (b) of Fig. 2.1. It was found experimentally in 1873 by\nWilloughby Smith that the conductivity of the metal selenium\nincreases when light is incident on it. This effect can\ntherefore be put to use in sensing and recording the presence\nof light, in the manner shown in the figure. The greater\namount of light incident on the selenium cell results in a\ncorrespondingly greater amount of current flowing through the\ncurrent meter. When no light is incident on the photoconduc-\ntive element, there is under normal conditions a small known\namount of current (the dark current) flowing in the circuit.\nThe full understanding of the photoconductive effect on a\nmicroscopic level was achieved only recently using the quantum-","VOL. II\n4\nRADIOMETRY AND PHOTOMETRY\nlight\nphoto emissive\nelement\nmeter\nlight\nelectromotive force\nphoto voltaic\n(a)\nelement\nlight\nphoto conductive\nelement\nmeter\n(c)\nmeter\nelectromotive force\n(b)\nFIG. 2.1 The Basic types of photoelectric cells\nbased theory of semiconductors. On the basis of this under-\nstanding, one can test and use all manners of semiconductors\nas possible photoconductive materials.\nA photovoltaic cell is schematically depicted in part\n(c) of Fig. 2.1. The photovoltaic element consists of two\ndissimilar substances in close contact (shown slightly sepa-\nrated, for clarity). Light incident on the photovoltaic ele-\nment generates a difference of electric potential between the\ntwo basic parts of the element and as a consequence a current\nflows in the circuit. This current is measured by a current\nmeter included in the circuit. When no light is incident on\nthe element, no electromotive force is normally produced in\nthe parts of the element, and consequently no current flows\nin the circuit. Generally, the greater the amount of incident\nlight on the element of the cell, the greater the resultant\npotential, and the greater the ensuing current in the circuit.\nThe photovoltaic effect antedates both other effects discussed","SEC. 2.1\nRADIANT FLUX\n5\nabove. It appears that Edmond Becquerel first observed it in\n1839 when a liquid electrolyte containing two immersed elec-\ntrodes connected through a galvanometer was irradiated by sun-\nlight. Becquere1 eliminated the possibility of a thermal vol-\ntaic effect generated by differential heating of the elec-\ntrodes and thereby was led to believe that the light itself\ngave rise to an electric potential between the electrodes\nwhich in turn gave rise to a current in the galvanometer.\nThe theory of the photovoltaic effect requires the\nquantum picture of the structure of matter for its complete\nformulation. However both the photoconductive and photovol-\ntaic effects can be intuitively pictured as being something\nlike weaker versions of the photoemissive effect: on the one\nhand, in the case of the photoconductive cell, instead of\nknocking electrons completely free of an area of selenium sur-\nface, the incident light on the surface merely gives them\nenough energy to skim through the lattice of the positive nu-\nclei of the selenium atoms. If there is an existing voltage\nin the metal, the footloose electrons in the irradiated re-\ngion are then more readily moved along in a more or less or-\nganized manner by the potential difference. On the other\nhand, the mechanism of the photovoltaic effect is relatively\ncomplex. For our descriptive purposes here it may be ex-\nplained in terms of the effects generated by inherently dif-\nferent electromotive forces of the chemical elements. When\ntwo substances of different electromotive force are placed in\nclose proximity (e.g., the dotted and solid elements schemat-\nically shown in part (c) of Fig. 2.1) the pull exerted by the\npositive nuclei of the atoms of one of the substances on elec-\ntrons is greater than that of the corresponding pull by the\nother substance. As a result some electrons are swapped from\nthe 'weaker' to the 'stronger' substance when the substances\nare placed into close contact. However, the electrons cap-\ntured by the stronger substance can be relatively easily dis-\nlodged by irradiation of the boundary between the substances,\nand thus be caused to move in the resultant electric field\nnaturally existing between the two substances. The magnitude\nof the potential of this field under irradiation is very near-\n1y the difference in the electromotive forces of the sub-\nstances.\nOperational Definition of Radiant Flux\nWe now present the operational definition of radiant\nflux. The brief preliminary excursion into the basic photo-\nelectric effects just completed will endow the definition pro-\ncedures below with a measure of realism that perhaps may not\nhave been possible had we not paused to make some contact with\nphysical reality. However, the logical basis of the defini-\ntion of radiant flux and its manifold properties discussed\nsubsequently are quite independent of what radiation measur-\ning devices are used in practice. Indeed, the concepts of\nradiometry as used in practice are all constructable in terms\nof the basic notion of radiant flux and appropriate geometri-\ncal notions such as surface areas and solid angles. The con-\ncept of radiant flux in turn and its few basic geometric","VOL. II\nRADIOMETRY AND PHOTOMETRY\n6\nproperties are now so well established that they can actually\nbe axiomatized for the purpose of developing a self-contained\ndiscipline of geometrical radiometry. In the present devel-\nopment we shall steer a middle road between these extreme al-\nternatives. We shall not go so far as to develop in complete\ndetail an axiomatic theory of radiometry, but we shall indi-\ncate the fundamental properties of radiant flux that would\noccur in such a formulation. The notion of radiant flux will\nfor the most part be handled as an empirically-base concept.\nHowever, we shall not, beyond the general suggestions given\nin the discussion of photoelectric devices above, fix in any\ndetail the form of the device which is used to sense and re-\ncord the incident flow of radiant energy. In sum, we shall\nhenceforth agree that we have some light-sensitive device\nwhich can accurately, quickly, and repeatedly reproduce a\nquantitive measure of the instantaneous flow of radiant ener-\ngy onto some well defined surface which acts as a collecting\nsurface for the incident energy. Except for some suggestive\nremarks in Sec. 2.2, the notion of 'radiant energy' will re-\nmain undefined in this work. We take it as given.\nFigure 2.2 depicts in more detail, and on a schematic\nlevel, the basic form of a widely used type of radiant flux\nmeter. The sequence of events leading to a radiant flux meas-\nurement with the radiant flux meter is generally as follows.\nRadiant energy is incident on the filter of the meter. This\nenergy is funneled in from the environment through a set D of\ndirections. The filter ideally transmits a set F of frequen-\ncies of the incident energy and does not transmit any other\nfrequencies. The transmitted frequencies then pass on to a\nplane collecting surface S. This surface acts to collect a\nrepresentative amount of the transmitted flux from each\ncollecting directions D\nincident radiant\n(variable, but not\nenergy of frequencies F\nexceeding a\nhemisphere)\nphotoelectric\nelement\nfilter\n-collecting surface S\ndial R\nFIG. 2.2 Schematic detail of a radiant-flux meter","SEC. 2.1\nRADIANT FLUX\n7\ndirection in D and to pass it on to the photosensitive ele-\nment of some type of radiant energy sensor. The sensor is\npart of a circuit of a photoelectric cell, and the presence\nof the radiant energy flow on the filter thus becomes manifest\nin a dial reading R of the current meter in the photoelectric\ncell's circuit (see Fig. 2.1).\nIn order to obtain a usable measure of the flow of ra-\ndiant energy, there is basically only one additional require-\nment on the radiant flux meter assembly, above and beyond the\nusual requirements on its components demanded by good mechan-\nical and electrical engineering practice. The additional re-\nquirement is that its collecting surface S collect energy in\na manner which is effectively independent of the direction of\nincidence of the energy on S. Thus, suppose a narrow beam of\nradiant energy is incident normally on S, and note the asso-\nciated reading R of the dial of the meter. Then let the\nbeam's incident angle vary slowly away from normal incidence,\nkeeping the beam always to fall within the surface S. An\nideal collecting surface will accept, diffuse, and pass on\nthe energy of this varying beam to the sensor below so that\nthe dial reading R remains fixed. When a collecting surface\ncomes within some preassigned distance of this ideal, we shall\ncall it a cosine collector. The reason for this terminology\nwill become clear after the study of the concept of irradiance\nbelow. Briefly, it derives from the fact that if the collec-\ntor is completely bathed in the flux of a homogeneous cylin-\ndrical beam, then the recorded flux will vary as the cosine\nof the angle the normal to the collector makes with the axis\nof the beam of flux. Henceforth, it will be assumed that the\ncollecting surface of the radiant flux meter is a cosine col-\nlector.\nWe now can state the operational definition of radiant\nflux. We assume that the radiant flux meter, outfitted with\na cosine collector S, has been calibrated against some radio-\nmetric standard with a known rate of radiant energy (radiant\nflux) output (see Chapter 6, Ref. [3]). Then we imagine that\nwe have taken the meter into some radiometric environment\nsuch as the depths of some lake or ocean, or perhaps to some\npoint in the atmosphere. The meter is then oriented so that\nat time t the surface S accepts through the set D of direc-\ntions radiant flux comprised of a set F of frequencies, with\na resultant associated reading R of the meter's dial. The\ncalibration of the dial permits the assignation to this read-\ning R of a radiant flux in the form of a nonnegative number\ndenoted by \"$(S,D,t,F)\". The reading R is thereby associated\nwith this particular S,D,t, and F in the radiometric environ-\nment. Thus \"$(S,D,t,F)\" denotes the radiant flux of frequen-\ncies in F which are incident on S, through D, at time t. The\ndimensions of radiant flux are energy/time, or synonymously,\npower, and convenient units are joules/sec, or synonymously,\nwatts. This pairing process therefore generates a function,\nthe radiant flux function denoted by \"\", which assigns to\neach collection (S,D,t,F) of surface, direction, frequency\nand time parameters the nonnegative number $(S,D,t,F) in the\nmanner just described.","RADIOMETRY AND PHOTOMETRY\nVOL. II\n8\nThe definition of radiant flux given above is an oper-\national definition in the sense that it may be translated in-\nto a definite sequence of physical operations with a specific\ninstrument in a given environment and which culminate in a\nunique nonnegative number $ (S, D, t,F). This type of definition\ncan be made to stand out in bold relief from still another\ntype which may also be used as effectively as the operational\ndefinition in establishing the theory of radiometry. This\nalternative definition is known as the constitutive definition\nof radiant flux which uses only the concepts of the mathema-\ntical framework within which radiometry is modeled. In a con-\nstitutive definition there is no immediate appeal to physical\noperations with a specific instrument in a given environment.\nFor an example of a constitutive definition of radiant flux\nand the other radiometric quantities, the reader may consult\nSecs. 109 and 131 of reference [251].\n2.2\nThe Meaning of 'Radiant Flux'\nIt will be helpful during the discussions of this and\nsubsequent chapters to have in mind some visualizable con-\nstruct of radiant flux. By having the reader picture in a\nrelatively concrete manner the meaning of the term 'radiant\nflux', the various principles and laws of radiative transfer\nused throughout this work will become more readily understood\nand applied. We have already given the term 'radiant flux'\na relatively concrete meaning by adopting an operational def-\ninition of the term. In this section we shall go one step\nfurther and suggest three ways in which one may visualize ra-\ndiant flux directly. What we shall offer, then, are concep-\ntual frameworks within which to view the notion of radiant\nenergy and which, especially during theoretical discussions\nof radiative transfer, one may use in a heuristic manner.\nOne manner in which radiant flux may be visualized is\nby a means similar to that used in geometrical optics. In\norder to discuss the theory of lenses within geometrical op-\ntics one may use the method of ray tracing. The heart of this\nmethod resides in the concept of the \"light ray\" and a few\nsimple rules of construction of a ray of light through a lens.\nCorresponding to this notion we have in geometrical radiometry\nthe notion of a line of flux. One may thus visualize\n$(S,D,t,F) as proportional to the number of straight or curved\nlines having directions lying within the set D where they ter-\nminate on the surface S. The time t and set F of frequencies\nare usually fixed or understood during a discussion so that\nthe lines of flux constitute a representation of the geomet-\nric construct of $(S,D,t,F). In this representation, the mag-\nnitude of (S,D,t,F) is proportional to the number of such\nlines of flux, the proportionality factor being some fixed\nnumber of lines per unit of radiant flux. One may thus imag-\nine the radiant energy as a fluid travelling along the lines\nof flux. The closer together the lines are within some re-\ngion, the greater the radiant flux (i.e., radiant energy flow)\nthrough that region. The lines of flux are to be determined\nusing the same formulas of geometrical optics as used in ray\ntracing. Whenever scattering takes place, however, some lines","SEC. 2.2\nMEANING OF 'RADIANT FLUX'\n9\nof flux undergo abrupt changes in direction. Between the\npoints of these abrupt changes in direction the structure of\nthe lines of flux are again governed by the ray tracing for-\nmulas of geometrical optics (and the lines are, in most prac-\ntical instances, merely straight line segments).\nAnother manner in which radiant flux may be visualized\nis by means of the Poynting vector of electromagnetic theory.\nBesides serving to generate an alternative semantic dimension\nto the term 'radiant flux', such a visualization if carefully\ndone serves to establish an analytic link between electro-\nmagnetic theory and the concepts of radiometry. Thus, con-\nsider the electric and magnetic vector fields E and H. The\nvector product EXH is called the Poynting vector field and\nis usually denoted by \"P\". A dimensional analysis of P shows\nthat it has the dimensions of radiant flux per unit area.\nThe direction of this flow is along the direction of P, and\nthe magnitude of the flow is that of P, and takes place across\na unit area normal to the direction of P. Generally, P varies\nin both magnitude and direction many times a second at a given\npoint. Thus the quantity $(S,D,t,F) may be viewed as a time\naverage of the magnitude of the Poynting vector P confined to\nthe directions within D over the surface S and during some\nshort time period T around time t. The analytic details of\nthe connection between P and $(S,D,t,F) beyond those alluded\nto here will have no application in the present work. The\nreader wishing to study this connection in more detail may\nconsult Sec. 124 of Ref. [251].\nOne further manner of visualizing the concept of radi-\nant flux is by means of the notion of photons, that is by\nmeans of 'particles' of light. Monochromatic radiant flux,\ni.e., radiant flux of a single frequency, say, v may be asso-\nciated with the flow of a set of photons each of energy hv,\nwhere \"h\" denotes Planck's quantum of action (per photon)\nIn more detail, let* \"n(x,5,t,v)\" (the phase-space density)\ndenote the number of v-frequency photons per unit frequency\ninterval and per unit volume at each point X over the surface\nS, and moving with speed V in a unit solid angle along the\ndirection E within the bundle D of directions at time t.\nThen:\nIII\nn (x,5,t,v) V (x,E,t) v (x,E,t) E.K(x) dA(x)d&(E)d1(v) (1)\nDSF\ngives the time rate of flow of radiant energy of frequencies\nin F crossing surface S, within directions D at time t. k(x)\nis the unit inward normal to S at X. This is then the seat\nof the meaning of (S, D, t, F) in terms of the photon concept.\nIn the preceding integral, \"A\", \"56\" and \"1\" denoted area,\nsolid angle, and frequency measures, respectively; and these\n*\nFor reference purposes, we observe here that the connection\nbetween the density n and the radiance N is: N(x,5,t,v) =\nhvvn(x,5,t,v). (See (5a) of Sec. 2.5.)","RADIOMETRY AND PHOTOMETRY\nVOL. II\n10\nwill be explained in greater detail in the subsequent sec-\ntions of this chapter.\nAs a special case of the preceding connection, let\nn(x,5,t,v) be constant of magnitude n over S and over a nar-\nrow bundle of directions D normally incident on S and let F\nconsist of discrete frequencies. Then, using the photon in-\nterpretation of $ (S,D,t,F) just described, we can write:\n(2)\n'$(S,D,t,F)\"\nhvnA(S).(D)\nfor\nv\nVEF\nIn summary, we have discussed three possible aids to\nvisualizing the meaning of 'radiant flux'. There is the\ngeometric-optics notion of lines of flux, the electromagnetic-\ntheoretic construct of the Poynting vector, and the quantum-\ntheoretic construct of the moving photon. A composite pic-\nture may be made by joining all three of the preceding con-\ncepts. Thus, one may visualize the photon not as a particle\n(i.e., a mathematical point) but rather as a spatially small\nwave train of electromagnetic waves of predominantly a single\nfrequency and moving along the lines of flux. This concept\nallows light to have at least intuitively, the properties of\nboth particles and waves.\nFundamental Geometric Properties of Radiant Flux\n2.3\nIn this section we shall assemble the six properties\nof $(S,D,t,F) on which geometrical radiometry may be based.\nThese six properties summarize precisely and explicitly those\nmacroscopic properties of light which are customarily impli-\ncitly assumed in radiometry, and which are based on extended\nexperience with the operational definition of radiant flux.\nBy explicitly recognizing and isolating these six properties\nwe may attain a unified and relatively rigorous development\nof geometrical radiometry. This fundamental group of six\nproperties falls naturally into three pairs of properties,\ncorresponding to the frequency, surface, and direction para-\nmeters occurring in $(S,D,t,F).\nWe begin with the properties of 8 associated with the\nfrequency parameter F. For every two disjoint sets F1 and F2:\n$ (S,D,t,F1) + $(S,D,t,F2) = $(S,D,t,F1 U F2)\n(1)\nand\nif\n1(F) = 0, then (S,D,t,F) = 0\n(2)\n.\nThese properties hold for arbitrary S,D, and t. The first of\nthese is the F-additivity property of &. The symbol \"U \" will\nbe used often below to denote the union of two sets of things.\nHere \"F1 UF2' denotes the set of all frequencies in either F1\nor F2. By \"disjoint sets\" we shall mean sets of things which\nhave no elements in common. Thus by \"two disjoint sets F1\nand F2\" we mean that F1 and F2 have no frequencies in common.","SEC. 2.3\nFUNDAMENTAL GEOMETRIC PROPERTIES\n11\nFor example the set F1 of frequencies in the interval from\n10 9 sec to 101 /sec inclusive is disjoint from the set F2 of\nfrequencies from 1011/sec to 10 12 /sec, inclusive. Thus\nF1UF2 is now the set of all frequencies which are either in\nthe interval F1 or F2. The second property (2) above is the\n(absolute) F-continuity property of D.\nThe meaning of the F-additivity property is quite sim-\nple: imagine radiant flux incident on S through D at time t\nand consisting of frequencies in F1 F2. This flux could be\nirradiated onto S through D simultaneously from two separate\nsources of frequencies F1 and F2, or by means of suitably\nchosen filters. Alternatively, the flux can be presented\nfirst for the frequencies of F1 and then for the frequencies\nof F2. In essence, (1) states that on the macroscopic level\nthere is no interference of two or more wave trains occupying\nthe same region of space. The F-continuity property asserts\nthat, on the macroscopic level and with S,D and t fixed and\nall other things being equal, the less the number of frequen-\ncies in F, the less the radiant flux amount of (S,D,t,F).\nIn particular, frequency intervals of zero length contain\nzero radiant flux.\nFrom the F-additivity and F-continuity properties of\nwe derive the concept of monochromatic radiant flux. Thus\nlet us write:\n(S,D,t,F)\n\"P(S,D,t,v)\"\nfor\nlim\n(3)\n1(F)\nF+{v}\nIt is precisely the properties (1) and (2) which permit\nthe limit P(S,D,t,v) to exist. * When v is understood, we may\ndrop it from the notation to write:\n\"P(S,D,t)\"\nfor\nP(S,D,t,v)\nand even further, we may write:\n\"P(S,D)\"\nfor\nP(S,D,t,v)\nwhen both V and t are understood. We call (S, D, t, v) the\nmonochromatic (or spectral) radiant flux of frequency v over\nS within D at time t, per unit frequency interval. The func-\ntion P which assigns to (S,D,t,v) the number P(S,D,t,v) is\n*\nThe mathematical reader may consult Refs. [216] and [103] for\nthe study of the existence of such limits and their use in\nradiative transfer theory. For simplicity in exposition we\nhave displayed finite additivity of &, rather than the full\ncountable additivity used in the cited references. Similar\nremarks pertain to subsequent additivity properties stated\nin this work.","12\nRADIOMETRY AND PHOTOMETRY\nVOL. II\nthe monochromatic (or spectral) radiant flux function. It\nfollows from (3) and a theorem of calculus that:\n(S,D,t,F)\nP(S,D,t,v)\nd1(v)\n(4)\nIf\n.\nThe symbol \"1\" denotes the length measure along the\nfrequency spectrum. Thus for the interval F consisting of\nall frequencies from frequency V1 to frequency V2, where\nV1 VI V2, we have (F) = V2-V1. Thus in practical computations\none can write \"dv\" for \"d1(v)\" and (4) is then understood to\nbe a Riemann integral. This is the intended interpretation\nof (4) for use in this work. However, general discussions\nare occasionally greatly facilitated by the retention of the\nlength measure, as shown in (4). The symbol \"1\" is also in-\nterpretable as the length measure along the wavelength spec-\ntrum. Furthermore, since both line spectra and continuous\nspectra are represented by the set of nonnegative real num-\nbers, 1 can be used to denote either the Lebesgue or Riemann\nmeasure on that set if continuous spectra are envisioned, or\nthe Stieltjes measure, if line spectra are considered. The\nparticular choice of the nature of 1 will be clear by conven-\ntion or from the context in each case. Thus, unless specifi-\ncally stated otherwise, 1 is to be considered as the usual\nRiemann type length measure used in ordinary calculus, and we\nconventionally consider continuous spectra. For integrations\nover wavelength space, use the transformation (32) of Sec.\n2.12.\nThe second pair of properties of $ is associated with\nsurfaces. For every two disjoint surfaces S1 and S2,\n(S1,D,t,F) + $(S2,D,t,F) = $(S1 U D, t, F)\n(5)\nand\nIf A(S) = 0, then $(S,D,t,F) = 0\n(6)\nThese properties hold for arbitrary D, and F. The first of\nthese is the S-additivity property of &. The second is the\nS-continuity property of &\nThe S-additivity property is understood as follows.\nSuppose the radiant flux meter has a variable collecting sur-\nface S, so that at one time it can be of extent S1 and at\nanother time (very soon after) it can be of extent S2, such\nthat S1 and S2 are disjoint. Then (5) states that the sum of\nthe two separate readings associated with S 1 and S2 equals\nthe reading associated with the union S1 U S2 of these surfaces.\nThis experimental fact is generally valid, provided of course,\nthat D, t, and F are fixed as closely as practicable through-\nout all three measurements Statement (5) is the ideal indi-\ncated by accumulated empirical findings. Statement (6) is\nalso intuitively clear: positive amounts of flux can only be\nrecorded over surfaces of positive area. This relatively in-\nnocuous pair of properties of comprises the logical root of\nthe concepts of irradiance and radiant emittance, to be","SEC. 2.3\nFUNDAMENTAL GEOMETRIC PROPERTIES\n13\nconsidered later.\nWe now turn to the third and final pair of fundamental\nproperties of the radiant flux function $ . These are associ-\nated with the direction set D. For every two disjoint direc-\ntion sets D1 and D2:\n$(S,D1,t,F) + § (S,D2,t,F) = (s, D1 U D2, t, F)\n(7)\nand\nIf\nso(D)\n0,\nthen\n$(S,D,t,F)\n0\n(8)\n=\n=\nThese properties hold for arbitrary S,t,F. The first of these\nis the D-additivity property of $, the second is the D-contin-\nuity property of $ . These properties along with the preceding\nfour will lead to the rigorous basis for the discussions of\nradiance, irradiance and related radiometric concepts.\nThe meaning of the D-additivity property is perhaps\nthe most interesting of all the additivity properties, for it\nshows most clearly that on the level of reality within which\nradiometry conventionally takes place, the interference phe-\nnomena of light are not discernable: the light fields are\ncomprised of incoherent electromagnetic fields. Examples are\nabundant on the microscopic level of light phenomena which\nillustrate the negation of (7), namely that for some S, t and\nF, there exist disjoint sets D1 and D2 such that:\n$(S,D1,t,F) + (S,D2,t,F) # (S, U D2, t, F)\n.\nTherefore, the left side can be either > or < the right side.\nFurthermore, the negation of (8) holds on the microscopic\nlevel, too. That is, for some S, D, t, F, we have:\ns(()) = 0\nand\n$(S,D,t,F) # 0\n.\nThe first of these inequalities may be illustrated by\nmeans of any diffraction arrangement; the second arises every\ntime Maxwellian electromagnetic theory is applied to a plane\nelectromagnetic wave. In such a case D consists of a single\ndirection--the direction of propagation - and $(S,D,t,F) is\ncomputed by means of the Poynting vector (cf. Sec. 2.2). It\nis principally through the properties (5) - (8) that one may\ndiscern the differences between the electromagnetic and phe-\nnomenological views of light, as far as logical form is con-\ncerned. That is, if we assume (5)-(8), then with a few addi-\ntional physical-process and logical requirements which are\ncommon to both the electromagnetic and phenomenological views\nof light, the fundamental equations of radiative transfer\ntheory are logically deducible. This may be seen, for example,\nby studying the results of Ref. [251]. Thus the electromag-\nnetic and the phenomenological views of light necessarily\npart ways in (5) - (8) In a similar manner the phenomenologi-\ncal and quantum views of light differ at the same two points\nas above and possibly also at property (2). For, in the quan-\ntum theory, as in electromagnetic theory, radiant flux of a","14\nRADIOMETRY AND PHOTOMETRY\nVOL. II\nsingle frequency as carried by a single photon (or a pure\nmonochromatic wave of light) exists in principle. Therefore,\nit is possible that $(S,D,t,{v}) >0 for some set F consisting\nof a single frequency v, and that (F) = 0 at the same time.\nThis follows from use of the usual measure of length on the\nfrequency domain. If one redefines length on the frequency\ndomain by adopting a Stieltjes measure, e.g., so that isolated\nsingle frequencies are given nonzero (usually unit) length\n(instead of the zero length we conventionally assigned them by\nthe usual continuum measure) then (2) would hold on the elec-\ntromagnetic and quantum levels too, and (5) - (8) remain as the\nsource of the fundamental distinctions between the microscopic\nand the macroscopic views of light.\n2.4\nIrradiance and Radiant Emittance\nWe now turn to the task of defining the radiometric\nconcepts used in radiative transfer in general and hydrologic\noptics in particular. The first two of these are the concepts\nof irradiance and radiant emittance. These concepts describe\nthe flow of radiant energy per unit area across a surface.\nThat is, they describe the area-density of radiant flux. Ir-\nradiance describes the flow onto a unit area; radiant emit-\ntance describes the flow from a unit area. From a strictly\ngeometric point of view, this is the only distinction between\nthe two concepts. However, radiant emittance occasionally\nhas an additional physical connotation associated with it,\nnamely that of a flow of radiant flux from a unit area of sur-\nface which encloses an emitting source of radiant flux, i.e.,\na region manufacturing radiant energy. However, within the\noperational definitions of these concepts, this additional\nconnotation does not exist; the connotation exists only in\nthe mind of the experimenter. We now turn to the detailed\ndefinitions of these concepts.\nDefinition of Irradiance\nWe begin with the concept of irradiance. Imagine a\nradiant flux meter transported to a point X in a natural hy-\ndrosol, or in the atmosphere. Let the collecting surface S\nof the meter be placed so that x falls within its small ex-\npanse, and orient the set D of directions of the meter as de-\nsired. A filter is fitted on the meter so as to pass mono-\nchromatic radiant flux of given frequency V. Hence, the me-\nter can be made to read P(S,D) directly (with V and t and\ntheir units understood) Let \"A(S)\" denote the area of the\ncollecting surface S. Then we shall write:\n\"H(S,D)\" for P(S,D)/A(S)\n(1)\nand call H(S,D) the (empirical) irradiance over S within D.\nIn full notation for the unpolarized context, we would write:\n\"H(S,D,t,F)\"\nfor $ (S,D,t,F)/A(S)\n(2)","SEC. 2.4\nIRRADIANCE AND RADIANT EMITTANCE\n15\nor\n\"H(S,D,t,v)\" for P(S,D,t,v)/A(S)\n(3)\n.\nHowever, in most discussions of radiative transfer in hydro-\nlogic and meteorologic optics the light field is steady in\ntime, and is studied frequency by frequency. Hence we shall\nuntil further notice hold t and V (or F) fixed and so exclude\ntheir symbols and units from the notation, as in (1).\nNext, we let S become smaller and smaller, such that\nit always contains the point X and such that the flow of ra-\ndiant energy is onto S along the fixed set D of directions.\nThen we write:\n\"H(x,D)\"\nfor\nlim\nH(S,D)\n(4)\n.\nS+{x}\nThe existence of this limit is guaranteed by the S-\nadditive and S-continuity properties of $ postulated in Sec.\n2.3. The irradiance H(x, D) is the (theoretical) irradiance\nat x within D. The dimensions of both empirical and theoret-\nical irradiance are radiant flux per unit area (per unit fre-\nquency interval) ; convenient units are watts/(meter) 2 (per\nunit frequency interval).\nIt is of interest to see the meaning of H(x,D) in terms\nof the radiant flux function $ of Sec. 2.2. Thus, , from (4)\nand (3) (making v and t explicit for the moment)\nH(x,D,t,v) = lim\nP\n(S,D,t,v)/A(S)\nS+{x}\nFrom Sec. 2.3 this becomes:\n$(S,D,t,F)\nH(x,D,t,v) = lim\nlim\nS+{x}\nF+{v}\n1(F) A(S)\n(S,D,t,F)\n1im S+{x} lim\n(5)\n=\nF+{v}\n1(F)\nIt follows from (3) above and a theorem of calculus that:\nP(S,D,t,v) = / H(x,D,t,v) A(x)\n(6)\nand hence from (4) of Sec. 2.3 that:\n$(S,D,t,F) = 11\nH(x,D,t,v) d1(v) dA(x)\n(7)\nS F","16\nRADIOMETRY AND PHOTOMETRY\nVOL. II\nIt is easy to see that these integrals can be generalized to\nthe case where D in H(x,D,t,v) may vary with x, and we shall\nunderstand that (6) and (7) hold in such cases.\nIn actual practice, the size of the collecting surface\nS, which serves to accept, diffuse, and transmit the incident\nflux on to the photoelectric element of the meter, ranges\nfrom the size of a pinhead to that of a dinner plate. These\nextremes are not intended to be precise limits; rather they\nare reprèsentative of the extremes that may be encountered in\nnatural radiometric environments under ordinary working con-\nditions. The lower limit cited above begins to approach the\nsize where, for very sensitive photoelectric elements, effects\nof diffraction may be noticeable. For example, an ordinary\nhousehold stickpin or a human hair held in a pencil-thin shaft\nof sunlight will cast a shadow with a diffraction pattern\nclearly discernable by the unaided human eye. Hence a very\nsmall radiant-flux meter collecting-surface can pick up such\nirradiance variations over the shadow. The upper limit cited\nabove is dictated by the fact that natural lighting variations\nbecome noticeable over such relatively large areal extents:\nchanges of lighting with depth in ponds or oceans, shadows\ncast by leaves or fish, edges of dense cloud shadows on the\nground or a sea surface, etc. Hence by staying within these\nlimits and choosing the size of S accordingly, good empirical\nestimates of irradiance can usually be made using the defini-\ntion (1).\nThe Meaning of 'Irradiance'\nIt is occasionally helpful in both theoretical and ex-\nperimental considerations to keep in mind the various mean-\nings of 'radiant flux' discussed in Sec. 2.2. These may be\napplied directly to the concept of irradiance. Thus H(S,D)\nmay be imagined as proportional to the number of lines of\nflux incident per unit area over S and whose directions at\ntheir points of intersection with S lie within the set D.\nFurther, using the Poynting vector interpretation of radiant\nflux, we see that the dimensions of the vector are precisely\nthose of irradiance. Finally, H(S,D) may be viewed as a meas-\nure of the number of photons per unit area per unit time on S,\nfunneling down onto S along the directions of D. In particu-\nlar, using (2) above and (2) of Sec. 2.2 for a monochromatic\nset of n photons over a small collecting area, and incident\nwithin a small set D of directions normally on S, we have:\nH(S,D,t,{v}) = $(S,D,t,{v}) = hvvns(D)\n(8)\nA(S)\nA further insight into the concept of irradiance is\nobtained by considering some of the typical magnitudes of ir-\nradiance encountered in natural environments. Table I lists\nsome of these values. They are order-of-magnitude estimates\nand are not to be used beyond establishing an intuitive feel-\ning for the meaning of irradiance (see also Sec. 1.2).","SEC. 2.4\nIRRADIANCE AND RADIANT EMITTANCE\n17\nTABLE 1\nENVIRONMENT\nTYPICAL ORDER OF\nMAGNITUDE OF IRRADIANCE\nAt sea level, on surface S\nnormal to sun's rays, clear\nday*\n103 watt/m2\nAt sea level, slightly overcast\n102 watt/m2\ndays, horizontal surfaces\nAt sea level, heavily over-\ncast day, horizontal surface\n10 watt/m2\nS (sunset)\nLighted interiors: walls,\n1 watt/m2\nceilings, floors\nAt sea level, clear night, high\n10-3 watt/m2\nfull moon, horizontal suface S\nAt sea level, clear night, flux\nfrom 1st magnitude (highly vis-\nible) star, on surface S normal\n10-9 watt/m²\nto star's rays\nAs another base for comparison and also to extend our\nintuitive feeling for irradiance and its connection with the\nphoton picture of light, let us calculate the number of pho-\ntons per unit volume, of wavelength A, required to produce\nH watt/m2 at a point of some surface. To fix ideas, suppose\na thin pencil of photons arrives at each point x of a surface\nS in the direction of its inward normal E, and that each pen-\ncil is of the same density comprised of photons of a single\nfrequency V. It follows that the photon density n(x,5,t,v)\nhas the form\nn(x,5',t,v') = no(x,t) S(E'-5) s(v'-v)\nwhere S is the Dirac delta function and where E is the inward\nnormal to S, and v is the frequency associated with 1. When\nused in (1) of Sec. 2.2, this equation yields:\n*\nAccording to Moon, Ref. [185], at sea level, for sun zenith,\nclear dry air, the irradiance is nearly 1200 watt/m2. See\nalso [296] for a survey of solar irradiation measurements.","18\nRADIOMETRY AND PHOTOMETRY\nVOL. II\nhvv III n (x,t)(-E)(v'-) o dA(x) d1(v')\nD S F F\n= hvv / n (x,t) dA(x) = hvvn.A(S)\nas the radiant flux crossing S normally at time t. Hence\nhvvno is the irradiance produced by each pencil. Setting\nhww. = H watt/m2\nwe have:\nphotos\n3\nm\nor\nnov = H photons\nhv sec x m ²\nor\nnov=Ha photons\nsec x m ²\nFor example, setting H = 1 watt/m2, 1 = 550 mu, and recalling\nthat h = 6.6 x 10-34 Joule sec/photon and V = 3 x 10 m/sec,\nwe have\nnov = number of photons of wavelength 550 mu\nper sec. normally incident per square\nmeter which produce one watt\n$ 5.5 10-7\n=\n6.6 x 10-34x3 x 10 8\n= 2.8 x 1018\nFrom Table 1 we see that the normal irradiance of a\nfirst magnitude star is on the order of 10-9 watt/m². If we\nassume this flux to be comprised of photons all of wavelength\nl = 550 mu, then the number of associated photons producing\nthis irradiance is 2.8 x 101 x 10-9 = 2.8 x 109 photons per\nsecond normally incident per square meter = 2.8 x 105 photons\nper second normally incident per square centimeter. Now a\nhuman eye's pupil is on the order of a tenth of a square cen-\ntimeter in area. Hence when our eyes are directed toward a\nfirst magnitude star such as the present one, about 2.8 x 10\nphotons per second enter each eye to produce the visual sensa-\ntion of brightness in the brain.","SEC. 2.4\nIRRADIANCE AND RADIANT EMITTANCE\n19\nTerrestrial Coordinate Systems\nIrradiance measurements and other radiometric measure\nments of hydrologic and meteorologic optics during careful ex-\nperimental investigations are usually made with respect to\neither one of two terrestrially-based frames of reference.\nEach reference frame uses the usual euclidean three-dimension-\nal coordinate system with its familiar xyz-axes. The two ter-\nrestrially-based reference frames are primarily distinguished\nby the way they anchor the directions of the X and Z axes in\neach case. See Fig. 2.3. The sun-based frame directs the\nplane determined by the x-axis and z-axis (the xz plane) so as\nto contain the center of the sun. (The north-based frame di-\nrects the xz plane so as to lie in the plane of the local me-\nridian circle on the earth.) In each frame the z-axis is\nparallel to the local vertical direction, (i.e., the local\ngradient of the gravitational field). In meteorologic optics\nZ is measured as increasing in the upward direction, i.e., the\nunit vector k along the z-axis. In hydrologic optics it is\nmore convenient to measure Z as increasing in the downward\ndirection -k, as shown in Fig. 2.3. In meteorologic optics,\n\"z\" (or other symbols) denotes altitude, in hydrologic optics,\n\"z\" (or other symbols) denotes depth, when specific reference\nto terrestrial coordinate frames is made.\nThe concept of direction within a reference frame es-\ntablished for a natural optical medium such as the atmosphere\nor the sea is of central importance in hydrologic optics and\nranks equally in importance with the notion of location. In\nview of this importance it will be well to define with care\nprecisely what is meant by \"direction\", and to develop some\nof the more frequently occurring concepts associated with it.\nNow, to locate an object within a terrestrially-base\nreference frame, it suffices to give the x,y and Z coordinates\nin terms of meters, say. Thus, in the hydrologic optics ref-\nerence frames, the triple of numbers (1, 10, 100) locates a\npoint in a natural hydrosol by going one meter along the di-\nrection i from the origin, then 10 meters along the direction\nj, and then 100 meters vertically downward. (Recall that in\nnatural hydrosols, Z is measured positive in the downward di-\nrection, i.e., in the direction -k.) Now this point obviously\nlies in a well defined \"direction\" from the origin of the ref-\nerence frame. We observe that this \"direction\", however, has\nnothing to do with the distance of (1, 10, 100) from the ori-\ngin. Indeed, the points (1/2, 5, 50) and (2, 20, 200) which\nare, respectively, half and twice as far from the origin as\nthe original point, all lie in the same \"direction\" from the\norigin. A convenient measure of this common \"direction\" of\nall three points then would be established if we chose a\npoint some standard fixed distance from the origin and which\nshares the same \"direction\" as they do. The obvious choice\nis the point a unit distance from the origin. Thus, if\n(x,y,y) is a point in a terrestrial frame of reference, then\n(x,y,y) / (x2+y2+z2) 1/2 is a point a unit distance from the ori-\ngin. We call this latter point the direction of (x,y, z) from\nthe origin.","RADIOMETRY AND PHOTOMETRY\nVOL. II\n20\n+z\nsun\n(x1,x2,x3) or (x,y, z)\n0\nII\nk\n4\ny\nmeteorologic optics\nsun-based frame\nX\ngravitation field falling\ndirection is downward\nsun\ndirection\n0\nk\n4\ny\nhydrologic optics\nsun -based - frame\nx\n+Z\n+\n(x1,x2,x3) or (x,y,z)\nE.\nFIG. 2.3 Sun-based terrestrial frames of reference for\nmeteorologic optics and hydrologic optics.","SEC. 2.4\nIRRADIANCE AND RADIANT EMITTANCE\n21\nIn many of our discussions we shall not need to spec-\nify explicitly the three coordinates of a point. In such\ncases we will simply write:\n\"x\"\nfor\n(x,y,y)\nor\n\"x\"\nfor\n(X1,X2,X3)\nwhere the ordered triplets are the three coordinates of point\nFurther we shall correspondingly write:\nX.\n\"E\"\nfor\n(x,y,z)/(x2+y2tz2)\n1/2\nor\n(X1,X2,X3)/ 2 +X3 2) 1/2\n\"E\"\nfor\nHence, throughout this work the letter \"x\" (in either\nlightface or boldface type, as emphasis requires) is generally\nto designate a location and the letter \"E\" is generally to\ndesignate a direction. The denotation of the components of\nx and E will vary so as to permit simplicity and clarity of\nexpression. We have already used the three special directions\ni, j, k, which we have agreed to be the points (1,0,0),(0,1,0),\nand (0,0,1), respectively.\nWe will also wish to consider collections of direc-\ntions in addition to single directions. For example, certain\nsets D were already encountered in our discussions above. In\nparticular, let us denote by \"E\" the set of all directions\nabout the origin. Clearly (1) is a sphere of unit radius with\norigin as center. Observe that we use an upper case Greek Xi\n(the Greek counterpart to the English letter \"X\") to designate\nthe set of all directions. There are two more sets of direc-\ntions of very frequent occurrence in practice. First, there\nis the set of all upward directions, i.e., the set of all di-\nrections E such that E and k make an angle of less than nine-\nty degrees. We shall designate this set by \"E+\". Second,\nthere is the set of all downward directions, i.e., the set of\nall directions E such that E and k make an angle of greater\nthat ninety degrees. We shall designate this set by\nThe\n\"++\" and 11_11 are convenient mnemonics which help distin-\nguish one set from the other. The reader may recall from vec-\ntor analysis at this point that if E is in E+ then E.K > 0,\ni.e. , the dot (or scalar) product of the vectors E and k is a\npositive number; and that if E is in E., then E.K is a nega-\ntive number. This is the reason for the plus and minus signs\nin the names \"E+\" and \"E_\". Indeed, it would be well to re-\ncall that for every direction E,\nE.K = COS 0\nwhere 0 is the angle between the lines along which E and k\nlie. See Fig. 2.3. For convenience we reproduce below the\ndefinition of the dot product of two unit vectors E1 and E2.\nSuppose we have written:","VOL. II\n22\nRADIOMETRY AND PHOTOMETRY\n\"E1\" for (a1,b1,c1)/(a12+b12+c12)12\n\"E2\" for (a2,b2,c2)/(a22b22+c22)1\nThen we write:\naja2+b1b2+c,cz\n\"51.52\"\nfor\nFrom analytic geometry it is known that:\n51.52 = cos 2\nwhere I is the angle between E1 and 52.\nThe representation of a unit vector E as an ordered\ntriple of numbers takes on deeper meaning when we observe the\nfollowing geometric fact. Let \"(a,b,c)\" denote the ordered\ntriple representation of E. Then compute the dot product of\nE with i, j, and k in turn. By the cosine law cited above we\nhave:\nE.i = cos 24 =\n5.j = = cos V2\nE.K = cos 2, =\nwhere V1, V2, and V3 are, respectively, the angles between\nE and the positive x, y and Z axes. Using the ordered triple\nrepresentations of E, i, j, and k, and the definition of the\ndot product, we have:\nEi = a\ns.j = b\nE.K = C\nHence the components a,b,c of the direction E are simply the\ncosines of the angles that E makes with the positive x,y and\nZ axes, i.e. :\na = cos 2, =\nb = cos 2\nc = cos 23 =\nThis leads to the representation:","SEC. 2.4\nIRRADIANCE AND RADIANT EMITTANCE\n23\nE = (cos 2 1, cos 2l 2 cos 23)\n.\n= i cos v 1 + j cos V2 + k cos V3\nwhere we have written:\n\"i\" for (1,0,0)\n\"j\" for (0,1,0)\n\"k\"\nfor\n(0,0,1)\nThere is an alternate mode of representation of a\nunit vector E. This alternate mode attains its greatest util-\nity in actual calculation. This is the representation of E\nby two especially constructed angles 0 and $ , found as fol -\nlows. By studying the schematic representation of (1) in Fig.\n2.4 it is clear that since each direction of (1) has fixed\nknown length (namely a unit length) it suffices to uniquely\nspecify E by the angle it makes with the z-axis and the angle\nthe plane determined by E and k makes with the X Z plane. Sup-\npose we designate the former angle by \"0\", the latter by \"o\",\nand agree to set 0 = 0 when E = k. Further, we agree to have\n0 increase to TT when E = -k. Further, we agree to set If 0\nwhen E is in the XZ plane and to have increase to 2 as\nthe plane of E and k rotates from the XZ plane to the yz plane.\nWe let increase in like manner through the next three quad-\nrants, and finally have it measure 2 radians after one com-\nplete turn in this manner. To summarize this alternate mode\nZ\nk\nIII +\nC\n0\nb\nII\ny\na\nX\nFIG. 2.4 Angle and direction definitions","24\nRADIOMETRY AND PHOTOMETRY\nVOL. II\nof representation of E, we agree to write:\n\"(0,0)\nfor\nE\nwhenever E is in E, and whenever E = (a,b,c), 0 = arc cos C,\nand = arc tan b/a, and where the quadrant containing is\nfixed by the signs of a and b. The angle 0 is the polar (or\nzenith) angle of E, and the azimuthal angle of 5.\nRepresentation of Irradiance in Terrestrial Frames\nLet us return now to apply these geometrical results\nto the task of specifying irradiance in natural optical media\nsuch as the atmosphere or the sea. It has become clear after\nmuch theoretical and experimental work in natural aerosols\nand hydrosols that the type of irradiance which is used most\noften in practice is the irradiance on a horizontal surface S\nat point x with a set D of directions which constitutes either\nthe hemisphere E+ or in of the unit sphere E. To specify\nsuch irradiances, we return to the definition in (4), and re-\nplace \"D\" first by \"E,\" and then by \"E\" (or by \"+\" and \"-\").\nThus H(x,E+) (or H(x,+)) is the irradiance at point X induced\nby upward flowing radiant energy in the directions of E+, and\nH(x,E_) (or H(x, is the irradiance at point X induced by\ndownward flowing radiant energy in the directions of E.\nA further specialization in notation can take place\nwhen the medium is stratified. Now, a natural optical medium\n(or a light field) with a terrestrially-based reference frame\n(Fig. 2.3) is said to be stratified if and only if the optical\nproperties of the medium ( or light field) as functions of co-\nordinates x, y, z, are independent of the coordinates x and y.\nThus for stratified light fields we may, for brevity and with-\nout loss of information, replace the general point name \"x\"\nin H(x,E+) by \"z\", the depth-parameter name. Thus, let us\nagree henceforth in stratified natural optical media to write:\n\"H(z,+)\"\nfor\nH(x,3+)\n(9)\nand\n\"H(z,-)\"\nfor\nH(x,E_)\n(10)\nWe call H(z,+) the upward irradiance and H(z,-) the downward\nirradiance.\nThe next most frequently occurring type of irradiance\nH(x,D) after the types H(z,+), is that for which D is an ar-\nbitrarily oriented hemisphere. Thus, let us denote by \"E(E)\"\nthat part of (1) consisting of all unit vectors E' such that E'\nand E subtend an angle less than ninety degrees. Hence, after\nadapting definition (4) to the case where D is E(E), we have\nH(x,E(E)) as the irradiance at point X on a collecting sur-\nface S with unit inward normal E, such that the irradiance is\nproduced by radiant flux incident on S at X along the direc-\ntions within E(E). See Fig. 2.5. Observe that the irradiance\nH(x,E(k)) is simply H(x,E+) considered earlier, since\nE(k) = E+; and similarly H(x,E(-k)) = H(x,E). Now a useful\nfact about such sets of directions as E(E) is that they are","SEC. 2.4\nRADIOMETRY AND PHOTOMETRY\n25\n(E)\nFIG. 2.5 Defining the hemisphere E(E) determined by the\ndirection E\nuniquely specified by giving the single vector E. We take\nadvantage of this observation to shorten the irradiance nota-\ntion by agreeing henceforth in (4), for the case D = E(E), to\nwrite:\n\"H(x,5)\"\nfor H(x,E(E))\n(11)\nIf we restrict attention to a fixed point x, then the totality\nof all values H(x,E) as E varies over (1) is called the irradi-\nance distribution at X. If the light field is stratified we\nfurther agree to write:\n\"H(2,E)\" for H(x,5)\n(12)\nThus in stratified light fields, one may specify irradiances\nby giving a depth Z and the unit inward normal E to a (hypo-\nthetical or real) collecting surface at that depth.\nIf one prefers to use the mode of representation of\nE by means of polar and azimuthal angles 0 and , then it will\nbe agreed to write:\n*\nWhenever wavelength dependence and time dependence is to be\nshown explicitly we would use \"H(x,E(E), ,A)\", or \"H(x,E(E), t)\"\nor \"H (x, (1) (E),t,1)\" as the case may be, and in contracted 5-\nnotation, as desired.","VOL. II\nRADIOMETRY AND PHOTOMETRY\n26\n\"H(x,0,0)\" for H(x,5)\n(13)\nor\n\"H(z,0,0)\"\nfor\nH(z,5)\nwhen the light field is stratified. It should be re-empha-\nsized that the direction E (and hence (0,0)) refers to the unit\ninward normal to the collecting surface S in the operational\ndefinition of (13) and that the flow of photons is onto S at\nx along the directions of E(E). This is the convention we\nshall adopt when discussing irradiance measurements by col-\nlecting surfaces on a theoretical level; for the transport\nequations for H(z, to be introduced later (Chapter 8) are\nwritten down in an intuitively natural manner if this conven-\ntion is adopted. The convention may be altered if need be\nfor empirical discussions. However, it is perhaps needless\nto point out that the fewer such conventions actually adopted\nfor radiometers, the smaller will be the chance of conceptual\nchaos in practice.\nOne final definition, and then we shall be ready for\na discussion of the cosine law for irradiance. We agree to\nwrite:\n\"H(x,E)\" for H(x,5) - H(x,-5)\n(14)\nand call H(x,5) the net irradiance at X in the direction E.\nThe Cosine Law for Irradiance\nWe now consider the property of irradiance which is\nits most important and most frequently used theoretical prop-\nerty. This is the cosine law for irradiance. The law is\nbased on the simple geometric fact that the apparent area of\na small plane surface at a fixed distance along one's line of\nsight varies as the cosine of the angle between the line of\nsight and the normal to the surface. If now we direct a\nswarm of photons along the line of sight toward the small sur-\nface then, all other things being equal, the area will inter-\ncept a number of photons proportional to the apparent area,\ni.e., proportional to the cosine of the angle between the di-\nrection of the beam of photons and the surface's normal.\nHence the area density, i.e., the irradiance of the photons\non the surface will vary as the cosine of this angle. The\nformal statement of this observation is the cosine law for\nirradiance. We now translate this verbal derivation of the\ncosine law into symbolic form.\nIn Fig. 2.6 a small plane surface is denoted by \"S\".\nAn amount P(S,D) of radiant flux is incident on S and arrives\nat each point of S through a very narrow fixed conical solid\nangle D such that the central direction of D is normal to S.\nSince the radiant flux is limited to a relatively narrow bun-\ndle of directions, essentially all the lines of flux are con-\nfined to a cylindrical volume C in the immediate neighborhood\nof S. Let \"S'\" denote a section of C generated by a plane\nwhose normal makes an angle 2 with the axis of C and such that","SEC. 2.4\nIRRADIANCE AND RADIANT EMITTANCE\n27\nD\nX\nC\nS\nD'\n0\ny\nS'\nFIG. 2.6 Geometry of the cosine law for irradiance.\nthe plane goes through some point x on S. The area A(S') of\nS' is clearly related to the area A(S) of S by the relation:\nA(S') = A(S) sec I\nAssuming no intervening sources or sinks of radiant flux in\nthe region of C between S and S' the flux P (S,D) then also\ncrosses S' . Thus we can write:\nP(S,D) = P(S',D)\nBy definition, the area density H(S',D) of radiant flux across\nS' is:\nH(S',D) P(S',D)/A(S')\nIn view of the preceding flux conservation statement and the\ngeometric relation between A(S') and A(S) we can write:\nH(S',D) = P(S,D)/(A(S) sec 2)\nBy definition H(S,D) is P(S,D)/A(S) and we therefore arrive\nat the statement:\nH(S',D) = H(S,D) cos V\n(15)","RADIOMETRY AND PHOTOMETRY\nVOL. II\n28\nThis is the empirical form of the cosine law for irradiance.\nA theoretical form of the law is obtained by letting S+ {{})\n(and hence S'+{x}). The result is:\nH(x,5') H(x,5) E.E. ,\n(16)\nHere we have used the fact that D was sufficiently narrow so\nthat in the limit H(S, D) goes to (xx, E) as S goes to the set\n{x} consisting of point X. Further, H(S' D) goes to H(x,5')\nas S goes to {x}. of course (16) is to be understood to ap-\nply to a set D of directions with a small but finite solid\nangle. The limiting case for D+ { E } can be handled naturally\nonly after the concept of radiance has been introduced. Fur-\nther we have replaced \"cos 2\" by \"E.E\" in going from (15)\nto (16). After the introduction of the concept of vector ir-\nradiance (Sec. 2.8), (16) can readily be generalized to the\ncase where the set of incident directions D is arbitrary.\nRadiant Emittance\nWe close this section with a few comments on the con-\ncept of radiant emittance. As already noted in the introduc-\ntory remarks to this section, the concept of radiant emittance\nis nearly identical to that of irradiance, differing from the\nlatter geometrically only by the sense of flow of the radiant\nenergy across a surface S. Fig. 2.7 schematically depicts\nthe geometrical distinction between irradiance and radiant\nirradiance\nS\nradiant\nemittance\nFIG. 2.7 Conceptual distinction between irradiance and\nradiant emittance.","SEC. 2.4\nIRRADIANCE AND RADIANT EMITTANCE\n29\nemittance; a given parcel of radiant energy flowing onto a\nsurface S generates irradiance on S : the same parcel flowing\nfrom the surface S generates radiant emittance of S. To em-\nphasize this distinction and to have appropriate notation\navailable when needed, we need only write \"d- (S,D)\" to denote\nradiant flux onto S and to write (S,D)\" for radiant flux\nfrom S. Then we extend this notation to radiant flux by means\nof \"p- (S,D)\" and \"p+ (S,D) \". Thus, the definition (1) of em-\npirical irradiance may be written as:\n\"H(S,D)\"\nfor\nP\n(S,D)/A(S)\n(17)\nfor emphasis of the \"onto\" interpretation of the flux; and we\nnow go on to write:\n\"W(S,D)\" for p+ (S,D)/A(S)\n(18)\nfor contrast of the two notions. We call W(S,D) the (empiri-\ncal) radiant emittance over S within D. From consideration\nof Fig. 2.7 it is clear that in the context of that figure:\nP*(S,D) = P* (S, )) =\n(19)\nso that\nW(S,D) = H(S,D)\n(20)\nAnother distinction between W(S.D) and H(S,D) for a\ngiven S and D lies on the physical rather than the geometric\nlevel. Indeed, it is on this level that the concept W(S,D)\nwas originally conceived and arose in connection with the der-\nivation of the complete (or Planckian) radiator wherein radi-\nant flux is generated within a body and then emitted through\nits boundary. This interpretation will be used in Sec. 2.12\nduring the transition from radiometry to photometry.\nWe conclude by observing that every auxiliary geo-\nmetric definition and geometric law considered above for ir-\nradiance now holds analogously for radiant emittance. We\nshall henceforth apply the analogous notation for W(S,D) (such\nas W(x, D) W(x,5), etc.) without further explicit definitions.\nThus for example we write:\n\"W(x,D)\"\nfor\nlim\nW(S,D)\n(21)\nS+{x}\nand\n\"W(x,5)\"\nfor\nW(x,E(E))\n(22)\nand so on.","30\nRADIOMETRY AND PHOTOMETRY\nVOL. II\n2.5 Radiance\nWe now define the radiometric concept of radiance,\ndiscuss some of its various forms, and study some of its basic\ngeometrical and physical properties of particular use in hy-\ndrologic optics.\nFor those who are studying the concept of radiance\nfor the first time, we may introduce it by saying that radi-\nance is designed to yield a simple mathematical representation\nof the percept of brightness experienced by the human eye as\nthe eye is directed along various paths of sight. Some intro-\nspection will show that when one directs visual attention to\na point in his environment, such as a point on a desk or a\nwall, the brightness sensations of neighboring points of the\npoint under scrutiny can be willfully suppressed. The result\nis a possible conscious comparison of \"brightness\" of succes-\nsive neighboring points in one's environment. Now when one\nattempts to simulate this sensation of brightness by means of\nradiant flux meter, one must introduce a mechanical means\na\nof directing the 'attention' of the collecting surface along\na narrow bundle of directions. The collecting surface by it-\nself is obviously incapable of the complex and partly auto-\nmatic process that takes place in the eye-brain circuits with-\nin a human head when visual attention is directed along a nar-\nrow bundle of directions. Some sort of \"blinder\", usually in\nthe form of a long narrow circular cylinder, must be fitted\naround a circular collecting surface so that its axis is nor-\nmal to the plane of the collecting surface. The result is a\nradiant flux meter with a relatively narrow conical set D of\ndirections along which radiant flux may be incident on a\nplane circular collecting surface S. Such an assembly is de-\npicted in Fig. 2.8, and is called a radiance meter.\nThe operational definition of the radiometric concept\nof radiance can be given in terms of a radiance meter as fol-\nlows. The radiance meter is taken to a point x in a natural\noptical medium such as the atmosphere or a natural hydrosol.\nThe center of the collecting surface is placed so as to be at\npoint X. The axis of the cylindrical collecting tube of the\nmeter is directed along a direction E so that radiant flux\nfrom the field of view is funneled along the set D in the\ngeneral direction of E. The sensor component of the radiance\nmeter records an incident radiant flux P(S,D) on the collect-\ning surface. The area A(S) of S and the solid angle so(D) of\nD are known instrumental constants. The quotient:\nP(S,D)/A(S) (?(D)\nis called the (empirical) radiance at X along E. Radiance,\ntherefore, is a nonnegative number which is paired with the\ndimensions of power per area per solid angle (per unit fre-\nquency interval), and with convenient units such as watts per\nsquare meter per steradian (per unit frequency interval). We\nwill write:\n\"N(S,D)\" for P(S,D)/A(S) (2(D)\n(1)","SEC. 2.5\nRADIANCE\n31\ncollecting directions D\n(fixed)\nE\ncollecting tube\n0\ncollecting\nsurface S\nX\nfilter\nphoto-\nelectric\nelement\ndial R\nFIG. 2.8 Schematic details of a radiance meter\nor in more complete notation:\n\"N(S,D,t,v)\" for (S,D,t,v)/A(S)s(D)\nSince A(S) and So(D) are fixed numbers for a given radiance\nmeter, the radiant flux reading can be calibrated directly in\nterms of N(S,D). Experimentation with variously proportioned\nradiance meters indicates that those meters with solid angle\nmagnitudes sb (D) on the order of 1/30 steradians, and with col-\nlecting areas A(S) on the order of that for a circular sur-\nface of a centimeter in diameter, are adequate for radiometry\nin most natural optical media. of course, the smaller A(S)\nand s(D), the sharper are the radiance maps obtainable (while\nstill remaining above the level where diffraction and general\ninterference effects set in).\nRecall the definition of empirical irradiance H(S,D)\nin (1) of Sec. 2.4. It follows from (1) above that:\nN(S,D) = H(S,D)/S2(D)\n(2)\nCorresponding to (4) of Sec. 2.4 we shall write:\n\"N(x,D)\"\nfor H(x,D)/S2(D)\n(3)\nand\n\"N(S,E)\"\nfor\nlim\nH(S,D)/S(D)\nD+{E}","32\nRADIOMETRY AND PHOTOMETRY\nVOL. II\nwhere the central direction of D is normal to the plane of S\nat X. We are then led to write:\n\"N(x,5)\"\nfor\nlim\nN(x,D)\n(4)\n.\nD+{E}\nWe will occasionally use the more complete notations\n\"N(S,D,t,F)\", \"N(x,E,t,F)\", \"N(x,5,t,v)\", etc., for radiance\nwhen time and frequency parameters are explicitly required.\nThe time symbol \"t\" and the frequency symbol \"v\" may be in-\ncluded or omitted as needed. In the case of the first of\nthose listed above we agree to write:\n\"N(S,D,t,F)\"\nfor (S,D,t,F)/A(S)S(D)\n.\nThe quantity N(x,5) is the (theoretical) radiance at X in the\ndirection E. It exists as a mathematical entity by virtue of\nthe D-additivity and D-continuity properties of & cited in\n(7) and (8) of Sec. 2.3.\nIt is instructive to disassemble the definition of\ntheoretical radiance layer by layer until the primitive con-\ncept of radiant flux $ is recovered. Thus, beginning with\n(4) and using (3)\nN(x,5) = H(x,D)/&2D\nD+{E}\nThen by means of (4) and (1) of Sec. 2.4:\nP(S,D)/A(S)Q(D)\nN(x,5)\nS+{x}\nD+{E}\nFinally, by means of (3) of Sec. 2.3 we have (in full nota-\ntion)\n[\n(S,D,t,F)\nN(x,5,t,v) = lim\nlim\nlim\nS+{x}\nF+{v}\nA(S)&(D)1(F)/\nN(S,D,t,F))\nlim\nlim\nlim\n(5)\n.\nS+{x}\nF+{v}\n1(F)\nThis is the basis for the fact that, in the last analysis, all\nradiometric concepts are reducible to the primitive physical\nconcept of radiant flux embodied by and the appropriate geo-\nmetrical and analytical notions of limit and measure. Hence\nall equations of pure and applied radiative transfer are re-\nsolvable into expressions containing only one primitive phys-\nical notion, namely $ (S,D,t,F), and auxiliary geometrical and\n*\nanalytical concepts.\n*\n[Those who desire radiant energy as the most primitive phys-\nical notion, may then start with U(S,D,T, F) where T is a fi-\nnite time interval, so that (S,D,t,F)=1im, T+{t} U (S, D, T, ,F)/1(T).\nIn Vol. I, U was taken as a primitive concept; in this and\nsubsequent volumes, U will be derived from $ as in (17) of\nSec. 2.7, e.g.]","SEC. 2.5\nRADIANCE\n33\nNow that the definition of radiance has been estab-\nlished it is an easy matter to return to the definition of\nthe phase density n(x,5,t,v) of photons in (1) of Sec. 2.2\nand by energy and dimensional arguments conclude that\nN(x,5,t,v) = hvvn(x,5,t,v)\n(5a)\nwhich together with (5) connects N(x,E,t,v) with some of the\nmore basic constructs of radiometry ($, and photons) State-\nment (5a) can be cast into terms of wavelength 1 by using the\ntransformation (32) of Sec. 2.12.\nTo gain some insight into the magnitudes of radiances\nfound in nature, we append Tables 1,2, which are constructed\nfrom the graphs in parts III, IV of [26] and which form part\nof a four-part series of compilations of sky (field) radiance\ndistributions. The skies in the present tabulation were morn-\ning (0800 hours) skies at sea level, covered 40% with scat-\ntered clouds. Two regions of the spectrum are considered:\nTable 1 gives orders of magnitude of field radiance in the\nwavelength interval [400, 500] mu, and Table 2 is for the in-\nterval [580,700] mu. The main purpose of the tables is to\ncomplete the statement: \"daylight skies (away from the sun)\nhave radiances on the order of 10n watts/ (m ² x steradian)\nwhere n = ?\" It is clear that the answer is around n= -1,0,1.\nBy way of contrast, recall that the radiance over the sun's\ndisk is on the order of 2 x 107 watts/( (m ² x steradian) as seen\njust outside the atmosphere, and for a wavelength interval\n[0,00] mu (cf., (3a) of Sec. 1.2). Hence N in the vicinity of\nthe sun runs from 10° to 107 units of radiance. The data\nwere taken June 21, 1958 in balloon flights over central Min-\nnesota. For angle conventions, see part (a) of Fig. 2.3.\nTable 1\nSample Radiances, Morning Skies\n400-500 mu, Sea level, sun zenith angle = 70°\nwatts/(m2 x steradian)\nAzimuthal\nsun's azimuth\n= 0°\n= 80°\n= 180°\nPolar 0\n0 = 0°\n1.5\n1.5\n1.5\nzenith\n0 = 45°\n5.3\n2.1\n2.0\n0 = 90°\n7.0\n2.2\n1.5","34\nRADIOMETRY AND PHOTOMETRY\nVOL. II\nTable 2\nSample Radiances, Morning Skies\n580 - 700 mu, Sea level, sun zenith angle = 70' o\nwatts/(m2 x steradian)\nAzimuthal\nsun's azimuth\nPolar 0\n= 0°\n= 80 o\n= 180\n0 = 0°\n.45\n.45\n.45\nzenith\n0 = 45° o\n3.0\n.62\n.45\n0 = 90°\n5.5\n2.0\n1.2\nRadiance Distributions\nWe have seen how the operational definition of radi-\nance leads to the theoretical radiance displayed in (4)\nThis in turn leads to the construction of a function N which\nassigns to each point x in an optical medium and direction E\nat that point, a radiance N(x, E) of the natural light field.\nN(x,5) is the number of watts of radiant flux incident per\nunit solid angle, in the direction E normal to a unit area at\nX. Implicit in the notation is the time t of the measurement\nand the frequency V of the energy passed by the filter of the\nmeter. The totality of all values N(x,5) paired to (x,5) as\nX ranges over all points of a selected optical medium X and\nas E ranges over the unit sphere (1) is called the radiance\nfunction on X X E and is denoted by \"N\". If attention is re-\nstricted to an arbitrary fixed point x and the totality of\nvalues N(x,5) are considered for all E in E, then that total-\nity of values is called the radiance distribution at x and is\ndenoted by \"N(x,.)\".\nThe radiance function is the most important radiomet-\nric function in geophysical optics and in particular, in hy-\ndrologic optics. For an exhaustive empirical study of radi-\nance distributions in a natural hydrosol, see the classic\nwork of Tyler [298]. The importance of the radiance function\nrests in the fact that from knowledge of the radiance func-\ntion alone, all other radiometric quantities are relatively\neasily calculable. This fact will become increasingly appar-\nent as the discussion of this work proceeds, and we begin be-\nlow with a first example of this fact. (See also Figs. 1.23-\n1.25)","SEC. 2.5\nRADIANCE\n35\nIrradiance from Radiance\nAs an illustration of the use of the concept of radi-\nance, and to aid the reader to fix in mind its definition, we\nshall derive the relation between irradiances of the form\nH(x,5) introduced in Sec. 2.4 and radiances (x, 5) just de-\nfined above. More detailed examples are reserved for Sec.\n2.11.\nWe begin with the empirical connection between H(S,D)\nand N(S,D) established as a matter of course in equation (2).\nIf N(S,D) is known, we can compute H(S,D) using:\nH(S,D) If N(S,D)R(D)\nIt should be recalled that D is a narrow conical set of direc-\ntions associated with the radiance meter, and that the central\ndirection of the cone is normal to the surface S.\nWe now apply this general relation to the following\nproblem, which is formulated with the aid of Fig. 2.9. A\nsurface S with inward normal is irradiated by n distinct\nsources of flux such that the i-th flux has radiance N(Si,Di)\nand is incident on the points of S through a small conical\nset Di of directions centered on direction Ei. The sets Di\nare pairwise disjoint (i.e., no two overlap) and all lie on\nthe same side of S. What is the resultant irradiance H(S,D)\nproduced by this given set of incident irradiances?\nDj\n(Si,S moved away\nfrom S for clarity)\nDi\nX\nSj\nX\nSi\nS\nX\nBi\nFIG. 2.9 Setting up the connection on going from radi-\nance to irradiance","36\nRADIOMETRY AND PHOTOMETRY\nVOL. II\nThe first step in relating H(S, D) to the n radiances\nis to observe that by successive applications of the D-addi-\ntivity property of radiant flux (equation (7) of Sec. 2.3) we\ncan write:\nH(S,D) = H(S,D1) + H(S,D2) +...+\nH(S,Dn)\n,\nwhere \"H(S,D)\" denotes the irradiance on S produced by radi-\nant flux incident within the set of directions Di. The sec-\nond step consists in using the cosine law for irradiance\n(equation (15) of Sec. 2.4) to relate H(Si,Di) and H(S,Di),\nfor i = 1,..., n. Thus:\nVi\n,\nwhere 11 v. 11 denotes the angle between the unit inward normal\nE to S and . We have chosen S small enough so that the con-\nditions of the derivation of the cosine law (15) or (16) of\nSec. 2.4, are satisfied. Furthermore, we use (4) above to\npermit slight adjustments of the choice of the Si as may be\nrequired to meet the cosine law derivation conditions without\nnoticeably changing the value of the radiance of the flux on\nSi through Di. Thus, by definition, for every i = 1,...,n:\nand this constitutes the third and final step. By assembling\nthe results of these three steps we have the desired connec-\ntion:\nH(S,D) = .{ N(S,Di) cos\n(6)\ni=1\nWhen n = 1, we have the intuitively useful special case of\n(6) :\nH(S,D) N(S',D)&(D) cos\n,\nwhere we have written \"S\" for S1, and now D = D1 in Fig.2.9.\nThe connection (6) is a useful relation in practical\nsituations where knowledge of radiance distributions is ap-\nplied to find irradiances on arbitrarily oriented surfaces.\nBy using terrestrially based coordinate systems (Sec. 2.4)\nequation (6) can be translated into a workable standard com-\nputation procedure. This task is facilitated by first estab-\nlishing the theoretical counterpart to (6). Thus, let S+{x},\nso that also Si+{x}. Then H(S,D)+H(x,D) and N(Si,Di)+N(x,Di),\naccording to (3). Equation (6) then becomes:","SEC. 2.5\nRADIANCE\n37\nn\nH(x,D) = E N(x,Di) cos re i\n(7)\nWe now apply (7) to the case where (1) is divided up into n dis-\njoint pieces Ei, and we then let the number n increase indef-\ninitely so that each E goes to zero. In this way we arrive\nat the integral counterpart to (7):\nH(x,5) =\n(x,\n(8)\nE(5)\nRecall that the symbol \"E(E)\" denotes that hemisphere of (1)\nconsisting of directions 5' which make an angle less than\nninety degrees with E. Further, E is the unit inward normal\nto the collecting surface S at X. Recall also that E.E. by\nthe discussion of 2.4, equals the cosine of the angle I be-\ntween E and 5'. Thus (8) may be rewritten in terms of v.\nBefore this can be done with complete clarity, we must express\ndo(s) in terms of polar and azimuth angles. This we shall\ndo, taking the opportunity to explicate at the same time the\nnotion of \"solid angle\".\nToward this end, let us consider a set D of directions\non the unit sphere E. Fig. 2.10 depicts a typical set D OC-\ncurring in practice, i.e., one which consists of a single\nZ\nday\ndb\nD\ny\nX\nIII\nFIG. 2.10 The unit sphere of directions as the natural\nsetting for solid angle measurements.","RADIOMETRY AND PHOTOMETRY\nVOL. II\n38\nconnected part of E. It is quite natural to characterize the\n\"amount of opening\" of the set D by specifying the amount of\narea that D occupies on the unit sphere. Thus, we denote by\n\"So(D)\" the number representative of the area of D on E. This\nis the standard definition of the measure of a set of direc-\ntions D, the one which we have been using informally up to\nthis point. (For a further discussion of solid angle measure,\nsee Note (h) to Table 3, Sec. 2.12.)\nWe can now go one step further and characterize (D)\nin terms of the polar and azimuthal angles 0 and (measured\nin radians) Clearly a small rectangular patch on (1) about\nthe point specified by (0,0) and of lateral extents da and db\nis very nearly of area da db. But since the sphere (1) has\nunit radius, db = de and da = sin 0 do. Hence:\nd.(5) = sin 0 do do\n(9)\nwhere (0,0) is associated with the direction E (see Sec. 2.4).\nFrom (9) we obtain:\nI\ns((D)\nds(5) =\nsin 0 de do\n(10)\n=\nD\nD\nIt should be clear that the radius of (1) plays no es-\nsential role in determining n(D). In general, if we write\n\"S((D)\" for A(D)/r2, , where A(D) is the area determined by the\nset D on a sphere of radius r, then equation (10) results\nonce again for so(D). We leave the ranges of integration in\n(10) undetailed, as the mode of specification of D varies\nwidely. The number (D) is customarily dimensionless. How-\never, when dimensions of So (D) are needed, the system in Table\n3 of Sec. 2.12 may be adopted. The standard unit of a solid\nangle is the steradian. It is important for a thorough under-\nstanding of solid angle, to make the distinction between the\nset D of directions and its measure (D) : D is a set of\npoints on E, s(D) is a number describing the size of that\nset.\nAs an example of the use of (10), consider the spher-\nical cap D on (1) consisting of all directions E with polar\nangles less than or equal to 0. See Fig. 2.11. Then:\n0\n2\n(D) =\nsin 0' de' do'\n=0\n= 2(1-cos 0)\n(General 0)\n(11)\n.\nThis formula is frequently used. It is also the ba-\nsis for the following well-known estimate of s((D) for small\n0. In (11) let 0 be small so that 0 2 is much smaller than","SEC. 2.5\nRADIANCE\n39\nZ\nspherical cap\n0\n8,\nO2\ny\nx\nspherical\nsegment\nspherical\nrectangle\nFIG.\n2.11 Solid angle measures of some simple shapes\n(negligible compared to) 0. Then we can approximate cos 0 by\n1-(02/2), by truncating the series expansion of cos 0 at its\nsecond term. Under this assumption, (11) becomes:\nso(D) = 2\n(Small 0)\n(12)\nFrom (11) we also obtain the solid angle measures for various\nspecial parts of E1) which can be made up from spherical caps.\nThus:\nSo(D) = 2\n(0 = /2, D is\n(13)\na hemisphere)\ns((D) = 4\n(0 = II, D is E)\n(14)\nso(D) = 2(cos 01-cos O2)\n(01 VI O2, D is a\n(15)\nspherical segment)","RADIOMETRY AND PHOTOMETRY\nVOL. II\n40\nThis last formula is, incidentally, a generalization\nof (11), for (11) is recovered by setting 01 = 0. Equation\n(15) can be used to obtain the solid angle measure of a spher-\nical rectangle bounded by two latitude circles and two longi-\ntude circles of the unit sphere. Thus, if 1 and 2 are the\nbounding meridians with 1 Q2, then the rectangle bounded by\nthem takes up the fraction (02-01)/211 of the spherical seg-\nment area bounded by latitude circles at 01 and O2. Hence\nfrom (15):\nso(D) = (02-01) (cos 01-COS 02)\n(01 O2, O2, (16)\nD is a spherical\nrectangle)\nEquation (16) is a further generalization of (11). The latter\nmay be obtained by assuming 2 = 01 + 2 and 01 = 0. Of\ncourse (16) can also be obtained by direct appeal to (10).\nWe now return from the preceding digression on solid\nangles and conclude our discussion of the computation of ir-\nradiance H(x,E), given a radiance distribution N(x,5). It\nremains to cast (8) into 0,0 notation. Using (8), and (13)\nof Sec. 2.4 we have:\n(N(x,e',+')\ncos I sin e' de' do',\n(17)\nH(x,0,0) =\nE(0,0)\nwhere \"E(0,0)\" is simply another name we shall use for (E),\nwhen (0,0) is explicitly associated with E. Further, v is\nthe angle between E and 5', where the latter direction is as-\nsociated with (0',0'). Now, cos il can be represented by\nmeans of (0,0) and (0',0') as follows. Recall first of all\nfrom Sec. 2.4 that if E is a unit vector, then:\nE = i cos U1 + j cos V2 + k cos v 3\n,\nwhere and V3 are the angles between E and the unit\nvectors i, j, and k, respectively. Once again, now for E':\nE' = i cos V1' + j cos V2' + k cos V3'\nThen by the observations in Sec. 2.4:\ncos v= E.E'= cos 1COS V1'+ COS V2COS V2 + cos V3COS V3' (18)\nBy means of Fig. 2.4, or Fig. 2.10, we see that:\ncos v = sin 0 cos\ncos N2 = sin 0 sin\n(19)\ncos v3 = COS 0","SEC. 2.5\nRADIANCE\n41\nThere are three precisely similar equations for the v as-\nsociated with 5'. In this way (17) is cast into a well-de-\nfined analytical formula involving only 0,0 and 0' ,'''' in an\nintegration over E(0,0). This completes the detailed unfold-\ning of equation (8). The result is a formula often used in\npractice to compute H(x,0,0), given N(x, ) ) at point X. We\nwill return to illustrate equation (17), and other formulas,\nin Sec. 2.11. For the present we continue with the develop-\nment of further properties of the concept of radiance.\nRadiance from Irradiance\nAs a further illustration of the interconnections be-\ntween the concepts of radiance and irradiance we now reverse\nthe considerations of the preceding discussion and show that\nfrom a given irradiance distribution at a point x in an opti-\ncal medium, one can compute the radiance distribution at that\npoint. As a consequence of this fact and the results of the\npreceding discussion, we see that radiance and irradiance dis-\ntributions share equal informational content. In addition to\nthis theoretical consequence, there is also one of experimen-\ntal import: it is possible, at least in principle, to measure\nirradiance distributions in natural hydrosols and aerosols\nand from this data deduce complete information about radiance\ndistributions. In other words, one can in principle complete-\n1y document the light fields in natural optical media solely\nby means of irradiance distributions.\nWe begin the illustration with the simplest possible\ncase: we are given that the irradiance distribution H(x,.)\nat point x is generated by a radiance distribution N(x,.)\nwhich is of uniform radiance N over a small conical set D' of\ndirections of solid angle magnitude s((') with central direc-\ntion 5' and with N(x,5') zero for all other directions. It\nis required to find N. Now from (8) we have very closely:\nH(x,5) = NEE'S(D')\n,\nwhence:\nN H(x,5)/5-5'0(D')\nwhere E is some specifically chosen vector such that .5'>0.\nSuppose next that the given irradiance distribution\nis generated by a radiance distribution which is of uniform\nmagnitude N1 over a narrow set D1' of directions with central\ndirection 51' and of uniform magnitude N2 over a narrow set\nD2' of directions with central direction E2' and such that\nD1 1 and D2' are disjoint and N(x,5') is zero for all other\ndirections. From (8) we have now:\nH(x,5) = N15.51'S (D1')x(5,51') + E2 (D2 X(5,52')\n(20)\nwhere X is a function with the property that X(5,5') = 1 or 0\naccording as E' is or is not in E(E), , respectively, and where\nE is any direction in E.","VOL. II\nRADIOMETRY AND PHOTOMETRY\n42\nNow we may choose E at will from a large collection\nof possibilities, and use the given values H(x,5) to try to\ndetermine the two radiances N1 and N2. Clearly we must gen-\nerally choose two directions E1 and E2 in order to determine\nN1 and N2. This is readily seen if we write:\n\"Cji\" for\nfor each i = 1,2, and j = 1,2, and furthermore, if we write:\n\"Hi\" for H(x,5j)\nfor each i = 1,2. Then equation (20) yields, for E=E1 and\nE=E2 the two equations:\nH1 = N1C11 + N2C21\nH2 = N1C12 + N2C22\nIf we write:\nC11 C12\n\"C\"\nfor\nC21\nC22\nthen the preceding set of equations can be written:\n(H1,H2) = (N1,N2)C\n(21)\nor, more succinctly, as:\n(22)\nH=NC\nWe have written:\n\"H\" for (H1,H2)\nand\n\"N\" for (N1,N2)\nThe solution of Equation (22) is that N for which\nH1\nC21\nN1 =\nH2\nC22\n(23)\nN2 = C11 H1 A-1\nC12\nH2\nwhere we have written:","SEC. 2.5\nRADIANCE\n43\n\"A\"\nfor\nC11C22- C12C21\n,\nand where the four Cij are such that A + 0. This solution\nmay be put into the form:\nN = HC- 1\n(24)\n,\nwhere\nC22\n-C12\nC-1 =\n-1\n(25)\n-C21\nC11\nThe pattern is now clear as to the means of obtaining\na radiance distribution from the generated irradiance distri-\nbution. For, generalizing the two simple cases just consid-\nered, we now suppose that a given irradiance distribution at\npoint x is generated by a radiance distribution at x which is\nuniform and of magnitude N over each of n narrow sets Di' of\ndirections such that Di' and Dj' are disjoint and with central\ndirection 5i' for each Di', i=1,...,n. Hence the set {D1',\nD2\nDn of subsets of (1) is an arbitrary partition of 3\ninto narrow bundles of directions. From (8) we have:\n(26)\n=\n,\nwhere E is any direction and X(E,5i') has the same meaning as\nbefore for the case n = 2.\nWe now choose n directions Ei, i= 1,2,...,n and write:\n\"Cji\" and Hi exactly as before, but now with i and j ranging\nover the general finite set (1,2,.) n} of integers. With\nthis notational convention (26) becomes:\nn\n(27)\nWriting:\n\"H\"\nfor\n(H1,...,\nHn)\n\"N\"\nfor\nNn)\nC11\nC12\nCin\n\"C\"\nfor\nC21\nC22\nC2n\nCn1\nCn 2\nCnn","44\nRADIOMETRY AND PHOTOMETRY\nVOL. II\nwe can then cast (27) into matrix form:\nH = NC\n(28)\nThe solution of (28) is given by:\nN = HC-1\n(29)\nwhere C-1 , the inverse matrix of C, generally exists upon\nsuitable choice of the\nThe proof of the existence of the general continuous\ncounterpart to C-1 is given in Example 15 of Sec. 2.11.\nThere the full equivalence of H(x, and N(x, is established.\nThe rigorous proof of the equivalence requires relatively ad-\nvanced concepts and for that reason is deferred to Sec. 2.11.\nHowever, the present discussion has been designed so that the\npractical details involved in the determination of N by H re-\nquire no tools beyond those of the elementary theory of al-\ngebraic equations.\nAs a result of the preceding discussion leading to\n(29), we can view in a new light the observation that \"radi-\nance is the most basic of radiometric concepts\". The radiance\nconcept is most basic in the sense that from it all other ra-\ndiometric quantities can be most conveniently derived; it is\nnot \"most basic\" in the sense that there is only a one-way\ncomputational path from it to every other radiometric quan-\ntity. This brings up the interesting question of: just which\nof the radiometric quantities discussed so far have informa-\ntional content equivalent to radiance? and: just what, in\nthe last analysis, characterizes a radiometric concept which\nhas equivalent informational content to radiance? These ques-\ntions will be briefly considered in Example 15 of Sec. 2.11.\nField Radiance vs Surface Radiance\nThere is a distinction that can be made in practice\nbetween two types of radiance, a distinction which is analo-\ngous to that made in Sec. 2.4 between irradiance and radiant\nemittance. This distinction is depicted with the help of Fig.\n2.12 which shows radiant flux across an hypothetical surface\nS in the indicated direction and within a narrow conical set\nD of directions around a direction E normal to S.\nNow, corresponding to the conceptual distinction es-\ntablished between W(S,D) and H(S,D) in (17) and (18) of Sec.\n2. 4, we can write:\n\"N+(S,D)\"\n11\nfor\nW(S,D)/S2D\n(30)\nand\n\"N\"(S,D)\" 11\nfor\nH(S,D)/S2D)\n(31)\n.","SEC. 2.5\nRADIANCE\n45\nN\n-\nS\n+\nN\nD\nFIG. 2.12 Conceptual distinction between field radiance\nN and surface radiance N+\nWe call N (S, D) the (empirical) field radiance, and N+ (S,D)\nthe (empirical) surface radiance (or specific radiance, or\nspecific intensity). It is quite clear that, in the general\ncontext of Fig. 2.12:\nN (S,D) = N (S,D)\n(32)\nDespite the numerical equality, the conceptual distinction\nbetween field and surface radiance is useful to maintain. In-\ndeed, some need for a conceptual distinction inevitably forces\nitself on the attention of careful students of applied radi-\native transfer theory where on the one hand emitting surfaces,\nreal or hypothetical, are characterized most naturally by sur-\nface radiance, and where measurements obviously result in\nfield radiances. The term \"surface\" in \"surface radiance\" is\na vestige of the days when surface radiance was associated\nwith the radiant emittance of real surfaces enclosing sources\nof radiant energy. The present interpretation of \"surface\",\nhowever, includes the possibility of hypothetical surfaces\nanywhere in an optical medium. The term \"field\" in \"field\nradiance\" denotes the sense of \"field of view\". In practice,\nwhenever possible, one of these two interpretations of radi-\nance is usually fixed and agreed upon throughout a given dis-\ncussion. Thus, we can omit the \"++\" (or \"-\") superscript from\n\"N\" when the type of radiance is understood.","46\nRADIOMETRY AND PHOTOMETRY\nVOL. II\n2.6\nAn Invariance Property of Radiance\nIn this section we shall discuss a property of the\nconcept of radiance which is of central importance from the\npoint of view of radiative transfer phenomena. This is the\nso-called n2 law of radiance which states that the quotient\nN/n2 does not change along a path of sight through a trans-\nparent medium in which there is a generally variable index of\nrefraction n. The importance of this law rests in the base\nline it establishes for comparison of the behavior of N/n2\nalong lines of sight in non-transparent media, i.e., media\nthat scatter and absorb radiant energy such as the atmosphere\nand the seas, and other natural optical media. The law also\nindicates a measure of success in our attempt to simulate the\nsensation of brightness by means of a simply defined radiomet-\nric concept. For it is a matter of daily experience that as\none approaches or recedes from an object along a line of sight\nthrough a very clear homogeneous stretch of atmosphere (so\nthat n is constant), the \"brightness\" of the object does not\nappear to change. For example, the brightness of a small part\nof a desk blotter does not change as we move away from it in\na room, keeping attention constantly directed toward the patch.\nof course, the total flux entering the eye and originating\nfrom the patch falls off rapidly with distance (very nearly\nas the square of the distance, as we shall eventually show)\nhowever, the brightness of the patch does not change with the\nobserver's distance. This phenomenon is reproduced in the\nspecial form of the -law where n is constant over the path\nof sight. We now show how the n2 law for radiance follows\nfrom the definition of radiance. We shall divide the discus-\nsion into two main parts. The first part considers the impor-\ntant case in which n is constant along the path of sight.\nThe second part considers the general case of a variable in-\ndex of refraction.\nThe Radiance-Invariance Law\nWe begin the derivation of the n2 law for the special\ncase where n is constant along a line of sight through a trans\nparent optical medium. This special case is of sufficient im-\nportance to be given a special name, the radiance-invariance\nlaw. We shall prove the radiance invariance law twice: first\nin as simple a way as possible so as to reveal the geometrical\nessence of the law; then the derivation will be repeated in\nslightly more detail, filling in steps and giving more expla-\nnations on the way.\nThe setting for the simple derivation is shown in Fig.\n2.13. Two holes S and S' of arbitrary shape and about the\nsize of collecting surfaces used in radiant flux meters are\ncut out of two large pieces of opaque cardboard. The pieces\nare then mounted so that they are parallel and separated a\ndistance r which is large compared with the linear dimensions\nof the holes. Light is then directed through S which flows\nalong straight lines in the transparent space between the\ncardboards and then on through S'. The holes are arranged so","SEC. 2.6\nRADIANCE INVARIANCE\n47\nS\ns'\nr\nFIG. 2.13 Illustrating the invariance of the radiance\nof a narrow bundle of light rays in a vacuum.\nthat for the most part, the lines of flux through both open-\nings are nearly perpendicular to the planes of the holes.\nThe observation is now made that the amount P of radiant flux\nacross S, associated with the common bundle of lines of flux\nthrough S and S' , is the same as that across S' . Thus the\nsame number of lines of flux go through both S and S' . With\nthis in mind we consider the number:\nP\nA (S) A (S') 1\nr 2\nin two ways. First as:\nP(S,D)\nA(S) A(S')\nr2\nand then as:\nP(S',D') ,\nA(S) A(S')\nr2\nIn the first case we observe that A(S')/r2 is essentially the\nsolid angle s((D) subtended by S' at each point of S. In the","48\nRADIOMETRY AND PHOTOMETRY\nVOL. II\nsecond case A(S)/r2 is the solid angle So(D') subtended by S\nat each point of S' . \"P(S,D)' \" and \"P(S',D')\" , both denote the\ncommon radiant flux P, , but now in an obviously suggestive way\nby recalling the meanings of S, S' D, D'. Therefore we have:\nN(S,D) N(S',D')\n(1)\nThis is the empirical form of the radiance - invariance law.\nThe form of the law is \"empirical\" because it is couched in\nterms of empirical radiances - - radiances directly measurable\nby real radiance meters.\nA somewhat more detailed derivation of the radiance-\ninvariance law will now be given. Part (b) of Fig. . 2.14 de-\n-\npicts a radiance meter G directed at a surface S at the end\nNr\nPr\n1\nSo\n(a)\nSr\nAo\nNo\nNr\n(b)\nS\nG\nNr\nSr\nPo\n,\n(c)\nAr\nI\nSr\nSr\nNr\n,\n(d)\nX\ny\nFIG. 2.14 A more detailed, and operational, study of\nthe radiance - invariance law","SEC. 2.6\nRADIANCE INVARIANCE\n49\nof a clear path of sight of length r. The surface S is nor-\nmal to the line of sight and has a uniform surface radiance\nNo over its extent in the direction of G. The meter G has\nits field of view completely filled by S. The resultant ra-\ndiance reading is Nr. We will show that, under these condi-\ntions we have, for every r, No = Nr. The basic idea of the\nproof is to examine the same diagram (b) from two distinct\npoints of view. These points of view are schematically de-\npicted in parts (a) and (c) of Fig. 2.14. We consider part\n(a) first. Here the radiance meter's reading Nr is seen to\nbe the quotient Pr/Aofo, where Pr is the radiant flux origin- -\nating on S and incident on the collecting surface of area Ao,\nand which has funneled through the solid angle of magnitude\nso, defined as shown. On the other hand, part (c) views this\nflux as an amount PO sent to area Ao in G and as emitted from\nthose points of Sr within G's field of view. The emitting\nsurface Sr comprising G's field of view is of variable magni-\ntude Ar, and the emitted flux from each point of Sr is within\nthe bundle of directions of solid angle defined as shown.\nHence, No is the quotient Po/Arsr. The definitions of Po and\nPr imply at once (as in the previous proof) that:\nPo = Pr\nFurther, we have the geometric observation that:\nA_A\nr\n= r r = So o\nr2\nOn the basis of these two facts, we see that, by virtue of\nthe defining equations:\n= Pr/Aon\nN = PO/ATST\nwe have:\nNo = N r\n(2)\nObserve how we have implicitly used the distinction between\nfield and surface radiance and the connection (32) of Sec.\n2.5 in order to interpret N operationally at surface S,\nwhich then is Nr for r = 0.\nThe Operational Meaning of Surface Radiance\nOne final matter must be resolved before the radiance-\ninvariance law is fully established. This is the matter of\nassigning a meaning to the surface radiance of a surface at a\npoint X in a direction other than the normal direction to S\nat X. Observe that this problem does not arise with field ra-\ndiance, since field radiance is defined by the convention of","VOL. II\nRADIOMETRY AND PHOTOMETRY\n50\nusing the fixed collecting area of a radiance meter, which is\nassembled so that it is perpendicular to the axis of the me-\nter. In the case of surfaces such as a portion of the earth's\nsurface, a desk top or a wall, or a given cloud boundary, how\nshall we assign a surface radiance to radiant flux leaving a\npoint on such surfaces in directions other than the perpendi-\ncular direction to the surface at that point? The path to\nthe answer is guided by the manner in which such surface-radi-\nance information is first of all to be interpreted and second-\nly how it is to be used. In the first case we really have\nvery little choice as to the manner of interpretation of the\nradiance information. We have already committed ourselves to\nwork solely with operational concepts: measurable fluxes,\nareas and solid angles. Hence, if we heard someone say: \"The\nsurface radiance of flux of wavelength 550 mu at point x on\nwall A is 2 watts/ (m 2 x steradian) in every direction 30° from\nthe normal to A at x\", our first impulse, after this data has\nbeen mentally assimilated, would be to attempt a verification\nby directing a radiance meter toward x on A so that the axis\nof the meter makes an angle of 30° with the normal to A at X.\nIf we were challenged to defend such a procedure, we would\ncite the argument leading to the radiance-invariance law a-\nbove. However, if the challenger were particularly tenacious,\nhe would point out that the argument establishing the law\nholds only for directions of sight normal to A at X. At this\njuncture we must concede that he is right.\nThe preceding objection to our justification for as-\nsigning an operational meaning to oblique surface radiance is\nlogically unassailable. However, we have one more matter to\nconsider which will add strength to the justification. We\nnow consider the second aspect of the question posed above,\nnamely: how is the information of oblique surface radiance\nto be used? The answer, based on considerable practical and\ntheoretical experience, is that such oblique surface radiance\ninformation is to be used to calculate irradiances, scalar,\nvector, or of the ordinary variety, at points optically acces-\nsible to the surface which emits the surface radiance. Or\nagain, the surface radiance information will be used to obtain\npath functions, and various attenuation functions used in hy-\ndrologic optics or meteorologic optics and these determina-\ntions will be made at points optically accessible to the sur-\nface. The pertinent fact that emerges as these uses of the\nsurface radiance information are paraded before the mind's\neye is the following: without exception, the information\nused can always be in the form of field radiance values of\nthe radiometric field in the direction of point x on surface\nA. In short, surface radiance per se while of great concep-\ntual and theoretical worth, is never really used in actual\npractical calculations--only the directly observable field ra-\ndiance values are used in such calculations. We are there-\nfore motivated to assign an operational value of surface ra-\ndiance to a surface A at x in the general outward direction\nE by means of the corresponding field radiance reading N(y,5)\nobtained when the radiance meter is at some point y and is\ndirected at x 80 that the unit inward normal to the collecting\nsurface of the meter is E. The point y is to be anywhere a-\nlong a clear path of sight from X in the direction E 80 that\nthe radiance-invariance law holds.","SEC. 2.6\nRADIANCE INVARIANCE\n51\nFor conceptual definiteness in the preceding conven-\ntion, one can imagine an hypothetical surface Sr normal to 5\nas in (d) of Fig. 2.14, which is assigned the surface radiance\nN(x,5) (as No in the derivation of the radiance-invariance\nlaw). Now while the radiance-invariance law allows us to con-\nclude that the radiance reading will remain unchanged as dis-\ntance r varies from 0 at Sr to larger values from Sr, there\nstill is a conceptual gap that must be filled between the sur-\nface radiance of Sr and the surface radiance of Sr', the pro-\njection of Sr onto the oblique surface under consideration\n(as, e.g., A above) And this gap, we have agreed, is to be\nfilled by means of the preceding convention. Equation (2)\nand every result deduced from it, shall henceforth be inter-\npreted with this convention implicitly understood.\nThe n2-Law for Radiance\nThe intuitive basis for the n 2 - law, to which we now\nturn, becomes clear upon consideration of Fig. 2.13. This\nfigure shows a narrow bundle of lines of flux coursing through\nempty space. The two holes in the cardboard arrangement used\nabove were so much inessential material scaffolding which can\nbe removed now that the idea of the derivation has been ex-\nplained. What is left after this is done is the concept of a\nnarrow bundle of lines of flux coursing through space in such\na way that at each section the product AS of the normal cross\nsectional area A of the bundle and solid angle So of the bundle\nis a fixed quantity. This invariance of AS is a purely geo-\nmetric concept. Physical considerations enter subsequently\nat the point where we assert the invariance of the radiant\nflux through a variable section of the bundle of lines of\nflux. By combining these physical and geometric considera-\ntions, the desired radiance-invariance law is obtained for a\nlight beam in a vacuum. We now inquire: how are these phys\nical and geometrical considerations to be modified in the case\nof a light beam coursing through matter such as air or water?\nThe physical considerations governing the radiant flux content\nof the beam must take into account the scattering and absorp-\ntion phenomena all along the extent of the beam. These phe-\nnomena affect the radiant flux content of the beam in complex\nand subtle ways. The full study of these effects is reserved\nfor the theory of Part Two of this work. We shall limit our\npresent inquiry to sets of adjoining transparent media. Any\nalterations of the radiant flux content of the beam are then\nlimited to the interfaces of these media. If we now repeat\nour query above for the case of contiguous transparent media\nwhich are distinguished from each other only by their various\nindices of refraction, then the answer to the query is given\nin the form of the n2-1aw for radiance. The derivation of\nthis law for the simplest case will now be given.\nFigure 2.15 depicts a beam of radiant flux lines inci-\ndent on the interface Y between two transparent optical media\nX1 and X2. Let us agree that the central axis of the beam is\nnormally incident on the interface, that it arrives from med-\nium X1, and that the beam passes on through the interface Y\nand enters medium X2. For example, X1 may be a part of the","RADIOMETRY AND PHOTOMETRY\nVOL. II\n52\n01\n01\nX\nY\ny\nX2\ne\nO2\nO2\n2\nFIG. 2.15 When a bundle of light rays is suddenly\nsqueezed into a narrower bundle without changing its flux\ncontent - - the radiance of the bundle increases proportionately.\nThis essentially is what happens, e.g., at the air-water sur-\nface of natural hydrosols (flux losses to one side).\natmosphere, and X2 a part of the hydrosphere, so that Y is\nthe air-water interface. In general, X1 has some index of re-\nfraction n1 and X2 an index of refraction n2 in the immediate\nneighborhood of the interface. Our current goal is to relate\nthe radiance of the beam in X1 to the radiance of the beam in\nX2 in the immediate vicinity of Y. Before going into the de-\ntails of the derivation it is instructive to anticipate the\nresult intuitively. We ask: which of the three main quanti-\nties P, A or So of the definition of radiance N will change\nfrom one side of Y to the other? Clearly, the radiant flux\ncontent P will be affected to some extent by reflection of\nsome of the lines of flux back into X1 The area A of the\nbeam will remain essentially unchanged arbitrarily close to\neach side of Y. Finally, the solid angle magnitude So of the\nbeam will change from one side to another because of refrac-\ntion of the lines of flux transmitted across the surface Y.\nFor the moment we ignore the change of P, this change yield-\ning a relatively small change in N and one which will even-\ntually vanish as the derivation proceeds to its final stages.\nHence the principal change in N that is wrought on the radiant\nflux is traceable to the abrupt change in the So of the beam as\nit crosses Y. For example, if X1 and X 2 are respectively, air\nand water, then as a glance at Fig. 2.15 would show, S1, the\nsolid angle magnitude of the beam in air, is greater than S62,\nthe solid angle magnitude of the transmitted beam in water.\nSince P and A are essentially unchanged during the passage of","SEC. 2.6\nRADIANCE INVARIANCE\n53\nthe flux across Y, the radiance N2 of the beam in water ex- -\nceeds its radiance in air. This yields the general observa -\ntion that, aside from reflection losses, as a narrow beam of\nradiant flux goes from a medium of smaller to greater index\nof refraction, its radiance changes from a smaller to a great\ner amount. The reverse change occurs for a reverse traversal.\nThe n2-law gives a precise quantitative description of this\nobservation.\nNow to the particulars. At each point y of the inter-\nface Y within the beam, there is a cone of incident lines of\nflux on y. This cone has a small half angle 01. The associ- -\nated solid angle magnitude So1 of the incident cone is, by (12)\nof Sec. 2.5, 012. Similarly 2 is the solid angle magni-\ntude S62 of the beam of transmitted lines of flux. Snell's\nlaw ((5) of Sec. 12.1) gives the connection between 01 and\nO 2 :\nn1 sin 0 1 = n2 sin O2\n.\nFor small 01 and O2, this becomes, very nearly:\nSquaring each side, we obtain:\nMultiplying each side by TT we obtain:\n=\nThat is:\n(3)\nNow starting with the equal irradiances H1 and H2 of the beam\non each side of Y, we have, for the reasons discussed above:\nP1 = H1 = H2 = P2\nA1\nWe now use (3) with this to get:\nThat is:\n(4)\nn12\nThis is the desired form of the n2-law - for radiance.","54\nRADIOMETRY AND PHOTOMETRY\nVOL. II\nIt is now an easy matter to gradually generalize (4)\nto the following more general settings: (a) passage through\nan arbitrary finite number of transparent contiguous media\nand going on to the limit of continuously varying n; (b)\noblique rather than normal incidence of the beam on Y; (c)\ninclusion of a transmittance factor to allow for scattering\nand absorption losses from P. Generalizations (a) and (b)\nresult in no change in the form of (4). Generalization (c)\nresults in a multiplicative factor T included on the left\nside of (4). This factor will be considered at great length\nin Sec. 3.10 in the discussions of beam transmittance. Spe-\ncific suggestions for these generalizations are given in Sec.\n12 of Ref. [251]. See also [98]. Henceforth, whenever radi-\nances are related within media of differing indices of refrao\ntion, it will be understood that N/n2 rather than N will be\nused, even though \"N\" only appears in the equations.\n2.7 Scalar Irradiance, Radiant Energy, and Related Concepts\nThe radiometric concepts studied in this section are\nthose of scalar irradiance, radiant energy, and related radi-\nometric concepts. The first of these concepts is designed to\nquantitatively describe the volume density of radiant energy\nin a way which is amenable to operational methods of deter-\nmination. In addition to the notion of scalar irradiance, we\nshall develop in this section several closely related notions\nwhich together with scalar irradiance comprise a set of useful\nmeasures of the volume density of radiant energy. The first\nof these is radiant density.\nRadiant Density\nThe notion of radiant density is one of several con-\ncepts designed to give a measure of the radiant energy per\nunit volume at a point. Consider a steady beam of radiant\nflux normally incident on surface S at point x at time t, as\nshown in Fig. 2.16. Let the field radiance of the beam at\nthis instant be N, its cross sectional area be A, and its\nsolid angle be S. The amount of radiant flux incident on S\nat time t is then NAS. An instant t later, the flux of the\nbeam will have moved on a distance r = vt, and the flux will\nhave swept out a cylindrical volume of magnitude V = Avt.\nDuring this time the beam has been steadily pouring an amount\nof radiant energy into the volume at the rate of NAS watts.\nHence the radiant energy content of the beam is NASt, and its\naverage content per unit volume is NASt/Avt = NS/V.\nSuppose that point X were simultaneously irradiated\nat time t by an arbitrary finite number of narrow beams of\nradiance Ni, i= 1,\nn, and corresponding solid angles Si.\nThen the radiant energy u(x,t) per unit volume at X is given\nat time t, by means of the D-additivity of (equation (7),\nSec. 2.3) :","SEC. 2.7\nRELATED RADIOMETRIC CONCEPTS\n55\narea A\nN\nSb\nX\nS\nFIG. 2.16 Setting up the connection between radiance\nand radiant density\nn\nu(x,t) =\n(1)\ni=1\nThe transition to the continuous case is immediate. Toward\nthis end, let us continue to write \"u(x,t)\" for the radiant\ndensity, i.e., we shall also write:\n1\n\"u(x,t)\"\nfor\nN(x,5,t) dn(E) .\n(2)\nv(x,t)\nThe units of u(x,t) are joules/m3. . We may use either the\nfield or surface interpretation of radiance in this defini-\ntion.\nScalar Irradiance\nLet us go on to write:\n\"h(x,t)\"\nfor\nN(x, , E,t) ds(s)\n(3)\n(1)","RADIOMETRY AND PHOTOMETRY\nVOL. II\n56\nh(x,t) is the scalar irradiance at x and time t. The field\nradiance interpretation of N is most often used in (3), and\nthis interpretation will be in force unless specifically\nnoted otherwise. The reason for singling out h(x,t) for spec-\nial consideration will be made clear in a moment. For the\npresent it suffices to note that in general:\nu(x,t)v(x,t) = h(x,t)\n(4)\nBy virtue of (3) it follows that in this equation the field\ninterpretation of u(x,t) is to be understood, and that while\nthe units of u(x,t) are joules/m3, those of h(x,t) are watts/m².\nHence, the term \"irradiance\" in the name \"scalar irradiance\"\nis appropriate. The reason for the modifier \"scalar\" will\nalso become clear subsequently after vector irradiance has\nbeen defined in (2) of Sec. 2.8. A generalization of (3) is\nobtained by replacing E by a subset D of E. In that case we\nwould write:\n/\nN(x,E,t) do(E)\n\"h(x,D,t)\"\nfor\nD\nThe radiant density associated with h(x,D,t) is u (x,D,t) and\n(4) holds for these two quantities.\nSpherical Irradiance\nWe shall now show why scalar irradiance is singled\nout as an alternate (and an actually preferred) description\nof the radiant density at a point in a radiant flux field.\nConsider the light field at a point x in a natural optical\nmedium at time t. Let N(x, be the radiance distribution at\nX. Now imagine a small spherical collecting surface S of ra-\ndius r in the field so that its center is at X. We then ask:\nwhat is the average amount of radiant flux incident per unit\narea over S?\nTo answer this question it is useful to conceptually\ndecompose the great number of radiant flux streams at X into\na discrete set of flows. Two such flows are shown in Fig.\n2.17. The lines of flux of one of these flows along the di-\nrection Ei have been fitted with little direction cones of\nsolid angle magnitude Si. Suppose the radiance at X in the\ndirection Ei is Ni. Then the irradiance at X on a plane nor-\nmal to Ei is Niki. If the sphere is small, say the size of a\nping pong ball, then for most natural light fields in the air\nand sea, Ni will not vary in the region of space taken up by\nthe volume of the sphere. From this we see that we can treat\nthe radiance function N as a constant with respect to loca-\ntion in the vicinity of the sphere and of value Ni for the\ndirection Si. It follows that the amount of radiant flux in-\ncident on the sphere contributed by the stream of flux in the\ndirection i is (nifi)nr2. This estimate is based on the","SEC. 2.7\nRELATED RADIOMETRIC CONCEPTS\n57\ns\nS\ns\nS\ni\ny\nr\nFIG.\n2.17 Computing the radiant flux intercepted by\na\nspherical collector in a general light field\nassumption that the amount of flux of a narrow beam intercept-\ned by a curved hemispherical surface is the same as the amount\nintercepted by the great-circle area associated with the hemi-\nsphere. The assumption is rigorously defensible for trans-\nparent media using the concepts of vector analysis and Stokes\nTheorem. For the present the reader's intuition will readily\nallow this assumption to stand even for the case of turbid\nmedia as long as r is kept very small. The \"line of flux\" in-\nterpretation will help the intuition considerably in this mat-\nter.\nThe main task in answering the above question has now\nbeen dispatched. It remains only to add up all the contribu-\ntions by the various beams of flux, using as justification\nEquation (7) of Sec. 2.3. The result is","RADIOMETRY AND PHOTOMETRY\nVOL. II\n58\nn\n{\n2\nnr\ni\ni=1\nThe average radiant flux per unit area of the sphere S is\nthen obtained by dividing this quantity by 4wr2 Let us des-\nignate this average by writing:\n1\n\"h4\"(x,t)\"\nfor\n(5)\n4\ni\ni=1\nand agree to call it spherical irradiance. We shall retain\nthis terminology and notation for the continuous formulation.\nThat is, we shall write:\n1/h(x,t)\n\"h4\" (x,t)\"\nfor\n(6)\nDefinition (6) is the basis for an operational determination\nof scalar irradiance using a spherical collecting surface S.\nFor the average radiant flux per unit area on S is readily\nmeasurable and this amount differs multiplicatively from\nu(x, t) by a fixed numerical factor. Hence, by only slight\nchanges in optical design, the same photoelectric devices\nused to determine H and N can be directed to obtain scalar\nirradiance h. Therefore it is spherical irradiance or scalar\nirradiance which is directly measurable by photoelectric de-\nvices. The concept of radiant density u(x,t) is by way of\ncontrast a theoretical concept related to the empirically-\nbased concept h (x, t) by means of (4).\nHemispherical Irradiance\nOne of the most useful mathematical models of light\nfields in natural waters is the exact two-flow model to be\nconsidered in detail in Chapter 8. A radiometric concept\nwhich arises in that theory, and one which also has been found\nof intrinsic interest to experimenters, is the concept of hemi-\nspherical scalar irradiance. We now discuss this concept.\nFigure 2.18 (a) depicts a small spherical collecting\nsurface S with center X which is exposed to flux from only\none hemisphere of E. Let N(x, ) be the radiance distribution\nat X. Let us say that light is incident on the sphere in the\ndirection of E(E). We ask: what is the average amount of ra-\ndiant flux incident per unit area over S? Clearly every point\nof S is in principle exposed to the light field over E(E).\nFig. 2.18 (b) shows how an obliquely incident beam with a di-\nrection in (E) can come close to illuminating the \"north\npole\" of the little spherical surface. If we divide up E(E)\ninto pieces analogously to the manner used in deriving the ex-\npressions above for spherical and scalar irradiance, then it\nbecomes clear that the integral of N(x, ) over (1) (5) yields\nthe appropriate scalar or spherical irradiance component.","SEC. 2.7\nRELATED RADIOMETRIC CONCEPTS\n59\n(a)\n(b)\n(E)\n(1))\nE\nFIG. 2.18 Details for a shielded spherical radiant flux\ncollector\nThus, using field radiance let us write:\n\"h(x,5,t)\"\nfor\nN(x,5',t) d2(5')\n(7)\nE(E)\nand analogously, we write:\n1/1\n1\n\"h4\"(x,5,t)\"\nfor\nN(x,5',t) do(E')\n(8)\nE(E)\nWe call h4t(x,E,t) the hemispherical irradiance at x, over\nthe hemisphere E(E), at time t. Further, h(x,e,t) is the as-\nsociated hemispherical scalar irradiance. It is clearly a\nspecial case of h (x, ,D,t) defined after (3) above. Methods of\nmeasuring hemispherical irradiances will be discussed in Chap-\nter 13. It follows immediately from (3) and (7) that:\nh(x,t)h(x,,t) + h(x,-5,t)\n(9)\nAn analogous connection to that displayed in (9) also holds\nbetween h4m(x,+5,t) and h4t((x,t). The introduction of\nh4 T (x,E,t) into the family of radiometric concepts is motivated\nexactly for the empirical reasons that motivated the intro-\nduction of its full spherical companion h4m(x,t).","60\nRADIOMETRY AND PHOTOMETRY\nVOL. II\nWhen we are working in stratified light fields (Sec.\n2.4) then it is possible to drop without loss of generality\nthe \"x\" and \"y\" coordinate symbols from the notation and re-\ntain only the depth coordinate symbol \"z\" in the notation.\nIn such contexts we agree to write:\n\"h(z,e,t)\" or \"h(z,0,4,t)\" for h(x,E,t)\n(10)\n.\nIn particular, if E is k or -k, which occurs in the important\ncase of the two-flow theory (Sec. 8.3), then we agree further\nto write:\n\"h(z,+,t)\" for h(z,+k,t)\n(11)\n,\nwhere we read upper signs together and then lower signs to-\ngether to obtain two separate definitions. As usual, when\nthe light field does not appreciably change in time, or when\ntime is understood, we shall drop \"t\" from the notation. Ap-\nplications of these concepts are taken up in Sec. 13.9.\nRadiant Energy over Space\nThe discussion of this section is now continued by\nofficially noting two interpretations of the term \"radiant\nenergy\". The first interpretation centers on the simple con-\nnection that exists between scalar irradiance and radiant en-\nergy. Suppose X is a subset of an optical medium over which\nat time t there is defined a scalar irradiance function h for\na given frequency V. Let \"U(X,t)\" denote the radiant energy\ncontent of X at time t. That is, by the definition of u(x,t),\nwe agree to write:\n(12)\n\"U(X,t)\"\nfor\nu(x,t) dV(x)\nX\nand from (4) :\n(h(x,t)/v(x,t)) dV(x)\n(13)\nU(X,t)\n=\nwhere \"V\" is the volume measure of the optical medium. As a\nspecial case, if v(x,t) and u(x,t) are independent of x and t\nthen (13) becomes:\nU(X) = (h/v) V(X)\n(14)\nwhere, for this case, we have written:\n\"U(X)\" for U(X,t)\n\"h\"\nfor h(x,t)\n\"V\" for v(x,t)","SEC. 2.7\nRELATED RADIOMETRIC CONCEPTS\n61\nIt is clear from (12) that U(,t) is V-additive and V-contin-\nuous. That is, for every two disjoint parts X1 and X2 of an\noptical medium:\nU(X1,t) + U(X2,t) = U(X1 U X2 t)\n(15)\n,\nand for every X and t:\nIf V(X) = 0, then U(X,t) = 0\n(16)\n.\nRadiant Energy over Time\nThere is still one more interpretation that can be\nmade of the term \"radiant energy\". The preceding interpreta-\ntion of (12) is associated with the energy content of a given\nregion X at time t. There is a complementary interpretation\nof the total energy incident on or leaving a surface S over\nan interval T of time. For this interpretation we write,e.g.:\n\"U\"(S,T)\"\nfor\nH(x,E(E),t) dA(x) dt\n(17)\nT S\ni.e., U\"(S,T) is the radiant energy incident on S over the\ntime interval T. The hemisphere of incident radiant flux at\neach X is E(E), with & normal to S at x, in the inward sense.\nA complementary definition can be made for U+(X,T) using ra-\ndiant emittance.\nIt is worthwhile isolating the important concept, OC-\ncurring in (17), of radiant flux across a general surface S\nrather than just a collecting surface of the kind encountered\nin the sections above. Thus we write:\nS\n\"P\"(S,t)\"\nfor\nH(x,E(E),t) IA(x)\n(18)\n,\nS\nwhere E is the unit inward normal to S at X. A similar def-\ninition of P+(S,t) can be phrased. As usual, the signs \"++\"\nand \" 1 1 can be dropped whenever no confusion results, and al-\nso the \"t\" can be omitted for brevity.\nScalar Radiant Emittance\nWe conclude this section with the definition of the\nnotion of scalar radiant emittance. This concept is the sur-\nface-counterpart to scalar irradiance h defined in (3). Thus,\nlet us write:\n\"w(x,t)\"\nfor\n(19)","62\nRADIOMETRY AND PHOTOMETRY\nVOL. II\nw(x,t) is the scalar radiant emittance at x at time t. This\nconcept is useful in describing certain sources of radiant\nflux distributed continuously over some region of an optical\nmedium. The emittance counterparts to hemispherical scalar\nirradiance emittance can now be defined for w(x,t). These\ndefinitions would exactly parallel those in (5), (6) (7),\n(8), (10), (11), and therefore need not be given in detail at\nthis time.\n2.8\nVector Irradiance\nThe radiometric concept of vector irradiance, which\nwill now be considered, constitutes an interesting and useful\ncomplementary concept to that of scalar irradiance. Whereas\nscalar irradiance in essence measures the volume density of\nradiant energy at a point and does so without emphasis on the\ndirections of incidence of the component flows but only their\nmagnitudes, vector irradiance in contrast gives a measure of\nthe direction of the preponderant flow of radiant energy at\nthe point without emphasis on the magnitude of the various\ncomponent flows. Besides serving to complement the geometric\nproperties of scalar irradiance in this way, vector irradiance\nforms a rigorous tool in deriving the transfer equations for\nscalar irradiance, and also a powerful means of measuring pre-\ncisely and directly the absorption properties of real optical\nmedia. The basis for the latter means (the divergence rela-\ntion for H) is considered in Chapter 8 and some of its appli-\ncations are discussed in Chapters 6 and 13. In this section\nemphasis will be on introducing and explicating the geometric\nand physical meanings of vector irradiance.\nA Mechanical Analogy\nThe notion of vector irradiance can be introduced by\nmeans of an analogy with the vectorial treatment of forces in\nstatic mechanics. Figure 2.19 (a) depicts a force diagram\nfamiliar to beginning students in static mechanics. A parti-\ncle at point P is subject simultaneously to two steady forces\nof magnitude F1 and F2 along directions E1 and E2. In order\nto establish equilibrium of the particle--i.e. to balance\nout F1 and F 2 so that the particle is stationary, another\nforce of magnitude F3' must be applied along direction 53'.\nThe magnitude F 3 and direction E3 of the equivalent force that\nmay replace F1 and F2 is found by means of the familiar para-\nllelogram of forces shown in Fig. 2.19 (b). The required bal-\nancing magnitude is then -F3' and its direction is -Es, which\nfollows directly from Newton's Third Law. The central obser-\nvation to be made here is that, for the purpose of static\nequilibrium, two forces F1 (= F151) and F2 (= F252) can be re-\nplaced by a single force F3 F353) which serves as a mechan-\nical equivalent of the set of forces consisting of F1 and F2\ntogether. Thus, F 3 is, for the purposes of an equilibrium\ncomputation, equivalent to F1 +F2.\nConsider now a point P irradiated by two beams of ra-\ndiant flux which are flowing along directions E1 and E2 with","SEC. 2.8\nVECTOR IRRADIANCE\n63\n(b)\n(a)\nF2\nF3\nF1\nE2\nE1\nF2\nP\nE3\nF1\nP\nF's\nFIG. 2.19 The parallelogram law in mechanics\nradiance N1 and N2 respectively, as in Fig. 2.20 (a). Each\nradiance has a fixed small solid angle So. Now whereas the\nmechanical context of Fig. 2.19 (a) is meaningful in terms of\nsets of directed forces and equivalent single forces, the con-\ntext of Fig. 2.20 (a) is meaningful in terms of sets of direc-\nted radiances and equivalent single radiances. In the mechan-\nical setting, a single force could, for the purpose of an e-\nquilibrium computation, replace the two given forces by a sin-\ngle force F3. We now ask: can we replace the two directed\nradiances N1 and N 2 by a single equivalent radiance N3?\nSome thought will show that, before the preceding\nquestion can even be entertained, the sense of \"equivalent\"\nmust be defined. Clearly, the replacing radiance can be\n\"equivalent\" in any one of several desirable ways. For exam-\nple, if it is required that the replacing radiance produce\nthe same scalar irradiance at P, then there are many possible\ncandidates for N3. If on the other hand it is required that\nthe replacing radiance produce the same net irradiance on an\narbitrary collecting surface at P, then there is generally\none and only one radiance N3 that can replace N1 and N 2 in\nthis sense. Observe that the replacing radiance N3 must be\nequivalent to N1 and N2 in this sense not just for one fixed\nposition of a collecting surface at P; if that were the case,\nthen N3 could be chosen from any of an infinite number of ra-\ndiances. Rather, N3 is to produce the same effect for all\npossible orientations of a collecting surface at P. The anal-\nogy here with the mechanical context is essentially exact: in\nthe mechanical context F 3 establishes the same net force on","64\nRADIOMETRY AND PHOTOMETRY\nVOL. II\n(a)\n(b)\nN\n3\nN2\nN\n&\nE2\nN1\nP\nN2\nS\nP\nFIG. 2.20 The parallelogram law in radiometry\nany particle at P as does F1 and F2; in the radiometric con-\ntext N3 is to establish the same net irradiance on any sur-\nface at P as does N1 and N2. It is a simple matter now to\nprove that the parallelogram law may also be used for the ra-\ndiometric context to solve the analogous problem in that set-\nting. Thus, the requisite replacing radiance N3 and its as-\nsociated direction E3 of flow follow from a parallelogram con-\nstruction as in Fig. 2.20 (b). In particular, if we write:\n\"N1\"\nfor\nE1N1\nand\n\"N2\"\nfor\nE2N2\nthen N1 +N2 is the requisite vector radiance provided it has a\nsolid angle S. For if E is the inward unit normal to a col-\nlecting surface S at P, then we have by (6) of Sec. 2.5:\nE.E.N1 + EE2N2\nas the expression for the total net irradiance on S produced\nby the two beams. This sum may be written:\nE.N.S. + EN2S\nor, as:\n(N1++22):\nThis representation suggests that if we direct a radiance\nbeam of solid angle So at P and along the direction of N1 + N2","SEC. 2.8\nVECTOR IRRADIANCE\n65\nand with the magnitude of N1 + N2, then this single beam will\nproduce the same net irradiance across S at P as the two given\nbeams. The vector N1 + N2 which we have denoted by \"N3\" in\nFig. 2.20 (b), is found exactly as in any vectorial addition\noperation.\nThe observations just made can be generalized to the\ncase of any finite set of beams irradiating a point X in a\nradiometric environment. Toward this end, suppose that the\nvarious beams have radiances N1 Nk along directions\nE1,..., Ek and that, for generality, they have generally dis-\ntinct solid angles, So1 SK, respectively. Then by repeat-\ned use of (6) of Sec. 2.5, the net irradiance produced on a\nsurface S with unit inward normal E at X is:\nE.E.N.S., + E.E2N2S62 +\n+\nSuppose we write:\nk\nfor E53Nj8j\n\"H(x)\"\nj=1\nClearly H(x) is a vector and its magnitude H(x) I has dimen-\nsions of irradiance. Furthermore H(x) has the property that:\nE.H(x)\nis the net irradiance on the surface S at X produced by the\nset N1 , Nk of radiances at X. In the introductory example\nconsidered above H(x) was ENN + E2N2S. We shall call H(x)\nthe vector irradiance associated with the discrete radiance\ndistribution N1\nNk.\nGeneral Definition of Vector Irradiance\nWe now can go one step further in the development of\nthe idea of vector irradiance. Instead of a discrete finite\nset of radiances N1 , Nk at x, we consider a general radi-\nance distribution N(x,.). Instead of the finite summation\nover the sets of directions of the radiances in (1), we use\nthe continuous counterpart to the sum, namely an integral\nover all the directions E at X. Thus let us write:\n\"H(x)\"\nfor\ndr(s)\n(2)\nand where, in turn, the integral uses field radiance and is\nto be understood as an ordered triple of integrals, as is cus-\ntomary in vector analysis. That is, we have written:\nif do(E)\" for","RADIOMETRY AND PHOTOMETRY\nVOL. II\n66\n(s N(x,8 cosvi da(e).[NCX.E) cosv2 dR(E)./NCX.5) cosv3 do(E)\n(3)\nand where V1,V2, and V3 are as defined in Sec. 2.5 (cf.,\ne.g., (18) of Sec. 2.5). We call H(x) the vector irradiance at\nX. The alternate form (3) of the integral in (2) is the form\nin which H (x) is computed in actual practice. The integral\nin (2) is a compact symbol for the ordered triple of integrals\nin (3). A researcher requiring the direction and magnitude\nof H(x) at x knowing N(x, .) at that point, computes the three\ncomponents of H(x) in accordance with (3). Thus, if we write:\n/\n\"H(x)\"\n(4)\nfor\nN(x,5) cos\nfor i = 1,2,3, then:\nH(x) = (H1 (x), H2 (x) ,H3 (x)\n(5)\nH(x) I of H(x) is:\nThe magnitude of\n(H120 (x) + H22(x) + H3 (x)) 1/2\n(6)\nand the direction of H(x) is the unit vector:\n(H1 (x) (x))/(H12(x) +H 3 (x)) 1/2\n(7)\nThe General Cosine Law for Irradiance\nThe cosine law for irradiance was introduced in Sec.\n2.4 in a rather special context. Our purpose here is to show\nhow the law can be given the status of a general theorem in\nradiometry. Thus we will free the cosine law in (16) of Sec.\n2.4 from the restrictions placed on it in that section. The\nmeans by which the generalization can be accomplished is the\nnotion of vector irradiance. The law may be stated as fol-\nlows: Let N(x, ) ) be a radiance distribution at point X in an\noptical medium. Let E be the unit inward normal to a surface\nS at X. Then the vector irradiance H (x) at X, as defined in\n(2) has the property that:\nEH()=H(x)cos I\n(8)\nwhere \"||H(x)| 1\" denotes the magnitude of H(x), as given by (6),\nand \"I\" denotes the angle between E, and the direction of","SEC. 2.8\nVECTOR IRRADIANCE\n67\nH(x) as given by (7). Furthermore,\n|H(x)| = max, H(x,5) =\n(9)\ni.e., |H(x) I is the maximum of the set of all net irradiances\nH(x,5) at x, where the net irradiance H(x,5) at X across S in\nthe direction of E is as defined in (14) of Sec. 2.4. The\nproof of (8) is immediate, since (8), stripped of all physi-\ncal connotations, simply constitutes an elementary theorem in\nvector analysis. Equation (9) is the more deep of the two\nand follows from the observation that each Hi (x), i = 1,2,3,\ncan be written out in full form as:\nN(x,5) cos 24 ds(E)\nH1 (x) =\nE(E)\nN(x,5) cos 21 do(E)\n+\nE(-E)\nH(x,i) - H(x,-i) = H(x,i)\n(10)\n,\nwhere the first equality results from writing (1) as the union\nof two disjoint hemispheres (5) and E(-E) and where the sec-\nond equality follows from two applications of (8) of Sec.\n2.5. In a similar way we show that:\nH2 (X) = H(x,j) - H(x,-j) = H(x,j)\n(11)\nH3(x) = H(x,k) - H(x,-k) = H(x,k)\n(12)\nIn this way we uncover the physical significance of the three\ncomponents H1 (x), H2 (x), and H3(x) of H(x). For example,\nH1 (x) is the net irradiance across a plane at X whose inward\nnormal is the coordinate unit vector i along the X axis. Con-\ntinuing on our way to establish (9), we now examine EH(x)\ndirectly:\nH(x)\nds(s)\nE.E'N(x,5') ds(s) +\n1\n(-E)*E'N(x,E') ds(E)\nE(5)\nE(-5)\n= H(x,5) H(x,-5) = H(x,E)\n(13)","68\nRADIOMETRY AND PHOTOMETRY\nVOL. II\nThis sequence of five equations is justified analogously to\nthe sequences culminating in (10), (11), and (12). Now, how-\never, we have included more detailed steps. Clearly, (13)\nsubsumes (10)-(12) - In view of (13), we may write (8) as:\nH(x,5) =\nH(x)\n(14)\ncos\nand now (9) follows immediately from this. The maximum value\nof H(x, E) occurs when v = 0. From this, we have (9). Thus\nH(x) I is simply the net irradiance across that surface S at\nX whose unit inward normal is the same as the direction of\nH(x).\nThe results (8) - (14) are of importance in both theo-\nretical and experimental radiative transfer. An intuitive\nfeeling for H(x) and for equations (8) and (14) may be ob-\ntained by imagining an experimental device of the kind sche-\nmatically depicted in Fig. 2.21. The device has two collect-\ning surfaces S+ and S_ placed so that S+ and S. together re-\nceive radiant flux from every direction in E. Further, the\nunit inward normal E to S+, may be represented by a wire with\na pointer welded to one end, and the whole arrow fastened to\nthe material collecting surfaces as shown schematically in\nFig. 2.21. The meter for the device is wired to read\nH(x,E)-H(x,-5), i.e., the recorded irradiance on S+ minus the\nrecorded irradiance on S.. A device so constructed is called\na subtracting janus plate, (where \"janus\" has the same etymol-\nogy as \"January\") and may be used to empirically determine\nH(x) in natural optical media. To operate the device, one\nS+\nS- -\nH(x,E)\nE\nFIG. 2.21 Schematic of a subtracting Janus plate, used\nin measuring the vector irradiance field.","SEC. 2.8\nVECTOR IRRADIANCE\n69\norients it at a point x so that the reading H(x,5) attains a\nmaximum. Then by (9) the magnitude of H(x) is this maximum\nreading, and the direction of H(x) is the direction of the\narrow fastened to the device. The full geometric and physical\nsignificance of (8) and (14) can now spring forth: regardless\nof the complexity of the radiance distribution N(x,.) --and\nas the eye is a witness, such complexity can be subtle and of\ninfinite variety in natural optical media--the meter reading\nof the janus plate varies precisely in a sinusoidal fashion\nas the meter's direction is varied and as ve increases from 0\nto T. By means of (x) we can develop a theory of the light\nfield which is similar in many respects to certain classical\nfluid flows in hydrodynamics. This analogy has been explored\nby Gershun [98], and by Moon and Spencer [187].\nEquations (8) and (14), aside from their intrinsic\nmathematical interest appear to have in store potential prac-\ntical applications. For example, (14) may be of use in facil-\nitating the practical task of computing and tabulating irra-\ndiances H(x,E) over all directions E at given points x in op-\ntical media. Such a task is encountered, for example, when\ncontrast calculations in submarine environments are desired.\nSpecifically, by (14), it is clear that at a point x a full\ntabulation of H(x,5) over all E in E need not require a cor-\nrespondingly full computation. Indeed, appropriate use of\nEquation (14) cuts the requisite work just about in half.\nThus, one can compute H(x,5) for each direction E over some\npre-selected hemisphere, say E(Eo), and also compute H(x),\nwhence H(x) I is obtained. Then from (14) we have:\nH(x,-5) = H(x,5) - /H(x) I cos it\n(15)\nwhere E is in E(E0), and so -E is in (1) (-Eo), the complement\nof E(Eo) with respect to E. Thus for every - 5 in E(-Eo) one\ncomputes H(x,-E) using the already tabulated value H(x,5),\nthe angle ve, and H(x)\nWe conclude the present discussion of the general\ncosine law for irradiance by casting its basic form (8) into\none which comes as close as possible to its special counter-\npart (16) of Sec. 2.4. Thus let \"m\" denote the unit vector\nassociated with H(x), i.e., m is the direction of H(x) as com-\nputed by (7) Then, by (14), we have H(x,m) = H(x) which\nis the maximum net irradiance at X. Further, in (14),\ncos I = E.m; hence (14) becomes:\nH(x,5) = H(x,m)m.5\n(16)\nClearly (16) of Sec. 2.4 is a special case of (16) above when\nradiant flux is incident on X in accordance with the restric-\ntions on the earlier equation.","70\nRADIOMETRY AND PHOTOMETRY\nVOL. II\nRadiant Intensity\n2.9\nThe concept of radiant intensity, the last of the set\nof basic radiometric concepts to be introduced in this chap-\nter, is designed to give a measure of the solid angle density\nof radiant flux. Thus radiant intensity is a dual concept to\nirradiance in the sense that the latter gives a measure of\nthe area density of radiant flux while the former gives a\nmeasure of the solid angle density of radiant flux. At one\ntime the concept of radiant intensity enjoyed the place now\noccupied by radiant flux. In the days of the candle and gas\nlamp there were very few artificial extended light sources.\nThe controlled artificial point source, such as a candle's\nflame, was the sole basis for radiometric standards and its\nradiant output was conveniently measured in terms of intensity.\nHowever, with the passing of years the numbers of extended\nartificial light sources increased and the classical mode of\nuse of radiant intensity has become correspondingly less fre-\nquent than that of radiant flux. Eventually radiance for the\nmost part usurped the once centrally disposed radiant inten-\nsity concept. Despite this displacement of radiant intensity's\nstatus, it appears that there will always exist times when its\nuse arises naturally. For example when emitting 'point\nsources' are considered, the use of radiant intensity seems\nautomatically indicated. This useful aspect of radiant inten-\nsity will be discussed during its systematic development, to\nwhich we now turn.\nOperational Definition of Empirical Radiant Intensity\nIn presenting the concept of radiant intensity we\nshall be guided by operational considerations so as to give\nthe concept a secure footing relative to the other radiometric\nconcepts already defined. Thus our first encounter with the\nnotion of radiant intensity is in the following context: in\nthe operational definition of P(S,D) (Sec. 2.3), a radiant\nflux meter with collecting surface S and collecting directions\nD and monochromatic filter passing a single frequency v re-\ncords an associated amount P(S,D) of radiant flux incident on\nS through the set of directions D. Once the datum P(S,D) is\nobtained, then one conceptual path leads, as in Sec. 2.4, to\nirradiance H(S,D), i.e., the area density of P(S,D) over S;\nanother path leads to J(S,D), i.e., the solid angle density\nof P(S,D) over D, where we have written:\n\"J(S,D)\" for P(S,D)/8(D)\n(1)\nWe call J(S,D) the (empirical) radiant intensity of P(S,D)\nover D on S. The dimensions of radiant intensity are radiant\nflux per solid angle (per unit frequency interval), and con-\nvenient units are watts/steradian (per unit frequency inter-\nval). In full notation for the unpolarized context, we would\nwrite:\n\"J(S,D,t,F)\"\nfor\n$(S,D,t,F)/S(D)","SEC. 2.9\nRADIANT INTENSITY\n71\nor:\n\"J(S,D,t,v)\" for P(S,D,t,v)/82D\n(2)\n.\nHowever, we shall need only to employ the briefer notation in\nmost of our discussions.\nAn examination of the operational definition of empir-\nical radiant intensity, summarized in (1), will show that\nthere is no restriction on the set of directions D. That is,\nD may be an arbitrary fixed set of directions along which the\nradiant flux funnels down onto the points of the collecting\nsurface S. In practice, however, a radiance meter is the de-\nvice used to estimate the radiant intensity of the light field\nat a point in an optical medium. In such an instrument, the\nset D is a relatively narrow conical bundle of directions\nwhose axis is perpendicular to the collecting surface S of the\nmeter. The connection between field radiance N(S,D), and ra-\ndiant intensity in such a context, follows from (1) and is\nreadily stated:\nJ(S,D) = N(S,D)A(S)\n(3)\nThe connection between J(S,D) and N(S,D) can be generalized\nto take into account radiant flux which crosses S obliquely\nwithin the narrow set of directions D. The geometric setting\nis essentially that depicted in Fig. 2.6, the setting for the\ncosine law for irradiance.\nTo establish the generalized version of (3), we re-\nturn to (1) and within the setting of Fig. 2.6, compute P(S',D)\ni.e. the radiant flux over S':\nP(S',D) = P(S,D) = N(S,D)A(S) & (D)\n.\nThe reason for the equality of P(S',D) and P(S,D) stems from\nthe hypothesized setting of Fig. 2.6, and the arguments pre-\nsented earlier. Therefore, from (1):\nJ(S',D)=P(S',D)/s(D)\nN(S,D)A(S)\nBut:\nA(S) = A(S') cos ve\nHence:\nJ(S', D) = N(S,D)A(S') COS I\n.\nBy the radiance invariance law:\nN(S',D) = N(S, D)\nHence:\nJ(S',D) N(S',D)A(S') cos 2l\n(4)","RADIOMETRY AND PHOTOMETRY\nVOL. II\n72\nwhenever the inward unit normal E' to S' makes an angle 2\nwith the central direction E of D, as in Fig. 2.6, Eq. (4)\nshould be compared with the special case (6) of Sec. 2.5. It\nis worthwhile re-emphasizing that relations (3) and (4) are\nrelations among empirical radiometric quantities, i.e., ra-\ndiometric quantities obtained with the use of a radiance me-\nter of finite solid angle opening s(D), and finite collecting\nsurface area A(S). The more finely-honed theoretical radio-\nmetric concepts come later with the help of the various addi-\ntive and continuity properties of § postulated in Sec. 2.3.\nThe empirical concepts serve to establish the bridge between\ntheoretical constructs and the immediately given physical\nrealities. The empirical concepts serve also to block out in\nrough form the incipient analytical structures of the theoret-\nical relations.\nField Intensity vs. Surface Intensity\nThere is a distinction that can be made in practice\nbetween two types of radiant intensity, a distinction that is\nexactly analogous to the distinction made in Sec. 2.5 between\nfield and surface radiance. Indeed, by referring to Fig. 2.12\nwherein is depicted the two types of radiance, N (S,D) and\nN-(S,D), which in turn are defined as in (30) and (31) of Sec.\n2.5, we are led to write:\n\"J+(S,D)\"\nP*(S,D)/&(D)\nfor\n(5)\nand\n\"J\"(S,D)\"\n(S,D)/S(D)\nfor\n(6)\nP\nin complete analogy to the definitions of W(S,D) and H(S,D)\nin (17) and (18) of Sec. 2.4. We call J-(S,D) the field in-\ntensity and J+(S,D) the surface intensity over D within S.\nThe utility of this distinction and the basis for the names\nof these concepts rest once again on the remarks for N+ (S,D)\nand - ( S , D in Sec. 2.5. In actual practice in natural opti-\ncal media it is the surface intensity which is used with\ngreatest frequency. However, in these settings it is the\nfield intensity (or rather radiance) which, in the final anal-\nysis, must be measured before the surface intensity is ob-\ntained. The basic quantitative connection between the two\ntypes of radiant intensity is analogous to that between sur-\nface and field radiance in (22) of Sec. 2.5:\nJ+(S,D) = (S,D)\n(7)\nIt follows from (30) of Sec. 2.5 and (5) above that:\n(S,D) = W(S,D)/S2D) = (S, D) /A(S)\n(8)\nand from (31) of Sec. 2.5 and (6) above that:\nN (S,D) = H(S,D)/S2D = J (S,D)/A(S)\n(9)\n.","SEC. 2.9\nRADIANT INTENSITY\n73\nHenceforth we shall drop the \"+\" and \" \" \" superscripts\nfrom the symbol \"J\" when it is clear from the context (or im-\nmaterial) which interpretation of radiant intensity is to be\nused in reading a statement using the concept of radiant in-\ntensity. Occasionally, however, especially for the purpose\nof emphasizing a delicate point in a discussion, the plus and\nminus appendages will be reattached to \"J\" In general, the\nfollowing rule may be observed in regard to the base symbols\n\"J\" and \"N\": whenever \"+\" and 11_11 are omitted from\n\"J\"\nand\n\"N\", then the associated statement or term in which \"J\" and\n\"N\" appear is valid under both surface and field interpreta-\ntions.\nTheoretical Radiant Intensity\nSuppose now that in the operational definition (1)\nthe set of directions D becomes smaller and smaller, such\nthat it always contains the direction E and such that the flow\nof radiant energy is onto S. Then write:\n\"J(S,E)\" for lim J(S,D)\n(10)\nD+{E}\nThe existence of this limit is guaranteed by the D-\nadditive and D-continuity properties of $ postulated in Sec.\n2.3. The radiant intensity J(S,E) is called the (theoretical)\nradiant intensity in the direction E on S. It is important\nto note that non-zero values of J(S,E) are necessarily asso-\nciated with surfaces S which have non-zero area A(S). This\nfact is based on the S-continuity property of recorded in\nSec. 2.3. Thus, by S-continuity and S-additivity of & we\nhave:\nlim_J(S,E) = 0 ,\n(11)\nS+{x}\nfor every x in S. However, once again by S-continuity and\nS-additivity of &, we have from (1), (4), (10) and the defi-\nnition of N(x,5)\nN(x,5) = lim\n(12)\nS'+{x}\nwhere E' (x) is the unit inward (or outward)* normal to S' at\nX. See Fig. 2.6.\nFrom (10) and the fundamental theorem of calculus we\nobtain:\nP(S,D) =\nJ(S,E)\ndo(E)\n(13)\nD\n* Recall our convention on field intensity and surface inten-\nsity stated above.","VOL. II\nRADIOMETRY AND PHOTOMETRY\n74\nFrom (4) and (12) we have for similar reasons:\n(14)\nJ(S,E) =\nS\nwhere E' (x) is the unit inward (or outward) normal to S at x\nand is in E(E).\nAt this point it would be instructive to view (12) in\nterms of (5) of Sec. 2.5. Furthermore, one can compare (14)\nwith its 'dual' in (8) of Sec. 2.5. This 'duality' stems\nfrom a comparison of what is held constant and what is varied\nin (8) of Sec. 2.5 and (14) above. In (8) of Sec. 2.5, x in\nS is held fixed while E varies over all directions in E(5).\nIn (14), E in (1) (5) is held fixed while x varies over all\npoints in S. Furthermore, while the integration in (8) of\nSec. 2.5 was limited for physical reasons to a hemisphere\nE(E) at x, the integration in (14) is limited, for similar\nreasons, to an S over which (x) also stays within E(E).\nHence the duality between H(x,E(E)) and J(S,E) is quite deep\nand complete.\nRadiant Intensity and Point Sources\nAs noted in the introductory statements of this sec-\ntion, radiant intensity first arose as a measure of the direc-\ntional radiant flux output of spatially very small emitters\nof flux. We shall now show that this feature of radiant in-\ntensity can still be employed within the operational point of\nview adopted in the present development of geometrical radio-\nmetry. The net result of this observation will be the recov-\nery of the original conceptual feature of radiant flux but in\na manner which will, it is hoped, now be operationally mean-\ningful.\nWe begin by defining the notion of a point source of\nradiant flux. A part Y of an optical medium X is a (radiomet-\nric) point source with respect to point X in X if the set\nD(Y,x) of directions subtended at X by the points of Y is such\nthat s(D(Y,x)) VI 1/30. The basis for this definition rests\nin two facts, one empirical, and the other theoretical.\nThe empirical fact is that radiance meters with solid\nangle openings such that (D) VI 1/30 have been found to be\nadequate for the practical purposes of geophysical optics to\ndistinguish the radiance variations occurring in natural opti-\ncal media. Hence any part Y of an optical medium X which can\nbe encompassed by the field of view of a radiance meter locat-\ned at point x in X is radiometrically a 'point source' of\nflux. It might be that Y is a ship or an extensive wheat\nfield, or a large patch of ocean surface, or a great cumulus\ncloud. As long as these objects (they can be either opaque\nsolids, surfaces, or certain well-defined nearly transparent\nvolumes of water or air) fall within the field of view of a","SEC. 2.9\nRADIANT INTENSITY\n75\nstandard radiance meter, they are considered 'point sources'\nwith respect to that meter.\nThe second fact on which the definition of 'point\nsource' is based is that a part Y of X, such that\ns(D(Y,x)) VI 1/30, has the property that the irradiance from Y\non a surface about point X will vary, to within one percent,\ninversely as the square of the distance from x to Y whenever\nY is some definitely localizable object such as a ship, or\npatch of sky or ocean surface, etc. In short, according to\nthe preceding definition, Y will be a point source of flux\nonly if the inverse square law and cosine law for irradiance\nholds with respect to it to within one percent. (See Example\n5, Sec. 2.11.) It might be of interest to take note of the\nlogical structure of the preceding statement. In particular,\nwe do not assert that \"if a part Y of X is such that the in-\nverse square law and cosine law hold with respect to it, then\nY is a radiometric point source\". By considering a spherical\nbody Y, the reason for this may be seen (cf., Example 4, Sec.\n2.11). Finally, we shall henceforth assume that in the deter-\nmination of the surface radiance of a point source Y, the solid\nangle opening of the radiance meter can be adjusted so as to\nfit exactly the set D(Y,x).\nConsider now a radiometric point source Y in a medium\nX. For definiteness, let the point source Y be a spherical\nregion of radius a within a vacuum and which steadily emits\nradiant flux. Further, Y is such that it can be observed\nfrom all directions. Suppose it is required to estimate the\nradiant flux output of Y but the measurements are constrained\nfor various reasons to take place a distance r not less than\na units from the center y of Y. Figure 2.22 (a) depicts the\npresent situation. By adjusting the meter's solid angle\nD\nY\nr\nI\nD'\n1\nS\nY\nFIG. 2.22 Operational definition of a point source","76\nRADIOMETRY AND PHOTOMETRY\nVOL. II\nopening so that the set D of directions from x to Y just fills\nthe field of view, the field radiance N(S,D) associated with\nY is read directly from the meter. Here S is the collecting\nsurface of the meter. By the radiance invariance law (1) of\nSec. 2.6, it follows that N(S,D) = N(S' ,D') where S' and D'\nare as shown in Fig. 2.22 (b), and are completely analogous\nto the observed surface and direction sets shown in Fig. 2.14.\nHence the radiant flux output of Y across the projected sur-\nface S' of Y and within the set D' of directions is estimable\nas:\nN(S',D')A(S')s(D')\nafter using the measured radiance N(S,D) for N(S',D').\nNow we have agreed to write:\n\"N(S',D')\" for P(S',D')/A(S')s(D')\nwhere P(S' D') is the desired radiant flux output of Y in the\ndirection D' Since this radiance may be written as:\nJ(S',D')/A(S') ,\nwe can now set:\nP(S',D') = J(S',D')&(D')\n.\nAt this juncture the reader should first observe how\nthe number N(S', D') 'belongs' to Y; that is, it is (by the\nradiance invariance law) independent of the mode of measure-\nment. Secondly, it should be noted that of the two numbers\nA(S') and S(('D'), the area A(S') 'belongs' to Y whereas s(('D')\ndepends on the mode of measurement (i.e., the distance r be-\ntween x and y). It follows that the product N(S', D') A(S')\n'belongs' to Y. But this product is simply J(S', D'), the ra-\ndiant intensity (watts per steradian) of S' in the directions\nwithin D' from X to y. Hence the number J(S',D') is an intrin\nsic property of Y in the sense that it is independent of the\nmode of measurement. Finally, by recalling that the dimen-\nsions of J(S',D') contain no linear (i.e., length) terms, it\nbecomes manifest that J(S',D') can be conceptually associated\nwith the radiant flux output of the point y (the center of Y)\nin the direction E (the central direction of D'). In this\nway we arrive at the classical conception of radiant intensity\nas the radiant flux emitted by a point X per unit solid angle\nabout a given direction E.\nWe\ncan now use (13) as a basis for the classical for-\nmula relating the radiant intensity and radiant flux output\nof the point source Y. Since Y is a sphere, the projected\narea A(S') of Y on a plane normal to a direction E is indepen-\ndent of the direction 5. More generally, in the point source\ncontext, we will agree to write:\n\"J(x,E)\"\nfor\nJ(S',5)\n\"P(x,D')\" for P(S',D')","SEC. 2.9\nRADIANT INTENSITY\n77\nwhenever S' is the projection of part (or all of) the bound-\nary of the point source Y on a plane normal to E, and X is\nsome point within Y. Then, with this understanding, (13) be-\ncomes:\nP(x,D')\nJ(x,5)\ndo(E)\nJ(x,0,0)\nsin\n0\nde\ndo\n(15)\nD'\nwhere (for terrestrially-based - coordinate systems) we have\nused (9) of Sec. 2.5, and have written \"(0,0)\" for E. If\nthe radiant intensity output of Y is independent of E over D',\nthen we can make the following statement:\nIf D' = E, then,\nP(x,D') = J(x)s(D')\n(16)\ni.e.,\nP(x) = 4 TJ(x)\n(17)\nwhere we have written:\n\"P(x)\" for P(x,E)\nand\n\"J(x)\" for J(x,5)\n.\nEquation (17) is the customary form of the connection\nbetween the radiant intensity J(x) of a (directionally) uni-\nformly emitting point source at x and its total power output\nP(x). By retracing the definitions of \"P(x)\" and \"J(x)\" the\nreader will see that the emitting object referred to is not a\ngeometric point but rather a small finite part Y of an optical\nmedium X, and that x is a point of X in or near Y. In this\nway we conceptually simplify the description of point sources\nto the form exhibited in (17) without contradicting the basic\ntenets of radiometry, in particular the S-continuity of\nin Sec. 2.3.\nCosine Law for Radiant Intensity\nThe cosine law for radiant intensity (Lambert's law)\ncan be stated as follows (cf., Fig. 2.6): If the surface ra-\ndiance N(S' , 5) of point source surface S' is independent of\ndirection E in E(E'), where E' is the unit outward normal to\nS' at y, then the surface intensity J(S',5) of S' varies as\nthe cosine of the angle between E' and E, i.e.:\nJ(S',5) J(S',E')E'.E\n(18)","78\nRADIOMETRY AND PHOTOMETRY\nVOL. II\nThe proof of statement (18) rests on (4). For by hy-\npothesis we now can write:\nN(S',D) = N(S',D')\nwhere D' is a narrow conical set of directions whose central\ndirection is is normal to S', as in Fig. 2.6. Hence (4) be-\ncomes:\nJ(S',D)=J(S',D') COS v\nLetting D and D' become smaller and smaller with limit {5}\nand {5'}, respectively, we arrive at (18).\nWe have deliberately retained the notation of Fig.\n2.6, despite the fact that (18) can be written with less\nprimes adorning \"S\" and \"5\", for the purpose of encouraging a\ndetailed comparison of (16) of Sec. 2.4 and (18) above. Close\nstudy will again reveal the interesting duality between inten-\nsity J and irradiance H already discerned by a comparison of\n(8) of Sec. 2.5 and (14) above. By dwelling on this recur-\nrent duality between J and H, one is moved to inquire whether\nthe cosine law for radiant intensity can be generalized to a\nform which would constitute a dual statement to the general-\nized cosine law for irradiance in the form of (8) or (16) of\nSec. 2.8. It turns out that an exact dual statement to (8)\nof Sec. 2.8 can indeed be made for radiant intensity. Now,\nsince the basis for the generalized cosine law for irradiance\ncan be viewed as embodied in (8) of Sec. 2.5, we should ex-\npect the basis for the generalized cosine law for radiant in-\ntensity to rest in (14) above. We now show that this expec-\ntation is correct. We begin with deriving a result, of inter-\nmediate generality, from (14), a result which provides an in-\nteresting insight into the structure of the classical Lambert\nlaw.\nLet Y be a region of an optical medium X. The region\nY may be of arbitrary shape. Suppose further that from van-\ntage point x, Y is a point source and that the observed sur-\nface radiance of its boundary surface is independent of the\ndirection of observation of Y. For simplicity, we assume\nthat the paths of sight from x to points of Y lie in a vacuum.\nThe current geometric situation is depicted in Fig. 2.23.\nLet N(S,D) and N(S', D') be the observed surface radiances\nseen from two arbitrary vantage points x and x' at both of\nwhich Y is a point source. The surfaces S,S' and direction\nsets D and D' are as shown in the figure. Thus S is the pro-\njection of Y on a plane normal to the axis of the radiance\nmeter located at X. Similarly for S'. Then by hypothesis\nand by the radiance invariance law:\nN(S,D) = N(S',D')\nThis radiance equality can be written in terms of radiant in-\ntensity:\nJ(S,D) - J(S', D')\n(19)\nA(S)","SEC. 2.9\nRADIANT INTENSITY\n79\nY\nS'\nD\nX'\nX\nFIG. 2.23 Establishing the Cosine law for radiant inten-\nsity in the context of point sources.\nSince the X and x' are arbitrary locations subject\nonly to the requirement that Y is a point source with respect\nto these points, we arrive at the following slight generaliza-\ntion of the cosine law for radiant intensity:\nIf a part Y of an optical medium X has uniform surface\nradiance for all directions and all points on the bound-\nary of Y, and Y is a point source with respect to points\nX in some subset X of X, and if the paths of sight from\npoints of X to Y lie in a vacuum, then the quotient\nJ(S,D)/A(S) is invariant for every point x in X, where\nS and D are defined as in Fig. 2.23.\nIt is clear how the classical form of Lambert's law\n(18) follows from this new statement and its analytic form\n(19); one now lets Y be a plane surface and lets X be all the","80\nRADIOMETRY AND PHOTOMETRY\nVOL. II\nappropriate points of X lying to one side of S.\nIt is of interest to note still one more variant of\n(19), one which has considerable intuitive value. Let \"N\" de-\nnote the hypothesized fixed radiance associated with Y.\nThen\n(19) implies that:\nJ(S,D) = NA(S)\n(20)\n,\ni.e., , that J(S,D) varies directly as the projected area A(S)\nof Y on a plane perpendicular to the central direction E of\nD. This may be compared with (3). From (20) we can deduce by\ninspection the direct square law--or area law--for radiant in-\ntensity which states that: for areas which are radiometric\npoint sources with respect to some observation point, the as-\nsociated intensity varies directly as the apparent (projected)\narea of the surface as seen from that point. If the area is\ncompared with geometrically similar areas, then the associated\nradiant intensity varies directly as the square of a common\ntransverse dimension of these areas. This observation brings\nto light still another facet in the duality between irradiance\nand radiant intensity, the dual law for irradiance in this\ncase being the inverse square law.\nGeneralized Cosine Law for Radiant Intensity\nThe preceding discussions on point sources and radiant\nintensity lead us to formulate several useful alternative ver-\nsions of the classical Lambert law for radiant intensity.\nDuring those discussions it was observed how the concepts of\nirradiance and intensity played the roles of dual concepts in\na sense made clear in those discussions. This duality of ir-\nradiance and intensity is capable of being expressed in a pre-\ncise fashion and on a level of generality comparable to that\nestablished for the general cosine law for irradiance (8) of\nSec. 2.8. We now pause briefly in our developments of geo-\nmetrical radiometry to establish this interesting generaliza-\ntion of Lambert's law. In doing so we round out and make for-\nmal the recurrent theme of duality between surface intensity\nJ and irradiance H encountered throughout this section.\nLet Y be a region of an optical medium X such that at\neach point x of the closed boundary surface S of Y the surface\nradiance is uniform over the hemisphere E (5' (x) where E' (x)\nis the unit outward normal to S at X. Let \"N(x)\" denote the\ncommon value of the uniform radiance distribution at x on X\nover the set (5'(x)). Observe that the variation of the val-\nues N(x) over S is left to be quite arbitrary. For the pres-\nent discussion the only restriction on the radiance function\nis that it be uniform over (5' (x) at each point X of S.\nSome approximate physical realizations of such a region Y are:\nan opaque irregularly shaped body painted with matte paints\nsuch that the paints have an arbitrary spatial pattern over S;\na luminous, dense region of space such as the sun which, for\npractical purposes, has a directionally nearly uniform radiance","SEC. 2.9\nRADIANT INTENSITY\n81\ndistribution at each boundary point, but which still may be\nmottled with lighter or darker regions; the moon's surface\nforms still another example. However, when each of these ob-\njects is examined with extreme accuracy of radiometric detail\nin mind, a wealth of departures from these ideals is encoun-\ntered.\nNow returning to Equation (14) and using the present\nradiance function in the integral, we consider the particular\nintegral:\n1.\n(x) E5'(x)\ndA(x)\n(21)\nS\nOur studies of the duality between J and H led us to\nbelieve that we may be able to do for J what we did for H\nwhen going from (8) of Sec. 2.5 to (2) of Sec. 2.8. There-\nfore, we are led to take (21) as a base and write:\n\"J(S)\"\nfor\nN(x) E'(x)\ndA(x)\n(22)\n.\nS\nWe call J(S) the vector intensity for S. The defini-\ntion of the integral is based on the notion of an ordered tri-\nple of integrals, using the form (3) of Sec. 2.8 as a model.\nNow J(S) is a bona fide vector. As such it has three\nreal numbers as x, y, and Z components, and so a non zero mag-\nnitude J(S) I and a direction J(S)/|J(S) This observation\nwill allow us to state succinctly the radiant intensity analog\nto (8) of Sec. 2.8. However, before doing so, we explore one\nfurther facet of J(S).\nFigure 2.24 depicts a typical region Y with boundary\nS for which J(S) is defined. If a direction E is chosen, then\nthe boundary S of Y can be partitioned into two parts S(E) and\nS(-E) (or \"S+\" and \"S-\" for short) with the properties that\nS(5) consists of all points x of S such that E.E' (x) > 0, and\nS(-E) consists of all points x of S such that E.E'(x) < 0.\nThere is generally, for all surfaces of use in practical radi-\nometry, a closed curve C on S such that E.E. (x) = 0 for every\nx on C. C is the boundary between S+ and S.. Observe how\nS(E) and (-5) in the present context have their counterparts\nin the sets E(E), E(-E) used in the vector irradiance context.\nReturning now to (14) we choose a E, determine the as-\nsociated S+ and S. as just described, and then evaluate:\nJ(S+,5) = I\nN(x)E'(x) dA(x)\n.\nS( E E\nSuppose we write:\n\"J(S,E)\" for J(S,,E)-J(S_,5)\n.","82\nRADIOMETRY AND PHOTOMETRY\nVOL. II\n((x)\nC\nx\nE(y)\nS(E)\nS(- E)\nE\nFIG. 2.24 Establishing the general Cosine law for radi-\nant intensity.\nThen from the definition of J(S) and J(S+,5) it follows that:\nE.J(S) = J(S,5) = J(S,,5)-J(S_,5)\n(23)\n.\nWe are now ready to state the generalized cosine law\nfor radiant intensity.\nLet N (x) be a uniform radiance distribution over the\nhemisphere E(E'(x)) at each point x of the boundary S\nof a region Y in an optical medium, where 5'(x) is the\nunit outward normal to S at X. Then the vector (sur-\nface) radiant intensity J(S) as defined in (22) has the\nproperty that:\nJJS\nE.J(S)\ncos v\n(24)\nwhere \" J(S) 11 denotes the magnitude of J (S) and v \"\ndenotes the angle between E and the direction of J(S).\nFurthermore:\n= max,J(s,5)\n(J(S)\n(25)","SEC. 2.10\nPOLARIZED RADIANCE\n83\nThe parallel of (24) and (25) with the irradiance case\nin (8) and (9) of Sec. 2. 8 is exact. In particular, from (23)\nwe can now write:\nJ(S,E) = J(S,m)m\n(26)\n,\nwhere m is the direction of J(S). This is the radiant inten-\nsity counterpart to (16) of Sec. 2.8. The special case (18)\nof Lambert's law now follows upon applying to (26) the condi-\ntions stated for (18). In particular Y now degenerates into\na plane, we let N(x) = 0 on S_, and N(x) be constant on S+.\nIt should be noted in passing that (24) holds for regions in-\ncluding non-point sources. The duality between J and H now\nbecomes clear upon comparison of, say (26) above with (16) of\nSec. 2.8: a point X in the irradiance context is replaced by\na surface S in the intensity context; the set E(E) in the ir-\nradiance context is replaced by the point E in the intensity\ncontext.\n2.10\nPolarized Radiance\nIn this section we shall develop an operational defi-\nnition of polarized radiance. The development shall take as\na point of departure the notion of empirical radiance intro-\nduced in Sec. 2.5. The details of the development shall be\nkept to a minimum, as we will not in this work make extensive\nuse of the concept of polarized radiance. For a somewhat more\ndetailed theoretical discussion of polarized radiance suitable\nfor geophysical applications, the reader is referred to Chap-\nter XII of Ref. [251].\nBefore going into the technical details of how to\nmeasure polarized radiance, a few comments may be made on the\nreason for wanting to measure polarized radiance in natural\noptical media. The first and most important reason is that\nthe systematic documentation of the state of polarization of\nsubmarine (and atmospheric) light fields increases our store\nof basic optical knowledge of the world in which we live. For\nthose of a more practical turn of mind, it may suffice to add\nthat knowledge of the kind and amount of polarization extant\nin a natural light field could yield efficient means of in-\ncreasing visibility in both the atmosphere and the sea. For,\nthe contrast of objects seen against a sky or underwater back-\nground is occasionally increased when viewed through a mater-\nial which can transmit polarized light in various amounts de-\npending on how the material is held and oriented. If we pos-\nsess systematic tabulations of polarized light fields and\nsome workable theoretical models of such fields, these empir-\nical observations can be more deeply explored and applied.\nFinally, there is the question, still not fully resolved--\nespecially for the hydrologic optics branch of geophysical op-\ntics--of whether and to what extent polarized light is used by\ncreatures in navigating, in foraging, and in their biological\ngrowth cycles.","84\nRADIOMETRY AND PHOTOMETRY\nVOL. II\nIn order to help resolve such questions and in order\nto add to our knowledge of the light fields in natural hydro-\nsols in a systematic manner, we must develop a precise but\nworkable means of measuring and theorizing about polarized\nlight fields. A small but definite beginning in this direc-\ntion will be attempted in this section and subsequently in\nChapter 4 wherein the equation of transfer for the polarized\nradiance vector is used to derive a theoretical model of po-\nlarized light fields in natural optical media.\nOne final comment is in order. It will be recalled\nthat our approach to hydrologic optics is through the tenets\nof radiative transfer theory and, as a consequence, we are\ncommitted to study the natural light fields on a phenomenolog-\nical level. In particular, as pointed out in Sec. 2.0, we\nhave agreed to adopt those instruments of investigation which\nmake quantitatively precise, all of the optical phenomena vis-\nible to the human eye. One may then--in view of this obser-\nvation--argue that in extending the capabilities of our in-\nstruments to detect and measure polarized light fields we are\ntranscending the bounds originally set down by us when we em-\nbarked on the development of the concepts of radiometry. It\nmay be observed, however, that whenever it is deemed necessary\nto extend the radiative transfer phenomena of concern to hy-\ndrologic optics in particular, or geophysical optics in gen-\neral, the extension will be made solely on its merits to add\nto the descriptive power of these branches of radiative trans-\nfer theory. * In the present discussion, the extension of the\nradiometric concept of radiance to the polarized level not\nonly fulfills this general criterion, but interestingly\nenough, still keeps the collection of radiometric concepts\nwithin that small, select circle of concepts which are direct-\nly observable by the unaided eye. For indeed, the polariza-\ntion of the light of the sky or a submarine light field is\ndirectly observable to the unaided (but practiced) human eye.\nThe physiological basis for this capability of direct obser-\nvation is the dichroic nature of either the material compris-\ning the yellow spot of the retina or perhaps that of certain\nof the optic nerve fibers themselves. (Dichroic materials are\nalso found in natural deposits, e.g., in the form of tourma-\nline crystals, and were already used in the early devices for\ndetecting polarized light.) It is the small but adequate\namount of dichroic material in the retina which thus permits\nthe unaided eye to detect and the brain to record the presence\nof linearly polarized light in a natural light field. This\ninnate ability of the eye to detect polarized light was re-\nported by Haidinger in 1846, and the elusive but yet visually\nobservable pattern seen by the eye is known as Haidinger's\nbrush. An informative description of how to facilitate the\ndetection of Haidinger's brush in skylight is given by Min-\nnaert in Ref. [182].\n'The case for an extension of the classical scalar theory to\nthe polarized level ultimately involves no less than the con-\nsistency of the classical scalar theory in the context of po-\nlarized light fields. See (17) of Sec. 13.11","SEC. 2.10\nPOLARIZED RADIANCE\n85\nE\nr\nD\ncollecting\nsurface S\n0\ncollecting tube\nwave plate W\nphoto-\npolarizer P\nelectric\nfilter\nelement\ndial R\nFIG. 2.25 Schematic details of a radiance meter fitted\nwith a polarizer P and a variable wave plate W, for measuring\npolarized radiance distribution.\nOperational Definition of Polarized Radiance\nThe operational definition of polarized radiance we\nshall adopt has been chosen for its inherent simplicity and\nits amenability to be linked with the classical Stokes vector\nfor polarized light. Thus the polarized radiance vector will,\non the one hand, be tied directly to observable qualities of\nnatural light fields and, on the other, be rigorously repre-\nsentable by means of concepts extant in the electromagnetic\npicture of light.\nWe begin with a radiance meter, as described in Sec.\n2.5, and adjoin to the meter, at the base of the tube, a po-\nlarizer P and a variable wave plate W. The order in which\nthe entering light encounters these devices is important and\nis depicted in Fig. 2.25: the light is to encounter the var-\niable wave plate first, and the polarizer second; then it\npasses on through the filter to the photoelement below.\nThis\nrelative placement of W and P is the essential point to ob-\nserve here; where the filter is relative to W and P is, how-\never, immaterial as far as ideal detectability of polarized\nflux is concerned.\nThe polarizer P, which is made from a dichroic crystal\nor a sheet of polaroid, is mounted so that it is rotatable\nabout the axis of the cylindrical tube of the radiance meter.\nThe orientation of the optic axis of the polarizer is impor-\ntant in what follows; therefore it is essential that some","86\nRADIOMETRY AND PHOTOMETRY\nVOL. II\nmeans be provided for the clear marking of the position of\nthe optic axis relative to the radiance meter's tube, or some\nother fixed part of the radiance meter. Further, if absolute\nradiance measurements are desired, the transmittance of P\nover the spectrum is required. The ideal transmittance of P\nis 1/2 for unpolarized light.\nThe wave plate can be made of some negatively doubly\nrefracting material such as calcite, and is assembled (at\nleast for the introductory discussion below) so that a wide\nrange of optical path lengths is available at the twist of a\nknob. For example, a Babinet compensator type of arrangement\nmay be employed. Later on, when the radiance meter is readied\nfor field use, the wave plate may be replaced by an attachment\nfashioned from a single sheet of some good grade of circularly\npolarizing material of known transmittance over the spectrum.\nThe ideal transmittance of W is 1, and that of circularly po-\nlarizing material, 1/2. The fact that a circular polarizing\nmaterial can be used in lieu of a variable wave plate will be-\ncome clear after the observable radiance vector has been de-\nfined.\nThe next step in the present operational definition of\npolarized radiance is to take the radiance meter, set for fre-\nquency v, to a point x in the environment, and direct it so\nthat flux enters the tube along the direction E at time t.\nThen one systematically varies the angle 4 of the polarizer's\noptic axis, starting from the vertical plane, with a given\nfixed retardation E 0 of the wave plate. (See Fig. 2.26)\nIn fact one varies & from 0 radians (so that the optic axis\nis in the vartical plane) and increases x clockwise (when\nseen looking into the tube from the front of the tube, i.e.\nlooking along the direction of travel of the photons) to TT\nradians (so that the optic axis is again in the vertical\nplane) As this is done, one should note how the recorded\nflux varies with 4 and that the variation is of period IT. In\nparticular, for E = 0 and a general light field, the variation\nof radiance turns out to be representable in the form:\nQ\n24\nU\nsin\n2v\n+\ncos\n+\n,\n2\nwhere we have written:\n\"I\"\nfor\n2N\n\"Q\"\nfor\n2AN cos 2v\n\"U\"\nfor\n2AN sin 2v\no\nwhere, in turn, we have written:\n\"N\"\nfor\n(N\n+ Nmin)/2\nmax\n\"AN\"\nfor\n(N\n- Nmin 1/2\n,\nmax","SEC. 21.0\nPOLARIZED RADIANCE\n87\nvertical axis\n(a)\nk\nX\nvertical plane\n(b)\no=45°\nvertical plane\nFIG. 2.26 How to measure and record the standard obser-\nvable radiance vector.\nand where \"Nmax\" and \"Nmin\" denote, respectively, the maximum\nand minimum radiance readings when is varied from = 0 to\n4 = TT. 40 is the angle of occurrence of the maximum reading\nNmax.\nFurther experimentation, with now a general e-setting\nW and with 4 varying over the interval [0,\"], shows the\non\nfull form of the radiance variation to be:\n1\n+ Q COS 24 + (U COS E-V sin E) sin 24\n(1)\nwhere V is determinable by a simple trigonometric analysis of\nthe recorded curves obtained by fixing u + 0 and varying E.\n(See, e.g., Ref. [193]). If this reading is obtained at point\nx, for the direction E at time t, for frequency v, and with a\nP-setting 4, and a W-setting E, then we will agree to denote","VOL. II\nRADIOMETRY AND PHOTOMETRY\n88\n\"N(x,5,t,v,v,e) \" or, if x, E, t, and v are understood,\nit\nby\nwe will denote it simply by \"N(v,e)\", for short. The devel-\nopment of the empirical basis of this quantity (using the \"S\",\n\"D\", \"F\" notation for radiant flux) is parallel to the un-\npolarized radiance case of Sec. 2.5, and therefore need not\nbe repeated here. Expression (1) constitutes the desired\noperational definition of the polarized radiance\nN(x, E, t, v, 4, e).\nThe Standard Stokes and Standard Observable Vectors\nThe operational definition (1) for polarized radiance\nand the experimental considerations leading to it draw out\nthe remarkable fact that the most general polarized radiance\nfield can be characterized by four functions, I,Q,U and V\nwhose values are determined once a selection of 4, E along\nwith x,E,t and v are given. In view of the potentially infi-\nnite variety of specific forms that polarized radiance fields\ncan assume, this is indeed a remarkably simple characteriza-\ntion and representation of the entire class of possible fields.\nThis theoretical characterization of the polarized light field\nby I,Q,U, and V was first systematically studied by Stokes in\n1852. We shall write:\n\"S\" for (I,Q,U,V)\nand call S the standard Stokes vector. S is a function which\nassigns to each choice of 4,E, along with x,E,t, and v, an\nordered quadruple of radiance numbers obtained as described\nabove.\nThere is an alternate method of quantitatively docu-\nmenting a polarized light field. Instead of obtaining I,Q,U,\nand V as described above, one may obtain four direct readings\nN(4,e) for the following four special pairs of settings w and\nE. We write:\n\"1N\"\nfor\nN(0,0)\nN(/2,0)\n2N\"\nfor\n'3N\"\nfor\nN(/4,0)\n\"4N\"\nfor\nN(T/4,/2)\nWe then go on to form an ordered quadruple from these numbers;\nwe write:\n\"N\"\nfor\n(1N,2N,3N,4N)\n,\nand call N the standard observable vector. N is a function\nwhich assigns to each choice of x,E,t, and V the four numbers\nshown. Observe how N requires use of W only for the setting\n4N. Readers familiar with the concepts of polarized light\nwill see that each N(T/4, /2) can be obtained by means of a\nsingle reading using a piece of circularly polarizing material.","SEC. 2.10\nPOLARIZED RADIANCE\n89\nWe shall see that S and N are equivalent descriptions of po-\nlarized light fields in the sense that knowledge of either\ndescription allows the deduction of the other. Since N is\noperationally the simpler of the two radiance vectors, we\nshall henceforth work with N. But before going on to the ex-\nlusive use of N, we shall establish some important and useful\nconnections between S and N. Then, since S is computable di-\nrectly from electromagnetic theory, we will have available a\ndirect tie between N and electromagnetic concepts, to be used\nwhen needed.\nAs an illustration of how N characterizes the commonly\noccurring polarized radiances, consider the following list of\nspecial cases where \"N\" denotes the relative magnitudes of\nthe components iN of N.\nObservable\nVerbal Description\nRadiance Vector\n1\nvertically linearly polarized radiance\n(2N,0,N,N)\n2\n1\nhorizontally linearly polarized radiance\n(0,2N,N,N)\n2\n1\nlinearly polarized radiance at +45 o\n(N,N,2N,N)\n2\n1\nlinearly polarized radiance at -45°\n(N,N,0,N)\n2\n1\nright circularly polarized radiance\n2 (N,N,N,0)\n1\nleft circularly polarized radiance\n2 (N,N,N,2N)\n1\nunpolarized radiance\n(N,N,N,N)\n2\nTo tie in these conventions with electromagnetic con-\nventions, recall first that the optic axes of P and W lie ini-\ntially in the standard preferred orientation, i.e., they lie\nin a vertical plane. Now the E-vector in vertically polarized\nlight by convention lies in a vertical plane as it crosses\nplanes perpendicular to the direction of travel. (Recall\nthat the E-field is a transverse field.) In +45° linearly po-\nlarized light the E-vector lies in a plane tilted +45° from\nthe vertical plane containing the direction of travel of the\nray associated with the electric field E. The + direction of\nthe +45° is measured clockwise as one looks along the direc-\ntion of travel of the ray. Finally, right circularly polar-\nized light is by convention that light associated with an E-\nvector whose tip describes a clockwise circular motion on a\nstationary plane perpendicular to its direction of travel as\nseen on the incident side of the plane. The most general\nlight field can be resolved into its linear and elliptical\ncomponents. This is the Polarization Composition theorem of\nStokes.","RADIOMETRY AND PHOTOMETRY\nVOL. II\n90\nAnalytic Link Between S and N\nThe connection between the two vectors S and N is eas-\nily established by means of (1). On the basis of (1) and the\ndefinition of iN, i = 1,2,3,4, we have:\nN - 1 1 1 - Q\nN - 1 1 1 - V]\nP\nFrom this set of equations we may construct the matrix\nwhich transforms S into N. Thus let us write:\n1\n1\n0\n\"p\"\nfor\n0 0 0 -1\nThen:\nN = SP\n(2)\nS=NP-1\n(3)\nwhere:\n1\n-1\n1\n-\n1 -1-1 1\nP-\n0020\n0 0 0 - 2\nAs an example of the use of (3), we obtain the following rep-\nresentation of vertically polarized radiance in terms of\nStokes vectors:\n1\n1\n1(2N,0,N,N)\n(N,N,0,0)\n.\n0020\n0 0 0 - 2","SEC. 2.10\nPOLARIZED RADIANCE\n91\nThe reader will find it instructive to use (3) to obtain a\nlist of Stokes vector representations of the seven special\nobservable vectors given above.\nStandard and Local Reference Frames\nUp to this point in the exposition of the polarized\nradiance vector all operational activity has implicitly taken\nplace in a terrestrial coordinate system. In particular the\nsetting of the polarized radiance meter was such that if\n4 = 0, then the optic axis of the polarizer P of the meter is\nin a vertical plane. (See Fig. 2.26 (a). Such a frame of\nreference for polarized radiance measurements we call a stan-\ndard reference frame. We now introduce a second reference\nframe--the local reference frame--whose main virtue and rea-\nson for being is that at each point in an optical medium it\npermits a simple means of experimental determination of the\npolarized volume scattering function. Furthermore, the intro-\nduction of the local reference frame considerably facilitates\nthe formulation and handling of the various forms of the\ntransfer equation in the polarized context.\nTo establish a 1ocal-reference frame at a point x in\nan optical medium, two directions E' and E must be given (in,\nsay, a terrestrial coordinate system) such that E' and E\nuniquely determine a plane. In other words, the only require-\nment on 5' and E is that they not be collinear. We shall call\nE' and E, respectively, the incident and scattered directions,\nand the plane they determine together with a point x, the\nplane of scattering. See Fig. 2.27.\nOnce a plane of scattering is determined by x,5 and E,\nthe incident polarized radiance is measured as follows: place\nthe radiance meter at x so as to allow the flux in the direc-\ntion 5' to enter the meter's collecting tube. With the 4-\nsetting at 0 with respect to the meter's tube, rotate the en-\ntire radiance meter around E' so that the optic axis of the\npolarizer P lies in that plane A' through E' and perpendicu-\nlar to the scattering plane. With the radiance meter so ori-\nented, perform the four operations leading to iN, i = 1,2,3,4.\nDesignate the local observable vector by \"No\" where \"o'\" de-\nnotes the angle through which the vertical plane through E'\nmust be rotated clockwise around E'--when looking along the\ndirection of E'--SO as to become coincident with the plane A'.\nThus ' varies from 0 to II; similarly with for the radiance\ndetermination in the scattered direction E. In general it\ncan be shown (Sec. 111, Ref. [251]) that the standard obser-\nvable vector N is related to a local observable vector No by\nmeans of the equation:\n(4)\nwhere we have written:","scattered\ndirection","SEC. 2.10\nPOLARIZED RADIANCE\n93\n\"L()\" for\nsin 2 sin2+12 sin 20 sin² 1/2 sin 2\n0\nsin2-12 sin 2 sin 2 o sin2 sin 2\n0\nsin 2\n-sin 2\ncos 2\n0\n0\n0\n0\n1\nIt is useful to observe that L(0) has the properties:\nL(01+02) = L(01) L(Q2)\n(5)\nL-1(d) = L(-0)\n(6)\nThus, in particular, the inverse L-1(0) of L(0) is obtained\nsimply by replacing by - - in L(0).\nThe following example will illustrate the use of (4).\nFig. 2.26 (b) depicts a beam of +45° linearly polarized radi-\nant flux proceeding along direction E at point X. Hence\nN = 1/2(N,N,2N,N). The reference frame is now switched from\nthe standard reference frame at X to a local reference frame\nat x defined by a rotation of magnitude +45° around E. In\norder to find the components of the given beam of polarized\nflux in this new frame, we first note that:\n0\n0\n0\n0\n0001\nHence:\n0\n1\n0\n0\n0\n0\n0001\n- 1(2N,0,N,N)","94\nRADIOMETRY AND PHOTOMETRY\nVOL. II\nthat is, N 45 is vertically polarized radiance in the new\nframe of reference, as was to be expected.\nThe primary advantage of introducing local reference\nframes and their corresponding local observable radiance vec-\ntors lies in the fact that the volume scattering matrix ob-\ntained in most natural optical media can be given a simple\nstandard form whenever such frames are used. The general re-\nlation (4) permits the resultant scattering matrix to be fit-\nted into discussions using the standard observable radiances.\nRadiant Flux Content of Polarized Radiance\nHaving extended the concept of radiance to the polar-\nized context, the question now arises as to the necessary con-\nnection that exists between the readings of a radiance meter\nwith and without polarization attachments. Specifically, let\nN be (1N, 2 N, 3N, N), the observable radiance vector (in either\nstandard or local form) at a point and a given direction at\nthat point. Further, let N be the simultaneous radiance read-\ning of the meter at the same point and same direction with\nthe polarizer P and wave plate W removed. What is the con-\nnection between N and N? This question, interestingly, can-\nnot be answered within the theoretical framework of radiative\ntransfer per se; of course it can be answered on the empirical\nlevel quite easily. However, to establish the desired theo-\nretical connection one must appeal to some relatively finer-\ngrained picture of light phenomena, such as electromagnetic\ntheory. On such a more fundamental level both N and the com-\nponents of N are representable in terms of the principal con-\nstruct on that level: the electromagnetic wave. The desired\nconnection can be established by suitably relating these elec-\ntromagnetic representations of N and the iN. On that level\nthe desired connection is readily forthcoming (see, e.g., Ref.\n[43]) and is of the form:\nN = 1N + 2 N\n(7)\n.\nThis relation is interpreted as described above and\nunder the assumption that P has ideal transmittance 1/2 for\nunpolarized flux and W has ideal transmittance 1.\nOf course these ideals are not attained in practice.\nHowever, with (7) as a starting point, the associated practi-\ncal version is readily established. The customary operational\ndefinition of the transmittance of the polarizer assembly is\nas follows. We write:\nfor max (N(v,0)/N)\n\"T\"\n(8)\nFurther, N(4,0) is, as the notation implies, the radiance\nreading with the W setting E equal to 0 (or W removed entire-\n1y) and with P in place and with optic axis rotated an amount\n4, in the usual way. For example, it follows from (1), in\nthe case of linearly polarized radiance that N(4,0) varies","95\nEXAMPLES\nSEC. 2.11\nideally as cos 2 4. Hence N(4,0) reaches its ideal maximum of\nN(0,0) = N at 4 = 0. In practice, however, N(0,0) < N, and\nso T<1. It is now easy to establish that the practical\ncounterpart to (7) is:\n(9)\nN = (1N + 2N)\nwhere 1N and 2N are measured with the same polarizer P as\nthat used to obtain T in (8) and with the identical disposi-\ntion of W as that used for (8), i.e., either W is in the tube\nand E = 0, or W is removed entirely.\nTo summarize, if a radiance tube is fitted with attach-\nments to allow the determination of the observable radiance\nvector (1N, 2N, 3N, N), then the associated reading N of the\nmeter without these accouterments is generally related to the\nvector components by means of (9), with T defined as in (8).\nIn this way the radiant flux content N of N is established.\nOn the basis of relation (9) or its suitable generali-\nzations, it is possible to use tabulated polarized radiance\ndata to compute all the usual unpolarized radiance, scalar\nirradiance and vector irradiance quantities, etc. formulated\nin the preceding sections, simply by replacing \"N\" everywhere\nin those formulas by \"(1N+2N)/T\". In this sense then we un-\nderstand polarized radiance data to be more general than un-\npolarized radiance data, for it includes the latter as a\nspecial case.\nExamples Illustrating the Radiometric Concepts\n2.11\nIn this section, we conclude our discussion of geomet-\nrical radiometry and, before going on to the discussion of\nphotometry, we consider some examples which may serve to il-\nlustrate in some depth the various radiometric concepts and\nthe relations among them. The contents of this section are\nintended to serve a multiple purpose. First of all we take\nthe opportunity of collecting together some worked examples\nin geometrical radiometry which illustrate the theory devel-\noped above; secondly, various special topics of only limited\ninterest to hydrologic optics per se are considered on the\nbasis of their intrinsic radiometric merits; and finally the\nsection serves as a repository for certain special radiometric\nresults needed as a matter of course in the later developments\nof this work.\nExample 1 : Radiance of the Sun and Moon\nWe illustrate the use of the empirical radiance defini-\ntion (1) of Sec. 2. 5 by using it to compute the empirical\nfield radiances of the sun and moon. Now in (1) of Sec. 2.5,\nS is the area collecting the flux P(S,D) funneling down the\nset D of directions from either the sun or the moon. Hence\nS may be chosen at will and we fix it in this example as a","96\nRADIOMETRY AND PHOTOMETRY\nVOL. II\n432,000 miles\nSun\napproximately\n93,000,000 miles\n100radian\nEarth\nFIG. 2.28 Approximate angular subtense of the sun at\nthe earth.\nsquare meter of plane surface just outside the atmosphere and\nwhose normal when extended goes through the center of the sun\nor moon. For the purpose of computing N(S, D) we choose D to\nbe the solid conical set of directions from any point on the\ncollecting surface to and within the limb of the sun or moon.\nSee Fig. 2.28. We consider first the case of the sun.\nThe sun is a nearly spherical body with diameter nearly\n864,000 miles and at a distance of about 93,000,000 miles from\nthe earth. It follows that the half-angle subtense 0 of the\nsun at the earth's surface is very nearly:\n0 = 4.32 x 10 5 /9.3 x 107\n-\n3\n= 4.65 x 10\nradians","SEC. 2.11\nEXAMPLES\n97\nHence, by (12) of Sec. 2.5, the solid angle subtense 8(D) of\nthe sun is:\ns(D) nx(4.65)2 x10-\n6.78 x 10-5\nsteradians\nNow Table 1 of Sec. 2.4 gives an order of magnitude estimate\nof 10 watts/m2 2 of the irradiance P(S,D)/A(S) produced by the\nsun's radiation over the whole spectrum and which is incident\nthrough D on a surface S normal to the sun's rays. A more\naccurate estimate of this irradiance produced outside the at-\nmosphere is 1396 watts/m²; see Ref. [128]. A still more mean-\ningful alternate estimate of H(S,D) can be made for the inter-\nval of wavelengths in the visible spectrum (approximately 400\nto 700 millimicrons). In this case H(S,D) estimates vary\nfrom 542 watts/m2 to 555 watts/m2 (p. 31, Ref. [185]), see al-\nso Ref. [128]). Using the first estimate we obtain*:\nN(S,D) = H(S,D)/&(D)\n= 5.42 x102/6.78x 10-\n8 x 10 2 watts/m2 x steradian\n(1)\nThis radiance is the overall average radiance of the sun's\ndisk as seen just outside the atmosphere and over the wave-\nlengths of the visible spectrum. (Hence the set F of fre-\nquencies of Sec. 2.3 now consists of all frequencies from ap-\nproximately 4 x to 7 x \"/sec.)\nA good rule of thumb for remembering the angular sub-\ntense of the sun is that its entire disk subtends an angle of\nabout 1/100 of a radian. The more exact estimate is given\nabove. In other words the sun subtends about the same angle\nas a disk of a centimeter diameter at a meter's distance.\nWe turn now to the case of the moon. The geometric\nand radiometric principles are the same as in the case of the\nsun. And again, the crucial point of the calculation rests in\nthe estimate of H(S,D). For this case we assume that the ir-\nradiance H(S,D) of the sun is on the order of 7 x 10 5 times\nthat of the full moon over the visible spectrum. (See, e.g.,\nFig. 1.12 and Table 2 of Sec. 2.12.) In other words we assume\nthat for the case of the moon, H(S,D) = 7.75 x 10-4 watts/m2.\nEstimates of this ratio vary considerably. The one just chos-\nen is an order of magnitude estimate only for the purposes of\nthe present example.\nThe moon is a nearly spherical body with diameter near-\nly 2100 miles and at a distance of about 240,000 miles. It\nfollows that the half-angle subtense 0 of the moon at the\nearth's surface is very nearly:\n= (1.05 x103)/(2.4 x105)\n0\n4.38 x 10-3 radians.\n*\nAt sea level under a clean dry atmosphere, H(S,D) on the or-\nder of 472 watts/m2. See also Table 2, Sec. 1.2.","VOL. II\n98\nRADIOMETRY AND PHOTOMETRY\nHence, by (12) of Sec. 2.5, the solid angle subtense s((D) of\nthe moon is:\ns(D)= T(4.38)2x10- If\n= 6.00 x10-5 = steradians\n.\nUsing the adopted estimate we obtain:\nN(S,D) = H(S,D)/S2(D)\n= 7.75x10 /6.00 x 10-5\n= 13 watts/m2 x steradian\n(2)\nThis may be used as an overall average radiance of the full\nmoon's disk as seen at sea level on a clear night and over\nthe wavelengths of the visible spectrum. An extensive litera-\nture exists with reference to lunar photometry and radiometry.\nSee, e.g., [8].\nIn conclusion we note that the rule of thumb adopted\nfor the angular size of the sun as seen from earth evidently\nalso holds for the moon. For more detailed radiometric infor-\nmation on the radiant energy output of the sun, the reader may\nconsult, e.g., Sec. 1.1 and Refs. [185] and [128]. Detailed\ndiscussion is made of the estimates of the solar irradiances\nin the latter references.\nExample 2: Radiant Intensity of the Sun and Moon\nThe present example illustrates the use of the concept\nof radiant intensity as defined in (1) of Sec. 2.9.\nWe begin by computing the radiant intensity of the\nhemisphere S of the sun visible from the earth. Let E be the\nunit vector pointing from the center of the sun to the center\nof the earth. Then the radiant intensity J(S,5) of S in the\ndirection E is given by (14) of Sec. 2.9, where N(x,5) is the\nsurface radiance of the sun in the direction E at a point x\non S. In Example 1 we estimated the field radiance of S for\nradiant flux in the wavelength interval from 400 to 700 milli-\nmicrons. Now, by the radiance invariance law (2) of Sec. 2.6,\nthe estimate of Example 1 may be taken as the surface radiance\nof the sun over S, the radiance N(x,5) being sensibly indepen-\ndent of x on S. Then if \"N\" denotes this fixed surface radi-\nance, (14) of Sec. 2.9 yields:\nJ(S,E) = N 5.5'(x) dA(x)\nS\nIf NA(S')\nwhere A(S') is the area of the projection S' of S on a plane\nperpendicular to E. The area A(S') is readily determinable.\nFrom the data in Example 1, we have:","99\nSEC. 2.11\nEXAMPLES\nA(S') ) = n (4.32)2x1010 = (miles) 2\n= (4.32) (1.6)2 x 106 x 10 10 (meters) 2\n= 1.5 x 10 ¹ (meters)\nUsing this estimate of A(S') and the estimate of N(S,D)\nfor the sun given in Example 1, we have:\nJ(S,)=8x106x1.5 x10 18\n1.2x1025 watts/steradian\n(3)\nas the radiant intensity of a hemisphere of the sun facing\nthe earth and over the visible spectrum.\nThe sun is radiometrically a point source (Sec. 2.9)\nwith respect to points on the earth and may thus be imagined\nto be compressed to its center X. Furthermore, we may evi-\ndently assume that J(S,E) is independent of 5. Hence (17) of\nSec. 2.9 is applicable, and we can estimate the total radiant\nflux output of the sun over the visible spectrum to be:\nP(x) = 4 TJJ(S,E)\n= 1.5 x 10 2 6 watts.\nTurning now to the case of the moon, we have a slightly\nmore interesting geometrical situation arising from the pos-\nsible phases of the moon. Fig. 2.29 depicts this situation.\nIf \"S'\" now denotes the projection of a lunar hemisphere on a\nplane normal to the direction E, then we have by means of (14)\nof Sec. 2.9:\ncos 0)\nwhere N is the surface radiance of the lighted hemisphere of\nthe moon, as estimated*, e.g., in Example 1, and 0 is the\nphase angle of the moon as described in Fig. 2.29. Thus at\nfull moon, 0 = 0 and J(S,E) is in particular NA(S). To esti-\nmate this product we first compute:\nA(S') = w(1.05)2x106 (miles)2\n= n(1.05)2x(1.6)2 x 106 x 10 (meters) 2\n= 8.9 x 10 1 (meters)\n*The precise analysis of the gradation of the radiance distri-\nbution over the sunlit hemisphere of the moon is a delicate\nproblem. The estimate here is deliberately kept simple in\norder to first emphasize the radiometric geometry essentials.\nA source reference on radiometry of the moon and planets is\n[8]. o","100\nRADIOMETRY AND PHOTOMETRY\nVOL. II\nSun's rays\nEarth\na\n0\n&\ns'\nMoon\nFIG. 2.29 Simple phase diagram for the earth-moon system\nUsing 13 watts/ (m ² x steradian) for N (justified by means of\nthe radiance invariance law) we have:\nNA(S') = 1.3 x10 x 8. 9 x 10 12 = 1.2x101 watts/steradian\nas the radiant intensity of the surface of the full moon over\nthe visible spectrum. Hence for any phase 0, the correspond-\ning radiant intensity of the lighted surface S of the moon in\ndirection E (Fig. 2.29) is:\nJ(S,E) = 0.6 x 10 1 (1 + cos 0) watts/steradian\n(4)\nWe conclude this example by computing the total radiant\nflux content of the reflected radiant flux from the moon, over\nthe visible spectrum. Using the radiant intensity estimate\njust made, and assuming N to be independent of direction S,\nand the moon to be a point source at its center x as seen from\nthe earth, we then integrate J (x, 5) over all directions to ob-\ntain the requisite radiant flux, according to (15) of Sec. 2.\n9.\nThus if \"x\" denotes the center of the moon and D' is now E,\nEquation (15) of Sec. 2.9 becomes:\nP(x) = J(x,5)\n= 2 TT\n(1+cos0) sin 0 de do If 2NA(S)\n=0\n0=0","SEC. 2.11\nEXAMPLES\n101\nTo perform this integration, a \"lunar based\" polar coordinate\nsystem was used with 0 measured as shown in Fig. 2.29 and\nmeasured from 0 to 2n around the axis a -a in a plane perpen-\ndicular to the page. We might have expected this relation on\nintuitive grounds: the radiant flux output of an entire\nsphere lighted uniformly all over, should be just twice that\ngiven off by one hemisphere. Thus P(x) is 2n times NA(S) in-\nstead of 4TH times NA(S'). Hence:\nP(x) = 2 x 1.2 x 10 14\n= 7.5 x 10 14 watts\nis the radiant flux output of the moon over the visible spec-\ntrum.\nExample 3: Radiant Flux Incident on Portions of the Earth\nIn this example, Equations (7) of Sec. 2.4 and (8) of\nSec. 2.5 will be illustrated. Now, from Example 1, we find\nthat at each point x just outside the atmosphere of the earth\nwe have (x, D,t,F) = 542 watts/m2 funneling down a narrow\ncone D from the disc of the sun and with wavelengths over the\nvisible spectrum F. Suppose S is some portion of the earth's\nsurface accessible to the sun's rays, as in Fig. 2.30. To\ncompute (S, D, t, F), we establish a polar coordinate system as\ndepicted in the figure. We first deduce that:\nSun's rays\nD\n0\nSubsolar\nHemisphere\nX\nS\nE(x)\nFIG. 2.30 Geometry for solar irradiation calculations","VOL. II\nRADIOMETRY AND PHOTOMETRY\n102\nH(x,D,t,F) = F H(x,D,t,v) d1(v)\n(5)\nwhich follows from (5) of Sec. 2.4 by means of a theorem of\nelementary calculus. Then by (7) of Sec. 2.4 we have:\nH(x,D,t,F) dA(x)\nNext, from (8) of Sec. 2.5:\n=\nD\nwhere we now explicitly use the fact that wavelengths are\nover the visible spectrum F. Since D is small and the sun's\nfield radiance is uniform of magnitude N over D we can esti-\nmate H(x,D,t,F) fairly accurately by means of the equality:\nH(x,D,t,F)=NE.E()s(D) =\nwhere N and s(D) were estimated for the sun in Example 1.\nFurthermore, E(x) is the unit inward normal to the earth's\nsurface at X, and E' , is the direction from the center of the\nsun to the center of the earth. Using this representation of\nH(x,D,t,F) in the preceding integral for $ (S,D,t,F), we ar-\nrive at the expression:\n(S,D,t,F) NS(D)\n(6)\nNA(S')&(D)\nwhere A(S') is the area of the projection S' of S on a plane\nnormal to the direction E' of the sun's rays.\nAs a specific example, we use N and So (D) as in Example\n1, and let S be the sub solar hemisphere of the earth. Then:\nA(S') = (4)2 x 106 (miles)2\n= =n(4)2x(1.6)2x 106 x 106\n= 1.3x101\"(meters)2\nHence:\n$(S,D,t,F) = 8 x 106 x 1.3 x 10 14 x 6.78x 10-5\n(7)\n= 7 x 10 16 watts","SEC. 2.11\nEXAMPLES\n103\nover the visible spectrum. The corresponding radiant flux\n& (S1 D,t, F) incident on any proper portion S1 of the entire\nsubsolar hemisphere is simply obtained by finding A(S1 ') /A(S),\nwhere now S1' is the projection of S1 on a plane normal to\nthe sun's rays, and then multiplying 7 x 1016 by this fraction;\nor alternatively, 542 watts/m2 by A(S1'). of course these\nestimates are somewhat crude, and serve only to illustrate\nthe correct mathematical use of the geometric radiometry for-\nmulas deduced above. The present estimate of $(S,D,t,F) omits,\ne.g., the effect of the atmosphere which at each point subtly\nattenuates and augments the solar influx by permitting absorp-\ntion, scattering, and interreflections with the earth below.\nExample 4 : Irradiance Distance-Law - for Spheres\nIn this and several of the examples below we shall ex-\nplore some interesting consequences of the irradiance integral\n(8) of Sec. 2.5.\nWe begin the investigations by considering a spherical\nsurface S of radius a with uniform radiance distribution of\nmagnitude N at each point. Suppose that S is viewed at a\npoint x a distance r from the center y of S. The lines of\nsight lie in a vacuum and the background radiance of S is\nzero. See Fig. 2.31 (a). We ask: what is the irradiance\nH(x,5) at point x? Here E is the direction from y to X.\nEquation (8) of Sec. 2.5 is readily applied to the\npresent situation. For the present case we may use the radi-\nance invariance law to say that N(x,5') = N for every E' in\nthe conical set D of directions subtended by S at X. Hence\n(8) of Sec. 2.5 becomes:\nH(x,5) = N 1 5.5' do(E')\nD\n2\n0\n1\n= N\ncos 0' sin 0 1 de' do'\n0\n2nN/0\ne'\nsin\n0'\nde\n1\ncos\n0\n= N sin 2 0\nIf TN (a/r) 2\nIf we write, ad hoc,\n\"HT\"\nfor\nH(x,5)\n,\nthen we have found that:","RADIOMETRY AND PHOTOMETRY\nVOL. II\n104\nD\nS\n0\nX\n(a)\nE\nr\nD\na\n8\nX\n(b)\ny\nE\nS\nFIG. 2.31 Deriving the Irradiance Distance-Law for\nspheres and disks\nHT = AN/r2\n(8)\nwhere we have written:\n2\n\"A\" for na\ni.e., , A is the area of a great circle of S; alternatively A\nis the area of projection of S on a plane perpendicular to E.\nFrom (14) of Sec. 2.9 applied to the present case, we may\nwrite:\nHT r = J/r2\n(9)\nwhere we have written:\n\"J\"\nfor\nAN\nIt is to be particularly noted that Hr varies precisely\nas the inverse square of the distance r, where a sr. If r = a,\nthen:\nN\n(10)\n.","SEC. 2.11\nEXAMPLES\n105\nExample 5 : Irradiance Distance-Law for Circular\nDisks; Criterion for a Point Source\nEquation (8) of Sec. 2.5 will now be used to derive\nthe law governing the irradiance produced by a circular disk\nS of uniform radiance. See Fig. 2.31 (b). In that figure is\ndepicted a circular disk of radius a and of uniform surface\nradiance N at each point. The disk is viewed at point x on\nthe perpendicular through the center y of S at a distance r\nfrom the center. The set D of the lines of sight from x to\nS\nlies in a vacuum and the background radiance of S is zero.\nWhat is the radiance H(x,5) at point x? Here E is the direc-\ntion from y to X.\nEquation (8) of Sec. 2.5 can be applied to the present\nsituation, as in the case of Example 4. Thus, (8) of Sec.\n2.5 becomes:\nH(x,5) = N\nD\n5/20/10\n2\n= N\ncos e' sin e' de' do'\n2nN\n0\nsin\ne'\nde\n1\n=\n= TN sin20\n= TN a2/(a2+r2)\n.\nIf we write, ad hoc:\n\"HT'\"\nfor\nH(x,5)\nand further, we write:\n2\n\"A\"\nfor\nTTA\nand:\n\"J\"\nfor\nAN\n,\nthen we have found that:\n(11)\nor:\nHy' = J/ (a2+r2)\n(12)","106\nRADIOMETRY AND PHOTOMETRY\nVOL. II\nFrom this we find first of all that Hr', unlike Hr of Example\n4, does not vary precisely as the inverse square of r, where\nr=0. However, in the special case of r = 0, , we have:\n(13)\n.\nFurther, in the other extreme, i.e., , when r is very much lar-\nger than a, Hr' varies very nearly as the inverse square of r.\nBy examining more closely the difference between Hr\nand Hr we arrive at the basis for the definition of a point\nsource given in Sec. 2.9. Suppose then we compare Hr and Hr'\nwhich are, respectively, the irradiances produced by a sphere\nof radius a and a circular disk of radius a both of uniform\nradiance N. Toward this end we form the difference:\nand then form the relative difference:\n- 1 If a2/r2\nThis relative-difference expression is the basis for\nthe following statements: The relative difference between\nthe irradiance Hr and Hr' is less than 1% whenever r>10a.\nMore generally: the irradiance produced by a finite object of\nuniform radiance decreases as the inverse square of the dis-\ntance from that object, within an error of 1 percent, when-\never the distance from the object is more than 10 times great-\ner than the object's largest transverse linear dimension.\nThis alternate statement follows readily from the preceding\nanalysis. Some further study is made in Example 6 of related\nquestions. Observe that the associated solid angle of the\ncircular cone of half angle 1/10 radian is very nearly\n(1/10) 2 = /100 = 1/30 steradian, in which lies the origin of\nthe solid angle number used in the point source criterion of\nSec. 2.9.\nExample 6: Irradiance Distance-Law for General Surfaces\nWe devote this example to the elucidation of the com-\nmon denominator of Examples 4 and 5; the net result being the\nformula for the irradiance distance-law for a general surface\nS of uniform radiance N viewed, as in Fig. 2.32, from an ex-\nternal vantage point X along a set of paths defined by a col- -\nlection D of directions, each path of which lies in a vacuum.\nThe derivation of the required H(x,5) begins, as in\nExamples 4 and 5, with (8) of Sec. 2.5, but now proceeds as\nfollows:","107\nSEC. 2.11\nEXAMPLES\ntransverse directions\nE longitudinal direction (normal to P)\ny\nE's\nO(P)\nX\nD\nC'\nP\nFIG. 2.32 Deriving the Irradiance Distance-Law for gen-\neral surfaces\nH(x,5) = N S\nD\n2 ft\n0\n()\nCOS e' sin 0 1 de' do\n= N\n0\n0\n2\n0(0)\nd (sin 0') do\n=\n0\n0\n()\ndo\n0\nLet us write, ad hoc:\n\"H\"\nfor\nH(x,5)\nWith this, we have attained the required result:\n() H\n(14)\n0","108\nRADIOMETRY AND PHOTOMETRY\nVOL. II\nThis formula for H reduces to the expressions for Hr and Hr'\nwhen the function 0 (.) is suitably prescribed for all from\n0 to In particular 0 is a constant function in the pre-\nceding two cases. More importantly, the reader should observe\nthe remarkable fact that the irradiance H depends only on the\nintegral over the outline C of S, as may be seen by studying\nthe central projection of S onto the background plane P (of\nFig. 2.32) which is perpendicular to E. Hence it is literally\nimmaterial to H what the longitudinal structure of S is as\nregards the computation of H at a fixed point x, as long as S\nhas the given outline C on P, and also has uniform radiance N.\nof course the shape of S is important when it is decided to\nlet x vary, and indeed the distance law for H(x,E) depends\ncritically on the longitudinal shape of S and in this context\ntakes its most general form displayed in the above equation\nfor H.\nAn alternate form of the distance-law for irradiance\nis obtained when we write:\n1\n\"0\"\nfor\n5' d&(5')\nD\nHence:\nH = NE.S\n(15)\nWhen the size So of D is small--e.g., when S is a point source\nat x, then we have, very nearly:\n8=58\nand in this special case (15) yields:\nH = No .\n(16)\nIf A is the projected area of S then in this case we\nhave very nearly:\nSo A r 2\n,\nwhere r is the distance from x to S. In this way we return to\nthe inverse square law for H in the limit of large r (or small\nA).\n.\nStill one more form for H, i.e., H(x,5) is obtainable\nusing the concept of vector irradiance introduced in Sec.2.8.\nThus we have\nH = E.H\n(17)\nwhere in the present case we have written:","SEC. 2.11\nEXAMPLES\n109\n1\n\"H\"\nfor\nNE' do(E')\nD\nAs a corollary we have:\nH = No\n(18)\nAn important and useful special case of (14) occurs when 0 ()\nis independent of . This happens when the surface S is a\nsurface of revolution about the direction E. (See Fig. 2.32).\nIn such a case (14) becomes:\nH = TN sin 2 0\nIn particular, if S is an infinite plane, then at all dis-\ntances r from S, S subtends a half angle 0 = /2. Hence\nH = TN for all r in such a case.\nExample 7 : Irradiance via Line Integrals\nThe present example is designed to let us investigate\nin greater depth the irradiance integral (14) of Example 6\nwhich showed that the irradiance produced at a fixed point X\nby an arbitrary surface of uniform radiance depended only on\nthe angular outline of S as seen at the point X. Our goal in\nthis example will be to cast equation (14) into explicit line\nintegral form over the curve C which defines the outline of S.\nFigure 2.33 (a) is a reconstruction of Figure 2.32\nwith surface S omitted. What is left is the geometric essence\nof the irradiance calculations done in Example 6. Specifi-\ncally, we have retained the central projection of S on plane\nP through point X. The boundary C of this projection of S on\nP is a closed curve characterized by means of the function\n(.) which assigns to each , 0 2 ft an angle 0 (), which\n0\ndetermines point y on C as shown in Fig. 2.33. We denote by\n\"0\" the foot of the perpendicular dropped on P from X. Fur-\nther, \"r(d)\" will denote the distance from the fixed point x\nto the variable point y on C.\nWith these preliminaries established, we can write (14)\nin the form:\n2\nH - N\n0(0)\nsin 0 () do\nr()\n0\nThe integral was rewritten this way to make use of the fact\nthat:\nr(d) sin 0 () do","110\nRADIOMETRY AND PHOTOMETRY\nVOL. II\n(0)\n(a)\nE'\no\na\nC\nP\ny\na\n(b)\n+\no\nC\n0\nA\nn(e)\n(c)\nB\nave\nFIG. 2.33 Setting for calculating irradiances via line\nintegrals.","SEC. 2.11\nEXAMPLES\n111\nis an element of length da along the direction of the unit\nvector a in P, normal to Oy, and at point y on C, as shown in\nFig. 2.33 (b), which is a plan view of P. The element of\nlength da is related to an element of arc length ds along C\nat y by a projection, and by definition of ds:\nda = cos a ds\nwhere a is the angle between a and the unit tangent vector s\nto C at y; hence:\nr(d) sin 0 () do = da = a°s ds\n.\nThe preceding integral then may be written:\nH - N/\nds\nr()\nC\nNext we observe, by means of Fig. 2.33, that:\na sin ( )\nwhere E is the unit outward normal to P (outward relative to\nE' 1 , i.e., , such that E'.) Hence:\nds\nr()\nThe triple box product of vectors in the integrand may be re-\narranged so that we obtain for C (or C'!) in Fig. 2.32:\n(19)\nC\nComparing this with (15) we deduce that:\n(20)\nEquation (20) displays a line integral representation of &,\nand (19) displays the desired line integral representation of\nH.\nAs an illustration of (20), let C' be the boundary of\na spherical lune L of angular opening 0, on a sphere of radius\na, as shown in Fig. 2.33 (c). . Thus L is now a specific in-\nstance of the general surface S of Fig. 2.32, and C may actu-\nally be taken as any outline of S (as, e.g., C' in Fig. 2.32).\nNote the present placement of point x and the direction E.\nThe contribution to of over the upper arc A of C is clearly:","112\nRADIOMETRY AND PHOTOMETRY\nVOL. II\nwhere n (0) is the unit normal to the plane containing arc A,\nand directed such that:\n,\nwhen A is traversed as shown in the figure. Further, r() =\na\nfor all . The contribution to of over the arc B of C is\nclearly:\nThe integrals over A and B were evaluated immediately by not-\ning that over A, s x 5' is a fixed unit vector, namely n(0);\nand over B, s x E' is the unit vector E, the unit inward nor-\nmal to the plane of arc B. The arc lengths of A and B are\neach aw. Hence for the present case:\n.\nObserve that if 0 = 0, then, n(0) = -E, and So = 0. If 0 = IT,\nthen n(II) = E, and of = TT. If L is of uniform radiance N,\nthen, by (15) or (19) :\nH End\n.\nExample 8: Solid Angle Subtense of Surfaces\nThe integral form of the solid angle subtense s((D) of\na set D of directions, as given in (10) of Sec. 2.5, will now\nbe recast into a form which arises when the solid angle sub-\ntense of specific surfaces (either real or hypothetical) are\nunder consideration. Thus, consider the surface S depicted\nin Fig. 2.34 (a) where S is shown viewed from an external van-\ntage point X. Let \"D(S,x)\" denote the set of all directions\nfrom points of S to X. Our present goal is to derive the ex-\npression for s(D(S,x)) (or \"s(s,x)\" for short) in the form of\na surface integral over S.\nWe begin by letting \"D\" in (10) of Sec. 2.5 be re-\nplaced by \"D(S,x)\". The result is:\ns(s,x) = 1\nsin 0 de do\nD(S,x)","113\nSEC. 2.11\nEXAMPLES\nD(s,x)\ny\nn(y)\n(a)\nS\n0\nX\nE(y,x)\nn(y)\ny\nr(y,x)\nE(y,x)\n(b)\nde\n0\n(do\nE\n(c)\nd\no\nsection through\n+\nsphere determined\nby points x,O,y.\nn(x)\nn(y)\n&\nE(y,x)\ny\nr(y,x)\nFIG. 2.34 Calculating the solid angle subtense of\n\"tangible\" surfaces.","114\nRADIOMETRY AND PHOTOMETRY\nVOL. II\nThe conventions for measuring 0 and are summarized in Fig.\n2.34 (a). In particular the details of the integration over\na part of S about a point y on S are depicted in part (b) of\nFig. 2.34. Points y and x determine a direction 5(y,x), as\nshown. It may be seen from part (b) of Fig. 2.34, that the\nrelation between a small patch of S of area A(y) about y is\nrelated to its projection's area A' on a plane perpendicular\nto 5(y,x) by the formula:\nA'(y) = A(y)n(y) (y,x) (=1 0, by choice of S)\nwhere n(y) is the unit outward normal to S at y. Hence the\nsolid angle subtense of the patch of S about y is:\nA'(y)\nn(y).E(y,x)A(y)\nre(y,x)\nThe entire solid angle subtense of S at x is obtained by add-\ning up all these solid angle subtenses of the component patches\ncomprising S :\ns(s,x) dA(y)\n(21)\nIt is of interest to observe that the set function so is\nnon-negative valued, S-additive and S-continuous (compare\nthese properties with those of the radiometric concepts in\nSec. 2.3). Thus for every X and pair S1, S2 surfaces with\ndisjoint sets D(x,S1) and D(S2,x) we have:\ns(S1,x) + S(S2,x) = S(S1US2,X)\nwhich is the S-additivity property; further:\nIf A(S) = 0, then s(s,x) = 0 .\nIn other words, the latter statement, the S-continuity prop-\nerty for s(°,x), asserts that s(s,x) > 0 only if A(S) > 0.* It\nfollows from these additivity and continuity properties of S\nand the calculus that the ratio s(s,x)/A(S) has a limit as\nS+{y}, where y is some point of S. Indeed:\nlim\n.\nr2(y,x)\nS+{y}\nThe converse clearly does not hold; thus, give a counterex-\nample for: If s(s,x) = 0, then A(S) = 0.","SEC. 2.11\nEXAMPLES\n115\nWriting \"dd(y,x)/dA(y)\" for the limit operation above, we can\nthen state that:\nda(y,x)\nn(y).5(y,x)\n(22)\nr2(y,x)\ndA(y)\nEquation (22) yields, for a given point X, the value\nof the general area derivative of the solid angle function\ns(,x) at point y of an arbitrary surface, where n(y) is the\nunit outward normal to the surface at y, r(y,x) is the dis-\ntance from x to y and 5(y,x) is the unit vector from y to X\nand such that the dot product of E(y,x) and n(y) is non-nega-\ntive (this fixes the sense of \"outward\" for n(y)). As an in-\nteresting exercise the reader should show that if x and y in\n(22) are on a spherical surface S of diameter d, (see Fig.\n2.34 (c)) then:\nds(y,x)\n1\ndA(y)\n[n(y) 2\nThe representation of s(s,x) in (21) is of particular\nvalue when the surface S is relatively concrete and has a\nspecific analytic description, (parts of spheres, walls, and\nrelatively tangible surfaces in general), whereas (10) of Sec.\n2.5 is of greatest use when no surface S is specifiable and\nwhen instead a set D of directions per se is to be assigned a\nsolid angle value. We close this example with the observation\nthat all of Euclid's Optics [36] can be placed on a solid mod-\nern mathematical foundation using (21) and its various logical\ncorollaries. (The translation of the first theorem in Euclid's\nOptics is given as a motto at the beginning of Volume I of\nthis work. The theorem thus has several levels of meaning.)\nExample 9 : Irradiance via Surface Integrals\nWe return now to the integral for irradiance given in\n(8) of Sec. 2.5 and cast it into that form which is most use-\nful when one must take into specific account the surface ra-\ndiance of some surface S producing an irradiance H(x,5) at\nsome point x outside of S. The geometric setting for the\npresent example is depicted by Fig. 2.34 (a), where at each\npoint y of the surface S, there is given a surface radiance\ndistribution N(y,.)). We assume that all directions in D(S,x)\nlie in a vacuum, that D(S,x) is less than a hemisphere, and\nthat the irradiance contributions to H(x,5) come only from S\nso that N(x,5') in (8) of Sec. 2.5 is zero for E' outside of\nD(S,x). Hence (8) of Sec. 2.5 may be written:\nH(x,5) = I\nD(S,x)\nIn the present study the dummy variable \"E\" is chosen to be\nthe name of the variable direction 5(y,x) used in Example 8.","116\nRADIOMETRY AND PHOTOMETRY\nVOL. II\nThen, in view of the radiance invariance law (2) of Sec. 2.6:\n,\nit is clear that H(x,5) is also represented by the integral:\n=\n.\nS\nThis may be written in the form:\nH(x,5) = 1 N(y,5')5.51 d&(y,x) dA(y) dA(y)\nS\nwhich, by (22) is reducible to:\nH(x,5) = N(y,E(y,x)) E.E(y,x)n(y).E(y,x) r2(y,x)\ndA(y)\n(23)\nEquation (23) is the desired surface integral representation\nof H(x,5). Suppose we write:\n1\nN(y,E(y,x)) E(y,x)n(y).E(y,x) dA(y)\n\"H(S,x)\"\nfor\nr2(y,x)\nS\nThis is the surface radiance counterpart to the field radiance\ndefinition of H(x) in (2) of Sec. 2.8. Then (23) becomes:\nH(x,)=g.H(S,x)\n(24)\n.\nEquation (24) suggests that the condition imposed at the out-\nset, namely that D(S,x) be less than a hemisphere, can evi-\ndently be relaxed. In that event (24) is generalizable to:\nH(x,5) = E+H(S,x)\n(25)\nthe proof of which is left to the reader.\nIf we assume that the point x is inside a closed sur-\nface S, then (23) still holds but with n(y) now being inter-\npreted, if desired, as an inward unit normal from y to X. In\nthat case, H(S,x) of (24) formally reduces to H(x) in (2) of\nSec. 2.8. These observations suggest that the true field ra-\ndiance counterpart to (25) is:\nH(x,5) = E+H(x, ),\n(26)","SEC. 2.11\nEXAMPLES\n117\nwhere we have written:\nI\n\"H(x,D)\"\nfor\nE'N(x,5') dd(E')\n(27)\nD\nThe connection between H(x,D) and H(x) of Sec. 2.8 is clearly\nthat:\nH(x)\n(28)\n.\nAn interesting special case of (23) arises when S is\npart of the inside of a spherical surface. In Fig. 2.34 (a)\nimagine x and y to be on the same spherical surface of diame-\nter d, and now E in (23) is to be the unit outward normal to\nS at x, i.e., E = -n(x). Under such conditions (see Fig.\n2.34 (c)) it follows that:\n5.5(y,x)\nand\nr(y,x) dn(y)(y,x)\nHence if N(y,E(y,x)) = N, over S, (23) yields:\nH(X,{) - NA(S)\nfor every X on the sphere, and arbitrary subset S of the\nsphere.\nExample 10: Radiant Flux Calculations\nThe irradiance integral (23) may be applied to the fol-\nlowing radiometric setting, depicted in Fig. 2.35, which arises\nfrequently in practice. A surface Y has a prescribed surface\nradiance N(y,.) at each point y. Surface X, which is disjoint\nfrom Y, receives an amount P(Y,X) of radiant flux from Y. It\nis required to express the amount P(Y,X) in terms of N(y,°)\nand the areas of X and Y, assuming the space between X and Y\nis a vacuum. Now from (23) we have for each x and E an ex-\npression for the irradiance H(x,5), so that we can immediately\ncompute P(Y,X) in terms of H(x,n(x)), using (6) of Sec. 2.4\n(in which D is now (1) (n(x)) ):\nP(Y,X) = H(x,n(x)) dA(x)\n,\nwhere n(x) is the unit inward normal to X at X. Hence\nP(Y,X) = (y,5(y,x)) n(x).E(y,x)E(y,x)n(y) dA(y) r2(y,x) dA(x)","118\nRADIOMETRY AND PHOTOMETRY\nVOL. II\nn(y)\nY\nE(y,x)\nD\n-X\nX\nn(x)\nFIG. 2.35 A radiant flux calculation for two disjoint\nsurfaces.\nIf we write\nn(x).E(y,x)E(y,x).n(y)\n\"K(y,x)\"\nfor\nr2(y,x)\nThen, more succinctly,\nP(Y,X) = 11 N(y,e(y,x))K(y,x) dA(y) dA(x)\n(29)\nObserve that K( .,.) is a symmetric function, i.e., for every\nx and y,\nK(x,y) = K(y,x)\nIf the areas A(X) and A(Y) of X and Y are small say when each\nis a point source with respect to any point on the other--then\n(29) yields the useful approximate relation:\nP(Y,X) = N(Y,X) K(y,x) A(Y) A(X)\n(30)\nwhere x and y are some fixed points of X and Y, respectively\nand \"N(Y,X)\" denotes the surface radiance of Y in the direction","SEC. 2.11\nEXAMPLES\n119\n5(y,x) of X. Writing, ad hoc:\nH(Y,X)\nfor\nP(Y,X)/A(X)\nand\n\"J(Y,X)\" for N(Y,X)A(Y)\n,\nEquation (30) becomes:\nH(Y,X)=J(Y,X) K(y,x)\n(31)\n,\nwhich is a highly compact formulation of several well-known\nradiometric laws, with built-in cosine laws for both irradi-\nance and radiant intensity, and furthermore, with built-in in-\nverse square law for irradiance and direct square law for ra-\ndiant intensity.\nExample 11: Intensity Area-Law for General Surfaces\nThis example serves to illustrate some further facets\nin the duality between irradiance and radiant intensity devel-\noped in various earlier sections throughout this chapter. In\nparticular we now direct attention to the intensity counter-\nparts of the relations (15)-(18) in Example 6 of this section.\nThus, starting with (22) of Sec. 2.9 as a conceptual\nbase, let us write:\n\"A\"\nfor\ndA(x)\nS\nThen if a surface S has a constant uniform surface radiance N\nover the part S(E) defined by a direction E (Sec. 2.9) then\nJ = NE.A\n(32)\nwhich is the exact intensity-counterpart to (15), and where\nwe have written:\n\"J\" for J(S,5)\nWhen the shape of S is nearly planar, e.g., when E' (x) varies\nwithin a solid angle 1/30 steradians over S, then we have,\nvery nearly:\nA = EA\nwhere A is the area of S. In this special case (32) yields\nthe present counterpart to (16):\nJ = NA\n(33)\n.\nIf S is a point source with respect to a point x, a distance\nr from S, then the apparent area of another point source S'","120\nRADIOMETRY AND PHOTOMETRY\nVOL. II\nsimilar in shape to S is related to that of S by the equation:\nA(S') C12\nwhere C is a constant and 1 is some given linear dimension of\nS. In this way we return to the direct square law for J in\nthe limit of large r (or small A). That is, using this esti-\nmate of A(S') in (33), we obtain the present counterpart to\nthe inverse square estimate of irradiance; and as it stands\nby itself, the preceding equation is the dual to the relation\nSo = A/r2 used to estimate solid angle subtenses of point\nsources.\nStill one more form for J, i.e. , J(S,E) , is obtainable\nusing the concept of vector area introduced in Sec. 2.9.\nThus\nwe have:\nJ = E.J\n(34)\nwhich is the dual to (17), and where in the present case we\nhave written:\nN/ 5'(x) da(x)\n\"J\"\nfor\nN\ndA(x)\nS\nAs a corollary we have:\nJ = NA\n(35)\nwhich is the dual to (18).\nExample 12: On the Possibility of Inverse nth Power\nIrradiance Laws\nThe cumulative evidence of the preceding examples, be-\nginning with Example 4, shows the predominant role played in\nradiometry by the inverse square law for irradiance. The law\nis evident in various guises in formulas (8) and (9) for\nspherical surfaces, in the point-source criterion of Example\n5, in the discussion of Example 6, in (21), (22), (29), and\nfinally, its dual (the direct square law for radiant intensity)\nis evident in the discussions of Example 11. A11 of this evi-\ndence appears to lead inexorably to the generalization that\nthe distance fall-off of irradiance produced by flux from all\nreal surfaces of uniform radiance must eventually (i.e., for\nlarge enough distance r) assume the inverse-square behavior\nwith r. This guess is essentially correct. However, the re-\nsult of Example 5 shows that for intermediate distances r, the\nirradiance decrease with r need not be exactly an inverse\nsquare type of decrease. A question of some interest now a-\nrises as to necessary conditions that may govern the rate of\nsuch decrease. For example, can a surface S be found such","SEC. 2.11\nEXAMPLES\n121\nthat the irradiance Hr over some interval of distances r from\nthe surface falls off exactly as C/r3, (where C is a con-\nstant)? Or, perhaps S can be found such that Hr behaves ex-\nactly as C/r over some interval of r. In general, can a sur-\nface S be found such that over some interval of r, H is of\nthe form C/rn, where n is any number? In this example we de-\nvote some attention to these questions as they are of intrin-\nsic interest and are of the kind which aid in forming a good\nintuition about the laws of geometrical radiometry.\nBefore attempting a systematic search for surfaces of\nthe kind discussed above, let us consider some special cases\nwhich may point the way to an appropriate methodical approach.\nFigure 2.36 (a) depicts a sphere S of radius a and uniform\nradiance N. The results of Example 4 let us conclude that\nthe irradiance Hr at distance r from the center of S is given\nby:\nHT=N(a2/r2)\nFig. 2.36 (c) depicts an infinite plane S of uniform radiance\nN. The results of Example 6 indicate that:\nfor every r. Further, Fig. 2.36 (b) depicts an infinite con-\nical surface S of half angle 0 Once again the results of\nExample 6 indicate that:\nHT N sin² 0 o\nfor every r, i.e., Hr in this case is independent of r but\nless than WN by the fixed factor sin 2 0 Finally, Fig.\n2.36 (d) depicts a general bounded closed smooth surface S of\nuniform radiance N which encloses a positive volume of space.\nSince S encloses a positive volume of space, there is a point\nX in S about which a sphere S can be drawn such that S1 lies\nwholly in S. Since S is bounded, there is a sphere S2 with\ncenter x, such that S lies wholly in S2. The spheres need\nnot be tangent to S. It follows that Hr, the irradiance pro-\nduced by S at any point outside S2 a distance r from x, must\nobey the following equalities:\n,\nwhere a1 and a2 are the radii of S1 and S2 respectively. This\nset of inequalities leads us to the following assertion: if S\nis any bounded surface enclosing positive volume and with uni-\nform surface radiance N, then associated with S is an irradi-\nance function Hr whose graph is bounded by two inverse square\ncurves. Thus for sufficiently large r, Hr is arbitrarily\nclosely described by an inverse square relation in r.\nIn view of the evidence just reviewed, the first main\nobservation toward resolving the question before us may be\nmade: If S is a surface with uniform radiance N and the A8-\nsociated Hr is of the form C/rn with n + 2 for every rza,\nwhere a is some nonnegative number, then S is either (a) not","122\nRADIOMETRY AND PHOTOMETRY\nVOL. II\nS\n(a)\nS\n0\n(b)\nE\nS\n(c)\nE\nS1\n(d)\nS\nFIG. 2.36 Attempting to generalize the inverse square\nlaw for irradiance.","SEC. 2.11\nEXAMPLES\n123\nbounded or (b) does not enclose a positive volume.\nLet us now attempt to find the actual shape of a sur-\nface S with the property that its associated irradiance varies\nprecisely as C/rn. Two conditions can be fixed at the outset\nin order to keep our initial search within reasonable con-\nfines. First we assume that n = 0. Secondly, we search only\nfor surfaces S which, like those in (a) - (c) of Fig. 2.36, are\nsurfaces of revolution about the direction E. It follows\nfrom (14) that the associated Hr is given by:\nHT = N sin 2 0 (r)\nwhere 0 (r) is the angle that the tangent to S from the point\nof observation, y, makes with the direction - -E. (See Fig.\n2.37.) Thus by specifying y and knowing S, we can in princi-\nple compute 0 (r) . Now use the hypothesized property of S to\nset up the following two equations:\nC r n = HT = wN sin 2 o(r)\nWe can evaluate the constant C by observing that: if S is a\nsmooth surface and r = a, then (r) = /2. This follows in- -\ntuitively, e.g., from the observation that the surface has\nthe appearance of an infinite plane for an observer at r = a.\nHence for smooth surfaces:\nC n = N\na\nO(ri)\nO(r)\ny\nx\nV\nyi\nE\na\nri\nr\nFIG. 2.37 Finding the shape of the luminous surface\nwhich has an inverse nth power irradiance law.","124\nRADIOMETRY AND PHOTOMETRY\nVOL. II\nso that:\nsin e(r) (a)n/2\n(36)\nis the necessary connection between r and e(r), where a is\nsome fixed length associated with S. In the case of a spher-\nical surface S and for the case of n = 2, a is simply the ra-\ndius of S. In general if the surface S divides all of space\ninto two separate parts (as, e.g., a plane or a paraboloid of\nrevolution) then we agree that a is the distance from the ver-\ntex V of S to some inside point x, the center of S, where y\nis, by agreement, outside of S.\nOn the basis of relation (36) a graphical construction\nprocedure can be evolved for the requisite surface S. First\nchoose n, with n > 0 and choose a, with a > 0. Then select\na\nset of distances r1, r2,\n, rk, such that a r1 < r2 <\n<\nrk.\nEquation (36) may now be used to compute the associated an-\ngles e(r1), 0 (r2),\n( (k). These angles are used in the\nfollowing manner. At each point yi which lies a distance ri\nfrom the center X along the axis of revolution of S, draw\ntwo straight lines making angles with the direction -E,\n(see Fig. 2.37). If the ri have been spaced sufficiently\nclosely together, then one may visually detect the envelope\nof the lines just drawn, i.e., the curve which is tangent to\neach straight line of the family just constructed. This en-\nvelope is the cross section of the desired surface S; i.e.\nby spinning this envelope around the direction E, the requi-\nsite S is formed.\nSome experimentation with the preceding construction\nprocedure yields information about how the surfaces S vary in\nshape as a function of the power n. Thus let the parameter a\nbe fixed. Then for every n in the range 0 n < 2, we find that\nthe associated surface s(n) is unbounded. The closer n is to\n0, the more of a conical shape is exhibited by S(n) about its\nvertex. The limiting curve (0) is a degenerate infinite cone\n00 = /2, of the kind depicted in (b) of Fig. 2.36. The\ncloser n is to 2, the more spherical is the shape of (n) in\nthe neighborhood of the vertex. The limiting curve S(2) is a\nsphere of radius a. The constructions of the surfaces (n)\nfor n >2 at first present rather puzzling anomalies. By choos-\ning the range of the values r1\nrk sufficiently small\nand having the ri closely packed together, one can construct\nthe surfaces S(3), S(4), within small regions around\ntheir vertices. In each case where n > 2, there is a critical\ndistance ro from the center X beyond which the envelope con-\nstruction degenerates. The larger n is, the smaller is the\ncorresponding critical distance ro.\nThese graphical experiments in constructing the surface\nS(n) for which the inverse nth power law for irradiance holds,\nespecially in the case of n >2, indicate the need for a rela-\ntively precise analytical approach to the problem of deter-\nmining S(n). We shall now briefly direct some attention to\nsuch an approach.","SEC. 2.11\nEXAMPLES\n125\nE\ny\n(((()\n-xy'(x) -\nc(n)\na\ny(x)\no\nincreasing\nX\nFIG. 2.38 Imbedding Fig. 2.37 in a cartesian frame.\nFigure 2.38 shows an xy coordinate frame in which the\ncross section of surface S(n) is depicted by curve c(n). The\nirradiance meter is imagined to be at a point on the y-axis\na distance r from the origin 0 of the frame. The unit inward\nnormal E to the collecting surface of the meter is directed\nalong the positive direction of the y-axis. The origin of the\nframe serves as the center of C (n) , and the vertex of C(n) is\na point on the y-axis a distance a from the origin. The curve\nC(n) is represented by some function y(). Our primary goal\nis to obtain the differential equation for the function y()\nof the curve C(n) . The starting point is equation (36) in\nthe form:\nsin² e(r) = (9)\"\nwhich, as we have seen, combines the inverse nth power re-\nquirement on the irradiance Hr with the general formula for\nHr. We now systematically replace r, and e(r), using the\nde-\n-\nscription of C(n) by y (.) Let us denote the derivative of\n(x) with respect to x by \"y'(x)\". First of all r is clearly\nthe algebraic sum of two terms: y (x) and -xy' (x), i.e.,\n= y(x) -xy 1 (x) , as a glance at Fig. 2.38 would show. Sec-\nr\nondly, it is also clear from the figure that:\ntan (r) = x/xy'(x)\n= 1/y'(x)","126\nRADIOMETRY AND PHOTOMETRY\nVOL. II\nFrom this:\nsin =\nHence:\n(y'(x))2 + 1 = = [y(x)-xy'(x)]\" (1/a\") .\n(37)\nWe have essentially reached our goal. Equation (37) is the\ndifferential equation for c(n). With an eye toward expediting\nthe solution of (37) we shall rearrange it into the form:\ny(x) xy'(x) + a a[1+(y'(x))2]1/1\n(38)\nEquation (38), as it stands, has the Gestalt of a Clairaut\nequation, an equation which is readily solvable in parametric\nform:\nx(t)\n(39)\ny(t) [1+t2]H\n(40)\nThis equation also has a singular solution of the form\n(40a)\nwhich represents straight lines of slope to. These singular\nsolutions evidently can yield the degenerate conical case\n00 = /2 depicted in (c) of Fig. 2.36.\nSetting n = 2 in (39) and (40), and eliminating the\nparameter t, we obtain:\nx2(t)+y2((t) = a2\nHence C(2) is a circle with center at the origin (0,0), and of\nradius a, as expected. Setting n = 1 in (39) and (40), and\neliminating the explicit appearance of the parameter t, we ob-\ntain\ny(t) = x2(t)/4a a","127\nSEC. 2.11\nEXAMPLES\nIn this case S(1) is a paraboloid of revolution with focus at\nthe origin (0,0), axis of symmetry along the y-axis, vertex\nat (0,a), and intercepting the x-axis at x = +2a. The surface\nS (1) is typical, as far as size (unboundedness) and general\norientation is concerned, of all S(n) with 0 0, a quadrant of\nthe circle S(2) is traced out and y(t)/x(t)+0. This tracing\nis depicted in (b) of Fig. 2.39. Further, if n = 1, then:\ny(t)/x(t) =\no\nIn this case, as t varies in the range 0 2, we see that\nlim y(t)/x(t) If -00, indicating that y (t) becomes much larger\nt->0\nthan x (t) as t->00. This in itself is not too informative, but\nwhen coupled with the observation that for large t,\n28 th\nx(t) behaves like\nand\n(1-2)th\n2\ny(t) behaves like\n,\nwe gain further insight into the behavior of the curves.\nFrom these observations we cull the following information:\nas t+00, ,\nfor 02: x(t)+0, y(t)+ +00\nThe behavior for the case n > 2 continues to present puzzling\nfeatures. Thus, when n > 2, x(t)+0 for large t, indicating\nthat x(t) I attains a maximum for some t. Examining (39) for\nthis possibility, we see that for C(n),","128\nRADIOMETRY AND PHOTOMETRY\nVOL. II\nn<2\n1=0 to t=00\n(a)\na\nr\n00\nn=2\n1=0 to 1=00\n(b)\na\n8\nB\nn >2\nt=0 to t=00\n(c)\nro\n(physically unrealizable)\nA\nrange\na\nFIG. 2.39 Some cross sections of surfaces which produce\nirradiance fall-off like 1/rn (see text).","SEC. 2.11\nEXAMPLES\n129\n/\nn\nmax /x(t)\noccurs at\nt\n=\nn-2\nFrom this relation, it is at once clear that we obtain observ-\nable maxima for n > 2, and no observable (i.e., real) maxima\nfor 0 2, S(n) ever intersects the x-\naxis as do its lower-n counterparts. Thus in (40) we set\ny(t) = 0, and find that:\nn\ny(t) = 0 when t =\n2-n\nHence if n > 2, then y(t) + 0 for every real t. This means\nthat C(n) does not meet the x-axis for n > 2.\nSummarizing the behavior of C(n) for n >2: as t varies\nfrom 0 to +00, a branch of C(n) is traced out which is of the\ngeneral form shown in (c) of Fig. 2.39. This figure explains\nthe source of our difficulties in the geometric constructions\nbased on equation (36). The construction is able to generate\na branch of S(n) from the vertex to the first point of inflec-\ntion at point A. Beyond A, the branch of c(n) and S(n) itself\nhas no conventional physical interpretation. Some interesting\nunconventional interpretations can be made; however, we leave\nit to the reader's initiative to interpret the meaning of C(n)\nbeyond point A. The tangent to C(n) at A meets the y-axis\nat a point B, which determines the critical range ro for which\nthe physically realizable inverse nth power law holds. Part\n(c) of Fig. 2.39 is drawn from computed data for the case\n= 3. The associated set of parametric coordinates are giv-\nn\nen in the table below, which was computed by Mrs. Judith\nMarshall.\nTable 1\nComputed Values for Part (c) of Figure 2.39\n(The case n = 3)\nt\n(t)\ny(t)\nx(t)\nt\ny(t)\n0\n-0\n1\n2.0\n-0.456\n0.797\n0.1\n-0.066\n0.997\n2.5\n-0.444\n0.826\n0.2\n-0.130\n0.987\n3.0\n-0.430\n0.802\n0.3\n-0.194\n0.971\n3.5\n-0.417\n0.906\n0.4\n-0.242\n0.954\n4.0\n-0.403\n0.958\n0.5\n-0.287\n0.933\n10.\n-0.307\n1.580\n1.0\n-0.420\n0.840\n20.\n-0.246\n2.43\n1.5\n-0.455\n0.798\n30.\n-0.213\n3.25\n1.6\n-0.458\n0.793\n40.\n-0.194\n3.91\n1.7\n-0.460\n0.790\n50.\n-0.180\n4.60","130\nRADIOMETRY AND PHOTOMETRY\nVOL. II\nSeveral general concluding comments can now be made on\nthe problem of the inverse nth power law for irradiance.\nFirst, there is the constantly recurring use of or reference\nto the integer 2 throughout the most general of the preceding\ndiscussions. Observe how critically 2 enters into the fol-\nlowing tabular classification of our main results:\nTable 2\nThe surface s(n) inducing\nRadius of\ncurvature at\nRange of\nH_ r = C/r\nn\nvertex\nvalidity\n0\nCurve with half-angle\n0 (= 0/2a)\nrza\n0 o = /2 (Fig 2.36 (c))\n1\nParaboloid of revolution\n1/2a\nrza\n(Fig. 2. 38)\n2\nSphere (Fig. 2.38)\n1/a (= 2/2a)\nrza\n3\n3rd order luxoid\n3/2a\nrzrza o\n(Fig. 2.39 (c))\nnth order luxoid\nn/2a\nn\nrozrza\n(Fig. 2.39 (c))\nIn this classification we encounter classical euclidean sur-\nfaces for all n, 0 kns2, but a definite break occurs at\n= 2, as has been repeatedly evident in the curve-tracing\nn\ndiscussion above. A11 this is apparently closely related to\nthe fact that we live in a three dimensional world, or at any\nrate, the radiometric laws above are represented most natural-\n1y in euclidean frames of dimension 3. The three dimensional\ngeometric frame has been used implicitly throughout our dis-\ncussions. We are thus led to conjecture that radiometry in a\ntwo dimensional world would have a ubiquitous inverse first\npower \"irradiance\" law and radiometry on a line would have\nits inverse zero power irradiance law. It is interesting to\nspeculate on the theory and utility of k-dimensional geometric\nradiometry in which very likely the \"irradiance\" in such a\ngeometry will obey an inverse k-1 power law, and to contem-\nplate the potentially rich multiplicity of irradiances, sca-\nlar irradiances, and radiant intensities and their manifold\ninterconnections latent in such geometries. Here the dualities\nbrought out between irradiance and radiant intensity in the\npreceding examples are likely to attain their deepest and\nbroadest meanings. These observations are commensurate with\nthe conclusions, already brought out in the studies in Sec.\n99 of Ref. [251], that radiometry and radiative transfer are","SEC. 2.11\nEXAMPLES\n131\nmeaningfully formulable in arbitrarily general spaces. These\ninteresting matters are left to the reader for further con-\nsideration.\nExample 13: Irradiance from Elliptical\nRadiance Distributions\nWe shall now illustrate the use of equation (17) of\nSec. 2.5 by computing the irradiance distribution associated\nwith an important type of theoretical radiance distribution,\nnamely the elliptical radiance distribution. The elliptical\nradiance distribution arises in the study of light patterns\nat great depths in oceans, lakes, and other natural hydrosols.\nIt is a convenient theoretical standard against which to com-\npare the light patterns that actually subsist in nature. The\nphysical basis for the elliptical radiance distribution will\nbe considered in Chapter 4. For the present we shall be con-\ncerned only with its geometric properties. In particular we\nshall investigate its structure with respect to direction, to\nvariation in eccentricity, and also compute its associated\nvector and scalar irradiances.\nWe begin by setting the geometric stage of the computa-\ntion. Figure 2.40 (a) depicts a terrestrial coordinate frame\n(Sec. 2.4) and the plane of a radiant flux collector oriented\nas usual by its unit inward normal E. Let the associated di-\nrection angles of E be 0 and . Thus if 0 = TT and = 0, say,\nthen E = -k and the collector receives flux from the upper\nhemisphere, i.e., receives flux flowing in the directions of\nE.. In general, when E is the unit inward normal to the col-\nlecting surface, the incident radiant flux is along the direc-\ntions of the hemisphere E(E), as defined in Sec. 2.4.\nNext we define an elliptical radiance distribution at x,\nof eccentricity E, Oses1, and magnitude N, to be a radiance\ndistribution of the form:\nN(x,5') = N/(1+e'k)\n(41)\nAn alternate mode of representation of N(x, E') is by means of\npolar and azimuthal angles. Thus (41) may be written\nN(x,0',0') = N/(1+e cos 0 )\n(42)\n.\nThe upper diagrams of Fig. 2.41 show four plots of elliptical\nradiance distributions N(x, ) of eccentricity E = .25,.50,.75,\n.95. For small values of E near 0 the associated distribution\nis predominantly spherical. For values of E near 1, the as-\nsociated distribution is long and narrow. When 0 = 0, the\nflow is upward and smallest; when 0 = II, the flow is downward\nand greatest, thus simulating, at least qualitatively, the\nreal flows in nature. (We are using surface rather than field\nradiance.) The \"size\" of the distribution is governed by N,\nbeing the radiance in the horizontal directions, i.e., E's\nwith angles of the form (/2,0). The ratio of downward (zen-\nith) to horizontal radiances in the present model is given by:","RADIOMETRY AND PHOTOMETRY\nVOL. II\n132\n(a)\nk\n(p')\nH\nunit inward normal\nplane of\nto collector\n(8,0)\n(8,)\ncollector\n0\n0\nI\n= (8,0)= (1)\nI\n2 -\nN\nvertical radiance\ndistribution\n4\n(b)\nk\n(unit inward normal to\ncollector)\n(8,)\n0\nplane of\n=0 for Ex. 14\ncollector\n= for Ex. 15\ni\nN\nrotated radiance\ndistribution\nFIG. 2.40 Some calculation details for irradiance from\nelliptical radiance distributions.","SEC. 2.11\nEXAMPLES\n133\nE=0.50\nE=0.75\nE= 0.95\nE=0.25\n0\n+\n+\n+\n1+E Cos A\nk\n0\n+\n+\n+\n2H(8)\n2H(0)\nH(A)\n2/5 H(8)\n2N\n2 N N\n2 NN\n2TN\nFIG. 2.41 Some representative irradiance distributions\nH(0) associated with elliptical radiance distributions. The\npoints are calculated from (48). The solid curves are cir-\ncles, showing a possible simplification of (48) for engineer-\ning calculation purposes.\n= 1/(1-e)\nThus, the nearer E is to 1, the greater is this ratio. The\nratio of zenith (downward) to nadir (upward) radiance is:\n1-E\nTurning now to the computation of H(x,0,0), we start\nwith (17) of Sec. 2.5:\nH(x,0,0) =\nN(x,0',0') cos 2 sin e' de 1 do'\n(43)\nE(0,0)","RADIOMETRY AND PHOTOMETRY\nVOL. II\n134\nwhere E(0,0) is the hemisphere E(E), as shown in part (a) of\nFig. 2.40. This range of integration in (43) may be given\nexplicitly:\nH(x,0,0)\ncosu sin e' de' do'\n=\n(44)\n0'=0\nwhere O(') is the angle between k and E' i.e., the variable\ndirection of integration in the collector's plane which has\nazimuth\n' O(') may be determined from the functional re-\nlation:\n0(0') = arc cot {-tan 0 cos (-) }\n(45)\n.\nThus, e.g., if 0 = 0, O(') = arc cot {0} = /2 for every ''\n05052m. Eq. (43) can be put into a more convenient form by\nusing the fact that, as far as the quantity H(x,0,0) is con-\ncerned, it is immaterial whether, on the one hand, the col-\nlector is tipped in the frame of reference of the radiance\ndistribution as in (a) of Fig. 2.40; or on the other hand,\nthe collector is held still and the radiance distribution is\nappropriately tipped in the frame of the collector as in (b)\nof Fig. 2.40. The computational merit of the arrangement in\n(b) is superior to that in part (a) of Fig. 2.40, and we shall\nadopt it in the present illustration. The salient change re-\nsulting in this switch of points of view is in the functional\nform of N(x,0',0'). Indeed, a glance at (a) and (b) of Fig.\n2.40 shows that the \"vertical\" radiance distribution in part\n(a) has undergone a rigid rotation to the \"tipped\" form in\npart (b), and rotated about the vertical axis so that k goes\ninto the unit vector whose angles are (e, N+)).\n.\nThe details of the transformation of N(x,0', ')') into\nits new form N' (x, e' ')') constitute a simple exercise in ana-\nlytic geometry and are left for the reader to formulate (re-\ncall (18) of Sec. 2.5). The resultant form is:\nN\nN'(x,0','') =\n1 + (-sin 0 sin e' cos ' + cos 0 cos 0')\n(46)\nin which we have set = 0 since the desired irradiance\nH(x,0,0) is independent of for the present radiance distri-\nbution, which is assumed symmetrical about the vertical. We\ncan partially check (46) by letting e' = 0 and = TT. The\nresultant radiance is:\nN'(x,O,)=N/(1+e)\nwhich is precisely the magnitude of N(x,0,0), as was to be\nexpected. Using (46), it is clear from (b) of Fig. 2.40 that\nthe desired irradiance H(x,0,0) is given by:","SEC. 2.11\nEXAMPLES\n135\nH(x,0,0) = '(x,0',0') cos e' sin 0' de'\ndo'\n(47)\nThe integration details of (47) are straightforward and are\ntherefore omitted. Writing, ad hoc:\n\"H(0)\" for H(x,0,0)\nwe then evaluate (47) to obtain:\n- - (cos0)1n\n(48)\nwhich is the desired functional form of the irradiance dis-\ntribution under an elliptical radiance distribution of eccen-\ntricity E and magnitude N. The reader should now show how to\nuse (48), without the need of further computation, for the\ncase where the axis of the elliptical distribution is origi-\nnally tilted at 00 from the vertical, and the angle between\nthis axis and the unit inward normal to the collecting surface\nis No.\nLet us study some of the properties of H(0). First of\nall, by setting 0 = II, we obtain the downward irradiance in-\nduced by the radiance distribution in (41) on the collecting\nsurface:\n(49)\n.\nThe upward irradiance is obtained by setting 0 = 0 :\n(50)\nThe net downward irradiance is therefore:\nH()=H()- H(O)\n(51)\nwhich is positive for all E, 0 < E < 1. H(T) is the magnitude\nof the vector irradiance H(x) associated with (41). The di-\nrection of H(x) is evidently -k. It may be verified directly\nfrom (48) that:","RADIOMETRY AND PHOTOMETRY\nVOL. II\n136\n(52)\n4\nwhere we have written\n\"H(0)\" for H(0) - H(V)\nand\n\"u\" for - 0\nEquation (52) is a specific example of (16) of Sec. 2.8.\nObserve next that H(0)+NN as E+O. This is to be expec-\nted since the elliptical radiance distribution becomes spher-\nical as -0, and as we now know, a spherical radiance distri-\nbution of magnitude N, induces an irradiance N. This fact\nabout the limit of H(0) as E+O may be seen relatively readily\nfor a special case by letting E+O in (49). Indeed, expanding\n1n (1-e) in a power series in E, we have, for very small E, as\nan approximation:\nH(T) =\nH(0) = (1-5-6)\n,\nwhence:\nH() = 4N/3\nThe scalar irradiance induced by the elliptical radi-\nance distribution (41) is also of interest. Using the repre-\nsentation (41) in (3) of Sec. 2.7 we have:\n(53)\nwhere we have written, ad hoc:\n\"h(e)\" for h(x,t)\n,\nNote that:\nlim = 4 NN\nE+O\nFor small E, we have, very nearly:\n= 4 TN\nComparing (53) and (51) we see that:","SEC. 2.11\nEXAMPLES\n137\n(54)\nThus, the magnitude of the vector irradiance associated with\nan elliptical radiance distribution of eccentricity E and\nmagnitude N is 1/€ times the difference between the scalar\nirradiance h(e) associated with the distribution and the sca-\nlar irradiance associated with a uniform radiance distribu-\ntion of magnitude N.\nFinally, we consider the hemispherical scalar irradi-\nances associated with (41) (see (7) of Sec. 2.7), writing ad\nhoc:\n\"h(e,-)\" for h(x,-k,t)\n\"h(e,+)\" for h(x,k,t)\nwe have for an elliptical radiance distribution:\n(55)\n(56)\nAdding these two and comparing the sum with (53) we have:\nh(e,+) + h(e,-) = h(e)\n,\nwhich illustrates (9) of Sec. 2.7. In the two-flow theory of\nlight fields, to be studied in Chapters 8 and 9, the follow-\ning ratios are of interest in that theory (see also (30) of\nSec. 10.7):\nh(e,-)/H() = E in(1-e)/(e+1n(1-e))\n(57)\n(58)\nThese ratios constitute convenient measures of the \"collima-\ntedness\" of the elliptical radiance distribution. Thus for\nthe case E = 0, the distribution is spherical and the very\nantithesis of collimatedness. In this case:\nlim h(e,-)/H() = 2\nE+0\nA similar limit, namely 2, holds for h(e,+)/H(0). In the\nother extreme, i.e., when E is near 1, the elliptical distri-\nbution of downward flux is relatively collimated. In this\ncase:\nlim h(e,-)/H() = 1\n.\nE+1\nOn the other hand, and somewhat unexpectedly, the upward ra-\ndiance approaches a certain shape for which:","138\nRADIOMETRY AND PHOTOMETRY\nVOL. II\nlim h(e,+)/H(0) = 1n 2/(1-1n 2)\n.\nE+1\nEquation (48) is plotted for the four values of E given in\nFig. 2.41. The plots of 2H(0)/2TN are shown in the lower line\nof that figure.\nExample 14: Irradiance from Polynomial\nRadiance Distributions\nThe present example is assigned the task of developing\na generalization of the elliptical radiance distribution con-\nsidered in Example 13, and of developing a formula for the\nassociated irradiance distribution. The main lesson of this\nexample is one not of importance to radiometry per se. Rath-\ner, it is designed to bring to the reader's attention the fact\nthat many of the techniques of classical polynomial and power\nseries theory are available to help obtain analytical repre-\nsentations of the radiance distributions measured in nature\nand on which, in turn, one can base practical methods of com-\nputing the associated irradiance distributions.\nSuppose then, that an empirical radiance distribution\nN(x, ) can be represented for each 0 and by the following\npolynomial in cos 0 :\nn\nN(x,0,0) = { ajPj (cos 0)\n(59)\nwhere Pj (cos 0) is the Legendre polynomial (in cos 0) of the\nfirst kind and of integral order j. The number n may be fi-\nnite or infinite, as required. Here we are assuming that\nN(x, .) is a radiance distribution symmetrical about the ver-\ntical but of a form which has a quite general e-dependence.\nAs in the case of the elliptical radiance distribution in Ex-\nample 13, we can use the fact that the irradiance produced by\nN(x,0,0) in (59) depends only on the angle v between its axis\nof symmetry and the inward normal to the collecting surface.\nTherefore we can use the results of this example, without fur-\nther effort, to compute irradiance on any collecting surface\nwhen the angle u between the axis of the symmetrical distri-\nbution and the unit inward normal to the collecting surface\nis known. Hence the assumption of the form (59) constitutes\nno loss of generality in this sense.\nWe first observe that the coefficients aj are readily\ndeterminable from the tabulated data N (x, 0, ). Indeed, using\nthe orthogonality properties of Pj (cos 0), we have from (59)\n(and cf. (3) of Sec. 6.3):\nN(x,0',')P (cos 0') sin e' de' do'","SEC. 2.11\nEXAMPLES\n139\ni.e.,\n0') sin 0' de' . (60)\nHence, if N(x,) is known, performing the operation on N(x,.)\nas defined in (60), yields ak for every k = 1,...,n. By\nwriting:\n\"u\" for cos 0'\nand\n\"N(u)\"\nfor N(x,0',0')\n,\nEquation (60) takes the relatively compact form:\ndu\n(61)\nEquation (61) can be evaluated by any of several avaliable\nnumerical quadratures, given the radiance data N(H). Having\nobtained the ak in this way, we now can go on to consider the\ncomputation details of H(x,.) associated with N(x,.)). With\n(17) of Sec. 2.5 as a starting point and using (59) we can\nwrite:\nH(x,0,0)\ncos 20 sin e' de' do'\n=\n(62)\nwhere v is the angle between the directions (0',0') and (0,0).\nThe preceding integral can be transformed into an alternate\nform by adopting the technique used in Example 13. (See Fig.\n12.40 (b).) Thus Equation (62) can be written:\n/2\nH(x,0,0)= cos 0' sin 0' de' do'\n(63)\nand which may be viewed as the present counterpart of (47),\nwherein:\ncos I = sin 0 sin e' cos ''' + cos 0 cos O'\n.\nEquation (63) now stands in a form which is readily evaluable.\nToward this end, observe that the sum of Legendre polynomial\nterms can be written in the form:","140\nRADIOMETRY AND PHOTOMETRY\nVOL. II\nn\nn\n(cos v') =\n(64)\nj=0\nwhere the numbers bj, j=\n0,\nn, are obtained by expanding\n,\neach P (cos v1) in the left side of (64) in powers of cos\nand collecting together like powers of COS vi Each \"bj\"\ntherefore denotes the coefficient of cos] u so obtained.\nHence from knowledge of the aj, the numbers bj are readily\ncomputed. Tables of Legendre Polynomials are available for\nthe determinations of the bj.\nNext write, ad hoc:\n\"x\"\nfor sin 0 sin 0'\n\"y\" for cos 0 cos 0'\nThen for every j, 1,...,n\n(cos ' + y)j\nj\nxj-i yi cosj-i '''\ni=0\nwhere ...) Ci 11 as usual denotes the combinatorial coefficient of\nthe ith term in the jth power of a binomial. Using this\nexpansion in (63), with the help of (64), we can rearrange\n(63) to read:\nH(x,0,0)=\ncosj-ip, cos O' sin 0' de' do'\nj=0 cos O' sin\n=\nde'\n(65)\n0'=0\nObserve that:","SEC. 2.11\nEXAMPLES\n141\n2\ndo'\n'=0\nwhere\nj-i is is odd even\nLet \"Iij\" momentarily denote the value of this integral of\ncosj-ipi.\nThen (65) becomes:\nH(x,0,0)\n/2\nsinj-i+1e cos1e cosi+1e, de'\n=\n0\nnj\n= sinj-ig costo de'\n(66)\n0\nObserve that:\n/2\ninj-i+1e cosi+1e, de'\n0\nwhere \"I(z)\" once again denotes the value of the gamma func-\ntion at z. Let us write \"Jij\" for the product of Iij and\nthe latter integral. Hence:\nr\nIf we write:\n\"C(e)\" for > ic, Jij sinj-ig cosie\ni=0","142\nRADIOMETRY AND PHOTOMETRY\nVOL. II\nand\n\"H(0)\" for H(x,0,0)\n,\nthen (66) becomes:\nn\nH(0) = [bjcj(8)\n(67)\nThis is the desired formula for the irradiance distribution\nassociated with the radiance distribution in (59). Observe\nthat the numbers ic. Jij are evaluable once for all, so that\nto use (67) with particular radiance distribution N(x,.) it\nis required only to evaluate the ak by means of (61) to the\ndesired degree of accuracy, and to obtain the bk using (64).\nIt is left as an exercise for the reader to evaluate iCjJij\nand to obtain explicit formulas for the bk in terms of\nthe ak for the first few values of k = 1, 2, n, and to\nmake a list of them so that the use of (67) is reduced to sim-\nply finding the ak for each new application.\nThe reader may verify that the scalar irradiance h as-\nsociated with a radiance distribution N(x, ) of the form in\n(59) is given by:\nh =\n(68)\nWe close this example by observing two special cases\nof the polynomial distributions. First we note that the set\nof polynomial radiance distributions discussed above contains\nas a special case the elliptical radiance distribution of Ex-\nample 13. To see this, in (64) choose the aj subject to the\ncondition that:\nfor every integer j 0, where OSE<1, and where N is a non-\nnegative number. Then:\nN(x,0,0) = N (E(e cos e) cos\nj=0\nN/ 1 + E cos 0)\n,","SEC. 2.11\nEXAMPLES\n143\nwhich is the form of (42). Secondly, an interesting radiance\ndistribution associated with heavily overcast skies is a spe-\ncial case of (59). This is the \"Moon-Spencer Sky\" representa-\ntion of radiance distributions and takes the following form.\nFor every 0, /2 SIT:\nN(x,0,0) = N(x,/2,) (1-2 cos 0)\n.\nThe empirical details of determining this distribution may be\nfound in Ref. [186].\nExample 15: On the Formal Equivalence of Radiance\nand Irradiance Distributions\nThe present sequence of illustrations of the radiomet-\nric concepts is concluded with a discussion of the theoreti-\ncal possibility of reversing the usual path between radiance\nand irradiance distributions. We shall show that, given an\nirradiance distribution H(x, ) it is possible, in principle,\nto deduce the associated radiance distribution N(x, .). This\ncourse of action is the reverse of that taken in the various\nExamples above, and in the discussion of Sec. 2.5. The theo-\nretical and experimental significance of this reversal of the\nusual computation procedure was touched on briefly in Sec. 2.5\nwherein also a practical scheme for obtaining N(x, ) ) from\nH(x, ) was suggested. The main purpose of this example is to\nshow that this reverse path is possible not only on a numeri-\ncal level, but also on an exact function-theoretic level.\nThis is tantamount to showing that (8) of Sec. 2.5, when\nviewed as an integral equation with unknown N(x,.)), has a u-\nnique solution in terms of the irradiance distribution H(x, ).\nWe shall discuss this point of view in detail, as it affords\nan opportunity to illustrate how the use of advanced vector\nspace concepts can facilitate the solutions of certain radio-\nmetric problems.\nWe can phrase the present problem in precise terms as\nfollows: Given: the irradiance distribution H(x,.) at a\npoint x in an optical medium. Required: the associated radi-\nance distribution N(x,*) Now, for every direction E and\npoint x we have, by (8) of Sec. 2.5:\n(69)\nH(x,5) =\n.\nE(5)\nLet us write:\nI\n1\n\"C(x)\"\nfor\n.\n2\nE(5)\nWe call C(x) the cosine operator, for obvious reasons. Then\n(69) can be written as:","144\nRADIOMETRY AND PHOTOMETRY\nVOL. II\nH(x,*) = N(x,+) ((c) =\n(70)\n,\n2\nwhere \"N(x, ) (C(x)\" means: \"operate on the radiance distribu-\ntion N(x, ) by substituting N(x,.) into the square bracket of\nthe integral operator C(x).\" For example, if at point x,\nN(x,.) is a uniform radiance distribution with magnitude N,\nthen for every E:\nN(x.+)((()\n= 2 N\n5.5' d&(E')\nE(E)\nN\n(71)\nIf\n2\nUp to this point in the present example our delibera-\ntions have been relatively elementary and were without excep-\ntion motivated by physical intuition. But now when we ask:\n\"Can we determine N(x, .) knowing H(x,.) and C(x)?\", we leave\nthe domain of physical intuition and are asking a purely math-\nematical question. Perhaps even in this general radiometric\nsetting some reader may see a physical reason for an affirma-\ntive answer to the query. For instance, by starting with the\nsimpler setting in (21) of Sec. 2.5 and by letting the number\nof equations of the type considered there increase indefinite-\n1y and by being assured at each step along such a course that\nN(x, ) is determinable from H(x, ), perhaps by following such\na line of thought one can be convinced of the general deter-\nminability of N(x, .) from H(x,.)). Indeed, it is most desir-\nable that some assurance be generated in such a manner. But\nat the present moment we are confronted by a mathematical\nquestion and in view of its important relevance to applica-\ntions we prefer to settle it using now the rules of mathemat-\nics.\nTo begin to answer the preceding question we generate\na mathematical setting in which the question suggests some\nfurther action toward the present goal. The appropriate set-\nting is obtained by considering the set n (x) of all radiance\ndistributions at point X. Next we observe the interesting\nfact that the sum of any two such radiance distributions is\nagain in the set n (x). For example, if N (x, .) and N' (x,)\nare in n(x), then the function:\nN(x, ) + N'(x, )\nis in n (x) and by definition assigns to each direction E at\nx the sum N(x,5) + N'(x,5) of the two radiances N(x,5) and","SEC. 2.11\nEXAMPLES\n145\nN'(x,5). The sum of two radiances is again a radiance*\nThat is, if each of N(x, ) and N'(x,.) is physically admis-\nsible then a lighting arrangement could be conceived so that\nN(x, ) + N' (x,.) was realizable. Observe, however, that we\nneed not introduce the preceding observation as an additional\njustification for the assertion about n (x) containing the\nsum of N(x, ) and N' (x, whenever it contains each. That\nassertion is simply the result of the present definition of\nn(x), and of the general definition of radiance. Next we ob-\nserve that if N(x,*) is in n (x), so is cN(x,.) where C is\na non negative real number. The physical plausibility of this\nassertion is obvious. As a result of these observations we\nsee that n(x) is part of the vector space V(x) of all func-\ntions f(x, at x over the domain E, and with dimensions of\nradiance. In fact n(x) forms what is referred to in mathe-\nmatical terminology as a non negative cone of V(x) which, by\ndefinition, is closed under formation of sums, and multipli-\ncation by non negative real numbers. (Take all the unit vec-\ntors in a subset E0 of E and form the set of all products CE,\nwith C , and E in Eo. Describe the geometrical appearance\nof this set.)\nNow what is the purpose of all this collecting together\nof huge families n (x) of radiance distributions? Simply\nthis: by collecting together the members of n (x) in the\nfashion just exhibited, the operator C(x) defined above takes\non the crucial role of a linear transformation from V(x) to\nV(x) and in this setting our original question, \"Can we ob-\ntain N(x, ) from H(x,.)?,\" takes a deeper and mathematically\nmeaningful cast.\nBefore rephrasing the question in the vector space ter-\nminology it may be well to include, simply for completeness,\na comment about what it means for C(x) to be a linear transfor-\nmation. It means this: If f(x, and g(x, ) are any two func-\ntions in V(x) and a and b are any two real numbers, then\n[af(x,\") + bg(x,.)) C(x) = a[f(x,*)(()) + b[g(x,)(x)]\nwhere f(x, ) (C(x) and g(x,)(x) are again members of V(x),\nbeing images of f(x, and g(x, ) under C(x). Observe that\nC(x) acting on a function with dimensions of radiance yields\nup once again a function with dimensions of radiance.\nWe can now ask our question about (x, .) and N(x,.) as\nfollows: \"Is the linear transformation C(x) from V(x) to V(x)\na one-to-one transformation when restricted to the part (x)\nof V(x)?\" By C(x) being \"one-to-one\", is meant that C(x)\nsends exactly one radiance function into each modified irradi-\nance function of the form: H(x, ) / 2 T defined in (70). Then,\nhaving given an irradiance distribution H(x,)), we are thereby\nassured that there is one and only one radiance function that\nit comes from (i.e., is associated with). Hence, whenever\n*This may be taken as intuitively obvious at this point of\nthe exposition. Formally, it is a consequence of the inter-\naction principle of Chapter 3.","146\nRADIOMETRY AND PHOTOMETRY\nVOL. II\nC(x) is one-to-one we are encouraged to find that unique radi-\nance distribution N(x, ) which, by (70), yields up H(x,.)).\nIt turns out that the linear transformation C(x) is in-\ndeed one-to-one when restricted to n (x) and we find its in-\nverse c-1(x) as follows. We begin by observing that:\n(x,5)/2) do(s) = (N(x, ) (C(x)) ds(5)\nd&(5)\n5'. ()\nda(s)\nE(E')\n=1/1\ndo(E')\nFor our present purpose, let us write:\n\"|f(x,.)|\" for\n,\nwhere f(x, is in V(x). In particular, if f(x,.) is in n(x),\nthen /f(x).)) I is the scalar irradiance associated with a gen-\neral radiance distribution f(x,.) at point X. The main thing\nthe preceding calculation has shown is that:\n(72)\n.\nThe significance of this equality for the present discussion\nis crucial, and we pause to make this significance clear.\nThe significance becomes clear when it is pointed out that the\nscalar irradiance h (x) acts as the \"length\" of the vector\nN(x, in V(x). Indeed, the bars around \"f(x, .)\" in the def-\ninition above are there to point up the easily verified fact\nthat f(x, .) I is analogous to the absolute value of a number\nor vector; and it may be shown that all the essential proper-\nties of length that we carry with us from euclidean space hold\nalso for the numbers N(x, 1. We call (i.e., h(x)\nin this case) the radiometric norm of N(x,.), to point up\nthis similarity between N(x, ) and the usual concept of norm\nor length of a vector.","SEC. 2.11\nEXAMPLES\n147\nThe significance of (72) can now be stated: the linear\ntransformation C(x) has the property that it maps a radiance\nfunction into one which has exactly half the norm (i.e.,\n\"length\") of the original radiance distribution. In short,\nC(x) has the norm contracting property with contracting factor\n1/2. The mathematical consequence of this fact is immediate:\nwe now can use the well known norm-contracting theorem of vec-\ntor space theory, as stated, e.g., in Ref. [251] for the radi-\native transfer context, to assert that the inverse C-1 (x) of\nC(x) exists, and that, indeed:\nc-1(x) = (11-c(x))3\n(73)\nwhere I is the identity transformation, i.e.,\nf(x,*)I = f(x,*)\nfor every f(x, .) in V(x). This identity transformation can\nbe written as an integral operator. Thus if we write:\n\"I\"\nfor\nE(E)\nit may be verified that I is the identity operator on V(x)\nwhenever s is the Dirac delta function (on the space with So as\nmeasure). Then if we go on to write:\n1/1/1\n\"D(x)\"\nfor\n-\nwe have the equivalent form for (73), where\nD(x) = (I-C(x))\nand\nC-1 (x) = D\n(74)\nand\n(75)\nwhere, in turn, we have defined D (x) recursively by writing:","148\nRADIOMETRY AND PHOTOMETRY\nVOL. II\n\"D°(x)\"\nfor\nI\n,\nand for every positive integer j:\n\"D' (x)\" for\nand where, finally, \"Dj-1 (x)\" denotes the customary inte-\ngral operation on Dj-1(x) as an integrand in D(x). Equation\n(74) yields the requisite inverse of C(x), and the solution\nof our present problem is summarized in (75).\nObserve that to use the norm-contracting theorem in\nRef. [251] we actually need the fact that I-C(x) is a norm-\ncontracting operator. The reader may now easily verify that:\nN(x,)(I-c(x))I(,\n,\nso that I-C(x) is, indeed, norm-contracting with contracting\nfactor 1/2, and the norm-contracting theorem statement yields\n(73) and hence (74).\nAside from the relatively advanced mathematical objects\ninvolved in (74) (namely, Dirac delta functions, and two-di-\nmensional iterated integration) the algebraic essence of (74)\nis identical to that of the formula used by every high school\nstudent summing a geometric series of the form:\n(1-x) + (1-x)2 + (1-x)3 +\nwhose value is clearly 1/x and where x is any number with ab-\nsolute value less than 1. Now, instead of squaring (1-x), i.e.,\nmultiplying (1-x) by itself, we are required to operate with\nI-C(x) on itself. Thus, e.g.,\nN(x,.) (I-C(x))2 =\n(2(E'-5)-5'-E) (2T8(E\"-)-\"()\nTo obtain the form for (I-C(x))2 itself, simply remove\n\"N(x, and \"N(x,5)\" where they occur in the preceding equal- -\nity. Thus, as in the case of computing the \"fraction\" 1/x by\nusing solely multiplication, addition and subtraction repeat-\nedly, so too can we compute \"1/C(x)\", i.e., , c-1(x) using solely\nintegration, multiplication, addition and subtraction, repeat-\nedly. The norm-contracting theorem states that by continuing\nsufficiently far, c-1(x) can be arbitrarily closely approxi-\nmated.\nThe error engendered by stopping the computation of\nc-1(x) in (74) at the kth term may be readily computed. Thus,\nwrite,","SEC. 2.11\nEXAMPLES\n149\nk\n\"N (k) (x, )\" 11 for 1 [H(x,.)D3(x)\nso that N (k) (x, ) serves as the kth order approximation to\nthe desired distribution N(x, ). . Then the radiometric norm\nof the difference between N(x,.) and N(k) (x, .) is:\nH(x,.)\nD3 (x)\n=\n((x,)\n=\n1/HCX,\"\n1\nThe reader may use as specific cases in (75) the formulas for\nH(x,0, ) in (48) and (67) of Examples 13 and 14 in order to\nrecover the associated radiance distributions of those exam-\nples. These will afford non trivial examples of (75).\nWe close this discussion with some general assertions\nto which one is naturally led after contemplating the lesson\nof the present example. The assertions concern the possibil-\nity of still further equivalences between radiance and other\nradiometric concepts which are natural generalizations of the\nconcept of irradiance. Recall that irradiance was defined\nempirically by specifying a small plane surface S onto each\npoint of which radiant flux could be incident within the set\nE(E), where E is the unit inward normal to S. If now we re-\nplace E(E) by any fixed conical set D (E) of directions of pos-\nitive solid angle content specified in some way with respect\nto E, then the generalized irradiance distribution H(x,D()),\nas defined in (4) of Sec. 2.4, is equivalent to N(x,) in the\nsame sense that H(x, .) and N(x, .) were shown to be equivalent\nin the present example. This is the first assertion to which\nwe are led. Its proof is left to the reader.\nThe lesson of the present example can be carried still\nfurther than the point reached in the preceding paragraph.\nLet \"S(x,5)\" denote a collecting surface S which is a convex\nsurface of revolution of fixed shape and size whose location\nand orientation in an optical medium X are uniquely specified","RADIOMETRY AND PHOTOMETRY\nVOL. II\n150\n( x x\nD(x')\nE\nx'\nXx\nC'\ns'\nS(x,E)\nFIG. 2.42 A diagram of a radiometrically adequate collec-\ntor. How many of them are there? (See text)\nby locating a fixed point x on (or within) S and giving the\ndirection E of the sensed axis of revolution of S. A typical\nsurface of the type S(x, E) is pictured in Fig. 2.42. Let S'\nbe a proper band of latitude circles on S, i.e., such that S'\nhas positive area and such that to the points x' of each lati-\ntude circle C' of S' there is assigned a right circular cone\nD(x') of directions whose axis direction E' lies in the plane\nof x' and E and makes a given angle with E, and of common pos-\nitive solid angle opening s(D(x')). We shall require that\nthe values E.E. . and s(D(x')) are fixed for the points x' on\neach latitude circle C' on S' but may vary from circle to cir-\ncle on S' . Let X(x) be the spherical region swept out by\nS(x,5) as X is held fixed and & allowed to vary through all of\nE. Finally, assume that a general radiance distribution of\nfixed structure is defined at each point within X(x). Then\nif \"P(S(x,5))\" denotes the radiant flux collected by S for a\ngiven x and E, we make the following plausible assertion with\nthe above conditions in mind: For every point x in the opti-\ncal medium X, the radiance distribution N(x,.) is equivalent\nto the radiant flux distribution P(S(x,.)) in the sense that\nthere is a one-to-one integral operator E(S,x) such that:\n(s(x,.)) If N(x,.) E(S,x)\n(76)","SEC. 2.12\nEXAMPLES\n151\nThe preceding assertion clearly contains the irradiance asser-\ntions above as special cases. For example, let S be a plane\ncircular surface of positive area, with unit inward normal E\nand center X. Let S' be one side of S such that D(x') = E(E)\nfor every x' in S'. Then under the conditions of the preced-\ning assertion, we have:\n(S(x,5)) = H(x,5) A(S)\nso that, according to (70) and (76)\nE(S,x) = 2C(x) A(S)\nwhere A(S) is the area of the plane circular surface S.\nThese examples do not exhaust the possibilities inher-\nent in (70) and (76); however, they will suffice for the pres-\nent to show that there is an infinite class of radiometric\nfunctions each member of which is equivalent to the radiance\nfunction in the sense of there being a one-to-one linear trans-\nformation between the vector spaces of radiance distributions\nand radiant flux distributions of such functions. Let us say\nthat an arbitrary convex surface S is a radiometrically ade-\nquate collector in an optical medium X if its associated radi-\nant flux distribution P(S(x, is equivalent, in the sense\nof the present example, to N(x, ) for every point X in X. We\nclose this example with the following problem directed to in-\nterested readers: Characterize the most general class of ra-\ndiometrically adequate collectors. (In other words: give\nthe necessary and sufficient conditions that a surface S be a\nradiometrically adequate collector.) We have shown in the\npresent example that plane circular surfaces, and more gener-\nally, have conjectured that surfaces of revolution such as\ncylinders, spheres, hemispheres, spherical caps, prolate and\noblate spheroids, etc., can be radiometrically adequate col-\nlectors. It is certainly clear, at least intuitively, that\nthe class of radiometrically adequate collectors is quite\nlarge and could, under suitable qualifications, contain sur-\nfaces not necessarily surfaces of revolution, such as the\nPlatonic \"solids\", rectangular parallelepipeds, convex sur-\nfaces, and even certain non convex surfaces. However, non\nconvex surfaces introduce self-interreflection complications\nwhich cannot be handled until the interaction principle (Chap-\nter 3) has been studied, and therefore for the present at any\nrate, will be omitted from the problem stated above.\n2.12 Transition from Radiometry to Photometry\nThe concepts of classical photometry, to which we turn\nour attention in this section, are designed to give quantita-\ntive measures of the capability of radiant flux to evoke the\nsensation of brightness in human eyes. These measures all\nrest in the single concept of the standard luminosity function\nthe key concept in the science of photometry. Photometry is\nprincipally concerned with the precise description of and the\ndeductions from the relative visibility of monochromatic radi-\nant flux as a function of wavelength and as embodied in the","RADIOMETRY AND PHOTOMETRY\nVOL. II\n152\nstandard luminosity function. The depth to which we shall\nstudy photometry will be only so far that the reader may gain\nan insight into the principal features of the subject and a\ncompetence in working with photometric concepts, in the forms\nthey commonly occur in the study of applied hydrologic optics.\nSuch interesting problems as the physiological basis of color\nvision, which lie at the base of the subject, transcend the\nscope of the present discussion.\nWe shall initially motivate the transition to the pho-\ntometric concepts by means of hypothetical experiments de-\nsigned to acquaint the reader with the main empirical features\nof photometry. The experiments described are to be understood\nas didactic devices and as such omit the wealth of detail re-\nquired for the implementation of their real counterparts.\nOnce the essential idea of the transition has been explained\nand the transition made from the concept of radiance to that\nof its photometric counterpart, luminance, then we shall em-\nbark on a systematic transition to geometrical photometry and\ncompile our results in compact tabular form suitable for con-\nvenient reference.\nThe Individual Luminosity Functions\nFigure 2.43 depicts an observer viewing a screen in a\nwell-lighted room. The screen is divided into two equal\nareas, and is devised so that on the left half a radiance of\nfixed amount N(10) of fixed wavelength 10 is constantly dis-\nplayed throughout the experiment. The magnitude of N(10) is\nchosen comparable to daylight radiances. The right half of\nVariable Radiance at some\nfixed 1, o OSAS 8\nFixed Radiance\nat 555mm\nObserver\nFIG. 2.43 A schematic setting for the empirical deter-\nmination of the individual luminosity function.","SEC. 2.12\nRADIOMETRY TO PHOTOMETRY\n153\nthe screen displays a radiance of variable amount N(Aj) of\nfixed wavelength 1j in some set L of wavelengths. The obser-\nver begins the experiment with N(Aj) =0 and, by means of a con-\ntrolling device, slowly increases N(Xj) from 0 until that\nmagnitude N(Aj) is attained for which he believes the bright-\nness sensation produced in his brain by N(Aj) is equal to\nthat produced by N(10). This decision process may require\nsome preliminary trials on the observer's part. Soon, how-\never, he makes a decision that N(Aj) and N(10) are of equal\n\"brightness\" and presses a button, thereby recording N(Aj) .\nThis procedure is now repeated for the other wavelengths over\nthe electromagnetic spectrum.\nThe experiment with observer a just described results\nin a set of radiance values N(Aj) with ^j in the set L. These\nvalues, it should be noted, are associated with the particular\nobserver aj used in the preceding experiment. To occasionally\npoint up this fact let us write \"N(ai, 1j)\" for the radiance\nthat matches N(10) as judged by observer ai in some set A of\nobservers which have performed the experiment. As a result of\nthese experiments, to each observer ai we may assign his par-\nticular luminosity function defined as follows. We write:\n\"y(a,\";j)\" for\n(1)\nfor every in L, and call y (ai,' the luminosity function\nfor observer ai. The value y(ai,dj is called the luminosity\nof the wavelength Aj, as judged by observer ai\nMatters have been arranged (on the basis of earlier\nexperiments with observer ai, not recorded here) so that wave-\nlength 10 was the wavelength of maximum luminosity for obser-\nver ai. To see what this means, recall that N(ai,dj) is\nchosen to be of such a magnitude as to match N(10) in its\ncapability of evoking the sensation of brightness. Since\nN(10), the radiance with wavelength 10 of maximum luminosity\nis fixed in magnitude, all other radiances N(aj,dj) must then\nbe increased to give the same brightness sensation to a as\ndid the radiance N(10). Hence a plot of y(ai,dj) versus 1j\nfor each observer ai in A will have a graph of the general\nform in Fig. 2.44. At 1j = 10, y(ai, 10) = 1. For every other\n1j, y(aj,dj) < 1. To point up the fact that 10 varies from\nobserver to observer, let us write, alternatively, \"10(ai)\"\nfor the 10 of observer ai.\nOnce each observer in the experimental group A has\nbeen assigned a luminosity function, this information could be\nused to predict the subjective sensation of brightness of a\ngiven sample of monochromatic radiant flux in the following\nsense. Suppose that observer ai encounters a radiance of mag-\nnitude N(Aj). Then by (1) we can predict that this radiance\nwould appear to him to have the same \"brightness\" as a sample\nof radiant flux of wavelength 10(ai) and radiance:\n(2)","RADIOMETRY AND PHOTOMETRY\nVOL.II\n154\nI\ny(01,11)\n10(a)\nIt\ny(a2,di)\n10(a2)\ny(ak,di)\n10(ak)\nL\nFIG. 2.44 Some individual luminosity functions (sche-\nmatic only).\nThe, term follows very simply and logically from (1) . But the\ninterpretation of this term, as just stated, is not compelled\nto follow from (1) by the laws of algebra. To make this inter-\npretation we must first make an assumption (preferably expli-\ncitly) that the subjective sensation of brightness that can\nbe produced by a radiance N(Aj) varies linearly with the mag-\nnitude of N(Aj). Thus if we were to double N(Aj), then the\nsensation would be the same as that produced by viewing ra-\ndiant flux of wavelength 10 and of double the radiance\nN(Aj)y(ai,Aj). The reasonab leness of this assumption rests\ncritically on the stability of ai's luminosity curve with re-\nspect to the absolute magnitude of N(101 (aj)) used in the ex-\nperiment, and on the general lighting level within the experi-\nmental room. Actual experimental evidence indicates that the\nluminosity function for ai is dependent to a measurable degree\non both N(10(aj)) and the background radiance. The description","SEC. 2.12\nRADIOMETRY TO PHOTOMETRY\n155\nof the hypothetical experiment above was careful to note that\nthe experiment took place in a well-lighted room. To point\nup this fact the resultant luminosity curves in Fig. 2.44 are\ncalled photopic luminosity curves. When the observers view\nthe screen in a darkened room with dark-adapted vision, it is\nfound that the luminosity curves shift en masse 50-60 mu to\nthe left with very slight overall change in shape. The resul-\ntant curves are called scotopic luminosity curves.\nTo summarize, a consistent, workable interpretation of\nthe meaning of (2) requires an explicit assumption of the lin-\nearity of subjective brightness sensations with respect to\nthe magnitude N(Aj). This assumption might be at slight var-\niance with experimental fact over some ranges of values of\nN(Aj), but it has the virtue of leading the way to a scientif-\nic basis of photometry. A science, it will be recalled, is an\norganized body of knowledge sustained within the webwork of a\nset of generally accepted conventions. To raise photometry\nto the level of a science, for better or worse, requires the\nexplicit statement of adopted conventions of the kind just\ndiscussed.\nWe return now to experimental subject ai, whose photop-\nic luminosity function has been determined, and attempt to pre-\ndict a new kind of response of ai to radiant flux. Suppose\nnow that ai is confronted with a radiance in the right half of\nthe screen in Fig. 2.43 which consists of a radiance which is\na superimposed mixture of two monochromatic radiances N(Aj),\nN(1k) from the set L, say of distinct wavelengths ^j and 1k.\nWere he confronted with each separately, we would be able to\npredict the equivalent sensation producing radiance of the\nwavelength (aj) by performing twice the operation in (2):\nonce for Aj and then again for 1k. In an attempt to predict\nthe sensation producing capabilities of radiance of wavelength\n10(ai) equivalent to that of the radiance mixture N(Aj)+N(Ak)\nwe are tempted by simple energy-addition arguments to say\nthat:\n(3)\n+\nis the requisite radiance. However, there appears to be no\nexperimental evidence to substantiate this attempt, although\npractical calculations based on (3), and physiological eye-\nmechanisms tend to lend some support of (3). In the absence\nof such experimental evidence and in the presence of a desire\nto progress to a scientific discipline, we must make an expli-\ncit assumption to the effect that: the radiance of wavelength\n10(ai), capable of producing the same sensation of brightness\nas a mixture of two radiances of wavelengths 1j and 1k, is\ngiven by (3) above. Clearly this is a generalization of the\nlinearity assumption above, the earlier form being obtained\nby setting Aj = 1k.\nOnce the preceding assumption - (or definition of equi-\nvalent radiance of wavelength 10, as it should preferably be\ncalled) is made, the path toward a sound basis for the sci-\nence of photometry is cleared of one further obstacle. In-\ndeed, it is but a formal step from (3) to the following gen-\neral definition for the relative luminance distribution","RADIOMETRY AND PHOTOMETRY\n156\nVOL. II\nassociated with a radiance distribution at a point x in an\noptical medium: Let N(x, t,1) be the radiance distribution\nat x at time t for wavelength 1. Then the associated relative\nluminance distribution with respect to observer ai is the\nfunction:\ndi\n(4)\n0\nwhich assigns to each E at x at time t, the relative luminance,\nwith respect to ai, of the integrated radiance distribution\nN(x, t, 1) dl. We shall denote the latter by \"N(x,\", t, A)\".\nA minor technical point should be noted here before\ngoing further, a point which concerns the integration of radi-\nance with respect to wavelength 1 rather than frequency V.\nIt will be recalled that the basis for integrating radiance\nover the spectrum of frequencies was established in Sec. 2.3,\nand that the possibility of such an operation is guaranteed\nby the additivity and continuity properties of $ with respect\nto frequency (cf., (1) and (2) in Sec. 2.2). By noting that\nVA = V implies dv = (v/l2) d, each integration with respect\nto v can be cast into an integration with respect to 1. (See\nnote (c) to Table 3 below.) Whenever such a change of varia-\nbles from V to 1 is made, we assume that the factor - (v/l2) is\nsuitably absorbed in the radiometric symbol, and the dimension\nof the radiometric concept, e.g., radiance, as far as the fre-\nquency component is concerned, is tacitly changed from \"per\nunit frequency length\" to \"per unit wavelength\".\nReturning now to (4), we attempt to interpret (4) after\nthe fashion of the interpretation of (3). A straightforward\nextension of the interpretation of (3) is the following: for\na given direction E, (4) is the amount of monochromatic radi-\nance of wavelength a(ai) which would produce an equivalent\nsensation of brightness in the brain of observer ai as would\nthe integrated radiance N(x,5,t,A), where A is the entire\nwavelength (or frequency) spectrum. In view of the preceding\nobservations, in the definitions (4) of Sec. 2.5, one can re-\nplace \"F\" by \"A\" and have:\nN(x,E,t,A) = d\n,\n0\nby virtue of (4), Sec. 2.3. It is to be particularly noted\nthat the preceding italicized interpretation is a formal inter-\npretation with no known empirical basis except for the sin-\ngle case where the given radiance distribution is monochro-\nmatic.\nWith the preceding interpretation of (4) in mind, we\nnext return to (1) and emulate that definition in the present\nheterochromatic setting of (4). Thus, we write:\n1\n\"y(a)\"\nfor\ndi\n(5)\nN(A)","SEC. 2.12\nRADIOMETRY TO PHOTOMETRY\n157\nand where for brevity we have written:\n\"N(X)\"\nfor\nN(x, , t,1)\nand\n\"N(A)\"\nfor\nN(x, , t,A)\nWe call y(ai) the relative luminosity of the radiance N()\nover A for the observer ai. In this way we come to one of the\nprincipal definitions of photometry:\nLet a be any radiometric concept of geometrical radi-\nometry (radiance, irradiance, radiant intensity, etc.), de-\nfined over the part M of the spectrum A. The relative lumi-\nnosity of R over M for an observer ai is the number\n(R M,a) where we have written:\n[/\ndX\nM\n\"Y(.,R,M,a.)\"\nfor\n(6)\n(Q(x)\ndi\nM\nThe Standard Luminosity Functions\nWe now re-examine the family of individual relative\nluminosity functions, depicted in Fig. 2.44, and attempt to\ndefine a single luminosity function which is representative of\nthe entire set A of individual observers. There are several\nways to go about this. For example in one method, we can go\nthrough the set of graphs of Fig. 2.44, note each 10 (ai) and\nmake a histogram, over A in 1, of the number of observers\nwhose maximum luminosity occurred at wavelength A. A typical\nhistogram that would result is shown schematically in part (a)\nof Fig. 2.45. All indications in real experiments and theo-\nretical considerations point to a gaussian distribution for\nthe ideal limit of such histograms as the number of members\nin the set A increases indefinitely. The peak of the distri-\nbution is found in actual experiments to occur near a A of\n555 mu. Next, a general wavelength A is selected and the\ngraphs of Fig. 2.44 are combed through with the specific goal\n-\nin mind of finding the spread of values of y(ai,1) over the\nai in A. This spread of values is then split up into inter-\nvals. Part (b) of Fig. 2.45 depicts a typical histogram with\nthe abscissas locating the observed values y(ai, X) occurring\nover the selected set of intervals, and the ordinates giving\nthe number of ai in each interval. Part (b) of Fig. 2.45 is\nadapted from Fig. .03a of Moon's treatise on Illuminating\nEngineering (Ref. [185]), which in turn is derived from actual\nexperimental results by Coblentz and Emerson who gathered\ndata from a set A of 125 observers. By means of (b) of Fig.\n2.45, the relative luminosity value of 0.1750 is assigned to\nthe standard observer for l = 640 mu.\nBy going through the entire spectrum in this way i.e.,\nby repeating the process summarized in (b) of Fig. 2.45, now","158\nRADIOMETRY AND PHOTOMETRY\nVOL. II\nNo. of\nai with\ndolai) in\ninterval\n400\n500\n600\n700\n555 1\n20\nl = 640\nNo. of aj\nwith y(a),1)\n15\nfor =640mu\n10\n5\no\n0.10\n0.20\n0.30\ny(a),1 = 0.1750\nFIG. 2.45 Towards determining the standard luminosity\nfunction y. (From [185], by permission)\nfor each 1 in a selected range of A's through A - - - the desired\nstandard luminosity function is obtained. A graph, to scale,\nof the standard photopic luminosity function y() is given in\nFig. 2.46, and a tabulation of y() is given in Table 1. A\nmore detailed tabulation of the values y(1) over the visible\nspectrum is given in Ref. [50].\nNow, , all that we did in the preceding discussion by\nmeans of individual luminosity functions y(ai,\") can be re-\npeated line for line for the standard observer a. Thus,\nwherever \"y(ai, .)\" appeared, we can write \"y(a, .)\" or, more\nsimply, \"y() \", for the standard photopic luminosity function,\nand where \"a\" stands for the hypothetical standard observer (a\ncreature who shares the same corner of conceptual reality with","SEC. 2.12\nRADIOMETRY TO PHOTOMETRY\n159\n1.2\n1.1\n1.0\nPHOTOPIC\n.9\nSCOTOPIC\n.8\n7\n6\n.5\n.4\n3\n.2\n.1\n0\n700\n800\n350\n400\n500\n600\nMILLIMICRONS\nFIG. 2.46 The solid curve depicts the standard photopic\nluminosity function for daylight adaptation. The standard\nscotopic luminosity function (for dark adaptation) is shown\ndashed.\nsuch entities as the \"average American male, age 30\"). Spe-\ncifically, we can now make the following definition which is\none of the principal definitions of photometry:\nLet R be any radiometric concept of geometrical radi-\nometry (radiance, irradiance, radiant intensity, etc.) defined\non the part M of the spectrum A. The relative luminosity of\na over M for the standard observer is the number Y(Q,M)\nwhere we have written:\n(X)(()) di\nM\n\"Y(R,M)\nfor\n(7)\nR\n(1) di\nM","160\nRADIOMETRY AND PHOTOMETRY\nVOL. II\nTABLE 1\nThe Standard Photopic Luminosity Function ()\nand its Indefinite Integral\nA (mu)\ny(x)\ny(x') dX'\n390\n4\n0 x 10 3\n390\n1. x 10\n400\n4.\n1\n410\n12.\n5\n420\n40.\n17\n430\n116.\n57\n440\n230.\n173\n450\n380.\n403\n460\n600.\n783\n470\n1,383\n910.\n480\n1,390.\n2,293\n490\n2,080.\n3,683\n500\n3,230.\n5,763\n510\n5,030.\n8,993\n520\n7,100.\n14,023\n530\n8,620.\n21,123\n540\n9,540.\n29,743\n550\n9,950.\n39,283\n560\n9,950.\n49,233\n570\n9,520.\n59,183\n580\n8,700\n68,703\n590\n7,570.\n77,403\n600\n6,310.\n84,973\n610\n5,030.\n91,283\n620\n3,810.\n96,313\n630\n2,650.\n100,123\n640\n1,750.\n102,773\n650\n1,070.\n104,523\n660\n610.\n105,593\n106,203\n670\n320.\n680\n170.\n106,523\n690\n82.\n106,693\n700\n41.\n106,775\n710\n21.\n106,816\n720\n11.\n106,837\n730\n5.\n106,848\n740\n3.\n106,853\n106,856\n750\n1.\n760\n0.\n106,857","SEC. 2.12\nRADIOMETRY TO PHOTOMETRY\n161\nPhotometric Bedrock: the Lumen\nWe now take the final step in the transition from ra-\ndiometry to photometry. This step consists in reaching an\nagreement on how to assign to every sample of radiant flux of\ngiven spectral composition a quantitative measure of the sam-\nple's capability of producing within the standard observer\nthe associated sensation of brightness. Now that the concept\nof the standard observer has been fixed, the remaining task\nconsists in finding a suitable standard radiant-flux emitter\nwhose wavelength dependence over the spectrum A is uniquely\ndefined within a rigid experimental and theoretical framework\nand which is precisely reproducible in practice. Once such a\nstandard is found it is assigned a preselected number of units\nof \"brightness\"- producing capability and all other radiant\nflux samples can then be given their amounts of brightness-\nproducing capability relative to the standard.\nSuch candidates as laser beams of given monochromati-\ncity, various flames of burning liquids or solids, incandes-\ncent gases of known spectral decomposition, the surface of\nthe sun, the surfaces of various molten metals--all these are\npossible candidates which can serve as photometric standards.\nThe traditional standard was a candle flame- the candle having\nbeen manufactured, set to burn, and observed in a rigidly con-\ntrolled manner. The current standard is the surface of a\npool of platinum which is at the precisely determinable tem-\nperature (see [51]) of its change from the solid to the liquid\nphase (2042° Kelvin). Once the metal has reached this temper-\nature within some thermally stable enclosure, its radiant\nemittance wb is precisely computable for each wavelength in\nthe spectrum using the laws of blackbody thermal radiation.\nThe surface radiance distribution of the platinum is uniform\nof magnitude N over all emergent directions from the surface\nat a point. Hence the relation Nb = Wb/TT exists between Nb\nand wb (cf., closing remarks of Sec. 2.4).\nThe key step is now taken: it is agreed that the sur-\nface radiance of the surface of freezing platinum is to be\nassigned a brightness sensation producing capability, a lu-\nminance, of 6 x 10 5 Zumens per square meter per steradian.\nThus, the unit of brightness-producing capability of radiant\nflux is called a Zumen. The lumen is the photometric counter-\npart to the radiometric watt. This convention is translated\ninto practical working formulas as follows: we observe that\nif Nb(1) is the radiance of the standard platinum surface as\ngiven by the blackbody thermal radiation laws, then on the\none hand, by (4), the relative (standard) luminance of the\nplatinum surface is:\n8\n[ 66 (A)T(1)\ndA\n(8)\n0\nand on the other hand, by fiat, the absolute (standard) 1u-\nminance of the platinum surface is:","162\nRADIOMETRY AND PHOTOMETRY\nVOL. II\nlumens\n6 x 105\n(9)\nsteradian\nThe units of (8) are watts/(m2 x steradian). Our agreement\nleads us to equate (8) and (9), after introducing a numerical\nconstant which will balance units in the resulting equation.\nLet us denote this number by \"Km\". Then we agree to write:\ndi\n(10)\nThe number Km so defined has units: lumens/watt. Its magni-\ntude is determined by explicitly introducing the functional\nform for the surface radiance Nb of the surface of a black-\nbody (a complete radiator or Planckian radiator) at tempera-\nture T:\nC2/(XT)\nin which we have set:*\nC1/TT = 2c2h = 1.1909 x 10 - 16 watts m²/steradian\nC2 = hc/k = 1.438010-2 °(Kelvin)\nT = 2042° °(Kelvin)\nIt follows, on numerical integration of Nb(1)y(1) over A,\nthat:\n8\ndi = 884 watts/(m2 x steradian)\n.\nHence, from (10):** **\nK = 6 x 105/884\n= 680 lumens/watt\n(11)\n*\nThe units of C1 are determined by specifically using the\nspectral density part of the dimensions of radiance. Thus\ndim [N] = watts/ (m ² x steradian x m), using wavelengths.\nUncertainties in the measured values of C1,C2 and in the nu-\nmerical integrations leading to the value 884 watts/(m2x stera-\ndian) lead to a corresponding uncertainty of Km of about 5 or\n6 units in the last digit. See, e.g., [51], [153].","SEC. 2.12\nRADIOMETRY TO PHOTOMETRY\n163\nThe only seemingly arbitrary feature in this final step from\nradiometry to photometry is the choice of the magnitude\n6 x 105. Actually, the choice of this particular magnitude is\nnot completely arbitrary; it is tied to the historical pre-\ncedent set for a lumen by the early international candle stan-\ndard. The historical details of these matters may be found\nin the standard treatises on photometry, Refs. [185], [311],\nor in Ref. [50]. See also [206], [51]. The current standard\nunit of luminous intensity (defined formally below) is the\ncandela which by definition is 1 Zumen per steradian. Hence\nthe convention in (9) may be read as 600,000 candelas per\nsquare meter.\nLuminance Distributions\nThe magnitude of the transition factor Km having been\ndetermined, we can go on to give a precise definition of the\nrequisite \"measure\" of the capability of a given sample of ra-\ndiant flux to evoke the brightness sensation. Thus let\nN(x,E,t,) be the radiance function which assigns to every A\nin the spectrum A a radiance N(x,5,t,1) at a fixed point x in\nthe fixed direction E at given time t. Then we call the num-\nber:\nxmJ\nN(x,E,t,A)y(1)\ndi\nthe luminance associated with the radiance function N(x,5,t,),\nand write:\n8\n\"B(x,5,t)\"\nfor\nN(x,5,t,1)y(1)\nK\ndi\n(12)\nIf x and t are fixed but E allowed to vary in N(x,5,t,1) then\nthe resultant function B(x,.,t) is called the luminance dis-\ntribution at x, at time t. Often the time t, or x, or even E\nare understood (as occurred e.g., in the radiometric context)\nand so may be dropped from the notation provided no confusion\nresults. Thus we agree that we can occasionally write:\n\"B(x,E)\"\n\"B(E)\"\n\"B\"\nfor\nB(x,5,t)\nor\nor\nThese definitions serve to fix B(x,.) as the photometric coun-\nterpart to the radiometric function N(x, studied in earlier\nsections of this chapter. The units of B(x,.) are lumens/\n(m2 x steradian).\nTable 1 of Sec. 2.4 can be used to construct a corres-\nponding table of radiance by assuming that the surfaces S re-\nferred to in Table 1 of Sec. 2.4 have uniform radiance. Then\nthe desired radiances are found by dividing each irradiance\nin the right hand column of that table by TT. A similar table\nfor the general order of magnitude of luminance of common","164\nRADIOMETRY AND PHOTOMETRY\nVOL. II\nnatural objects can be constructed. A sample of such a table\nis given below which is partly constructed from Fig. 1.12.\nTABLE 2\nLuminance (lumens/(m2 x steradian)\nSource\n2 x 10 9\nSurface of sun\nClear sky\n3200\n*\nSurface of moon\n6200 (full)\n1400 (half)\n2800 (average)\n0 (new)\nFurther illustrative examples of luminance are easily\nconstructed: suppose a source of monochromatic radiant flux\nhas a radiance N of 1000 watts/(m2 x steradian) per meter wave-\nlength for each 1 over an interval Al of 10 mu centering on\nwavelength A = 555 mu, and of zero radiance outside this in-\nterval. What is the luminance B of this source? Returning\nto (12) we see that in this case:\n8\nB =\ndi\nK_N(555)y(555)A>\n680 x1000 x1 x10- (1 mu = 10-9m)\n=\n= 6.8 x 10-3 lumens/(m2x steradian)\n.\nAs another example, consider a source of radiance\n1000 watts/ (m 2 x steradian) per millimicron wavelength at\nN\n=\nA = 450 mu over an interval A11 of 10 mu about this wavelength,\nand of radiance N = 500 watts/(m2 x steradian) per millimicron\nwavelength at A = 600 mu over an interval DA2 of 5 mu about\nthe latter wavelength. What is the associated luminance of\nthis source? By (12) we have:\n*\nThese luminances are computed directly from the full and half\nphase illuminances produced by the moon, as given in Fig. 1.12.\nFor half phase, the solid angle of the luminous surface was\ntaken as 3 x 10 - 5 steradians. Standard references give 2500-\n3000 lumens/(m2 x steradian) for the moon's luminance. The\nlighting geometry on the porous and craggy lunar surface is\npartly involved in this spread of values.","SEC. 2.12\nRADIOMETRY TO PHOTOMETRY\n165\n8\nB\nK\nd\n=\n= 680 (1000 x.03810 + 500 x . 631 x 5)\n= 680 (380 + 1580)\nIf 1.34x106 lumens/(m2 x steradian)\n.\nHere y (450) and y(600) may be estimated from Fig. . 2.46 or\ntaken directly from Table 1. Note the two different ways of\nspecifying the spectral density of radiance in these two ex-\namples.\nAs a final example, let a source be of constant radi- -\nance N (per millimicron wavelength) over the region of the\nspectrum from 390 mu to 760 mu (the part of the spectrum over\nwhich y(1) is defined in Table 1, and zero outside this region.\nWhat is the luminance of the source? By (12) we have:\n760\nK\nNy(X) di\nm\n390\n760\n= K m N\ny(X) di\n390\n= 680 x 107 x N = 7.3x 10\" N\nHere we have integrated () over the range A = 390 mu, to\n1 = 760 mu in steps of 10 mu using the values of Table 1\nabove. The result is:\n760\ny(x) di = 106.857 (millimicrons)\n390\nThis value may, for all practical purposes, be taken as the\nintegral of y() from 1 = 0 to A = 80. 0\nTransition to Geometrical Photometry\nThe transition from geometrical radiometry to geomet-\nrical photometry has so far been made between two points, i.e.,\nbetween the radiance and luminance concepts by means of (12),\nand with the help of (10) and (11). This choice of the radi-\nance-luminance bridge rather than any other means was governed","166\nRADIOMETRY AND PHOTOMETRY\nVOL. II\nby the relative visualizability of these concepts as contrast-\ned with other radiometric-photometric pairs, say with the vis-\nualizability of hemispherical irradiance and its counterpart\nhemispherical illuminance (to be defined below). But now that\nthe bridge has been constructed with suitable attention to in-\ntuitive motivations and visualization, we return to its site\nand start anew with the purpose in mind of constructing the\nbridge once again, but now in a logically more satisfying way.\nBy undertaking this reconstruction we are given the opportun-\nity to re-emphasize and make formal the additivity assumption\nwe had encountered on our way to the relative luminance dis-\ntribution in (4). This formalized additivity assumption will\nsubsequently take its place among the other basic assumptions\nof radiometry which we isolated for the radiant flux function\nin the discussions of Sec. 2.3.\nThe transition from radiance to luminance, as summa-\nrized in (12), may now be emulated systematically for each ra-\ndiometric concept. That is, for every part M of the spectrum\nA we first define a general integral the radiometric-photo-\nmetric transition operator by writing:\n\"Y(,M)\"\nfor\ndi\n(13)\n.\nM\nThen it follows from (12) that:\nB = Y(N,A)\n(14)\n,\nwhere \"B\" and \"N\" are the abbreviated names for the given lu-\nminance and radiance functions in (12). But we need not stop\nat (12). Indeed, let us go on and write:\n\"F (S,D,t)\"\nY(P*(S,D,t,+),A)\nfor\n(luminous flux,\n(15)\n(3) of Sec. 2.3;\ncf., (17) of Sec.\n2.4)\nY(H(x,E,t,),A)\n\"E(x,5,t)\"\nfor\n(illuminance (11), (16)\n(17) of Sec. 2.4)\nY(W(x,E,t,),A)\n\"L(x,E,t)\"\nfor\n(luminous emit-\n(17)\ntance, (22) of\nSec. 2.4)\n\"B-(x,5,t)\"\nY(N*(x,5,t,.),A)\nfor\n(luminance, (30), (18)\n(31) of Sec. 2.5)\n\"It(s,e,t)\"\nY(J*(s,5,t,.),A)\nfor\n(luminous inten-\n(19)\nsity, (7),(10) of\nSec. 2.9)\nThese are the definitions of the first five principal photo-\nmetric concepts under both the surface (+) and field (-) inter-\npretations. The names of the concepts are given to the right\nof each definition and reference is made to the appropriate\nradiometric ancestor of each concept. Thus, e.g., surface\nluminous flux (S,D,t) is derived from surface radiant flux","SEC. 2.12\nRADIOMETRY TO PHOTOMETRY\n167\nP+(S,D,t), which in turn is defined in (3) of Sec. 2.3, and\nwith the surface interpretation of p+(S,D,t) given in (17) of\nSec. 2.4. In this way the logical ancestry of each of the\npreceding eight photometric concepts is traceable back to the\nprimitive radiometric function &. The bridge to the ancestor\nin each case is the integral operator Y(,), defined in (13).\nThis integral operator now permits, without the necessity of\nany further geometrical arguments, all the radiometric connec-\ntions among P, H, W, N and J developed in the preceding sec-\ntions, to be carried over directly to the photometric context.\nAs an example, (8) of Sec. 2.5 is carried over to the photo-\nmetric context by applying Y(,) to each side of that equa-\ntion. Thus, by (16) above, (31) of Sec. 2.5, and (8) of Sec.\n2.5:\n(1\nE(x,5) Y(H(x,5),A) = Y\n(x,\nE(5)\nE(E)\n=\n(20)\nE(E)\nWhenever either 11+11 or 11_11 is understood, or an equation is\nvalid under both the field and surface interpretations, then\nthese signs may be dropped, if desired. For example, in the\ncase of (20), we know from (21) of Sec. 2.5 that H and N go\ntogether, so that dropping \" 1 \" 11 on the right sides of the equa-\ntions in (20), no confusion can result. Hence, every occur-\nrence of the signs \" 1 \" 11 may be dropped from (20) and left im-\nplicitly understood.\nThe roll-call of principal photometric concepts is con-\ntinued as follows. We shall write:\nQt(x,t)\" for Y(u*(x,t,*),A)\n(luminous energy in (21)\nregion X, at time t,\n(12) of Sec. 2.7)\nY(u+(s,r, ),A)\nfor\n(luminous energy\n(22)\nacross surface S\nover time interval\nT, (17) of Sec. 2.7)\nfor Y(u*(x,t,*),A)\n(luminous energy\n(23)\ndensity, (2) of Sec.\n2.7)\n\"e(x,t)\"\nY(h(x,t,),A)\nfor\n(scalar illuminance, (24)\n(3) of Sec. 2.7)","RADIOMETRY AND PHOTOMETRY\nVOL. II\n168\n\"1(x,t)\" for Y(w(x,t,*),A)\n(scalar luminous\n(25)\nemittance, (19) of\nSec. 2.7)\nWe illustrate again the fact that any linear relation\nbetween two radiometric quantities has a carbon copy in the\nphotometric context. Thus, consider (14) of Sec. 2.7; assum-\ning V is independent of A in X and applying the operator\nY(,) to each side we have:\nQ(X,t) = Y(U(X,t,),A) = Y (((u/v)V(X),A)\n= (V(X)/v)Y(u,A)\n= (q/v)V(X)\n(26)\nThere remains to be defined certain of the photometric\nconcepts such as the vector counterpart E to H,I to J, etc.\nHowever, instead of going on to explicitly exhaust all these\ntransitions, which are quite numerous, we state below a gen-\neral definition-scheme which covers all transitions just made,\nand any yet unmade.\nLet R be a radiometric function defined on A. Then\nY( R, , A) is the photometric counterpart to R. Let \" P \" de-\nnote this photometric counterpart. Then the following state-\nment is a definitional identity:\n=Y(R,A)\n(27)\nA definitional identity is a statement of the form \"A = B\"\nwhere \"A\" and \"B\" are the names of one and the same object\narising from a definition. Thus, e.g., \"Q(X,t)\" and\n\"Y(U(X,t,\"),A)\" are names of one and the same object, namely\nthe number:\n8\nu(x,t,A)y(A) a d\n0\nand so:\nQ(X,t) = Y(U(X,t,),A)\nand alternatively:\nd\nare definitional identities. For example, definitional iden-\ntities were used to start and end the series of deductions\nsummarized in (20) and (26). The significance of (27) is","SEC. 2.12\nRADIOMETRY TO PHOTOMETRY\n169\nsimply this: every photometric concept P is the image, under\nY(,A), of some radiometric concept R; thus, to define a new\nphotometric concept, first find its radiometric progenitor R.\nThen P is the desired concept Y(R,A).\nGeneral Properties of the Radiometric-Photometric\nTransition Operator\nThe integral operator Y( M) defined in (13) has sev-\neral properties built into it which are of critical importance\nin establishing the science of theoretical photometry. To\nrecognize and understand these properties is to recognize and\nunderstand the role of photometry as a descriptive science.\nTherefore we devote some attention to the isolation of these\nproperties.\nLet R1 and R2 be any two radiometric functions de-\nfined on a subset M of A and let C1 and C2 be any two real\nnumbers such that the sum (C1 R1 + R2 ) is defined. Then\nby (13) and the linearity of the mathematical integration pro-\ncess:\nY(C1 R1 + C2 R2, A) C1Y( a 1,A) ((R2,M)\n(28)\nThis is the linearity property of Y(, M), the formal vestige\nof the associated property of y(aj,1) discussed in (2).\nNext, for every radiometric function a defined on A,\nY ( M2) = Y a. M1) + Y (R,M2)\n(29)\nfor every pair of disjoint subsets M1 and M2 of A. This is\nthe additive property of Y ( R, ) and is the formal vestige\nof the property of y (ai 1) discussed in (3). Finally, for\nevery radiometric function R defined on 1,\nIf (M) = 0, then Y(R,M) = 0\n(30)\na,\nwhich is the M-continuity property of Y\n(\n)\nfor continuous\n.\nspectra. The length measure 1 and its general use was defined\nin (4) of Sec. 2.3.\nThe Mathematical Basis for Geometrical Photometry\nProperties (29) and (30) may be added to the set of\nsix additivity and continuity properties of discussed in\nSec. 2.3. In fact, in an axiomatic development of the mathe-\nmatical theory of photometry, statements (28), (29) and (30)","VOL. II\nRADIOMETRY AND PHOTOMETRY\n170\nwould constitute the essential starting point of the construc-\ntion of the theory, just as the properties of $ in Sec. 2.3\nconstitute the essential starting point of the theory of geo-\nmetrical radiometry. Indeed, for any radiometric function\ndefined on A, we may deduce from (29) and (30) alone the exis-\ntence of a function (.) on A such that:\nY(Q,A) =\n(31)\nEvidently F'() will turn out to be Kmy() discussed above.\nThe complete details of the mathematical justification of\nthis assertion lie beyond the scope of this work. Some of\nthe mathematical background of (31) will be covered as a mat-\nter of course in Sec. 3.16. The requisite mathematical basis\nof the assertion may be found in part in Sec. 56, in particu-\nlar theorem D, of Ref. [103]. The general measure-theoretic\napproach to foundations of radiative transfer theory, intro-\nduced in Ref. [216], can now, by (31), be systematically ex-\ntended to the domain of photometry. Hence, as far as the\nmathematical structure of photometry is concerned, it rests\non three pillars: (28), (29), and (30), and its framework\ncan be erected by means of the theorems of modern measure\ntheory and without the necessity of any further reference to\nphysical constructs. In other words, the epistemological con-\ntent of classical photometry rests in but three postulates,\nthe statements of the linearity, M-additivity and M-continuity\nof Y introduced above. We note in closing that the preceding\nobservations apply immediately to the representations of col-\nors by the tristimulus procedure of colorimetry; all that has\nbeen said for the function y, now applies, without essential\nchange, to the other two tristimulus functions X and IN (cf.,\nSec. 1.7). The mathematical setting in the colorimetric case\nwould be a three-dimensional vector space, and the measure-\ntheoretic aspects will be elevated from the scalar to the vec-\ntor level.\nSummary and Examples\nThe present discussion of radiometry and photometry\nwill be brought to a close with a summary of the main concepts\nintroduced in this chapter. The units and dimensions of the\nconcepts will be tabulated, discussed and illustrated, and a\nfew further illustrative examples will be given.\nTable 3 lists the main radiometric concepts by name,\nsymbol, units, dimensions, and reference to its definition in\nthe present work. A similar Table 4 lists the main photomet-\nric concepts in an exactly analogous way, as far as possible.\nExplanatory notes are appended to each table.","RADIOMETRY TO PHOTOMETRY\n171\nSEC. 2.12\nTABLE 3\nRADIOMETRIC CONCEPTS\nDEFINITION\nBASIC\nDIMEN-\nREFERENCES\nNAME\nSYMBOL\nSIONS\nMKS UNITS\np+\nSec. 2.1; (17)\nRADIANT FLUX\n$\nWATT\nand (18) of\n(general)\nSec. 2.4\n+\nWATT/mu\n(3) of Sec. 2.3\nRADIANT FLUX\nP\nP\n(17) and (18)\n(spectral)\nof Sec. 2.4\nPEASE\nWATT/(m2xsr)\n(1), (4), of\nRADIANCE (all\nN\nSec. 2.5\nradiometric con-\n(See note\ncepts here and\n(c) below)\nbelow may be\neither general\nor spectral)\nP A 1\nWATT/m2\n(1), (17) of\nIRRADIANCE\nH\nSec. 2.4\nPTA-1\nWATT/m2\n(2) of Sec.2.8\nVECTOR\nH\nIRRADIANCE\nPTA-1\nWATT/m2\n(3) of Sec.2.7\nSCALAR\nh\nIRRADIANCE\nP+A-1\nWATT/m2\n(18), (22) of\nRADIANT\nW\nSec. 2.4\nEMITTANCE\nP+A-1\nWATT/m²\nSee note (d)\nVECTOR RADIANT\nW\nbelow\nEMITTANCE\nP+A-1\nWATT/m2\n(19) of Sec.\nSCALAR RADIANT\nW\n2.7\nEMITTANCE\nptn-1\nWATT/sr\n(1), (10) of\nRADIANT\nJ\nSec. 2.9\nINTENSITY\n-1\nWATT/sr\n(22) of Sec.\nVECTOR RADIANT\nJ\n2.9\nINTENSITY\np+n-1\nWATT/sr\nSee note (d)\nSCALAR RADIANT\nj\nbelow\nINTENSITY","VOL. II\n172\nRADIOMETRY AND PHOTOMETRY\nTABLE 3 (Continued)\nNAME\nBASIC\nDIMEN-\nDEFINITION\nSYMBOL\nSIONS\nMKS UNITS\nREFERENCES\nPTT\n(12), (17) of\nRADIANT ENERGY\nU\nWATT-SECOND\nor JOULE\nSec. 2.7\nP+TV-1\nWATT-SECOND/\n(2) of Sec.2.\nRADIANT DENSITY\nu\nm 3 or\nJOULE/m 3\np+v-10-1\nWATT/ (m 3 xsr)\n(RADIANT) PATH\nN*\n(2) of Sec.3.12\n(8) of Sec.3.14\nFUNCTION\nor\nHERSCHEL/m\n(3) of Sec.13.3\nPIA\nN°\nWATT/(m2xsr)\n(1) of Sec. 3.12\nPATH RADIANCE\n(15) of Sec. 3.12\nor\nHERSCHEL\n(2) of Sec.13.3\nExplanatory Notes for Table 3\n(a)\nThe names and basic symbols are drawn, as far as pos-\nsible, from the current standard in nomenclature, namely that\nrecommended in 1953 by the American Standards Association\nSection Committee Z-58, sponsored by the Optical Society ([4],\n[49], also cf., p. 229, Ref. [50]). The basic symbols are\nused to construct names for various radiometric functions by\nplacing various modifiers after them. Thus, e.g., (S,D,t, F)\nis the value of the function $ which assigns to each set F\nof frequencies the radiant flux incident on collecting sur-\nface S through the set D of directions at time t. Further ex-\namples are found throughout the preceding sections of this\nchapter. It might be well to observe here that the symbols\nand names for the concepts in such a venerable subject as geo-\nmetrical radiometry are still in a state of change. However,\nthere is currently some effort being made in the direction of\nestablishing an international standard of terminology in radi-\nometry and photometry (see, e.g., Ref. [130]). . It may be\nnoted that the terminology and notation listed in Tables 3 and\n4 have withstood the severe tests of use in courses and re-\nsearch studies by the author and his colleagues over the past\ntwenty years, and have been found adequate for the purposes\nof radiative transfer studies in natural optical media. (See\nalso p. 6, [177]. )\nIt now appears possible to attain a systematic and bas-\nic terminology for radiometry and photometry by combining the\nbest features of Table 3 and Table 4 (below) and the sugges-\ntions by Jones in Ref. [130]. Toward this end we observe that","SEC. 2.12\nRADIOMETRY TO PHOTOMETRY\n173\nJones extracts the idea of flux, as the basic concept whose\ntask is to describe the flow of a generalized 'substance'\nThe 'substance' may be radiant energy, luminous energy (the\nphotometric counterpart to radiant energy) or even entropy.\nThe suggested term for the 'ometry' which studies general\nflows is 'phluometry' (\"phluo\" = \"to flow\") There are five\nsuch 'ometries' suggested at present:\nName of the Phluometry\nPhluometric\nUnit of Flux\nModifier\nRadiometry\nRadiant\nWatt\nPhotometry\nLuminous\nLumen\nErgometry\nEnergic\nJoule\nErgophotometry\nErgolumic\nLumen-second\nEntropometry\nEntropic\nWatt/degree\n(b)\nThe basic radiant flux dimensions p+, , p- are associated\nwith flux leaving and incident on a surface, respectively.\nThe idea of 'radiant flux' is the central physical idea of\ngeometrical radiometry. However, it is found useful in theory\nand practice to distinguish between emitted and incident ra-\ndiant flux. This distinction has been placed into the dimen-\nsions for appropriate use, if needed, and its geometrical sig-\nnificance is summarized in Fig. 2.47. (See also Fig. 2.12.)\nIf the distinction is not needed, or is understood, the occur-\nrences of \"+\" or 11_11 may be omitted. Further discussion of\ndimensions is made in note (h).\n(c)\nThe spectral radiant flux P has units of WATT/mu if\nwavelengths in millimicrons (mu) are used, or has units of\nS\nv\no\nP-(x, = p+ (x, II (E) )\nE unit inward normal to\nunit outward normal to\nsurface S at X\nsurface S at X\nFlux incident on S\nFlux leaving S\n[\n[\nField interpretation\nSurface interpretation\nFIG. 2.47 Field and surface interpretation of radiant\nflux.","174\nRADIOMETRY AND PHOTOMETRY\nVOL. II\nWATT/sec\"1, if frequency is in units of sec 1 . The dimensions\n\"WATT/mu\" are used often in practice; hence their inclusion in\nthe table. The radiant flux dimension P of any radiometric\nquantity below P in the table can be either spectral (hence\nPA) or general (hence P). For simplicity, only the general\nradiant flux dimension is given. When working with spectral\nradiant flux, it is occasionally necessary to explicitly use,\nduring a given discussion, both wavelength and frequency di-\nmensions for radiant flux. The radiometric quantities can\nthen be given a \"X\" or a \"v\" subscript for the duration of\nsuch discussions. In general, however, such explicitness is\nnot needed and the dimension of the spectral flux is under-\nstood implicitly, and (except for specific numerical examples)\nwill so be understood throughout this work. In theoretical\nradiative transfer discussions, e.g., the frequency dimension\nis usually preferred over wavelength (and this preference is\nimplicit in the notation) because frequency of radiation is\ninvariant along a path with variable index of refraction. The\ngeneral (definitional) connection between P1 and Px is ob-\ntained by writing:\nd\nV\n\"P v\n11\nfor\ndv\nd\n1\n\"P A 11\nfor\nd\nWhence:\ndv\nax\nV\nP=Pv\nP\n(32)\n=\n=\n.\nv\n2\n(d)\nTable 3 is divided into five natural groupings of con-\ncepts. First in order are the three main concepts - - &, P, N.\nThen comes the irradiance group, the radiant emittance group,\nand the radiant intensity group. These are followed by the\nenergy group, and the radiative transfer group consisting of\nN* and N* In principle, the irradiance group and the radiant\nemittance groups may be coalesced into a single group by using\nexplicitly the surface (+) and (-) concepts. However, his-\ntorical precedent has fixed the distinction between these\ngroups by means of the generic letters \"H\" and \"W\", and we\nsee no reason at the present time to change such established\nnotation to \"H+ for W and \"H-\" for H merely on the grounds\nof esthetic reasons. However, esthetic reasons (in particular\nthe desire for symmetry) are responsible for the inclusion of\ntwo concepts in Table 3 which-- if the practical photometric\nworker had a ay--would normally be omitted. These are the\ntwo concepts W and j. The distinction between W and H is very\nfine conceptually and non-existent vectorially. For we define\nas follows. We write:\n\"W(x)\" for\n(33)\nwhere, as noted, the integral uses the surface radiance. We\ncall w(x) the vector radiant emittance at X. Thus (33), by","SEC. 2.12\nRADIOMETRY TO PHOTOMETRY\n175\n(32) of Sec. 2.5, and (2) of Sec. 2.8, yields the equality\nw(x) = H(x) Further, whenever S is a surface and N(x) is a\nuniform radiance distribution at X on S (either field or sur-\nface) then we may write:\n(34)\n\"j(s)\"\nfor\nN(x) dA(x)\nS\nand call (S) the scalar radiant intensity. By including (33)\nand (34) we round out each of the radiant emittance and radi-\nant intensity families to a full threesome, and maintain the\ninteresting duality between irradiance and radiant intensity\nbrought out in the main discussions above (where j (S) is now\nthe dual to h (x) ) .\nThe unit name \"herschel\" for radiance is adopted from\n(e)\na suggestion by Moon (Ref. [184]). However, the unit, as used\nhere, is left \"unrationalized.\" This means simply that for\nuniform radiance distribution, we have H = N. Hence if N = 1\nherschel, then H = TT watts/m², and numerical computations are\nnot jeopardized by not remembering what to do with \"II\". Fur-\nthermore, the TT serves to keep tabs on the dimensions of H\nand N in calculations. It is clear that something other than\nthe relatively lengthy \"WATT/(m2 x sr)\" is desirable, at least\nin verbal discussions, where \"sr\" stands for \"steradian\", and\n\"m2\" as usual denotes \"square meters\".\nThe final group consisting of path radiance N* and path\n(f)\nfunction N* is included for convenience of reference. These\nare the only two additional radiometric concepts needed in\nthe general studies of radiative transfer in natural optical\nmedia. Actually these concepts are mutually dependent and\nonly one is needed. The full discussion of this matter is re-\nserved for Chapter III.\nThe only radiometric concepts omitted from Table 3 and\n(g)\nwhich are of some importance, are the spherical and hemispher-\nical scalar irradiances defined in Sec. 2.7. These concepts,\nespecially the latter, are primarily indigenous to plane-paral-\nlel (or one parameter) geometries, whereas all the listed con-\ncepts pertain to general geometries. Not defined at all were\nthe spherical and hemispherical scalar emittances. For the\nsake of completeness (cf., (6) of Sec. 2.7), we write:\n1/w(x,t)\n(35)\n\"W4\"(x,t)\"\nfor\nand (cf. (7) of Sec. 2.7) :\nI\nda(s')\n(36)\n\"w(x,E,t)\"\nfor\nE(E)\nand (cf. (8) of Sec. 2.7):\nw(x,E,t)\n(37)\n\"W4m(x,5,t)\"\nfor\nand these are called, respectively, spherical radiant","176\nRADIOMETRY AND PHOTOMETRY\nVOL. II\nemittance, hemispherical scalar radiant emittance, and hemi-\nspherical radiant emittance.\n(h)\nThe theory of dimensions of radiometric and photometric\nconcepts has received relatively little systematic attention.\nWe shall devote a few comments to this matter in the present\nnote. The dimensional system chosen for Tables 3 and 4 is\nconstructed from two basic physical dimensions and one basic\ngeometrical dimension. These are the dimensions of radiant\nflux P, time T, and length L. The general radiant flux func-\ntion & is assigned the dimension P; this dimension is consid-\nered irreducible in the radiometric context. In other con-\ntexts, P need not be irreducible. Thus, in the electromagnet-\nic context P is representable in terms of the dimensions of\nforce, length and time: (force) x (length) x (time) or\nas\n(mass) x (length) 2 x (time) -3 The \"+\" and \" \" \" superscripts\non\n\"P\" do not change its dimension; they merely serve as conven-\nient mnemonics for the surface and field interpretations of\nradiant flux.\nAs already made clear in note (c) above, the dimensions\nPV or PX are reducible to PT or PL-1, , respectively. Specifi-\ncally, in Table 3 we have implicitly written:\n\"P\" \" =\"\nfor\nPT\nand for the wavelength case we have explicitly written:\n\"P14\"\nP+L-1\nfor\nNow just as we find it convenient to append \"+\" and\n\" \" \" to the basic symbol \"P\" to denote the geometric sense of\nthe flow of radiant flux, so too is it helpful to distinguish\nbetween two types of length in geometrical radiometric dis-\ncussions. Following Moon [184], we write: \"Lt\" to denote the\ndimension of length measured in a direction transverse (i.\nperpendicular) to a given direction E; and \"Lr\" to denote the\ndimension of length along the given (radial) direction E. As\nin the case of p+, attaching \"t\" and \"r\" to \"L\" does not\nchange the dimension; rather it serves as a conceptual remind-\ner of the transverse and radial interpretations of length.\nThen in the table we have written:\n2\n\"A\"\nfor\nL\nt\n\"V\"\nfor\nL\nLt24-2\n-2\n\"8\"\nfor\nThus in the present dimensional system, area has the dimen-\nsions of transverse length squared-- most natural dimension\nwithin radiometry since we perceive areas as two-dimensional\nextensions of space in the transverse directions to a line of\nsight. Volume has dimensions of ALT, i.e., (transverse) area\ntimes (radial) length--again a most natural combination of","SEC. 2.12\nRADIOMETRY TO PHOTOMETRY\n177\ndimensions for the radiometrist. Finally, solid angles are\nmeasured using the steradian concept. In the present system\nthe dimension of solid angles does not vanish from view, but\nrather is expressed as the product of Lt 2 and LT-2, as indi-\ncated above. Since Lt and Lr are conceptually distinct, this\nproduct is conceptually not dimensionless. In this way the\noccasionally bothersome problem of the vanishing dimensions\nof solid angle can be solved. (Ordinarily the dimensions van-\nish, but like the smile of the Cheshire cat, the units remain.)\nIt should be noted that the conventional dimensions of the ra-\ndiometric concepts are recovered by dropping \"+\", \"-\", \"t\",\nand \"r\" from P and L wherever they occur.\nAs an illustration of the use of these dimensions, ob-\nserve that the dimension of path function can be written as\n(PA-1-1) LT-1, so that the path function concept is seen to\nhave the dimensions of radiance per unit of radial length.\nThe full significance of this interpretation will become clear\nin Sec. 3.12, wherein the path function concept is formally\nintroduced. On the other hand, we may rearrange the path\nfunction dimensions as follows: (P+s-1)v-1, and thereby dis-\ncern another facet of this concept, namely that it may be\nviewed as a radiant intensity per unit volume (cf. (7), (10)\nof Sec. 13.6). The radiance concept itself may be viewed via\nthe dimensional arrangement (P+A-1)8-1 as irradiance (-) or\nradiant emittance (+) per unit solid angle on the one hand,\nand via the arrangement (P+s-1)A as field (-) or surface (+)\nradiant intensity per unit area, on the other hand.\nA general guide to the fixing of dimensions of radio-\nmetric concepts and their manifold derivates in practice is\nas follows. Let us refer to \"area\", \"length\", \"time\", etc.\nby the generic term \"measure\", and use the generic symbol \"m\"\nfor a measure. Let us write \"dim(m)\" for the dimension of m.\nThus if A is an area measure, then A(S) is the area of a sur-\nface S, and dim(A) = Lt2 Further, if 1 is a length measure\nalong paths of sight, then 1(p) is the length of a path p,\nand dim (1) = LT. Now, according to our development of geo-\nmetrical radiometry in this chapter, every radiometric concept\nQ is definable first on the empirical level and then on the\ntheoretical level. The empirical level of definition is sim-\nply the level on which the measures are used directly. Thus,\ne.g., recall that empirical irradiance H(S,D) is P(S,D)/A(S),\ni.e., the quotient of incident radiant flux over a surface S\nby the area of S. The corresponding theoretical definition is\nobtained by going to the appropriate limit (e.g., S+{x} in\nthe case of irradiance). In going from the empirical level\nto the theoretical level, it is desirable to have the dimen-\nsions remain unchanged. Hence the definition on the empirical\nlevel already fixes the dimension of a radiometric concept.\nSuppose then that R is a radiometric concept and its empiri-\ncal definition is such that we write:\nom 1 m\n1\n\"R\"\na\nfor\nm1...mb\nwhere\ni\n1,\na, and \"mj\", j = 1,...,b, denote meas-\n=\nures and denotes the radiant flux function, which is also\na measure with dimension dim(d) = P. Then the dimension of","178\nRADIOMETRY AND PHOTOMETRY\nVOL. II\nRis:\n(dim x lim(ma')\ndim(m1) x\ndim(mb)\nx\nThe preceding reduction of a dimension to simpler terms is\nfacilitated by adopting the following conventions for the di-\nmention operator dim. Let x and y be any two measures or\nphysical concepts. Then:\n(i)\ndim(xy) = dim(x) dim(y)\n(ii)\ndim(x/y) = dim(x)/dim(y)\n(iii) If dim(x) = dim(y), then dim(x) If dim(x+y)\n(iv) If {xn} is a sequence of terms of common\ndimension d, and if lim n *n = y, then\ndim(y) = d.\nIn our development of radiometry, the basic dimensions\nare P, L, and T. In order to use rules (i) - (iv) , we agree\nthat these dimensions obey the same rules of addition and mul-\ntiplication as real numbers. This is implicitly assumed in\nthe tables and in the various manipulations above. In addi-\ntion to the four dimensions above, we introduce one more,\nnamely 1, which has the property that:\ndl = 1d = d\nfor every dimension d, and\ndi/d2 = 1\nfor every pair of dimensions d1 and d2 such that d1 = d2\n.\nThus \"1\" denotes the dimensionless concept.\nExplanatory Notes for Table 4\n(a)\nThe notes and comments for Table 3 apply also to this\ntable except where explicit references to frequency or wave-\nlength concepts are made. Observe that Tables 3 and 4 cor-\nrespond item for item, except that there is naturally no lu-\nminous counterpart to the general radiant flux function $, ,\nthe primitive radiometric function from which all others\nspring. The unit of luminance, the (unrationalized) blondel,\nis adapted from a suggestion by Moon (ref. [184]). The lu-\nminous counterparts to (35) - (37) are obtained by means of the\ngeneral definition scheme of (27). In (35) and (37) \"w\" is\nreplaced by \"1\", and \"radiant\" replaced by \"luminous\", to\neffect the definitions. We assign to the lumen the basic di-\nmension F. Hence, in particular, dim(Km) = FP-1 . By (15)\nand property (iv) of the operator dim in note (h) for Table 3,\nwe have, e.g., dim(F*(S,D,t)) = F+. In this case, the limit","SEC. 2.12\nRADIOMETRY TO PHOTOMETRY\n179\nTABLE 4\nPHOTOMETRIC CONCEPTS\nBASIC\nDIMEN-\nDEFINITION\nNAME\nSYMBOL\nSIONS\nMKS UNITS\nREFERENCES\nF+\nLUMINOUS FLUX\nF\nLUMEN\n(15)\nF--1\nLUMINANCE\nLUMEN/(m2xsr)\nB\n(18)\nILLUMINANCE\nF A 1\nLUMEN/m2\nE\n(16)\nVECTOR\n(2) of Sec.\nF A 1\nLUMEN/m2\nILLUMINANCE\nE\n2.8 and (27)\nSCALAR\nF A - 1\nLUMEN/m2\nILLUMINANCE\n(24)\ne\nLUMINOUS\nF+A-1\nLUMEN/m2\nEMITTANCE\nL\n(17)\nVECTOR LUMINOUS\nF+A-1\nLUMEN/m2\nEMITTANCE\nL\n(33), (27)\nSCALAR LUMINOUS\nF+A-1\nLUMEN/m2\nEMITTANCE\n1\n(25)\nLUMINOUS\nLUMEN/sr or\nF+0-1\nINTENSITY\nI\nCANDELA\n(19)\nVECTOR LUMINOUS\nLUMEN/sr or\n(22) of Sec.\nF+0-1\nINTENSITY\nI\nCANDELA\n2.9 and (27)\nSCALAR LUMINOUS\nLUMEN/sr or\nF+n-1\nINTENSITY\ni\nCANDELA\n(34), (27)\nF+T\nLUMINOUS ENERGY\nQ\nLUMEN-SECOND\n(21), (22)\nor TALBOT\nF+TV-1\nLUMEN-SECOND/m3\nLUMINOUS DENSITY\nq\n(23)\nor TALBOT/m3\nLUMEN/(msr)\n(LUMINOUS) PATH\n(2) of Sec.\nF+v-1s-1\nFUNCTION\nB*\nor BLONDEL/m\n3.12 and (27)\nLUMEN/(m2x 5r)\nPATH LUMINANCE\n(1) of Sec.\nB*\n*\nor BLONDEL\n3.12 and (27)","180\nRADIOMETRY AND PHOTOMETRY\nVOL. II\noperation is that used in the definition of the integral op-\nerator (13).\nWe conclude this discussion of the photometric concepts\nwith a few examples.\nExample 1. Using the luminance of the sun as given in Table\n2, compute the corresponding illuminance on a plane normal to\nthe rays of the sun. To find the requisite illuminance, re-\ncall from (2) of Sec. 2.5 that we can write:\nH(S,D) = N(S,D)&(D)\nApplying the transition operator Y(,), as defined in (13),\nto each side of this equation, we obtain:\nY(H(S,D),A) = Y(N(S,D)G(D),A)\ns(()) Y(N(S,D),A)\nUsing (16) and (18) and the general definition scheme (27) to\ndefine the empirical counterparts of radiance and irradiance\nthe preceding equation yields:\nE(S,D)= B(S,D)S(D)\n,\nwhich is the desired connection between empirical luminance\nand empirical illuminance.\nFrom Table 2,\nB(S,D) 2 x 10' blondels or candelas/m²\nor lumens/m²: x sr\nand from Example 1 of Sec. 2.11:\ns(D) = 6.78 x 10-5 steradians\nHence*:\nE(S,D) = 2 x 10' x (6.78 x 10-5)\n136,000 lumens/m²\nExample 2. If the sun in the context of Example 1 is at\n0 = 50° from the zenith, and surface S' is the projection on\na horizontal plane of the surface S used in Example 1, what\nis the illuminance E(S',D) produced by the sun's rays on S'?\nTo find this illuminance, recall (15) of Sec. 2.4:\nH(S',D)-H(S,D) cos\nwhere the symbols are explained in detail in Sec. 2.4, and\n*A relatively recent estimate (Ref. [128]) of E(S,D) is\n136,000 lumens/m². See also [296] for a survey of measure-\nments of the solar constant.","SEC. 2.12\nRADIOMETRY TO PHOTOMETRY\n181\nwhich now apply directly to the present context. In partic-\nular now v = 50°, , and D is the cone of directions subtended\nby the sun's disk. Applying the operator Y(,A), as defined\nin (13) to this equation, we have:\nY(H(S',D),A) = Y(H(S,D) cos V,A)\nUsing the definition scheme (27) now for empirical illumi-\nnance, this becomes:\nE(S',D) = E(S,D) cos v\n.\nHence:\nE(S',D) = 136,000 x cos 50°\n= 136,000 x.643\n= 87,500 lumens/m²\nExample 3. Using the illuminance of Example 1, compute the\nilluminance produced by the sun's rays normally incident on\nsurfaces in the orbits of the planets Venus and Mars. The\nrequisite illuminances can be found by means of the inverse\nsquare law for irradiance deduced in Example 4 of Sec. 2.11.\nThus, from (8) of Sec. 2.11, if r and S are two distances\nfrom the sphere's center, we deduce that:\nHar2 = Hss2\nUsing the operator Y(,) and the definition scheme (27),\nthis equation becomes:\n2 - Ess2\nLet Er be the illuminance at the earth as given in the form\nE(S, D) in Example 1. Hence r = 93 x 10 ' miles. In the case of\nVenus, S = 67 x 106 miles. Hence:\nEs = Ex(r/s) 2\n= 136,000 x (93/67)2\n= 136,000 x 1.93\n= 262,000 lumens/m²\nIn the case of Mars, S = 142 x 106 miles. Hence:\n= 136,000 x(93/142)2 =\n= 136,000 .430\n= 58,500 lumens/m²","RADIOMETRY AND PHOTOMETRY\nVOL. II\n182\nExample 4. Compute the number of lumens F incident on a plane\nsurface S of area A(S), every point of which is illuminated by\na luminance distribution of constant magnitude B incident over\ndirections within a conical solid angle D of half angle 0 and\nwhose axis is normal to S. The requisite relation for the lu-\nmens incident on S is obtained by beginning with (14) in Ex-\nample 6 of Sec. 2.11:\ndo\n,\nand applying the operator Y(,A) to each side to get:\nY(H,A) d$,A)\ndo Y(N,A)\nwhich, via the definition scheme (27), can be written:\ndo\n.\nIn the present case, 0(0) = 0 for every , 05052W.\nHence:\nE = jB sin ² e\n.\nNext, from (6) of Sec. 2.4 we have:\nP(S,D,t,v) = H(S,D,t,v) A(S)\nApplying the operator Y(,), to each side of this equation,\nwe have:\nY(P(S,D,t,),A)\n.\nFrom (15) and the definition scheme (27) applied to\nempirical irradiance, we consequently have:\nF(S,D,t) = E(S,D,t)A(S)\n.\nConsidering references to S,D, and t as understood for the\npresent discussion, we distill this to:\nF = EA .\nThus we are led to the desired relation:\nF = BA sin20","SEC. 2.13\nGENERALIZED PHOTOMETRIES\n183\nAs a specific example, let 0 = 30°, B = 120 blondels,\nand A = 4 m². Then:\nF = (3.14) x (120) x 4 x (1/2) 2\n= 378 lumens.\nExample 5. A red-orange appearing filter is known to have a\nband pass of 10 mu, but it is not known precisely what wave-\nlengths of radiant flux it transmits. An experiment is sug-\ngested and tried in which it is inferred that an irradiance\nof 2 watts/m² over the transmission interval Al = 10 mu pro-\nduces an illuminance of 1360 lumens. Can the transmission\nwavelengths of the filter also be inferred from this informa-\ntion? To answer this, consider the following observations.\nFrom the definitional identity:\nE = Y(H,A)\nd\n0\nand the fact that H(X) = 0 outside the interval AA about the\nunknown 1, we have very nearly:\nK_H(A)y(A)A)\nHence:\n= 1360/(680 x 2 x 10)\n= 1/10 = .10\n.\nFrom Table 1, by linear interpolation, we infer that 1 = 472 mu\nor 652 mu. From the given general appearance of the filter's\ncolor, we infer that A = 652 mu.\n2.13 Generalized Photometries\nWe conclude this chapter with a few observations on\nthe necessary forms of certain generalized photometries which\narise in an attempt to extend the salient ideas of classical\nphotometry. The directions of extension to which we subject\nthe ideas of photometry in this discussion are toward a more\ngeneral class of 'luminosity' functions. The class we envi-\nsion here is to contain not only the classical luminosity\nfunctions of human eyes, as briefly discussed in 2.11, but\nalso irradiation-response functions describing photographic,\nphototransmissive, photovoltaic, photoemissive, and photo-\ncurrent phenomena. In short, we attempt to sketch in broad\nterms certain possible generalizations of the 'lumen' concept","184\nRADIOMETRY AND PHOTOMETRY\nVOL. II\nwith reference to irradiations which can be measurably effec-\ntive on both organic and inorganic levels. Our discussion\nwill consider in turn linear and nonlinear generalized photom-\netries.\nLinear Photometries\nLet us begin with the simpler of the two generaliza-\ntions: the linear photometry. The classical photometry dis-\ncussed in Sec. 2.12 is an instance of a linear photometry.\nUsing that discussion as a suitable motivation and background,\nwe can initially and broadly define theoretical linear photom-\netry to be the study of the properties of the effects z(A,M),\non some physical object, of radiometric causes R over a wave-\nlength set M, and under the premise that the numbers Z(R,M)\nhave certain postulated general properties. Specifically,\nfor a given physical object (eye, skin, selenium cell, etc.),\nlet (, ) be a function which assigns to each radiometric\nconcept R and part M of the spectrum A a real number Z(A,M)\nwith the following properties*:\na -Linearity: For every two radiometric con-\n(i)\ncepts R1 and R2 and nonnegative real numbers C1\nand C2 for which C1 Rs C2Q2 is defined, and for\nevery part M of the spectrum A,\n2(c).G1+C2.G2,M) = c1z(R1,M)+c22(R2,M)\n(ii)\nM-Additivity: For every radiometric function\nand every two disjoint parts M1 and M2 of A,\nZ(R,M,UN =\n(iii) M-Continuity: For every radiometric function\nif 1(M) = 0, then Z(R,M) = 0.\n.\nAn example of 2(Q,M) would be the amount of reddening\n(suitably measured) of human skin under irradiation (so that\nR can be irradiance H) over a certain portion of the far in-\nfrared (so that M consists of all wavelengths from, e.g.,\n1 = 800 mu to A = 850 mu). Another example of Z(R,M) would\nbe the rate of oxygen production by a leaf of some type of\nvegetation under irradiation (so that R can be scalar irra-\ndiance h) and over some part M of the spectrum. Marine bio-\nlogical contexts appear also to present potential areas for\ngeneralized photometries.\nAt any rate, the landmarks of an incipient linear\nphotometry are properties (i), (ii), (iii) above. The concept\nof a linear photometry is certainly not empty since we have\nThe footnote to the discussion of (3) and (4) of Sec. 2.3\napplies also to the present discussion and should be consulted\nbefore proceeding.","SEC. 2.13\nGENERALIZED PHOTOMETRIES\n185\nclassical photometry; and many additional photometric phenom-\nena appear to be linear over given ranges within A. One of\nthe useful facts about a linear photometry is the provable\nexistence of a generalized luminosity function z(.) within\nthat photometry with the property that:\nR(X)((A)\n2(Q,M)\ndi\n(1)\nM\nWe call this the canonical representation of 2(,) for a\nlinear photometry. The mathematical basis for this fact rests\nin general measure theory (the Radon-Nikodym theorem), and was\nalluded to earlier in (31) of Sec. 2.12 in connection with\ny().\nIn sum then, it is possible to carry over to any gen-\neral linear photometry the useful notion of a general 'lumi-\nnosity function' which describes a general 'relative luminos-\nity' of R over M (cf. (6) and (7) of Sec. 2.12). As a result\nit also appears possible to generate the concept of general-\nized 'lumens', so that one can initially place on a firm sci-\nentific footing such generalized linear photometries.\nNonlinear Photometries\nTurning now to consider the prospects of forming a\nfoundation for nonlinear photometries we are faced with the\nusual arresting fact about nonlinear phenomena: there are so\nmany types of them. Were the world built so that there was\nonly one type of nonlinearity - say of the power-exponential\ntype or the sinusoidal type, etc. then the problem of repre-\nsenting nonlinear phenomena would long ago have been thorough-\nly subdued, analytically speaking. However, since man's fi-\nnite amount of attention must be spread over an apparently\ninfinite class of nonlinear phenomena, this layer of attention\nmust be nearly 'monomolecular' in depth wherever it exists.\nTo make a small start into the wilderness of nonlinear\nphotometries, let us consider the first and logically the\nsimplest types of departure from linearity. The preceding\nthree statements (i)-(iii), constituting the defining proper-\nties of a linear photometry, may not all hold for given\nphotometric phenomena. The three main types of departure from\nlinearity would be:\nType I nonlinearity:\n(i) does not hold; (ii) and\n(iii) hold\nType II nonlinearity:\n(ii) does not hold; (i) and\n(iii) hold\nType III nonlinearity: (i) and (ii) do not hold:\n(iii) holds\nThis choice of classification is based on the plausible feel-\ning that: \"if 1(M) = 0, then (R,M) = 0\" will always hold in\nany reasonable designed measure 2(,0) of a radiometric effect.\nTherefore, if a nonlinearity is encountered, it is likely to","VOL. II\nRADIOMETRY AND PHOTOMETRY\n186\nbe traceable to a violation of either (i) or (ii), or both.\nEach of the three types of nonlinearity will now be briefly\ndiscussed with the purpose in mind of suggesting possible\nroutes toward linearization.\nOne very promising mode of approach to Type I nonlin-\nearities is to find a function f which would linearize 2(,M)\nfor every M. Specifically, we suggest finding a real valued\nfunction f, defined on the real numbers, such that:\n(iv) = c1f(2(R1,M))+c2f(2(R2,M))\nMany logarithmic and power nonlinearities are linearized away\nin this manner by the time-tested technique of plotting on\nlogarithmic or exponential, or power coordinates. Whenever\na linearizing function f can be found so that (iv) holds, then\nwe say that the Type I nonlinearity is removable. The func-\ntional composition f°Z of the linearizer f and the Z suffer-\ning a Type I removable nonlinearity, is now linear. Thus (i)-\n(iii) hold for foZ and so the canonical form (1) is available\nfor use with foZ. Summarizing: whenever a Type I nonlinear-\nity of a photometric measure Z(,M) is removable by a linear-\nizer f such that (iv) holds, then the composition foZ(,M)\nhas a canonical representation (1).\nLet us consider now the Type II nonlinearity. We ask:\nif (ii) does not hold, in what way is it most likely not to\nhold? Imagine an erythemal phenomenon: a bit of living ani-\nmal tissue is irradiated simultaneously by two distinct sets\nof radiation of non-overlapping wavelength sets M1 and M2.\nThe effect Z C(R,M,UN M2) is noted. Then a biologically equi-\nvalent piece of tissue is irradiated in turn by samples of\nwavelength sets M1 and M2, and Z (R,M1) and Z(R,M2) are noted.\nSince M1 and M2 are allowed to be active separately, more\neffect-activity say, may take place in the tissue under each\nirradiation by Mi than when they act simultaneously. Thus,\nit may be that while the effects are not additive, they are\nM-subadditive:\nZ (a,M, M2) VI Z(R,M1) + Z(R,M2)\n(v)\nWhenever a Type II linearity is encountered so that\n(ii) does not hold, it may be the case that M-subadditivity\nsubsists. If subadditivity is indicated in a Type II nonlin-\nearity, then it may be shown (cf. [103]) that for every R\nthere exists an extended measure Z* (R, which is additive in\nthe sense of (ii). The net result we have reached may be\nstated as follows: Every photometric measure (R, which\nexhibits nonlinearity of Type II and which is subadditive\n(i.e., (v) holds) may be extended to a linear photometric\nmeasure Z* (R, for which a canonical representation (1) is\npossible.\nThe immediate attempt at linearization of a Type III\nnonlinearity is to seek a linearizer f such that (iv) holds.\nSome Type III nonlinearities will surely succumb to these\nvery general modes of attack. Beyond these few approaches\nlies an unknown field of potential modes of study of","SEC. 2.14\nBIBLIOGRAPHIC NOTES\n187\nnonlinear photometries.\n2.14\nBibliographic Notes for Chapter 2\nThis chapter is based in the main on unpublished lec-\nture notes (Refs. [210], [211]) in radiometry and photometry\ngiven in 1953 and 1954 at the Visibility Laboratory of the\nUniversity of California, San Diego. The characterization of\nthe foundations of radiometry in terms of a systematic use of\nadditivity and continuity properties of the radiant flux func-\ntion &, as given in Sec. 2.3, is derived from a similar treat-\nment given in Ref. [251], and which in turn is based on the\ngeneral easure-theoretic approach to radiometry and radiative\ntransfer theory introduced in [216]. An important paper on\nphotometry is that of Gershun, [98] who introduced and made\nprecise the concept of the light field (our vector illuminance\nE). Gershun also introduced the operational definition of ra-\ndiance in the form N = H/8 (re: (2) of Sec. 2.5). An impor-\ntant source of photometric wisdom may be found in the writings\nof Moon. In particular, the radiometric lectures cited above\ndrew inspiration from some of the ideas of Refs. [184] and\n[185], especially in connection with developing general pho-\ntometries. An old standard work on photometry and still val-\nuable is Walsh's treatise [311]. The work by Le Grand, Ref.\n[153], is a relatively modern work on the optical-physiologi-\ncal properties of human vision which may be used to supplement\nthe discussions of Sec. 2.12.","CHAPTER 3\nTHE INTERACTION PRINCIPLE\nGird up now thy loins like a man;\nfor I will demand of thee, and answer\nthou me.\nWhere wast thou when I laid the\nfoundations of the earth? declare, if thou\nhast understanding\nWhereupon are the foundations thereof\nfastened? or who laid the cornerstone\nthereof\nJOB XXXVIII, 3-6.\n3.0 Introduction\nRadiative transfer theory is distinguished by the fact\nthat it is one of the branches of theoretical physics that\ncan be made to rest on a single principle from which all the\nsalient structures of the theory may be systematically de-\nduced. In this sense it is a closed subsystem of electromag-\nnetic theory. The principle that permits this mode of con-\nstruction of radiative transfer theory is called the interac-\ntion principle. The interaction principle is a distillate of\nmany diverse conceptual constructions concerned with radia-\ntive transfer which have arisen during the past seven decades\nof evolution of the theory. In this chapter we shall state\nthe principle and present various instances of it for a selec-\nted range of physical situations customarily encountered in\npractical applications of radiative transfer theory. It will\nbe demonstrated that these physical situations can all be for-\nmulated within the theory in a uniform manner using a method\nwhich we call the method of the interaction principle. By\nmeans of examples we shall verify, on the one hand, that the\nsalient theoretical structures of the theory do indeed fall\nunder the domain of the principle, and, on the other hand, we\nshall prepare the groundwork for the various applications of\nthe principle in the subsequent chapters of this work.\nThe principle of interaction in its essential form is\na statement of the linearity of the classical radiative trans-\nfer processes. Thus radiative transfer theory, a complex web-\nwork of deductions following from the principle, is at its\ncore a linear theory of the interaction of light with matter\non a phenomenological level. The linearity of the theory\narises from the confluence of two main points of view adopted\n188","SEC. 3.0\nINTRODUCTION\n189\nby its principal developers and investigators since the turn\nof the century.\nThe first of these views is that the theory is con-\ncerned in the main either with radiant energy phenomena within\nthe relatively hair-thin visible wavelength interval from\n4 x 102 mu to 7 x 102 mu, or within a wider band of wavelengths\nfrom 10 to 10 5 mu. Radiant energy phenomena within this four-\norders of magnitude spread of the electromagnetic spectrum\nare, as we shall see below, associated with energies which\nbarely tap the electronic energy levels of common atomic struc-\ntures. The resultant interactions of radiant energy with mat-\nter are thereby limited essentially to elastic scatter activi-\nty, photoelectric effects, and simple absorption-emission phe-\nnomena. Inelastic scatter interactions of photons with matter\nare virtually ruled out within the 10-10 mu range of wave-\nlengths. Within this domain the radiant energy interactions\nare manifestly linear, and thereby set one part of the stage\nfor the linear structure of the interaction principle.\nThe second viewpoint adopted by the founders of the\ntheory is that the interaction of light with matter is to be\nviewed on the phenomenological level, i.e., on the macroscopic\nlevel, with instruments which mimic normal human vision in its\nessential geometric characteristics. Therefore the delicate\neffects of wave phenomena, such as diffraction, interference,\nand other coherence activities are automatically excluded, by\nfiat, from the domain of classical radiative transfer theory.\n(See problems I-V of Sec. 141, Ref. [251]) In adopting this\napproach, we have 'shut our eyes completely and have thought\nabout all that we have seen. The linearities resulting from\nthis predominantly geometrical viewpoint form the basis for\nthe various additive and continuity properties of radiant flux\ndiscussed and developed at length in Chapter 2. These two\nviews, one physical, the other geometrical, combine to act as\neffective linearization forces on the formulations of the con-\ncepts designed to describe radiative transfer processes in\ngeophysical optics and great stretches of astrophysical optics.\nThe Physical Basis of the Linearity\nof the Interaction Principle\nBefore going on to state and illustrate the interaction\nprinciple, it will be instructive to examine in more detail\nthe preceding physical assertions about the types of radiative\nprocesses limited to the purview of radiative transfer theory.\nIn contemplating the consequences of the modern view that ra-\ndiant energy is carried by quantized electromagnetic fields--\ni.e., by photons--we encounter a great number of possible\ntypes of interactions of photons with matter. Adopting a sug-\ngestion by Fano [90], we can usefully classify all of these\nvariations into five main types of photon interactions:\nI\nInteractions with atomic electrons.\nII\nInteractions with atomic nucleons (protons,\nneutrons).","190\nINTERACTION PRINCIPLE\nVOL. II\nIII Interactions with electric fields around charged\natomic particles (electrons, charged nucleons)\nIV Interactions with meson fields surrounding nucleons.\nV Interactions with other photons.\nThe effects of these interactions are also greatly\nvaried. But again for our present purposes, we need distin-\nguish only three broad types of effects:\nA. Outright absorption\nB. Elastic scatter\nC. Inelastic scatter\nA word or two on the meaning of these terms is in order. Sup-\npose we picture a photon as a small colored fuzzy ball, and\nan atom or a molecule of an optical medium as a relatively\nlarge complex spherical maze of thin, widely spaced fuzzy\nwires (electronic orbits or electron bonds) with tiny rela-\ntively dense central cores. Then in the case of effect A, the\ncolored ball either zooms into the wire cage and becomes en-\nmeshed in the maze of wires or is captured by a dense core,\nthere to stay for a period of time far greater than that nor-\nmally required to traverse the diameter of the cage at its\ninitial speed. If it is ultimately released, we say an emis-\nsion process has occured. In this captured state the ball,\nin effect, has been absorbed by the atom, and loses its iden-\ntity as such, resulting momentarily in a higher orbit of one\nof the atom's electrons or in a higher stationary energy state\nof a molecule or in an increase in kinetic heat energy of the\natom, or some combination of these. In the case of effect B,\nthe colored ball caroms off (or skims through) the electronic\nshells of the atom, the net effect being a change of direction\nof travel of the photon with no change of its color, and we\nsay that the photon is scattered without change in wavelength.\nIn the final case, C, the ball becomes very briefly enmeshed\nin the electronic shell, or glances off the dense core, with\ngreater or lesser wavelength than before, the net effect being\na change of color and direction of travel, and we refer to the\nphoton as scattered with change in wavelength.\nReturning now to the interactions and their effects,\nwe see that there are, in the present view, five possible\ntypes of interaction of a photon with matter and three pos-\nsible types of effect. There are then in all fifteen possible\ninteraction-effect pairs we can form: IA, IB, IC, IIA, IIB,\nIIC, VC. We shall call any of these fifteen interaction-\neffect pairs a radiative process. In Table 1 the fifteen gen-\neral radiative processes are displayed by their characteristic\ninteraction energies and by name whenever possible. For ex-\nample, the class of processes we know as Rayleigh scatter is\nsubsumed by the process IB. In this process a photon inter-\nacts with an atomic electron with the effect that it is scat-\ntered elastically. The inequalities that are indicated in the\nentries of the Table specify the interaction energies for","SEC. 3.0\nINTRODUCTION\n191\nTABLE 1\nGENERAL RADIATIVE PROCESSES\nPhotons Inter-\nOutright\nElastic\nInelastic\nacting with\nAbsorption\nScatter\nScatter\nA\nB\nC\nAtomic\nPhotoelectric\nRayleigh\nCompton\nI\nElectrons\nEffect\nScatter\nScatter\nVI 0.1 Mev\n0.1 Mev\n0.1 Mev\nAtomic\nNuclear Photo-\nNuclear\nNuclear\nII\nNucleons\nelectric\nScatter\nResonance\nEffect\n10 Mev\nScatter\n10 Mev\n10 Mev\nElectric field\nPair\nDelbruck\nDelbruck\naround Elec-\nProduction\nScatter\nResonance\ntrons, Nucleons\n1 Mev\n3 Mev\nScatter\nIII\n3 Mev\nMeson field\nMeson\naround Nucleons\nProduction\nIV\n150 Mev\n150 Mev\n150 Mev\nOther Photons V\nPair\nProduction\n1 Mev\n= 1 Mev\n1 Mev\nwhich the associated process takes place. For example,\n11 VI 0.1 Mev\" means that the associated process takes place at\n0.1 million electron volts or lower. Further, \"z0.1 Mev\"\nmeans that the associated process takes place at 0.1 million\nelectron volts or higher. The unnamed processes and some of\nthe other processes (IIIC, IVB, IVC, and the photonic inter-\nactions) have not been observed at this time of writing.\nIt will be instructive to correlate the Mev means of\nspecifying the energy of a photon with its associated wave-\nlength. By doing so, we shall be able to see clearly where\nthe interaction energies common to radiative transfer theory\nstand in the arena of all this activity. To facilitate com-\nparisons, we convert Mev units to wavelength units. The tran-\nsition from Mev to wavelength is made by first recalling that\nthe basic quantum of energy E associated with a photon of fre-\nquency v is\nE = hv","192\nINTERACTION PRINCIPLE\nVOL. II\nwhere the frequency is related to wavelength A by:\nAv = V\nand where \"v\" denotes the speed of light. If we let V = C =\n=3 x 10° m/sec, and recall that h = 6.625 x 10-27 ergsec, then\nfrom the preceding relations:\nA = hc/E meters\nor\nA = 1.24x10-3 =\nmu\n,\nwhere E is in Mev units. Thus if E = 1, then the associated\nenergy is one million electron volts. The form in which we\nrequire this formula is:\nE = 1.24 x 10 -\nMev\nA\nwhere A is in mu, i.e., millimicrons (10- meter), or as they\nare also called, nanometers. Assuming that our present inter-\nests lie mainly with processes in the wavelength range\n10 mu, we can now estimate the associated energies of\ninteraction. Then by looking over the table of processes we\ncan judge which of the areas of the main interaction arena are\nof primary interest. Thus we are interested in the energy\nrange:\n1.24 x 10-'\nSE S 1.24 x 10-3\n10 S\ni.e.,\n1.24 x 10 - 8 VI E VI 1.24 x 10 Mev\nIn particular, green light (555 mu) is on the order of\n2 x 10- Mev.\nWhat a tiny corner of the interaction arena we find\nourselves in. A glance at the table shows that our world of\nradiant phenomena lies well within classes IA and IB. We\nshall call IA and IB the classical radiative processes. The\nclassical radiative processes are, of course, replete with\nspecial radiative processes which include the various well-\nknown absorption and scattering processes such as Raman, Ray-\nleigh, Tyndall and resonant scatter; also fluorescence, and\nphosphorescence.\nThe simple calculation just performed shows that we\nneed not be overly concerned in this work with such phenomena\nas Compton scatter - - a relativistic phenomenon; pair production\n--a - quantum electrodynamics phenomenon; or scattering of light\nby light- a quantum relativistic phenomenon. Even if we ex-\ntend our interests down three orders of magnitude to","SEC. 3.0\nINTRODUCTION\n193\nwavelengths of the order of 10- mu, we still remain essen-\ntially within parts IA, IB, and IC of the interaction arena.\nThe classical radiative process region is the domain of the\nclassical Maxwell equations. We need not at present use any\nother models of the light field such as the Schrödinger or\nDirac models, or those of general relativity to describe the\nactivity in that part of the interaction arena in which we\nhave found our current interests to lie.\nIt follows from the preceding analysis that the Maxwell\nequations in quantized, special relativistic form will suffice\nfor most conceivable applications of radiative transfer theory\nin geophysical settings. Actually, it has been found that\nthe classical (non quantified, non relativistic, linear) Max-\nwellian theory of electromagnetic fields may for all usual\npurposes encountered at present, serve as the nearest point\non the mainland of physics to which radiative transfer theory\nmay be adjoined when desired (see Chap. XIV of Ref. [251])\nIn this way is set the predominantly linear cast of the phe-\nnomenological theory, at the base of which may be found the\ninteraction principle.\nPlan of the Chapter\nThe plan of the remaining part of this chapter is as\nfollows: we present in the next section a preliminary example\nof the interaction principle. This will serve to focus atten-\ntion on a relatively concrete but yet typical instance of the\nuse of the principle. From the example we shall extract the\nessence of the principle and state and discuss the result in\nSec. 3.2. Beginning with Sec. 3.3, further examples of the\ninteraction principle will be given. The examples of applica-\ntion will proceed in a systematic manner from relatively sim-\nple cases to progressively more complex cases until all the\nmain tools of radiative transfer, as needed in the present\nwork, have been formed.\nThus in Sections 3.3 to 3.5 we apply the interaction\nprinciple to the development of the reflectance and transmit-\ntance operators for plane and curved surfaces, with detailed\nexamples presented to help fix the main ideas of the deriva-\ntions and applications. In Sections 3.6 and 3.7 the reflec-\ntance and transmittance operators for plane-parallel media\nare developed and applications are given. The next step in\nthe ascending scale of applications is taken in Sections 3.8\nand 3.9 in which the interaction operators for general media\nare defined, functional relations governing the resulting op-\nerators are derived, and applications of the operators illus-\ntrated. Then the sequence of five sections 3.10-3.14 goes on\nto apply the preceding theory to the problem of constructing\nthe basic inherent optical properties and radiance functions\nof radiative transfer theory (volume attenuation function,\nvolume scattering function, path function, path radiance) and\nin Sec. 3.15 these are all assembled into the fundamental in-\ntegral equation for radiance. At this point all the main\ntools of radiative transfer theory will have been constructed\nby means of the methodical use of the interaction principle.\nThis use of the interaction principle is systematized and","INTERACTION PRINCIPLE\nVOL. II\n194\nsummarized in Sections 3.16-3.18 in such a way as to aid the\nstudent of radiative transfer theory in attempting further\napplications and development of the method.\nThroughout all the examples of this chapter regardless\nof their level of complexity-- - runs a common thread of method:\nthe method of the interaction principle. This method begins\nto form in Example 1 of Sec. 3.4; crystallizes in Example 2 of\nthat section; and then recurs repeatedly, in the manner just\noutlined, through all the remaining illustrations of the\nchapter.\n3.1\nA Preliminary Example\nWe shall develop an example of the interaction princi-\nple in this section with the purpose in mind of fixing, on a\nrelatively simple intuitive level, the salient features of\nthe principle preparatory to stating the principle in its full\nform.\nEmpirical Reflectances and Transmittances for Surfaces\nA prerequisite for the development of the example is\nthe definition of the empirical reflectance of a small plane\nsurface S. Figure 3.1 depicts such a surface S with unit out-\nward normal k, which is irradiated at each point by radiant\nflux* through a narrow solid angle D', the flux passing\nthrough a hypothetical collecting surface S' on its way to S.\nThe observed (empirical) field radiance of the incident flux\nis N(S',D') and the observed (empirical) surface radiance--\narising from reflection of (S' D') by S in a narrow solid\nangle D--is N(S',D';S,D) . We write:\nN(S',D';S,D)\n\"r(S',D';S,D)\"\nfor\nN(S',D')s(D')\nand call r (S' , D' the (empirical) reflectance of surface\nS for the incident and reflected directions D' and D, respec-\ntively. Here S' is the projection of S on a plane perpendic-\nular to a direction E', the central direction of D'. . The func-\ntion which assigns to (S',D') and (S,D) the number r(S',D';S,D)\nis called the (empirical) reflectance function for S. For\nthe purpose of the present example, we assume r (S' , D' is\nknown for all pairs (D', D) of incident and response (reflected)\nFor simplicity in exposition, throughout this work all radi-\nant flux quantities will be assumed unpolarized, unless spe-\ncifically stated otherwise. For an outline of the task of ex-\ntending all results below to the polarized context, see Chap-\nter XII of [251]. The interaction principle, however, holds\nimplicitly for the polarized case. For the relative mathe-\nmatical consistency of the assumption of the unpolarized\nlight field with respect to the complete theory of the polar-\nized field, see Sec. 13.11.","SEC. 3.1\nPRELIMINARY EXAMPLE\n195\n(for reflectance)\nS'\nI\nD\nk\nS\nD\nD'\nus\n(for transmittance)\nFIG. 3.1 Setting for empirical reflectances and trans-\nmittances of surfaces.\ndirection sets such that E'OK< 0 and E.K> 0, respectively.\nHere E' 1 and E are arbitrary central directions of the sets D'\nand D, respectively. These two conditions merely require D'\nand D to lie on opposite sides of S, as in Fig. 3.1.\nNow the essential property of the response radiance\nN (S' D' S, , D) is that it is additive with respect to D' . More\nprecisely, experimental evidence indicates that we may assert\nthe following property of N(S', ,D' ; D,S). In each case let the\nsets D,D' of directions be circular, conical sets with cen-\ntral directions E,5', respectively. Then:\n(i) (D'-Additivity) If S is a surface in an optical\nmedium X and S is irradiated in turn by radiances\nN(S1', D1') and N(S2', D2'), , with N(S1 ', D1';S, D)\nand N(S2' D2 S,D) as the respective observed re-\nsponse radiances, then N(S1' D1 S, D)+N(S2', D2 S,D)\nis the observed radiance of the S under simulta-\nneous irradiation.\nFurthermore:\n(ii)\n(D'-Continuity) Let the geometric setting be de-\nfined as in (i). If so(D') = 0 and i # E, then\nN(S',D'; S,D) = 0.\nBy letting D lie on the same side of S as D' , these\ntwo empirically-based - properties of reflected radiant flux\nare readily turned into the corresponding laws for","VOL. II\nINTERACTION PRINCIPLE\n196\ntransmitted flux (see dotted direction cone in Fig. 3.1).\nOb-\nserve that, by virtue of (i), r (S' D' ;S, D) is independent of\nthe magnitude of N (S' , D'). It should be particularly noted\nthat (i) and (ii) are new laws which are independent of the\nD-additivity and D-continuity properties of in Sec. 2.3.\nThe present laws are intended to characterize a particular\ntype of interaction of radiant flux with matter, whereas the\nearlier laws were intended to characterize certain intrinsic\nradiometric (principally geometric) properties of radiant flux\nregardless of its interaction with matter. These two proper-\nties permit a limiting process to culminate in rigorously de-\nfined reflectance and transmittance functions for surfaces.\nThe details of such definitions will be considered in (6)-(9)\nof Sec. 3.3. For the present we use (i) and (ii) as they\nstand to help solve the following radiometric interaction\nproblem.\nThe Problem\nTwo plane surfaces, S1 and S2, in a vacuum are mutual\npoint sources. In addition, they are mutually visible and\nare irradiated by sources of radiance N1 o and N2° over solid\nangles, Do 1 and D02, respectively, as shown in Fig. 3.2.\nThese incident radiances initiate an interreflection process\nbetween S1 and S2 with resultant surface radiances N(S1, D12)\nand N(S2, D21). Here D12 is the set of directions from a point\nin S1 to every point in S2. Since S1 and S 2 are mutual point\nsources (i.e., each is a point source as seen from the points\nof the other), D12 does not vary appreciably as location is\nvaried over S1, and so may be assumed constant over S1.\nS2\nX2\nD21\nDo2\nDoi\nSI\nXIMI\nD12\nFIG. 3.2 Setting up an interaction calculation for sur-\nfaces S1 and S2.","SEC. 3.1\nPRELIMINARY EXAMPLE\n197\nSimilarly D21 is the set of directions from a fixed point of\nS2 to every point of S1; and D21 has the same general property\nas D12. With these preliminaries out of the way, we can now\nstate the present problem: Given: S1 and S2, as above, with\nknown empirical reflectance functions, r1 and r2, and initial\nirradiations, Required: the steady state empirical\nradiances N(S1,D12) and N(S2,D21).\nThe Present Instance of the Interaction Principle\nTo facilitate the present discussion let us write:\n\"N12\" for N(S1,D12)\n\"N21\" for N(S2,D21)\n\"r212\"\nfor\nr1(S12',-D12;S1,D12)\n\"r121\"\nfor\nr2(S21',-D21;S2,D21)\n\"r012\"\nfor\nri(S.1',Do1;S1,D12)\n\"r021\" for r2(S02',D02;S2,D21)\nwhere S12', e.g., is the projection of S1 on the plane normal\nto the direction from X1 to X2. Similarly, for S21', Soi',\nand S02'. In the case of Soi', e.g., imagine an external\nsource point XO. The set -D12 consists of all negatives of\ndirections in D12. Thus if E is in D12, then -D12 contains\n-5. Now by virtue of the definition of empirical reflectance,\nthe D'-additivity property (i) above, and the fact that the\nintervening space between S1 and S2 is a vacuum (so that the\nradiance invariance law can be used) we have:\nN12= Niro12801 + N21Y212862\n(2)\nN21= Nor021802 + N12F1218621\n(3)\nwhere we have written:\n\"doi\"\nfor\ni=1,2\nand\n\"Sij\" for s(Dij), i,j=1,2\nFor later purposes it is convenient to make one final\nset of definitions. We write:\n\"[i]j\"\nfor\nroijoi\ni,j=1,2\n\"[iji\" for Tijiji i,j=1,2.","INTERACTION PRINCIPLE\nVOL. II\n198\nThen (2) and (3) become:\nN12 If Niciz N214212\n(4)\nN21N221N12121 =\n(5)\nEquations (4) and (5) together constitute the algebraic\ncore of the statement of the present form of the interaction\nprinciple. In the present case we have two relatively small\nplane surfaces which are interacting radiometrically. Each\nsurface Si (i=1 or 2) is irradiated by two incident parcels\nof radiant flux in the form of the empirical radiances, Ni°\nand Nji, and Si itself has a resultant surface radiance Nij.\nTo the sets of such incident radiances, Ni° and Nji and re-\nsponse radiances Nij associated with Si (i=1 or 2), there cor-\nrespond four interaction operators (numbers in this case),\nnamely E.o and Ejij, such that (4) and (5) hold. The main\nrole of ij the interaction principle in the present case\nwould be to assert the existence of these operators and to\nyield the interaction equations (4), (5).\nSolution of the Problem\nThe interaction principle formulation (4), (5) of the\npresent problem leads to the solution of the problem by means\nof the theory of simultaneous algebraic equations. Thus, mul-\ntiplying each side of (4) by 1121:\nN12E121 NiE12E121 + N21E2122121\nand using this representation of N12121 in (5):\nN21=NOE21 = + N21E212E121)\nwhence:\n2212121\nThe radiance N12 can be found by permuting the symbols \"1\"\nand \"2\" in this equation. The complete solution of the system\n(4), (5) is then:\n(6)\n1 - E1212212\n(7)\n21212212","SEC. 3.1\nPRELIMINARY EXAMPLE\n199\nDiscussion of Solution\nA sufficient condition that N12 and N21 are determin-\nable via (6) and (7) is that the product 21212212 is less\nthan 1. We shall now show that a sufficient condition that\nthis latter property holds is that at least one of 121 and\n2212 is strictly less than 1. An examination of these defini-\ntions of 121 and 2212 shows that this condition may be based\non a particular form of the principle of conservation of en-\nergy. To see this in the case of 2212, we need only system-\natically unfold the definitions leading to it. Thus, 2212 is:\nr212802\nThe quantity r212 is:\nri(Si2,-D12;S1,D12)\nThis in turn is the value of the empirical reflectance func-\ntion for S1. By (1), and the fact that this value of r is\nindependent of the magnitude of the irradiating flux, we can\nselect any incident radiance, say N12 over -D12, and let N12\nbe the response (reflected) radiance over D12. Then:\nIf A(Si2) is the projected area of S1 on a plane normal to\nthe direction from X1 to X2 (see Fig. 3.2), and P(S1, -D12)\nand P(S1, D12) are the radiant fluxes associated with\nN12,\nthen the incident radiant flux is given by:\nP(S1,-D12) = Ni2A(S12)8812\nand the surface (response) radiant flux is given by:\nP(S1,D12) = Ni2A(Si2)8012\nHere we have used the fact that s(-D12) = S(D12) = 812, and\nalso the operational definition of surface radiance (Sec. 2.6).\nHence:\nP(S1,-D12)\nAt this point we choose the energy conservation principle in\nthe form which states that: if P is the total radiant flux\nincident on a given surface S and p+ is the total radiant flux\nleaving the surface S and p+ and p- are independent of time,\nthen p+. We shall assume this statement is true. From\nthis we deduce in particular that:\nVI 1\n9","INTERACTION PRINCIPLE\nVOL. II\n200\nso that:\n22121 .\nA similar inequality now follows for £121. These inequalities\nare the most we can say, without further qualifications, about\nany reflectance (or transmittance) operator occurring in the\ntheory of radiative transfer. Thus in a particular geometri-\ncal situation we must explicitly postulate or demonstrate\nthat at least one of 121 and 212 in (6) and (7) is strictly\nless than 1; and as our analysis has now made clear, this is\nsufficient condition that (6) and (7) uniquely determine\na\nN12 and N21.\nRelated Problems and their Solutions\nThe solutions (6) and (7) of the problem considered\nabove can be used to solve related problems centering on the\nradiometric interaction of S1 and S2. Suppose, for example,\nwe require the surface radiance of S1 in some set D1B of di-\nrections other than D12. Here \"B\" is associated with point\nxB which may be any point in the surrounding medium either in\nor not in S1 or S2. Toward this end we write:\n\"N1B\"\nfor N(S1,D1B)\n\"r21B\" for ri(Si2,-D12;S1,D18)\n\"ro18\"\nfor\nThen by the D' additive property (i) above and the radiance\ninvariance law we have:\n+ N21Y21B812\nIn an exactly similar manner we arrive at the surface radiance\nof S2:\nN2B = N2r028201 + N12F12B822\nOnce again we can contract these solutions into a fixed\nform which clearly reveals the underlying unity of the inter-\naction concept. Thus by writing:\n\"Eig\"\nfor\n,i=1,2\nroibloi\nand\n\"LijB\" for TijBdjj i,j = 1,2\nthe preceding equations become:","SEC. 3.1\nPRELIMINARY EXAMPLE\n201\nN18 = NPEIG + N212218\n(8)\n(9)\nwhere N12 and N21 are as given in (6) and (7). For the pur-\nposes of later comparison with the general statement of the\ninteraction principle we observe that: to the incident radi-\nances Ni° and Nji on Si (i = 1 or 2) and response radiance NiB\nthere correspond four interaction operators (numbers in this\ncase), namely i and LijB such that (8) or (9) hold. The\nindex B in (8) and (9) (and in the equations below) may be re-\nplaced by distinct indices, if desired. In other words, sur-\nfaces S1 and S 2 may give off radiances in distinct directions,\nwhich may be computed by (8), (9) by replacing index B in (9),\nsay, by a new index Y.\nAn Alternate Form of the Principle\nWe now abruptly change our conceptual orientation in\nFig. 3.2 from that of two radiometrically interacting surfaces\nS1 and S 2 to that of a single subset S of the optical medium\nirradiated from without by radiant flux. This change in ori-\nentation can be encouraged by imagining S1 and S2 in Fig. 3.2\nto be encircled by a closed dashed curve and to think of the\ncurve as holding a single subset S of space (that is, S is a\ndisconnected subset which happens to consist of two separate\nsurfaces, S1 and S2). This subset S is irradiated at two\nplaces by incident radiances N1° and N20, and the response of\nS is imagined in the form of two streams of flux characterized\nby N1B and N2B. This conceptual compression of S1 and S2 into\na single radiometrically responsive entity can be expressed\nsymbolically as follows. We first write the system (8) and\n(9) in matrix form (replacing \"B\" in (9) by \"r\", for general-\nity)\n(N1B,N2y) =\n(N21,N12)\n+\nFurther, from (6) and (7) we can write:\n(N21,N12) = (Ni,N2)\nwhere, in turn, we have written:","VOL. II\n202\nINTERACTION PRINCIPLE\nii2\n\"412\"\nfor\n(1-121212)\nE21\n\"421\"\nfor\n(1-121212)\nEizE121\n\"411\"\nfor\n(1-1212212)\nE21E212\n\"Y22\"\nfor\n(1-E121E212)\nThen going one step further and writing:\n\"YO\" for (99,991)\nand:\n\"By\" for\nand:\n221B\n0\n\"By\" for\nwe arrive at the following alternate representation of the\nsystem (8) and (9):\n(N1B,N2y) (Ni,No) (4gr + 404BY)\nLet us write:\n\"By\" for\nand thereby arrive at the desired form of the system (8), (9):\n(N1B,N2y) =\n(10)\nThe significance of (10) may be discerned as follows: for the\ngiven subset S we have shown that to an arbitrary pair of in-\ncident radiances (NP, NO) and response radiances (N1B,N2Y)\nthere corresponds a unique interaction operator (a 2x2 matrix","SEC. 3.1\nPRELIMINARY EXAMPLE\n203\nof real numbers in this case), namely BY' such that (10)\nholds.\nEquation (10) constitutes an alternate form of the in-\nteraction principle to that displayed in (4), (5) or in (8),\n(9). This alternate form is designed to give an indication\nof the potential internal complexity of an object S to which\nthe interaction principle may assign an interaction operator.\nIt is not too great an extension of ideas from the setting of\n(10) to the setting of an arbitrary finite number of inter-\nacting surfaces. However, the systematic study of such sys-\ntems of interacting surfaces or solids is the domain of dis-\ncrete space radiative transfer theory and lies far beyond our\npresent concerns. For those interested in pursuing this mat-\nter further, we observe that the complete theory of such sys-\ntems is developed in Ref. [251].\nThe Natural Mode of Solution\nWe conclude this preliminary example of the interaction\nprinciple by displaying an alternate mode of solution of the\nproblem of the radiometric interaction of the two surfaces S1\nand S 2 considered above. Our purpose is to show that this al-\nternate mode of solution and the interaction principle mode\nof solution are equivalent. As our developments proceed into\nthe next chapter, we shall also see that each mode of solution\npossesses a valuable conceptual kernel which is capable of ex-\ntension to quite wide domains of application in radiative\ntransfer theory in general, and hydrologic optics in particu-\nlar. This alternate mode of solution we call the natural\nmode of solution, for it appears to be conceptually the sim-\nplest and most natural approach to interreflection problems.\nThe natural mode of solution may be described quite\nbriefly as follows. We imagine a hyper-fast camera filming\nthe radiometric interaction of two surfaces, S1 and S2. The\nfilmed episode begins the instant the incident radiances N1 o\nand N2° simultaneously impinge on S1 and S2, respectively.\nIn a playback of the filmed episode in slow motion, we see\npart of N1 reflected from S 1 and start to travel toward S2.\nThis reflected flux eventually reaches S2 and part of it is\nredirected back toward S1. In the meantime N20 has been re-\nflected at S 2 and part of the reflected flux moves on to S1\nthere to be reflected and to have some flux begin to return\nto S2. As the film continues, the sources N1 and N2 o con-\ntinue to steadily pour flux on S1 and S2. After a while S1\nis being irradiated by photons, some of which come directly\nfrom N1°, some of which are making their first arrival from\nS2, and some their second arrival from S2 etc. By and by\nthe fluxing and interfluxing reaches a measurable steady state\n(while, in principle, however, there will always be some in-\nterreflection number which has not yet been attained) The\nfollowing argument develops the symbolic representation of\nthis steady state interreflection process.\nRetaining the notation of the preceding discussions,\nlet us go on to write:","INTERACTION PRINCIPLE\nVOL. II\n204\n\"N12\"\nNiciz\nfor\nand\n\"N11\"\nNOE21\nfor\nFurther, for every j = 2,3,...,\nwe write:\n\"N12\"\nj-1\nN21\nfor\n2212\nj-1\n\"N21\"\nfor\nN12\n121\nBy recalling the moving-picture allusion it is easy to see\nthat N2jis interpretable as the surface radiance of S in\nthe directions of S2 consisting of radiant flux having under-\ngone precisely j reflections. Again, by means of the analogy,\nwe are led to write:\n\"N12\"\nfor\n(11)\nN\n\"N21\"\nfor\n(12)\nj=1\nThe numbers N1 2 and N21 obtained in this way are called the\nnatural solution of the present problem of the radiometrically\ninteracting surfaces S 1 and S2. That N12 and N21 are indeed\nsolutions of the steady-state - interreflection problem associ-\nated with S1 and S 2 will now be shown. By starting with the\ndefinitional identity arising from (11):\nN12 - [Nia\nwe deduce the following chain of equalities:","SEC. 3.2\nINTERACTION PRINCIPLE\n205\nN12 = Ni2\nj\n1\nN12\nj=2\n1\nj-1\n[212)\n(Exd.)\nNiciz + N21212\n(13)\nThe last equality follows from (12) and the preceding defini-\ntion of Ni . . . By comparing (13) and (4) we see that the nat-\nural mode of solution implies the interaction mode of solu-\ntion. Evidently the steps in (13) are reversible, so that\nthe interaction mode of solution implies the natural mode of\nsolution. Thus the two modes of solution are equivalent in\nthis case. Since the interaction mode of solution clearly\nrepresents the solution of the interreflection problem of S1\nand S2, the natural mode of solution therefore is also, by\nvirtue of the preceding equivalence, a solution of the inter-\nreflection problem. This equivalence actually holds in very\ngeneral settings and has been established in detail for these\nsettings, in Ref. [251]. We shall have occasion to study and\nuse once again this equivalence of the two techniques later\nin the present work. Finally, we observe that the sums in\n(11) and (12), being reducible to a simple geometric series\nwith ratio E121E212 and initial term of the form\n(Nono + (i=1, j=2 for (11) ; i=2, j=1 for (12)),\nare readily evaluated; these sums are given by (6) and (7).\n3.2 The Interaction Principle\nWith the preliminary example complete, we turn now to\nthe statement of the central principle of radiative transfer\ntheory:\nThe Interaction Principle: For every X, S, A, B, m and\nn, if X is an optical medium and S is a subset of X, and A\nAm) is a class of sets Aj consisting of incident\n(=\n(A1\nradiometric functions on S, and B (= (B\nBn)) is a class\nof sets Bj consisting of response radiometric functions on S,\nand m and n are positive integers, then there exists a unique\nset n} of linear (interaction)","206\nINTERACTION PRINCIPLE\nVOL. II\noperators Sij with domain Aj and range Bj with the property\nthat for every element (a am) of A there exists an ele-\nment (b)\nbn) of B such that:\nm\nbj = { ij\ni=1\nor in matrix form:\nb = as\nwhere we have written:\n\"a\"\nfor\n)\nm\n\"b\"\nfor\n(b\nb\n)\n1\n,\n,\nn\nS\nS\nS\n11\n12\nin\nS\nS\nS\n21\n22\n2n\n\"s\"\nfor\n.\n.\nS\nS\nS\nm1\nm2\nmn\nDiscussion of the Interaction Principle\nWe shall discuss in some detail the meanings of the\nvarious terms in the interaction principle. First of all,\nthe meaning of the term \"optical medium\" as used in the\nstatement is quite broad and, for example, is intended to have\nas real designata such parts of the world as lakes, oceans\nand various portions of the atmosphere. From the mathemati-\ncal point of view, \"optical medium\" may be interpreted simply\nas part of Euclidean three-dimensional - space such as the re-\ngion between two infinite parallel planes or the interior of\na sphere, etc., in which we assume that the principles of geo-\nmetric optics hold, in particular, Fermat's principle. There\nwill eventually evolve, as the studies progress and the basic\nconstructs assume their final form, a relatively technical\nversion of what we mean by the term \"optical medium\" in the\nfully developed theory (re: Def. 5 of Sec. 9.1). However,\nfor the present the term may have either of the simple mean-\nings suggested above.\nThe meanings of the terms A, B, and Sij in the princi-\nple can be illustrated using the preliminary example of Sec.\n3.1. Let us return to the setting summarized by Eqs. (4) and\n(5) of Sec. 3.1. In that setting the optical medium was some\n(physically) vacuous region X of Euclidean three-space con-\ntaining two plane surfaces S1 and S2. We concentrate atten-\ntion on S1 Then S1 is an instance of S in the principle.\nConsider the set of all incident radiances like N1 o on S1.","SEC. 3.2\nINTERACTION PRINCIPLE\n207\nThis set of incident radiances becomes the set A1 in the prin-\nciple. Consider the set of all incident radiances like N21\non S1. This becomes the set A2 in the principle. Together,\n(A1,A2 constitute the incident class A in the principle, so\nthat m = 2. It should be noted that A1 and A2 are each closed\nunder the operations of forming sums and products by nonnega-\ntive numbers (Zinear closure). Thus if N1 and N2 are in A1,\nthen so is cN1 + dN2 where C and d are nonnegative numbers.\nThis feature of A1 and A2 comes automatically with the requi-\nsite linearity of the Sij The class B of response functions\nS1 consists of one set B1, with N12 as a typical element.\nTherefore in the case of S1 we have m = 2, and n = 1, with\n221 and 2212 as the present instances of S11 and S21, respec-\ntively. Hence one invocation of the interaction principle\nfor the case of S1 yields (4) Another and distinct invoca-\ntion in the case of S2 yields (5).\nThe alternate example summarized in (10) of Sec. 3.1\nprovides a further illustration of the principle's linear al-\ngebraic statement. In (10) of Sec. 3.1, X is the same space\nas above. Now, however, S1 and S2 are considered parts of\none and the same subset, say S of X. Consider the set of all\nordered pairs of incident radiance on S like (N1,No) This\nbecomes A1 in the principle. Consider the set of all ordered\npairs of response radiances of S like (N1B,N2Y). This becomes\nB1 in the principle. Therefore in the present case of S, we\nhave\n1,\nand S11 is § As we select any new incident\nm\n=\nn\n=\nBY\no\npair (N1, N2) there corresponds the associated response pair\n(N1B, N2Y) given by (10). Clearly (10) is the present instance\nof the matricial form of the principle's algebraic statement.\nAs we progress along the line of examples of the inter-\naction principle we shall be gradually less explicit in point-\ning out the particular parts of the current form of the inter-\naction principle, leaving the details of correlation more to\nthe reader as he becomes familiar with the principle. In all\nthe subsequent uses of the principle, we shall look upon it as\nconvenient working principle, i.e., a rule of action for the\na\nformulation of subordinate principles, the various laws, and\neveryday problems of radiative transfer theory. The practical\nuses of the principle are directed to determining the light\nfield in natural optical media by finding the interaction op-\nerator Sij, supplied by the basic principle, for a given med-\nium. The determination of the structure of the operators sij\nand the various functional equations they satisfy constitutes\none of the more interesting and challenging problems of modern\nradiative transfer theory. We shall begin the investigation\nof these operators in the present chapter and continue it in\nChapter 7.","INTERACTION PRINCIPLE\nVOL. II\n208\nThe Place of the Interaction Principle in\nRadiative Transfer Theory\nIt is not intended that the interaction principle ca-\ntegorically replace all classical instances of itself such as\nthe principles of invariance and the invariant imbedding re-\nlation, or other classical instances that occur in the liter-\nature or that arise during the subsequent developments below.\nRather, it is intended that the principle be viewed by its\nusers simply as a working principle of radiative transfer the-\nory, and to be used (and perhaps refined) by those students\nof the subject who prefer to envision the theory as governed\nby and derivable from a single idea. The place of the inter-\naction principle in radiative transfer theory and in the main-\nstream of physics may be summarized by the following diagram:\n(Mainland of Physics\nMathematical Measure Theory\nELECTROMAGNETIC\nAXIOMS OF RADIATIVE\nTHEORY\nTRANSFER\nINTERACTION PRINCIPLE\n(Radiative Transfer Theory)\nPrinciples\nInvariant\nEquation\nof\nof\nImbedding\nTransfer\nInvariance\nRelation\nAs the diagram indicates, radiative transfer theory\nmay join the mainland of physics via electromagnetic theory\n(see, e.g., Chapter XIV, Ref. [251]) or the theory may be\nmade completely autonomous using an axiomatic formulation\nmade elsewhere (Chapter XV, Ref. [251]). Direct interconnec-\ntions also exist between the three principal parts of the\ntheory (indicated in the diagram below the interaction prin-\nciple) In fact the internal ties on the level of the general\nequation of transfer, the general principles of invariance,\nand the general invariant imbedding relation are so strong\nthat these ties are effectively logical equivalences. The de-\ntails of the pursuit of these connections are mainly mathe-\nmatical and are beyond the scope of the present work. For\nfurther details on this matter, the reader is referred to the\nvarious chapters of Ref. [251].\nLevels of Interpretation of the Interaction Principle\nThe great practical range and depth of the interaction\nprinciple arises from the levels of interpretation on which\nit may be applied. There are generally four main levels of","SEC. 3.2\nINTERACTION PRINCIPLE\n209\ninterpretation of the principle: the point, line, surface,\nand space levels. Of these, the surface and space levels of\ninterpretation are operationally the most meaningful. The\npoint and line interpretations are special theoretical arti-\nfices which increase the range of the principle in specific\nsettings. The preliminary example above is an instance of\nthe surface level of interpretation. In general, the surface\nlevel interpretation of the interaction principle subsists\nwhen one interprets the subset S of a space X as a subset of\none less dimension than X. For three-dimensional spaces X, S\nwould have two dimensions. For two-dimensional spaces X\n(which arise in certain mathematical models) S would have one\ndimension, etc. In general the space-level interpretation of\nthe interaction principle subsists when one interprets the\nsubset S of a space X as a subset of the same dimension as X.\nPlane-parallel slabs, spherical solids in Euclidean three\nspace are settings for the space-level interpretation. For\ntwo-dimensional spaces X, the subset S would have two dimen-\nsions, etc.\nof the remaining two levels of interpretation of the\nprinciple, the point level interpretation is the more widely\nused. In fact the point-level interpretation covers so much\nground that it is convenient to regard it from two separate\naspects. The general point-level interpretation of the inter-\naction principle subsists when X is a general space whose\npoints are arbitrary. The general point-level interpretation\nis of most use in the development of general discrete-space\ntheory (Ref. [251]). The special point-level interpretation\nof the interaction principle subsists when S is a point or an\noptically small three-dimensional subset of space (i.e., e.g.,\na point source) in which single scattering processes are to\nbe dominant relative to multiple scattering processes. This\nspecial interpretation is commonly used to establish in an\nintuitive fashion the concept of the volume scattering func-\ntion, which plays a key role in the theory (see Sec. 13.4).\nAn alternate establishment of the volume scattering function\ncould take place strictly and rigorously in the space-level\ninterpretation (see Sec. 3.14). The special point-level in-\nterpretation is also a useful and defensible ploy in setting\nup radiative transfer theory and is thereby retained and giv-\nen a special status. (See, e.g., Example 1, Sec. 3.17.)\nThe final level of interpretation to be discussed is\nthe line-level interpretation of the interaction principle.\nThe line-level interpretation subsists when one interprets\nthe subset S of a space X as a one-dimensional subset of X.\nThe line-level interpretation is not operationally meaningful\nas are the surface, space and special point-level interpreta-\ntions. However, it is retained because it favors useful math-\nematical artifacts, as does the special point-level interpre-\ntation. Furthermore, like the special point-level interpre-\ntation, the use of the line-level interpretation is rigorously\ndefensible by means of limit aguments starting with the space-\nlevel interpretation; for that reason it is retained as a use-\nful technical device. We shall use it below in viewing the\npath radiance as the response of a path in real optical medium\nto the incident path function radiances along the path. (Ex-\nample 2, Sec. 3.17.)","210\nINTERACTION PRINCIPLE\nVOL. II\nUnless specifically noted otherwise, we shall hence-\nforth mean by \"optical medium\" any three-dimensional part X\nof Euclidean three-dimensional space. This then will auto-\nmatically set the dimensionality of S in the various inter-\npretations of the interaction principle. (A formal definition\nof optical media, as they are studied in radiative transfer\ntheory, is given in Sec. 9.1.)\n3.3\nReflectance and Transmittance Operators for Surfaces\nIn this section we begin the sequence of constructions\nof the concepts needed for the description of the manifold ra-\ndiative transfer phenomena encountered in the practice of ra-\ndiative transfer theory. In particular in this section we\nshall use the interaction principle as a base for the con-\nstruction of the more commonly used surface reflectance and\ntransmittance concepts. Some work has already been done in\nthis direction in Sec. 3.1. In fact the empirical reflectance\nfunction was defined in that section as a necessary prerequi-\nsite for the construction of the preliminary example of the\ninteraction principle. We now return to that setting for the\npurpose of establishing systematic definitions for the family\nof reflectance and transmittance operators for surfaces.\nGeometrical Conventions\nFigure 3.3 (a) depicts a general surface Y in an opti-\ncal medium X and a relatively small part S of Y about point X\non Y. We are interested in the reflectance and transmittance\nof Y in the region S about X. Now the terms \"transmittance\"\nand \"reflectance\" become meaningful only after adequate ref-\nerence frames have been established at given points X of Y\nwithin which one can unambiguously establish conventions about\nthe notions of \"inwardness\", \"outwardness\", \"upwardness\",\n\"downwardness\", \"forwardness\", \"backwardness\", etc. Suppose\nthen we affix to point x of Y a unit vector k(x) and call it\nthe unit outward normal to Y at X. Perhaps some readers\nwould prefer to call -k(x) the unit outward normal to Y at X.\nThis is perfectly admissible for our present purposes, and\nthe reader may therefore turn around the arrows in parts (a) -\n(d) of Fig. 3.3 and read the following discussion as it\nstands. The point being made here is that what one calls\n\"outward\", etc., is immaterial. What does matter is what one\nsubsequently does with the concept and that, within a given\ndiscussion, a measure of consistency is sustained in the use\nof the concept once the convention is made.\nDuring the present discussion, let \"D'\" and \"D\" denote\nnarrow circular conical solid angles of central directions E'\nand E, respectively. S is a small collecting surface on Y,\nand x is a point of Y in S. Let \"S \" denote the projection\nof S on a plane normal to E'. (See parts (c) and (d) of Fig.\n3.3.) D' is the set of incident directions; D is the set of\nresponse directions. Both D' and D will always lie completely\nwithin it (k(x ) or E_(k(x)) where E+ is the set of all\ndirections E' such that E'.K(x) > 0, and E (k(x)) is the set","SEC. 3.3\nOPERATORS FOR SURFACES\n211\n(x)\nxx\n(a)\nS\nY\nIII + (x)\nk(x)\nX\n(b)\nS\n=-(x)\nk(x)\ns'\n(c)\nS\nD\nD'\nS'\nk(x)\nD\nS\n(d)\nD'\nFIG. 3.3 Setting for reflectance and transmittance op-\nerators for surfaces.","212\nINTERACTION PRINCIPLE\nVOL. II\nof all directions 5' such that E'.k(x) < 0. (See part (b) of\nFig. 3.3 and compare with Sec. 2.4, so that (k(x))=E(k(x))\nand (1) (k(x))==(-k(x)). We shall also write for brevity:\n\"E4(x)\"\nfor\n(k(x))\nThe notation \"E+(x)\" finds its best use when specific\nsurfaces are under consideration, while the notation\n\"E(+k(x))\" finds its greatest use when (as in Sec. 2.4) purely\nradiometric arguments are in effect as no specific surfaces\nare being discussed.\nThe Empirical Reflectances and Transmittances\nWith these preliminaries established we can define\nwith some measure of precision the empirical reflectance and\ntransmittance function. Emulating (1) of Sec. 3.1 we write:\nN(S',D';S,D)\n\"s(S',D';S,D)\"\nfor\n(1)\nN(S',D')8(D')\nwhere all terms on the right side of the definition are as\ndescribed in Sec. 3.1, but now with and D as\nspecified above. The notation in (1) does not tell us spe-\ncifically on which side of S the sets D' and D lie. By spe-\ncifying this information, the values s (S',D';S,D) take on the\ncharacteristics of reflectances and transmittances. Thus let\nus write:\n\"I(S',D';S,D)\" for s(S',D';S,D), if D'CE,(x) and DCE_(x)\n(2)\n\"r_(S',D';S,D)\" for s(S',D';S,D), if D'CE_(x) and DCE.(x)\n(3)\n\"t,(S',D';S,D)\" for s(S',D';S,D), if D'CE((x) and DCE(x)\n(4)\n\"t_(S',D';S,D)\" for (S',D';S,D), if D'CE((x) and DCE(x)\n(5)\nHere \"D'CE,(x)\" is an inclusion statement which means that\nD' is contained in E+(x). Similar interpretations hold for\nthe other three inclusion statements. For example, part (c)\nof Fig. 3.3 depicts the geometrical arrangement for\nt (S',D';S,D), and part (d) of Fig. 3.3 depicts the arrange-\nment for r. (S' ,D';S,D). Definitions (2) and (4) cover the\noutward (or upward or forward) empirical reflectance and trans-\nmittance of Y over S. Properties (i) and (ii) in Sec. 3.1\nhold for the rt and tt just defined.\nWe could have arrived at the preceding four empirical\nreflectance and transmittance functions just above by direct\nappeal to the interaction principle. Thus, with X and S as\ngiven, let m = n = 1 and A be the set of all outward directed\nincident radiances N(S',D') on S (i.e., D' contained in\nit (x) ), and let B be the set of all inward directed response","SEC. 3.3\nOPERATORS FOR SURFACES\n213\nradiances of S (i.e., D contained in E. (x)). Then the inter-\naction principle asserts the existence of a linear interaction\noperator S11--call it \"r+ So (.) with the property\nthat for every N(S',D') in A there exists an N(S,D) in B such\nthat:\nN(S,D) = N(S',D')r, (S', ;S,D)&(D')\n.\nHence the interaction operator in this instance is a real\nvalued function of four variables (S',D',S,D) which assigns to\neach choice of these variables a number--the reflectance of Y\nover S under the indicated conditions. If instead of incident\nradiances, we chose incident scalar irradiances over D' for\nthe set A, then itself would have been obtained.\nIf we had chosen incident irradiances instead, then\nwould have been obtained. This shows the\npotential flexibility of the principle in supplying a great\nvariety of \"reflectances\", depending on what set of radiomet-\nric quantities are chosen for A and for B.\nThe Theoretical Reflectances and Transmittances\nBy letting S approach {x} D' approach {E' }, and D ap-\nproach {5} in the limit, definitions (2)-(5) yield definitions\nof the corresponding theoretical reflectances and transmit-\ntances of Y at X. Thus by performing the indicated limit op-\nerations, we arrive at:\nr+(x;5';5) E'EE (x) and EEE_(x)\n(6)\nr_(x;5';5) if E'EE_(x) and EEE(())\n(7)\nt+(x;5';5) E'EE (x) and\n(8)\nt_(x;5';5) if E'EE_(x) and EEE(E)\n(9)\nHere \"E'EE+(x)\" means that E' is a direction in E+(x), etc.\nIt is a simple matter to show how these theoretical re-\nflectance and transmittance functions for surfaces follow di-\nrectly from the interaction principle. The technique of ob -\ntaining r+ or t+ is similar to that discussed in Sec. 2.13\nfor obtaining the generalized luminosity function z(.) Spe-\ncifically, we would use the interaction principle to supply a\npositive linear function with the property that it acts on\nincident radiance distributions and yields reflected or trans-\nmitted radiance distributions. Interested mathematical read-\ners may pursue this matter further in Sec. 3.16. To develop\nthis application of the interaction principle in the present\nsection would be to digress too far from the chosen scope of\nthe present discussions. We give only the results of such\nan excursion into measure theory. Thus, we write:","214\nINTERACTION PRINCIPLE\nVOL. II\n(10)\n\"r+(Y)\"\nfor\n4()\n(11)\n\"t+(Y)\"\nfor\nThese are the general reflectance and transmittance integral\noperators associated with an arbitrary surface Y with outward\nunit normal k (x) at each point x of Y. The domain of inte-\ngration in each operator is of the form E+ (Y) or E_ (Y) and is\nknown once x in Y is specified. Thus, if N(x,.) is an inward\nincident radiance distribution at X in Y, then:\nI\nN(x,5') r_(x;5';5)dn(E')\n(12)\n(x)\nis the outward reflected radiance at X in the direction E in\nresponse to N(x, .). In general, if N(x,.) and r+ and t+ are\ndefined over just part a of Y, then we use \"N-(a)\" to denote\ninward incident or response radiance distributions over part a,\nand \"N+ (a) \" to denote outward incident or response radiance\ndistributions over part a. For example, if x is a point of a\nand E is an outward direction, then N+ (a) assigns to X and E\nthe response radiance N(x,5). If we let x in (12) range over\nall points of part a of Y, then we see that (12) defines the\nresponse function N+ (a) of a. Hence N+ (a) in this instance\nis a general reflected radiance distribution resulting from\noperating on N_ (a) by r. (a). This fact we write in the form:\nN+(a) N_(a)r_(a)\n(13)\nwhere we have written:\nN (14)\n\"N_(a)r_(a)\"\nfor\n11) (a)\nThe radiance distribution appearing in the integral is (by\nnoting that the range of integration is E. (a)) an inward ra-\ndiance distribution incident on a at a general (unspecified)\npoint. The definition (14) can be repeated for the three\nother general cases associated with a, namely N+(a)r+ (a),\nN_ (a) t. (a), N+ (a) t+ (a) Equation (13) gives the integration\noperation an algebraic appearance, a feature which, as we\nshall see, is most conducive to rapid and creative manipula-\ntions during theoretical radiative transfer computations.\nThis algebraization of radiative transfer theory is fostered\nby the interaction principle whose salient character is itself\nbasically algebraic (rather than, say, analytic or geometric).","SEC. 3.3\nOPERATORS FOR SURFACES\n215\nVariations of the Basic Theme\nSome attention will next be given to the possible var-\niations the preceding definitions of r and t+ may undergo as\nshifts are made in the choice of types of radiometric inci-\ndent and response quantities. A few specific instances will\nsuffice to show the potentially great number of variations\npossible.\nTo begin, suppose that the radiometric quantities in\nthe incident set A are to be irradiances and those in the re-\nsponse set B to be radiances. Then, e.g., in the expanded\nrendition of (13):\nN(x,5) =\n,\nE( ) x)\nwe rearrange matters so:\nN(x,5) = 1\nr-(x;5';5)\nN(x,5')|5'.k(x)\ndr(E)\n(15)\n15'.k(x)\n(x)\nwith the result that the new reflectance operator has a ker-\nnel with values of the form:\nr-(x;5';5)\n(16)\n15'.k(x)\nWe shall not devise notation to cover this case or the multi-\ntude of alternate cases possible. The notation is best set-\ntled by those who must work repeatedly with the specialized\nconcepts. A semblance of order and universality is attained\nin such matters, however, if some set of functions such as\nthose defined via (6)-(9) is taken as a fixed base of opera-\ntions from which to proceed to new territory.\nReflectance functions of the form displayed in (16)\nare used in practice where the surfaces under study are often\nconsidered ideally or nearly uniform (or lambert) reflectors.\nFor suppose a surface Y at x has the property that there is a\nreal number r_ such that:\nr_(x;5';5) r.\n(17)\n15'.k(x)\nfor every E' in E_(x) and every 5 in e (x) . Then (15) becomes:\nN(x,5) E_(x)\n-- H(x, E (x))","216\nINTERACTION PRINCIPLE\nVOL. II\nHence the reflected radiance distribution N(x,.) is uniform\n(independent of E) of magnitude N(x), say. Then the associ-\nated radiant emittance is:\n= r_H(x,E_(x))\n,\nas one would expect by the way r. is defined. If the inci-\ndent radiance distribution itself was uniform, of magnitude\nN' '(x) then\nH(x,E(x))= wN'(x)\nFrom this and the preceding equation we have:\nN(x) = r_N' (x)\n,\nagain as one would expect of the new version of the reflec-\ntance function and a lambert reflector.\nAs another example, suppose that the incident radio- -\nmetric quantities in A are radiances and those in B are radi-\nant emittances. Specifically, let (15) be used as starting\npoint and operate on each side of (15) with an integration of\nthe kind:\nI\n=\n(x)\nE.K(x)dd(E) . (18)\nIt is clear that the integral on the left yields the requisite\nradiant emittance W(x,E+(x)) (cf. (22) of Sec. 2.4) which thus\nis obtained by operating on the incident radiance distribution\nN(x, ) ) with the integral operator\nI\nE.(x) L E.(x)\nNow it is quite natural when using irradiance and radiant\nemittance in this way for us to assign to the quotient\nW(x,E,(x))/H(x,E_(x))\nthe meaning of a reflectance (an albedo) of the surface Y at\nX. Thus if we write:","SEC. 3.4\nAPPLICATIONS TO PLANE SURFACES\n217\n\"r_(x)\" 11\nfor\n-\nI\n1\nN(x,5')r (x;5';5)ds(E')\nE.K(x) do(E)\n.\nH(x,E(()))\n(x)\nthen we can go on to rearrange (18) into the form:\nW(x,E,(x)) H(x,E_(x))r_ (x)\n(19)\nThis definition of r. (x) (and the three analogous definitions\nr+ (x), t+ (x)) is motivated by the need for working with numer-\nical irradiances and radiant emmittances, and numerical re-\nflectances rather than the analogous functional and operator-\nial concepts which must be used in certain full treatments of\ninterreflection problems. In the next section, we shall il-\nlustrate in more detail the use of (13) and (19).\n3.4 Applications to Plane Surfaces\nIn this section we shall illustrate the application of\nthe reflectance and transmittance operators for surfaces,\nconstructed in Sec. 3.3, for several types of frequently en-\ncountered plane-surface settings in radiative transfer theory.\nThroughout this section and, indeed, the remainder of this\nchapter, one of the principal goals is the demonstration of\nthe systematic use to which the interaction principle may be\nput in formulating the concepts and problems of radiative\ntransfer theory.\nExample 1: Irradiances on Two Infinite Parallel Planes\nLet \"a\" and \"b\" denote two infinite parallel plane sur-\nfaces separated by a vacuum, as in Fig. 3.4. The coordinate\nsystem used is the terrestrial system defined in Sec. 2.4.\nEach plane has assigned reflectance and transmittance func-\ntions as developed in Sec. 3.3 which are to be constant over\na and b. However, the directional structures of the reflec-\ntance and transmittance functions are otherwise arbitrary.\nAn interreflection process between a and b is initiated and\nsustained by a steady downward field radiance distribution\nNo(b) on plane b which has the same structure at all points\nof b. Our present goal is to compute the resultant steady\nstate irradiances on a and b, that is the upward irradiance\nH+ (a) on a and the downward irradiance H_ (b) on b.\nThe interaction principle applied to a and b in turn\nyields the requisite irradiance reflectance operators. Thus\nfor a the set A of incident radiometric functions consists of\nirradiances like H. (a), and the set B of response radiometric\nfunctions of a consists of downward radiant emittances W (a)\nwhich by the hypothesized vacuum between a and b have magni-\ntudes equal to H_ (b). (Cf. also Example 12 of Sec. 2.11\nshowing independence of Hr with r in the case of infinite","218\nINTERACTION PRINCIPLE\nVOL. II\nk\nk\nIt\na\n/H\n=+\nW_(a)\nH+(a)\nW+(b)\nH_(b)\nNo(b)\nE.\nb\nH+(a)\n(b)\n0\nW_(a)\nW+(b)\nH_(b)\nX\nFIG. 3.4 Two interreflecting parallel planes.\nplanes). . The interaction principle then asserts the existence\nof a reflectance (a number) r+ (a) such that:\nW_(a)=H_(b)=H(a)r+(a) = =\n(w+(b)r+(a)\n(1)\nThe last equality uses the radiance invariance law which im-\nplies that W+ (b) = H+(a). The closing example of Sec. 3.3\nshows the necessary form of r+ (a). Thus, following the pat-\ntern (19) of Sec. 3.3 we have written:\n\"r+(a)\"\nfor\nand:\n\"H+(a)\"\nfor\nwhere N(x, ) is now the upward surface radiance distribution\nof b at X which, with r+ (x;5';5), is independent of X. By\nnoting that the iterated integration amounts to finding W_ (a),\nthe downward radiant emittance of a, we see that we are simply\nwriting:","APPLICATIONS TO PLANE SURFACES\n219\nSEC. 3.4\nW-(a)\n\"r+(a)\"\nfor\nH+(a)\nHowever, r+ (a) is now precisely determinable as shown in the\niterated integration whenever N(x,5) and r+ (x;5';5) are known\nfor every x,5', and E. Even if the surface radiances N(x,5)\nof b (and also a) are not known in absolute magnitude, but\nonly in relative magnitude (i.e., its shape but not the size\nis known) the present goal can be attained, as we shall see.\nContinuing, we apply the interaction principle to\nplane b, which has two sets of incident functions and one set\nof response functions. For, the given downward surface radi-\nance No(b) on b gives rise to a known incident irradiance\nHo(b). Irradiances like HO (b) comprise the set A1 of incident\nradiometric functions for b. Irradiances like H_ (b) comprise\nthe set A2 of incident radiometric functions for b. The set\nB1 of response functions of b consists of radiant emittances\nW+ (b) numerically equal to H+ (a), (via the radiance invariance\nlaw once again). The interaction principle then yields two\nreflectances (numbers) ro(b) (for A1 and B1) and r. (b) (for\nA2 and B1) such that:\nW+(b) = H+ (a) = °(b)r°(b) + H_(b)r_(b)\n(2)\ne\nThe numbers (b) and r_(b) are defined exactly analogously\nto r+ (a). Equations (1) and (2) together determine H+ (a).\nThus, using (1) to eliminate H_ (b) from (2), we have:\n= H°(b)r_(b) + H.(a)r1(a))r_(b)\n,\nwhence:\n= (a) = Ho(b)ro(b)/[1-r(a)r_(b)]\n(3)\nand so:\n(4)\n= H (b) =\nThese solutions exist provided that the product r+(a)r_(b) is\nless than 1. This provision is reminiscent of a similar pro-\nvision for 121 and 2212 encountered in the preliminary exam-\nple of Sec. 3.1, and may also be handled via the energy con-\nservation law if desired. It is clear that (3) and (4) are\nusable in practice once reasonable estimates of r+ (a) and\nr_ (b) are made. Such estimates can be based either on empir-\nical data in the form of measured ratios such as W_ (a)/H+(a),\nor by means of integral computations knowing the values\nr+ (x;5';5) and the shape of the reflected radiance distribu-\ntions. For example one can assume the perennial favorite:\na uniform radiance distribution, or other readily integrated\nproducts of the form N(x,5' r+ (x;5';5)","220\nINTERACTION PRINCIPLE\nVOL. II\nk\nE+\n+\nE-\nW+(a)\nHo(a)\na\nH+(a)\nW-(a)\nHq(a)\nH_(b)\nW+(b)\nHo(b)\nb\nW-(b)\nHq(b)\nFIG. 3.5 Systematic details for an interreflection cal-\nculation between two parallel planes.\nExample 2: Irradiances on Two Infinite\nParallel Planes, Reexamined\nIn this example, we systematize the procedure and re-\nsults of Example 1. In that example the radiometric details\nwere kept at an absolute minimum so that the algebraic work-\nings of the interaction principle could be readily followed.\nNow that the algebraic details of the interaction formulation\nhave been demonstrated, we return to that simple setting and\npull out nearly all the radiometric stops and turn on all the\nlights- - - so to speak. Specifically, we now let plane a be ir-\nradiated by two external sources, (i.e., , origins of flux\nother than a and b) which produce downward HO (a) and upward\nHo(a) irradiances; similarly, b is irradiated by two external\nsources which produce HO (b) and HO (b) as schematically shown\nin Fig. 3.5. Our present goal is to use the interaction prin-\nciple to formulate the equations governing the four quantities:\nW. (a), W+ (b) , i.e., , the upward (+) and downward (-) radiant\nemittances of a and b induced by the interreflection inter-\naction between a and b and the incident external sources on\na and b. We direct attention first to plane a and list all\npossible incident radiometric quantities on a:\nA1: all irradiances like HO (a)\nA2: all irradiances like H° O (a)\nA3: all irradiances like H+(a)","APPLICATIONS TO PLANE SURFACES\n221\nSEC. 3.4\nA1 and A2 are self explanatory; A3 is the set of irradiances\ninduced by the presence of plane b below a. Next, the set of\nall response radiometric quantities of a are enumerated as\nfollows:\nB1: all radiant emittances like w+(a)\nB2: all radiant emittances like w_(a)\nThus, in the case of plane a, m = 3, n = 2, and the six ab-\nstract interaction operators Sij supplied by the interaction\nprinciple are in the form of reflectance and transmittance\nnumbers as follows:\nS11 -- ro(a)\nS12 -- to(a)\nS21 -- to(a)\nS22 -- ro(a)\nS31 -- t+(a)\nS32 -- r + (a)\nThe six numbers r.(a),...,r (a) are defined exactly\nanalogously to r+(a) in Example 1 and come ultimately from\nthe interaction principle as outlined in Sec. 3.3. The su-\nperscripts \"0\" set off the external incident sources from the\ninternal sources. Then, according to the interaction princi-\nple W+ (a) and W (a) are given by:\nw+(a) = H°(a)r°(a) + ((a)to(a)\n(5)\n(a) Ho(a)to(a) Ho(a)r(a) + +(a)r+(a)\n(6)\n.\nBy repeating this process of application of the interaction\nprinciple to plane b we arrive at the analogous pair of state-\nments:\nHo(b)t((b) + Ho(b)ro(b) + H_(b)r_(b)\n(7)\n+ (°(b)t°(b) (b)t_(b) .\n(8)\nWhen we append the following two equations:\nW+(b) = H+(a)\n(9)\nW_(a) = H_(b) ,\n(10)","222\nINTERACTION PRINCIPLE\nVOL. II\nwhich follow from the hypothesized vacuum between a and b and\nthe radiance invariance law, the resulting system (5)-(10) is\nself-contained and in principle solvable. In particular when\nW_ (a) and W+ (b) in (6) and (7) are eliminated via (9) and (10),\nthe resultant pair of equations is autonomous:\nH_(b)=A_H(a)r,(a) =\n(11)\nH+(a)=B H_(b)r_(b)\n(12)\nwhere we have written:\n\"A_\" for Ho(a)to(a) + Ho(a)r(a)\n\"B+\" \" for H4(b)t((b) + Ho(b)r°(b)\nUsing (11) eliminate H_(b) from (12):\n+ [A_ + (a)r (a)]r_(b)\n,\nwe readily solve for H+(a) :\n= [B+\n(13)\n.\nThen by (9) we obtain W+(b). From (13) and (11) we find H_(b)\nand so by (10), W_ (a). Equations (5) and (8) then yield W+(a)\nand w_(b). In this way all four radiant emittances W+(a) and\nW+(b) are determined.\nA First Synthesis of the Interaction Method\nThis example is valuable in pointing up the systematic\nuse to which the interaction principle may be put in formulat-\ning and solving a radiative transfer problem associated with\na subset S of an optical medium X. The essential steps of\nthis method exhibited by the preceding example are as follows:\n(i) Isolate the subset S of the optical medium X.\n(ii) Enumerate the incident radiometric quantities ai\non S. This determines A1,...,Am.\n(iii) Enumerate the requisite response radiometric\nquantities bj of S. This determines B1,...,Bn.\n(iv) Enumerate the mn operators Sij, 1,...,m,\nj=1,...,n supplied by the interaction principle.\n(v) Write the interaction\nequation =\nfor j = 1,...,n.","SEC. 3.4\nAPPLICATIONS TO PLANE SURFACES\n223\n(vi) Append auxiliary equations connecting various\nchosen a and bj, in as much detail as required\nto solve the system in (v) for the bj.\nStep (vi) in the present example occurred in (9) and (10)\nabove. Invariably, the additional auxiliary equations in (vi)\nare equations which match radiances on adjoining subsets of X\nand use one or the other of the following laws:\n(a) The radiance invariance law\n(b) The equality of field and surface radiance at\na given point and for a given direction.\nThe six steps (i) - (vi) together with (a) and (b) above\nwill be used time and again in the following examples. These\nsteps appear to lead to systematic formulations of radiative\ntransfer problems in a manner similar to that used in the\nformulation of the problems of statics and dynamics in mechan-\nics, i.e., by using the technique which begins with the time-\nhonored injunction to: \"isolate the body\", then categorically\nadding up all forces on the isolated body, and finally apply-\ning one or all of the three basic Newtonian laws of mechanics\nto the isolated system. It is somewhat amusing and perhaps\nof interest to observe that the three Newtonian laws even ap-\npear to have their explicit radiometric counterparts in the\nform of (a) above for the First Law, (v) above for the Second\nlaw, and (b) above for the Third law. We shall call the meth-\nod of formulation summarized in (i) - (vi) and (a), (b) above\nthe method of the interaction principle, or simply the inter-\naction method.\nExample 3: Irradiances on Finitely Many\nInfinite Parallel Planes\nWhat we have done above for two plane surfaces we can\nin principle do again for any finite number and even an in-\nfinite number of plane surfaces. We now consider the case of\nfinitely many parallel planes mainly for the novel problems\nof solution it presents subsequent to the invocation of the\ninteraction principle. This will serve to show that the\nIn studies of linear hydrodynamics subsequent to the comple-\ntion of the present work, I have found that the interaction\nmethod is capable of unifying this field in an elegant and\npractical manner, and that it leads to detailed numerical de-\nscriptions of scattered fields of surface water waves. Fur-\nther studies in water wave-guide theory show similarities to\nthe scattering matrix method in e.m. wave-guide propagation.\nA11 of this is not surprising, as the wording of the inter-\naction principle is quite wide, and will apply to these other\ncontexts by changing \"radiometric\" along with \"optical medium\"\nappropriately. See, e.g., Preisendorfer, R.W., \"Surface-wave\ntransport in nonuniform canals,\" Report NOAA/JTRE-80, Hawaii\nInstitute of Geophysics, 1972.","224\nINTERACTION PRINCIPLE\nVOL. II\nk\n= +\nE-\na i-1\nW-(ai-1)\nH_(a))\nHo(a)\nW+(aj)\nai\nH+(a)\nHi(aj)\nW-(a)\nW+(ai+1)\na\nFIG. 3.6 Interaction calculation details for finitely\nmany parallel planes.\ninteraction principle can lead even the most assiduous inves-\ntigator only so far: there will always be a need for effec-\ntive solution procedures of the more complex formulations\nsupplied by the method of the interaction principle.\nFigure 3.6 depicts three adjacent, parallel planes\nai - 1, ai, and ai+1 in a family of p parallel planes separated\nby vacua. Hence 1 * 0 at the outset and then solve for the k which will yield\nthat E. Thus, from (88), we set:\ns (a)y_(b)(7_(a)r_(b))k+1\n=\n[1 - y (a) y_(b)]\nwhence we find k by means of the relation:\nIn(Y+(a)r_(b))\nThis formula is associated with the particular geometric ar- -\nrangements of the present example. It is a relatively simple\nmatter to extend this result to other formulas in connection\nwith related problems, one of which will be discussed next.","254\nINTERACTION PRINCIPLE\nVOL. II\nQuantum-Terminable Calculations\nIn closing Example 7 we remarked that the preceding\nmethod of determining the value of k, which goes with a par-\nticular E, may be extended to certain interesting extreme\ncases. For example, suppose that the average number n of\nphotons of a given frequency v incident per second per unit\narea per unit solid angle on a surface falls below some num-\nber no, say no = 10-2, or no = 10-3, etc. Suppose that this\nmagnitude of no is so small that it is operationally meaning-\nless to theorize about or experiment with the radiance No\nproduced by no. That is, No is not measurable using available\nradiance meters because it is below their threshold of sensi-\ntivity. Suppose E = No/N, where N is some fiducial magnitude\nfor radiance--say that of the order of magnitude of the sun's\nmaximum spectral radiance. This value of E will then deter-\nmine a corresponding finite value of k, say k(e), after the\nmanner illustrated for the special case above. This value\nk(e) in turn can evidently be used in defining a terminable\nresponse radiance calculation. For example:\nk(e)\nN+(b;k(e)) = N°(a)t_(a)r_(b) (r+(a)r_(b))\nj\nwould define a terminable calculation for N (b;k(e)). This\nwould in turn give rise to a terminable calculation for\nN- (a;k(e)).\nSuch calculations, which are terminable by introducing\nquantum concepts in the way just indicated, are called\nquantum-terminable calculations and provide a basis for a\nstrong physical argument in favor of the study of terminable\ncalculations in radiative transfer theory. Terminations\ntherefore need not be arbitrary; but can be based on real\nphysical limitations of the apparatus on which rest the phe-\nnomenological foundations of the discipline. A systematic\nstudy of quantum-terminable calculations appears to hold cer-\ntain interesting theoretical challenges (for example, can a\nconsistent finite algebra of operators be developed on the\nbasis of quantum-terminable calculations?). This study, how-\never, is beyond the scope of the present work and is left for\nthe interested reader to pursue.\nExample 8: Two Interacting Finite Plane Surfaces\nIn the present example we return to the setting of the\npreliminary example in Sec. 3.1 and reformulate the problem\nof that section using now theoretical radiances and the method\nof the interaction principle. Fig. 3.13 reconstructs the es-\nsential features of the setting of Fig. 3.2 in anticipation\nof the use of the appropriate forms of the integral operators\n(Y) and t+ (Y). The unit outward normals k1 and k2 for the\ntwo plane surfaces S1 and S2 fix the outward E+(S) and","SEC. 3.4\nAPPLICATIONS TO PLANE SURFACES\n255\nD(S2,x)\nN°(S)\nX\nS\nI\nN:(S1\nN+(S)\nNI(S2)\nN=(S2)\nk2\nS2\ny\nN°(S2)\nD(S),y)\nFIG. 3.13 Two interacting finite plane surfaces.\ninward E. (S) hemispheres on Si, i = 1 or 2. Thus E+(S)\nconsists of all unit vectors E such that E.Ki > 0 and E. (S)\nconsists of all unit vectors E such that E.Ki < 0. This con-\nvention of fixing outward and inward hemispheres of inter-\nacting surfaces is to be distinguished from the corresponding\nconvention for collecting surfaces used in Sec. 2.4. For col-\nlecting surfaces it is sometimes more convenient to refer the\ndirections of incident flux to a unit inward normal. For a\nsurface which is explicitly considered to interact with anoth-\ner, the outward unit normal is occasionally a more convenient\nreference direction to use. We do not intend, however, to\npermanently fix such conventions. Rather we shall choose be-\ntween the conventions as a given situation favors one or the\nother. With the direction coordinate frames anchored to S1\nand S2 in the above manner we now require that for every y\nin\nS2 the set D(S1,y) of all directions from points of S1 to y\nto lie in E. (S2). Conversely, we require for every X in S","256\nINTERACTION PRINCIPLE\nVOL. II\nthe set D(S2,x) of all directions from points of S2 to x to\nlie in E-(S1). See Fig. 3.13. These conditions amount to\nthe simple requirement that each surface lie above the other's\nhorizon. This is not an essential restriction; it serves\nonly to shorten the number of cases considered in the analysis\nbelow. We require that the reflectance and transmittance\nfunctions of S1 and S2 be known and that the space between S1\nand S2 be a vacuum. The two surfaces are irradiated by inci-\ndent radiance distributions No(S), i = 1 or 2. It is then\nrequired to formulate and solve the interreflection problem\nassociated with S1 and S 2 under these conditions. In partic-\nular we require the response (surface) radiance distributions\nN+(Si) of Si, i = 1 or 2. Thus N+(S1), e.g., is a function\nwhich assigns to each x in S1 and E in E+(S1) a surface radi-\nance N+(x,5). As usual when the need arises to distinguish\nbetween field and surface radiances for S1, the appropriate\nsuperscripts 11.11 or \"+\" respectively, will be appended to \"N\".\nThus \"N+(S)\" will denote field (incident) radiance distribu-\ntions over Si, i = 1 or 2, and \"N+(S)\" will denote the sur-\nface (response) radiance distributions of S\nWe isolate surface S1 and enumerate the sets of inci-\ndent radiometric functions on S1:\nA1: A11 field radiance distributions like N°(S1)\nA2: A11 field radiance distributions like No(S1)\nA3: A11 field radiance distributions like N_(S1)\nThe set of response radiometric functions for S1 are:\nB1: A11 surface radiance distributions like N+(S1)\nB2: A11 surface radiance distributions like N*(S1)\nThus, in the case of surface S1, m = 3, n = 2, and the six in-\nteraction operators Sij supplied by the interaction principle\nare in the form of reflectance and transmittance integral op-\nerators as follows:\n--\nS12 -- to(s)\nS21 --t+(s1)\nS22 -- r4(s1)\nS31 -- r_ (S1)\nS32 -- t_(S1)\nThe six operators r°(S1)\nt-(S1), are instances of defini-\ntions (10) and (11) of Sec. 3.3. They are handled using the\ntechniques illustrated in Example 5. Then, according to the\ninteraction principle the response radiance distributions are\ngiven by:","SEC. 3.4\nAPPLICATIONS TO PLANE SURFACES\n257\nN+(S1) = + + N_(S1)r_(S1) (89)\nN*(S1) = N°(S1)t_(S1) + + N_(S,)t_(s1).(90)\nBy repeating this process of application of the interaction\nprinciple to surface S2, we arrive at the analogous pair of\nstatements:\nN*(S2) = N°(S2)r_(S2) + N°(S2)t+(S2) + (S2)r (S2) (91)\nN*(S2) i°(S2)t°(S2) + N°(S2)I+(S2) + 1_(S2)t_( (S2). (92)\nAn inspection of (89) and (91) shows that these equations are\nidentical in structure; similarly for (90) and (92). The\npresent choice of coordinate frames has rendered the formula-\ntion completely symmetrical with respect to S1 and S2. It is\nof interest to note that the domain of integration of the op-\nerator r_ (S1), e.g., may be limited at each X of S1 to\nD(S2,x), and that of r_ (S2), may be limited to D(S1,y) at\neach y of S2. Similar observations hold for t. (Si), i = 1 or\n2.\nThe solution procedure of the system (89) - (92) will\nnot be exhibited; it is similar in all essential respects to\nthat for the system (76) -(79) - Those who wish to solve (91) -\n(92) explicitly should observe that the present counterparts\nto (80), (81) are given by the symmetric pair of auxiliary\nequations:\n(93)\nNI(S1)=N_(S2)\n(94)\nwhere the domain of the distributions are suitably restricted.\nThus, e.g., (93) is understood to state that\nN*(y,e) = N_(x,5) =\n(95)\nfor every x in S1 and y in S2 such that\nE (x-y)/|x-y\nBy allowing S1 and S2 to be mutual point sources as in the\npreliminary example of Sec. 3.1, and by setting No(S) = 0,\ni = 1 or 2, the reader may easily show that (89) can be re-\nduced to (2) of Sec. 3.1 and that (91) can be reduced to (3)\nof Sec. 3.1. In this reduction, observe that D12 in Fig. 3.2\nis now replaced by -D(S2,X), and D21 by -D(S1,y). of even\nmore interest is the fact that the present formulation con-\ntains as a limiting case all the preceding examples on infi-\nnite parallel planes (set S1 and S2 parallel, and let them\nbecome arbitrarily large).","258\nINTERACTION PRINCIPLE\nVOL. II\nX\n(y\nk(y)\n(x)\nN°(,)\nX\n=_(x)\n= + (x)\nN ( x , E )\nN$(x,E)\nk(x)\nFIG. 3.14 Geometric conventions for radiometry on open\nconcave surfaces.\n3.5 Applications to Curved Surfaces\nThe distinguishing feature of curved surfaces for ra-\ndiometry in general and the interaction principle in particu-\nlar is the fact that such surfaces, unlike plane surfaces,\nmay interact radiometrically with themselves. In this sec-\ntion we illustrate the application of the interaction princi-\nple to curved surfaces with this feature of self-interaction\nparticularly in mind.\nExample 1: Open Concave Surfaces\nAs a first illustration, consider a smooth open con-\ncave surface S in an optical medium X which is otherwise a\nvacuum. S is of finite extent and, as depicted in Fig. 3.14,\nhas the general appearance of a dish or bowl. Each point X\non S is visible to every other point y on S. At each point X\nof S we erect a unit outward normal k (x) which automatically\ndetermines the outward hemisphere: E+ (x) ; and inward hemi-\nsphere: E (x) , at X. Instead of going into complete analyti-\ncal specifications of the sense of \"outward\", we let Fig.\n3.14 help fix the sense which is intended: the angles between\nk(x) and the directions to every other y in S from X are less\nthan 90° . Here \"outward\" direction at x, as usual, means\n\"away from S\" in the immediate vicinity of X along some speci-\nfied direction. By traveling in an outward direction from a\nplane surface, one is carried ever farther from the plane.","SEC. 3.5\nAPPLICATION TO CURVED SURFACES\n259\nIn the case of a curved surface such as S however, by trav-\neling far enough along some outward directions from x, one\ncan eventually reach S again at a point y and make contact\nwhile traveling along an inward direction at y. This elemen-\ntary observation is a key observation needed in the formula-\ntion of the present interreflection problem. We let S be ir-\nradiated at each point by steady inward and outward incident\nradiance distributions (S) which are conveniently thought\nof as originating at places other than points on S. Thus the\nvalue of No (S) at x and in the direction E in E+ (x) is\nNo(x,E). For example, if S is a portion of the sea surface\nat an instant of time, then N° (S) is the radiance distribution\nover that part of the sky visible at the point of interest,\nand No(s) is the radiance distribution of that part of the\nunderwater scene visible at the point of interest. These\nsources initiate and sustain an interreflection process on S\nwhere it is now possible for the immediate neighborhoods of\nevery pair of points X and y of S to interact radiometrically.\nReturning to Fig. 3.14, let \"N+(S)\" denote the resultant re-\nsponse radiance distribution over S. Thus for every X in S\nand E in E+(x), N+(x,5) is the resultant surface radiance of\nS at x in the direction E. As usual, the superscript \"+\" de-\nnotes surface radiance, and the subscript denotes that E is\nin (x). Furthermore let \"N_(S)\" denote the resultant field\nradiance distribution over S. Thus for every X in S and E in\n(1) (x), NI(x,E) is the resultant field radiance at x in the\ndirection E. In summary, then, (S) plays the role of an in-\ncident radiance on S, and N+(S) that of a response radiance\nof S.\nThe connection between N+ (S) and N_(S) is readily es-\ntablished using the radiance invariance law. We have for eve-\nry distinct pair x, of points in S :\nN*(y,5) = N_(x,5)\n(1)\nwhenever x and y are two points whose common line lies in a\nvacuum, and:\nE = (x-y)/|x-y|\nThe reader will find it of interest to compare (95) of Sec.\n3.4 and (1), and dwell on the points of similarity between\nthe formulations of Example 8 of Sec. 3.4 and those of the\npresent example. In particular he may ask: which of the two\nproblems is more general (in the usual sense that the more\ngeneral problem yields as a special case the less general\nproblem) ?\nEquation (1) can be stated in a more useful manner by\nfirst letting \"D(S,x)\" denote the set of all directions & from\npoints y in S to the fixed point X. Thus D(S,x) is analogous\nto the sets D(S2,X) and D(S1,y) of Example 8 of Sec. 3.4.\nThen (1) holds at x in S for every & in D(S,x), where y= x-rE,\nand r = /x-y/. Observe that D(S,x) is part of E_(x) for every\nX.\nWe are now ready to use the interaction principle to\nformulate the present problem. We isolate S and then enumer-\nate the sets of all incident radiometric functions on S.","260\nINTERACTION PRINCIPLE\nVOL. II\nA1: all field radiance distributions like N°(S)\nA2: all field radiance distributions like N°(S)\nA3: all field radiance distributions like N_(S)\nEnumerating all the sets of response radiometric functions\nfor S:\nB1: all surface radiance distributions like N*(s)\nB2: all surface radiance distributions like N*(s)\nIn the present case m = 3, n = 2, and the interaction princi-\nple yields the following six interaction operators Sij:\n$11 -- ro(s)\nS12 -- to(s)\nS21 -- to(s)\nS22 -- r4(s)\nS31 -- r_(S)\nS32 -- t_(S)\nThe six operators ro(s),..., (S) are instances of defini-\ntions (10), , (11) of Sec. 3.3. Then, according to the inter-\naction principle, N+(S) are given by:\nN*(s) = N°(S)r°(s) +\n(2)\nN*(s) = No(s)to(s) +\n(3)\nThis pair of interaction equations and the auxialiary equa-\ntion:\nN+(s) = N_(S)\n(4)\nform an autonomous system of equations. The latter equation\nis understood in the sense of (1). The order of solution of\nthe equations is dictated by (2): it must be solved first.\nThus using (4) in (2) we have:\n(5)\n=\nwhere we have written:\n\"A+(S)\" for +","261\nSEC. 3.5\nAPPLICATIONS TO CURVED SURFACES\nand where we have written:\n\"r'(s)\"\n(6)\nfor\nD(S,x)\nThe prime (') on the square bracket denotes retardation of the\nargument of the radiance function on which r-(S) operates.\nThus, if NIC (S) is to be evaluated at x and for E in E+(x),\nthen (5) becomes:\nN+(x,5) = (x;5';5 ds(s)\nD(S,x)\nwhere \"A+ (x,)\" denotes the value of A+ (S) at x and E. It\nis\nclear that r'(s) can operate on any element in the response\nset B1. It is particularly to be noted that A+ (S) is an ele-\nment of B1. Thus, solving (5), we have:\nNt(s)=A(s)I-r(s)]-\n(7)\nand the inverse of [I - r_(s)] exists provided -(S) is norm\ncontracting (cf. (60) of Sec. 3.4).\nThe prime on the operator r' (S) is adequate to commun-\nicate the idea of a retarded argument in (6) and (7) to the\ngeneral reader whose insight into our intentions fortunately\ncan lighten our expository task. However, if (7) is to be\nprogrammed for evaluation on an automatic computer, then\nanother- a more mechanical expedient must be devised to com-\nmunicate the idea of retarded arguments of a function. For\nexample we could define a mapping to (S) which assigns to eve-\nry x in S and E in E. (x) the point x - r 5 where r is the dis-\ntance from x to S along the direction -E. Knowing the ana-\nlytic description of S, it is in principle possible to com-\npute this r for each x and E in E. (x) and hence to construct\nto(s). Then \"N$(S)to(S)\" will denote the function which as-\nsigns to every x in S and E in E_(x) the radiance NE(x,5)\n(= N°(x-r,5)) where x - r is on S and E is in E+(x-r).\nWith this definition of to(s), we can rewrite (4) as:\n(8)\nwithout the need of further qualifications as was necessary\nin qualifying (4) by (1). Then using (8) in (2), the more\ndetailed version of (5) is:\n= A+(S) + Nt(s)to(s)r_(s)\nfrom which the more detailed version of (7) follows:\n(9)\n=","262\nINTERACTION PRINCIPLE\nVOL. II\nIt is easy to see that if r. (S) is norm contracting, then so\nis to(s)r.(s), where we have written:\nto(s)]r_(x;5';5) do(s) (10)\n\"to(s)r_(s)'\nfor\nD(S,x)\nHere any function on which t°(S)r. (S) operates automatically\nhas its argument x,5 (5 in E_(x)) first retarded to x-r, and\nE respectively (5 now considered in E+ (x-r)) With this def-\ninition of (s), (9) and (7) are equivalent ways of indicat-\ning the computation of the response function N+(S). The com-\nputation of *(s) can be performed using (8), (9) and (3).\nIn closing we note that one can also view the object\nto (S) as a mapping from response set B into incident set A\no\nThis interpretation is based on (8). Such mappings occur\nnaturally in the strictly mechanical formulations of the aux-\niliary equations arising from step (vi) in the interaction\nmethod (cf. Example 2, Sec. 3.4).\nExample 2: Closed Concave Surfaces; the\nIntegrating Sphere\nIn the present example we allow the rim of the surface\nS in Fig. 3.14 to diminish in diameter while leaving the area\nof S greater than some fixed constant. Thus S becomes a\nclosed concave surface (as seen from within). It is the pur-\npose of this example to point out that the formulations of\nExample 1 remain unchanged as the open concave surface be-\ncomes a closed concave surface. Indeed, as a review of Exam-\nple 1 would show there is no essential use made at all of the\nopenness of S as depicted in Fig. 3.14. The only important\nchange to note is that D(S,x) is now exactly (x) for every\nX in a closed concave surface S. Hence (9) holds also for\nclosed concave surfaces. We shall now illustrate (9) for the\nmost useful case of a closed concave surface: the integrating\nsphere.\nFigure 3.15 depicts a spherical surface S of diameter\nd enclosing a vacuous region. Incident source radiance is re-\nstricted to a general part a of S. For simplicity we let the\nincident source radiance be represented by No(S) over part a\nof S. Hence we will write \"No(a)\" for N°(S) and set No(S) = 0\nin A+ (S) of Example 1. NO (a) is of arbitrary directional\nstructure but is independent of location over a with respect\nto the local direction frame, determined at each point y by\nk(y). Then (9) becomes:\nN+(s) = No(a)to(a) [I - t (s)r_(s)] - -\n(11)\n.\nWe next adopt the classical assumption that the inside sur-\nface of S is a lambert reflector. In addition we assume a is\na lambert transmitter. That is, we are assuming (cf. (17) of\nSec. 3.3):","SEC. 3.5\nAPPLICATIONS TO CURVED SURFACES\n263\nN+(a)\nof\ny\na\n(y)\nr(y,x)\n=-(y)\n=(y)\nd\n=+(x)\n(x)\nk(x)\nX\nS\nFIG. 3.15 Illustrating the radiometric self-interaction\nof a closed concave surface. The case of the integrating\nsphere.\nfor every X in S, E' in E_(x), and in E+(x); and with\nOsr<< 1. Further we assume:\nfor every X in a, E' in E+(x), in E+ (x). Then:\nwhere fa is a function on S such that fa (x) = 1 if X is in a\nand fa(x) = 0 if X is not in a. Further, HO is the constant","VOL. II\nINTERACTION PRINCIPLE\n264\nirradiance on a produced by No(a). Hence No(a)to(a) is a uni-\nform radiance distribution over a and is a member of set B1,\n(see Example 1) which we shall denote by \"No\".\nWe now write (11) in the form:\n= +\nN°\n(12)\nConsider the first term:\nN°to(S)r_(s)\nof the infinite series. We write \"N1\" for Noto(s)r_(S).\nThen by (10)\nN°(x,5) =\nfor every x in S and E in E+(x). Since the incident source\nNo vanishes outside part a of S and is of constant magnitude\nwithin a, the domain of integration shrinks from D(S,x) to\nD(a,x), and\nD(a,x)\nTHE\n(5'.k(x)\ndo(E')\n0\nD(a,x)\nNow by means of an observation following (22) of Sec. 2.11,\nthis integral is readily evaluated:\nD(a,x)\nHence\nfor every X in S and E in E+ (x). This result was grouped in\nthe indicated manner to show the effect of the operator\nto (S)r_ (S) on N°. The effect is to multiply the value of N°\nby [r_ A(a)/d2] = r- A(a)/A(S).\nThe uniform surface radiance distribution N° is now\nacted on by to(s)r_(s) to yield the second term of the series.","SEC. 3.5\nAPPLICATIONS TO CURVED SURFACES\n265\nObserve that N° is constant-valued over S, whereas N°\nis constant-valued over a and zero over the remainder of S\noutside a. Thus the second term of the series is:\nN1to(s)r_(s) =\nN°(to(s)r_(s))2\n=\nLet us write \"N2\" for N1t(s)r_(S). Then:\n=\nD(S,x)\n=\nWith this second iterate of (S)r_(S), the pattern begins to\nform. We first note that:\n,\nD(S,x)\nfor every x in S. Hence:\n=\nThus if we write: for j=1,2,...,\n,\nthen for j = 2,3,..., :\nwx(\n=\n=\nD(S,x)\n= Nj-1(x,5)\nr_\nSince\n=\n,","266\nINTERACTION PRINCIPLE\nVOL. II\nfor every x in S and E in E+(x), we then have:\nfor j = 2,3,....\nHence:\nj=1\nNA\nIt follows from (12) that\n[1-r_]\nHence using the explicit values of N°, we have for every x in\nS and E in E+ (x)\nN(x,E)\n(13)\nwhere as noted before fa (x) = 1 if X is in a, and fa(x) = 0\nif x is not in a. This formula shows that N(x,5) is of uni-\nform directional structure over E+(x) at each X in S, and is\nindependent of X over a and over the part S-a of S outside of\na. However, the radiance distributions over a exceed those\nin S-a by an amount t+Hq/, which is precisely the transmitted\nradiance through the \"window\" a of the integrating sphere S.\n(Observe that no essential use has been made of the sphericity\nof the surface S. Hence we should expect to extend (13), with\nonly minor changes, to the case of an arbitrary closed surface\nwith lambert properties.)\nExample 3: Open and Closed Convex Surfaces\nThe need to illustrate in great detail the interaction\nprinciple for the case of open and closed convex surfaces is\nobviated by the observation that concavity of surfaces is a","SEC. 3.5\nAPPLICATIONS TO CURVED SURFACES\n267\nrelative property, that is, a property relative to the van-\ntage point of the observer. Thus the surface S in Fig. 3.14\nis concave relative to an observation point inside the space\nenclosed by S--i.e., within the bowl of S. On the other hand\nit appears convex when viewed from below S in the Figure.\nThe interaction equations automatically adjust, without alter-\nation of their general forms, to these two points of view and\nequations (2), (3), (4) hold also for the convexity interpre-\ntation. The only changes in (2) - (4) that might occur are\nthose associated with a reversal of direction of k(x). Accord-\ningly, if the user deems to introduce this change, so as to\nstudy convex surfaces, then all subscripts \"+\" and \"_\" in (2)-\n(4) and their logical descendants may be interchanged: every\noccurrence of subscript \"+\" may be replaced by \"-\" and con-\nversely.\nExample 4 : General Two-Sided Surfaces\nIn this example we ascend one more rung with respect\nto the generality of the type of surface considered: we shall\napply the interaction principle to general self-interacting\none-piece, two-sided surfaces which may be either locally con-\ncave or convex, i.e., have alternating hollows or hills. The\nsurfaces may be closed in the sense of enclosing a volume, or\nopen. We assume their reflectance and transmittance proper-\nties are known at each point and that they are embedded in a\nvacuum. As a concrete illustration that may be kept in mind\nduring the following discussion, the instantaneous configura-\ntion of a dynamic wind-blown air-water surface will serve well.\nAn application of this example of the air-water surface is\nmade in Sec. 12.10.\nParts (a) - (g) of Figure 3.16 depict some particular\ninstances falling within the scope of the present discussion.\nAn examination of these general surfaces reveals two features\nwhich were not present in the cases of concave or convex sur-\nfaces considered above. First, for some point X of S there\nmay be points y of S such that x and y are not mutually visi-\nble. That is, on the straight line between X and y there\nlies at least one other point of S. Second, for some points\nX of S there may be points y of S such that X and y are mu-\ntually visible but are on opposite sides of S.\nThe interaction principle is immediately applicable to\ngeneral surfaces such as those depicted in Fig. 3.16, and\nwhich have the two additional features just described. In or-\nder to display the application so that it may be useful in\npractice and be subject to mechanical manipulation, it is de-\nsirable to introduce some preliminary geometric concepts.\nFirst we assign, as usual, a unit outward normal k(x) to S at\neach X. This fixes (x) at each X and arbitrarily determines\nthe outside and the inside of S. Further, we let \"D(S,z)\"\ndenote in general the set of all directions from points y of\nS to a point Z. D(S,x) consists of directions E in either\nE+(x) or E. (x). Further if E is extended to meet S at y,\nthen E may be also in either E+(y) or (1) (y). See, e.g., parts\n(a) and (e) of Fig. 3.16. It will be necessary to distinguish\nbetween such members of D(S,x) which are in E+ (y) or E-(y).","268\nINTERACTION PRINCIPLE\nVOL. II\n=+(x)\n=+(x)\nk (_)\nE_(x)\nk(y)\nX\ny\n(a)\n(b)\nk(x)\nk(x)\nk(y)\nx\nM\nx\ny\nyy\n(c)\n(d)\nk(y)\nFy\n-k(y)\ny\n-k(y)\nk(x)\nk(x)\n(f)\nx\n(e)\nc\nb\n(g)\nFIG. 3.16 Radiometric self-interaction of two-sided\nsurfaces.","SEC. 3.5\nAPPLICATIONS TO CURVED SURFACES\n269\nWe shall do this in the following way. To begin, for every\nx in S and E in D(S,x) we let \"Im(x,5)\" or simply \"rm\" denote\nthe least non negative r such that x-r is a point of S. The\ngeometrical significance of rm is clear: if at x in S we go\nalong a straight line in direction -E, then eventually we may\nreach S, and since some surfaces are corrugated (as in parts\n(a) or (b) of Fig. 3.16) we may reach S again and again if we\ncontinue to travel along the straight line in the direction\n-E. The distance rm is the distance to the first of such\nmeetings with S. Next we let (S)\" denote the function\nwhich assigns to each x in S and E in D(S,x) the point x-rm5\non S and the direction E. That is, tm (x,5) = (x-rm5,5) for\nevery x in S and 5 in D(S,x). Hence tm(s) is a mapping which\nis an immediate generalization of the mapping to(s) introduced\nin Example 1. Finally we construct two functions X+ (S) and\nX- (S) with the following properties: for every x in S and E\nin D(S,x), X (x, E) = 1 if E is in E+(x) and X+(x,E) = 0 if E\nis not in E+(x). Finally, for every x in S and E not in\nD(S,x), X+(x,5) = 0. With these geometric preliminaries com-\npleted, we can now go on directly to the application of the\ninteraction principle.\nLet No(S) be steady incident source radiance distribu-\ntions on S. These incident distributions generally initiate\nand sustain a self-interreflection process over S. Let N+(S)\nand N+ (S) be the resultant surface and field radiance distri-\nbutions over S.\nWe first isolate S and then enumerate the incident ra-\ndiometric functions on S :\nA1: all field radiance distributions like N°(S)\nA2: all field radiance distributions like No(S)\nA3: all field radiance distributions like N_(S)\nA4: all field radiance distributions like N+(S)\n.\nThe sets of response functions of S are:\nB1: all surface radiance distributions like N*(s)\nB2: all surface radiance distributions like N*(s)\n0\nIn the present case m = 4, n = 2, and the interaction princi-\nple yields the following eight interaction operators Sij:\nS11 -- r o (s)\nS12 -- to(s)\nS21 -- t+(s)\nS22 -- r (s) --\n$31 -- r_(S)\n- -\nS32 -- t_(S)","270\nINTERACTION PRINCIPLE\nVOL. II\nS41 -- t.(s)\nS42 -- r+ (s)\nThe preceding eight operators (°(s) r+ (S) are instances\nof definitions (10), (11) of Sec. 3.3. Then, according to\nthe interaction principle, NI(S) are given by:\nN+(s) = N°(S)r°(s) + N°(s)to(s) + N_(S)r_(S) +\n(14)\n= N°(s)to(s) + N°(s)r(s) + N_(S)t(S) + N°(S)r.(S).\n(15)\nFrom the radiance invariance law and the definitions of the\ngeometric functions X+(S), tm(s), we have the two auxiliary\nequations:\nN_(S) =\n(16)\n(17)\n= +\nThese auxiliary equations together with (14) and (15) consti-\ntute an autonomous system of integral equations governing the\nsurface radiance distribution (N+(S),N_(S)) over an arbitrary\ntwo-sided surface S. The dots denote multiplication of func-\ntions, and the multiplication is done after the operation\ntm(s) is applied to X+(S) and NI(S).\nAs an illustration of the use of (16), let x be a\npoint of S and let E be in D(S,x). Then the value of tm(S)\nat (x, E) is (x-rm5,5) and this is used in the argument of\nX- (S). It follows that one of the two values (x-rm,5) or\nX_ (x-rm5,5) must be 0, and the other 1. Say the former is 0\nand the latter is 1. Then the value of NI(x,5) of NI(S) at x,\nis:\nN_(x,5) =\nHence the downward field radiance [(x,5) comes from the down-\nward surface radiance at x-rms. Thus X and x-rms are on op-\nposite sides of S, and so S must be curled like that in (e)\nor (f) of Fig. 3.16. This illustration shows that X. (S) tm(s)\nis to be interpreted as the composition of the functions\nX. (S) and tm(s).\nThe integral operations in (14) and (15) are all gen-\nerally alike. For the purposes of illustration we take\nNi(s)t+ (S) as typical. Then, according to (11) of Sec. 3.3\nwith Y = S1, the value of N+(S)t+ (S) at x in S and E in\nE+(x) is:\nE.(x)","SEC. 3.5\nAPPLICATIONS TO CURVED SURFACES\n271\nIt is clear that we may replace E+ (x) by E+ (x) n D(S,x), i.e.,\nby that part of D(S,x) in E+ (x) Furthermore, if we use (17)\nto replace N+(x,E') in the integrand, the integral becomes:\n1\nD(S,x)\nt+(x;5';5) da(E')\nThe reader should now examine the set (14) - (17) with\nthe purpose in mind of noting that the set contains as special\ncases the convex and concave examples above. Plane surfaces\nare also covered; for then D(S,x) has zero solid angle meas-\nure for every x on S and the last two terms in (14), (15) van-\nish by virtue of (16), (17) and the definitions of X+. The\npreceding illustration bears this out in part. Furthermore,\nby invoking a certain amount of geometric-radiometric trickery,\nthe set (14) - (17) can also yield, in the limit, the cases of\na set of finite or infinite parallel planes. For example, S\nmay have the three part configuration as in part (g) of Fig.\n3.16, with parts a and b parallel planes and part C having\nzero reflectance and unit transmittance. This would yield\nthe case of parallel finite planes. These observations will\nmake plausible the assertion that the system (14)-(17) - actual-\nly constitutes the interaction equations for a general col-\nlection of two-sided surfaces S, where S is either in one\npiece or several distinct pieces, and of concave, convex, or\nmixed curvature. It is not intended, however, that the set\n(14)-(17) - itself always be reduced to each case as it arises.\nWe have exhibited the preceding interaction equations mainly\nto show the scope of the interaction method and the mechanical\nease with which it can formulate radiometric interaction prob-\nlems. It is desirable, rather, especially for students of\nthe subject, that each specific instance of an interaction\nproblem be derived anew from the principle and that simplifi-\ncations be made and auxiliary equations invoked which are mo-\ntivated by the particular features of the individual case.\nExample 5 : General One-Sided Surfaces\nIn this example we apply the interaction method to the\nformulation of the interreflection light field over one-sided\nsurfaces. Before going into the details it is of interest to\nobserve that one-sided surfaces are mathematical objects\nwhich arose originally in critical studies of the classical\nsurface integration theorems of Stokes and Gauss. One sided\nsurfaces were used principally as counterexamples to show the\nlimitations of the classical forms of these theorems. It is\nbecause of this predominantly negative role played by one-\nsided surfaces in the early training of physics and mathe-\nmatics students, and because of the spectacular and intuitive-\nly bizarre claims made for these surfaces, that a student\neventually carries away with him the general impression that","272\nINTERACTION PRINCIPLE\nVOL. II\none-sided surfaces are conceptual beasts which are inferior\nto their more applicable two-sided cousins, and are better\nleft alone. This impression is, for the most part, defensible\nsince the classical surface integration theorems in the usual\nphysical applications of mathematics pertain only and impli-\ncitly to two-sided surfaces. The implicitness of the two-\nsided condition is eventually forgotten, and the ingrained\navoidance of one-sided surfaces prevents their use in the ap-\nplication of the usual theorems. However, with a little ad-\nditional effort the one-sided surfaces can occasionally be\nbrought into physical discussions and their physical proper-\nties compared-usually with deeper resultant insight--with\nthe corresponding properties of two-sided surfaces. In this\nexample we shall perform this service for the radiative trans-\nfer context. We shall briefly consider the interaction prin-\nciple applied to the most notorious of one-sided surfaces,\nthe Möbius Strip. What we shall find out in this application\nwill be typical of the radiometric properties of one-sided\nsurfaces in general, and no more dramatic than the simple but\nuseful insight that it takes exactly one half the number of\nequations to formulate the radiometric interaction equations\nfor one-sided surfaces as it does for two-sided surfaces.\nHence the four general equations of Example 4 will be reduced\nto two for the most general one-sided surface.\nTo help fix ideas we shall adopt as the prototype of\none-sided surfaces the Mobius strip depicted in Fig. 3.17.\nThe Möbius strip S is shown in plan view in part (b) of Fig.\n3.17 and its mode of generation is shown in perspective in\n(a) of Fig. 3.17. To generate S, one can imagine first of all\na circle Co of radius ro in a plane. Then a line segment L of\nlength 2a is placed so that its center is on Co and so that\nthe line segment, extended, goes through the center of Co. If\nL is moved around C with this orientation maintained, and re-\nmaining in the plane of Co, L will sweep out a circular annu-\nlus of radii ro+a and ro-a. To generate the Mobius strip S\nitself, instead of keeping L in the plane of Co, now, keeping\nL perpendicular to Co, let L rotate with its center always on\nCo, and at a uniform angular speed so that as L makes one cir-\ncuit of Co, it will turn 180° about its point of contact with\nCo. The equations for this Mobius strip are given in para-\nmetric form using cylindrical coordinates as shown in (b) of\nFig. 3.17:\np cos ( / / 2 )\n=\n= p sin (/2)\n-aspsa, osa0,\nand \"E_(x)\" will denote the set of all directions E in (1) such\nthat 5.K(x) <0. The directions in E+ (x) are called the outward\n(+) or inward (-) directions at X. Radiance distributions\nN(x, ) at points X of the boundary Y are split, as usual, into\ntwo parts: the outward radiance distribution N+(x, ) and the\ninward radiance distribution N_ (x, If \"a\" denotes a part\nof Y, then \"N+(a)\" and \"N_ (a)\" denote the outward and inward\nradiance distributions of Y restricted to part a. The part a\ncan vary from a set {x} consisting of one point X of Y, up to","SEC. 3.8\nOPERATORS FOR GENERAL SPACES\n315\nk(x)\n=+ (x)\n(x)\na\nX\nY\nX\nFIG. 3.22 Direction convention for interaction opera-\ntors on general spaces.\nY itself.\nThe geometrical conventions for the empirical quanti-\nties D' , D, S' , S, , established in Sec. 3.3, will also hold below.\nAnother geometric convention we shall require is that\nbased on the process of converification of a concave optical\nmedium. This process will allow in many instances both con-\nvex and concave media to be treated alike during a given dis-\ncussion. Let any of parts (a), (b), (c) of Fig. 3.23 repre-\nsent an optical medium which has a concave boundary. This, it\nwill be recalled, means that some points of the boundary can\nbe joined by straight lines lying partially outside the sur-\nface Y. To be specific, we have pictured solid subsets X for\nthe present discussion. It should be noted that all that we\nsay below can be applied, mutatis mutandis to surfaces also.\nNow imagine a rubber sheet to be neatly applied all around X,\nenclosing X like a tight-fitting cocoon. On those parts of X\nwhere its surface is convex, the rubber sheet will cling and\nfollow the contours of the original surface. On those parts\nof X where the surface is concave, the rubber sheet will soar\nas a plane surface across the concave hollow and will thereby\nestablish a smooth convex surface enclosing X, of minimal pos-\nsible area. Thus the step-like concavity of X in (b) of Fig.\n3.23 will be ideally bridged by the rubber coating as sketched\nby the dashed lines in the figure, and the hollows and holes\nof (a) and (b) of the figure will be enclosed likewise. The\nnet result will be a new region X' containing X with the rub-\nber sheet as a convex boundary of the newly encased volume X'","316\nINTERACTION PRINCIPLE\nVOL. II\n(a)\nX\nP\nP\n(b)\nX\nP\n(c)\nX\nFIG. 3.23 Illustrating the convexification of concave\nmedia.\nThe new surface so formed is called the convex hull of X. In\nshort, X' is the smallest convex solid containing X. So far\nwe have engaged in pure geometry.\nNext we introduce a radiometric element into the dis- -\ncussion. We consider all the regions which comprise the dif-\nference P between X and its convex hull X' including any\nholes inside X. For example, the triangular prism region P\nin Fig. 3. 23 is one such region, and the hole in (a) and the\nhemisphere in (b) are further examples of the difference P be-\ntween X' and X. It is found that certain theoretical consid-\nerations of X are facilitated by considering all such regions\nlike P as filled either (a) by a hypothetical vacuum of unit\ntransmittance and zero reflectance, or (b) by its antithesis:\na hypothetical black material of zero transmittance and zero\nreflectance. In the case (a), we use X' and say that X has\nbeen white convexified and in case (b) we use X and say that","SEC. 3.8\nOPERATORS FOR GENERAL SPACES\n317\nk (x)\n=+(x)\nE_(x)\nb\nY\nk(x')\n(x')\nS\nD\nD'\nX\nFIG. 3.24 Details for defining empirical scattering\nfunctions on arbitrary optical media.\nX has been black convexified. It is obvious that if X is con-\nvex to begin with (and hence also with no holes), then either\nits black or white convexification results in X once again.\nIt is perhaps needless to add that a convexified X\n(either way) is still a conceptual object which can be consid-\nered irradiated or probed at will at any point of its surface\nor interior. However, the definitions of convexified media\nhave an operationally meaningful cast which, if the necessity\never arose, could quite possibly be realized in many instances.\nThe Empirical Scattering Functions\nThe empirical scattering functions will now be estab-\nlished for a general optical medium X. The medium X may be\nconvex or concave. If X has a concave boundary Y then we\nshall consider X to have been either white or black convexi-\nfied. The present discussion is independent of the particular\nchoice of these convexifications and hence we need not distin-\nguish between them.\nConsider two parts a and b of the boundary Y. Let S\n1\nbe a small patch of part a around point x', , and S be a small\npatch of b around point x, as in Fig. 3.24. Thus the present\ngeometric situation is similar - as far as the present general\ngeometry will allow - - to Fig. 3.18. Let an amount N (S' , D') of\nradiance be incident over S' and within the narrow conical\nsolid angle D' which lies wholly inside E. (x') . This is the\nonly source of irradiation either in or on X. (Again \"S'\",","INTERACTION PRINCIPLE\nVOL. II\n318\nas in Sec. 3.6, should be replaced in \"N(S' D')\" by the name\nof the projection of S' on a plane normal to the axis of D'.\nHowever, brief notation wins out over logical notation once\nagain.) Let N(S', D' S, D) be the resultant radiance of S with-\nin the conical solid angle D which lies wholly inside E+ (x).\nThen let us write:\nN(S',D';S,D)\n(1)\n\"S(X;S',D;S,D)\"\nfor\nThe non negative valued function S (X;\n) is the standard\n,\n(empirical) scattering function for X.\nOccasionally it is convenient to know if X has been\nwhite or black convexified, and when it is necessary to expli-\ncitly note this fact in the symbol for the standard empirical\nscattering function we shall write:\n\"S\"(X;S',D';S,D)\"\n\"\nfor\nS(X;S',D';S,D)\n(2)\nif X has been white convexified and:\n\"S\"(X;S',D';S,D)\"\nfor\nS(X;S',D';S,D)\n(3)\nif X has been black convexified.\nAt about this point in the corresponding developments\nof Secs. 3.3 and 3.6, it was customary to observe that the\ncounterparts to (X; ) obeyed D and S additivity and\ncontinuity properties. The observance of this procedure is\nnow well established and, therefore, in order not to repeat\nunnecessarily, these facts need only be alluded to here with\nan observation that these properties are stated in detail in\nSec. 18 of Ref. [251]. of course, while we are currently\ngiving slight attention to these properties, this does not in\nany way mitigate their supreme importance in allowing the rig-\norous deduction from the interaction principle of the standard\nand u operators below, and hence, ultimately, all of ra-\ndiative transfer theory on discrete or continuous optical me-\ndia. At any rate, the formal establishment of all these func-\ntions in 3.3, 3.6 and the present function, starting from the\ninteraction principle, will be discussed in detail in Sec.\n3.16. In particular, it will be shown in that section that\neach of the various S'-additivity and D'-additivity properties\nwill take its formal place as an appropriate property of the\ninteraction measure, and the various D' and S' continuity\nproperties will be formulated as the so-called AC property of\nthe interaction measure.\nThe Theoretical Scattering Functions\nLet us write:\n\"S(X;s',D';x,E)\"\n\"\nfor\nlim\nS(X;S',D';S,D)\n(4)\nS+{x}\nD+{E}","SEC. 3.8\nOPERATORS FOR GENERAL SPACES\n319\nand\n\"S(X;x',5';x,5)\" for\nlim\nS(X;S',D';x,5) , .\n(5)\nS'+{x'}\nD'+{5'}\nThese limits exist by virtue of the various D and S additivity\nand continuity properties of the empirical scattering function.\nIf black or white convexification is to be explicitly noted,\nthen \"b\" and \"W\" subscripts are inherited, appropriately,\nfrom (2) ((3) . We go on to write:\n11\n\"S(X;a,b)\"\nfor\ndo(E')da(x')\na E_(x')\n(6)\nwhere a and b are parts of the original surface Y of X and X\nis in b, and E is in E+ (x). Further, we write:\n11\n\" 2((X;a,b)\"\nfor\na E_(x')\n(7)\nwhere a and b are parts of the (original) surface Y of X and\nX is in b and & is in E+ (x) . (X;a, b) (or \"& \" for short\nwhen X,a,b are understood) is the standard -operator for X\nover a and b. u (;a,b) (or \"u\" for short) is the standard\nu -operator for X over a and b.\nTo see the relative roles played by\nand u we ob -\nserve that & is to a black convexified X as u is to a white\nconvexified X. The theoretical connections between & and u\nfor a given concave space X have been given in Sec. 25 of Ref.\n[251]. It suffices to say that this connection is intricate\nand its applications have not yet been completely explored.\nof the two, the standard S-operator is by far the more useful\nin the immediate generalizations of classical radiative trans-\nfer theory, especially in the theory of one-parameter carrier\nand general spaces (Examples 4,5 in Sec. 3.9). The operators\n& promise to help organize and systematize the theory on the\nmore general spaces which have little or no symmetry or reg-\nular structure.\nIt will be instructive for the reader to give simple\nverbal proofs, based on the appropriate definitions, of the\nfollowing statements:\n(a) For every X, a, b, if X is convex and a and b\nare parts of the boundary of X, then\n(X;a,b) = (XXa,b).\n(b)\nFor every X, a, b, if X is concave and a and b\nare parts of the boundary of X, then\n|N_(a) x P(x;a,b) /s/N_(a) u (X;a,b) |\n.\nIn statement (b) above, we have used the definition of radio-\nmetric norm (Example 5, Sec. 3.4) extended to curved surfaces","320\nINTERACTION PRINCIPLE\nVOL. II\nY. Thus, we write in general:\n\"|N+(Y)|\"\nfor\nN(x,5) ds(E) dA(x).\n(8)\nFurthermore, we have written in statement (b):\n\"N_(a) & (S;x,a,b)\" for\n11\nN da(E')da(x')\n.\na E_(x')\nA similar definition holds for the term N_(a)2((X;a,b).\nSince we have defined the radiometric norm for radiance\ndistributions over surfaces Y bounding general optical media\nX, it is natural to try to extend the definition of the norm\nof a reflectance operator, as given in Sec. 3.4, to a more\ngeneral object such as the & -operator for a medium X. The\nrequisite sequence of definitions for the norm of & (X;a,b) is\npatterned closely after (44)-(49) of Sec. 3.4, and proceeds\nas follows. First we agree that if X has a boundary Y of fi-\nnite area A(Y) then we normalize all radiometric norms of ra-\ndiance distributions N+ (a), defined over parts a of Y, with\nrespect to A(Y) rather than with respect to A(a). Thus on a\nfixed finite boundary surface Y of an optical medium X we\nagree to write:\n\"IN+(a)ly\"\nfor\nN(x,5) do(E) dA(x) (9)\nIf A(Y) if infinite, then, as in Sec. 3.4, we employ a limit\nprocess to define the norm. In practice, when working with a\nfixed boundary Y, then \"Y\" may be dropped from the norm nota-\ntion, for brevity.\nNext we write:\n\"Sb(X;x',5';x)\"\nSb(X;x',5';x,5)\nfor\nds(E)\n(10)\nwhere x' and X are in Y and E' is in E_(x') . We have chosen\nto work with Sb simply to be specific. All that follows be-\nlow holds also for Sw. Further, we agree to write:\nE_(x')\n\"B(X,N;x',x)\"\nfor\nI\nN(x',5') dd(E')\nE_(x')\n(11)","SEC. 3.8\nOPERATORS FOR GENERAL SPACES\n321\nNext, we write:\na N(x',E') dA(x')\n\"B(X,N;x)\" for\nValence\",\nN(x',5') dR(E') dA(x')\n(12)\nAnd finally:\n\"B(X,N)\" for dA(x)\n(13)\nb\nThe motivation for this sequence of definitions is\nmade clear by computing the norm N.(a) S(X;a,b)ly. Thus:\n|N_(a) &(X;a,b)ly =\nS(x;a,b)\nds(s)\ndA(x)\nds (5') dA ds(E)dA(x)\nb'E+(xja' E_(x')\ndA(x')dd(x)\ndA(x')dA(x)\nE_(x')\n/\ndA(x)\nB (X,N) |N_(a) ly\n(14)\nAs in the case of the norm of the surface reflectance opera-\ntors r+ (a), t+(a) (Sec. 3.4) it can be shown with the help of\nthe energy conservation principle that:\n(15)\nfor every X,a,b on X, and every radiance function N. For a\ngiven X, we write\n\"B(X)\" for\n(16)\n,","INTERACTION PRINCIPLE\nVOL. II\n322\nwhere the maximum operation is taken over the set of all ra-\ndiance functions on Y. Then the conclusion in (14) implies:\n|N_(a) (S(X;a,b) VI (X) N (a) ly\n(17)\nS(x;a,b) is norm contracting if:\nWe say that\n(18)\n0 < B(X) < 1\n.\nVariations of the Basic Theme\nThe operators & or u can be used as a basis for fur-\nther definitions of operators which work with radiometric\nquantities other than radiance. Thus, following the patterns\nestablished in Secs. 3.3, 3.6, we could redesign so as to\nmap radiance into radiant emittance, or irradiance into radi-\nance, etc. These brief comments will suffice to make the\nreader aware of the potential variations he himself may wring\nfrom & and u as the occasion may arise.\nIt should be noted in conclusion that the operators\n(X;a,b) and u (X; b) serve the capacities of both reflec-\ntance and transmittance operators depending on the relative\ndisposition of parts a and b over the boundary of X. Thus we\nagree to call 8(x;a,b) or U(X;a, a reflectance operator\nwhenever a = b, and call it a transmittance operator whenever\na and b are disjoint, i.e., have no points in common. This\nconvention attains its greatest conceptual utility when X is\nvery irregular and no simple directional conventions are pos-\nsible, such as are available in the case of plane-parallel\nmedia. Observe, that if X is a plane-parallel medium X(a,b),\nthen our present convention essentially reduces to that es-\ntablished earlier for a plane-parallel medium X(a,b) with up-\nper boundary a and lower boundary b. (See, e.g., (8) -(11) of\nSec. 3.6).\n3.9 Applications to General Spaces\nThe applications of the interaction principle will now\nbe extended to general optical media. We will begin with\nsome relatively simple but important extensions of the prin-\nciples of invariance to curvilinear media such as spherical,\ncylindrical and toroidal media. Then the abstract versions\nof these media--one-parameter carrier spaces - are considered,\nand finally the illustrations culminate in the principles of\ninvariance for completely arbitrary media which are not repre-\nsented explicitly as one-parameter media. Throughout this\nsection, the proceedings may best be viewed once again from\nthe two vantage points defined and discussed in the introduc-\ntion to Sec. 3.7. In regard to these vantage points, Sections\n3.4-3.8 and the present section begin to illustrate the effi-\ncacy of the interaction principle, not only as a theoretical\ntool, but as one which shows promise in fostering novel meth-\nods of numerical computations in radiative transfer problems.","SEC. 3.9\nAPPLICATIONS TO GENERAL SPACES\n323\n(a)\naxyzb\n(b)\nb\nX\naxyzb\na\n(c)\nxyz\nb\naxyzb\nFIG. 3.25 Illustrating some applications of the inter-\naction principle to various optical media.","INTERACTION PRINCIPLE\nVOL. II\n324\nT\n(d)\nC\na\nX\ny\nZ\nb\nd\nr\n(e)\na\n=\na\nX\nb\ny\nZ\nb\nFIG. 3.25, concluded","SEC. 3.9\nAPPLICATIONS TO GENERAL SPACES\n325\nExample 1: Principles of Invariance on Spherical\nCylindrical, Toroidal Media\nOur present goal is to use the interaction principle\nto formulate the principles of invariance on three common\ntypes of curvilinear media. Figure 3.25 depicts four in-\nstances of a curvilinear inhomogeneous optical medium X and\none linear inhomogeneous optical medium. Part (a) depicts a\nspherical medium in the form of a spherical shell with inner\nradius a, and outer radius b. Adjacent to the schematic cut-\naway of the spherical shell is a diagram showing a partition\nof X into concentric spherical shells of radii x,y,z, with\nasxsyszsb. Similar descriptions can be made of the hollow\ncylindrical medium X in part (b) of Fig. 3.25, the hollow tor-\noidal medium in part (c), the rectangular parallepiped medium\nin part (d), and the solid vertical cylindrical medium of\npart (e). In the case of the hollow cylindrical medium, its\naxial length may be finite or infinite. In the case of the\nparallelepiped, it may be of infinite extent in one or both\nlateral dimensions. In all five cases we may have a = 0.\nHowever, for the present illustration, we consider for gener-\nality a > 0.\nWe shall use as a prototype for the present formula-\ntions, the four principles of invariance derived in Example 3\nof Sec. 3.7 for the case of plane-parallel media. As in that\nearlier example, we shall for brevity use the letters \"a\",\n\"x\", \"y\", etc., as names for both the parameter of the asso-\nciated surface and the surface itself. Each medium in Fig.\n3.25 will be designated by the name \"X(a,b)\", and subsets of\nX(a,b) as \"X(x,z)\", etc., just as in the plane-parallel case.\nEach medium is irradiated over surface a and b by incident ex-\nternal radiance distributions; N_(a) for a, (b) for b. No\nother sources are incident on or within X(a,b). The direction\nconventions are also analogous to the plane-parallel cases:\nwe agree that at each point X on a parameter surface, the unit\nnormal k (x) is directed toward the direction of decreasing\nparameter values.\nNow, isolating X(a,y) and considering it black convex-\nified, we enumerate the sets of incident radiance distribu-\ntions:\nA1: all field radiance distributions like N_(a)\nA2: all field radiance distributions like N+(y)\n.\nEnumerating the response radiance distributions, we have:\nB1: all surface radiance distributions like (a)\nB2: all surface radiance distributions like\n.\nThe four interaction operators S ij are:","326\nINTERACTION PRINCIPLE\nVOL. II\nS11 - R(a,y)\nS12 T(a,y)\nS21 --T(y,a)\nS22 - ( ( R y , a)\nThese four operators are instances of the standard\nS-operator S(X;a,b) in (6), where X is now X(a,y) and \"b\"\nis replaced by \"a\" where y is now a spherical surface in\nX(a,b). . For the standard reflectance operator R(a,y) we have,\nexplicitly:\n1\nwhere X is in spherical surface a, 5 is in E+ (x), and X is\nX(a,y). Similar constructions are made for the remaining\nthree standard R and T operators. The R-T notation has been\nchosen so as to be uniform with the plane-parallel case of\nSec. 3.7.\nThe interaction principle then states that:\nN+(a) = N_(a)R(a,y) +\n(1)\nN_(a)T(a,y) +\n(2)\nBy repeating this process now for X(y,b) we arrive at the\nanalogous pair of statements:\nNI(y) = N](b)T(b,y) + N_(y)R(y,b)\n(3)\nN*(b) = N](b)R(b,y) + N_(y)T(y,b)\n(4)\nThe similarity of (1) - (4) with (15)-(18) - of Sec. 3.7 is un-\nmistakable: the interaction principle unifies all these in-\nstances. When we append the following two auxiliary equations:\nN+(y) = N](y)\n(5)\nN°(y) = N_(y)\n(6)\nthe set (1)-(6) - becomes autonomous, as usual. The remaining\ndiscussion of Example 2 of Sec. 3.7 now is--virtually un- -\nchanged--including the definition (27) of iterated operators.\nNow, however, we use the standard S-operator. It is not nec-\nessary to rewrite the principles of invariance I-IV of Example\n3\nof Sec. 3.7. They apply, as they stand to the present con-\ntext. The only salient change is in the basis of the R and T\noperators: we now use the standard S-operator, as defined\nin (6) of Sec. 3.8, as a basis. As in the plane-parallel\ncase, the four principles of invariance are instrumental in\nallowing one to solve for N (y) for every y, asysb, assuming","SEC. 3.9\nAPPLICATIONS TO GENERAL SPACES\n327\nthe standard R and T operators are known. These, in turn,\nare obtained from solutions of functional equations of the\nkind to be studied in Chapter 7.\nExample 2: Invariant Imbedding Relation for\nOne-Paramenter Media\nThe comprehensiveness of the principles of invariance,\nas extended from their classical plane-parallel settings by\nmeans of the interaction principle, begins to emerge as the\nfive specific media in Example 1 are re-examined. In this\nexample we systematically extend the results of Example 1 to\ntheir immediate logical limit. To do this, we ask: what is\ncommon to all the specific instances of Example 1? The an-\nswer is that these media are all constructed by assembling\nlayer upon layer of surfaces of geometrically similar shapes.\nIn part (a) of Fig. 3.25, we can imagine the hollow sphere to\nbe built up from spherical surfaces of radii y, asysb, much\nin the way an onion is built up layer by layer. Parts (b)\nand (e) of Fig. 3.25 show that the cylindrical medium can be\nbuilt up from cylindrical surfaces or circular plane surfaces.\nThis two-way slice can be done for every instance shown in\nFig. 3.25, and many others not shown. In each of the five\ninstances displayed in Fig. 3.25, the medium X(a,b) may be\nimagined to consist of a set of geometrically similar surfaces\nX with asxsb, i.e., with X a point in the interval [a,b] of\nreal numbers. Thus we may set:\nX(a,b) [a,b]}\n(7)\ni.e., X(a,b) is equal to the set of all geometrically similar\nsurfaces Xx, each being indexed (identified) by a single par-\nameter X drawn from an interval [a,b] of real numbers.\nThe examples of Fig. 3.25 only begin to illustrate\nfirst of all the great number of three-dimensional subsets of\nEuclidean space which are one-parameter spaces and available\nfor study, and secondly the multiplicity of ways in which a\ngiven solid can be represented as a one-parameter space (viz.\n(b) and (e) of Fig. 3.25). Indeed, as can readily be verified\nany solid of Euclidean three-space may be represented as the\nunion of a one-parameter family of two-dimensional surfaces,\nand in many distinct ways! Despite this great variety of\nshapes and sizes for each set X(a,b) and each source-free\nsubset X(x,a of X(a,b), we can isolate X(x,z) consider\nX(x,z) black convexified if it is concave, and enumerate the\nsets of incident radiance distributions on X(x,z):\nA1: all incident (surface) radiance distributions like N+(z)\nA2: all incident (surface) radiance distributions like N_(x)\nwhere we are now following the pattern established in Example\n4 of Sec. 3.7 and using surface radiances throughout (see,\ne.g., (21) - (24) of Sec. 3.7). The sets of response functions\nof interest are","INTERACTION PRINCIPLE\nVOL. II\n328\nB1: all response (surface) radiance distributions like N (y)\nB2: all response (surface) radiance distributions like N_(y)\nIn the present enumerations, N+(z) is the outward radiance\ndistribution over the parameter surface Xz, aszsb. The unit\noutward normal k(p) at point p on Xz is in the direction of\ndecreasing parameter values.\nThe interaction principle then asserts the existence\nof four interaction operators Sij:\nS11 -- T(z,y,x)\nS12 -- R(z,y,x)\nS21 -- a(x,y,z)\nS22 -- T(x,y,z)\nThese four operators are not instances of the operators de-\nfined in (6) of Sec. 3.8. Rather, they are exactly analogous\nto the complete reflectance and transmittance operators (40)-\n(43) of the plane-parallel case in Example 4 of Sec. 3.7.\nThe interaction principle now yields the two statements:\nN+(y)=N(z)J(z,y,x) = +\n(8)\nI.\nN (y) = N+(2). a(z,y,x) + N_(X)J(x,y,y)\n(9)\nII.\nwhich we can write as:\n(N+(y),N_(y)) = (N+(z),N_(x))m(x,y,z)\nwhere we have written:\nT(z,y,x) R(z,y,x)\n\"m(x,y,z)\"\nfor\nR(x,y,z) T(x,y,z)\nThe preceding equation is the invariant imbedding relation for\none-parameter media. It is exactly analogous to (36) of Sec.\n3.7. On the strength of this analogy, we summarize the pre-\nceding results as follows:\nLet X(= {Xx: X E [a,b]}) be a one-parameter optical medium\nwhere [a,b] is a closed interval in the extended real-number\nsystem. For every y E [a,b], there is a pair (N+( (y) ,N-(y))\nof (real or vector valued) response functions on Xy. Let \"N\"\ndenote the set of all ordered pairs (N+ (z) N - (x)) of incident\nfunctions, [x,z] ( a, b, with subsets N+ and n. defined as\n{N+(2): Z E [a,b]} and {N_(x) : X E [a,b]}, respectively.\nThen for every x,y,z with y E there exists an in-\nteraction operator M(x,y,z) of n into n such that:\n(N+(y),N_(y)) =\n(10)","SEC. 3.9\nAPPLICATIONS TO GENERAL SPACES\n329\nwhere we have written:\nJ(z,y,x) R(z,y,x)\n\"M(x,y,z)\" for\n(11)\n(a(x,y,z) T(x,y,z)\nin which R(z,y,x), Q(x,y,z) are the complete reflectance op-\nerators with domains n+,n. and ranges n.,N., respectively;\nT (2,y,x),J(x,y,2) are the complete transmittance operators\nwith domains n+, and ranges n.,N., respectively. In\naddition, T(x,z,z) = T(x,z) is the standard transmittance op-\nerator for X(x,z) and T(x,x,z) = I, the identity operator;\nQ(x,x,2) = R(x,z) is the standard reflectance operator and\nQ(x,z,z) = 0, the zero operator.\nThe preceding statement of the invariant imbedding re-\nlation is essentially that given in Ref. [233]. It is now a\nsimple matter to deduce from (10) the semigroup properties.\nT(a,z,b) = T(a,y,b)J(y,2,b)\n(12)\nR(a,z,b) =\nfor complete transmittances (cf. (52), (53) in Example 5, Sec.\n3.7). Furthermore, the principles of invariance for one-par-\nameter media are readily forthcoming from (10)--or the equiv-\nalent set (8), (9). Indeed, setting X = y in (8):\nI.\nN.(y) 1+(2)T(z,y) + N_(y)R(y,z)\n.\nSetting z = y in (9) :\nII.\n(y) = N_(x)T(x,y) + N+(y)R(y,x)\n.\nPrinciples III and IV now follow from I, II, as in Example 3\nof Sec. 3.7. The present instances of the principles are i-\ndentical in form to those in Sec. 3.7 and therefore need not\nbe repeated in detail here. Furthermore, the representations\nof the present complete reflectance and transmittance opera-\ntors in terms of the standard operators are identical in form\nto those given in (40)-(43) of Sec. 3.7 for the plane-parallel\nsetting. Furthermore, the properties (44)-(47) also are easi-\nly shown to hold for the present complete reflectance and\ntransmittance operators. The present forms of the standard R\nand T operators are important enough to repeat here. Thus\nfor an arbitrary one-parameter optical medium X(a,b) we write:\n11\n\"R(a,b)\"\nfor\n[]Sg(X;x',5';x,5)\nd&(E')dA(x')\na E_(x')\nif x is in a and E is in E+(x).\n11\n[ ]Sf (X;x',5',x,5) dd(E')dA(x')\n\"T(a,b)\"\nfor\na E_(x')\nif X is in b and E is in E_(x).","INTERACTION PRINCIPLE\nVOL. II\n330\n11\nds(E')da\n(x')\n\"R(b,a)\"\nfor\nb E+()\nis in b and E is in E_(x).\n11\n[]Sf(x;x',5',x,5)\nds(E')dd(x')\n\"T(b,a)\"\nfor\nb E ( x ')\nif X is in a and E is in E+(x).\nExample 3: One-Parameter Media with Internal Sources\nIn this example we show how the interaction principle\nmay be used in the task of formulating the equations governing\nthe radiance distribution N(y) over a parameter surface Xy\nin\na one-parameter optical medium X(a,b) which has internal\nsources generally distributed over an internal parameter sur-\nface Xs, asssb. To see at the outset the essential struc-\nture of the resultant equations, we assume that no other\nsources are incident on X(a,b).\nFigure 3.26 depicts the one-parameter optical medium\nX(a,b) with the incident source (field) radiance distributions\nNo(s) and No(s) over level S in X (a,b). We imagine No(s) to\nirradiate X(a,s) and No(s) to irradiate X(s,b). Thus, it is\na\ny\nN°(s)\nS\nN°(s)\nb\nFIG. 3.26 Taking into account internal sources in gen-\neral one-parameter media.","SEC. 3.9\nAPPLICATIONS TO GENERAL SPACES\n331\nas if the incident source radiance distribution N° (s)\n(= (No(s),No (s)) ) were placed (like a thin transparent lu-\nminous vanilla filling) into X(a,b) after the latter had been\nmomentarily sliced open (like a layer cake) along XS. It fol-\nlows that the light field generated by this source may be\nviewed as being distinct from N° (s) We assume N° (s) to vary\nfrom point to point over XS, and to be of arbitrary direc-\ntional structure at each point of XS. Thus in particular,\nN° (s) could consist of a narrow pencil of radiation at one\npoint only, or it could be of uniform radiance over all direc-\ntions at each point, etc. As usual X (a,b) is generally in-\nhomogeneous. The only requisite regularity in X(a,b) is its\ngeometric one-parameter structure (and even this can eventually\nbe relaxed) optical properties and radiometric properties are\nleft unconstrained--except for a modicum necessary to define\nintegration and to have the usual additivity and continuity\nproperties on which to build the operator algebra.\nThe given internal source over XS suggests a partition\nof X(a,b) into two parts X(a,s) and X(s,b). In order to in-\nvoke the interaction principle we could employ the usual no-\ntation \"N+ (y) 11 for surface radiance of Xy, and 'N-(y)' \" for\nfield radiance over Xy, asysb; however, now that some spe-\ncific examples have shown how to systematically use surface\nradiance, we shall limit our use mainly to that kind of radi-\nance. When \"N\" has no superscript, surface radiance is under-\nstood. The outward and inward directions over Xy for radiance\ndistributions are as defined in Example 2.\nIsolating X(a,b) and enumerating the sets of incident\nradiance distributions on X(a,b) we have:\nA1: all radiance distributions like No(s)\nA2: all radiance distributions like N°(s)\nEnumerating the sets of response radiance distributions:\nB1: all radiance distributions like N+(y)\nB2: all radiance distributions like N_(y)\nThen m = 2, n = 2, and the interaction principle yields four\ninteraction operators Sij such that:\nS11 \"++(s,y)\nS12 -- Y (s,y)\nS21 - - (s,y)\nS22 Y (s,y)\nThe fact that these four operators belong to the medium\nX(a,b) is implicit in the notation. Occasionally it will be\ndesirable to explicitly denote this fact (see, e.g., Sec.\n7.13) and we shall then write \"Y++ (s,y:a,b)\" for ++ (s,y);\n\"Y+ (s,y:a,b)\" for . (s,y), etc. The interaction principle\nthen states that, for every pair of levels y,s in X(a,b):","VOL. II\n332\nINTERACTION PRINCIPLE\n= No(s)4++(s,y) + N°(s) 4_+(s,y)\n(13)\n(y) = No(s)4+_(s,y) + N°(s) 4__(s,y)\n(14)\n.\nIn matrix form, (13) and (14) become:\nN(y) N°(s)4(s,y)\n(15)\nwhere we have written:\n\"N(y)\" for (N+(y),N_(y))\n\"N°(s)\" for (No(s),No(s))\n[\n\"Y(s,y)\"\nfor\nWe next show how the four operators 4+(s,y), ,4__(s,y)\ncan be represented in terms of the standard operators associ-\nated with the space X(a,b) and its subsets X(x,2). The deri-\nvation of the representation will proceed in two parts. The\nfirst part obtains a representation of ((s,s). The second\npart obtains the representation of 4(s,y) with S # y.\nWe turn now to the case of (s(s). Consider, for ex-\nample, the subset X(a,s). Isolating this subset and enumerat-\ning its incident functions and response functions under the\npresent hypothesized conditions, we have No(s) and the surface\nradiance N+ (s) of X(s,b) as incident functions which both act\non the lower boundary of X(a,s). These are the only incident\nfunctions on X(a,s). Hence by principle of invariance II in\nExample 2 (with z=y = S, X = a) we have:\nN_(s) = (No(s) + N.(s))R(s,a)\n(16)\n.\nSimilarly, for subset X(s,b) and principle I of Example 2\n(withxys,zb):\n=(No(s) + N_(s))R(s,b)\n(17)\nFrom (16) and (17)\nN+(s) = [N°(s)R(s,b) + No(s)R(s,a)R(s,b)][I-R(s,a)R(s,b)]-\n(18)\nN_(s) = [No(s)R(s,a) + No(s)R(s,b)R(s,a)][I-R(s,b)R(s,a)]\n(19)","APPLICATIONS TO GENERAL SPACES\n333\nSEC. 3.9\nComparing (18) and (19) with (15) (in which S = y) and recal-\nling that No(s) are arbitrary, we deduce for the case*\nasssb:\n4+(s,s) = R(s,a)R(s,b) [1-R(s,a)R(s,b)]\n(20)\n4+ (s,s) = R(s,a) [I-R(s,b)R(s,a)]\n(21)\n\"-+(s,s) = R(s,b)[I-R(s,a)R(s,b)]-1\n(22)\n(s,s) = R(s,b)R(s,a)[I-R(s,b)R(s,a)] -1\n(23)\nWe now go on to the second part of the representation deriva-\ntion for Y(s,y) with S # y. For definiteness we first assume\nasy 0 if f is a non negative valued\nmember of F(S) (both examples W and P above are examples of\npositive linear functionals) Then the requisite theorem\n(which is a general form of the Riesz representation theorem)\ngoes as follows:\nTheorem A. If L is a positive linear functional on\nF(S), then there exists a (Borel) measure H on S\nsuch that for every f in F(S)\nL(f)\ndu\n(x)\n=\nA complete general development of this theorem may be found,\ne .g., in Sec. 56 of Ref. [103].\nThe second theorem we shall need concerns measures\nwhich are absolutely continuous with respect to other measures.\nA measure H is absolutely continuous with respect to a measure\nV on a space X if u(E) = 0 whenever V (E) = 0 for every meas-\nurable subset E of X. This ostensibly forbidding-sounding de-\nscription hides a very simple idea which may be illustrated\nas follows. To each subset X of ordinary Euclidean three-\nspace assign the radiant energy content U(X) of that subset,\nas, e.g., we did in (14) of Sec. 2.7. Now it is obvious from\nthe relation (14) of Sec. 2.7 that U(X) = 0 whenever V(X) = 0.\nThat is, the radiant energy content of a set X of zero volume\nis zero. Using the present terminology we say that the radi-\nant energy measure U is absolutely continuous with respect to\nthe volume measure V. Other common examples may be found:\nmass measure, heat measure, etc., are absolutely continuous\nwith respect to volume measure. Now the next theorem we have\nin mind says that for the case of U and V, for example, there\nis a function f on X such that:\nU(X)\nf(x)\ndV(x)\n(5)\n=\nIn other words, the theorem guarantees the existence of an\nenergy density function f which when integrated over X gives\nthe radiant energy content U(X) of X. In the other two cases\ncited we have the existence guaranteed of the mass density\nfunctions and heat density functions. Another way of writing\nf above is as:\ndu\ndV\npointing up the nature of f as a volume derivative of energy.\nWe could then write the preceding integral as:\nU(X) = du dV\n.","376\nINTERACTION PRINCIPLE\nVOL. II\nThe function f above is a special instance of the general con-\ncept of a Radon-Nikodym - derivative of one measure H with re-\nspect to another V. This derivative of H exists whenever H is\nabsolutely continuous with respect to V. The general state-\nment is as follows:\nTheorem B. Let S be a subset of Euclidean n-space\nXn and let V be a finite valued measure on S. Let\nH be a finite valued measure on S which is absolutely\ncontinuous with respect to V. Then there exists a\nfinite valued function f on S such that\n(E)\nf(x)\ndv\n(x)\n=\nE\nfor every subset E of S.\nThe wording of this theorem, whose full version may be\nfound in Sec. 31 of Ref. [103], has been deliberately simpli-\nfied - references to fixed measure spaces and fixed families\nof measurable sets and functions have been suppressed and are\nto be implicitly understood. We are concerned here with only\nthe essential conceptual content of Theorems A and B, what\nmathematical things they yield up for use, and their perti-\nnence to the physical radiative transfer context. In the con-\ntext of Theorem B we shall write\n\"du\"\nfor\nf\n(6)\n.\ndv\nThe final theorem we shall need has been anticipated\nby the integral representation of U(X) above. Its statement\ngoes as follows:\nTheorem C. Let S be a subset of Euclidean n-space\nXn. If and v are finite valued measures on S and\nH is absolutely continuous with respect to V and\ng is a function on S such that SEg dv is defined\nfor every subset E of S, then:\nE g du = E dv dv\nfor every subset E of S.\nAgain the wording of this theorem has been mercifully\nsimplified so that one is encouraged to follow its physical\napplications below. Its unexpurgated and generalized version\nmay be found in Sec. 32 of Ref. [103].","SEC. 3.16\nINTEGRAL STRUCTURE OF OPERATORS\n377\nInteraction Operators for Surfaces\nThe preceding mathematical theorems will now be applied\nto the case of reflectance and transmittance operators for sur-\nfaces. Let us return to Fig. 3.3 of Sec. 3.3. We are inter-\nested in particular in the interaction properties of the sur-\nface Y depicted in part (a) of that figure. If a is any sub-\nset of Y, (e.g., a could be S of the figure) then the inter-\naction method yields an operator r. (a) such that:\n= N_(a)r_(a)\nfor the incident downward radiance distribution N. (a) on a\nand the reflected upward radiance distribution N+ (a). Recall\nthat N (a) is a function which assigns to each point X on a\nand direction E in E. (x) the incident (field) radiance\nN_ (a) (x,5), called \"N_(x,5)\" for short. The set E-(x) is de-\npicted in (b) of Fig. 3.3.\nAccording to the interaction principle, r_(a) is a pos-\nitive linear functional which, for a fixed choice x in a, and\nE in E+ (x) assigns to each N. (a) in the set of incident radi-\nance functions on a the reflected radiance N+ (a) (x, 5) or\n\"N+(x,5)\" for short. More specifically, the set S in Theorem\nA is the set E_ (x). F(S) is now the set of all incident ra-\ndiance distributions N_(x,) at x on a. Hence, by Theorem A,\nthere is a measure 11, depending on the current fixed choice\nof x and E, such that:\n1\nN+(x,5) =\nN\n(x,5')\ndu(x;5';E)\n(7)\n(x)\nHere we have written the \"H\" in Theorem A with sufficient no-\ntational paraphernalia so as to completely identify and keep\ntrack of it. The variable E' is like the x in the theorem.\nThe variables x, E remind us that we have momentarily limited\nthe values of N+(a) to x and E in a and E+(x), respectively.\nNow the measure (x; ;5) just obtained from Theorem A\nis defined on E-(x). That is, it assigns to subsets D' of\nE. (x) a number whose geometric and physical significance be-\ncomes clear by letting N-(x,) be uniform valued with value 1\nover subsets of (x). For example, if D' is a subset of\nE. (x) over which N-(x, has value 1 and has value 0 outside\nD' , then from (7):\nv+(s) =\nN_(x,5') du(x;5';5)\n_(x')\nu(x;5';5)\nD'\nu (x;D';E)","INTERACTION PRINCIPLE\nVOL. II\n378\nThe construction of H(x;D';5) in the present case is such\n(according to the proof of Theorem A) that for every D' if\nSo(D') = 0, then = 0 under all natural physical con-\nditions. This means that a unit radiance distribution inci-\ndent on surface a through solid angles of zero measure will\ninduce zero radiance N+ (x,5) Hence u(x;;5) is to be abso- -\nlutely continuous with respect to the solid angle measure S.\nWe are now ready to use Theorem B. The subset S is the\nsame set just used in Theorem A. The measure v is now solid\nangle measure So and H is u(x;;5). Hence Theorem B says that\nthere is a finite valued function f--in this case call it\n\"r_ (x;*;5)\", such that:\n1(x;D';5) r\nD'\nfor every subset D' of E_(x). In other words:\nds\nTheorem C completes the derivation when we observe that\ng is now to be N-(x,)), u is now 1((;;;;5), and v is S. We\ntherefore have from (7):\nN+(x,5) =\nN_(x,5')\n)du(x;5';E)\nN (x,5')r_(x;5';E) do(E')\n(8)\n=\n.\nE_(x)\nSince x and E were arbitrary, (8) holds for every x in a and\nE in E+ (x), and the deduction of the form:\nI\n]r_(x;5';5) d&(E')\nE_(x)\nfrom the interaction principle is complete. The integral rep-\nresentations of the remaining three operators r+ (a), t+(a) in-\ntroduced in Sec. 3.3 are obtained similarly.\nInteraction Operators for General Media\nWe go on now to consider a general optical medium X,\nbypassing the operators for plane-parallel media as being\nmerely a special case of the present setting. Our goal is to\nderive the integral form of the linear operator & (X;a,b) in\n(6) of Sec. 3.8.\nThe interaction method yields a linear operator\nS(x;a,b) such that","SEC. 3.16\nINTEGRAL STRUCTURE OF OPERATORS\n379\nN+(b)= N_(b)&(x;a,b)\n(9)\nwith the geometric conventions as defined in Sec. 3.8. The\nradiance distribution N (b) is one of a family F(S) of inci-\ndent radiance distributions on S, where S is the set a x E_\nconsisting of all pairs of points (x,5) with X in a and E in\nE-(x). F(S) is an instance of an incident set A1 in the in-\nteraction principle and N+ (b) is a member of the set B1 of\nresponse functions. Hence m = n = 1 for the interaction prin-\nciple yielding (9).\nThe interaction principle implies that S(x;a,b) in-\nduces a positive linear functional over F(S) in the following\nway. We choose a fixed point X on b and fixed direction E in\nE+ (x) and consider the value N+(x,5) of N+(b) at (x,5). The-\norem A then allows us to write:\nv+(s) = (N_0 S(x;a,b)) (x,5)\nN (x',5') du(X;x',5';x,5)\n(10)\nS\nwhere now is the measure on a X E_, , denoted by \"S\"\nin (10), whose existence is asserted in Theorem A. Again we\nhave lavishly embellished of Theorem A with identifying\nvariables: X,x,E. Further (x' E') in (10) acts like X in\nthe theorem.\nA glance at (10) shows that u(X;*;x,5) is a measure on\nS (= a x E_) and so the integral is a double integral over S.\nThe geometric measure V over S is the product of the area\nmeasure A over a and solid angle measure Sb over E. If we let\nN. (a) be of uniform unit value over a subset S' of S and zero\noutside S', then (10) implies:\nN+(x,5) =\nu(X;S';x,E)\n(11)\nAs in the case of the reflectance operator for surfaces, we\nrequire, for obvious physical reasons, the radiance N+ (x,\nto be zero when the measure V of a subset S' of S is zero.\nThat is, we require u(X;S';x,E) = 0 whenever v(s') = 0 and we\nshall assume that this is true.\nNow we have:\nv(S) =\nS\nS\n11\nd&(E')dA(x')\n(12)\n.\na E_(x')","VOL. II\nINTERACTION PRINCIPLE\n380\nRecall that a general subset S' of S is a collection of or-\ndered pairs (x' , E') such that x' is in a subset a' of a and\n5' is in E-(x'); hence (12) is a special case of:\nds(E')da(x')\n(13)\n=\n.\na' E ( x ')\nWe now return to Theorem B which asserts that there is in\nthis case a finite valued function f on a x E -call it\nS(X;,;x,)--such that:\nS(X;x',5';x,5) dv(x',5')\nu(X;S';x,5)\na' E (x')\nfor every subset S' of S. In other words:\ndu(X;x',5';x,5)\n=\nd(s) x A)\nTheorem C allows us to complete the derivation when we\nobserve that g is to be N_ (a), u is now and V is\nSo x A, the Cartesian product of the solid angle and area meas-\nures. We therefore have from (10):\nN+(x,5) = S du(X;x',5';x,E)\ndd(E')dA(x') . (14)\na E_(x')\nSince X and E were arbitrary, (14) holds for every X and E in\nb and E+ (x) , respectively, and the deduction of S(X;a,b) in\nits integral operator form:\n11\n]S(X;x',5';x,E)\n,\nE_(x')\nfrom the interaction principle, is complete.\nInteraction Measures and Kernels\nThe features common to the two discussions just com-\npleted will now be summarized so as to extract the salient\nsteps that must be generally taken in deducing from the inter-\naction principle the requisite integral operator describing a\ngiven radiative transfer interaction.\nSuppose a particular discussion using the interaction","SEC. 3.16\nINTEGRAL STRUCTURE OF OPERATORS\n381\nmethod has reached the stage where the interaction principle\nyields for a subset S of an optical medium X the operator\nequation:\nb = as\n(15)\nwhere a and b are elements of the incident and response clas-\nses A and B of radiometric functions, respectively (cf. Sec.\n3.2). The functions a and b are quite general and may be\neither number-valued or vector-valued, or matrix-valued, etc.,\nwith domains of space, directional, frequency, time variables,\nsingly or in combination. Let C be the domain of a. For the\npresent discussion we shall view the class A explicitly as a\nset A(C) of continuous non negative valued functions on C.\nSimilarly B is viewed explicitly as a set B(D) of non negative\nvalued functions on some set D. (For example, in the case of\nthe surface reflectance operators the medium X was three-\nspace, S was a surface a, C was (x), and D was E+ (x) for a\nfixed point x on surface a. A(C) was the set of all radiance\ndistributions of the form N_ (x, ) ), and B (D) was the set of\nall radiance distributions of the form N+ (x, ).) Finally,\nthe subset C is generally assumed to have some pertinent meas-\nure V. (For example in the preceding discussion of the sur-\nface reflectance operators v was So the solid angle measure on\n(x)\nWith these preliminaries established, the general meth-\nod proceeds by selecting an arbitrary fixed point y in D.\nWith this fixed y, we see that, the interaction operator S of\n(15) becomes a positive linear functional S (y). That is, if:\nb1(y) = a1s(y) =\nand:\nb2(y) = azs(y) =\nthen:\nab1 (y) + Bb2(y) = (aa1 + Ba2)s(y) 20\n,\nwhere a and B are non negative numbers, and where the bi(y)\nare images of the ai under s(y), i = 1,2.\nBy Theorem A, there is a measure H (S,.,y) depending on\nthe subset S of X and the point y in D such that:\nS = [ ] (S,.,y)\n(16)\nC\nso that for every y in D, (15) may be represented as:\n(x)\ndu\n(S,x,y)\n(17)\nC\nLet us say that u(s,,y) has the AC property (with re-\nspect to v) whenever v and H(S,,y) are such that if:\nv(E) = 0 then (S,E,y) = 0 for every subset E of C. This\nproperty, it should be noted, is not asserted to hold univer-\nsally. We view it as a regularity property whose validity\nmust either be postulated (as an axiom, say) or demonstrated","VOL. II\nINTERACTION PRINCIPLE\n382\nin each situation. Thus the properties of each operator must\nbe suitably stated so that the AC property holds (see remarks\non the Stages of the Interaction Method in Sec. 3.18). The\nAC property of u(s, ,y) is the abstract version of all the S'\nand D' continuity statements made in Secs. 3.3, 3.6, and 3.8.\nThe additive property of the measure u (S, ,y) is the abstract\nversion of the S' and D' additivity properties stated in these\nsections. The initials \"AC\" stand for \"absolute continuity\".\nThe next step in the general method is to postulate\n(or verify) the AC property of u (S, . ,y) so that we may go on\nformally to Theorem B which asserts that there exists a func-\ntion K(S,,y) on C such that:\nu(S,E,y) = / K(S,x,y) dv (x)\n(18)\n,\nE\nfor every subset E of C. For example, S(X;x 1,5';x,5) , defined\nin the preceding example on general media is the special case\nof K (S,x,y) for a general optical medium X. In the case of\nS(X;x', E' ;x, 5), \"x\" (in (18)) plays the role of \"(x',5')\",\nand \"y\" plays the role of (x, 5)\", and of course \"S\" (in (18))\nplays the role of X. Hence we have:\nK(S,x,y) = du(S,X,Y)\n(19)\nWe call K(S, . ,y) the interaction kernel for the subset S of X\nand u(s,,y) the interaction measure for S.\nAn application of Theorem C then completes the general\nmethod by allowing us to write:\ndv\n(x)\nb(y)\n=\ndv(x)\nThat is, for every y in D, (15) may now be written:\nb(y)\nx)K(S,x,y)\ndv(x)\n(20)\n=\nso that:\n(21)\n]K(S,x,y)\ndv(x)\nS\n=\nC\nEquation (21) is the requisite integral representation of the\ninteraction operator s, , associated with the subset S of the\noptical medium X.","SEC. 3.17\nFURTHER EXAMPLES\n383\n3.17 Further Examples of the Interaction Method\nWe conclude the illustrations of the interaction method\nin this chapter with a brief listing of some further important\nradiative transfer phenomena which can be methodically sub-\nsumed under the interaction principle. We begin with two con-\ncepts which we have already studied: the path function and\nthe path radiance (cf. Sec. 3.12). . Now we approach these fa-\nmiliar concepts in perhaps the most interesting way of all.\nExample 1: The Path Function Operator\nThe equation connecting a radiance distribution N(x,.)\nat a point in an optical medium X and the associated path\nfunction distribution N*(x,.) at the same point in X was fi-\nnally attained in Sec. 3.14 after a relatively laborious strug-\ngle which first had to bring into the light of day the concept\nof volume scattering function. We now connect N* (x, .) and\nN(x,) in an alternate and less arduous way. However, what we\ngain in elegance and mathematical insight by taking the pres-\nent approach, we lose in physical meaning. The earlier route\ntaken, however long and detailed, has the virtue that it sug-\ngests operational means of measuring o in situ, i.e., within\nan optical medium. The present approach has the virtue of\nshowing the logical structure of the relation between N (x,\nN(x,.)), and o (x;;), and does so with unprecedented clarity.\nLet X be an optical medium and let the present subset\nS of X be a singleton {x}, i.e., a one-point subset of X.\nHence we will be using the special point-level interpretation\nof the interaction principle (re: Sec. 3.2). Let the set A1\nof incident radiometric functions on {x} be radiance distri-\nbutions like N(x, ) . Let the set B1 of response functions be\nthe path functions like N°(x,) and defined using (3) of Sec.\n3.12. Then m = n = 1 in the interaction principle of Sec.\n3.2, and there exists an interaction operator R such that\nN*(x,*) = N(x, ) F\n(1)\nIn the terminology developed in the closing paragraph of Sec.\n3.16, in particular with reference to (15) of Sec. 3.16, b is\nnow N* (x, .), , a is now N(x, , and S is now R. The sets C and\nD are each now the unit sphere E, and V is solid angle measure\non E. R gives rise for each fixed E in E1) to a positive\nSo\nlinear functional, so that for a particular fixed 5 in D (= E)\nwe obtain by Theorem A of Sec. 3.16, an interaction measure\nH(x;;5) such that\nN+(x,5)\nN(x,E')\nlu(x;5';5)\n(2)\n=\nclearly has the AC property with respect to S. Hence\nTheorem B of Sec. 3.16 there is an interaction kernel\nby\no (x;;5) such that for every subset D' of E:","384\nINTERACTION PRINCIPLE\nVOL. II\n1(x;D';5) =\n(3)\n.\nEquation (2) corresponds to (17) of Sec. 3.16; Equation (3)\ncorresponds to (18) of Sec. 3.16, in which y is now E. In\nthe present instance the interaction kernel K for {x} is the\nvolume scattering function o. The present specific instance\nof (20) of Sec. 3.16 is obtained by means of Theorem C of Sec.\n3.16:\nN*(x,5)\n(4)\n=\nand which is to be compared with (8) of Sec. 3.14. Thus we\nhave:\nR =\nds(E)\n(5)\nWe call R the path function operator.\nExample 2: The Path Radiance Operator\nThe equation which represents the path radiance\nNr ( 2, 5) over a path Pr(x,5) in an optical medium in terms of\nthe path function N*(,5) defined over Pr(x,5) was obtained\nin (15) of Sec. 3.12 after some rather delicate analysis but\nin which each step was completely meaningful physically. We\nnow establish (15) of Sec. 3.12 using the interaction princi-\nple in a radically different way; one that exhibits the logi- -\ncal interrelation of these concepts with a minimum of direct\nappeal to physical meaning.\nWe begin by choosing a path Px(x,5) in an optical med-\nium X (see Fig. 3.33). This path is a lone-dimensional subset\nof X, and so we will be using the line-level interpretation\nof the interaction principle (Sec. 3.2). We let A1 be the\nset of all incident radiometric functions on Pr(x,E), , in this\ncase all path functions like N*(,5). We let B1 be the set of\nall path radiances like Nr(2,5), where Z = X + Er (see Fig.\n3.33). Then the interaction principle yields an interaction\noperator T such that:\n(6)\nIn the terminology of Sec. 3.16, in particular (15) of Sec.\n3. 16, b is now NT(2,E), a is now N*(,5), and S is now T.\nThe set C is Pr(x,5), and the set B(D) is the set of all\npath radiance values (non negative real numbers) NT(2,5) on\nthe set D {(z,E)}. The measure v is now the length measure\n1 along Pr(x,E).","SEC. 3.17\nFURTHER EXAMPLES\n385\nT is a positive linear functional, so that we obtain\ndirectly from Theorem A of Sec. 3.16 an interaction measure\n14(Pr(x,5),,,2) such that:\nN+(x',5) o\n(7)\nPx(x,5)\n4(Pr(x,5),,,2) clearly (i.e., on physical grounds) has the\nAC property with respect to 1(,) the measure which assigns\nto x' in Pr(x,E) the distance 1(x' z) = r' between X and x' .\nHence by Theorem B of Sec. 3.16 there is an interaction kernel\nK(Pr(x,5),x,z) such that:\n(8)\nfor every subset E of Pr(x,E). In the present instance, the\ninteraction kernel is none other than the beam transmittance\nfunction such that:\n(9)\nso that in particular:\n=1\n(10)\nBeing able to write down (9) at this time is of course the re-\nsult of hindsight. Were we Martians developing radiative\ntransfer theory for the first time, having been given only\nthe interaction principle and none of the developments of Sec.\n3.10, we would retain and work only with \"K(Pr(x,5),x',z)\" ,\nand perhaps eventually deduce in our own way the multiplica-\ntive, identity and contraction properties of the beam trans-\nmittance function (Sec. 3.10), the differential governing it,\nand finally the volume attenuation function.\nFinally, by means of Theorem C of Sec. 3.16 we come to:\nN*(2,5)\nd1(x',z)\nPx(x,5)\nr\ndr'\n(11)\nby virtue of (9). Hence\ndr'\n(12)\nWe call T the path radiance operator associated with the path\nPr(x,5).","386\nINTERACTION PRINCIPLE\nVOL. II\nE\nE\nR Radiometrically relates two\ndirections in a given point\nR:\nE\nT Radiometrically relates two\nT :\npoints in a given direction\nx\nFIG. 3.34\nBefore closing this example we wish to point out an in-\nteresting geometrical duality between the path radiance op-\nerator T and the path function operator R. This duality is\nbest described in ideographic form in Fig. 3.34. In other\nwords, if we interchange the words \"direction\" and \"point\" in\nthe description of R, we obtain that of T, and conversely.\nExample 3: The Volume Transpectral Scattering Operator\nWe now formulate the definition of an important exten-\nsion of the volume scattering function- - - the volume transpec-\ntral scattering function. As its name implies this new scat-\ntering function relates incident radiance of frequency v' at\na point X to resultant scattered radiance at X of frequency\nV. In short we shall now consider scattering of flux not only\nfrom one direction to another, but also from one frequency to\nanother.\nThe use of the interaction method has been illustrated\noften enough by now so that it will suffice in this and the\nremaining examples to be somewhat less detailed in the expla-\nnations.\nStage one: Construct a function NS (x, , . , v) (the transpec-\ntral path function) at point X in medium X such\nthat its value at E is the radiance of frequency\nV generated by inelastic (transpectral) scat-\ntering at X, of an incident radiance distribu-\ntion N(x, vv) of frequency v'. . This stage cor-\nresponds to the definition of the path function","SEC. 3.17\nFURTHER EXAMPLES\n387\nusing (3) of Sec. 3.12.\nStage two: Proceeds exactly as in Example 1 above. In-\nstead of (1), we now have from the interaction\nprinciple:\nThe resultant integral representation is:\nR =\n(13)\n.\nThe interaction kernel in (13) is called the volume transpec-\ntral scattering function. This function is a proper general-\nization of the monochromatic volume scattering function as\ncan be seen by setting v' = V.\nAlternate Stage two: Proceeds analogously to Example 1 but\nnow the incident radiance distributions have\ntwo free variables E' and v', so that the prin-\nciple yields the operator equation:\nThe resultant integral representation is:\n= d1(v')dd(E')\n(14)\nwhere A is the spectrum. The operator R in (14) is called\nthe standard transpectral scattering operator.\nThe undecomposed transpectral scattering operator com-\nbines R of (14) and R of (5) :\n+ 6(x;5';5;v',v)] d1(v')dd(E') (15)\nwhere s is the Dirac delta function. The dimensional distinc-\ntion between & and the two o's should be noted. We shall al-\nso call to the volume transpectral scattering function. Oper-\nator (13) is useful when only a finite number of discrete\nfrequency transitions are considered. Operator (14) is a nat-\nural choice when continuous frequency transitions are con-\nsidered.\nMiscellaneous Examples\nWe leave the applications of the interaction principle\nopen-ended at this stage and merely list some further possi-\nbilities for consideration by interested students of the sub-\nject:\n(i)\nInteraction Operators for Internal Sources (cf. (37)\nof Sec. 3.9). .","VOL. II\nINTERACTION PRINCIPLE\n388\nPath Function Operator for Polarized Radiance (and\n(ii)\nhence the genesis of the volume scattering matrix\n- -see Sec. 112 of Ref. [251])\n(iii) The Path Radiance Operator for Polarized Radiance\n(and hence the genesis of the beam transmittance\nmatrix--see Sec. 112 of Ref. [251]).\nTime Dependent Operators - the time dependent versions\n(iv)\nof all the kinds of operators considered so far.\n(See (4) of Sec. 3.15 and Sec. 127 of Ref. [251]).\nThe Photometric Operators Y(a,M), Z(Q,M). (See\n(v)\n(13) of Sec. 2.12 and (1) of Sec. 2.13.)\nThe Operator C(x). (See Sec. 2.11.)\n(vi)\nThe Operators of the Mueller Phenomenological Alge-\n(vii)\nbra (Refs. [192], [193], [194], and Sec. 137 of\nRef. [251]).\n3.18 Summary of the Interaction Method\nThe interaction method is a method of formulating\nradiative transfer problems by means of the interaction prin-\nciple. After some preliminary examples, the steps of the\nmethod were listed following Example 2 of Sec. 3.4. The meth-\nod was then extensively applied throughout the remaining part\nof the chapter. In this section we summarize the method as\ndeveloped throughout this chapter and include the steps of\nSec. 3.17 leading to the integral representation of the in-\nteraction operators used in the method. The section concludes\nwith some observations on the relative roles played by the in-\nteraction principle in this work and in Ref. [251].\nSummary of the Interaction Method\nThere are three main stages of the Interaction Meth-\nod. Let X be an optical medium and S be a subset of X. Then:\nIsolate the subset S of the optical medium.\nStage I\n(i)\nIf S is concave decide how S is to be con-\nvexified (Sec. 3.8).\nEnumerate the incident radiometric quanti-\n(ii)\nties ai on S. This determines Aj,\ni = m. (Sec. 3.2)\n(iii) Enumerate the requisite response radiometric\nquantities bj on S. This determines Bj,\n}\nn. (Sec. 3.2)\n=\nEnumerate the mn operators sij, i= 1,...,m ;\n(iv)\nn, supplied by the interaction\nj = 1\nprinciple (Sec. 3.2).\nWrite the interaction equations:\n(v)","SEC. 3.18\nSUMMARY OF THE METHOD\n389\nm\nbj =\ni=1\nfor = 1\n,n.\n(vi)\nAppend to (v) any auxiliary equations con-\nnecting various chosen ai and b 80 that it\nis possible to algebraically solve the sys-\ntem of n functional equations in (v) for\nthe requisite response functions bj. Invar-\niably, these auxiliary equations may be\nbased on one or the other of the following\nradiometric laws:\n(a) The radiance invariance law over a\npath in a vacuum (Sec. 2.6).\n(b) The equality of field and surface\nradiance distributions at a point\nin a general optical medium (Sec.\n2.5).\nStage II\nIf the structure of A,B, and S indicate the\npossibility of an integral representation\nof the interaction operators sij, then use\nthe technique of the interaction measures\nand interaction kernels of Sec. 3.16 to ob-\ntain:\n1\ndv\n(x)\nSij\n=\nC\nfor i = 1, , m j = 1 n.\nStage III\nDetermine by means of suitable functional\nrelations the explicit structure of the\nmn interaction kernels Kij, if they exist,\nand use the results in (vi) of Stage I to\nobtain a solution of the interaction prob-\nlem.\nRemarks on the Stages of the Interaction Method\nStage I was fully illustrated in the present chapter;\nhowever, some further aspects of the details of Stage II and\nStage III beyond those covered in Sec. 3.17, remain to be ob-\nserved. As regards Stage III, the interaction kernels aris-\ning in homogeneous plane-parallel media and their governing\nfunctional relations have been exhaustively studied by Chan-\ndrasekhar (Ref. [43]) Further functional relations were given\nin Refs. [13], [14], by Bellman and Kalaba for inhomogeneous\nmedia. The functional relations for the complete set of four\ninteractions kernels in non homogeneous one-parameter media,\n(i.e., the four reflectance and transmittance functions R and","VOL. II\nINTERACTION PRINCIPLE\n390\nT) were introduced and derived in Refs. [233] and [234].\n(See also Sec. 7.1.) The functional relations governing the\ninteraction kernel for the general operators (X;a,b were\nderived in Ref. [251]. The general procedures for the solu-\ntion of the functional relations governing the operators\nR(a,b), T(a,b) , R(b,a), T(b,a) for general one-parameter opti-\ncal media for (X;a,b) are given in Chapter 7 of this work.\nNow that the conceptual structure of radiative trans-\nfer theory has been elucidated by the interaction principle,\nand its mathematical foundations established (ref. [251]) it\nremains to solve the important mathematical problems of mod-\nern radiative transfer theory centering around the functional\nrelations governing the interaction kernels (see problem VIII,\nSec. 141 of Ref. [251]).\nOne final remark on Stage II must be made. This con-\ncerns the AC property of an interaction measure. If the AC\nproperty is valid for a given interaction measure, then the\ninteraction kernel of that measure is, according to Theorem\nB of Sec. 3.16, the Radon-Nikodym derivative of that measure.\nIn this regard the development of interaction kernels will be\noccasionally simplified if the transmittance-type operators\nare decomposed into their residual and diffuse parts, i.e.,\ninto parts which, respectively, describe radiant flux which\nhas not been scattered (i.e., beam transmitted) and which has\nbeen scattered. It turns out that transmittance operators\nfor diffuse flux always have the AC property. (Reflectance-\ntype operators generally have the AC property outright since\nthey describe only diffuse flux.) The basis for these re-\nmarks rests in Ia, Ib of Sec. 23, Ref. [251], which, in the\npresent work, may be taken as basic postulated regularity\nproperties of interaction kernels. A model for this proced-\nure of decomposing operators will be found in Sec. 7.1. The\ndecomposition of the light field, which is a natural prereq-\nuisite to the decomposition of interaction operators, can\neasily be done in general since the concept of scattered and\nnon scattered radiant flux is now rigorously definable by\nmeans of the path function and path radiance operators of Ex-\namples 1 and 2 of Sec. 3.17. This decomposition will be\nstudied as a matter of course in Chapter 5. The net result\nof Stage II of the interaction method will be that the diffuse\ncomponent of a transmittance operator (rather than the unde-\ncomposed operator) will be passed on to Stage III for the de-\ntermination of its kernel. The prototype of this procedure\nmay be found in Ref. [43], and in Refs. [234], [235].\nThe Interaction Method and Quantum Theory\nWe append here some final observations on the gener-\nal methodology of the interaction method, an observation which\nwill point up some points of similarity between the interac-\ntion method and two basic methods of solving dynamical prob-\n1ems in classical and modern physics. The observation is de-\nsigned to be of especial interest to physicists, rather than\nradiative transferists per se. Nevertheless, since radiative\ntransfer is ultimately derivable from quantum mechanics, the\nlatter workers may peruse the following with some profit.","SEC. 3.18\nSUMMARY OF THE METHOD\n391\nThe first point of similarity was noted in the discussion fol-\nlowing Example 2 of Sec. 3.4 where a comparison was made be-\ntween the Newtonian laws of motion and the interaction princi-\nple, and note was made of the applicability of the method to\nlinear hydrodynamics and general wave guide phenomena. We\nneed not repeat it here. The second point of comparison ap-\npears to be even deeper than the first when we note the simi-\nlarity between the interaction method and the formulation of\nthe quantum mechanics of many-state atomic systems. To facil-\nitate the comparison, the reader may consult, e.g., [92].\nHere are the parallel correspondents: to an atomic or molecu-\nlar system we pair an optical medium (step (i)) To the var-\nious base states of the atomic system we pair the sets of in-\ncident and response radiometric quantities (steps (ii), (iii)).\nTo the Hamiltonian matrix of the atomic system we pair the set\n(sij) of interaction operators (step (iv)) To the transition\nprobability equation (the linear superposition of amplitude\nfunctions) we pair the interaction equation (step (v)) To\nthe finding of either the Hamiltonian matrix (using conserva-\ntion laws and auxiliary physical arguments) or S-matrix, we\npair the finding of the interaction operators (step (vi) and\nStages II and III). The mystery of this remarkable similarity\nbetween the quantum mechanical and radiative transfer formal-\nisms is only apparent and is resolved by noting that each dis-\ncipline is founded (for its own particular experimental rea-\nsons) on a set of linear superposition principles. Hence both\nmethodologies come under the single unifying framework of vec-\ntor space theory. The salient difference between the two for-\nmalisms is that the possibility of interference of amplitudes\nexists in quantum theory, whereas interference of radiant\nfluxes is ruled out by fiat from radiative transfer theory\n(cf. Sec. 2.2) In the preceding point by point parallelism\nof the mathematics of quantum theory and radiative transfer\ntheory lie the keys to the solutions of the basic problems II,\nand IV in Sec. 142 of Ref. [251]. It may be noted in passing\nthat the applications of the linear interaction principle to\nquantum mechanics, linear hydrodynamics, acoustics, and elec-\ntromagnetic theory, e.g., introduce complex-valued interfer-\ning amplitudes, and on this level the theoretical and numeri-\ncal methodologies of all these fields are strikingly alike.\nThe Interaction Principle as a Means and as an End\nThroughout this chapter there have been several OC-\ncasions to refer to the developments of radiative transfer\ntheory in Ref. [251] and in particular to the interaction\nprinciple in that work. A few words may be in order to help\nplace in perspective the relative roles of the interaction\nprinciple in these two works.\nThe interaction principle in Ref. [251] was the end\nof a long series of generalizations and abstractions starting\nmainly with the work of Schuster, on through classical prin-\nciples of invariance of Ambarzumian and Chandrasekhar, and up\nthrough the principle of invariant imbedding of Bellman and\nKalaba as applied to transport phenomena, and finally on to\nthe invariant imbedding relation, and the interaction principle","VOL. II\nINTERACTION PRINCIPLE\n392\nitself. It was shown, in particular, how all these principles\ncould be deduced from the classical equation of transfer, and\nhow the equation of transfer could itself be viewed as a 10-\ncal form of the principles of invariance. Hence, in a word,\nthe interaction principle was viewed in Ref. [251] as an end\nof a set of long conceptual and deductive trails, the main\ntrail starting from Schuster's initial insight in 1905. Thus\nin [251] the roots of interaction principle were established\nin the classical origins of the subject along with electro-\nmagnetic and axiomatic bases of the principle. With this in\nmind we have taken the alternate view in the present chapter\nthat the interaction principle is a basic means of formulating\nradiative transfer theory, a single working principle from\nwhich the salient algebraic structures of the theory may be\ndeduced. The thirty-eight enumerated examples throughout this\nchapter, starting in Sec. 3.4 and ending in Sec. 3.17, have\nshown that the interaction principle can indeed be used as a\nstarting point for the construction of the principles of in-\nvariance on all types of three-dimensional media, the various\nclassical interreflection problems of surfaces, the beam\ntransmittance function for paths, the classical attenuation\nand scattering functions of the media used in the equation of\ntransfer, and the equation of transfer itself.\nConclusion\nIn sum, then, the work of the monograph [251] consti-\ntuted a necessary prerequisite for the establishment of the\ninteraction principle. The present work no longer views the\ninteraction principle as an end of research but rather as a\nmeans of application and a source of new research in radiative\ntransfer theory and general linear transport theories (even\nbeyond radiative transfer, as in hydrodynamics, acoustics, e.\nm. wave propagation, etc.). The first application of the in-\nteraction principle was to the development of the discrete-\nspace theory of radiative transfer in Ref. [251]. These ap-\nplications are continued in this chapter, and the following\nchapters of the present work.\n3.19 Bibliographic Notes for Chapter 3\nThe interaction principle as given in Sec. 3.2 was\nfirst formulated in Ref. [251], the end result of an extended\nseries of generalizations. A historical sketch of the evolu-\ntion of the main lines of radiative transfer theory (not its\nmanifold applications) which are pertinent to the interaction\nprinciple is given cumulatively in the bibliographic notes\nfor the chapters in Ref. [251]. The formulation of the inter-\naction method, as summarized in Sec. 3.18, is new.","HYDROLOGIC OPTICS\n393\nBIBLIOGRAPHY FOR VOLUME II\n1.\nAmbarzumian, V.A., \"Diffuse reflection of light by a\nfoggy medium, \" Compt. rend. (Doklady) Acad. Sci.\nU.R.S.S. 38, 229 (1943).\nAmbarzumian, V.A., \"On the problem of the diffuse re-\n2.\nflection of light,\" J. Phys. Acad. Sci. U.S.S.R.\n8, 65 (1944).\nAmerican Institute of Physics Handbook (McGraw-Hill,\n3.\nNew York, 1957).\n4.\nAmerican Standards Association, \"Nomenclature for ra-\ndiometry and photometry (258.1.1-1953),\" J. Opt.\nSoc. Am. 43, 809 (1953).\n8.\nBarber, E., Radiometry and Photometry of the Moon and\nPlanets, (Literature Search No. 345, Nat. Aero. and\nSpace Adm., Contract NASw-6, Jet Propolusion Labo-\nratory, California Institute of Technology, Pasa-\ndena, September 1961).\nBellman, R., and Kalaba, R., \"On the principle of in-\n13.\nvariant imbedding and propagation through inhomo-\ngeneous media,\" Proc. Natl. Acad. Sci. 42, 629\n(1956).\nBellman, R., and Kalaba, R., \"On the principle of in-\n14.\nvariant imbedding and diffuse reflection from cylin-\ndrical regions,\" Proc. Natl. Acad. Sci. 43, 514\n(1957).\nBoileau, A.R., Atmospheric Optical Measurements During\n26.\nHigh Altitude Balloon Flight, parts I, II, III, IV\n(Scripps Inst. of Ocean. Refs. 59-32, 61-1, 61-2,\n61-3, University of California, San Diego, December\n1959, July 1961, July 1961, December 1961).\nBurton, H.E., \"The optics of Euclid,\" J. Opt. Soc. Am.\n36.\n35, 357 (1945).\nChandrasekhar, S., \"On the radiative equilibrium of a\n42.\nstellar atmosphere, II,\" Astrophys. J. 100, 76\n(1944).\nChandrasekhar, S., Radiative Transfer (Oxford, 1950).\n43.\nCommittee on Colorimetry (Optical Society of America),\n49.\n\"The psychophysics of color,\" J. Opt. Soc. Am. 34,\n245 (1944).\nCommittee on Colorimetry (Optical Society of America),\n50.\nThe Science of Color (Thomas Y. Crowell Co., New\nYork, 1953).","394\nHYDROLOGIC OPTICS\nVOL. II\n51.\nCondas, G.A., \"Maximum spectral luminous efficiency,\"\nJ. Opt. Soc. Am. 54, 1168 (1964).\n90.\nFano, V., \"Gamma ray attenuation, part 1, basic pro-\ncesses,\" Nucleonics 11, 8 (August 1953).\n92.\nFeynman, R.P., Leighton, R.B., and Sands, M., The Feyn-\nman Lectures on Physics (Addison-Wesley Pub. Co.,\nReading, Mass., 1965), vol. III, Quantum Mechanics.\n98.\nGershun, A., \"The Light field,\" (trans. by P. Moon and\nG. Timoshenko), J. Math. and Phys. 18, 51 (1939).\n103. Halmos, P.R., Measure Theory (D. Van Nostrand, New\nYork, 1950).\n128. Johnson, F.S., \"The solar constant,\" J. Met. 11, 431\n(1954).\n130. Jones, R.C., \"Terminology in photometry and radiometry,\"\nJ. Opt. Soc. Am. 53, 1314 (1963).\n150. Landé, A., From Dualism to Unity in Quantum Physics\n(Cambridge Univ. Press, 1960).\n151. Landé, A., New Foundations of Quantum Mechanics (Cam-\nbridge Univ. Press, 1965).\n153. Le Grand, Y., Light, Colour and Vision (John Wiley and\nSons, New York, 1957).\n177. Middleton, W.E.K., Vision Through the Atmosphere (Univ.\nof Toronto Press, 1952).\n182. Minnaert, M., The Nature of Light and Color in the Open\nAir (Dover Pub., Inc., New York, 1954), trans. by\nH.M. Kremer-Priest, revised by K.E. Brian Jay.\n184. Moon, P., \"A system of photometric concepts,\" J. Opt.\nSoc. Am. 32, 348 (1942).\n185. Moon, P., The Scientific Basis of Illuminating Engineer-\ning (Dover Pub., Inc., New York, 1961), rev. ed..\n186. Moon, P., and Spencer, D.E., \"Illumination from a non-\nuniform sky,\" Illum. Eng. 37, 707 (1942).\n187. Moon, P., and Spencer, D.W., \"Theory of the photic\nfield,\" Jour. Franklin Inst. 255, 33 (1953).\n192. Parke, N.G., Statistical Optics: I Radiation (M.I.T.\nResearch Laboratory of Electronics Report 70, Cam-\nbridge, Mass., 1948).\n193. Parke, N.G., Statistical Optics: II Mueller Phenomeno-\nlogical Algebra (M.I.T. Research Laboratory of\nElectronics Report 119, Cambridge, Mass. 1949).","BIBLIOGRAPHY\n395\n194. Parke, N.G., \"Optical algebra,\" J. Math. and Phys. 28,\n131 (1949).\n206. Pivovonsky, M., and Nagel, M.R., Tables of Blackbody\nRadiation Functions (Macmillan Co., New York, 1961).\n208. Pontrjagin, L., Topological Groups (Princeton Univ.\nPress, Princeton, N.J., 1946).\n210. Preisendorfer, R.W., Lectures on Photometry, Hydrologic\nOptics, Atmospheric Optics (Lecture Notes, vol. I,\nVisibility Laboratory, Scripps Inst. of Ocean.,\nUniversity of California, San Diego, Fall 1953).\n211. Preisendorfer, R.W., A Preliminary Investigation of the\nTransient Radiant Flux Problem (Lecture Notes, vol.\nII, Visibility Laboratory, Scripps Inst. of Ocean.,\nUniversity of California, San Diego, February 1954).\n216. Preisendorfer, R.W., \"A mathematical foundation for ra-\ndiative transfer theory, \"J. Math. and Mech. 6,\n685 (1957). .\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). .\n251. Preisendorfer, R.W., Radiative Transfer on Discrete\nSpaces (Pergamon Press, New York, 1965).\n264. Rickart, C.E., , General Theory of Banach Algebras (D.\nVan Nostrand, New York, 1960).\n296. Thekaekara, M.P., \"The solar constant and spectral dis -\ntribution of solar radiant flux,\" Solar Energy 9,\n7 (1965).\n298. Tyler, J.E., \"Radiance distribution as a function of\ndepth in an underwater environment,\" Bull. Scripps\nInst. Ocean. 7, 363 (1960).\n311. Walsh, J.W.T., Photometry (Dover Pub., , Inc., , New York,\n1965).","INDEX\n396\ndirectly transmitted radiance,\nabsolutely continuous\nmeasure, 375\n347\nAC property, 381\nelectromagnetic, view of light\nAC property, 381\n(vs phenomenological), 13\nalbedo, 216\nenergy conservation principle\nalgebra\nof reflectance and trans-\n(for radiometry), 199\nequation of transfer for radi-\nmittance operators, 230\nnormed, 243\nance, 368\ntime dependent and polarized\nBanach, 244\naltitude (in a reference\nform, 371\nframe), 19\nflux (see 'radiant flux' or\nAmbarzumian's principle,\n'luminous flux')\n228\nangle, polar, azimuthal, 24\nframes (of reference, e.g.\napparent radiance, 362\nterrestrial), 24\napparent-radiance equation,\n361\ngroup structure\nof natural light fields, 307\nBanach algebra, 244\nHaidinger's brush (in polarized\nBeam Transmittance function,\n344\nlight), 84\nvarious properties, 348\nhemisphere (E+), 24\nblondel, 179\nherschel, 172\nboundary\nreflecting, 340\nilluminance, 166\ninherent radiance, 363\ncandela, 163, 179\nintegrating sphere, 262\nClairaut's equation (for in-\nintensity\nverse nth power irradi-\nradiant, 70\nance law), 126\nluminous, 166\ncomplete (Planckian) radia-\ninteraction method\ntor, 162\na first synthesis, 222\nconstitutive definitions, 8\nin other fields, 223 (foot-\ncontraction property (of\nnote)\nbeam transmittance), 348\nsummary, 388\nconvexification\nand quantum theory, 390\nwhite, black, 316\ninteraction operators\ncoordinate systems, terres-\nintegral structure, 372\ntrial, 19\nkernel, 380\ncosine collector, 7\nsee also 'operators'\ncosine law, for irradiance,\ninteraction principle\n26\nphysical basis, 189\nbasic statement, 205\ndefinitions: constitutive,\nplace in radiative transfer\noperational, 8\ntheory, 208\ndepth (in a reference frame),\nlevels of interpretation, 208\n19\nAmbarzumian's principle, 228\ndichroic material (and po-\napplications to plane sur-\nlarized light), 84\nfaces, 217\ndiffraction, limits on ra-\napplications to curved sur-\ndiometry, 16\nfaces, 258\ndirection, defined, 19\napplications to plane-paral-\nupward, downward, 21\nlel media, 285\nunit inward, 25\napplications to general\nfor reflectance and trans-\nspaces, 322\nmittance, 212\non one-parameter media with\nsources, 330","INDEX\n397\nas basis for beam transmit-\niterated operators, 236\ntance and volume attenua-\ntion function, 344\nJanus plate, 68\nas basis for path function,\njoule, 172\npath radiance, 351\nas basis for volume scat-\nkernal, interaction, 380\ntering function, 364\nas basis for equation of\nlambert\ntransfer, 368\ntransmitter or reflector, 262\nas a means and as an end,\nlinear functional, 373\n391\npositive, 375\ninterfaces\nline of flux, 8\nreflecting, 340\nlumen, 161, 179\ninterreflection calculations\nluminance\nterminable and non-\ndistribution, relative, 156\nterminable, 248\ndistribution, general, 163,\ninvariant imbedding relation\n166\non plane-parallel media,\ntypical magnitudes, 164\n297\npath, 179\nhistorical notes, 299\nluminosity\ngeneralized form, 301\nof a wavelength, 153\nfor one-parameter media,\nluminosity function\n327\nstandard, 151, 157, 159, 160\nin general media, 339\nfor individuals, 153\nirradiance\nphotopic, 155, 158\ndefined, 14, 171\nscotopic, 155\nmeaning, 16\nrelative (of radiance), 157\ntypical orders of magni-\ngeneralized, 184\ntude, 17\nluminous emittance, 166, 179\nin terrestrial frames, 24\nscalar, 168, 179\nupward, downward, 24\nvector, 179\nnet, 26\nluminous energy, 167\ncosine law for, 26, 66\nluminous energy density, 167\n(from radiance), 35, 131,\nluminous flux, 166\n138\nluminous intensity, 166\nradiance from, 41\nluxoid (via inverse nth power\nscalar, 54\nfor irradiance), 130\nspherical, 56\nhemispherical, 58\nmu (millimicron), 192\nvector, and mechanical\nmeasure\nanalogy, 62\nRiemann, Lebesque, Stieltjes,\njanus plate (for net), 68\n12\n-distance law (for spheres),\nillustration of, 373\n103\nbasic theorems, 375\n-distance law (for circular\ninteraction, 380\ndisks), 105\nmedia\n-distance law (for general\nplane-parallel, cylindrical,\nsurfaces), 106\netc., see Table of Contents\nvia line integrals, 109\nmeter\nvia surface integrals, 115\nradiance, 30\nlaws of the form 1/rn, 120\nfor polarized radiance, 85\ndistributions, equivalence\nmillimicron (mu), 192\nof with radiance distribu-\nMöbius strip, 271\ntions, 143\nmoon\ncomputation for parallel\nradiance of, 98\nplanes, 217\nradiant intensity of, 99\non plane-parallel media,\nluminance of, 164\n286","398\nINDEX\nmultiplicative property (of\nphotoelectric effects (photo-\nbeam transmittance), , 348\nelectric cells, photoemis-\nsive, photoconductive,\nnanometer (= 10-9 m) =\nphotovoltaic), 3\nmillimicron (mu), 192\nphotometry\nnatural (mode of) solution,\ngeometrical, 2, 165\n203, 247\ngeneralized, 183\nnorm-contracting property\nnon linear, 185\nof C operator, 147\nphoton\nof R, T operators, 236\nas an aid to visualization,\nof operators, 322\n10\nnormed operator algebra, 243\nentering an eye from a star,\nnorth-based reference frame,\n18\n19\nPlanckian (complete) radiator,\n162\noperational definitions, 8\nPlanck's quantum of action, 9\nplane of scattering, 91\noperators\nalgebra, 230\nplane-parallel media, see\niterated, 236\nTable of Contents\nalgebra and radiative\npoint source\ntransfer, 241\noperational definition, 75\nalgebra, normed, 243\ncriterion for, 105\nu, S, 319\npolarization convention, 194\ninteraction, structure of,\n(footnote)\n372\npolarized equation of trans-\nvolume transpectral, 386\nfer, 371\npath function, 383\nPoynting vector field, 9\npath radiance, 384\nPrinciples of Invariance\ninteraction for general\non plane-parallel media,\nspaces, 314, 378\n294\ninteraction for surfaces,\non spherical, cylindrical,\n377\ntoroidal media, 325\nmiscellaneous examples,\non general media, 336\n387\noptical properties\nquantum-terminabel calcula-\ninherent and apparent, 349\ntions, 254\nquantum theory and interaction\npath function\nmethod, 370\nradiant, 172\nluminous, 179\nradiance\nderivation 351\nempirical definition, 30,\nconnection with path radi-\n171\nance, 354\nmeter, 30, 85\nintegral representation,\ntheoretical, 32\n367\nvia photon density, 33\noperator, 383\ntypical values, 33, 34, 97\npath luminance, 179\ndistributions (on E), 34\npath radiance, 172\nfunction (on XXE), 34\nderivation, 351\nirradiance from, 35\nconnection with path func-\n(from irradiance), 41\ntion, 354\nfield vs surface, 44\noperator, 384\ninvariance property, 46\nphase space density, 9\nradiance-invariance law, 46\nphenomenological, view of\noperational meaning of sur-\nlight (vs electromagnetic),\nface radiance, 49\n13\nn2- law, 51\npolarized, 83\nstd. Stokes vector, 88","INDEX\n399\nstd. observable vector, 88\narea-law for general sur-\npolarization composition\nfaces, 119\ntheorem, 89\nradiative process\nlocal observable vector, 91\ndefined, 190\nradiant flux content of po-\nradiative transfer theory\nlarized, 94\nas based on the interaction\ndistributions, elliptical,\nprinciple and operator al-\n131\ngebra, 188\ndistributions, polynomial,\nradiator\n139\nPlanckian or complete, 162\ndistributions, equivalence\nradiometrically adequate col-\nof with irradiance distri-\nlector, 151\nbutions, 143\nradiometric norm, 146, 232\npath, 172\nradiometric-photometric trans-\nD'-additivity (surfaces),\nition operator, 166, 169\n195\nradiometry\nD'-continuity (surfaces),\ngeometrical, 2\n195\ntransition to photometry,\nof parallel planes, 244\n166\nD'-additivity (slabs), 282\nmathematical basis of, 169\nD'-continuity (slabs), 282\nRadon-Nikodym\non plane-parallel media,\ntheorem, 185\n290\nderivative, 376\ndirectly transmitted, 347\nreference frame\nresidual, 347\nterrestrial, 20\nunattenuated, 347\nstratified, 24\nradiant density, 54, 172\nlocal vs standard (in po-\nradiant emittance, 28, 171\nlarized context), 91\nempirical, 29\nreflectance\nscalar, 61\nempirical, for surfaces,\nvector, 171\n194\nradiant energy, 54, 172\noperators, for surfaces,\nover space, 60,\n210\nover time, 61\ntheoretical, for surfaces,\nradiant flux\n213\noperational definition, 2,\nlambert, 215\n7, 171\nalgebra of operators, 230\ncalculations, 117\noperators for plane-parallel\nmeaning, 8\nmedia, 279\nF-additivity, 10\nsemigroup properties, 300\nF-continuity, 11\nresidual radiance, 347\nmonochromatic (or spectral),\nphenomenological interpre-\n11\ntation, 349\nfinite vs countable addi-\ntivity, 11\nscalar illuminance, 167\nS-additivity, 12\nscalar irradiance, 54\nS-continuity, 12\nscalar luminous emittance, 168\nD-additivity, 13\nscattering\nD-continuity, 13\nplane of, 91\nradiant intensity\nstandard function, 318\nempirical definition, 70,\nscatter processes (Rayleigh,\n171\nCompton, etc.), 191\nfield vs specific, 72\nsemigroup properties\ntheoretical, 73\nof reflected and transmitted\npoint sources and, 74\nradiant flux, 300\ncosine laws for, 77, 80\nof 2 (a,b), 309\nvector, 81\nof 3 (a,b), 298, 301\nof 4 (a,b), 303, 312","400\nINDEX\nconnections among T3, T4,\nvolume attenuation function,\n313\n349\nof beam transmittance, 348\nvolume scattering function,\nsolid angle, 37\n364\nsubtense of surfaces, 112\nvolume transpectral scatter-\nS-additivity property, 114\ning operator, 386\nS-continuity property, 114\nand the foundations of\nwatt, 171\nEuclid's optics, 115\nsources\nin one-parameter media,\n330\nspace light (= path radiance),\n363\nspecific intensity (see ra-\ndiance, surface), 44\nspecific radiance, 44\nspherical irradiance, 56\nsteradian, 38\nStokes vector (for radiance),\n88\nsun\nradiance of, 97\nradiant intensity of, 98\nluminance of, 164\nsun-based reference frame,\n19\nsurfaces\nsolid angle subtense of,\n112\ngeneral two-sided, 267\ngeneral one-sided, 271\ntalbot, 179\ntime dependent equation of\ntransfer, 371\ntransmittance\nempirical, for surfaces,\n194\noperators, for surfaces,\n210\ntheoretical, for surfaces,\n213\nlambert, 262\nalgebra of operators, 230\noperators for plane-paral-\n1e1 media, 279\nsemigroup properties, 300\ntruncation error estimates,\n250\nunattenuated radiance, 347\nunified atmosphere-hydro-\nsphere problem, 343\nviews of light (phenomeno-\nlogical vs electromagnetic),\n13\n* U.S. GOVERNMENT PRINTING OFFICE: 1976-677-882/ 37 REGION NO. 8"]}*