IT/2 (upwelling light).\nThe amount of departure of the go(0) for a real medium\nfrom the standard ellipsoid may be taken as a measure of the\nanisotropy of scattering in the medium.","OPTICAL PROPERTIES AT EXTREME DEPTHS\nVOL. V\n252\nTABLE 1\nDistribution and reflectance factors\nfor standard ellipsoid\nD(-)\nD(+)\nRoo\nE\n2.0319\n0.8750\n0.100\n1.9664\n0.200\n1.9286\n2.0622\n0.7640\n0.6642\n0.300\n1.8881\n2.0911\n0.400\n1.8438\n2.1185\n0.5733\n2.1143\n0.4895\n0.500\n1.7943\n0.600\n1.7381\n2.1692\n0.4110\n0.700\n1.6722\n2.1928\n0.3361\n0.800\n1.5906\n2.2157\n0.2622\n0.900\n1.4775\n2.2377\n0.1841\n0.950\n1.3911\n2.2483\n0.1379\nThe Determination of E\nThe quantity E (= koo/a) is functionally related to p.\nIn the case of isotropic scattering the relation is well\nknown and of a particularly simple structure (cf. Ref. [43]).\nIn general, E is determined by viewing it as an eigen-\nvalue of the integral equation (20). There is an alternate\nway, however, to characterize E which, while not the most\nanalytically direct way, is perhaps of greatest value in\ngenerating an insight into the physical signficance of E\nand also of supplying a link between E and the directly\nobservable quantities of the light in real media. This\nalternate characterization of E stems from the following\nfunctional relation which holds between K (z, +) and the\nvarious scattering and absorption function of an arbitrary\nmedium ((31) of Sec. 9.2):\nb(z,-)\nb(z,+)\nK(2,+) + a (z,+)\nAs depth is increased each term, as a result of the\nasymptotic property of the light field, tends toward a well-\ndefined limit, so that as Z-00 , the above relation tends\nto:\nb(-)\nb(+)\n1\n=\nk + D(+)a\nThis may be rewritten as:\nB(-)\nB(+)\n(32)\n1\n=\nE + (1 - p)D(+)\n-","SEC. 10.7\nSOME PRACTICAL CONSEQUENCES\n253\nwhich is the general characteristic equation for E. . Here\nwe have written:\ndn\n\"B( )\" for\n+\nIn the case of isotropic scattering:\n,\nand (32) reduces to the following simple form after the ex-\nplicit expressions for D(+), as given by (30), are substituted\nin it:\nTHE\n(33)\nThis is the well-known characteristic equation for E in the\nisotropic case. As p varies from 0 to 1, E varies from 1\nto 0. Hence, for all P, 0 /2 , it follows immediately from (34) that\nK(z,e,o) = k.\nfor all 0 > /2. This means that the shape of the downwell-\ning radiance distribution becomes fixed at great depths. It\nfollows from the principles of invariance that the reflected\nupwelling radiance distribution also becomes fixed, so that\nthe shape of the entire radiance distribution becomes fixed\nat great depths.\nA Criterion for Asymptoticity\nAccording to (34), K(z,0,0) approaches Koo with least\nspeed when 0 = TT (i.e., for the directly downward direction,\nas in Fig. 10.15). Hence when K(z,, 0) has come within a\ngiven distance of koo, we can conclude that the other values\nK(z,0,0), 2 < LO00, of the K - -\nfunctions K (z, ) k(z) for irradiance and\nscalar irradiance, respectively.\nD( ) are the limits, as Z+00, of the distribution\nfunctions D(z,","OPTICAL PROPERTIES AT EXTREME DEPTHS\nVOL. V\n256\nis the limit, as Z - 00 , , of the reflectance\nR\nfunction R(z,-) = H(2,+)/H(2,-). .\nH(z,-)\n= H(z,-) - H(z,+), , the net downward irradiance\nat any depth Z > 20, where 20 is the depth\nbelow which the light field has essentially\nattained its asymptotic structure.\nThe details of the derivation of I, II, and II, will\nnow be given.\nShort Derivation of I\nThe short derivation of I starts with (1) , and the fact\nthat there exists a depth ZO below which the logarithmic\nderivatives of H(z,-), H(z,+), and h (z) are constant and\nequal to a common value k (see (22), (24), and (35) of Sec.\n10.7). Therefore:\ndH(z,+) = dH(z,+) - dH(z,-)\ndz\n= - H(z,+) + _H(z,-) = k_H(z,-) , (2)\nfor all\nHence:\na\nLong Derivation of I\nThe long derivation of I is essentially an exercise in\nthe use of the integrated form of the divergence relation for\nthe light field vector ((33) of Sec. 8.8)\nP(s,-) = avv(M) ,\n(3)\nwhere M is any regularly or irregularly shaped region of\nthe optical medium, S is its boundary, and P(S,-) is the\nnet inward flux across S into M. U(M) is the radiant\nenergy content of M, , V is the speed of light in M, , and\na is the required value of the volume absorption coefficient.\nIt is interesting to observe that (3) yields a value of\nin any homogeneous medium, regardless of the structure of\na\nthe light field:\n(4)\nMUM","SEC. 10.8\nSIMPLE FORMULAS\n257\nThe numerator of (4) can be obtained by traversing the\nboundary of M with flat plate collectors or other flux-\nmeasuring devices. The denominator is obtained by probing\nthe interior of M with a spherical collector (to find h (p)\nat each point p) and integrating the values over M.\nIn the present case, the extreme regularity of the as-\nymptotic light field allows one to estimate U(M) knowing\nonly one value of the scalar irradiance at a boundary point\nof M. This fact holds also for P(S,-). Specifically,\nconsider a region M in the form of a vertical column of\nunit cross section, and bounded by two parallel planes at\ndepth Z 1 , and Z2, such that The medium\nis homogeneous and stratified; hence:\n+\n(5)\nThe net fluxes over the vertical sides of the column cancel\nby virtue of the stratified light field. By hypothesis, we\nhave:\n(6)\n,\nso that:\n5(s,-)\n(7)\n.\nFurthermore:\nv ( U M ) =\ndz\n(8)\nInserting (7) and (8) into the general formula (4), we\nhave the desired result\n,\nDerivation of II\nThe formula II can be obtained directly from I by\nrecalling that:","OPTICAL PROPERTIES AT EXTREME DEPTHS\nVOL. V\n258\nH(z,-)H(z,-)-H(z,+)\n,\n(z)=h(z,-)+h(z,+)\n,\nand invoking the definitions of R(z,-), and D(z,-). That\nis, , in general:\nH(z,-) = H(z,-)-R(z,-H(z,-) = H(z,-) )[1-R(z,-)]\nand\nz)D(z,-)H(z,-) + D(z,+)H(z,+)\n;\nso that when we have:\n,\nwhich is the desired alternate formula. We observe in passing\nthat II is a limiting form of the exact formula:\n-\n(9)\n,\nthe basis for which is (25) of 9.2. Clearly, as Z+00 , equa-\ntion (9) takes the limiting form II. Furthermore if we assume\nD(+) = 2, as is done in the classical one-D two-flow theory\nof the light field, then II reduces to the relation:\n(10)\nApplied Numerology: A Rule of Thumb\nFormula III is to be taken as a convenient rule of\nthumb, and as such, is subject to possible revision whenever\nspecific optical media are under study. Yet for many pur-\nposes it is quite adequate, a fact which is based on the\nfollowing observed regularities in the values of Roo and\nD(+) in natural waters: Roo is usually found to be in the\nneighborhood of 0.02, give or take 0.01 for wavelengths near\n500 um . Furthermore for the same wavelength vicinity,\nD( ) appears to be such that the sum D (+) + D(-) is usually\nvery nearly equal to 4; and the ratio D(+)/D(-) is usually\nvery nearly equal to 2, over great ranges of depths and in\nmany media. Solving these two simultaneous equations yields,\nto two significant figures:\nD(-) = 4/3\n(11)\nD(+) = 8/3\nwhich agrees very well with experimental results (cf., e.g.,\nTable 1 of Sec. 8.5). It follows that, to the nearest rational","SEC. 10.9\nBIBLIOGRAPHIC NOTES\n259\nnumber with small integers for numerator and denominator, we\nhave from II:\n3\na = 4 k co\n(12)\nor:\na\n(13)\nAny similarity between the appearance of the fraction 4/3 in\n(13) and the index of refraction of water must be viewed as\nan amusing coincidence. Equation (13), , incidentally, points\nup once again the kinship of k with the absorption mechan-\nisms in optical media (see the discussion of (5) of Sec. 9.2\nand (29) of Sec. 9.3).\n10.9 Bibliographic Notes for Chapter 10\nThe developments of Secs. 10. 1 to 10.4 are based on the\nwork of [245].\nThe problem of the asymptotic light field in natural\nhydrosols was first clearly recognized by Whitney (re: [315]\nand [316]). The mathematical formulations and solutions of\nthe problem as in Secs. 10.5, 10.6, and 10.7 are based on the\nresearches in [224], [225], [244], and [226], respectively.\nImportant references to the asymptotic radiance hypothesis in\nthe hydrologic optics context may be found in [107], [108],\nand [209]. References to the asymptotic radiance hypothesis\nin the astrophysical context may be found in [43] and [147];\nreferences to the neutron diffusion setting are made in [62].\nSection 10. 8 is based in the main on [230].\nExperimental data in [298] exhibit clearly the asymp-\ntotic property of radiance fields in a real optical medium\nand were instrumental in the empirical establishment of the\nhypothesis.","","CHAPTER 11\nTHE UNIVERSAL RADIATIVE TRANSPORT EQUATION\n\"AZZ these examples, which might be\nmultiplied by the millions, are cases in which\na long, laborious, conscious, detailed process\nof acquirement has been condensed into\none.\nFactors which formerly had to be considered one\nby one in succession are integrated into what\nseems a single simple factor. \"\n(From: \"The Miracle of Condensed Recapitulation\"\nin the Preface of Back to Methuselah\nBernard Shaw)\n11.0 Introduction\nThe present chapter concludes the development of the\nbasic theory of radiative transfer in the present work with\na survey of the manifold transport equations for the radio- -\nmetric concepts introduced during Parts I, II, and the pre-\nceding chapters of Part III. The main purpose of the survey\nis to bring to light, especially for those readers interested\nin the theoretical aspects or radiative transfer, a recurrent\nsymbolic theme which runs through every transport equation\nconsidered so far, and to go on to capture its essence in the\nform of a \"universal radiative transport equation.\"\nThe universal radiative transport equation is an equa-\ntion which, by suitable choice of its parameters, yields in\nturn such equations as the general equation of transfer for\nradiance, the general two-flow transport equations for irradi-\nance, the transport equation for scalar irradiance, and the\ntransport equations governing the apparent optical properties\nof an optical medium.\nThe primary purpose of the universal radiative trans -\nport equation is to formulate in a single mathematical pack-\nage all the important transport equations which have evolved\nduring the past seventy years in the theoretical studies of\nthe steady state transfer of radiance energy through scattering-\nabsorbing media of the stratified plane-parallel type. In\nthis way a recapitulation of the evolutionary process of the\ntransport equation's growth is achieved and a unification of\nall these important transport equations is attained. We shall\nillustrate the scope of the equation by selecting thirty-four\n261","OPTICAL PROPERTIES AT EXTREME DEPTHS\nVOL. V\n262\ntypes of transport equations discussed in this work or implied\nby the discussions of their principal functions, and showing\nhow these various types may be uniformly subsumed under the\nregime of the universal transport equation.\nA second purpose of the universal transport equation is\nto provide a new useful tool in the study of radiative trans-\nfer theory. For example, certain special forms of the uni-\nversal transport equation have already been successfully used\n(Secs. 10.5 and 10.7) to obtain a solution to the long-standing\npractical problem of the existence of the asymptotic light\nfield in deep stratified hydrosols, a mathematical task which\nappears to be simplified, and given interesting physical sig-\nnificance with the introduction of the general type of func-\ntions associated with the universal transport equation. Fur-\nther evidence of the usefulness of the universal transport\nequation as a tool which leads to new practical results will\nbe illustrated below.\nBefore we go into the details of how the universal\ntransport equation can achieve a semblance of unity in the\nclassification of modern radiative transport equations, and\nof how it leads in some cases to new results which are beyond\nthe immediate capabilities of the classical transport equa-\ntions, it may be of help to the reader to indicate the steps\nin the development of modern radiative transfer theory which\nhave led to the idea of the universal transport equation.\nWith such information in mind the reader can then easily fol-\nlow the steps of the synthesis.\nThere are four well-defined steps in the development of\nmodern radiative transfer theory which form the immediate\nbackground to the formulation of the universal transport equa-\ntion. These are, in chronological order: The adoption of the\ngeneral equation of transfer for radiance and the development\nof the notion of equilibrium radiance [279], [111], and [43]\nthe development of the unified two-flow irradiance equations\nand the notion of equilibrium irradiance as recorded in Chap-\nter 8; the development of the canonical equation of transfer\nand the notion of the radiance K -function as recorded in\nChapter 4; the development of the theory of the asymptotic\nlight field and the transport equation for the radiance K -\nfunction as recorded in Chapter 10.\nIn the following two sections we will illustrate these\nsteps in detail and add still further illustrations which\nhave been uncovered subsequent to the time of the fourth step.\nIn this way we will systematically build up evidence for the\nexistence of a universal transport equation and for the equi-\nlibrium principle (described below) with which it is closely\nassociated. After these concrete examples of the various\ntransport equations have been assembled, the genotype of the\nuniversal transport equation is extracted from them and dis-\nplayed ((1) of Sec. 11.3). The chapter closes with a brief\nsurvey of less common but equally important examples of trans-\nport equations which are also subsumed by the universal trans-\nport equation.","SEC. 11.1\nEQUATIONS FOR RADIOMETRIC CONCEPTS\n263\n11.1 Transport Equations for Radiometric Concepts\nIn this section we will present the transport equations\nfor the following six radiometric quantities used in the study\nof plane-parallel media: radiance function N(z,0,0), upwell-\ning and downwelling irradiance functions H(z,+), upwelling\nand downwelling scalar irradiance functions h(z,+), and the\nscalar irradiance function h (z) .\nEach of these transport equations is cast into a form\nwhich explicitly exhibits a certain attenuation function and\nequilibrium function associated with the radiometric concept\nit governs. It is the isolation and emphasis of these two\nconcepts which is the earmark of the universal radiative\ntransport equation. Thus, for example, the customary form\nof the equation of transfer for radiance is recast so that\nit explicitly exhibits the special attenuation function\n-a(z)/cos 0 and the equilibrium function No (z,0,0) = N*(z,o,o)\n/a(z). Similarly, the unified irradiance equations governing\nH(z, are recast into forms which explicitly exhibit the\ncorresponding attentuation functions I [a(z,+) + b (z,=) ] and\nequilibrium functions Ha (z, ). These two reformulations for\nthe transport equations of N(z,0,0) and H(z,+) are already\nknown (see Sec. 10.7 for the case of N, and Sec. 8.: for\nthe case of H) ; however, the reformulations are now viewed\nwith the purpose of seeing what mathematical and physical\ncharacteristics are held in common by these transport equa-\ntions. It turns out that the common characteristics are the\nattenuation and equilibrium functions associated with each of\nthe radiometric concepts governed by these equations and that\neach of these transport equations is but a special case of a\nmore general equation, to be determined.\nThe discussion of the present section continues with\nthe derivation of the exact transport equations for (z, =\nand h(z). It is shown that each of these functions also may\nhave associated with it an attenuation function and an equi-\nlibrium function. In this way we show that the six radiomet-\nric quantities used in the study of plane-parallel media have\nan important set of properties common to all: The notion of\nan associated attenuation function and an associated equilib-\nrium function, and finally that the transport equation for\neach of these six radiometric concepts, is subsumed under one\ngeneral equation.\nWe now proceed to substantiate the preceding assertions\nby considering in turn each of the six radiometric concepts\nand its associated transport equation.\nEquation of Transfer for Radiance\nThe equation of transfer for radiance ( (3) of Sec. 3.15)\nin source-free stratified plane-parallel media is of the form:\ncos 0 dN(z,0,0) dz = - a(z)N(z,o,o) + N+ (z,0,0) (1)\n-\n,","VOL. V\nOPTICAL PROPERTIES AT EXTREME DEPTHS\n264\nwhere:\nNa(2,6,4) - | = des\nEquation (1) is the most basic of all transport equations and,\nas we have seen repeatedly in the preceding chapters, can\noften be used in its full generality in the several different\nbranches of applied radiative transfer theory such as astro-\nphysical optics, and in the two subdisciplines of geophysical\noptics: hydrologic optics and meteorologic optics.\nThe reformulation of (1) which is of immediate interest\nis obtained by using the notion of equilibrium radiance:\n(2)\n,\nfor by means of this function, (1) may be written:\n(3)\n=\n-\ndz\nEquation (3) is the desired reformulation of (1) * For our\npresent purposes we draw special attention to the two func-\ntions:\n(i) - cos 0 a(z)\n(4)\n(ii) N q (z,0,0)\nFunction (i) is the attenuation function for N(z,0,0) for a\nfixed direction (0,0). Function (ii) is the equilibrium\nfunction for N(z,0,0) for a fixed direction (0,0).\n*An alternate formulation of (3) is possible by adopting\nthe optical depth parameter I (= sz a(z') dz') . Such a formu-\nlation using T has been found of especial use, e.g., in\nChapter 10. However, for our present purposes, Eq. (3) is\nmore appropriate.","SEC. 11.1\nEQUATIONS FOR RADIOMETRIC CONCEPTS\n265\nTransport Equations for H(z,=)\nThe transport equations for H(z,=) (or more accurately\nthe two-flow equations for the irradiance field) are of the\nform (Chapter 8):\ndH(z,=) = [a(z,+) + b(z,+)]H(z,+) +\n(5)\nIt\nAssociated with H(z,-) and H(z,+) are the equilib-\nrium functions H (z,-) and Ha (z,+), respectively. These\nequilibrium functions are defined by writing:\nb(z,7)H(z, 7)\n\"Hq (2, ) \"\n(6)\n.\nBy means of these functions the equations in (5) may\nbe written:\nIt dH(z,=) dz = [a(z,+) + (z,=) [H(z,+) - Ha(2,=)]\n(7)\nThe equations in (7) are the desired reformulations of (5).\nFor our present purposes we draw special attention to the\ntwo sets of functions:\n(i) I [a(z,=) + (z,=)\n(8)\n(ii) Hq(2,=)\nSet (i) gives the attenuation function for the upwelling\n(+) and downwelling (-) irradiances H(z,=). Observe that,\nby (11) of Sec. 8.3, the terms in (i) can be represented by\n+t(z,=). Set (ii) gives the equilibrium function for the\nupwelling (+) and downwelling (-) irradiances H(z,4).\nTransport Equations for h(z,=)\nThe exact transport equations for h(z,+) and H(z, -)\napparently have never been even remotely discussed in the\nliterature. The reason for this gap in the family transport\nequations for the common radiometric concepts is two-fold.\nFirst, and perhaps most important, in the classical one-D\ntheory, there has never been an explicit need for the trans-\nport equations for h(z,=); the ordinary irradiances H(z,\nwere considered adequate in the early studies of the light\nfield in stratified media. However, with the advent of\nmore precise and detailed studies of the irradiance field\n(Chapters 8, 9, 10), the functions h(z,=) have finally","266\nOPTICAL PROPERTIES AT EXTREME DEPTHS\nVOL. V\nassumed a legitimate and useful role in modern radiative\ntransfer theory. Second, there is no simple or intuitively\nobvious way of obtaining the exact transport equations for\nh (z, =) from first principles (that is, obtaining de novo\nderivations starting only with the definition of h (z,-) and\nthe basic volume absorption and volume scattering functions)\nas is the case for the irradiances H(z, +). Neither is there\nany simple way of obtaining the requisite transport equations\ndirectly from the equation of transfer for radiance (again in\ncontradistinction to case for H(z,=)). In the present para-\ngraph we derive the exact transport equations for h(z,=) by\na simultaneous use of (a) : The connections between these\nfunctions and H(z, +) , provided by the distribution functions\nD(2,=);and (b): t the exact transport equations for H(z,4).\nWe begin with the derivation of the transport equation\nfor h (z,-). By definition of D(z,-),\nh(z,-) = D(z,-)H(z,-)\n(9)\nTaking the derivative of each side with respect to Z :\nth(2,)((( +\n.\nBy means of (5), this may be written:\ndh dz = ]H(z,- +\nH (z,-) dd(z,-)\n.\nUsing the definitions of D(z,-) and D(z,+) (= (z,+)/H(z,+))\nand denoting the derivatives with respect to Z by a prime\n(which will be used interchangeably with d/dz in all that\nfollows), the preceding equation may be written:\nh' (z, = h(z,-)\nD(z,-) 2,-b(z,+)h(2,+)\n(10)\nD(2,+)\n,\nwhich is the general transport equation for h(z,-).\nNow, as in the case of N(z,0,0) and H(z,=), we may\nassociate with h (z,-) an equilibrium function g(z,-) where\nwe write:\nD(z,+)\n\"ha(z,-)\"\nfor\n[a(z,-) + b(z,-)] - DIEB\n(11)\n.","SEC. 11.1\nEQUATIONS FOR RADIOMETRIC CONCEPTS\n267\nAn alternate representation of hq(2,-) is:\n0°(2,-)b(2,+)h(z,+)\n=\nWith this definition of hq(z,-), the transport equation\n(10) may be written:\nh (z, - )\n=\n+\ndz\n(12)\nEquation (12) is the reformulation of (11) which is of central\ninterest in the present study, and as before we call special\nattention to the two functions\n(i) + -\n(13)\n(ii) hq(2,-)\nThe function (i) is the attenuation function for h(z,-). The\nfunction (ii) is the equilibrium function for h(z,-).\nThe derivation of the transport equation for h(z,+)\nproceeds in a similar manner to that leading to (12) and (13)\nin the case of h(z,-). Therefore, the reader may easily\nverify first of all that:\n- =\n(14)\nNext, if we write:\n\"hq(2,+)\" for\n(15)\nthen we have also:\nD°(2,+)b(2,-)h(z,-)\nso that (14) may be written:","OPTICAL PROPERTIES AT EXTREME DEPTHS\nVOL. V\n268\ndh(z,+) dz 2- = -\n(16)\nwhich is the desired reformulation of (14) We draw special\nattention to the functions:\n(i) - [a(z,+) + b (2,+)] - D1(2,4)\n(17)\n(ii) hq(2,+)\nThe function (i) is the attenuation function for h (z, +). .\nThe function (ii) is the equilibrium function for h(z,+).\nWe pause to observe the similarity of the functions in\n(8) (the set for H(z,=)) and with those in (13) and (17) (the\nset for These sets coincide when D' (z,=) = 0, i.e.,\nwhen H(z, and h(z,=) differ multiplicatively by a constant\nfactor. That is, under this condition, (i) of (8) reduces to\n(i) of (13) and (17), and\nHa(z,=)\n= D(+), , for all Z.\nThe physical significance of the condition D' = 0 is\nnow clear from the study of the two-D model for irradiance\nfields in Chapter 8, in particular from the introductory dis-\ncussions of Sec. 8.5.\nTransport Equation for Scalar Irradiance\nTo obtain the transport equation for the scalar irradi-\nance function h (z), , we begin by decomposing h (z) into its\nupwelling and downwelling components:\nh(z)=h(z,+)+h(z,-) =\n*\nThen by using the definitions of the distribution func-\ntions:\nD(z) = hlz,\n,\nh (z) may be represented in terms of D(z,=) and H(z,\nh ( z ) = D(z,-)H(z,-) + D(z,+)H(z,+)\n.\nTaking the derivative of h (z), we have","SEC. 11.1\nEQUATIONS FOR RADIOMETRIC CONCEPTS\n269\n+\n+\ndz\nWe now make use of the exact transport equations for H(z,+)\n)( +\n+ H(z,-)D'(z,-)\n+\nThe next step is to convert the products D(z,=)H(z,+)\ninto the equivalent functions h(z, and write h'(z) as a\nlinear combination of h (z,+) h(z,-)\ndz = +\n+\n+ -\n*\nCollecting coefficients of h(z,\n+\n(18)\n,\nwhere we have written:\n\"A - (z)\" for - [a(z,-) + b(z,-)) + D'(z,-)- D(z,+)b(z,-)\nD(z,-)\nand:\n\"A+(z)\" for [a(z,+) + +\nD(z,+)\nEvidently (18) is unchanged if we write:\n+\n+ t(z)h(z,+)A(z)h(z,-\n- [A_(z)h(z,+)+A(z)h(z,)]","OPTICAL PROPERTIES AT EXTREME DEPTHS\nVOL. V\n270\nBut then this equation may be reduced to:\ndh(z) + A (z) ]h (z) - [A (z)h(z,+) + A+(2)h(z,-)]\n(19)\nwhich is the transport equation for h(z).\nBy writing:\nA_(z)h(z,+)+A(z)h(z,-)\n\"hq(2)\"\nfor\nA_(2) +A+(2)\n,\nEquation (19) is expressible as:\n(20)\nFor our present purposes, Equation (20) is of central\ninterest, and we mark for future reference:\n(i) - [A_(z) + A+ (z) ]\n(21)\n(ii) hq(2)\nExpression (i) is the attenuation function for h ( z) . Expres-\nsion (ii) is the equilibrium function for h (z).\nPreliminary Unification and Preliminary\nStatement of the Equilibrium Principle\nWe have now reached a point in our discussion where we\nmay consolidate the results obtained so far. The consolida-\ntion will serve two purposes: It will yield a preliminary\nview of the structure of the universal transport equation,\nand secondly, it will prepare the way for a discussion of the\ntransport equations for the apparent optical properties to be\ntaken up in the next section.\nWe turn now to the transport equations discussed so far,\nin particular the equations (3) , (7), (12), (16), , and (20).\nThese six equations have a common mathematical structure, and\nthe various components of the structure are associated with\nphysical concepts common to the respective radiometric con-\ncepts. Specifically, let the general symbol \"P(z)\" denote\nany one of the following six radiometric concepts:\nN(z,0,0)\nH(z,+)\n(z):\nh(z,+)\nh (z)","SEC. 11.2\nEQUATIONS FOR OPTICAL PROPERTIES\n271\nFurthermore, let \" Pa(z)\" denote the associated atten-\nuation function for @(2). Finally, let \"Po (z)\" denote the\nassociated equilibrium function for P(z). Then each of the\nsix transport equations developed above is precisely of the\nform:\nd@(z) = Pa(z) [@(2)\n(22)\nWe now may make a key observation on the dynamic be-\nhavior of the five radiometric concepts which are associated\nwith a general direction of flow (h (z) is the only one of the\npreceding concepts which, by definition, is not associated\nwith any particular directed pencil of radiation or general\nhemispherical flow). If \"P(z)\" stands for any one of these\nfive concepts: N(z,0,0), H(z,=), h(z,+), then it is easy to\nto verify that on the basis of (22) :\nd@(z)\nIf (z) q (z), , then\n0\n<\n,\nand:\n(23)\nd@(z)\nP\nif\n(z)\nthen\n0\n>\n,\nq\n,\nd@(z)/d\nwhere the symbol\nis defined as follows, we\nwrite:\nif (z) is associated\nd@(z)\"\nd@ (z)\nwith the direction of\nfor\ndz\ndz\nincreasing Z (downwell-\ning direction)\nand:\nif P(z) is associated\n\"dP(z)\"\ndP(2)\nwith the direction of\nfor\ndz\nd(-z)\ndecreasing Z (upwell-\ning direction)\nIn other words, the equations (23) simply state that as\nthe geometric form of the radiation represented by (z)\ntravels in its assigned direction, the magnitude of @ (z)\nalways changes in such a way that it tends to approach the\nmagnitude of its equilibrium function q (z). This observa-\ntion forms the core of the general equilibrium principle\nformulated below.\n11.2 Transport Equations for Apparent Optical Properties\nThe notion of \"apparent optical property\" is discussed\nin detail in Chapter 9. The following list consists of the\nten more important apparent optical properties associated\nwith plane-parallel media, as developed in Chapter 9 :","OPTICAL PROPERTIES AT EXTREME DEPTHS\nVOL. V\n272\nK(z,5)\nK(z,\nR(z,\nD(z,\nk(z,\nk(z)\nWe shall show in this section that a transport equation\nmay be assigned to each of the above K - functions. As noted,\nwe can assign a transport equation to either of the reflec-\ntance functions R(z,=) and to the distribution functions\nD(z,=), and in fact we will exhibit the transport equation\nfor R(z,-) and go on to deduce, by means of this equation,\nan interesting property about the depth behavior of R(z,-).\nWe will not, however, exhibit the transport equation for\nR(z,+) and D(z,=) for the following reasons: By definition,\nR(z,+) = 1/R(z,-), so that once a transport equation is ob-\ntained for R(z,-), one for R(z,+ +) would be superfluous.\nThe reason for not obtaining transport equations for the op-\ntical properties D(z, is more subtle and may be inferred\nfrom the preceding formulations by recalling that the trans-\nport equations for H(z,=) , h (z,=) make implicit or explicit\nuse of the distribution functions. If we were to deduce the\ntransport equations for D(z, we would see that the quanti-\nties H(z, or h(z, would be explicitly involved in them.\nTherefore, a logical circularity would creep into the final\nset of transport equations if we insisted on obtaining trans-\nport equations for D(z,=) in addition to those of H(z,=)\nand h(z,=). In order to avoid such a circularity we must\ndecide on the elimination of one of the three sets of quanti-\nties: H(z,+), h (z, ) D(z, Such a decision is easy to\nreach after we note that H(z, and ((z,=) are the funda-\nmental observables in natural light fields, and that the\nD(z, simply act as analytical liaisons between these quan-\ntities. Therefore, we will agree that D(z,=) are to con-\ntinue to act as the connecting links between the irradiance\nand scalar irradiance concepts, and that they are to enter\ninto the calculations solely in the capacity of dimensionless\nmathematical parameters. Their usual physical interpretation\nwill, of course, be retained, namely that they are measures\nof the directional variation of the radiance distribution at\na general depth Z . (In this connection, see Sec. 8.5.)\nCanonical Forms of Transport\nEquations for K Functions\nThe procedure for obtaining the the transport equation\nfor the six K-functions is facilitated by the preceding re-\nsults, in particular by means of the six transport equations\nfor (z, 0, ()) H(z,=) h (z,=), and h (z). If \"P(z)\" denotes\nfor any of these six functions, then the corresponding K -\nfunction K(P) is defined by writing:","SEC. 11.2\nEQUATIONS FOR OPTICAL PROPERTIES\n273\n\"K(@)\" for @(z) __PP(z)\n(1)\nUsing the generic equation (22) of Sec. 11.1 and the\ndefinition (1), we have:\n.\nSolving this for P(z), we obtain the canonical form of\nthe transport equation for :\n(2)\nThe canonical form for the radiance function of Chapter\n4 is thus extended to wider contexts (see, e.g., (5) of Sec.\n4.7)\nThis canonical form of the transport equation serves as\nthe common starting point for the derivation of the equations\ngoverning individual K - - functions. Thus, by taking the for-\nmal logarithmic derivative of each side of (2) :\ndz =\n,\nand writing, in analogy to (1)\nfor\n\"Kq(\")\nfor\n(3)\n,\ndz\nwe have:\nd\nK(P)\ndz\nPa(2)\nK(P)\nwhence:\ndz d = kq())\n(4)\nAs it stands, (4) may be taken as the transport equation for\nK (P). However, by suitable transformations of variables, we\ncan reduce (4) to the general form of the universal transport\nequation. We next consider such transformations.","OPTICAL PROPERTIES AT EXTREME DEPTHS\nVOL. V\n274\nDimensionless Transport Equation for K (P)\nAt the present point of the discussions (namely (4)),\nwe have two alternative routes open to a universal transport\nequation: One route starts with the adoption of a general-\nized notion of optical depth defined by writing:\n\"I(z)\" or \"I\" for\nalong with a relativization of K(P) and K\nwith\nrespect\nto Pa(2); thus we write:\nK\n(P)\n\"K(@)\"\nfor\n,\nP\na\nand:\nK\n\"Kq(P)\"\nfor\nP\na\nThen (4) may be written in the dimensionless form:\ndK(P)\n(5)\n=\n-\ndT\nEquation (5) has the advantage of simplicity of structure and\nis therefore ideal for formal work. For example, the dimen-\nsionless form of (5) for the case (z) = N(z,0,0) was used\nin the proof of the asymptotic radiance hypothesis in Sec.\n10.5. However, (5) has the disadvantage of not showing the\nexplicit effects on the associated K - - functions produced by\ninhomogeneities of the medium nor of the way in which the\nK - functions vary with geometrical depth, the natural measure\nof depth used in experimental work. Therefore, we will actu-\nally take the second route which consists in adopting geomet-\nrical depth and which uses unrelativized K - functions. This\nresults in a mathematically more cumbersome transport equa-\ntion but is actually of greater use in practical applications.\nBy adopting the alternative route, we are now obliged to con-\nsider each of the K - - functions in turn. The common starting\npoint is (2) in which the explicit forms of Pa (z) and Pq(z)\nfor the various concepts have been substituted.\nTransport Equation for K(z,0,0)\nFrom (2) we have\nNa(2,0,0)\nN(z,0,0)\n(6)\n=\n+ cos 0 K(z,0,0) a(z)","SEC. 11.2\nEQUATIONS FOR OPTICAL PROPERTIES\n275\nin which we have set Pal (z) = a(z)/cos 0, =\nso that K (P) = K (z,0,0) and K. (@) = using the defi-\nnitions in (4) of Sec. 11.1. Taking`the logarithmic deriva-\ntive of each side of (6) and solving for dK(z,e,o)/dz:\nd(K,0,0) dz +\nKq(2,0,4)\nThe right-hand side of this equation may be factored\ninto the product of two functions yielding the desired form\nof the transport equation for K(z,0,0)\nK(2,0,0) dz = [K(z,0,0) - (2,0,0)\n(7)\nwhere K. and kg are defined in context by the following\ntwo equations :\n(8)\nThe functions Ka(z,e,o) and Kg (z,0,0) appearing in (7) are,\nrespectively the attenuation and equilibrium functions for\nK(z,0,0). They are defined as shown by the pair of simultan-\neous equations in (8), whose solutions are:\n2 Kg\n= - a + +\n2K\nThe quantities Kq and Kg should not be confused with each\nother. Kq is the logarithmic derivative of N, (see defini-\ntion (3)) whereas Kq is the sought-for equiliBrium function\nfor K(z,0,0) in the general context. Observe, however, that\nif the medium were homogeneous, then:\n= - a\n=","OPTICAL PROPERTIES AT EXTREME DEPTHS\nVOL. V\n276\nMore generally, in eventually homogeneous media (i.e.,\nmedia in which a' (z) 0 as 2 - 00)\nThis follows from the asymptotic radiance theorem and its\nvarious consequences discussed in Chapter 10.\nTransport Equations for K (z, =\nThe appropriate form of (2) in the case of the K -\nfunctions K(z, is obtained by substituting the attenuation\nand equilibrium functions for H(z,=) in (2):\nHq(2,=)\nTaking the logarithmic derivatives of each side, solv-\ning for K' (z,=), and factoring the quadratic in K(z,=), we\nhave\n(9)\n= - ,\nik(z,+) dz = [K(z,+) - Ka(z,+)\n(10)\nFor K (z, +) the functions Ka(z,+), Kq(2,+) are defined in\ncontext by the equations\n+ b(z,-)]+Kg(2,-) (1n[a(z, -) + b(z,-)])\n+\nSimilarly, for K(z,-):\n+ (1n[a(z,+)+b(z,+)])'\nThese simultaneous equations may be solved to obtain\nexplicit expressions for the respective Ka's and Kg's\nWe will not do this here, but simply point out that, in all\neventually homogeneous media, as 2 - 00 ,\nKa(z,)+a(z,)+b(z,)]\n,","SEC. 11.2\nEQUATIONS FOR OPTICAL PROPERTIES\n277\n.\nThis follows from the asymptotic radiance theorem and its\nvarious consequences studied in Chapter 10. As in the case\nof Kq(z, 0, , ) and Kg(2,0,0), care should be taken so as not\nto confuse Kg (z, ) with Kg (z, =). The former is defined in\n(3), the latter by the preceding simultaneous equations.\nTransport Equations for k(z, and k(z)\nStarting with the general canonical equation (2), we\nhave for (z):\nA\n(z)\nSimilarly, for h(z,\n- D(z,+)\nThe existence of these canonical equations for (z, =\nand h (z) is sufficient to prove the existence of the appro-\npriate transport equations for k(z, and k (z) by following\nthe procedure illustrated in the preceding two paragraphs.\nThe results are\n, (11)\n- ,\n(12)\nThe exact forms for the respective Ka's and 's\nwill not be worked out; this may be left as an exercise for\nthe interested reader. The important point to observe is\nthat we have now proved that for all six K - functions, the\ngeneric transport equation is:\n(13)\nEquations (22) of Sec. 11.1 and (13) form the two major\nsets of transport equations considered in this chapter. These","VOL. V\nOPTICAL PROPERTIES AT EXTREME DEPTHS\n278\ntwo equations cover all twelve transport equations for P\nand K(() considered so far.\nAs in the case of (22) of Sec. 11.1, it is easy to\nverify on the basis of (13) that:\ndK (\nIf K(P) > KQ, then\n0\n<\ndz\n,\n(14)\nand\ndK I\n(P) < Kg, then\n> 0\nif\nK\n,\nwhich show that K (P) always tends toward* its equilibrium\nfunction Kq(P).\nWe now turn to consider the last of the standard trans-\nport equations, namely that for R(z,-).\nTransport Equation for R(z,-)\nBy definition of R(z,-):\nTaking the logarithmic derivative of each side, and applying\nthe definitions of K(z,+) and K(z,-), we have:\nUsing the following representations (18) and (19) of Sec. 9.2\nof K(z, )\nK(z, = I [a(z,=) + b(z,=)) + (b(z,7)R(z,=)\n,\nthe derivative of R(z,-) may be cast into the form:\n+ [a(z,-) a(z,+)+b(z,-)+b(z,+)].\nThe right-hand side, which is a quadratic in R(z,-),\nmay be factored:\ndR(z,-) 2-1-b(z,)R(z,-) - -\n*The term \"tends toward\" has a precise meaning here: If\nf1 and f2 are two real-valued functions defined on some\ncommon domain D of the reals then f1 tends toward f\nat\nX E D if sign [f2(x) - f, (x)] = sign fi(x) where \"sign\" means\nthe same as \"sign of.\" As an earlier example of this, see (4)\nof Sec. 9.4.","SEC. 11.3\nEQUILIBRIUM PRINCIPLE\n279\nEquation (15) is the required transport equation for R(z,-)\nin which R, (z,-) is the attenuation function for R(z,-) and\nRq (z,-) is the equilibrium function for R(z,-) (compare with\n(2) of Sec. 9.4). These functions are defined in context by\nthe following system of simultaneous equations :\n(16)\n.\nAs in the case of the K - functions, these may be\nsolved for Ra (z,-) and Rq (z, -) :\n(z,\n2R\nR((\n=\n-\n1/2\n+\n(17)\nR goes with the plus sign, Rq with the minus sign.\nWe observe that, in eventually homogeneous media, as\nZ -00:\n(18)\n, (z,-)\nR(z,-)\nR\noo\nThese facts follow from (17) and the asymptotic radi-\nance theorem of Sec. 10.7.\n11.3 Universal Radiative Transport Equation and the Equi- -\nlibrium Principle\nFor the purposes of this section, let us refer to the\nthirteen quantities studied so far as the standard concepts\n(namely N(z,0,0), H(z,=), h(z,+), , h(z), , K(z,e,o), , K(z,\nk(z, k(z), and R(z,-)). A directed standard concept is\nany of the preceding standard concepts except h(z) and K(z).\nThe evidence gathered in the preceding discussions may\nnow be assembled in the form of:","VOL. V\nOPTICAL PROPERTIES AT EXTREME DEPTHS\n280\nTHE UNIVERSAL RADIATIVE TRANSPORT EQUATION AND THE\nEQUILIBRIUM PRINCIPLE. Let X be an arbitrarily\nstratified source-free plane-parallel medium with\narbitrary incident lighting conditions. Let \"Cl2)\"\ndenote any one of the standard concepts. Then asso-\nciated with C(2) are two functions Ca(2) and C(2),\nthe attenuation and equilibrium functions for C(2),\nrespectively. The standard concept C(2) together\nwith Ca(2) and Ca (2) satisfy the functional rela-\ntion:\ndC(z) = ,\n(1)\nwhere ( 2 ) and s are known parameters depending on\nC(2). The relation (1) is the universal radiative\ntransport equation. (It is degenerate if s = 0 ;\nand normalized if u = 1, S = 1.)\nIf ((z) is a directed standard concept and u(z) > 0 ,\nthen:\nde(z) dz AV 0 whenever C(z) (Cq(2) ;\n(2)\nand if ((2) is any standard concept, and X\nis\neventually homogeneous, then:\n'a( 00) = lim Co (z)\nexists,\n(3)\n= lim '2+00Cq(2)\nexists,\nand:\n(4)\nlim =\nThe proof of the statements (1), (2), (3), and (4)\nhave essentially been covered in the preceding discussions\neither directly (as in the case of (1)) , or indirectly by\nreferences to the appropriate portions of the present work\n(as in the case of (2)-(4)). - Table 1 below gives the ex-\nplicit forms of (z) and S for the thirteen standard con-\ncepts: An examination of Table 1 shows that if R(z,-) is\nremoved from the list of standard concepts, a considerable\nsimplification is effected in the form of (1). However, in\nthe interests of completeness we have included R(z,-) with-\nin the purview of (1), and we note that by a change of Z -\nscale, the equation is normalizable.","SEC. 11.4\nUNIVERSAL TRANSPORT EQUATION\n281\nTABLE 1\nStandard Cases of the Universal\nRadiative Transport Equation\nStandard Concept\nValues of u, 8\nN(z,0,0)\nu(z) = 1\nH(z,\nS = 0\nh(z,+)\n(degenerate)\nh ( z)\nK(z,0,0)\nu(z) = 1\nK(z,=)\nS = 1\nk(z,=)\n(normalized)\nk(z)\nu(z) = - b z , + =\nR(z,-)\ns = 1\n(normalizable)\n11.4 Some Additional Transport Equations Subsumed by the\nUniversal Transport Equation\nThe standard transport equations enumerated in Table 1\nof Sec. 11.3 constitute the most frequently used equations in\ngeneral radiative transfer theory. This list, however, by no\nmeans exhausts the various ramifications of the universal\ntransport equation as given by (1) of Sec. 11.3. An addition-\nal set of transport equations which fall under the domain of\nthe degenerate universal transport equation will now be men-\ntioned. This set is associated with less frequently used--\nbut no less important - - radiometric concepts than those of the\nstandard type. We will consider in particular the following\nradiometric quantities:\nNn\n(i) n-ary radiance\n(ii) n-ary radiant energy Un\n(iii) path function\nN*\n(iv) vector irradiance\nH\n(i) The transport equation governing Nn in plane-\nparallel media is (1) of Sec. 5.2) :\n- cos 0 dN' (2,0,0) = - a(z): N°(2,0,0) + Nn(2,0,0)\n(1)\ndz","282\nOPTICAL PROPERTIES AT EXTREME DEPTHS\nVOL. V\nwhere (re: (6) of Sec. 5.1):\nNn(2,0,0) = Nn-1 . (2)\nHere Nn, = 1, is the n-ary scattered radiance\n(re: (11) of Sec. 5.1), i.e., radiance consisting of photons\nhaving been scattered precisely n-times with respect to those\ncomprising the initial radiance N° entering the medium. In\nany particular problem, it is assumed that N°(z,0,0) is given.\nFrom this, N1 (z,0,0) is obtainable by means of (2). Then\nN1 (z,0,0) is known, and (51) becomes a differential equation\nin N1 (z,0,0), which is easily solved in principle. This\nsolution, we recall, is the basis of the definition (4) of\nSec. 5.1. Numerical solutions of N1 (z,e o) may be readily\nobtained by means of a computer programmed for (1) (cf. con-\ncluding remarks in Sec. 5.6). Once N1 (z,0,0) is known for\nall Z and (0), (2) yields N2(2,0,0) and (1) may be\nsolved for N2(2,0,0). By repeating this process, we are led\nto obtain 1(z,0,0) knowing Nn-1 (z,0,0). The total (observ-\nable) radiance N(z,0,0) is defined by writing ((3) of Sec.\n5.2):\n\"N(z,0,0)\"\nfor\nn=o\nFor our present purposes we write:\n\"Na(2,8,4)\" for Nn(2,0,0) a(z)\nso that (1) may be written:\n(3)\n-\n.\nWhen written in this form, (3) closely parallels the form of\n(3) of Sec. 11.1, so that we conclude, as in (4) of Sec. 11.1:\n(a) COS A a(z) is the attenuation function for N°(2,0,0)\n(b) (z,0,0) is the equilibrium function for N°(2, 0, 0)\nIn this way the transport equation for N°(2,0,0) is subsumed\nby (1) of Sec. 11.3, in which u (z) = 1, S = 0.\n(ii) The time-dependent transport equation governing\nUn\nin a medium with no net flux across its boundary is usually\nwritten in terms of a time parameter t instead of a space\nparameter Z ((24) of Sec. 5.8):","SEC. 11.4\nUNIVERSAN TRANSPORT EQUATION\n283\n(4)\nwhere Ia = 1/va, Ts = 1/vs. However, if \"v\" denotes the\nspeed of light in X, then we may introduce a new variable\nr by writing\n\"r\" for vt,\nso that (4) becomes:\n= + sun-1(r)\n(5)\nThe symbol 11 Un (r)\" denotes the n-ary radiant energy content\nof a sphere of radius r about a point source (in a space X )\nwhich emits radiant flux in some prescribed manner starting\nfrom time t = 0. The space is assumed homogeneous so that\na(z) = a for every Z in the space. By writing:\n\"un(r)\" for sun-1(r)\nand:\n\"un(r)\" for\nwhere \"p\" denotes s/a, (5) may be then written:\n(6)\n=\nHence:\n(a) a is the attenuation function for Und\n(b) un is the equilibrium function for under\nand (6) is subsumed by (1) of Sec. 11.3. .\n(iii) The transport equation governing N * has the form\n(re: (9) of Sec. 5.2):\n- + (7)\nwhere\nN**(z,e,o) = 1s\n(8)\n.","284\nOPTICAL PROPERTIES AT EXTREME DEPTHS\nVOL. V\nEquation (7) holds in all homogeneous isotropic media\n(thus the reason for explicity dropping Z - notation in a\nand o; on the other hand, O may be arbitrary) If we write:\nNt+(2,0,0)\n\"N*q(2,0,0)\"\nfor\nthen:\ndz = - ; (9)\ntherefore\n(a)\nis the attenuation function for N*\n-\ncos\n0\n,\n(b) N. * q is the equilibrium function for N . . .\n(iv) The steady-state, source-free transport equation\nfor vector irradiance H has the form (the case n = 2 in\n(55) of Sec. 8.6):\nd\n=\n(10)\nHere we write:\n\"H(z,n,E_)\" for\n,\nwhich is the component of H(z,E0) along the direction of the\nunit inward normal n to a unit area at depth Z. H(z,Eo)\nis the vector irradiance generated by radiant flux at Z arriv-\ning from the general subregion 30 of the unit sphere E. If\nE0 = E, then the usual vector irradiance at Z.\nThe quantity H(z,n,e') is the associated (net) irradiance on\nthe unit area contributed by the complement E of Ed with\nrespect to E. Because of the assumed stratification,\nH(z,Eo) (and hence all its components) depends only on Z .\nBy writing:\nb(2,n,E')H(z,n,E)\n(11)\n,\nwe may write (10) as:\ndH(z,n,E\ndz = - [a(z,n,E ) +\n(12)\nso that:","285\nSEC. 11.5\nBIBLIOGRAPHIC NOTES\n(a) a(z,n,= (z,n,E\nis the attenuation function\nfor H(z,n,E ),\n(b) Ha(2,n,=\nis the equilibrium function\nfor H(z,n,E.).\nThe transport equation (10) is a generalization of the\nstandard two-flow equations for H(z,+) and H(z,-) (Chapter 8).\n(In the latter case, for example, E0 = E_ , the downward hemi-\nsphere, and n = - k, where k is the unit outward normal to\nthe plane-parallel medium.)\nEach of the four preceding transport equations may be\ncast into a canonical form (see (2) of Sec. 11.2) by introduc-\ning the appropriate K - function for the associated radiometric\nquantity (see general definition (1) of Sec. 11.4). Therefore,\na transport equation for each of these K - function exists,\nand is of the form (1) of Sec. 11. 3. The equilibrium princi-\nple holds for Nn, N*, H, and Un.\nSummary and Conclusion\nTo summarize, the domain of applicability of the univer-\nsal transport equation (1) of Sec. 11.3 is quite wide. In fact\nits domain covers the totality of radiometric functions used\nand known to date in radiative transfer theory (the 17 dis-\ntinct types of radiometric concepts and their corresponding\nK - functions discussed above--34 concepts in all). By means\nof it, the general mathematical structure of the light field\nin plane-parallel media can be contained in a single unifying\nframework, and the necessity of invoking individual discussions\nand principles for each of the many radiometric quantities, at\nleast on a logical level, is now obviated. A11 this from the\ninteraction principle. Hence:\nFrustra fit per plura quod potest fieri per pauciora. *\nWilliam of Ockham (ca. 1300-1347)\n11.5 Bibliographic Notes for Chapter 11\nThe concept of a universal radiative transport equation\nwas introduced in [240]. The mathematical vehicle of the uni-\nversal radiative transport equation is that of a Riccati dif-\nferential equation in factored form:\n-\n*It will be futile to employ more [principles] when it is\npossible to employ fewer.","286\nOPTICAL PROPERTIES AT EXTREME DEPTHS\nVOL. V\nand with each radiometric quantity f (or K - function, or\nreflectance function, etc.) the equation associates two aux-\niliary functions fa, fa which, as was seen in the text,\nplay the roles of attenuation and quilibrium functions, re-\nspectively. For an elementary discussion of the nondegener\nate Riccati equation, see, e.g., [116]. For modern develop-\nments of the theory of nondegenerate Riccati equations perti-\nnent to possible radiative transfer applications, see the\nwork of Redheffer [254], [255], [256], [257], and also Reid\n[261] and [262]. These mathematical studies are also of po-\ntential applicability to the operator equations for the R\nand T operators in Chapter 7, as noted in Sec. 7.15. In\nview of the work of this chapter and that of Chapter 7, along\nwith the results of the work of Redheffer and Reid, it is\nclear that the Riccati differential equation enters a produc-\ntive new phase in mathematical physics.","BIBLIOGRAPHY\n287\nBIBLIOGRAPHY FOR VOLUME V\nAmbarzumian, V. A., \"Diffuse reflection of light by a\n1.\nfoggy medium,\" Compt. rend. (Doklady) Acad. Sci.\nU.R.S.S. 38, (1943).\nBeach, L. A. et al., Comparison of Solutions to the\n11.\nOne-Velocity Neutron Diffusion Problem (Naval Res.\nLab. Report 5052, December 23, 1957).\nBenford, F. \"Reflection and transmission by parallel\n18.\nplates, \" J. Opt. Soc. Am. 7, 1017 (1923).\nBenford, F. , \"Transmission and reflection by a diffus-\n19.\ning medium,\" Gen. Elec. Rev. 49, 46 (1946)\nBenford, F., \"Radiation in a diffusing medium,\" J. Opt.\n20.\nSoc. Am. 36, 524 (1946).\nBoldyrev, N. G. , and Alexandrov, A., , \"On the lightfield\n27.\nin lightscattering media,\" (in Russian), Trans.\nOpt. Inst. Leningrad 6, (No. 59), 1 (1931), and\n11 (No. 99), 56 (1936).\nBrand, L., Vector and Tensor Analysis (John Wiley and\n30.\nSons, New York, 1947).\nChandrasekhar, S., \"On the radiative equilibrium of a\n42.\nstellar atmosphere, II,\" Astrophys. J. 100, 76\n(1944).\nChandrasekhar, S., , Radiative Transfer (Oxford, 1950)\n43.\nChannon, H. J., Renwick, F. F., and Storr, B. V., \"The\n44.\nbehaviour of scattering media in fully diffused\nlight,\" Proc. Roy. Soc. A 94, 222 (1918).\nDavison, B. Neutron Transport Theory (Clarendon Press\n62.\nOxford, 1957).\nDietzius, R., \"Die lichtdurchlassigkeit und die albedo\n65.\nvon nebel und wolker,\" Beitr. z. Physik d. Frei.\nAtm. 10, 202 (1922).\nDuntley, S. Q. , \"The optical properties of diffusing\n69.\nmaterials, \" J. Opt. Soc. Am. 32, 61 (1942).\nDuntley, S. Q., \"The mathematics of turbid media,\" J.\n70.\nOpt. Soc. Am. 33, 252 (1943).\nDuntley, S. Q., \"The reduction of apparent contrast by\n71.\nthe atmosphere, J. Opt. Soc. Am. 38, 179 (1948).\nDuntley, S. Q., \"The visibility of distant objects,\"\n72.\nJ. Opt. Soc. Am. 38, 237 (1948).","288\nPROPERTIES\nVOL. V\n78.\nDuntley, S. Q., \"Light in the sea,\" J. Opt. Soc. Am.\n53, 214 (1963).\n80.\nDuntley, S. Q., Boileau, A. R., and Preisendorfer,\nR. W., \"Image transmission through the troposphere,\nI, 11 J. Opt. Soc. Am. 47, 499 (1957).\n82.\nDuntley, S. Q., and Preisendorfer, R. W., The visibil-\nity of Submerged Objects (Final Report N5ori-07854,\nVisibility Laboratory, Massachusetts Institution\nof Technology, 31 August 1952).\n94.\nFresnel, A. J., Oeuvres Completes (Paris, 1866), vol.\n10, p. 640.\n97.\nGershun, A., \"The passage of light through a plane\nlamina of lightdiffusing material,\" Trans. Opt.\nInst. Leningrad 11, (No. 99), 43 (1936).\n98.\nGershun, A. \"The light field,\" (trans. by P. Moon\nand G. Timoshenko), J. Math. and Phys. 18, 51\n(1939).\n102.\nGurevic, M. M., \"On a rational classification of light\nscattering media,\" (in Russian), Trans. Opt. Inst.\nLeningrad 6 (No. 57), 1 (1931).\n103.\nHalmos, P. R., Measure Theory (D. Van Nostrand, New\nYork, 1950).\n107.\nHerman, M. and Lenoble, J., \"Asymptotic Radiation in\na Scattering and Absorbing Medium,\" Proceedings\nof the Symposium on Interdisciplinary Aspects of\nRadiative Energy Transfer, 24-26 February 1966\n(Cosponsors: General Electric Co., Joint Inst. for\nLab. Astrophys. of Univ. of Col., and Off. of Nav.\nRes.).\n108.\nHerman, M., and Lenoble, J., , \"Etude du Régime Asymp-\ntotique dans une Milieu Diffusant et absorbant,\"\nRevue d' Optique 43, 555 (1964).\n111.\nHopf, E., Mathematical Problems of Radiative Equilib-\nrium (Cambridge Univ. Press, 1934).\n114.\nHulburt, E. 0. \"Propagation of radiation in a scat\ntering-absorbing medium,\" J. Opt. Soc. Am. 33, 42\n(1943).\n116.\nInce, E. L. Ordinary Differential Equations (Dover\nPublications, Inc., New York, 1956).\n138.\nKing. L. V., \"On the scattering and absorption of\nlight in gaseous media, with applications to the\nintensity of sky radiation,\" Phil. Trans. A 212,\n375 (1913).\n141.\nKoschmieder, H. \"Theorie der horizontalen sichtweite,'\nBeitr, Z. Phys. d. Frei. Atm. 12, 33 (1924).","BIBLIOGRAPHY\n289\n142.\nKottler, F. , \"Turbid media with plane-parallel sur-\nfaces,\" J. Opt. Soc. Am. 50, 483 (1960)\n145.\nKubelka, P., \"New contributions to the optics of in-\ntensely light-scattering materia.' J. Opt. Soc.\nAm. 38, 448 (1948) and 44, 330 (1954).\n146.\nKubelka, P., and Munk, F., \"Ein beitrag zur optik\nder farbanstriche,\" Z. F. Tech. Physik 12, 593\n(1931).\n147.\nKuscer, I., \"Milne's problem for anisotropic scatter-\ning,\" J. Math. and Phys. 34, 256 (1955).\n154.\nLenoble, J., \"Angular distribution of submarine day-\nlight in deep water,\" Nature 178, 745 (1956)\nMecke, R., \"Über zerstreuung und beugung des lichtes\n174.\ndurch nebel und wolken, Ann. der Physik 65, 257\n(1921).\n177.\nMiddleton, W. E. K., Vision Through the Atmosphere\n(Univ. of Toronto Press, 1952).\n178.\nMiddleton, W. E. K., \"The color of the overcast sky,\"\nJ. Opt. Soc. Am. 44, 793 (1954).\n180.\nMilne, E. A., \"Thermodynamics of the stars,\" Handbuch\nder Astrophysik (Springer, Berlin, 1930), vol. 3,\nChap. 2.\n187.\nMoon, P. and Spencer, D. E., \"Theory of the photic\nfield, : Jour. Franklin Inst. 255, 33 (1953).\n188.\nMoon, P., and Spencer, D. E., \"Some applications of\nthe photic field theory,\" Jour. Franklin Inst.\n255, 113 (1953).\n209.\nPoole, H. H., \"The angular distribution of submarine\ndaylight in deep water,\" Sci. Proc. Roy. Dublin\nSoc. 24, 29 (1945)\n210.\nPreisendorfer, R. W., Lectures on Photometry, Hydrol.\nogy Optics, Atmospheric Optics (Lecture Notes,\nvol. I, Visibility Laboratory, Scripps Inst. of\nOcean., University of California, San Diego, Fall\n1953).\n220.\nPreisendorfer, R. W., The Divergence of the Light\nField in Optical Media (Scripps Inst. of Ocean.\nRef. 58-41, University of California, San Diego,\n1957.\n221.\nPreisendorfer, R. W., Unified Irradiance Equations\n(Scripps Inst. of Ocean. Ref. 58-43, University\nof California, San Diego, 1957)","290\nPROPERTIES\nVOL. V\n222.\nPreisendorfer, R. W. Directly Observable Quantities\nfor Light Fields in Natural Hydrosols (Scripps\nInst. of Ocean. Ref. 58-46, University of Califor-\nnia, San Diego, 1958).\n223.\nPreisendorfer, R. W., Canonical Forms of the Equation\nof Transfer (Scripps Inst. of Ocean. Ref. 58-47,\nUniversity of California, San Diego, 1958).\n224.\nPreisendorfer, R. W., A Proof of the Asymptotic Radi-\nance Hypothesis (Scripps Inst. of Ocean. Ref.\n58-57, University of California, San Diego, 1958).\n225.\nPreisendorfer, R. W., On the Existence of Character-\nistic Diffuse Light in Natural Waters (Scripps\nInst. of Ocean. Ref. 58-59, University of Cali-\nfornia, San Diego, 1958).\n226.\nPreisendorfer, R. W., Some Practical Consequences of\nthe Asymptotic Radiance Hypothesis (Scripps Inst.\nof Ocean. Ref. 58-60, University of California,\nSan Diego, 1958).\n227.\nPreisendorfer, R. W., Photic Field Theory for Natural\nHydrosols (Scripps Inst. of Ocean. Ref. 58-66,\nUniversity of California, San Diego, 1958).\n228.\nPreisendorfer, R. W., and Harris J. L., The contrast\nTransmittance Distribution as a Possible Tool in\nVisibility Calculations (Scripps Inst. of Oceanog-\nraphy Ref. 58-68, University of California, San\nDiego, 1958).\n230.\nPreisendorfer, R. W., and Richardson, W. H., Simple\nFormulas for the Volume Absorption Coefficient in\nAsymptotic Light Fields (Scripps Inst. of Ocean.\nRef. 58-79, University of California, San Diego,\n1958).\n237.\nPreisendorfer, R. W., A Study of Light Storage Phenom-\nena in Scattering Media (Scripps Inst. of Ocean.\nRef. 59-12, University of California, San Diego,\n1959).\n240.\nPreisendorfer, R. W. The Universal Radiative Trans-\nport Equation (Scripps Inst. of Ocean. Ref. 59-21,\nUniversity of California, San Diego, 1959).\n242.\nPreisendorfer, R. W., The Covariation of the Diffuse\nAttenuation and Distribution Functions in Plane-\nParallel Media (Scripps Inst. of Ocean. Ref. 59-\n52, University of California, San Diego, 1959).\n243.\nPreisendorfer, R. W., , Principles of Invariance for\nDirectly Observable Irradiances in Plane-Parallel\nMedia (Scripps Inst. of Ocean. Ref. 59-73, Univer-\nsity of California, San Diego, 1959).","291\nBIBLIOGRAPHY\n244.\nPreisendorfer, R. W. \"Theoretical proof of the\nexistence of characteristic diffuse light in\nnatural waters,\" J. Mar. Res. 18, 1 (1959).\n245.\nPreisendorfer, R. W., On the Structure of the Light\nField at Shallow Depths in Deep Homogeneous Hydro-\nsols (Report 3-5, Task 3, Contract NObs-72039,\nVisibility Laboratory, University of California,\nSan Diego, March 1959).\n246.\nPreisendorfer, R. W., General Analytical Representa-\ntions of the Observable Reflectance Function\n(Report 5-1, Task 5, Contract N0bs-72039, Visi-\nbility Laboratory, University of California, San\nDiego, November 1959).\n247.\nPreisendorfer, R. W. \"Application of radiative trans-\nfer theory to light measurements in the sea,\"\nSymposium on Radiant Energy in the Sea, Interna-\ntional Union of Geodesy and Geophysics, Helsinki\nMeeting, August 1960 (L'Institute Geographique\nNational, Monograph No. 10, Paris, 1961).\n251.\nPreisendorfer, R. W. Radiative Transfer on Discrete\nSpaces (Pergamon Press, New York, 1965).\n254.\nRedheffer, R., \"On solutions of Riccati's equation as\nfunctions of the initial values,\" J. Rat. Mech.\nand Anal, 5, 835 (1956).\n255.\nRedheffer, R. , \"The Riccati equation: Initial values\nand inequalities,\" Math. Ann. 133, 235 (1957).\n256.\nRedheffer, R. , \"Inequalities for a matrix Riccati\nequation,\" J. Math. and Mech. 8, 349 (1959).\n257.\nRedheffer, R. , \"Supplementary note on matrix Riccati\nequations,\" J. Math and Mech. 9, 745 (1960).\n261.\nReid, W. T., \"Solutions of a Riccati matrix differen-\ntial equation as functions of initial values,\" J.\nMath. and Mech. 8, 221 (1959)\n262.\nReid, W. T., , \"Properties of solutions of a Riccati\nmatrix differential equation,\" J. Math. and Mech.\n9, 749 (1960).\n263.\nRichardson, W. H., Determination of the Non-Zero\nAsymptote of an Exponential Decay Function\n(Scripps Inst. of Ocean. Ref. 58-36, University\nof California, San Diego, 1958).\n270.\nRyde, J. W., and Cooper, B. S., \"The scattering of\nlight by turbid media,\" Proc. Roy. Soc. London\n131. A, 451 (1931)\n273.\nSchmidt, H. W., \"Uber reflexion und absorption von\nB - strahlen,\" Ann. der Physik 23 (series 4), 671\n(1907).","292\nPROPERTIES\nVOL. V\n279.\nSchuster, A. , \"Radiation through a foggy atmosphere,\"\nAstrophys. J. 21, 1 (1905).\n281.\nSchwarzschild, K. \"Ueber das gleichgewicht das son-\nnenatmosphäre, Gesell. Wiss. Gottingen, Nachr.\nMath.-phys. Klasse, p. 41, (1906)\n285.\nSilberstein, L., \"The transparency of turbid media,\"\nPhil. Mag. 4, 1291 (1927).\n287.\nSliepcevich, C. M. Churchill, S. W. Clark, G. C. ,\nand Chiao-min Chu, Attenuation of Thermal Radia-\ntion by a Dispersion of Oil Particles (Army Chem.\nCorps Contract No. DA18-108-CML-4695, AFSWP-749\nERI-2089-2-F, Eng. Res. Inst., University of Mich-\nigan, An Arbor, 1954).\n291.\nStokes, G. G., , \"On the intensity of the light reflected\nfrom or transmitted through a pile of plates,\"\nMathematical and Physical Papers of Sir George\nStokes (Cambridge Univ. Press, 1904, vol. iv, p.\n145.\n298.\nTyler, J. E. , \"Radiance distribution as a function of\ndepth in an underwater environment,\" Bull. Scripps\nInst. Ocean. 7, 363 (1960).\n305.\nTyler, J. E., , and Preisendorfer, R. W. \"Light,\" in\nThe Sea (Interscience Pub., New York, 1962), M.\nN. Hill, ed., vol. I, Chapter 8.\n315.\nWhitney, L. V. \"The angular distribution of charac-\nteristic diffuse light in natural waters,\" J. Mar.\nRes. 4, 122 (1941).\n316.\nWhitney, L. V. , \"A general law of diminution of light\nintensity in natural waters and the percent of\ndiffuse light at different depths,\" J. Opt. Soc.\nAm. 31, 714 (1941).\n319.\nWick, G. C. , \"Uber ebene diffusionsprobleme,\" Z. f.\nPhys. 121, 702 (1943).","INDEX\n293\nabsorption function (for\nnondegenerate\nirradiance), 12. 97\ndegenerate\nsimple formula for in\nfirst, second kind\nasymptotic light field, 255\nforbidden, 202\napparent optical property, 106\nclassical models for irra-\n178, 181\ndiance fields, 19\ntransport equations for 271\ncollimated-diffuse light\nasymptotic form of light\nfields, 19\nfields, 95\nisotropic scattering\ncriterion for asymptoticity,\nmodels, 23\n254\nconnections with diffusion\nasymptotic radiance hypothesis,\ntheory, 24\n212\nclassical theory, inade-\nmain mathematical proof\nquacies, 115\n(using canonic equation for\ncomplete (general) solution\nK), 213-227\nof irradiance equations, 42\nintegral equation for limit\ncomplete reflectance for ir-\nradiance distribution, 228\nradiance, 4, 62, 64, 79\n237, 245\ncomplete transmittance for\ntheoretical and experimental\nirradiance, 4, 62, 64, 79\nexample, 229, 230\ncontrast, 165\nsimple proof (using exponen-\ncontrast transmittance, 162\ntiality of h(z)), 230\nproperties, 168\npractical consequences, 238\nin canonical equation for\nK characterization of\nradiance, 170\nhypothesis, 242\nalternate representations,\ncritique of Whitney's\n171\n\"general law\", 248\nas an apparent optical\nheuristic proof (using dif-\nproperty, 172\nferential equation for K),\neffect of shadows on, 174\n253\ncontravariation of K and D,\nattenuation functions (for\n144\nirradiance) 11\nconvariation of K and D,\ndepth dependence, 25\n128, 136, 140\nfor radiance, 264\nrule of thumb, 145\nfor irradiance, 265\nfor scalar irradiance, 270\ndecomposed (diffuse) irradi-\nfor reflectance, 149, 279\nance, 14\nfor K-function, 275\ndiffuse absorption coeffi-\nfor other concepts, 281\ncient, 111\nattenuating functions\ndiffuse (decomposed) irradi-\ndepth dependence, 25\nance, 14\ntransmittance for, 17\nbackward scattering functions,\ndiffusion theory, 24\n11, 141\nDirac matrices, 8\nboundary effects, 46, 71\ndirectly observable optical\nproperties, 109, 178\ncharacteristic equation\ndiscrete-space radiative\nfor second order differ-\ntransfer, 51\nential equation, 39\ndistribution function, origin,\nfor K - function, 123, 252\n10\ncanonical forms of transport\nfor diffuse irradiance, 15\nequation for K functions,\nrepresentative values, 26\n273\nrepresentation via radiance\ncanonical representation of\ndistribution, 27\nirradiance fields, 98\ndepth dependence, 26\ncatalog of K - figurations\nin canonical equation for\n(shallow depth theory), 201\nirradiance, 99","INDEX\nVOL. V\n294\ndistribution function--Cont'd irradiance fields--Cont'd.\nexperimental, 115\nequilibrium, 13\ncontravariation (with K-func-\ndecomposed (diffuse) 14\ntion), 144\nresidual, 15\ncovariation (with K-function)\nclassical models, 19\n128, 136, 140\ntwo-D models, 25\nphysical and geometrical\nprimary scattered, 43\nfeatures, 128\nboundary effects, 46\nasymptotic limits, 246\nmethod of modules, 80\ninternal source, 37, 55, 81\nequilibrium radiance function,\nvector model for, 87\n151\ncurl and divergence of, 91\nfor radiance, 151\nglobal properties, 97\nfor irradiance, 13, 265\ncanonical representation,\nfor scalar irradiance, 270\n98\nfor reflectance, 149, 279\nshallow depth theory, 187,\nfor K-functions, 275\n193\nfor other concepts, 281\nfine structure hypotheses,\nequilibrium-seeking theorem\n199\nfor R, 152\nin asymptotic setting, 238\nfor N, H. h, 270\nisotropic scattering (in vec-\nand universal radiative\ntor light field model), 94\ntransport equation, 279\nmodels, 23\nequivalence theorem for\nR(z,-), 159\nK-function\ntwo-D model, 31, 39\nfine structure of light field\none-D model, 53, 56\nhypotheses, 199\nin canonical equation, 100\nspecial relations, 208\ntheoretical forms, 111\nfirst standard solution\ndiffuse absorption coeffi-\na two-D model\ncient, 111\nforward scattering functions,\nexperimental, 115\n11, 141\nsignificance of sign, 120\nforward scattering media, 140\ncharacteristic equation\nfundamental optical property,\nfor, 123\n107, 178\nconnections among (rradi-\nance K-functions), 123\nglobal optical property, 107,\ngeneral forms, 125\n180\nfor radiance, 125\nintegral representations,\ninequalities (among optical\n126\nproperties), 114, 119\nin spherical coordinates,\ninherent optical property,\n127\n106, 178, 180\ncontravariation (with dis-\ninteracting media (invariant\ntribution function), 144\nimbedding operators for), 76\ncovariation (with distribu-\ninternal sources and irradi-\ntion) 128, 136, 140\nance fields, 37, 55, 81\nphysical and geometrical\ninvariant imbedding relation\nfeatures, 128\nfor irradiance fields, 2, 61\nabsorptionlike character\nin two-D model, 36\n(for irradiance), 138\nincluding boundary effects,\ngenealogy of configurations\n50\nfor shallow depths, 201\nfor interacting media, 76\nasymptotic limits, 246\nirradiance (two-flow) equa-\ncanonical form of transport\ntions, 6\nequations for, 273\nirradiance fields,\ninvariant imbedding rela-\nlocal optical property,\ntion for, 2\n107, 179, 180","INDEX\n295\nlocal transmittance and\nreflectance--Cont'd\nreflectance, 7\ndifferential equations, 65,\n79, 123, 148\nmany-D models, 57\nRoo formulas, 113\nmethod of modules for irradi-\nexperimental, 115\nance fields, 80\nconnections with attenuat-\nmonotonicity condition on\ning functions, 118\nradiance distribution, 28\nanalytic representation,\n146\none-D models for irradiance\nequilibrium-seeking prop-\nfields, 51\nerty, 150\nfor undecomposed fields, 52\nintegral representations\ninternal sources, 55\nof R(z,-), 156, 196\nfor decomposed fields, 56\nequivalance theorem for\nconnections with observable\nR(z,-), 159\nfields, 160\nequilibrium and attenuation\nontogeny (family roots) of\nfunctions, 149\ntwo flow equations, 13\nasymptotic limits, 246\noptical medium, definition,\nregular neighborhoods of\n108\npaths, 166\noptical properties,\nresidual (reduced) irradi-\ninherent, apparent, 106\nance, 15\nlocal, global, 107\nfundamental, 107\nscattering functions (for\ngeneral definition, 109\nirradiance)\ndirectly observable, 109\nforward, backward, 11, 141\nclassification, 178\ntotal, 12\nin asymptotic light fields,\nfor decomposed irradiance,\n238\n16\nforward scattering media,\nPauli matrices, 8\n140\npolarity of R,T ractors, 34\nsecond standard solution\nprimary scattered irradiance,\nof two-D model, 34\n43\nsemigroup properties (third\nprinciples of invariance\norder) for reflectance and\nglobal (for irradiance), 2\ntransmittance factors, 67\nlocal (for irradiance), 7\nsemigroup property for con-\nfor diffuse irradiance, 18\ntrast transmittance, 169\nshallow depth theory (of ir-\nquantum mechanics\nradiance field)\nformal similarity with, 8\nexperimental basis, 187\nquasi-irrotational light\nformulation, 193\nfield, 88\ncomparison with experiment,\n197\nRoo formulas, 113\nsimple model for radiance dis-\nradiance\ntributions, eventual exact-\nequilibrium, 151\nness, 249\ntransmittance, 164\nstandard ellipsoid (for radi-\nmultiplicity of representa-\nance distribution), 250\ntions, 177\nsubmarine light field, general\nreduced (residual) irradiance,\nrepresentation, 93\n15\nreflectance\ntotal scattering functions,\nfor irradiance (undecom-\n12\nposed), 3\ntransmission line equations\nin two- D model, 33, 35, 45\nformal similarity with, 8\ncomplete (two-D), 62\ntransmittance, for irradiance\ncomplete (one-D), 64\n(undecomposed), 3","VOL. V\nINDEX\n296\ntransmittance- -\nfor reduced and diffuse\nirradiance, 17\nin two-D model, 33, 35, 45\ncomplete (two-D), 62\ncomplete (one-D), 64\ndifferential equations,\n65, 79\nfor radiance, 164\ntwo-D models for irradiance\nfields, 25\nfor undecomposed fields, 30\nfirst standard solution, 31\nsecond standard solution, 34\nfor internal sources, 37\nfor decomposed fields, 43\ninadequacies, 115\neventual exactness, 247\ntwo-flow equations (for irra-\ndiance), 6\nundecomposed form, 8\ndecomposed form, 14\nequilibrium form, 13\nontogency, 13\nfor reduced irradiance, 17\ntwo-D (undecomposed) model,\n30\nstandard solutions, 31-34\ncomplete (general) solution,\n42\nfor decomposed irradiance,\n43\nboundary conditions\n(effects), 46\none-D (undecomposed) model\n52\nmany-D models, 57\nexact vs. two-D, 115\nasymptotic behavior, 247\nuniversal radiative transport\nequation, 263\nfor radiometric concepts, 263\nfor apparent optical prop-\nerties, 271\nand equilibrium principle,\n279\nstandard cases, 281\nadditional cases, 281\nvector irradiance field, model\nfor, 87\nWhitney's \"general law\" of\ndiminution of light field\nwith depth, 248"]}