{"Bibliographic":{"Title":"Hydrologic optics. Volume I: Introduction ","Authors":"","Publication date":"1976","Publisher":""},"Administrative":{"Date created":"08-17-2023","Language":"English","Rights":"CC 0","Size":"0000422297"},"Pages":["GB\n665\n.P645\nv.1\nHYDROLOGIC OPTICS\nVolume I. Introduction\nR.W. PREISENDORFER\nU.S. DEPARTMENT OF COMMERCE\nNATIONAL OCEANIC & ATMOSPHERIC ADMINISTRATION\nENVIRONMENTAL RESEARCH LABORATORIES\nHONOLULU, HAWAII\n1976","GB\n665\nP645\nV.1\nDEPARTMENT OF COMMUNITY\nHYDROLOGIC OPTICS\n*\n*\nwith\nSTATES OF\nVolume I. Introduction\ncedolph\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\n76\n24.17","ii\nNothing that is seen is seen at once\nin its entirety\nEUCLID\n(First theorem, The Optics of Euclid [36])","iii\nCONTENTS\nVolume I\nPART I BASIC PRINCIPLES\nChapter 1\nIntroduction to Hydrologic Optics\n1. 0\nHydrologic Optics: Definition, Domain and\n1\nDesiderata\nThe Problems of Hydrologic Optics\n2\nThe Aims and Desired Goals of This Work\n3\nThe Plan and Scope of This Work\n4\n1.1\nA Primer of Geometrical Radiometry and Photometry\n6\nThe Nature of Radiant Flux\n6\nThe Unpolarized-Flux Convention\n7\nGeometrical Channeling of Radiant Flux\n9\nOperational Definitions of the Densities\n10\nField and Surface Interpretations of Radiant\nFlux and its Densities\n12\nOperational Definitions of Field and Surface\nQuantities\n12\nSummary of Concepts and Some Principal Formulas of\nGeometrical Radiometry\n14\nn2 - Law for Radiance\n18\nThe Bridge to Geometrical Photometry\n18\n1.2\nA Survey of Natural Light Fields\n22\nThe Solar Constant\n22\nGeneral Irradiance Levels at Earth's Surface\n24\nGeneral Illuminance Levels at Earth's Surface\n25\nGross Features of Atmospheric Radiative Transfer\n27\nRadiative Transfer Across the Air-Water Surface\n28\nGlitter Patterns on the Air-Water Surface\n32\nSubsurface Refractive Phenomena\n33\nThe Decay of the General Light Field with Depth\n37\nBehavior of Radiance Distributions with Depth\n39\nThe Asymptotic Radiance Hypothesis\n41\nUnderwater Irradiance Distributions\n42\nSubsurface Contrast Reduction by Scattering and\nAbsorbing Effects\n44\nSubsurface Contrast Reduction by Refractive\nEffects\n48\nThe Polarization of Underwater Light Fields\n50\nBiological Sources of Submarine Light Fields\n53","CONTENTS\niv\n55\nThree Simple Models for Light Fields\n1.3\n55\nThe Two- - Flow Model\n58\nThe Radiance Model\n61\nThe Diffusion Model\n66\nSome Deductions from the Light Field Models\n1.4\nThe Decay of the General Light Field with Depth\n66\nReflectance and Transmittance of Finitely Deep\n68\nHydrosols\n71\nInvariant Imbedding Relations for Irradiance\nA Theoretical Basis for the Law:\nN, (z,0) = N+ * (0,0) e - kz\n81\n*\n84\nComputing Radiances from the Simple Model\nDerivation of the Contrast Transmittance Law\n89\nand the Radiance Difference Law\nContrast Transmittances for General Backgrounds\n92\nThe Multiplicative Property of Contrast\n93\nTransmittance\n96\nTheory of the Secchi and Duntley Disks\nTheory of Absorption Measurements in Natural\n103\nHydrosols\nSome Properties of Artificial Light Fields in\n1.5\n109\nNatural Waters\n109\nThe Pure Absorption Case\nDerivation of the Semi-empirical Diffusion\n110\nModel for Point Sources\nTwo Examples of the Empirical Diffusion Model\n112\nRadiance Distribution Produced by a Submerged\n113\nUniform Point Source\nAn Empirical Study of Light Fields Produced by\n114\nCollimated Sources\nInherent and Apparent Optical Properties of\n1.6\n118\nHydrosols\nOperational Definitions of the Inherent Optical\n119\nProperties\n119\nThe Volume Attenuation Function\n122\nThe Volume Scattering Function\nVolume Total Scattering Function and Volume\n123\nAbsorption Function\nSelected Physical Measurements of the Inherent\n125\nOptical Properties\nOperational Definitions of the Apparent Optical\n135\nProperties\nPreliminary Observations on the Classification\n138\nof Natural Hydrosols\nSome General Modes of Classification of Natural\n1.7\n139\nOptical Media\n139\nModes of Classification","CONTENTS\nV\n1.8\nColorimetric Radiative Transfer\n142\nThe Quantitative Description of Color\n143\nAn Example of Experimentally Determined Chroma-\nticity Coordinates\n149\nOn the Use of Simple Models for Theoretical\nPredictions of Chromaticity Coordinates\n151\n1.9\nApplications of Hydrologic Optics to Underwater\nVisibility Problems\n154\nIntroduction to the Nomographs\n154\nA. Selection of the Proper Chart\n156\nA.1 Introduction\n156\nA.2\nNatural Illumination\n156\nA.3\nEffect of Depth and Water Clarity\n157\nStratified Water\n157\nEffect of Sea-State\n157\nExamples\n160\nA.4\nAdaptation Level\n160\nInclination Factor\n160\nBottom Influence\n162\nA.5\nCalculation of Adaptation Luminance\n162\nChart Selection\nA.6\n163\nB.\nUsing the Nomographs\n163\nB. Introduction\n163\nObjects on the Bottom\n163\nObject Size and Shape\n163\nVertical Path of Sight\n163\nInclined Paths of Sight\n165\nB.3\nThe Secchi Disk\n169\nB.4\nTarget Markings\n170\nThe Measurements of Target Reflectance\nB. 5\n170\nB. 6\nHorizontal Paths of Sight\n170\nB.7\nThe Roo Correction\n172\nB.8 Correction of the Sighting Range\n172\nC.\nInterpretation of Sighting Range\n193\nC.1 Introduetion\n193\nC.2 Effect of Lack of Warning\n193\nC.3\nEffect of Observer Training\n193\nC.4 Effect of Observer Visual Capability\n194\nD.\nVisualization of Water Clarity\n194\nD.1 Introduction\n194\nD.2\nEstimation of Sighting Range\n194\nEstimation of Adaptation Luminance\nD.3\n195\nD.4\nEstimation of a and K\n195\nCharacterization of Natural Waters\nD.5\n195\n1.10\nApplications of Hydrologic Optics to the Food-\nChain Problem in the Sea\n196\nThe General Exponential Law of Change\n197\nThe Volterra Prey-Predator Equations\n198\nThe General Food-Chain Equations\n199\nAn Illustration of the Food-Chain Theory with a\nRadiant Energy Term\n201","vi\nCONTENTS\n1.10\nApplications of Hydrologic Optics to the Food-\nChain Problem in the Sea (Continued)\nThe General Three-Term Equations\n202\nThe Quasi-Steady State Equations\n202\nThe Equilibrium Solutions\n203\nSome General Properties of Equilibrium Solutions\n204\n1.11\nFuture Problems of Hydrologic Optics\n205\nProblem One: To Establish Theoretically the\nPhysical Basis of the Inherent Optical Proper-\nties of Natural Hydrosols\n206\nProblem Two: To Establish Complete Empirical\nClassifications of Natural Hydrosols\n207\nProblem Three: To Establish a Unified Automatic\nComputation Program for Prediction Computations\nand Data Reduction Computations in Geophysical\nOptics (the GEOVAC)\n208\n1.12\nBibliographic Notes for Chapter 1\n208\nVolume II\nChapter 2\nRadiometric and Photometric Concepts\n2.0\nIntroduction\n2.1\nRadiant Flux\nBasic Photoelectric Effects\nOperational Definition of Radiant Flux\n2.2\nThe Meaning of 'Radiant Flux'\nFundamental Geometric Properties of Radiant Flux\n2.3\n2.4\nIrradiance and Radiant Emittance\nDefinition of Irradiance\nThe Meaning of 'Irradiance'\nTerrestrial Coordinate Systems\nRepresentation of Irradiance in Terrestrial\nFrames\nThe Cosine Law for Irradiance\nRadiant Emittance\n2.5\nRadiance\nRadiance Distributions\nIrradiance from Radiance\nRadiance from Irradiance","CONTENTS\nvii\n2.5 Radiance (Continued)\nField Radiance vs Surface Radiance\n2.6 An Invariance Property of Radiance\nThe Radiance-Invariance Law\nThe Operational Meaning of Surface Radiance\nThe n2-Law for Radiance\n2.7 Scalar Irradiance, Radiant Energy, and Related\nConcepts\nRadiant Density\nScalar Irradiance\nSpherical Irradiance\nHemispherical Irradiance\nRadiant Energy over Space\nRadiant Energy over Time\nScalar Radiant Emittance\n2.8 Vector Irradiance\nA Mechanical Analogy\nGeneral Definition of Vector Irradiance\nThe General Cosine Law for Irradiance\n2.9 Radiant Intensity\nOperational Definition of Empirical Radiant\nIntensity\nField Intensity vs Surface Intensity\nTheoretical Radiant Intensity\nRadiant Intensity and Point Sources\nCosine Law for Radiant Intensity\nGeneralized Cosine Law for Radiant Intensity\n2.10 Polarized Radiance\nOperational Definition of Polarized Radiance\nThe Standard Stokes and Standard Observable\nVectors\nAnalytic Link Between S and N\nStandard and Local Reference Frames\nRadiant Flux Content of Polarized Radiance\n2.11 Examples Illustrating the Radiometric Concepts\n1.\nRadiance of the Sun and Moon\n2. Radiant Intensity of the Sun and Moon\n3. Radiant Flux Incident on Portions of the\nEarth\n4. Irradiance Distance-Law for Spheres\n5. Irradiance Distance-Law for Circular Disks;\nCriterion for a Point Source\n6.\nIrradiance Distance-Law for General Surfaces\n7.\nIrradiance via Line Integrals\n8. Solid Angle Subtense of Surfaces","viii\nCONTENTS\n2.11 Examples Illustrating the Radiometric Concepts\n(Continued)\n9. Irradiance via Surface Integrals\n10. Radiant Flux Calculations\n11. Intensity Area-Law for General Surfaces\n12. On the Possibility of Inverse nth Power\nIrradiance Laws\n13. Irradiance from Elliptical Radiance\nDistributions\n14. Irradiance from Polynomial Radiance\nDistributions\n15. On the Formal Equivalence of Radiance and\nIrradiance Distributions\n2.12 Transition from Radiometry to Photometry\nThe Individual Luminosity Functions\nThe Standard Luminosity Functions\nPhotometric Bedrock: the Lumen\nLuminance Distributions\nTransition to Geometrical Photometry\nGeneral Properties of the Radiometric-\nPhotometric Transition Operator\nThe Mathematical Basis for Geometrical\nPhotometry\nSummary and Examples (Tables of Radiometric\nand Photometric Concepts)\n2.13 Generalized Photometries\nLinear Photometries\nNonlinear Photometries\n2.14 Bibliographic Notes for Chapter 2\nChapter 3\nThe Interaction Principle\n3.0 Introduction\nThe Physical Basis of the Linearity of the\nInteraction Principle\nPlan of the Chapter\n3.1 A Preliminary Example\nEmpirical Reflectances and Transmittances\nfor Surfaces\nThe Problem\nThe Present Instance of the Interaction\nPrinciple\nSolution of the Problem","CONTENTS\nix\n3.1 A Preliminary Example (Continued)\nDiscussion of Solution\nRelated Problems and Their Solutions\nAn Alternate Form of the Principle\nThe Natural Mode of Solution\n3.2 The Interaction Principle\nDiscussion of the Interaction Principle\nThe Place of the Interaction Principle in\nRadiative Transfer Theory\nThe Levels of Interpretation of the Interaction\nPrinciple\n3.3 Reflectance and Transmittance Operators for\nSurfaces\nGeometrical Conventions\nThe Empirical Reflectances and Transmittances\nThe Theoretical Reflectances and Transmittances\nVariations of the Basic Theme\n3.4\nApplications to Plane Surfaces\nExample 1: Irradiances on Two Infinite Parallel\nPlanes\nExample 2: Irradiances on Two Infinite Parallel\nPlanes, Reexamined. A First Synthesis of the\nInteraction Method\nExample 3: Irradiances on Finitely Many Infinite\nParallel Planes\nExample 4 : Irradiances on Infinitely Many Infinite\nParallel Planes\nExample 5: The Algebra of Reflectance and\nTransmittance Operators for Planes. Radiometric\nNorm. Iterated Operators. Operator Algebras\nand Radiative Transfer.\nExample 6: Radiances of Infinite Parallel Planes\nExample 7 : Terminable and Non Terminable Inter-\nreflection Calculations. A Terminable Calculation.\nTruncation Error Estimates. Quantum-\nterminable Calculations.\nExample 8: Two Interacting Finite Plane Surfaces\n3.5 Applications to Curved Surfaces\nExample 1: Open Concave Surfaces\nExample 2: Closed Concave Surfaces; the\nIntegrating Sphere\nExample 3: Open and Closed Convex Surfaces\nExample 4 : General Two-Sided Surfaces\nExample 5 : General One-Sided Surfaces\n3.6 Reflectance and Transmittance Operators for\nPlane-Parallel Media\nGeometrical Conventions\nThe Empirical Reflectances and Transmittances","CONTENTS\nx\n3.6 Reflectance and Transmittance Operators for\nPlane-Paralle1 Media (Continued)\nThe Theoretical Reflectances and Transmittances\nVariations of the Basic Theme\n3.7 Applications to Plane-Parallel Media\nExample 1: Irradiances on Plane-Parallel Media\nExample 2: Radiances in Plane-Paralle1 Media\nExample 3: The Classical Principles of\nInvariance\nExample 4 : The Invariant Imbedding Relation\nExample 5 : Semigroup Properties of Transmitted\nand Reflected Radiant Flux\nExample 6: The Generalized Invariant Imbedding\nRelation\nExample 7: Group-Theoretic Structure of Natural\nLight Fields. Group Theory, Radiative Transfer\nand Quantum Theory.\n3.8 Interaction Operators for General Spaces\nGeometrical Conventions\nThe Empirical Scattering Functions\nThe Theoretical Scattering Functions\nVariations of the Basic Theme\n3.9 Applications to General Spaces\nExample 1: Principles of Invariance for Spherical,\nCylindrical, Toroidal Media\nExample 2: Invariant Imbedding Relation for\nOne-Parameter Media\nExample 3: One-Parameter Media with Internal\nSources\nExample 4 : Principles of Invariance for General\nMedia\nExample 5 : Invariant Imbedding Relation in\nGeneral Media\nExample 6: Reflecting Boundaries and Interfaces\nExample 7 : The Unified Atmosphere-Hydrosphere\nProblem\nExample 8: Several Interacting Separate Media\n3.10 Derivation of the Beam Transmittance Function\n3.11 Derivation of the Volume Attenuation Function\n3.12 Derivation of Path Radiance and Path Function\nThe Path Radiance\nThe Path Function\nThe Connection Between Path Function and Path\nRadiance\n3.13 Derivation of Apparent-Radiance Equation","CONTENTS\nxi\n3.14 Derivation of the Volume Scattering Function\nRegularity Properties of o\nThe Integral Representation of the Path\nFunction\n3.15 The Equation of Transfer for Radiance\nSteady State Equation of Transfer\nTime Dependent and Polarized Equations of\nTransfer\n3.16 On The Integral Structure of the Interaction\nOperators\nThe Mathematical Prerequisites\nInteraction Operators for Surfaces\nInteraction Operators for General Media\nInteraction Measures and Kernels\n3.17 Further Examples of the Interaction Method\nExample 1: The Path Function Operator\nExample 2: The Path Radiance Operator\nExample 3: The Volume Transpectral Scattering\nOperator\nMiscellaneous Examples\n3.18 Summary of the Interaction Method\nSummary of the Interaction Method\nRemarks on the Stages of the Interaction Method\nThe Interaction Method and Quantum Theory\nThe Interaction Principle as a Means and as an\nEnd\nConclusion\n3.19 Bibliographic Notes for Chapter 3\nVolume III\nPART\nII\nTHEORY OF LIGHT FIELDS\nChapter 4\nCanonical Forms of the Equation of Transfer\n4.0 Introduction\nRadiance in Transparent Media\n4.1","xii\nCONTENTS\n4.2\nRadiance in Absorbing Media\n4.3\nKoschmieder's Equation for Radiance\n4,4 The Classical Canonical Equation\n4.5 The General Canonical Equation for Radiance\nCanonical Representation of Apparent Radiance\nThe Canonical Form for Stratified Media\n4.6 Canonical Representation of Polarized Radiance\nA Simple Model for Polarized Light Fields\nExperimental Questions\n4.7 Abstract Versions of Canonical Equations\n4.8 Bibliographic Notes for Chapter\n4\nChapter 5\nNatural Solutions of the Equation of Transfer\n5.0 Introduction\n5.1 The n-ary Radiometric Concepts\nn-ary Radiance\nn-ary Scalar Irradiance\nn-ary Radiant Energy\nGeneral n-ary Radiometric Functions\n5.2 Equation of Transfer for n-ary Radiance, Diffuse\nRadiance, and Path Function\n5.3 Canonical Equations for n-ary Radiance\nConcluding Observations\n5.4\nThe Natural Solution for Radiance\n5.5 Truncated Natural Solutions for Radiance\n5.6 Optical Ringing Problem. One Dimensional Case.\nGeometry of the Time-Dependent Light Field\nThe Equation of Transfer\nOperator Form of the Equation of Transfer\nThe Natural Solution\nAn Example\nConcluding Observations","CONTENTS\nxiii\n5.7 Optical Ringing Problem. Three Dimensional Case.\nThe Characteristic Ellipsoid\nTime Dependent R and T Operators and the\nNatural Solution\nTruncated Natural Solution\n5.8 Transport Equation for Residual, Directly\nObservable, and n-ary Radiant Energy\nResidual Radiant Energy\nTransport Equation for Residual Radiant Energy\nThe Attenuation Time Constant\nGeneral Representation of Residual Radiant\nEnergy\nTransport Equation for n-ary Radiant Energy\nTransport Equation for Directly Observable\nRadiant Energy\nThe Natural Solution for Directly Observable\nRadiant Energy\n5.9 Solutions of the n-ary Radiant Energy Equations\nNatural Integral Representations of n-ary\nRadiant Energy\nNatural Closed Form Representations of n-ary\nRadiant Energy\nGeneral Integral Representations of n-ary\nRadiant Energy\nStandard Growth and Decay Formulas for n-ary\nRadiant Energy\n5.10 Properties of Time Dependent n-ary Radiant\nEnergy Fields and Related Fields\nSome Fine-Structure Properties of n-ary\nRadiant Energy\nScattered, Absorbed, and Attenuated Radiant\nEnergies\nRepresentations of U(t;a), U(t;s), , and U(t;a)\n5.11 Dimensionless Forms of n-ary Radiant Energy\nFields and Related Fields\nConversion Rules for Dimensionless Quantities\nDimensionless Forms for UO (t)\nDimensionless Forms for Un(t)\nDimensionless Forms for U* (t)\nDimensionless Forms for U(t)\nDimensionless Forms for U(t;a), U(t;a)\nSome Graphical Representations of Solutions\nin Dimensionless Form\nA Discussion of Time Constants\n5.12 Global Approximations of General Radiance\nFields\nGlobal Approximations of Higher Order","CONTENTS\nxiv\n5.13 Light Storage Phenomena in Natural Optical\nMedia\nEveryday Examples of Light Storage\nStorage Capacity\nMethods of Determining Storage Capacity\nExample\n5.14 Operator-Theoretic Basis for the Natural\nSolution Procedure\n5.15 Bibliographic Notes for Chapter 5\nChapter 6\nClassical Solutions of the Equations of Transfer\n6.0\nIntroduction\n6.1 The Bases of the Spherical Harmonic Method\nPhysical Motivations\nAn Algebraic Setting for Radiance\nDistributions\n6.2 Abstract Spherical Harmonic Method\nFinite Forms of the Abstract Harmonic\nEquations\n6.3 Classical Spherical Harmonic Method; General\nMedia\nThe Orthonormal Family\nProperties of the Orthonormal Family\nGeneral Equations for Spherical Harmonic\nMethod\n6.4 Classical Spherical Harmonic Method: Plane-\nParallel Media\nA Formal Solution Procedure\nA Truncated Solution Procedure\nVector Form of the Truncated Solution\nSummary\n6.5 Three Approaches to Diffusion Theory\nThe Approach via Fick's Law\nThe Approach via Spherical Harmonics\nRadiance Distributions in Diffusion Theory","CONTENTS\nXV\n6.5 Three Approaches to Diffusion Theory (Continued)\nApproaches via Higher Order Approximations\nThe Approach via Isotropic Scattering\n6.6 Solutions of the Classical Diffusion Equations\nPlane-Parallel Case\nPoint Source Case\nDiscrete Source Case\nContinuous Source Case\nPrimary Scattered Flux as Source Flux\nHigher Order Scattered Flux as Source Flux\nTime Dependent Diffusion Problems\n6.7\nSolutions of the Exact Diffusion Equations\nInfinite Medium with Point Source\nInfinite Medium with Arbitrary Sources\nSemi-infinite Medium with Boundary Point Source\nSemi-infinite Medium with Internal Point Source\nObservations on the Functional Relations\nfor f\nand\nf\nC\no\n6.8 Bibliographic Notes for Chapter 6\nVolume IV\nChapter 7\nInvariant Imbedding Techniques for Light Fields\n7.0 Introduction\n7.1 Differential Equations Governing the Steady\nState R and T Operators\nLocal Forms of the Principles of Invariance\nThe Differential Equations for R and T\nDiscussion of the Differential Equations\nFunctional Relations for Decomposed\nLight Fields\n7.2 Differential Equations Governing the Time\nDependent R and T Operators\nTime Dependent Local Forms of the Principles\nof Invariance\nTime Dependent Invariant Imbedding Relation\nIntegral Representation of Time Dependent\nRand T Operators\nTime Dependent Principles of Invariance\nDifferential Equations for the Time Dependent\nR and T Operators","CONTENTS\nxvi\n7.2 Differential Equations Governing the Time\nDependent R and T Operators (Continued)\nDiscussion of the Differential Equations\n7.3 Algebraic and Analytic Properties of the R and\nT Operators\nPartition Relations for R and T Operators\nAlternate Derivations of the Differential\nEquations for R and T Operators\nAsymptotic Properties of R and T Operators\n7.4 Algebraic Properties of the Invariant Imbedding\nOperators\nThe Operator M(x,z)\nThe Connections Between M(x,z), m (x, z) , and\nm (z,x)\nInvertibility of Operators\nRepresentations for the Components of m(x,z),\nm(z,x)\nThe Isomorphism Between T2(a,b) and G2 (a,b)\nThe Physical Interpretation of the Star Product\nThe Link Between m (a,x,b) and m (a,y,b)\nRepresentation of m(x,y,z) by Elements of\nT2(a,b)\nA Constructive Extension of the Domain of\nm (x,y, z)\nRepresentation of m (v,z;u,y) by Elements of\nT2 (a,b) and 3 (a,b)\nThe Connection Between 4(x,y) and m(s,y)\nA Star Product for the Operators n(x,y,z)\nPossibilities Beyond M(v,x;u,w)\nPossibilities Beyond 2 (a,b)\n7.5 Analytic Properties of the Invariant Imbedding\nOperators\nDifferential Equations for m(x,y)\nDifferential Equations for m(x,y,z)\nDifferential Equations for M(v,x;u,w)\nDifferential Equations for M(x,y) and 4(s,y)\nAnalysis of the Differential Equation for\nR(y, b)\n7.6 Special Solution Procedures for R(a,b) and T(a,b)\nin Plane-Parallel Media\nThe General Equation for R(a,b;5',5)\nThe Isotropic Scattering Case for\nA Sample Numerical Solution for r(x;u',v)\nThe General Equation for T *(a,b;E';E)\nThe Isotropic Scattering Case for T*\n7.7 General Solution Procedures for R(a,b) and T(a,b)\nin Plane-Parallel Media","CONTENTS\nxvii\n7.8 The Method of Modules for Deep Homogeneous\nMedia\nThe Invariant Imbedding Relation for Deep\nHydrosols\nThe Module Equation\nEmpirical Bases for the Use of the Module\nEquations\n7.9 The Method of Semigroups for Deep Homogeneous\nMedia\nThe Semigroup Equations for T (z)\nThe Infinitesimal Generator A\n7.10 The Method of Groups for Deep Homogeneous Media\nThe Return of the Group 2 (0,00)\nThe Infinitesimal Generator of 2 (0, co)\nThe Exponential Representation of m (y) and\nN(y)\nThe Exponential Representation of a (y)\nNumerical Procedures of N(y) : The Exponential\nTechnique\nThe Characteristic Representation of N(y)\nAsymptotic Property of N(y)\nAsymptotic Properties of Polarized Radiance\nFields\n7.11 Method of Groups for General Optical Media\nAnalysis of the Group Method: Initial Data\nAnalysis of the Group Method: Limitations of\nthe Equation of Transfer\nAnalysis of the Group Method: Summarized\nThe General Method of Groups\nObservations on the Method of Groups\nThe Method of Groups and the Inner Structure\nof Natural Light Fields\n7.12 Homogeneity, Isotropy and Related Properties of\nOptical Media\nLocal Concepts\nGlobal Concepts\nSummary\nConclusion\n1.13 Functional Relations for Media with Internal\nSources\nPreliminary Relations\nIntegral Representations of the Local\nY-Operators\nIntegral Representations of the Global\nY-Operators\nIncipient Patterns and Nascent Methods","xviii\nCONTENTS\n7.13 Functional Relations for Media with Internal\nSources (Continued)\nDual Integral Representations of the Global\nY-Operators\nLogical Descendants of 4(s,y;a,b)\nDifferential Equations for the Dual Operators\nA Colligation of the Component Y-Operator\nEquations\nAsymmetries of the Y-Operator\nA Royal Road to the Internal-Source Functional\nEquations\nSummary and Prospectus\nFinal Observations on the Relations Between the\nOperators M(v,x;u,w) and 4(s,y;a,b)\n7.14 Invariant Imbedding and Integral Transform\nTechniques\nAn Integral-Transform - Primer\nTime-Dependent Radiative Transfer\nHeterochromatic Radiative Transfer\nMultidimensional Radiative Transfer\nConclusion\n7.15 Bibliographic Notes for Chapter 7\nVolume V\nChapter 8\nModels for Irradiance Fields\n8.0\nIntroduction\n8.1 Invariant Imbedding Relation for Irradiance\nFields\n8.2 General Irradiance Equations\n8.3 Two-Flow - Equations: Undecomposed Form\nEquilibrium Form of the Two-Flow Equations\nOntogeny of the Two-Flow Equations\n8.4 Two-Flow - Equations: Decomposed Form\nPrinciples of Invariance for Diffuse Irradiance\nClassical Models for Irradiance Fields\nCollimated-Diffuse - Light Field Models\nIsotropic Scattering Models\nConnections with Diffusion Theory","CONTENTS\nxix\n8.5 Two-D Models for Irradiance Fields\nOn the Depth Dependence of the Attenuating\nFunctions\nTwo-D Model for Undecomposed Irradiance Fields\nTwo-D Models for Internal Sources\nTwo-D Model for Decomposed Irradiance Fields\nInclusion of Boundary Effects\n8.6 One-D and Many-D Models\nOne-D Models for Undecomposed Irradiance\nFields\nOne-D Model for Internal Sources\nOne-D Model for Decomposed Irradiances\nMany-D Models\n8.7\nInvariant Imbedding Concepts for Irradiance\nFields\nExample 1: R and T Factors in Two-D Models\nExample 2:\nand T Factors in One-D Models\nExample 3: Differential Equations for R and T\nFactors\nExample 4 : Third Order Semigroup Properties\nof R and J Factors\nExample 5 : Systematic Analyses of Boundary\nEffects\nExample 6: Invariant Imbedding Operator for\nInteracting Media\nExample 7: Differential Equations Governing\na and T Factors\nExample 8: Method of Modules for Irradiance\nFields\nExample 9: Method of Semigroups for Irradiance\nFields\nExample 10: Irradiance Fields Generated by\nInternal Sources\n8.8 A Model for Vector Irradiance Fields\nThe Quasi-Irrotational Light Field in Natural\nWaters\nInterpretations of the Integrating Factor\nThe Curl and Divergence of the Submarine\nLight Field\nGeneral Representation of the Submarine Light\nField\nExample 1: The Case of Isotropic Scattering\nExample 2: Asymptotic Form of the Light Field\nGlobal Properties of the Irradiance Field\n8.9\nCanonical Representation of Irradiance Fields\n8.10 Bibliographic Notes for Chapter 8","CONTENTS\nXX\nPART III THEORY OF OPTICAL PROPERTIES\nChapter 9\nGeneral Theory of Optical Properties\n9.0\nIntroduction\n9.1 Basic Definitions for Optical Properties\n9.2 Directly Observable Quantities for Light Fields\nin Natural Hydrosols\nIntroduction\nClassical Two-Flow Theory: The Theoretical\nK-Functions\nDiffuse Absorption Coefficient k\nThe R-Infinity Formulas\nThe Inequalities\nObservations on Inadequacies of Classical\nTheory\nExact Two-Flow Theory: Experimental K-Functions\nand R-Functions\nThe Basic Reflectance Relation\nThe Exact Inequalities\nThe Significance of the Condition 0 VI K(z,+)\nRelative Magnitudes of H and K-Functions\nCharacteristic Equation for K(z,\nThe Depth Rate of Change of R(z,-)\nConnections Among the K-Functions\nK-Function for Radiance\nGeneral K-Functions\nIntegral Representations of the K-Functions\nIntegral Representations of the Irradiance\nand Radiance Fields\n9.3 The Covariation of the K-Functions for Irradiance\nand Distribution Functions\nSome Elementary Physical and Geometrical\nFeatures of K(z,- and D(z,-)\nThe General Law Governing K(x,-) and D(z,-)\nThe Absorption-Like Character of K(z,-)\nForward Scattering Media\nThe Covariation Rule for K(z,-) and D(z,-)\nIllustrations of the Rule\nThe Contravariation of K(z,+) and D(z,+)\nA Covariation Rule of Thumb\n9.4 General Analytical Representations of the\nObservable Reflectance Function\nThe Differential Equation for R(.,-)\nUnfactored Form","CONTENTS\nxxi\n9.4 General Analytical Representations of the\nObservable Reflectance Function (Continued)\nThe Differential Equation for R(,-) , : Factored\nForm\nSecond-Order Form of Differential Equation\nfor R( . 9 -)\nThe Equilibrium-Seeking Theorem for R(,-):\nPreliminary Observations\nThe Equilibrium-Seeking Theorem for R(,-)\nObservation 1\nObservation 2\nThe Integral Representations of R(z,-)\nApplications\nSpecial Closed Form Solution\nDifferential Analyzer or Digital Solutions\nSeries Solutions\nEquivalence Theorem for R(.,-)\nConnections with the Two-Flow - Theory\nSummary\n9.5 The Contrast Transmittance Function\nThe Concept of Contrast\nRegular Neighborhoods of Paths\nContrast Transmittance and Its Properties\nAlternate Representations of Contrast\nTransmittance\nContrast Transmittance as an Apparent Optical\nProperty\nOn the Multiplicity of Apparent Radiance\nRepresentations\n9.6 Classification of Optical Properties\n9.7 Bibliographic Notes for Chapter 9\nChapter 10\nOptical Properties at Extreme Depths\n10.0 Introduction\n10.1 On the Structure of the Light Field at\nShallow-Depths: Introductory Discussion\n10.2 Experimental Basis for the Shallow-Depth Theory\nSummary of the Experimental Evidence\n10.3 Formulation of the Shallow-Depth Model for\nK- and R-Functions\nFormulas for H(z,","CONTENTS\nxxii\n10.3 Formulation of the Shallow-Depth Model for\nK- and R-Functions (Continued)\nFormulas for K (z, +)\nFormula for R(z, -)\nComparisons of Experimental Data with\nCalculations Based on the Model\nHypotheses on the Fine Structure of Light\nFields in Natural Hydrosols\n10.4 Catalog of K-Configuration for Shallow Depths\nSome Special Fine Structure Relations\nConclusion\n10.5 A General Proof of the Asymptotic Radiance\nHypothesis\nIntroduction\nPreliminary Definitions\nFormulation of the Problem\nThe Functions P,Q,R\nThe Limit of kg(,,)\nThe Limit of K(,,)\nNotes and Observations\n10.6 On the Existence of Characteristic Diffuse Light:\nA Special Proof of the Asymptotic Radiance\nHypotheses\nIntroduction\nPhysical Background of the Method of Proof\nThe Proof\nThe Equation for the Characteristic Diffuse\nLight\n10.7 Some Practical Consequences of the Asymptotic\nRadiance Hypothesis\nBasic Formulas: The Irradiance Quartet\nThe D- and R-Functions\nThe K-Functions\nThe K-Characterization of the Hypothesis\nThe Basic Transfer Equations\nConsequences for Directly Observable Quantities:\nThe Equation for the Asymptotic Radiance\nDistribution\nThe Limits of the K-Functions\nThe Limits of the D- and R-Functions\nConsequences for Some Simple Theoretical Models:\nThe Two-D Model for Irradiance Fields\nCritique of Whitney's \"General Law\"\nThe Simple Model for Radiance Distributions\nFurther Consequences of Asymptoticity\nThe Standard Ellipsoid\nExpressions for D( ) and R\nThe Determination of E\nAn Heuristic Proof of the Hypothesis\nA Criterion for Asymptoticity","CONTENTS\nxxiii\n10.8 Simple Formulas for the Volume Absorption\nCoefficient in Asymptotic Light Fields\nIntroduction\nShort Derivation of I\nDerivation of II\nApplied Numerology A Rule of Thumb\n10.9 Bibliographic Notes for Chapter 10\nChapter 11\nThe Universal Radiative Transport Equation\n11.0 Introduction\n11.1 Transport Equations for Radiometric Concepts\nEquation of Transfer for Radiance\nTransport Equations for H(z,\nTransport Equations for h (z,\nTransport Equation for Scalar Irradiance\nPreliminary Unification and Preliminary\nStatement of the Equilibrium Principle\n11.2 Transport Equations for Apparent Optical\nProperties\nCanonical Forms of Transport Equations for\nK-Functions\nDimensionless Transport Equation for K(@)\nTransport Equation for K(z,0,0)\nTransport Equations for K(z,\nTransport Equations for k(z, and k(z)\nTransport Equation for R(z,-)\n11.3 Universal Radiative Transport Equation and the\nEquilibrium Principle\n11.4 Some Additional Transport Equations Subsumed\nby the Universal Transport Equation\nSummary and Conclusion\n11.5 Bibliographic Notes for Chapter 11","xxiv\nCONTENTS\nVolume VI\nChapter 12\nOptical Properties of the Air-Water Surface\n12.0 Introduction\n12,1 Reflectance and Transmittance Properties of\nthe Static Surface\nThe Geometric Law of Reflection\nThe Geometric Law of Refraction\nThe Fresnel Laws for Reflectance\nThe Fresnel Laws for Transmittance\nExample 1: Reflectance Under Uniform\nRadiance Distributions\nExample 2: Reflectance Under Cardioidal\nRadiance Distributions\nExample 3: Reflectance Under Zonal Radiance\nDistributions\n12.2 Radiative Transfer and the Static Surface\nIrradiance Interaction Between the Surface\nand the Hydrosol\nThe Three-fold - Irradiance Interaction: Aerosol,\nAir-Water Surface, and Hydrosol\nThe Three-fold Radiance Interaction: for the\nStatic Surface\nContrast Transmittance Formulas for the Static\nSurface\nContrast Transmittance Formulas for Extended\nPaths Across the Static Air-Water Surface\n12.3 Elementary Hydrodynamics of the Air-Water\nSurface\nThe Fluid Transfer Process\nPhysics of the Fluid Transfer Process\nGeneral Equations of Motion of a Fluid\nSpecial Equations of Motion for the Air and\nWater Masses\nSurface Kinematic Condition\nSurface Pressure Condition\nSinusoidal Wave Forms\nLinearized Equations of Motion\nClassical Wave Model\nKelvin-Helmholtz Model\nKelvin-Helmholtz Instability\nCapillary and Gravity Waves\nEnergy of Waves\nSuperposition of Waves\nSpectrum of the Air-Water Surface","CONTENTS\nXXV\n12.4 Harmonic Analyses of the Dynamic Air-Water\nSurface\nThe Roots of Harmonic Analysis\nHarmonic Synthesis vs. Harmonic Analysis\nIntegrals vs. Series in Harmonic Analysis\nFourier Series Representations of the\nAir-Water Surface\nHydrodynamic Basis for Harmonic Analysis of\nAir-Water Surfaces\nThe Periodogram Basis of the Energy Spectrum\nFourier Integral Representations of the\nAir-Water Surface. Case 1 : The Surface is\nAperiodic\nFourier Integral Representations of the\nAir-Water Surface. Case 2 : The Surface is\nPeriodic or Random\nA Working Representation of the Dynamic\nAir-Water Surface and its Directional Energy\nSpectrum\nGeometrical Applications of the Directional\nEnergy Spectrum\n12.5 Wave Slope Data\nThe Logarithmic Wind Profile Model\nVisual Observations on Wave Slopes\nHulburt's Observations of Wave Slopes\nDuntley's Immersed-Wire Measurements of Wave\nSlopes\nIntuitive Picture of the Gaussian Slope\nDistribution\nThe Wave-Slope Wind-Speed Law (Duntley)\nCox and Munk's Photographic Analysis of the\nGlitter Pattern\nThe Wave-Slope Wind-Speed Law (Cox and Munk)\nSchooley's Flash Photography Measurements of\nWave Slopes\n12.6 Wave Generation and Decay Data\nGeneration of Waves: Shallow Depths, Small\nFetches\nGeneration of Waves: Deep Depths, Large Fetches\nDecay of Waves\n12.7 Wave Spectrum Data\nWave Spectra by Aerial Stereo Photography\nWave Spectra by Floating-Buoy Motion\nWave Spectra from Submarine Echo Recordings\n12.8 Empirical Wave Spectra Models\nThe Neumann Spectrum\nDerivation of the Neumann Spectrum\nThree Laws Derived from the Neumann Spectrum\nAlternate Forms of the One-Dimensional Spectrum","CONTENTS\nxxvi\n12.8 Empirical Wave Spectra Models (Continued)\nGeneral Properties of Gamma Type Spectra\nWind Speed, Wave Length, and Wave Energy\n12.9 Theoretical Wave Spectra Models\nThe Wave Elevation Distribution\nThe Wave Slope Distribution\nThe Wavelength Distribution\nThe Bretschneider Spectrum\nThe Wave Height Distribution\nModels of Wind-Generated Spectra\nSpectral Transport Theory\n12.10 Instantaneous Radiance Field Over A Dynamic\nAir-Water Surface\nThe Geometrical Setting\nThe Integral Equation for the Instantaneous\nSurface Radiance N+(S)\n12.11 Time Averaged Radiance Field Over A Dynamic\nAir-Water Surface\nDirect and Indirect Radiance Averages\nThe Stationarity Condition\nThe Independence Condition\nThe Weighting Functions\nThe Time Averaged Integral Equation for N+ (S)\nStructure of the Weighting Functions\nThe Instantaneous and Time Averaged Equations\nfor N+ (S)\n12.12 Instantaneous and Time-Averaged Radiance\nFields Within A Natural Hydrosol\nTwo Types of Time-Averaged - Radiance Fields\nEquations of Transfer for Time-Averaged\nRadiance Fields\nConnection Between Fixed Depth and Cosurface\nTime-Averaged Radiances\n12.13 Synthesis of Time-Averaged Radiance Fields\nComparison with the Static Case\n12.14 Observations on the Theory of Time-Averaged\nRadiance Fields for Dynamic Air-Water\nSurfaces\nA Hierarchy of Approximate Theories\nIllustrations of Some Classical Partial\nTheories\nConcluding Observations","xxvii\nCONTENTS\n12.15 Simulation of the Reflectance of the Air-Water\nSurface by Mechanical Devices\nThe Central Idea of the Sea State Simulator\nErgodic Hypothesis\nThe Discrete Case\nThe Continuous Case\nSome General Observations on the Ergodic Cup\nDevice\nSea Simulator Devices Beyond the Ergodic Cup\n12.16 Bibliographic Notes for Chapter 12\nChapter 13\nOperational Formulations of Concepts for\nExperimental Procedures\n13.0 Introduction\n13.1 Operational Definitions of the Principal\nRadiometric Concepts\n13.2 Operational Definition of Beam Transmittance\nGeneral Two-Path Method\nGeneral One-Path Method\n13.3 Operational Definitions of Path Radiances\nand Path Functions\nOperational Formulation of Path Radiance\nOperational Formulation of Path Function\n13.4 Operational Definition of Volume Attenuation\nFunction\n13.5 A General Theory of Perturbed Light Fields,\nwith Applications to Forward Scattering Effects\nin Beam Transmittance Measurements\nIntroduction\nGeneral Representation of a Perturbed Light\nField\nLinearized Representation of Slightly\nPerturbed Light Fields\nApplication to Bright-target - Technique\nApplication to Dark-target Technique\nAn Outline of Possible Experimental Procedures\nof a in Perturbed Light Fields\nOrder of Magnitude Estimates\nSummary and Conclusions","xxviii\nCONTENTS\n13.6 Operational Definition of Volume Scattering\nFunction\no-Recovery Procedures\nDetermining the Volume Scattering Matrix\nin the Polarized Case\n13.7 Direct Measurement of the Volume Total\nScattering Function\nThe General Method\nObservations\nTwo Special Methods\nCylindrical Medium\nSpherical Medium\n13.8 Operational Definition of Volume Absorption\nFunction\nProcedures for Stratified Light Fields\nProcedures for Deep Media\nGeneral Global Method\nFurther Procedures for General Media\n13.9 Operational Procedures for Apparent Optical\nProperties\nThe Fundamental Irradiance Quartet\nDiscussion of the Distribution Functions\nDiscussion of the K-Functions\n13.10 Theory of Measurement of Local and Global\nR and T Properties\nExample 1: R and T Factors in Homogeneous\nPolarity-Free - Settings\nExample 2: Homogeneous Media with Polarity\nExample 3: Forward and Backward Scattering\nFunctions\nExample 4 : R and T Operators for Radiance\nGeneral Observations on Inverse Problems\nin Hydrologic Optics\n13.11 On the Consistency of the Operational\nFormulations\nOn the Relative Consistency of the Unpolarized\nand Polarized Theories of Radiative Transfer\n13.12 Bibliographic Notes for Chapter 13","xxix\nPREFACE\nIn this work I conclude my studies of radiative trans-\nfer theory begun in the monograph, \"Radiative Transfer on\nDiscrete Spaces.\" In that monograph the main goal was the\nfounding of the interaction principle underlying the phenom-\nenological theory of light in scattering-absorbing media. In\nthis treatise, I systematically construct from the interac-\ntion principle those basic laws and formulas of the disci-\npline of radiative transfer that pertain to hydrologic optics.\nThus while the first work was concerned with the gathering\ntogether of many single threads of theory converging on the\nnotion of the principle of interaction, the present study\nstarts with the principle as a base, deduces the superstruc-\nture of general radiative transfer theory, and applies it to\nthe special case of light in the sea. This task is essen-\ntially carried out in Chapter 3 and culminates in the classi-\ncal principles of invariance and in the equation of transfer\nfor radiance. Concurrent with this is the deduction of the\nexistence of the fundamental optical properties used in the\nequation of transfer, namely the volume attenuation and vol-\nume scattering functions. Some of the remaining chapters of\nthe book (Chapters 4, 5, 6, 7, 8, 11) are devoted to deduc-\ntions from the principles of invariance and the equation of\ntransfer of those laws of radiative transfer and those prop-\nerties of natural optical media which are particularly suited\nto the study of radiant energy transfer in the sea and other\nnatural bodies of water. Actually, many hydrologic optics\nprinciples discussed in this work can also describe radiative\ntransfer phenomena in general optical media, such as those\nencountered in both the astrophysical and geophysical (in-\ncluding industrial) settings. However these principles have\noften been deliberately phrased for use within the context of\nhydrologic optics in order to retain the concreteness and\npractical utility of the theory. The quest for generality\nwas fulfilled in the discrete-space monograph.\nIn completing the preceding task, I brought to a close\na long and almost circular conceptual odyssey which began for\nme during a summer eighteen years ago (1950) when I was a\nstudent at the Massachusetts Institute of Technology. I was\ngiven the problem of determining the reduction of visibility\nof submerged objects as seen along inclined paths of sight\nthrough the wind-crinkled, air-water surface. The odyssey\nwas 'circular' in the sense that my preoccupations in this\nfield began and ended essentially with the problem of radia-\ntive transfer through the wind-blown air-water surfaces of\nnatural hydrosols (Chapter 12). Between these end points con-\ncerned with the initial and final studies of this problem, I\ntravelled a conceptual journey which for long periods was","PREFACE\nXXX\noccupied with the search for the most basic principles and\nconcepts underlying the solution of this and related problems\nof light in the sea. As explained in the preface of the\nfirst work, that search was guided by a personal interest in\ncarrying the theory of hydrologic optics to its highest level\nof geometric and algebraic perfection.\nDuring the past eighteen years the theory was most in-\ntensively pursued within the period of seven years from 1953\nto 1960 and during a brief period around 1964-1965. The re-\nmaining periods of time were occupied at first with student\nstudies and later with writing, teaching, travels, and ap-\nplied and pure mathematical studies in other fields. In par-\nticular, the manuscript for the present work was first\ndrafted in rough outline in the spring of 1958. Successive\ndrafts were enriched as additional theory was created. The\nmotivations of these additions were through the experimental\nfindings of my colleagues and my own imperfect applications\nof the rough theory. The roots of the present work extend\nback to a series of lectures I gave on hydrologic and atmos-\npheric optics in the fall of 1953 and the spring of 1954,\nand earlier still to the joint work in 1950-1952 with Duntley\nsummarized in the first four chapters of \"The Visibility of\nSubmerged Objects.\" The final and main manuscript of the\npresent work was essentially completed in the summer of 1965,\nafter approximately 20 months of writing which was begun hard\non the heels of finishing my monograph. During this period\nlarge parts of Chapters 2, 3, 6, 7, and 12 were originated as\nthe writing proceeded. In general, every chapter had new\nmaterial of some kind added at this time. The present work\nthen lay dormant for nearly three years, awaiting final proof-\nreading, while I was occupied with new teaching and research\nresponsibilities. On recently re-reading the manuscript and\nteaching from parts of it, I find that the fundamental theory\nhas mellowed well; it has reached a stage of internal com-\npleteness which will be adequate to the needs of all advanced\nexperimental and theoretical work in the forseeable future.\nThose points in the present study where contact is made\nwith physical reality, in the form of useful illustrative\nexperimental data on the radiance of submerged light fields\nand in instructive listings of optical properties of various\nseas and lakes, are due principally to the labors of my col-\nleagues Dr. S.Q. Duntley and Mr. J.E. Tyler. Their key meas-\nurements of the basic radiometric quantities and optical prop-\nerties of these media provided some of the original impetus\ntoward my construction of the theory of hydrologic optics.\nThe construction was undertaken as an attempt to conceptually\nsort and order the many empirical laws of light in the sea\nwhich their probings uncovered. My indebtedness to these men\nactually is deeper than this, and I would like to record here\nthe following observations in this regard.\nTo Dr. Duntley I owe much of the support of my work dur-\ning all the past years through his various contracts with the\nBureau of Ships and the Office of Naval Research of the\nUnited States Navy. The early years were interspersed with\nconversations and working sessions in which I received from\nhim some of my first glimpses of a possible theory of","PREFACE\nxxxi\nhydrologic optics. In the summer of 1950 at the Diamond Is-\nland Experimental Station in Lake Winnipesaukee, New Hamp-\nshire he described his important empirical discovery of the\nelliptical hydrologic range law made during some underwater\nexperiments. The hint of theoretical order in that experi-\nmental polar plot of hydrologic range versus downward angle\nof sight inspired me subsequently to fathom first the physi-\ncal and then the mathematical laws underlying that phenomenon.\nThe ensuing summer was spent happily in my sun-baked cabin on\nthat tiny island as I tackled my first independent scientific\nstudies. These resulted in the deduction of the elliptical\nhydrologic range law and also the simplest radiance-propaga-\ntion laws for lines of sight through air-ruffled water sur-\nfaces and along inclined paths of sight through deep regions\nof seas and lakes. Duntley's influence on my studies occur-\nred not only in the experimental quarter, but also on first\nreading his distinguished contributions to the Schuster two-\nflow theory: I recall the train ride through New Hampshire\ncountryside from Boston which began that summer of 1950 and\nwhich is forever linked with the conceptual revelations ex-\nperienced as I read his two papers on \"Optical Properties of\nDiffusing Materials\" and \"The Mathematics of Turbid Media.\"\nThe first paper pointed the way toward the improvement of the\nSchuster two-flow theory. The latter paper was eventually to\nprovide an instance of the interaction principle in the form\nof Schuster's \"principle of self-illumination.\" A dozen\nyears were to pass and a score or more of distinct manifesta-\ntions of the principle of interaction were to be discovered\nbefore its universality was to become manifest in my mind.\nIt was also Duntley's exposition of L.V. King's integral equa-\ntion method and especially the closing remarks in the latter\npaper that eventually encouraged me to create the discrete\nspace theory of radiative transfer. This theory on the one\nhand retains the generality of the integral equation approach\nand on the other leads without modification to numerical de-\nterminations of light fields in general optical media. The\nrequisite procedure is given by the Categorical Analysis\nMethod in my monograph.\nI wish also to note in some detail the profound influ-\nence of the work of Tyler on my constructions of hydrologic\noptics theory. Unquestionably his experimental measurements\non the \"Radiance distribution as a function of depth in an\nunderwater environment\". was for me a watershed of at least a\ndozen incipient theoretical laws of hydrologic optics. It\nprovided, for example, the definitive experimental data\nneeded to verify L.V. Whitney's conjecture on the existence\nof \"characteristic diffuse light\" deep below the surface of\nevery natural optical medium and which belongs exclusively to\nthat medium regardless of the lighting conditions above its\nsurface. These findings encouraged my search for theoretical\nexpressions of the fundamental properties of real light\nfields far from the boundaries of deep optical media. It was\nalso Tyler's accumulation of data by means of ever more pre-\ncise radiometric measurements in oceans and lakes that led us\nboth to realize the inherent limitations of the classical\nSchuster two-flow (one-D) model of the light field in hand-\nling such data: his measurements of upward and downward ir-\nradiance flows, for example, were uncovering new kinds of","xxxii\nPREFACE\ndepth behavior of the diffuse attenuation and reflectance\nfunctions of such subtle and delicate forms that they lay far\nbeyond the descriptive powers of the classical theory. This\nstate of affairs eventually led me to formulate the theory of\ndirectly observable optical properties of light fields in\nreal stratified media. These formulas for directly observ-\nable properties were subsequently applied by Tyler and his\ncolleagues in various papers, and particularly in the \"Method\nfor obtaining the optical properties of large bodies of water.\"\nThe present account must also take cognizance of many conver-\nsations with Tyler on the puzzles of practical radiometry in\nthe sea. These discussions gave me insight into the needs of\nthe experimenter in hydrologic optics and for whom in turn\nChapters 9, 10, and 13 are specifically written. In the\ncourse of the years the contents of these chapters arose in\nvarious attempts to cast into a mathematically self-consis-\ntent array of operationally meaningful forms all the funda-\nmental concepts of radiative transfer in the sea, such as the\nvolume attenuation, scattering, absorption, and the diffuse\nattenuation functions for all radiometric concepts. These\nconcepts in other branches of radiative transfer, notably as-\ntrophysical optics, were either nonexistent or in the form of\nunrealizable mathematical abstractions of no use to one with\ndirect instrumental access to the interior of the optical\nmedium of interest; in our case, the sea. Finally, I grate-\nfully acknowledge that a large part of the writing of this\nwork was generously supported by portions of Tyler's National\nScience Foundation Grants (G 11668 and G 289).\nThe preceding description of the background of the pre-\nsent work has implicitly referred to the contents of all the\nchapters except the first two. The first chapter may serve\nas a self-contained 'short-course' on hydrologic optics. In-\ndeed it has been used as a base for the first course on 'Ra-\ndiative Transfer in the Sea' given at Scripps Institution of\nOceanography in the fall of 1967. Particular attention is\ndirected toward the three simple models for light fields in\nnatural waters given in Chapter 1. These models constitute\nthe minimal theoretical tools for anyone who enters the field\nof hydrologic optics and wishes to do productive work therein.\nIn particular for one who plans to do experimental studies,\nsome guidelines are necessary to first of all measure the\nquantities of hydrologic optics in a consistent manner and\nsecondly, to measure something that will be useful to others\nin the same field. These models and the constructs from\nwhich they are fashioned supply the requisite guidelines. As\none's needs for precision and comprehensiveness of concepts\nevolve, then the theoretical developments comprising the re-\nmaining chapters of the work will be of help in filling these\nneeds. Attention is also directed to the section of the\nfirst chapter dealing with practical nomographs for predict-\ning the range of visibility available to underwater swimmers\nin various natural hydrosols such as harbors, lakes and seas.\nThese nomographs are based on the work of Duntley, which\ncombines the properties of the human eye with one of the\nthree models of the light field referred to above. Also of\ngeneral interest are the many samples of magnitudes of light\nfields and optical constants found in natural waters. These","PREFACE\nxxxiii\nsamples are based mainly on the field work of Tyler, Duntley,\nand Jerlov and serve to fix one's intuition for the sizes of\nthe optical constants found in nature. This in turn allows\nintelligent derivations of new approximate formulas based on\nthe light field models alluded to above. Finally, the pres -\nence of Chapter 2 is almost self-explanatory, being concerned\nwith the scientific language of radiative transfer: geomet-\nrical radiometry. Students of geometrical radiometry may\nfind the various novel formulas and laws developed throughout\nthe chapter of independent interest. However, the chapter\nfinds its place in this work by providing the radiometric\nconcepts and formulations needed in the applications of the\ninteraction principle to hydrologic optics.\nThe main drafts were expertly typed by Mrs. Lynn White\nand by Mrs. Judith Marshall. Mrs. Marshall also assisted in\nthe preparation of various tables and graphs, and the typing\nof the final draft for photocopy.\nR.W.P.\nSan Diego\nDecember 1968\nThe final draft was completed while undertaking new re- -\nsearches in hydrodynamics with the Tsunami Research Effort\n(J.T.R.E.), which is part of the Environmental Research Lab\n-\noratories of the National Oceanic and Atmospheric Administra\ntion. I am grateful to the Director of J.T.R.E., Dr. Gaylord\nMiller, for making available the Graphic Arts facilities at\nthe Institute of Geophysics of the University of Hawaii, and\nparticularly to Mr. Brad Evans for his art work on the\nfigures.\nR.W.P.\nHonolulu\nJanuary 1972","CHAPTER 1\nINTRODUCTION TO HYDROLOGIC OPTICS\n1.0\nHydrologic Optics: Definition, Domain, and Desiderata\nAs the earth swings round the sun, it continuously turns its\natmosphere, its lands and its seas to face into the steady\ntorrent of energy streaming from that radiant star. of the\nnearly 65,000,000 watts of radiant power of all wavelengths\nemitted from each square meter of the sun's surface, about\n1,400 watts are incident on each square meter of the upper\nlevels of the earth's atmosphere directly facing the sun,\nthere to initiate and sustain the complex chains of meteoro-\nlogic and hydrologic events among which are the important\nbiologic links evolving in the atmosphere and the seas. In\nthe meteorologic domain, the radiant flux from the sun is\npartly absorbed to warm the earth's gaseous mantle so as to\ngenerate winds and habitable climes; and partly scattered so\nas to help grow plants and light the ways of the creatures of\nthe air and earth below. In the hydrologic domain the radi-\nant flux, when in sufficient abundance, is partly absorbed to\nhelp keep the seas and lakes and other natural hydrosols in\ntheir fluid state, and is partly scattered about in their\nupper levels so as to light the ways and help provide suste-\nnance for the creatures of these watery domains.\nHydrologic optics is the quantitative study of the in-\nteraction of radiant energy with hydrosols, especially the\nnatural hydrosols of the earth such as its seas, lakes, ponds,\nrivers, and bays. Hydrologic optics is part of a broader\ndiscipline known as geophysical optics which studies the com-\nmon physical and geometrical principles governing radiant en-\nergy fields in both the meteorologic and hydrologic domains.\nGeophysical optics together with astrophysical optics--in\nwhich the emission, absorption and scattering of radiant en-\nergy within general planetary and stellar atmospheres is of\nprimary concern - - fall under the aegis of radiative transfer\ntheory, which is defined as the quantitative study, on a\nphenomenological level, of the transfer of radiant energy\nthrough media that absorb, scatter, or emit radiant energy.\nRadiative transfer theory, in turn, is viewable as a logical\ndescendent of electromagnetic theory, and in this way hydro-\nlogic optics, and more generally radiative transfer theory,\nmay take its place among the theories of modern physics.\nThese interrelations are summarized in Fig. 1.1.","INTRODUCTION\nVOL. I\n2\nELECTROMAGNETIC THEORY\nINTERACTION PRINCIPLE\nGENERAL RADIATIVE\nTRANSFER THEORY\nASTROPHYSICAL\nGEOPHYSICAL\nOPTICS\nOPTICS\nHYDROLOGIC\nMETEOROLOGIC\nPLANETARY\nOPTICS\nOPTICS\nOPTICS\nOCEANOGRAPHIC\nLIMNOLOGIC\nOPTICS\nOPTICS\nFIG. 1.1 Hydrologic optics as a logical descendant of\nradiative transfer theory and electromagnetic theory.\nThe Problems of Hydrologic Optics\nThe theoretical and empirical studies comprising hydro-\nlogic optics arise in the attempts to answer several diverse\ntypes of questions such as the following. How much radiant\nenergy of a given wavelength is reflected from a sea or lake\nsurface, and how much penetrates this surface and reaches\neach depth of the sea or lake? How does the amount trans\nmitted depend on the surface winds and other factors affect-\ning the physical, geometric, and dynamic state of the moving\nsurface? Does the light penetrate the body of the ocean or\nlake in some general and predictable manner as regards depth\ndependence and directional dependence of the light distribu-\ntion? If so, what are the pertinent physical measurements\nthat must be made to facilitate such predictions? What ef-\nfects on the light field are engendered by the proximity of\nthe shores, bottoms and other boundaries of the hydrosols?\nWhat are the pertinent optical properties of natural hydro-\nsols by which oceanographers and limnologists can character-\nize these waters? How may these scientists usefully employ\nthese concepts in the pursuit of their special interests such\nas marine biology, geology, and hydrodynamics? How far can a","SEC. 1.0\n3\nDEFINITION, DOMAIN, DESIDERATA\ndiver or submariner expect to see a given submerged object\nas he maneuvers in the submarine world of blue-green lights\nand shadows? How far can one expect to communicate under-\nwater by means of given types of light sources such as lasers,\npoint sources, etc.? Of what significance is the polarized\nlight field to the denizens of the deep and to enterprising\nhumans interested in navigating through the submarine world\nby unconventional means? These summarize some of the basic\ntypes of questions with which hydrologic optics is concerned.\nThe questions have many variations and their resolutions are\noften of great difficulty, so that the theory of radiative\ntransfer which underlies hydrologic optics is often taxed\nto\nits limits in the attempts to provide quantitative or even\nqualitative answers. As the discussion proceeds, we shall\nmake clear the present status of the solutions to the general\nproblems listed above.\nThe Aims and Desired Goals of This Work\nIn this work we shall be concerned with the systematic\ndevelopment of the basic physical principles and mathematical\nprocedures of radiative transfer theory which have been found\neffective in solving the general types of problems cited\nabove. The reason for selecting the domain of hydrologic op-\ntics for specific study rather than meteorologic optics or\nany other branch of general radiative transfer rests simply\nin the fact that it is in this domain that most of the prac-\ntical experience of the author lies.\nIt should be emphasized at the outset that our primary\nconcern is with the principles of hydrologic optics rather\nthan the detailed numerical and experimental aspects of the\nstate of the art of the discipline. These latter procedures,\nas important as they are in the various stages of securing\nour knowledge, both theoretical and empirical, are in the\nlast analysis meaningful and efficacious only if they are\nbased on sound physical principles and mathematical tech-\nniques. Repeated direct experiences of the author in pur-\nsuing complete or partial solutions of problems of the types\nlisted above, have demonstrated the importance of having a\nwell-grounded knowledge of the principles of radiometry and\nradiative transfer theory during the search for the solutions.\nIt would seem to follow that anyone faced with similar prob-\n1ems and armed with a comparable battery of principles and\nlaws of the subject will also eventually find his way to his\nown desired experimental or theoretical goals. This, then,\nleads to the primary aim of the present work: to give a sys-\ntematic development of the fundamental principles and proce-\ndures of radiative transfer theory which may be employed by\nstudents of the subject in the pursuit of solutions of their\nparticular theoretical and experimental problems of geophys-\nical optics, and especially hydrologic optics. It has also\nbeen the experience of the author that both the theoretical\nand experimental practitioners of the arts of radiometry and\nradiative transfer are singularly independent individuals,\neach in his own way, and in view of this it would be somewhat\nfutile to preoccupy the potential student and researcher with","4\nVOL. I\nINTRODUCTION\nanything but the most pertinent and general principles and\nprocedures. This observation is cited to reinforce our aim\nenunciated above.\nThe Plan and Scope of This Work\nIt is in the nature of the theory of hydrologic optics\nthat the full founding and delineation of its basic princi-\nples is tantamount to a full founding and delineation of the\nbasic principles of radiative transfer theory itself. This\nfact rests on the observation that the physical-geometric\nproblem of completely describing the structure of the scat-\ntered light field in a sea or lake is just as complex a task\nas that of describing the light field in the atmosphere, or\nfor that matter in any real medium that emits or scatters\nlight. This realization dawned very early in the author's\nstudies of oceanographic and limnologic optics and in his\ntheoretical excursions into the problems of meteorologic op-\ntics. It was eventually realized that the appropriate direc-\ntion of study was not a problem-by-problem horizontal advance\nthrough the everyday jungle of examples, cases, and counter-\nexamples, but rather the direction required a sharp vertical\ntack, straight up into the heights of abstraction, from\nwhence one could most economically view the radiometric\nscenes spread out below from horizon to horizon. This at-\ntempt to escape into the thin air of general constructs and\nguiding principles was made as often as the exigencies of\ndaily problems and consultations would allow, and eventually\nas reports and papers accumulated, there emerged a pattern of\nprinciples and procedures which could be seen to apply to all\nthe special principles and special procedures accumulated to\nthat time. Interestingly, it was found that the abstract\nprinciples could be phrased and assembled using very meager\namounts of advanced mathematical machinery. This, coupled\nwith the author's classroom experience that the basic con-\nstructs of radiative transfer, namely radiant flux, scatter-\ning, absorption, volume, area, and length are all readily\nvisualizable, resulted in a theoretical framework which was\nreadily understood and applied once a small number of academ-\nic prerequisites had been dispatched, namely the equivalent\nof\na one year course in advanced calculus, which includes\nvector analysis, and first and second order ordinary differ-\nential equations.\nFor all these reasons it was decided in the planning\nstages of this work that its scope be widened to embrace,\nwhenever possible, the completely general principles of radi-\native transfer theory, and to attempt a systematic develop-\nment of the subject by starting from a single fundamental\nprinciple, namely that which eventually came to be called the\ninteraction principle (Sec. 3.2). For, it would be ineffi-\ncient and unesthetic to base a science on many seemingly un-\nrelated principles when it is possible to employ merely one.\nAccordingly, in Chapter 3, after a thorough grounding in geo\nmetrical radiometry, the reader is lead through a methodical\nconstruction program of general radiative transfer theory.\nThe elaboration of the details of this task will occupy most","DEFINITION, DOMAIN, DESIDERATA\nSEC. 1.0\n5\nof the remainder of the work, with several important chapters,\nincluded as integral parts of the main discussion, which are\ndevoted to the richer theoretical details made possible by\nadopting the plane-parallel settings indigenous to hydrologic\noptics.\nIt was found possible to adopt the preceding form of\ndevelopment of radiative transfer theory provided some care\nwas taken at the outset to equalize the backgrounds and in-\ntuitions of potential students of the subject. It is to such\nstudents and to the general reader that we devote this chap-\nter. In the following sections we shall acquaint these read-\ners with the general outlines of hydrologic optics by sup-\nplying representative radiometric examples of natural light\nfields and typical magnitudes of optical properties encoun-\ntered in natural hydrosols. We shall also present three of\nthe simplest models of light fields which are capable of des-\ncribing a very wide number of situations encountered in prac-\ntical hydrologic optics. We shall in addition illustrate the\nuse of these models by means of explicit deductions and cal- -\nculations. We shall also present graphs and tables based on\nthese models which have been found useful in practice. Then\nwith these introductory developments completed, we shall feel\nfree to start from scratch in Chapter 2 and proceed rigorous-\nly with the systematic construction of the modern theory of\nCHAPTER AND VOLUME INTERDEPENDENCE\n2\nI\nVOL.\nVOL.11\n3\n7\n8\n4\n5\n6\nVOL.IV\nVOL.III\n9\nVOL.V\nFIG. 1.2 Interdependence of the chapters of this work.","6\nINTRODUCTION\nVOL. I\nradiative transfer. The results will embody powerful exten-\nsions which appear to be capable of solving-- principle and\nin practice--every known current problem of applied radiative\ntransfer theory in the domains of the air and the sea.\nAs an aid in studying the present work Fig. 1.2 indi-\ncates the logical interdependence of the various volumes and\nchapters. Actually every chapter is connected in some way\nwith every other; however, some connections are stronger than\nothers, and these are shown in the diagram. Thus the prereq-\nuisite most essential to understanding a given chapter is the\nchapter (or chapters) which stand immediately above it via\nthe horizontal and vertical lines in the diagram. For exam-\nple Chapter 11 depends directly on 4,5, 7 and 10, while 6\ndepends directly only on 3. Furthermore, the chapters whose\ncontexts are developed on the level of general radiative\ntransfer theory (Fig. 1.1) are outlined in heavy boxes; those\nthat are more directly concerned specifically with hydrologic\noptics (or the theory of stratified plane parallel media) are\noutlined in the dashed boxes.\n1.1 A Primer of Geometrical Radiometry and Photometry\nAfter the solar radiant energy incident on the upper\nlevels of the atmosphere has rapidly percolated down through\nthe atmosphere and redistributed itself via scattering pro-\ncesses throughout the lower reaches and in the upper layers\nof the seas and lakes, its flow within these media assumes an\nintricate, and relatively steady geometric pattern. A parti-\ncularly useful mode of representation of this flow of scat-\ntered radiant energy is possible by means of the concepts of\ngeometrical radiometry, whose definitions and interrelations\nwe shall now briefly study. A relatively complete and de-\ntailed study of geometrical radiometry and photometric con-\ncepts is reserved for Chapter 2.\nThe Nature of Radiant Flux\nThe radiant energy streaming in from the sun is under-\nstood to be electromagnetic energy. The atomic radiative\nprocesses of the sun generate a wide range of frequencies (or\nwavelengths) of electromagnetic energy, only a small part of\nwhich is visible to the human eye, or detectable by human\nskin, or usable by the plants and animals of the earth. The\npart of the electromagnetic spectrum visible to normal human\neyes lies essentially in the range from 400 to 700 millimic-\nrons wavelength, the 400 mu light being deep blue-violet, the\n700 mu light being deep red, with all the colors of the rain-\nbow ranging continuously between these extremes. The wave-\nlength of electromagnetic energy evoking the greatest sensa-\ntion of brightness is the yellow-green at 555 mu under normal\ndaylight conditions. If radiant energy of wavelengths much\nless than 400 or much greater than 700 mu fall on normal re-\ntinas, there is relatively no conscious awareness of such an\nevent by the associated brain, though--in some extraordinary\ncases, some ultra violet (380 mu) and some infra red (780 mu)","SEC. 1.1\nRADIOMETRY AND PHOTOMETRY\n7\nphenomena are still within the range of detectability by the\nhuman visual organs. By and large, however, the human visual\nsensor system effectively samples and reacts to only the min-\nute portion of the whole outpouring of radiant energy by the\nsun between 400 and 700 mu--much in the way that a taut wire\nof given length and diameter resonates most sharply to a sin-\ngle acoustic frequency and less sharply to the frequencies in\na small interval surrounding the central frequency, outside\nof which the wire is essentially insensitive to the vibra-\ntions. Figure 1.3 depicts the place of the visible portion\nof the spectrum within the electromagnetic spectrum, along\nwith schematic diagrams of those portions of which we are a-\nware by means of various devices used to detect and measure\nradiant energy. (Current manufacturer's catalogs should be\nconsulted for precise details of individual devices.) Any\nobservable part of the electromagnetic spectrum, observable\nnot only as visible light but also by suitable technical\nmeans, falls under the aegis of geometrical radiometry.\nThe central construct of geometrical radiometry is\nradiant flux which we define generally as the time rate of\nflow of radiant energy of given wavelength (or frequency) a-\ncross a given surface. (It has dimensions of (radiant) ener-\ngy per unit time per unit frequency.) Thus radiant flux is a\ntime density* of radiant energy. For our present purposes\nand in the exposition of radiative transfer theory, we may\nimagine the flow of radiant energy to be in the form of mu-\ntually non-interfering swarms of tiny colored particles\nwhich we call photons. While this may not correspond in all\naspects to physical reality, it nevertheless is a helpful\nconstruct in practical work. Each photon contains a well\ndefined amount hv-- a quantum of radiant energy associated\nwith its color, or frequency V. This means of picturing ra-\ndiant energy for the purposes of geometrical radiometry is\nquite useful and correct within the modern framework of phys-\nics. It will make the exposition of the notions of geomet-\nrical radiometry a relatively simple task, and the visuali-\nzations of the various concepts an almost trivial matter. In\nthe terminology of electromagnetic theory, we shall work with\nelectromagnetic fields produced by mutually incoherent\nsources and which are studied on a macroscopic level, i.e.,\nwhere the dimensions of the detectors are very large compared\nto the observed wavelengths.\nThe Unpolarized-Flux Convention\nThe radiant flux always will be assumed unpolarized,\nunless specifically noted otherwise. This will result in\nsimplified working formulas of relatively great practical val-\nue and of adequate accuracy in the pursuit of most applica-\ntions of hydrologic optics. Whenever it is necessary to in-\ndicate how the theory may be elevated to the polarized level,\n*\nBecause most of our discussions center on an arbitrary fre-\nquency (or wavelength) of radiant flux, the reference to the\n\"per unit frequency\" part of the dimension of radiant flux\nwill be omitted, unless specifically noted otherwise.","VOL. I\nINTRODUCTION\n8\n106m\n105 mu\n104 mu\nINFRARED\n103 mu\n1000mu\n(3x1014 cycles)\n800-red\n600\nsec.\ngreen VISIBLE\n400\nblue\n200\n102 mu\n100mu\n(3x1015 cycles)\nULTRAVIOLET\nsec.\n10 mu\nFIG. 1.3 The electromagnetic spectrum and the ranges of\nsome typical radiant energy detector domains.","RADIOMETRY AND PHOTOMETRY\n9\nSEC. 1.1\nnotes will be made to that effect. The general theory of po-\nlarized radiative transfer is outlined in Sec. 114 of Ref.\n[251], and the problem of the relative consistency of the po-\nlarized and unpolarized theories is examined in Sec. 13.11,\nbelow.\nGeometrical Channeling of Radiant Flux\nOnce the nature of radiant flux is clarified, as above,\nthe descriptions of the remaining concepts, theorems and pro-\ncedures of geometrical radiometry are essentially geometric\nin nature. There are only two distinct, ideal modes of des-\ncribing a flow of particles past a point in three dimensional\nspace, and these are shown in Fig. 1.4. In part (a) of the\nfigure a parallel flow of photons is described in terms of\nthe passage of particles through a small region S on a plane\nnormal to the flow around a point p on the plane. A comple-\nmentary mode of the flow is in terms of the passage of parti-\ncles through a small set D of directions around a given di-\nrection E and through the point p. Considering these two\nmodes in a given flow of photons, let P(S) and P(S) be the\nradiant fluxes in each of these cases, with A(S) the area of\nS and s(()) the solid angle content of the bundle D of direc-\ntions. Further, let the central direction 5 of the bundle D\nbe normal to S at p. Then we write:\n\"P(S)/A(S)\" for the area density of radiant flux\n\"P(D)/8(D)\" for the solid angle density of radiant\nflux\n(a)\n(b)\nD\np\nFIG. 1.4 Two geometric modes of describing radiant flux.","10\nINTRODUCTION\nVOL. I\nIt is convenient in geometrical radiometry to call P(S)/A(S)\nsimply a (radiant) flux density and P(D)/s(D) a (radiant)\nintensity.\nThese are the two basic modes of conceptually channel-\ning the flow of photons in space or matter. There is an im-\nportant third mode which is the result of the direct union of\nthese two modes. If we reconsider the setting of Fig. 1.4\nand imagine a narrow bundle of directions D around a central\ndirection E normal to S at each point p of S, then there\nwould be an associated flow P(S,D) of radiant energy across\nthe combined set S X D of the surface set S and the direction\nset D. We write:\n\"P(S,D)/A(S)s(D)\" for the phase density of radiant\nflux\nThe term \"phase density\" is simply a convenient descriptive\nterm for the combined areal and directional densities, and it\ncan be related to the phase space concept of classical sta-\ntistical mechanics, though there is no need to do so here.\nThe conventional term for phase density of radiant flux, the\none we adopt for use in this work is radiance; it is radiance\nwhich is used to describe the monochromatic brightness of\nradiant flux.\nOperational Definitions of the Densities\nAn operational definition of radiance and its companion\ndensities is effected by means of a radiant flux meter, de-\npicted schematically in (a) of Fig. 1.5. A radiant flux\nmeter forms the heart of the radiance meter, as shown in (b)\nof Fig. 1.5, and may embody any one of several means of meas-\nurement of radiant flux, such as photoconductive, photoemis-\nsive, or photovoltaic devices (see Sec. 2.1). Before the ra-\ndiant flux reaches the collecting surface S of the radiance\nmeter, it is filtered to the desired wavelength and is also\nconfined to flow onto S about point x through a narrow cir-\ncular conical bundle D of directions whose central direction\nE is normal to S. A good radiance meter will have D so that\nSo (D) is as small as practicable. A magnitude of (D) < 1/30\nsteradians serves well for most geophysical optics tasks. If\nthe reading of the radiant flux meter is P(S,D) when it is\nlocated at X and oriented by E (see Fig. 1.5), then the as-\nsociated radiance is P(S,D)/A(S) ( (D), which we can denote by\n\"N(x,5)\". Here \"x\" denotes where the flow is, and \"E\" de-\nnotes its direction. The associated radiant intensity is\nP(S,D)/S(D) and the radiant flux density is P(S,D)/A(S).\nThese operational definitions reduce to a practical level the\nideal situations pictured in Fig. 1.4. They are ideal be-\ncause in (a) of Fig. 1.4 the flow was assumed to be along a\nsingle direction and in (b) the flow was assumed to be through\na single point. The operational definitions give workable\napproximations to these ideals and form the basis for a rigor-\nous transition to the ideal limit, which will be made in\nChapter 2.","SEC. 1.1\nRADIOMETRY AND PHOTOMETRY\n11\nphotosensitive surface (s)\nflux\nE\nRADIANT FLUX\nX\n(a)\nMETER\ndial\n(b)\ncollecting surface (S)\nflux o\nfilter\nincident direction\nRADIANCE\nbundle (D)\nMETER\nX\nshading tube\nFIELD RADIANT\nFLUX P(S,D)\nD\n(c)\nX.\nS\nE\nIRRADIANCE\nP(S=(8)) = H(x,E)\nS\n(d)\nIN (E)\nA(S)\nFIELD INTENSITY\nP(S,D) = J(x,E)\n(e)\nSL(D)\nS\nDL\n(f)\nFIELD RADIANCE\nP(S,D) = N(x,E)\nS\nA(S)S((D)\nX\n&\nFIG. 1.5 Operational definitions of the radiometric con-\ncepts.","12\nINTRODUCTION\nVOL. I\nField and Surface Interpretations of Radiant Flux\nand its Densities\nIn Fig. 1.4 one important fact about the radiant flux\nwas omitted, namely its sense of flow. In practice we often\nfind it useful to distinguish between the flow of radiant en-\nergy onto a surface S and from the surface S. When we do so,\nthe three central densities introduced above each have either\none of the two possible interpretations, according as the ra-\ndiant flux comprising the density is viewed as flowing onto\nor from a surface. When radiant flux comes from the radio-\nmetric field and falls onto the collecting surface S of the\nradiance meter we call the associated radiance the field ra-\ndiance. When the radiant flux is seen to leave a surface\n(either real or imaginary) for the surrounding radiometric\nfield we use the term surface radiance. Similarly for radi-\nant flux density: when radiant flux falls onto a surface we\nspeak of the radiant flux density as the irradiance of the\nflux at a point, and when the radiant flux density leaves S,\nwe speak of the radiant emittance of the radiant flux at a\npoint. Similarly also for (radiant) intensity: we have sur-\nface (radiant) intensity and field (radiant) intensity. The\nparenthesized \"radiant\" indicates that this adjective can be\nomitted when radiant flux is understood to be the flux of in-\nterest.\nOperational Definitions of Field and Surface Quantities\nWe may summarize the preceding definitions in parts (c)-\n(f) of Fig. 1.5. These diagrams emphasize the operational\nprocedures used to measure the various quantities in actual\nradiometric environments.\nThus field radiant flux can be defined over the surface\nS of the radiant flux meter for an incoming bundle D of direc-\ntions. The heavy arrows give the general sense of the flow.\nWhen the meter is oriented so that at point X the inward unit\nnormal to its collecting surface is E, and D is opened up to\nbe the hemisphere E(5) of all directions E' such that\nE.E'= COS 0 0 0 then by definition we measure the irradiance\nat X for the orientation E of the collector. The field (ra-\ndiant) intensity J(x,5) and the field radiance N(x,E) are de-\nfined analogously. It is important to emphasize that the so(D)\nin the latter two cases should be on the order of 1/30 of a\nsteradian or smaller for best results. The 'surface' coun-\nterparts to the preceding 'field' quantities may be pictured\nby reversing the flux arrows in parts (c) to (f) of Fig. 1.5.\nFigure 1.6 shows the details of how a surface radiance\nmay generally be assigned to a real or imaginary surface. We\nuse the radiance invariance law (Sec. 2.6) to assign to the\ndirection E at point p on S the radiance N(x, 5) when p is\nviewed by a radiance meter oriented as shown. This is a con-\nsistent assignation since the radiance-invariance law states\nthat for a fixed E, N(x,5) is independent of y along a","SEC. 1.1\nRADIOMETRY AND PHOTOMETRY\n13\np\nn\ny\nFIG. 1.6 The method of assigning radiances to real or\nimaginary surfaces.\nvacuous path between X and p. In this way each E at p in the\noutward hemisphere E(n) of directions at p can be assigned a\nradiance.\nA useful property of irradiance is the cosine law,\nwhich follows directly from the present operational consider-\nations. Fig. 1.7 shows a thin collimated steady stream of\nphotons incident normally on a small hypothetical plane sur-\nface S. If P (s, D) is the radiant flux produced on S by this\nstream, then this same flow P (S' , D) exists across the surface\nS' whose unit normal is tilted 0' from the direction of the\nstream. The connection between the two irradiated areas is:\nA(S') COS 0' = A(S)\nD\nS\nX\ns'\n0\nFIG. 1.7 Deriving the cosine law for irradiance.","14\nINTRODUCTION\nVOL. I\nRADIOMETRIC CONCEPTS\nIRRADIANCE H\n(onto a surface)\nAREA\nDENSITY\n(watt/m²)\nRADIANT\n(from a surface)\nEMITTANCE W\nFIELD\n(onto a surface)\nINTENSITY J\nELECTRO\nRADIANT\nRADIANT\nSOLID ANGLE\nMAGNETIC\nENERGY U\nFLUX P\nDENSITY\nENERGY\nSURFACE\n(watt/sr)\n(from a surface)\nINTENSITY J\n(joules)\n(watt)\nFIELD\n(onto a surface)\nRADIANCE N\nPHASE\nDENSITY\n(watt/m2sr) 2\nSURFACE\n(from a surface)\nRADIANCE N\nFIG. 1.8 Logical lineage of the radiometric concepts.\nHence the connection between the irradiances on S , and S pro-\nduced by the stream is:\nH(x,5') = P(S',D)= = P(S,D) cos 0' H(x,5)\n= cos e'\nA(S')\nA(S)\nThat is ,\nH(x,5') = H(x,5) cos e'\nwhich is a form of the cosine law for irradiance (the general\nlaw is given in Sec. 2.8). The companion law to this for the\nradiant emittance of S' is:\nW(x,5') = W(x,5) cos 0'\nSummary of Concepts and Some Principal Formulas\nof Geometrical Radiometry\nA schematic diagram of radiometric concepts, developed\nin the manner described above, which summarizes the geometric\nderivatives of radiant energy, along with their mks units,\nand current standard symbols, is given in Fig. 1.8. The\nnames of the six concepts above, and their designating sym-\nbols may come and go with the years, but the logical lineage","SEC. 1.1\nRADIOMETRY AND PHOTOMETRY\n15\nof the concepts depicted above, with their tap root in the\nconcept of radiant energy and indicated branching structures,\nwill withstand the rigors of time. For while the names in the\nboxes are transient conventions, the arrangement of the boxes,\nand the underlying concepts for which the boxes stand are\nsimply manifestations of the way we naturally view radiant\nenergy and the flow of radiant energy in space and time. In\nthis sense the indicated conceptual scheme in Fig. 1.8 is im-\nmutable. The full developments of the analytical connections\namong the radiometric concepts are not needed in this intro-\nductory chapter, and are reserved for Chapter 2. However, a\nbrief survey of some of the main formulas of geometrical ra-\ndiometry is given here for convenient reference during the\nremainder of this chapter's discussions.\nThe primary concept of geometrical radiometry in prac-\ntice is the phase density concept, namely radiance. We find\nit possible to describe all other concepts in terms of this\ndensity. Thus for example in the case of the flux density\nconcept:\nH(x,5)\n=\n(with field\n(1)\nradiance)\nW(x,5)\nN(x,E)E()\n=\n(with surface\n(2)\nradiance)\nH(x,5) is the irradiance at X on a surface whose inward nor-\nmal is the direction E. The basis for (1), (2) rests in the\ncosine law for irradiance and the possibility of the linear\nsuperposition of radiant fluxes. The symbol \"E(E)\" stands\nfor the hemisphere of all directions E' such that 51.5 > 0,\n(hence E(-5) is the hemisphere of all directions E' such\nthat E'. (-E) > 0, i.e. E'. E < 0). Here \"ds(E')\" is short for\n\"sin 0' de' do' where (0',0') define E' in some reference\nframe. Of course 5'.5 is the scalar or dot product of the\ndirections E' and E. The representations of the solid angle\ndensity in terms of radiance are not needed at present and\nmay be found, along with many related concepts, in Sec. 2.9.\nWe shall also find it convenient to introduce at this time\ntwo cousins of the flux density concept, namely scalar and\nvector irradiance, defined, respectively, by writing:\n\"h(x)\"\nfor\n(watt/m2)\n(3)\nand :\n\"H(x)\"\nfor\nN(x,5')E'\nda(E')\n(watt/m2)\n(4)\nHere (1) is the set of all unit vectors (directions) in euclid-\nean three space. The scalar irradiance h(x) is the total ra-\ndiant flux per square meter coursing through point X in all\ndirections. It is related to radiant energy per cubic meter","VOL. I\nINTRODUCTION\n16\nu(x) (the radiant density: Joules/m3 by means of the for-\nmula:\nv(x) u(x) = h(x)\n(5)\nwhere v(x) is the speed of light at x (in m/sec). The quan-\ntity H(x) is a vector; the indicated equation is really three\nequations: one for each of the x, y, Z components of H(x),\nas given by the corresponding components of is . The vector\nH(x) also has units of watts per square meter: its magnitude\nis the maximum net irradiance attainable as one samples all\npossible directions E of flow about X. The direction of H(x)\ndefines this direction of maximum net irradiance. The net\nirradiance H(x,5) at x in the direction E is defined as\nH(x,5)-H(x,-5); see Sec. 2.8 for complete details.\nIt will be necessary in this introductory chapter to\nalso consider hemispherical scalar irradiance, defined by\nwriting:\nN(x,5') da(E') (watt/m²)\n(6)\n\"h(x,5)\"\nfor\nE(E)\nN(x,5') do(E') (watt/m²)\n(7)\n\"h(x,-E)\"\nfor\nE(-E)\nwhere, by (3),\nh(x) h(x,5) + h(x,-5)\n(8)\nfor every E in E. A convenient terrestrial reference frame\nin hydrologic optics is that depicted in Fig. 1.9. We will\noften use the special case of (6), (7) where E = k, and we\nshall write\n\"h(z, ))\" for h(p,+k)\n(9)\nwhere we retain only the depth variable Z of the usual\n(x, y, z) - coordinates of the point p. Corresponding to h(z,+)\nwe have the companions from (1) in which E = +k; we write\n\"H(z,+)\" for H(p,+k)\n(10)\nIrradiances associated with plus signs are upwelling (or up-\nward) irradiances; those with minus signs are downwelling (or\ndownward) irradiances. A11 these irradiances have units of\nwatt/m². In natural hydrosols H(z,+) can be measured by\nhorizontal flat plate collectors, while (z,+) can be meas-\nured by spherical collectors, suitably shielded (see Sec.\n2.7). Some useful special cases of the preceding formulas\nare the following.\nLet N(x,5) be uniform, i.e., independent of E at some\nx and of magnitude N; then by (1)","SEC. 1.1\nRADIOMETRY AND PHOTOMETRY\n17\nZ\nE'=(a,B,Y)\nupward direction\nB=sing'sin\ndownward direction\ny=cos8'\n(a2+B2+y2=1\nk\n0\n0\ny\nX\nE\np= (x,y,z)\nZ\n(measured positive\ndownward)\nFIG. 1.9 The standard terrestrially-based coordinate sys-\ntem in hydrologic optics.\nH(x,E)= N (E) N 2 /2 0' sin e'\nde'do'\n= N\n(11)\nwhich holds for all E at X. . The computation was made with\nthe k axis momentarily shifted parallel to E. Further, from\n(2), in the same way:\nW(x,5) =\n(12)\nfor all E at X. Next, by (3):\nh(x)= N lo(E') N TT e'=0 sin 0 1 de do'\n= 4 NN\n(13)","INTRODUCTION\nVOL. I\n18\nBy (4)\n(14)\n= N = 0\nBy (6)\nh(x,) = 2 NN\n(15)\nObserve the effect of the cosine in the integrand: for a un-\niform radiance distribution at x, h(x,5)=2H(x,E) , for every\nE. Further examples are given in Sec. 2.11.\nn2-Law for Radiance\nWe mention in passing an important law of geometrical\nradiometry concerning radiance: If P is an arbitrary photon\npath through a transparent optical medium within which the\nindex of refraction n varies continuously with location, then\nphoton flux along the path P having radiance N moves such\nthat N/n2 is invariant along the path (cf. Sec. 2.6). This\nis the n2-law for radiance.\nThe Bridge to Geometrical Photometry\nThe conceptual bridge from geometrical radiometry to\ngeometrical photometry is built on the empirical fact that\nnot all wavelengths of radiant flux invoke the same sensation\nof brightness in the human eye. The green-yellow wavelength\n555 mu is the brightest. In fact one would require, e.g.,\nabout 2 watts of blue-green light of 510 mu or 2 watts of\norange light of 610 mu to produce the same sensation of\nbrightness as one watt of green-yellow light of 555 mu. The\nphotopic luminosity curve depicted in Fig. 1.10 summarizes a\nquantitative measure y(1) of the brightness-sensation - produc-\ning capabilities of a wavelength 1 in the electromagnetic\nspectrum. Observe that for wavelengths A below 400 mu and\nabove 700 mu, electromagnetic radiation no longer is seen by\nnormal human eyes. A fuller discussion of this curve is gi-\nven in Sec. 2.12. See also Sec. 1.8.\nThe conversion rule from a radiometric concept to its\nphotometric counterpart is based on the photopic luminosity\ncurve and is given as follows:\nLet be any radiometric concept (e.g., U, P, H, W, J,\nor N) which is defined over the electromagnetic spectrum.\nThen the photometric concept L (namely Q, F, E, L, I, or B,\nrespectively) associated with R is given by\nL= 680/ Q(A) ((1) di","SEC. 1.1\nRADIOMETRY AND PHOTOMETRY\n19\n1.0\n0.8\n0.6\n0.4\n0.2\no\n700\n400\n500\n600\n(MILLIMICRONS)\nFIG. 1.10 The photopic luminosity function.\na\nhas units watt/(*), then L has units lumen/(*), , where\nIf\n\"(*)\" stands for (meter) or (steradian) or various permissible\ncombinations of these geometrical units. For example,\nB(x,E)= 680 di ,\nlumens/m² sr\nThis gives the luminance (loosely, the \"brightness\") produced\nby a given sample of radiance. This is what, in essence, we\ncan see as a result of the radiant flux of photons at x in\nthe direction E. Again, for example, illuminance is:\n, lumens/m²\ndi\nThe logical interrelations among the photometric concepts pre-\ncisely parallel those of radiometry. . Thus, starting with lu- -\nminous energy Q, which, according to the rule above, we de-\nfine as:","VOL. I\nINTRODUCTION\n20\nPHOTOMETRIC CONCEPTS\nILLUMINANCE E\n(onto a surface)\nAREA\nDENSITY\n(lumen/m2)\nLUMINOUS\n(from a surface)\nEMITTANCE L\nFIELD\n(onto a surface)\nINTENSITY I\nELECTRO-\nLUMINOUS\nLUMINOUS\nSOLID ANGLE\nMAGNETIC\nENERGY Q\nFLUX F\nDENSITY\nENERGY\nSURFACE\n(from a surface)\n(lumen/sr)\nINTENSITY I\n(talbots)\n(lumens)\nFIELD\n(onto a surface)\nLUMINANCE B\nPHASE\nDENSITY\nSURFACE\n2\n(lumen/m2sr)\n(from a surface)\nLUMINANCE B\nFIG. 1.11 Logical lineage of the photometric concepts.\ndi\nwe then can construct a diagram similar to that in Fig. 1.8.\nThis is shown in Fig. 1.11. Consequently, everything we can\nsay about the geometrical properties of the radiometric con-\ncepts, we can also say about the corresponding properties of\nphotometric concepts.\nWe mention in passing some classical alternate sets of\nphotometric units:\nfoot candle = 1 lumen/ft2 (area density of flux)\n1\n(16)\n1 candela = 1 lumen/sr (solid angle density of flux) (17)\n1 (centimeter) lambert = 1/2 lumen/cm 2 sr\n(phase\n(meter) lambert = 1/2 lumen/m² sr\n1\ndensity\n(18)\nof flux)\n1 (foot) lambert lumen/ft2\nsr","SEC. 1.1\nRADIOMETRY AND PHOTOMETRY\n21\nFrom (17) we can compactly express luminance generally in\nterms of candelas/m² when using the mks system (the preferred\nsystem). The lambert unit arises as follows: let a surface,\nwhich has both unit reflectance with respect to irradiance for\neach wavelength and also a directionally uniform reflected\nradiance for each wavelength, be called a perfectly diffusing\nsurface, for short. By definition, a perfectly diffusing sur-\nface irradiated by one lumen has a luminance of one lambert.\n(Use Eq. (12) ) However, the conversion rules above in (18)\nare by convention now used under arbitrary directional and re-\nflectance conditions.\nThus we have the general rule: To convert B(x,5) lu-\nmens/m2 sr to meter lamberts, multiply B(x,5) by IT. (This fol-\nlows from the fact that as defined above the meter lambert is\nabout 1/3 of a lumen/m² sr; so it takes about 3 meter lamberts\nto every lumen/m2 sr to describe the same scene.)\nWith due respect to the historical origins of the pre-\nceding terms, it is felt that the continued employment of\n\"foot candle\" and \"lamberts\" will serve no logical purpose.\nTheir mention here simply serves to keep open the passageway\nto the classical literature of photometry and radiative trans-\nfer theory to which we must refer now and then during this\nwork. New students are advised to use the lumen, meter, ste-\nradian system of units in photometry, along with the watt,\nmeter, steradian system in radiometry in their future studies.\nA convenient abbreviated mks unit of radiance is the (unra-\ntionalized) * herschel:\n1 herschel = 1 watt/m2 2 sr\n(19)\nand an mks unit of luminance is the (unrationalized) blondel:\n1 blondel = 1 lumen/m2 sr\n(20)\nThese abbreviations should be used only when the sheer fre-\nquency of mention of \"watt/m2 2 sr\" or \"lumen/m2 2 sr\" becomes so\ngreat in a given discussion that facile communication is im-\npaired; otherwise they simply should be spelled out in full\nusing watts, meters and steradians. Further discussion of\nthe foundations of photometry is given in Sec. 2.12.\nAn unrationalized radiance (or luminance) unit is one for\nwhich a uniform radiance distribution of magnitude N produces\nan irradiance of TN. A rationalized unit would associate to\na uniform N the irradiance N. An unrationalized radiance\nunit is thus logically simpler than a rationalized unit. The\nterm \"rationalized\" here means \"removed m-factor\". It is ir-\nrational to rationalize radiance units just because it is too\ntiresome to carry around a T-factor which arises in calcula-\ntions with radiance distributions which in fact do not occur\nin practice in real environments in the first place! (namely\ndirectionally uniform distributions).","22\nINTRODUCTION\nVOL. I\n1.2 A Survey of Natural Light Fields\nThe intricate chain of radiative transfer processes\nwithin the air and seas of the earth begins with the influx\nof solar radiant energy at the upper levels of the atmosphere\nand partially ends in the depths of the seas and lakes. We\nshall now briefly survey the main features of the light field\nin the meteorologic and hydrologic domains. We conduct the\nsurvey with the purpose of establishing the general orders of\nmagnitudes of the set of radiometric phenomena in natural op-\ntical media which the theory of radiative transfer has been\nevolved to describe and predict.\nThe Solar Constant\nThe solar (irradiance) constant is the total irradiance\nproduced by solar radiant energy of all wavelengths at a\npoint located outside the earth's atmosphere at the mean dis-\ntance of the earth from the sun and on a plane normal to the\ndirection of the sun's center:\nsolar (irradiance) constant = 1396 watt/m2\n= 2.002 gm cal/cm2min\n(1)\nwhere\n1 joule = 0.2389 gm cal\nThe quantity (1) is based on the results summarized by John-\nson [128], and actually pertains to wavelengths in the range\n220 to 7000 mu. For a survey of solar constant measurements\nand some theoretical bases for them, see [296]. . Table 1\ngives a wavelength by wavelength analysis of the solar (irra-\ndiance) constant in watts/m² 2 xmillimicron. In the table, p(X)\nis the percentage of the total solar constant included in the\nwavelength range from 0 to 1. It is interesting to note that\nthis distribution of H(X) with l is very close to the radiant\nemittance curve of a 6000°K complete radiator. The solar\n(illuminance) constant, i.e., the photometric counterpart to\nthe solar (irradiance) constant is obtained by computing\nE = 680 di\n(2)\nin accordance with the general rules of photometry laid down\nin Sec. 1.1 1. We find:\nsolar (illuminance) constant = 136,700 lumens/m²\n(3)\n= 12,700 footcandles","SEC 1.2\nNATURAL LIGHT FIELDS\n23\nTABLE 1\nSolar Spectral Irradiance Data\nWavelength in millimicrons. H(X) in watts/m2mu.\n1\nH(X)\np(1)\nl\nH(X)\np(x)\nl\nH(X)\np(x)\n220\n0.030\n0.02\n420\n1.92\n11.7\n640\n1.66\n42.1\n225\n0.042\n0.03\n425\n1.89\n12.4\n650\n1.62\n43.3\n230\n0.052\n0.05\n430\n1.78\n13.0\n660\n1.59\n44.5\n235\n0.054\n0.07\n435\n1.82\n13.7\n670\n1.55\n45.6\n240\n0.058\n0.09\n440\n2.03\n14.4\n680\n1.51\n46.7\n245\n0.064\n0.11\n445\n2.15\n15.1\n690\n1.48\n47.8\n250\n0.064\n0.13\n450\n2.20\n15.9\n700\n1.44\n48.8\n255\n0.10\n0.16\n455\n2.19\n16.7\n710\n1.41\n49.8\n260\n0.13\n0.20\n460\n2.16\n17.5\n720\n1.37\n50.8\n265\n0.20\n0.27\n465\n2.15\n18.2\n730\n1.34\n51.8\n270\n0.25\n0.34\n470\n2.17\n19.0\n740\n1.30\n52.7\n275\n0.22\n0.43\n475\n2.20\n19.8\n750\n1.27\n53.7\n280\n0.24\n0.51\n480\n2.16\n20.6\n800\n1.127\n57.9\n285\n0.34\n0.62\n485\n2.03\n21.3\n850\n1.003\n61.7\n290\n0.52\n0.77\n490\n1.99\n22.0\n900\n8.95\n65.1\n295\n0.63\n0.98\n495\n2.04\n22.8\n950\n0.803\n68.1\n300\n0.61\n1.23\n500\n1.98\n23.5\n1000\n0.725\n70.9\n305\n0.67\n1.43\n505\n1.97\n24.2\n1100\n0.606\n75.7\n310\n0.76\n1.69\n510\n1.96\n24.9\n1200\n0.501\n79.6\n315\n0.82\n1.97\n515\n1.89\n25.6\n1300\n0.406\n82.9\n320\n0.85\n2.26\n520\n1.87\n26.3\n1400\n0.328\n85.5\n325\n1.02\n2.60\n525\n1.92\n26.9\n1500\n0.267\n87.6\n330\n1.15\n3.02\n530\n1.95\n27.6\n1600\n0.220\n89.4\n335\n1.11\n3.40\n535\n1.97\n28.3\n1700\n0.182\n90.83\n340\n1.11\n3.80\n540\n1.98\n29.0\n1800\n0.152\n92.03\n345\n1.17\n4.21\n545\n1.98\n29.8\n1900\n0.1274\n93.02\n350\n1.18\n4.63\n550\n1.95\n30.5\n2000\n0.1079\n93.87\n355\n1.16\n5.04\n555\n1.92\n31.2\n2100\n0.0917\n94.58\n360\n1.16\n5.47\n560\n1.90\n31.8\n2200\n0.0785\n95.20\n365\n1.29\n5.89\n565\n1.89\n32.5\n2300\n0.0676\n95.71\n370\n1.33\n6.36\n570\n1.87\n33.2\n2400\n0.0585\n96.18\n375\n1.32\n6.84\n575\n1.87\n33.9\n2500\n0.0509\n96.57\n380\n1.23\n7.29\n580\n1.87\n34.5\n2600\n0.0445\n96.90\n385\n1.15\n7.72\n585\n1.85\n35.2\n2700\n0.0390\n97.21\n390\n1.12\n8.13\n590\n1.84\n35.9\n2800\n0.0343\n97.47\n395\n1.20\n8.54\n595\n1.83\n36.5\n2900\n0.0303\n97.72\n400\n1.54\n9.03\n600\n1.81\n37.2\n3000\n0.0268\n97.90\n405\n1.88\n9.65\n610\n1.77\n38.4\n3100\n0.0230\n98.08\n410\n1.94\n10.3\n620\n1.74\n39.7\n3200\n0.0214\n98.24\n415\n1.92\n11.0\n630\n1.70\n40.9\n3300\n0.0191\n98.39","VOL. I\n24\nINTRODUCTION\nTABLE 1 (Continued)\n1\nH(X)\np(x)\np(x)\nA\nH(X)\np(X)\nl\nH(X)\n3400\n0.0171\n98.52\n4400\n0.0067\n99.29\n4900\n0.0044\n99.48\n3500\n0.0153\n98.63\n4500\n0.0061\n99.33\n5000\n0.0042\n99.51\n3600\n0.0139\n98.74\n4600\n0.0056\n99.38\n6000\n0.0021\n99.74\n3700\n0.0125\n98.83\n4700\n0.0051\n99.41\n7000\n0.0012\n99.86\n3800\n0.0114\n98.91\n4800\n0.0048\n99.45\n3900\n0.0103\n98.99\n4000\n0.0095\n99.05\n4100\n0.0087\n99.13\n4200\n0.0080\n99.18\n4300\n0.0073\n99.23\n(From [128], by permission)\nBy dividing the solar constant by the approximate solid\nangle subtense of the sun at the mean distance of earth from\nsun, Sl= 6.8 x 10 steradians, we obtain the approximate solar\nradiance and luminance constants:\nN = 2 x 10' watts/m2sr\n(3a)\nB=2 x 10 9 lumens/m²s\nGeneral Irradiance Levels at Earth's Surface\nThe irradiance levels at the earth's surface can vary\nrelatively widely because of correspondingly wide variations\nof atmospheric clarity and elevation differences of locales\nabove mean sea level. Hence the magnitudes to be offered\nhere are not as unique or invariable as the solar constant\ngiven above, and must be understood as general indicators of\ntypical irradiance levels at the earth's surface. Table 2 is\nadapted from one given by Moon [185]. The solar constant val-\nues in the indicated ranges have been computed from Table 1\nabove and included for comparison. The column marked \"405 to\n704 mu\" is of especial interest since it gives the irradi-\nances in the visible portion of the spectrum. By an odd nu-\nmerical fluke, the solar irradiance constant 555 watts/m 2 o-\nver the visible spectrum numerically equals the wavelength\n(in mu) at which the photopic luminosity curve has its maxi-\nmum. It is instructive to study the tabulated effects of\nmoisture content of the air and altitude on the irradiance as\ngiven in Table 2. (The totals have been rounded out so as\nnot to appear misleadingly accurate.) Quite a battery of em-\npirical models have been evolved to predict the effects of\nmoisture, dust, elevation of sun and of observer on the meas-\nured irradiances on the earth's surface. An excellent sum-\nmary of these models may be found in [96]. Another reference,\nof interest to oceanographers, would be [173]. For a recent\nsurvey of solar irradiation measurements, see [296].","SEC. 1.2\nNATURAL LIGHT FIELDS\n25\nTABLE 2\nIrradiance Data at Earth's Surface\n(in watts/m2 on a plane normal to sun's rays, within\nindicated portions of the electromagnetic spectrum)\nConditions\nWavelength Range\nBelow\n346 to\n405 to\nAbove\n346 mu\n405 mu\n704 mu\n704 mu\nTotal\nMountain tops,\nsun at zenith,\n23\n47\n484\n668\n1220\ndry clean air.\nMountain tops,\nsun at zenith,\n16\n43\n466\n534\n1060\nmoist dusty air.\nAt sea level,\nsun at zenith,\n16\n42\n472\n665\n1200\ndry clean air.\nAt sea level,\nsun at zenith,\n4\n30\n375\n425\n834\nmoist dusty air.\nSolar (irradi-\ndiance) Constant\n58\n76\n555\n707\n1396\n(for comparison)\n(From [185], by permission)\nGeneral Illuminance Levels at Earth's Surface\nAn extensive photometric survey of illuminance at sea\nlevel on a horizontal plane under various sky conditions was\nmade by Brown [35], part of which is summarized in Fig. 1.12.\nThe graphs in Fig. 1.12 give a detailed photometric portrait\nof the extremes of variation and the modes of variation of\nnatural illumination generated by the light from the sun and\nthe moon. We have seen in (3) that the solar (illuminance)\nconstant is 12,700 footcandles, which corresponds to a solar\ndisk luminance of 2 x 109 blondels. This level of illumina-\ntion is approached by the \"unobscured sun\" curve in Fig. 1.12\nfor zenith sun. Notice how little the average overcast con-\nditions affect the general order of magnitude of the sea lev-\nel illuminance. Inexperienced bathers who think they will be\nsafe from sunburn under overcast skies will do well to take\nnote of this fact which follows from Fig. 1.12: one can get\nbaked just as severely under overcast skies as in bright di-\nrect sunlight. Moonlight bathing is harmless--photometrically\nspeaking - - for, the average level of full moonlight illuminance","INTRODUCTION\nVOL. I\n26\n90'\n70°\n80'\n10'\n20\"\n30°\n40'\n50'\n60'\n90\n40°\n30\"\n20\n-10\"\n0\nSUNRISE\nTWILIGHT\nCIVIL\nBASIC CURVE\nNAUTICAL\nTWILIGHT\nTWILIGHT\n10,000\nASTRONOMICAL\nCONDITIONS,\n10,000\nCLOUD\nBLACK STORM CLOUD\nCONDITIONS\n1,000\n1,000\n100\nCENTER OF SUNS DISC IS AT HORIZON 68 fa\n100\nSUNRISE 0.8 42 fc\n42 fc UPPER LIMB OF SUN APPEARS AT HORIZON\nUPPER LIMIT OF ALL TWILIGHT\nD\n10\n10\n1\n1\nLOWER LIMIT OF CIVIL TWI.\nSOLAR AND LUNAR\n6' 3.16X10 fc\nC\nALTITUDES\n10\n10\nPHASE L 0°\nPHASE & 60°\n2\n10\nPHASE L 90\nLOWER LIMIT OF\nNAUTICAL TWI\nPHASE L 120\n-12° 9x10 4\n10\n10\nLIGHT\nDISC\nTHE\nLOWER LIMIT OF\nASTRONOMICAL TWI.\n10\n10\n18 6.05X10 & fc\nA\nREB\nBASIC DATA AND CURVE BY BUREAU OF SHIPS CODE 174\nCOR\n10\n10\n,\n10'\n20'\n30'\n40'\n50'\n60'\n70°\n80'\n90'\n90\n40'\n30'\n-20'\n-10'\n0\nALTITUDE\nFIG. 1.12 Illuminances on a horizontal surface at sea le- -\nvel under indicated conditions. (From [35], by permission)","SEC. 1.2\nNATURAL LIGHT FIELDS\n27\nis about five orders of magnitude less than corresponding\nsunlight conditions. Typical clear sky luminances away from\nthe sun are on the order of 3000 blondels, with very heavily\novercast skies on the order of 300 to 1000 blondels at the\nzenith. For further details on the use of the graph in Fig.\n1.12, one should consult the discussion given in [35].\nGross Features of Atmospheric Radiative Transfer\nThe tables and graphs of the irradiance and illuminance\nsurveyed above show the great temporal and spatial variations\npossible in the magnitudes of these quantities. Therefore to\ntry to assign specific numbers to the reflectance and trans-\nmittance of the atmosphere at any given time is seldom an in-\nstructive activity. However, discernable patterns and stable\npercentages emerge when the daily variations of the reflec-\ntances and transmittances are averaged over long times and\nover great areas. Such averages begin to show the general\nfeatures of the radiative transfer processes extant in the\natmosphere, and help us form an initial picture of the ra-\ndiant energy budget of the atmosphere-surface system. Con-\nsider, for example, the average yearly irradiance (of all\nwavelengths) on an average horizontal surface just outside\nthe atmosphere over the entire northern hemisphere. On purely\ngeometrical grounds, this amounts to about one quarter of the\nsolar constant or 340 watts/m² (about 0.485 gm cal/cm2 min)\nover one year.\nThe annual radiant energy budget may be analyzed as\nfollows: for easy visualization, we normalize the 340 watts/\nm²\nand start with 100 watts/m². Thus, if 100 units of irra-\ndiance on the average are incident on the upper atmosphere,\nthen the general radiative transfer activities in the atmos-\nphere at steady state are reflection, absorption, and trans\nmission, which take up, respectively, 34, 19, and 47 of these\n100 incoming units as shown in (a) of Fig. 1.13. Part (b) of\nFig. 1.13 breaks the reflected and transmitted fluxes down\neven further. Thus, of the 34 units reflected, 25 of these\nare by the clouds, and 9 by the clear atmosphere. Of the 47\nunits transmitted, 24 of these are directly transmitted (with-\nout scattering), and 23 are transmitted via scattering. of\nthese 23 transmitted units 17 are transmitted by the clouds,\nand 6 by the clear atmosphere.\nNow the 47 transmitted units are received in turn by\nthe earth (terra firma + terra infirma), are chewed up and\nare eventually given back via heat radiation (14 units), or\nlatent heat of evaporation in cloud formation (23 units) or\nvia convection-conduction activity between the atmosphere and\nthe earth's surface (10 units) This is shown in (c) of Fig.\n1.13.\nAn exact mathematical formulation of these interactions\ncan be written down using the principles of invariance for\nirradiance, as described generally in Sec. 8.7, assuming,\ne.g., a three-layer system (atmosphere + clouds + earth's\nsurface) ; see in particular Examples 5 and 6 of Sec. 8.7. The\nnumbers cited above, however, are not theoretical, but rather\nbased on actual observations and are patterned after the","VOL. I\n28\nINTRODUCTION\n100 UNITS IRRADIANCE\n34 UNITS REFLECTED\nINCIDENT\nE\nATMOSPHERE\n19 UNITS ABSORBED\n(NON RADIANT ENERGY)\n47 UNITS TRANSMITTED\n(a)\nEARTH\n25 UNITS REFLECTED BY CLOUDS\n9 UNITS REFLECTED\nBY CLEAN AIR\n34 UNITS\nREFLECTED\n47 UNITS TRANSMITTED\n23 UNITS TRANSMITTED\nVIA SCATTERING\n24 UNITS TRANSMITTED DIRECTLY\n(6 BY CLEAN AIR, 17 BY\nCLOUDS)\n(b)\n10 UNITS RETURNED VIA CONVECTION-CONDUCTION ACTIVITY BETWEEN\nEARTH'S SURFACE AND ATMOSPHERE\n14 UNITS RETURNED VIA HEAT RADIATION\n23 UNITS RETURNED VIA\nLATENT HEAT OF EVAP-\n47 UNITS TRANSMITTED\nORATION IN CLOUD\nFORMATION\n(c)\nFIG. 1.13 The average yearly radiant flux budget over the\nsunlit hemisphere of earth. (From [96], by permission)\nmagnitudes summarized in [96].\nRadiative Transfer Across the Air-Water Surface\nThe still air-water surface acts like an imperfect mir-\nror which reflects only about 2% of an unpolarized light beam\nnormally incident on it from the air side, and transmits\nabout 98% of the incident flux of the beam into the water be-\nlow. As the beam is tipped and all other factors the same,\nthis reflectance stays fairly constant until, at about 45°\nfrom the vertical, the reflectance curve begins to soar to a\ncomplete reflectance of unity at grazing incidence to the air-\nwater surface. The functional dependence of this reflectance\nis quite well known and is governed by Fresnel's formulas, to\nbe studied in Sec. 12.1.","SEC. 1.2\nNATURAL LIGHT FIELDS\n29\nWhen the air-water surface is ruffled by capillary waves\ninduced by the wind, or when the surface is heaving with grav-\nity waves, the average amount of flux reflected from a verti-\ncal light beam incident on the moving surface over a given\ntime can be computed, once again by means of the Fresnel re-\nflectance function, but now with that function's values\nweighted by numbers between 0 and 1 which are the fraction of\nthe given time interval the surface is tipped away from the\nhorizontal by a given angle between 0 and 90°. The determi-\nnation of these weighting factors required in such a computa-\ntion is at present principally an empirical matter, and one\nof the first such determinations made in hydrologic optics is\ndepicted in Fig. 1.14. This curve, based on the experimental\nresearches by Duntley in [82], gives the number of times the\nwater surface normal at a fixed point was observed to tip\nover by an amount ,\n< 90° , during a given time period.\nThe solid curve is for the case where the normal was observed\nwithin the up-down wind plane; the dashed curve is for the\ncross-wind plane case. There is very little difference be-\ntween the two cases. A steady wind of 18 knots (about 9 m/sec)\nwas blowing and maintaining a steady capillary wave and small\ngravity wave complex. It was found that the number no of\ntimes the wave surface normal was tipped $ from the vertical,\nduring the experiment was very nearly expressible as:\ntan 2\n20 2\n(4)\nn o e\nCOUNTS no\n200\n-100\nWIND\n18 KNOTS\n10°\n20°\n30°\n-30°\n-20°\n-10°\no\n-\nSLOPE (Z = tan\nFIG. 1.14 Relative frequency of occurrence of a given\ntilt of a water wave facet.","INTRODUCTION\nVOL. I\n30\nIn other words, no was found to vary in a gaussian manner\nwhen tan (rather than ) was used as an independent vari-\nable. The quantity o is the usual standard deviation of the\nobserved slopes (the mean slope tan was zero). It is clear\nthen, that the relative number of times the wave slopes were\ntipped at tan , is given by no/no. For the 18 knot wind, it\nturned out that o was 0.162, which may be pictured as the tan-\ngent of a standard deviation angle of inclination of the sur-\nface normal of about 9.2 degrees from the vertical. It was\nalso found that the square of o, i.e., o2, varied nearly lin-\nearly with the surface wind speed generating and sustaining\nthe steady wave complex. A flat calm surface clearly has a o\nof 0. The preceding gaussian distribution was also found by\nCox and Munk [56] in their study of the glitter patterns on\nthe sea surface.\nThe preceding statistical type of description of the\ndynamic air-water surface can be used, under suitable condi-\ntions, to estimate the time averaged reflectance and trans-\nmittance of the air-water surface over a given time interval\nat a certain point; or dually, to estimate the space averaged\nreflectance and transmittance of the surface over a given re-\ngion at a certain time instant. Table 3 displays three re-\nflectances computed under the indicated conditions.\nTABLE 3\nIrradiance Reflectance (0, +) /H of the\nAir-Water Surface for Sky Light\nSky\nAir-Water Surface\nSmooth (o=0)\nRough (0=0.2)\nClear, sun at 60 o\n(no wind)\n(13-18 knot wind)\nfrom zenith\n.100\n.071-.088\nUniform\n.066\n.050-.055\n. 043-.044 -\nOvercast\n.052\n(From [58], by permission)\nThus under a clear sky with the sun at 60° from the zenith, a\nsmooth sea surface will reflect about 10% of its total irra-\ndiance ( (0, -) ) back into the sky, whereas, under the same\nsky condition, a sea driven by a steady 13-18 knot wind would\nreflect a slightly less amount of about 7 to 9% of the total\nirradiance (over the whole spectrum). This is in reasonable\naccordance with an intuitive estimate based on the Fresnel\nreflectance function for the air-water surface. In all dis-\nplayed cases in Table 3, the irradiance reflectance decreases\nwhen the wind starts to blow over the surface and hence rough-\nens the surface. As Cox and Munk observe, this fact has an\nimportant oceanographic significance, namely that in summer\nthe open stretches of the Arctic Ocean surface (or any","SEC. 1.2\nNATURAL LIGHT FIELDS\n31\n(a)\n(b)\nNo\nN\nFIG. 1.15 Contrast reduction by time-averaged refraction\nat the air-water surface.\nroughened surface for that matter) will reflect less and\ntransmit more radiant flux than has been previously estimated\nusing simple unweighted Fresnel reflectances (cf. [58]).\nMore exact values of the reflectance for o = 0 are given in\nTable 4 of Sec. 12.1.\nA complete theory of the reflectance and transmittance\nof both the static and dynamic air-water surface is developed\nin Chapter 12 below.\nBesides oceanographic applications there are also visi-\nbility applications of the observed gaussian structure of the\nruffled air-water surface slopes. Thus while it is common-\nplace that the visibility of a submerged object below a wind\nblown surface as seen through the surface is less than when\nthe surface is calm, due to the blurring action of the refrac-\nting processes at the surface, it is possible actually to\nmake quantitative predictions of the time-averaged apparent\ncontrast of a given submerged object against its background\nas a function of the size of the object and the standard de-\nviation o of the wave slopes through which the line of sight\nis directed. Part (a) of Fig. 1.15 depicts the basis of such\npredictions when the surface is flat and horizontal at the\npoint of intersection with the line of sight, and when the\ncenter of the submerged object (here a circular disk) is ob-\nserved to have an apparent radiance No. When the surface is\ntipped, as in (b) of the figure, the refracted line of sight\npicks up the apparent radiance N of the background of the ob -\nject. The still water apparent contrast C of the center of","32\nINTRODUCTION\nVOL. I\nthe object with respect to its water background is by defini-\ntion (No-N)/N. If the time-averaged apparent contrast of the\nobject against its background is C when the surface slopes\nhave a standard deviation of o, then it can be shown that:\ntan 2\n-\n2g 2\n(5)\nC\n1-e\n=\nwhere the object has an angular radius of 4. Observe that for\nC > 0, if o increases, then C decreases for a given 4, as\nwould be expected. Further, for given o, the time-averaged\ncontrast C increases as 4 increases; again as would be expec-\nted, but now in a definite quantitative way. For small ob-\njects or rough seas (or both) the preceding formula yields\nthe rule of thumb :\nC=C ( tan 202 2\n(6)\nThese formulas, which describe the contrast reduction by time-\nvarying refraction effects, will be developed in detail in\nSec. 12.14.\nGlitter Patterns on the Air-Water Surface\nSunlight reflected from a still air-water surface can\nbe seen, by each observer, as a circular image lying angularly\njust as far below the observer's horizon as the sun lies above\nthat horizon. A slight breeze disturbs the water and the sin-\ngle image splits into two or more irregularly shaped randomly\nmoving images of the sun. The breeze continues and the few\nimages ignite into a dazzling glitter pattern. To a poeti-\ncally inclined observer, the glitter pattern invokes very un-\ngeometrical and unhydrodynamical thoughts. In Russian, for\nexample, the glitter pattern is sometimes referred to as the\n\"road to happiness\". However, to analytically inclined ob -\nservers, the glitter pattern contains a wealth of information\nabout the geometrical structure of the surface, the statisti-\ncal distribution of wave slopes and, as we have seen above,\nimportant consequences for the radiative transfer processes\nacross the air-water surface.\nAs an illustration of these more technical ideas consi-\nder the problem of finding the greatest occurring slopes on a\nrough sea surface at a given time. It is seemingly impossible\nto do this visually or even with photographs or other optical\nmeans until certain geometrical features of the sun's glitter\npattern come under scrutiny. Then it becomes clear that in\norder for an observer to see the instantaneous reflected\nimage of the sun in a wave facet, the three participants in\nthis phenomenon, namely the sun, the facet, and the observer,\nmust subtend very precise geometrical relations. These re-\nlations are readily calculated using a bit of analytic geom-\netry. Figure 1.16 (adapted from Minnaert [182], in turn","SEC. 1.2\nNATURAL LIGHT FIELDS\n33\n60\no\n35\n30°\n25\n40°\n40°\n20°\n45°\na\n15\n50°\n55°\n20°\n10\n60°\n65°\n70°\n0°\no\n10\n20\n30\nw\nFIG. 1.16 How to find the tilt of a sun-reflecting water\nfacet's normal knowing the sun altitude a and the horizontal\nangle w of the facet from the vertical plane containing the\nsun. (Based on Hulburt's calculation) (From [113], by per-\nmission)\nderived from [113]] summarizes one such calculation, and may\nbe used as follows to estimate the required maximum tilt of\nwave facet-normals on an air-water surface which has a glit-\nter pattern. First estimate the angular half-width w of the\npattern, and estimate the altitude a of the sun above the\nhorizon. Suppose, e. w = 15° and a = 30°. Then the curve\ngoing through the grid point (15°, 30°) is labeled \"30°\" and\nthis is the requisite maximum tilt of the normals to the\nglittering facets. When a grid point (such as (20° 40°)\n)\nfalls between two curves, one must visually interpolate to\nfind the requisite maximum tilt (about 32° in this case).\nThese and related calculations are studied further in Sec.\n12.5.\nIt is of interest to observe that the graphs in Figure\n1.16 may be used to estimate the amount of tilt of any ob-\nserved reflecting air-water facet; furthermore the object re-\nflected in the facet need not be the sun--any point source\nwhose distance from the facet is several times greater than\nthe observer-facet distance may replace the sun.\nSubsurface Refractive Phenomena\nOnce one descends below the ir-water surface a new\nrealm of relatively strange radiative transfer phenomena is\nencountered. At the very instant light passes that incredi-\nbly thin air-water film the radiance function receives a jolt\nin the form of an abrupt increase in radiance of the sky in\neach observable direction. The increase is by a factor of\n(4/3) 2 or 16/9. This is a purely geometric effect due to the","34\nINTRODUCTION\nVOL. I\nD\nAIR\nWATER\nD'\nFIG. 1.17 The effect which gives rise to the n° - law - for\nradiance.\ngeneral narrowing of a bundle of refracted light rays as they\nenter the more dense water from the air (see Fig. 1.17). It\nis interesting to note that this phenomenon, as such, is not\ndetectable by the unaided eye since the apparent radiance as-\nsociated with a bundle of light rays depends (scattering ef-\nfects aside) only on the indices of refraction at the begin-\nning and end of the light bundle's path. Since the bundle\nbegins in air and ends on the retina inside the eye, the in-\ntermediate water domain has no effect in this special geomet-\nrical sense. The full effect, however, can be measured by\nsimple radiance meters, if they are suitably built.\nThe optical distortions attendant upon the refraction\nof the light rays at the surface are quite marked. For ex-\nample as one slowly descends into a body of water with a rel-\natively calm surface and continues to look upward, one is\nstruck with the impression that he has just descended downward\ninto a room with a circular hole a \"manhole\" in its ceiling.\nThrough this manhole one sees the objects above the surface\nbecome visually compressed the closer their images lie to the\nrim of the hole (Fig. 1.18) Just to one side of the hole\nthe underside of the air-water surface appears as a slightly\nundulating perfect mirror, in which nearby fish or other ob -\njects may be imaged--upside down. Also, if the bottom is\njust below the observer, he can see it mirrored on the sur-\nface above him around the rim of the manhole. As one des-\ncends further the manhole's outline is slightly dimmed by the\nscattering and absorbing effects of the water, but it contin-\nues to subtend the same angular radius - - about 48°, the angle\nbeyond which, according to Snell's law of refraction, total\ninternal reflection takes place.\nIf the air-water surface is not calm, but ruffled with\nwavelets, then the ideal geometric reflection pattern is re-\nplaced by something relatively complex. Beebe [12] gives the","SEC. 1.2\nNATURAL LIGHT FIELDS\n35\n48°\n48\nFIG. 1.18 The swimmer's optical manhole to the outside\nworld.\nfollowing interesting account:\n\"As to the opacity of the ceiling, I thought it abso-\nlute until I threw my head back as far as I dared, [he was in\nan old fashioned iron helmet rig exploring Haiti Bay, in 1927]\nand saw, almost directly overhead, facets of clarity, appear-\ning and vanishing, showing me an instant's patch of sky, a\nmomentary glimpse of friend or boat of that world to which\nit seemed at this moment inconceivable that I belonged. But\nanywhere except straight above me, the ceiling of the bay was\nwatered gauze.\"\nIf the underwater observer now directs his attention\ndownward, he may see in relatively shallow water a moving mo -\nsaic of bright and dark areas on the bottom, produced by the\nrefracted sun's rays converging and diverging at various\npoints on the bottom. When two bundles of rays are refracted\nso as to momentarily converge at a point A on the bottom (Fig.\n1.19) the irradiance at A abruptly increases and is seen by\nthe swimmer as a bright spot. On the other hand, rays could\nbe diverted away from a point such as at B in Fig. 1.19,\nwhereat it will be momentarily relatively dark. By knowing\nthe statistics of the air-water surface slopes (as discussed\nabove) it is possible to determine the statistics of the irra-\ndiance pattern on the bottom. The problem has recently been\nstudied, e.g., by Redmond [260], and Schenck [272].\nAs one descends still farther, and if the water has a\nmodicum of suspended and dissolved material which scatters\nlight, the refracted rays of sunlight are then seen to form a\npattern of moving beams and weaving, lighted, curtain sur-\nfaces very much like a watery aurora borealis or like the\nshafts of sunlight one sees directed earthward from rifts be-\ntween clouds. These beams die away relatively quickly with\ndepth in natural waters, at least as compared to the decay of\nthe general diffuse light originating from the sky and clouds.","36\nINTRODUCTION\nVOL. I\nAIR\nWATER\nA\nB\nFIG. 1.19 Generating light patterns on shallow bottoms.\nWe shall look into this phenomenon in some detail later in\nthis section.\nOne final subsurface refractive phenomenon we shall\nnote here is that associated with the thermocline in natural\nhydrosols. The thermocline is the region of abrupt tempera-\nture change, (usually taking place in an extensive thin hor-\nizontal layer) found in most all natural waters, which sepa-\nrates a warmer layer from a cooler layer of water below it.\nIt is detectable by means of a submersible thermometer known\nas a bathythermograph. Accompanying this temperature change\nis a corresponding density change of the water, and with this\noccurs a change in the refractive index of the water. There-\nfore we would expect some interesting refractive optical phe-\nnomena at the thermocline. Some observations of optical\nthermocline phenomena were made by Limbaugh and Rechnitzer\n[160] and are schematically summarized in Fig. 1.20, which is\nadapted from their paper. When the thermocline occurs in its\nmore frequent guise, as a thin, horizontal, nearly motionless\nlayer below the surface (as in the upper third of Fig. 1.20)\none can actually see the thermocline from below as a smooth,\nnearly flat mirror-like plane boundary between the two water\nlayers of differing temperature - and it generally manifests\nitself very much in the way the air-water surface does, even\nto the extent of having its own manhole into the warmer layer\nof water above. (Would one expect this manhole to subtend\nthe same angular radius as the surface manhole?) Occasionally\nsome rather unusual refractive phenomena may be observed when\na moving tongue of cold water snakes its way through a warmer\nregion on the bottom, (as in the lower left third of Fig.\n1.20). The convex boundary of the tongue is visible all\nalong its extent at grazing incidence, and its general appear-\nance is reminiscent of the intertwining portions of two mis-\ncible liquids, such as clear alcohol and clear water. Finally,\nLimbaugh and Rechnitzer observed the optical thermocline ef-\nfect in small isolated pools of relatively cold water resting","SEC. 1.2\nNATURAL LIGHT FIELDS\n37\nTURBID\nwarm\ncold\nCLEAR\nwarm\ncold\n\\warm\ncold\nTEMPERATURE\nTHERMOCLINE\nBATHYTHERMOGRAPH TRACING\nFIG. 1.20 Three interesting subsurface refractive phenom-\nena. (From [160], by permission)\non the bottom in the midst of warmer water. These cool pools\nreflected light at their surfaces much in the way the still\nair-water surface reflects light for an observer above it.\nThe Decay of the General Light Field with Depth\nPerhaps one of the most striking and outstanding fea-\ntures of the light field in deep natural waters is that it\ngets dark fast with increasing depth. For example infrared\nradiation (which comprises about half the irradiance at sea\nlevel on sunny noon days) is essentially absorbed in the first\nmeter or so of most natural waters. There is a reasonably\nprecise and simple law of darkening of the light field in\nthis regard: the light field of any wavelength generally\nfalls off or decays exponentially with depth. That is, if\nh(z) is the scalar irradiance at depth Z in a homogeneous,\ndeep lake or portion of the sea, then:\nh(z) = h(0) e-Kz\n(7)\nThis type of law, namely the exponential type, is unquestion-\nably the most ubiquitous of all types of natural laws in geo-\nphysics: it describes thermal and radioactive decay in sol-\nids and liquids, evaporation rates of falling rain droplets,\ngrowth rates of plant and animal species, fall off of atmos -\npheric density with altitude, only to mention a few. In our","38\nINTRODUCTION\nVOL. I\n100\n80\n60\n50\n40\n30\n20\n10\n8\n6\n5\n4\n3\n2\no\n10\n20\n30\n40\n50\n60\n70\nDEPTH (FEET)\nFIG. 1.21 Showing how scalar irradiance decreases expo-\nnentially with depth. Experiment by Duntley, Lake Winnipe-\nsaukee, N. H. , September 1948. (Fig. 30, left diagram, from\n[78] by permission.)\npresent studies, it describes not only the decay of the nat-\nural light field with depth, but generally the decay of a\nbeam of light with distance along its path. In the present\ncase, the decay rate K depends on the wavelength A of light\nconsidered (h (z) depends on X; however for brevity, as usual\nwe omit \"X\") and of course the clarity of the water consid-\nered. Indeed, as we shall see later, in Sec. 1.7, we may use\nthe wavelength dependence of K to help classify the optical\nproperties of natural hydrosols.\nFigure 1.21 illustrates a sample experimental determi-\nnation (taken from [78]) of the depth dependence of scalar\nirradiance in a deep clear lake (Lake Winnipesaukee, N.H.)\nover a depth range of 60 feet or 18.3 m. The crosses indi-\ncate the experimental points. The straight line is the best\nstraight line for the data, and is plotted on semilog paper.\nThe magnitude of the constant K is: K = .066/ft. = .216/m,\nfor green light.\nIn view of the preceding observations there is no need\nat present of giving further graphs of h (z) vs depth Z in\ndeep homogeneous media; for as the saying goes, 'if you have\nseen one, you have seen them all', the prototype being that\ndisplayed in Fig. 1.21. What is more worthwhile at present,\nis to raise such questions as : how is the exponential decay\nlaw affected if the medium is not deep, or if the bottom is\nclearly visible? What effects do inhomogeneities of the me-\ndium have on the exponential law? Does h (z) decay at the","SEC. 1.2\nNATURAL LIGHT FIELDS\n39\nA (looking up)\n.01\n.00I\nB (looking down)\n.0001\n.00001\no\n20\n40\n60\nDEPTH (meters)\nFIG. 1.22 Two experimental determinations of radiance by\nTyler, Pend Oreille Lake, Idaho, April 1957. Note the gener-\nal exponential decrease. Note, also, the slight buildup of\nradiance for the upward looking path near the surface. (From\n[298], by permission)\nsame rate at H(z,=) ? (cf. (9) and (10) of 1.1) . Does the\nexponential law hold right up to the surface, or is there a\nboundary effect? These and other questions are readily an-\nswered in detail by the theories developed in Chapter 8.\nSome simple answers are given in Sec. 1.4.\nBehavior of Radiance Distributions with Depth\nIf we fix attention on the zenith radiance as we des\ncend into the sea, then, aside from the effect on the radi-\nance induced by a change of index of refraction (discussed\nabove) there is observable a general build-up - of radiance in\nthe first meter or so below the surface. This build-up of\nlight is depicted by Curve A of Fig. 1.22 (adapted from [298])\nand is quite analogous to the increase in the light field one\nexperiences as an airline passenger during the initial stages\nof the airliner's descent into a thick cloud layer lighted\nfrom above by the sun. We are observing in either case the\nstorage of scattered radiant energy within the medium. In\nthe case of the sea this increase in radiance is observable\nnot only at the zenith, but in all upward looking directions,\nbut is occasionally obscured by the refracted sunlight beams\nand other surface phenomena. The depth at which the maximum\nradiance occurs is predictable in theory and varies with the","40\nINTRODUCTION\nVOL. I\n10\n106\n105\n4.24 METERS\n104\n16.6\n3\n10\n29.0\n10\n2\n41.3\n10\n53.7\nO\n10\n66.1\n10\n-180\n-120\n-60\nO\n60\n120\n180\nZENITH ANGLE (DEGREES)\nFIG. 1.23 Radiance distributions, in the vertical plane\ncontaining the sun, on a clear sunny day, at the indicated\ndepth, in Lake Pend Oreille, Idaho, as measured by Tyler,\nApril 1957. Observe how the shapes of the curves become sim-\nilar as depth increases. (Fig. 26, from [78], by permission)\ndirection of sight and the clarity of the medium (cf. (12) of\nSec. 4.4). .\nAfter the maximum radiance occurs in a given direction,\nthe radiance in that direction begins to fall off rapidly\nwith depth and soon assumes the exponential behavior that\n(z) universally exhibits. This trend to exponentiality is\nseen quite clearly in the nadir curve B of Fig. 1.22, or more\ngenerally in Fig. 1.23, which is adapted from [78]. Fig.\n1.23 is designed to show how the shapes of the radiance dis-\ntributions vary with depth in the hydrosol. The\ngraphs in particular\nFig. 1.23 are adapted from [78] and represent the\nlight field measured in Lake Pend Oreille, Idaho by Tyler\n[298]. The radiance is associated with a wavelength of 480\n+ 64 mu, in water with a K of about 170/m and (for future\nreference) an a of . 370/m. Two important and universal prop-\nerties of underwater radiance distributions are discernable\nin this set of curves: (i) the decrease in peakedness of the\ncurves with depth, accompanied by a trend toward a limiting\nshape as depth increases, and (ii) the shift of the radiance\nmaxima toward the zenith with increasing depth. Near the\nsurface the peaks are pointed toward the refracted image of\nthe sun; but this orientation is lost as depth increases.\nThis trend toward a stable vertically-oriented smooth distri-\nbution is shown in more detail in Fig. 1.24, wherein the ze-\nnith angles of the maxima in Fig. 1.23 are plotted as a","SEC. 1.2\nNATURAL LIGHT FIELDS\n41\nDEPTH (ATTENUATION LENGTHS )\no\n4\n8\n12\n16\n20\n24\n25\n20\n15\n10\n5\no\no\n20\n40\n60\n80\n100\n120\nDEPTH (METERS)\nFIG. 1.24 Plot of zenith angle of the maxima of the curves\nof Fig. 1.23. The maxima shift toward the zenith with in-\ncreasing depth. This figure and Fig. 1.23 present graphic\nevidence of the validity of the asymptotic radiance hypothe-\nsis. (Fig. 28 from [78], by permission)\nfunction of depth. The problem of the description of the\ndepth dependence of the radiance distribution in natural hy-\ndrosols is one of the principal tasks of hydrologic optics\nand to which much of this work is devoted.\nThe Asymptotic Radiance Hypothesis\nThe fact that the shapes of the radiance distributions\nin deep hydrosols approach limiting forms with increasing\ndepth is observable in Both Figs. 1.23 and 1.24. In the for-\nmer figure all the radiance curves eventually steady in shape\nwith increasing depth. This means that eventually all radi-\nances are decreasing at the same exponential rate with depth.\nHence the evidence points to the fact that radiance distribu-\ntions eventually assume certain stable shapes and these dis-\ntributions subsequently shrink down exponentially in size with\nincreasing depth, all the while preserving those shapes.\nThe\ngeneral statement of the existence of such limiting shapes in\nall homogeneous natural hydrosols is the asymptotic radiance\nhypothesis which was first clearly enunciated by Whitney [315]\non the basis of experimental findings, and subsequently\nproved mathematically in [225]. . The validity of the asymp-\ntotic radiance hypothesis has important consequences for the\ndevelopment of simple theoretical models of the light field","42\nINTRODUCTION\nVOL. I\nin the sea and in deep lakes, rivers and harbors. For exam-\nple the scattering and absorption functions in the general\ntheory depend in part on the shape of the radiance distribu-\ntions. If these distributions do not vary too much with\ndepth, vast simplifications of the general theory are pos-\nsible. These matters will be pursued at some length in Chap-\nters 6, 8 and 10.\nUnderwater Irradiance Distributions\nThe studies of visibility and biological problems--as\nfar as they are concerned with the radiometric environment-\nare facilitated by knowledge of the irradiance distributions\nH(z,-) at each depth Z in the medium of interest. Figure\n1.25, plotted from the tables in [304], illustrates such a\ndistribution as a function of orientation of the collecting\nsurface's outward normal direction (0,0) and also of depth,\nfor a sun zenith angle of 33.4°. This graph keys in with\nthat of Fig. 1.23, being the irradiance distribution computed\nfrom the radiance distributions in Fig. 1.23, using (1) of\n1.1. The role of (0,0) is depicted in Fig. 1.26.\nIt is of both practical and theoretical interest to\nknow that an irradiance distribution H(z,.) at a depth Z con-\ntains just as much information as the radiance distribution\nN(z,.) at that depth. This will be shown in Ex. 15 of Sec.\n2.11, wherein knowledge of N will be used to deduce knowledge\nof H, and conversely. The bridge between N(z,.) and H(z,.)\nis easily traversed in the direction N+H but is somewhat more\ndifficult to traverse numerically in the direction H+N, and\nuntil an efficacious numerical scheme to bridge the latter\ngap is devised, the radiance distribution will continue to be\nmeasured and be the favored means of cataloging natural light\nfields.\nSome practical features of irradiance distributions are\nas follows. Every irradiance distribution satisfies the exact\ncosine law:\nH(z,5) = H(z,m) cos 0\nwhere H(z,E) is the net irradiance in the direction E, m is\nthe direction of greatest net irradiance (cf. (14) of Sec.\n2.8), and 0 is the angle between E and m. This law shows\nthat we need only plot or tabulate irradiance distributions\nH(z, for directions E not greater than 90° away from some\narbitrary fiducial direction, say the vertical direction k.\nTo see this, suppose that we have H(z,k) and H(z,-k) and that\nwe know m. Then by the exact cosine law:\nH(z,k) = H(z,k) - H(z,-k) = H(z,m) cos 0\n(8)\nm\nwhere is the angle between k and m. From this we can com-\npute H(z,m). Now suppose we know H(z,-5) and that we want to\nknow H(z,5), where E is less than 90° from k. Then the","SEC. 1.2\nNATURAL LIGHT FIELDS\n43\n0°=0\n4\n30°\n10\n60°\n0°= 0 90°\n30°\n3\n10\n60°\n120°\n150°\n90°\n180°\n0°=0\n30°\n60°\n102\n120°\n150°\n90°\n180°\n120°\nZ\nplane of sun\n10\nk\n150°\noutward\nnormal\n180°\n0\n= 29.0m\ny\n= 41.3m\n= 53.7m\nX\ncollecting surface\nO\n40\n80\n120\n160\n200\n240\n280\nIRRADIANCE DISTRIBUTION (H) CLEAR, SUNNY SKY\nFIG. 1.25 Irradiance distribution on a clear sunny day at\nthe indicated depths, in Lake Pend Oreille, Idaho, 28 April\n1957, as computed by Schaules and Tyler from Tyler's data.\nFIG. 1.26 The collecting surface receiving the irradiance\nrecorded in Fig. 1.25.","44\nINTRODUCTION\nVOL. I\ncosine law yields:\nH(z,5) = H(z,-5) + H(z,m) cos 0\n(9)\nTherefore knowledge of m and H(z,m) together with H(z,.) over\none hemisphere of directions, will yield H(z,.) over the re-\nmaining hemisphere.\nAnother practical aspect of the irradiance distribution\nis that is can be used to compute one of the basic optical\nproperties--namely the volume absorption function, a --of nat-\nural optical media, by using the divergence law:\ndH(2,K) = a(2) h(z)\n(10)\ndz\nfor the vector irradiance (cf. (1) of 13.8, and Sec. 1.4 be-\nlow). Thus knowledge of H(z,.) leads to H(z,k) and to the\nlatter's derivative by straightforward computations. This,\ntogether with auxiliary determinations of h, yields estimates\nof a.\nSubsurface Contrast Reduction by Scattering\nand Absorbing Effects\nUnderwater scenes in seas, lakes and harbors are char-\nacteristically dim and blurry. The sharp outlines and stark\ncontrasts above the surface are relatively absent from under-\nwater scenes. Even in the clearest swimming pools, distant\nobjects no longer have sharp edges, and contrasts are slightly\nbut yet noticeably decreased. If one looks a bit closer at\nthese contrast-reduction phenomena, one outstanding and fun-\ndamental fact soon becomes manifest: on the one hand, as the\nobserver recedes from a relatively bright object, its lumi -\nnance rapidly falls off and soon melts into the background\nluminance; on the other hand, if the object is relatively\ndark, its luminance rapidly increases with viewing distance\nand eventually also melts into the background luminance. Is\nthere some order and regularity in these changes of apparent\ncontrast with viewing distance? In other words is there some\ngeneral law followed by these changes in apparent contrast of\ndistantly viewed objects in underwater scenes? The answer is\n'yes', provided a judicious scientific choice is made in the\nselection of the notion of contrast.\nIf tNr is the apparent (surface) radiance of an object\n(the target) viewed at a distance r underwater, and bNr is\nthe apparent (surface) radiance of its background, then we\nwrite\n\"CT\"\nfor\n(11)\nand call CT the apparent contrast of the target with respect\nto its background. The geometry of this situation is pictured\nin Fig. 1.27. If r=0, we call C the inherent contrast of the","SEC. 1.2\nNATURAL LIGHT FIELDS\n45\nTARGET\nNr\nbNr\nBACKGROUND\nFIG. 1.27 The apparent contrast of a target against its\nbackground.\ntarget with respect to its background.\nFigure 1.28 shows an experimental arrangement, devised\nby Duntley [78], to study contrast reduction phenomena in\nLake Winnipesaukee, N.H.. A telephotometer (i.e., a radiant\nflux meter attached to a telescope) was mounted on a small,\nhooded glass-bottomed - boat which looked at a flat white tar- -\nget at depth r. At the time of the experiment (sometime in\nSeptember 1948) the water was calm, the sky was clear, with a\nlow sun. For later reference we will note that the lake at\nthat time had a K of 0.216/m and an a of 0.594/m, for green\nlight. The observation of interest at the moment is recorded\nin Fig. 1.29, in its original form, which shows the sought-\nfor law governing Cr vs distance r in feet. This clearly\nshows an exponential decrease of Cr with r, in this case depth\nbelow the bottom of the boat. In fact it was found, on con-\nr\nverting to meter lengths, that:\ncr = Co o e - . 810 r\nThis finding of the exponential law is in itself a remarkable\none; however, the really exciting fact lay in the nature of\nthe number 810/m (= 247/ft), the exponential decay rate of the\napparent contrast. It was found that:\n. 810 =\n.594 +\n.216 = a + K\n(per meter)","46\nINTRODUCTION\nVOL. I\nHOOD\nTELEPHOTOMETER\nGLASS BOTTOM\nTARGET\nBACKGROUND\n(IMAGINARY PLANE)\nFIG. 1.28 Physical set-up for Figs. 1.29, 1.30.\nTo see the significance of this, recall our earlier observa-\ntions on the general mode of decay of the natural light field\nin the water. The depth rate of decay is given by K. The a\non the other hand, gives the depth rate of decay of a beam of\nlight in the water. Therefore there are two mechanisms in-\nvolved here in giving rise to contrast reduction. These are\nsummarized by K and a, and are generally distinct. These will\nshare our attention later. But for the moment we quietly rev-\nel in the presence of discerned order in at least one feature\nof the underwater radiometric environment. It was perhaps\nthis experimental finding and the ones immediately following\nit, shown in Fig. 1.30, that contributed more than any others,\nto inspire Duntley and one of his students (the present au-\nthor) to turn to the problem of explaining these interesting\n(and then, mysterious) manifestations of order in the sub-\nmarine light field, and relating them to the general radiative\ntransfer phenomena in scattering-absorbing media.\nWhat is shown in Fig. 1.30 (which holds for the same\nsetting as above) is an extension of the findings in Fig.\n1.29, and once again in the original form given by Duntley.\nThe new figure shows several things. First, it shows that\nthe apparent contrast of an object is exponentially attenuated\nwith target distance at the same space rate for both light and\ndark targets. Second, this space rate is independent of azi-\nmuth of the line of sight (here, the direction of motion of\nthe photons) which in this experiment was inclined at an angle\n0 of 30° away from vertically upward, or an amount 0= 150°\nfrom vertically downward. (See Fig. 1.31) In particular the\nazimuths, measured from the vertical plane of the sun, are","SEC. 1.2\nNATURAL LIGHT FIELDS\n47\n30\n-1.0\n100\n-0.8\n80\n20\nWHITE TARGETS\n-0.6\n60\n-0.5\n50\n-04\n40\n10\n-0.3\n30\n8\n6\n-0.2\n20\n5\n4\n3\n-0.10\n10\n-0.08\n8\n2\n-0.06\n6\n-0.05\n5\n-0.04\n4\nBLACK TARGETS\n1.0\n-0.03\n3\n0.8\n0.6\n-0.02\n2\n0.5\n0.4\n0.3\n-0.01\no\n6\n12\n18\n24\n30\no\n5\n10\n15\n20\nDEPTH (FEET)\nTARGET DISTANCE (FEET)\nFIG. 1.29 Duntley's classic experiment showing the expo- -\nnential law of decrease of apparent contrast along a vertical\npath in a natural hydrosol (Lake Winnipesaukee, N.H., Autumn,\n1948. See also Figs. 1.28, 1.30) (Fig. 30, middle diagram,\nfrom [78], by permission)\nFIG. 1.30 Further experimental evidence for the exponen-\ntial apparent contrast law. (See Figs. 1.29, 1.31) (Fig. 30,\nright diagram, from [78], by permission)\n= 0° (circled points), = 45° (crosses), , = 95° (diamonds)\nand = 135° (squares). The dashed straight lines are drawn\nparallel to help judge the slope and linearity of the data\nand have a natural logarithmic slope of about . 781/m. Once\nagain this exponential decay rate is a source of surprise\nwhen it is observed that\n.781\n.594\n.216 cos 30 (per meter)\n=\n+\nThis would lead one to conjecture that paths of sight inclined\ngenerally at 0 from the vertical in homogeneous stratified\nmedia, as shown in Fig. 1.31, would have an apparent contrast\nCr associated with them of the general form\ncos r\n(12)\nThe conjecture was confirmed and a simple theoretical model\nunderlying this contrast reduction law was soon evolved. The\nmodel will be discussed further in Sec. 1.4, in Chapter 4,\nand Chapter 9.","VOL. I\nINTRODUCTION\n48\nWATER SURFACE\nOBSERVER\nII 0\nSEES Cr\nHERE\n0\nTARGET\nHAS INHERENT\nCONTRAST\nCo HERE\nFIG. 1.31 The geometrical details for Fig. 1.30, in which\n0 = 30°\nSubsurface Contrast Reduction by Refractive Effects\nWhen one looks across an extensive flat stretch of the\nearth's surface such as a meadow or stretch of ocean on a\nsunny or very windy day, distant objects seem to be blurred\nnot only by the usual atmospheric haze, but also by a rapidly\nvarying shimmering or \"heat wave\" effect. This phenomenon is\nproduced by inhomogeneities of the refractive index of the air\nalong the line of sight and is associated with cells of air of\ndifferent density. These in turn are related to uneven tem-\nperature distributions in the air mass or simply to the local\nmechanical compression of the air in gusts of wind on windy\ndays. The same mechanism makes the stars twinkle at night.\nIt may come as a mild shock to some observers to occa-\nsionally see this same twinkling, heat-wave like effect in the\notherwise cool depths of an incompressible fluid like a sea or\na lake. Nevertheless, the effect exists, and on closer exam-\nination, sanity prevails: the underlying mechanism is seen\nto be refractive, but produced by myriads of tiny transparent\nplankton, whose indices of refraction differ very slightly\nfrom that of water. In some south sea waters, it is said\nthat the concentration of such plankton is so great, the spac-\ning between a swimmer's toes cannot be distinguished by him,\nthough the foot is visible with high contrast against its\nbackground. A somewhat less dramatic but similar phenomenon\nwas observed and recorded by Duntley at the Diamond Island\nField Station in Lake Winnipesaukee, N.H.. Figure 1.32, from\n[78], shows a photograph of the light distribution on a camera","SEC. 1.2\nNATURAL LIGHT FIELDS\n49\nSCREEN\nUNDERWATER COLLIMATED LAMP\nCAMERA\nWITH NO LENS\nWATERTIGHT\nENCLOSURE\nOPTICALLY FLAT WINDOW\nFIG. 1.32 Swarming plankton photographed in the light of\na strong collimated beam, as observed by Duntley in Lake Win-\nnipesaukee, N.H., 22 August 1961. Plankton swarms such as\nthese may contribute to contrast reduction along underwater\npaths of sight. (Fig. 22 from [78], by permission)\nFIG. 1.33 Arrangement for plankton photograph, Fig. 1.32.\n(Fig. 21 from [78], by permission)\nfilm produced by a collimated light beam after having travel-\nled through a horizontal 3 m water path shown in Fig. 1.33.\nThe time of year was late August (1961) and the exposure time\nwas 1/50 sec. on an Eastman Plus-X film with a normal D-76\ndevelopment. The beam had a diameter of about 5 cm and a","50\nINTRODUCTION\nVOL. I\nspread of about 0.01° The a of the water was .585/m, in\ngreen light. The water path between the lamp and camera was\nswarming with plankton, and the bright collimated beam has\nlimned some of these on the photographic film. To judge the\nsize of these tiny organic refractive cells, the diameter of\nthe black circular border (caused by the camera opening) was\nmeasured to be 3.3 cm on the negative.\nA theory for the loss of contrast of objects seen\nthrough atmospheric boil was evolved some time ago by the au-\nthor and some of his colleagues [81]. This theory appears to\nbe applicable also to the contrast reduction phenomenon des -\ncribed above. The effect, however, is generally mild when it\ndoes occur, and may for virtually all practical purposes be\nignored in the problem of predicting underwater visibility.\nHowever, in passing we may note that in a natural hydrosol\nwhich has such transparent plankton distributed uniformly and\ndensely along a path of sight of length r the theory predicts\nthat the magnitude of the blur (the standard deviation of the\nangular displacement of a typically straggling light ray from\nobserver to object plane) increases like r 1/2 and the apparent\ncontrast of fine details in an object against the general\nbackground decreases like 1/r3 Thus the contrast reduction\nlaw produced by refractive inhomogeneities in a medium is, on\nthe one hand, quite different from that produced by scattering-\nabsorbing mechanisms in that medium, and summarized in (12)\nOn the other hand, as a perusal of [81] would show, the theory\nof the present effect is quite close to that used to derive\n(5)\nThe Polarization of Underwater Light Fields\nUp to now we have been describing those optical effects\nin natural hydrosols that have very little directly to do with\nthe fact that photons, in their pristine state, are viewable\nas particles with observable spins--i.e., with an observable\nproperty we usually call polarization. If we now invoke the\nquantum theoretical wand of complementarity and imagine the\nphoton to be not a small, hard, colored ball but, rather a\nrelatively compact packet of electromagnetic waves whose E\nand H vectors vibrate in fixed mutually orthogonal planes as\nthe packet moves along (see Fig. 1.34), then we add a new di-\nmension to the description of radiometric phenomena. No lon-\nger is it sufficient to merely describe the unpolarized radi-\nance of the light field, but rather we must go on to describe\nradiance carried by those photons at x in the direction E\nwhose E vector is oriented by the general angle x with respect\nto some reference frame.\nSuppose we place a polarizer into the radiance tube, as\nshown in Fig. 1.35. (Compare with (b) of Fig. 1.5.) This may\nbe made from some commercially available polaroid material.\nThen if we fix x and E as usual, and rotate the polarizing\nelement, we can detect the presence of polarized radiance by\nthe varying output of the radiant flux meter's dial. Suppose\nwe turn the polarizer one full turn. Let Nmax(x,E) and\nNmin 1(x,5) be the maximum and minimum radiances so obtained.","SEC. 1.2\nNATURAL LIGHT FIELDS\n51\n(a)\ndirection of\npropagation\nH\nE\npath of E - vector\nreference frame\nH\n(b)\nE\nshade tube of radiance meter\nfilter\noptic axis of polarizer (i.e., analyzer)\nradiant flux meter\ndial\nFIG. 1.34 A linearly polarized E-vector. -\nFIG. 1.35 The placement of a polarizer in a radiance tube\npreparatory to measuring the polarization of a light field.\nThen we write\nN\n(x,5)\n- N min (x,5)\nmax\n\"p(x,5)\" or \"p\" for\nN max (x,5) + N min (x,5)\np(x, 5) is called the polarization of the light field at X in\nthe direction E, and is a useful measure of how much polari-\nzation is present in the light field at X.\nNow if we train such a polarized radiance meter at a\nclear sky, we find that the sky radiance is most noticeably\npolarized in all directions which lie in a plane normal to the\ndirection of the sun's rays. If we go below the air-water\nsurface we find that the light field is still polarized but to\nlesser extent. The shafts of sun and skylight beaming down\na\ninto and around the manhole (described above) are scattered\ninto the line of sight by the water in a manner completely\nanalogous to the sunlight streaming into and scattering within\nthe upper atmosphere. Furthermore, the underwater light field\nmay also be reflected into the line of sight by the underpart","VOL. I\nINTRODUCTION\n52\nmanhole\nair\nwater\npolarized reflected\ncontribution\npolarized scattered\ncontribution\npolarized radiance meter\nFIG. 1.36 The observed underwater polarized radiance can\ncome from the sky via refraction through the manhole, or from\nthe underwater domain via air-water surface reflection outside\nthe manhole.\nof the air-water surface outside the manhole (see Fig. 1.36).\nThese two mechanisms, the scattering and reflection of under-\nwater light, contribute the principal polarized parts to the\nunderwater light field. On purely theoretical grounds (which\nneed not concern us here) one would expect the scattered light\nto be predominantly linear, and the reflected light to be el-\nliptical, and hence the general underwater light field to be\na mixed inear-elliptical polarized field (see Sec. 2.10 and\nthe Stokes Polarization Composition theorem).\nThe general features of polarized submarine light fields\nmay be summarized, according to Ivanoff and Waterman [117],\n[118], as follows. In general for a fixed direction E the po-\nlarization p(x, E) is greatest near the air-water surface, and\ndiminishes rapidly with depth down to about 10-20 attenuation\nlengths and then settles down to an asymptotic value, which\ndoes not change with further increase of depth (this is remi-\nniscent of the asymptotic radiance theorem described earlier;\nand in Sec. 4.6 the potential connections between these two\nideas will be outlined). Furthermore, the limiting p value\ndepends on the water clarity, and we would expect on theoreti-\ncal grounds that it eventually be independent of surface and\nbottom effects provided the medium is deep enough. It is\nnoted that, all other factors remaining fixed, polarization\nincreases rapidly with transparency from turbid to moderately\nclear waters, but the increase slows down as waters become\nmore and more transparent. In oceanic hydrosols p may vary,","SEC. 1.2\nNATURAL LIGHT FIELDS\n53\ne.g., from .60 at the surface to .30 as an asymptotic value.\nIn a horizontal sweep, with low sun, the azimuth dependence\nof p is generally such that in directions normal to the ver-\ntical plane of the sun p is greatest, less for directly away\nfrom the sun and least of all looking toward the sun. For\nhigher suns or for more turbid waters a horizontal sweep of\nthe radiance tube may find little variation in p. The wave-\nlength dependence of.p is such that, with all other factors\nremaining fixed, p attains a minimum at the blue-green wave-\nlengths (450 mu) --i.e., just about where in the spectrum nat-\nural waters transmit best. This ties in with the observations\ncited just above about turbidity dependence of p. (Remember\nthe proviso, \"all other factors remaining fixed\".) Thus both\nends of the spectrum should yield higher p values, and hence\nmore pronounced polarized fields in reddish and bluish light\n--of what there is to measure. The polarization of under-\nwater light fields decreases when diffuseness of the field in-\ncreases. For example, when depths are shallow, overhead\ncloudiness will tend to increase the diffuseness and hence de-\ncrease the polarization. Under best conditions, the ellipti-\ncal component of the underwater radiance field reaches about\n10% of the total radiance, and about 50% of the linear com-\nponent. At very great depths the light is predominantly hor-\nizontally linearized (because the predominant flow is down -\nward; and recall the analogy with scattered skylight).\nFurther details will be found in [117], [118], and also\nin Tyler's article [301]. A simple model for polarized light\nfields in the sea is developed, along with the general theory,\nin Sec. 4.6. Sec. 2.10 develops the essentials of the radi-\nometry of polarized light.\nBiological Sources of Submarine Light Fields\nHow many have ever seen the unforgettable sight of lu-\nminous bow waves of a ship plowing through nighttime tropical\nand semitropical waters? Many types of marine animals large\nand small are known to emit radiant energy when disturbed-- a\nsort of pale cold light, obviously of chemical (quantum) rath-\ner than thermal origin. Other organisms seem to flash on and\noff under their own volition, deep in the sea or in nighttime\nwaters nearer the surface.\nAn important study of such self-regulative radiometric-\nbiologic phenomena was made by Kampa and Boden [133] in which\ndetailed and careful measurements of the radiant flux output\nof a certain type of luminescent creatures (Euphasia pacifica)\nwere made both in situ in the San Diego Trough, and in the\nlaboratory. The presence of these creatures is generally not-\ned by sonar operators because the creatures form a sonic-scat-\ntering layer in the water. By lowering a bathyphotometer (a\nradiant flux meter tightly encased for deep water work) down\ninto the layer, day and night recordings of the output of the\nEuphasia were made.\nIt was observed that the creatures emitted flashes hav-\ning a mean irradiance of about 1.1 x 10 microwatts/cm2\nthroughout the day. The output was in the form of flashes","54\nINTRODUCTION\nVOL. I\nwhich varied in frequency as a function of time of day great-\nest (42/min) during twilight when the Euphasia migrated up-\nward, least (10-24/min) during midday when they were at rest\nin the depths, and intermediate (32/min) during the night.\nThe color of the luminescence was blue-green, with maximum\noutput near 478 mu, and a secondary maximum near 520 mu.\nKampa and Boden postulate that the time dependence of the\ndepth of the Euphasia scattering layer is photoregulated; that\nis, the creatures constantly monitor the environmental level\nof irradiance and according raise or lower themselves to a\ndepth at which the total irradiance ( H(z,+) + H(z,-) ) is on\nthe order of 10 microwatts/cm2. A11 this activity trans-\npires along with the flashing at the above-mentioned mean ir-\nradiance and frequencies. The type of flashes are temporally\nhighly peaked and these peaks were observed to be one to two\norders of magnitude greater than the total environmental irra-\ndiance (see Fig. 1.37). It appears that this is an optical\nmeans of assuring togetherness during the vertical migrations,\nfor the eye pigment of the Euphasia has a greatest photosen-\nsitivity to the predominant color of its flashes.\nUsing the irradiance models developed in Chapter 8, it\nis a relatively straightforward task to describe and predict\nthe light field generated in the sea by extensive layers of\nthe Euphasia or other stratified biological sources of radi-\nant flux. The photoregulative activities of these creatures\ncoupled with the general food chain activities in the seas\nH watts/cm 2\nTOTAL IRRADIANCE\n10-6\n10-4\n10-2\n100\n102\n4\n10\no\n50\n100\n150\n200\n250\n300\n350\nFIG. 1.37 Depth dependence of downward irradiance in\nwhich discrete flashes of light generated by Euphasia pacifica\nare evident at the depths around 300-350 meters, as observed\nby Kampa and Boden in the San Diego Trough, 20 February 1956.\n(From [133], by permission)","SEC. 1.3\nTHREE SIMPLE MODELS\n55\npresents a challenging problem to hydrologic optics in the\ndescription of the dynamical interactions of plants, animals\nand photons in seas and lakes. We shall briefly reconsider\nthis problem in Sec. 1.10.\nThree Simple Models for Light Fields\n1.3\nHow do we seek order in all that we have encountered\nabove? How do we incorporate those few evidences of order,\nalready glimpsed, into some greater scheme, satisfying for\nits accuracy, comprehensiveness, and relevance to the main\nstream of modern physical theory? The number of effects to\nbe described is great, and their intricacy has a tendency to\ninitially intimidate those who attempt a precise description:\nnature's ways are orderly but infinitely complex, the theo-\nrists are few and finite; therefore, each stage of theoreti-\ncal knowledge inevitably rests on chosen compromises. Three\nsuch theoretical compromises are selected for study here;\neach is designed to describe one facet of the radiometric\ncomplex encountered in the seas and lakes of the earth: the\nfirst two describe the light fields generated by sunlight and\nskylight and give simple models for the radiance distributions\nand two-flow irradiance fields; the third describes artificial\nlight fields set off in the water by man-made point sources\nand extended artificial sources of radiant flux.\nThe Two-Flow Model\nThe two-flow model of the light field pictures the ra-\ndiant flux in a natural hydrosol X, free of internal sources,\nas divided into two streams at each depth Z below the bound-\nary: a downward stream of radiance H_ and an upward stream\nof irradiance H+ (see Fig. 1.38). The primary purpose of the\nmodel is to predict H+ and H_ at each depth z, given H+ and\nH_ at the upper boundary, or more generally, given H+ at some\ndepth and H_ at another (possibly the same) depth. The hy-\ndrosol, therefore, is viewed by this model as a plane-parallel\nmedium, i.e., an infinite region of space caught between two\nhorizontal parallel planes, which are the boundaries of the\nmedium. The physical properties of the hydrosol are described\nin the present model by means of two optical properties a, b;\nand the geometrical flow of the radiant energy is described by\nmeans of a distribution factor D. These three concepts are\ndefined in detail as follows. We write:\nthe amount of irradiance absorbed from a\n\"a\"\nfor\nnarrow vertical beam of radiant flux of\nunit irradiance as it crosses a horizontal\nlayer of unit thickness in X.\nthe amount of irradiance back scattered\n\"b\"\nfor\nwithout change in wavelength from a given\narbitrary stream of radiant flux of unit\nirradiance as it crosses a horizontal layer\nof unit thickness in X.\nFinally, if h+, h_ are the scalar irradiances associated with","56\nINTRODUCTION\nVOL. I\nair\nT\nwater\n(measured\nX\npositive\nZ\ndownward)\nH-\nAZ\nthin layer\nH+\nbottom\nFIG. 1.38 Setting for the two-flow model for irradiance.\nthe two given streams of radiant flux in X, we write:\n\"D+\" for h+/H +\nD+ give the mean distances traversed by each stream through a\nhorizontal layer of unit thickness. They are also convenient\nmeasures of the diffuseness or collimatedness of the flows.\nThis latter interpretation can be made plausible by a few ex-\namples. If the downward stream, say, is collimated, i.e., in\nthe form of a narrow beam which makes an angle 0 with the ver-\ntical, then from (9), (10) of 1.1 it is easy to see that\nD_ = sec 0. Further, if the downward radiance distribution\nis uniform, then by (11), (15) of Sec. 1.1, we have D_ = 2.\nIn the model currently under study, it is assumed that:\nD+ = D_\n(1)\nand we shall write \"D\" for this common value. (On the basis\nof this assumption, we occasionally call the resultant two-\nflow model the one-D (two-flow irradiance) model.) It is\neasy to see that the amount of irradiance lost by absorption\nfrom a flow of unit irradiance and of distribution factor D,\nas it traverses a unit thickness layer in X, is aD. On the\nother hand the amount of loss by backscattering is simply b,\nwith the quantity D not appearing explicitly. The reason why\nabsorption is treated differently than scattering in the above\nsense, rests in the fact that these processes manifest them-\nselves differently geometrically : when flux is absorbed it\ndisappears from the scene; when it is scattered, it must","SEC. 1.3\nTHREE SIMPLE MODELS\n57\nstill be contended with in the radiometric scene. This is\ndiscussed further throughout Chapter 8, along with precise\ndefinitions of D and b.\nWe are now ready to derive the basic differential equa-\ntions of the two-flow model.\nConsider the downward stream of radiant flux as it\npasses through a horizontal layer of thickness Az, where Z\nis\nmeasured positive in the downward direction. (Fig. 1.38) As\nthe stream progresses through the layer, it is partially ab-\nsorbed and partially scattered backwards to join the upward\nstream of flux. The total amount of irradiance lost from H_\nby these two processes is, according to the definitions of a\nand b:\naDH_Az + bH Az\nOn the other hand, H_ will be increased by that amount of\nflux, namely bH+Az, scattered backwards from the upward\nstream. The net change AH_ of the downward irradiance, after\ntraversing the layer of thickness Az, is therefore:\nAH = - (aD + b)H_ Az + bH Az\n(2)\nIn the same way we find that for the upward stream of radiant\nflux, which moves through the same layer (so that its asso-\nciated Az is negative) the net change AH+ of H+ is:\nAH = - (aD + b) H+ (-Az) + bH (-Az)\n(3)\n.\nDividing each side of (2) by Az, and each side of (3) by -Az,\nand letting Az-0, we have:\ndH_ = - (aD + b)H + bH\n(4)\n,\ndz\ndH+ = - (aD + b) H + bH\n(5)\ndz\nThese equations constitute the two-flow model for light fields\nin homogeneous stratified natural hydrosols. This model (the\none - D model), in undecomposed form, in essence goes back to\nSchuster in 1905 who first formulated similar equations in\nthe astrophysical context. In Chapter 8 we review the high\npoints of the model's history and place it on a sound physical\nand mathematical basis. For the present, however, we indulge\nin a relatively uncluttered derivation and solution of the\nmodel, in order to point up its central ideas and its simple\nbeauty.","58\nINTRODUCTION\nVOL. I\nThe solution of the system (4), (5) is\nH(z,-) = m+g_e kz + m_g+e-kz -kz\n(6)\nH(z,+) = m+g+e kz + m_g_e-kz\n(7)\nwhere m+, m_ are arbitrary constants to be fixed by specify-\ning either one of H+ and H_ at each of two chosen depths (dis-\ntinct or not) and where we have written:\n\"g+\"\nfor\n(8)\n,\nand:\n\"k\"\nfor\n[aD(aD + 2b)]\n(9)\nThis completes the construction of the two-flow model. We\nshall put it to work in Sec. 1.4.\nThe Radiance Model\nThe radiance model connects the radiances at the begin-\nning and end of an arbitrary path, such as AB, in a natural\nhydrosol X (Fig. 1.39). Thus, given the radiance at A in the\ndirection E, the model yields the radiance at B in the same\ndirection E. This model is quite general, for we can choose\npoint A to be on the upper or lower boundary of X and so the\nradiance at the end B will give the apparent radiance of the\nboundary; and this is just the radiance one sees or measures\nat B with a radiance meter.\nIn order to construct such a model we need to know what\nhappens to the radiance as it travels along a straight path\nin the water. If we imagine the radiance to be generated by\na swarm of photons travelling along the path, then on the one\nhand we would expect this swarm to lose some members via\nscattering and absorption at each point along the path. Ac-\ncordingly, let us write:\n\"a\"\nfor the amount of radiance absorbed from a\nnarrow beam of radiant flux of unit ra-\ndiance travelling a unit distance along\na path.\nand\n\"s\"\nfor\nthe amount of radiance scattered without\nchange in wavelength from a narrow beam of\nradiant flux of unit radiance travelling a\nunit distance along a path.\n*H(z,+) is the value of the function H+ at depth Z. Similar-\n1y, H(z, -) is the value of H_ at Z. The functional notations\n\"H+\" and \"H(,=) \" are to be considered synonymous and may be\nused interchangeably.","SEC. 1.3\nTHREE SIMPLE MODELS\n59\nk\nunit sphere\nair\nwater\nX\nZo\no\n0\nA\nZ\nZ'\n- r COS 0\nB\nA\nforward scatter\ndirection on lobe\n0\nscattering into path (gain)\ngiving rise to N\n*\nabsorption from path\n(loss) giving rise\nto - -a N, r\nscattering out of path (loss)\ngiving rise to - -sNr\nFIG. 1.39 Setting for the radiance model.","INTRODUCTION\nVOL. I\n60\nWe note in passing that the volume absorption function a for\nX just defined is identical with that defined for the two-\nflow model The function S is the volume total scattering\nfunction for X.\nNow, on the other hand, we would expect the swarm of\nphotons to gain new members from the surrounding environment\nsimply as a result of some of the nearby photons being scat-\ntered into the swarm as it passes along a small segment of its\npath. Thus, let us write:\nfor the amount of radiance scattered without\n\"N*\"\nchange in wavelength into a narrow beam\nof radiant flux travelling a unit dis-\ntance along a given path past a given\npoint.\nIf No is the inherent radiance, of the path, i.e., , the begin-\nning radiance at point A in Fig. 1.39, and Nr is the apparent\nradiance of point A as seen at point B a distance r along the\npath, then according to the above remarks the change ANT of\nNr in the next increment of distance Ar along the path is\nexpected to be:\nANT==(a + s)N.Ar N * Ar\nDividing by Ar and letting Ar-0, we arrive at\n(10)\ndr =\nwhere we have written\n\"a\" for a + S\n(11)\nEquation (10) is the equation of transfer for radiance. It\nis the central equation of radiative transfer theory. We\ncall a the volume attenuation function and N the path func-\ntion. The equation is used to connect the value Nr ( 2, 0) of\nNr at depth z, in the direction 0 with the value No (20,0) of\nNo at depth 20 in the direction 0. (See Fig. 1.39.)\nAs it stands, (10) looks like a simple differential\nequation, and, indeed, it is readily integrated if we know a\nand N along the path. We shall assume a to be constant\nalong the path, and N* to be given along the path, and that\nN varies only with depth. Then it is easily verified that\nthe general solution of (10) is (see, e.g., (1) - (3) of Sec.\n3.15)\nr(z,0)\n(12)\nz'= cos 0","SEC. 1.3\nTHREE SIMPLE MODELS\n61\nThe simple model we are interested in at present rests on the\nassumption that N (z,0) in optically very deep media depends\nonly on depth Z in X, in the manner:\nN*(z,0) N+(0,0)e-Kz\n(13)\nwhere K is the empirical depth rate of decay of the general\nlight field in X. For example it may be taken as the empir-\nical K in (7) of 1.2, or the theoretical k in (9) above en-\ncountered in the two-flow model (cf. (61) of Sec. 1.4). At\nany rate, using (13) in (12), performing the integration and\nsimplifying, we have:\n)1\n(14)\nThis is the requisite simple model for radiance. We shall\nstudy it later to see if it helps us understand some of the\nobserved properties of the underwater light field surveyed in\nSec. 1.2. It is a simple matter to generalize (14) to the\ncase where N* (z,6) depends also on the azimuth angle . (See\nChapter 4.) For the present we can think of (14) holding in\nan arbitrary given azimuth plane.\nThe Diffusion Model\nThe diffusion model is designed to describe the spatial\nvariation of scalar irradiance in a natural hydrosol. This\nmodel together with the two-flow model for irradiance, and\nthe model for radiance, forms a reasonably exhaustive battery\nof elementary descriptions of most of the natural and artifi-\ncial light fields encountered in everyday practice.\nA simple and instructive route to the diffusion model\ncan be made via the two-flow model (4), (5), as follows. Let\nus add together, term by corresponding term, the two equations\n(4), (5). We find:\n(15)\n=\nNow, according to (8) of Sec. 1.1 and the definition\nof net irradiance H(z,+), which is defined by writing:\n\"H(z,+)\" for H(z,+) - H(z,-)\n,\nor more briefly:\n\"H+\"\nfor H+ H\n\"H_\"\nfor H_ H\n,","VOL. I\nINTRODUCTION\n62\nwe can cast (15) into the form\ndH_\ndH+\n(16)\nah\n=\n=\n,\ndz\ndz\nusing the definition of the distribution factor D, and (1).\nThis states that the depth rate of change of the net upward\nirradiance at a point is jointly proportional to the volume\nabsorption coefficient and the scalar irradiance at that\npoint.\nReaders familiar with the rudiments of vector analysis\nwill see that either derivative term on the left side of (16)\nis simply the negative of the divergence of the vector irra-\ndiance H (cf. (4) of Sec. 1.1). The other two (the x,y) de-\nrivatives of the components of H are missing from (16) be-\ncause the two-flow model applies to stratified media, i.e.,\nmedia whose properties are constant over horizontal planes in\nthe hydrosol. However, this recognition of the nature of the\nleft side of (16) permits us to write:\n(17)\nV.H = -ah\nin place of (16).\nEquation (17), despite the route we have just taken, is\na quite general law which holds in source-free media of arbi-\ntrary shape and inhomogeneities and whose light fields are of\narbitrary spatial and directional structure. We have in this\nway made a leap from the special to the general by making a\nsimple observation on the mathematical form of the divergence\nof a vector field. (For further details, see (5) of Sec. 2.8\nand (15) of Sec. 8.8.) An even more general form can be ob-\ntained if we allow the presence of sources in the medium:\n(18)\nV.H = -ah+h n\nwhere hn is the radiant flux generated per unit volume by in-\nternal sources.\nNow, the diffusion model we are interested in springs\nfrom (18) once we have made a special assumption about the\nbehavior of the light field and the nature of the term hn.\nThe requisite assumption is concerned with the scattered\nlight field in the medium of interest, so that we shall look\nonly at the components of H and h which consist of radiant\nflux having been scattered at least once. In order to point\nthis up in the notation, it can be shown that we may write\n(17) in a form quite analogous to (18) :\nV.H* = -ah*+h+ 1\n(19)","SEC. 1.3\nTHREE SIMPLE MODELS\n63\nThis star notation is standard notation for scattered radiant\nflux. To indicate how we may arrive at (19), we first observe\nthat the full vector and scalar irradiance fields are repre-\nsented as:\nH=HO + B*\n(20)\nh = h o h n\n(21)\nwhere H°, h o consist of residual radiant flux directly trans-\nmitted from the sources and boundaries. When written in this\nform, we say that the light field H has been decomposed into\nits residual and scattered parts. This mode of decomposition\nis not new to our discussions in this chapter. For we have\nin effect represented the apparent radiance Nr in (12) in pre-\ncisely this way. Indeed, if in the context of (12) we write\n\"NO\"\nNoe-ar\nfor\n(22)\nand\nfor (\"Nae-a(r-r')dr'\n''N*''\n(23)\n,\nthen the equation (12) for apparent radiance NT becomes (in\nfunctional form) :\n(24)\nwhere NOT is the residual radiance and Nr the path radiance.\nThis form is completely analogous to (20), (21) In fact, all\nwe have to do to get (20), (21) is integrate (24) over all\ndirections and apply (3), (4) of Sec. 1.1 (cf. Secs. 6.5 and\n6.6). Hence if we integrate each side of (10) over all di-\nrections in this manner, we can obtain (19) quite rigorously.\nThe complete details of this derivation may be found in the\nderivation of (63) of Sec. 6.6.\nWe return to (19), and make the assumption about H\nwhich invokes the desired diffusion model. The assumption is\nsimply this:\nH* shall be proportional to -Vh*\n(25)\nHere Vh* is the gradient of h*. For example, in a stratified\nplane-parallel medium, this amounts to saying that:\ni.e that the scattered irradiance vector--which in the sea\nclearly points downward in the direction of greatest net irra-\ndiance is simply the derivative of the scattered scalar irra-\ndiance times the unit downward vector (-k), i.e., the vector\npointing along the direction of increasing Z. It is inter-\nesting to note that this is a sort of backwards version of","VOL. I\nINTRODUCTION\n64\n(16), obtained from the latter essentially by moving the de-\nrivative operation from its left to its right side. Notice\nthat H* is required by (25) to point in the direction of de-\ncrease of h. In natural waters dh/dz is negative (with in-\ncreasing Z measured downward as usual). We shall use the con-\nventional symbol \"D\" for the diffusion constant of propor-\ntionality. Notice that its dimension is that of a length.\n(We use the letter \"D\" here without fear of confusion with\nour distribution coefficients.) Hence assumption (25) can be\nwritten as an equality:\n*\nH* = -DVh\n(26)\nand when this assumption is used in (19) we have:\nv.(-Dvh*) = -ah* + h1\nor, since D is a constant we have, finally:\n(diffusion\n-DV2 * + ah* = h\n(27)\nequation for\ndecomposed\nlight field)\nwhich is the present desired form of the diffusion model.\nThe symbol \"D2\" is the laplacian operator used in vector anal-\nysis. In this model we assume that the source term h1 des-\ncribes the origin of the scattered scalar irradiance h* and\nthereby is of the form:\nh 1 = hos\n(28)\nwhere S is the volume total scattering coefficient defined in\nthe preceding radiance model discussion and ho is the scalar\nirradiance associated with the residual flux from the source\nand boundaries. The diffusion model takes its name from the\nassumption (26), which is Fick's law of diffusion, now applied\nto the diffusion of photons.\nEquation (27) as it stands constitutes a reasonably\ngood model of the scattered (or diffuse) scalar irradiance in\nboth natural and artificial light fields. By way of contrast,\nwe observe that it is more accurate than the diffusion model\nthat comes from applying (26) (without the stars) to (18),\ninstead of (26) to (19). For in the former case, i.e., when\napplying (26) (without the stars) to (18) we find\n(diffusion\n2\n(29)\n-DV h + ah = hn\nequation for\nundecomposed\nlight field)\nand even though the mathematical forms of (27) and (29) are\nthe same, an essential difference between them arises by vir-\ntue of the nature of the source term hn. In the case of (29),\nhn for artificial point sources is a Dirac delta function,\nwhereas in (27), as we see by (28), , h1 is a relatively","SEC. 1.3\nTHREE SIMPLE MODELS\n65\nsmoothly varying function throughout the medium. Since dif-\nfusion models become more accurate the smoother the spatial\nvariation of the source terms, the superiority of (27) over\n(29) is quite clear.\nHowever, it takes correspondingly more effort to solve\n(27) than it does (29). The formal solution of (29) for a\npoint source is straightforward, and takes the form:\n(undecomposed\nh(r) = Joe-kr\nh, and point\n(30)\nsource)\nwhere we have written\nPO\n\"Jo\"\nfor\n(31)\n4TT\nand PO is the radiant flux output of the point source, as-\nsumed to be uniform in all directions. Furthermore r is dis-\ntance from the observation point to the point source, and we\nhave written:\na\nPla\n\"K\"\nfor\n(32)\n,\nwhere a is the volume absorption coefficient for the medium,\nand D is the diffusion constant (cf. (27) of Sec. 6.5)\nThe general solution of (27) is now forthcoming by\nmeans of (30) and a straightforward integration. To see this,\nwe imagine that at each point x' of the Medium X (which is an\nextensive region without perturbing boundaries) the residual\nscalar irradiance ho (x') is scattered, there to give rise to\nan entirely new point source problem whose solution at an ob-\nservation point x is described by (30), now written in the\nform:\n(33)\nwhere\n|x-x'\n(34)\nr\nand\n(35)\nHence if the original point source is at the origin (i.e., at\nx=0), and of a relatively mild directional output, then the\nscalar irradiance field h(x) at x is given very nearly by:\nh(x) = h o (x) + h*(x)\n(36)","VOL. I\nINTRODUCTION\n66\nwhere\nh*(x)=\n(37)\n=\nand\n(38)\nand where\nr'=\n(39)\nand s(x') is the solid angular subtense of the point source\nas measured at x'. The source is actually a small finite\nsphere of surface radiance N° in the direction E' = x/|x'\nV is the volume measure in X. We shall not go into further\ndetails here. See (66) of Sec. 6.6 in particular, and Sec.\n6.6 in general for complete details.\n1.4 Some Deductions from the Light Field Models\nThe three models for natural and artificial light fields\nderived above allow us to explain and interrelate many of the\nobserved features of light fields in natural hydrosols. We\nshall consider here and in subsequent sections a small rep-\nresentative sample of such activity, based on simple deduc-\ntions from the three models.\nThe Decay of the General Light Field with Depth\nWe shall now show how (7) of Sec. 1.2 follows from the\ntwo-flow model for light fields. Toward this end, we let the\nscattering medium X be infinitely deep and be absorbing, i.e.\na>0. Then we compute the net downward irradiance at a general\ndepth, using (6), (7) of Sec. 1.3.\nH(z,-)H(z,-)-H(z,+)\n(1)\n=\nNow from (16) of Sec. 1.3 we find, by integrating between\ndepths 0 and z, and noting that h(z) is a non negative quan-\ntity for all Z :\ndz's\nHence for all 2 :\nH(z,-) (,-)\n(2)","SEC. 1.4\nSOME DEDUCTIONS\n67\nThis shows that the net downward irradiance is bounded. In-\ndeed, from Tables 2, 3 of Sec. 1.2 we can estimate an upper\nbound of H(z,-) as 1396 watts/m², and infer that H(z, -) > 0\nin real optical media. It follows that (2) and (1), along\nwith a A 0, force m+ to be zero; otherwise we could find a\ndepth Z at which (2) would be violated. Some further general\ninequalities related to (2) are given in Sec. 9.2.\nHaving established that m+=0 in infinitely deep absorb-\ning media, (6), (7), of Sec. 1.3 yield the requisite forms of\nH(z, for every Z :\nH(z,-) = m_g.e-kz =\n(3)\n(4)\n=\nFrom (3), (4) we have, on setting z=0:\nH(0,-) = m_g =\nH(0,+) = m_g_ =\nLet us write\n\"Roo\"\nfor\nH(0,+)/H(0,-)\nClearly, we then have from (3), (4) :\n(5)\nThis shows that the reflectance Roo of the medium is indepen-\ndent of depth and determinable once a, k, and D are known.\nHence for every z,\nH(z,+) = H(z,-)R. =\nwhere\nThus we have shown, among other things that:\nH(2,+) = H(0,+)e-kz =\n(6)\nfor all Z.\nFurthermore, by definition of the distribution factor D\n(cf. (1) of Sec. 1.3) we have, with the help of (8) of Sec.\n1.1:","VOL. I\n68\nINTRODUCTION\nh(z)h(z,+)h(z,-)\n=D(H(Z,+)+H(z,-) )\nD( H(0,+) + H(0,-) e-kz\n(7)\nwhich is the theoretical basis for (7) of Sec. 1.2.\nObserve how the assumption that a>0, is needed in vari-\nous parts of the arguments above. This assumption is quite\nreasonable in terrestrial settings; indeed, in such settings\nthe condition a=0 for every wavelength is never observed.\nWhat would the light field look like in an infinitely deep\nmedium in which a=0? Equation (1) shows us that if a=0 for\nall wavelengths, then: since gu g+=1,\nH(z,-) = 0\nso that\nH(z,-) = H(z,+)\nat all depths Z and for all wavelengths. The sea would be of\nthe same general brightness and color of the sky in this case\n--at every depth!\nReflectance and Transmittance of\nFinitely Deep Hydrosols\nThe simple two-flow model allows us to estimate the re-\nflectances and transmittances of finitely deep layers of\nwater. We return to (6), (7) of Sec. 1.3 and consider a fi-\nnitely deep homogeneous layer whose upper surface is at 0 and\nwhose lower surface is at Z. The upper surface is irradiated\nwith a given irradiance H(0,-) and we set H(z,+)=0, which sim-\nulates zero irradiation at the lower boundary (Fig. 1.40 (a)).\nWe then find the m+, m_ corresponding to these two given ir-\nradiances, and solve for H(0,+). Thus, if under these condi-\ntions we write\n\"Ry(T)\" for H(0,+)/H(0,-)\nthen Ry(T) is the reflectance of the slab of (diffuse) optical\ndepth* T = kz, and Ry(T) is found to be of the form:\ne-T\n-\n(8)\n*There are many 'optical depths' possible in radiative trans-\nfer theory; one for each scattering or absorbing concept. In\nthe present case we use k as a base for optical depth.","SEC. 1.4\nSOME DEDUCTIONS\n69\n(b)\n(a)\nH(O,-)\nH(O,+)\nH(O,-)\n(GIVEN)\n(REQUIRED)\n(GIVEN)\nH(Z,+)=0\nH(Z,-)\nH(Z,+)=0 (GIVEN)\n(REQUIRED)\n(GIVEN)\nT is optical depth corresponding to Z\nFIG. 1.40 Boundary conditions for the reflectance and\ntransmittance of finitely deep layers in a hydrosol.\nwhere we have written:\naD\n\"y\"\nfor\n(9)\nk\nThe transmittance Ty (T) of the slab of optical depth T can be\nfound in an analogous manner (Fig. 1.40 (b) ) by now seeking\nH(z,-) under the same conditions. Thus if we write:\n\"Ty(T)\" for H (z, -) /H(0, -)\n,\nthen it follows that:\n4Y\nI(T) = -\n(10)\nOne should see that, because the medium is homogeneous, Ry(T)\nand Ty (T) depend spatially only on the optical depth T, so\nthat (8) and (10) pertain to any slab of thickness T in the\nmedium regardless of its vertical location within the medium.\nIt will also be interesting to look at some of the lim-\niting values of Ry (T) and Ty(T) for various extreme values of\nT and Y. For example, one may verify that:","VOL. I\nINTRODUCTION\n70\n(11)\n(12)\n(13)\n(14)\n(15)\n(16)\nFrom (15) we see that the reflectance of very thin slabs is\nproportional to the backscattering coefficient b. Indeed,\nso that:\n(17)\nRy(T) 12 bz\nfor small T. From (16) we see that the transmittance of very\nthin slabs is:\n(18)\nTy(t) = 1 - (aD+b)z\n.\nFrom (17), , (18) we conclude that for thin slabs:\nand if in general we write:\n\"Ay(T)\" for 1-[R,(+) + Ty(+)\n(19)\nwe see that in particular for thin slabs:\n(20)\nAy(7) = (aD) 2\n.\nClearly Ay(T) for general T is the amount of irradiance ab -\nsorbed by a slab of optical thickness T and with optical prop-\nerties a, b, and D. From (19) we have the general conserva-\ntion law:","SEC. 1.4\nSOME DEDUCTIONS\n71\nAy(T) Ry(T) + Ty(T) = 1\n(21)\nFigs. 1.41, 1.42 represent Ry (T) and Fig. 1.43 represents\nTy(T) for a selected set of Y and T values. Values of k and\nY can be obtained by direct computation from the definitions\nof k and Y, or by their graphs in Figs. 1.44, 1.45. The com-\nputations were done by Mrs. Judith Marshall.\nInvariant Imbedding Relations for Irradiance\nWe now wish to investigate a particularly interesting\nproperty of the reflectance and transmittance functions Ry (T)\nand Ty (T) defined above. This property will allow us to\nwrite down Eqs. (6), (7) of Sec. 1.3 by sight for homogeneous\nmedia with transparent boundaries. We shall fix attention on\nan arbitrary medium X whose upper boundary is at optical\ndepth 0 and whose lower boundary is at optical depth C (= zk),\nwhere Z is the geometric depth of the medium. Since X is\nfixed throughout the present paragraph, we can drop the \"y\"\nfrom the R and T notation. Furthermore, to emphasize the geo-\nmetric limits of X we shall denote it by \"X(0,c)\".\nNow suppose X(0,c) is irradiated at the upper boundary\nonly. Then by definition of R(c) and T(c) we have:\nH(0,+)=H(0,-)R(c)\n(22)\nH(c,-) H(0,-) T(c)\n(23)\nThis is a simple application of (8) and (10) and the basic\nmeanings of R(c) and T(c). Next, assume that X(0,c) is ir-\nradiated only on its lower boundary. Then, by the same token:\nH(0,+) = H(c,+) T(c)\n(24)\nH(c,-) = H(c,+) R(c)\n(25)\nThese formulas follow rigorously using the pattern of deriva-\ntion leading to (8) and (10). However, they should be intui-\ntively clear simply on the basis that T(c) and R(c) are trans-\nmittances and reflectances of homogeneous slabs of scattering\nabsorbing material of optical thickness C in which complete\nsymmetry of the light field has been assumed (in the form of\n(1) of Sec. 1.3).\nFurthermore, and this is a crucial step, because the\nbasic differential equations of the two-flow model are linear,\nhave at our beck and call the mathematical principle of\nwe\nlinear superposition of solutions of these equations. Thus,\nif X(0,c) is irradiated simultaneously at levels 0 and C,","VOL. I\nINTRODUCTION\n72\n1.0\nY\nOI\n05\n.80\n.IO\n15\n.20\n60\n25\n30\n35\n40\n40\n45\n.50\n60\n20\n.70\n80\nO\n12\n80\n10\n825\n08\n85\n875\n.06\n.90\n04\n925\n.95\n.02\n96\n97\n.98\n99\nO\nO\n5\n1.0\n1.5\n2.0\n2.5\n3.0\nT\nFIG. 1.41 Calculated reflectance R (T) versus T, for\nY\n.01 .80.\nFIG. 1.42 Calculated reflectance R (T) versus T, for\n.80 ISQ. FT.\nFIG. 1.84\n10-2 SQ.FT.\n1111 111\n150\nHIIIIII\n10-1SQ.FT.\n100\n100 120\n0.7\nK\n3\n2\n4\n5\nI\n2\n3\n4\n5\n6\n7\n.02\n.03\n.04\n.08\n.05\n.06\n.07\n.09\n.01\n.12\n.13\n.14\n10\n.11\no\nD","VOL. I\nINTRODUCTION\n162\nterms of the numerical example in the preceding paragraph, his\nadaptation luminance is 2x2 = 4 foot-lamberts. The theoreti-\ncal basis for the inclination factor is (68) of Sec. 1.4.\nBottom Influence\nIf the swimmer is near the bottom, his adaptation may\nbe affected, depending (i) on how greatly the bottom differs\nin reflectance from 1/50, (ii) on the clarity of the water,\nand (iii) upon its distance from the swimmer. Generally\nspeaking, dark mud bottoms have little or no effect on adap-\ntation and light-colored bottoms have negligible influence\nwhen the sighting range is the order of 3/K or greater. Even\nat a sighting range of only one diffuse attenuation length\n1/K, few bottoms are white enough to affect the swimmer's\nadaptation significantly. Generally speaking, therefore, the\ninfluence of the bottom upon adaptation can be neglected in\ncalculating visibility by swimmers. It should be noted, how-\never, that the reflectance of the bottom may have a major\neffect on the inherent contrast of the object and, therefore,\nupon its visibility, as discussed in Section B. 2 below.\nA.5 Calculation of Adaptation Luminance\nThe foregoing discussion can be summarized and illus-\ntrated by concrete examples: let it be required to find the\nadaptation luminance for a swimmer 60 feet beneath the sur -\nface of deep water characterized by a diffuse attenuation CO-\nefficient K of 0.10 per foot, or 0.328 per meter which, as we\nhave seen in Tables 7 and 8 of Sec. 1.6, is on the order of\nK-values found in clear lake water. It is also a value typi-\ncal of coastal water. Let it be assumed that the sun is 16.8\ndegrees above the horizontal plane on a clear sunny day.\nReference to Figure 1.12 shows that the illuminance on\nthe sea-surface is 2000 lumens/ft2. Inspection of the line\nmarked K = 0.10 per foot in Figure 1.82 shows that the hori-\nzontal plane containing the swimmer receives 2.5 x 10 3\nas\nmuch downward light as does the sea-surface, or 2000 x 2.5\nx\n10- = 5 lumens/ft2\nIf the swimmer looks straight downward his adaptation\nluminance will be 5 x 1/50 = 0.1 foot-lamberts if there is no\nbottom influence.\nIf the swimmer looks along a downward slant path having\na zenith angle of 110 degrees in a plane at right angles to\nthe azimuth of the sun, the inclination factor graph in Fig.\n1.84 shows that his adaptation luminance is 2.5 times greater\nthan if he looks straight down. Along this inclined path of\nsight the swimmer's adaptation luminance is, therefore,\n0.10 x 2.5 = 0.25 foot-lamberts. The user of Figure 1.84\nshould verify that the \"across sun\" curve is applicable by\nnoting that the depth (60 feet) of the swimmer is 6/K, since\nK = 0.10 per foot, and that this depth lies between limits\nspecified in the figure.\nHad the solar elevation been 65 degrees, Figure 1.12","SEC. 1.9\nUNDERWATER VISIBILITY PROBLEMS\n163\nshows that the illumination at the sea-surface would have\nbeen 10,000 lumens/ft2 and the adaptation luminances of the\nswimmer at 60 feet would, therefore, have been five times\nhigher; i.e., 0.50 foot-lamberts when looking straight down\nand 1.25 foot-lamberts when looking at right angles to the\nazimuth of the sun along a downward path of sight having a\nzenith angle of 110 degrees.\nA.6 Chart Selection\nParagraph B below (in Figs. 1.89-1.106) contains nine\npairs of nomographic charts, each pair representing a decimal\nvalue of adaptation luminance, as follows: 1000, 100, 10, 1,\n10-1 10-2, , 10-3, 10-\", 10-5 foot-lamberts. One member of a\npair is for low clarity, the other for high clarity water.\nAfter the adaptation luminance of the swimmer has been calcu-\nlated the chart closest to this level is selected. If the\nadaptation luminance is not close to any decimal value, sight-\ning range for the visual target should be calculated by means\nof charts for higher and lower light levels respectively in\norder to bracket the desired answer and provide for interpo-\nlation between these sighting ranges.\nB. Using the Nomographs\nB.1 Introduction\nOnce the adaptation luminance for the swimmer has been\ndetermined and the proper nomographic chart selected, sight-\ning ranges can be predicted. The calculation procedures are\nslightly different for each type of visual task and, there-\nfore, they will be discussed separately. The basic nomo-\ngraphs are given in Figs. 1.89-1.106. However, for illustra-\ntive purposes, two charts have been excised from that group\nand appear in Figs. 1.84 and 1.85. This is the low-clarity,\nhigh-clarity pair for 10-1 foot-lambert adaptation.\nB.2 Objects on the Bottom\nThe nomographic visibility charts can be used to calcu-\nlate the sighting range of flat, horizontal objects of uniform\nreflectance lying on the bottom.\nObject Size and Shape\nThe size of the object is measured by its area, expres-\nsed in square feet; the shape of the object is unimportant un-\nless it is an extremely elongated form (10:1 or greater) and\nunless adaptation luminance is 10 foot-lamberts or greater.\nEven in such unusual cases the effect of object shape on\nsighting range is usually small.\nVertical Path of Sight\nSighting range calculations are simplest when the path\nof sight is vertically downward. Each nomograph requires","VOL. I\nINTRODUCTION\n164\nfive items of input data: target area, target reflectance,\nbottom reflectance, the volume attenuation coefficient a, and\nthe diffuse attenuation coefficient K. The coefficients a\nand K must be for the water between the swimmer and the tar-\nget.\nThe vertical scales on the nomographs are labeled\n\"a-K cos 0\". (The use of a minus sign here, relative to the\nuse of a plus sign in Sec. 1.3 wherein the theory of the sim-\nple radiance model was developed, is to facilitate the direc-\ntion specifications by the swimmer. In other words we adopt\nhere field luminances and the swimmer-centered direction con-\nvention.) A downward vertical path of sight has a zenith an-\ngle 0 = 180 degrees, and cos 180 = -1. A point representing\nthe sum of a and K, expressed in reciprocal feet, is marked\non the left vertical scales.\nThe right vertical scales of the nomographs are labeled\n\"target reflectance minus bottom reflectance\". The algebraic\nsign of this difference is of no importance; if the bottom is\nmore reflective than the target the difference will, of\ncourse, be a negative number; disregard the negative sign and\nplot the magnitude of the difference on the right vertical\nscale. Reflectance must be expressed as a decimal; i.e., as\n0.06, not as six percent. Bottom reflectance should be meas-\nured at the sea-bottom with great care to avoid disturbing\nany fine silt which may be present. Bottom samples cannot be\nbrought to the surface for measurement without disturbing the\nmaterial sufficiently to alter its reflectance. Target re-\nflectance may be measured at the sea-bottom or on ship-board\nby means of a technique described in paragraph B.5 of this\nsection.\nThe curved lines which cover the upper right corner of\nthe nomographic visibility charts represent visual threshold\ndata for the target area with which each curve is identified.\n(The refractive effect of the swimmer's flat face-plate has\nbeen allowed for in constructing these nomographs.) Curves\nrepresenting decimal values of target area are marked accord-\ningly. Intermediate unmarked curves refer respectively to 2,\n4, 6, and 8 times the decimal value except in those cases\nwhen only a single line appears between decimal curves; in\nthis case the unmarked curve related to 5 times the decimal\nvalue.\nSpecial Charts for Water of Low-clarity. Two series of\nnine nomographic charts are presented below. In the first\nseries, the scales have been optimized for use in clear oce-\nanic and coastal waters where sighting ranges of 20 feet to\n100 feet or more often occur. The second series of charts\nare designed for waters of poor to medium clarity where\nsighting ranges of 1 foot to 20 feet or more prevail. Either\nseries of charts may be used for any problem having input\ndata within the range of its scales, but experience will even-\ntually indicate which chart is best suited for any given prob-\nlem.\nSighting Range Calculations, Clear Water. To calculate\nsighting range, connect the appropriate points on the left","SEC. 1.9\nUNDERWATER VISIBILITY PROBLEMS\n165\nand right vertical scales by a straight line and note its in-\ntersection with the curve corresponding to the area of the\ntarget. From this intersection proceed vertically to the\nsighting range scale. The following numerical example will\nillustrate this procedure with the aid of Figure 1.84.\nLet the following input data be assumed:\nAdaptation luminance = 10 1 foot-lamberts\nTarget: flat; horizontal; on the bottom\nTarget area = 10 square feet\nTarget reflectance = 0.080; non-glossy\nBottom reflectance = 0.030\nVolume attenuation coefficient = a = 0.073 per foot\nDiffuse attenuation coefficient = K = 0.027 per foot\nFrom these data, (recalling that paths of sight at pres-\nent are vertical) a + K = 0.100, and target reflectance minus\nbottom reflectance is 0.050. The solid line drawn on Figure\n1.84 intersects the curve marked \"10 square feet\" at the ver-\ntical line denoting a sighting range of 47.6 feet. The same\nline drawn on Figure 1.84 indicates that a swimmer looking\nstraight down under the assumed conditions can sight a 0.1\nsquare foot object at 43 feet, an object of 1 square foot at\n46 feet, and all objects of area 100 square feet or more when\nhe is 48.5 feet or less from the bottom.\nSighting Range Calculations, Low-clarity Water. The\nsame example may be solved by means of the low-clarity chart\n(Figure 1.85) and corresponding sighting ranges obtained, but\nwith far less precision.\nIn an hypothetical water of lesser clarity, character-\nized by a = 0.43 per foot and K = 0.17 per foot, the sum a+K\nis 0.60 per foot. If all other input data remain unchanged\nthe high-clarity nomograph (Figure 1.84) cannot readily be\nused because its left vertical scale goes only to 0.14. Ac-\ntually, this chart can be adapted by extending the left ver-\ntical scale linearly downward to 0.60 and constructing a di-\nagonal line from that point to 0.05 on the right vertical\nscale, but such a procedure is unnecessary because the low-\nclarity nomograph (Figure 1.85) is available. The straight\nline drawn on that figure indicates by its intersection with\nthe lower-most curve that flat horizontal objects of all\nsizes greater than 1 square foot can be seen by a swimmer\nlooking straight down under the assumed conditions when he is\n8 feet or less from the bottom. The same line shows by other\nintersections, that he must descend to within 7.5 feet of the\nbottom to see an object 10-2 square feet in area and to 5.5\nfeet from the bottom before a tiny object of area 10 square\nfeet can be seen.\nInclined Paths of Sight\nThe nomographic visibility charts can be used for the\ncalculation of sighting range along inclined paths of sight.\nThree additional items of input data are necessary: (1) the\napproximate azimuth of the path of sight relative to the sun,","INTRODUCTION\nVOL. I\n166\nNO1108 - 3ONV1937338\n(100s add) 0 soo X-D","SEC. 1.9\nUNDERWATER VISIBILITY PROBLEMS\n167\n(2) the depth of the swimmer expressed in units of 1/K, and\n(3) the zenith angle of the path of sight.\nThe first two items of data are used to estimate the in-\ncrease in adaptation luminance associated with the inclined\npath. This is accomplished by means of the inclination fac-\ntor curves in the lower left corner of Figure 1.84. (Identi-\ncal curves appear on all of the nomographic visibility charts.)\nA continuation of the numerical example begun in the preceding\nsection will illustrate this step:\nLet the following input information be assumed:\n(1) Azimuth of the path of sight: at right angles\nto the azimuth of the sun; i.e., the path of sight is \"across\nsun\".\n(2) Depth of swimmer = 2.7/K. This would be the\nif his depth is 100 feet and K = 0.027. The depth,\ncase\n2.7/K, falls within the range for which the \"across sun\" curve\napplies.\n(3) Zenith angle of the path of sight = 120 de-\nrees.\nEffect of Zenith Angle on Adaptation. Reference to the\n\"across sun\" inclination factor graph discloses that the in-\nclination factor for this zenith angle is 1.9. This means\nthat the adaptation luminance is 1.9 times as great as that\nexperienced by the swimmer when looking vertically downward;\ni.e., , 1.9 x 10-1 = 0.19 foot-lamberts. Since this adaptation\nluminance falls between the nomograms for 1 and 10-1 foot-\nlamberts, both charts should be used in order to bracket the\nsighting range. The effect of adaptation on sighting range\nwill be discussed further in a later part of this section and\nillustrated by Figure 1.86.\nEffect of Zenith Angle on Left Vertical Scale. The ze-\nnith angle of the path of sight (120 degrees) affects the val-\nue plotted on the left vertical scale of the nomograph:\na-K cos 0 = 0.43 - (0.17)(-0.50) = 0.51 .\n(A table of cosines is available in Table 7 of Sec. 12.1) Use\nthe relation cos 0 = -COS (180-0) for 0 in the range\n90 VI 0 <180.\nEffect of Zenith Angle on Effective Area. The effec-\ntive area of the object depends on the observer's line of\nsight; thus A cos (180-0) = 10 X 0.50 = 5 square feet. In-\nspection of the curves in Figure 1.85 shows that, in this\ncase, no sighting range will be lost by the foreshortening be-\ncause all targets having an effective area greater than 1\nsquare foot are visually detectable at the same distance un-\nder the conditions assumed in this numerical example.\nEffect of Inclination Factor on Right Vertical Scale.\nThe inclination factor affects the value plotted on the\nright vertical scale as follows: the difference between tar-\nget reflectance and bottom reflectance must be divided by the\ninclination factor before the number is plotted. Thus,","VOL. I\nINTRODUCTION\n168\n100\n10\nI\n10\n10-2\n10-3\n10-4\n10-5\n2\n4\n6\n8\n10\n12\no\nSIGHTING RANGE (FEET)\nFIG. 1.86 The effect of adaptation on sighting range (see\ntext) .","SEC. 1.9\nUNDERWATER VISIBILITY PROBLEMS\n169\n(0.050/1.9) = 0.026. The inclination factor curves which ap-\npear on each chart have been plotted on an inverted logarith-\nmic scale having the same modulus as the right vertical scale\nof the nomograph in order that the division can be accom-\nplished graphically. Draftsman's dividers can conveniently\nbe used for this purpose: measure downward from the top bor-\nder of the figure to the inclination factor curve and trans-\nfer this setting to the right vertical scale of the nomograph,\nusing it to reduce the plotted value of target reflectance\nminus bottom reflectance.\nCalculation of the Sighting Range. The broken line on\nFigure 1.85 shows that the sighting range would be 8.1 feet\nfor the inclined path if the adaptation luminance was 10-1\nfoot-lamberts Since, as shown above, the adaptation lumi -\nnance is 1.9 x 10- foot-lamberts a minor correction to the\nsighting range should be made in the following manner:\nEffect of Adaptation on Sighting Range. Since the lu-\nminance to which the swimmer's eyes are adapted is 0.19 foot -\nlamberts, an interpolation should be made between the sight-\ning range 9.1 feet indicated by the nomograph for 1 foot-lam-\nbert and the sighting range 8.2 feet indicated by the nomo-\ngraph for 10- foot-lambert. By linear arithmetic interpola-\ntion, 8.2 + (9.1-8.2) (1.9 x 10-1) = 8.4 feet. This value\ncompares with the sighting range of 8.5 feet found by the\ngraphical interpolation provided by Figure 1.86, which illus-\ntrates the effect of adaptation on sighting range in this il-\nlustrative example. Figure 1.86 has been prepared by assum-\ning successively all decimal values of adaptation luminance\nand plotting the resulting sighting ranges given by the en-\ntire series of nomographic charts. * Linear arithmetic inter-\npolation of sighting range between adjacent decimal levels of\nadaptation luminance suffices for the needs of most problems.\nImplication of the Sighting Range. Although the sight-\ning range for the inclined path (8.5 feet) happens to be only\nslightly longer than the sighting range for the vertical case,\nit should be recognized that the swimmer must be within 4.25\nfeet of the bottom in order to see the target at this inclin-\nation angle.\nB.: 3 The Secchi Disk\nThe underwater sighting range of a flat horizontal sur-\nface of uniform reflectance, suspended in (optically) deep\nwater, e.g., a Secchi Disk, can be calculated by means of the\nnomographic visibility charts. Ordinarily, Secchi Disk read-\nings are obtained by an observer above the surface of the sea\nThe discontinuity in curve slope at about 4.4 x 10 foot-\nlamberts results from a change from central fixation to avert-\ned vision on the part of the swimmer, in order to achieve max-\nimum sighting range in the dim light; this change of fixation\nis built into the nomographs.","170\nINTRODUCTION\nVOL. I\nwho must look downward through the surface (see the analysis\nof the Secchi Disk theory in Sec. 1.4). Sky reflection and\ncomplex refractive effects resulting from water waves greatly\ncomplicate the interpretation of the greatest depth at which\nthe disk can be seen. If, however, a swimmer lowers a Secchi\nDisk beneath him and observes its disappearance, the sighting\nrange can be predicted by means of the nomographic visibility\ncharts if a and K are known. Conversely, the observed sight-\ning range can be inserted in the nomograph in order to find\nthe sum of the attenuation coefficients, a+K.\nLet it be assumed that the water is so deep beneath the\ndisk that the bottom has no significant effect upon the light\nfield. The nomographs are so constructed that they will cor-\nrectly predict the sighting range of the disk if the right\nvertical scale of the nomograph is imagined to be labeled\n\"Secchi Disk reflectance minus 0.02\". All other details of\nthe calculation are identical with those described in the pre-\nceding paragraphs of this section which deal with objects on\nthe sea-bottom. Attention is called, however, to subject mat-\nter of Section B.7, entitled \"The Roo Correction\".\nB.4 Target Markings\nThe preceding paragraphs of this section have dealt\nwith the sighting ranges of the whole target. It is some-\ntimes required to calculate the sighting ranges of certain de-\ntails or markings on a target. This is readily accomplished\nby means of the nomographic visibility charts. The only mod-\nifications of the procedure described in the preceding para\ngraphs are (i) to imagine the right vertical scale to be la\nbeled \"reflectance of marking-reflectance of target\", and (ii)\nto use the curve which applies to the area of the marking.\nB.5 The Measurement of Target Reflectance\nThe reflectance of painted surfaces differ, often mar-\nkedly, when dry and when wet. The values of target reflect-\nance required for use in the nomographic visibility charts\nare those which would be measured by a water-filled reflect-\nometer submerged with the target. This submerged reflectance\ndiffers from reflectances measured by conventional laboratory\nreflectometers even if the painted surface is wet.\nTarget reflectance may be measured at the sea-bottom,\nor, with greater convenience, it may be measured on ship-\nboard by means of a technique developed by the Visibility\nLaboratory of the University of California (San Diego) and\ndescribed in [82]. Excerpts from that report have been as-\nsembled and are reproduced in Fig. 1.87.\nB.6 Horizontal Paths of Sight\nThe visibility nomographs can be used for calculating\nsighting ranges along horizontal paths of sight provided the\ninherent contrast of the object against its horizontal water\nbackground is known. Such contrasts are determinable in any\nof several ways. For example, one may use irradiance","UNDERWATER VISIBILITY PROBLEMS\nSEC. 1.9\n171\n0.10\n0.05\n0.01\n0.005\n0.001\n0.0005\n0.0001\n0.0005\n0.001\n0.005 0.01\n0.05 0.10\nAPPARENT REFLECTANCE OF WET OBJECT (W)\nFIG. 1.87 Graphical means of determining the reflectance\nRO of a submerged surface given its wet reflectance. The\ntechnique involves wetting the sample with a thin film of wat-\ner, irradiating it with a beam at 45°, , and viewing it normally,\nsay with a conventional refractometer. This determines the\nabscissa of the graph. The associated ordinate yields Ro.\nThis scheme was designed by Duntley, and the plotted points\nare the results of his experimental check of the graph.\ndistributions of the kind shown in Figs. 1.25, 1.26, for the\ngeneral class of medium (specified by Mode III of Sec. 1.7)\nunder study. Such irradiance distributions are also readily\nmade from radiance distributions obtained via a Mode IB clas-\nsification of media. Finally, one may use the simple radiance\nmodel of Sec. 1.3 to provide such estimates.\nFor the calculation of horizontal sighting ranges, the\nright vertical scale should be imagined to be labeled \"inher-\nent contrast 50\" Thus an inherent contrast of +1 plots at\nthe point marked 0.02 on the right vertical scale.\nFor horizontal paths of sight the zenith angle 0 = 90","VOL. I\nINTRODUCTION\n172\ndegrees and, since COS 90 = 0, the left vertical scale in-\nvolves only the volume attenuation coefficient a. The areas\nassociated with the curved lines on the nomograph refer to\nthe projected area of the target as seen from the position of\nthe swimmer.\nSighting ranges are calculated by connecting the right\nand left vertical scales with a straight line, and reading\nthe sighting range from the scale division directly above the\nintersection of this line with the curve which applies to the\ntarget area. When the nomographic charts are used in this\nmanner for horizontal sighting range calculations no approxi-\nmations are involved so that neither of the corrections des-\ncribed in the next two sections of this report are required.\nB.7 The Roo Correction\nIn nearly all optically deep natural waters and at all\ndepths approximately 50 times more illuminance reaches any\nhorizontal plane from above than from below. The ratio of\nthe illuminance from below to the illuminance from above is\ndenoted by the symbol \"Roo\". This notation implies that the\n(optically) infinite deep water beneath any horizontal plane\nin the sea could be replaced, for optical purposes, by a sur-\nface of reflectance Roo. This quantity is often measured by\nmeans of two photoelectric cells mounted back-to-back and\nfacing upward and downward respectively.\nBecause Roo = 0.02 for most natural waters of moderate\nto high clarity, the nomographic visibility charts have this\nvalue built into their scales. If Roo is known to be different\nthan 0.02 in any specific instance, this information can be\nentered in the calculation by dividing the value of \"target\nreflectance - bottom reflectance\" by 50 Roo before plotting the\npoint on the right vertical scale of the nomograph. Alterna-\ntively, the \"Roo CORRECTION\" scale printed on the nomograph can\nbe used to apply a correction after the point has been plotted\nbut before the line is drawn across the chart. Draftsman's\ndividers are a convenient tool for this purpose: set one leg\nof the dividers at the circled point on the \"Roo CORRECTION\"\nscale and adjust the other leg to the known value of Roo.\nTransfer this setting to the right vertical scale, maintaining\nthe direction of the correction indicated by the \"Roo CORREC-\nTION\" scale; i.e., the plotted point on the right vertical\nscale is moved downward when Roo exceeds 0.02, and upward when\nRoo is less than 0.02.\nB.8 Correction of the Sighting Range\nThe nomographic visibility charts involve certain alge-\nbraic approximations which may lead to invalid sighting rang-\nes when the indicated value of sighting range is short and\nwhen the reflectance of the bottom departs markedly from 0.02.\nFigure 1.88 is provided as a means for testing any indicated\nsighting range for error and indicating the needed correction.\nThe following numerical example will illustrate the use of\nFigure 1.88.\nA sighting range of 4 feet is indicated by the","SEC. 1.9\nUNDERWATER VISIBILITY PROBLEMS\n173\nnomographic visibility chart when:\nAdaptation luminance is 10-1 foot-lamberts\nTarget area = 10 3 square feet\na-K cos 0 = 0.50\nBottom reflectance = 0.10\nInclination factor = 2\nTarget reflectance - bottom reflectance = 0.010\nSince the sighting range is short and the reflectance\nof the bottom departs markedly from 0.02, the predicted sight-\ning range should be tested and corrected by means of Figure\n1.88 as follows:\nOn the vertical scale of Figure 1.88, labeled \"sighting\nrange x (a-K cos 0)\" enter the data 4 x 0.5 = 2. On the hor-\nizontal scale labeled \"bottom reflectance/inclination factor\"\nenter the data 0.10/2 = 0.05. These entries locate a point\non Figure 1.88 which falls close to the curve marked \"0.83\".\nThe factor 0.83 is to be applied to the value of \"target re-\nflectance - bottom reflectance\" Thus, 0.010 x 0.83= 0.0083.\nIf this corrected value is plotted on the right vertical\nscale on the nomographic visibility chart the corrected sight-\ning range is 3.72 feet.\nExcept in extreme cases the corrected sighting range\nwill differ but little from the uncorrected value. In most\ncases the point on Figure 1.88 will plot above the highest\ncurve, indicating thereby that no correction is required.\nLike the nomographic visibility charts, Figure 1.88 has\nbeen constructed with the assumption that Roo = 0.02. If Roo\nis known to differ from this value this information can easily\nbe inserted in the correction process by dividing the value\nof \"bottom reflectance/inclination factor\" by 50 Roo before\nplotting the point on Figure 1.85. Thus, in the foregoing ex-\nample if Roo = 0.0154, the horizontal coordinate of the point\non Figure 1.85 is 0.050/(50 x 0.0154) = 0.065, and the cor-\nrection factor is 0.77 rather than the value 0.83 obtained\nbefore the insertion of the Roo information, and the new cor-\nrected sighting range is 3.60 feet.","","SEC. 9.1\nUNDERWATER VISIBILITY PROBLEMS\n175\nNO1108 -\n(100s dad 0 SOO X-0","0.0002\n0.0005\n0.0004\n0.0003\n0.004\n0.003\n0.005\n0.002\n0.001\n0.03\n0.02\n0.05\n0.04\n0.01\n0.4\n0.3\n0.2\n1.0\n0.5\n0.1\nO\nFIELD FACTOR 2.4\n20\nAcos(180-0)\nADAPTATION LUMINANCE 100 FOOT LAMBERTS\n30\n10-2 SQ.FT.\nA\nSIGHTING RANGE (FEET)\n0\n10-3 SQ FT\nCORRECTION\n10\n10 4 SQ. FT.\n0.03\n0.04\n0.05\n0.06\n0.07\n0.01\nR 00\n120 100\nAWAY FROM SUN\n9\nK\nINCLINATION FACTOR\nDEPTH\nZENITH ANGLE 0\n150\n180\n50000300\n9\nK\nQ.7\nFIG. 1.90\nK\nACROSS SUN\nTOWARD SUN\nA>10SQ.FT.\n< DEPTH\n150\nDEPTH =\nISQ.FT.\nO-ISQ.FT.\n100 120\n500000\n0.7\nK\n3\n4\n5\n2\n3\n4\n5\n6\n7\n2\nI\nI\n.13\n.14\n.12\n.08\n.09\n.10\n.11\n.06\n07\n04\n.05\n02\n.03\n.01\no\nx\no","0.0005\n0.0004\n0.0002\n0.0003\n0.004\n0.003\n0.005\n0.002\n0.001\n0.05\n0.04\n0.03\n0.02\n0.01\n1.0\n0.4\n0.3\n0.2\n0.5\n0.1\no\n10 SQ.FT.\nFIELD FACTOR 2.4\n10-2 SQ. FT.\n20\n10-3 SQ.FT.\nAcos (180 0 )\nADAPTATION LUMINANCE 10 FOOT LAMBERTS\n10 4 SQ.FT.\n40\nA\nSIGHTING RANGE (FEET)\n50\n0\n60\nCORRECTION\n10\n0.03\n0.04\n0.05\n0.06\n0.07\n0.01\naO\nR\n120 100\nAWAY FROM SUN\n9\nK\nINCLINATION FACTOR\nDEPTH\nZENITH ANGLE 0\n150\n180\n9\nK\n0.7\nK\nACROSS SUN\n< DEPTH<\nTOWARD SUN\nFIG. 1.91\n150\nDEPTH\nAZ100 SQ.FT\nIOSQ.FT.\nISQ.FT.\n120\n0.7\nK\n100\n3\n4\n2\n5\n2\n3\n4\n5\n6\n7\nI\nI\n.08\n.12\n13\n.14\n.09\n.02\n.03\n.04\n.05\n06\n.07\n.10\n.11\n.01\no","VOL. I\nINTRODUCTION\n178\nNO1108 -\n(100g add) 0 500 -","SEC. 9.1\nUNDERWATER VISIBILITY PROBLEMS\n179\nNO1108 -\n(100g add) 0 SOO X-D","180\nINTRODUCTION\nVOL. I\nW01108 -\n03\n(100s 83d) 0 SOO -","SEC. 9.1\nUNDERWATER VISIBILITY PROBLEMS\n181\nNO1108 --\n(100g 83d) e SOO X-D","VOL. I\nINTRODUCTION\n182\nNO1108 -\n(100g add) 0 soo X-D","0.0005\n0.0004\n0.0003\n0.0002\n0.004\n0.003\n0.005\n0.002\n0.001\n0.04\n0.05\n0.03\n0.02\n0.01\n0.4\n1.0\n0.5\n0.3\n0.2\n0.1\no\nSQ.FT\nSQ.FT.\n10-3\nO\nFIELD FACTOR 2.4\n20\n10-5 FOOT LAMBERTS\n-SQFT\nAcos(180-8)\n10\n10-2SQ.FT.\n&O\nA\nSIGHTING RANGE (FEET)\n50\n0\n60\nCORRECTION\n10\nADAPTATION LUMINANCE\n0.03\n0.04\n0.05\n0.06\n0.07\n0.01\nR\n8\n120 100\nAWAY FROM SUN\n9\nK\nINCLINATION FACTOR\nDEPTH=\nZENITH ANGLE 0\n150\n180\n50000300\n9\nK\nA>100SQ.FT.\n0.7\nK\nACROSS SUN\nTOWARD SUN\n< DEPTH<\nFIG. 1.97\n1111\nIOSQ.FT\n150\nDEPTH\nISQ.FT.\n100 120\n0.7\nK\n3\n4\n2\n3\n4\n5\n6\n2\n5\n7\nI\n.08\n.09\n.12\n13\n14\n.04\n.10\n11\n.02\n.03\n.05\n06\n07\n.01\no","184\nINTRODUCTION\nVOL. I\nNO1108 - 3ONV1037338\n9\n(100g 83d) 0 soo -","SEC. 9.1\nUNDERWATER VISIBILITY PROBLEMS\n185\nNO1108 -\nI\nos\n(100g 83d) 0 soo X-D","186\nINTRODUCTION\nVOL. I\nN01108 -\ns\n60\n03\nOS\n(100g 83d) 0 SOO X-D","0.0005\n0.0004\n0.0002\n0.0003\n0.004\n0.003\n0.005\n0.002\n0.001\n0.04\n0.05\n0.03\n0.02\n0.01\n0.4\n1.0\n0.3\n0.2\n0.5\n0.1\no\nFIELD FACTOR 2.4\n2\nAcos (180-6 0)\nADAPTATION LUMINANCE I FOOT LAMBERT\nSIGHTING RANGE (FEET)\nA\n8\nCORRECTION\n0.03\n0.04\n0.05\n0.06\n0.07\n0.01\nR BB\n100\nAWAY FROM SUN\n10-2 SQ.FT.\n120\nDEPTH 9\nK\n10-3 SQ.FT.\nINCLINATION FACTOR\n_11111\n1111\n4 SQ. FT.\nZENITH ANGLE 0\n20\n150\n180\n30\n10\n9\nK\nQ.7\nK\n5040\nACROSS SUN\nTOWARD SUN\n< DEPTH.\nFIG. 1.101\n150\nA>ISQ.FT.\nDEPTH\n10-1 SQ.FT.\n100\nIIIII\n100 120\n50000\n0.7\nK\n3\n4\n2\n5\n2\n3\n4\n5\n6\nI\n7\n.08\n.02\n.03\n.04\n.05\n.06\n.07\n.09\n.12\n.13\n.14\n.10\n.11\n.01\no\nx","VOL. I\nINTRODUCTION\n188\nN01108 -\n03\n(100g add) 0 SOO X-D","0.0005\n0.0004\n0.0002\n0.0003\n0.004\n0.003\n0.005\n0.002\n0.001\n0.05\n0.04\n0.03\n0.02\n0.01\n1.0\n0.4\n0.5\n0.3\n0.2\n0.1\nFIELD FACTOR 2.4\nAcos (180-8\nADAPTATION LUMINANCE 10 -2 FOOT LAMBERTS\nSIGHTING RANGE (FEET)\nA\n0\nCORRECTION\n0.03\n0.04\n0.05\n0.06\n0.07\n0.01\nR\n120 100\n10-3 SQ.FT.\nAWAY FROM SUN\n10-4 SQ.FT.\nDEPTH 9\nK\nINCLINATION FACTOR\nZENITH ANGLE 0\n20\n150\n180\n30\nA>ISQ.FT.\n9\nK\n0.7\n10 SQ.FT.\nACROSS SUN\nK\n< DEPTH<\nFIG. 1.103\nTOWARD SUN\n10 2 SQ.FT.\n150\nDEPTH\n100 120\n50000\n0.7\nK\n3\n4\n2\n5\nI\nI\n2\n3\n4\n5\n6\n7\n.02\n.03\n.04\n06\n.07\n.08\n.09\n.12\n.13\n.14\n.01\n.05\n10\n.11\no\nto","0.0002\n0.0005\n0.0004\n0.0003\n0.004\n0.003\n0.005\n0.002\n0.001\n0.05\n0.04\n0.03\n0.02\nQ.OI\n0.4\n1.0\n0.3\n0.2\n0.5\n0.1\nFIELD FACTOR 2.4\nADAPTATION LUMINANCE 10-3 FOOT LAMBERTS\nAcos(180-8)\na\nA\nSIGHTING RANGE (FEET)\n0\n10-3SQ.FT.\nCORRECTION\n6\n101 4SQ.FT.\n0.03\n0.04\n0.05\n0.06\n0.07\n0.01\n8\nR\n120 100\nAWAY FROM SUN\nDEPTH= 9\nK\nINCLINATION FACTOR\nZENITH ANGLE 0\n150\n20\n180\nAZISQ.FT.\n9\nK\n0.7\nACROSS SUN\nK\nFIG. 1.104\n< DEPTH<\nTOWARD SUN\nIO-ISQ.FT.\n150\nDEPTH\nSQ.FT.\n120\n0.7\nK\n100\n3\n4\n3\n2\n5\n2\n4\n5\n6\nI\n7\n.08\n07\n.09\n.12\n13\n14\n.02\n.03\n.04\n.05\n.06\n.10\n.11\n.01\nO","SEC. 9.1\nUNDERWATER VISIBILITY PROBLEMS\n191\n-\n9\n(100g ddd 0 SOO X-D","VOL. I\n192\nINTRODUCTION\nNO1108 -\n(100s 83d) 0 SOO X-D","SEC. 1.9\nUNDERWATER VISIBILITY PROBLEMS\n193\nC. Interpretation of Sighting Range\nC.1 Introduction\nThe sighting ranges calculated by means of the nomo-\ngraphic visibility charts are the limiting distances at which\na swimmer will be aware of seeing the object. It is assumed\nthat he is fully familiar with the underwater environment,\nwell acquainted with the objects for which he looks, and pos-\nsessed of perfect vision. It is not assumed, however, that\nhis training has included a lengthy special training period\ndevoted to maximizing his ability to produce long sighting\nranges.\nIt is assumed that the swimmer knows the direction in\nwhich to look and that he expects to see the visual target.\nIn other words, the swimmer is not required to search his vis-\nual field and there is no problem of vigilance.\nThe above described interpretation of \"sighting range\"\nis indicated on the nomographic visibility charts by the in-\nscription \"field factor 2. 4\". This notation, meaningful only\nto specialists in visibility calculations, implies that nomo-\ngraphic charts can be constructed to depict other levels of\nobserver performance, i.e., other values of \"field factor\"\nA general discussion of visual search, field factors, and ob-\nserver characteristics is out of place in this work, but three\ncommon effects will be discussed in simplified form in the fol-\nlowing paragraphs.\nC.2 Effect of Lack of Warning\nWhen an underwater object is encountered by a swimmer\nwithout warning, the sighting range will be somewhat shorter\nthan otherwise. This is to say that unexpected objects will\nbe less well detected initially than will those whose exis-\ntence is known and whose appearance is expected. This effect\nis independent of training, experience, or visual capability.\nIts effect upon the sighting range can be allowed for by di-\nviding the value of \"target reflectance minus bottom reflec-\ntance\" by 1.2 before entering the right vertical scale of the\nnomographic charts.\nC.3 Effect of Observer Training\nExtensive practice in sighting underwater targets at\nlimiting distances will enable good observers to exceed slight-\nly the normal sighting range. A correction for the effect of\ntraining can be made by multiplying the value of \"target re-\nflectance - bottom reflectance\" by a training factor between\n1 and 2 before entering the right vertical scale of the nomo-\ngraphic charts. A training factor of 1.0 represents the usual\ncapability of experienced swimmers who are fully familiar with\nthe underwater environment and are well acquainted with the","VOL. I\nINTRODUCTION\n194\nobject for which they look; this value (unity) should ordi-\nnarily be used. If an experienced swimmer is considered to\nbe unusually good at underwater sightings a training factor\nof 1.2 is recommended. * Laboratory experience indicates that\nonly after many thousands of careful attempts to achieve\nsightings at maximum range can even the most experienced per-\nsonnel achieve a training factor of 2.\nC.4 Effect of Observer Visual Capability\nAll human eyes are not created equal with respect to\ntheir capability to detect underwater objects at limiting\nrange; this is not a matter of training but represents subtle\nphysiological differences between men which are beyond detec-\ntion by ordinary eye-examinations. The nomographic charts\nhave been drawn to represent the performance of average \"per-\nfect\" young eyes. Some estimate of the effect on sighting\nrange of the spread in visual capability within the popula-\ntion of \"perfect\" observers can be obtained by successively\ndoubling and halving the value of \"target reflectance minus\nbottom reflectance\" before entering the right vertical scale\nof the nomographic chart.\nD. Visualization of Water Clarity\nD. 1 Introduction\nThe clarity of natural waters can be visualized directly\nin terms of the attenuation coefficients a and K on the basis\nof experience gained through the use of the nomographic visi-\nbility charts. It will be found that most objects can be\nsighted at 4 to 5 times the distance (a-K cos 0) unless the\nadaptation level is low; exceptions to this rough rule-of-\nthumb are common but they can easily be categorized. Alter-\nnatively, a convenient conceptualization of the appearance of\nany underwater environment can be obtained from 1/a and 1/K.\nD.2 Estimation of Sighting Range\nThe rough rule-of-thumb stated in the preceding para-\ngraph is illustrated by the examples in paragraph B.2 of this\nsection. In the first (clear water) case 1/ (a-K cos 0) =\n1/0.10 = 10 feet and the vertical sighting range of the large\n(10 square feet) object is 48 feet, or 4.8 times 1/ (a-K cos 0).\nIn the second (low-clarity water) case 1/(a-K cos 0) = 1/0.60\n= 1.67 feet, and the vertical sighting range of the same tar-\nget is 8.0 feet, or 4.8 times 1/ (a-K cos 0).\nIt will be recognized that the factor 1/1.2 for lack of warn-\ning and the training factor 1.2 cancel; thus the nomographic\ncharts as drawn apply without correction to the case of the\nexperienced, highly trained swimmer who comes upon objects\nwithout warning.","SEC. 1.9\nUNDERWATER VISIBILITY PROBLEMS\n195\nThe value 4.8 is not universal; it will be altered by\nchanging target size, adaptation luminance, zenith angle, tar-\nget reflectance, etc. For example, it was noted in paragraph\nB.2 that in the low-clarity water the vertical sighting range\nof a small target 10 square feet in area is 5.5 feet or 3.3\ntimes 1/(a-K cos 0). If, however, the reflectance of the ori-\nginal 10 square foot visual target had been 0.330 (instead of\nthe value 0.080 assumed in paragraph B.2), thus forming a\nhigh inherent contrast with the dark (0.030) bottom, its ver-\ntical sighting range is found to be 11.0 feet or 6.6 times\n1/ (a-K cos 0). In summary, small values of target size or\nlow. values of adaptation luminance (or both) will produce\nsighting ranges shorter than 4 times 1/ (a-K cos 0) whereas\nhigh values of \"target reflectance minus bottom reflectance\"\nmake large objects visually detectable at ranges in excess of\n5 times 1/ (a-K cos 0).\nAn important and common special case is that of large\ndark objects viewed horizontally. In this case 1/ (a-K cos 0)=\n= 1/a, since cos 90° = 0, and the sighting range will be ap-\nproximately 4 times 1/a unless the adaptation luminance is\nlow.\nD.3 Estimation of Adaptation Luminance\nInspection of Figure 1.12 will enable convenient order-\nof-magnitude values of illuminance on the surface of the sea\nto be noted for, say, noon and sunset, clear and cloudy.\nTranslation of these values to the approximate illuminance at\nthe depth of the swimmer is often facilitated by noting that\nthe illuminance, and, therefore, the adaptation luminance is\nreduced by a factor of 1/10 for each (1n 10) /K of depth.\nFigures 1.82 and 1.83 provide convenient illustrations of\nthis concept.\nD.4 Estimation of a and K\nIn some, but by no means all, waters the distance 2.3/K\nis about 50% greater than the distance 4/a; i.e., about 6/a;\nthus the natural illuminance (and the adaptation level) may\ndecrease by a factor of 1/10 for each unit of depth equal to\n1.5 times the horizontal distance at which a swimmer can see a\nlarge dark object at high light levels. If measured values\nof a and K are not available, these constants can be esti-\nmated by means of the relations a = 4/d and K = 1.5/d, where d\nis the horizontal sighting range for large dark objects at\nhigh light levels. The estimate of a is more reliable than\nthe estimate of K. Rules of thumb such as these can be given\na better basis after more extensive Mode III classifications\nof natural hydrosols have been made (cf. Sec. 1. 7; see also\n(11)-(13) of Sec. 10.8).\nD.5 Characterization of Natural Waters\nFor purposes of easy visualization, it is possible for\nnatural waters to be characterized by the distances 4/a and","INTRODUCTION\nVOL. I\n196\n2.3/K, though the numbers 1/a, and 1/K can do just as well.\nIn the clearest known natural waters* these distances 4/a\nand 2.3/K are believed to be less than 230 feet and 340 feet\nrespectively. In the first numerical example given in para-\ngraph B.2 the distances were found to be 4/a = 55 feet and\n2.3/K = 85 feet; in the second, 4/a = 9.3 feet and 2.3/K =\n13.5 feet.\n1.10 Applications of Hydrologic Optics to the Food-Chain\nProblem in the Sea\nIn this section we shall discuss, from the point of\nview of radiative transfer theory, the problem of food-chain\nrelations in the ocean. The theory of food-chain relatations\nattempts to describe, in quantitative terms, the distribution\nin time and space, within a given oceanic region, of the food\nsupply of the main animal populations of that region. The\nfood supply is an essentially self-sustaining collection of\nbiological organisms, inorganic matter, and radiant energy.\nAside from radiant energy, the chain consists principally of\nthe following four links: nutrients (e.g., phosphate), phy-\ntoplankton, herbivores, and predators. This set of interact-\ning organisms is arranged so that each item in the list con-\nstitutes the food of the next item in the list, and in this\nsense forms a food-chain in an oceanic region. This food-\nchain is initiated and sustained by solar radiant energy pene-\ntrating into the sea. The radiant energy sustains the photo-\nsynthesis within the phytoplankton and the life processes of\nthe herbivores and predators. Furthermore, the continued de-\ncomposition into nutrient material of each of the last three\nlinks in the chain also contributes to its maintenance. Thus,\nany complete theory of food-chain relations in the ocean must\ntake into explicit account, among other things, the role of\nradiant energy in the food-chain relations. A survey of the\npresent state of the theory (ref. [265]) indicates that the\nsystematic inclusion of radiant energy terms into the food-\nchain relation has been avoided because of the additional dif-\nficulties attendant on such an inclusion in an already complex\ntheory. In the present discussion, it will be shown how the\ngeneral inclusion of radiant energy terms into the descrip-\ntion of the food-chain relations can be carried out in such a\nway that the attendant increase in the complexity of the the-\nory will not render the result altogether impracticable. Fur-\nthermore, it will be shown that the resultant formulations\npoint to some novel, detailed descriptions of the depth dis-\ntributions of the light field in a region containing the mem-\nbers of the food-chain. By doing so, the main purpose of the\ndiscussion will be fulfilled, namely, to round out the clas-\nsical Volterra prey-predator equations [309] which describe\nProbable values:\na = 0.017 per foot = 0.056 per meter\nK = 0.0067 per foot\n= 0.022 per meter at 480 millimicrons\nCompare this a with that in Table 1 of Sec. 1.6.","SEC. 1.10\nFOOD CHAIN PROBLEM\n197\nfood-chain relations, by including one more equation which\nspecifically-- and in a manner uniform with the other equa-\ntions--incorporates the photons of the light field into the\nlist of interacting members of the the food-chain. The manner\nin which light particles can generally be considered as \"prey\"\nor \"predator\" will become clear as the discussion proceeds.\nThe General Exponential Law of Change\nThe simple differential law:\ndA\n= KA\n(1)\ndt\nhas been found to describe a wide variety of natural phenome-\nna, among which are: growth of yeast cultures and bacterial\ncultures, decay of radioactive substances, growth and decay\nof animal populations, damped or resonating oscillations of\nmechanical and electrical systems, and the darkening of light\nfields with depth in scattering-absorbing media, to name a\nfew. Up until now we have been concerned in this work prin-\ncipally with the latter use of the exponential law. As we\nshall see in the latter stages of this discussion, we may\nvery well view (1), under suitable interpretations, as the\nalpha and the omega--that is, the beginning and the end--of\nthe general theory of the food-chain relations. However, for\nthe present, we view (1) as the ostensibly simple equation it\nappears to be, with constant coefficient K, and thereby ob -\ntain the general solution of (1) in the form:\nKt\nA(t) = A(0)e\n(2)\nwhere A(t) is the amount at time t of the entity under con-\nsideration. When K is positive, then there is growth of A(t);\nwhen K is negative, there is decay of A(t), as time t in-\ncreases.\nThe description of natural growth and decay processes\nsummarized in (1) and (2) is known as the exponential law and\npertains as it stands basically to isolated and relatively\nsimple systems. When the systems are no longer isolated or\nno longer simple in internal structure, then (1) is replaced\nby a correspondingly modified equation. For example, by re-\nmoving the isolation restriction, two new features appear:\na\nsource term An may be added to the right side of (1); and the\npossibility arises of a non-constant growth rate term K. From\nthe present point of view, the inclusion of a source term An\npresents no essential modification of the equation (1), and\nso will not be studied in this discussion. However, the prac-\ntical and theoretical possibilities inherent in a non-constant\ngrowth rate term K are endless, and some of them hold the key\nto the solution of the general problem of the food-chain re-\nlation; some of these possibilities will now be considered.","VOL. I\n198\nINTRODUCTION\nThe Volterra Prey-Predator Equations\nA theory of food-chains can be made to rest in the clas-\nsical equations postulated by Volterra [309] which govern the\nevolution in time of the number P of prey and number A of\npredators feeding on the prey. Thus, for example, if P is the\nnumber of plants and A the number of animals in a symbiotic\nrelation, then their evolution in time may be governed by gen-\neral equations of the form:\ndP\n(3)\ndt = KPF\ndt dA\n(4)\nwhere we have written\n\"Kp\" for p-bA\n(5)\nand\n\"KA\" for CP-a\n(6)\nThat is, the growth rate term Kp for the prey is the sum of\nthe intrinsic growth rate p for the prey population and the\ninteraction decay term -bA, where b is a coupling constant be-\ntween the populations of A and P. Similarly a is the coeffi-\ncient of decay of the predator population, and C is the coup-\nling constant between A and P in this instance. The coupling\nconstants b and C are usually taken as equal or as connected\nby some given relation.\nNow each equation (3), (4) is of the general type as (1)\nand, assuming Kp and KA known as functions of time along with\nthe initial values P(0) and A(0) of P and A, are directly in-\ntegrable:\n(\"XACE\")\nA(t) = A(0) exp\ndt\nP(t) P(0) exp {6 t dt'\n=\nHowever, equations (3) and (4) are generally coupled (i.e.,\nb # 0 and C # 0) so that the preceding solutions, while for-\nmally correct, are of no immediate practical use, since know-\nledge of Kp and K A is tantamount to knowledge of P and A them-\nselves.\nThe equations (3) and (4), despite their analytically\nunpleasant nonlinear coupling, form a workable starting point\nin the quantitative description of the food-chain relation.","SEC. 1.10\nFOOD-CHAIN PROBLEM\n199\nIt is clear, however, that the equations as they stand de-\nscribe only the herbivore and predator components of the\nchain and so cannot adequately describe the complete food-\nchain relation as defined above. The other members of the\nchain, namely the phytoplankton and the nutrients (which also\nconstitute a prey-predator pair), along with the radiant en-\nergy, are excluded from (3), (4).\nThe General Food-Chain Equations\nWe turn now to a formulation of the Volterra-type prey-\npredator equations which goes beyond that of (3) (4) and\nwhich takes into account the interactions of all five members\nof the food-chain relation. To keep the geometric and phys-\nical variables down to a comfortable minimum at the outset,\nwe shall assume that all quantities of the chain depend on\ndepth and time only, over the oceanic region of interest. Thus\nlet:\nU(z,t)\nbe the radiant density (radiant energy per\nunit volume) at depth z, time t\nP(z,t)\nbe the number of phytoplankton per unit\nvolume at depth z, time t\nB(z,t)\nbe the number of herbivores per unit volume\nat depth z, time t\nC(z,t)\nbe the number of carnivores per unit volume\nat depth z, time t\nN(z,t)\nbe the amount of nutrient per unit volume\nat depth z, time t\nWe postulate a food-chain ordering among the members of\nthe food-chain, and which is schematically summarized below:\nC\nB\nP\nN\nU\nC\n0\n+\n+\n+\n+\nB\n0\n+\n+\n-\n+\nP\n0\n-\n+\n(7)\n-\n+\nN\n+\n+\n+\n0\n+\nU\n0\n-\n-\n-\n-\nThis ordering is to be interpreted as follows: consider the\ncarnivore row. Carnivores in the present hierarchy are under-\nstood to grow at the expense of most other members of the\nchain (hence the + signs in the row). Herbivores, on the\nother hand, grow at the expense of phytoplankton, nutrients\nand radiant energy (hence + signs) but are preyed upon by car-\nnivores (hence - sign). The zero entries indicate that in the\npresent model, members of the chain do not increase or de-\ncrease at the expense of their own numbers. (In mathematical","VOL. I\nINTRODUCTION\n200\nterms the food-chain ordering relation in (7) is an irreflex- -\nive, asymmetric, transitive relation.) The double signs (+)\nin the nutrient row indicate that at times, N may increase (+)\nin the direct presence of the other members and at other\ntimes may decrease (-) in the direct presence of the other\nmembers.\nThe food-chain ordering associated with each pair of\nthe food-chain is given a quantitative measure by assigning\ninteraction functions to each pair of members of the chain.\nThus to the pair (C,B) we assign a function KCB which on the\nbasis of the food-chain - ordering relation tabulated above, is\npositive for all Z and t. Similarly to (C,P) we assign the\ninteraction function KCP which is also positive-valued. - Con-\ntinuing in this way we assign to the pair (U,N) the function\nKUN which is negative-valued for all z,t. The functions KCC,\nKBB, etc. are all zero-valued, and KNC may be positive, zero\nor negative-valued for various z, and t.\nOnce a food-chain ordering has been established and the\n20 non zero interaction functions have been assigned, the Vol-\nterra interaction equations can be written down:\n(8)\nAND\nwhere we have written\n\"KC\" for kc + KCBB + KCPP + KCNN + KCUU\n\"KB\" for KB + KBCC + KBPP + KBNN + KBUU\n(9)\n\"Kp\" for kp + KPCC +\n\"KN\" for KN + KNCC + KNBB + NP KNUU\n\"KU\" for ku + KUCC + KUBB + KupP + KUNN","SEC. 1.10\nFOOD-CHAIN PROBLEM\n201\nThe five functions kc,\n, KU are inherent growth-decay rates,\nwhich are operative independently of the presence of other\nmembers of the chain. Furthermore, the differentiation opera-\ntor d/dt in (8) is a total derivative operator, i.e., , we have\nwritten\n\"d/dt\" for a/at + v(a/az)\n(10)\nwhere in each case V is an averaged speed of propagation in\nthe Z direction. In the case of U it is the speed of light.\nIn the case of C and B, it is variable with time and space\naccording to the vertical movements of the animals. In the\ncase of P and N, V represents rate of rising and sinking, plus\neddy diffusion rates. The theoretical basis for the equation\ngoverning U in (8) which is one of the novel features of (8),\nrests in the general theory of K-functions for directly observ-\nable radiometric quantities as developed in Chapter 9 below.\nFor practical purposes, one may, however, use (7) of Sec. 1.4\nwith each side divided by V (recall (5) of Sec. 1.1).\nOnce the interaction functions are known and the initial\nstates C(z,0), K(z,0) U(z,0) are known over all depths Z\nin the region of interest, the system (8) is in principle solv-\nable by iteration techniques. Thus, for example, by writing\n\"A\"\nfor\n(C, B, P, N, U)\n(11)\nand\nK\n0\nC\n\"K\"\nfor\nKB\n(12)\n0\nK\nU\nThe system (8) becomes transformed into the vector equation:\ndA ==AK\n(13)\ndt\nwhich may be solved by any of several modern iteration tech-\nniques (see, e.g., [23]) using large scale computers. It is\ntherefore no longer necessary to limit the generality of a\nfood-chain theory because of the possible intractability of\nthe analytic solution procedure (e.g., the impossibility of\nobtaining closed forms for the integrations).\nAn Illustration of the Food-Chain Theory with\nA Radiant Energy Term\nAs a simple illustration of the general theory outlined\nabove, let us consider a three-member food-chain consisting of\nphytoplankton, herbivores, and radiant energy. Hence we will\nstudy the effect of adding to the classical prey-predator equa-\ntions (3), (4), another equation which specifically includes","VOL. I\nINTRODUCTION\n202\nradiant energy in the prey-predator interactions. The follow-\ning discussion is actually independent of the number of mem-\nbers in the food-chain, so that a reader following the general\nline of argument developed below may extend the arguments and\ntheir results to arbitrarily large food-chains.\nThe General Three-Term Equations\nThe requisite equations for the present illustration\nare:\n(14)\n(photons)\n+ + (herbivores)\n(15)\ndt (phytoplankton)\n(16)\nThe Quasi-Steady State Equations\nWe shall be interested for the present in a quasi-steady\nstate solution of the preceding system of equations. By\n'quasi-steady state' we mean that the time rates of change of\nthe magnitudes of P and B are negligible compared to that of\nU, so that the light field U adjusts to and settles down to\nsteady state almost instantly in accordance to the prevailing\nspatial distributions of P and B at time t. Therefore, in\n(14) we may drop the time derivative and consider only change\nof U in depth for fixed t and adjust the definitions of the\nK-functions to absorb the speed constant V; and in (15) and\n(16) we may drop the spatial derivatives, and consider only\nthe change of B and P in time for a fixed depth Z :\n(17)\ndz\n(18)\n+\n(19)\nThis set of equations like the general equations, is readily\nsolvable in principle for given arbitrary constants ku, KUB,\netc., , and initial conditions. The steady state spatial dis-\ntributions of U, P, B are of especial interest, and we shall\ndevote the remainder of this section to the study of these\ndistributions","SEC. 1.10\nFOOD-CHAIN PROBLEM\n203\nThe Equilibrium Solutions\nWhen = 0 for every Z at a given time t, the exist-\ning spatial distribution of B is called the equilibrium popu-\nlation and denoted by \"Bq\"; similarly for P. The equilibrium\npopulations of P or B are readily characterized in terms of\nthe spatial distribution of the radiant energy. Thus from\n(18) we have:\nwhich implies\nKB + KBPPa+KBUU = 0\nso that\n=\n(20)\nSimilarly from (19), for steady state:\nso that\nKp+Kpg = 0\nwhence\n=\n(21)\nEquations (20) and (21) show that if the steady state radiant\nenergy distribution U is known, the equilibrium P and B dis- -\ntributions are determinable over the range of depths of in- -\nterest.\nWe now show that the relations (20) and (21) together\nwith (17) uniquely determine the steady state radiant energy\ndistribution through the medium so that Pq and Bq are uniquely\ndeterminable, in turn. Substituting Pq and Bq as given by\n(20) and (21) into (17), and rearranging, we have:\ndz du = Kpu4) -\nU\nThat is:\ndU\n=\n(22)\ndz","VOL. I\nINTRODUCTION\n204\nwhere we have written:\n(23)\n\"a\" for KU -\nand\n(24)\n\"b\"\nfor - KBP\nIf \"U(0)\" denotes the initial value of U at some fiducial\ndepth (here Z = 0), then (22) resolves into:\nau(0)e\n(25)\nU(z) =\n-bu(0)e [bU(0) + a]\nThis solution may now be used in (20) and (21) to obtain de-\ntailed descriptions of the depth distribution of the steady\nstate populations of P and B. The solution (25) exhibits\nsome interesting mathematical properties for various choices\nof a and b. For b = 0, we have simple exponential growth\n(a>0) or decay (a < 0). For a = 0, by a limiting argument,\nwe have\nU(2)\n1-bU(0)z\nSome General Properties of Equilibrium Solutions\nThe equilibrium solutions found above have several in-\nteresting practical properties, one of which we isolate for\nparticular attention here. This is the property of predict-\ning a possible band of depths below the ocean surface outside\nof which the P and B populations cannot exist. To find the\nlimits of this band of depths, we return to equations (20)\nand (21) and require that Pq 0 and Bq 0. These conditions\nmerely state that real distributions of phytoplankton and her-\nbivores must not have negative populations. The non negativ-\nity condition applied to (20) yields:\nFrom the interaction table (7) we find that KBP At 0, so that\nwhence\nU S -KK/KBU","SEC. 1.11\nFUTURE PROBLEMS\n205\nSimilarly, from (21) with the help of the nonnegativity con-\ndition we find:\nHence a necessary condition for the existence of steady state\nP and B equilibrium distributions at depth Z is that\n(26)\nIt is to be noted that (26) are necessary conditions (i.e.,\nif a band exists, then it must be such that (26) holds) and\nnot sufficient conditions, except insofar as the steps can be\nretraced from (26) to (20) and (21). This can be done if KPB\nand KBP are strictly negative and strictly positive, respec-\ntively, and if the left side of (26) is indeed less than the\nright side.\nNow according to (25), U(z) is under certain conditions\na decreasing function of Z (for negative a). Thus if U(0) is\ngreater than -kB/KBU, then (26) shows that no steady state\npopulation should exist for depths Z = 0 down to where\nU(z) = -kB/KBU. Then there is expected a band of depths with-\nin which P > 0 and B > 0. Since U(z) decreases monotonically,\nthere will be depths below which the left side of (26) no\nlonger holds, so that P = 0 and B = 0 in those depths. It ap\npears then that the present model can in principle predict a\neuphotic zone in natural hydrosols in which the food-chain is\nin a quasi-steady state condition.\nWe have reached the main goal of the discussion, namely\nto supplement the classical Volterra prey-predator equations\nwith a third equation governing the flow of radiant energy in\nthe sea, and to briefly explore the consequences of the inter-\nactions of the prey-predator-photon system.\n1.11 Future Problems of Hydrologic Optics\nThe present introductory chapter to hydrologic optics\nis brought to a close with a small, carefully selected list\nof important problems which are as yet only partially resolved.\nThe list is deliberately kept small so as not to overwhelm\nprospective students of the subject with a mass of more or less\nobvious types of applicational problems they soon would en-\ncounter in their own fashion as their studies proceed. Rather,\nwe have selected for presentation and discussion here three\narchetype problems which, if eventually satisfactorily re-\nsolved, would elevate the discipline of hydrologic optics to\nthe level of a mature science which could predict and des-\ncribe, in the fullest sense of these terms, all aspects of\nthe transfer of radiant energy in the seas, lakes and other\nnatural hydrosols of the world.","VOL. I\nINTRODUCTION\n206\nProblem One: To Establish Theoretically the\nPhysical Basis of the Inherent Optical Properties\nof Natural Hydrosols\nThe two main inherent optical properties a, o, of the\nhydrosols, and of optical media in general defined in Sec.\n1.6, together with the equation of transfer ((10) or (12) of\nSec. 1.3) form the core of modern radiative transfer theory.\nThis theory is by definition (i.e. by actual considered\nchoice) predominantly phenomenological in outlook, and accord-\ningly the optical properties a, o are left unspecified in the\ngeneral theory. The theory thus contains no formalism which\npredicts the values of a and o in a given medium in terms of\nthe inherent physical structure of that medium. It is impor-\ntant to understand the significance of this observation. It\ndoes not maintain that the theory of radiative transfer is in-\ncapable of providing procedures to measure a and o in natural\noptical media. The operational procedures in Sec. 1.6 and in\nChapter 13 below supply abundant methods for arriving at a\nand o in given media. Rather, what is intended is the obser-\nvation that the connections between a and o and the electro-\nmagnetic structure, and more basically, the molecular struc-\nture of these media is beyond the ken of the principles of\nthe theory. The purpose of Problem One is to establish theo-\nretical connections between a and o and the physical proper-\nties of an hydrosol the properties of a given solution\nor suspension (or both) of substances in H20. One such con-\nnection is possible on the electromagnetic level wherein a\nand o could be related theoretically to the permittivity, per-\nmeability, and conductivity functions of the hydrosol. Such\nconnections have received initial attention in Chapters XIV\nand XVI of Ref. [251], and the results there suggest further\ndirections in which to pursue this problem. Observe that the\napproach in [251] is not the approach of the Mie theory of\nscattering, since the latter applies only to single scatterers.\nThe suggested approach attempts to obtain a basis for o as\nactually measured in situ. The motivation for Problem One is\nquite clear: if this problem is solved, it may someday be\npossible to predict, by calculation, the a and o of an hydro-\nsol, given its physical analysis; and conversely, from a spec-\ntral radiometric analysis of a and o, to determine the physi-\ncal components of the hydrosol. It may then also be possible\nto resolve once and for all the quantitative and conceptual\nproblems of the nature of forward scattered light for very\nsmall and very large angles of scatter (see Sec. 1.6, in par-\nticular Fig. 1.72; Sec. 18 of Ref. [251], and [78]) and also\nto provide a rational basis for such interesting findings as\ndisplayed in Table 4 and Fig. 1.73 of Sec. 1.6, of the uni-\nformity of shape of O. Furthermore, by solving Problem One,\nwe may also resolve such questions as the existence of spec\ntral windows in the sea which even though seemingly settled\non an empirical level (cf., Sec. 1.6) will continually nag at\nthe analytically inclined individual who would prefer such an\nimportant question to be resolved in a way which rests on nec-\nessary inferences drawn from established physical principles;","SEC. 1.11\nFUTURE PROBLEMS\n207\nprinciples which are, incidentally, on a more fundamental 1e-\nvel than those on which radiative transfer theory is made to\nrest. Still further, the problem of the structure of o in\nthe polarized context (using the matrix p) may be solved (see\nSec. 13.11). Last, but not least, the resolution of the pres-\nent problem will securely anchor the discipline of hydrologic\noptics, and radiative transfer in general, to the mathemati-\ncal and physical bedrock of the mainland of modern physics.\nProblem Two: To Establish Complete Empirical\nClassifications of Natural Hydrosols\nThe discussion of this problem was essentially presented\nin Sec. 1.7, and so need not be repeated here. It should per-\nhaps be emphasized that this problem is unquestionably the\nsingle most important problem facing experimenters in the\nfield of hydrologic optics. A moment's reflection will show\nthe experimenter (who is for example bent on the problem of\nthe connections between the ideal photosynthesis in a region\nand the measurement of radiant energy in that region) that\nthis problem is essentially one of classification of an opti-\ncal medium in either of the three main modes (Modes I, II,\nIII) described in Sec. 1.7. Or again, a scientist concerned\nwith the problem of underwater optical communication or visi-\nbility will benefit from complete empirical classifications\nof the media of interest. Even theoreticians, on descending\nfrom their ivory towers after making some inroads into Prob-\n1em One above, will require corroboration of the kind that\nonly a truly exhaustive solution of the present problem can\nsupply.\nPerhaps we can put the nature of the present problem\ninto perspective by enjoining the prospective experimenter on\nwhat not to do if his work is to contribute to the solution of\nProblem Two and is to be of lasting worth and importance to\nthe discipline of hydrologic optics:\n(i) Do not omit to mention the spectral range and\naccuracy of your determinations of the optical properties.\n(ii) Avoid broad-band measurements whenever narrow-\nband measurements are possible, even if considerably more\neffort is entailed for the latter.\n(iii) Do not measure a alone or o alone; measure them\ntogether (Mode IA), over at least the visible spectrum.\nAlternatively:\n(iv) Do not measure a alone or K alone; measure them\ntogether (Mode III), over at least the visible spectrum.\nAlternatively:\n(v) Do not measure H(z,- -) alone or (z) alone; measure\nall four irradiances: H(z, and (z, together (Mode II),\nor preferably N(z,.) (Mode IB), over at least the visible\nspectrum.","208\nINTRODUCTION\nVOL. I\nOf course with these don'ts go important positive obser-\nvances of the usual kind, especially for alternatives (iv)\nand (v) : recording of lighting conditions above the air-\nwater surface, the state of the air-water surface, the proxim-\nity and state of the bottom, the state of polarization, and\nso on.\nProblem Three: To Establish A Unified Automatic\nComputation Program for Prediction Computations\nand Data Reduction Computations in\nGeophysical Optics (the GEOVAC)\nThe theory of radiative transfer is now well founded\nwith many excellent means of solution of the equations of the\ntheory, as explained at length at appropriate points through-\nout the remainder of this work, or in Ref. [251], and in other\nworks on the subject. In need at present are workable compu-\nter programs which will take a and o and boundary lighting\nconditions (either unpolarized or polarized) and yield inter-\nnal radiance distributions throughout the medium of interest,\nregardless of its shape and size. In other words we envision\na hardware realization of the Mode IA classification of natur-\nal optical media. Conversely, the computation programs\nshould be able to convert experimental documentations of the\n(unpolarized or polarized) radiance distributions (or at least\nirradiance quartets), as a function of wavelength and depth,\ninto the appropriate determination of the inherent and ap-\nparent optical properties of the medium. In this way we can\nalso achieve a hardware realization of the Mode IB (or, re-\nspectively, the Mode II) classification of natural optical\nmedia. The applications of such a program-complex to the\nproblems cited in the opening remarks of Sec. 1.0 are mani-\nfold, and many uses of such a program are undoubtedly yet to\nbe conceived. The geophysical optics automatic variable com-\nputer--the1GEOVAC program envisioned above will serve to tie\ntogether efforts on both Problems One and Two above, as well\nas help solve the everyday problems arising in the engineering\napplications of meteorologic and hydrologic optics.\n1.12 Bibliographic Notes for Chapter 1\nIn addition to the mention of various references given\nat the appropriate points in the discussions of this chapter,\nthe following references are noted for especial attention, as\nthey form a relatively immediate point of entry into the do-\nmain of hydrologic optics, either directly or via their ref-\nerences. First there is the survey article of light in the\nsea by Duntley [78] which covers the gist of the hydrologic\noptics work of the Visibility Laboratory of the University of\nCalifornia over the twenty year period 1944-1964. Contempo-\nrary and earlier work in hydrologic optics by other organiza-\ntions and individuals is surveyed in the annotated bibliog-\nraphy on transmission of light in water by Du Pré and Dawson\n[84]. This bibliography covers approximately 650 abstracts","SEC. 1.12\nBIBLIOGRAPHIC NOTES\n209\nby over 400 authors in more than 150 European and American\njournals, extending over the period from 1818 to 1959. Two\nsymposia on radiant energy in the sea resulted in published\npapers relevant to hydrologic optics: the Helsinki meeting\nof I.U.G.G. in August 1960 is summarized in [124]; and papers\npresented at the Hawaiian meeting of the tenth Pacific Science\nCongress are in [303]. Reference [109] contains a summary of\na small amount of theory and a relatively larger amount of\npractical experimental results along with descriptions of in-\nstrumentation used in hydrologic optics. Reference [109],\naccordingly, is a useful supplement to the present work. The\npaper and recent book by Jerlov [125] also surveys recent de-\nvelopments in the field. Of some historical interest in the\ndevelopmental aspects of the field of hydrologic optics are\nChapters I-IV of [82] which are the synthesis of the experi-\nmental work by Duntley and the early theoretical work of the\nauthor. The roots of this chapter trace back in part to some\nearly studies presented in [210]. The basis of the subsequent\nchapters of this work are given in the bibliographic notes\nappended to each chapter.\nThe numbering of the bibliography items in this volume\nand succeeding volumes follows that of the master bibliog-\nraphy given in the final volume (VI) of the present work.","210\nINTRODUCTION\nVOL. I\nBIBLIOGRAPHY FOR VOLUME I\n6. Atkins, W.R.G. and Poole, H.H., \"The angular scattering\nof blue, green and red light by sea water,\" \" Sci.\nProc. Roy. Dublin Soc. 26, 313 (1954).\n7. Austin, R.W., Water Clarity Meter, Operating and Main-\ntenance Instructions, (Scripps Inst. of Ocean. Ref.\n59-9, University of California, San Diego, 1959).\n12. Beebe, W., Beneath Tropic Seas (Blue Ribbon Books, New\nYork, 1928).\n23. Birkhoff, G., and Rota, G., Ordinary Differential Equa-\ntions (Ginn and Co., New York, 1962).\n35. Brown, D.R.E., Natural Illumination Charts (Report No.\n374-1, Proj. NS 714-100, Bureau of Ships, Dept. of\nthe Navy, September 1952).\n36. Burton, H.E., \"The optics of Euclid,\" J. Opt. Soc. Am.\n35, 357 (1945).\n50. Committee on Colorimetry (Optical Society of America),\nThe Science of Color (Thomas Y. Crowell Co., New\nYork, 1953).\n56. Cox, C., and Munk, W., \"Measurement of the roughness of\nthe sea surface from photographs of the sun's glit-\nter,\" J. Opt. Soc. Am. 44, 838 (1954).\n58. Cox, C., and Munk, W., \"Some problems in optical ocean-\nography,\" J. Mar. Res. 14, 63 (1955).\n63. Dawson, L.H., and Hulburt, E.O., \"Angular distribution\nof light scattered in liquids,\" J. Opt. Soc. Am. 31,\n554 (1941).\n68. Drummeter, L.F., and Knestrick, G.L., A High Resolution\nInvestigation of the Relative Spectral Attenuation\nCoefficients of Water--Part I: Preliminary (Report\n5642, U.S. Naval Research Laboratory, May 24, 1961).\n74. Duntley, S.Q., Examples of Water Clarity, I, II (Visi-\nbility Laboratory, Univ. of California, San Diego,\nReport 3-7, Task 3, Contract NObs-72039 Bureau of\nShips, May 1959; and Report 5-7, Task 5, same con-\ntract, June 1960).\n75. Duntley, S.Q., Improved Nomographs for Calculating Visi-\nbility by Swimmers (Natural Light) (Visibility Lab -\noratory, Univ. of California, San Diego, Report 5-3,\nTask 5, Contract NObs-72039, Bureau of Ships, Feb-","BIBLIOGRAPHY\n211\n76. Duntley, S.Q., Measurements of the Transmission of Light\nfrom an Underwater Point Source (Visibility Labora-\ntory, Univ. of California, San Diego, Report 5-11,\nTask 5, Contract NObs-72039, Bureau of Ships, Octo-\nber 1960).\n77. Duntley, S.Q., Measurements of the Transmission of Light\nfrom an Underwater Source Having Variable Beam-\nSpread (Scripps Inst. of Ocean. Ref. 60-57, Univ.\nof California, San Diego, 1960).\n78. Duntley, S.Q., \"Light in the Sea\", J. Opt. Soc. Am. 53,\n214 (1963).\n79. Duntley, S.Q., Underwater Communication by Scattered\nLight (Visibility Laboratory, Univ. of California,\nSan Diego, Report 5-12, Task 5 (Final Report), Con-\ntract NObs-72039, Bureau of Ships, September 1963).\n81. Duntley, S.Q., Culver, W.H., Richey, F., and Preisendor-\nfer, R.W., \"Reduction of contrast by atmospheric\nboil,\" J. Opt. Soc. Am. 53, 351 (1963).\n82. Duntley, S.Q., and Preisendorfer, R.W., The Visibility\nof Submerged Objects (Final Report N5ori-07864,\nVisibility Laboratory, Massachusetts Institute of\nTechnology, 31 August 1952).\n83. Duntley, S.Q., Tyler, J.E., and Taylor, J.H., Field\nTest of a System for Predicting Visibility by Swim-\nmers from Measurements of the Clarity of Natural\nWaters (Scripps Inst. of Ocean. Ref. 59-39, Univ. of\nCalifornia, San Diego, 1959)\n84. DuPré, E.F., and Dawson, L.H., Transmission of Light in\nWater: An Annotated Bibliography (U.S. Naval Re -\nsearch Laboratory, Bibliography No. 20, April,1961).\n96. Gates, D.M., Energy Exchange in the Biosphere (Harper\nand Row, New York, 1962).\n109. Hill, M.N., ed., The Sea (Interscience Pub., New York,\n1962), vol. I, Physical Oceanography.\n113. Hulburt, E.O., \"The polarization of light at sea,\" J.\nOpt. Soc. Am. 24, 35 (1934).\n115. Hulburt, E.O., \"Optics of distilled and natural water,\" \"\nJ. Opt. Soc. Am. 35, 698 (1945)\n117. Ivanoff, A., and Waterman, T.H., \"Elliptical polariza-\ntion of submarine illumination,\" J. Mar. Res. 16,\n255 (1958).\n118. Ivanoff, A., and Waterman, T.H., \"Factors, mainly depth\nand wavelength, affecting the degree of underwater\npolarization,\" J. Mar. Res. 16, 283 (1958).","VOL. I\n212\nINTRODUCTION\n122. Jerlov, N.G., \"Optical studies of ocean water, \" Reports\nof the Swedish Deep-Sea Expedition 3, 1 (1951).\n123. Jerlov, N.G., \"Optical measurements in the eastern\nNorth Atlantic,\" Discovery II exp. of August and\nSeptember 1959, Medd. Oceanogr. Inst. Göteborg 30,\n1 (1961).\n124. Jerlov, N.G., ed., Symposium on Radiant Energy in the\nSea, International Union of Geodesy and Geophysics,\nHelsinki Meeting, August 1960 (L'Institute Géo-\ngraphique National, Monograph No. 10, Paris, 1961).\n125. Jerlov, N.G., \"Optical oceanography,\" Oceanogr. Mar.\nBiol. Ann. Rev. 1, 89 (1963). See also: Optical\nOceanography (Elsevier Publishing Co., N.Y., 1968).\n126. Jerlov, N.G., \"Factors influencing the colour of the\noceans, 11 in Studies on Oceanography (1964), p. 260.\n127. Jerlov, N.G., \"Optical classification of ocean water,\"\nin Physical Aspects of Light in the Sea (Univ. of\nHawaii Press, 1964), J.E. Tyler, ed.\n128. Johnson, F.S., \"The solar constant,\" J. Met. 11, 431\n(1954).\n132. Kalle, K., \"What do we know about the 'Gelbstoff'?,\"\nSymposium on Radiant Energy in the Sea, Interna-\ntional Union of Geodesy and Geophysics, Helsinki\nMeeting, August 1960 (L Institute Géographique\nNational, Monograph No. 10, Paris, 1961).\n133. Kampa, E.M., and Boden, B.P., \"Light generation in a\nsonic-scattering layer,\" Deep Sea Res. 4, 73 (1957).\n144. Kozlyaninov, M.V., \"New instrument for measuring the\noptical properties of sea water,\" Trudy Inst.\nOkeanol. Akad. Nauk S.S.S.R. 25, 134 (1957), (trans.\navailable Off. Tech. Serv., U.S. Dept. of Comm.,\nWashington, D.C.).\n160. Limbaugh, C., and Rechnitzer, A.B., \"Visual detection of\ntemperature-density discontinuities in water by\ndiving,\" Science 121, 1 (1955).\n173. Malkus, J.S., \"Large scale interactions,\" in The Sea\n(Interscience Pub., New York, 1962), M.N. Hill, ed.,\nvol. I, Chapt. 4.\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.","BIBLIOGRAPHY\n213\n185. Moon, P. The Scientific Basis of Illuminating Engineer-\ning (Dover Pub., Inc., New York, 1961), rev. ed.\n210. Preisendorfer, R.W., Lectures on Photometry, Hydrologic\nOptics, Atmospheric Optics (Lecture Notes, vol. I,\nVisibility Laboratory, Scripps Inst. of Ocean.,\nUniv. of California, San Diego, Fall 1953).\n225. Preisendorfer, R.W., On the Existence of Characteristic\nDiffuse Light in Natural Waters (Scripps Inst. of\nOcean. Ref. 58-59, Univ. of California, San Diego,\n1958).\n251. Preisendorfer, R.W., Radiative Transfer on Discrete\nSpaces (Pergamon Press, New York, 1965).\n260. Redmond, P.M., Light Refraction by a Free Ocean Surface\n(AIAA paper No. 65-238, Am. Inst. of Aeronautics\nand Astronautics, New York, March 1965).\n265. Riley, G.A., \"Theory of food-chain relations in the\nocean,\" in The Sea (Interscience Pub., New York,\n1963), M.N. Hill, ed., vol. II, Chapt. 20.\n271. Sasaki, T., Okami, N., Oshiba, G., and Watanabe, S.,\n\"Angular distribution of light in deep water,\"\nRecords of Ocean. Works in Japan 5, 1 (1960).\n272. Schenck, H., \"On the focusing of sunlight by ocean\nwaves, \" J. Opt. Soc. Am. 47, 653 (1957)\n283. Secchi, P.A., Relazione della Esperienze Fatte a Bordo\ndella Pontificia Pirocorvetta L'Immacolata Con-\ncezione per Determinare la Trasparenza del Mare\n(circa 1866) [Reports on Experiments Made on Board\nthe Papal Steam Sloop L'Immacolata Concezione to\nDetermine the Transparency of the Sea] (Translation\navailable, Dept. of the Navy, Office of Chief of\nNaval Operations, O.N.I. Trans. No. A-655, Op-923\nM4B, 21 Dec. 1955).\n290. Spilhaus, A.F., Observations of Light Scattering in Sea\nWater (PhD Thesis, Dept. of Geology and Geophysics,\nM.I.T., February 1965, prepared under ONR Contract\nNonr 1841 (74). NR 083-157).\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).","VOL. I\n214\nINTRODUCTION\n299. Tyler, J.E., , An Instrument for the Measurement of the\nVolume Absorption Coefficient of Horizontally Strat-\nified Water (Report No. 5-4, Task 5, Contract NObs-\n72039, Bureau of Ships Project NS 714-100, Visibili-\nty Laboratory, Univ. of California, San Diego,\nFebruary 1960).\n300. Tyler, J.E., \"Scattering properties of distilled and\nnatural waters,\" Limnology and Oceanography 6,\n451 (1961).\n301. Tyler, J.E., \"Estimation of percent polarization in deep\noceanic water,\" J. Mar. Res. 21, 102 (1963)\n302. Tyler, J.E., \"Colour of the ocean,\" Nature 202, 1262\n(1964).\n303. Tyler, J.E., ed., Physical Aspects of Light in the Sea,\nA Symposium at the Tenth Pacific Science Congress,\nHonolulu, Hawaii, August 1961 (Univ. of Hawaii\nPress, Honolulu, Hawaii, 1964).\n304. Tyler, J.E., and Shaules, A., \"Irradiance on a flat ob-\n-\nject underwater,\" Applied Optics 3, 105 (1964).\n305. Tyler, J.E., and Preisendorfer, R.W., \"Light,\" in The\nSea (Interscience Pub., New York, 1962), M.N. Hill,\ned., vol. I, Chapt. 8.\n309. Volterra, V. , Theory of Functionals and of Integral and\nIntegro-differential Equations (Dover Pub., Inc.,\nNew York, 1959).\n315. Whitney, L.V., \"The angular distribution of character-\nistic diffuse light in natural waters,\" J. Mar. Res.\n4, 122 (1941).","INDEX\n215\na (alpha), 60\nComplete transmittance (for\na (ay), 55, 58, 60\nirradiance), 79\nAbsorbed flux, 55, 58\nConsistency, check for inher-\nAbsorption (of a finitely\nent optical properties, 124\ndeep slab of water, Ay),\nContrast; apparent, inherent,\n70; measurements, 103 ;\n44; transmittance law, 89,90,\nlength, 110\n99; multiplicative (semi-\nAdaptation level (for visi-\ngroup) property, 95\nbility), 160\nContrast reduction; subsurface,\nApparent optical properties,\nby scattering and absorption,\n118; defined, listed, 135\n44; by refractive effects,\nApparent radiance, 60\n48\nAstrophysical Optics, de-\nConventions (used in this work)\nfined, 1\nnature of radiant flux, 6;\nAsymptotic radiance hypo-\nunpolarized flux, 7; fre-\nthesis, 41\nquency density (footnote)\nAtmospheric radiative trans-\nCosine law, for irradiance, 13;\nfer, gross features, 27\nfor radiant emittance, 14\nAttenuation length, 90\nDecomposed (light field), 63\nBack scattered flux, 55\nDiffusion constant (D), 64; in\nBeam transmittance, 120\nterms of K, 111\nBeebe, L., 143, 153\nDiffusion equation (for h), 64\nBiological sources, under-\nDiffusion length, 135\nwater light field, 53\nDiffusion model, 61; for point\nBlondel, 21\nsources, 110; empirical ex-\nBoundaries, 55\namples, 112\nBrightness (monochromatic)\nDistribution factor, 55\nof radiant flux, 10;\nDivergence law, 44; for vector\nBrightness' is an untech-\nirradiance, 62\nnical term for the precise\nDominant wavelength, 149\nconcepts of radiance or\nDuntley Disks, 96\nluminance (as the case may\nbe)\nEquation of transfer, 60\nEquilibrium radiance, 85\nCandela, 20\nEquilibrium solutions (food\nCarnivores (in food chain),\nchain), 203\n199\nExponential law of change (gen-\nChromaticity (color), 146;\neral), 197; differential\ncomponents, 148; plane,\nform, 201\n147; diagram, 149; coor-\ndinates, 149\nFick's law (of diffusion), 64\nClassification of natural\nField interpretations of radi-\nhydrosols, 138\nant flux, 12\nCollimated flux, scattering\nFinitely deep hydrosols, re-\nfunctions for, 83; produced\nflectance and transmittance,\nby sources, 114\n68\nColor, 146; components, 146;\nFlux density (radiant), 10\npurity, 149; dominant wave-\nFood chain problem (in the sea),\nlength, 149\n196\nColorimetric radiative trans-\nFoot candle, 20\nfer, 142\nFrequency density convention\nComplete reflectance (for ir-\n(in this work), 7\nradiance), 79","VOL. I\nINDEX\n216\nphotometry. The meaning in-\nGeophysical Optics, defined,\ntended for the term 'light'\n1\nwill be implicit in each\nGEOVAC (geophysical optics\ncontext of its use. Thus\nvariable automatic computer),\n'light field' may, e.g.,\n208\ncorrespond informally to\nGlitter patterns, on air-wa-\n'radiant energy', 'radiant\nter surface, 32\nflux', 'radiance distribu-\ntion', 'irradiance function',\nHerbivore (in food chain),\n'luminous energy', 'luminous\n199\nflux', 'luminance distribu-\nHerschel (luminance unit), 21\ntion', 'illuminance func-\nHomogeneity (of o), 82\nHydrologic Optics, defined, 1;\ntion', etcetera.\nLight field, decay with depth,\nfuture problems, 205\n37, 66; polarization, under-\nHydrologic range, 90\nwater, 50; biological sources,\nIlluminance, 19; measured at\n53; artificial, 109; decom-\nposed, 63\nearth's surface, 25\nInherent optical properties,\nLumen, 19\n118; defined, listed, 119\nLuminance, 19\nLuminosity function (photopic),\nInherent radiance, 60\nIntensity (radiant), 10;\n145\nLuminous energy, 19\nfield, 12; surface, 12\nInteraction principle, 4\nInterdependence (Plan) of\nManhole (optical), 34\nMelanoidines (Gelbstoff), 133\nchapters in this work, 5\nModes of classification of\nInvariant Imbedding Relation\nnatural hydrosols, 140\n(for irradiance), 71, 80\nMultiplicative (semigroup)\nIrradiance, 12; scalar, 15,\nproperty, of contrast trans-\n106; hemispherical scalar,\nmittance, 93; of beam trans-\n16; vector, 15; net, 16,\nmittance, 120\n61; upwelling (upward), 16,\n55, 58, 106; downwelling\nNatural hydrosols, classified,\n(downward), 16, 55, 58,\n138; characterization (for\n106; measured at earth's\nvisibility), 195\nsurface, 24; reflectance\nNatural illumination, 156\nof air-water surface, 30;\nNomographs for underwater vis-\nreflectance in deep water,\n67; invariant imbedding\nibility, 154\nNutrient (in food chain), 199\nrelations, 71\nIrradiance distributions,\nOne-D (two-flow irradiance)\nunderwater, 42\nIsotropy (of o), 82\nmodel, 56\nOperational definitions of the\nK (kappa) (k-function or dif-\ndensities, 10\nOptical properties, inherent,\nfuse attenuation function\nfor diffusion model), 65\napparent, 118\nk (little kay) 58; inter-\nPath function, 60\nchangeable with K (big\nPath radiance, 63\nkay), 83\nPerfectly diffusing (surface),\nLambert, 20\n21\nPhase density, of radiant\nLight. This term is used\nthroughout the present work\nflux, 10\nPhotometry, geometrical, 18\nas an informal correspon-\nPhotons, as viewed in this\ndent to any one of the de-\nfined concepts of geometri-\nwork, 7\ncal radiometry and","INDEX\n217\nPhotopic luminosity curve,\nRadiative transfer theory\n18, 145\n(con't.)\nPhytoplankton (in food chain),\ncolorimetric, 142\n199\nRadiometric concepts, opera-\nPolarization, defined, 51;\ntional definitions, 11\nunderwater properties, 52\nRadiometry, geometrical, 7\nPlane-parallel medium, 55\nRationalized units (polemic\nPrey-predator equations, 198\non), 21\nPrinciples of invariance for\nReflectance, for irradiance at\nirradiance, 73, 79\nair-water surface, 30; for\nProblems of hydrologic optics,\ninfinitely deep homogeneous\n2, 205\nwater, 67; for finitely deep\nhomogeneous water, 68; com-\nQuantum, 7\nplete (for irradiance), 79\nQuasi-steady state (food\nRefraction, subsurface, 33\nchain), 202\nResidual radiance, 63, 120\nRiccati Equation, in food\nRadiance, 10; field, 12;\nchain, 203\nsurface, 12; n 2- law, 18,\nR-infinity (Roo), 67; correc-\n87; inherent, 60; appar-\ntion in visibility, 172\nent, 60; equilibrium, 85;\n-difference law, 92; re-\no (sigma), 122\nsidual, 120; path, 63\nS (ess), 58\nRadiance distribution, behav-\nScattered flux, 58, for col-\nior with depth, 39; asymp-\nlimated flux, 83; forward,\ntotic hypothesis, 41; by\nbackward, 124\nsubmerged point source, 113\nSchuster, A., 57\nRadiance model, 58\nSecchi Disk, 96; in visibility\nRadiant density, 16\ncalculations, 169\nRadiant emittance, 12\nSemigroup (multiplicative)\nRadiant energy. In this work\nproperty, of contrast trans-\nradiant energy is the unde-\nmittance, 95; of beam trans-\nfined, primitive concept,\nmittance, 120\ntaken as given by nature\nSighting range (interpreta-\nand axiomatized by radio-\ntions), 195\nmetrists as their primary\nSimple model for radiance, 61\nphysical notion. In other\nSolar (irradiance) constant,\n22; (illuminance), 22\nfields, such as electromag-\nnetics, it can be made to\nSource term (for scalar irra-\ndiance), 64\nrest on one step lower: on\nthe constructs (E,D,B,H) of\nSpectrum locus, 149\nthe electromagnetic field.\nStratified media, 62\nThese steps into physical\nSubsurface refractive phenom-\nprimitivity descend even\nena, 33\nlower. But this nether re-\nSurface interpretation of ra-\ngion is of no concern to us\ndiant flux, 12\nin this work.\nRadiant flux, defined, 7;\nThermocline phenomena, 36\nmonochromatic brightness of,\nTransmittance, for irradiance\n10; field and surface inter-\nat air-water surface, 30;\npretations, 121\n(t=1-r), for finitely deep\nRadiant intensity, 10; field\nhomogeneous water, 68; com-\nand surface, 12\nplete (for irradiance), 79;\nRadiative transfer theory,\nbeam, 120; contrast, 93\ndefined, 1; basic con-\nTristimulus functions, 144\nTwo-flow (irradiance) model,\nstructs, 4; atmospheric\nfeatures, 27; across air-\n55-57\nwater surface, 28;","INDEX\n218\nUnpolarized-Flux convention\n(in this work), 7\nVector analogy with color, 146\nVisibility underwater, 154;\neffect of depth and water\nclarity, 157; use of nomo-\ngraphs, 163; along inclined\npaths of sight, 165; hori-\nzontal paths of sight, 170\nVolterra prey-predator equa-\ntions, 198\nVolume absorption function,\n60; measurement, 103; oper-\national definition, 124\nVolume attenuation function,\n60; operational definition,\n119; empirical, 120\nVolume backward scattering\nfunctions, 124\nVolume forward scattering\nfunctions, 124\nVolume scattering function,\n122\nVolume total scattering func-\ntion, 60; operational defi-\nnition, 123\nWater clarity (visualization),\n194\nWavelength, dominant, 149\nWhite light, 149\nWindow (spectral), 134\n* U.S. GOVERNMENT PRINTING OFFICE: 1976-677-881/ 36 REGION NO. 8"]}