HYDROLOGIC OPTICS Volume V. Properties R W. PREISENDORFER U.S. DEPARTMENT OF COMMERCE NATIONAL OCEANIC & ATMOSPHERIC ADMINISTRATION ENVIRONMENTAL RESEARCH LABORATORIES HONOLULU, HAWAII 1976 Digitized by the Internet Archive in 2012 with funding from LYRASIS Members and Sloan Foundation http://archive.org/details/hydrologicopOOprei M. ^res o* * «/5 D HYDROLOGIC OPTICS Volume V. Properties R.W Preisendorfer Joint Tsunami Research Effort Honolulu, Hawaii 1976 U.S. DEPARTMENT OF COMMERCE National Oceanic and Atmospheric Administration ^ Environmental Research Laboratories •g Pacific Marine Environmental Laboratory 11 The importance of light in the sea is apparent when it is recalled that solar radiation supplies most of the energy input to the ocean and supports its ecology through photosynthesis. The biological productivity of an acre of ocean has been estimated to be, on a worldwide average, comparable to that of an acre of land. . . . All these aspects of light in the sea can be treated by describing the optical nature of ocean water.- S. Q. DUNTLEY Light in the Sea [78] CONTENTS iii 1 VOLUME V Chapter 8 Models for Irradiance ■ ... 1 \ ^ J FieldX^^TS C^X Introduction 1 1 Invariant Imbedding Relation for Irradiance Fields 2 2 General Irradiance Equations 5 3 Two- Flow Equations: Undecomposed Form 8 Equilibrium Form of the Two-Flow Equations 13 Ontogeny of the Two-Flow Equations 13 8 . 4 Two-Flow Equations: Decomposed Form 1 4 Principles of Invariance for Diffuse Irradiance 18 Classical Models for Irradiance Fields 19 Collimated-Diffuse Light Field Models 19 Isotropic Scattering Models 23 Connections with Diffusion Theory 24 8.5 Two-D Models for Irradiance Fields 25 On the Depth Dependence of the Attenuating Functions 2 5 Two-D Model for Undecomposed Irradiance Fields 30 Two-D Models for Internal Sources 37 Two-D Model for Decomposed Irradiance Fields 43 Inclusion of Boundary Effects 46 8.6 One-D and Many-D Models 51 One-D Models for Undecomposed Irradiance Fields 52 One-D Model for Internal Sources 55 One-D Model for Decomposed Irradiances 56 Many-D Models 57 8. 7 Invariant Imbedding Concepts for Irradiance Fields 61 Example 1: /( and O" Factors in Two-D Models 62 Example 2: ^ and ZT Factors in One-D Models 64 Example 3: Differential Equations for R and T Factors 65 Example 4: Third Order Semigroup Properties of J% and -T Factors 67 Example 5: Systematic Analyses of Boundary Effects 71 iv CONTENTS 8. 7 Invariant Imbedding Concepts for Irradiance Fields - -Continued Example 6: Invariant Imbedding Operator for Interacting Media 76 Example 7: Differential Equations Governing^ and y Factors 79 Example 8: Method of Modules for Irradiance Fields 80 Example 9: Method of Semigroups for Irradiance Fields 81 Example 10: Irradiance Fields Generated by Internal Sources 81 8.8 A Model for Vector Irradiance Fields 87 The Quasi-Irrotational Light Field in Natural Waters 88 Interpretations of the Integrating Factor 89 The Curl and Divergence of the Submarine Light Field 91 General Representation of the Submarine Light Field 93 Example 1: The case of Isotropic Scattering 94 Example 2: Asymptotic Form of the Light Field 95 Global Properties of the Irradiance Field 97 8.9 Canonical Representation of Irradiance Fields 98 8.10 Bibliographic Notes for Chapter 8 101 PART III THEORY OF OPTICAL PROPERTIES Chapter 9 General Theory of Optical Properties 9.0 Introduction 105 9 . 1 B asic Definitions for Optical Properties 106 9. 2 Directly Observable Quantities for Light Fields in Natural HydrosoLs 109 Introduction 109 Classical Two-Flow Theory: The Theoretical K Functions 111 Diffuse Absorption Coefficient k 111 The R- Infinity Formulas 113 The Inequalities 114 Observations on Inadequacies of Classical Theory 115 Exact Two-Flow Theory: Experimental K Functions and R Functions 115 The Basic Reflectance Relation 118 The Exact Inequalities 119 The Significance of the Condition < K(z,+) 120 CONTENTS 9. 2 D irectly Observable Quantities for Light Fields in Natural Hydrosols - -Continued. Relative Magnitudes of H and K Functions 121 Characteristic Equation for K(z,±) 123 The Depth Rate of Change of R(z,-) 123 Connections Among the K Functions 123 K Function for Radiance 125 General K Functions 125 Integral Representations of the K Functions 126 Integral Representations of the Irradiance and Radiance Fields 126 9. 3 T he Covariation of the K Function for Irradiance and Distribution Functions 128 Some Elementary Physical and Geometrical Features of K(z,-) and D(z,-) 128 The General Law Governing K(x,-) and D(z,-) 136 The Absorption-Like Character of K(z,-) 138 Forward Scattering Media 140 The Covariation Rule for K(z,-) and D(z,-) 141 Illustrations of the Rule 143 The Contravariation of K(z,+) and D(z,+) 144 A Covariation Rule of Thumb 145 9- 4 General Analytical Representations of the Observable Reflectance Function 146 The Differential Equation for R(-,-): Unfactored Form 148 The Differential Equation for R(«,-): Factored Form 149 Second-Order Form of Differential Equation for R(-,-) 149 The Equilibrium-Seeking Theorem for R(*,-): Preliminary Observations The Equilibrium-Seeking Theorem for R(',-) Observation 1 Observation 2 The Integral Representations of R(z,-) Applications Special Closed Form Solution Differential Analyzer or Digital Solutions Series Solutions Equivalence Theorem for R(»,-) Connections with the Two-Flow Theory Summary 9- 5 The Contrast Transmittance Function The Concept of Contrast Regular Neighborhoods of Paths Contrast Transmittance and Its Properties Alternate Representations of Contrast Transmittance Contrast Transmittance as an Apparent Optical Property On the Multiplicity of Apparent Radiance Representations 9- 6 C lassification of Optical Properties 150 152 154 155 156 157 157 159 159 159 160 162 162 165 166 168 171 172 177 178 vi CONTENTS 9.7 Bibliographic Notes for Chapter 9 182 Chapter 10 Optical Properties at Extreme Depths 10.0 Introduction 183 10.1 O n the Structure of the Light Field at S hallow Depths: Introductory Discussion 184 10.2 Experimental Basis for the Shallow-Depth Theory 187 Summary of the Experimental Evidence 192 10.3 F ormulation of the Shallow-Depth Model for K and R Functions 19 3 Formulas for H(z,±) 193 Formulas for K(z,±) 194 Formula for R(z,-) 196 Comparisons of Experimental Data with Calculations Based on the Model 197 Hypotheses on the Fine Structure of Light Fields in Natural Hydrosols 199 10.4 C atalog of K Configurations for Shallow Depths 201 Some Special Fine Structure Relations 208 Conclusion 211 10.5 A_Ge neral Proof of the Asymptotic Radiance H ypothesis 212 Introduction 212 Preliminary Definitions 213 Formulation of the Problem 218 The Functions P,Q,R, 219 The Limit of K q (-,y,) 222 The Limit of KC*,y,j 226 Notes and Observations 227 10.6 O n the Existence of Characteristic Diffuse L ight: A Special Proof of the Asymptotic R adiance Hypothesis 2 30 Introduction 230 Physical Background of the Method of Proof 231 The Proof 233 The Equation for the Characteristic Diffuse Light 237 10.7 Some Practical Consequences of the Asymptotic R adiance Hypothesis 2 38 Basic Formulas: The Irradiance Quartet 239 The D and R Functions 240 The K Functions 241 The K Characterization of the Hypothesis 242 The Basic Transfer Equations 243 CONTENTS vii 10.7 Some Practical Consequences of the Asymptotic Radiance Hypothesis - -Continued Consequences for Directly Observable Quantities: The Equation for the Asymptotic Radiance Distribution • 244 The Limits of the K Functions 245 The Limits of the D and R Functions 246 Consequences for Some Simple Theoretical Models: The Two-D Model for Irradiance Fields 247 Critique of Whitney's "General Law" 248 The Simple Model for Radiance Distributions 249 Further Consequences of Asymptoticity 249 The Standard Ellipsoid 250 Expressions for D(±) and Roo 251 The Determination of e 252 An Heuristic Proof of the Hypothesis 253 A Criterion for Asymptoticity 254 10.8 S imple Formulas for the Volume Absorption Coefficient in Asymptotic Light Fields - 2 55 Introduction 255 Short Derivation of I 256 Long Derivation of I 256 Derivation of II 257 Applied Numerology: A Rule of Thumb 258 10 . 9 Bibliographic Notes for Chapter 10 2 59 Chapter 11 The Universal Radiative Transport Equation 11.0 Intorduction 261 11.1 T ransport Equations for Radiometric Concepts 26 3 Equation of Transfer for Radiance 263 Transport Equations for H(z,±) 265 Transport Equations for h(z,±) 265 Transport Equation for Scalar Irradiance 268 Preliminary Unification and Preliminary Statement of the Equilibrium Principle 270 11.2 Transport Equations for Apparent Optical Properties 2 71 Canonical Forms of Transport Equations for K Functions 272 Dimensionless Transport Equation for K((P) 274 Transport Equation for K(z,6,) 274 Transport Equations for K(z,±) 276 Transport Equations for k(z,±) and k(z) 277 Transport Equation for R(z,-) 278 11.3 Universal Radiative Transport Equation and the Equilibrium Principle 2 79 viii CONTENTS 11.4 Some Additional Transport Equations Subsumed by the Universal Transport Equation 2 81 Summary and Conclusion 285 11.5 Bibliographic Notes for Chapter 11 285 PREFACE The major emphasis in this volume of Hydrologic Optics is on the optical properties of tne sea tnat govern the penetration of natural light into its depths. (The optical properties of artificial light fields were discussed in Sec- tion 1.5.) Because seas and lakes are largely horizontally stratified, we begin our studies with a simple irradiance model for light fields which describes the downward and up- ward flow of radiant energy in terms of a pair of coupled ordinary differential equations. This system of equations goes back in essence to the modern originator of our subject, Arthur Schuster [279]. In one sense the solution of these equations would constitute a simple exercise in an elemen- tary differential equations course; in another and more pro- found sense these equations hold the conceptual keys to the subject of radiative transfer in scattering-absorbing media. It is not the mathematical simplicity of these equations that we shall exploit, but rather their conceptual content. Thus, throughout Chapter 8, we have an example of how to mine an unexpectedly deep vein of physical ideas centering on a sys- tem of equations ((8) of 8.5) that could otherwise be easily solved, and then forgotten in a few minutes, by an impatient young physical oceanographer who has visions of sun-glittered curling breakers awaiting him at the ocean's shore. In Chapter 9 we step carefully into deeper waters to search for and distinguish between those optical properties of the sea that are either inherent or apparent. We begin with the apparent properties given us by the Schuster two- flow equations of Chapter 8. Then in ever more comprehensive terms we generalize these concepts and study their behavior under varying lighting conditions and physical settings, reaching a general classification in the closing section of Chapter 9. In Chapter 10 we return again to the stratified light field, so prevalent in oceanography and limnology, and ex- plore in detail the special and interesting behavior of the light field and its attendant apparent optical properties at both small and great depths in the sea. Ms. Louise F. Lembeck typed the camera ready manuscript and assisted in editorial matters. The camera ready manu- script is an unchanged version of an earlier draft written in 1965. Parts of the chapters have been used in various course and seminar lectures over the years at Scripps Insti- tution of Oceanography, La Jolla; the Naval Postgraduate School at Monterey, California; and at the Hawaii Institute of Geophysics, University of Hawaii. R.W.P. Honolulu, August 1975 CHAPTER MODELS FOR IRRADIANCE FIELDS 8. Introduction In this chapter we shall develop some simple models of light fields in natural optical media, models which have been found to be most useful in the study of radiative transfer in the seas and the atmosphere. These models are built around the concept of irradiance and derive their simplicity and utility directly from the simplicity and utility of irradiance itself For irradiance is the concept which describes radiant flux per unit area on a surface, and as such utilizes a single number rather than an infinite set of numbers as in the case of the radiance distribution studied throughout Chapter 7. The introduction of irradiance fields into the study of nat- ural light fields is also encouraged by the following con- venient geometric structure of the light fields found in the atmosphere, the sea, lakes and other natural hydrosols: over relatively great distances in all directions within hori- zontal planes in the sea or air, the natural light fields are often found to vary essentially very little, so that an irradiance value at one point of a plane matches that at other quite remote points on the plane; the points being 'remote' in the sense that they are separated by distances vast compared with the attenuation lengths of the media. This latter feature is especially noted in the seas and other laterally extensive natural hydrosols. Because of this gen- tle almost imperceptible variation over such planes, the radiometric field over these planes may be characterized by the single irradiance magnitude common to all points of a given plane. As a consequence the description of the flow of radiant energy down into the depths of the sea or into the atmosphere can often be reduced to the description of an irradiance flow along any one of the straight lines nor- mal to the family of horizontal planes comprising that por- tion of the air or sea of interest. In short, the appro- priate introduction of an irradiance field into a plane parallel medium reduces the description of the radiative transfer problem within that medium to a one-dimensional problem. One of the more fascinating features of irradiance fields in plane-parallel media, especially for the theo- retically inclined investigator, is the fundamental simi- larity between the basic equations for irradiance fields MODELS FOR IRRADIANCE FIELDS VOL. V and those for the radiance distributions as studied, e.g., in Chapter 7. The similarity is a thorough-going one which may, indeed, be used as a heuristic guide in pursuing either subject matter, using the other as a base. In this connec- tion, it may be noted that the original forms of the invar- iant imbedding relations and internal source relations studied in Chapter 7 were tentatively found in the irradi- ance context by means of informal scratch pad calculations; the simplicity of their derivation and resultant forms in that context encouraged the rigorous search for the asso- ciated full fledged operator equations for radiance sprinkled throughout that chapter. We shall be guided in our discussions in the initial sections of this chapter (8.1, 8.2) and once again in Sec- tion 8.7 by the close conceptual connections between the functional equations for irradiance and radiance fields. In this way we can effect a smooth transition from the gen- eral formulations of Chapter 7 to the simpler settings of this chapter and at the same time gain some insight into the unity of the theory engendered by the invariance concepts. However, for the main sections of the chapter (8.3-8.6) the discussions will for the most part dwell on specific models which have been tried and found useful in the daily tasks of obtaining numerical estimates and rule-of- thumb algebraic approximations to the magnitudes of light fields, and the properties of their associated optical media. Throughout this chapter we shall work with an arbitrary plane-parallel medium X(a,b) and adopt the reference frame for X(a,b), as defined in Section 2.4. Furthermore the steady state irradiances H(z,±) defined in Section 2.4, along with their attendant radiometric concepts, will be adopted without further explanation. The medium X(a,b) will be assumed free of internal sources unless specifically noted otherwise, and arbitrarily stratified with a stratified light field, so that both radiometric and optical properties depend only on depth z within X(a,b) , a <_ z _< b . Arbi- trary sources are incident on the upper and lower boundaries of X(a,b) . (In real media, however, it is customary in practice to have no sources incident on the lower boundary.) The explicit retention of a source on the lower boundary will have the effect of keeping the resultant theoretical imbed- ing relations in their full symmetric form. 8 . 1 Invariant Imbedding Relation for Irradiance Fields Our point of departure for the present discussion is the set of principles of invariance (7), (8) of Section 3.7. We recall that these statements were deduced from an appli- cation of the interaction principle to an arbitrary subslab X(x,z) of a plane parallel medium of the type X(a,b), sche- matically depicted in Fig. 8.1. The results may be written: I. H(y,+) = H(z,+) T(z,y) + H(y,-)R(y,z) (1) II. H(y,-) = H(x,-)T(x,y) + H(y,+)R(y,x) (2) SEC. 8.1 INVARIANT IMBEDDING RELATION FIG. 8.1 The setting for the principles of invariance governing irradiance fields on plane-parallel media. where a£x<_y£z^b. The four numbers T(x,y), R(y,x) , T(z,y), R(y,z) are the various transmittances (T) and reflectances (R) of the pieces X(x,y) and X(y,z) of the partitioned slab X(x,z). The present goal is to derive the invariant imbedding relation for X(a,b) in the irradiance context. Our activity will parallel very closely that in examples 4 and 5 of Section 3.9, thereby casting light on those earlier computations and in turn adding evidence to the belief that the manner of approach to the invariant imbedding relation, at least on the algebraic level, is independent of the geometry of the medium and the radiometric concepts used in the approach. Most of the work toward attaining the invariant imbed- ding relation is already contained in the results (9) , and (10) of Section 3.17; for if we now apply those equations to the subslab X(x,z) of the present setting (by letting z = x, b = z) and write : "tffx v zV for T(x,y)R(y,z) f31 ^(x,y,zj tor 1 _ R(y>x)R(y>z) UJ "!rtx,y,zr for 1 . H l%f R \ ytZ) (4) X( Z ,y,x)" for I I |f^ (5) MODELS FOR IRRADIANCE FIELDS VOL. V \T(z,y,x)" for T(z,y) l-R(y,x)R(y,z) then those equations can be written: H(y,+) = H(z, + )y(z,y,x) + H (x , - )(?(x,y , z) H(y,-) = H(x,-)^(x,y,z) + H(z,+)#.(z,y,x) (6) (7) (8) Equ for the r bedding r this reas in Chapte to explor concepts . The above are factors , cases of ations (7) and (8) present an excellent opportunity eader to become acquainted with the invariant im- elation in a relatively simple setting. It was for on that the irradiance example was presented first r 3. In this chapter we shall have more opportunity e the irradiance context of the invariant imbedding operators ^ and °f defined in (3) through (6) the complete reflectance and complete tvansmittance respectively. Observe that the following special <%. and "/ hold: ft(x,x,y) = R(x,y) y(x,y,y) = T(x,y) #.(x,y,y) = :/0,x,y) = 1 (9) (10) (11) (12) These statements may be obtained directly from (3) and (4) upon suitable substitutions, and by appeal to (13) and (14) of Sec. 3.7. A complementary set of four equations can be obtained from (5) and (6) . It follows that the invariant imbedding equations (7) and (8) contain the principles of invariance (1) and (2) as special cases. The equations (7) and (8) can be cast into matrix form by writing: '^(x,y,z) for jT / (z,y,x)(^(z,y,x) ^(x,y,z)T(x,y,z) so that we have: (H(y,+),H(y,-)) = (H (z , x) ,H (x , - ) M(x ,y , z) This irra ing nota Chap a gr tive of r ant ing is the diance in the tion fo ter 3 f eat eco notati adiativ imbeddi from th requ conte irrad r s ta or th nomy on, a e tra ng te is pr ired xt. iance ndard e rad of te nd se nsfer chniq actic znvar It sh cont and iance rmino rves theo ues . e, fo zant ould ext , compl (ope logy, to st ry at Ther r it imbeddi be obse without ete ope rator) retain rengthe tained e shall very ra ng relatio rved that essential rators use context, s a useful n the cone by means o be no con rely happe n fo we a cha d ea This and eptu f th fusi ns t (13) (14) r the re adopt nge, the rlier in affords sugges- al unity e invari on aris- hat the SEC. 8.2 GENERAL IRRADIANCE EQUATIONS irradiance field and the radiance field are simultaneously under study in a given one-parameter medium, since these are two very distinct levels of description of given radiative transfer phenomena: the irradiance description presently under study is a simple numerical description of a light field while the radiance description is a more detailed functional descrip- tion of the light field. As it stands, the invariant imbedding relation (14) is the general form of the solution to the radiative transfer problem in X(a,b) for the irradiance field: knowledge of #|[x,y,z) for every three successive levels x,y,z in a sub- slab X(x,z) of X(z,b) allows one to compute the irradiance field (H(y,+) ,H(y , -) ) at every level y in X(x,z) knowing the incident radiance field (H(z ,+) ,H(x, -) ) on X(x,z). The complete reflectance and transmittance operators (%_ and IT in 7f\S.x,y,z) depend in turn on the standard operators R and T as shown in (3) through (6) . Therefore the complete solu- tion of the irradiance transfer problem in X(a,b) devolves on knowledge of the standard R and T factors, in exact analogy to the radiance transfer problem studied in detail in Chapter 7. Consequently, knowing the R and T , or better still the A and *¥ factors, for a medium X(a,b), we can write down by sight the answer to every question about H(y,±) for every level y in X(a,b). We shall in the course of this chapter obtain methods for the determination of the standard R and T factors and the complete factors ft and y. For the present we go on to formulate further equations governing the irradiance field. 8. 2 General Irradiance Equations The global description of the irradiance field in a plane parallel medium X(a,b) as given by the invariant im- bedding relation (14) of Sec. 8.1 will now be supplemented by a local description in the form of a pair of differential equations for the irradiances H(z,±) as a function of depth z in X(a,b). The approach we shall take at present is through the principles of invariance (1) and (2) of Sec. 8.1. The idea of the derivation is quite simple: We isolate for attention the subslab X(x,z) of X(a,b) and form the difference quotient: H (z,-) - H(x,-) z - x l J We then let x approach z and determine the associated limit of (1) . The physical meaning of this activity should be carefully noted at the outset, as it will repeatedly sug- gest the subsequent moves in the sequence of explorations below. Thus (1) is the average rate of change of the down- ward irradiance field over the depth interval from x to z. As z - x is made smaller and smaller (and hence X(x,z) thinner and thinner) we increasingly localize the factors governing this average rate of change until, in the limit, we should have a completely local description of the change of the downward irradiance field at depth z . MODELS FOR IRRADIANCE FIELDS VOL. V Following the program just outlined, we set y=z in (2) of Sec. 8.1, the result being: H(z,-) = H(x,-)T(x,z) + H(z,+)R(z,x) This representation of H(z,-) is used in (1) to obtain: H(z,-) - H(x,-) = f , [T(x,z) - 1] + R(z,x) (2) Now as x approaches z , the difference z - x approaches zero, and the slab X(x,z) becomes increasingly thinner, so that its downward transmittance T(x,z) approaches 1 and its upward reflectance R(z,x) approaches zero (cf. (9)- (12) of Sec. 7.3). Therefore the quotients on the right side of (2) have a chance of going to well-defined limits. Indeed, our discussion in Sec. 7.3 (see e.g., (9)-(12) of that sec- tion) prepares the ground for the following definitions; we write : t(z,-)" for lim. p(z,+) M for lim. T(x,z) - 1 R(z,x) (3) (4) Then, if we write as usual: "dH(z,-) n r \ * - for In dz equation (2) yields: H(z,-) - H(x,-) ^dV" 3 = t(z,-)H(z,-) + p(z,+)H(z,+) (5) (6) This is the equation governing the downward irradiance field H(z,-). In a similar way, setting y = x in (1) of Sec. 8.1 forming the difference quotient: H(x,+) - H(z,+) H(z,+) and writing: m t(z,+) m for lim "p(z,-) M for lim T(z,x) - 1 z - X H(x,-) R(x,z) T(z,x) - 1 x ■* z z - X R(x,z) x ■> z z - X and dH(z, + ) M r , . H(x, + ) - H(z,+ ^ — - — J — for lim — L — '- — i — ! — ■ dz x -*■ z x - z SEC. 8.2 GENERAL IRRADIANCE EQUATIONS the preceding equation yields " dH( A >+) = T(Z,+)H(Z,+) + P(Z,-)H(2,-) dz (7) This is the equation H(z,+). The notation i fundamental similari o£ invariance (1) an similarity (6) and ( principles of invari local transmittance tively for upward (+ of the invariant imb and (7) are compared on the basis of this governing the upward irradiance field n (6) and (7) is designed to point up the ty of these equations to the principles d (2) of Sec. 8.1. On the basis of this 7) are also called the local forms of the ance, and t(z,±) and p(z,±) are the and local reflectance factors^ respec- ) and downward (-) irradiance. The unity edding approach is underscored when (6) with (5) and (6) of Sec. 7.1. In fact comparison, we are moved to write: "H(z) n for (H(z,+) ,H(z,-)) "- t(z,+) p(z,+) "^(z)" for _- p(z,-) t(z,-) _ so that (6) and (7) can be written: (8) (9) dH(z) dz H(z)^(z) (10) Equation (10) is the vector form of the general irradiance equations, and is the irradiance counterpart to (9) of Sec. 7.1. The practical distinction between (10) above and (9) of Sec. 7.1 should be kept firmly in mind: (10) is a vector equation whose components are numbers while (9) of Sec. 7.1 is a vector equation whose components are functions and thus one level higher in the conceptual hierarchy. However, both the functional and numerical components obey many of the same algebraic relations and, generally speaking, whenever a functional equation deduced from (9) of Sec. 7 . 1 is valid 3 then there exists a corresponding valid counterpart deducible from (10). It is almost as if the local forms of the theorems of irradiance fields are the one dimensional shadows of the corresponding theorems of radiance fields; similarly for de- ductions from the global forms of the principles of invariance and their irradiance correspondents. Caution should be exer- cised in attempting to extend results the other way, i.e., from the irradiance level (10) to the radiance level (9) of Sec. 7.1, or from (1) and (2) of Sec. 8.1 back to the princi- ples in example 3 of Sec. 3.7. While no universal rule exists to guide extensions from the irradiance to the radiance level, MODELS FOR IRRADIANCE FIELDS VOL. V it is clear that if essential use is made of the commutativ- ity of the numerical factors R and T while gaining a result, then the associated result need not exist on the radiance level, since commutativity of the R and T oper- ators does not hold in general. Some examples using this observation were discussed in Sec. 7.13 (see (91) of Sec. 7.14) . We shall turn to the applications of (10) in the dis- cussions of Sec. 8.7; for the present we continue to explore its analytic structure. Before going on to do so, we pause and note one rather interesting similarity between (10) above and the fundamental dynamical equation of quantum mechanics: ih ^p = | ^ >H (10a) Vfhere | ip > is a state vector which pairs with our H(z) and H (or - (i/"h) H) is the Hamiltonian matrix operator which pairs with our ~^C(z) . As a result of this pairing we see that the mathematics of time-dependent atomic systems is homomorphic to (i.e., of the same kind as) that for steady irradiance fields in stratified media (cf. also (46) of Sec. 8.6 and the remarks following (91) of Sec. 3.7). One more connection between (10) above (and its generalization (46) of Sec. 8.6) and the mathematical structure of different fields of physics may be noted. This concerns the formula- tions by Brillouin* of the transmission line equations for two phase and polyphase electric fields. The use of Pauli and Dirac matrices to compactly represent circuit equations for such fields can evidently be carried over with only slight modifications to the radiative transfer context. How- ever, in the present work we shall develop the algebra of radiative transfer on the basis of the invariant imbedding point of view introduced in Chapter 7. 8. 3 T wo-Flow Equations: Undecomposed Form We now add another block to the foundations for the model constructions of irradiance fields to be given below by deriving the classical two-flow equations for irradiance which correspond to (6) and (7) of Sec. 8.2. The primary distinction between (6) and (7) above and the two-flow equa- tions below lies in the structure of the coefficients of H(z,±) in the respective equations. In order to arrive at the two-flow equations we shall analyze the local transmit- tance factors t(z,±) and the local reflectance factors p(z,±) into further parts and relate these parts directly to the radiance distributions and the inherent optical proper- ties a , a of the optical medium X(a,b). In this way we will be able to make direct contact with certain well-known models of irradiance fields starting with the Schuster pro- genitors of classical radiative transfer theory, down through the variations wrought by Ryde , Gurevic, Duntley and others during the decades that followed. The historical details of *Brillouin, L. , Wave Propagation in Periodic Structures. Dover Publications, New York (1953). SEC. 8.3 TWO-FLOW EQUATIONS: UNDECOMPOSED < the manifold forms of the two-flow equations are reserved for discussion in the bibliographic notes appended to this chapter. For the present we go on to a modern development and analysis of the two-flow equations. Our starting point for the present derivations is the steady state equation of transfer for radiance in a source- free isotropic stratified optical medium, i.e., we begin with (3) of Sec. 3.15 in the form:* C-VN(z,£) = -a(z)N(z,C) + f N(z,T )a(z; V ;&tt(V ) (1) Here "z" denotes depth in the present stratified plane parallel medium X(a,b), and the term "£»V" is defined in (6) of Sec. 3.15. If ever sources are to be taken into account, one need only add fI N n (z,£) " to the right side of (1) . The net result on all subsequent equations is the addi tion of irradiance terms H (z,±) to the right sides of the respective equations. The time-dependent version of (1) is obtained, as usual, by adding the time derivative term (1/v) 3N/3t to the left side. The next step of the derivation is to integrate the terms of each side of (1) over the set 5 + of upward direc- tions. Taking the terms one by one, we begin with the deriv ative term: j £-VN(z,)dn(5) = - [ €-k ^ N(z,5)dn(0 _d_ dz J^ 5-kN(z,Odfl(0 = - dH( d z ; + ) (2) The first equality rests on the form of V in a Cartesian coordinate frame in which z is measured positive in the direction - k , as is the case in the terrestrial reference frame for hydrologic optics presently in use.** The strati- fied light field condition is also used to obtain the first equality, for under that condition the x and y derivatives of N vanish. The last equality is based on (9) of Sec. 2.4 and (8) of Sec. 2. 5. *The isotropic assumption plays no essential role in this derivation in the sense that the structure of the re- sultant formulas (9) and (10) below are the same for the anisotropic case. **If V is to be used in a coordinate frame other than Cartesian then, in general (2) yields V«H(x,+), where H(x,+) is that contribution to H(x) by radiances in the directions of E + . See [221], 10 MODELS FOR IRRADIANCE FIELDS VOL. V Starting now on the right side of (1) , we integrate the linear term over H+ : L a(z)N(z,Odfl(0 = a(z)h(z,+) (3) in which we have used (7) and (11) of Sec. 2.7. Next, the integrated integral term in (1) becomes: N*(z,£)dn(£) = l = fl N(z,^)o(z;£«;O^U') dfi(0 f J, N(z,?')a(z;C';C)dn(C') f. N(z,S')a(z;£' ;£)dflU') dfi(£) (4) In the last equality, over E into two par Equations (2) , blindly and mechanica of the two- flow equat tion of the goal, nam irradiances H(z,±). the right track; but form some analytic le from h(z,+); and equ analytic trickery mus Some experimentation we have merely split the integration ts : over 5 + and over H_ . (3), and (4) are as far as we can go, lly. The next step in the derivations ions requires a strong sense of direc- ely two differential equations for the Now, equation (2) shows us we are on equation (3) shows us that we must per gerdemain in order to obtain H(z,+) ation (4) shows us that the requisite t be thoroughgoing and boldly done, shows that we may profitably write: D(z,±) u for h(z,±) H(z,±) (5) "a(z,±) n for a(z)D(z,±) and: (6) M f(z,±)" for uJT^J {„ |„ N(z,£')a(z;£;';a^U') dfl(S) (7) SEC. 8.3 TWO- FLOW EQUATIONS: UNDECOMPOSED 11 "b(z,±) M for H(z,±) N(z,S')a(z,£';£)daU') do(£) (8) It follows at once that (3) becomes: J_ a(z)N(z,Odfi(0 = a(z,+)H(z,+) and (4) becomes: J_ N*(z,€)dfl(0 = f(z,+)H(z,+) + b(z,-)H(z,-) so that with (2) , these results assemble into the requisite equation for H(z,+): dH ^ > + ) = [£(z,+) - a(z,+)]H(z,+) + b(z,-)H(z,-) (9) Integrating (1) over E_ , and using similar tactics to those described, we have: dH ^ , " ) = [f(z,-0 " a(z,-)]H(z,-) + b(z, + )H(z, + ) (10) Equations (9) and (10) are the requisite two-flow equations for the irradiance fields H(z,±). Comparison with (6) and (7) of Sec. 8.2 yields the important connections: t(z,±) = f (z,±) - a(z,±) p(z,±) = b(z,±) (11) (12) f(z,±) and b(z,±) are respectively, the forward and backward scattering functions for the irradiances H(z,±); a(z,±) are the attenuation functions for H(z,±), respectively. The functions D(z,±) are called the distribution functions for H(z,±). We also note the interesting connection between the values a(z,±) and those of a(z,±) and s(z,±) where we have written: a(z,±)" for a(z)D(z,±) s(z,±)" for s(z)D(z,±) (13) (14) 12 MODELS FOR IRRADIANCE FIELDS VOL. V The connection of interest is: a(z, ±) = a(z ,±) + s (z, ±) (15) which follows at once from the definition of the volume absorption function a(z,£) given in (4) of Sec. 4.2; a(z,±) and s(z,±) are, respectively the absorption and total scattering functions for the irradiances H(z,±). Equation (15) parallels the basic connection: a(z) = a(z) + s (z) (16) among the volume attenuation, absorption and total scattering functions in X(a,b). Furthermore, from (3) of Sec. 4.2 and (7) and (8) above we have: s(z,±) = f(z,±) + b(z,±) which, combined with (15) yields: a(z,±) = a(z,±) + f(z,±) + b(z,±) (17) (18) Equation (18) shows that the attenuation function for H(z,±) is generally the sum of three terms: the absorption, forward and backward scattering functions. Using this connection, the two-flow equations (9) and (10) may be cast into their alternate forms : dH ( z , ± ) dz [a(z,±) + b(z,±)]H(z,±) + b(z,+)H(z,+) (19) We pause to examine the meanings of the terms in (19) and to sample the strong intuitive flavor of the two-flow equations Choosing the upper signs in (19) , we have the differential equation for H(z,+) which states that the rate of change of the upward flow of radiant energy per unit area consists of three terms representing the simultaneous activity of the following three processes in X(a,b): (i) The decrease of H(z, H(z,+) per unit leng (ii) The decrease of H(z, of H(z,+) per unit 1 (iii) The increase of H(z, of H(z,-) per unit 1 A similar interpretation may be radiance field H(z,-) by replac (iii) above. The minus sign be adjusts the vertical measuremen that equation, and is the vesti to measure r positive in the d equation of transfer (3) of Sec +) by absorption of th of travel. +) by backscattering ength of travel. +) by backscattering ength of travel. assigned to the downward ir- ing "+" by "-" throughout (i) fore the derivative of -H(z,+) ts to be positive upward for ge of the general convention irection E, in the general . 3.15. SEC. 8.3 TWO- FLOW EQUATIONS: UNDECOMPOSED 15 Equilibrium Form of the Two-Flow Equations We can cast the two-flow equations (19) into a form which points up even more strikingly the intuitive features of the irradiance field and which underscores still further their similarities to the radiance equations. We have in mind the introduction of the irradiance counterparts to the equilibrium form of the equation of transfer (4) of Sec. 4.3. Toward this end let us write: "H (z,±) q v ' J for b(z,+)H(z,+) a(z,±) + b (z,±) (20) so that (19) becomes: dH(z,±) dz [a(z,±) + b(z,±)] [H(z,±) H (z, q^ (21) H q (z,±) son for Conside of H(z H q (z,+) is the equilibrium irradiance for H(z,±). The rea- this nomenclature is obvious on inspection of (21) . r H(z,-): If H(z,-) < H q (z,-), then the derivative ,-) is positive and H(z,-) is increasing toward On the other hand, if H q (z,-) < H(z,-), then the ive of H(z,-) is negative and H(z,-) is decreasing Hq(z,-). Thus H(z,-) relentlessly pursues Hg(z,-). the race between N and N a along a horizontal line derivat toward Unlike of sigh in real our discussions in Chapters 9, 10, and 11. t, that between H(z,-) and H q (z,-) is never finished hydrosols. Reasons for this will become clear during Ontogeny of the Two-Flow Equations Before going on to further derivations and to the solu- tion procedures for the two- flow equations we pause to take note, for the student of radiative transfer theory, of the ontogenetic features of the two-flow equations. Historically, they trace back to Schuster's basic paper of 1905 (Ref. [279]). Logically, they rest on the interaction principle of Chapter 3. The logical route to them may be traversed in two distinct ways, The following diagram illustrates these routes schematically: Interaction Principle Example 1 of Sec. 3.7 ((17) and (18) of Sec. 3.7) Equation of transfer (3) of Sec. 3.15 Principles of invariance (1) and (2) of Sec. 8.1 Irradiance equations (6) and (7) of Sec. 8. 2 (11) and (12) of this sec- tion Irradiance equations (19) of this section 14 MODELS FOR IRRADIANCE FIELDS VOL. V The bridge between the two forms of the irradiance equations is the pair of relations (11) and (12) which can be rigo- rously proved from their definitions and the interaction principle. However the simple visual match between the set (9) and (10) and (6) and (7) of Sec. 8.2 which suggests (11) and (12) will suffice for the purposes of this work. In- terested students may attempt the direct proof of (11) and (12). It should be noted that the rigorous proof is not trivial and, if successfully done, has important related results in alternate modes of approach to radiative transfer theory (e.g., see step seven in Sec. 126 of Ref. [251]). 8. 4 Two-Flow Equations: Decomposed Form In this section we retrace the main steps of the pre- ceding section with the goal in mind of deriving the two-flow equations for the decomposed light field in X(a,b). The immediate basis for the derivation rests in (7) of Sec. 5.2. See also (19) through (22) of Sec. 5.1 wherein are also de- fined the notions underlying the idea of a decomposed light field. A suitable prerequisite for the present derivations are the discussions between (1) and (7) of Sec. 5.2, and between (56) and (62) of Sec. 6.6. The ultimate basis of the present discussion is (5) of Sec. 3.13. Starting with (7) of Sec. 5.2: £-VN*(z,£) = -a(z)H*(z,U + j^N*(z,Da(z;r ;^)dfi(^') + N ;(z,a (1) where Nl(z,£) = J_ N°(z,e)a(z^';adttU') (2) we integrate each side of (1) over ~ + . The derivative term becomes : [ C-VN*(z,5)da(S) * " ^ {„ 5*kN*(z,Od0(O This motivates us to write: "H*(z,±) M for J | £«k | N-*(z,£)dn(0 (3) We call H*(z,±) the diffuse upward (+) or downward (-) irra^ anoe. Similarly for a < z < b , we write: H°(z,±)" for | £.k | N°(z,£)dfl(0 , (4) SEC. 8.4 TWO- FLOW EQUATIONS: DECOMPOSED 15 so that by (5) of Sec. 3.13: H(z,±) = H°(z,±) + H*(z,±) (5) We call H (z,±) the residual (or reduced) upward ( + ) or downward (-) irradiance. Equation (5) exhibits the decompo- sition of the irradiance fields. The definitions (5) through (8) of Sec. 8.3 can now be repeated for the diffuse and resid- ual irradiances by the simple expedient of placing a star (*) or zero (o) superscript on the radiometric quantities involved. For example in the case of diffuse irradiance we write: "D*(z,±) n M a*(z,±) M for for h*(z,±) H*(z,±) a(z)D*(z,±) f*(z,±) n for H*(z,±) /„ j |„ N*(z,C')a(z;r ;.5)dn(D (6) (7) dnco (8) "b*(z,±) M for H*(z,±] j. ■»] N*(z,C')a(z;^';^)dfi(C , )| dfl(5) (9) Of course in (6) we have written: "h*(z,±) u for | N*(z,£)dft(0 (10) In this regard, see (19) and (22) of Sec. 5.1, which prevent unnecessary listings of definitions associated with n-ary concepts . Continuing with the methodical integration of the terms of (1) over H + , the first term on the right becomes: I. a(z)N*(z,£)dft(£) = - a(z)h*(z, + ) = - a(z , + )H* (z , + ) (ID The integral term becomes: J_ J^N*(z,^)G(z;£'U)cUHC) dfi(C) = f*(z,+)H*(z,+) + b*(z,-)H*(z,-) 16 MODELS FOR IRRADIANCE FIELDS VOL. V Finally, in like manner (and using the notation conventions agreed upon above) : [J. N u (z,£')cj(z;^;£)d^U') dfl(5) = f°(z,+)H°(z,+) + b°(z,-)H°(z,-) Assembling these results, we have: dH * d ( z 2, + ) = [f*(z, + ) " a*(z, + )]H*(z, + ) + b*(z,-)H*(z,-) + f°(z,+)H°(z, + ) + b°(z,-)H°(z,-) (12) In a similar manner we derive dH *^ , " ) = t £ ^ z »") " a*(z,-)]H*(z,-) + b*(z, + )H*(z, + ) + f°(z,-)H°(z,-) + b°(z,+)H°(z,+) (13) Analogously to (15) and (17) of Sec. 8.3, we have: a*(z,±) = a*(z,±) + s*(z,±) a°(z,±) = a°(z,±) + s°(z,±) s*(z,±) = f*(z,±) + b*(z,±) s°(z,±) = f°(z,±) + b°(z,±) From the preceding four equations, we have: a*(z,±) = a*(z,±) + f*(z,±) + b*(z,±) a°(z,±) = a°(z,±) + f°(z,±) + b°(z,±) In view of (18) , equations (12) and (13) may be written alternatively as: (14) (15) (16) (17) (18) (19) _ dH*(z,±) o _ [a * (z>±) + b*(z,±)]H*(z,±) + b*(z,+)H*(z,+) f°(z,±)H°(z,±) + b u (z,^)H u (z,^) (20) The interpretation of the terms of the differential equations (20) for the diffuse component of the light field is analogous SEC. 8.4 TWO- FLOW EQUATIONS: DECOMPOSED 17 to that for (19) of Sec. 8.3. Now, in addition to the in- crease of H*(z,±) from backscattering of the other stream there are two additional terms representing increases: the forward and backward scattering effects of the residual irradiance flows. To round out the system (20) , we deduce from (2) of Sec. 5.2 the following pair of equations governing the residual irradiance fields: dH°(z,±) dz a°(z,±)H°(z,±) (21) This shows that the system (20) is a pair of nonhomogeneous equations with known "source terms" as represented by the scattered residual irradiances. By (21) we can solve for H°(z,±) directly, provided a°(z,±) are known. These attenu- ation functions are known once a(z) for X(a,b) is given a < z < b , and the shapes of the incident radiance distribu- tions on the boundaries of X(a,b) are specified. The shape of the incident radiance distribution determines the distri- bution functions D°(z,±). Hence we have at once from (21): H°(y,-) = H°(x,-) exp (z\-)dz' (22) H°(y,+) = H°(z,+) exp / - j" a°(z',+) dz (23) for a<_x<_y<_z<_b. Let us write: "T°(x,y) n for exp < - fa (z , ,-)dz' (24) and: T°(y,x)" for exp a (z ' , + )dz' (25) whenever a <_ x < y < b . T (x,y) is the transmittance factor for residual irradiance. From these definitions we can con- struct the transmittance factors for diffuse irradiance. For we need only write: 'T*(x,y) for T(x,y) - T u (x,y) (26) whenever a < x < y lb. This defines the transmittance factor for diffuse downward irradiance . A similar definition is read- ily phrased for the upward diffuse irradiance. Definition (26) is the irradiance counterpart to (41) of Sec. 7.1. From (26) follows : 18 MODELS FOR IRRADIANCE FIELDS VOL. V T(x,y) = T u (x,y) + T*(x,y) (27) which is the decomposition of the transmittance factor T(x,y) into its residual and diffuse parts. Principles of Invariance for Diffuse Irradiance Starting with the principles of invariance (1) and (2) of Sec. 8.1 for irradiance, and using the decomposition (27) of the irradiance transmittance factors along with the irrad iance decomposition (5) , we can deduce the principles of in- variance for the diffuse irradiance field. Thus, from (1) of Sec. 8.1: H°(y,+) + H*(y,+) = (H°(z,+) +H*(z,+))(T°(z,y) +T*(z,y)) (H°(y,-) +H*(y,-))R(y,z) and using (23), we have: H*(y, + ) = H*(z,+)T(z,y) +H*(y,-)R(y,z) + H°(z, + )T*(z,y) + H°(y,-)R(y,z) (28) Similarly II* H*(y,-) = H*(x,-)T(x,y) + H* (y , +) R(y ,x) + H°(z,-)T*(x,y) +H°(y,+)R(y,x) (29) where a] dfi(C) (41) The corresponding attenuating functions for the residual irradiance field associated with the collimated radiance distributions are obtained by using the residual- flux ana- logs to (7) through (19): a°(z,-) = a(z)/ U°-k| (42) f°(z,-) U°-k I a(z;5 ;5)dA-(0 (43) b u (z,-) = k I cr(z;S ;Odfl(C) (44) From the residual counterpart to (16) : s°(z,-) = s(z)/ U° -k and similarly: a°(z ) = a(z)/ U° -k | (45) (46) We have retained the " z " in the notation, even though X(a,b) is assumed homogeneous, to show that arbitrary depth dependence of a , a does not destroy the otherwise simple relations holding among the attenuation functions for the residual and diffuse components of the light field. In this way we show that it is the directional structures of the radiance distributions which complicate the form of the asso- ciated attenuating functions, and not the depth dependence of the attenuating functions. The early principal work on the attenuating functions was done by Ryde and Cooper in Ref. [270]. However, no account was taken of the dependence of these functions on the intricate directional structure of the radiance distri- butions in real media. As can be seen by an examination of (8) and (9) , all dependence of f* and b* on N* is wiped away by the assumption (33). Furthermore in the absence of a rigorous general definition of f and b , as given in (7) and (8) of Sec. 8.3 or (8) and (9) of this section, one was unable to deduce with rigor the various important properties of these functions, and occasionally inaccuracies arose. For example, one of Ryde's principal conclusions about the prop- erties of f and b was that (in the present notation) : 22 MODELS FOR IRRADIANCE FIELDS VOL. V that f*(z,±) + b*(z,±) = f°(z,±) + b°(z,±) s*(z,±) = s u (z,±) However, it is at once clear from (14) of Sec. 8.3 that for the diffuse and residual cases: s*(z,±) = s(z)D*(z,±) = 2s(z) (47) s°(z,±) = s(z)D°(z,±) = s(z)/ U Q -k | (48) so that s° = s* if and only if 2 = 1/ | £ • k | . Thus the distribution function plays an essential role in the correct study of the directional structure of radiance distributions and of the dependencies of the various attenuating functions on the radiance distributions. The connections among the functions a(z,±), s(z,±), a(z,±), f(z,±), b(z,±) and the directional structure of the light fields was not clearly understood in the early papers of the two-flow theory. The investigators were invariably preoccupied with obtaining a soluble differential equational for some particular special practical problem and sparse attention was addressed to the general logical and physical aspects of the equations. It was not until the work of Duntley, Ref. [69], that a reasonably clear indication was obtained of the possible existence of a full family of attenuating functions that may be associated with the two- flow equations. Duntley added a new attenuating function to Ryde's list, namely (in our no- tation) a°(z,-). Under the usual assumptions on N° and N* it follows from (13) of Sec. 8.3 that: and that a°(z,-) = a(z)/ U° • k | (49) a*(z,-) = 2a(z) . (50) Duntley concluded that a°(z,-) and a*(z,-) should differ by virtue of the difference in directional structure of N° and N* . However, the simple connection: a*(z,-) = [2 U° k | ] a°(z,-) (51) that existed between these two absorption functions was not given, for lack of availability of the concept of the distri- bution function. A connection between a and a* was noted by Hulburt in Ref. [114] for the special case where E, = - k , so that a*(z,-) = 2a°(z,-). The preceding relation (51) and Hulburt's special observation are special cases of the general relations: a*(z,+) = [D*(z,±)/D°(z,±)] a°(z,±) (52) SEC. 8.4 TWO- FLOW EQUATIONS: DECOMPOSED 23 s*(z,±) = [D*(z,±) /D°(z,±)]s°(z,±) and these in turn are special cases of: (53) a*(z,±) = [D*(z,±) /D°(z,±)]a°(z,±) (54) These observations show the importance of the systematic use of the distribution function concept in the two- flow theory of irradiance fields. We shall use this concept repeatedly in subsequent discussions. Isotropic Scattering Models The next class of special two-flow equations to be considered is distinguished by the assumption of isotropy imposed on a . Thus it is assumed that: a(z;£' ;£) = s/4tt (55) for every £' , g in E . Hence all directional structure of o is suppressed in such models. Let us examine the conse- quences of such an assumption. In what follows, we shall allow the directional structure of the light field to be arbi- trary. We begin with the undecomposed light field. Using (55) in (7) and (8) of Sec. 8.3 we have: f(z,±) = £ D(z,±)s (56) b(z,±) = ± D(z,±)s (57) Further a(z,±) = D(z,±)a s(z,±) = D(z,±)s (58) (59) So that a(z,±) = D(z,±)a (60) From this we see that, under the assumption (55), the burden of the depth dependence of the light field is carried by the distribution functions. The associated forms of (19) of Sec. 8. 3 are: (61) 24 MODELS FOR IRRADIANCE FIELDS VOL. V Since D(z,±) clearly depends on the unknown structure of the radiance distributions in X(a,b) , equation (61) as it stands has unknown variable coefficients. If the usual assumption is now made that D(z,±) are known (or that they vary in some relatively innocuous manner) then the preceding system is solvable. The original Schuster equations were of the form (61) in which the irradiances were diffuse only and such that D*(z,±) = 2 , and with source terms h (z,±) added. The transition from (61) to its decomposed form is then at- tained by simply starring all quantities and adding the appro- priate source terms (cf. (20)). Connections with Diffusion Theory It is of interest to observe that (61) is just two steps away from a steady state diffusion equation for photons. The first step toward the diffusion equation is taken by adding the members of (61) , term by term. Thus the left side becomes: - A [H(z, + ) - H(z,-)] = V • H(z) By virtue of (10) through (12) of Sec. 2.8 and the stratified light field condition the x and y derivatives in the di- vergence operation vanish. Furthermore, the sum of the first terms on tne right becomes : - j [2a + s][D(z,+)H(z,+) + D(z,-)H(z,-f| = - i Q2a + s]h(z) The sum of the second terms on the right is: \ s[D(z,+)H(z,+) + D(z,-)H(z,-)] = \ sh(z) The first step concludes as we reduce these sums still fur- ther so that the net result is: V • H(z) = - ah(z) . (62) The second step toward the diffusion equation is to assume tnat Fick's law (5) of Sec. 6.5 is valid and of the form: H(z) = - DVh(z) This assumption, as we saw in Chapter 6, holds relatively accurately in decomposed light fields. Combining this with (62) we have: D V 2 h(z) - a h(z) = (63) which is the steady-state emission- free version of (7) of Sec. 6.5. Now that we have seen the connection between (61) and the diffusion equation, we are led to wonder if (19) of Sec. 8.3 has the same property, i.e., of being connectible to SEC. 8.5 TWO- D MODELS 25 diffusion theory via the two generic steps just taken for (61). The answer is in the affirmative. However, we shall leave this matter until Sec. 8.8 in which the vectorial as- pects of the irradiance field will be studied in detail. 8. 5 Two-D Models for Irradiance Fields We arrive now at the heart of the theory of irradiance fields in natural optical media, namely the two-D models in such media. The "two-D" aspect of the models refers to the radiance distributions over E ± being assigned fixed shapes so that in turn the distribution functions D(z,±) are as- signed two arbitrary fixed values: D(±) for all z . As a result, the exact two-flow equations (19) of Sec. 8.3 and (12) of Sec. 8.4 have known depth- independent attenuation functions and so the solution procedures of those equations reduce to straightforward applications of the theory of second order ordinary differential equations with constant coeffi- cients. The routine solution procedure of the equations can be enriched with digressions into the physical meanings of the various basic terms arising in the procedure, and we shall expend most of our efforts in the present section in such activity. On the Depth Dependence of the Attenuating Functions The first matter we shall take up is the depth depen- dence of the functions f(z,±), b(z,±), a(z,±), and s(z,±), in natural optical media. The observations we shall make are designed to lay the ground work for the two-D theory. Thus our present goal is to show that the attenuation functions listed above vary relatively little with depth in homogeneous media. To begin, we consider the depth behavior of a(z,±) and s(z,±), as this behavior is relatively simple to analyze into its physical and geometrical components. The case of a(z,±) is typical, so that we can limit our attention to it. According to (6) of Sec. 8.3 a(z,±) is the product of two factors: a(z) and D(z,±). Hence the depth-variation of a(z,±) is tied to that of a(z) and D(z,±). The depth vari- ation of a(z) constitutes the physical component of the depth variation of a(z,±) and the distribution function D(z,±) constitutes the geometric component of the depth variation of a(z,±). If the medium X(a,b) is homogeneous, then a(z) is independent of z , so that any depth variation of a(z,±) is contributed by that of D(z,±). Now it is intuitively clear that there is generally a variation of the shape of the radiance distribution with depth z in natural waters or altitude z in the atmosphere. This variation in shape is reproduced by D(z,±) in its own char- acteristic manner. It turns out that in most natural hydro- sols, for example oceans and lakes, the depth variation of D(z,±) is quite small, and what variation exists is quite regular or mild with depth. Table 1 exhibits a typical set of cases of the depth variation of D(z,±) on clear and over- cast days. These values were computed from some experimental data collected by J. E. Tyler (Ref. [298]) taken in a homog- eneous part of Pend Oreille Lake, Idaho, for wavelength 480 2b MODELS FOR IRRADIANCE FIELDS VOL. V TABLE 1 Experimentally Determined Distribution Functions CI ear Sunny Day Completely Overcast Day Depth z (meters) D(z,+) D(z,-) Depth z (meters) D(z,+) D(z,-) 4.0 2.67 1.25 3.0 2.75 1.22 10.0 2. 70 1.26 12.0 2.82 1.32 16.0 2. 79 1.28 24.0 2. 85 1.31 28.0 2. 76 1.31 36.0 2.93 1.33 40.0 2. 78 1.31 49.0 2.86 1.33 53.0 2. 77 1.30 ± 64 my . In particular, we used of Sec. 1.4, and rounded depths t observation we can make about the tively small amount of depth vari sunny and overcast conditions. I D(z,+) is approximately twice tha sum hovers in the immediate vicin numerical relations are quite uni stated conditions and we shall st regularities in Chapter 10. For fact that there is an empirical b about natural hydrosols to be mad It remains to make some obs ior of the forward and backward s b(z,±). According to (7) and (8) pendence of both of these functio tion of a physical component cont geometric component associated wi separate the geometric and physic variation of f(z,±) and b(z,±) a(z ,±) and s(z,±). However, w vations that will be of help in b First of all we note that i distribution N(z,«) is arbitrary depth, then D(z,±) are independe the data recorded in Table 1 o integral values . The main data of Table 1 is the rela- ation in D(z,±) under both t is also to be noted that t of D(z,-) and that their ity of 4. These interesting versally observed under the udy and apply such numerical the present we rest with the asis for the two-D assumption e below. ervatio catteri of Sec ns is a ributed th N(z al comp as simp e can s uilding f the s but fi nt of d ns on the ng functi . 8.3, th complica by a(z; ,5'). We onents of ly as in till make the two- hape of t xed as z epth. Th N(z,£) f(z)g + U) if K is in f(z)g_U) if C is in depth behav- ons f (z ,±) , e depth de- ted composi- £' ;£) and a thus cannot the depth the case of a few obser- D theory. he radiance varies with us suppose: (1) Then: SEC. 8.5 TWO-D MODELS 2 7 h(z,±) - N(z,Odfl(0 f(z) g + (S)dfi(0 and: H(z,±) = J^ N(z,£) U -k | dfl(S) - + = f(z) {_ g ± CO I C -k | dn(0 Hence: J. •. (OdfiU) D(z,±) = — (2) | n g ± (0 U k | dfi(S) This shows that, under the fixed-shape assumption (1) on N(z,£) the distribution functions are independent of depth. Now what about the converse of this observation: If the distribution functions D(z,±) are independent of depth, are the associated radiance distributions fixed in shape as a function of depth? If this were so, then Table 1 would supply the empirical evidence necessary to assert the depth independence of the shape of radiance distribution. To answer the preceding question, let us consider what con- straints are imposed on N(z,«) when we require D(z,-) to be independent of depth, say with fixed magnitude D(-) . All we need know about D(-) at the moment is that it is not less than 1, as a perusal of its definition would show. Thus we have, directly from (5) of Sec. 8.3: \, N(z,£)dn(0 D(-) J_ N(z,0 | 5 ' k | dfiU) For D(-) to be fixed is evidently a rather restrictive con- dition on N(z,')- This condition can be rewritten in the form: 28 MODELS FOR IRRADIANCE FIELDS VOL. V {„ N ( Z >^[ D (-) |S -k | - 1 dQU) (3) This form makes it quite clear that, despite the condition imposed on it by (3), N(z,£) can still vary in shape with depth. For suppose that we partition E_ into m pieces tially constant. m , over each of which Then (3) becomes: N(z,£) is essen- m I c- i=l c i (4) where N-j_ is the constant value of N(z,«) over A where we have written: and for [D(-) U • k | - i]dn(o A. 1 (5) It is evident from equation (4) that there is an infinite number of ordered m- tuples (N 1 , N 2 , ..., N m ) which satisfy it, even if the m- tuples are constrained to have nonnega- tive components, as required in the present case by the non- negativity of N . While this may cut down on the number of multiples which satisfy (4) we certainly cannot work with the residual infinite number of possible solutions for which the constant property of D(z,±) generally holds. However we still have at least one trick to play in the present algebraic game with (4) , one that is based on the fact tnat the numbers N^ in (4) are not to be drawn at ran- dom from the real number system but are to represent physical radiances typical of those found in natural waters. There- fore it is fair to impose a further condition on the N^ , other than that of nonnegativity . This additional condition is what we shall call a monotonicity condition and it may be stated as follows: Let " 7[ m " denote the collection cf all multiple solutions of (4) . Then we say that 7l m obeys the monotonicity condition if the members of ^ m can be arranged in a sequence, ordered by real numbers, such that if (N 1 (z) , \ The the in radi In N m (z)) and (N, (y) , N m (y)) are two members of with y"' : z , then N^(y) > N^(z) for i = 1 , ..., m, physical origin of the monotonicity condition is clear, real number indexing the multiples corresponds to depth X(a,b) . The greater the depth z , the smaller the m ance components of the multiple (N 1 (z) , ... , N m (z)). fact the components are to decrease monotonically with depth. We now return to (4) armed with the monotonicity condi- tion and require the collections ^ m of solutions of (4) to obey this additional condition. Toward this end observe that the coefficients c^ partition into two groups: those that are positive, and those that are negative. If any of the c^ are zero, clearly a partition { A 1 , ... , A m } of 5_ can SEC. 8.5 TWO-D MODELS 29 be re-chosen with only minor changes so the associated c^ is not zero. The sum in (4) is rearranged so as to collect together all positive terms in one group and all negative terms in another. Hence as depth z is increased the monotonicity condition requires the m-tuples in the posi- tive group to uniformly decrease and (4) requires the sum to be zero. Hence the members of the negative group must also decrease and in such a manner as to preserve the balance of (4). A moment's reflection will show that this still leaves many solutions satisfying (4) but it is clear that the variation of the shapes of these distributions has been severely restricted by the imposition of the monotonicity condition. It therefore appears that, on a practical level, the depth independence of D(z,±) entails that of the shape of N(z,») in real optical media. We shall rest the matter of the converse property of the distribution functions at this stage, having made it plausible that the observed depth independence of D(z,±) in natural waters will imply a corresponding depth independence of the shape of the radiance distribution in natural waters because of the conditions such as nonnegativity and monoto- nicity imposed on the radiance distributions which are based on auxiliary physical reasons. Another condition on N(z,*) which may be imposed is that of convexity of the shape of the distribution. The preceding discussion has shown that the problem of the converse property of the distribution function is not fully resolved and it is left to interested students of the subject to pursue. Briefly the problem is this: What condi- tions on N(z,») in addition to (3) and the monotonicity con- dition must be imposed so that the shape of N(z,*) is to be depth independent? A more practical problem of comparable mathematical difficulty is: Describe the limits within which the shape of N(z,») may vary when N(z,«) is subject to con- dition (3) and the monotonicity condition. Our preliminary analysis above showed that these limits may be quite narrow. Returning now to the question of the depth independence of f(z,±) and b(z,±), we see that in pursuing this question we are led along essentially the same analytic and algebraic path as in the case of the distribution function just con- cluded, so that any solution to the converse distribution problem defined above should shed light and be directly appli- cable to the associated depth independence problem of f(z,±) b(z,±). In particular, we can reach the corresponding con- clusion that a relatively small depth dependent variation in the shape of the radiance distribution can be expected if the depth variations of f(z,±) and b(z,±) are small, whenever we are in homogeneous natural hydrosols. It turns out that the converse distribution problem out- lined above is needed only for relatively shallow depths in homogeneous media (on the order of three or four attenuation lengths) for below such depths the asymtotic radiance distri- bution begins to take hold (cf. (3) of Sec. 7.10, and Chapter 10) and the distribution function becomes essentially depth independent along with f(z,±) and b(z,±). For if (1) holds, then in addition to (2) , we can conclude that in homogeneous media: 50 MODELS FOR IRRADIANCE FIELDS VOL. V j„ {„ g ± (^') a ( z ' V ;Z)dto(V) f(z,±) = g + (0 U • k | dfi(a dfl(C) (6) and b(2,±) = L L g * K ' )a(z;£' ;Od$UO dfl(0 f g ± U) U • k I dfl(5) (7) which fo is homog to empha £(z,±) a We and theo of the t and b(z cal medi media, tions ar tions in in the s purposes pair of Hows fro eneous, t size the nd b(z,± have now retical , wo-D theo ,±) are e a over th Furthermo e workabl natural hape of r be succi numbers m (7) he " homog ) are arra to ma ry, n ssent e gre re we e ind water adian nctly D(z,± and (8) of S z " may be dr eneity assump then depth i yed enough ev ke plausible amely that D ially depth i ater part of have shown t ices of the s s, so that th ce distributi compressed i ec. 8.3. Si opped from ' tion and the ndependent . idence, both the working (z,±) along ndependent i the depth ra hat the dist hape of radi e relatively ons can for nto and repr nee the medium 'a(z;£';0 " fact that empirical assumptions with f(z,±) n natural opti- nges of such ribution func- ance distribu- complex changes many practical esented by the Two-D Model for Undecomposed Irradiance Fields The two-D model for undecomposed irradiance fields takes as its foundation the general two-flow equations (19) of Sec. 8.3 and adopts the following additional assumptions. Let X(a,b) be a plane-parallel medium such that: (i) , b = z with < z (ii) X(0,z 1 ) is separable and its boundaries are transparent. (iii) The radiance distributions in X(0,z 1 ) satisfy condition (1) . (iv) H(0,-) is an arbitrary irradiance, and H(z 1 ,+) = 0. Assumption (i) merely sets the stage on a convenient slab in euclidean space with a terrestrial coordinate frame for hydro- logic optics. The upper boundary is at depth and the slab SEC. 8.5 TWO-D MODELS 31 may be either finite or infinite in de says several things at once; first of that the ratio o / a is independent o over the range of depths <_ z <_ z 1 . of the depth variable from geometric t can be rendered homogeneous in the opt system. We shall assume that this is may be independent of depth in what fo no restrictions are made on the relati and a , or on the directional structu quired transparency of the upper and 1 X and eliminates the need to c effects between these planes and the b an interref lection calculation is neat of the interaction principle and is re stages of the present discussion. The diately implies, via (5) through (7), that all the coefficient functions in constants with respect to depth. We s consequence of (iii) throughout this d 11 a(±) M and "b(±) n for the coeffic equations (cf. (7)) and M D(±)" for distribution functions as given by (2) (iv) limits the incident source flux t ance on the upper boundary of X(0,z 1 ) of Sec. 8.3 obtained under assumptions known as the first standard solution o undecomposed irradiance fields. Equations (19) of Sec. 8.3 reduc ditions, to: pth. Assumption (ii) all, separability means f depth (cf. Sec. 7.12) Therefore, by a change o optical depth, X(0,z ) ical depth coordinate 1 done so that a and a Hows. Other than (ii) ve magnitudes of a re of o . The re- ower boundary planes onsider interref lection ody of X(0,zj . Such ly dispatched by means served for the latter assumption (iii) imme- and (13) of Sec. 8.3 (19) of Sec. 8. 3 are hall emphasize this iscussion by writing ients of the two-flow the fixed values of the Finally, assumption o an arbitrary irradi- . The solution of (19) (i) through (iv) is f the two-D model for under the above con- dH(z,±) _ dz [a(±) +b(±)]H(z,±) +b( + )H(z,+) (8) The general solution of system (8) may be cast into the form: H(z,±) = m +g+ (±)e k+z + m_g_(±)e k-z (9) where m+ are two constants which will be determined by as sumption (iv) , and where we have written: "g + (±) " for 1 ± ^iii (±) " for 1 + a£ii (10) (11) and: for H [a( + ) +b( + ) - a(-) - b(-)] ± [(a(+) +b( + ) + a(-) +b(-)) 2 - 4b( + )b(-) ] i/s\ (12) 52 MODELS FOR IRRADIANCE FIELDS VOL. V It is easily seen from (12) that: if a > , then k_ < < k + (13) if a = and b( + ) = b(-), then k_ = = k + (14) if a = and b(+) > b(-), then k_ = and k + = b( + )-b(-) (.15) if a = and b( + ) < b(-), then k_ = b( + )-b(-) and k + = (16) In all cases, then, we have k_ <_ <_k + . This permits us to view the term in (9) with k + in the exponential as a general growth term, while that with k_ may be viewed as a general decay term, when z is measured positive downward into X(0,z 1 ). We shall henceforth assume the medium to be nondegenerate, i.e., k + f k_ , which limits our considera- tions to media with properties (13), (15), and (16). Degen- erate media have trivial radiative transfer properties, as may be inferred from the solutions displayed below. Putting assumption (iv) to work, we require of (9) that (-) k+z 1 ( ^^ k_ z. 1 + m g ( + ) e 1 H(0,-) = m +g+ (-) ■ = m +g+ (+)e From (17) and (18) we find that: m ± = + H(0,-)g_( + )eV 1 /a( Zi ) where we have written: "A(z )" for g + (+)g_(-)e k+Zl - g + (- ) g_ (+) e k " z (17) (18) (19) (20) Therefore the first standard solution of the two-flow equa- tions for X(0,z ) consists of the following two equations: H(z,±) = H(0,-) r .N rj .«, k + z n +k_z , ^, r .> k+z + k-z-, + ( + )g_(±)e + 1 - g_ (+)g + (±)e (21) Writing the upward and downward irradiance equations sepa rately, we have: H(z, + ) = ffi^ g + (+)g_(+) k+z, + k z k+z + k_ z7| e 1 - e 1 [22) SEC. 8.5 TWO-D MODELS 33 and H(z,-) A( Zi ) [ g + ( + )g.(-)e k+Zl +k - Z -g + (-)g_(-)e k+Z + k (23) Observe how the boundary conditions (iv) are now built into (22) and (23). For by setting z = z in (22), we obtain H(z 1 , + ) = . Setting z = in (23) yields an identity; as expected. For all intermediate depths < z < z 1 , equa- tions (22) and (23) give the values of the irradiance fields at those depths. It is of interest to examine the standard solutions (22) and (23) in the light of the invariant imbedding rela- tions (7) and (8) of Sec. 8.1. Thus in (8) of Sec. 8.1, set x = , y and so that: H(z,-) = H(0,-)r(0,z, Zl ) (24) Furthermore, in (7) of Sec. 8.1 with the same substitutions: H(z,+) = H(0,-)^(0,z, Zi ) (25) From (24) and (23) we have at once an explicit representation of D"lo ,z,z^) for X(0,z ) and from (25) and (22) we have an explicit representation of ^(0,z,z.,) for X(0,z 1 ). Further- more, since: ^(0,0, Zi ) = R(0, Zi ) it follows that: g + ( + )g_( + ) R(0,z ) - ^ ' 1 J A(z ) k+z i - e k - z i (26) Similarly, since: it follows that: r(0,z ,zj = T(0,zj (k + +k_)zj T (°' z ^ = A(z ) g + ( + )g-^) (-)g_( + (27) We note that for infinitely deep media, i.e., for the case z 1 = °° , (26) and (27) become, respectively: 34 MODELS FOR IRRADIANCE FIELDS VOL. V R = C-kJ - a(-) ("k ) +a( + ) T = A question that often arises in practical applications of the principles of invariance to plane-parallel media con- cerns the polarity of the R and T factors (or R and T operators, if radiance is used) (cf. Sec. 7.12). It turns out that the two-D theory allows very detailed studies to be made of this question in the irradiance context. Thus, in the present setting the R and T factors by definition possess polarity if: R(0,z ) + R(z ,0) or As w when pres is i X(0, no e we n theo ator R(0, role exam b(±) seco diti T(0, Zi ) + T( Zi ,0) e saw in Sec. 7.12 the phenomenon of polari X(0,z 1 ) is anisotropic or nonseparable . ent medium is separable, the only way polar n the case that the inherent optical proper z.,) exhibit anisotropic structure. We have ssential use of the isotropy of X(0,z ). ow assume the medium to be isotropic. From rem of Sec. 7.12 we know then that the R s do not possess polarity. But what of the zj and T(0,z.,)? Since a(±) and b(±) pi s for H(z,±) as a and a do for N ± (z), ine this question with specific reference t ty can arise Since the ity can arise ties of as yet made Suppose that the polarity and T oper- f actors ay the same we should o a(±) and In preparation for the answer, let us generate the nd standard solution by replacing (iv) above by the con on: (v) H(z ,+) is an arbitrary irradiance, and H(0,-) =0. Therefore we are to irradiate X(0,z 1 ) from below. Since the procedure for fixing m+ is now clear, we merely state that under condition (v) with (i) through (iii) above in force, we have : m ± = ±H(z i , + )g_(-)/A(z i ) It follows that the second standard solution is: (28) H(z,+) H(z i>+ ) A(zJ ; + ( + )g_(-)e k+z (-)g ( + )e J ■3 (29) and SEC. 8.5 TWO-D MODELS 35 H(z .+) H ( z >-) = TfTT" g + ( " )g - ( - } k+z k_z e + - e (30) Being guided by the invariant imbedding relation once again we arrive at: R(z ,0) = (-)g.(-) Kx Z 1 K_Z 1 e + 1 - e 1 (31) (32) It is now of interest to compare (26) with (31) and (27) with (32). In the case of the reflectance factors we see that: R(0, Zi ) R(z ,0) if and only if g + (-)g_(-) g + ( + )g_( + ) (33) and that T(0,z ) = T(z ,0) if and only if k + k = (34) The conditions on the right of each statement are each implied by a single statement; namely: a( + ) = a(-) and b( + ) = b(-) (35) In this way we can find an answer to the question about polarity of the R and T factors. Toward this end, we shall agree that the medium X(0,z 1 ) is anistropic with respect to the irradiance field if (35) does not hold, i.e., if its negation: a(+) + a(-) or b( + ) * b(-) (36) is true. Thus, irradiance, we and T factors (v) a full desc X(0, z 1 ) by mean factors: R(0,z (27) , (31) , ; and The basic source-free irr lutions of the giving rise to if X(0 f z 1 ) is anisotropic with respect to are generally to expect polarity of the R , so that under standard conditions (i) through ription of the radiative transfer process in s of the irradiance field requires the four J, R(z.O), T(0,z ), T(z,0), as given by (26), (32). 1 1 1 equations of the two-D theory for undecomposed adiance fields have now been derived. The so- special forms of the two-flow equations (8) the two-D equations are conveniently grouped 56 MODELS FOR IRRADIANCE FIELDS VOL. V into parts: the first and second standard solutions given, respectively by (22), (23) and (29), (30). These two groups can be assembled into one compact package by means of the invariant imbedding relation. Toward this end and in view of (24) and (25) we deduce: #(0,z,zj = 7(0, z,z) g+( + )g_( + ) k + z 1 + k_ z k+z + k_ z. e + 1 - e + 1 ~^TJ (+)g_(-)e k+Z i - g + (-)g_( + )e k_z k+z + k.z 1 (37) (38) In a similar way we deduce the present forms of the complete reflectance and transmittance factors from (29) and (30). As a result we have: <3Uz ,z,0) g + (-)g_(") A(z ) k+z k_z (39) '^V^ - ^fy ;+( + )g_(-)e k+z + (-)g_( + )e k_z (40) The prec that the generali brackete terms of However final an structur are thos invarian right si numerica material are. In braic st eding expre square-bra zed forms o d quantitie hyperbolic such conden alysis, qui es of the i e given by t imbedding des of (37) 1 magnitude how simple view of (7 ructures ar ssions can be c cketed quantiti f A(x) for sui s in (37) and ( functions, aft sations are rel te inessential, rradiance field the invariant i relation to wh through (40) s s of the ti\ an or complex the ) and (8) of Se e the following ondensed sli es in (38) a table x . 39) can be r er suitable atively comp since the t in plane-pa mbedding fac ich they bel erve merely d 7" factor se symbolic c. 8.1, the ghtly by noting nd (40) are Furthermore the epresented in rearrangements . lex and, in the rue algebraic rallel media tors and the ong. Thus the to supply the s. It is im- arrangements essential alge- H(z,+) = H(z i , + )r(z i ,z,0) + H(0,-)6U0,z, Zi ) (41) H(z,-) = H(0,-)y(0,z, Zi ) + H(z i , + )^(z i ,z,0) (42) or their matricial form: (H(z,+),H(z,-)) = ((H(z i ,+),H(0,-))^(0,z V (43) Therefore, once a tabulation of the entries of the matrix "^(OjZjZ.,) is made for a medium X(0,zj, (43) supplies the SEC. 8.5 TWO-D MODELS 37 irradiance field H(z,±) at each depth z in X(0,z 1 ) in terms of the incident irradiances on boundaries X Q and X z 1 . A convenient means of tabulation is given by the set (5) through (6) of Sec. 8.1, One can first build up an inde pendent tabulation of four standard R and T factors for X(0,zj using (26), (27) and (31), (32). Then the # and J/ factors may be obtained in any detail as desired. The particular forms of (3) through (6) of Sec. 8.1 required in the setting of X(0,z ), are obtained by making the substi- tutions x ■»■ , y •*■ z , and z -> z . However, before extensive tabulations are considered, the observations throughout Sec. 8.7 should be studied, especially those which show how the R and T factors and their invariant imbedding counterparts and generalizations, can be obtained by direct integration procedures and semigroup calculations. Two-D Models for Internal Sources The basic equations for the two-D theory of irradiance fields as given in (8) are written explicitly for source-free media. It is a simple matter to extend these equations to include continuously distributed internal sources in X(0,z ), and we shall now devote some attention to this matter. Be- fore doing so, we note that the case for a finite number of discrete internal sources in X(0,z ) is readily solved using the results of Example 3 of Sec. 3.9. (See (38) of Sec. 3.9, and replace (N+ (y) , N- (y)) by (H(y, + ), H(y,-)), etc.) These results, though written for the radiance functions, adapt immediately to the irradiance case. The R and T factors (26), (27), (31), and (32) and the ^ and T factors (37) through (4) are now used to construct the ¥ - factors by means of (20)-(23) and (31)-(34) of Sec. 3.9. The equations (8) are adapted to the continuous internal source problem by simply adjoining the source terms h n (z,±) to the respective equations, thus: _ dH(>,±) = T(±)H(Z>±) + p( + )H (z,+) + h n (z,±) (44) Here we have used (11) and (12) of Sec. 8.3 to help write the present compact version of the two-D equations. The present version is the irradiance counterpart to the local forms of the principles of invariance. The connection of h n (z,±) with the source function in X(0,z 1 ) is simply this: we have written: h n (z,±)" for I N n (z,Odfl(0 (45) and in the process of going from the equation of trans- fer to the two- flow equations , as explained in Sec. 8.3, N n (z,£) is converted to h n (z,±). We may note in pass- ing that we have used the field radiance interpretation of N n in (45) so as to use h^ . The alternate interpre- tation of N^ as a surface radiance would have resulted in 58 MODELS FOR IRRADIANCE FIELDS VOL. V w n (cf. (19) of Sec. 2. 7). To solve (44) we can make use of the solutions ob- tained earlier in the source-free case. The homogeneous solutions of that case will be used. In order to find the particular solutions of (44) we proceed as follows. First observe that the system (44) may be converted into two second order differential equations in H(z,±). Thus, let us write D" for -f- dz Then (44) becomes: [D+ x(+)]H(z,+) [D- t(-)]H(z,-) p(-)H(z,-) - h n (z,+) p(+)H(z,+) + h (z,-) (46) (47) Operating on (46) with the operator [D - t(-)], we have: [D-t(-)] [D+t(+)]H(z,+) = -p(-) [D-T(-)]H(z,-)-[D-T(-)]h n (z,+) = -p(-)[p( + )H(z, + )+h n (z,-)] - [D-T(-)]h n (z,+) The second equality follows from use of (47). Simplifying this result we have: [D 2 +(t(+)-t(-))D+ P (+)p(-)-t(+)t(-)]H(z,+) = -[D-T(-)]h n (z,+) -p(-)h r] (z,-). Next, operating on (47) with [D + t(+)], we have: [D 2 + (t( + ) - t(-))D + p( + )p(-) - t( + )t(-)]H(z,-) = = [D+ T( + )]h n (z,-) - p( + )h n (z,+) (48) (49) (50) Let us write: "Y(z,±)" for +[D +T(+)]h n (z,±) - p(+)h n (z,+) and "JD" for [D 2 + (t( + ) - t(-))D + p( + )p(-) - t( + )t(-)] (51) Then (48) and (49) can be written as: f)H(z,±) = Y(z,±) (52) SEC. 8.5 TWO-D MODELS 39 The source-free, two-D equations (8) are a special case of (52). Thus (8) is equivalent to: £>H(z,±) (53) and is obtained by setting h n (z,±) = . The characteristic equation associated with the differential operator £> is: k" + [t( + ) - T(-)]k+ [p( + )p(-) - tO)t(-)] = (54) whose solutions are: k, = i{ [T(-) - T(+)] ± [(T(+) +T(-)) 2 - 4p(+) P (-) (55) These roots k+ , k_ are precisely those given in (12) , but are now represented in terms of the local transmittance and reflectance factors t(+) , p(+). The earlier forms can be recovered by means of (11), (12), and (18) of Sec. 8.3. We are now ready to find the general solutions H(z,+) of (52) . First we observe that two linearly independent so- lutions of the homogeneous equations (53) are available in the forms: k+z Indeed, each function satisfies (53) and the Wronskian W(ek +Z , e " z ) of these two functions is of the form: Wz- Kj. z k _ z n (e + , e ) k + z k_z kKj. Z i K _ Z + e + k_e = (k_ - k + ) exp ((k + + kjz) for every z in the interval [0,z.,]. Since we have agreed to work in nondegenerate media (cf. (13) -(16) of Sec. 8.5), it follows that k + f k_ , and so the Wronskian does not van- ish on [OjZ^, thereby indicating the linear independence of the solutions. By means of the method of variation of parameters, par ticular solutions of (52) are found to be of the form: i f z e k+(z-s) _ e k_(z-s) ] Y(s,± ) ds (56) which we will denote by H H p (z,±) M . 40 MODELS FOR IRRADIANCE FIELDS VOL. V It follows that the general solutions of (52) are of the form: H(z,±) = m +g+ (±)e k+z + m_g_(±)e k - z + H (z,±) (57) where m + are constants of integration. These constants may be determined by suitable choices of the values H(y,±) at any two distinct levels in XC0,z 1 ) > or by knowing H(y,±) at a given depth and dH(y,±)/dy at the same, or generally some other depth. Still another and quite general requirement would be imposed by simultaneously specifying H(0,-), H(z 1 ,+), exactly as in the source-free case considered above (re: (19), (28)). This we shall do, our ultimate goal being a representation of H(z,±) by means of an invariant imbedding relation, as in (43) , but now taking cognizance of source terms in (44) and hence in (57). Thus, we require: H(0,-) = m +g+ (-) + m_g_(-) H(z if +) = m +g+ (+)e k+Zl + m_g_(+)e k - z i + H (z^ ,+) The requisite values of m+ are: A(z ) ^^^-HpCz^lgJ-)- )]gj- (58) These values are now returned to (57) and some algebraic reductions made, keeping in mind the present goal. The results may be written as: (H(z, + ),H(z,-)) = (H(z i , + ),H(0,-))7^(0,z,z i ) - (H p (z i ,-H),0)»l(0,z,z i ) + (H p (z, + ),H p (z,-)) (59) This is the desired set of general solutions of the equations (52). Observe that we return to the source-free case (43) if h (z,±) = for every z in [0,z.,]. The new parts compris- ing the present solution are generated by the presence of sources in X(0,z 1 ). The members of the 2x2 matrix ty(0,z,z ) were evaluated in (37) through (40), so that, as it stands, (59) is ready for numerical computations. Equation (59) can be recast into an alternate form so as to achieve greater symmetry of form and also to bring out explicitly the intuitive features of the role played by the continuous sources h n (z,±) within X(0,z 1 ). The concept of the continuous ^ - operator , introduced in (37) of Sec. 3.9, guides the reformulation of (59) towards this new goal. First we observe that each particular solution H p (z,±) may be written as a sum of two terms, using the explicit form of Y(x,±) given in (50): SEC. 8.5 TWO-D MODELS 41 H p (z,+)=-j Q(z-s)[D-T(-)]h n (s,+)ds - J Q(z-s)p(-)h n (s,-)ds (60) H (z,-) =[ Q(z-s)[D+x(+)]h (s,-)ds - j Q(z-s)p( + )h n (s, + )ds J o ' o (61) Note in particular how the derivative operators [D-t(-)1 and [D+t(+)] are to be applied t first integrals in each of t For brevity, we have written [D+t( + )] are to be applied to h r) (s, + ) and h (s,-) in the first integrals in each of the representations (60) and (61) . "Q(z-s)" for (k+ ] k) [e k ^ z " s ) - e k -^- s j]. (62) The two equations (60) and (16) can be succinctly written in matrix form and in such a way as to make contact with the ¥ - operator concepts of example 3, Sec. 3.9. Thus: (H p (z,+),H p (z,-)) = (h n (s,+),h n (s,-)) [D-x(-)] P(-) P( + ) [D+t( + )]J Q(z-s)ds Let us denote the matrix in the preceding integrand by "+ ),0) = (H p (z i ,+),H p (z i ,-))C + where C+ is one of a pair C ± of 2x2 matrices defined in (4) and (5) of Sec. 7.4. Now the I ± occurring in C ± reduce to the number 1 in the irradiance context. Hence, by the same reasoning as before: (H p ( Zi , + ),0) = j 1 (h n (s, + ),h n (s,-))f > (s)C + Q(z i -s) ds . (64) 42 MODELS FOR IRRADIANCE FIELDS The integral in (63) can be given z 1 as an upper limit by adopting the function x defined on the real line, where X(x) if x < 1 if x > so that (H p (z,+),H p (z,-)) = J (h n (x, + ),h n (x,-)) < z <_ z, . Therefore X(0,z 1 ) has a collimated source of radiant flux incident on its upper boundary along the direction £° in H_ . No other sources are incident 44 MODELS FOR IRRADIANCE FIELDS VOL. V on or in X(0,z 1 ). As stated in (vi) , the boundary condi- tions H*(0,-) = and H*(z , + ) = are also in force. In regard to related uses of these conditions, see (30) and (31) of Sec. 8.4. With the preceding assumptions in force, (20) of Sec. 8. 4 reduces to : t dH *^ ,±) = " [a*(±) +b*(±)]H*(z,±) + b*(+)H*(z,+) dz~ exp { - az/y }a_(y ) (67) where we have written: "o + (\i o y< for y o f u (z,-) £ i- n -- • It is this term in (71) which gives the dependence of H*(z,±) on y , and which permits the simulation of general incident lighting conditions. By tabulating the values C ( |j , +) for SEC .5 TWO-D MODELS 45 a few typical choices of a and a , a useful set of tables of H*(z,±) can in turn be constructed, and these can be used via superposition calculations to simulate general incident conditions. The constants m ± in (71) are as yet the only undetermined quantities in (71) . However the remaining light ing conditions on H* in (vi) rigidly fix the structures of m+ and m_ . Thus it may be shown that: N' A(z^ h (-)C(u ,+)e QLZj\\ _ g_( + )C(y ,-)e Kz (73) Explicit expressions for H*(z,±) or for H(z,±) are now derivable from (71) using (73). (See, e.g., (1) and (2) of Sec. 10.3.) In particular, expressions for ^(0,z,z,,) and CX* (0, z, z 1 ) are now readily forthcoming. However, we shall not take the space here to display such representations of the complete reflectance and transmittance factors. We shall be content to exhibit, for later purposes, the standard reflectance and transmittance factors R(0,z 1 ) and T*(0,z 1 ). Toward this end, by means of (30) of Sec. 8.4 and suitable substitutions, we have: Since H*(0,+) = H°(0,-)R(0,z J H°(0,-) = N°y (71) and (73) combine to yield: R(0, Zi ) C(y,-)g + (+)g_( + ) y A(z ) o 1 J [ k + z 1 k_z e * 1 - e C(u 3 °l2. [aw e^^/iio . ri p L*?v J (74) Further, from (31) we have: H*(z if -) = H°(0,-)T*(0,z i ) which, with (71) and (73), combine to yield: T*(0, Zi ) C(y ,+)g + (-)g.(-) y A( Zi ) c(y.,-) a( Zi ) k+z, - e k - z i i e " az i /y ACQ) .(k+tkOZi . p-az^y (75) 46 MODELS FOR IRRADIANCE FIELDS VOL. V There is generally one other reflectance- transmittance pair of factors for upward incident flux on X(0,z ). The pro- cedures used to find (31) and (32) may serve as a guide to find the factors for the present case. Inclusion of Boundary Effects We conclude this section with a description of how to include the effect of reflecting upper and lower boundaries on the medium X(0,z ). The general problem of interreflec- tions between the body of a medium and its boundary, includ- ing the possibility of internal reflecting interfaces, was discussed in Example 6 of Sec. 3.9 and applied to the unified atmosphere-hydrosphere problem in Example 7 of that section. We shall now repeat the essential ideas of those examples, but in the irradiance context, and, so as to keep the dis- cussion simple, we shall assume no internal interfaces. The simplest case will be considered first: the medium X(0,z 1 ) has only one reflecting boundary namely that at level 0, and which we shall denote by "X ". The boundary at level z 1 will be assumed transparent. In practice, when z 1 is relatively great, the present case may be freely used even though X(0,z,) has a reflecting lower boundary. Further- more, the only source in X(0,z 1 ) will be the downward irradiance H (0,-), incident on the upper boundary. We will work with the undecomposed irradiance field. With these conditions, we have established the present interaction problem as that between two media: a plane X and a slab X(0,z 1 ). The reflectance and transmittance fac- tors for X are developed in their full generality in Sec. 3.3. In particular we use r ± (x) and t ± (x) as developed in (19) of Sec. 3.3. Examples of the use of these factors in the irradiance context are given in Sec. 3.4. Hence we can employ the interaction method in the present problem without further explanation. We direct attention first to plane X Q , and enumerate all incident irradiances: A : all irradiances like H (0) A : all irradiances like H fO) 2 + The set A 1 is the set of all irradiances on the upper side of X . The set A 2 consists of all irradiances on the underside of X . The set of response radiant emittances of X are: B : all radiant emittances like W (0) B : all radiance emittances like W fO) The four associated interaction operators for X are simply the reflectance and transmittance factors r+(0), t ± (0) asso- SEC. 8.5 TWO-D MODELS 47 s t (0) 11 - v J s r (0) 12 s r.(0) 2 1 + s t (0) 22 + V ' We next direct attention to the slab X(0,z ). The class of all incident irradiances on X(0,z ) is : A : all irradiances like H(0,-) The class of all response radiant emittances is: B : all radiant emittances like W(0,+) B : all radiant emittances like W(z ,-) 2 V 1 ' The requisite response operators s and s 12 are in this case the numerical reflectance R(0]z ) given by (26) and T(0,z,,) given by (27). According to the interaction principle applied to X , we have: W + (0) = H_(0)r_(0) + H + (0)t + (0) (76) W_(0) = H_(0)t_(0) + H + (0)r + (0) (77) The interaction principle applied to X(0,z 1 ) yields: W(0,+) = H(0,-)R(0,z ) (78) W(z ,.-) = H(0,-)TC0,z ) The auxiliary equations for the present problem are: H_(0) = H°(0,-) (79) W_(0) = H(0,-) (80) H + (0) = W(0,+) (81) We are interested in the responses of X and X(0,z 1 ) as a result of their radiometric interaction induced by H°(0,-), and so use the auxiliary equations (79) through (81) to remove as many incident quantities as possible from (76) through (78). The results are: W + (0) = H°(0,-)r_(0) + W(0,+)t + (0) (82) W_(0) = H°(0,-)t_(0) + W(0,+)r + (0) (83) W(0,+) = W_(0)R(0,z ) (84) W(z if -) = W_(0)T(0, Zi ) Equations (83) and (84) are autonomous, so that: 48 MODELS FOR IRRADIANCE FIELDS VOL. V W_(0) = H°(0,-)t_(0) + [W_(0)R(0,z i )]r + (0) and we have: W (0) H u (0,-)t (0) 1-R(0,z )r + (0) From this and (84) : W(0, + ) = H u (0,-)t (0)R(0 >Zl ) 1-R(0,z )r + (0) (85) (86) From this and (82) : W + (0) = H°(0,-) t,(0)R(0,z 1 )t (0) - C j 1-R(0,z )r + (0) (87) These results may now be used to obtain the internal irradiances H(z,±) in X(0,z 1 ). Indeed, by the invariant imbedding relation (43) with H(z 1 ,+) = and H(0,-)=W_(0) now given by (85), we have: (H(z,+),H(z,-)) = (0,H(0,-))7)[(0,z,z i ) . In particular: ^^ * lYcuizVg) ^0,z, 2i ) Hf7 , H°(0,-)t_(0) ^ H(Z '' } - l-R(0,z i )r + (0) yC ^^,) (88) (89) It is interesting and instructive to pause here and view (88) and (89) in the light of the semigroup relations (52) and (53) of Sec. 3.7 which certainly apply in the ir- radiance context. Toward this end, observe that the factors before % and y in (88) and (89) comprise basically a complete transmittance factor. Thus suppose we write: "7(-l,0,z )" for t-(0) 1-R(0,z )r + (0) SEC. 8.5 TWO-D MODELS 49 and M H(-1,-)" for H°(0,-) Then (88) is of the form: H(Z,+) = H(-l,-)~(-l, 0,2^(0,2,2^ and (89) is: H(2,-) = H(-l,-)r(-l,0,2 )7(0,2,2 ) (90) (91) The semigroup relations show that we can write (90) and (91) as : H(2,+) = H(-l,-H(-l,z, Zi ) H(2,-) = H(-l,-)r(-l,2,2 ) (92) (93) The conceptual unity affo point of view is illustra version of (88) and (89) quence we can see that th boundary X and X(0,2 1 from that between any two X(0,z 1 ). In view of this analysis so that we need ing with - 1 , but rather However, the present nota workable in the discrete shall be retained. The reader who has of (92) and (93) can now medium X(0,2 1 ) which is aries X , X Z1 . Indeed for one-parameter source- juncture without the nece interaction method. We m (12) of Sec. 3.9 and the of that section applied t of the union of X Q , X(0 To see how such an establish some notation, are as listed above. Tho as established generally are the four standard R above. Thus each of thre and X Zl is generally as two reflectances and two together, as a class, wil further, we write "H(-l, ward and upward incident then the irradiance field is given by: rded by the invariant imbedding ted quite strikingly by this con- into (92) and (93) . As a conse- e interaction problem between the ) does not differ algebraically subslabs X(0,y), X(y,zJ of , we could rework the preceding not count depths in X(0,2 1 ) start- with some other fiducial depth* tion has been found convenient and space setting (Ref. [251]), and followed the preceding derivation readily extend these results to the endowed with two interacting bound- , the invariant imbedding relation free media can be invoked at this ssity of a fresh application of the erely use the semigroup relations invariant imbedding relation (10) the one parameter medium made up , z ) and X z . application proceeds, let us first The interaction factors for X Q se for X z are r + (z ) , t ± (z ) , in Sec. 3.3. Those'for X(0,2 1 ) and T factors found in detail e interacting entities X Q , X(0,2 1 ), signed four interaction operators: transmittances . The three media 1 be denoted by M X 3 (0,2 1 ) M . If, -)" and "H(z 1 +l,+) n for the down- irradiances on the space X 3 (0,z 1 ), H(2,±) at any depth 2 , < 2 < z^ 50 MODELS FOR IRRADIANCE FIELDS VOL. V (H(z,+),H(z,-)) (H(z i+ l, + ),H(-l,-))^(-l,z,z i+ l) (94) where the four components of the matrix A!(-l,z,z +1) are found by decomposing them using the semigroup properties (12) of Sec. 3.9. Two examples of such decompositions have already been given in (90) and (91) for the one-boundary downward flux case. In the present two-boundary case, we have for example: y(-l,z,z +1) = !T(-l,0,z +lpt0,z,z +1) (95) The geometric setting for (95) is depicted in Fig. 8.2. Usinj this figure as a guide, and turning to (40) through (43) of Sec. 3.7, we see from (42) of Sec. 3.7 that: T(-l,0,z +1) = T(-l,0)[l-R(0,z +1)R(0,-1)] (96) FIG. 8.2 Geometric scheme for including boundary effects of boundaries X and X Zl ,of medium X(0,z 1 ). Hypothetical levels labeled " -1 " and " z. + 1 " are introduced as places external to X(0,z 1 ) from which radiant flux may irradiate X(0,z ) or at which emergent radiant flux may be measured. SEC. 8.6 ONE -D AND MANY-D 51 Here, of course, T(-1,0) = t_(0) (97) R(0,-1) = r + (0) (98) Furthermore, by (15) of Sec. 7.3, we have: R(0,z i+ 1) = R(0, Zi ) + ^(0^^+1)1^,0) (99) By the semigroup property of <^: ^(0,Z 1 ,Z 1 +1) = T(0,z i ,z i +1)R(Z 1 ,z i+ l) (100) Here we have: R(z ,z +1) = r (z ) (101) and once again by (42) of Sec. 3.7: y(0, Zi ,Z i+ l) = T(0,z i )[l-R(z i ,0)R(z i ,z i+ l)]" 1 (102) In this way we can completely and systematically analyze the factors on the right in (95) using only the main semigroup properties of y and for some positive constant c . (iv) H(0,-) is an arbitrary irradiance, and H(z 1 ,+)=0. A comparison of these assumptions with the corresponding set ( i) through (iv) in Sec. 8.5 just before (8) of that section shows that the main change is in the uniformization of the radiance distribution: It is now to have the same shape over E + and 5_ . The shape may be spherical, elliptical, or any arbitrary shape imaginable; however, the crucial point is that the shapes are the same in the upper and lower hemispheres of The immediate consequences of assumption (iii) are the following. First we have: D(+) = D(-) , (1) which comes from (2) of Sec. 8.5. Suppose we write " D " for this common value of D(±). As a result, (6), (13), and (14) of Sec. 8.3 imply: a( + ) = a(-) = aD (2) a(+) = a(-) = aD (3) s( + ) = s(-) = sD (4) Finally, (6) and (7) of Sec. 8.5 show that: f(+) = f(") (5) b( + ) = b(-) (6) We shall write " f " and " b " for these common values of f (±) , b(±). These observations make it clear that the gen- eral effect of (iii) is to induce a systematic collapse of complexity throughout Sec. 8.5. Some of the resultant con- densations will now be surveyed. The equations for the irradiance field (8) of Sec. 8.5 now take the form: SEC. 8.6 ONE-D AND MANY-D 53 dH(z,±) = dz [aD + b]H(z,±) + bH(z,+) (7) The first discernible effect of the one-D assumption on the solution of these equations is in the structure of k + and k_ , as given in (12) of Sec. 8.5. We now have: k + = ± [(aD + b) b 8]l/8 [aD(aD + 2b) ] (8) If the radiance field is assumed spherical (as it occurs very nearly deep inside media with high s/a ratio) then D = 2 and: k + = ± 2[a(a + b)] l/2 . If the radiance field is assumed nearly collimated vertically (as it occurs very nearly deep inside media with low s/a ratio) then D = 1 and: k + = ± [a(a + 2b)] 1 (9) Observe that for the one-D theory, having a = requires k+ = . However, in the two-D theory, matters need not be so simple (cf. (13) through (16) of Sec. 8.5). For brevity we shall henceforth write: for k so that k = - k (10) Observe that in the one-D theory, whenever b < f , we have from (8) : k < aD (11) which represents a fundamental inequality throughout radia- tive transfer theory (cf. (7) of Sec. 6.6 and (26) of Sec. 9.2). We next find that g ± (±) in (10) and (11) of Sec. 8.5 take the forms: M*) = i + -f g_(±) = i aD (12) (13) In other words: g + ( + ) = g.(-) = 1 + aD (14) 54 MODELS FOR IRRADIANCE FIELDS VOL. V the common value for which we shall write " g + " ; and we shall write " g_ " for the common value of: aD g + (") = g.( + ) = 1 " if (15) From this and (9) of Sec. 8.5, we see that the solutions of (7) can be written: H(z,±) = m +g+ e kz + m_g_e~ kz (16) From this in turn we see that the system determinant A(z.,) in (20) of Sec. 8.5 becomes: A(z ,>- {^^Y^ - (i-^) 2 e" kZl (17) so that: 4(0) - i£ The irradiance fields, as given by (22) and (23) of Sec. 8.5 now take the forms: H(z,+) _ H(0,-) f a¥] f X 2 ! -2 ) - e - k ( Z -,- Z ^ (18) H < z >-) = %tt a (V z ) (19) An important constant in the one-D theory is the reflectance of an infinitely deep medium X(0,°°). This constant may be obtained from (26) of Sec. 8.5 by going to the limit as z 1 -*«> . Thus if we write: "R " for lim R(0,zj oo 2. -*■ °° ' (20) and observe that: lim e kz ^ 1_ A(z ) 2 then (26) of Sec. 8.5 implies: i ♦ 4^ k 1 - a( + ) Alternatively, in view of (14) and (15) this may be written: SEC. 8.6 ONE-D AND MANY-D 55 R = aD k k-aD ., aD k+aD 1 "F (21) Another important feature of tne one-D model for X(0,°°) consists in its representations of H(z,±). To ob- tain these representations, it is sufficient to observe that A(z -z) lim . -, 1 , — = lim z -*■ °° A(z ) z a k(z,-z) g + e _ -kz A(z ) " e Then (19) supplies the requisite representation for H(z,-) in X(0,~): H(z,-) = H(0,-)e -kz (22) and from (18) and (22) ; H(z,+) = H(z,-)R = H(0,+)e kz (23) which holds in X(0,°°). One-D Model for Internal Sources The reductions from the two-D to the one-D model in the of internal sources proceeds generally as the undecom- light field discussion just completed. Of course, the eduction features should be the one-D forms of the par- ar source solutions (56) of Sec. 8.5. To illustrate reductions, suppose that there is a set of sources rmly distributed throughout X(0,z 1 ), so that h n (z,±) case posed new r ticul these unifo is independent of depth and that h n (z,+) = h n (z,-). „ h ii comes denote the constant value. Then (50) of Sec. B< rx Let .5 be Y(s,±) = (t -p)h so that (56) of Sec. 8.5 reduces to: h^(T - p) rZ o Hp(z,±) = - 2k (24) j z [e k( Z -s) . e -k( Z - S)] ds (25} 56 MODELS FOR IRRADIANCE FIELDS VOL. V One-D Model for Decomposed Irradiances The reduction of the two-D equations for decomposed ir- radiance fields proceeds similarly to the undecomposed case, starting with the same conditions (i) through (iii) now ap- plied to N* . In particular, the attenuation coefficients become: a*(+) = a*(") = aD* . (26) a*(+) = a*(-) = aD* (27) s*(+) = s*(-) = sD* (28) where " D* " denotes the common value: D*( + ) = D*(-) (29) and £*(+) = f*(-) (30) b*(+) = b*(-) (31) k ± = ± [aD*(aD* + 2b*)] l/s (32) These equalities show that the decomposed case develops in a manner that is exactly parallel to the undecomposed case, as far as the homogeneous part of the solution goes. The par- ticular part of the solution, as embodied in C(y ,±) (cf. (72) of Sec. 8.5), is now such that: o,(\i )b* + a_(y ) fa* + b* + (a/y J] c(y Q ,±) - -=— \ . (33) Furthermore {$ ■ *-) Finally, we note that: g + ( + ) = g.(") = 1 + ^ , (34) the common value of which may be denoted by " g + " . Further, we denote by " g_ " the common value of: an* g + (") = g.( + ) = 1 " T" (35) Equation (20) of Sec. 8.4 then reduces to: SEC. 8.6 ONE-D AND MANY-D 57 dH*(z,±) dz [a* + b*]H*(z,±) + b*H*(z, + ) + f°H°(z,±) + b°H°(z,+) (36) The general solution of the homogeneous part of (36) is iden- tical in form to (16). The case of X(0,°°) is also of inter- est for decomposed irradiance fields. Suppose we denote by " Roo(u ) " tne present counterpart to R^, in (20). Then, parallel to (21), we have: C( V -) (u 1 - g_ ccvf CC V -) C(y ,-) \ C(M 0> -) where Roo now uses starred distribution coefficients. The respresentation of the irradiance field in X(0,°°) is readily obtained by using (73) of Sec. 8.5 to observe that: lim m z -*-<» ■ = and: lim m z -* 00 1 .o C so that (71) reduces to H*(z, + ) = N' R oo C(y o ,-)e" kz - C(y o , + )e" az/y H*(z,-) = N u C(y Q ,-) -kz _ ^-az/y Q (37) (38) These equations show that eventually the ratio H* (z ,+) /H* (z , -) approaches Roo . This ratio settles to R^ as soon as the effects of the collimated light, due to No(0,£), have died away. It is noteworthy that the structures of (37) and (38) are identical to their counterparts in the full two-D theory (cf. also (1) and (2) of Sec. 10.3). Many-D Models There are generally two ways in which to gain insight into a physical theory: to work out numerical examples or special cases of the theory, and second, to generalize the theory to see it from a broader perspective. In this and the preceding section we have gained insight into the two- flow 58 MODELS FOR IRRADIANCE FIELDS VOL. V theory by pursuing its special ramifications in the various two-D and one-D models. We close this section with an activ- ity of the second kind, that is, by finding the immediate generalization of the two-D theory to the so-called "many-D" theory. The point of departure for the two-flow equations was the steady-state source-free version of the transfer equation (1) of Sec. 8.3. We may use this equation once again as the starting point for the present discussions. However, very little extra effort will be expended if instead we start with the time-dependent transfer equation with sources, and derive the many-D theory from that. This we shall do. Let X be an arbitrary optical medium. First we write down: £- NC ^' t} + E-VN(x,5,t) = = - a(x,t)N(x,£,t) + jjN(x,S',t)a(x;5';S,t)dfiU')+ N n (x,£,t) (39) Next, we partition E into n disjoint subsets, E 1 , ... , H n , n > 2 . For example if n = 2 and E., = H+ and E z = E_ , then we would return to the usual two-flow setting. If n=2 and E 1 = { E, : £ • n > } = E(n), and E 2 = { £ : i • n < } = E(-n) , then we would have a two-flow setting based on the partition around n instead of around the unit vector k along the z - axis . (See Fig. 8.3(a).) Thus, assuming a general parti- tion { E 1 , E 2 , . . . , E n } of E with respect to a unit vector n, as in Fig. 8.3(b), we proceed to generalize (5) through (8) of Sec. 8.3 and related concepts to these. First we write: for each j , 1 < j < n : "hj(x,t) M for j_ N(x,5,t)dfl(5) (40) "j and "H.j(x,t)" for J_ N(x,5,t05dQ(O (41) ~j The vector Hj(x,t) is the irradiance vector induced at x at time t by the radiance distribution restricted to Ej . The net irradiance induced by this vectorial flux on a sur- face with inner normal n at x and time t is: n • HjCx.t) (42) and which we shall designate by "H . (x,n, t) " . Next we write h (x,t) "V X ' n ' t} " £ ° r H. ' lx,n,t) C 4 « SEC. 8.6 ONE-D AND MANY-D 59 FIG. 8.3 Illustrating a two-flow decomposition of direc- tion space 5 , as in part (a) ; and a many-flow composition as in part (b). Each gives rise to a set of irradiance equa tions at point x in an optical medium. The attenuating functions cu (x,t) , sj(x,t), aj(x,t), j = l ... , n are defined analogously to (t>) , (13), and (14) of Sec. 8.3. Thus, e.g., we now write: "a.(x,n,t) n for a(x, t)D. (x,n, t) (44) The four forward and backward scattering forms of the two- flow theory sublimate into the following n 2 quantities H k (x!n,t) L [L N(X '^ t)a(x;£';?,t)dftU') dfl(S) (45) 60 MODELS FOR IRRADIANCE FIELDS VOL. V which we shall denote by " Sj^CXjii^c) " for j = 1 k = l, ... , n. Finally, integrating (39) over -J , , n ; we find: - ^r Td-H. I + V • H. n = - a-H. + I s.,H J J k=l j " 1 > + h jk k n,j (46) where, for brevity, space and time coordinates have been suppressed, and where in general we have written: 'h - (x t)" for ]., N n (x,€,t)dfi(S) J (47) The system (4 one parameter family a one-parameter opti cal properties are c and cylindrical medi to which (46) may be usually be establish case, so that the di tive of a suitable c will find it instruc cylindrical coordina flow setting in thes The general s form more nearly res Sec. 8.3. To see th 6) is most useful when there exists a of space-filling surfaces (in order words, cal medium) over which the Hj and opti- onstant valued. For example, spherical a are used occasionally in practice, and applied. Then a coordinate system can ed in the medium, as in the plane-parallel vergence term reduces to a single deriva- omponent of H-; . For example, the reader tive to obtain V • H in spherical and te systems, and illustrate (46) for a two- e two coordinate systems. ystem can be cast, as it stands, into a embling the homogeneous terms of (19) of is, write: jk when j ± k ; also write: for for s jk (48) •b." for I s 3 k^j kj (49) where " £ " stands for the sum over all n indices k except j . k^j Furthermore, write: "a." for aD- J J ii., 1 1 for aD . s - " for sD . J J (50) (51) (52) Then it follows that: SEC. 8.7 CONCEPTS FOR IRRADIANCE 61 f. + b. J J s . a . = a. + s - J 3 J a. + £ ♦ b. (53) (54) and (46) can then be cast into the form: (55) which establishes the final generalization. By letting n+«> such that max { fi(Hj) : j = 1 , ..., n } -► , (55) returns to the equation of transfer (39), and the circle is complete. 8. 7 Invariant Imbedding Concepts for Irradiance Fields The formulations of the preceding sections of this chapter can be placed into deeper perspective when viewed from the general standpoint of the invariant imbedding con- cepts developed in Chapter 7. In this section we select some of the results obtained in this chapter to be so viewed. The main purpose of this activity is to indicate the conceptual and the numerical advantages gained by adopting the invariant imbedding point of view: Unsuspected symmetries and connec- tions between the solutions of the two- flow equations spring into view when led in their general directions by the alge- braic equations stored up in Sec. 7.4; furthermore the exis- tence of quite general differential equations collected in Sec. 7.5 are now seen to hold also among the components of the irradiance field, and of the R and T factors. Further- more, the semigroup and group- theoretic methods of Sec. 7.8 through Sec. 7.10 are awaiting their systematic translation into the irradiance context. In general, according to our observations in Sec. 8.2, all the functional equations de- rived in Chapter 7 from the local or global forms of the principles of invariance also hold for the irradiance context, Therefore, to be thorough, we could, in principle, repeat vir- tually all of Chapter 7 in the present section. But this is a lavish and unnecessary task for the purposes of the present work. By leaving it undone, we allow room for the few perti- nent remarks made below and, more importantly, encourage stu- dents of the subject to explore such matters on their own and perhaps find new and interesting facets to develop and use. The selected examples below will indicate a few of the possi- ble modes of approach. 62 MODELS FOR IRRADIANCE FIELDS VOL. V Example 1: (f{ and D"" Factors in Two-D Models Equations (24) through (43) of Sec. 8.5 represent the results of a tentative, initial excursion into the invariant imbedding domain of the reflectance and transmittance con- cepts associated with the two-D theory. We now develop the full structure of these (% and y factors, being guided by the basic equations of the invariant imbedding relation in the irradiance case, namely (7) and (8) of Sec. 8.1. Our main goal is to represent the four (R. and J* factors for an arbitrary subslab X(x,z) of X(a,b) in terms of the local scattering and absorbing properties of X(a,b). We begin with the general solution (9) of Sec. 8.5 for the irradiance field at level y in X(a,b). Equations (7) and (8) of Sec. 8.1 state that the irradiances H(y,±) are a linear combination of the upward irradiance at level z and the downward irradiance at level x , the coefficients of the combination being the complete reflectance (<7£) and transmittance (¥ ) factors for X(x,z). Evidently, m+ and m- in (9) of Sec. 8.5 hold the key to determining ^(x,y,z), 3t x >y> z )> ^?(z,y,x'), and -^(z,y,x). Therefore, since (9) of Sec. 8.5 holds for all depths z in X(a,b) we have: H(x,-) = m +g+ (-)e k+X + m.g_(-)e k - x H(z,+) = m +g+ (+)e k+z + m_g_ ( + )e k_z (1) (2) These two equations can be solved for m + , m_ . The results are: H(z, + )g_(-)e k + X - H(x,-)g_( + )eVJ /a(x,z) (3) where we have written: "A(x,z)" for g + ( + )g_(-)e k+z + k " x - g + (-) g_ ( + ) e k+x + k " z (4) These solutions are the full symmetric forms of (19) and (28) of Sec. 8.5. The earlier forms are obtained by set- ting to zero the appropriate irradiances in (3). Now, from (9) of Sec. 8.5, we can write H(y,±) = m +g+ (±)e k+y + m_g_(±)e k ' (5) with m ± as given in (3). Therefore, after some algebraic reductions to the forms (7) and (8) of Sec. 8.1, we find that for az) g + (-)g.(-) tf((z,z,x) = R(z,x) = A(x>z) k + z + k_x k+x + k_ z e - e (10) k+z + k_x k. x + k_ z e + - e + (11) 7tx,z,z) = T(x,z) = ^|| y C z,x,x) = T(z,x) = Mf^i Furthermore from (6) through (9) we see that: ^(x,z,z) = Z) ■ [t(-) + p( + )R(x,z)]T(x,z) (38) The differences between (35) and (38) and their opera- tor correspondents in Sec. 7.1 (starting with (18) of Sec. 7.1) immediately strike the eye: the presence of signed p and x factors and the rearrangement of terms, both in the present set. Both differences are superficial and can be erased with a few strokes of the pen and some accompanying reasons. The signed p and x factors reflect the two-D nature of the light field and summarize the fact that the local optical properties of X(a,b), with respect to irradi- ance, are accordingly anisotropic, as discussed at some length in Sec. 8.5. If, in the developments of Sec. 7.1, we chose to explicitly consider aniostropic media, then the four operators P+(y) , T+(y) would have been used throughout that discussion (see note after (4) of Sec. 7.1). However, for simplicity and for practical reasons, namely that aniso- tropic media are seldom encountered in practice , the develop- ments of Sec. 7.1 took their present form. Readers may view the anisotropic versions of the differential equations for general R and T operators in Sec. 25 and Sec. 125 of Ref. [251]; furthermore, Sec. 7.7 and Sec. 7.13 contain differ- ential equations which incorporate signed local reflectance and transmittance operators. As far as the order of the terms within I 1 through IV above is concerned, we need only note that we are now working with real valued functions of real numbers rather than with operators, so that commutativ- ity of the present multiplications is in force. As a result we have been able to rearrange the differential equations of R and T to look "more natural" to the eye. The solutions of (35) through (38), namely R(x,z) and T(x,z) and their companions R(z,x), T(z,x) are readily SEC. 8.7 CONCEPTS FOR IRRADIANCE 67 obtained. However, we need only note that they are of the form (10) through (13). The advantage of the differential equations (35) through (38) is that they may be integrated even when p and t vary with depth, so that they potential- ly transcend the two-D theory in versatility. The reader should now derive the completely general differential equa- tions for R(x,z) and T(x,z) starting with (1) through (4) and with no assumptions on the homogeneity of X(a,b). The results should be of the form: I' - 8R 3 ^ ,Z) = P(x,-) + [t(x, + ) + t(x,-)]R(x,z) + P (x, + )R 2 (x,z) (39) II' - T ^' Z) = Ct(z,-) + p(z,-)R(z,x)]T(x,z) (40) HI' 3R ^' Z) = T(x,z)p(z,-)T(z,x) (41) IV - 9T ^ ,Z) = [t(x,-) + p(x,+)R(x,z)]T(x,z) (42) Equations (39) and (42) are the key relations here. Using this we can find R(x,z), T(x,z) for X(x,z), with initial conditions R(z,z) = 0,T(z,z) = 1. The reader should now develop the equations for R(z,x) ,T(z,x). (See bibliographic notes.) Example 4: Third Order Semigroup Properties of +X++ys^^^^ Y x b FIG. 8.6 A five-part optical medium consisting of three reflecting- transmitting surfaces (heavy lines) and two diffus ing media (dotted). X (y,b)" for X(y,b)UX 1 (62) "X 3 (y,b)'» for X y UX 2 (y,b) 'X 4 (a,b) for X(a,y)UX fy,b) (63) (64) This mode of construction of X (a,b) is patterned after the invariant imbedding process used in Sec. 7.13 (re: Fig. 7.25), We shall develop a discrete-space approach to the present in- variant imbedding process. We are now ready for the analysis, Stage 1 of the present invariant imbedding process (Fig. 8.7(a)) consists in finding the invariant imbedding operators for X^ . These are simply the factors r ± (b), t±(b). To establish a systematic notation which will hold in all stages of the analysis we write: '^(b,b,b+l) for r_(b) 'y(b-l,b,b) n for t_(b) '^(b,b,b-l)" for r + (b) ■^(b+l,b,b) M for t + (b) (65) (66) (67) (68) This notation is patterned after that of general discrete SEC. 8.7 CONCEPTS FOR IRRADIANCE 73 (a) (b) (d) (e) X(y,b) (c) X(y,b) X(a,y) X(y,b) X(a,y) X(y,b) i n. Xy X b Stage I X b Stage 2 Xy Stage 3 Xb Stage 4 Stage 5 FIG. 8.7 A systematic analysis (and resynthesis) of the medium in Fig. 8.6. space theory, (Ref. [251]). Hence, for Stage 1, invariant imbedding relation associated with X^ the pertinent is : (H(b,+),H(b,-)) = (H(b + l, + ),H(b-l,-)W(b-l,b,b + l) (69) Here T\(b- 1 ,b ,b+l) is the 2x2 matrix made up from the four factors defined in (65) through (68) . By the simple notation- al device of adding 1 to b and subtracting 1 from b , we can conveniently denote the incident irradiances on X^ con- sidered as an isolated medium. This tactic will be used re- peatedly below and is the signal that a discrete-space calcu- lation is in progress. Stage 2 of the present invariant imbedding process finds the operators for X 2 (y,b). The ^-operator for X 2 (y,b) is ~7»£(y,z,b + l) , a 2x2 matrix. Thus, by (52) and (54), adapted to the present one-parameter medium, we have: 74 MODELS FOR IRRADIANCE FIELDS VOL [(y,z,b + l) = T(b + l,b,y)T(b,z,y) J(b + l,b,y)^(b,z,y) ;y,z,b) + ^(y,b,b+l)r(b,z,y) 7(y, z ,b) +^(y,b,b + lK(b,z,y) The preceding matrix can be analyzed into factors of the in variant imbedding type as follows: 7)[(y,z,b + l) = - [C_ + 7^(y,b,b+l)C + ]^(y,z,b) a < y < z < b (77) SEC. 8.7 CONCEPTS FOR IRRADIANCE 75 where C + and C_ are defined in (4) and (5) of Sec. 7.4. Now 1+ occurring in C ± are simply the number 1 in the present irradiance context. Equation (77) shows how the in- variant imbedding operators of two contiguous media can be algebraically combined to yield the invariant imbedding oper- ator for their union. In this case the media are the slab X(y,b) and the surface X^ . Stage 3 of the present invariant imbedding process finds the "7t - operator for X (y,b). The nr l- operator for X 3 (y,b) is the 2x2 matrix 7»$y-l , z ,b+l) . Thus, by (53), and (55), adapted to the present one-parameter medium, we have: <^(y-l,z,b + l) = r(y-l,y,b + l)^(y,z,b + l) (78) r(y-l,z,b + l) = 7(y-l,y,b + l)7(y,z,b + l) (79) which hold for y £ z <_ b . The two factors ^(y,z,b + l) and 7(y,z,b+l) are known from the preceding stage of the analysis The remaining factor is obtained by means of (42) of Sec. 3.7 !T(y-l,y,b + l) = T(y-l,y)[I - R(y ,b+l)R(y ,y- 1) ] " 1 (80) Here "T(y-l,y)" and "R(y,y-1)" are simply other names for t- (y) and r + (y), respectively, used in discrete-space theory Finally, R(y,b+1) = ^(y,y,b+l) , (81) which is known from the preceding stage of analysis. The remaining two components of TJ[(y-l, z ,b + l) are found by means of (52) and (54) ; ^(b+l,z,y-l) = ^(b+l,z,y) + ^(b + l,y,y-l)7(y,z,b+l) (82) r(b+l,z,y-l) =T(b+l,z,y) + ^(b + l ,y ,y- l)^(y , z ,b+l) (83) which holds for y <_ z <_ b . Here all factors are known from Stage 2 except for <^(b+l ,y ,y- 1) , which can be obtained using the semigroup relation: (^(b + l,y,y-l) = ^(b + l,y,y-l)^(y,y,y-l) Finally IT(b+l ,y ,y- 1) is found by means of (43) of Sec. 3.7. Stage 3 may be summarized by the following equation: 7^(y-l,z,b+l) = ~y(b+l,z,y)+^(b+l,y,y-l)^(y,z,b+l) ^(b+l,z,y) + ^(b+l,y,y-l)a(y,z,b+l) ir(y-l,y,b+l)^(y,z,b+l) T(y-l,y,b+l)7(y,z,b+l) 76 MODELS FOR IRRADIANCE FIELDS VOL. V In other words: 7)J(y-l,z,b + l) = [C + + ^(y-1, y,b+l)C_]^(y,z,b+l) y < z < b (84) The pattern that is forming is now sufficiently clear so that the reader may complete stages 4 and 5. Observe that (84), as it stands, solves the general boundary-effect problem of a medium X(y,b) with interref lecting boundaries X v and X^ . Example 6: Invariant Imbedding Operators for Interacting Media We consider next the problem of predicting the irradi- ance field within a medium X(a,c) composed of two contiguous media X(a,b), X(b,c). The media may be of infinite depth or they may be degenerate, i.e., they may be surfaces. For example, if X(a,b) is degenerate, then we write "X(a-l,a) M for X(a,b) and construct the reflectances and transmittances in the manner described in Example 5 above. If there is an interface at level b , let it belong to X(a,b). Our main purpose in this example is to present a unified algebraic approach to the problem of irradiance (or any other* radio- metric) fields in sets of contiguous plane-parallel media. We have developed sufficient background for the solution of this problem in the preceding Examples 4 and 5 and in Sec. 8.5 (re: (94) of that section) to permit the broad algebraic techniques of the present example to be followed without dif- ficulty. Figure 8.8 depicts the composite medium X(a,c) = X(a,b)UX(b ,c) . We assume that the operators (2 x 2 matrices) 7^(a,y,b) and 7>t(b,y,c) associated with the component media X(a,b) , X(b,c) are known. Our goal is to characterize 7^(a,y,c), the invariant imbedding operator for X(a,c), in terms of the operators "^l(a,y,b) and "^L(b,y,c). The analy- sis of the problem reduces to the two cases, depicted in Fig. 8.8. Consider case (a). The irradiance field (H(y,+), H(y,-)) at depth y , a <_ y <_ b may be viewed from two van- tage points: as an irradiance field in X(a,b), which is the response of the isolated medium X(a,b) to the incident ir- radiances H(b,+), H(a,-); or as an irradiance field in X(a,c) which is the response of X(a,c) to the incident ir- radiances H(c,+) ,H(a,-). The first interpretation is rep- resented as: *In the event that any of the present results are adapted to the radiance context, and media with different indices of refraction are considered, it will be understood that each radiance will be divided by the square of the index of refrac- tion of the medium to which it pertains (cf. (4) of Sec. 7.6), so that we use N/n 2 rather than simply N throughout any formula. SEC. 8.7 CONCEPTS FOR IRRADIANCE 77 FIG. 8.8 The response of the composite medium X(a,c) can be characterized algebraically in terms of the responses of its components X(a,b) and X(b,c). (H(y,+),H(y,-)) = (H(b , + ) ,H(a, -) )7^(a,y ,b) (85) The second interpretation is represented as: (H(y,+),H(y,-)) = (H(c,+),H(a,-)H(a,y,c) (86) Finally, the irradiance field (H(b ,+) ,H(b , -) ) may be repre- sented as: (H(b, + ),H(b,-)) = (H(c,+),H(a,-))^(a,b,c) (87) Using the contracting matrices C + , C_ in (4) and (5) of Sec. 7.4, it follows from (87) that: 78 MODELS FOR IRRADIANCE FIELDS VOL. V (H(b,+),0) = (H(c,+),H(a,-))%a,b,c)C + Further we have: (0,H(a,-)) = (H(c,+),H(a,-))C_ Adding these two equations, we obtain: (H(b,+),H(a,-)) = (H(c,+),H(a,-))[C_ + 7^(a,b ,c) C + ] (88) which, operated on by ^(a,y,b), would, according to (85), yield an alternate representation of (H (y , + ) ,H(y , -) ) to that given in (86). Hence, since (H (c , + ) ,H (a, -) ) is arbitrary, (89) Case (b) in Fig. 8.8 proceeds analogously, and the form of ^(a,y,c) in this case turns out to be: fl[(a,y,c) s = [C_ + ^(a,b,c)C + ]^(a,y,b) a < y < b < c fl[(a,y,c) = [C + + ^(a ,b ,c) C_ ]7IJ(b ,y , c) a < b < y < c (90) Equation (89) is the general form of (77) and (90) that of (84). Further, (89) and (90) implicitly contain the third- order semigroup relations (52) through (55) for the settings of Fig. 8.8. The generality of (89) and (90) resides in the po bility of either component X(a,b) or X(b,c) being its composite space made up any number of contiguous slabs internal boundaries (see, e.g., Examples 3 and 4 of Sec Equations (89) and (90) serve as guides in the construe of ^l(a,y,c), knowing the corresponding operators for i component spaces. Hence (89) and (90) constitute an in tive step in the construction of "^l(a,y,c) in precise a ogy to the customary inductive step used in the applica of the principle of induction in mathematical arguments where one proceeds from a statement P(n) associated wi integer n to statement P(n+1). One interesting appl tion of (89) and (90) would be to work out the complete tails of assigning distinct pairs D^(±) of distributio values to each slab X(ai,a^ +1 ) in a sequence of n si comprising X(a,b), so that (6) through (9) may be used actual numerical calculations based on (89) and (90). It remains to observe how 7KL(a,b,c) is found. Th reader may have noted, on the basis of the discussion i ssi- elf a and . 3.4) tion ts duc- nal- tion th an ica- de- n abs in SEC. 8.7 CONCEPTS FOR IRRADIANCE 79 Example 5, that equations (40) through (43) of Sec. 3.7 will serve adequately in this task. In this connection, we ob- serve that an elegant algebraic formulation of (40) through (43) of Sec. 3.7 is possible by using the star product for r 3 (a,b) introduced in (75) of Sec. 7.4. For by (76) of Sec. 7. 4 we have : Wa,b,c) = %a,b,b) * #?(b,b,c) (91) Continuing on this algebraic level, we can further resolve 7^(a,b,b) and 7^(b,b,c) by means of the M-operators (2) of Sec. 7.4. It is easy to see that: #((a,b,b) = R(b,a) T(a,b) [C + + M(a,b)C_] ; (92) n (b,b,c) T(c,b) R(b,c) I [M(b,c)C + + C_] . (93) Finally, the star product * for G 2 (a,b), as defined in (35) of Sec. 7.4 and studied in (37) and (38) of that section, may be used analogously to (91) to find the R and T operators of the union of contiguous media. Thus, e.g., the equation: M(a,c) = M(a,b) * M(b,c) (94) shows how to algebraically construct the standard R and T factors for X(a,c), knowing those for X(a,b) and X(b,c). Example 7: Differential Equations Governing 6{ and X Factors The preceding examples have shown the key role played by the complete $. and X factors in the determina- tion of the irradiance field in a variety of practical in- stances. It is of interest to observe that these 6L and 0" factors may be obtained directly by integrating the dif- ferential equation that governs them. Several such differen- tial equations were developed in (11) through (13) of Sec. 7.5. The one we select for attention here is (38) of Sec. 7.5: da(y) _ dy a(y)X(y) defined for each y in the depth interval [ a initial conditions: (95) b ] and with 80 MODELS FOR IRRADIANCE FIELDS VOL. V tf.(b) = [0,T(a,b)] (96) or: a(a) = [R(a,b),I (97) Here we have written: •4(y) for (^(a,y,b),!T(a,y,b)) and ^t(y) now has the form given in (9) of Sec. 8.2. For example, knowing ^i(y) and R(a,b) , one can find ^(a,y,b), T(a,y,b) directly by integrating (95) from a to y . The integrations may be theoretical or numerical, where appro- priate. Further observations on ^(y) , of interest to the irradiance context, are given in (11) through (15) of Sec. 7.10. Example 8: Method of Modules for Irradiance Fields We now present an illustration of the method of modules, as developed in Sec. 7.8, for the case of irradi- ance fields. Equations (14) of Sec. 7.8 are readily put to use in irradiance computations once J (d) is obtained. From (8) and (21) we have at once: red) lim Z +co^ > d > Z ) = lim A(d,z) z+°° A(0 , z) = lim Z~*"°° k_d A(z-d) +(k + +k_)d -aTTT e (98) Equations (14) of Sec. 7.8 then take the form H(jd, + ) = H(0, V k - d R (- 00 v H(jd, -.) = H(0, -)e* k - d (99) (100) where we have written R (-)" for lim R(0,z) (101) and which, by (10), has the representation: SEC. 8.7 CONCEPTS FOR IRRADIANCE 81 R (-) - g_( + ) l + 4^ k_ g.C-3 1 k a(-) - a( + ) (102) The latter equalities follow from (11) of Sec. 8.5 Example 9: Method of Semigroups for Irradiance Fields The method of semigroups, applied to general radiance fields in Sec. 7.9, yields formulas for H(y,±) in X(0,°°). Thus from (10) and (12) of Sec. 7.9, we now may write: H(y,-) = H(0,-) exp { (t(-) + P(+)R (-))y > (103) and as usual: H(y, + ) = H(y,-)R (-) (104) The present setting is X(0,°°) and we have used the two-D theory concepts t(-) , p( + ), and R 00 (-). The latter constant was defined in (101) . In view of (99) (which holds for arbitrary d , so let d = y , thereby fixing j as 1) we see that we must have: k = T(-) p( + )R (-) (105) It may be verified that this is consistent with (102) . This is the desired connection between k_ , t(-), p( + ), and R ro (-) . Equations (102) and (105) are the first views we have of an important set of general connections which hold between the theoretical and the exact observable counterparts to these concepts, and which are studied in detail in Sec. 9.2. Example 10: Irradiance Fields Generated by Internal Sources We devote tne final example of this trating some of the relations developed in internal sources in the special case of ir (i.e., one-dimensional) settings. As a re spective on the structure of the one-D and internal sources discussed in general in ( of Sec. 8.5, and in particular in (25) of According to the general conversion in the introductory remarks to this sectio tional equation developed in Sec. 7.13 may the present irradiance context. Because o devote most of the discussion to the task light some special relations for internal- irradiance fields which hold only in the i section to illus- Sec. 7.3 for radiance field suit we gain per- two-D moels for 44) through (66) Sec. 8.6. principle, stated n, every func- be converted to f this we shall of bringing to source generated rradiance context MODELS FOR IRRADIANCE FIELDS VOL. V Toward this end, consider the observations in (87) through (92) of Sec. 7.13 concerned with the asymmetry of the 'i'-operator . It was observed that the invariant imbedding op- erators, such as <^(s,y,b) and j^s^jb) were generally distinct from their duals ^ + (y,s,b) and J + (y ,s ,b) , respectively . An examination of the matter showed that if ^(s,y,b) and &i. (y, s,b) were ever equal in some setting, then the computation details of the internal source problem in that setting would be ef- fectively halved with respect to the general case. Therefore a search for a reciprocity property between ^(s,y,b) and <^(s,y,b) was launched. It did not require an extensive an- alysis to see that the requisite reciprocity property (i.e., the equality of <^(s,y,b) and ^ + (y,s,b)) is generally barred by polarity of the R and T operators and general noncommu- tativity of the integral operators. Since commutativity of 0{ and ^factors is now available, and since the R and T factors of one-D models do not possess polarity (re: (35) and (36) of Sec. 8.5, and also (24) and (25V) we return to the matter of reciprocity of ^(s,y,b) and (K (y,s,b) and reexamine some of the functional relations of Sec. 7.13 in the present simpler setting. Therefore for the remainder of this example, we shall work in a separable plane-parallel medium X(a,b) in which the one-D assumptions of Sec. 8.6 are in force. First we explicitly verify that: ^(s,y,b) = -k(s+y) R CO 1 R 2 R CO CO (118) There are many ways to arrange the final form of (118). For example, one such form is: ns,y:0,«>) = e k(y-s) R 1 [• p_ | e -k(y-s)_ e -k(s+y) L K Roo 1 (119) From this form we can pick off the four components of V(s,y:0,-): PR * ++ ( s ,y:o,~) = -3= ^ + _(s,y:0,°°) = ^ e -k(y-s) _ e -k(s+y) k(y-s) _ -k(s+y) pr: ^_ + (s,y:0,co) = R^ e- k ^'^ + -£ • r e -k(y-s) _ e -k(s + y)J (120) (121) (122) 86 MODELS FOR IRRADIANCE FIELDS VOL. V Y__(s,y:0 ,) = e- k ^" s ) pR -k(y-s) -k(s+y) ] (123) From these, in turn, we find the local Y- factors for one-D irradiance fields in X(0,°°): pR V ++ Cs,s:0,-) = ^ F 2ks v (s,s:0,°o) P 2k ¥_ + (s,s:O f «0 = R^ P R oo ^__(s,s:0,oo) = - IF The preceding set of e lent opportunity to illustra tions (31) through (34) of S complementary relations (39) thermore, the relations (6) also illustrated in perhaps for example, how (39) of 7.1 from (123), and how (14) of When the source level lower into X(0,°°), equations relatively simple forms: 1 - e 1 - e PR ] 1 + "2F CI " e -2ks| 2ks >] (124) (125) (126) (127) ight equations offers an excel- te the general functional rela- ec. 3.9 and especially their through (42) of Sec. 7.13. Fur- through (15) of Sec. 7.13 are their simplest settings. Observe, 3 guided the derivation of (127) Sec. 7.13 was used to find (126). s is allowed to sink lower and (124) through (127) go to the pR a lim *i\ ^(s,s :0,oo) = -^ (128) lim r (s,s:0,«>) = X- lim ¥ fs,s:0,°°) = R lim S -M» ¥ (s f s:0,«0 = PR "IF pR r ex (129) (130) (131) Furthermore, if both source level s and observation level y descend into X(0,°°) so that the difference d = y-s remains fixed, the observed boundary effects at level eventually die away and (120) through (123) yield: pR lim S-K» Y ++ (s,s + d:0,oo) = TF e lim _, Y (s,s + d:0,°°) c ->-oo + - v 7 ' J p Tk kd kd (132) (133) SEC MODEL FOR VECTOR H 87 lim s _ TO Y_ + (s,s + d:0,°°) = r^" -led lim^ Y__(s,s + d:0,°°) = e Ka kd 1 + pR r oc ~2Y 1 + P R co (134) (135) An unexpected dividend accrues from the preceding array of ^ -factor relations. Observe first that (128)-(131) agree with our intuitive ideas that the relation between the source irradiance H°(s,+) and the response field H(s,+), should be the same as that between H°(s,-) and H(s,-) when boundaries are far away from level s (because the medium is optically symmetric about very deep levels) . By the same intuitive expectations, ¥ + _ and y_+ of (126) and (130) should be nu- merically equal. Apparently, this does not seem to be the case. However, if we rely on the correctness of our princi- ples and algebraic manipulations, to yield up (129) and (130) then on the basis of our intuition we are led to conclude that: 2k = R pR 1 + or equivalently that: where p is the local tor for the one-D the decay rate and reflec fields in X(0,«). It (128)- (131) may be ch reader may establish argument by using the Sec. 7.3, which holds further connections b are available in Chap (136) reflectance (i.e., backscattering) fac ory and k and R are respectively the tance factor associated with irradiance follows that all the preceding results aracterized in terms of R^ only. The (136) independently of the preceding connections (105) above, with (32) of in the irradiance context also. Still etween k, p, R^, and related concepts ters 9 and 10. 8. A Model for Vector Irradiance Fields The purpose o theory of the irrad tering-absorbing op hydrosols consistin The application is plicit expressions vector in terms of of the optical prop discussion presents the quasi -potential analysis and which light fields. Thes f this section is to apply the vecto iance field to an important class of tical media, namely the class of nat g, e.g., of oceans, harbors, and lak of practical value in that it. yields for the depth-dependence of the irra its components at the surface and ce erties of these media. Furthermore, particularly simple interpretations and related functions, arising in ve are pertinent to the description of n e vector interpretations emerge natur r scat- ural es . ex- diance rtain the of ctor atural ally 88 MODELS FOR IRRADIANCE FIELDS VOL. V from the geometry and physics of the present application, and will be given as the discussion proceeds. In this way we add to the evidence that the formalism of the "photic field" as developed by Moon, Spencer, and others (Refs. [187], [188]) is of more than academic interest, and in fact provides an elegant tool for the study of the light vector H(s) in the practical settings encountered in the study of hydrologic optics . While the practical context of the present discussion is limited specifically to that of natural hydrosols, the mathematical arguments apply equally well to any arbitrary plane-parallel scattering-absorbing medium in which the light vector possesses a quasi-potential. The radiometric pre- requisites for the present discussion are given in (2) -(16) of Sec. 2.8. A useful text on vector analysis is Ref. [30]. The Quasi- Irrotational Light Field in Natural Waters We fix the stage of the present discussion by adopting a stratified plane-parallel medium X(0,b) with stratified light field. The present discussion makes use of the concept of a quasi-irrotational light field, i.e., a light field in which at each depth z of a natural hydrosol, the irradiance vector satisifes the condition: H(z) • [vxh(z)] = (1) In general, H , the vectov-ivradiance function , (or light field) , is defined at each point x (which stands for the ordered triple (x,y,z)) of an optical medium by writing: M H(x)" for £N(x,£)dftU) , (2) where 5 is the unit sphere (the collection of all unit vec- tors) in euclidean three-space E 3 , and N(x,*) is the radi- ance distribution at point x (see Sec. 2.8). As will be shown in (12) below, the justification of the use of the relation (1) rests on the following two vec- torial versions of well-known experimental facts about the spatial and directional distribution of light in natural hydrosols : (i) For every fixed z > in X(0,b), H(x,y,z) is independent of x and y . (ii) For every fixed pair (x,y), H(x,y,z) lies in a fixed vertical plane for all z > . Of course, some variations of H on horizontal planes, and some oscillations of the vertical plane containing H do occur in all natural hydrosols. However, properties (i) and (ii) summarize the two most readily apparent permanent gross features of the light field in natural waters, on which it is possible to develop a mathematical theory of the light vector H(x) . SEC. 8.8 MODEL FOR VECTOR H 89 Interpretations of the Integrating Factor Since our interests lie principally in the physical and geometrical aspects of natural light fields, it would be in- structive to develop some physical interpretations of (1) and concepts immediately related to it. This we now do. The general theory of vector fields asserts that to each quasi-irrotational light field one may associate two real-valued functions $ and Z, , defined on the appropriate subset of X(0,z.,) representing the optical medium. These functions have the property that: H(x) = ^j V* (x) . (3) $ is the quasi-potential function, and £ is the integrating factory unique to within a multiplicative constant, associated with $ . Equation (3) is the necessary and sufficient condi- tion that: H(x) • [V x H(x)] = at each x of the medium. (See, e.g., Sec. 105, Ref. [30].) In the present context the function £ has particularly simple and interesting geometrical and physical interpreta- tions. We begin with the geometric interpretation. Figure 8.10 defines a terrestrially based coordinate system usually adopted for the discussion of the light fields in natural hydrosols (re: Sec. 2.4). The fixed plane referred to in property (ii) is taken as the xz-plane, and thus lies in the plane of the figure. The standard unit vectors i and k are positioned as shown. The unit vector j along the posi- tuve y-axis is normal to the plane of the figure and directed away from the reader. Consider an arbitrary rectangular path ABCD in the xz- plane such that its sides are parallel to the coordinate axes According to (3) and properties (i) and (ii) of the light field in natural hydrosols, it follows that: ABCD lM*W ' ds so that: f C(x)H(z ) • ds = f C(x)H(z ) • ds ; AB 1 J DC The condition (i) suggests that c, can be independent of x and y . By (13) of Sec. 2.8: H(z,i) = H(z) • i 90 MODELS FOR IRRADIANCE FIELDS VOL. V +z \ collimated flux (Example I) \ i Surface \ +x H(Z,,i) D \ N \ \ -•— H(Z 2 ,i) \H(Z 2 ) FIG. 8.10 How to visualize a quasi- irrotational irradiance vector field. so that the integral equality above implies e(zJH(z .1) = C(zjH(z ,i) (4) From this and the fact that t, is determined only to within a multiplicative constant, we can set £(0) = 1 and so: C(z)H(z,i) = H(0,i) (5) for all z in [0,b]. Thus <; may be selected as a dimension- less quantity which stretches the horizontal component H(z,i) to H(0,i) at every depth z in X(0,b). Next we consider the physical interpretation of C(z). The invariance with depth of the product: ?(z)H(z,i) SEC. 8. MODEL FOR VECTOR H 91 shows that the depth dependence of C(z) is such that its logarithmic derivative is equal, to within an algebraic sign to the_logarithmic derivative of the net horizontal irradi- ance H(z,i). That is: 1 = , and where $ is an azimuth an in a horizontal plane from the positive x-axis. y is defined as in (70) of Sec. 8.5. Now it is easy to verify that the diffuse the light field (i.e., that part consisting of a flux scattered one or more times) is symmetrical z-axis. Hence the net horizontal irradiance rec tribution from the diffuse light field. Therefo H(z,i) = H(0,i)e az/y we will e. , that a h z . The ration of that scat- e upper O = - arc gle measured The number component of 11 radiant about the eives no con re : (25) so that K(z,i) in this case is represented by K(z,,i) = a/y^ (26) and C(z) in the general theory above reduces to: C(z) = az/y (27) SEC. 8 MODEL FOR VECTOR H 95 Example 2: Asymptotic Form of the Light Field In optically infinitely deep media (i.e., b = °° in X(0,b)) the values k(z) of the function k defined in (7) rapidly approach, with increasing z , a fixed magnitude k which is independent of the external lighting conditions and which depends only on the inherent optical properties of the medium. This and related facts we shall explore in some de- tail in Chapter 10. In view of this fact it is permissible, for most engineering calculations, to assume that there is a depth z _> below which k(z) = k . From the divergence rela- tion (18) we see that in general: H(z ,k) - H(z ,k) a(z)h(z) dz (28) so that in particular H(z,k) - H(0,k) -i a(z')h(z')dz' Furthermore, in the present case: H(z,k) - H(z ,k) = a h(z')dz' = ah(z .'i: -k(z'-z ), e v crdz ah(z )e kz o -kz -kz n e ° ah(z ) k(z-z ) It follows that the k-component of H(z) in (24) may be written: H(z,k) = H(0,k) + a(z')h(z')dz' = H(z Q ,k) + a(z')h(z')dz' ' z ■ H(z .k) + o ah(z ) k(z-z ) Now since H(z,k) ■*- as z+°° (see, e.g., the two-D models in Sees. 8.5 and 8.6), it follows that we may set: H(z Q ,k) = (?)*<'„> 96 MODELS FOR IRRADIANCE FIELDS VOL. V so that (24) reduces to: C(z-z ) + H(z) = iH(z ,i)e " v " "o J + kH(z ,k)e k(z-z ) (29) for z > z, A further simplification is effected if we can find a suitable approximation for the function C . Thus ob- serve that for z > z Q , the diffuse component of the light field is essentially symmetrical about the z-axis (Chapter 10) so that, as in the isotropic case (27) : K(z,i) = Da where we have for clear sunny skies with sun at 6 zenith, or for some fixed D° , with = - arc cos y n from the 1 < D w < 2 for overcast days (see (2) of Sec 5) . With these assump- tions, (24) takes the particularly simple approximate form: H(z) = iH(z Q ,i)e D a(z-z D ) + ^ >k)e "k(z v. y j , (30) where D° may take any of the above special values. For most engineering applications it is permissible to take z = in (29) or (30). We conclude this example by making a few observations on the limiting directions of H as z ■+ °° . First, if s ^ , then k < a , (see, e.g., (9) of Sec. 6.6). If, in addition, we also have a ^ then from (30) and the fact that H(a,k)<0, it is clear that: lim z-*-°° H(z) H(z) If, on the other hand, we have a = 0, then H(z,k) = and: for all z ; the direction being that of H(0,i). Finally, if s = , then the problem of the explicit determination of H(z) for all z reduces to a relatively trivial (although sometimes tedious) calculation. In this case the limiting direction of H depends in a simple way only on the external lighting conditions. If N°(0,C) represents the incident ra- diance distribution at z = 0, suppose | £ • k | is the largest value for which N (0,£ o ) f 0. Then the limiting direction of H(z) , as z -> °° , is along the line defined by E, . SEC. 8.8 MODEL FOR VECTOR H 9 7 Global Properties of the Irradiance Field The curl and divergence of the irradiance field H(x) show how the field behaves in the neighborhood of a point. In other words, the curl and divergence of H(x) are local properties of H(x). An interesting and important global property associated with the irradiance field comes from an application of the divergence theorem to (15) and we shall now derive that property. Let X be an arbitrary connected, bounded homogeneous subset of X(0,b), with boundary surface S and with steady light field. Then on the one hand from (15) : f 7-H(x)dV(x) = - j ah(x)dV(x) J X j X = - av I u(x)dV(x) J X = - avU(X) . (31) The latter two equalities rest on (4) and (12) of Sec. 2.7. On the other hand if n(x) is the unit inward normal to S at x , then by the divergence theorem, we have: I V-H(x)dV(x) = - j H(x) • n(x)dA(x) (32) J X J S and we shall denote the integral on the right side by "P(S,-) M which thereby represents the net inward radiant flux across S (cf. (8) and (9) of Sec. 2.8). Combining (31) and (32) we have: P(S,-) = avU(X) (33) This equation shows how the net inward flux P(S,-) across the boundary of S is related to the energy content U(X) of X and the volume absorption coefficient a of X . Equation (33) may have some practical interest in laboratory procedures of determining the volume absorption coefficient a . The radi- ant energy content U(X) of X may be obtained by systemati- cally probing X with an instrument which measures scalar irradiance h(x). By numerically integrating the measured values h(x) throughout X , the term vU(X) may be obtained. Further, the term P(S,-) may be obtained by traversing the boundary S of X with a subtracting j anus plate (re: Sec. 2.8) to find H(x,n(x)), and then integrating the measured values. It is clear that this method would be independent of the directional structure of the light field within X . This 98 MODELS FOR IRRADIANCE FIELDS VOL. V fact may be used in laboratory setups to prearrange the light field so as to require a minimal amount of measuring through- out X . If this can be achieved, novel and simple means of determining the volume absorption coefficient will thereby be attained. 8. 9 Canonical Representation of Irradiance Fields We close this chapter on models for irradiance fields with a derivation which parallels the canonical representa- tion of the radiance field given in (5) of Sec. 4.5. It is possible to derive the requisite relation so as to be a proper generalization of (5) of Sec. 4.5, and we shall now follow such a course. Let X be an arbitrary optical medium. Let x be an arbitrary point of X and to x associate a direction n(x) and a set £ (x) of directions. Let us write: o H(x,S fx))" for N(x,£Kdfi(£) . (34) H o (x) This is a generalization of the irradiance vector H(x). The latter is obtained by requiring H (x) = E (cf. (2) of Sec. 2.8, and also (4) of Sec. 2.4 for the numerical in- stance of (34); and (41) of Sec. 8.6 for an alternate ver- sion of (34)). Further, let us write: "H(x,n(x),E o (x))" or »H(x,n,H o )" for n(x) • H(x, H q (x)) (35) It is clear that H(x,n,H Q ) is the quantity measured by a sub- tracting janus plate (Sec. 2.8) whose collecting surfaces are exposed to the set E (x) of directions and whose pointer is directed along n (cf. Figs. 8.11 and 2.21). Associated with H(x,n,H ) is the scalar irradiance h(x,E ), where we have written: ,, h(x,5 o ) M for N(x,£)d^U) (36) h(x,- ) is measured in practice by a spherical irradiance collector exposed to the direction set H . We pause to observe that by suitable choice of H , H(x,n,H ) can generate the usual irradiances H(x,£) and the radiances N(x,£) (see Fig. 8.11). In the former case we need only set n(x) = £ and E (x) = 5 (£) . In the latter case, we let E (x) be a variable circular conical set with central direction E, . Then: H(x,£,S (x)) N(x,U = lim 5 ^ U} jj^Sj (37) SEC. 8.9 CANONICAL REPRESENTATION 99 FIG. 8.11 The radiometric constructs, as defined opera tionally, which are used in the canonical equation for irradiance. The mathematical basis for this rests in (4) of Sec. 2.5. Next we define the distribution function associated with the set E . We write: D(x,n(x),E (x)" for h(x,H o (x)) H(x,n(xj ,H q (x)) (38) This is a generalization of the distribution functions intro duced in (5) of Sec. 8.3. For example, D(x,+) is obtained from (38) by letting k = n(x) and H (x) = E+ . We are now ready to cast the equation of transfer into the canonical form for H(x,n,H ). The derivation model we shall follow is that of (l)-(5) of Sec. 4.5. Thus, let us write: 100 MODELS FOR IRRADIANCE FIELDS VOL. V ••K(x,n(x),~ o (x))" for - H(x ^ £ } V-H(x,E o ) (39) Integrating the equation of transfer: C*VN(x,0 = - a(x)N(x,5) + J^ N(x,r)a(x;£' ; K) dfl(D over 5 Q (x), we have: V.H(x,H o ) = - a(x)h(x,» o ) + J^ N*(x,£)dflU) ~o and using the preceding definitions, we have: K(x,n,5 o ) = a(x)D(x,n,H o ) - H(x> ^ >H } f N*(x,^)dflU) (40) A final arrangement yields: I. N,(x,adfi(U D(x,n,H )H(x,n,~ ) = [«w - mm] (41) which is the desired canonical representation of H(x,n,H ). We readily verify that (41) is a proper generalization of (5) of Sec. 4.5 by recalling (37) and observing that: J_ N*(x, 5)d«CO linu o^ } fl(S ) v cr N*(x,C) lim s +{ } D(x,c,5 ) = 1 O SEC. 8.10 BIBLIOGRAPHIC NOTES 101 8.10 Bibliographic Notes for Chapter 8 It is generally agreed that the history of the two-flow irradiance equations begins with the classic paper by Schuster [279]. The differential equations derived dealt with a pair of irradiance functions representing two counter- flowing streams of radiant energy (one outward and one inward) in a stellar atmosphere. In the hands of Schwarzschild [281], King [138], and Milne [180], Schuster's approach was developed into a relatively complete description of the light field by means of the equation of transfer for radiance. With the advent of the work of Hopf [111] , the problems of radiative transfer theory took a deeper mathematical turn, and with the physical insight of Ambarzumian [1], and the industry of Chandrasekhar [43], the notions of the principles of invari- ance were conceived and exhaustively developed for the sim- plest plane-parallel settings; and radiative transfer theory as it is known today took its early definitive form. On the other hand, there followed from Schuster's work another chain of studies which dwelled almost exclusively on his original pair of equations for irradiance; reshaping them, generalizing them, occasionally rediscovering them, and apply- ing them to all manners of optical media from paint and paper, to the atmosphere and the sea. These took, for the most part, the form of the one-D model of Sec. 8.6. The industrial re- searchers and the geophysicists took alternate turns in the formulations and applications, the results being typified by papers of Schmidt [273], Benford [18], Kubelka and Munk [146], Channon, Renwick, and Storr [44], Mecke [174], Dietzius [65], Silberstein [285], Ryde and Cooper [70], and Duntley [69]. Concurrently, certain Russian authors, notably Gurevic [102], Boldyrev and Alexandrov [27], and Gershun [97], made impor- tant contributions to Schuster's theory. The latter papers are curious mixtures of the archaic forms of the equations during that period along with a few isolated brilliant inno- vations which only much later came into widespread use. In- deed, in the papers of Gurevic [102], and Schmidt [273], for example, may be found the rudimentary but recognizable germ of the idea behind equations (40) and (41) of Sec. 8.7. The fundamental Riccati equation (39) of Sec. 8.7 did not appear in its full form, but as a recognizable primitive variant, and with the physical significances of the coefficient func- tions being obscure. Stokes [291] also obtained Riccati-type equations in his early researches. The invariant imbedding formulas, of a faint but noticeable variety, can be traced back to Fresnel [94] . With the formulation of neutron diffusion problems there arose a certain amount of mutually profitable cross- fertilization of techniques between neutron diffusion and radiative transfer theories, which stems principally from the papers of Wick [319], and Chandrasekhar [42]. The rami- fications of this interaction may be traced in neutron trans- port theory in [62]. In the Wick paper and subsequently in Chandrasekhar 's work, the Schuster two- flow equations were extended to handle n- flows with particular emphasis on the form of the coefficients most suitable to numerical analysis. 102 MODELS FOR IRRADIANCE FIELDS VOL. V rather than on their physical significance (as developed for example, in (5)-(8) of Sec. 8.3, (6)-(9) of Sec. 8.3). Some relatively recent works based on or relatable to Schuster's theory are contained in the papers of Benford [18], [19], [20], Whitney [316], Hulbert [114] , Kubelka [145] , Middle- ton [178] , a report by Sliepcevitch and associates [287] , and a paper by Kottler [142]. A fairly exhaustive bibliography of the two-flow theory may be compiled from the references of the above papers. The developments of this chapter are .drawn in the main from [221], in which the two-flow equations (19) of Sec. 8.3 were first rigorously derived from the equation of transfer and with particular emphasis on the structure of the functions f(z,±), b(z,±), a(z,±), a(z,±), s(z,±). Reference [221] is also the source of the two-D model and related concepts. The invariant imbedding relation and the principles of invariance for irradiance were developed in [243], along with a Green's function construction of the R and T factors. The latter construction is simply a special case of the method of the interaction principle, and reproduces in miniature the con- structions of Chapter XIV of Ref. [251]. Sec. 8.8 is based in the main on [227]. The divergence law of the light field (15) of Sec. 8.8, along with its most general form is given in [220]. A theory of the irradiance vector H(x) , and the corresponding divergence law in vacua is developed in [187] and [188]. Gershun [98] was first to explicitly recognize the importance of the photometric counterpart to the irradi- ance vector H(x) , and Milne [180] noted the divergence law's occurrence in astrophysical optics. This same law may be found in Chandrasekhar [43] . The canonical form for irradi- ance, as given in (40) of Sec. 8.9, was developed in [223]. The theory of internal sources given in Sec. 8.5 has been considerably extended in subsequent invariant imbedding studies of linear hydrodynamics. 1-2 These new techniques (de- rived in the hydrodynamic context) are directly applicable to the Schuster two flow equations with source terms. The determination of the four transfer functions R(x,z), T(x,z), R(z,x), T(z,x) (cf. (39)-(42) of Sec. 8.7) can be made simultaneously via integration of so called Riccati quartets of differential equations, as developed recently in linear hydrodynamics. 3 1. Preisendorfer , R. W. , Forcing Long Surface Waves Through Two-Port Basins. I. Circuit Variables, NOAA- JTRE- 156 , HIG-76-11 Hawaii Institute of Geophysics, November 1976 . 2. Preisendorfer, R. W. , Forcing Long Surface Waves Through Two-Port Basins. II. Two-Flow Variables 3 NOAA- JTRE- 157 , HIG-76-12 Hawaii Institute of Geophysics, November 1976 . 3. Preisendorfer, R. W. , Multimode Long Surface Waves in Two- Port Basins, NOAA- JTRE- 125 , HIG-75-4 Hawaii Institute of Geophysics, January 1975. PART III THEORY OF OPTICAL PROPERTIES CHAPTER 9 GENERAL THEORY OF OPTICAL PROPERTIES 9 . Introduction This chapter opens Part III of the present work wherein we shall be concerned with the theoretical study of the main optical properties of natural optical media, with particular emphasis on the properties of natural hydrosols such as seas, harbors, and lakes principally under natural lighting condi- tions. Some of the optical properties to be considered have been introduced as a matter of course in the earlier develop- ments of Part II. Other properties will be defined and stud- ied here for the first time. For example, we have already encountered the volume attenuation function a (Sec. 3«H)> the volume scattering function a (Sec. 3.14), the volume absorption function a (Sec. 4.2). The functions a and o constitute the inherent local optical properties of radiative transfer theory which are fundamental in the sense that they are sufficiently complete to allow the construction, in prin- ciple, of all other optical properties of radiative transfer theory. They, however, are not unique in this property: There are other collections of inherent optical properties which are fundamental, i.e., which allow a similar construc- tion of the class of optical properties used in radiative transfer theory. It is the purpose of this chapter to define and classify the optical properties normally encountered in radiative transfer discussions, to derive and display some of the manifold interconnections existing among them, and to list some of the fundamental sets of optical properties (such as a , a mentioned above) . We shall begin in the following section with some gen- eral definitions which will help to initially classify the collection of optical properties associated with a given me- dium into four main groups: the local, global, inherent, and apparent properties. Since the settings of hydrologic optics are most naturally plane-parallel media, we shall for the most part develop and illustrate the optical properties and their interconnections in such media. Thus in Sec. 9.2 the most important apparent optical properties of hydrologic optics will be introduced and some general patterns of geometric and radiometric behavior of these properties will be derived in Sees. 9.3 to 9.5. Section 9.6 collects together for refer- ence the principal optical properties of radiative transfer theory in plane-parallel media. 105 106 GENERAL THEORY OF OPTICAL PROPERTIES VOL. V 9 . 1 Basic Definitions for Optical Properties The optical properties of a natural optical medium, such as the sea, or a lake, or a portion of the atmosphere, may be broadly grouped into two classes: those that are inherent optical properties and those that are apparent opti- cal properties. In simple terms, an inherent optical prop- erty of a medium is independent of the various possible light- ing conditions that may occur in the medium, whereas an ap- parent optical property varies with the lighting conditions but usually in such a manner that its other regular proper- ties justify its assignation of the title "optical property." More precisely, and in the terms we have used in the closing remarks of Sec. 3.10, we can make the following: Definition 1 : An optical property P of a subset S of optical medium X (P in the form of a number, function, or operator) such that P is independent of the incident radi- ance distributions on S will be called an inherent optical property of S ; otherwise, P is an apparent optical property of S . Here are some examples of inherent optical properties: the volume attenuation function a, as defined in Sec. 3.11; and the beam transmittance function T r as defined in Sec. 3.10. The independence of these properties from the shape and magnitude of ambient radiance distributions is evident at once from an inspection of their definitions. For example, in the case of T r , it is seen from (2) and (3) of Sec. 3.10 that T r is independent of the radiance distributions occur- ring along the path + ) (18) (19) which show how the experimental K- functions are linked to the absorption and scattering properties of the medium. From these, we immediately obtain the exact counterparts to (7) and (8) : (20) R(z,-) = b(z,-) a(z, + ) + b(z,+) + K(z, + ) (21) While the overall resemblance to (7) and (8) is quite evident , care should be taken to distinguish the relatively subtle roles now played by the functions associated with each flow. The counterparts to (9) and (10) are obtained by making use of the general identity ((18) of Sec. 8.3): a(z,±) = a(z,±) + s(z,±) = a (z , ±) + f (z , ±) + b (z , ±) Thus, in general: R(z,-) = a(z,-) f(z,-) - K(z,-) b(z,+) (22) R(z,-) = b(z,-) a(z,+) - f (z, + ) + K(z, + ) (23) 118 GENERAL THEORY OF OPTICAL PROPERTIES VOL.' V The Basic Reflectance Relation The general counterpart to (6) is singled out for special attention because it is the most useful representa- tion of relfectance functions in practice. It relates the six direct observables: the two K-functions, the two dis- tribution functions, the volume absorption function, and the reflectance function. It therefore may replace (6) by pro- viding an exact formula to check the consistency of the experimentally determined values of these functions. Further- more, from a theoretical point of view, the general counter- part to (6) is closely related to the divergence relation for the light field (re: (18) of Sec. 9.8) in fact directly de- rivable from it, as shown below. Alternate derivations, of course, can be made directly from the system (14). Now, for the derivation at hand, the divergence relation for the light field in stratified media may be written in the form: ^^i= a(z)h(z) (24) where H(z, + ) = H(z,+) - H(z,-), and where h(z) is the scalar irradiance at depth z , and a(z) is the value of the volume absorption function at depth z . Rewriting this with the help of (9) of Sec. 2.7, as: dHU^i. M^i. a(z)[h(z, + ) + hCz,-)] , and dividing each side by, say H(z,-), we have: H(z,+) 1 dH(z,+) 1 dH(z,+) H(z,-J H(z,+) dz _ dH(z,-) _ H(z,-) dz h(z,+) H(z,+) h(z,-) H(z,+) H(z,-) H(z,-) = a(z) Applying the appropriate definitions, this may be rewritten: - R(z,-)K(z,-) + K(z,-) = a(z)[D(z,+)R(z,-) + D(z,-)] = a(z,+)R(z,-) + a(z,-) Solving for R(z, -) : Rf 7 ^ - K(z,-) - a(z,-) R(z ' } " K(z,+) + a(z,+) (25) which is the desired exact experimental counterpart to (6) SEC. 9.2 OBSERVABLE QUANTITIES FOR LIGHT FIELDS 119 The Exact Inequalities To obtain the exact counterparts to the classical inequalities (11), we encounter a reversal of difficulty: the counterpart to k < a is relatively simple to estab- lish, and its validity is completely general; the counter- part to a* < k requires additional assumptions, but of a kind which are a consequence of the generality of the present formulations rather than their shortcomings. The first member of (14) may be written as: dH(z,-) 3i a(z,-)H(z,-) N*(z,Odfi(0 so that K(z,-) = a(z,-) - 1 H(z,-) N*(z,S)dn(0 This representation may be obtained by using the definition of K(z,-), recalling (18) of Sec. 8.3, and the derivation leading to (9) of Sec. 8.3. Since the subtracted member of the right side is never negative, we have immediately: K(z,-) < a(z,-) for all z . Finally, whenever £ K(z,+), we have (see note below) from (25) : a(z,-) < K(z,-) which establishes the desired inequalities: a(z,-) < K(z,-) < a(z,-) (26) or equivalently : and alternately ^ inff^Ti ^ ijyff^} - a < z) ± s < z > A corresponding set of inequalities for the upwelling stream may be obtained in a similar way: a(z,+) < - K(z,+) < a(z,+) (27) 120 GENERAL THEORY OF OPTICAL PROPERTIES VOL. V or equivalently : and alternately a ^ 1 " §fft^} - aCz) < - n ( r Z ' + A - a(z) < s(z) D(z,+) The right-hand side of (27), as that of (26), holds in gen- eral; the left side of (27) holds whenever K(z,-) £ , a condition completely symmetric to the condition, <_ K(z,+) used to establish the left side of (26) . The Significance of the Condition <_ K(z, + ) The significance of the condition, <_ K(z, + ), used to establish the left side of (26), is quite important; and the adoption of this condition raises some interesting questions. First of all, we observe that if the condition holds, then the denominator of (25) is positive. Since R(z,-) is posi- tive, this then requires the numerator of (25) to be positive, from which the desired inequality follows. But one may ask: is this condition ever violated? In other words, can we ever have: K(z,+) < 0? Before answering this, we recall that K(z,+) < means that K(z,+) is negative, and physi- cally this means that the function H(»,+) is increasing with increasing depth at z . A similar interpretation exists for the condition K(z,-) < . The preceding question may then be put very concretely as follows: As one measures up- welling and downwelling irradiances in a real stratified op- tical medium, is it ever possible to observe an increase in these irradiances as depth is increased? The answer is yes. There are two possible mechanisms which generally allow such a phenomenon to be observed. The first mechanism is that associated with self-lumin- ous sources within the medium. For example, light-giving organisms distributed in a horizontal layer of water clearly make it possible for increases of irradiance to be observed as the irradiance collectors approach the layer and pass downward through the layer. These layers can occur at quite large depths. Other examples are given by various physical emission processes, fluorescence, i.e., e.g., scattering with change in wavelength, et cetera. The latter mechanisms, as noted several times, ordinarily play a subordinate role in many natural hydrosols,* but in the atmosphere, they can be important. If the emission terms h r) (z,±) are included in the two- flow equations, this phenomenon can be represented *When biological processes in natural hydrosols are of interest in hydrologic optics studies, fluorescence can play import roles in the associated radiative transfer process. In this case transpectral scattering theory (cf. Sec. 19 of [251] is the appropriate theory to use. SEC. 9.2 OBSERVABLE QUANTITIES FOR LIGHT FIELDS 121 explicitly, and a suitable parallel theory can be built up around such a phenomenon. The second mechanism is that of simple scattering processes: the redirection of radiant flux without change in wavelength. This results in an effective storage of radiant energy within scattering media. It seems plausible that if an increase of irradiance is induced by this mechan- ism in natural water, the increase should noticeably occur at depths near the surface of the medium. For in these re- gions of small depth the diffuse light field is still build- ing up in magnitude, and just the right kind of inhomogene- ities may possibly contribute to the effect. Generally, in optically deep scattering media, the downwelling diffuse radiant flux is zero at the surface, increases with increas- ing depth, reaches a maximum at some depth, and then falls off in a more or less exponential way ever afterward. But the diffuse irradiance H*(z,-) is not directly observed. Superimposed on it is the reduced irradiance H°(z,-), which clearly must decrease continuously with depth, starting right from the upper surface. Thus, whether or not the directly observable irradiance H(z,±) = H°(z,±) + H*(z,±) exhibits any increase with increasing depth clearly depends upon the magnitudes and relative rates of change of each of its components. A theoretical discussion of the conditions which govern the growth of the light field in stratified media is out of place here. We merely note in passing that many such condi- tions can be extracted from expressions like (14), (18), or (19) given above; furthermore, various approximate models of the light field such as the two-D theory discussed in Chap- ter 8 give very explicit, if only approximate, criteria for the growth of the light field. Some of these possibilities will be explored in Chapter 10. Finally, the possibility of the growth of radiance values has been predicted by a simple model for radiance distributions in natural hydrosols. This prediction has been verified by experiment. The model is based on the canonical form of the equation of transfer; see in particular (12) of Sec. 4.5. Relative Magnitudes of H and K Functions For source-free stratified media, we can make several general observations about the relative magnitudes of the observable H- values and K- values. These have been of help in checking and collating experimental data. First of all, from the integrated divergence relation (24) , we deduce that: z H(z 2 ,+) - H(z i>+ ) = j ~ a(z)h(z)dz > z i so that: 122 GENERAL THEORY OF OPTICAL PROPERTIES VOL. V (28) which demonstrates that the net downward irradiance function H(",-) ( = H( - ,-) -H(-,+)) never increases with depth. Here z 1 and z 2 are any two depths, z 2 being the greater. If the medium is, in particular, finitely deep with a bottom surface whose reflectance r is such that <_ r <_ 1 , (re: discussion of solutions s (6) and (7) of Sec. 3.1) then (28) immediately implies that, for all depths z , R(Z > } " H(z,-) - 1 (29) If the medium is optically infinitely deep, we have a simi lar result. For in this case, we have for all depths z : H(z,-) = a(z')h(z')dz ? > ' z so that H(z,+) < H(z,-) from which (29) follows once more. We can derive a correspondingly general inequality that must hold between K(z,-) and K(z,+). From (25): K(z,-) - K(z,+)R(z,-) = a(z,-) + a(z,+)R(z,-) > ; whence : or equivalently : K(z,+)R(z,-) < K(z,-) dH(z, -) dH(z, + ) dz dz (30) This relation throws some light on the question raised above. Relation (30) shows that if K(z,-) is negative, then necessarily K(z,+) is negative, too. Conversely, if K(z,+) is positive, then so must K(z,-) be positive. Finally, (30) hints at real situations in which K(z,+) may well be nega- tive while K(z,-) is positive. Ideal examples of each of these three situations are easily found; however, occurrences in real media have not yet been sought. SEC. 9.2 OBSERVABLE QUANTITIES FOR LIGHT FIELDS 123 Characteristic Equation for K(z,±) The classical two-flow theory gives a convenient ex- pression for k in terms of absorption and scattering coef- ficients as in (2). There is a remarkable corresponding formula which characterizes K(z,-) and K(z,+) in addition to (18) and (19). This exact counterpart to (2) is obtained by eliminating R(z,-) from (20) and (21). The result is: 1 = b(z,-) K(z,-) - a(z,-) b(z, + ) K(z,+) + a(z,+) (31) That this is the general counterpart to (54) of Sec. 8.5 may be verified for example by setting, as such a verification requires, b(z,-) = b(z,+) = b*, a(z,-) = a(z,+) = a* , and K(z,-) = K(z,+) = k . When this is done, (31) reduces to (2). The Depth Rate of Change of R(z,-) Since the experimental counterpart to Roo generally varies with depth, it is of interest to characterize the variation in terms of the experimental K- functions. The desired formula follows immediately from the definition (16) of R(z,-): (32) From this we see that the constancy of R(*,-) is equivalent to the equality of K(»,-) and K(»,+). In other words, R(»,-) is constant over any interval (z 1 ,z 2 ) of depths when and only when K(z,-) = K(z,+) for every depth z in the interval (z 1 , z 2 ) . Connections Among the K Functions In this paragraph we will briefly discuss the connec- tion between the k of the classical theory (as in (2)) and the exact K-functions introduced in (5) above. First of all we observe the simple connection that exists between the scalar irradiances h(z,±) and the irradiances H(z,±) with- in the framework of the classical theory. Recall that both the diffuse and reduced radiance distributions in both the upper and lower hemispheres are assumed uniform; therefore, for all depths z , Dfz +^ = h (z,±) = ? UlZ '- J H(z,±) Z 124 GENERAL THEORY OF OPTICAL PROPERTIES VOL. V This means that h(z,±) and H(z,±) differ multiplicatively only by a fixed factor 2. Thus the operational definition (5) for k lets us conclude: k = - 1 dH(z,±) H(z,±) dz 1 h(z _ dh(z,±) ±) dz In o esti ance (12) nece prac cuss loga that Thus ther words, the classica mated equally well from s or ordinary irradiance and (13)) distinctions ssary not only in theory tice. Consequently, whe the growth or decay of, rithmic derivative is ge of H(« ,-) . A similar we are led to consider 1 theory says that k may be measurements of scalar irradi- s. As demonstrated above (see between h and H are often but in careful experimental n it becomes necessary to dis- say, h(*,-) with depth, its nerally considered distinct fron statement is true for h(«,+). operations of the kind: 1 dh(z,±) h(z,±) dz and to distinguish these from the operations 1 H(z,±) we write: "k( Z ,±r for dH(z,±) dz 1 dh(z,±) h(z,±) dz (33) As a mnemonic, we observe tnat in the exact theory for real media, the little k's go with little h's and big K's go with big H's. In real media the connection between these is (34) ablish: of ,±) is uation lassi- 0,±). nde- epth ), dia , ry to (2). e of e dis- We may now state the connection we set out to est k(*,±) and K(*,±) are equal over some interval (z 1 ,z 2 ) depths when and only when the distributon function D(* constant over that interval. This is precisely the sit that holds for optically infinitely deep media in the c cal theory so that we have in that setting: k(*,±) = K Furthermore, in this case, R(*,-) = R^ , and thus is i pendent of depth. From (32) we may then conclude that K(*,-) = K(»,+). The net conclusion is that for each d z , the four quantities: k(z,+), k(z,-), K(z,+), K(z,- which are generally four distinct quantities in real me are constrained in the one-D model of the two-flow theo be identical, their common value being k , as given by In this way we justify the generally interchangeable us k and K in any discussion which has the constancy of th tribution functions in the background. SEC. 9.2 OBSERVABLE QUANTITIES FOR LIGHT FIELDS 125 K-Function for Radiance The K- functions discussed throughout this section are associated with an exact formulation of the two- flow analysis of the light field. They are the little k's for the little h's, and the big K's for the big H's. In more detailed ex- perimental studies of the light field, namely those that document the radiance distribution values N(z,9,), a corresponding K-function has been found extremely useful in theoretical work (re: (20) of Sec. 4.5 and Sees. 10.5 and 10.6) and in graphical and tabular representations of these distributions. It is de- fined by writing: »K(z,e f <10 (35) *■ ' yrj N(z ,0 ,) dz K J No confusion should arise from the continued use of the letter "K": (35) will always explicitly exhibit three variables or places for them when clarity is threatened, the other K's only one. We note in passing that this function, analogously to the other K-functions discussed above, has several interesting theoretical consequences in addition to its immediate experimental uses. However, a discussion of these matters is deferred until Chapter 10. General K Functions To round out the discussion of the experimental K- functions, we note that all of the K-functions defined above fall into a specific class, each member of which is defined by an operation of the kind: 1 dA " Adz where A could be any of the functions: H(*,±), h(*,±), N( # ,9,(J>). Some further possibilities for A are, h( # ), H(*,+). Furthermore we observe, by means of the divergence relation (21) or more generally by (15) of Sec. 8.8, that the basic volume absorption function may be defined as the result of the operation, 1 dH h dz on the two types of irradiances shown. Finally, the K-func- tion (1) of Sec. 4.5 should be noted. On the basis of these examples, it appears that the most general notion of an ex- perimental attentuation function (i.e., a general K-function) is definable by an operation of the kind, 1 dB VB f ~,, Adl > 0r X (36) on any two observable radiometric quantities A ar»d B. 126 GENERAL THEORY OF OPTICAL PROPERTIES VOL. V Integral Representations of the K Functions The K-function for radiance is basic in the same way that radiance itself is basic; that is, as a fountainhead of representations of the various radiometric concepts. Thus, it is easily shown that: /. N(z,5)K(z,5)5 • nd^U) (37) N(z,€K * nd^U) 5 N(z,OK(z,€)dfl(§) k(z,±) = -I£- (38) i N(z,OK(z,Od0(O — (39) N(z,Odn(0 where k(z) is the negative logarithmic depth derivative of scalar irradiance h(z). Equations (37) through (39) are indicative of the type of integral representations of the various K-functions defined in (36). Integral Representations of the Irradiance and Radiance Fields It follows at once from the definitions of the various K-functions introduced above that the directly observable irradiances H(z,±) can be given the following integral re- presentations : H(z,±) = H(x,±) exp f - | K(y,±) dy j (40) where x,y,z are three depths in stratified plane-parallel media X(a,b) such that a^x^y^z^b. Similarly: N(z,Q = N(x,0 exp ( - J" K(y,U dy j (41) SEC. 9.2 OBSERVABLE QUANTITITES FOR LIGHT FIELDS 127 Since K(z,±) and K(z,£) are thus observed to play the gen- eral roles of absorption functions analogously to a and k , we can alternately refer to them as absorption functions for H or N , as the case may be. (See (29) of Sec. 9.3.) As an example of the use of (40) , let us determine the K-function belonging to a spherically symmetric light field about a point source, imbedded in a natural or laboratory hydrosol. The two- flow equations governing radiative trans- fer across spherical surfaces of radius r and concentric with the source are governed by (46) of Sec. 8.6 in which now V • H(z,±) takes the form: V • H(z,±) = -L j- (r 2 H(r,±)) where H(r,+) is the centripetal flux and H(r,-) is the centrifugal flux at radius r . Hence for a steady spheri- cally symmetric light field, the negative logarithmic de- rivative of H(r,-) with respect to r is: K(r,-) = a(r,-) + b(r,-) - b(r,+)R(r,-) + | . (42) A similar formula holds for the centripetal flux by suitably changing signs in the arguments (cf. (19)). If r and s are any two radii with r < s , then (40) and (42) yield the formula: . fr] z T a (r,s)T b (r,s) (43) H(r,-) where we have written: {•i: 'T a (r,s) u for exp { I a(u)D(u,-)du "T b (r,s)" for exp f- J* [b (u, -) - b (u, + ) R(u, - ) ] du Here T a and T^ are special transmittances built up from absorption and backscatter coefficients, respectively. For media with relatively small b(u,±) it follows that T^ = 1 and so a practical rule of thumb for K in spherically sym- metric fields is: K(r,-) - a(r,-) + f = a(r)D(r,-) + \ and (43) becomes 128 GENERAL THEORY OF OPTICAL PROPERTIES VOL. V I |] T (: = H(r,-) exp I - a(u)D(u,-)du > A simple, but approximate, operational procedure for measur- ing volume absorption function in spherically symmetric fields is forthcoming from this. Since £n f r*H(r,-) \ = f S a(u)D(u,-)du , I s 2 H(s,-) J J r on holding r fixed and varying only s : a(s) - r^-T ■ A *n ( ^^) (44) D(s,-J ds Vs 2 H(s,-)J where D(s,-) is the value of the distribution function D(«,-) at radial distance s from the source, and a(s) the required value of the volume absorption function at the same radial distance s . This method of determining the volume absorption function supplements those discussed in Sec. 13.8. Finally, the preceding example shows how, with only minor modifications, all the exact two-flow theory formulas for stratified plane-parallel fields can be used to obtain their correspondents in stratified spherical f ields- -i . e. , spheri- cally symmetric fields of irradiance. 9. 3 The Covariation of the K Function for Irradiance and Distribution Functions The purpose of this section is to establish the theorem that at arbitrary fixed depths z the attenuation function value K(z,-) and the distribution function value D(z,-) . vary directly (but not necessarily linearly) one with the other, in all steady state stratified real plane-parallel media whose volume scattering functions are predominantly forward scattering. In this way we establish a useful cri- terion for the behavior of K(z,-) in terms of the intuitive- ly simpler concept D(z,-). The theorem is expected to find its greatest use in natural hydrosols. By way of background to these results we now discuss in some detail the physical significance of K(z,-) and D(z,-). Some Elementary Physical and Geometrical Features of K(z,-) and D(z,-) It is a well-known fact in hydrologic optics that the amount of light in a natural hydrosol such as an ocean or deep lake decreases essentially in an exponential manner with depth from the surface of the hydrosol. The simplest SEC. 9.3 COVARIATION OF K-FUNCTIONS 129 models of the light field exhibit this fact (cf. (22) of Sec. 8.6). This fact may be expressed succinctly as: H = H e" kz (1) z o v J where H z is the amount of radiant flux of a given wavelength falling downward on a unit horizontal area (i.e., the irradi- ance) at depth z , and K is a constant determined by measur ing the slope of a semilog plot of measured values of H z versus depth z . The quantity K has dimensions: per unit length, and is usually called the attentuation coefficient for irradiance, for the medium under study. Like H z the quantity K depends implicitly on a specific wavelength A of the radiant energy penetrating the hydrosol. For many engineering questions, questions of underwater photography, television and visibility, and for many purposes of marine biologist, the story of H z and K may appropriately end with equation (1). 'However, for the more demanding purposes of geophysicists charged with the tasks of determining the fundamental optical constants of natural hydrosols, and for those who must make sense out of the experimental data lead- ing to the numerical determination of the fundamental con- stants, the story of H z and K has only begun to be told with equation (1). Confronting these latter investigators in their quest for precisely measured radiometric quantities which must be interrelated by consistent rules of calculation, is a wealth of intricate and nonlinear detail in the depth behavior of H z . The simple exponential behavior of H z as summarized in equation (1) must now as a matter of experimental expedi- ency be discarded and in its stead be made to appear a more detailed formula exhibiting the same general outlines of (1) , but containing now all the potential variations which may be uncovered in a careful documentation of the light field in real hydrosols. As we saw in (40) of Sec. 9.2, this more detailed formula can tc.ke the form: H(z,-) = H(0,-) exp f - j K(z',-)dz' j (2) Equation (2) is the physicists' generalization of the mathematician's simple model of the light field expressed in equation (1). Let us examine (2) in detail and thereby un- cover its similarities and dissimiliarities with (1). First, H(z,-) represents the measured irradiance at depth z >_ in the medium produced by downwelling radiant flux on a unit horizontal area. Hence H(0,-) is the downwelling irradiance of such flux on such a surface at depth z = measured just below the air-water film. Second, suppose we plot the general equation (2) on semilog paper with depth z as abscissa and H(z,-) as ordinate. Equation (2) gives the value of H(z,-) at a general depth z : Hence equation (2) may then also be used to give the value of H(z,+Az,-), i.e., the downwelling irradiance at a depth z+Az, where Az is any finite posi- tive increment in depth. The appropriate formula for this is: 130 GENERAL THEORY OF OPTICAL PROPERTIES VOL. V Az H(z+Az,-) = H(0,-) exp { - K(z ! ,-)dz' (-1: Now, as in elementary calculus, we may approximate the slope of the semilog plot of H(z,-) at depth z by letting z grow by an increment Az , by finding the new value H(z + Az,-) and then by performing the operation: £nH(z + Az,-) - &nH(z,-) s ("slope of £nH(z,-)l f . Az L at depth z J ^ } The smaller the magnitude of Az , the more accurate is the estimate of the slope of the curve by this operation. Let us now perform the operation on the values of H(z,-) and H(z+Az,-), as given by (1) and (2). From (1): £nH(z,-) = &nH(0,-) - K(z',-)dz' (4) ■' o and from (2) : rZ + AZ £nH(z+Az,-) = J>nH(0,-) - I K(z',-)dz' . (5) ' o Inserting these values in (3) we have: rZ rz + Az K(z' ,-)dz' - K(z* ,-)dz' Islope of £nH(z,-)| _ }_o ' o I at depth z J Az (6) We now recall two elementary facts from integral calculus the first is: ra + A ra ra + A f dz = I f dz + I d dz A ra f a + A (7) a which shows how the range (0,a + A) of integration may be broken into two parts: (0,a) and (a,a + A) for any function integrable over (0,a + A); and the second fact that: a + A f dz - A f(a) , (8) a whenever A is small and f is continuous at a . Applying these two facts to (6), where K(*,-) now takes the place of f in (7) and (8), we have, on application of (7) to (6): SEC. 9.3 COVARIATION OF K-FUNCTIONS 131 slope of £nH(z,-) at depth z z + Az K(z' ,-)dz' (9) Az Then, applying fact (8) to (9) and letting Az ^ we have: ■ K(z,-) . (10) slope of £nH(z,-) at depth z Equation (10) tells us that K(z,-) is the negative of the slope of the semilog plot of H(z,-) versus depth z . Another way of obtaining (10) is to directly differen- tiate (2), the result being: dH(z,-) dz and then solve for K(z,-) K(z,-) = - K(z,-)H(z,-) _ dH(z,-) ) dz H(z d £nH(z,-) dz (11) This latter method is more elegant than the preceding pedes- trian method, and thereby brings out more succinctly the geometric meaning of K(z,-). We may perform the same operation on H z as given in equation (1). Hence, either by the z - method or the simple derivative scheme shown above, (1) yields: dH (12) K = slope of £n(h z ) at depth z 1 II dz Thus both K and K(z,-) are the negative slopes of the semi- log plots of H z versus depth z . In this way they are similar. But the point where they differ is in the fact that K is independent of depth and that K(»,-) is not indepen- dent of depth. And in this difference lies precisely the difference between mathematical fiction and physical reality. Is there a simple explanation for this gap in terms of the accumulated concepts of hydrologic optics? We now consider in detail an explanation of this difference in terms of the currently accepted concepts of radiative transfer theory, as applied to hydrologic optics. A careful examination of the experimental evidence leading to H z determinations shows that there are actually two mechanisms in source-free media which may give rise to the gap between the simple classical K and the modern K(z,-). We have up to this point slanted the discussion to bring out only one of these mechanisms, the one which we may term the physical (or dynamical) mechanism of the variation of K(*,-). Thus, in the preceding discussion we centered 132 GENERAL THEORY OF OPTICAL PROPERTIES VOL. V attention on the depth behavior of H(z,-), a behavior which is discerned only as an irradiance probe moves downward into a natural hydrosol and continuously records the magnitude of H(z,-) as z continuously increases. Now if the external lighting conditions on the upper boundary of the source-free medium are arbitrary but fixed in time, that is stationary, and if the hydrosol' s fundamental inherent optical properties (namely the volume attenuation a and volume scattering func- tion a ) are arbitrary but stationary, then the depth depen- dence of K(»,-) is an indicator of the natural interaction of the penetrating photons with the material of the medium; it provides a running account of the depth-rate of attenua- tion of the number of the downwelling photons streaming past the level z . This dynamical aspect of the variations of K(*,-) will be studied in an elementary but detailed manner in Sees. 10.3 and 10.4. For our present purposes we may thus consider the physical (or dynamical) mechanism behind the depth behavior of K(»,-) to be fairly well understood. There remains, however, the problem of the geometric mechan- ism which also gives rise to variations of !((•,-) to which we now turn, and which forms the central problem under study in the present section. If, instead of continuously moving the irradiance probe vertically downward in a natural hydrosol, we hold the depth z fixed and let H(z,-) be recorded as a function of time, we would expect in general, because of the continuous ly chang- ing external lighting conditions above any natural hydrosol, a time dependence of H(z,-) for the fixed depth z . For example, if the probe is set at a depth of five meters at 0600 hours local time in a certain lake, we would expect H(z,-) generally to increase as the sun rises to reach a maximum around 1200 hours, and then to descend at 1800 to a reading comparable to the 0600 reading. On this basically regular diurnal variation of H(z,-) there is superimposed a relatively more rapid variation in H(z,-), induced for ex- ample by the movement of clouds or cloud layers between the sun and the hydrosol' s surface. Even though the time rates of these latter changes in the external lighting conditions are thousands of times greater than those associated with the more stately diurnal changes, they nevertheless are far too small to cause any true transients in the natural light field. For this reason all these changes in the light field are of a quasi-stationary character. Now, suppose a probe were to be sent quickly downward to accurately record the stationary light field all the while the sun is covered by a cloud and then, when the cloud has just passed away from the sun and allows it to shine with full strength on the hydrosol, another quick but pre- cise probe uncovers the irradiance's depth profile during this sunny condition. If this were done then we would dis- cern upon careful examination of the two semilog plots of H(z,-) that H(z,-) at each given depth on the overcast plot would differ from the value at that same depth on the sunny plot. Actually, experimenters need not go out of their way to uncover this phenomenon; it crops up with exasperating inevitability in any painstaking (and thus time-consuming) SEC. 9.3 COVARIATION OF K-FUNCTIONS 133 mapping of the irradiance depth-profile in natural hydrosols. For in a particularly deep hydrosol the determination of the vertical depth-profile may take on the order of a half hour; and when the probe ascends past one of the relatively shal- lower depths in a check re-run, the sun may have moved as much as five to ten degrees, cloud covers may have changed, thus producing (with most modern equipment) easily measured changes in the structure of the light field in the interim. When plots of H(z,-) are made of each of these runs, the logarithmic slope K(z,-) on each plot may be noticeably nonconstant with depth; furthermore, and this is the crux of the matter at hand, the K(z,-) value at a fixed depth z for the downward run may differ from that for the upward run. The experimenter would, at this juncture, if he considers this difference in K(z,-) from one plot to another with care, soon realize that the difference may be the result of a super- position of two basically different physical mechanisms: On the one hand there is of course the possibility that the inherent optical properties of the medium may have changed during the interval between the times that the probe has visited the given depth z ; on the other hand there is the possibility that the external lighting conditions have changed during this period and that this change has in some way become manifest in the difference in the K-values for the given depth z . If the investigator had made provisions to record dur- ing the same period, and over the same depth interval, the radiance distributions within the medium and the inherent optical properties of the medium, then he may be able to quantitatively, at least in principle, ascertain, by means of the representations (18) and (19) of Sec. 9.2 and the knownstructures of a(z,±), b(z,±), those parts of the dif- ferences in the K(z,-) values which are traceable to the changes in the inherent optical properties. Thus, once again we are in possession of sufficient knowledge to under- stand and cope with the physical aspects of the behavior of K(z,-). There remains, however, that component of the change of K(z,-) at a given depth z which is traceable to the change of the external lighting conditions. In order to relate the way in which K(z,-) changes with external lighting conditions we must have some means of specifying in a precise manner the concept of "lighting con- dition." Clearly, in choosing a precise characterization of this concept the absolute amount of the incident radiant flux is of no essential importance. Of critical importance, how- ever, is the relative amounts of radiant flux which arrive on the upper boundary of the medium, or on some internal hori- zontal plane, from the infinite number of possible directions in the hemisphere of incidence. One obvious, and incidental- ly the most complete, characterization would be by means of the radiance distribution N(z,*) at depth z . While this means may be of considerable use in other contexts x it re- quires of the experiments a prodigious auxiliary effort to provide the necessary measuring and recording apparatus to obtain this large number of readings. In the interests of experimental expediency, what is needed is a characterization of the relative values of N(z,») without having to measure 134 GENERAL THEORY OF OPTICAL PROPERTIES VOL. V each of the infinitude of values N(z,£)» U= where 5 is the set of all unit vectors in euclidean space. Since relative values are of primary interest, the ideal charac- terization would then involve not more than two readings of some simple kind. In searching for two simple radiometric measurements which would be capable of characterizing the relative values of the downwelling radiance distribution, it would be of some convenience to the experimenter if he could make use of his existing data, namely the values of H(z,-). To see what possible choices remain if H(z,-) is adoped as one of the two radiometric measurements which will characterize the relative magnitudes of N(z,»)> l et us express H(z,-) in terms of these values. By definition, we have: H(z,-) = N(z,5K ' ndfl(0 where n is now the unit inward normal to the hydrosol. In practice, N(z, # ) is usually determined by a suitably chosen finite set { £., , ..., £ n } of downward directions, and H(z,-) is then computed by the rule (re: (6) through (17) of Sec. 2.5): n l 1=1 H(z,-) = I N(z,5 i )C i • nA^ i where Afi^ is the solid angle associated with the direc- tion £i . The quantities A^i are subject to no specific restrictions except that they be small enough so that N(z,») is fairly uniform over the associated direction sets and that of course E A ft^ = 2tt . Now the relative magnitudes of the quantitites N(z,£i), i = 1, ..., n may be obtained by choosing any one of them, say N(z,5i)» and forming the quotients N(z ,£^)/N(z, £ ) , which we shall denote by: "g( z >£i) n - Then the quotient: N(z,£ ) A ft 1 1 H(z,-) I g(z,? i )C i • nAU i would serve as a measure of the way in which the g(z,£j_) are distributed over the downwelling hemisphere H_ . How- ever, such a measure falls short of being satisfactory for several reasons: First, we have isolated a particular value N(z,£i), and therefore have distinguished it with artificial importance; actually any one of the n - 1 other values would serve just as well. Secondly, in order to measure N(z,£ ) we would require the services of a specially designed radi- ance meter, or bring into use for extremely restricted pur- poses the actual radiance distribution measuring apparatus- - which might as well then be used to determine a working sam- ple of N(z,»)- Finally, we would prefer to measure an amount of flux comparable in magnitude to H(z,-); for N(z,£.,) would generally be a far smaller number than H(z,-) SEC. 9.3 COVARIATION OF K-FUNCTIONS 135 which must then be divided into or divided by N(z,£ t ), there- by setting the stage for the disruption of numerical accuracy in the data reduction tasks that follow. The cumulative effect of these observations is to lead to the choice of the sum n l 1=1 h(z,-) = I N(z,? i )Afl i as the most logical choice for the second radiometric mea- surement. Its integral representation is: ->-i. h(z,-) = I N(z,Odfl(5) In this way we are led to consider the ratio which we have encountered before ((15) of Sec. 8.3 and (1) through (7) of Sec. 8.5) and which we have termed the dis- tribution function. In terms of the finite-summation repre- sentation of h(z,-) and H(z,-), D(z,-) becomes: n J N(z,? i )Afi i D(z,-) = ijp (14) J N(z,5 i )C i • nA^ i or, in terms of their integral representations: I N(z,£)d Q D(z,-) = -I: . (15) N(z,£K ' n d n . The quantity h(z,-) may be measured by simple devices, (see [305] and Chapter 13 below). We have seen in Chapter 8 how D(z,-) serves to characterize the distribution of the irradi ating flux. Thus, let n = 1 in (14), i.e., let the flux come from any single solid angle A Q in the general direc- tion £ ; the distribution function then is: D(z,-) = -^ . (16) Hence if the irradiation is incident vertically at depth z , £ = n and D(z,-) = 1 . In general, the more obliquely in- cident the pencil of radiation^ the smaller the dot product E, • n , and the larger is the distribution function. This 136 GENERAL THEORY OF OPTICAL PROPERTIES VOL. V conclusion, although obtained for a special case, is never- theless valid for any number of pencils. In what follows we will show in what manner the con- cepts K(z,-) and D(z,-) are qualitatively interdependent. This will be done by noting the behavior of the intuitively more simple and predictable quantity D(z,-) which, as we have repeatedly seen, serves as a convenient characterization of the lighting conditions. In this way we may gain insight into the dependence of the more complex quantity K(z,-) on given lighting conditions. The General Law Governing K(z,-) and D(z,-) We now have sufficient background established so as to derive with some clarity the general law which governs the exact interrelation of K(z,-) and D(z,-). We start with the canonical representation of K(z,-) derived from the equation of transfer in (40) of Sec. 8.9. This representa- tion- of K(z,-) now takes the form: 1. N*(z,Odft K(z,-) = a(z,-) - — - . (17) H(z,-) To obtain (17) from (40) of Sec. 8.9, we set E Q = »_ and set n = ~ k (see Fig. 8.11). An alternate derivation may be obtained by following the steps leading to (26) of Sec. 9.2. To keep the present discussion self contained, we ob- serve that the various terms in (17) are defined as follows: (i) a(z,-) = a(z)D(z,-), where, of course, a(z) is the value of the volume attenuation function at depth z . (ii) N*(z,£) is the value of the path function at depth z for the direction E, . Its analytical representation is: N*(z,€) = N(z,?)a(z;5 , ;5)dn(5 l ) , (18) where a(z;£';£) is the value of the volume scattering function at depth z for the inci- dent £' and the scattered direction £. In order to extract from (17) the desired explicit connection between K(z,-) and D(z,-), we must reformulate the second term of (17) in such a way as to cause D(z,-) to appear explicitly in that term. Toward this end we first recall that D(z,-) is be definition the ratio of h(z,-) to H(z,-). Now the appearance of the integral in the second term of (17) has all the external earmarks of an h(z,-) type quantity; this is suggested by observing that N*(z,») is a radiance function and it is integrated, like N(z,*)> over SEC. 9.3 COVARIATION OF K-FUNCTIONS 137 analogy with h(z,-), we write: "h*(z,-)" for f N*(z,£)dfi(0 (19) Physically, h*(z,-) is the downwelling scalar irradi- ance generated by the radiant flux scattered in a unit volume at depth z . The analogy need not end with (19) ; in fact we can extend it quite naturally to include the * -counterparts to h(z,+), and to scalar irradiance h(z) itself. Thus in analogy to h(z,-) we write (as in (11) of Sec. 2.7): 'h(z, + ) M for N(z,£)dfi(£) . (20) which is the upwelling scalar irradiance at depth z (and which forms the basis for defining D(z, + ) = h (z ,+) /H (z ,+) for the upwelling stream); we further write: "h*(z,+)" for J_ N*(z,€)dft (21) which represents the upwelling scalar irradiance generated by the radiant flux scattered in a unit volume at depth z . Finally, the scalar irradiance h(z) at depth z , being de- fined by writing (as in (3) of Sec. 2.7): "h(z) M for [ N(z,£Ddfl , (22) has its scattered analogy in the form h*(z) where we have written: r or N Thus corresponding to: h(z) = h(z,-) + h(z,+) , (24) which is based on (9) of Sec. 2.7, we have: h*(z) = h*(z,-) + h*(z,+) . (25) An extremely useful and unexpectedly simple relation sub- sists between h(z) and its scattered counterpart h*(z), namely that the ratio of h :% (z) to h(z) is precisely the value of the volume total scattering function s : h*(z) s(z) - ^^ . (26) 13; GENERAL THEORY OF OPTICAL PROPERTIES VOL. V The derivation of this relation along with some sugges- tions for its use in practical direct determinations of the values of s are given in (7) of Sec. 13.7. We have now assembled all the required concepts needed for a complete formulation and discussion of the general law governing K(z,-) and D(z,-). Starting with (17) and using definition (19) we may write: K(z,-) = a(z,-) - h*(z,-) H(z,-) (27) By means of the definition of D(z,-) this may be written: K(z,-) = a(z,-) h*(z,-) h(z,-) D(z,-) and, finally, using the definition of a(z,-), we may recast this into the presently desired form of the law governing K(z,-) and D(z,-) : K(z,-) = [a(z) h*(z,-) h(z,-) ] D(z,-) (28) The Absorption-Like Character of K(z,-) Before presenting the general proof of the covariation of K(z,-) and D(z,-) which will be based on some observa- tions of the structure of (28) , we pause to discuss the gen- eral radiative transfer nature of the function K(»,-). We will show by general arguments and also by means of a simple example that K(z,-) is essentially an "absorption coeffi- cient," i.e., it serves as an analytical bookkeeping device for the depth rate of absorption of the stream of downwelling photons as the stream passes a general depth z . The heart of equation (28) resides in the difference of the two bracketed terms. The first term is a value of the volume attenuation function which shows that K(z,-) first of all takes cognizance of the simultaneous loss of photons by means of both scattering and absorbing mechanisms. Thus, as a stream of photons crosses an hypothetical surface at depth z , the stream suffers a loss by having some of the photons scattered in all directions about the point of cross- ing and also by having some of its photons of (implicitly stated) wavelength A converted into photons of (generally) longer wavelength or into nonradiant energy. Now the second term, involving h A (z,-) and being subtracted from a(z), in effect returns to the downwelling stream all photons which have been scattered in the "forward direction"- -the direction of motion of the downwelling stream. The net loss to the downwelling stream is then represented by this difference; it represents the amount of radiant flux per unit area that either has been scattered back into the upwelling stream, or which has suffered true absorption. The first of these al- ternate possible activities (scattering back into the stream) SEC. 9.3 COVARIATION OF K-FUNCTIONS 139 we now consider two extreme examples of optical media that is purely absorbing, the other purely scattering; we will show that in each of these two extremes the values of !((•,-) tend either immediately or eventually to become di- rectly proportional to the given values a(z) of the volume absorption function. These arguments will be general ver- sions of those leading to the demonstration of the absorption- like character of k in (5) of Sec. 9.2. First, let s(z) = for all z , and let a(z) be arbitrary. We then have the extreme case of a purely absorb- ing medium. By hypothesis, it follows that N*(z,£) = for each z and for all £ . Furthermore, since a(z) = a(z) + + s(z) it follows that a(z) =a(z). With these observations, equation (28) reduces immediately to the simple form: K(z,-) = a(z)D(z,-) (29) Here D(z,-) generally depends on depth along with a(z). However if depth z and a(z) are held fixed and the external lighting conditions are varied so that D(z,-) is changed, it is quite clear that K(z,-) varies directly and linearly with these changes in D(z,-). This is the first and simplest instance of the covariation of K(z,-) and D(z,-). The essentially absorption- like character may be seen by holding D(z,-) fixed. Then K(z,-) varies directly with the value a(z) of the absorption function. Next, let a(z) = for all z in an optically in- finitely deep medium in which s(z) = s > for all z . We then have the other extreme case of a purely scattering medium. In such a medium, according to (15) of Sec. 8.8, the divergence V • H(z) of the net- irradiance vector van- ishes at each depth z . In particular, in a plane-parallel medium such as that around which the present discussion is centered, this divergence relation takes the simple form: dH(z,-) m az where H(z,-) = H(z,-) - H(z,+), H(z,+) being the upwelling irradiance at depth z . It follows that, for every z , H(z,-) = c where c is a constant. Since the medium is in steady state and no photons which enter it are ever lost by absorption, we expect that the time rate of emergence H(0,+) of photons per unit are at the surface just equals the time rate of incidence per unit area H(0,-). Hence H(z,-) = H(z,+) 140 GENERAL THEORY OF OPTICAL PROPERTIES VOL. V for all z . Now the g more alike optically a level z , and the more the equal irradiances ary at depth z . (See of Sec. 8.7.) Thus we symmetry above and bel to conclude that h(z, great depths in the pr that the angular struc radiance distributions and in fact (by symmet of these observations reater the depth z in the medium, the re the regions above and below the alike are their optical responses to impinging on the common internal bound- , e.g., the discussion of (128)-(131) are led, by considerations of increasing ow horizontal planes at great depths, -) -> h(z,+) as z -> °° . Hence at esent medium, D(z,-) •*- D(z, + ), so ture of the upwelling and downwelling become equal (cf. (2) of Sec. 8.5), ry) uniform. An immediate consequence is the fact that: h*(z,-) h*(z,-) + h*(z,+) h*(z) h(z,-) h(z,-) + h(z,+) h(z) in which (26) was used for the last equality. Hence, since equation (28) for the present case of a(z) = takes the form: K(z,-) = we conclude that: h*(z,-) -. K(z,-) -> 0(= a(z)) (30) as z ■> oo . Thus, in the both e one is purely absorbing and the diffuse attenuation function wa which are directly proportional absorption function for these m that we understand the absorpti Another instance (36) supportin K(z,-) will be encountered as a eluding observations of this se sent a practical rule of thumb K(z,-) and D(z,-). xtreme types o other purely s shown to ten to the values edia. It is i on-like charac g this interpr matter of cou ction. There based on the c f media, where scattering, the d toward values of the volume n this sense ter of K(z,-) . etation of rse in the con- we shall pre- ovariation of Forward Scattering Media A necessary prerequisite to the establishing of the general statement of the covariation rule between K(z,-) and D(z,-) is the introduction of the notion of a forward scattering medium. Briefly, a forward scattering medium is one for which the volume scattering function has a predomi- nant forward scattering lobe as compared to its backward scattering lobe. We shall assume that the medium is iso- tropic. For a precise definition, let k be the unit outward normal to the plane-parallel medium, and let £ be an arbi trary element of 5 ; then we write: + (z,£) for a(z;5;Ddn(C) (31) SEC. 9.3 COVARIATION OF K-FUNCTIONS 141 whenever E, e H_ and: "s + ± (z;0" for J^ o(z;Z;V)dQ(V) (32) whenever £ e H . Some general properties of these s-func- tions are easily deduced. For example, if the medium is iso- tropic, then s+_(z;£) = s_ + (z;£ f ), provided that E,»k=-E,''k, Further, for these E, , £ ' : s + +(z;£;) = s._ (z ' £' ) . ■ For ex- ample, if £ • k = , then s_ _ (z ; £) = s(z)/2 since the axis of the scattering lobe would then lie in the horizontal plane at depth z and the region of integration would be over pre- cisely half the scattering lobe (see Fig. 9.1). Furthermore, for every E, e H s__ (z;£) + s_ + (z;-£) = s(z) The connection between s++(z;^) and the forward and backward scattering functions for collimated irradiance is quite close and should be noted (see (43) and (44) of Sec. 8.4). The value s__(z;^) has the interpretation of a forward scatter- ing function for the direction E, while a_+(z;£) has that of a backward scattering function. Physically s._(z;£) gives the fraction of flux of a beam of downward direction E, that is scattered in all downward (forward, with respect to i ) directions (see Fig. 9.1). If now we write : " 9 " for arc cos ( - £ • k) we may write M s__(z,£)" as "s__(z,0) M and we finally may define a forward scattering medium as one for which s__(z 3 Q) decreases mono tonic ally with increasing in the range < 9 £ tt/2. Clearly, (32) implies s + + (z,a + s + _ (z^) = s(z) for all E, e H_ , and analogously to s_ + (z,£) we define s++(z,H) as the forward (+) or backward (-) scattering func- tion for the upward direction E, in S+. Since s(z) is independent of E, (the medium has been assumed isotropic) , we have alternate means of characterizing a forward scatter- ing medium now using s_+(z,9), which of necessity is mono- tonically increasing with 9 in any forward scattering medium. The Covariation Rule for K(z,-) and D(z,-) We may now state the covariation rule: Let X be an arbitrarily stratified plane-parallel forward scattering medium with given fixed inherent optical properties. If z >_ is any fixed depth in X 3 then K(z 3 -) and D(z 3 -) increase 3 remain constant 3 or decrease together. Thus if 3 in particular 3 over a certain time period K(z 3 -) and D(z 3 -) exhibit increments in their values of magnitude t\K(z 3 -) and t\D(z 3 -) 3 then these increments must be simultaneously positive 3 zero 3 or negative . The complete proof of the rule is tedious because it requires an analysis of the total light field throughout the 142 GENERAL THEORY OF OPTICAL PROPERTIES VOL. V FIG. 9.1 Defining forward and backward scattering functions for the proof of the covariance of K(z,-) and D(z,-)- medium. However, the outline of a proof an that, under the condi is positive and AD(z in the magnitude of sion of the propertie derance of the radiat angle with the unit i at depth z . Now the a weighted mean of th o(z ; •• ; wherein the radiance components. D(z,-) requires that preponderance of the with larger values following argumen d makes the rule p tions of the hypot ,+) is zero. This D(z,-) which, by o s of D(z ,-) , impl ion has now shifte nward normal to an term h*(z,-)/h(z e volume scatterin weighting factors As observed above the radiance compo downwelling flux t in the arguments o t forms the lausible. hesis above implies an ur previous ies that th d to make a hypothetic ,-) in (28) g function are the do , the incre nents carry o be now as f forward s essential Suppose , AD(z,-) increase discus- e prepon- greater al surface is simply values wnwelling ase in ing the sociated cattering SEC. 9.3 COVARIATION OF K-FUNCTIONS 143 function s__(z,0); since the medium is forward scattering, this weighted mean then generally experiences a decrease in magnitude; the net result being an increase in the bracketed quantity in (28). The total change of K(z,-) is the combi- nation of the increase of the bracketed quantity and the in- crease in the factor D(z,-); that is, K(z,-) experiences an increase in magnitude. Summarizing: An increase in D(z,-) is attended by an increase in K(z,-) all other things remaining fixed in the forward scattering medium. A similar argument may be applied to the assumption that D(z,-) ex- hibits a decrease. With these two facts established it is then a necessary consequence of continuity in all physical situations that AK = whenever AD = 0. Illustrations of the Rule Example 1 . Most natural hydrosols are forward scatter- ing media; in fact s__(z,0), when 0=0, occasionally is on the order to ten to twenty times the magnitude of s__(z,tt/2) over the visible spectrum. The values o(z;£;£;) and cr(z;£;5r) » where £ • ^ 1 = 0, often subtend a ratio of forward to side scattering on the order of 100 to 1, over the visible spectrum. Even more dramatic ratios > 100:1 are indicated in Figs. 1.72 and 1.73. The rule may saTely be extended even to natural aerosols, these media being predominantly forward scattering; even the borderline case of Rayleigh atmospheres wherein s__(z,0) = s(z)/2 for in [0,tt/2], are subject to the rule, since the square bracketed quantity in (28) does not generally change magnitude, with a change in D(z,-). Example 2 . As a specific illustration, suppose the sky above a lake is completely overcast and that the downwelling distribution and diffuse attenuation functions at some rela- tively shallow depth z have values D and K , respec- tively, under this overcast condition. Suddenly, the near- zenith sun breaks through the clouds. The resulting value D 1 , of the distribution function, is expected to be less than D : D 1 < D , which follows from the fact that the predominant portion of the radiation now comes from general- ly less oblique directions. It follows that AD < , that is the increment of D(z,-) is negative, and thus the covaria- tion rule requires that K(z,-) is negative, so that the new value K 1 of the diffuse attentuation function is less than K : K., < K . Example 3 . As a final illustration, suppose that we fix attention on a relatively shallow depth in a natural hydrosol which is irradiated by a clear sunny sky for an en- tire afternoon. As the sun descends, D(z,-) clearly in- creases because the direction of the predominant portion of the irradiating flux supplied by the sun increases its angle with the vertical (i.e., 1/ | £ • k | increases). The covaria- tion rule would then require that K(z,-) exhibit a corre- sponding increase. 144 GENERAL THEORY OF OPTICAL PROPERTIES VOL. V The Contravariation of K(z,+) and D(z,+) Up to this point in the discussion we have excluded any detailed mention of the upwelling irradiance H(z,+). However, all that has been discussed for the downwelling stream of radiant energy may be applied mutatis mutandis to the upwelling stream, i.e., by replacing minus signs by plus signs in a systematic manner, etc. Therefore, the discussion of the interdependence of K(z,+) and D(z,+) may be con- ducted relatively quickly by pointing up only the basic dif- ferences between the two cases. Now the first of these main differences between the upwelling and downwelling streams H(z,+) and H(z,-) lies in their magnitudes over the visible portion of the spectrum; the ratio of their magnitudes, H(z ,+) /H(z , - ) known as the reflectance of the medium at depth z : R(z,-) = H(z,+) H(z,-) is very nearly and almost universally in the neighborhood of 0.02. Thus H(z,-) is on the order of 50 times the size of H(z,+). Secondly, a fairly constant tie exists between the two streams by virtue of the ratio and sum of their distribu- tion functions. It is found that almost universally over the visible portion of the spectrum (see, e.g., Table 1 of Sec. 8.5). DITTO " 2 ' D(z,-) + D(z,+) = 4 (33) (34) expressed to the nearest integer, which then requires that D(z,-) = 4/3, and D(z,+) = 8/3. The + stream counterpart to (17) is: - K(z, + ) = a(z, + ) + H (z,+) > which may be reduced to a corresponding expression to (28) : (35) In much the same manner as K(z,-), the dynamical and geometric mechanisms giving rise to the depth and temporal changes of K(z,+) may be discussed in complete detail. The only precautionary observation that should be made here is that the dynamical mechanisms governing K(z,+) should be examined as depth z decreases , this being the natural SEC. 9.3 COVARIATION OF K-FUNCTIONS 145 direction of flow of the upward stream. Finally, owing to the negative sign in front of K(z,+), the signs of the in- crements in K(z,+) and D(z,+) are opposite. Thus there is what we may term as a contravariation in the magnitudes of K(z,+) and D(z,+). This is the final distinction that must be made between the two streams, for the present. Whence does this striking difference in the relative variations of the magnitudes of K(z,+) and D(z,+) arise? In what light should this difference be viewed? The answer to the first question is that the difference arises in the definitions of K(z,+) and K(z,-); each is defined by writing: "KT7 +v» for 1 dH(z,±) K(z ' ±} for " h(z,±) — ai — * Now a plane-parallel medium representing a natural hydrosol, by its very physical nature and usual coordinate system, normally invites the choice of the downward direc- tion as the direction of increasing z values. Thus the spatial evolution of quantities associated with the downwell- ing stream are treated in a natural way, i.e., so that the natural unfolding of radiant energy in a downward direction takes place in the direction of the natural unfolding of the coordinate system, i.e., along with increasing z coordi- nates. The upwelling stream on the other hand naturally evolves spatially in the direction of decreasing z values, hence the contravariation, or topsy turvy interdependence of K(z,+) and D(z,+). This contravariation therefore is not an essential phenomenon and so can be erased and converted to a covariation if we reinterpret the derivative dH/dz as dH/d(-z) when considering the upward- flowing case. In an- swer to the second question, all that can reasonably be done is to view this state of affairs as an inessential perversi- ty of standard coordinate systems, and to understand that it is the inevitable result of an attempt to depict an inherent- ly three-dimensional process by an artificial two-dimensional symbolism designed by a basically one-dimensional thought process . A Covariation Rule of Thumb The general law (28) governing the interdependence of K(z,-) and D(z,-), while of extreme of importance in estab- lishing the exact relationship between these two quantities, is somewhat unwieldy for use in quick estimates of the rela- tive magnitudes of their increments. We conclude the present section with the derivation of a simple rule of thumb, based on experimental evidence, which relates in a linear manner the relative magnitudes of K(z,-) and D(z ,-), and also their increments. We begin with the exact expression for R(z,-) in terms of K(z,±) and a(z,±) as given in (25) of Sec. 9.2: R(z,-) _ K(z,-) - a(z,-) K(z,+) + a(z,+) 146 GENERAL THEORY OF OPTICAL PROPERTIES VOL. V Here a(z,±) = a(z)D(z,±). Now it is an experimental fact that the difference between the magnitudes of K(z,+) and K(z,-) over the visible spectrum is very small for many practical purposes, being on the order of five percent of the magnitude of K(z,-). (This may be seen in Sec. 10.4.) Hence we may replace K(z,+) by K(z,-) in the preceding formula and solve for K(z,-). The result is: K(z,-) = a(z) [ & (z,-)D(z,+) + D(z,-) 1 - R(z,-) (36) Equation (36) may be taken, as it stands, as a rule of thumb connecting K(z,-) with a(z). This equation serves to under- score the conclusion reached earlier in this work that K(z.-) is basically an absorption- like optical property of a medium. To arrive at the desired rule of thumb we now make use of the experimental-numerical relations (33) and (34) between D(z,+) and D(z,-) and also of the fact that R(z,-) is of the order 0.02. The result is: K(z,-) = a (z) [ 1 - R(z,-) J ulz > = a (z) [ 50 x2+1 1 " TO — ] D(z,-) A2 49 a(z)D(z,-) = 1.06 a(z)D(z,-) Hence AK(z,-) = 1.06 a(z)AD(z,-) (37) which is the desired rule of thumb relating the covariation of K(z,-) and D(z,-). 9.4 General Analytic Representation of the Observable Reflectance Function The concept studied in this section is the observable reflectance function R(»,-) whose value at a depth z in an arbitrarily stratified plane-parallel optical medium is given by: R(z,-) = H(z,+) H(z,-) where, as usual, the quantities H(z,±) are the observed up- welling (+) and downwelling (-) irradiances at depth z in the medium (re: (16) of Sec. 9.2). Several representations of the function R(*,-) are established which will, (a) SEC. 9.4 OBSERVABLE REFLECTANCE FUNCTION 147 explicitly exhibit in terms of differential equations and definite integrals the dependence of R(«,-) on the inherent optical properties of the medium, as far as this is possible; (b) illustrate the dynamic equilibrium-seeking tendency of R(*,-) which appears to hold in all plane-parallel media; and finally, (c) suggest some methods of solving the problem of predicting the depth-structure of R(«,-) in general media. To place the present discussions in their proper perspective for the general reader, we prefix the following observations. The reflectance function R(»,-) is one of a set of seven main apparent optical properties introduced in Sec. 9.2. This, set consists of the functions R(*,±), K(*,±), k(*,±), and k , and is defined along with D(°,±) in terms of the four directly observable radiometric functions: H(*,±), h(»,±), where h(«,±) are the upwelling (+) and downwelling (-) scalar irradiance functions. The theory of the measure- ment of these latter radiometric quantities and a discussion of the salient physical characteristics of this extremely useful set of apparent optical properties was briefly sketched in Sec. 2.7 (see Fig. 2.18). Further discussion of these properties is given in Chapter 13. Section 9.6 contains a classification of the optical properties of an optical medium into the classes of inherent and apparent optical properties, and the necessary distinctions that must be made between them, in both experimental and theoretical procedures. The main fundamental set of local inherent optical prop- erties of any scattering-absorbing optical medium consists of the functions a and a , the volume attentuation and volume scattering function, respectively. These functions are by definition independent of the ambient light field. The appar- ent optical properties, however, depend jointly on the inher- ent optical properties and the ambient light field. Specifi- cally, the apparent optical properties depend on a , a , and the radiance distributions N(z,*) in the medium. Despite this dependence of the R , K , D , and k functions on ephemeral lighting conditions, they exhibit a behavior in both space and time of such a strikingly regular and generally predictable kind, that each is dignified with the appellation: "optical property." However, we point out the fact that this is a matter of first appearances only, and that, under incisive analytical and experimental scrutiny, their regularities are seen to be at the mercy of variable boundary lighting conditions and the internal distribution of the values of a and o in an optical medium. To emphasize this fact, the qualification "apparent" has been put before "optical property." Detailed examples of the regular behavior of the appar- ent optical properties are given in Sees. 9.2 and 9.3, in the following sections, and in the remaining chapters of Part III. The present section adds to this store of knowl- edge of the apparent optical properties by developing in detail the exact differential and integral representations of the reflectance function R(»,-) and drawing some theoreti- cal and practical conclusions from them. 148 GENERAL THEORY OF OPTICAL PROPERTIES VOL. V The Differential Equation for R(*,-): Unfactored Form The phys is a plane-par ent optical pr this depth dep medium may hav arbitrary boun cident lightin Assume t of depths betw that at each o are taken, z 1 each depth is ical sett allel sou operties endence i e either dary refl g conditi hat an ir een z' f the dep < z <_ z" . determine ing for the deri rce-free optical spatially depend s assumed arbitr finite or infini ectance properti ons on its upper radiance probe s and z" , where ths in this rang Then the refle d by: vations that follow medium whose inher- only on depth, and ary. Further, the te optical depth with es and arbitrary in- and lower boundaries. weeps through a range £ z ' £ z" <_<*> , and e readings H(z,±) ctance R(z,-) at R(z,-) = H(z,+) H(z,-) and its depth rate of change dR(z,-)/dz for downward motion or for upward motion is easily found and is given as (re: (32) of Sec. 9.2) : dR(z,-) _ dz R(z,-)[K(z,-) - K(z,+) We may now introduce the exact representations of K(z,±) established earlier ((18) and (19) of Sec. 9.2): +K(z,±) = [a(z,±) + b(z,±)] - b(z,+)R(z,±) where a(z,±) = a(z)D(z,±), and b(z,±) are the values of the absorption and backward scattering functions at depth z for the upwelling (+) and downwelling (-) streams of radiant flux. Substituting these representations of K(z,±) in the above derivative, the result is: where This unfac ernin equat ting of th of (1 ly ha to ob (39) dR(z,-) dz b(z,+)R 2 (z,-) - c(z)R(z,-) + b(z,-) c(z) = [a(z,-) + a(z,+) + b(z,-) + b(z,+)] (1) is t tore g th ion and e pr ) is s no serv of S he des d form e obse within forms operti that neleme e the ec. 8. ired diff j and is rvable re the pres the basis es of R( of a gene ntary sol striking 7, keepin erenti the ba f lecta ently for a -,-)• ral Ri utions resemb g in m al equat sic diff nee func chosen g 11 our s The mat ccati eq . The r lance be ind (11) ion fo erenti tion. eneral ubsequ hemati uation eader tween and ( r R( al eq It i phys ent d cal s , whi shoul (1) a 12) o • ,-) in uation gov s an exact ical set- eductions tructure ch general d not fail bove and f Sec. 8.3 SEC. 9.4 OBSERVABLE REFLECTANCE FUNCTION 149 We shall return to consider this resemblance later, in the equivalence theorem. The Differential Equation for R(*,-): Factored Form The basic differential (1) may be factored by observing that its right-hand side is a quadratic in R(z,-) for each depth z . Thus for a given z , the roots of the quadratic equa- tion: b(z, + )t 0, N* lim N = — r -+ oo r a The quantity N*/a , usually denoted by "Nq" , is the equilibrium radiance of the path of sight, and in this case is simply the observable horizontal radiance. It is depen- dent on both the local inherent optical properties of the medium ( a , a) and the lighting conditions along the path (N*) . The term equilibrium radiance is understood in the following sense: For any initial choice of N , N r tends toward and eventually attains the value N q . Thus if N exceeds Nq , then N r decreases from N Q to Nq as r goes from to °° . On the other hand, if N is less than Nq , then N r increases from N to N q as r goes from to °° . This phenomenon of the equilibrium-seeking tendency of the apparent radiance actually holds for an arbitrary path of sight in an arbitrary optical medium along which there are no sources and along which a > . This may be seen by taking the general transfer equation for radiance: aN dN dT = and writing it in the form: §--a[N-N q ] , where we have written " Nq " for N^/a , as in Sec. 4.3. It must be emphasized that this equation is completely general; hence a may change from point to point along a path, N* (and hence Nq) may depend on direction about a fixed point, and the angular dependences of N* at two different points may be quite distinct. Now select any path of sight in the medium, and choose an initial point of the path. At this point suppose the value of N is given. If then N > Nq , the above equation immediately shows that dN/dr < , so that N tends toward the value of N q at this point as r in- creases. On the other hand, if N < N q , then dN/dr > , and N tends toward Nq once again as r increases. Now it is quite possible that N q may change from point to point along the path. But the important fact to observe is that 152 GENERAL THEORY OF OPTICAL PROPERTIES VOL. V at every point of the path, regardless of the relative sizes of N and N q , the tendency of N at that point is to change its value so as to decrease the absolute value of the difference N - N q at that point. The phenomenon of the equilibrium- seeking tendency of radiance in an arbitrary medium gives rise to a host of equi- librium theorems for various other radiometric quantities and even for the apparent optical properties. These equilibrium theorems are explored in detail in Chapter 11 where it is shown that no less than 34 radiomet- ric and related concepts are subject to a single equilibrium principle. The following discussion of the equilibrium theorem for R(»,-) is patterned after that exhibited for N above. Furthermore, this special discussion will pave the way for some interesting observations of the properties of the re- flectance function as seen in the light of the principles of invariance. These observations will be given later in this section. The Equilibrium-Seeking Theorem for R(* ,-) To establish the present equilibrium theorem for R(»,-), consider an arbitrarily stratified source-free plane-parallel medium over the depth interval <_ z ' < z" <_ °° in which scattering takes place, i.e., a ± . The medium may be optically shallow or deep; its boundary reflectances are arbitrary, as are the boundary lighting conditions. The present setting, therefore, is of maximum generality. Im- agine a reflectance meter at depth z in the medium. The reading R(z,-) gives the complete reflectance of the ma- terial between the level z and the lower boundary, inclu- sively. This number is a complex combination of the effects of the standard reflectance of the medium in that depth in- terval (i.e., the standard reflectance R(z,z") of the slab X(z,z")), the interreflections between X(0,z') and X(z',z M ), and the angular structure of the downwelling incident flux at level z . The angular structure of the downwelling flux, at level z in turn depends on the inherent optical proper- ties of the medium throughout its extent. However, despite this complex situation, there exists at every level z along with R(z,-), the values R a (z,-) and R q (z,-) of the attenu- ation function and equilibrium function associated with R(*,-), which guide the evolution of R(z,-) as z increases. In view of the active role played by Rg(z,-) and R q (z,-) in determining the depth behavior of R(z,-), we pause to examine their structure. It turns out that R a (z,-) is of central interest. We shall first establish the fact that R a (z ,-) > 1 for all z . We deduce the fact that R a (z,-) > 1 for all z on strictly analytical grounds, starting from the defining equation: R {7 , _ c(z) + [c 2 (z) - 4b(z,-)b(z,+)] l/g a U '" j 2b(z, + ) " SEC. 9.4 OBSERVABLE REFLECTANCE FUNCTION 153 Observe first that the value R a (z,-), when considered as determined solely by the magnitude of c(z), monotonically increases with c(z), in the sense that we hold b(z,±) fixed and let c(z) increase. Thus in particular if for some special value c of c(z) we can show that R„(z,-) >. 1 , then for all c(z) > c we will certainly have R a (z,-)_>1 . Now c(z) = a(z,-) + a(z,+) + b(z,-) + b(z,+). Hence (since all a ' s and b ' s are nonnegative) : c(z) > b(z,+) + b(z,-) in fact the strict inequality holds in all real media. Let us denote b(z, + ) + b(z,-) by "c ". Then c o + (V - 4b(z,-)b(z, + )] 1/2 2b(z,+) = X ' It follows that R^Cz,-) >_ 1 for <_ z <_ z" , in every plane-parallel optical medium. We recall at this point the fact that: R(z,-) < 1 in all real optical media. (See (29) of Sec. 9.2; in fact the strict inequality holds in such media.) From these in- equalities we deduce the fact that the difference R(z,-) - R a (z,-) in all real media is negative for all z . Continuing with the development of the theorem, sup- pose that we now measure R(z,-), z > , and then move the reflectance meter a small distance in the upward direction (maintaining, of course, its horizontal collection-orienta- tion throughout the move) . What we are in effect doing by such a move is increasing by a small amount the material of the medium below the level occupied by the meter. It turns out that this upward motion is the natural direction of mo- tion one should go in order to discern the equilibrium-seek- ing behavior of R(»,-), just as the natural direction of motion of the observer in the equilibrium theorem for N was such that it increased the amount of scattering-absorbing material between the observer and the initial point of the path (Figs. 9.2-9.3). In this connection see also the dis- cussion of the contravariation of K(z,+) and D(z,+) pre- sented in Sec. 9.3. Therefore, to analytically describe the result of this motion the derivative term of equation (2) is now read as dR(z , - ) /d(- z) . The final steps in in the proof may now readily be taken. Suppose that R(z,-) < R q (z,-) at the depth under consideration. (See Figs. 9.2, 9.3.) Hence, R(z , - ) - R q (z , -) is negative. By the preceding observations, it is known that R(z,-) - R a (z,-) is invariably negative in all real media. Thus the derivative dR(z ,- )/d(- z) is positive , indicating that R(z,-) tends toward the value of the equilibrium re- flectance Rq(z,-) at this depth, as z decreases. On the other hand, if R(z,-) > R q (z,-), then R(z,-) - R q (z,-) is positive, and since R(z,-) - R a (z,-) is invariably negative it follows that in this case dR(z , - )/d(- z) < , so that 154 GENERAL THEORY OF OPTICAL PROPERTIES VOL. V 1 b(-) yR a (Z.-) Roo b < + > 1 **"*> /R(Z,-) dR(Z, d(-2 ?>°" dR d( -Z) u dR(Z r ) d(-Z) Z —►00 FIG. 9.2 As one moves upward through the hydrosol, in the direction of decreasing z , observe how the slope of R(z,-) is always directed so as to decrease the gap between R(z,-) and the equilibrium reflectance Rq(z,-). This is the process referred to by the equilibrium-seeking theorem for R(z,-). R a (z,-). This completes once again R(z,-) tends toward the proof of the theorem. We summarize the equilibrium-seeking theorem symboli- cally as follows: rdR(z,-) i S1 § n L d(-z) J = sign [R q (z, ) - R(z,-) (4) We may now make several observations on this equilibrium- seeking property of R(*,-). Observation 1 By returning to the basic premises of the present dis- cussion, we observe that the condition a ^ was imposed. This condition has both physical and mathematical relevance to the conclusion (4). Mathematically, R a (z,-) and Rq(z,-) are prima facie undefined for the case o = . Physically, the reflectance of a purely absorbing medium or a vacuum is SEC. 9.4 OBSERVABLE REFLECTANCE FUNCTION 155 _l_bH Roo b(+) oo ■R a (Z,-) R q (Z,-) R(Z,-) d(-Z) dR(Z,-) d(-Z) >0 m FIG. 9.3 Further variations on the equilibrium- seeking theorem for R(z,-). trivially zero. To lift the veil of mathematical indeter- minacy of. R(z,-) in the case of a = , we return to equa tion (1). Under the present conditions (1) reduces to: dR(z,-) d(-z) c(z)R(z,-) where in this case c(z) = a(z,-) + a(z,+) > . Hence if the initial value of R(*,-) at z" is R(z",-) > , then clearly: R(z,-) = R(z",-) exp {■/: c(z')dz in the quadratic equation governing q(z,-) = 0. The preceding formula for By letting b(z,±) -* Rq(z,-) we see that R R(z,-) shows that, for 'media with a(z) > for all z , and o = on [z',z"], limr Z M _i ^ooRC 2 ? - ) = . Hence the eq librium seeking tendency of R(»,-) is borne out for this case also. Observation 2 What about the opposite case to that just considered? Namely that a(z;£';S) 4 for all z,£',£, and a = 0? It follows that R a (z,-) = 1 and that R q (z,-) = b (z , -)/b(z , + ) 156 GENERAL THEORY OF OPTICAL PROPERTIES VOL. V for each z . Now suppose that R(z,-) < 1 at some depth z in the interval [z',z"]. This implies that H(z,+) 1 or in other words : b(z,-) R ( Z >") < oW) = R q (Z '" } ' By the equilibrium- seeking theorem, it follows that dR(z, - )/d(- z) is positive. Hence as depth is decreased in the purely scattering medium, the values R(z,-) increase monotonically. But since the preceding supposition, namely R(z,-) < 1 , was for an arbitrary R(z,-) magnitude less than unity, it follows from (2) that for every z , lim ( Z "-Z) -*«> R (Z>") = 1 • Thus we have in observations 1 and 2 above proved (or outlined proofs for) some outstanding folklore about the elementary properties of R(»,-) in plane-parallel media. These proofs were arrived at by reasoning strictly from the various exact differential equations governing R(*,-). In this way we hope to illustrate the power inherent in that approach to radiative transfer problems under development in this chapter, which discards particular mathematical models and which concentrates on the study of directly observable quantities of the light field. It is to be emphasized that the reasoning in this approach proceeds directly from the exact forms of the equations of transfer. The Integral Representations of R(z,-) Starting with the factored form (2) of the transport equation for R(»,-) we make use of the separation of vari ables that exists within it, and we write: SEC. 9.4 OBSERVABLE REFLECTANCE FUNCTION 157 dR(z,-) [R(z,-) - R a (z,-)JLR(z,-j R q (z,-)J = b(z, + )d(-z) If we formally integral repres value z 1 , to R(« ,-) at these depth range of so that this is trivial over th integration can each , R(* ,-) is correspondence of the interval integrate each entation by le z 2 > z 1 . Let depths be dis interest can a true; otherwi is depth range be subdivided monotonic and between the va With these side, we obtain the desired tting -z range from some the corresponding values of tinct: R(z 1 ,-) f R(z 2 ,") ; the lways be partitioned into pieces se the problem of R(*,-) is In other words, the range of into intervals so that over so that there is a one to one lues of R(*,-) and the points observations we may then write: R(z ,-) RU ,-) dt [t - R ft -)][t - R ft -)J b(t,+)dt , (5) which is the desired integral representation of R(* 3 -). The variable t in the integrals acts as a dummy variable of inte- gration, oriented as in the equilibrium-seeking theorem. An alternate integral representation of R(«,-) may be obtained from (1) in which the variables are also convenient- ly separated. The same general arguments used to establish (5) may now be directed to the equation (1). The result is: R(z ,-) R(z .-) dt b(t,+)t £ - c(t)t 2 + b(t,-) " (z 2 " z i^ . (6) Applications We now discuss two methods of evaluating R(*,-) by means of its differential and integral equation representa- tions given above. We illustrate the use of (5) for a very simple case, which is a useful approximation to reality, namely the case in which R a («,-) and R q ( • , - ) are constant functions. The second method is based directly on (1) or (3) and promises to yield a means of determining R(«,-) under realistic conditions. Special Closed Form Solution If over some depth interval the functions a(*,±) and b(*,±) are constant, then the functions R a (*,-) and R (•,-) are constant functions over the same arbitrary depth interval, 15, GENERAL THEORY OF OPTICAL PROPERTIES VOL. V say [z 1 , z g ] , z 2 > z , in a plane-parallel medium. Under these conditions, (5) is immediately integrable, and the definite integrals take the forms: l in 3, R - R ■ Ln t-R 1 a a R(z ,-) b(+)(z -zj From the definitions of R q and R a , we see that R a - R q > and in fact: R R - [C 2 - 4b(-)b( + )] l/2 V q bTO where we have written: "c" for a(-) + a( + ) + b(-) + b( + ) and where a(±) and b(±) are the assumed constant values of the functions a(«,±) and b(«,±) over the depth range [z 1 , z g ]. In the present method all four of these quantities may be distinct. Applying the limits to the left integral, we have: R(z ,-)"R R(z ,-)~R In Hence, _R(z ,-)-R \ : z q LR(z ,-)-R r^j - i"[ R (z , :o-R q ] - [ c 2 -4b(-)b( + )] 1/2 (z 2 -z i ) ,-)-R - r 1 ' J a ^ R(z^ ,-)-R„ - Rlz c 2 -4b(-)b( + )] l/2 (z 2 - Zi ) If we write: " c(z .-) " for R(z ,-) - R R(z ,-) - R then: R -R c(z -)exp{-[c 2 -4b(-)b(+)] l/2 (z -z )} R( Z ,-) = _2 — 2 ] 5 ] 2 t ,- n rr2. l .x l ..-,1 / 2 1 c(z .-)exp{-[c -4b(-)b(+)]"*(z - z )} (7) Hence if the four constants a(±) and b(±) are known or estimable over an interval [z, ,z 2 ] and R(z-),-) is known then R(z 2 ,-) is determinable. Observe that if we let (Zg-zJ ■* °° , thereby simulating an infinitely deep layer then R(z 2 ,-) -* R q . Hence R q in this instance is the Roo - quantity of the classical theory. SEC. 9.4 OBSERVABLE REFLECTANCE FUNCTION 159 The points of contact with the classical theory may be increased by observing that if we set: a(+) = a(-) = a* , and b( + ) = b(-) b* , then: [c 2 -4b(--)D( + )] l/2 = 2[a*+2b*)] l/2 = 2k where k is the diffuse absorption coefficient of the clas- sical one-D model of the two-flow theory (see, e.g., (8) or (32) of Sec. 8.6). By partitioning an inhomogeneous medium into essen- tially homogeneous contiguous layers, successive applica- tions of (7) will yield a useful practical formula for the reflectance of the entire medium. The solution (7) auto- matically includes the effects of interref lections between the partition pieces. Thus suppose the medium, which ex- tends over an interval [z ,z n ], is partitioned in n homo- geneous layers defined by the depths: [z ,zj, [z 1 ,z g ], ... , [z nM , z n ] . If R(z n ,-) is known (this may be the reflectance of the bottom boundary of the layer [z n _ 1 ,z n ]), then by (7) we find R(z n _ 1 ,-). Another application of (7) with R(z n _ 1 ,-) as the initial reflectance then yields R( z n- 2 >")> an d so on to R(z Q ,-) which then is the reflec- tance associated with the medium over the depth interval [z > z n]- Differential Analyzer or Digital Solutions Equations (1) and (3) as they stand, are suitable for determinations of R(»,-) by means of differential analyzer (and analog) or digital techniques, especially when the func tions a(*,±) and b(*,±) vary extensively over the medium. Series Solutions By means of series solution techniques, equations (1) and (3) may also be used to solve the difficult problem of determining R(»,-) over some interval [z 1? z ] when a(*,±) and b(*,±) are nonconstant and known over this interval. By expanding the coefficients of R 2 (z,-) and R(z,-), and the b(z,-) term in (1) in terms of infinite series in z , recursion formulas may be obtained for the coefficients in the infinite series expansion of R(z,-) over [z 1} z 2 ]. Equivalence Theorem for R(*,-) Comparison of the differential equation (1) with (39) of Sec. 8.7 shows that the observable reflectance function R(*,-) and the standard reflectance function R(*,z 1 ), both defined in a given slab X(0,z 1 ), satisfy the same differ- ential equation within X(0,z 1 ). This is a somewhat arrest- ing fact since the interpretation of the two numbers R(z,-) and R(z,z.,) are quite different conceptually. Briefly, 160 GENERAL THEORY OF OPTICAL PROPERTIES VOL. V R(z,-) is obtained as a ratio of H(z,+) to H(z,-) deep within X(0,z.,); and R(z,z.,) is the ratio of H(z, + ) to H(z,-) when X(z,z.,) is thought of as an isolated slab of X(0,z.,). This is not to assert that these functions need always agree in value at each point. Indeed, for z = z 1 , we have R(z 1 ,z 1 ) = , since the reflectance of a slab of zero thickness is zero. On the other hand R(z 1 ,-) is gen- erally not zero: its magnitude being the reflectance of the lower boundary of X(0,z 1 ). However, if X(0,z.,) has no lower boundary and no sources there or one whose reflectance and transmittances are and 1 , then R(z 1 ,-) = and we would expect R(z,-) = R(z,z.,) for every z in [0,z.,], since both functions now satisfy not only the same differential equation but also the same initial condition. The principles of invariance for irradiance help clari- fy this somewhat unexpected relation between R(z,-) and R(z,z.,). From (1) of Sec. 8.1 we have for the present medium X(z, Zl ): H(z,+) = H( Zi ,+)T(z;,z) + H(z,-)R(z,Z 1 ) If X(z,z.,) has no upward irradiance at level z , so that H(z 1 , + ) = , then: 1 H(z,+) = H(z,-)R(z, Zi ) on the other hand, we have, by definition of R(z,-): H(z,+) = H(z,-)R(z,-) whence follows the equality of the two functions R(*,-) and R(*,z 1 ) over [0,z ]. We shall summarize these observations as follows: Equivalence theorem for reflectances: Let X(0 3 z^) be an arbitrary stratified source- free plane-parallel opti- cal medium with arbitrary boundary irradiances H(0 S -) and H (z 1 3 +) . Then the observable reflectance function R(* 3 -) and the standard reflectance function R(' 3 z^) for a general subslab X(ZjZ^) of X ( 3 z ^ ) 3 satisfy the same differential equation [(1) above, or (39) of Sec. 8.7]. If E(z^ 3 +) = 3 then R(z 3 -) = R(z 3 z ) ) for every z in [0 3 z 1 ] . The reader may gain still further insight into the connections between R(»,-) and RC^z,,) by contemplating the connections between R(«,-) and the complete reflectance functions #(z,z',z ), z < z' < z\ in X(0,z.,); and also the relation (35) of Sec. 7.5. Connections with the Two-Flow Theory The equivalence theorem cited above permits a simple bridge to be constructed between the two-flow theories of Chapter 8 and the directly observable quantities H(z,±) of the present chapter. The classical one-D model of the two- flow theory of the light field describes the irradiances in a boundaryless , sourceless, isotropically scattering SEC. 9.4 OBSERVABLE REFLECTANCE FUNCTION 161 homogeneous slab over an interval [0,z.,] irradiated at the upper level (z = 0) by a directionally uniform radiance distribution and with H(z i + ) = (Sec. 8.6). The theory proceeds on the assumption that b(z,-) = b(z,+) = b* and and absorption functions for each stream are pairwise iden- tical and have the constant starred values over the slab). We can immediately deduce the values R(0,z) and T(0,z) associated with this slab on the basis of the present gener- al theory. To do this, we recall the statement of the equivalence theorem for reflectance equations proved above. This allows us to use the expression for R(z ,-) given in (7). We need only observe that, under the present setting, we should make the following pairings of variables: z < -> 1 Z * ► Z R(z ,-) = R R = 1 a q R + R = c/b* a q and finally we observe that: [c 2 -4b(-)b(+)] l/2 = 2[a*(a*+2b*)] l/2 = 2k where k is the diffuse absorption coefficient of the one-D model. Hence (7) reduces to: [l-exp{-2kz }] RCO.z,) == D _ P Q _ r _^, ; (8) R -R exp/-2kz \ a q r l 1 s b* sinh kz (a*+b*)sinh kz +k cosh kz 1 1 (9) Equation (9) gives the usual form for the reflectance of a slab of depth z 1 . (As a check, let z .,-*<» , and compare result with (8) of Sec. 9.2.) The remaining form for T(0,z 1 ) can now be deduced im- mediately from (42) of Sec. 8.7, but the point of this dis- cussion has essentially been made: The classical two-flow theory is an elementary special degenerate case of the pres- ent theory of directly observable quantities in real light fields. For convenience of reference, the transmittance T(0,z 1 ) is given by: T(0,zj = r- x Al x, . « J* n- ~ , < (10) (a*+b 5% )sinh kz +k cosh kz 162 GENERAL THEORY OF OPTICAL PROPERTIES VOL. V Summary This section develops several differential and integral formulas governing the observable reflectance function R(»,-). Methods of using the formulas are outlined. Thus R(«,-) may be obtained directly without first solving for the irradiance functions H(«,±) as has been necessary previously. The methods discussed are general enough to allow the determina- tion of R(*,-) if the absorption and backward scattering functions a(*,±), b(*,±), respectively, for each stream in an arbitrarily inhomogeneous stratified medium are known. A general equilibrium-seeking theorem for R(«,-) is also demon- strated, the substance of which is the fact that the deriva- tive of R(»,-) at each depth z invariably has an algebraic sign so as to decrease the absolute magnitude of the differ- ence R(z,-) - Rq(z,-) between R(z,-) and the value Rq(z,-) of the equilibrium reflectance function. An equivalence theorem is proved which shows that the observable reflectance function R(*,-) and the generally nonobservable standard re- flectance function R(«,z.,) for slabs within X(0,z.,) both obey the same differential equation, thereby establishing an important link between these theoretical (R(»,z )) and em- pirical (R(«,-)) concepts. 9. 5 The Contrast Transmittance Function The list of transmittance concepts associated with a path (? r (z,^) in an optical medium will be completed in this section with the introduction of a companion transmittance concept to the beam transmittance function introduced in (3) of Sec. 3.10, and the radiance transmittance function intro- duced in (13) of Sec. 4.5. This new transmittance concept is called the contrast transmittance function. By way of introduction to the contrast transmittance function, we review briefly the general types of transmit- tance functions studied so far in the present work. Suppose X is an arbitrary optical medium with boundary S . Then associated with X itself is the standard qJ - operator: ^(X;a,b), introduced in Sec. 3.8, whose physical signifi- cance is depicted in (a) of Fig. 9.4. When the incidence region a on the boundary S coincides with the response region b , then ^(X;a,a) is interpreted as a general re- flectance operator. When a and b are disjoint, s^p(X;a,b) is interpreted as a general transmittance operator. Three general transmittance operators may be associated with the light field in X : X(X;a,b) J*(X;a,b) J(X;a,b) The circled and starred operators are associated with trans- mitted residual and diffuse radiance respectively; the un- adorned v/-operator describes the undecomposed or directly observable light field. In Sec. 3.8 J(X;a,b) was defined in detail. */°(X;a,b) is manufactured readily using the geomet- ric structure of X , a Dirac-delta function, and the beam transmittance function T r for X , much in the way T°(x,z) SEC. 9.5 CONTRAST TRANSMITTANCE FUNCTION 163 FIG. 9.4 Three settings for transmission operators: (a) general media, (b) one-parameter media, (c) plane-parallel media. is defined in (32) of Sec. 7.1 for plane-parallel media. >/* follows by subtraction of J° from J (cf. (41) of Sec. 7.1). It follows that: J(X;a,b) = J°(X;a,b) + J*(X;a,b) (1) Much in the same way the complete transmittance operator 3"(x,y,z) associated with a subset X(x,z) of a one parameter medium X(a,b) can be rendered into its residual and diffuse parts : J~(a,y,b) = Cr°(a,y,b) +J"*(a,y,b) and which is depicted in (b) of Fig. 9.4. Similarly, the decomposition of the standard transmittance operator: (2) 164 GENERAL THEORY OF OPTICAL PROPERTIES VOL. V T(a,b) = T°(a,b) + T*(a,b) (3) is but a special case of (1) or (2) and is depicted in (c) of Fig. 9.4. All of the preceding examples pertain to transmittances of three dimensional subsets of an optical medium. One- dimensional subsets can be endowed with similar sets of transmittances. Thus if (? r (x,£) is a path in X , we have in our earlier work associated with it the beam transmit- tance T r (x,£) (Sec. 3.10). In order to emphasize, during the present discussion, that the beam transmittance is asso- ciated with residual radiant flux, i.e., flux transmitted over (? r (x,£) without being scattered or absorbed, let us place a circle superscript on the transmittance symbol so: "Tj(x,0." Furthermore, corresponding to ^(X;a,b), :J(a,y,b), and T(a,b) above, that is, corresponding to the transmittance operators for undecomposed (i.e., observable) radiance, we have the radianoe transmittance operator asso- ciated with (? r (x,£), as introduced in Sec. 4.5. Let us denote this operator by n T r (x,^)." The fundamental roles played by T£(x,£) and T r (x,^) may be seen by comparing their effects on the initial radiance N (x,£) of the path (?r(x,U: N°(z.O = N q (x,OT°(x,U (4) N r (z,0 = N o (x,5)T r (x,£) (5) where N r (z,£) is the radiance at end point z of u r (x,£) in the direction £ . We shall consider only straight paths (?r(x,£) in this discussion, so that we have the simple rep- resentation: z = x + rE, . We interpret (4) as usual by saying that Nj(z,£) is the residual radiance of N (x,£) transmitted over (? r (x,£). We interpret (5) in the sense that T r (x,£) is simply the number which when multiplied into N (x,£) gives N r (z,£) where N r (z,£) and N (x,£) are the two radiances as they exist at points x and z in the actual light field in X . In other words, the current definition of T r (x,£) consists precisely in agreeing to write : N r (z,0 "T r (x,0» for IT - T ^ T (6) with respect to a path (j r (x,£) in X . Thus (5) is an ele- mentary consequence of (6j . It is obvious that T r (x,£) is immediately obtainable as a consequence of the interaction principle applied to (? r (x,£). We can now readily round out the roster of transmit- tance concepts, associated with a path (p r (x,^) in X , and in such a way as to be uniform with the other basic transmit- tance equations (l)-(3). Thus let us write: "TJ(x,O m for T r (x,S) - T°(x,0 (7) SEC. 9.5 CONTRAST TRANSMITTANCE FUNCTION 165 Then: (8) By the work of Sec. 3.11 and Sec. 4.5 we have conve- nient integral representations of Tp and T r . Thus, by (3) of Sec. 3.11: T^(x,?) = exp f - ; adr* | 1 (P r (x.O J (9) and by (2) of Sec. 4. 5: T r (x,^) = exp tf r (x,0 Kdr' j (10) In terms of the compact notation of Sec. 4.5, these may also be written: T = T [ - a r r L T r = T r [ - 5 • K ] (11) (12) It is clear from the definitions that T r is an inherent optical property of ^(x,^) and T r (and hence Tf) is an apparent optical property of (? r (x,t). The Concept of Contrast We are now ready to formulate the c and develop some of its basic properties state that the sense in which we use the radiative transfer theory is in the rela two radiances. For example if one direc points 1,2 on a distant mountainside which have apparent radiances N 1 and trast of the first point relative to the quantitatively by the difference quotien N 1 = N 2 , then there is zero contrast,, r ing between the two points. If it happe then we say that there is a positive con respect to point 2 . Conversely, since negative, we say that point 2 has negati spect to point 1. One of the fundamenta bility theory in natural optical media i the contrast of a visual target against The problem is solved if the contrast tr path of sight from the observer to the t shall now develop the theory of the cont of an arbitrary path of sight in a natur oncept of contrast To begin, we can idea of contrast in tive difference of ts attention to two or submerged scene N 2 , then the con- second is given t: (N 1 -N 2 )/N 2 . If adiometrically speak- ns that N 1 > N 2 , trast of point 1 with (N 2 -N 1 )/N l is now ve contrast with re- 1 problems in visi- s the prediction of its background, ansmittance of the arget is known. We rast transmittance al optical medium. 166 GENERAL THEORY OF OPTICAL PROPERTIES VOL. V noti radi ance in a stat nont X ( " N s with ence resp By on of ometr s whi n opt e the extracti contras ic contr ch in tu ical med followi Definition rivia Fig. resp quot ect t 1 paths 9.5), th 2 )]/N s (y ect to ient is o N (x 2 ng the salient fe t given above, we ast is characteri rn are associated ium. Therefore w ng: i. if GUxi.sj (T.e. , r f , s en the difference a ,£ 2 ) is the appa N s (y 9 .5 2 ). I f r the inherent eont atures of the intuitive see that, at its core, zed in terms of two radi- with two paths of sight e can quite generally and (? s (x 2 ,£ 2 ) are two f 0) in an optical medium quotient [N r (y ,£,) - rent contrast of N-^y,,,^) = s = , then the differ- rast of N (x 1 ,CJ with Regular Neighborhoods of Paths The notion of contrast attains virtually all its use in situations where the two paths of sight are in some suit- able sense optically close neighbors of each other. The following definition isolates the essence of the requisite concept of "closeness" of paths: Definition 2 . Let A and B be two subsets of an optical medium S , and let C(A,B) be a collection of paths in S such that the initial points of the paths are in A and the terminal points of the paths are in B . (See Fig. 9.6.) Then C(A,B) is called a regular neighborhood of paths FIG. 9.5 Two paths and their associated contrasts as in definition 1. SEC. 9.5 CONTRAST TRANSMITTANCE FUNCTION 167 FIG. 9.6 A regular neighborhood of paths. in S if and only if the paths in C(A,B) have a common length r , a common beam transmittance T^ , and a common path radiance NJ . Here are some examples of regular neightborhoods of paths: Let A and B of the definition be the two boundary planes of a stratified plane-parallel medium X(a,b) , and let C(A,B) be the set of all paths parallel to a given direction £ . Then with stratified lighting conditions in X(a,b) C(A,B) is a regular neightborhood of paths in X(a,b) . Ob- serve that two paths of C(A,B) may be extremely remote, spatially. As another example, imagine two points x and y in a natural hydrosol, say the sea. It is clear, on in- tuitive grounds at least, that we can always find two spheri- cal regions A and B or two small parallel plane regions, A and B , about x and y as centers, respectively, such that the associated set C(A,B) of all paths with initial points in A and terminal points in B is, for all practi- cal purposes, a regular neighborhood. Our intuition in this matter is grounded in the general appearance of spatial con- tinuity (but not necessarily uniformity) in the inherent op- tical properties and light fields of real media. Finally, it is clear that if A and B consist of any two single distinct points, then C(A,B) is trivially a regular neigh- borhood of paths. The importance of the idea of regular neighborhoods of paths in an optical medium rests in the following observa- tion. Observation 1 . If C(A,B) is a regular neighborhood of paths in an optical medium 'X , and (? r (x ? , £) and $r ( x i > O ar e two paths of z ,z , respectively, then: C(A,B), with terminal points 168 GENERAL THEORY OF OPTICAL PROPERTIES VOL. V [ *7v^ J = l~ vw TTHTTIT (13) or more briefly: T°(x ,5) C r (z i)5 ) - C o (x i)5 ) T ^ vg) where l, C r (z 1 ,5) M and "C Q (x l ,^) n denote the apparent and inherent contrasts occurring in (13). This observation fol- lows readily from use of the fact that if both N r (z g ,£ s ) and N r (z ,£ ) are decomposed: N (z ,U = N°(z ,£ ) + N*(z ,£ ) r^ 2*^2^ r v 2'^2^ r v s'^2 N r (z i ,C i ) = N°(z i} ^) + N*( Zi ,^) and their difference taken, then, since C(A,B) is a regular neighborhood: WU - MVU ■ N°( V « 8 ) - N°(z i>?i ) = CW^ - N o (x i ,5 i ))T°(x i , ? ) Furthermore N r (z ,5 ) = N o (x i ,C 1 )T r (x i ,C) so that (13) follows. On the basis of this observation, we are led to make: Observation 2 . If C(A,B) is a regular neighborhood of paths in an optical mediuam X , then to each path Q T (x,E,) there is assignable the quotient T£(x,£)/T r (x,£) where x is in A and r and £ are such that x + r£ is in B , and this quotient is an apparent optical property and is generally dependent on x , r and £ . Contrast Transmittance and Its Properties The preceding two observations point up the fact that the quotient of transmittances in (13) is a number which can be assigned to each member of a regular neighborhood C(A,B) of paths. Since the quotient incorporates the apparent opti cal property T r (x,£), it i- s also an apparent optical proper ty in general. In view of this and observation 1, we can state: SEC. 9.5 CONTRAST TRANSMITTANCE FUNCTION 169 Definition 3 . Let C(A,B) be a regular neighborhood of paths in an optical medium X. The quotient Tf(x,£)/ T r (x,£), where x is in A and x + r is in B, is called the contrast transmittance of the path ^(x,^) in C(A,B), and shall be denoted by "JVCx^)" or "J^." Thus with the preceding definition (13) may be written succinctly as: C r (z if £) = C o (x i ,5)Jr(x 1 ,0 or C = C J r or Observation 3 . If (? r (x,£) is any path in a general optical medium X , with integrable a and K then: (14) The proof is immediate, using (11) and (12) and the multi- plicative property (8) of Sec. 4.5. Therefore *X T is deter- mined for a path of sight once a and K are known over that path. Another result which devolves on the multiplicative property (8) of Sec. 4.5 is: Observation 4 . If 6? r (x,C) and ^(y,5) are any two contiguous subpaths of a path C? r+S (x,^) (see Fig. 9.7), and 3~ T , 3" s , and J~x+ S are their respective contrast transmittances , then: J- r+s (x,£) = J! (x,OJ~(x+n,£) (15) or, briefly: «/ r+s xT r J\. In other words, the contrast transmittance , group property along with t£ , T r and the mittance operators ^T(x,y,z). The semigroup holds even for paths along which the index o nonconstant, and even discontinuous (see, e. 12.2). The value of (15) lies in the fact t transmittance of an extended path of sight i once the contrast transmittances of its part Some important special cases of (14) a case of stratified plane-parallel media. Ac of Sec. 4.5, we have: enjoys the semi- complete trans- property (15) f refraction is g. , (25) of Sec. hat the contrast s determinable s are known. re found in the cording to (18) T [ - (a + K cos 6)] (16) in such settings. Moreover, if we adopt a two-D model of the light field in an infinitely deep medium X(0,°°), then (16) reduces to: 170 GENERAL THEORY OF OPTICAL PROPERTIES VOL. V J exp [ " (ct k_ cos e)r } (17) where K now becomes the diffuse absorption coefficient k of the two-D theory (Sec. 8.5). In the case of one-D theory k_ = (Sec. 8.6). Observation 5 . If X is an arbitrary optical medium with mtegrable a and K, then the canonical form of the apparent radiance N r associated with a path (? r in X can be written in the form: (18) where 3~ r is the contrast transmittance of (? r . This observation follows at once from observation 3 and (15) of Sec. 4. 5. FIG. 9.7 The semigroup property for contrast transmittance. SEC. 9.5 CONTRAST TRANSMITTANCE FUNCTION 171 Alternate Representations of Contrast Transmittance In view of the fact that the component transmittances making up contrast transmittance ( definition 3) are them- selves expressible as ratios of radiances, it follows that the contrast transmittance of a path may also be represented as such. Indeed, from definition 3 and (4) and (5) : (19) for every path (? r (x,£;) with terminal point z = x + r£ . Hence ^T r may be usefully thought of as the ratio of the residual radiance to the apparent radiance of the associated path. The manner in which N r is generated along @ r may be through either scattering or reflection or both. The passes through interfaces , Since N, latter mechanisms occur when (P T such as the air-water interface. Since N r (z,£) is decom- posable into the sum of Nj(z,£) and N$(z,£), (( 5 ) °f Sec. 3.13), (19) yields up still other useful forms, such as the following : J~ (x r O = l - N r (z,?) (20) or j;(x,£) = 1 + NJ(z,C) N^(z,U J (21) or T r (x,0 - N r (z,5) - N*(z,5) N°(z,5) + N*(z,C) (22) or J" r (x,U = 1 N^(z,C) N*(z,U + 1 (23) or: 172 GENERAL THEORY OF OPTICAL PROPERTIES VOL. V X( x ^3 = rN r (z,0 I L N*(z,0 1J LN r (z,£) + 1 (24) In virtue of the fact that radiance values are invari- ably nonnegative, and that decomposition property (5) of Sec. 3.13 holds, it follows that: N°(z,C) £ N r (z,0 or: (25) N*(z,?) < N r (z,?) for every path "r(x,£)> so that (19) or any one of the pre ceding representations shows that: T r (x,0 < 1 (26) for every path (? r (x,£) in a general optical medium X . Some necessary and sufficient conditions that ^(x,^) be or 1 are clearly discernible from the preceding representa- tions. Observe in particular that: ^ T [ r L (27) if K = over (? r , as is readily apparent from (14). Condition (27) holds for example when the path C? r is a horizontal uniformly lighted path in an extensive homogene- ous portion of the sea or atmosphere. Furthermore, 3~ = 1 if and only if N* = . It is clear that N* = if and only if the index of refraction over the path is continuous at the point of consideration of N* (for an example where 5~ f 1 , see (20) of Sec. 12.2). A contrast transmittance £~ Q which is distinct from 1 is called a singular contrast transmittance, and the associated path (p a singular path. Contrast Transmittance as an Apparent Optical Property It was observed above (observation 2) that contrast transmittance is an apparent optical property, i.e., that it depends on the radiance field in the medium of interest. If the lighting along the path G > r is uniform, then we obtain such a corresponding extreme case as (27). On the other hand if the lighting along (? r is irregular, for example when (? r is directed through shadowed and sunlit regions, then the SEC. 9.5 CONTRAST TRANSMITTANCE FUNCTION 173 values of \T r manifest these shadowings and lightings in a regular and predictable way, as the following two examples will show. To fix ideas we shall initially choose a very simple setting. For the purposes of the first example, part (a) of Fig. 9.8 depicts a path £? r in a plane-parallel medium X(a,b) which is parallel to the boundaries of X(a,b) and at some depth y . The medium is stratified and has a strati- fied light field except in the shaded region shown, which simulates a shadowed part of the medium. For example, such a shadow may be produced by an isolated cloud over the ocean, or a ship shadow, etc. It is of interest to relate the con- trast transmittance of £) 1 N r (z,£) and N°(z,£) is unchanged, we have: 3r(z,0 iJ- r (z,5) ( (? is shadowed internally) r (28) for a shadowed path C? r as in (a) of Fig. 9.8. In the second example which is depicted, in case (b) of Fig. 9.8, both Nj(z,£) and N_(z,£) are affected and exhibit a de- crease. However n£( z >£) is decreased more than N r (z,£) so that: JrCz.o i?;(z,a ( (? nas shadowed background) (29) for the path (? r shadowed as in (b) of Fig. 9.8. This may be seen by inspection of (20). For, by hypothesis, N$(z,£) is unaffected (no shadow from x to z) and N r (z,£) is de- creased, so that the quotient in (20) is increased, the dif- ference in (20) thereby decreased, which was the effect to be shown. The apparent radiance formula (5) of Sec. 3.13 may be used to obtain quantitative estimates of the increase or de- crease of and "T°" will be short for T°(v,£). Then for the portion of path (? r in (a) of Fig. 9.8 extending from v to z , we have from (5) of Sec. 3.13: N(z) = N(v)T° + N* Further, for the segment from shadowed: to before the path is N(v) = N(u)T£ + N* where we have written "N(u)" for N(u,£) now shadowed, then N* decreases to, say write (ad hoc) : for N*/N t If the path is Let us SEC. 9.5 CONTRAST TRANSMITTANCE FUNCTION 175 We can now estimate the magnitude of the shadowed apparent radiance N # (z) of (? r as follows: Let " N # (v) " denote the shadowed apparent radiance at v . We then have: N # (v) = N(u)T° + N* Cu)T° cN t = N(u)T° + N* + (c-l)N* = N(v) + (c-l)N* Returning to N (z) , we have: N # (z) = N # (v)T° + N* = [N(v) + (c-l)N*] T° + N* = N(z) + (c-l)T°N* u Hence the desired representation of N(z) - N (z) is" N(z) - N # (z) = (l-c)T°N* (30) Now in general, the smaller t is , the smaller Nj will be and so by (30), the less effect the shadow will have. Further, the farther away the shadow is (i.e., the greater s is) from the point of observation, the less effect the shadow will have, since Tg decreases as s increases in real media. A simi- lar analysis can be made for part (b) of Fig. 9.8, and it is left to the reader to show that, in this case: N(z) - N # (z) = (l-c)T°N* (31) From (30) and (31) we see that both cases are covered by the same type of formula. It is interesting to observe that if the shadow region from u to v in Fig. 9.8 straddles point x in just the right way, the contrast transmittance of the path ^(x,^) is unchanged, even though, as (30) and (31) in- dicate, there are definite changes in the lighting conditions. An examination of the preceding arguments would show that no essential use was made of the plane-parallel struc- ture of the medium depicted in Fig. 9.8, nor of its strati- fied light field. Furthermore, the shadowing factor c may just as well have been a lighting factor (i.e., c > 1) , with- out affecting the algebraic structure of the resulting equa- tions (30) and (31). We use these observations to generalize the results and to summarize our findings as follows. 176 GENERAL THEORY OF OPTICAL PROPERTIES VOL. V FIG. 9.9 The general setting for Fig. 9.8. Let (P r (x,£) be a path in a general optical medium X , and let (? t (u,£;) be a proper oollinear path with respect to (? r (x,^) i.e., for some positive or negative number s , u = x + s£ and r :> s + t (see Fig. 9.9 in which distance is mreasured positive from x to z , and negative from z to x) . Suppose that the path radiance N*(v,£) of 6 > t (u,Q (where v = u + t£) changes by a factor c . This change generally affects the apparent radiance N r (z,£), where z = x + rE, . Let M Nf(z,£) " denote this new apparent radi- ance, then: N r (z,5) Nj(z,£) = Cl-c)T;. (s+t) (v,5)N*(v,« (32) If the proper collinear path (? t (u,£) is wholly contained within 6r(x,£) (i.e., s > 0) then the contrast transmittance SEC. 9.5 CONTRAST TRANSMITTANCE FUNCTION 177 -CT r (x,£;) corresponding to the factor c is greater or less than ^(x,^) according as c is < 1 . On the other hand, if the proper collinear path (? t (u,£;) is behind and disjoint from (? r (x,£) (i.e., s<0, and s + t < 0) then the contrast transmittance £T* (x,£) corresponding to the factor c is less or greater than £T r (x,£;) according as c is < 1 . A quantitative estimate of ^r|(x,^) may be based on (32) by solving for N r (z,£) and using any one of (19) -(24). In case s£0, then N£(z,£) is also changed, and must be com- puted accordingly, using (32), and (5) of Sec. 3.13. In general three cases should be distinguished for ^rj(x,5): the proper collinear path (? t (u,£) is either (a) contained in 5\.(x,C), (b) behind and disjoint from 0. On the Multiplicity of Apparent Radiance Representations One final observation may be made at this point on the use of (5) of Sec. 3.13 in deducing the preceding properties of contrast transmittance, and which helps cast some light on an interesting general fact about the concept of apparent ra- diance. The attentive reader will have noticed that there appears to be an infinite number of choices in the manner of representing a radiance N(x,£) as an apparent radiance asso- ciated with some path (? r (x,£;). This fact was essentially observed in Sec. 3.13 in the discussion of (5) of that sec- tion. However, the present applications of that equation in deducing (30), (31) , and (32) bring home the multiplicity of the representations with renewed force; and we shall now formalize this fact for future reference. The observation may be phrased generally as follows. Suppose N(z,£) is the radiance along the direction £ at some point z in an op- tical medium X . If now we place a path (an imaginary con- struct) in X with direction £ so that x is its initial point and z = x + vE, is the terminal point (Fig. 9.7), then we may immediately reorient our conception of N(z,£) from that of a primitive radiance in X to that of an apparent radiance of the medium at x associated with the path ^r (**£)• That is, by (5) of Sec. 3.13, we can write: N(z,Q = N(x,OT°(x,£) + N*(z,S) (33) and to point up the fact that N(z,£) and N(x,£) are viewed in the framework of the path (? r (x,£), we may, as is custom- ary, attach " r " and " o " subscripts to them, respectively. By imagining still another path £? s (y,S) with direction £ and initial and terminal points y and z = y + s £ , we can represent the same N(z,£) in (33) as: N(z,€) = N(y,5)T°(y,0 + N*(z,0 (34) This phenomenon of the multiplicity of possible representa- tions of a given measurable radiance N(z,£) with respect to 178 GENERAL THEORY OF OPTICAL PROPERTIES VOL. V imaginary paths in an optical medium is reminiscent of, and actually logically related to, the freedom of choice of the parameters x and z (Holding y fixed) in the statements of the principles of invariance (e.g., in Example 3 of Sec. 3.7) or the invariant imbedding relation (e.g., in Example 4 of Sec. 3.7). 9. 6 Classification of Optical Properties We conclude this chapter with a summary of the main optical properties introduced and developed in the present work. We shall classify the properties in several ways, according to the dimension of the medium to which they pri- marily apply, and according to whether they are local, glo- bal, inherent or apparent optical properties. TABLE 1 Generic Inherent Optical Properties for One- , Two- , and Three-Dimensional Media Dimension Optical Property Section 1 (paths) 2 (surfaces) 3 (solids) R , T r + , t + 3.17 3.3 3.8 The term "generic," used in Table 1 to describe the listed optical properties, refers to their ability to gener- ate all the secondary optical properties associated with the respective media, as explained in the various sections of Chapter 3. Thus, e.g., ^(X;a,b) can generate all the stan- dard reflectance and transmittance operators for plane- parallel media. Below are tables of optical properties for plane- parallel (or generally one-parameter) optical media, the media of principal interest in hydrologic optics. The prop- erties above each level may be used to deduce those on that level in the manner explained in the notes or references ac- companing each table. Several explanatory comments on Table 2 can be made. First, the operators P+(z) and t+(z) were defined in (3) and (4) of Sec. 7.1. The functions f(z,£) and b(z,£) are added to complement f(z,±), b(z,±) of Table 4 below, and are defined by writing: f(z,S)" for a(z;?;£')dfiU') : + U) (1) "b(z,C) M for f a(z;S;€*)dfi(£') m (2) SEC. 9.6 CLASSIFICATION OF OPTICAL PROPERTIES 179 TABLE 2 Local Inherent Optical Properties for Plane-Parallel Media a(z) o{z\V ;0 P + (z) b(z f O t + (z) s(z) I a(z) transition to global level transition to global level J?(X;a,b) Here H ± (£) is the set of all directions £ ' such that £ • £' _> (for +) or £ • £• < (for - ). If the medium is isotropic, then f(z,£) and" b(z,£) are independent of £. The dashed line in the diagram of Table 2 represents the end of Table 2 and serves first of all to indicate the transition to the global level of Table 3, and is also in- cluded in order to show how Tables 2 and 3 are related, and finally to emphasize the fact that the pair (a , a) is funda- mental in the sense of Definition 3 of Sec. 9.1. The proof of this feature of (a , a) is given in Sec. 22 and Sec. 23 of Ref [251]. The operators R and T are given in (5) and (12) of Sec. 3.17, and J is introduced in (6) of Sec. 3.8 and studied in Example 4 of Sec. 3.9. Table 3 performs a double task of describing both the inherent and apparent global properties of plane-parallel media. The inherent properties are in force in Table 3 when radiance is being used; the apparent properties are in force when irradiance is being used. The operators 180 GENERAL THEORY OF OPTICAL PROPERTIES VOL. V TABLE 3 Global Inherent or Apparent Optical Properties for Plane-Parallel Media *f(X;a,b) f fy(u,x;v,z) ¥(x,y) 1 m (x,y) 7??(x,y,z) transition to local level 6lO,y,z) I R(x,z) P + (z) I ^7(x,y,z) I T(x,z) T + (Z) transition to local level a(z) o(z;V JO ' /? 2(u,x; v, z) and ^(x,y) are introduced in Examples 6 and 7 of Sec. 3.7. The operator ¥(x,y) is introduced in Example 3 of Sec. 3.9. The invariant imbedding operator "^(x,y,z) is in- troduced in Example 4 of Sec. 3.7, along with its component operators $.(x,y,z) and ^tx,y,z). The standard operators R(x,z) and T(x,z) are introduced in Sec. 3.6. The transi- tion to the local level is indicated by a dashed line which signals the end of Table 3 and is also included in the table to emphasize the fact that Q?(X;a,b) is a fundamental optical property (in the sense of Definition 3, Sec. 9.1). The proof that ,0(X;a,b) is a fundamental optical property is based on the derivations given in Sec. 125 of Ref. [251]. The optical properties of Table 4 are written for undecomposed irradiance fields. By appending star or circle superscripts,* the properties for diffuse or residual *The basis for this notational device is described in Sec. 8.4. SEC. 9.6 CLASSIFICATION OF OPTICAL PROPERTIES 181 TABLE 4 Local Apparent Optical Properties for Plane-Parallel Media a(z) o{z\V,K) i i transition to the level ' D(z ,±) i P(z,±) a (z T(Z,±) ,±) b(z,±) s(z a(z transition to the f(z',±) ,±) ,±) level ' R(z,±) k(z,q k(z,±) K(z k(z) irradiance are obtained. In the interests of brevity, these additional concepts are not diagrammed. The primary level of optical properties in Table 4 for the most part parallel their inherent counterparts in Table 2. They may be thought of as hybrids resulting from the union of inherent optical properties and the light field. The secondary apparent level summarizes some optical properties which are on the border- line between local and global properties. For example, R(z -) is the reflectance at level z , and thus, being defined at a point at level z , is ostensibly local in nature. However, the value of R(z,-) is intimately tied to the values of the light field and the inherent optical properties of the medium at all levels above and below level z . The classical coun- terparts to the secondary optical properties of Table 4 arose in the solutions of the one-D two-flow models of the light field. The secondary properties appearing in Table 4 are the exact, directly observable counterparts to the earlier one-D model concepts. The various primary properties occurring in Table 4 are defined throughout Chapter 8, the secondary prop- erties (including K(z,£)) are defined in Sec. 9.2. 182 GENERAL THEORY OF OPTICAL PROPERTIES VOL. V 9. 7 Bibliographic Notes for Chapter 9 The development of the directly observable quantities for light fields in natural hydrosols, as presented in Sec. 9.2, is based on Ref. [222]. A published version of this reference was made available in the literature in Ref. [247] The covariation of K(z,-) with D(z,-) is based on the re- sults in Ref. [242]. The analytical representation of the observable reflectance function developed in Sec. 9.4 is drawn from Ref. [246]. The concept of contrast in the form of a relative dif- ference of two radiances occurs in the writings of Mecke [174] and was applied by Koschmieder [141] in his classical studies of 1924. Systematic use of the concept of contrast was made by Duntley [71,72] in 1948 in the study of visi- bility in the atmosphere. Further uses of contrast in the atmosphere are given in Middleton's [177] work. Applica- tions of the contrast concept to hydrologic optics were re- ported [82] and systematically generalized [210]. These generalizations were subsequently applied to the atmospheric context in [80] and in [228]. The discussion of Sec. 9.5 is an outgrowth of the work in [210]. The optical properties of plane-parallel media were first studied via the one-D model (the Schuster two-flow theory) discussed in Sec. 8.10. The definitions and classi- fications of the optical properties in Sec. 9.1 and Sec. 9.6 are based on the unifying concept of the interaction princi- ple and are for the most part new. Some preliminary classi- fications preparing the way to those in Sec. 9.1 and 9.6 are given in [210] and [247]. CHAPTER 10 OPTICAL PROPERTIES AT EXTREME DEPTHS 10.0 Introduction In this chapter we examine some theoretical and experimental evidence for possible regular behavior of apparent optical properties at deep and shallow depths in natural bodies of water. These extreme depths in a natural hydrosol are the settings of interesting and complex radiometric interactions of light from the sky with the body of the medium. The ob- served interactions at these depths are exaggerated either because of the extreme proximity of the air-water boundary or because of its extreme remoteness and thus present to both theoretical and experimental workers a challenging puzzle to unravel and bring conceptual order to the understanding of the light field observations at these depths. In the attempt to understand the radiometric phenomena at extreme detphs , we shall be led to supplement the collection of laws governing the optical properties derived in Chapter 9 (which hold for all depths in a natural hydrosol). Furthermore, by concen- trating on the extreme depths in the present chapter, we can extract correspondingly more detailed behavior of the observ- able K-functions of both the radiance and irradiance fields. Our investigations are based on the general equation of trans- fer and on the exact two-flow equations for irradiance of Chapter 8; in particular we shall make extensive use of the accurate two-D model for irradiance fields developed in Chap- ter 8. We shall begin with the study of irradiance fields at shallow depths in media with calm surfaces, our primary aim being to discern from both theoretical and experimental clues, the precise nature of the depth-behavior of the upward and downward irradiance fields H(z,±) at these depths. By exam- ining these fields via their K-functions K(z,±) we shall, as it were, be examining them under a powerful magnifying glass by means of which every tendency of decay or growth of H(z,±) is limned in bold relief against a conceptual background of algebraic signs and magnitudes of derivatives. The second half of the chapter is devoted to the study of the natural light fields at great optical depths as they occur in oceans, harbors, and deep lakes. At remote distances from the generally disturbing air-water boundary of the medium, the radiance distributions eventually attain a smooth, characteristic shape independent of the external lighting conditions and dependent only on the inherent optical prop- erties a , a of the medium. The problem of the nature of the 183 184 OPTICAL PROPERTIES AT EXTREME DEPTHS VOL. V light field at great depths in natural waters has been com- pletely solved only recently. As a result of having a theo- retical basis for the existence of the regular behavior of the light field at great depths (for most practical purposes, beyond 10-20 attenuation lengths), we can justify certain simplifications of the models for light fields below such depth intervals. In particular, in such regions the classi- cal canonical form of the equation of transfer (Chapter 4) for radiance, and the two-D model for irradiance (Chapter 8) can be demonstrated to serve as accurate tools with which to quantitatively predict the magnitudes of the natural light fields. 10.1 On the Structure of the Light Field at Shallow Depths : Introductory Discussion In this section experimental determinations of the up- welling and downwelling irradiances are studied with the purpose of explaining certain observed regular nonlinear trends in the semilog plots of these irradiances, principally at shallow depths in media with flat calm surfaces. We shall develop a mathematical model from the general equation of transfer which describes these irradiances in great detail over the shallow-depth range. The model explains the observed phenomena in terms of the inherent optical properties of the medium and its external lighting conditions. On the basis of experimental evidence, cited below, and on the basis of supporting theory, the following hypothesis about light fields in all homogeneous natural hydrosols is proposed: (a) The ratio of the upwelling irradiance to the downwelling irradi- ance, i.e., the observable reflectance function R(z,-), is invariably monotonic increasing or decreasing at shallow depths with increasing depth (depending on the medium) and approaches a limit which is independent of the external light- ing conditions and which depends only on the inherent optical properties of the medium. (b) The logarithmic derivatives, i.e., the K-functions K(z,±), of the upwelling and downwell- ing irradiance at shallow depths are monotonic increasing or decreasing with increasing depth (depending on the medium) and approach a common limit which is independent of the ex- ternal lighting conditions and which depends only on the in- herent optical properties of the medium. In this way we arrive at a fairly detailed understanding of the light field at extreme depths (shallow and deep) in all homogeneous natu- ral hydrosols. To set the stage for the general reader, the following observations on the empirical roles of the K and R func- tions will be helpful. For many practical purposes in applied hydrologic optics the downwelling irradiance H(z,-) at a depth z in a natural hydrosol can be represented by the fol- lowing simple formula H(z,-) = H(0,-)e" Kz , (1) where K is a fixed number which characterizes the overall flux transmitting properties of the hydrosol. A similar for- mula may be used to determine the upwelling irradiance H(z,+) at any depth z : SEC. 10.1 STRUCTURE AT SHALLOW- DEPTHS 185 H(z,+) = H(0,+)e~ Kz , (2) where--again for many practical purposes--K is a fixed number and in fact identical to the one appearing in (1) . Simple models of the light field, such as the two-D and one- D models of Sees. 8.5 and 8.6 supply detailed formulas of the kind (1) and (2) . Still another practical formula is the one which de- scribes the depth dependence of scalar irradiance h(z) at each depth z : h(z) = h(0)e" Kz , (3) where K is the same number as that appearing in (1) and (2). In practice H(z,+) and H(z,-) are measured by suit- ably designed horizontal flat plate collectors exposed to the appropriate hemisphere, and h(z) is measured by a suit- ably designed spherical collector. In view of (1) , (2) , and (3) , relatively quick estimates of a K for a particular natural body of water can be obtained by measuring any one of these three radiometric quantities at two distinct depths, and using the formula: where " A(z) " stands for any one of the three quantities: H(z,+), H(z,-), or h(z) at depth z. The practical procedures of hydrologic optics, sum- marized in formulas (l)-(4), are quite analogous to the fol- lowing well-known procedure used in applied heat conduction studies to estimate the temperature T(t) of a cooling spheri- cal body at time t immersed in an infinite bath of zero temperature: -kt T(t) = T(0)e KZ , (5) where k is a known fixed number which characterizes the overall heat conducting properties of the material comprising the spherical body. Conversely, (5) may be used to estimate k by measuring T(t) at two distinct times and using a for- mula exactly analogous to (4). The specialists who use (5) are aware of the fact that it is a useful approximate formula which becomes an exact formula for T(t) in the limit as t •*■ °° . They also realize that (5) becomes quite inadequate for relatively accurate estimates of T(t) whenever t is small, and must resort in such estimates to more general forms representing T(t). These more general forms, of which (5) is a special limiting case, are well known and are solidly founded in the general 186 OPTICAL PROPERTIES AT EXTREME DEPTHS VOL. V theory of heat conduction, and experimental fact, and may be found in standard treatises on heat conduction. Equations (l)-(3) are regarded by the specialists in hydrologic optics in much the same way as (5) is regarded in its own discipline: They are useful approximate relations which can be shown to become exact formulas for H(z,±) and h(z) in the limit as z -+ °° in deep homogeneous plane paral- lel optical media (see, e.g., Sec. 10.7). Perhaps what is not well known--or at any rate not fully realized--is that, like (5) these equations do not exactly represent H(z,±) or h(z) for small values of z , even in homogeneous hydrosols with uniform external lighting conditions and perfectly calm air-water surfaces. Thus, for relatively accurate estimates of H(z,±) and h(z) such as those required in basic scien- tific studies of the light fields in natural hydrosols, (1)- (3) are quite inadequate. They do not represent the small but experimentally demonstrable departures from linearity of the semilog plots of H(z,±) and h(z). What is required at present in the discipline of hydro- logic optics is a set of more general formulas which can accurately represent the quantities H(z,±) and h(z) in the small z ranges and which reduce to these simpler formulas in the limit as z -»■ °° . One of the two purposes of the following sections is to present a set of formulas for H(z,±) which yield a closer approximation to reality than (1) and (2). The search for these formulas was motivated by the results of recently per- formed accurate measurements in the light field in real natu- ral hydrosols, and their derivations are founded on the tenets of general radiative transfer theory. The second purpose of the present discussion is to ex- amine the resulting formulas for indications of possible gen- eral qualitative rules that may be hypothesized about the fine structure of shallow-depth light fields, and to put the hypotheses into forms which will be amenable to further theo- retical study or experimental verification. On the basis of the models constructed below it was possible to formulate three such hypotheses about the quantities: K( 7 + i - 1 dH(z,±) ffi . and introduced in Chapter 9. These hypotheses governing the functions K(z,±) and R(z,-) are presented in detail below in Sec. 10.3, but for the present we shall undertake some preliminary discussion of the roles of these functions in the study of natural light fields. As noted in Sec. 9.3, the quantities K(z,+) and K(z,-) are simply the slopes of the semilog plots of H(z,+) and H(z,-). According to the simple formulas (1) and (2), these slopes do not change with depth and in fact are of the form: SEC. 10.2 EXPERIMENTAL BASIS FOR THEORY 187 K(z,+) = K(z,-) = K , where K is defined in (1) and (2) . Careful experiments show, however, that K(z,+) and K(z,-) are generally dis- tinct numbers that do change with depth. Furthermore, we shall see in Sec. 10.7 that, in homogeneous media, lim K(z,+) = lim K(z,-) = k where koo is a number which depends only on the inherent optical properties of the medium and is completely independ- ent of the external lighting conditions on the upper boundary of the medium. One of the goals of the present chapter is to find out something about the nonlinear behavior of K(z,±) at relatively small depths. The quantity R(z,-) summarizes the flux transmitting and reflecting properties of the medium both above and below the hypothetical plane at depth z . According to the simple formulas (1) and (2) RU ' ] ' H(0,-) a fixed number for all z . Careful experiments show, how- ever, that R(z,-) changes with depth; and in all homogeneous media it will be shown (Sec. 10.7), to approach a well-defined limit as z* °° : lim R(z,-) = R , where Roo is a number which depends only on the inherent op- tical properties of the hydrosol and is completely independ- ent of the external lighting conditions on the upper boundary of the medium. Another of our present goals is to understand the nonlinear behavior of R(z,-) for relatively small values of z . 10.2 Experimental Basis for the Shallow Depth Theory To prepare the groundwork leading to the theory of the light field at shallow depths, we now consider some experi- mental data which supplies graphic evidence of the nonlinear depth behavior of K(z,±) and R(z,-) in near-surface regions of a specific hydrosol. The experimental evidence presented in this and the following sections has been computed from the data obtained in Lake Pend Oreille, Idaho, by J. E. Tyler [298]. Figure 10.1 depicts the semilog plots of H(z,+) and H(z,-) over the range of depths 5 < z <_ 55 meters; H(z,±) are associated with a wavelength interval of width 64 my centered at 480 my. This depth range corresponds to a range of about 20 optical depths, so that the light field in the vicinity of 50 meters should have for all practical purposes attained the asymptotic limit- -assuming complete homogeneity of the medium. 188 OPTICAL PROPERTIES AT EXTREME DEPTHS VOL. V 100,000 • i 1 1 i 1 1 r PLOTS OF : \ UPWELLIN6 IRRADIANCE H(Z,+) DOWNWELLING IRRADIANCE H(Z,-) AS FUNCTIONS OF DEPTH Z 10 20 30 40 50 DEPTH Z METERS 60 FIG. 10.1 Irradiance plots (480 my) from Tyler's study of Lake Pend Oreille, Idaho, spring 1957. Just how close is t geneous? To answer this, volume attenuation functi 10.2 shows a plot of a An optical medium is by d constant function within a plot of Figure 10.2 th homogeneous at the time o 12 percent variation in t depth range. In several range, there is a relativ percent in the values of these changes are negligi poses of the present disc interest. In addition to ume absorption function a he present hyd we must know on a within versus depth f efinition homo that medium, at the hydroso f the experime he values of places , in par ely abrupt cha a . For many ble. However, ussion, these a (z) , the va re plotted for rosol to being homo- the values of the the medium. Figure or the present medium. geneous if a is a It is clear from the 1 was not strictly nt. There is about a a over the indicated ticular the bracketed nge of the order of 5 practical purposes for the specific pur- changes are of extreme lues a(z) of the vol- several depths. SEC. 10.2 EXPERIMENTAL BASIS FOR THEORY 189 D(Z,+) 2.9 2.7 D(Z,-) 1.4 1.2 ( a /m 0.460 0.440 0.420 0.400 ■D(Z,+)* »♦■ . . . t - . — DISTRIBUTION FUNCTIONS D(Z,±) — -D(*Z,-)*' ■ ...J !___! 1 L i i i i i i — i — i — i — i — i — i — i — i_i — j — i — ■ i 10 20 30 40 50 _i 71 VOLUME ATTENUATION FUNCTION a(Z) (+) «5i~; VOLUME ABSORPTION FUNCTION a(Z) (e) T. . Vf. L-»» % 0.140 OI20 0.100 10 20 30 DEPTH Z METERS 40 50 FIG. 10.2 Optical properties a(z), a(z), and D(z,±) of Lake Pend Oreille, Idaho (for 480 my) as determined by Tyler in spring 1975. Observe how tne values a(z) tend to follow the changes in a.(z). This feature will be noted again later in this study when the mathematical model is being discussed. With this background information in mind we may now turn to a detailed examination of the plots of H(z,±). Observe that each of the plots in Fig. 10.1 exhibits a small but noticeable nonlinearity. The curves are slightly concave upward, indicating relatively steep slope at shallow depths and less steep slopes at greater depths. To facili- tate the examination of the logarithmic slopes, they have been plotted as functions of depth in Fig. 10.3. Both K(z,+) and K(z,-) exhibit a uniform downward trend toward a common asymptote defined by the horizontal line across the figure at ordinate k^ = 0.178/meter. This value was obtained using the fact, cited earlier, that K(z,±) has a common limit and then performing a suitable extrapolation based on this fact (see Ref. [263]). The uniform approach of K(z,±) to this common limit appears to be interrupted in the neighborhood of 40 meters. There appears to be some optical disturbance in the medium within the immediate depth range (bracketed in Fig. 10.3) that results in a marked deviation of these curves from their expected paths. We can explain this anomalous behavior on the basis of our observations of the depth dependence of a(z) in Fig. 10.2. The abrupt change in the values of a in the same depth range appear to hold the key to the explanation when the following formulas are examined: 190 OPTICAL PROPERTIES AT EXTREME DEPTHS VOL. V 0.220 0.200 - +1 * 0.180 - 0.160 20 30 DEPTH Z METERS 50 FIG. 10.3 K- function values (at 480 my) for Lake Pend Oreille, Idaho, spring 1957, computed from Tyler's data. K(z,-) = a(z,-) K(z,+) = - a(z,+) I. N*(z,5)dfi(0 H(z,-) H(z,+) (1) (2) These are exact formulas relating K(z,±) to the values of a and the angular distribution of the light field at depth z . These formulas need not be derived here; they readily follow from the representations of the K-functions in (18) and (19) of Sec. 9.2. See the derivation of (26) of Sec. 9.2. Moreover, the definitions of the quantities a(z,±) : a(z,±) = a(z)D(z,±) are given in Sec. 8.3. Here D(z,±) are the values of the distribution functions for the downwelling (-) and upwelling (+) streams of radiant flux at depth z . Figure 10.2 shows a plot of D(z,+) and D(z,-) for the present hydrosol. It is evident that these quantities are nearly constant over the entire depth range under study. SEC. 10.2 EXPERIMENTAL BASIS FOR THEORY 191 On the basis of (1) , whenever there is an abrupt change in the values a(z) over some small depth interval and when- ever D(z,-) is relatively fixed over this depth interval (so that the integral term is relatively constant) , we predict that K(z,-) must exhibit a change in the same direction as that, of a(z). Thus if a(z) abruptly decreases, K(z,-) is expected to exhibit a decrease in value. On the basis of (2) , on the other hand, under the same conditions, we should expect a simultaneous change of K(z,+), but in the opposite direction as that of a(z). Thus if a(z) abruptly decreases, K(z,+) is expected to exhibit an increase in value. Returning now to Figs. 10.3 and 10.2, these predictions are apparently borne out by the portions of the a , and K curves in the bracketed depth range. Therefore the abrupt inhomogeneity in the structure of the hydrosol in this depth range appears to induce the observed interruption of the or- derly trend of the K- curves toward their limit. One might inquire why the comparable change in a in the vicinity of 20 meters does not produce a similar marked effect on the K - curves. The answer lies in the fact that the light field in the vicinity of 5-30 meters (2-12 attenua- tion lengths) is evidently still in the process of settling down and attaining a spatial steady state configuration, so that changes in K - values are naturally relatively great in this region; changes in a - values thus have relatively little additional influence on the fine structure of the K - functions, and their effects are obscured by the settling- down changes taking place. However, at around 40 meters (about 16 attenuation lengths) the light field has begun to assume its asymptotic angular structure. Any abrupt change in a(z) would now cause the entire smoothing process to re- commence; in particular the K - values are now very close to their limit, and the effect of any inhomogeneity would be relatively magnified. As a final step in the examination of the experimental evidence, we turn to Fig. 10.4 which exhibits a plot of R(z,-) versus depth. In this case there is a uniform upward trend, as depth increases, of the values of R(z,-) toward the limiting value Roo = 0.0278. This limiting value may be found from the formula: where k -a(-) k +a(+) a(±) = aD(±) (3) The quantities D(±) are the limits of D(z,±) as z + °° , and are found from the plots of D(z,±) in Fig. 10.2. The values used were D( + ) = 2.77, D(-) = 1.33 ; koo is as defined in Fig. 10.3 and a was taken as the value of the volume absorp- tion function at depth 42 meters: a(42) = 0.123 per meter. The basis for (3) will be established in Sec. 10.7. However, 192 OPTICAL PROPERTIES AT EXTREME DEPTHS VOL. V R(Z,-) 0.030 0.028 0.026 0.024 0.022 0.020 . . - .- n - — • *^_ - - R Ct> _**•»♦ ""^""^ ■**■»»- - ^♦"*""*^ - - - ^^ - .R(Z,-U^ OBSERVED R-VALUES - - L_J 1 1 1 J 1 1 1 1 1 i i i i i i i i i i i — i — i — 10 20 30 40 50 DEPTH Z METERS FIG. 10.4 Reflectance function data (at 480 my) for Lake Pend Oreille, Idaho, spring 1957, computed from Tyler's data. we already have a version of it given us by the two-D model for irradiances in (102) of Sec. 8.7. As in the case of the plots of K(z,±), the plot of R(z,-) exhibits a change in trend in the bracketed depth range discussed earlier. From the exact representation: R(z,-) = K(z,-) - a(z,-) " K(z,+) + a(z,+) of the function R(z,-) (given in (25) of Sec. 9.2), and the observed changes in K(z,+) and K(z,-), we see that the ob- served anomaly, namely the downward trend exhibited by R(z,-) in the vicinity of 40 meters, is traceable directly to the abrupt change of a(z) in this vicinity. Another way to see the cause of the change of slope of R(z,-) in Fig. 10.4 is to examine (32) of Sec. 9.2 in tne neighborhood of the cross- ing of the K-curves in Fig. 10.3. Summary of the Experimental Evidence We may now summarize the preceding observations: (a) Over the depth ranges where the hydrosol is prac- tically homogeneous, the magnitudes of the K- functions ex- hibit a monotonia decrease with increasing depth, with K(z,-) > K(z,+). It appears that if the medium were homogeneous and infinitely optically deep, the monotonic decrease would con- tinue indefinitely toward a common limit k^ . (b) Under the same conditions as in (a) , the values R(z,-) appear to exhibit a monotonia increase toward a well- defined limit R m . SEC. 10.3 FORMULATION OF MODEL 193 (c) The distribution functions D(z,±) are practical- ly constant with depth. (dj The ratio of a(z) to a(z) and hence the ratio s(z)/a(z), where a(z) = a(z) + s(z) , appears to be prac- tically constant with depth. 10. 3 Formulation of the Shallow-Depth Model for K and R Functions On the basis of the Sec. 10.2, in particular s the two-D model of the lig The equations forming the plored in detail throughou remains to solve the equat lar context at hand. We s composed light fields as g the assumptions leading to emphasize that the optical (i) Optically infin (ii) Separable. (Th with depth--see Sec. 10.2.) (iii) Irradiated by a of magnitude N the normal to i experimental evidence summarized in tatements (c) and (d) , we may adopt ht field as developed in Sec. 8.5. basis of this model have been ex- t Chapter 8. Therefore it simply ions of this model for the particu- hall adopt the two-D model for de- iven in Sec. 8.5. In addition to (67) of Sec. 8.5, we specifically medium is: ite deep. e ratio of s(z)/a(z) is invariant experimental statement (d) of collimated radiance distribution incident at an angle O from ts upper boundary. Formulas for H(z,±) It follows from the two-D theory, in particular (71)- (73) of Sec. 8.5, that under the present conditions the ex- pressions for H(z,±) (= H°(z,±) + H*(z,±)) are: H(z,-) = n' H(z, + ) = N C(y Q ,-)e ° |c(u ,-)R c Koo Z -ko=z [M -C(M ,-) az/y C(y o ,+)e- az/ ^o (1) (2) The quantities C(u ,±), k^ , and R^ and their com- ponent parts are defined in detail in Sec. 8.5. It is of interest to compare (1) and (2) with (37) and (38) of Sec. 8.6. For convenience we repeat basic formulas of Sec. 8.5; (72) of Sec. 8.5 is of the form: o + (y )b*( + ) + a + (yj C ^o>- + ) = a*(+) + b*(+) + -11 where y = cos (cf. (70) of Sec. 8.5). Furthermore: (3) 194 OPTICAL PROPERTIES AT EXTREME DEPTHS VOL. V a*( + ) + b*( + ) - a*0) " b*(-) (a*( + ) + b*(+) + a*(-) + b*(-)) - 4b*(-)b*(+) (4) and we have chosen to adopt the notation " koo ," to point up the infinite depth of the medium. Hence: k < a (5) Finally from (102) of Sec. 8.7: k +a*( + ) (6) Because of assumption (iii) , the H(z,±) depend implicitly on the quantity y and to explicitly note this we would write "H(z , ± ;y ) • " If the medium is irradiated by an arbitrary radiance distribution N°(u,cj>) then the associated irradiances are found by an appropriate integration of the normalized forms of (1) and (2) (set N° = 1 in (1), (2)): 2TI H(z,±) = f j H(z,±;y )N (u,c|>)dyd o' y- o However, the present results can be deduced by a direct examination of (1) and (2) for an arbitrary y with- out having to consider the general y - effects as summarized in the preceding equation. Formulas for K(z,±) Using the definitions of the K - functions and the for- mulas for H(z,±) given in (1) and (2), we have -k z K(z,-) = C(y o ,-)e" k ~ Z + (%- c ^c-)) -az/y (7) or K(z,-) = — A(y ,-)e o — - k 1 - A(y Q ,-)e (t - k -) (8) SEC. 10.3 FORMULATION OF MODEL 195 where we have written: "A(u o ,-) M for C(y Q ,-) - y -cur— y- (9) Furthermore: K(z,+) = k C(p ,-)R e" k - z - -5L C (y , + )e" az/y o O C(y ,-)R e~ k °° Z - C(y ,+) e ~ az/y ° (10) or K(z,+) = k - — A(y , + )e 00 y o o " I"" - k ) 1 - A(y , + )e V^o ° ' (11) where we have written: 1 C ^o> + ) (12) Formulas (8) and (11) may be used to predict the depth depen- dence of K(z,±). We deduce immediately from these equations the general fact that: lim K(z,±) = k 7 ->-oo v * J c (13) Furthermore the K - functions approach this limit in a mono- tonic manner, as can be seen by taking their derivatives with respect to z : dK(z,±) _ dz f-SS--k I A(y ,±)e K m) 1 - A(y ,±)e " (~ ' k ^ z ] (14) The monotonic behavior of K(z,±) follows from the ob- servation that the derivatives dK(z,±)/dz are of constant sign for all z and for arbitrary choice of y . In par- ticular, the model predicts that: dK(z,±) > if A(y Q ,±) > K increasing ,^^ H concave downward and: 196 OPTICAL PROPERTIES AT EXTREME DEPTHS VOL. V dK(z, and: = if A(y ,±) = \ K constant / H linear (16) dK(z,±) the e mariz and tied spect incre ing t ing c lient (summ tions turn dz It t xperi ed in K(z,- by ev ively ase o o the ondit opti arize of t to th if A(u o ,±) K decreasing H concave upward (17) hus ment (a) ) in alua , an r de pre ions cal d by his e co appears that ally observe of Sec. 10. crease or de ting the qua d applying t crease of th sent model, (summarized properties o a , k M , A fact are giv nsideration the model d depth be 2. The qu crease wit ntities A he criteri e K-funct by the nat by the pa f the medi (y .±)). en below, of R(z,-) qual havio estio h inc (^o> + a (15 ions ure o ramet urn us Some For itativ r of n of w reasin ) and )-(17) is gov f the ed in specif the pr ely r K(z,± hethe g dep A(U . CI erned exter ) and two-D ic il esent eproduces ) as sum- r K(z,+) th is set- ,-) , re- early the , accord- nal light the sa- models lustra- , we Formula for R(z,-) Using the definition of R(z,-) and the formulas for H(z,±) given in (1) and (2), we have: R(z,-) C(y ,-)Rooe" kooZ - C(y , + ) e " az/y ° o v o 7 ' C(y Q ,-)e °° + (y o -C(u o ,-)e (18) or (19) Formula (19) may be used to predict the depth depen dence of R(z,-). We deduce immediately that, in general lim R(z,-) = R i *« * J e and that R(z,-) approaches this limit in a monotonic manner, as can be seen by taking the derivatives of (19) with respect to z : SEC. 10.3 FORMULATION OF MODEL 197 dR(z,-) di -°L-k R m - R(0,-))v o C(y o ,-)e ^- k -) C(y o ,-) + (y o C(y ,-))e (£ - H (20) or dR(z,-) dz I I [A(y o , + ) - A(y o ,-)]e ± - k 1 - A(y ,-)e U ' (21) The monotonic behavior of R(z,-) follows from the observa- tion that dR(z,-)/dz is of constant sign for all z , and for arbitrary fixed choice of y . It appears that the model can qualitatively reproduce the experimentally observed depth behavior of R(z,-) as summarized in (b) of Sec. 10.2 above. In particular, the model predicts that: R increasing (22) ^fl > if A(y o , + ) > A(y o ,-) dR(z,-) if A(y o ,+) = A(y Q ,-) constant (23) ^f 1 < if A(y o , + ) < A(y o ,-) R decreasing (24) The increase or decrease of R(z,-) with increasing depth is therefore governed by the relative magnitudes of A(y Q ,±), according to the criteria (22)- (24). Comparisons of Experimental Data with Calculations Based on the Model Figure 10. 5 shows mentally determined K - the calculated values o the formulas (8) and (1 gives a numerical compa between experimental da Figure 10. 6 shows mentally determined R - calculated values of th formula (19) deduced fr numerical comparison of the computed and measur a graphical comparison of the experi- ■ function values (the crosses) with f these functions (solid curves) using 1) deduced from the model. Table 1 rison of the values. The agreement ta and theory appears to be good. a graphical comparison of the experi- • function values (crosses) with the ese functions (solid curve) using the om the model. Table 1 includes a the values. The agreement between ed values is excellent in this case. A word about the calculation procedure may be in order. The following values of the optical properties were used: k^ = 0.178/meter, a = 0.430/meter. The quantities A(y Q ,±), for curve- fitting purposes, may be considered as constants of 19. OPTICAL PROPERTIES AT EXTREME DEPTHS VOL. V K/m 0.220 0.210 0.200 0.190 0.180 0.170 0.160 k^ =0.178 OBSERVED K-VALUES (♦) COMPUTED K-VALUES (Solid Curve) J I I I I I I L 10 20 30 DEPTH Z METERS 40 50 FIG. 10.5 Comparison between experimental and theoretical K- function values, as given by equations (8) and (11). R(Z,-) 0.028 0.026 0.024 0.022 0020 R(Z,-) + OBSERVED R-VALUES (+) COMPUTED R-VALUES (Solid Curve) J I I I I I L i i i 10 20 30 DEPTH Z METERS 40 50 FIG. 10.6 Comparison between experimental and theoretical reflectance function values, as given by (19). integration. Their values were therefore determined in the present by using the following boundary conditions: K(12. 2,-) = 0.216/meter K(12.2,+) = 0.206/meter SEC. 10.3 FORMULATION OF MODEL 199 TABLE 1 Comparison of Calculated and Measured K and R Functions z meters K(z,-) K(z,+) R(z,-) Data Calculated Data Calculated Data Calculated 6.10 _ - 0.0221 0.0221 12.20 0.216 0.216 0.206 0.206 - 18.30 0.206 0.204 0.198 0.196 0.0250 0.0249 24.41 0.196 0.195 0.191 0.189 - 30.52 0.189 0.188 0.185 0.185 0.0266 0.0266 36.64 0.183 0.184 0.179 0.182 - 42. 76 0.180 0.182 0.182 0.180 0.0279 0.0274 48.88 0.178 1.180 0.184 0.179 - 54.99 - 0.0258 0.0277 Note: y = 1.582 A(y , + ) = - 1.337 A(y ,-) = - 2.141 R = 0.0278 k = oo oo 0.178/meter a = 0. 430/meter A = 480 ± 64 my A value z = 12.2 an effec mated ex at the b (stated only to collimat integrat (6) , and passed. so found ted in ( values o dieted v meters tive y ternal oundary before media w ed radi ion pro which . 583 was found of the experi in the sens lighting condi Recall that (8) of Sec. 8. ith nonreflect ance distribut cess of the ki is strictly ne by computing the slope at mental K(z,-) curve. This is e that it simulates the nonaolli- tions and interref lection effects assumption (ii) of the model 5) makes it strictly applicable ing boundaries irradiated by a ion. In this way the lengthy nd described above (following cessary) was conveniently by- When the values of the constants A(y ,±) and y , at the single depth z = 12.2 meters, were substitu- 8) , (11) , and (19) , these formulas predicted a set of f K(z,±) and R(z,-) for all other depths. These pre- alues are shown in Table 1. Hypotheses on the Fine Structure of Light Fields in Natural Hydrosols We have seen (Sec. 10.2) that there is experimental evidence of a regular nonlinear trend in the logarithmic de- rivatives and the ratios of the upwelling and downwelling ir- radiances in near-homogeneous natural hydrosols. On the basis of this evidence, and the ability of the present mathe matical model of the light field for homogeneous natural hydrosols to quantitatively reproduce these effects, we con- clude that these nonlinearities are effects which may be 200 OPTICAL PROPERTIES AT EXTREME DEPTHS VOL. V expected to be observed in all homogeneous natural hydrosols. We are thus led to tentatively propose the following hypoth- eses about the fine structure of the light field in all homo- geneous source- free natural hydrosols. The part of the hy- potheses concerned with the limiting behavior of the K and R functions will be proved in detail in Sec. 10.7 but is included here for completeness. I. The ratio of the upwelling to the downwelling irradiance, i.e. 3 the observable reflectance function R(z 3 -) 3 is monotonic increasing or decreasing with increasing depth z; R(z 3 -) always approaches a limit R m 3 which depends only on the inherent optical properties of the hydrosol and which is independent of the external lighting conditions at the upper boundary of the medium. II. The logarithmic derivatives K(z 3 ±) i.e. 3 the K functions for the upwelling and downwelling irradiances are monotonic increasing or decreas- ing with increasing depth z ; K(z 3 ±) always approaches a common limit k^ which depends only on the inherent optical properties of the hydrosol and which is independent of the external lighting conditions at the upper boundary of the medium. On the basis of the experimental evidence cited above, and on an examination of the mathematical model of the ob- served phenomena, we can propose an additional hypothesis which goes on to state more specifically the depth behavior of the K- and R- functions. III. In all homogeneous source- free natural hydrosols : (a) K(z,-) > K(z,+) for all z > . (b) dK(z,±)/dz < for all z > . We immediately deduce that: 4M^l > o , (25) which follows from (a) and general relation ((32) of Sec. 9.2) dR( dV" } = R(*>-)[K(z,-) - K(z,+)] . (26) Hypothesis III is more specific than hypotheses I and II: (b) implies hypothesis II, and Equation (25) asserts that the reflectance function R(z,-) monotonically increases with in- creasing depth, thus (a) implies a specific alternate in I. The hypothesis cited in III is actually but one of a score of possibilities. It has, however, a relatively high probability of occurring. The sense of this "probability" will be made clear below in Sec. 10.4. SEC. 10.4 CATALOG OF K- CONFIGURATION 201 10.4 Catalog of K Configurations for Shallow Depths In order to facilitate further theoretical studies of hypothesis III in Sec. 10.3 and to anticipate alternate possi- bilities , we shall develop in this section a catalog of all possible K- function configurations, as predicted by the two-D model. The catalog of K- configurations in Figs. 10.7-10.12 is a graphical listing of all ways in which K(z,-) and K(z,+) may approach their common limit koo in various homogeneous source-free natural hydrosols. It would be of interest to try to reproduce each of the possible configuration under controlled laboratory conditions. A K-con figuration is defined as an ordered quadruple of the four quantities: k^, K(0,+), K(0,-), and 0. The catalog NONDEGENERATE CONFIGURATIONS CI K(0,-) K(0,+) k oo C2 K(0,-) k oo K(0,+) C3 K(0,-) k co K(0,+) 7 C4 k oo K(O r ) K(0,+) FIG. 10.7 Figures 10.7-10.12 catalog the totality of distinct depth-dependent configurations possible between the three K-functions K(z,±), k(z) in homogeneous stratified plane-parallel optical media, as deduced from the two-D theory for irradiance fields. See text for further details. 202 OPTICAL PROPERTIES AT EXTREME DEPTHS VOL. V NONDEGENERATE CONFIGURATIONS, CONTINUED k, C5 'CO K(0,-) K(0,+) C6 C7 ^oo °7 K{O r V K(0,+) K(0,+) K(O r ) ^00 C8 K(0,+) koo K(O r ) FIG. 10. consists of 25 configurations. These configurations are divided into three main classes: 1. Nondegenerate Configurations (9 members) 2. Degenerate Configurations (a) First Kind (8 members) (b) Second Kind (3 members) 3. Forbidden Configurations (5 members) The nondegenerate configuration is defined as one in which: t k^ + K(0,-) + K(0,+) f k (1) SEC. 10.4 CATALOG OF K- CONFIGURATION 203 NONDEGENERATE CONFIGURATIONS, CONCLUDED C9 DEGENERATE CONFIGURATIONS, FIRST KIND (k^X)) (0,-) K(O,+)=k 0O D.I D,2 K(0,-)= K(O,+)=k 0O K(0,+)=k D,3 CO K(0,-) FIG. 10.9 A degenerate configuration is defined as one in which at least one of the inequalities in (1) is replaced by an equality. A forbidden configuration is one in which the following basic inequality of the general theory ((30) of Sec. 9.2) is violated: K(0,+)R(0,-) < K(0,-) (2) Observe that the K-conf igurations are defined in terms of the values of the K-functions for z = 0. This is possible because the monotonic behavior of the K and R functions, as established in 10.3, fixes their relative behavior at all depths once their initial values are known. For example, in CI of Fig. 10.7: K(0,-) > K(0,+) > k, > . 204 OPTICAL PROPERTIES AT EXTREME DEPTHS VOL. V DEGENERATE CONFIGURATIONS, FIRST KIND (k >0) CONTINUED K(0-)=K(0,+) D,4 CO D,5 GO K(0,-)=K(0,+) D,6 K(0,-)=k GO K(0,+) D,7 K(0,+) K(O,-)=k 0O FIG. 10.10 Then since K(z,±) must always decrease or always increase toward its limit we must have, in the present example, a decrease of both K(z, + ) and K(z,-) toward koo . Further- more, since R(z,-) also exhibits a fixed monotonic behavior for all z , CI must have--by virtue of (26) of Sec. 10. 3-- K(z,-) > K(z,+) for all z. Similar arguments show that all the configurations are well-defined in terms of the ini- tial values of the K-functions. Knowing the initial magni- tudes of K(0,+) and K(0,-) therefore fixes each configura- tion for all z . The general relation (26) of Sec. 10.3 may be used to determine whether R(z,-) increases or decreases for a particular K- configuration. We now give evidence that a configuration in which K(z,-) > K(z,+) is preferred to one with K(z,-) < K(z,+). We begin by noting that the most unlikely configuration is D 2 3 which is associated with nonabsorbing (purely scattering) SEC. 10.4 CATALOG OF K-CONFIGURATION 205 DEGENERATE CONFIGURATIONS, FIRST KIND (k^O) CONCLUDED D,8 K(O,-)=k 0O K(0,+) 7 DEGENERATE CONFIGURATION, SECOND KIND (k^O) K(0,-)=K(0,+) D 2 I koo = D 2 2 K(0,-)= K(0,+)= D 2 3 K_ . The preceding formula implies that K(z,+) = K(z,-) for all z > , whence by (26) R ( Z >") = R oo 1 1 ^or all z > . Now from the fixed value of H(z,+) and (1^ 206 OPTICAL PROPERTIES AT EXTREME DEPTHS VOL. V Fl F2 F3 F4 F5 FORBIDDEN CONFIGURATIONS K(0,+) k a> K(0,-) k/^ K(0,+) K(0,-) k oo /^ ' mm K(0,+) K(O t -) f^ K(0,-) = K(0,+) K(0,+) = k m ' CD K(0,-) F] [G. 10.12 and (2) of Sec. 10.3, it must follow that R = 1 , k > and C (y ,-) = C(y oo ' oo_ v O O Hence we have the following representations of H(z,±): H(z,-) H(z, + ) - N' C(u ,-)e C(y o ,-)e k z k z sz/y C(y , + )e C(y Q ,+)e sz/ -•] that is H(z,+) = H(z,-) for every z ^ . Moreover, a study of (14) -(16) of Sec. 8. shows that if k + = k_ (as must be the case in view of K(z,+) = K(z,-) deduced above), then necessarily koo = . (Here we are identifying K(z,±) with ± k + respectively. The basis for this is (31) of Sec. 9.2.) It follows that, under the SEC. 10.4 CATALOG OF K-CONFIGURATION 207 present circumstances we must have: H(z,±) = N sz/y C(y o ,-) - C(y o ,+)e Now, in forward scattering media, an examination of (3) of Sec. 10.3 shows that it is more likely to have than ^o > C ^o P o < C(y o> -) Hence if a medium is such that a = , and that it exhibits backward scattering, tnen H(z,±), by the preceding represen- tations for H(z,±), increase with increasing z , so that K(0,+) = K(0,-) < . This unlikely state of affairs is represented by D 2 3 in Fig. 10.11; hence the configuration D 2 3 is very low on the list of likely configurations encoun- tered in nature, as was to be shown. Next, if the volume absorption coefficient is very slightly increased from to a > , then the resulting effect is such that k > Furthermore, by (13) of Sec. 8.5, we would expect that K(z,-) ? K(z,+). Hence we would expect that k > > K(0,-) > K(0,+) which is represented by configuration C6 in Fig. 10.8 (see configuration F3 which shows that the reverse inequality between K(0,+) and K(0,-) is impossible). As the value of the volume absorption coefficient a is allowed to increase a bit more, K(0,-) and K(0,+) move upward on the vertical axis, maintaining the above inequality as they approach and assume configuration C5. At this point, as a is further increased, the configuration assumed depends on the external lighting conditions and the volume scattering function a . If a is highly anisotropic with high forward scattering values and small backward scattering values, as is the case in most natural hydrosols, then the configurations C4, C3, C2, and CI are most likely to be realized on the basis of various simple models obtained by assuming the appropriate forms for a . Thus the phrase "configuration X is more probable than configuration Y ," means that the values of the optical parameters associated with configuration X are more likely to be observed in nature than those associated with Y . This ordering of the likelihood of occurrence of optical parameters is suggested by experimental evidence. The configurations therefore are roughly in the order of decreas- ing probability of occurrence. 08 OPTICAL PROPERTIES AT EXTREME DEPTHS VOL. V The discussion will not go into further detail on the catalog of these configurations. We merely mention that various special models can be obtained by assuming specific, but simple, forms of a . These easily yield most of the 20 possible configurations. These forms for a are: (i) a U;c') = a + 6(c-C') fixed constants, and tion (Stick model). + o 5 (c+S 1 ) , where a_ are is the Dirac delta func (ii) a(c ; £') = s/4tt where s is the volume total scattering coefficient (Ball model) . Some Special Fine Structure Relations The models developed in Sec. 10.3 may be used to answer questions of the following kind. 1. What quantitative estimates can be made of the differences : K(0,-) - K(0,+) K(0,-) - koo K(0,+) - k.. ? 2. What quantitative estimates can be made of the differences R(0,-) - R^ knowing either R^ or R(0,-)? 3. What can be said about the relative magnitudes of H(Z,: What 4. If K :) (respect estimates C(z,+), K(z,- this a ma of (0,±) < 0, ively) has can be made the f and the e preci shoul the ties catal the d after The irst R(z, xpec se m d be H , and og o etai numer three -) in ted er easure gener K , an their f the led de Answer to simplif ic ical exampl questions the hydroso rors of the ments of th ally expect d R plots classificat observed In- scription o Question 1 ation, we h es g show 1 of obs e li ed t ; th ion conf f th ave ) , and k(z)? implies that in real media ximum at some depth z max . z max ? iving the that the Sec. 10. erved dat ght field o exhibit e presenc (by means iguration e hydroso rom (8) a general answers to variations of K(z,±) 3 are not less than a. Therefore, detailed in natural hydrosols the nonlinearity in e of these nonlineari- of the appropriate ) could form part of Is. nd (11) of Sec. 10.3, K(0,-) - K(0,+) — - k A(y ) " A(y Q ,-) [ A(u >") ] [■ A(y., + ) (3) As an example of the use of this formula we use the values of A(u ,±) obtained in the computations for Table 1 in Sec. 10.3 above, whence: SEC. 10.4 CATALOG OF K -CONFIGURATION 209 K(0,-) - K(0,+) = 0.010/meter In addition to (3), we have: K(0,±) - k = ^-k •) A( ^o 1 - A(y o ,±) (4) For the case of the present medium, we estimate that: K(0,-) - koo = 0. 064/meter K(0,+) - k^ = 0.054/meter. Answer to Question 2. From (19) of Sec. 10.3 we have: R R(0,-) A(y Q ,-) - A(y o , + )_ 1 - A(y Q ,-) In the present case, we know R to the estimate: and A(y ,±). This leads R(0,-) - R = - 0.007 for the present medium. Therefore the spread R(0,-) - Roo of R(z,-) values in the present hydrosol is on the order of 30 percent of R(0,-) . Answer to Question 3. The definition of k(z) is exactly analogous to the definition of K(z,±) : km 1 dh(z) k(z) = h(F) —XT' ' To obtain an expression involving K(z,±) and k(z), we use the notion of the distribution function which links H(z,±) and the corresponding components h(z,±) of h(z): h(z) = h(z,-) + h(z,+) where h(z,±) (as in (11) of Sec. 2.7) is the scalar irradi- ance associated with the downwelling (-) or upwelling (+) stream of radiant energy. From the definitions of D(z,±): D(z,±) = h(z,±) H(z,±) and that of K(z), we have: 210 OPTICAL PROPERTIES AT EXTREME DEPTHS VOL. V k(z) = - 1 d M7T oT D(z,-)H(z,-) + D(z,+)H(z,+) 1 h(z) D(z,-)H(z,-)K(z,-) + D(z,+)H(z,+)K(z,+) dp C z >-) H(z,-) dz v ' * t dD^H H( j This is the exact representation of k(z) in terms of the D and K functions. According to the present model, however, dD(z,±) _ ai U This assumption is in good agreement with experimental fact (re: Fig. 10.2). For the present medium, dD(z,±) dz ^ 0. 001/meter , as may be verified from Fig. 10.2. The number 0.001 is an upper limit of the derivative values over the indicated depth range. Since K(z,±) are usually determined to about 10" 3 per meter, the contribution to k(z) by the terms containing the derivatives of D(z,±) is not significant. Hence we may write: k(z) = y(z)K(z,-) + [1 - y(z)]K(z, + ) where nzJ h(z) i This representation of k(z) shows that k(z) is expected to be between K(z,-) and K(z,+), regardless of the algebraic signs of K(z,+) and K(z,-). As an example, let z = 30 meters. Computations from the data yield the value k(30) = = 0.187/meter. Therefore we have, as expected, 0.188/meter - K(30,-) > 0.187 = k(30) > K(30,+) 0.185 Answer to Question 4. Configurations C3, C5, C6, D 8 and D 3 exhibit all possible ways in which either K(0,+) or K(0,-) may be negative. All except the last configuration exhibits a finite depth max at which K(z, or j . - • — » max ------- point on the corresponding H-curve (observe that -max K(z max ,-) = 0. In these cases, z ma ^ is the abscissa of the maximum z max ma y differ for K(z,+) and K(z,-)). An estimate of z max for the upwelling (+) or downwell ing (-) stream can be made directly from (8) and (11) of Sec. 10.3 by setting: K(z ,±) = v max ' J SEC. 10.4 CATALOG OF K-CONFIGURATION 211 and solving for z max . Thus, from (8) and (11) 'max = K(z ,±) v max* J iL A ( M ,±)e" U" " U ) 1 - Afu +)e V^o max k I z oo / max from which = k Uq o koo z max Solving for z max z (±) max^ ^ In A(y„,±) -5L - k where the plus sign refers as usual to the upwelling stream and the minus sign refers to the downwelling stream. The criterion for the existence of a positive z max value is evidently: k ii A(y ,±) > 1 If, in particular, the argument of the natural logarithm is positive but less than unity then Z max has a negative y sign, which means that H(z,+) (or H(z,-)) has no maximum value. In this case the irradiance simply decreases mono- tonically for all depths z > . The conditions under which the preceding inequality holds have yet to be fully explored. Conclusion The discussions of the preceding four sections have shown that there exist in natural optical media certain well ordered, calculable tendencies of growth and decay of the ir- radiance fields, and their K- functions, with respect to depth Furthermore, only a few of a large set of theoretical possi- bilities (configurations CI, C2, C3, and C4)aremost likely to be observed in nature, and then principally in shallow depth regions (less than 20 attenuation lengths) of homogene ous hydrosols with calm surfaces. 212 OPTICAL PROPERTIES AT EXTREME DEPTHS VOL. V 10.5 A General Proof of the Asymptotic Radiance Hypothesis In this and the following section we present two proofs of the asymptotic radiance hypothesis, that is, the conjecture concerning the shape of the radiance distribution and its behavior at great depths in natural hydrosols. The two proofs differ in their starting assumptions. The proof given in this section is completely general and uses only the mildest assumptions concerning the functions used to describe natural light fields and their supporting media. The proof is based on the natural mode of solution of the equation of transfer (Chapter 5) and thereby has, despite its analytic complexity and length, the virtue of using only intuitively simple con- structs in its development and which, furthermore, begins and ends with directly observable concepts, namely the radi- ance K- function, the volume attenuation function a , and volume scattering function o . The proof given in Sec. 10.6 is considerably simpler than that offered in this section because it is assumed at the outset that scalar irradiance h(z) in natural media eventually decreases exponentially with depth, a fact which is not assumed and is eventually derived in the present longer proof. This is quite a reasonable as- sumption, however, being born and sustained on inspection of much experimental evidence, and most students of the subject will thus be content with the shorter arguments of Sec. 10.6. However, those who wish to see the argument developed from first principles, are invited to read on below. The reader wishing only an overview of the arguments of the present proof need only read on through the discussion of (18) below. At the conclusion of the chapter we will have accumu- lated four essentially distinct types of proof of the asym- totic radiance hypothesis, namely that given via the charac- teristic representation of N in Sec. 7.10, those of the present and subsequent section, and finally that in the clos- ing observations of Sec. 10.7. Further, perhaps more elegant proofs should be forthcoming from the functional equations developed throughout Chapter 7. These are left for interested students to pursue. Introduction The asymptotic radiance hypothesis was formulated in the field of experimental radiative transfer dealing with the penetration of natural light into the oceans and deep lakes. It may be stated as follows: The form of the radiance distri- bution about a point in an optical medium approaches, with in- creasing depth, a characteristic form which is independent of the external lighting conditions at the upper boundary of the medium and which depends only on the inherent optical proper- ties of the medium. Some relatively early references to the hypothesis may be found in the experimental papers of Whitney [315], [316], Poole [209], and Lenoble [154]. Some recent theoretical discussions for particular cases may be found Her- man and Lenoble [107], [108]. Subsequently, the mathematical problem underlying the hypothesis took on meaning in a wider set of contexts such as astrophysical optics and neutron trans- port theory. However, the statement of the hypothesis for these contexts is essentially the same. SEC. 10.5 PROOF OF ASYMPTOTIC RADIANCE 213 In this section a proof of the hypothesis is given for a rather wide class of inhomogeneous spaces known as even- tually separable spaces, a term. which is defined in detail below. The discussion is designed so that the main results are also applicable to the astrophysical and neutron con- texts. The approach used is direct in the sense that it is based on a study of the natural mode of solution of the equa- tion of transfer rather than first solving the equation for particular cases and then inspecting the properties of the resulting solutions. Furthermore, the quantities introduced in the study are for the most part directly observable quan- tities, a feature which reflects the experimental origins of the problem and which keeps sight of possible practical appli- cations of the asymptotic radiance hypothesis. In this way the discussion complements a different approach to the prob- lem, namely the formal-solution approach initiated by Chandra- sekhar [43] and extended by Kuscer [147] in the radiative transfer context, and also considered, for example by Davison [62] in the neutron transport context. In particular, the present discussion shows in terms of directly observable quantities that when an asymptotic radiance distribution exists in a medium, it is represented by a formal-solution distribution and is approached in a continuous way by the natural distributions as depth is increased in the medium. Two illustrations of this fact are given. One is based on tables compiled from theoretical calculations made in the neutron transport context, the other is drawn from an experi- ment which documented the light field in a natural hydrosol. These illustrations will be considered later. The practical consequences of the asymptotic radiance theorem are many. They take on especial utility in the field of geophysical optics. While an exhaustive discussion of these consequencies is out of place here (see Sees. 10.7 and 10.8), we should observe that the classical two-flow equations of the light field (Chapter 8) , are accurate and become exact with increasing depth whenever the hypothesis holds. This re- sults in an enormous simplification of the standard experiment- al procedures dealing with the determination of the optical properties of natural hydrosols. Finally, the present method allows a means of estimating, with respect to a given pre- assigned criterion, the optical depth at which the asymptotic distribution has been attained (Sec. 10.7). Preliminary Definitions We begin with the general source-free equation of trans fer for the radiance function N on a general isotropic* space X as used in geophysical optics ((14) of Sec. 3.15), *The arguments developed below go through with minor changes also for nonisotropic media. However, such additional generality does not add materially to the theoretical or prac- tical consequences of the asymptotic radiance hypothesis, and is therefore not postulated at this time. 214 OPTICAL PROPERTIES AT EXTREME DEPTHS VOL. V dN(x,5) = g • VN(x,0 = " a(x)N(x,C) + N*(x,£) (1) and recast it into a form which will be most suitable for the present discussion, and which will insure the widest domain of applicability of the present results to related fields such as as trophysical optics and neutron transport theory. Here: N*(x,0 = J_ N(x,€ f )a(x;r;£)dfl(r) a(x;5;S f )N(x,S»)dn(5») represents the path function N* ; and is written twice, as shown, so as to point up the isotropy of X (re Definition 3 of Sec. 7.12). The volume scattering function is a , the unit sphere of direction vectors £ is H , and the attenu- ation function is a . The present problem is meaningful only in the steady state case, and is most immediately concerned with emission- free arbitrarily stratified plane-parallel media with con- stant index of refraction. These conditions have been adopt- ed in (1) . The introduction of the plane-parallel geometry into (1) results in the usual equation: - 5 ■ k dN( d Z z ' e,(|)) = - a(z)N(z,e,) + N*(z,0,) N q (x,y ,) (3) where we have written: "N q (x,y,cf>) for J, [ p(x;y,;y' , f )N(x,y f , ' ) dy ' d which defines the equilibrium radiance function Nq . Thus (1) reduces, under the above assumptions- -which will be con- sidered in force in the sequel- -to the standard form for the equation of transfer in plane-parallel media. The discussion will require consideration of the follow- ing scattering-order decomposition of (3) : y dNJ( d V y ^ ) = NJ(T,y,*) - N;>CT,y,40 J = 1 (4) where we have written 'N;I(t, V,) for jr w. p(x;y, ' ) dy ' d< (5) and where N J and Ni are positive valued radiance functions on Z x e which refer to radiant flux which has been scattered precisely j times, so that (as in (1) of Sec. 5.4) we have the definitional identity: N( T ,y, , is most conveniently studied by means of the associated func- tions K(x,»,«) on H defined by writing (re: (35) of Sec. 9.2) : 1 dN(i,y,(j)) N(x,y,) di K(x,y,cf>y for (7) The present discussion will also require consideration of the function K a (x,*,«) on H K q (x,y,)" for defined by writing N q d,y "*T dN (x,y,)" for and: K^(T,y,cj)) M for dN J (T,y,<})) N J (x,y,cj)) dT r dNJ(T f y,4>) Ni(T,y, , (9) , j = 1,2,..., t > . (10) Finally, corresponding to: N (n) (T,y,cjO" for I N;>(T,y,cjO (11) we write »r^(T,y.*r for --pjj where, of course dN (n) (x,y,cj)) Nq J (T,y,(J)) dx / n . ' I N^(T,y,^))KJ(T,y,(J)) 3'1 4 4 V - 1 I N^(x,y,(f)) (12) lim (n) n-^oo q (x,y,c{)) = N (x,y,) K(T,y,c|0 . (14) SEC. 10.5 PROOF OF ASYMPTOTIC RADIANCE 217 Furthermore, from (4) and the definition of KJ and K^ dK 3 (T,y,(J)) _ dl K J (T,u, (15) We now can give the motivation for the preceding adop- tions of the K - functions in the present approach to the asymptotic radiance problem. Suppose there is some depth in the medium below which the functions K(t,«,«)> t 2l t o are constant functions on E , and whose values are equal to a fixed number ko> • Then we may write, for all t _> t q : N(t,u,) di ' - J K(T»,y ,)dx A N(T Q ,y,) exp (t - t )k Thus if we write: "g(y, cj))" for N(T Q ,y,) exp ( - ikj (16) Relation (16) is the formal procedures referred mination of a specific rad the point of view of the p incidental end rather than cerned with the determinat the radiance distributions of the kind summarized in determine a class of space for the asymptotic radianc The preceding heuris the motivation for the fol radiance distribution: An is said to exist if (i) 1 by "KooCy ,)") exists for constant function on S . starting point of the to above which lead to iance distribution g resent approach, howeve a means. That is, we ion of a class of space tend continuously to a (16) , and thus as a mat s in which such a forma e distribution is meani tic argument leading to lowing definition of an asymptotic radiance di im T _H„K(T,y ,) (hencefor each (y,) eE and (ii) classical the deter- on H . From r, (16) is an will be con- s in which structure ter of course,, 1 procedure ngful. (16) supplies asymptotic stribution th denoted Koo(* , •) is a 218 OPTICAL PROPERTIES AT EXTREME DEPTHS VOL. V It is quite possible for condition (i) of the preceding definition to hold, while condition (ii) does not hold. This state of affairs is encountered, for example, in space in which s(T)/a(x) = for all x > , where = j_ a(x; s(x) = a(x;y' ,' ;u ,<{>)du »d . Furthermore, we will require that the boundary radiance function N°(0,*,*) on H_ be a nonnegative valued, nontrivial integrable function with respect to the measure ft . Here, we have 5 : 5 • k < = (y, ; y ' ,

;y ! ,4>' ) + (t;u, fj>;y',') such that poo is independent of x and not identically zero on E x 5 , and (ii) tf> ■* (the zero function) uniformly on S x E , as x ■+ °° . SEC. 10.5 PROOF OF ASYMPTOTIC RADIANCE 219 We can now state the main result: an asymptotic radi- ance distribution exists in every plane-parallel medium if and only if the medium is eventually separable . This state- ment is understood to hold in media whose equation of trans- fer is given by (3), and whose boundary conditions are given as above. All of the effort below will be devoted to prov- ing the sufficiency of the eventually separable condition. Simple counter examples show that if a space is not eventually separable, the asymptotic radiance distribution necessarily does not exist. (For example, stack in alternate layers pure- ly absorbing and purely scattering plane parallel media.) We close this preliminary discussion by making some observations on the K - functions which will be required below. First we observe that from (3), if y = , then N(x,0,cJO = N q (T,0,(J>) . Hence, for all t>0, K(x,0,) - K q (T,0,) , Secondly, for each j,x > , NJ (t , • , • ) is bounded away from zero, is continuous on the compact set E , and hence is uniformly continuous on 5 and H_ . A similar observation holds for NJ(x,-,-)> J = 1 , 2 , It follows that K q (x,-,«) and 5 for ^ft V. «• > » J > J ■*■ > «- » .... -L L ^_ V-, _L _L W V» ^ UiUt iv q and KA (!,•,♦) are uniformly continuous on E a all x > and j = 1 , 2 , .... .Finally, from and the definitions of K and KJ , (3) and (4) K(T,y,40 + \ < , (17) K J (x,y,c}>) + i < , j = , 1 , ... , (18) for all (y, . Properties (17) and (18) are particularly useful in conjunction with (14) and (15). For example if K(x,y,(j>) > K q (x,y,), then by (14) and (17) it follows that dK(x,y ,cf>)/dT J , showing that in general K always tends toward the equilibrium function K q . This is a useful fact in practice. Whether or not K -> K q as t -*■ °° depends on the relative sizes of K q (y ,(J)) = lim m K(t , £ ,cj)). and - (1/y) . It follows directly from the properties of the Riccati equation (cf. e.g., [116, p. 312]), that K+min{K q (y ), - (1/y)} assuming of course that K q (y,cj>) exists. A similar set of remarks holds for (15) and (18). The Functions P,Q,R In order to insure that the main sequence of arguments is uninterrupted by the development of certain required auxiliary relations, these auxiliary relations are gathered here for ready reference. The first relation needed below gives the connection between the downwelling j - scattered flux at level x > and the upwelling scattered flux at level x : N j + 1 (x,y,;u',cj>*r for ^| p (x ' ;y , cj> ;p ',cf>' ) 1 _ 1 y y ' exp < - (x' - t) dx (20) If the space were separable, i.e., p were independent of t (or, in the present case, the phase function component 9=0) then writing "Poo" for the limit of p as t ■* °° , we have: P oo (y,({);y',(j)») = ± ^-^ Poo (y ,') . (22) uniformly on n+ x H_ , and that: lim T ^ co P(T;y,(j);y f ,(j)') = P^Cy ,cf>;y ' ,cj>' ) uniformly on 5+ x h_ and finally, that: lini K 3 + 1 (T,y,(tO = J_ P(x;y,4);y' ,4)')N j (x,y' ,cf)')K ;j (T,y f ,cf)')dy' = lim j, P(x;y'4>;y' ,4>*)N J (x,y' ,(f> » ) dy • dcj) ' (23) for (y,(j)) eE + . A similar expression holds for Kq ( x,y , ) which follows from (5) and (10) : lim K^ + 1 (T,y,) = x->oo q V J r- J T V lim P(x;y,(j);y',cj)')N j (x,y' , ' P(x;y,(j);y' , cj) ' ) N J (x,y' ,<£') dy ' d ' (24) SEC. 10.5 PROOF OF ASYMPTOTIC RADIANCE 221 The next relation required below makes use of the forms of the principles of invariance in generally nonseparable media, in particular, use will be made of:* ,u,« = lf o R N(x,y,cf)) = i R(T,«>;y,(j);u' ,' )N(x,y ' ,' )dy »d(j> ' where (y,f) e 5 + and R(t, «>;•;•) on H + x H_ is the reflec- tance function associated with the subset of Z x 5 below level t >0 , i.e., with X(t,«>) (see (31) of Sec. 3.7). If the medium were separable, then for all pairs (t 1 ,t 2 ) of depths : R(t i ,«;y,^;y f , ! ) = R(x g ,«»;y ,<|>;y ' ,' ) In the present case it is possible to verify on the basis of the differential equations for the R and T operators of Sec. 7.1 that: l im dR(T,«>;y,4>;y',(f>') . X-yoo dl uniformly on H+ x 5_ , and that: lim T ^ oo R(T,oo;y, ( |);y« ,(()') = R^ (y , $ ; y • , (J, • ) uniformly on 5+ x H_ , where R^ on 5+ x H. is the reflec- tance of a homogeneous medium X(0,°°) with phase function Poo • (Roo may be found using the methods of Sec. 7.6.) Finally, the integral operator: [ ] Q(T,°o;y,(j>;y« ,(f> ' ) dy ' d(j> ' (26) will be used. This operator maps the function N(t,*,*) on H_ into the function N Q (x,',») on ~_ . The kernel Q is defined by writing: "Q(T,~;y,')" for p( T ;y ,cj>;y * ,' ) - f p(T;y,(();y",({)»')R(T,-;y",^••; h i^ct) , ) ^ df *This section is adapted from Reference [224] with a minimum of notational change. Hence the reversal of the positions of N and R (and the primed and unprimed argu- ments) from that throughout the remainder of this work. 222 OPTICAL PROPERTIES AT EXTREME DEPTHS VOL. V The operator (26) is a positive operator.* From the defini tion of Q , it follows once again from the differential equations for the R operators of Sec. 7.1 that: lim dQ(i,w;y,(l);y' ,<$>') dx = and that: lim T _ >oo Q(T,°°;y,cf);y ) = P^CUj^jy 1 ,') dp" Pjy,;y , \) We now begin the main steps of the proof. The object of the present discussion is to show that the function K^ ( • , • ) on E defined by writing: K q (u,oo K (i,y,(J)) exists and is continuous almost everywhere on E . The dis- cussion begins with some observations on the functions KJ , Kj , K( n ) . In particular, we observe that for (u,(j))e E and every ^t > , Kq(T,y,) j_ p(T;y,^;y',cj)')N (T,y',cj)') ^A J^ p'(T;y,(|);y',4)')N (T,y',(t)')dy'd({)' J_ p(x;y,cj) ,y',(J)')N°(T,u , ,cj)')dy»d(J)' *For the present discussion, an operation T is said to be positive if T maps nonnegative functions into nonnegative functions and Tf = (the zero function) implies f is the zero function, where f is a nonnegative valued function on H_ , and the vanishing of f is taken in the sense of Lebes- gue (cf. , e ~. , [111] , p. 25) . SEC. 10.5 PROOF OF ASYMPTOTIC RADIANCE 223 where p' denotes the derivative of p with respect to x . The function N°(t,«,*) on H_ is related to the boundary radiance distribution by N°(T,y,40 = N°(0,y,cf>)e T/y Hence each integrand in (27) is integrable on H_ , so that Kq(T,y,) exists and is well defined for every t^O and (u,)eS . Furthermore each integrand in (27) satisfies the hypothesis of Lebesgue's bounded convergence theorem, so that by (24): K^(u,<(0 = lim T +oo K q( T >^) C28) exists for every (u,)eE and in fact KA ( • , • ) is continuous and therefore bounded on E . The values of IP ( • , • ) are readily determinable for specific choices of N° (0, •,•)• F° r example, if we adopt the standard discrete boundary radiance distribution defined by: N°(0,y,cf>) = N°6(y-y o )6(4)-0) o ) , -1) - - ~ , (y,(D) e 3 (29) n N (0,u,cj)) = I N°(y 0<5(u-y.)6(-cj>.) i = o x 1 - 1 < y. < and if: then y o = mm \ y. : i = o, 1 K q(n»40 = " — , (y,4>) e E . (30) Other simple examples of N°(0,-,-) may be given, such as step-function representations, various simple continuous functions on 5. , but (29) and (30) will suffice to illus- trate the general procedure. In particular they help to illustrate the use of (15) which is required in the next step of the proof and which runs as follows: By means of (15) and (28), we see that for each (p^jsE. such that K q (y,*) > - i (31) 224 OPTICAL PROPERTIES AT EXTREME DEPTHS VOL. V we have: lim K 1 (t ,y ,d>) = - - Since Kq(«,») is bounded, the subset of y ' s for which (31) holds is a relatively open subset of [-1,0] exclud- ing 0. Finally, from (15) and (28), for each (y,)eH_ such that: Kq(u,40 i - ^ , (32) we have: lim^K 1 (T,y,) = K^(y,(',0 on H_ defined by writing: "K^(y,)" for lim^^K^T,^,^) exists for all (u,) 1 - i (33) on E_ . The discussion of Ki( • , • ) is completed by showing that it exists and is continuous on H + . This is done by applying the preceding arguments to (23). As an example, one may consider (29) once again, which yields Ki(u,cjO = - l/u for all (u ,c)0 £ E + . We now take the general inductive step, that is, we assume that K j ( • , • ) , j >_ 1 is continuous on 5 and in particular, KJ,(u,cJ0 1 ~ 1/y f° r every (y,<£) e H_ . Then by means of (24) and the previously cited convergence arguments, we find that: KJ + 1 (y , (j)) , where we write: "K j + 1 (y,ct>) M for lim K^ + 1 (t , y , ) (34) exists for every (y,) e E_ , and K^ 1 (*,*) is continuous on 5_ ; and in particular,* K^ +1 (m,4>) < - i , (y,) = lim T ^K j + 1 (T f u,40 ■ K^ 1 (y ,4>) (36) on E_ . Finally, from (23), K^ + 1 ( • , • ) exists and is continu- ous on 5+ and moreover, Since the induction hypothesis has been demonstrated for the case j+1 assuming the case j , and it is true for j = 1 , the conclusions (34) -(37) then hold for all integers j = 1 , 2 , .... It follows from (12) and the preceding results that: K^ n) (y,cj)) = lim T ^ n) (T,y,(l)) < - I , (38) exists and is continuous on E , for n = 2 , 3 , .... Now consider the function gi n ^ (• ,y ,) on Z defined for every (u,cj))£ E by writing: fnl N (n) (T,w,*D »gW(T, H ,«» for g (T>li> ^ , n = 1,2,.... (39) Clearly (g(( n ) (• ,y ,<]>) ) is an increasing sequence of func- tions on Z , such that for every (y,)e E , lim n^q Il)( ' ,P '* ) = 1 ' the unit function on Z . From this and the definitions of KC n ) and Kq , we conclude, first of all, that the sequence {Kj. n -) (• ,u ,$) } of functions converge in the mean to K q (*,y,4>) on Z . This in turn implies that the convergence to KqC*,u,cj)) is almost uniform on Z for some subsequence (K^ n k) (• , y , , and subset Z £ of Z, (n k ) (n ) lim n v - lim T-ooK q (T,y,cD) - lim^lim K (T,y,cj>) = lim T ^ TO K(T,y,(j)) on Z ' = Z - Z £ , where di f < e J Z e It follows that the function Kq(«,») on 5 has the prop- erty: 226 OPTICAL PROPERTIES AT EXTREME DEPTHS VOL. V Ok) 1 K q (y,) in E. Finally, from (24), (15), (23), and (12) (in that order), we establish the fact that (KJ- n ) (•,•)} is a sequence of continuous functions whose essential suprema form a nonincreasing sequence of real numbers. It follows that Ki n )(*,») converges uniformly a.e., on E to K (•,•)> and that Kq ( • , • ) is continuous on 5 - E Q where E Q is a subset of - such that ft(E ) = . Hence K (•,•) is con- tinuous almost everywhere on 5 . The Limit of K(- ,y ,) The proof is now concluded by showing that K(*,y,) (41) for all (y,cj)) e E_ . Hence by our preceding result on K a ( • , • )> K 00 (*,*) is continuous almost everywhere on E_ . It follows [103, p. 242, problem (3)] that there is, -for every e > , a compact subset E_ (e) of E_ - E Q such that fi[(E_ - E ) - E_(e)] < e on which K 00 (',*) is continuous. We use this fact to establish the existence of a minimal value of K 0O (*,*) on E_(e). Let (y.,, <]>.,) be any direction in E_(e) (there is at least one) defined by the condition: (y,40 : (u,) e 5 _( £ ) > for N(T,y,(j)) K (y ,4> ) - = inf and then write "g(i ,v,40" and observe that g on "g(y 40" N(x,y a , ) 5_(e) defined by writing for lim T ^ oo g(i,y,(})) is at least bounded and measurable (hence integrable) on E_ (e) Then by means of the operator defined in (26), we have: Q oo (y,4);y' , cj) ' ) g(y' ^'JKJy' ,(J)') dy ' dcf) ' K q (y,cf>J ■ , (42) f Q oo (y,4);y' ,cf)')g(y' , ' ) dy ' d(J) ' 5_(e) SEC. 10.5 PROOF OF ASYMPTOTIC RADIANCE 227 for all (y,)e 5_(e) . In particular, (41) holds for (y, ,,)£ H_ (e) . Using (41) , (42) may be rewritten as: f Q oo (y i ,(}) i ;u',(f)')g(y',(l)') [^(y',4)') - ^(y^^^jdy'dcp' = '5_(e) This operator Qoog , as that in (26) , being a positive opera- tor, requires that the everywhere nonnegative valued function: on 5_ (e) be the zero function almost everywhere on H_(e) . Writing "koo" for K 0O (y 1 ,(J) 1 ) we have: for almost every (y ,cj)) eH_ (e) . Since e is arbitrary, this result holds almost everywhere on 5_ - E Q and, by extension, everywhere on E_ . An application of (25) to the definitions of K(x,y,cj)) and Koo(y,), yields the result that K (y,cf)) = k for almost every (y,) on H + , so that Koo(*,*) is a constant function almost everywhere (and, by our agreed extension, everywliere) on 5 . This concludes the proof. We observe finally, that, by means of the definition of N q and (8) , K q Qi,40 " KJy,^ = k ro everywhere on Notes and Observations We now make an observation on the physical significance of the number k^ • We observe that the scalar irradiance function h on Z defined by writing M h(T)" for J N(T,u, has, in analogy to N , a K -function defined by writing: M k(T)" for dh(T) h(i) dx which, as we saw in (39) of Sec. 9.2, is represented in terms of K(t,*,«) by the formula: 228 OPTICAL PROPERTIES AT EXTREME DEPTHS VOL. V k(x) = N(T,y,)dyd [ N(T,y,)dyd)K oo (y,)dyd) N q (T,y,4Q 1 + yK(T,y,) which is the canonical form of the equation of transfer for the slab geometry (Chapter 4). The limit of the canonical form as t+°° is, by the preceding results (and recall (16)): g(y,')dy'dct>' 1 + yk (43) which is the general form us cedures discussed in the int now takes on the additional value of an eigenvalue probl integral equation for g on conditions adopted in the pr noted before, stem from the totic radiance problem--the negative, and in fact, <_ k tion of g) . As a final observation ical findings to some indepe measurements of asymptotic r 10.13 shows the depth depend directions (y,)eS. The as cal separable half-space, ir collimated neutron flux, in ed in the formal- solution pro- roduction. The real number k^ significance of being an eigen- em associated with the above E . For the kind of boundary esent section- -which as we have geophysical origins of the asymp resultant values of k^ are non oo < 1 (cf. (17) and the defini- , we relate the present theoret- ndent computations and empirical adiance distributions. Figure ence of K(»,y,(J0 for several sociated medium is a hypotheti- radiated by normally incident which scattering is isotropic SEC. 10.5 PROOF OF ASYMPTOTIC RADIANCE 229 .600 .500 400 i-300 .200 .100 PLOTS OF K(t,/i,0) vs DEPTH t (for neutron flux) isotropic scattering, s/a=0.9 111^^.^00^^ = 0.525 4 6 OPTICAL DEPTH r 8 10 FIG. 10.13 A theoretical example of the asymptotic radiance theorem. and s/a = 0.9 . These plots tations of N(x,y,(j>) (for neut The plots snow clearly that as ly attained at x = 10, for at K(10,-,«) is essentially const Figure 10.14 shows the d for several directions (y,)eE natural hydrosol, namely Lake the time of measurement of N( from a clear sunny sky (angle 40°, hence the associated y Q found to be highly anisotropic increasing t , a constant val the medium was eventually sepa on experimental determinations [298]. All N - measurements w microns. The plots show that ly approached at depth x = 20 has been fixed at 0°, which through the sun. Plots for to asymptoticity for depths at cal K - scale has been exagger Fig. 10.13) in order to more c transition to asymptoticity. are based on theoretical compu ron flux) compiled in [11]. ymptoticity has been essential this depth the function ant on 5 . epth dependence of K(-,u,cj0 The associated medium is a Pend Oreille, Idaho, which at x,y,(j)), was irradiated by light of sun from zenith was about was - 0.77); scattering was and s/a approached, with ue of about 0.7, indicating rable. These plots are based of N(x,u,) recorded in ere made at about 480 milli- asymptoticity is being marked- and below. The azimuth angl-ie denotes the vertical plane f 0° indicate similar trends t = 20 and below. The verti- ated (relative to that of learly show the details of the 230 OPTICAL PROPERTIES AT EXTREME DEPTHS VOL. V 500- i 400- 300- 10 20 OPTICAL DEPTH r FIG. 10.14 An experimental example of the asymptotic radiance theorem. 10.6 On the Existence of Characteristic Diffuce Light: A S pecial Proof of the Asymptotic Radiance Hypothesis In this section we return to the problem of the asymp- totic radiance hypothesis and, as outlined in the introduc- tory remarks to Sec. 10.5, we approach the hypothesis from a basically simpler, more empirical point of view. We shall therefore reintroduce the problem in the following paragraphs from this alternate point of view, and carry out the discus- sion so that it is virtually independent of that in Sec. 10.5. Introduction Recent experimental the basis for fresh suppo that the radiance distrib deep natural hydrosol app characteristic form which lighting conditions and t the medium, and which dep properties of the medium, given its first definitiv [316] , who referred to th as characteristic diffuse evidence, recorded in [298], forms rt of the long-standing conjecture ution about a point in an optically roaches, with increasing depth, a is independent of the external he optical state of the surface of ends only on the inherent optical This conjecture was apparently e formulation by Whitney [315] , e asymptotic radiance distribution light. (We shall use these two SEC. 10.6 SPECIAL PROOF OF ASYMPTOTIC RADIANCE 231 names interchangeably in what follows.) In this section we complement the experimental evidence in favor of this conjec- ture by supplying a simple proof of the existence of charac- teristic diffuse light in all homogeneous optically deep natural waters. The discussion concludes with a derivation of the integral equation governing the angular structure of the characteristic diffuse light and a brief discussion of an interesting and tractable example for the case of iso- tropic scattering. We note in passing that since the time of the formula- tion of the asymptotic radiance hypothesis by Whitney, its domain of applicability has been widened considerably. The problem of a limiting angular distribution has since been encountered in modern neutron transport theory but basically as an abstract mathematical problem rather than experimental phenomenon. A similar type of problem has long been extant in astrophysical radiative transfer. A general proof of the existence of an asymptotic radiance distribution which covers all these contexts is given in Sec. 10.5. Despite the widening of the domain of applicability of the hypothesis, it will retain its greatest usefulness in the context of geophysical optics, and in particular in hydro- logic optics. For in this field, unlike the others mentioned above, the trend to a characteristic limiting form is a di- rectly observable phenomenon. The existence of such a form is of inestimable importance to all experimental research work dealing with the determination of the optical properties of natural waters. In many important instances, knowledge that an asymptotic radiance distribution exists will obviate the necessity for experimental probings to extremely large depths; for such knowledge will allow, by means of relatively simple formulas, the accurate prediction of the geometrical structure of the light field in the great-depth ranges. Some of these practical consequences of the asymptotic radiance hypothesis are developed in Sees. 10.7 and 10.8. Physical Background of the Method of Proof The argument used by Whitney in establishing experimen- tal evidence for the asymptotic radiance hypothesis went ba- sically as follows: He showed that when the experimentally obtained plots of radiance distributions at various large depths were all blown up to the same size (more precisely, the zenith radiances were all normalized to a common value) , they formed a set of nearly congruent figures. Now, an in- teresting feature of such radiance distributions is that they assume the same shape , and decrease in size with increasing depth at very nearly the same exponential rate. This fact can be stated precisely as follows: N(z,6,(j>) = g(6,cj))e" kz . (1) From this we see that the asymptotic radiance hypothe- sis is equivalent to the statement that the directional and depth dependence of radiance distributions multiplicatively uncouple at great depths. That is, the radiance function N 232 OPTICAL PROPERTIES AT EXTREME DEPTHS VOL. V may be represented as the product of two functions: The function g gives the shape or directional structure common to all the distributions, and the exponential function gives the depth dependence of the distributions. Each factor on the right hand side of (1) has special physical significance. The function g eventually defines the angular form of tne characteristic diffuse light. The exponent k of the exponential function has the following interesting interpretation: We define the scalar irradianoe h(z) at depth z as usual by writing: 2tt f-u cf>)sin 6 d6 d4) . (2) (L\\ Ml "h(z) n for N(z, The quantity h(z) is a measure of the volume density of radiant energy at depth z . Measurements of h(z) over the years in many hydrosols have shown that h(z) varies essen- tially in an exponential manner with depth. That is, semi- log plots of h(z) versus depth show an unmistakable trend toward linearity as depth increases. In any event, h(z) may be accurately represented by a formula of the type: I k(z')< ■} where k(z) is seen to be the logarithmic derivative of h(z). As depth increases, the experimental evidence is that k(z) approaches a constant value. Let us denote this limit value by "koo" . Now assuming that an asymptotic radiance distribution is approached by the radiance distributions in a particular body of water, we see from (1), (2), and (5), that: h(z) = h(z Q )e k (z-z ) -kz r 2TT r 71 = e KZ g(e,c|))sin e de dcj) , (4) ; =0 J

), we write as in (35) of Sec. 9.2: SEC. 10.6 SPECIAL PROOF OF ASYMPTOTIC RADIANCE 233 "Vf7 ft AV« for - I QELlAAl (^ K(z,0,) for N(z>e ,cj)) Zi * (6} Then, in analogy to (3), N(z,6,$) at any depth z may be represented exactly by: N(z,6,) = N(0,6,(J))exp Now suppose there is some depth z below which we have K(z,e,cj)) = k^ for all directions (6,)dz f - K (z • , 9 , (J>) dz • = N(z o ,e,c^)exp/- kjz - z q ) If we write: { "g(e,). Furthermore, this limit, in accordance with the preceding discussion, should be none other than the limit koo of k(z), as defined in (3). Hence- forth, we explicitly assume that koo , defined as the limit of k(z) as z-*» , exists as a nonnegative number. The Proof We make use of the steady state source-free equation of transfer for radiance: dN(Z d ' r 9 '^ " - aN(z,e,») » N.(z,B,4>) (9) where, as usual: 234 OPTICAL PROPERTIES AT EXTREME DEPTHS VOL. V r 2TT rTT N*(z,e,)sin O'de'dcJ)' (10) J cj> f = o J e' = o represents the path function N* ; a is the volume scatter- ing function (which governs the law of scattering in the water) , and a is the volume attenuation coefficient. The formal solution of (9) is readily obtained and is simply the integral form of the equation of transfer (re: (6) of Sec. 3.13) : N(z,e,40 = N°(z,e,40 + f N*(z» ,e,(f))e" a(r " r,) dr' . (11) The first term is the residual radiance which represents the component of N consisting of unscattered light. The second term is the path radiance which represents the space light over the path of length r , (Fig. 10.15). This path radiance has been generated by light scattered into the path of sight all along its extent. The formal solution (11) has been written for a general downward direction of flow of light (see Fig. 10.15), so that N°(z,e,c{)) is interpreted as the residual radiance transmitted from the upper boundary of the medium and is of the form: where N°(z,0,) clearly exceeds its path radi- ance component at all depths, we can write: N(z,6,40 > f N A (z',e,4))e" a(r " r,) dr' •> o Second, using the definition of N* , we strengthen the inequality when we write: a(r-r') , , N(z,6, a m . n I h(z')e dr o where o m i n is the minimum value of the volume scattering function; that is, we have used (10) to deduce that /■2TT rTT Mz,e,40 >° ■ N(z,e',(j)')sin e'de'de' = a. h(z) o j e ' = o SEC. 10.6 SPECIAL PROOF OF ASYMPTOTIC RADIANCE 235 surface of the water N°(O,0,<£) ncident radiance Radiance tube measuring N(z,0,<£); The plane of the figure is in the vertical plane defined by FIG. 10.15 Setting for a simplified proof of the asymptotic radiance theorem. Finally, since h(z) generally decreases with increas- ing depth (i.e., koo is positive), we certainly strengthen the inequality by writing: N(z,e,cJ0 > a . h(z) v > >yj min V J -a(r-r') d r* That is , we have: N(z,.6,cj>) > -^ h(z)(l - e" ar ) (12) for all depths z . From this we see that as depth z in- creases indefinitely, the exponential rate of decrease 236 OPTICAL PROPERTIES AT EXTREME DEPTHS VOL. V K(z,0,) of the radiance cannot eventually exceed, and remain larger by any finite amount, the k(z) of the scalar irradi- ance. For if it did, the plot of N would eventually fall and remain arbitrarily far below that of h . In other words, the ratio N (z , , ) /h(z) would go to zero, with increasing z , contrary to (12). The conclusion of this observation may be stated as follows: liin Kfz.e.dO < lim k(z)=k (13) for all downward directions (0,4)). We now show that strict equality must hold in (13). We achieve this by initially assuming the contrary, that is, we assume that there is a set of directions E Q with positive solid angle measure over which: lim Kfz.e.dO < k - e 2 ->-oo v ' * T J — oo where e is an arbitrary small positive number. Then it is clear that the radiances in this set of directions decreases at a definitely smaller rate than the scalar irradiance, so much smaller, in fact that, by our assumption, it is true that for some depth z , we must have N(z,0,cj))sin ed0d(}) > h(z ) However, this conclusion clearly contradicts (2) (a part can- not exceed the whole). We have reached a contradiction which leaves only one other possibility, namely that: lim Kfz,e,). In the light of the pre- ceding discussions (cf. (8)) this means that the shapes of the radiance distributions impinging on the upper boundaries of deep layers of water eventually assume a fixed form. But it is known that the shape of the reflected radiance distri- bution at the upper boundary of a scattering layer is deter- mined by the shape of the incident radiance distribution at that boundary (e.g., principle of invariance III of Example 3 of Sec. 3.7, with N+ (b) = in the medium X(0,°°)). Hence if the incident radiance distribution approaches a fixed shape, so does that of the reflected distribution. This com- pletes the proof. We observe that the present proof can also be applied in all natural waters which eventually become homogeneous. That is, the preceding arguments are basically unchanged if the medium is inhomogeneous over any initial finite depth range below the surface. Even more general situations exist which allow asymptotic radiance distributions, namely media in which the ratio a/a. eventually becomes independent of depth (Sec. 10.5). SEC. 10.6 SPECIAL PROOF OF ASYMPTOTIC RADIANCE 237 The Equation for the Characteristic Diffuse Light Using the equation of transfer, the definition (6) , and the relation between z and r , we can write the equa- tion of transfer in the following canonical form (Chapter 4) N(z,e,)cos 6 (15) From (14) and (8) we see that the limiting form of (15) (as depth increases' indefinitely) is g(e,) r 2TT rT\ | i g(0',(f)')G(e',(j)' ;6,)sin e» J (j)t =0 J e' =0 de 1 d ! a + k cos 9 oo (16) which is the equation governing the angular form of the char- acteristic diffuse light (cf. (43) of Sec. 10.5). It is a property of equations of the type shown in (16) that the func- tion g is independent of for all real physical situa- tions. Thus the characteristic diffuse light is always rep- resented by a surface of revolution whose axis of symmetry is vertical . The theory of the solution of such equations as (16) is fairly well understood (see e.g., [62]). The present discus- sions, therefore, will not consider in any detail the general solutions of (16). However, there is one simple special case which is immediately solved and which can shed much light on some of the salient details of the structure of the asymptot- ic radiance distributions. This is the case of isotropic scattering, where the volume scattering function a is in- dependent of direction and has the form: a(9,;e',(f>») -^ (17) where s is the total scattering coefficient. To see the resulting structure of the asymptotic radi ance distribution, it is convenient in the present case to turn to (15). With the assumption (17) and the definitions (2) and (10) , we have N(z,e,) _s h(z) 4tt a + K(z ,0 , ) cos which at great depths approaches the form: koo(z-z Q ) 1! " U ' = '^f h(z )e v o J m cos (18) 238 OPTICAL PROPERTIES AT EXTREME DEPTHS VOL. V Here z is the depth below which h(z) is essentially of exponential behavior. Comparing (18) with (8) , we see that for the present case, h(z )e k °° z ° *- ' rj 4tt la 1+ I — I cos (19) en (19) in the indicated form to point up trie fact: A polar plot of g(6,) approach, with increasing z , a common fixed value koo for all di- rections (0,); further, the logarithmic derivatives of scalar irradiance h(z) , its upwelling and downwelling com- ponents h(z,+) and h(z,-), along with the derivatives of the upwelling and downwelling irradiances H(z,+) and H(z,-) all approach the common limit k^ as depth increases. Fur- ther consequences are that the two-D model for the irradiance field in natural waters (Sec. 8.5) becomes exact with in- creasing depth. These and related results are illustrated by examples drawn from the special case of isotropic scattering. Finally, a formula is developed which allows an estimate of the depth at and below which the actual radiance distribu- tions differ from the asymptotic distribution by no more than a preassigned amount. Thus, the formula may constitute a criterion for asymptoticity in natural hydrosols. In order to keep the present discussion essentially self contained and useful for references purposes, we shall review the definitions and properties of the various concepts to which the asymptotic radiance hypothesis will be applied. Basic Formulas: The Irradiance Quartet As is demonstrated in Chapter 2, the radiance function N is a basic radiometric quantity in terms of which all others can be defined. In particular the downwelling and upwelling irradiances H(z,-) and H(z,+) at depth z in a natural hydrosol are given by: H(z,-) = - N(z,e,(j>) cos G dfi , (1) and H(z, + ) = N(z,9,({))cos 6 dfi , (2) where, as usual, we define E_ as the collection of all downward (or inward) directions (0,) : tt/2 < 6 , < < 2tt , where is measured as usual from the outward normal k to the medium. We have thus partitioned the set E of all directions into its upper ( + ) hemisphere and its lower ( - ) hemisphere . For brevity we have written, " dft " for sin d0 dd> , in (1) and (2). In addition to H(z,+) and H(z,-), underwater optical experiments usually consider the following downwelling and upwelling scalar irradiances: 240 OPTICAL PROPERTIES AT EXTREME DEPTHS VOL. V h(z,-) N(z,e,)dft (3) and h(z,+) ■J, N(z,0 ,)dft (4) and their sum h(z): h(z) = h(z,-) + H(z,+) (5) which is to the pr u(z) at d cussed in The formulati the nucle to docume a complet atic dete depths z is seen i tet of ir ful radio the oduc epth det fou on o us o nt t e do rmin an n th radi metr scalar irradian t of the speed z) . These co ail in Chapter r quantities H f the two-flow f a set of mode he light field cumentation is ation of the ra d over all dire e developments ances contains ic information ce at depth z (h(z) is equal of light v and radiant density ncepts were introduced and dis- 2. (z,±), h(z,±), so useful in the theory of Chapter 8, also form rn experimental quantities used in natural waters. Of course, obtained only through a system- diance values N(z,6,) at all ctions (0,) the basic quartet of irradiance functions defined above can be used to derive most of the information needed for the solu- tion of underwater visibility problems, and image and flux transmission problems in general. In particular, an excel- lent index of the shape of the radiance distributions at depth z is given by the distribution functions D(z,±) represented as: D(z,±) = h(z,±) H(z,±) (6) Furthermore, information about the reflectance properties of the water at depth z is furnished by a study of the ratio: R(z,-) H(z,+) H(z,-) (7) which is the experimental counterpart to the classical Roo formula as given by classical one-dimensional two-flow anal- ysis of the light field. In fact, the D and R functions defined above, and the functions defined below are all either modern experimental counterparts or logical extensions of the SEC. 10.7 SOME PRACTICAL CONSEQUENCES 241 tools provided by the classical two-flow theory of the light field in natural hydrosols. As noted above, the background of these particular radiometric quantities is considered in detail in Chapters 8 and 9 so that the present discussion need not dwell further on their definitions and interrela- tions. We are concerned here only with the behavior of these quantities at great depths in media satisfying the requirement of the asymptotic radiance hypothesis. The K Functions The essentially exponential behavior of the irradiance quantities supplies the motivation for the definitions of the K functions (Chapter 9) assembled here for convenience: K(Z ' ±} " " H(z,±) ^~ dH(z, ±) dz dh(z, + ) ^--FTIJ^ ■ ■ Cio) If the various irradiance quantities vary exactly in an exponential manner at all depths, then the corresponding K functions would be constant functions each assuming a fixed value at all depths. In general, however, (Sees, 10.1- 10.4) the depth-dependence of these quantities is nonconstant and it is only after 10-20 attenuation lengths that the ex- ponential features eventually emerge. The preceding repre- sentations, however, are designed to characterize the depth- dependence of the irradiances under all conditions. One of tne main consequences derived from the asump- totic radiance hypothesis is that the five K functions represented above all tend to a common limit with increasing depth. We prepare the groundwork leading to this conclusion by reintroducing the K function for the radiance function itself (re (34) of Sec. 9.2): Thus we write: uv f a ami r 1 dN(z,0,(f)) nn n K(z,6,4))" for - rr-7 , . W "-*- . (11) v ' * rj N(z ,6 ,) dz v ' Just as each of the various irradiance quantitites may be expressed in terms of radiance, so can its corresponding K function be expressed in terms of the K function for radiance : / N(z,6,(}))K(z,e,(j))cos dft K(z,±)=-— , (12) N(z,6,(j))cos dft I. 242 OPTICAL PROPERTIES AT EXTREME DEPTHS VOL. V N(z,e,cf>)K(z,e,(j>)dft , (13) N(z,e,)dft + /, N(z,0,)K(z,8 ,) for radiance is of funda- mental importance in the present discussion of the asymptotic radiance hypothesis. In fact it is the function which gives rise to that form of the hypothesis which is most amenable to exact mathematical analysis. The desired characterization reads as follows ((16) of Sec. 10.5) for each fe^Je H ^ the function K( z 3 Q s §) has a limit, as z ■* °° 3 and this limit is independent of (B 3 $). In symbols: k = lim K(z,0,) may be represented exactly by K N(z,0,) = koo , a fixed number for all (6,). Then N(z,9,(j)) = N(O,0,) dz ' = N(z o ,0,)exp / - k^Cz - z Q ) Write "gCz ,e.,_ z , N(z,6,) has a fixed angular structure given by g(z 01 9,). The Basic Transfer Equations One final relation needed below is the reformulation of the equation of transfer in terms of the K function for radiance. This is easily obtained from the standard form of the transfer equation for stratified source- free plane-parallel media: _ C os e gJilljJLii) = _ a (z)N(z,e,cjo + N*(z,e,cjo where : N*(z,6,(J0 = | N(z,0' ,^ ! )a(z;e' , ! ;0,(|))dfi By means of the definition of K(z,0,), the above equation may be rewritten in its canonical form (Chapter 4) : N(».e,») ■ a(z) ;*^'^^ )cos 9 . d6) The equation of transfer governing K(z,0,) is also easily found. From (16), the definition of K(z,0,), and the following definition of an analogous K function: V 2 ' 9 '*) = " n (z,e,») ' Sz dN q (z,e,») (17) where N q (z,e,d>) = N 5% (z,e,4>)/a(z) , (18) we have: dK(z,e,(j)) dz [K(z,9,(J)) - K (z,e,)][K(z,e t ) in which N q (z,0,) is used: 244 OPTICAL PROPERTIES AT EXTREME DEPTHS VOL. V dN(Z dz 9,(10 = [ N ( z > 6 >^ " N q ( z > e >40H + a(z)sec 0)] . These formulations point up the following physical significance of the equilibrium radiance N q and its K function K q : for > (tt/2) we > observe that if N(z,0,)> $ Nq(z,0,) then dN (z , , <$>) /dz < . This follows immediate- ly from the preceding equation. Thus N , > tt/2 , always tends toward the equilibrium radiance N q . Now a similar phenomenon exists between K and Kq . To see this, we observe that the second factor on the right in (19) has the property that: K(z ,0 ,) + a(z)sec < , for all z and all downward directions (6,(J>). Therefore if K(z,6,c|>) > Kq(z,e,4>) then dK(z ,0 ,)/dz < 0, showing that K always tends toward Kq for these directions. This property of the function K(z,6,) provided the key to the rigorous proof of the existence of an asymptotic radiance distribution given in Sec. 10.5. A further example of such a use of (19) is given in the final sections below. Consequences for Directly Observable Quantities: The Equation for the Asymptotic Radiance Distribution An application of the asymptotic radiance hypothesis to (16) yields the formula for the asymptotic radiance distribu- tion g . In view of the heuristic discussion leading to (15) and the statement of the hypothesis in terms of K(z,0,cf>), we see that lim z ^ 0O g(z,6,(})) = lim z ^ oo N(z,0,(J))exp ( k B z ) exists for all (0,). We shall denote this limit by "g(0, ®" Hence multiplying each side of (16) by exp { k^ z } and passing to the limit as z-»-°° , we have: ~ J_ g(0' ,cj)')p(0' ,' ;e,«j))dfi g(0,oo 4Tro(z;0' , )/a(z) k = lim K(z,0,) SEC. 10.7 SOME PRACTICAL CONSEQUENCES 245 and* a = lim a(z) The integral equation (20) has the property that the values of its solution g are independent of . Thus we may write: "g o (8) M for 27Tg(e,(f>) and (20) may be simplified to read: , (21) where we have written; "p (o) (6' ;6)" for if p (6 ' , ' ; ,cj>) d • The function g Q descri form of the asymptotic radianc g is clearly a surface of re (in the coordinate system of t Furthermore, the structure of are completely determined by t plays the role of an eigenvalu Thus g is determined comple properties of the medium by me fore is independent of the ext extension of (21) to the polar starting the preceding derivat of transfer (8) of Sec. 4.6 fo ance vector. bes the essential ge e distribution. A g volution with vertic he plane-parallel me g and the value o he phase function p e of the equation (2 tely by the inherent ans of equation (21) ernal lighting condi ized case is immedia ion with the canonic r the standard obser ometric raph of al axis dium) . f k„ (koo/a D). optical and there- tions. The te, upon al equation vable radi- The Limits of the K Functions From the relations (12) -(14), and the statement of the hypothesis, we conclude that: *For most practical situations, the medium is homoge- neous or eventually homogeneous, so that this limit exists. Actually, as shown in Sec. 10.5., the asymptotic radiance distribution exists whenever lim ZHK30 a/a exists, without necessarily requiring that the individual limits lim z -M»cr and lim. exist. 246 OPTICAL PROPERTIES AT EXTREME DEPTHS VOL. V lim K(z,±) = k , (22) lim k(z,±) = k , (23) £->-00 *• ' J OO ' V J lim k(z) = k . (24) The limit (24) is interpreted as follows: The loga- rithmic derivative (with respect to z) of scalar irradiance h(z) eventually approaches the common limit k of the loga- rithmic derivatives of radiance distribution N(z,0,). The limit (23) shows that the logarithmic derivatives of the up- welling and downwelling irradiances (which as we saw in Sec. 10.2, are measurably distinct at all small depths z) approach a common value, namely koo . A similar interpretation holds for the K functions of the upwelling and downwelling scalar irradiances . The Limits of the D and R Functions From (6) and the hypothesis, we have immediately: + ° 1, g (0)sin 6 di (0) [± cos 0]sin lim z _D(z,±) = — , (25) which we shall denote by "D(±) n . Furthermore in (25) we have written: "r" fir/ 2 "r" fir for , and for j + J 6 = J- J 6 = tt/2 In (25) of Sec. 9.2 it was shown that R(z,-) can be represented quite generally in terms of the K functions and the distribution functions as follows: R(z,-) K (z,-) - a(z,-) K(z,+) + a(z,+) wnere : a(z,±) = D(z,±)a(z) and where a(z) is the value of the volume absorption func tion of the medium at depth z . Let us write: "R " for lim R(z,-) 00 2 -*" 00 SEC. 10.7 SOME PRACTICAL CONSEQUENCES 247 It follows from the preceding representation for R(z,-) that Roo exists and is given by: (26) where: lim a(z :) = lim z ^D(z,±)a(z) = D(±)a and where we have written: "a(±) M for D(±)a Further limit relations may be determined by systemati- cally going through the set of directly observable quantities discussed in Chapter 9. The preceding results will serve to illustrate the general procedure of obtaining the desired limit expressions. We observe that (26) is similar to the classical ex- pression for R^ as given by the two-D model for the irradi- ance field ((102) of Sec. 8.7). This similarity is not coin- cidental; it is, rather a consequence of the fact that under the asymptotic radiance hypothesis, the general two-D model becomes exact with increasing depth. We now consider this fact in more detail. Consequences for Some Simple Theoretical Models: The Two-D Model for Irradiance Fields In Chapter 8 a study of the classical two-flow equa- tions for H(z,+) and H(z.-) showed that these equations were exact if and only if the distribution functions D(z,+) and D(z,-) were independent of depth (Sec. 8.5). Under the asymptotic radiance hypothesis it was seen in (25) that the distribution functions become independent of depth at great depths. It follows that the two-D equations for undecomposed irradiance H(z,+) and H(z,-) become exact at great depths whenever the hypothesis holds. In Chapter 8 a formulation of the equations for H*(z,+) and H*(z,-) (the decomposed irradiances associated with dif- fuse light) was made in which each stream of flux was assigned a fixed distribution factor D*(+), D*(-) (the two-D theory for decomposed irradiance). This formulation was justified on the basis of experimental evidence which showed that D(z,+) and D(z,-) were essentially fixed (generally distinct) constants. In the light of the present analysis, the use of the two-D theory for decomposed irradiance is also given fur- ther justification on theoretical grounds whenever the asymptotic radiance hypothesis holds. The two-D model gives explicit formulas for H*(z,+) and H*(z,-). In view of the preceding observations, these 248 OPTICAL PROPERTIES AT EXTREME DEPTHS VOL. V expressions become exact with increasing depth z . Using the equations developed in Sec. 8.6 it may be shown that for every depth z in an infinitely deep medium X(0,°°): H*(z,~) N°C(y ,-) H*(z,+) = N' C(y ,-) j^-v az/ ^oJ , (27) + ^ e -az/y (+) -k z e Ko ° Z - C(y (28) Observe that we have set koo = - k_ , where k_ is given in (12) of Sec. 8.5. The physical setting associated with (27) and (28) is an infinitely deep plane-parallel slab irradiated by collimated flux incident at the upper boundary at an angle O = arc cos y from the outward unit normal The response to an arbitrary incident distribution is ob- tained by integrating (27) and (28) over S_ (see Sec. 10.3) C(y ,±) are constants for each y , determined by the op- tical parameters and boundary conditions; and: g.(±) = 1 * am where a(±) are as defined in (26). The observable irradi ances H(z,±) are, by definition, H(z,±) = H°(z,±) + H*(z,±) where : H°(z,+) = H°(z,-) = N°y e" az/y o The expressions for the observable irradiances are given in (1) and (2) of Sec. 10.3. The preceding model yields the following prediction of the limit of R(z,-): R = lim R(z,-) = lim oo 2j ~ Jyco Z _ *"°° H(z, + ) 8-JM k ~- a ^ H(z,-) "" ^TFT '" k +a( + ) which agrees with (26) , the exact limit given by general radiative transfer theory. These observations show that in any medium in which the asymptotic radiance hypothesis holds, if discussions are restricted to the class of all possible two-flow models of the light field, the model which attains maximal accuracy is that given by the two-D theory. Critique of Whitney's "General Law" After conjecturing that the radiance distributions assume a fixed shape at great depths, L. V. Whitney made use of the conjecture to deduce a so-called "general law of the SEC. 10.7 SOME PRACTICAL CONSEQUENCES 249 diminution of light intensity in natural waters" (cf. ref. [316]). An examination of the differential equations formu- lating this law reveals that they are incomplete: They fail to account for the contribution to the downwelling irradiance by the backscattered fraction of the upwelling irradiance. As a result, the solutions of the differential equations are generally inadequate to cope with the contribution from one half of the light field, namely the component associated with the upwelling flux. Furthermore, some (convenient, but in- correct) assumptions were made about the depth rate of change of the mean free path for unscattered light at various depths. On this basis the equations were integrated, holding fixed the mean free path for directly transmitted light. Both of these inadequacies of an otherwise satisfactory theory have been remedied in the two-D tneory of the light field. The equations (27) and (28) (or their observable counterparts (1) and (2) of Sec. 10.3) represent the concomitant effects of both upwelling and downwelling streams. Finally, the awkward- ness stemming from the change with depth of the mean free path of directly transmitted light has been avoided by considering only collimated incident flux of radiance N° at the upper boundary. The Simple Model for Radiance Distributions In (2) of Sec. 4.4 a simple model for radiance distri- butions was derived in the form of the canonical representa- tion of the apparent radiance. One key assumption in the establishment of the model was the depth- independence of the K function for radiance. In view of the preceding observa- tions, it is concluded that the simple canonical model be- comes exact with increasing depth in all media in which the asymptotic radiance hypothesis holds. Further Consequences of Asymptoticity We conclude with some examples drawn from the case of a plane-parallel medium which exhibits isotropic scattering and in which the asymptotic radiance hypothesis holds. In this way we obtain some general ideas about the shape of g , as governed by (21) , and the order of magnitudes of the quan- tities D(±), Roo , and koo one may expect in veal media. Finally, it is possible to give, in the present context, a simple heuristic proof of the hypothesis, and at the same time derive a formula which will provide a means of deter- mining the depth in a medium below which asymptoticity has essentially been attained. We shall now consider these mat- ters in turn. 250 OPTICAL PROPERTIES AT EXTREME DEPTHS VOL. V The Standard Ellipsoid When scattering is isotropic, the phase function takes on the form: p(9' ,' ;9,) = p = s/a where s is the volume total scattering coefficient and 11 p" is an abbreviation of "s/a", and both denote the scat tering-attenuation ratio. Using this phase function in (21) we see that g (6) takes on a particularly simple form, g, 1 + e cos W = % • T^^ C29) where we have written and " e " for k /a L g ° g " for g (6)sin dG Physical significance can be attached to g by returning to the definition of g(6,cj)) and integrating over H (see (15) and (20)). The result is: g = lim h(z)e k °° z to o z->°° Hence if there is some depth z below which one may con- sider that for practical purposes asymptoticity has been attained, then the preceding relation can be written: :„ = h(z o )e Koo Z o Expression (29) represents a prolate spheroid of revo- lution whose axis of symmetry is vertical. The eccentricity of the ellipsoid is e = koo/ct . This ellipsoid may serve as a convenient reference against which distributions from real media may be compared. To effect a comparison one must know the p and e of the medium. Since e and p are gener- ally related, it suffices in principle to know only p and the phase function. This is illustrated below after a neces- sary preliminary discussion of D(±) and Roo . SEC. 10.7 SOME PRACTICAL CONSEQUENCES 251 Expressions for D(±) and R ro By means of (25) and (29) we find that (see also (57) and (58) of Sec. 2.11) : r = e ln(l ± e) UUJ £ i ln(l ± e) * ^ UJ Furthermore, from (7) and (29) (i.e., (29) replaces N(z,6,) in (7)), we have: . ln(l + e) -e (31) °° ln(l - e) + e The same result could be obtained by using (26) and the pre- ceding form for D(±). Values of D(±) and Roo as functions of e , < e < 1 are given in Table 1. It is easy to verify that for the ex- treme values and 1 of e the corresponding values of D(±) and R^ are: lim £ ^ D(±) lim c+l D ^ = CTT2 = 2 ' 259 lim^ 1 D(-) lim £ ^ Rc lim -.Roo = Table 1 of Sec. 8.5 gives values of D(z,±) for a real medium under varying external conditions. A comparison of these real values with those summarized in Table 1 below re- veals the following information: The D(z,-) values are significantly less than the standard D(-) values; the D(z,+) values are significantly greater than the standard D( + ) values. Since all natural waters exhibit anisotropic scattering we can infer the following features of the struc- ture of asymptotic radiance distributions in all natural waters: When compared with the standard ellipsoid, the plots of g (9) for real media must necessarily be narrower in the angular range 9 > tt/2 (upwelling light). The amount of departure of the g ( e ) for a real medium from the standard ellipsoid may be taken as a measure of the anisotropy of scattering in the medium. 252 OPTICAL PROPERTIES AT EXTREME DEPTHS VOL. V TABLE 1 Distribution and reflectance factors for standard ellipsoid e D(-) DO) Roo 0.100 1.9664 2.0319 0.8750 0. 200 1.9286 2.0622 0.7640 0.300 1.8881 2.0911 0.6642 0.400 1.8438 2.1185 0.5733 0. 500 1. 7943 2.1143 0.4895 0.600 1.7381 2.1692 0.4110 0. 700 1.6722 2.1928 0.3361 0. 800 1.5906 2.2157 0.2622 0.900 1.4775 2.2377 0.1841 0.950 1.3911 2.2483 0.1379 The Determination of The quantity £ (= k<»/a) is functional In the case of isotropic scattering the rel known and of a particularly simple structur In general, e is determined by viewing i value of the integral equation (20). There way, however, to characterize e which, wh analytically direct way, is perhaps of grea generating an insight into the physical sig and also of supplying a link between e an observable quantities of the light in real alternate characterization of e stems fro functional relation which holds between K( various scattering and absorption function medium ((31) of Sec. 9.2): ly related to p ation is well e (cf. Ref. [43]), t as an eigen- is an alternate ile not the most test value in nficance of e d the directly media. This m the following z,±) and the of an arbitrary 1 = b(z,-) b(z,+) K(z,-) - a(z,-) K(z,+) + a(z,+) As depth is increased each term, as a result of the asymptotic property of the light field, tends toward a well defined limit, so that as z^°° , the above relation tends to: b(-) b( + ) 1 k - D(-)a k Oa OO V J CO K J This may be rewritten as: 1 = B(-) (1 - P)D(-) 3( + ) + (1 " P)D(+) (32) SEC. 10.7 SOME PRACTICAL CONSEQUENCES 253 which is the general characteristic equation for £ . Here we have written: i |_ J_ g o (e , )p(e , ,(j) , ;e,(i))dfi' d^ M 3(±)" for J B M 9 ') ± I g (e » )cos 9' dft + In the case of isotropic scattering: 3(±) = § D(±) and (32) reduces to the following simple form after the ex- plicit expressions for D(±) , as given by (30), are substituted in it : P = U r- • (33) -m This is the well-known characteristic equation for e in the isotropic case. As p varies from to 1, £ varies from 1 to 0. Hence, for all p, 0

, then the use- ful inequality koo < a holds. Actually, the inequalities £ koo/ot <_ 1 hold in general (Sec. 10.6). This fact is made plausible by an inspection of (21) keeping in mind that the function g is bounded in all physically meaningful situa- tions, so that the denominator cannot vanish. An Heurisitc Proof of the Hypothesis We now present a brief argument which makes plausible the assertion of the hypothesis, namely that K(z,0,) ■»■ k TO for all (6,cj)). For simplicity we will assume that the space is homogeneous and that scattering is isotropic. The result ing line of argument, while restricted to this special set- ting, can be made completely rigorous. The setting is that depicted in Fig. 10.15. Under the present assumptions, we see that (18) may be written N q (z,6,) = -^ ph(z) so that: K q (z,0,(j)) = k(z) 254 OPTICAL PROPERTIES AT EXTREME DEPTHS VOL. V Thus (19] reduces to dK (z,0,4>) = [K(z,0,cfO - k(z)] [K(z,e,<(0 + a sec 0] . The preceding discussion of this equation showed that K(z,6,) always tends toward k(z) for downward directions. Hence if k(z) approaches a limit, K(z,9,) whose general solution is: k + asecQ C exp { (k + asec0)z} K O> >^ = 1 - C exp((k + a°sec 0)z) (34) where: Since K(O,0,cj)) - k M C " K(O,0, tt/2 , it follows immediately from (34) that lim K(z,0,d)) = k 1 ->-oo v ' * r J oo for all > tt/2 . This means that the shape of the downwell- ing radiance distribution becomes fixed at great depths. It follows from the principles of invariance that the reflected upwelling radiance distribution also becomes fixed, so that the shape of the entire radiance distribution becomes fixed at great depths. A Criterion for Asymptoticity According to (34), K(z,G,) approaches koo with least speed when = tt (i.e., for the directly downward direction, as in Fig. 10.15). Hence when K(z,7r,) has come within a given distance of k^ , we can conclude that the other values K(z,6,°° , of the K- functions K(z,±), k(z) for irradiance and scalar irradiance, respectively. D(±) are the limits, as z+°° functions D(z ,±) . of the distribution 256 OPTICAL PROPERTIES AT EXTREME DEPTHS VOL. V R^ is the limit, as z+°° , of the reflectance function R(z,-) = H(z , +) /H (z , -) • H(z,-) = H(z,-) - H(z,+), the net downward irradiance at any depth z > z Q , where z Q is the depth below which the light field has essentially attained its asymptotic structure. The details of the derivation of I, II, and II, will now be given. Short Derivation of I The short derivation of I starts with (1) , and the fact that there exists a depth z below which the logarithmic derivatives of H(z,-), H(z,+), and h(z) are constant and equal to a common value k (see (22) , (24) , and (35) of Sec. 10.7). Therefore: dH(z,+) = dH(z,+) _ dH(z,-) cfz dz dz k H(z,+) + k H(z,-) = k H(z,-) , (2) for all z >_ z Q . Hence: a- koo *£^l h(z) Long Derivation of I The long derivation of I is essentially an exercise in the use of the integrated form of the divergence relation for the light field vector ((33) of Sec. 8.8) P(s,-) = a v U(M) (3) where M is any regularly or irregularly shaped region of the optical medium, S is its boundary, and P(S,-) is the net inward flux across S into M . U(M) is the radiant energy content of M , v is the speed of light in M , and a is the required value of the volume absorption coefficient It is interesting to observe that (3) yields a value of a in any homogeneous medium, regardless of the structure of the light field: (4) SEC. 10.8 SIMPLE FORMULAS 257 The numerator of (4) can be obt boundary of M with flat plate measuring devices. The denomin the interior of M with a sphe at each point p) and integrati In the present case, the ymptotic light field allows one only one value of the scalar ir of M . This fact holds also f consider a region M in the fo unit cross section, and bounded depth ained by traversing the collectors or other flux- ator is obtained by probing rical collector (to find h(p) ng the values over M . extreme regularity of the as- to estimate U(M) knowing radiance at a boundary point or P(S,-). Specifically, rm of a vertical column of by two parallel planes at and , , such that z <_ z 1 is homogeneous and stratified; hence: The medium P(s,-) = H(z ,-) + H(z .+) (5) The net fluxes over the vertical sides of the column cancel by virtue of the stratified light field. By hypothesis, we have : H(z a ,±) = H(z -,±)e + ^" k oo(Z 2 - Zj (6) so that P(s,-) = H(z ,-) 1 - e" k ~ (z 2 ' z i } (7) Furthermore: vU(M) = | * h(z)dz z Mz^ | "' e" koo(z - z 1 ) dz h( Zi ) 1 - e ■Koo C Z 2 - Z 1 J (8) Inserting (7) and (8) into the general formula (4) , we have the desired result a = k » h( Zi ) > - ° Derivation of II The formula II can be obtained directly from I by recalling that: 258 OPTICAL PROPERTIES AT EXTREME DEPTHS VOL. V H(z,-) = H(z,-) - H(z,+) h(z) = h(z,-) + h(z,+) and invoking the definitions of R(z,-), and D(z,-). That is, in general: H(z,-) = H(z,-) -R(z,-)H(z,-) = H(z,-)[1 - R(z,-)] and h(z) = D(z,-)H(z,-) + D(z,+)H(z,+) ; so that when z >_ z , we have: a = k r D(-) + RJH + ) which is the desired alternate formula. We observe in passing that II is a limiting form of the exact formula: am _ K (z,-) - R(z,-)K(z,+) f . UJ " D(z,-) + R(z,-)D(z,+) » W the basis for which is (2-5) of 9.2. Clearly, as z+°° , equa- tion (9) takes the limiting form II. Furthermore if we assume D(±) = 2, as is done in the classical one-D two-flow theory of the light field, then II reduces to the relation: k 1 - R 2 1 + R (10) Applied Numerology: A Rule of Thumb Formula III is to be taken as a convenient rule of thumb, and as such, is subject to possible revision whenever specific optical media are under study. Yet for many pur- poses it is quite adequate, a fact which is based on the following observed regularities in the values of Roo and D(±) in natural waters: Roo is usually found to be in the neighborhood of 0.02, give or take 0.01 for wavelengths near 500 ym . Furthermore for the same wavelength vicinity, D(±) appears to be such that the sum D( + ) +D(-) is usually very nearly equal to 4; and the ratio D(+)/D(-) is usually very nearly equal to 2, over great ranges of depths and in many media. Solving these two simultaneous equations yields, to two significant figures: D(-) = 4/3 (ID D(+) = 8/3 which agrees very well with experimental results (cf., e.g., Table 1 of Sec. 8.5). It follows that, to the nearest rational SEC. 10.9 BIBLIOGRAPHIC NOTES 259 number with small integers for numerator and denominator, we have from 1 1 : (12) (13) Any similarity between the appearance of the fraction 4/3 in (13) and the index of refraction of water must be viewed as an amusing coincidence. Equation (13), incidentally, points up once again the kinship of koo with the absorption mechan- isms in optical media (see the discussion of (5) of Sec. 9.2 and (29) of Sec. 9.3). 10.9 Bibliographic Notes for Chapter 10 work hydr and the res e Impo the and in t ref e Sect The of [ The oso 1 s [316] prob 1 arche rt ant hydro [209] he as rence ion 1 developments of Sees. 10.1 to 10.4 are based on the 245] . problem of the asymptotic light field in natural was first clearly recognized by Whitney (re: [315] ). The mathematical formulations and solutions of em as in Sees. 10.5, 10.6, and 10.7 are based on the s in [224], [225], [244], and [226], respectively, references to the asymptotic radiance hypothesis in logic optics context may be found in [107], [108], References to the asymptotic radiance hypothesis trophysical context may be found in [43] and [147]; s to the neutron diffusion setting are made in [62]. 0.8 is based in the main on [230]. Experimental data in [298] exhibit clearly the asymp- totic property of radiance fields in a real optical medium and were instrumental in the empirical establishment of the hypothes i s . CHAPTER 11 THE UNIVERSAL RADIATIVE TRANSPORT EQUATION "All these examples 3 which might be multiplied by the millions 3 are cases in which a long 3 laborious 3 conscious } detailed process of acquirement has been condensed into... one. Factors which formerly had to be considered one by one in succession are integrated into what seems a single simple factor." (From: "The Miracle of Condensed Recapitulation 1 in the Preface of Back to Methuselah Bernard Shaw) 11.0 Introduction The present chapter concludes the development of the basic theory of radiative transfer in the present work with a survey of the manifold transport equations for the radio- metric concepts introduced during Parts I, II, and the pre- ceding chapters of Part III. The main purpose of the survey is to bring to light, especially for those readers interested in the theoretical aspects or radiative transfer, a recurrent symbolic theme which runs through every transport equation considered so far, and to go on to capture its essence in the form of a "universal radiative transport equation." The universal radiative transport equation is an equa- tion which, by suitable choice of its parameters, yields in turn such equations as the general equation of transfer for radiance, the general two-flow transport equations for irradi- ance, the transport equation for scalar irradiance, and the transport equations governing the apparent optical properties of an optical medium. The primary purpose of the universal radiative trans- port equation is to formulate in a single mathematical pack- age all the important transport equations which have evolved during the past seventy years in the theoretical studies of the steady state transfer of radiance energy through scattering absorbing media of the stratified plane-parallel type. In this way a recapitulation of the evolutionary process of the transport equation's growth is achieved and a unification of all these important transport equations is attained. We shall illustrate the scope of the equation by selecting thirty-four 261 262 OPTICAL PROPERTIES AT EXTREME DEPTHS VOL. V types of transport equations discussed in this work or implied by the discussions of their principal functions, and showing how these various types may be uniformly subsumed under the regime of the universal transport equation. A second purpose of the universal transport equation is to provide a new useful tool in the study of radiative trans- fer theory. For example, certain special forms of the uni- versal transport equation have already been successfully used (Sees. 10.5 and 10.7) to obtain a solution to the long-standing practical problem of the existence of the asymptotic light field in deep stratified hydrosols, a mathematical task which appears to be simplified, and given interesting physical sig- nificance with the introduction of the general type of func- tions associated with the universal transport equation. Fur- ther evidence of the usefulness of the universal transport equation as a tool which leads to new practical results will be illustrated below. Before we go into the details of how the universal transport equation can achieve a semblance of unity in the classification of modern radiative transport equations, and of how it leads in some cases to new results which are beyond the immediate capabilities of the classical transport equa- tions, it may be of help to the reader to indicate the steps in the development of modern radiative transfer theory which have led to the idea of the universal transport equation. With such information in mind the reader can then easily fol- low the steps of the synthesis. There are four well-defined steps in the development of modern radiative transfer theory which form the immediate background to the formulation of the universal transport equa- tion. These are, in chronological order: The adoption of the general equation of transfer for radiance and the development of the notion of equilibrium radiance [279], [111], and [43]; the development of the unified two- flow irradiance equations and the notion of equilibrium irradiance as recorded in Chap- ter 8; the development of the canonical equation of transfer and the notion of the radiance K -function as recorded in Chapter 4; the development of the theory of the asymptotic light field and the transport equation for the radiance K - function as recorded in Chapter 10. In the following two sections we will illustrate these steps in detail and add still further illustrations which have been uncovered subsequent to the time of the fourth step. In this way we will systematically build up evidence for the existence of a universal transport equation and for the equi- librium principle (described below) with which it is closely associated. After these concrete examples of the various transport equations have been assembled, the genotype of the universal transport equation is extracted from them and dis- played ((1) of Sec. 11.3). The chapter closes with a brief survey of less common but equally important examples of trans- port equations which are also subsumed by the universal trans- port equation. SEC. 11.1 EQUATIONS FOR RADIOMETRIC CONCEPTS 263 11. 1 Transport Equations for Radiometric Concepts In this section we will present the transport equations for the following six radiometric quantities used in the study of plane-parallel media: radiance function N(z ,6 ,), upwell- ing and downwelling irradiance functions H(z,±), upwelling and downwelling scalar irradiance functions h(z,±), and the scalar irradiance function h(z). Each of these transport equations is cast into a form which explicitly exhibits a certain attenuation function and equilibrium function associated with the radiometric concept it governs. It is the isolation and emphasis of these two concepts which is the earmark of the universal radiative transport equation. Thus, for example, the customary form of the equation of transfer for radiance is recast so that it explicitly exhibits the special attenuation function -a(z)/cos and the equilibrium function Nq(z,9,) - N^ (z , 9 ,(j>) /a(z). Similarly, the unified irradiance equations governing H(z,±) are recast into forms which explicitly exhibit the corresponding attentuation functions + [a(z,±) + b(z,±)] and equilibrium functions Hq(z,±). These two reformulations for the transport equations of N(z,8,) and H(z,±) are already known (see Sec. 10.7 for the case of N , and Sec. 8.3 for the case of H) ; however, the reformulations are now viewed with the purpose of seeing what mathematical and physical characteristics are held in common by th'ese transport equa- tions. It turns out that the common characteristics are the attenuation and equilibrium functions associated with each of the radiometric concepts governed by these equations and that each of these transport equations is but a special case of a more general equation, to be determined. The discussion of the present section continues with the derivation of the exact transport equations for h(z,±) and h(z). It is shown that each of these functions also may have associated with it an attenuation function and an equi- librium function. In this way we show that the six radiomet- ric quantities used in the study of plane-parallel media have an important set of properties common to all: The notion of an associated attenuation function and an associated equilib- rium function, and finally that the transport equation for each of these six radiometric concepts, is subsumed under one general equation. We now proceed to substantiate the preceding assertions by considering in turn each of the six radiometric concepts and its associated transport equation. Equation of Transfer for Radiance The equation of transfer for radiance ((3) of Sec. 3.15) in source-free stratified plane-parallel media is of the form: - cos e dN(z d ? z 6>(|)) = - a(z)N(z,e,cj>) + N*(z,6,) , (1) 264 OPTICAL PROPERTIES AT EXTREME DEPTHS VOL. V where : N*(z,e,) j„ N(z > 0' ,' )a(z;0 ' ,' ;0,c|)) dfi Equation (1) is the most basic of all transport equations and. as we have seen repeatedly in the preceding chapters, can often be used in its full generality in the several different branches of applied radiative transfer theory such as astro- physical optics, and in the two subdisciplines of geophysical optics: hydrologic optics and meteorologic optics. The reformulation of (1) which is of immediate interest is obtained by using the notion of equilibrium radiance: N q (z,0,cjO N«(z,8,) for a fixed direction (0,). Function (ii) is the equilibrium function for N(z,0,) for a fixed direction (9,). *An alternate formulation of (3) is possible by adopting the optical depth parameter t(= /?• a(z')dz') . Such a formu- lation using t has been found of especial use, e.g., in Chapter 10. However, for our present purposes, Eq. (3) is more appropriate. SEC. 11.1 EQUATIONS FOR RADIOMETRIC CONCEPTS 265 Transport Equations for H(z,±) The transport equations for H(z,±) (or more accurately the two-flow equations for the irradiance field) are of the form (Chapter 8) : dH(z,±) + Tz [a(z,±) + b(z,±)]H(z,±) + b(z,+)H(z,+) (5) Associated with H(z,-) and H(z,+) are the equilib- rium functions Hq(z,-) and Hq(z,+), respectively. These equilibrium functions are defined by writing: ■H (z,±)" for b(z,+)H(z,+) a(z,±) + b(z,±) (6) By means of these functions the equations in (5) may be written: dH(z,±) dz" [a(z,±) + b(z,±)][H(z,±) - H (z,±) • (7) The equations in (7) are the desired reformulations of (5) . For our present purposes we draw special attention to the two sets of functions: (i) + [a(z,±) + b(z,±)] (ii) H q (z,±) (8) Set (i) gives the attenuation function for the upwelling (+) and downwelling (-) irradiances H(z,±). Observe that, by (11) of Sec. 8.3, the terms in (i) can be represented by ±t(z,±). Set (ii) gives the equilibrium function for the upwelling (+) and downwelling (-) irradiances H(z,±). Transport Equations for h(z,±) The exact transport equatio apparently have never been even r literature. The reason for this equations for the common radiomet First, and perhaps most important theory, there has never been an e port equations for h(z,±); the o were considered adequate in the e field in stratified media. Howev more precise and detailed studies (Chapters 8, 9, 10), the function ns for h(z,+) and H(z,-) emotely discussed in the gap in the family transport ric concepts is two-fold. , in the classical one-D xplicit need for the trans- rdinary irradiances H(z,±) arly studies of the light er, with the advent of of the irradiance field s h(z,±) have finally 266 OPTICAL PROPERTIES AT EXTREME DEPTHS VOL. V assumed a legitimate and useful role in modern radiative transfer theory. Second, there is no simple or intuitively obvious way of obtaining the exact transport equations for h(z,±) from first principles (that is, obtaining de novo derivations starting only with the definition of h(z,±) and the basic volume absorption and volume scattering functions) as is the case for the irradiances H(z,±). Neither is there any simple way of obtaining the requisite transport equations directly from the equation of transfer for radiance (again in contradistinction to case for H(z,±)). In the present para- graph we derive the exact transport equations for h(z,±) by a simultaneous use of (a) : The connections between these functions and H(z,±), provided by the distribution functions D(z,±);and (b): the exact transport equations for H(z,±). We begin with the derivation of the transport equation for h(z,-). By definition of D(z,-), h(z,-) = D(z,-)H(z,-) . (9) Taking the derivative of each side with respect to z : dh ^--) - D(z,-) dH t z >-) + H(z,-) dD ^'- } . dz *• ' J dz v ' J dz By means of (5) , this may be written: dh(z,-) _ dz D(z,-) f [a('z,-) + b(z,-)]H(z,-) +b(z, + )H(z, + )j H(z,-) dD(z,-) ol Using the definitions of D(z,-) and D(z,+) (= h(z , + )/H(z , + ) ) and denoting the derivatives with respect to z by a prime (which will be used interchangeably with d/dz in all that follows), the preceding equation may be written: '(z,-)= (- [a(z,-) +b(z,-)] + D D ' ( ( Z Z ;; ) ) j h(z,-) l[l]' + ] b(z, + )h(z, + ) , (10) which is the general transport equation for h(z,-). Now, as in the case of N(z,6,) and H(z,±), we may associate with h(z,-) an equilibrium function h q (z,-) where we write: h (z,-)" for ^}b(z, + )h(z, + ) a^ 9 J D* Tz -"1 q [a(z,-) + b(z,-)] " d(zV-) * (U) SEC. 11.1 EQUATIONS FOR RADIOMETRIC CONCEPTS 267 An alternate representation of h a (z,-) is: h (7 , _ D"(z,-)b(z,+)h(z,+) V Z '" J " D(z, + )D(z,-)[a(z,-) + b(z,-)] " D'(z,-)D(z, + ) With this definition of hq(z,-), the transport equation (10) may be written: dh(z,-) _ cTz" (a(z,-) + b(z,-)) + ^ 1 l [h(z,-) h (z )] (12) Equation (12) is the reformulation of (11) which is of central interest in the present study, and as before we call special attention to the two functions: (i) + [a(z,-) + b(z,-)] - ^^1 (ii) h q (z,-) (13) The function (i) is the attenuation function for h(z,-)- The function (ii) is the equilibrium function for h(z,-). The derivation of the transport equation for h(z,+) proceeds in a similar manner to that leading to (12) and (13) in the case of h(z,-). Therefore, the reader may easily verify first of all that: dh(z dT *-{ [a(z, + ) + b(z, + )] + D\(z, + A D(z,+) J h(z,+) D(z,+) D(z,-) b(z,-)h(z,-) Next, if we write: (14) "h (z,+) n for q v. > j Plf4jMz,-)h(z,-) [a(z, + ) + b(z, + ) ] + %^j- then we have also: K ( 2 > + ) = (15) D*(z, + )b(z,-)h(z,-) D(z, + )D(z,-)[a(z, + ) +b(z, + )] +D»(z, + )D(z,-) so that (14) may be written 26 OPTICAL PROPERTIES AT EXTREME DEPTHS VOL. V dh(z,+) _ dz (a(z, + ) +b(z-, + )) * D d( ( z Z ; + + ) } h(z,+) - h (z,+) (16) which is the desired reformulation of (14). We draw special attention to the functions: (i) - [a(z,+) + b(z,+)] (ii) h q (z,+) D'(z,+) D(z,+) (17) The function (i) is the attenuation function for h(z,+). The function (ii) is the equilibrium function for h(z,+). We pause to observe the similarity of the functions in (8) (the set for H(z,±)) and with those in (13) and (17) (the set for h(z,±)). These sets coincide when D'(z,±) = , i.e., when H(z,±) and h(z,±) differ multiplicatively by a constant factor. That is, under this condition, (i) of (8) reduces to (i) of (13) and (17) , and H (z,±) q v ' J h (z,±) = D(z,±) = D(±) for all z . The physical significance of the condition D'(z,±) = is now clear from the study of the two-D model for irradiance fields in Chapter 8, in particular from the introductory dis cussions of Sec. 8.5. Transport Equation for Scalar Irradiance To obtain the transport equation for the scalar irradi- ance function h(z), we begin by decomposing h(z) into its upwelling and downwelling components: h(z) = h(z,+) + h(z,-) Then by using the definitions of the distribution func- tions : Dfzl = M^) DUJ H(z,±) h(z) may be represented in terms of D(z,±) and H(z,±): h(z) = D(z,-)H(z,-) + D(z,+)H(z,+) Taking the derivative of h(z) , we have SEC. 11.1 EQUATIONS FOR RADIOMETRIC CONCEPTS 269 dz v ' ' dz v ' J dz + D(Z; + ) dH(z, + ) + ( j dDJz^+i v ' ^ dz ' dz We now make use of the exact transport equations for H(z,±) ^f- - D(z,-) f- [a(z,-) +b(z,-)]H(z,-) +b (z, + )H(z, + ) + H(z,-)D'(z,-) + H(z,+)D'(z,+) + D(z,+) |[a(z, + ) +b(z, + )]H(z, + ) - b(z,-)H(z,-)j The next step is to convert the products D(z,±)H(z,±) into the equivalent functions h(z,±) and write h'(z) as a linear combination of h(z,+), h(z,-): ^f- = - [a(z,-) + b(z,-)]h(z,-) + n[l' 9 l] b(z, + )h(z,+) + D'(z,-) hfz , + D'(z,+) D(z,-) h{ < z > } D(z,+) + [a(z, + ) + b(z, + )]h(z, + ) - S(z|-) b (z,-)h(z,-) Collecting coefficients of h(z,±): ^i = A_(z)h(z,-) + A + (z)h(z,+) , (18) where we have written: »A.(z)» for - [a(z,-)+b(z,-)] ♦ "' (^O^DCz.^bCz,-) and: D(z,+) "A + (z)" for [a(z,+) + b(z, + )] + p, (z> + ) + g(z,-)b(z,+) Evidently (18) is unchanged if we write: Q£f- = A_(z)h(z,-) + A_(z)h(z,+) + A + (z)h(z, + ) + A + (z)h(z,-) - [A_(z)h(z,+) + A + (z)h(z,-)J 270 OPTICAL PROPERTIES AT EXTREME DEPTHS VOL. V But then this equation may be reduced to: ^f- = [A.(z) + A + (z)]h(z) - [A_(z)h(z,+) +A + (z)h(z,-)] (19) which is the transport equation for h(z). By writing: A_(z)h(z,+) + A + (z)h(z,-) h (z) M for q^ ; A (z) + A fz) Equation (19) is expressible as: ^1 = [A_(z) +A + (z)] [h(z)-h (z)] (20) For our present purposes, Equation (20) is of central interest, and we mark for future reference: (i) - [A_(z) + A + (z) (ii) h fz) (21) Expression (i) is the attenuation function for h(z). Expres sion (ii) is the equilibrium function for h(z). may tion view and tran take in p Thes the phys cept any Preliminary Unification and Preliminary Statement of the Equilibrium Principle We have now reached a point in our discussion where we consolidate the results obtained so far. The consolida- will serve two purposes: It will yield a preliminary of the structure of the universal transport equation, secondly, it will prepare the way for a discussion of the sport equations for the apparent optical properties to be n up in the next section. We turn now to the transport equations discussed so far, articular the equations (3), (7), (12), (16), and (20). e six equations have a common mathematical structure, and various components of the structure are associated with ical concepts common to the respective radiometric con- s. Specifically, let the general symbol " (P(z) n denote one of the following six radiometric concepts: (S>c z) N(z,e,(j)) H(z,±) h(z,±) h(z) SEC. 11.2 EQUATIONS FOR OPTICAL PROPERTIES 271 Furthermore, let "<3? a (z) M denote the associated atten- uation function for 0(z) . Finally, let "6? (z)" denote the associated equilibrium function for 0(z) . Then each of the six transport equations developed above is precisely of the form: (22) We now may make a key observation on the dynamic be- havior of the five radiometric concepts which are associated with a general direction of flow (h(z) is the only one of the preceding concepts which, by definition, is not associated with any particular directed pencil of radiation or general hemispherical flow). If "(^(z)" stands for any one of these five concepts: N(z,6,cf>), H(z,±), h(z,±), then it is easy to to verify that on the basis of (22): If @{z) 9 (z), q^ ' then d(?(z) h-ttt o and (23) if 0(z) < where the symbol ' d$(z)/d | z | ' is defined as follows, we write : "*»(')," for ^5i d z dz if @ (z) is associated with the direction of increasing z (downwell- ing direction) and: "d0fz) M for d ^ z ) d(-z) if 0(z) is associated with the direction of decreasing z (upwell- ing direction) . In other words, the equations (23) simply state that as the geometric form of the radiation represented by ), H(z,±), h(z,±), and h(z). If "(P(z) u denotes for any of these six functions, then the corresponding K- function K((?) is defined by writing: SEC. 11.2 EQUATIONS FOR OPTICAL PROPERTIES 273 K(0)" for 1 d(?(z) WUJ dz (1) Using the generic equation (22) of Sec. 11.1 and the definition (1), we have: - 0(z)K( d dz" l - ^3 whence: d dT r Key) 1 . KCgj ~¥Tz) K(). However, by suitable transformations of variables, we can reduce (4) to the general form of the universal transport equation. We next consider such transformations. 274 OPTICAL PROPERTIES AT EXTREME DEPTHS VOL. V Dimensionless Transport Equation for K ((?) At the present point of the discussions (namely (4)), we have two alternative routes open to a universal transport equation: One route starts with the adoption of a general- ized notion of optical depth defined by writing: t(z)" or "t" for f ^')dz' along with a relativization of K (£?) and K ((?) with respect to {? [z) ; thus we write: ^ "K () w hypothesis i e of not show functions pro he way in whi h y the natura erefore, we w ts in adoptin d K - functio some transpor ractical appl re now oblige The common s of <^ a (z) a tituted. cture and e dimen- as used n Sec. ing the duced by ch the 1 measure ill actu- g geomet- ns. This t equa- ications . d to con- starting nd 6? q (z) Transport Equation for K(z,6,4>) From (2) we have N(z,6,), so that K((P) =K(z,6,(f)) and K (0) = K q (z,6,) /dz : dK(z^e,^ , K s (z>e>| ^ otfz) „ , - , •* , 1 da(z) cos 6 q^ ' irj a(z) dz q v » ,y ^ cos K(z,e,(j») a(z) The right-hand side of this equation may be factored into the product of two functions yielding the desired form of the transport equation for K(z,0,cf>): dK(z,e,)] (7) where A and ? q are defined in context by the following two equations: 7i (z,e,). They are defined as shown by the pair of simultan- eous equations in (8), whose solutions are: cos K + (lna) ' q J f\2 41* coT-ff-Vana)']*- 4K a q The quantities Kq and "fy, should not be confused with each other. Kq is the logarithmic derivative of N q (see defini- tion (3)) whereas 7^ is the sought-for equilibrium function for K(z,0,) in the general context. Observe, however, that if the medium were homogeneous, then: ^(z.9,*) = - a cos •^ (z,e,(j>) = k (z,0,4>) 276 OPTICAL PROPERTIES AT EXTREME DEPTHS VOL. V More generally, in eventually homogeneous media (i.e., media in which a' (z) ■*■ as z -*•<») *a< COS ^ q (z,e,tf>) - K (z,e,40 - k^ . This follows from the asymptotic radiance theorem and its various consequences discussed in Chapter 10. Transport Equations for K(z,±) The appropriate form of (2) in the case of the K - functions K(z,±) is obtained by substituting the attenuation and equilibrium functions for H(z,±) in (2): H(z,±) = H^z,±) 1 ± K(z,±) [a(z,±) + b(z,±)] Taking the logarithmic derivatives of each side, solv- ing for K'(z,±), and factoring the quadratic in K(z,±), we have dK( d V } = [K(z,-) - ^(z,-)][K(z,-) - \{z,~) dK( d V + ) = [K(z, + ) - \(z,+)][K(z,+) - if (z, + ) , (9) . (10) For K(z, + ) the functions ^ a (z, + ), ^L(z, + ) are defined in context by the equations: ^(z, + ) + ^(z, + ) = [a(z,-) + b(z,-)] + K q (z,-) - (ln[a(z,-) + b(z,-)]) ' # a (z, + )^ q (z, + ) = [a(z,-) + b(z,-)]K q (z,-) Similarly, for K(z,-): f (z,-) + T q (z,-) =- [a(z, + ) + b(z, + )] +K q (z, + ) - (ln[a(z,+)+b(z, + )])' l a (z,-)/f q (z,-) = - [a(z, + ) +b(z, + )]K q (z, + ) These simultaneous equations may be solved to obtain explicit expressions for the respective ~% a 's and ^ Q 's . We will not do this here, but simply point out that, in all eventually homogeneous media, as z ■* °° , /f (z,±) - + [a(z,±) * b(z,±)] SEC. 11.2 EQUATIONS FOR OPTICAL PROPERTIES 277 # q (z,±) » K q (z,±) - k TO . This follows from the asymptotic radiance theorem and its various consequences studied in Chapter 10. As in the case of Kq(z,6,(J>) and \, (z , 8, ) , care should be taken so as not to confuse Kq(z,±) with ^q(z,±). The former is defined in (3), the latter by the preceding simultaneous equations. Transport Equations for k(z,±) and k(z) Starting with the general canonical equation (2) , we have for h(z) : h(z) - \ (Z) i + k(z) 1 [A + (z) * A_(z)J Similarly, for h(z,±): h(z,±) = 1 ± - h (z,±) k(z,±) [a(z,±) + b(z,±)] - "> VV D(z,±) The existence of these canonical equations for h(z,±) and h(z) is sufficient to prove the existence of the appro- priate transport equations for k(z,±) and k(z) by followinj the procedure illustrated in the preceding two paragraphs. The results are dk( d Z z ,±} = [k(z,±) - k a (z,±)][k(z,±) - k q (z,±)] dk(z) [k(z) - k a (z)][k(z) - k q (z)] , (11) . (12) The exact forms for the respective 1i a ' s and \„ ' s will not be worked out; this may be left as an exercise for the interested reader. The important point to observe is that we have now proved that for all six K -functions, the generic transport equation is: (13) Equations (22) of Sec. 11.1 and (13) form the two major sets of transport equations considered in this chapter . These 278 OPTICAL PROPERTIES AT EXTREME DEPTHS VOL. V two equations cover all twelve transport equations for @ and K($) considered so far. As in the case of (22) of Sec. 11.1, it is easy to verify on the basis of (13) that: If K((?) > ^(0), then ^^ < , and (14) if K(6P) < f q (0), then ^^ > , which show that K((?) always tends toward* its equilibrium function AqO^) • We now turn to consider the last of the standard trans- port equations, namely that for R(z,-). Transport Equation for R(z,-) By definition of R(z,-): RU ' J H(z,-) Taking the logarithmic derivative of each side, and applying the definitions of K(z,+) and K(z,-), we have: ^If 1 = R(z,-)[K(z,-) - K(z,+)] Using the following representations (18) and (19) of Sec. 9.2 of K(z,±): K(z,±) = ? [a(z,±) + b(z,±)] ± b(z,+)R(z,±) the derivative of R(z,-) may be cast into the form: dR ^~ ] ' ~ b(z,+)R 2 (z,-) + [a(z,-) + a(z,+) +b(z,-) + b(z,+)]. •R(z,-) - b(z,-) The right-hand side, which is a quadratic in R(z,-), may be factored: dR( dV" } = " b(z,f)[R(z,-) - R a (z,-)][R(z,-) - R q (z,-)].(15) *The term "tends toward" has a precise meaning here: If f 1 and f 2 are two real-valued functions defined on some common domain <£> of the reals then f., tends toward £ 2 at x e <£) if sign [f 2 (x) - f (x) ] = sign f J (x) where "sign" means the same as "sign of." As an earlier example of this, see (4) of Sec. 9.4. SEC. 11.3 EQUILIBRIUM PRINCIPLE 279 Equation (15) is the required transport equation for R(z,-), in which R a (z,-) is the attenuation function for R(z,-) and Rq(z,-) is the equilibrium function for R(z,-) (compare with (2) of Sec. 9.4). These functions are defined in context by the following system of simultaneous equations: R (z,-) + R (z q 1 - a(z,-) + a(z , + ) + b(z > b(z, + ) ) +Mz, + ) (16) V'"^ 1 ' = ofel As in the case of the K- functions, these may be solved for R (z,-) and R q (z,-) 2R (z,-) 2R (z,-) -{^>-^wb7^7-Hh7 [K (z,-) - K(z,+) 4 Mz,-j (17) R goes with the plus sign, R q with the minus sign. We observe that, in eventually homogeneous media, as R ( 7 1 - 1 b(z,-) _1_ bH V Z '" J R(z,-) b(z, + ) R m b(+) (18) R (z,-) q^ ' ; R(z,-) - R These facts follow from (17) and the asymptotic radi ance theorem of Sec. 10.7. 11.3 Universal Radiative Transport Equation and the Equi- librium Principle For the purposes of this section, let us refer to the thirteen quantities studied so far as the standard concepts (namely N(z,e,cf>), H(z,±), h(z,±), h(z), K(z,0,), K(z,±), k(z,±), k(z), and R(z,-)). A directed standard concept is any of the preceding standard concepts except h(z) and K(z) The evidence gathered in the preceding discussions may now be assembled in the form of: 280 OPTICAL PROPERTIES AT EXTREME DEPTHS VOL. V THE UNIVERSAL RADIATIVE TRANSPORT EQUATION AND THE EQUILIBRIUM PRINCIPLE. Let X be an arbitrarily stratified source- free plane-parallel medium with arbitrary incident lighting conditions . Let " £( z ) " denote any one of the standard concepts . Then asso- ciated with (?(z) are two functions d^z) andCq(z) 3 the attenuation and equilibrium functions for l(z) 3 respectively . The standard concept C(z) together with C a ( z ) and C^ ( z ) satisfy the functional rela- tion: ^f- - y(z)[6 , then: ^j^l \ o whenever oo q^ * and. ^\^C(z) = C q M • (4) The proof of the statements (1), (2), (3), and (4) have essentially been covered in the preceding discussions either directly (as in the case of (1)), or indirectly by references to the appropriate portions of the present work (as in the case of (2)- (4)). Table 1 below gives the ex- plicit forms of y(z) and 6 for the thirteen standard con- cepts: An examination of Table 1 shows that if R(z,-) is removed from the list of standard concepts, a considerable simplification is effected in the form of (1). However, in the interests of completeness we have included R(z,-) with- in the purview of (1) , and we note that by a change of z - scale, the equation is normalizable. SEC. 11.4 UNIVERSAL TRANSPORT EQUATION 281 TABLE 1 Standard Cases of the Universal Radiative Transport Equation Standard Concept Values of u,6 N(z,0,cfO H(z,±) h(z,±) h(z) y(z) = 1 6 = (degenerate) K(z f e,4>) K(z,±) k(z,±) k(z) U(z) = 1 6 = 1 (normalized) R(z,-) y(z) = - b(z,+) 6 = 1 (normalizable) 11.4 Some Additional Transport Equations Subsumed by the Universal Transport Equation The standard trans of Sec. 11.3 constitute general radiative transf means exhausts the vario transport equation as gi al set of transport equa the degenerate universal tioned. This set is ass but no less important--r standard type. We will radiometric quantities: port equations enumerated in Table 1 the most frequently used equations in er theory. This list, however, by no us ramifications of the universal ven by (1) of Sec. 11.3. An addition- tions which fall under the domain of transport equation will now be men- ociated with less frequently used-- adiometric concepts than those of the consider in particular the following (i) n-ary radiance N J (ii) n-ary radiant energy U (iii) path function (iv) vector irradiance N. (i) The transport equation governing parallel media is ((1) of Sec. 5.2): N n in plane- cos dN n (z,6,<})) - a(z)N n (z,e,) + N^z^cj)) (1) 282 OPTICAL PROPERTIES AT EXTREME DEPTHS VOL. V where (re: (6) of Sec. 5.1): N*(z,e,cj)) = ( N n " 1 (z,e' ,cj>')a(z;0' ,)dft (2) Here N n , n = 1 , 2 , (re: (11) of Sec. 5. having been scattere comprising the initi any particular probl From this , N* (z ,0 , N 1 *(z,e ,(j)) is known, in N 1 (z ,0 ,(})) , whic solution, we recall, Sec. 5.1. Numerical obtained by means of eluding remarks in S all z and (0 , ) , solved for N 2 (z,0,cj) to obtain N n (z ,0 ,) able) radiance N(z, 5.2): ... , is the n-ary scattered radiance 1) , i.e., radiance consisting of photons d precisely n-times with respect to those al radiance N° entering the medium. In em, it is assumed that N°(z,0,) is given. ) is obtainable by means of (2) . Then and (51) becomes a differential equation h is easily solved in principle. This is the basis of the definition (4) of solutions of N 1 (z,0,) may be readily a computer programmed for (1) (cf. con- ec. 5.6). Once N 1 (z,0,cf)) is known for (2) yields Ng(z,0,cj>) and (1) may be ). By repeating this process, we are led knowing N n ~ 1 (z ,0 ,) . The total (observ- 0,) is defined by writing ((3) of Sec. "N(z,e,)] . (3) When written in this form, (3) closely parallels the form of (3) of Sec. 11.1, so that we conclude, as in (4) of Sec. 11.1: (a) — ^-^q is the attenuation function for N (z,0,) (b) N (z,0,) is the equilibrium function for N (z, 9, <{>). In this way the transport equation for N (z,0,(J>) is subsumed by (1) of Sec. 11.3, in which u(z) = 1 , 6 = 0. (ii) The time-dependent transport equation governing U in a medium with no net flux across its boundary is usually written in terms of a time parameter t instead of a space parameter z ((24) of Sec. 5.8): SEC. 11.4 UNIVERSAN TRANSPORT EQUATION 283 dU n (t) U n (t) U n " 1 (t) 3t T T a s (4) where T a = 1/va , T s = 1/vs. However, if " v " denotes the speed of light in X , then we may introduce a new variable r by writing for vt so that (4) ^becomes dU n (r) aU n (r) sU n_1 (r) (5) The symbol " U n (r) " denotes the n-ary radiant energy content of a sphere of radius r about a point source (in a space X ) which emits radiant flux in some prescribed manner starting from time t = . The space is assumed homogeneous so that a(z) = a for every z in the space. By writing: "uJtOO " for sU n -'(r) and "ITJ(r) " for J&til (.,#->) dU u (r) _ dr U n (r) - U*(r) (6) Hence: (a) a is the attenuation function for U (b) U is the equilibrium function for U , and (6) is subsumed by (1) of Sec. 11.3. (iii) The transport equation governing N $ has the form (re: (9) of Sec. 5.2) : - cos 6 d Mz> 9 >40 = - aN*(z,6,cjO + N**(z,0,cj)) (7) where N**(z,e,) = N*(z,6 , ,<|) l )a(e , ,(|) , ;e,<|))dfi (8) 284 OPTICAL PROPERTIES AT EXTREME DEPTHS VOL. V Equation (7) holds in all homogeneous isotropic media (thus the reason for explicity dropping z - notation in a and a ; on the other hand, a may be arbitrary). If we write: M N^(z,e,(j)) " for N ^ ( ^ 9 ^ } then: *q " ' ' ■ ' a dN«(z,9,(f>) = [N*(z,6,) - N* fz,e,)] ; (9) dz COS 9 L*i*^,v»'r; '*q therefore (a) - K is the attenuation function for N* , v J cos J * (b) N* is the equilibrium function for N* . (iv) The steady-state 3 source- free transport equation for vector irradiance H has the form (the case n = 2 in (55) of Sec. 8.6) : ££ H(z,n,H o ) = - [a(z,n,H o ) + b(z,n,H Q ) ]H(z ,n,E Q ) + b(z,n,H;)H(z,n,H^) (10) Here we write: "H(z,n,E )" for n • H(z,H ) v ' o o which is the component of H(z,5 ) along the direction of the unit inward normal n to a unit area at depth z . H(z,H Q ) is the vector irradiance generated by radiant flux at z arriv- ing from the general subregion H Q of the unit sphere 5. If E = 5 , then H(z,S ) = H(z) the usual vector irradiance at z. The quantity H(z,n,S') is the associated (net) irradiance on the unit area contributed by the complement E' of H with respect to 5 . Because of the assumed stratification, H(z,5 ) (and hence all its components) depends only on z . By writing: b(z,n,5') H(z,n,HM "V^'V" *<" a(z,n,E o )°+b(z,n,H o ) > &» we may write (10) as: dH(z,n,H ) dz ° = - [a(z,n,H Q ) + b(z,n,H o )][H(z,n ,S q ) - H q (z,n,H Q ); (12) so that: SEC. 11.5 BIBLIOGRAPHIC NOTES 285 (a) a(z,n,- ) + b(z,n,5 ) is the attenuation function for H(z ,n, 5 ) , (b) H (z,n,~ ) is the equilibrium function q ° for H(z,n,H ). The transport equation (10) is a generalization of the standard two-flow equations for H(z,+) and H(z,-) (Chapter 8). (In the latter case, for example, H Q = 5_ , the downward hemi- sphere, and n = - k , where k is the unit outward normal to the plane-parallel medium.) Each of the four preceding transport equations may be cast into a canonical form (see (2) of Sec. 11.2) by introduc- ing the appropriate K - function for the associated radiometric quantity (see general definition (1) of Sec. 11.4). Therefore, a transport equation for each of these K- function exists, and is of the form (1) of Sec. 11.3. The equilibrium princi- ple holds for Nn Summary and Conclusion To summarize, the domain of applicability of the univer- sal transport equation (1) of Sec. 11.3 is quite wide. In fact its domain covers the totality of radiometric functions used and known to date in radiative transfer theory (the 17 dis- tinct types of radiometric concepts and their corresponding K- functions discussed above- -34 concepts in all). By means of it, the general mathematical structure of the light field in plane-parallel media can be contained in a single unifying framework, and the necessity of invoking individual discussions and principles for each of the many radiometric quantities, at least on a logical level, is now obviated. All this from the interaction principle. Hence: Frustra fit per plura quod potest fieri per pauciora.* William of Ockham (ca. 1300-1347) 11.5 B ibliographic Notes for Chapter 11 The concept of a universal radiative transport equation was introduced in [240]. The mathematical vehicle of the uni- versal radiative transport equation is that of a Riccati dif- ferential equation in factored form: ^1 = y( X )[6f(x) - f a (x)][f(x) - f q (x) *It will be futile to employ more [principles] when it is possible to employ fewer. 286 OPTICAL PROPERTIES AT EXTREME DEPTHS VOL. V and with each radiometric quantity f (or K- function, or reflectance function, etc.) the equation associates two aux- iliary functions f a , f q which, as was seen in the text, play the roles of attenuation and quilibrium functions, re- spectively. For an elementary discussion of the nondegener- ate Riccati equation, see, e.g., [116]. For modern develop- ments of the theory of nondegenerate Riccati equations perti- nent to possible radiative transfer applications, see the work of Redheffer [254], [255], [256], [257], and also Reid [261] and [262]. 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INDEX 293 absorption function (for irradiance) , 12. 97 simple formula for in asymptotic light field, 255 apparent optical property, 106 178, 181 transport equations for 271 asymptotic form of light fields, 95 criterion for asymptoticity , 254 asymptotic radiance hypothesis, 212 main mathematical proof (using canonic equation for K) , 213-227 integral equation for limit radiance distribution, 228 237, 245 theoretical and experimental example, 229, 230 simple proof (using exponen- tiality of h(z)) , 230 practical consequences, 238 K characterization of hypothesis, 242 critique of Whitney's "general law", 248 heuristic proof (using dif- ferential equation for K) , 253 attenuation functions (for irradiance) , 11 depth dependence, 25 for radiance, 264 for irradiance, 265 for scalar irradiance, 270 for reflectance, 149, 279 for K-function, 275 for other concepts, 281 attenuating functions depth dependence, 25 backward scattering functions, 11, 141 boundary effects, 46, 71 characteri for seco ential for K - canonical equation 273 canonical irradian catalog of (shallow stic equation nd order differ- equation, 39 function, 123, 252 forms of transport for k functions, representation of ce fields, 98 K - figurations depth theory) , 201 nondegenerate degenerate first, second kind forbidden, 202 classical models for irra- diance fields, 19 collimated-dif fuse light fields, 19 isotropic scattering models, 23 connections with diffusion theory, 24 classical theory, inade- quacies, 115 complete (general) solution of irradiance equations, 42 complete reflectance for ir- radiance, 4, 62, 64, 79 complete transmittance for irradiance, 4, 62, 64, 79 contrast, 165 contrast transmittance, 162 properties , 168 in canonical equation for radiance, 170 alternate representations, 171 as an apparent optical property, 172 effect of shadows on, 174 contravariation of K and D, 144 convariation of K and D , 128, 136, 140 rule of thumb, 145 decomposed (diffuse) irradi- ance, 14 diffuse absorption coeffi- cient, 111 diffuse (decomposed) irradi- ance, 14 transmittance for, 17 diffusion theory, 24 Dirac matrices, 8 directly observable optical properties, 109, 178 discrete-space radiative transfer, 51 distribution function, origin, 10 for diffuse irradiance, 15 representative values, 26 representation via radiance distribution, 27 depth dependence, 26 in canonical equation for irradiance, 99 294 INDEX VOL. V distribution function- -Cont ' d. experimental, 115 contravariation (with K- func- tion) , 144 covariation (with K-function) 128, 136, 140 physical and geometrical features , 128 asymptotic limits, 246 equilibrium radiance function, 151 for radiance, 151 for irradiance, 13, 265 for scalar irradiance, 270 for reflectance, 149, 279 for K- functions, 275 for other concepts, 281 equilibrium-seeking theorem for R, 152 for N, H. h, 270 and universal radiative transport equation, 279 equivalence theorem for R(z,-), 159 fine structure of light field hypotheses, 199 special relations, 208 first standard solution a two-D model forward scattering functions, 11, 141 forward scattering media, 140 fundamental optical property, 107, 178 global optical property, 107, 180 inequalities (among optical properties), 114, 119 inherent optical property, 106, 178, 180 interacting media (invariant imbedding operators for), 76 internal sources and irradi- ance fields, 37, 55, 81 invariant imbedding relation for irradiance fields, 2, 61 in two-D model, 36 including boundary effects, 50 for interacting media, 76 irradiance (two-flow) equa- tions , 6 irradiance fields, invariant imbedding rela- tion for, 2 irradiance fields- -Cont f d. equilibrium, 13 decomposed (diffuse) 14 residual, 15 classical models, 19 two-D models , 25 primary scattered, 43 boundary effects, 46 method of modules, 80 internal source, 37, 55, 81 vector model for, 87 curl and divergence of, 91 global properties, 97 canonical representation, 98 shallow depth theory, 187, 193 fine structure hypotheses, 199 in asymptotic setting, 238 isotropic scattering (in vec- tor light field model) , 94 models, 23 K-function two-D model, 31, 39 one-D model, 53, 56 in canonical equation, 100 theoretical forms, 111 diffuse absorption coeffi- cient, 111 experimental, 115 significance of sign, 120 characteristic equation for, 123 connections among (rradi- ance K- functions) , 123 general forms, 125 for radiance, 125 integral representations, 126 in spherical coordinates, 127 contravariation (with dis- tribution function) , 144 covariation (with distribu- tion) 128, 136, 140 physical and geometrical features, 128 absorptionlike character (for irradiance) , 138 genealogy of configurations for shallow depths, 201 asymptotic limits, 246 canonical form of transport equations for, 273 local optical property, 107, 179, 180 INDEX 295 local transmittance and reflectance, 7 many-D models, 57 method of modules for irradi- ance fields, 80 monotonicity condition on radiance distribution, 28 one-D models for irradiance fields, 51 for undecomposed fields, 52 internal sources, 55 for decomposed fields, 56 connections with observable fields, 160 ontogeny (family roots) of two flow equations, 13 optical medium, definition, 108 optical properties, inherent, apparent, 106 local, global, 107 fundamental, 107 general definition, 109 directly observable, 109 classification, 178 in asymptotic light fields, 238 Pauli matrices , 8 polarity of R,T ractors, 34 primary scattered irradiance, 43 principles of invariance global (for irradiance) , 2 local (for irradiance) , 7 for diffuse irradiance, 18 quantum mechanics formal similarity with, 8 quasi- irrotational light field, 88 Roo formulas, 113 radiance equilibrium, 151 transmittance, 164 multiplicity of representa- tions, 177 reduced (residual) irradiance, 15 reflectance for irradiance (undecom- posed) , 3 in two-D model, 33, 35, 45 complete (two-D) , 62 complete (one-D) , 64 reflectance- -Cont ' d. differential equations, 65, 79, 123, 148 R^ formulas, 113 experimental, 115 connections with attenuat- ing functions, 118 analytic representation, 146 equilibrium-seeking prop- erty, 150 integral representations of R(z,-) , 156, 196 equivalance theorem for R(z,-), 159 equilibrium and attenuation functions, 149 asymptotic limits, 246 regular neighborhoods of paths, 166 residual (reduced) irradi- ance, 15 scatterin irradi forward total, for dec 16 forward 140 second st of two- semigroup order) transmi semigroup trast t shallow d radianc experim formula compari 197 simple mo tributi ness, 2 standard ance di submarine represe g functions (for ance) , backward, 11, 141 12 omposed irradiance, scattering media, andard solution D model, 34 properties (third for reflectance and ttance factors, 67 property for con- ransmittance , 169 epth theory (of ir- e field) ental basis , 187 tion, 193 son with experiment, del for radiance dis- ons , eventual exact- 49 ellipsoid (for radi- stribution) , 250 light field, general ntation, 93 total scattering functions , 12 transmission line equations formal similarity with, 8 transmittance, for irradiance (undecomposed) , 3 296 INDEX VOL. V transmittance- -Cont * d. for reduced and diffuse irradiance, 17 in two-D model, 33, 35, 45 complete (two-D) , 62 complete (one-D), 64 differential equations, 65, 79 for radiance, 164 two-D models for irradiance fields, 25 for undecomposed fields, 30 first standard solution, 31 second standard solution, 34 for internal sources, 37 for decomposed fields, 43 inadequacies, 115 eventual exactness, 247 two-flow equations (for irra- diance) , 6 undecomposed form, 8 decomposed form, 14 equilibrium form, 13 ontogency, 13 for reduced irradiance, 17 two-D (undecomposed) model, 30 standard solutions, 31-34 complete (general) solution, 42 for decomposed irradiance, 43 boundary conditions (effects) , 46 one-D (undecomposed) model 52 many-D models, 57 exact vs. two-D, 115 asymptotic behavior, 247 universal radiative transport equation, 263 for radiometric concepts, 263 for apparent optical prop- erties , 271 and equilibrium principle, 279 standard cases, 281 additional cases, 281 vector irradiance field, model for, 87 Whitney's "general law" of diminution of light field with depth, 248 PENN STATE UNIVERSITY LIBRARIES ADDDD7D L|5ESb