{"Bibliographic":{"Title":"Hydrologic optics. Volume IV: Imbeddings","Authors":"","Publication date":"1976","Publisher":""},"Administrative":{"Date created":"08-17-2023","Language":"English","Rights":"CC 0","Size":"0000382720"},"Pages":["GB\n665\n.P645\nv.4\nHYDROLOGIC OPTICS\nVolume IV. Imbeddings\nR.W. PREISENDORFER\nU.S. DEPARTMENT OF COMMERCE\nNATIONAL OCEANIC & ATMOSPHERIC ADMINISTRATION\nENVIRONMENTAL RESEARCH LABORATORIES\nHONOLULU, HAWAII\n1976","GB\n665\nP645\nv.4\nOF COMMUNITY\nHYDROLOGIC OPTICS\nSTATES OF\nVolume IV. Imbeddings\nR.W Preisendorfer\nJoint Tsunami Research Effort\nHonolulu, Hawaii\n1976\nATMOSPHERIC SCIENCES\nLIBRARY\nAUG 30 1976\nN.O.A.A.\nU. S. Dept. of Commerce\nU.S. DEPARTMENT OF COMMERCE\nNational Oceanic and Atmospheric\nAdministration\nEnvironmental Research Laboratories\nPacific Marine Environmental Laboratory\n16 3420","ii\nWe owe to Ambarzumian the first introduction\nof a principle of invariance in the treatment\nof transfer problems.\nS. Chandrasekhar\nRef. [43]\nThese ideas were further developed and exten-\nsively generalized by Chandrasekhar\nR.E. Bellman, R.E. Kalaba\nM.C. Prestrud Ref. [15]\nIn three successive generalizations the\ntheory ascended from Ambarzumian's concept of\nthe invariance of a visual impression of\nbrightness up to the concept of the invariance\nof a set of radiance functions under the appli-\ncation of each member of a set of transformations\nassociated with a given set of optical media.\nRef. [251], p. 167","iii\nCONTENTS\nVolume IV\nChapter 7\nInvariant Imbedding Techniques for Light Fields\n7.0\nIntroduction\n1\n7.1 Differential Equations Governing the Steady State\nR and T Operators\n2\nLocal Forms of the Principles of Invariance\n2\nThe Differential Equations for R and T\n4\nDiscussion of the Differential Equations\n8\nFunctional Relations for Decomposed Light Fields\n9\n7.2 Differential Equations Governing the Time Dependent\nR and T Operators\n17\nTime Dependent Local Forms of the Principles of\nInvariance\n18\nTime Dependent Invariant Imbedding Relation\n19\nIntegral Representation of Time Dependent\na and Operators\n21\nTime Dependent Principles of Invariance\n23\nDifferential Equations for the Time Dependent\nR and T Operators\n24\nDiscussion of the Differential Equations\n26\n7.3 Algebraic and Analytic Properties of the R and T\nOperators\n27\nPartition Relations for R and T Operators\n27\nAlternate Derivations of the Differential Equations\nfor R and T Operators\n31\nAsymptotic Properties of R and T Operators\n33\n7.4 Algebraic Properties of the Invariant Imbedding\nOperators\n35\nThe Operator M(x,z)\n35\nThe Connections Between M(x,z) , m 7(x,z), , and m(z,x)\n37\nInvertibility of Operators\n39\nRepresentations for the Components of m(x,z), m(z,x)\n41\nThe Isomorphism Between 2(a,b) and G2(a,b)\n44\nThe Physical Interpretation of the Star Product\n46\nThe Link Between m (a,x,1 b) and m(a,y,b)\n47\nRepresentation of m(x,y,z) by Elements of T2(a,b)\n49\nA Constructive Extension of the Domain of m (x,y,y,)\n50\nRepresentation of m (v,z;u,y) by Elements of 2 (a, b)\nand r 3 (a, b)\n51\nThe Connection Between 4(x,y) and m(s,y)\n52\nA Star Product for the Operators m(x,y,z)\n54\nPossibilities Beyond M(V,x;u,w)\n58\nPossibilities Beyond 2 (a,b)\n61","CONTENTS\niv\n7.5 Analytic Properties of the Invariant Imbedding\nOperators\n68\nDifferential Equations for m (x,y)\n69\nDifferential Equations for m (x,y,y)\n71\nDifferential Equations for m(v,x;u,w)\n76\nDifferential Equations for M(x,y) and Y(s,y)\n79\nAnalysis of the Differential Equation for R(y,b)\n80\n7.6 Special Solution Procedures for R(a,b) and T(a,b)\nin Plane-Paralle1 Media\n83\nThe General Equation for R(a,b;E',5)\n86\nThe Isotropic Scattering Case for R\n86\nA Sample Numerical Solution for r(x;u',v)\n90\nThe General Equation for T* (a,b;5';E)\n*\n93\nThe Isotropic Scattering Case for T'\n95\n7.7 General Solution Procedures for R(a,b) and T(a,b)\nin Plane-Parallel Media\n97\n7.8 The Method of Modules for Deep Homogeneous Media\n103\nThe Invariant Imbedding Relation for Deep Hydrosols\n104\nThe Module Equation\n106\nEmpirical Bases for the Use of the Module Equations\n106\n7.9 The Method of Semigroups for Deep Homogeneous Media\n108\nThe Semigroup Equations for T (z)\n109\nThe Infinitesimal Generator A\n112\n7.10 The Method of Groups for Deep Homogeneous Media\n114\nThe Return of the Group 2 (0,00)\n115\nThe Infinitesimal Generator of 2 (0,00 )\n116\nThe Exponential Representation of m(y) and N(y)\n117\nThe Exponential Representation of 2(y)\n117\nNumerical Procedures of N(y) : The Exponential\nTechnique\n119\nThe Characteristic Representation of N(y)\n122\nAsymptotic Property of N(y)\n127\nAsymptotic Properties of Polarized Radiance Fields\n128\n7.11 Method of Groups for General Optical Media\n129\nAnalysis of the Group Method: Initial Data\n129\nAnalysis of the Group Method: Limitations of the\n130\nEquation of Transfer\nAnalysis of the Group Method: Summarized\n135\n135\nThe General Method of Groups\n138\nObservations on the Method of Groups\nThe Method of Groups and the Inner Structure of\nNatural Light Fields\n141\n7.12 Homogeneity, Isotropy and Related Properties of\nOptical Media\n143\n144\nLocal Concepts\n148\nGlobal Concepts\n151\nSummary\n152\nConclusion","CONTENTS\nV\n7.13 Functional Relations for Media with Internal Sources\n152\nPreliminary Relations\n153\nIntegral Representations of the Local\nY-Operators\n154\nIntegral Representations of the Global\nY-Operators\n162\nIncipient Patterns and Nascent Methods\n164\nDual Integral Representations of the Global\nY-Operators\n167\nLogical Descendants of 4(s,y:a,b)\n171\nDifferential Equations for the Dual Operators\n173\nA Colligation of the Component Y-Operator Equations\n176\nAsymmetries of the Y-Operator\n179\nA Royal Road to the Internal-Source Functional\nEquations\n181\nSummary and Prospectus\n186\nFinal Observations on the Relationships Between the\nOperators m (v,x:u,w) and 4(s,y:a,b)\n188\n7.14 Invariant Imbedding and Integral Transform\nTechniques\n188\nAn Integral - Transform Primer\n191\nTime Dependent Radiative Transfer\n194\nHeterochromatic Radiative Transfer\n197\nMultidimensional Radiative Transfer\n198\nConclusion\n200\n7.15 Bibliographic Notes for Chapter 7\n200","","vii\nPREFACE\nIt is a relatively rare occurrence in applied mathematics\nthat one encounters a method of solution of a given type of\nequation that is both effective numerically and rich in physi-\ncal imagery. With the advent of the principles of invariance\ninto radiative transfer theory, the equation of transfer un-\nderlying the theory received its natural solution companion.\nTogether the principles of invariance and the equation of\ntransfer form a combination which illustrates that rare occur-\nrence alluded to above.\nIn the present work we use this combination to explore\nthe transfer of radiant energy through general optical media\n(exemplified by the atmosphere and the sea) and develop nu-\nmerically effective procedures of strong intuitive content.\nThe Method of Groups is a case in point. It is summarized in\nEquations (6) - (9) of Sec. 7.11 and shows in outline how the\ncomplex problem of radiant energy scattered in a general\nthree-dimensional medium (such as a cloud) may be reduced to\nan ostensible one-dimensional sweep method--the hallmark of\nthe invariant imbedding idea.\nOver the years this useful combination of an equation\nof transfer and the principles of invariance has been extended\nto other fields of physics. In linear hydrodynamics, e.g.,\nthe counterpart to the equation of transfer is the set of dy-\nnamic and continuity equations. Instead of radiance (upward\nand downward into the sea) we have water wave elevation and\nfluid volume flux over the surface of a fluid basin, such as\nthe sea. The principles of invariance go over essentially\nunchanged into the hydrodynamic setting. Consequently, all\nof the visualizable physical notions of invariant imbedding\nare transferable intact to hydrodynamics, such as the trans-\nmittances and reflectances of bodies of water--canals, bays,\noceans. Moreover, the numerical efficiency of the imbedding\ntechnique is once again realized in this new setting. Work\nin this direction has proceeded far enough to show the thor-\noughgoing analogy between radiative transfer of light in op-\ntical media and the linear transport of water waves in natural\nbodies of water. 1-5\nThe development of the invariant imbedding idea contin-\nues in still other fields and may be pursued in a recently\ncompiled bibliography. 6\nThe work in this volume was essentially done in the peri-\nod 1964-1965 while I was with the Visibility Laboratory at\nScripps Institution of Oceanography, and has been essentially\nunchanged in its conversion to manuscript form. My recent\napplication of invariant imbedding to linear hydrodynamics\nhas served to check the correctness of the theory below, and\nto reinforce my confidence in its universal applicability to\nall linear transport phenomena including light, ocean wave,\nelectromagnetic or acoustic fields.","PREFACE\nviii\nThe final manuscript was typed by Ms. Judith Marshall.\nR.W.P.\nHonolulu, Hawaii\nJanuary 1976\n1. Preisendorfer, R.W., Surface-Wave Transport in Nonuniform\nCanals NOAA-JTRE-80, HIG-72-18 Hawaii Institute of\nGeophysics July 1972.\n2. Preisendorfer, R.W. and F.I. Gonzalez, Jr., Classic Canal\nTheory NOAA-JTRE-83, HIG-73-14 Hawaii Institute of\nGeophysics August 1973 (vols. 1 and 2).\n3. Preisendorfer, R.W., , Multimode Long Surface Waves in Two-\nPort Basins NOAA-JTRE-125, HIG-75-4 Hawaii Institute\nof Geophysics January 1975.\n4. Preisendorfer, R.W., Directly Multimoded Two-Flow Long\nSurface Waves NOAA-JTRE-131, HIG-75-12 May 1975.\n5. Preisendorfer, R.W., Marching Long Surface Waves Through\nTwo-Port Basins I. Continuous Circuit Case NOAA-JTRE-\n132 HIG-76-1 January 1976.\n6. Scott, M.R., , A Bibliography on Invariant Imbedding and\nRelated Topics SLA-74-0284 Sandia Laboratories\nAlbuquerque, N.M. June 1974.","CHAPTER 7\nInvariant Imbedding Techniques for Light Fields\n7.0 Introduction\nWe return in this chapter to the general circle of ideas\nintroduced in Chapter 3, our purpose being to give detailed\nderivations of functional relations holding among the various\ninteraction operators introduced there and discussions of how\nthose operators can be evaluated in practice. For expository\nreasons we shall at first limit the discussions for the most\npart to the case of light fields in isotropic plane-parallel\nmedia. However, the techniques displayed are all extendable\nin principle to light fields in arbitrarily shaped anisotropic\nmedia. By carrying out the present program we not only add\nto the store of solution techniques for light fields discus-\nsed in Chapters 4-6, but illustrate within the domain of radi-\native transfer an important procedure of modern theoretical\nphysics, the invariant imbedding solution technique. This\ntechnique gives rise to functional equations governing vari-\nous physical processes by means of certain general group-\ntheoretic and limit-theoretic arguments. These functional\nequation representatives of the physical processes give in-\nsight into the processes and occasionally result in useful\nnumerical methods of determination of the processes. Some of\nthese methods will be illustrated in this chapter.\nFrom the great number of results on functional relations\nfor various radiative transfer operators obtained in recent\nyears by means of invariant imbedding techniques, we select\nthe following for exposition in the present chapter: first,\nthe derivations of the differential equations governing the\nreflectance and transmittance operators R and T for plane-\nparallel media. The steady state version of the derivation is\ngiven in Section 7.1, the time-dependent version is given in\nSection 7.2. A particularly interesting feature of these der-\nivations is the statement of the local forms of the principles\nof invariance and their conceptual relation to the usual (glob-\nal) forms of the principles of invariance. In Sections 7.3- -\n7.5 it is shown how new and possibly useful functional rela-\ntions can be discovered for the various interaction operators\nby treating the operators as algebraic entities and the equa-\ntions in which they appear as algebraic statements which are\noccasionally subject to simple limit arguments. As a result\nof these heuristic manipulations three novel means of deter-\nmining light fields in natural optical media, which occur in\nSections 7.4-7.5, are selected for further study in Sections","INVARIANT IMBEDDING TECHNIQUES\nVOL. IV\n2\n7.6-7.8. An example of an actual numerical computation of\nthe R and T operators based on the functional relations of\nSection 7.1 is given in Section 7.9 for the case of homogene-\nous source-free plane-parallel media with isotropic scatter-\ning. This numerical method is generalized in Section 7.10.\nIn Section 7.11 the preceding results are consolidated and\ngeneralized. Section 7.12 is concerned with the conditions\nof\nhomogeneity and isotropy and related ideas, which will\nhelp simplify theoretical and numerical work and help classify\noptical media in general. Section 7.13 develops some deep\nconnections among the various standard and invariant imbed-\nding operators within media with internal sources. Finally,\nin Section 7.14, it is observed how the Laplace and Fourier\ntransform techniques, which have proved so useful in the\nclassical formulation of the transport phenomena, can be com-\nbined with the invariant imbedding approach to simplify the\nfunctional relations of the latter approach and to encourage\ntheir applications to time-dependent problems, point source\nproblems, and other transport problems which ordinarily in-\nvolve higher numbers of variables.\n7.1 Differential Equations Governing the Steady State, R and\nT Operators\nIn Sections 3.6 and 3.7 we saw how the R and T operators\nof plane-parallel (and other) media were used in both theory\nand practice to determine light fields in natural optical\nmedia. In this section we show how the four R and T operators\ngenerally associated with stratified plane-parallel media may\nbe determined from knowledge of the volume scattering and\nvolume attenuation functions within the medium. This will be\ndone by deriving the differential equations governing the op-\nerators as a function of the thickness of the medium. Thus,\nif we know the R and T operators for a given layer of material\nthe differential equation will show how the operators change\nby addition of a very thin layer of the material to the given\nlayer. By letting the given layer grow continuously from\nsome given thickness, we will therefore know how its R and T\noperators evolve from their given values, and how they may be\ncomputed in both theory and practice. We turn now to the de-\ntails of the derivations.\nLocal Forms of the Principles of Invariance\nWe begin the derivations by casting the equation of\ntransfer for a stratified plane-parallel medium into a pair\nof equations which are strongly reminiscent of the two main\nprinciples of invariance for such media (Ex. 3, Section 3.7)\nthe main difference being the presence of derivatives of N in\nthe new equations. Thus under the assumption that all func-\ntions (radiance distributions and optical properties) depend\nonly on depth y in the medium (cf. Fig. 7.1) Equation (3) of\nSec. 3.15 becomes:","SEC. 7.1\nSTEADY STATE OPERATORS\n3\n(-E.K) dN(y,5) -a(y)N(y,E) + N*(y,5)\n(1)\ndy\nwhere\nN*(y,5)\ndo(E)\n(2)\n=\nHere k is the unit outward normal to the medium, E is an arbi-\ntrary direction in E, and asysb.\nTo obtain the requisite form of the equation of trans- -\nfer we restrict the radiance distribution N(y,.) to the two\nhalves E+ and E_ of E, (cf. Fig. 7.1, and Sec. 2.4). We de-\nnote the restriction of N(y, ) to E+ as usual by \"N+ (y)\", and\nthe restriction of N(y,) to E_ by \"N_(y)\"; asysb. Next we\nwrite:\n\"p(y)\"\nfor\nda(E')\n(3)\nin which E is in E_; and\n\"I(y)\"\nfor\n(4)\nin which E is in E+, and in both of which asysb. Further-\nmore, we assume the medium to be isotropic, so that\n0(y;5';5) depends only on the value 51.E for each choice of\nE' and E. Hence for each E,Y, the values of the integrals in\neach definition in (3) and (4) are unchanged if E+ is replaced\nby E-, and E_ by E+ throughout. The operator p(y) is the lo-\ncal reflectance operator and T (y) is the local transmittance\noperator. In discussions where it is necessary to consider\nthe possibility of anisotropic media, the operators p(y) and\nT(y) must be defined with specific reference to the domains of\nintegration in (3) and (4). Thus \"p(y)\" in (3) becomes \"p+(y)\"\nand \"p_ (y)\" denotes the same kind of integral but over E.\nwith E in E+. Similarly (4) will define what we will call\n\"It (y)\" and a similar integral over E. with E in E. will be de-\nnoted by \"I-(y)\". (See, Ref. [251], and Sec. 7.13 below.)\nThe local operators p(y) and T(y) are used as follows:\nIn (1) let E be in E+, so that N(y,5) = (N+(y))(E). Further-\nmore, writing N ( y , E) in (1) as:\nN ( y , 5 )\nds(E') + do(E') ,\nwe divide through each side of (1) by 15.K apply (3) and (4),\nand end up with:","4\nINVARIANT IMBEDDING TECHNIQUES\nVOL. IV\n- dN (y) = +\nN+(y)T(y) N_(y)p(y)\n(5)\ndy\nSimilarly, now with E in (1) , so that N(y,5) = (N_(y))(E), we\nobtain:\ndN_(y)\n= N_(y)t(y) + N+(y)p(y)\n(6)\ndy\nEquations (5) and (6) are the desired local (or integrodif- -\nferential) forms of the principles of invariance for plane- -\nparallel media. The striking similarity between the pair (5),\n(6) and the pair I, II of Ex. 3, Sec. 3.7, is evident once\n\"T\" is paired with \"I\" and \"R\" with \"p\". These equations can\nbe put into a more compact form by first writing:\nItem\n-T(y) p(y)\n\"K(y)\"\nfor\n(7)\n-p(y) t(y)\nand\n\"N(y)\" for (N+(y),N_(y))\n(8)\nThen (5) , (6) become:\ndN(Y) = N(y) K(y) =\n(9)\ndy\nwhich is an alternate and equivalent rendition of the equation\nof transfer (1) via the local forms (5), , (6) of the principles\nof invariance. We shall return to this form of the equation\nof transfer in subsequent sections, wherein it will play an\nimportant role in determining the radiance functions. For\nthe present, we continue the derivation of the desired func-\ntional relations for the R and T operators.\nThe Differential Equations for R and T\nThe main step in the derivation of the differential\nequations for the R and T operators will now be taken. We be-\ngin with the operator R(y, b) for an arbitrary subslab X(y,b)\nof the plane-parallel - medium X(a,b), asysb. (We now are\nusing the notation of Section 3.7). We let N (a) be an arbi-\ntrary incident radiance function over the plane upper boundary\nof X(a,b) at level a. We set (b) = 0, and assume that no\nsources of radiant flux are within X(a,b). The two main prin-\nciples of invariance for an arbitrary subslab X(x,z),\nasxsyszsb, of X (a, b) are as given in Ex. 3, Sec. 3.7:","SEC. 7.1\nSTEADY STATE OPERATORS\n5\nI. 1,(y) = N+(z)T(z,y) + N_(y)R(y,z)\nII. N_(y) + N (_)())(x,y) + N.(y)R(y,x)\nWe next set Z = b in principle I, which with the present\nboundary lighting conditions becomes:\nN,(y) = N_(y)R(y,b)\n(10)\nEquation (10) states that the upward radiance distributions\nat level y in X(a,b) consist of the reflected flux from X(y,b)\ninduced by the downward radiance distributions entering X(y,b)\nat level y. We next take the derivative of each side of (10)\nwith respect to y, thus:\ndN_(y)\ndR(y,b)\nR(y,b)\n(y)\n(11)\n=\ndy\nwhere we have written:\n\"dR(y,b)\"\nfor\n(y,b;E';E) ds(E) (12)\ndy\nand where R(y, b;5';5) is defined in Example 3 of Sec. 3.7 (cf.\nalso (8) - (11) of Sec. 3.6). Therefore dR(y,b)/dy in (11) is\nan integral operator acting on N-(y). Further, R(y,b) in (11)\nacts on the function dN_(y)/dy. Thus all terms of (11) are\nwell defined. Now, we are interested in R(a,b), which we\nmay envision as the limit of R(y,b) as y+a. Hence in (11) we\nlet y approach a. Thus we are led to consider\nlim\nwhich by (5) is given as:\nin = [N+(a)t(a) + N_(a)p(a)]\n(13)\n.\nBy principle III of Example 3, Sec. 3.7, which we repeat here\nfor convenience:\nIII. N+ (a) = N (b)T(b,a) = + N_(a)R(a,b)\n,\nequation (13) becomes:\ndN+(y)\n-N_(a) [R(a,b)t(a) + p(a)]\n(14)\n=\ndy\ny+a","6\nINVARIANT IMBEDDING TECHNIQUES\nVOL. IV\nwhere we have used the boundary condition that N+(b) = 0.\nIn a similar manner, we find that the limit of the remaining\nderivative of N.(y) in (11) can be represented via (6) and\nprinciple III as:\ndN_(y)\nlim dy = N_(a)[T(a) + R(a,b)p(a)]\n(15)\n.\ny+a\nLet us agree to write:\nlim dR(y,b)\n\"aR(a,b)\"\nfor\n(16)\nda\ny+a dy\nThen applying the limit operation, lim\nto each side of\n,\n(11), we have:\ny+a\n-N_(a) [R(a,b)t(a)+p(a)] =\nN_(a)[T(a)+R(a,b)p(a)]R(a,b) + N (a) aR(a,b)\n(17)\n.\nda\nThis equation holds for every incident radiance function\nN_ (a). Hence we can formally cancel \"N_ (a)\" from each side.\nAfter rearranging the resultant operator equation, we have:\nI'\n-aR(a,b)= p(a)+t(a)R(a,b)+R(a,b)t(a)+R(a,b)p(a)R(a,b)\nda\n(18)\nEquation I' is the requisite differential equation for R(a,b)\nas a function of the depth parameter a. Observe that I' has\nthe form of a Riccati equation for the operator R(a,b) with\nknown operators p(a) and T (a). Thus, (18) is in principle\nsolvable for R(a,b) with the initial condition: R(a,b) = 0\nwhenever a = b, (cf. (30) of Sec. 3.7). Hence from I' we\nhave:\n(19)\nfor a = b, showing that the initial rate of growth of R(a,b)\nis given directly by the local reflectance operator, p(a),\ni.e., the integral operator with the volume scattering func-\ntion as kernel.\nThe determination of the differential equation for\nT(a,b) may be made next, starting with principle II in which\nX = a, the result being:\n(20)\nN_(y) = N_(a)T(a,y) + N+(y)R(y,a)\n.\nTaking the derivative of each side with respect to y:","SEC. 7.1\nSTEADY STATE OPERATORS\n7\ndN_(y) dy = dy R(y,a) + N+ dR(y,a) (21)\nHere we have written:\n\"dT(a,y)\"\nfor\ndy\n(22)\nand where T(a,y;5';5) is defined in Example 3 of Sec. 3.7 (cf.\nalso (8) - (11) of Sec. 3.6). Thus dT(a,y)/dy in (21) is an\nintegral operator acting on N_ (a). Now in (21) we consider:\nwhich by (6) is given as:\n= [N_(b)t(b) + N+(b)p(b)]\n(23)\n.\nFrom principle IV of Example 3 Sec. 3.7:\nIV N (b) = N_(a)T(a,b) + (b)R(b,a),\nwhich, applied to (23) yields:\n(24)\n=\nIn a similar way we obtain for the derivative of N+(y) in\n(21):\n-N_(a)T(a,b)p(b)\n(25)\n=\nWriting:\n\"JT(a,b)\"\nfor\n(26)\nab\nwe here apply the limit operator lim to each side of (21),\nthe result being:\ny+b\n(27)\n+\nin which we have used the fact that N_(a) is arbitrary. This\nshows that once R(a,b) is known, the operator T(a,b) is","INVARIANT IMBEDDING TECHNIQUES\nVOL. IV\n8\nobtainable by a simple quadrature.\nThe pattern of derivation of the differential equations\nis now clear. By using next the second versions of principles\nIII and IV in Example 3 of Sec. 3.7, together with (5) and (6)\nwe arrive at:\naR(a,b)\nT(a,b)p(b)T(b,a)\n(28)\nIII'\n=\nab\naT (a,b)\n(29)\nIV\n=\n-\nda\nDiscussion of the Differential Equations\nStatements I' - IV' above are the desired differential\nequations for the R and T operators associated with the plane-\nparallel medium X(a,b). Observe how I' is autonomous with re-\nspect to R(a, b) Thus, as we already observed, I' in princi-\nple can yield R(a,b), starting with the initial condition\nR(a,b) = 0. By reversing \"a\" and \"b\" in I', (i.e., by liter-\nally turning X(a,b) upside down in the given coordinate sys-\ntem) we can also obtain R(b,a). Observe that if p(y) and ty\nvary with depth y in X(a,b), we generally will have R(a,b)#\nR(b,a), i.e., the R operator will exhibit polarity. If X(a,b)\nis homogeneous, then R(a,b) = R(b,a) and clearly depends only\non the difference b-a of the depth parameters. (Recall, we\nhave assumed at the outset that X(a,b) is isotropic.) Once\nR(a,b) and R(b,a) have been found, T(a,b) and T(b,a) both fol-\nlow from II' using first R(a,b) then R(b,a) by reversing \"a\"\nand \"b\" in II'. If polarity is the case for R-operators,\nthen generally, the T operator will possess polarity also.\nThus I' and II are in principle sufficient to determine the\nfour R and T operators. However, it is interesting to note\nthat I'-IV' are sufficient, as they stand, to determine in\nprinciple all four operators R(a,b), T(a,b), T(b,a), R(b,a)\nin that order, by successively using I', IV', III', II', in\ncorresponding order. Alternatively, the equations may be\nsolved in the order I', IV', II', III' For the general forms\nof these observations, in the context of general media the\nreader may consult section 25 and other relevant sections of\nRef. [251].\nEquations I'-IV' constitute a wealth of intuitive infor-\nmation about light fields in scattering media neatly summa-\nrized in symbolic form, and which the reader is invited to\ndiscover. Thus I' and III' considered together show the two\ndistinct modes of growth of R(a,b) when the medium is altered\nby varying the parameter a, and then the parameter b. In oth-\ner words R(a,b) grows differently when layers are added to\nX(a,b) from below, than when layers are added from above. The\nprecise manner of growth in each case is clearly discernable\nfrom each differential equation and can be pictured in terms\nof the interaction of X(a,b) with an infinitely thin layer\nadded to X(a,b), e.g., at level a, whose reflectance and","SEC. 7.1\nSTEADY STATE OPERATORS\n9\ntransmittance are p(a) and T(a), respectively. The growth of\nR(a,b) when a is varied is far more complex than when b is\nvaried. Unfortunately this more complex growth is necessary\nto contend with in the task of determining R(a,b). Remarks\nof a similar nature can be made about the general growth pat-\nterns of T(a,b) using II' and IV' In the case of T(a,b) the\ndifference of growth rates, depending on whether a or b is\nvaried, are less subtle than that of R(a,b), and rest mainly\nin the order of application of the operators in the square\nbrackets with respect to T(a,b).\nFunctional Relations for Decomposed Light Fields\nSome radiative transfer investigations are simplified if\none is able to treat separately the reduced and diffuse com-\nponents of the radiance field. Thus in the classical re-\nsearches of Chandrasekhar, the computations were limited to\ncomputing the diffuse radiance transmitted through plane-\nparallel media. In addition, our discussions of the point\nsource problem were facilitated in Sec. 6.6 by adopting for\nstudy not N but the diffuse component N* of N. Furthermore,\nas noted in the Remarks on the Interaction Method in Sec.\n3.18, the AC property of general interaction operators is eas-\nily shown to hold for those operators whose response functions\ndescribe diffuse radiant flux, i.e., radiant flux which has\nbeen scattered at least once. With such observations in mind\nwe are motivated to study some of the salient properties of\nthe decomposed R and T operators for plane-parallel media, in\nparticular the principles of invariance (both local and glob-\nal) which govern them, and the differential equations they\nsatisfy. The extensions of the results of the present dis-\ncussion to more general geometries is straightforward and the\npresent techniques are presented so as to readily serve as\nthe prototype for such extensions.\nWe shall work with the setting already established and\nused in carrying out the discussion from (1) to (29) above.\nThus Fig. 7.1 will represent the present geometrical setting.\nNow, the first step in the decomposition of the R and T opera-\ntors is to decompose the light field at general level y into\nits reduced and diffuse components. The basis for this de-\ncomposition rests in (5), (6) of Sec. 3.13.\nThus, N_(y), e.g., may be written:\n_(y) N°(y) + (y)\n(30)\n,\nfor every y, where No (y) is the reduced (or residual)\nradiance distribution over the directions of E at level y.\n(y) is the diffuse radiance distribution over the same di-\nrection set and at the same level. A similar decomposition\nholds for N+ (y) The incident radiance distributions N (a)\nand N+ (b) on the slab X(a,b) will, by convention, be of re-\nduced form, i.e., we will assume (a) = 0 and *(b) = 0.\n(See the Principle of Relative Scattering Order in Sec. 22 of\nRef. [251]) Hence N° (a) and No(b) serve as the incident ra-\ndiance distributions on X(a,b).","10\nINVARIANT IMBEDDING TECHNIQUES\nVOL. IV\n=+\n+\nE-\n-\nN_(a)\na\nN_(x)\nX\nN+(y)\ny\nN_(y)\nZ\nN+(z)\nb\n/\nN+(b)\nFIG. 7.1 Plane-parallel setting for the local and global\nforms of the Principles of Invariance.\nThe connection between the initial radiance function\nNo (a) over the upper boundary of X(a,b) and the residual ra-\n-\ndiance function No(y) over level y within X(a,b) is readily\nestablished, using the results (4) of Sec. 3.10 and (3) of\nSec. 3.11. Thus we have in general:\nN°(z) = N°(x)T°(x,z)\n(31)\n,\nwhere we have written:\n\"T°(x,z)\"\nfor\n, (32)\nfor and where p' is a point (i.e., an ordered\ntriple of real numbers) in level x, and q is a point in level\nZ such that:","SEC. 7.1\nSTEADY STATE OPERATORS\n11\n(33)\nwhere r is determined by:\n|z-x|/|5.k\nA companion equation to (31), written for No (x), , is readily\nstated. To see the way in which (32) is used, suppose X =\na\nand Z = y, and that the value of No(a) at p' and E' is spe-\ncifically of the form No(p',5'). Further, let No(q,5) be the\nresidual radiance at point q induced by No(a). Then T° (a,y)\nacting on No(a) yields the radiance:\nN°(9,5) =\nN°(p,5)T,(p,5)\n(34)\nwhere p = q-rE, in which the distance r is determined by\nr = |y-a|/|E.k.\nRecalling that y is the depth parameter for X(a,b), i.e.,\nthe distance to the upper boundary of X(a,b), it follows from\n(31) and (2) of Sec. 3.11 that:\n(35)\ndy\nSuppose we write:\nfor\n\"T°(y)\"\n(36)\n.\nThen (35) becomes:\n(37)\ndy\nA similar equation may be shown to hold for No(y):\n(38)\nThe number °(y) defined in (36) (and which acts as a\nmultiplicative operator on radiance, as in (37), (38)) is\ncalled the local residual (or reduced) transmittance operator.\nObserve the analogous roles played by TO (y) and T (y) in (37),\n(38) and (5), (6). This observation prompts us to write:","12\nINVARIANT IMBEDDING TECHNIQUES\nVOL. IV\n\"T*(y)\"\nt(y) - t°(y)\nfor\n(39)\n-\nwhich we call the local diffuse transmittance operator.\nIn view of (39), we have:\n(40)\nwhich is the decomposition of the local transmittance operator\ninto its residual and diffuse parts. This should be compared\nwith (4), so that T* (y) is seen to be the integral operator\npart of (4).\nWe see from (3) and (4) that the local reflectance oper-\nator p(y) is already in diffuse form, i.e., that it already\nconsists of just an integral of o over E+. This fact lies at\nthe base of the fundamental distinction between reflectance\nand transmittance operators whenever decomposed light fields\nare considered. This distinction may be carried on up to the\nglobal level where R(a,b) is necessarily already in diffuse\nform and where T(a,b) may be rendered into reduced and diffuse\nparts by writing in general:\n\"T*(x,z)\" for T(x,z) - T°(x, 2)\n(41)\nfor asxszsb so that:\nT(x,z) = T°(x,z) + T*(x,z)\n(42)\nA similar definition holds for upward transmittances. It\nfollows immediately from (42), (32), and from (29), (30) of\nSec. 3.7 that:\nT*(y,y) = 0\n(43)\nfor\nevery y, asy 0. Since T(a,b) is positive, (27) of Sec. 7.1 then im-\nplies (29) above. However, by slowly increasing o from 0 to\nsmall positive values, the inequality (29) clearly persists\nfor a while; and indeed, in all natural optical media, (29)\ncan be shown to hold with only mild regularity properties im-\nposed. From (29) and (28) of Sec. 7.1 we now can see that:\nJR(a,b)\nb+00\nab\nso that, by (27) above (i.e., since R(a,b) = R(b,a)):\np + TR(a,00) + R(a, + = 0\n(30)\nwhere we have written:\n\"R(a,\") for lim R(a,b)\n(31)\nb+00\nEquation (30) shows that R(a,00) is independent of a since p\nand T are. This property was formally used by Ambarzumian in\n[1] to derive some of the earliest forms of the integral equa-\ntions indigenous to the invariant imbedding point of view of\ntransfer phenomena. When certain reciprocity conditions hold\nfor the medium, we have:\nand\nR(a,)ppR(a,0)\ni.e., we have commutativity of the T, R, T and p operators.","SEC. 7.4\nINVARIANT IMBEDDING ALGEBRA\n35\nUnder such conditions (which hold, e.g., when scattering is\nisotropic) (30) becomes:\np + 2TR(a,00) + (a,00) 0\n(32)\nThe solutions of (30) or the special case (32) yield the form\nfor R(a,00). A numerical procedure leading to R(a,b) for a\nrange of finite b (from which R(a,00) is estimable) will be\ngiven in Sec. 7.6 for spaces X(a,b) in which scattering is\nisotropic. We shall return to these heuristic operations with\nthe R and T operators in Sec. 8.7. The operations just per-\nformed can be redone in the irradiance context and can be made\nfully rigorous without the need for advanced mathematical tech-\nniques. See (35) - (38) and (39) - (42) of Sec. 8.7.\n7.4 Algebraic Properties of the Invariant Imbedding Operators\nThe various invariant imbedding operators introduced in\nexamples 4-7 of Sec. 3.7 will now be studied in greater de-\ntail. Our main purpose in the present section will be to dem-\nonstrate the fact that the collection T2(a,b) of operators of\nthe form M(x,y), which we found in Example 4 of Sec. 3.7 to\nconstitute a partial group, may be used as basic building\nblocks to systematically construct, via simple algebraic pro-\ncedures, all other operators of the collections T ( a , b) and\nT4(a,b), and hence all R and T operators and their simple\ncombinations. The net result of these possible constructions\nwill be novel procedures for solving transfer problems in\nplane-parallel and, indeed, general optical media. In other\nwords, we shall demonstrate that the operators M(x,y) can\nserve as the computational work horses on both theoretical\nand practical levels in the theory of radiative transfer and\nthereby have them earn their right to reside among the giants,\nthe elements of 4 (a,b), which in turn serve to unify the\ntheory and to link the theory with the interaction principles.\nThroughout this section, unless stated otherwise, we\nshall work with an arbitrarily source-free plane-parallel medi-\num X(a,b), asb, with arbitrary incident radiance distributions\nN (a) and N+ (b) over the upper boundary Xa and lower boundary\nxb respectively. Generalizations of the indicated results to\ngeneral one-parameter media are immediate; generalizations to\narbitrary media can be patterned after the discussions of Sec.\n25, Ref. [251]. Throughout the discussion all reference to\nvarious regularity properties required for inverse operations,\ndifferentiations, integrations, etc., has been avoided so as\nto bring out the highly intuitive flavor of the operator al-\ngebra.\nThe Operator M(x,z)\nThe simplest interaction operator associated with a gen-\neral plane-parallel medium X(a,b), sb is that which maps (or\ntransforms) the pair N_ (a)) of incident radiance dis-\ntributions on X(a,b) into the pair N+(a),N_(b)) of response\nradiance distributions for X(a,b). It is a simple exercise in","INVARIANT IMBEDDING TECHNIQUES\nVOL. IV\n36\nthe use of the principles of invariance for X(a,b) to deter-\nmine this operator. Thus, from principle III, in Example 3\nof Sec. 3.7, we have:\n= T(b,a) + N_(a)R(a,b)\n;\nand from principle IV we have:\nN_(b) = N_(a)T(a,b) + N+(b)R(b,a)\nThe matricial form of this system of equations is:\nT(b,a) R(b,a)\n(N+(a),N_(b)) = (N+(b),N_(a))\nThe displayed matrix of R and T operators is the requi-\nsite interaction operator. More generally, let X(x,z) be an\narbitrary plane-parallel subset of X(a,b), as x s 2 5 b, and\nsuppose (N+(2), N (x)) and N+(x), N_(z)) are, respectively,\nthe incident and response radiance distributions on X(x,z) as\nthey exist in the medium X(a,b) which is irradiated by an ar-\nbitrary set N+ (b), N (a) of radiance distributions on its\nlower and upper boundaries, respectively. (See Fig. 7.1.)\nThen principle I in Example 3 of Sec. 3.7 yields for the case\nx = y:\nN+(x)N(z)T(z,x) + N_(x)R(x,z)\n.\nSimilarly, principle II yields: for the case y = z :\nN = +\n.\nThe matricial form of this system of equations is:\nR(z,x)\nT(z,x)\n(N+(x),N_(z)) = (N+(2),N_(x))\nR(x,z)\nT(x,z)\n(1)\nLet us write:\nR(z,x)\nT(z,x)\n(2)\n\"M(x,z)\"\nfor\nR(x,z) T(x,z)\nwhere asxszsb. Thus M(x,z) is a 2x2 operator matrix which\nis defined for depth variables x, Z such that the preceding\nequalities hold. Some experimentation with (1) will show why\nthis restriction (namely X S Z) is necessary if we are to re-\ntain the useful convention of always writing radiance distribu-\ntion pairs with the upward (+) distributions as the first mem-\nber of the pair. Another advantage in preserving the fixed\norder of variables x, Z in M(x,z) shows up in the detailed com-\nputations below wherein it will always be clear whether an op-\nerator on an upward or downward flow in X(a,b) is being repre-\nsented. Thus in all that follows, M(x,z) with x s z is a use-\nful conceptual anchor whose components have simple physical","SEC. 7.4\nINVARIANT IMBEDDING ALGEBRA\n37\nsignificance. Let us denote by \"G2 (a,b)\" the set of all op-\nerators M(x,z), asxszsb.\nThe Connections Between M(x,z), M(x,z), and M(z,x)\nWe now establish the connections between the operator\nM(x,z) and the operators M(x,z), M(z,x) in the setting of an\narbitrary sub-medium X(x,z) in X(a,b). (Recall (78) of Sec.\n3.7.) Once this connection is established, we will have an\neffective means of computing M(x,z) and M(z,x) in terms of\nthe standard R and T operators for X(x,z): and conversely,\nthe operator M(x,z) will be directly representable in terms of\nthe operators M(x,z), m(z,x). This latter representation\nwill be a prototype of more general representations of the\nmembers of 3 (a,b) and T4(a,b) to be derived subsequently, and\nwill be instrumental in developing novel methods of solution\nof light fields in X(a,b), later in this chapter.\nTo establish the requisite connections we require the\npartition of the identity operator I on r (a,b):\n(3)\nI = C+ + C .\nwhere we write:\n:\n(4)\n\"C+\"\nfor\n0\nand\n0\n0\n(5)\n\"C_\"\nfor\nI\nIn C+ and C_, I+ is the identity operator on the set of all*\nupward radiance distributions and I_ is the identity operator\non the set of all downward radiance distributions associated\nwith X(a,b). No confusion will result if in the subsequent\ndiscussions we drop the signed subscripts from the identity\noperators (their positions in the matrices provide adequate\nidentification). The general working properties of C+ and C.\nare obtained by direct computation:\nC2 = C+\n(6)\nC2 = C_\n0\n(a)\n=\n*\nWe drop +, - on I when direction is clear.","38\nINVARIANT IMBEDDING TECHNIQUES\nVOL. IV\n[.....\n(a,b)C, = (a,0)\n(a,b)C_= (0,b)\nHence, via suitable pre- and post-multiplications by C+ or C_,\nvarious elements of a matrix of operators or of a vector can\nbe isolated as needed.\nNow, equation (1) holds for all incident radiances\nN+(z),N-(x)) on X(x, z). From the definition of the operators\nM(x,z) and m(z,x) and the partition operators of I, we have:\nAdding, we have:\n(N+(x),N_(z)) = +\n(7)\nFurther:\nAdding, we have:\n(N+(2),N_(x)) =\n(8)\nCombining (1), (7) and (8),\n(N+(2),N_(2)) [c++ m(z,x)c_]M(x,z)=(Ne(z),N_\nThis holds for every incident light field on X(x,z) Hence:\n[C++M(2,X)C_]M(x,2) = [M(z,x)C+ + c_]\n(9)\nwhence\n. (10)\nM(x,z) =\nOn the other hand, solving (9) for m(z,x), we have:","SEC. 7.4\nINVARIANT IMBEDDING ALGEBRA\n39\nM(2,X)[C_M(x,z) - c ] - = [c_ - c+M(x,z)]\nwhence:\nm(z,x) = [c_ - cM(x,z)][c_M(x,z)- c]-1\n(11)\ne\nOn taking inverses of each side of (11):\nM(x,z) = [C_M(x,z) - c+][Cc - C M(x,z)\n(12)\nThis may be solved for M(x,z) to yield a companion formula to\n(10):\n(13)\nM(x,z) = +\nEquations (10)-(13) are the desired connections between\nthe operators M(x,z), m(x,z), and m(z,x) for levels x,z in\nX(a,b) with X S2.\nInvertibility of Operators\nThe inverse operators in the preceding representations\ncan be examined in detail so as to allow us to establish some\nconditions sufficient to insure their existence. The inverses\ngenerally encountered in computations with (10)-(12) - are of\nthe form:\n[C++AC_]-'\n[AC++C_]-'\n[c_ - c+A]\n[C_A - c.]-'\nwhere:\n\"A\" denotes either the m or M matrices so that a, b, C, and\nd are generally operators on radiance distributions. To eval-\nuate these inverses consider for example the first; we require\na 2x2 matrix with elements a, B, Y, S such that:","VOL. IV\nINVARIANT IMBEDDING TECHNIQUES\n40\nb\nI\n0\nB\na\nd\nFrom this are obtained the four equations:\na + by = I\nB + bo = 0\ndy=0\ndo = I\nwhich in turn determine the elements of the inverse:\na = I\nB==bd-1\nY=0\nS=d-1\nHence:\n-bd-1\n(14)\nThe remaining three inverses are obtained similarly:\n(15)\n[c- - -a-1b I\n(16)\n(17)\nFrom an inspection of this collection of inverses it is\nclear that their existences depend in turn on the existences\nof the inverses of the component operators a and d in A.\nWhen A is M(x,z), this requires the transmittance operators\nT(x,z) and T(z,x) to have inverses. In most natural optical\nmedia (oceans, atmosphere), the volume scattering function o\nand volume absorption function a are positive throughout the\nmedia. This property of o and a generally insures the norm\ncontraction property of -T(x,z) or I-T - (z, x) so that under\nthese conditions the inverses of T(x,z) and T(z,x) exist. of\ncourse in any specific instance, it is good practice to have","SEC. 7.4\nINVARIANT IMBEDDING ALGEBRA\n41\nthe invertibility of the transmittance operators verified in\ndetail. In the present discussions our interest is solely in\nthe algebraic structure of and interconnections between the\nvarious interaction operators, and the discussion proceeds on\nthe assumption that all required regularity properties are in\nforce.\nRepresentations for the Components of Mx,2),M(z)\nBy means of the functional equations (11), (12) for\nm(z,x) and M(x,z) we can find explicit formulas for the com-\nponents of these operators in terms of the four standard R\nand T operators for X(x,z). Thus let us write:\nfor m(x,z)\n(18)\nthereby defining, in context, four operator components of\nM(x,z). A similar definition is made for M(z,x). Next we\nobserve that the two factors comprising (z,x) in (11) may be\nwritten:\nI\nand, by (17):\n( T-1(x,z) 0\nWith these specific representations of the factors in (11),\nwe have:\nT(z,x)-R(z,x)T-1(x,)R(x,z) -R(z,x)T-1(x,z)\nT-1(x,z)\nwhence\nM++(z,x) = T(z,x) - R(z,x)T-1(x,z)R(,z)\n(19)\nm+-(z,x) = -R(z,x)T-1(x,2) =\n(20)\n(21)\nm\n(22)\nNext, we use (12) to find the component operators of m(x,z).\nThe first factor in (12) is:","42\nINVARIANT IMBEDDING TECHNIQUES\nVOL. IV\n[\n0\nT(x,z)\nThe inverse operator is evaluated by means of (16) :\n-T-1(2,x)R(2,x)\nI\nThen (12) becomes:\nT-1(z,x)\nT-1(2,x)R(z,x)\nm(x,z)\n-R(x,z)T-1(z,x)\nwhence\nM++(x,z) = T-1(2,x)\n(23)\n(24)\nm-+(x,z) = -R(x,z)T-1(z,x)\n(25)\n(26)\nm__(x,z) = T(x,z) -\nThe components of M(x,z) may be represented in two equiv-\nalent ways, depending on whether (10) or (13) is used. Using\n(10), the factors are, explicitly:\n[\n=\nand from (14)\nI\nThen\nM(x,z)=\nM=1(z,x)\nFrom this:\n(27)\nT(z,x) = M++(z,x) -\n(28)\nR(z,x) =","SEC. 7.4\nINVARIANT IMBEDDING ALGEBRA\n43\nR(x,z) = M-1(2,x) M.+(2,x)\n(29)\nT(x,2) = M-1(2,x) =\n(30)\nAlternatively, the factors in (13) are:\nM+_(x,z)\n=\nM__(x,z)\nand from (15):\n[\nM-1(x,z)\n0\nI\nThen\nM++(x,z)\nM(x,z)\nM\nFrom this:\nT(z,x) = m-1(x,2)\n(31)\nR(z,x) =\n(32)\nR(x,z) =\n(33)\n= (34)\nThe connections between the two sets of representations\n(27)-(34) - of M(x,z) rest on the fact that M(x,z) and m(z,x)\nare mutual inverses. The four component equations harbored by:\nM(x,z)M(z,x) = = I\nprovide the necessary explicit link between the two preceding\nsets of representations. It is interesting to observe that one\nmay go from one set of representations to another by simultan-\neously interchanging the arguments \"x\" and \"z\" along with the\nsubscripts \"+\" and \"-\" This interchange rule also works for\nthe sets (19)-(22) and (23) - (26), and also for the functional\nequations (10)-(13) - (leaving M(x,z) inviolate). The physical\nbasis of this rule is that such interchanges applied to the\nradiance vector (N+(2),N-C (x)) and the matrices M(x,z), m(z,x),\neffectively reverse the incident and response radiances and\nthe operators applied to them.","VOL. IV\n44\nINVARIANT IMBEDDING TECHNIQUES\nThe Isomorphism Between (a,b) and G2(a,b)\nThe algebraic links just established between the opera- -\ntors M(x, z) and M(x,z) suggest a close overall structural re-\nsemblance between the members of the set G 2 (a,b) (i.e., , all\nM(x, z), with a b) and the members of the partial group\n2 (a,b) (i.e., , all Mlx, z) , asxsb, a (szsb). We can use\nthis strong tie between the two sets to induce a means for\nmultiplying together members of G 2 (a, b) in a way that faith-\nfully mirrors the natural multiplication of elements of\n2 (a,b). The practical utility of the newly formed multipli- -\ncation process will become clear as this discussion nears its\nclose.\nLet us denote by \"d(m(x,2))\" the operator M(x,z) found\nfrom m ((x,z) using (31)-(34) - and let \"d-1(M(x,z))\" denote\nthe operator m(x,z) obtained from M(x,z) using (23)-(26) -\nm(x,z)\nO\nm(z,x)\nO\nx\n2(a,b)\n+\nM(x,2)\nG2(a,b)\nm(x,z)= m(x,y) m(y,2)\n2(a,b)\nm(y,z)\n(=s'(M(y,z ))\nm(x,z)\nom(x,y)\nM(x,z) = 3(m(x,z))\nM(x,z)\n=M(x,y)*M(y,z)\nM(y,z)\nM(x,y)\nG2(a,b)\nFIGS. 7.3 , ,7.4 The meaning of the isomorphism between\n2 (a,b) and G2 (a,b).","45\nSEC. 7.4\nINVARIANT IMBEDDING ALGEBRA\nwhere, In this manner we define in context a\nfunction on part of 2 (a,b) (call it the upper triangle of\n2 (a,b) onto G2(a,b). (That is, we do not define for all\npairs x,z, but only those such that x s z . ) This function is\none to one in the sense that to each M(x,z) in G2(a,b) there\nis assigned at most m (x, z) in the upper triangle of 2 (a,b)\nfor any choice of levels x,z in X(a,b), where X S Z . The term\n\"upper triangle\" of 2 (a,b) is suggested by the fact that in\na cartesian coordinate plot of the pairs depths (x,z), those\npairs such that z, lie above the diagonal line. (See shaded\nregion in Fig. 7.3.) An alternate one to one mapping 4 from\nthe lower triangle of (a,b) onto G2 (a,b) is possible using\nthe systems (19)-(22) and (17)-(20). Either mapping or 4\nwill suffice for our present purposes. We choose to work with\nas far as possible. With this choice of (12), (13) may be\nrewritten as:\nM(x,z) = (M(x,2)) = [m(x,z)C+ + c.]-'[c++m(x,2)C.]\nm(x,z) = -1 (M(x,z)) = [c_M(x,z)\nThe induction of the multiplication process on G2 (a,b)\nis now carried out as follows. Let M(x,y) and M(y,z) be any\ntwo elements of G2(a,b), provided that they have a depth level\nin common (e.g., y, as shown). It seems natural to require\nthat their \"product\" be such that the usual matrix product of\nthe corresponding operators -1 (M(x,y)) and o-1 (m (y,z)) in\n2 (a,b) maps back, under , to the required \"product\". (See\nFig. 7.4). Thus we agree to write:\n(35)\n\"M(x,y)*M(y,z)\" for .\nBy definition of o-1 and the one to one properties of o:\nm(x,y) -1(M(x,y))\nand:\nHence:\n(M(y,z))\nTherefore an alternate way of expressing (35) is:\n= (m(x,y)) *(M(y,z))\n(36)\nThis alternate form of describing the star product of elements\nof G2 (a,b) defined in (35) shows how the structure of multi-\nplication in G2 (a,b) mirrors that of 2 (a,b). In modern al-\ngebra the function which induces operations such as the op-\neration * is called an isomorphism, the etymology of the word\nin this physical case being most appropriate (iso = same;","46\nINVARIANT IMBEDDING TECHNIQUES\nVOL. IV\nmorph = form). . Under the introduction of the star product,\nG2 (a,b) becomes a partial semigroup, with an identity opera-\ntor of the form M(x,x), and with the associativity property\nand inverse properties holding.\nThe Physical Interpretation of the Star Product\nThe star product on G2 (a,b) introduced above has a most\ninteresting physical interpretation. It is worthwhile to pur-\nsue this interpretation as it will permit us to tie together\nthe territory covered so far in this section with that of\nSection 7.3. Since M(x,y) describes the reflectance and trans-\nmittance properties of X(x,y), and M(y,z) describes those of\nX(y,z), we ask: What physical description, relative to X(x,z),\ndoes the star product M(x,y)*M(y,z) represent? The clue to\nthis description is given by examining (35) The right side\nof the definition is simply the image, under , of M(x,z).\nHence we see that:\nM(x,z)=M(x,y)*M(y,z)\n(37)\nTherefore the star product of M(x,z) and M(y,z) is the opera-\ntor M(x,z) associated with the union (the sum) of the two con-\ntiguous slabs X(x,y) and X (y, z) (as depicted e.g., in (b) of\nFig. 7.2).\nLet us find the components of the star product\nM(x,z) *M(y,z) directly in terms of the components of the fac- -\ntors M(x, z) and M(y,z). We begin the derivation with (35)\nThus, by (23)-(26):\nT-1(y,x)\nT-1(y,x)Ry,x)\n-1 (M(x,y))\n.\n-R(x,y)T-1(y,x) T(x,y)-R(x,y)T-1 (y,x)R(y,x)\nSimilarly:\nT-1(z,y)\nT-1(2,y)R(z,y)\n-1 (M(y,z))\n-R(y,z)T-1 (z,y) T(y,2)-R(y,z)T-1 (z,y)R(z,y)\nThe product of these matrices is:\n(M(x,y)) (M(y,z)) = M(x,z)\nand where:\nM++ (x, z) = T-1(y,x)T-1(2,y) - T-1(y,x)R(y,x)R(y,z)T-1(z,y)","47\nINVARIANT IMBEDDING ALGEBRA\nSEC. 7.4\n+T-1(y,x)R(y,x)[T(y,z)-R(y,z)T-1(z,y)R(z,y)]\nM+(x,z)=-R(xy)T-1(y,x)t-1(z,y) -\n-[T(x,y)-R(x,y)T-1(y,x)R(y,x) [R(y,z)T\"-(2,y)]\nm (x,z)= -R(x,y)T-1(y,x)T-1(z,y)R(z,y) +\n+[T(x,y)-R(x,y)T-1(y,x)R(z,y)][T(y,x)-R(y,z)T-1(z,y)R(z,y))\nEach of these may be reduced considerably if we use algebraic\nformulas developed earlier. For example:\nM++(x,2) T-1(y,x)[I - R(y,x)R(y,z)]T-1(z,y)\n=T-1(2,x)\n9\nwhen the last inequality is based on (18) of Sec. 7.3. (See\nalso (8) of Sec. 7.3.) In a similar (but slightly more ardu-\nous) manner the remaining components may be reduced so that\nthey may be used in (31)-(34). - The net result of the mapping\nback to M(x,z) from m(x,z) is:\nM(x,z) = M(x,y)*M(y,z) =\n[J(2,Y,X)T(y,x)\nR(z,y)+Q(z,y,x)T(y,z)\nR(x,y)+Q(x,y,z)T(y,x) T(x,y,z)T(y,2)\n(38)\nIn this way the representation of the star product is rendered\ninto a mathematically self-contained form by means of the par-\ntition relations developed in 7.3. The representation is made\nparticularly meaningful physically by using the complete re-\nflectance and transmittance operator for X(x,z), so that each\ncomponent of the product can be read directly in terms of re-\nflectances and transmittances. We summarize (38) by saying\nthat: the star product of M(x,y) and M(y,z) is the mathema-\ntical form of the partition relations (15)-(18) of Sec. 7. 3\nfor the medium X(x,z), and therefore contains all the infor-\nmation for determining the standard reflectance and transmit-\ntance operators of the union X(x,y) U X(y, z) of two contiguous\nmedia, knowing the respective operators of each component of\nthe union.\nThe Link Between M(a,x,b) and M(a,y,b)\nTwo invariant imbedding operators for X(a,b), such as\nM(a,x,b) and M(a,y,b), may be linked by the operator m(x,y)\nas follows. The definition of the invariant imbedding opera-\ntor yields the equations:","48\nINVARIANT IMBEDDING TECHNIQUES\nVOL. IV\n(N+(y),N_(y)) = (N+(b),N_(a)) M(a,y,b)\n(N+(x),N_(x)) = (N+(b),N_(a)) M(a,x,b)\nSince\n(N+(y),N_(y)) =\nit follows at once from these three equations that:\n(N.(b),N_(a))(m(a,x,b)m(x,y))= = (N+(b),N_(a))m(a,y,b)\no\nThe incident radiance distributions being arbitrary, we have:\nM(a,y,b) =\n(39)\nfor every x, y in (a,b). If the inverse of M(a,x,b) exists,\nwe find:\nm(x,y)=m-1(a,x,b)m(a,y,b) =\n(40)\nwhich shows how M(x,y) is represented in terms of the third\norder invariant imbedding operators.\nIt is interesting to view (39) not as representing a\nstatic link between members of T3(a,b) but as depicting the\ntransformation of the interval [a,b] into the set T3(a,b).\nThis new view is obtained by first fixing level x, a s x s b.\nThen for each choice of y in the interval [a,b] equation (39)\nassigns to y the operator M(a,y,b) in 3 (a,b). In this way\nm(x,.) serves as a mapping or transformation from [a,b] to\nTs(a,b).\nBuilding on the preceding viewpoint, equation (39) may\nbe envisioned as stating four \"principles of invariance\" for\nthe complete a and Toperators. Thus, unfolding (39) compo-\nnent by component:\nT(b,y,a) = T(b,x,a)M++(x,y) + Q(b,x,a) M-+(x,y)\nQ(b,y,a) = J(b,x,a) m+(x,y) +\nR(a,y,b) = Q(a,x,b)M++(x,y) + T(a,x,b) M.+x,y)\nT(a,y,b) = R(a,x,b) M+_(x,y) + T(a,x,b) m__(x,y)\nIn the present point of view the Q and I operators act\nthe role the radiances did in the final statements (e.g., Ex.\n3, Sec. 3.7) and the components of m(x,y) act like transmit-\ntance and reflectance operators: those with like signs are\ntransmittance operators, those with unlike signs are reflec-\ntance operators. This analogy is exact in the sense that an\noperatorial theory for the a and T operators can be developed","SEC. 7.4\nINVARIANT IMBEDDING ALGEBRA\n49\nwhich is essentially parallel to the radiometric theory for\nN+(y). This and still other analogies (some of which are\nbrought to light below), open up vistas in algebraic radiative\ntransfer theory which are beyond the scope of this work but\nwhich are potential areas of basic research in the theory.\nSee Problem X, Sec. 141, Ref. [251].\nRepresentations of M(x,y,z) by Elements of 2 (a,b)\nIn view of the success in representing the basic opera-\ntors M(x,y) by means of the imbedding operator m (x,y) (See\n(10)-(13)) - we are led to seek still further representations\nof interaction operators by members of the partial group\n2 (a,b). We shall find that the set 2 (a,b) is an extremely\npowerful set of operators in the sense that virtually all op-\nerators in modern radiative transfer theory are representable\nby suitable algebraic combinations of members of 2 (a,b). In\nthe next few paragraphs we shall assemble some evidence in\nthis direction. The formulas so gathered will be employed in\nSec. 7.5 to find various differential equations governing the\ninteraction operators, equations which should suggest novel\nsolution procedures in radiative, neutron, and generally\nlinear transport theory.\nOn the one hand the light field at level y in X(a,b) is\nobtained from arbitrary incident light fields at levels x and\nz, xsysz, by the relation:\n(N+(y),N_(y)) = (N+(z),N_(x))m(x,y,z)\n(41)\n0\nOn the other hand those on levels x and Z are related\nby that on level y by using the following operators:\n(N+(z),0) = (N_(y),N_(y)) M(y,2)C,\n(0,N_(x)) =(N.(y),N_(y))m(y,x)C =\nAdding these equations:\n(N+(z),N_(x)) =\nand using (41):\n(N+(z),N_(x)) - = (N+ (z),N_(x)) M(x,y,z) [m(y,z)c+ +((M(y,x)c\nwhich, in view of the arbitrary nature of the incident distri-\nbutions, yields the desired representation:\n(42)\n=\nIt is interesting to speculate what would happen if we\nallowed the variables x,y, Z in (42) to take on any three val-\nues in the depth interval [a,b]. The derivation of (42) by","50\nINVARIANT IMBEDDING TECHNIQUES\nVOL. IV\nconvention (but not essentially) is performed only for the\ndepths x, y, z, in the usual order X y V Z within X(a,b). But\nsince the operators M(y,z) and M(y,x) are defined for all\npairs of depths, and since the inverse of the indicated linear\ncombination of these operators should exist just as often as\nthose in more orthodox settings, there now is a way, as indi-\ncated by (42), of formally extending the domain of definition\nof the invariant imbedding operators.\nA Constructive Extension of the Domain of (x,y,y,z)\nThe preceding observations of the potential extensibility\nof the domain of definition of the invariant imbedding opera-\ntor M(x,y,z) is reinforced by recalling equation (39), in\nparticular the interpretation of the equation as implicitly\ndefining a mapping which, in effect assigned to each y in the\ninterval [a,b] an operator M(a,y,b), as explained above.\nSuppose then we write, ad hoc:\n\"m(x,u,z)\"\nfor\nM(x,y,z)M(y,u)\n(43)\n.\nIt follows that, as long as we have X S U S Z , the operator\n(x,u,z) is, by (39), simply (x,u,z). But the product of\nthe operators in (43) is certainly compatible for any u, given\neach factor associated with that u. In this way, then, we can\nformally extend the domain of M(x,y,z) so that the parameters\nmay fall outside of the subinterval [x,z] in [a, b]. Once the\nextension is fully and unambiguously made, the bar above \"m\"\nin (43) may be dropped in practice.\nThe extension just made is a constructive extension of\nM(x,y,z) in the sense that, given M(x,y,z) and mly,u) there\nis a definite construction procedure that may be followed in\nthis case, a simple matrix product effecting the extension.\nIt should be recalled, of course, that M(x,y,z) is in \"al-\nready extended\" form as it is cut directly from the more com-\nprehensive mold of the generalized invariance imbedding rela-\ntion. (See the discussion of (76) of Sec. 3.7.) Thus we may\nsimply write:\n\"M(x,y,z)\" for M(x,y;z,y)\n(44)\n,\nwhere x,y,z are any three levels in X(z,b), and study\nM(x,y,z), so formed, as a special instance of the generalized\ninvariant imbedding operation. Thus (43) without the bar over\nm is in the last analysis simply a consequence of the semi-\ngroup property (84) of Sec. 3.7. Further, if one returns to\nthe derivation of (42) or repeats its derivation, now using\nthe definition (44) for M(x,y,z), the same functional rela-\ntion (42) would be obtained, and the speculations on the ex-\ntension of (42) to general parameters x,y,z, now have a solid\naffirmative basis.","51\nSEC. 7.4\nINVARIANT IMBEDDING ALGEBRA\nRepresentation of Mv,z;u,y) by Elements of 2 (a,b) and (a,b)\nWe begin the derivations by representing M(v,z;u,y) as\na product of two simpler operators by means of the semigroup\nrelation (84) of Sec. 3.7.\nM(v,z;u,y) = M(v,x;u,x) M(x,z;x,y)\n(45)\nin which we have set x = W. With this simple identification\nof x and W we have managed to represent M(v,z;u,y) as a prod-\nuct of two operators of the extended type M(x,y,z). Thus,\nthe first factor M(v,x;u,x) in (45) is simply an extended in-\nvariant imbedding operator M(v,x,u) as defined in (44). The\nother factor appears to be the inverse of such an extended\noperator. Indeed, using the semigroup relation (84) of Sec.\n3.7 once again, it is clear that:\nM(x,z;x,y) M(z,x;y,x) = I\n(46)\n.\nHence:\n(x,z;x,y) = m-1 (z,x;y,x)\n= m-1(z,x,y)\n(47)\nIt remains only to return to (42) and make the appropriate\nsubstitution of variables to obtain the desired representation\nof M(v,z;u,y). Thus from (42):\n(v,x;u,x) = M(v,x,u) =\nOnce again from (42) :\nM(z,x;y,x) = m(z,x,y) = [m(x,y)c+ +(M(x,2)C_]-'. (49)\nIn view of (47), equation (45) therefore becomes:\n(v,z,u,y) =\n(50)\nwhich is the requisite representation of an arbitrary member\nof 4 (a, b) by members of T2(a,b), and which holds for every\nu,v,x, y, Z in [a,b], provided, of course, that the inverse\noperator in (50) exists in a given setting. An alternate form\nof (50), using the generalized invariant imbedding operator,\nis:\nm(v,z;u,y) = m(v,x,u)m-1(z,x,y)\n(51)\nIn equations (50) and (51) the depth variable X is free to be\nchosen anywhere in [a,h b]. Observe in (51) how the first fac-\ntor, as a generalized invariant imbedding operator, maps","INVARIANT IMBEDDING TECHNIQUES\nVOL. IV\n52\n(N+(u),N_(v)) into (N+ (x) ,N_ (x) and then how the inverse\nfactor maps the latter radiance distribution into\n(N+(y),N_(z)) . The composite mapping of these functions is\nprecisely that performed by M(v,z;u,y). Thus one could al-\nmost write down (51) by sight if the various ranges and do-\nmains of the operator are kept in mind. It is of interest to\ncompare (51) with (40) which yields a representation of\nm(x,y) in a similar vein to that of (v,z;u,y) above in (51).\nThe representation (50) may be used to yield at once,\nunder suitable confluence of the variables u,v,y,z, the entire\nfamily of interaction operators considered so far in this sec-\ntion. This is left as an exercise for the interested reader.\nThe derivation of the following alternate representation of\nm(v,z;u,y) by extended members of 3 (a,b) is also left to\nthe reader:\nm(v,z;u,y) = m (v,y,u) + 7(v,z,u)C.\n(52)\nThe Connection Between 4(x,y) and M(s,y)\nThe interaction operators for media with internally\ndistributed sources of radiant flux differ fundamentally from\nthose designed to describe radiative transfer in source-free\nmedia. The origin of this difference was pinpointed in the\nequation (31) of Sec. 3.9 for the operator Y (s, y) ; and the\nsubsequent discussion of this operator showed that it was dis-\ncontinuous at the point (s,s) of its domain, a property not\npossessed by operators of the source-free kind. The operator\nY(s,y) introduced in Example 3 of Sec. 3.9 (of which 4+(s,s)\nis one of four components) is specifically designed to de-\nscribe light fields in a media which have internally distrib-\nuted sources. Since we have apparently reached in this sec-\ntion a culmination point in the discussion of source-free\nmedia, it would be of interest to relate the operator 4(s,y)\nto the basic operator m (x,y) for source-free media.\nWe now momentarily abrogate the standing condition a-\nbout source-free media X(a,b) made at the outset of this sec-\ntion. We postulate instead a source of flux arbitrarily dis-\ntributed over level S in X(a,b), asssb. The source is rep-\nresented as an arbitrary radiance distribution N°(s), where\nN° (s) is conceptually partitioned into the pair (No (s) , N (s))\nof upward (+) and downward (-) radiance distributions Then\nthe radiance distribution N(y) (= (N+ (y) N_(y)) at any level\ny in X(a,b) is given, according to (15) of Sec. 3.9, by:\n(N+(y),N_(y)) = (N°(s),N°(s))Y(s,Y\nWhat we must do next is to use the operator m(s,y),\nwhich is designed for use in source-free contexts, to relate\nthe radiance distribution at level S to that at level y. It\nis important, therefore, to renew acquaintance with the manner\nin which the source radiance function N° (s) is viewed in radi-\native transfer theory. A re-reading of the opening paragraph","SEC. 7.4\nINVARIANT IMBEDDING ALGEBRA\n53\nof Example 3, Sec. 3.9 will serve this purpose. We see that\nthe source is pictured very much like a thin transparent lay-\ner of pure light sandwiched between the media X(a,s) and\nX(s,b). For true internal sources, we require a~~y in accordance with\nthe jump property of 4(s,y) at S = y. Ct are defined in (4),\n(5).","INVARIANT IMBEDDING TECHNIQUES\nVOL. IV\n54\nA Star Product for the Operators M(x,y,z)\nWe end the present section on three ascending general\nnotes, of which the present discussion sounds the first: we\nwish to extend the concept of the star product of the opera-\ntors M(x,z), as developed in (35), to the invariant imbedding\noperators M(x,y,z). This product, which we found to be the\nalgebraic essence of the partition relations (15)-(18) - of\nSec. 7.3, serves to show how to combine the interaction prop-\nerties of two contiguous media X(a,b) and X(b,c) to find the\ncorresponding interaction properties of their union X(a,c).\nWe now attempt to do the same for the complete reflectance\nand transmittance operators of any two adjacent media.\nFigure 7.5 depicts the present setting. We imagine a\nplane-parallel optical medium X(a,b) to be the union of two\narbitrarily overlapping sub-media: X (a,z) and X(x,b). Let y\nbe any level in X (a, b) such that xsysz. The problem before\nus is: to represent Q(a,y,b), T(a,y,b), and m(a,y,b) in\nterms of a suitable algebraic combination of the complete a\nand T operators associated with X(a,z) and X(x,b). The pres-\nent problem is geometrically slightly more general than its\ncounterpart posed in Sec. 7.3 for the R and T operators in\nthe sense that we require not contiguity of X(a,z) and X(x,b)\n(so that necessarily X = z), but merely intersection of the\nmedia (so that S Z).\nThe incident radiance distributions N_ (a) and N (b) on\nX(a,b) generate a light field at general levels x,y,z in\nX(a,b) which may be computed several ways depending on which\nmedium one envisions the levels to be in, , i.e., as light\nfields in X(a,b), or in X(a,z) or in X (x, b) . Thus (y), as\na\nX\ny\nZ\nb\nFIG. 7.5 The setting for the star product in T3(a,b).","SEC. 7.4\nINVARIANT IMBEDDING ALGEBRA\n55\nradiance distributions in X(a,b) are given by:\nN+(y)=N_(a)R(a,y,b) = + Ny(b)J(b,y,a)\n(57)\n=N_(a)J(a,y,b) = + (b)Q(b,y,a)\n(58)\nwhich follows at once from the invariant imbedding relation\nfor X(a,b). On the other hand, the distribution N+(y) con-\nsidered as being in X(a,z), is given by:\nN+(y)= N_(a)B(a,y,z) + N+ (z) T(z,y,a)\n(59)\nand N (y), considered as being in X(x,b) is given by:\nN_(y) = =N_(x)J(x,y,b) + N+ (b)R(b,y,x)\n(60)\nwhich are the results of applying the invariant imbedding op-\nerators of X(a,z) and X(x,b), respectively. Now the distri-\nbutions N+(z) and N_ (x) appearing on (59) and (60) can be\nfound by solving the system:\nN(z)N_(x)R(x,z,b) + N+ (b) T(b,z,x)\n(61)\nN_(x)=N_(a)T(a,x,z) = + N+(z) R(z,x,a)\n(62)\nwhich is derived similarly to (59), (60) by considering level\nx as occurring in X(a,z) and level Z as occurring in X(x,b).\nThe solutions are:\nN+(z) =\n(63)\nN_(x)\n-1\n(64)\nThese equations should be compared with (9), (10), (25) and\n(26) of Sec. 3.7 and (49), (50) of Sec. 3.9 for structural\nsimilarities.\nNext consider the two alternative ways of describing\nN+(y) in (57) and (59). If these two expressions for N+ (y)\nare equated and if N+ (z) as given in (63) is used, then since\nN+ (b) and N (a) are arbitrary, we derive the following two\noperator equations as a result:\nR(a,y,b)\n=\n(a,y,z)+J(a,x,z)R(x,z,b) [I-R(z,x,a)a(x,z,b)]-J(z,y,a)\n(65)\nJ(b,y,a) = J(b,z,x) [I-Q(z,x,a)Q(x,z,b)]J(z,y,a) (66)","INVARIANT IMBEDDING TECHNIQUES\nVOL. IV\n56\nThese two equations constitute a rather interesting generali-\nzation of (15) and (18) of Sec. 7.3. For by letting y = Z in\n(65) and (66) we have:\nR(a,z,b) = T(a,x,2) R(x,z,b) [I- a(z,x,a) (R(x,z,b)]-'\n(67)\nand\nT(b,z,a) =\n(68)\nand these representations may be plowed back into (65) and\n(66) to yield the following compact forms of (65) (66) :\nR(a,y,b) = R(a,y,z) + A(a,2,b)J(2,y,a)\n(69)\nJ(b,y,a) = J(b,z,a) T(z,y,a)\n(70)\nThe latter equation is simply the semigroup property (52) of\nSec. 3.7 for the J operator. However, (69) is a relatively\nnovel equation, much in the way (15) of Sec. 7.3 was a new-\ncomer to the semigroup scene in that setting. Equation (69)\nwill be used at crucial points of the investigation in 7.13.\nThe analogy between the present derivation and those leading\nto (15) and (16) of Sec. 7.3 appears to be a throughgoing\none, on the strength of which we can write down the remaining\ntwo correspondents of (16) and (17) of Sec. 7.3:\nJ(a,y,b) = T(a,x,b)J(x,y,b)\n(71)\nR(b,y,a) = R(b,y,x) + Q(b,x,a)J(x,y,b)\n(72)\nwhere\nJ(a,x,b) = T(a,x,z) [I-R(x,z,b) a(z,x,a)]th\n(73)\na (b,x,a) = J(b,z,x) R(z,x,a) [I-R(x,z,b)R(z,x,a)]1 (74)\nThe requisite star product for the invariant imbedding\noperator M(a,y,z) and M(x,y,b) associated with the sub-\nmedia X(a,z) and X(x,b) may then be defined as follows. We\nwrite:\n(a,y,z)*m(x,y,b)\" for m(a,y,b)\n(75)\nwhere m(a,y,b) in (75) is constructed from the operators of\nm(a,y,z) and M(x,y,b) using (69)-(72) in which R(a,z,b),\nT(b,z,a), T(a,x,b), and R(b,x,a) are as given in (67),(68),\n(73) and (74), respectively. Thus:\nM(a,y,b) = M(a,y,2)**Y(x,y,b) =\nQ(b,y,x)+a(b,x,a)J(x,y,b)\nT(b,z,a)J(2,y,a)\n=\nQ(a,y,z)+Q(a,z,b)J(z,y,a) T(a,x,b)J(x,y,b)\n(76)\nThis star product will be used subsequently in the\nstudy of irradiance fields in interacting media (cf. (91) of\nSec. 8.7).","SEC. 7.4\nINVARIANT IMBEDDING ALGEBRA\n57\nA\nB\nZ\nFIG. 7.6 The star product of invariant imbedding op-\nerators can be defined for arbitrary media.\nRecall that the depth variables x and Z in (75) are\narbitrary, subject only to the condition XSYSZ, i.e., that\nthe media X(a,z) and X(z,b) overlap, and that y be chosen in\nthe intersection of these media. Equation (76) is to be com-\npared with (39).\nThe power of the present algebraic approach to radia-\ntive transfer theory can be appreciated in some detail if we\nnow turn to the general invariant imbedding relation (51) of\nSec. 3.9 and observe that all activity we have gone through\nto reach (76) can be repeated for the general medium X of\nexamples 4 and 5 of Sec. 3.9. Thus if we have a medium X\nwith two overlapping submedia A and B of a one-parameter medi-\num as in Fig. 7.6, and more generally, if we have two media\nX and Y which intersect in a region Z as in Fig. 7.7, then we\ncan form a star product of the invariant imbedding operators\nof X and Y to obtain the invariant imbedding operator of their\nunion XUY, in exact analogy to (76).\nIn view of these observations, the possibilities for\nfurther exploration of the algebraic theory of radiative\ntransfer are clearly mounting in number and in depth. The\npossibilities branch off into topological and algebraic direc-\ntions which, if kept bound together by suitably defined con-\ncepts, will raise the theory of radiative transfer the re-\nmaining distance to its logical haven: a possible general\ntheory of linear transport processes. Such a pursuit is un-\nfortunately beyond the scope of the present work, and we rest\nthe matter here.","58\nINVARIANT IMBEDDING TECHNIQUES\nVOL. IV\nY\nX\n2\nFIG. 7.7 A general setting for the star product of\ninteraction operators in general optical media.\nPossibilities Beyond M(v,x;u,w)\nIn this the penultimate note of the present section,\nthe possibility of operators more comprehensive than those in\n4 (a, will be considered. We shall show that such possibil-\nities of arbitrarily great comprehensiveness are easily con-\nstructed. However, in a sense, such generality is no longer\nneeded now that operators like M(x,y) harnessed in parallel\nhave been shown to have sufficient computational power (cf.\n(50), (51) and (52)) to do everything M(v,x;u,w) can do.\nFor simplicity, we shall remain in the setting of one-parame-\nter media during the present discussion.\nTo see what direction we may take in generalizing\nm (v,x;u,w) let us return to its definition in (56) of Sec.\n3.7. Recall that the primary motivation for m(v,x;u,w) was\nthe need for an operator which would take as input the pair\n(N+ (u),N_(v)) of radiance distributions on arbitrary levels\nof u and V in X(a,b) and yield as output the pair (N+(w),N_(x))\non still two more arbitrary levels W, X in X(a,b). In this\nway we achieved a comprehensive, symmetric setting for all\nclassical operators. In particular, these choices of input\nand response distributions constitute the natural generaliza-\ntion of the classical type of inputs and responses of M(x,y)\n(cf. (2) above) and the general invariant imbedding operator","SEC. 7.4\nINVARIANT IMBEDDING ALGEBRA\n59\nM(x,y,z). Having thus extended the input and output types\nto a reasonably general kind (there is still room beyond here\ntoo--consider, - e.g., partitioning E+ and E. into many and\nsundry pieces) we turn to consider the effect of an increase\nin the number of levels. Thus, suppose we ask for an operator\nwhich takes as input the 2m component radiance vector:\n... ; N+(um),N_(vm) (77)\nand yields as output the 2n component radiance vector:\n(N+(w1),N_(x1);N(w2),N_(x2); ; N+(w)),N_(x_)) (78)\nwhere U1, and V1, are 2m arbitrary levels in\nare 2n arbitrary levels in\nX(a,b). N+ (ui) is as usual the upward radiance distribution\nover level ui. Similarly with the other radiances. Then the\ninteraction principle supplies an operator\n(79)\nM(Vmm;\nwhich is a 2m x 2n matrix of operators of which 2mn are\nT-like and 2mn are R-like (which we need not display here)\nand which clearly reduces to M (v,x;u,w) by setting m = n = 1.\nWe shall now show that the operator (79) can be represented\nas a linear combination of generalized invariant imbedding\noperators of the form m Then, in view of (50),\nthe algebraic representation of (79) in terms of members of\nthe partial group 2 (a, b) will stand established.\nThe key to the desired representation of (79) rests in\nthe following two partitions of identity operators:\n(80)\n(81)\nwhere Ci is a 2m x 2 matrix and Dj is a 2n x 2 matrix of the\nform:\nI\n(82)\n2m\nI\n= C\n2n\nDj\n=\n0\n0\nand where \"0\" denotes the 2 x 2 zero matrix and \"I\" the 2 x 2\nidentity matrix considered earlier (e.g., in (3)). The","60\nINVARIANT IMBEDDING TECHNIQUES\nVOL. IV\nnotation \"trn Ci\" denotes the transpose of Ci, i.e., 2 x 2m\nmatrix obtained by turning Ci on its side so that the identity\noperator I'm in Ci is the ith matrix counting from the left as\nusual. That (80) and (81) represent, respectively, the\n2m x 2m and the 2n x 2n identity matrices is readily estab-\nlished, and is left as an exercise to the reader. Observe\nalso that [trn Ci]Ci is the 2 x 2 identity matrix I, for\nevery i.\nThe operators Ci have the useful properties that:\n(N+(uj),N_(vi))\n(83)\nfor lsism, ljn, and where \"a\" and \"b\" denote (77) and\n(78), respectively. We shall continue to use these abbrevia-\ntions \"a\" and \"b\" in the remainder of this discussion.\nNow, we know how to relate (N+(ui), N_(vi)) and\n(N+(wj),N_(xj)). Such relating is the specific task of\nM(Vi,Xj;Uj,wj). Thus:\nN+(wj),N_(xj)) = (84)\nIn other words (84) states that:\n(85)\nwhere we have written, ad hoc:\n\"Mij\" for\n(86)\nEquation (85) therefore suggests that we start with:\nb = a m\n(87)\n,\nwhere \"m\" at present denotes (79), and insert the 2m x 2m\nidentity operator Im, in the form (80), between a and m in\n(87) to obtain:\nb = a E\n(88)\nOnce this is done we operate on each side of (88) with Dj to\nobtain:\n(89)\n.\nIt is clear from (89) and (85) and the fact that these equa-\ntions hold for every incident radiance vector a, that:\n(90)","SEC. 7.4\nINVARIANT IMBEDDING ALGEBRA\n61\nIn this way we see the 2x2 operator matrix Mij is a special\ncase of m. Going on with the present analysis of (87), we\noperate on each side of (89) with trn Dj and sum over all j.\nThus:\nn\nb=b = a j=1 Dj]\n(91)\n.\nAgain, since a is arbitrary, we have, from (87) and (91):\n(92)\n=\nj=1 i=1\nBy means of this equation, we see that m is representable as\nan mxn block matrix with Mij as the element in the ith row\nand jth column.\nEquation (92) is the desired representation of m in\nterms of m ij? i.e , in terms of the members of T4 (a,b).\nThus we see that m may be represented by a suitable algebraic\ncombination of elements of P2 (a, b) , using (50).\nPossibilities Beyond 2 (a,b)\nWe conclude this section with some observations on the\npossible direction in which the notion of the partial group\nT2 (a,b) for a plane-parallel medium X(a,b) may be extended.\nAn immediate extension of 2 (a,b) may be made to a\none-parameter three-dimensional optical medium in which \"a\"\nand \"b\" are indices of the two-parameter surfaces bounding\nthe general curvilinear medium X(a,b). The resultant alge-\nbraic structures are isomorphic, (i.e., algebraically identi-\ncal) to that of the plane-parallel case and so will not be\nexplicitly considered.\nAn extension of 2 (a,b) beyond one-parameter media\nwould be to an arbitrary connected medium X in which \"x\" and\n\"y\" in m(x,y) now denote two arbitrary points of X or possi-\nbly small subsets of X. We shall call X a point in either\ncase in what follows. This extension is of great physical\ninterest and we pause to examine it using formal operations\nin just enough detail to see how the generalization may go.\nLet X be an optical medium in three-dimensional Euclid-\nean space, i.e., the space which represents an ordinary every-\nday world. Within X we can simulate portions of the earth's\natmosphere, or its seas and lakes. Let N° be the incident\nradiance function on X and N the associated response radiance\nfunction on X. Then the interaction principle supplies an\ninteraction operator m(x) which maps N° into N:\nN = N° M(X)\n(93)","62\nINVARIANT IMBEDDING TECHNIQUES\nVOL. IV\nThe reader will recall that N is a function which\nassigns to each X in X and E in E the radiance N(x,5) at X in\nthe direction E. Thus, from N we can obtain the radiance\ndistribution N(x) at point X. Let E(x) be the operator which\nassigns to point X in X the radiance distribution N(x) at\npoint x as induced by the radiance function N. Thus:\nN(x) = NE(x)\n(94)\nIn other words, E(x) is a continuous (or generalized)\nversion of C or D introduced in (82). Conversely, from\nknowledge of N(x) at each point x of X, we can reconstruct N.\nLet \"trn E(x)\" be the operator such that:\n[trn E(x)] dV(x)\n(95)\nwhere I is the identity operator (transformation) on the vec-\ntor space V(X) of all radiance functions defined on X. (The\nuse of vector space concepts was introduced in an earlier\ndiscussion; see Example 15 of Sec. 2.11.) We shall not go\ninto the details of construction of the operator trn E(x).\nIt will suffice to note that it is intended to be analogous\nto the transpose operators discussed in (80) and (81) and may\nbe constructed using theorems A,B,C of the interaction method\nin Sec. 3.16. Using this partition of I in (93) we have:\nN = N°1 M(X) = / N°E(X) = [trn E(x)] m(x) dV(x) . (96)\nApplying the operator E(y) to each side of (96) we have:\nN(y) = NE(y) = N°E(x) [trn E(x)] m(x) E(y) dV(x). (97)\nLet us write:\n\"m°(X;x,y)\" or \"m°(x,y)\" for\nSu\n[trn E(x)]m(x) E(y) dV(x)\n(98)\n.\nX\nThen (97) can be written as:\nN(y) N°(x) m°(x,y)\n(99)\nwhere:\nN°(x) = N°E(x)\n(100)\nWe shall now assume that the operator mo(x,y) is one to one\nfor every pair (x,y) of points in X, in the sense that two\ndistinct incident radiance distributions N1° (x), N2° (x) al-\nways are mapped into distinct corresponding response radiance","SEC. 7.4\nINVARIANT IMBEDDING ALGEBRA\n63\ndistribution functions N1 (y), N2 (y) using (93). . By distinct\nradiance distribution N1, N2 it shall be understood that for\nsome set E0 of directions in E, SEN)()0\nMatters can usually be arranged so that an optical medi-\num X can be partitioned into pieces Xi over each of which the\noperator mo (Xi;x,y) is one to one. Hence no essential loss\nin generality will be engendered in what follows if we assume\nmo (X;x,y) is one to one over an arbitrary optical medium X.\nThe one to one property of mo(x,y) is used to insure that\nthe inverse (mo(x,y)) of mo(x,y) exists. For, once this\ninverse is available, we can directly relate any two radiance\ndistributions in X. Thus, from (99) used twice:\nN(y) = N°(x) M°(x,y)\nN(z) = N°(x) MO(X,2)\nwhence:\nN(y)\nwhence again:\nwhich holds for every x in X, so that if we write:\n\"m(y,z)\" for\n(101)\nwe have:\nN(2)=N(y)m(y,z)\n(102)\nfor every pair y, of points in X. In this way we generalize\nthe invariant imbedding operator m(u,x) of Sec. 3.7 to a wid-\ner geometric setting, i.e., to one in which X and y are not\nsurfaces, but possibly points or subsets of X. We retain the\nnotation \"m(x,y)\" without fear of confusion with the simpler\nconcept in the present discussion. Recall that x and y are\nnow points or subsets of X rather than depth parameters for\nsurfaces. We shall denote the set of all m(x,y) with X and\ny in X, by \"T2 (X) . \"\nIt follows at once from (101) that the operators\nm(x,y) in 2 (X) form a partial group in the sense explained\nin the discussion around equation (79) in Sec. 3.7. Hence\n2 (X) is a proper generalization of 2 (a,b).\nSeveral directions of further development of (102) are\npossible at this exploratory stage of the analysis. For ex-\nample, using Stage II of the interaction method we can repre-\nsent M(y,z) as an integral operator over E. Alternatively,\nwe could partition m(y,z) into a 2x2 matrix analogously to\nthe partition in (18) for the plane-parallel case, and develop\na theory for m(y,z) analogous in every detail to that be-\ntween (18) and (92) above, but now for the general medium X.\nSince this is representative of a nontrivial extension of the\ninvariant imbedding group 2 (a,b) to more general settings,\nwe shall now explore the initial details of such an extension.","64\nINVARIANT IMBEDDING TECHNIQUES\nVOL. IV\nIn the plane-parallel case we had the terrestrially-\nbased coordinate system as a frame of directional reference\nfor the partition of (x,z) as shown in (18). In the present\ncase there is no preferred or pre-existing coordinate frame\nfrom which to launch the construction of the present counter-\nparts to M++ (x,z), m+-(x,z), M- (x, z) , and (x,z).\nTherefore for the first stage of the present extension we\nsimply assign to each point x in X a partition E1 (x), E2 (x)\nof E into two parts. This partition can follow any rule, so\nthat Ej (x) need not be a hemisphere. Once the partition is\nspecified at each X in X, the radiance distribution N(x) is\nrestricted to E1 (x) and E2(x) resulting in N1 (x) and N2 (x),\nrespectively-- in complete analogy to the N+ (y) and N_ (y) of\nthe plane-parallel case. This partitioning of N(x) at each X\nin X into the pair (N1 (x), N2 (x) in turn induces a cleavage\nof m(y,z) into a 2x2 operator matrix such that:\nM11(y,z)\nM12(y,z)\nm(y,z) =\n(103)\nM21(y,z)\nM22(y,z)\nThe details of this partitioning of m(y,z) are very much like\nthose used to establish mo P(y,z) from m(x) above or Mij\nfrom m in (90), except now (95) is replaced by a formula like\n(80):\n2\n(104)\ni=1\nwhere Ci, i If 1,2, is the operator which assigns N (y) to\nN(y), so that the Ci are like C+ and C in (4), (5). Indeed,\nthe partition (103) is obtained in precise analogy to (92)\nfor the case m = n = 2. Hence we may refer the reader to\nequations (80)-(92) for the general outline of the details.\nWith this decomposition (103) of M(y,z), equation\n(102) may be written:\n[\nM11(y,z)\nM12(y,z)\n(N1 (z),N2(z)) =\nM21(y,z)\nM22(y,z)\n(105)\nAs a specific instance of (105), let P(y,z) be a smooth di-\nrected path in X connecting point y to point Z (in that order).\nOnce P(y,z) is specified then any point X along it is located\nby a single parameter - the distance of X from y along the\ncurve, and the tangent to the curve is given the usual sense\nat X. See Fig. 7.8. At each point X of P(y,z), let (x) be\nthe tangent to the curve. Then let E1(x) and E2 (x) of the\ngeneral discussion above be -(5(x)), respectively,\nwhere E+(E(x)), it will be recalled (Sec. 2.4), is the hemi-\nsphere of (1) consisting of all directions E' such that\nE' - E (x) = 0, and E_ (5 (x)) ) consists of all E' such that\nE' E (x) VI 0. With these assignations of E1 (x), E2 (x), the\nformula (105) takes the form:","SEC. 7.4\nINVARIANT IMBEDDING ALGEBRA\n65\nX\nZ\nP (y,z)\nE(x)\ny\nX\nE(x)\n(x)\nt\n(x)\nIN\nFIG. 7.8 Extending the m(x,y) operators to general\ngeometries.\nM++y,z) M+_(y,z)\n(N+(2),N_(2)) = (N+(y),N_(y))\n(106)\n,\nm__(y,z)\nwhere N+(x) and N (x) are now the restrictions of N(x) to\nE+ (E(x)) and in (E(x)), respectively. Thus (106) is formally\nindistinguishable from its plane-parallel - counterpart; fur-\nthermore the algebraic properties of m(y,z) in (106) are\nidentical to those of its algebraic counterpart and (106) re-\nduces to the stratified plane-parallel case when P (y,z) is\nthe straight path from level y to level Z and such that\nP(y,z) is perpendicular to the parallel planes of X(y,z) in\nX(a, b). As we shall see in Sec. 7.11, (106) reduces the type\nof solution procedures used for light fields in a general me-\ndium X to those used in plane-parallel media, with arbitrary\nlighting and optical conditions.\nIn the preceding explorations of the possibilities be-\nyond 2 (a, b) there is a general pattern forming for one such\nfamily of extensions. We conclude these explorations with a\nsummary and review of the incipient pattern for the case of\nan arbitrary subset S of a medium X. The formation of the\nextensions begins with an invocation of the First Stage of\nthe interaction method. This yields the generic equation:\nN = N°M(S)","66\nINVARIANT IMBEDDING TECHNIQUES\nVOL. IV\nwhere S may be all of X or a proper part of X. Furthermore\nN may now be radiance functions for polarized light, and may\ndepend explicitly on scattering with change in wavelength,\netc. Hence X may be more than three-dimensional. Let us as-\nsume X is n-dimensional. (See opening remarks, Sec. 99 of\nRef. [251].) Using the technique of decomposing the identity\noperator, as in (80), (81), (95), or (104), the basic equa-\ntion (107) can be systematically taken apart leaving an op-\nerator which forms a member of a new partial semigroup 2 (S).\nThe ways in which (107) may be so analyzed are manifold. The\nexamples cited above show that the partition of the identity\noperator may be over spatial variables (as in the case of\n(80), (81), and (95)) or over directional variables (as in\nthe case of (104)). The work of Ref. [251] shows how the\npartition of the identity operator may in other contexts be\nover the location space of a discrete optical medium (Sec. 90,\nRef. [251]) with the resultant generation of the local opera-\ntor 40 analogous to mo in (98). In addition, the technique\nof partitioning the identity operator is applicable to the\npolarized radiance context (Sec. 114, Ref. [251]) and also\nthe heterochromatic radiative transfer and even the general\nMarkov-process context of general radiative transfer of equa-\ntion VII, (Sec. 119 of Ref. [251]) With these examples in\nmind let us assume a quite general partitioning of the identi-\nty operator I on the vector space of radiance functions on S,\nthus:\n]C(x)\n[trn\nC(x)]\ndV(x)\n(108)\n,\nwhere now x is a point of the subset S of the n-dimensional\nspace X, and V is the volume measure on X. (The various di-\nmensions of X may arise from the various parameters needed to\ndescribe N--location variables, direction variables, polari-\nzation parameters, wavelength parameters, etc.). The opera-\ntors C(x) are analogous to E(x) in (95). Therefore we write:\n\"N(x)\" for NC(x)\n(109)\nin complete analogy to the earlier special cases of N(x).\nNext we insert I, in the form given by (108), between\nN° and m (S) in (107) to obtain:\nN = = (N°C(x) [trn C(x)]M(S) dV(x) .\n(110)\nBy (109) we have:\nN°(x) = N°C(x)\n(111)\nand in analogy to (98) we write:\n\"m°(s;x,y)\"\n\"m°(x,y)\"\nfor\nor\n1x11\n[trn\nC(x)\nm(s)c(y)\ndV(x)\n(112)\n.","SEC. 7.4\nINVARIANT IMBEDDING ALGEBRA\n67\nTherefore, upon operating on each side of (110) with c(y) we\nhave:\nN(y)\n(113)\nAssuming the integral operators m°(x,y) to be one to one for\nevery x and y in S we define:\n\"M(y,z)\" for\n(114)\nin analogy to (101). The collection T2(S) of all operators\nM(y,z), with y, Z in S is seen to be a partial group as in\nthe earlier instances; so that for every y, Z in S :\n(115)\nwhich holds for an arbitrary radiance field N in S.\nFurther partitions of the identity can now be made on\nthe vector spaces of radiance distributions with elements\nN(x), x fixed in S. For example, if X is simply the spatial\nvariable then further partitioning of the direction space or\nwavelength space can be made if desired. Thus in general,\nlet Da(x) and DB(x) be operators such that:\n[trn Da(x)] du(a)\n(116)\nis the partition of the identity operator on the vector space\nof functions N(x) at x in X. The space A is the space (either\ndiscrete or continuous) which is being partitioned and is\nthe measure, and could be direction space or wavelength space,\netc. Let us write:\n\"Na(x)\" for N(x)Da(x)\n(117)\nand\n\"MaB(S;s,y)\" or \"MaB(x,y)\" for [trn Pa(x)]M(x,y)Dg(y)\n(118)\nThe functions Na(x) with a in A; and m aB (x,y) are generali-\nzations of Ni(x), and m ij (x,y) in (105), where now the space\nA is quite arbitrary. See also the discrete example (85) of\n[118]. Then to see how far these generalizations can go, we\nreturn to (115) and observe that we may write:\nN(z) = N(y)I(y)Mly,z)\n=\n.","68\nINVARIANT IMBEDDING TECHNIQUES\nVOL. IV\nOperating on each side with DB(y):\n=\n.\nHence:\n(119)\n=\nfor every a,B in A. Operating on each side of (119) with\n[trn DB(2)] and integrating over A :\nN(z) = (A\"B(t) du (B) =\n(y,z) [trn DB(2)] du (a) du(B)\n.\nSince N(y) is arbitrary, we have from this and (115) :\nm(y,z) = [trn DB(2)] du(a) du(B)\n(120)\nwhich is one of the possible generalizations of the type to\nwhich (92) belongs. This concludes the summary and overview\nof a possible general method of constructing partial groups\n2 (S) of operators on the subset S of the optical medium X.\nThe problem of generalizing 2 (a,b) to 2 (S) will be consid-\nered once again in Sec. 7.11.\n7.5 Analytic Properties of the Invariant Imbedding Operators\nWe now continue the work, begun in Sec. 7.1, of deriv-\ning the differential equations governing the main invariant\nimbedding operators. In particular we shall derive the var-\nious functional differential equations governing the opera- -\ntors m(x,y), , M(x,y,z), and M(v,x;u,w). Since these opera-\ntors are in turn 2x2 matrices of operators, each such differ-\nential equation is a potential plethora of differential equa- -\ntions for its component operators. Such a superabundance of\noperator differential equations would constitute an embarrass-\nment of riches for the theory were it not for the insight\ngained into such operators in the preceding section. Indeed,\nour studies there showed that the operators of the form\nm(v,x;u,w) could be studied in terms of those of the form\nM(x,y,z), and the latter in terms of those of the form\nm(x,y). Hence the operators m(x,y) emerge as the undisputed\nvictors in any contest of conceptual simplicity and inherent\npower of representation. In summary, then, it was shown how\nthe members of T2 (a,b) could represent, via simple algebraic\nformulas, all the other invariant imbedding operators of\nT3(a,b) and T4(a,b), plus the operators of G2 (a,b), and even\nthe classical R and T operators. Hence the multitude of","SEC. 7.5\nINVARIANT IMBEDDING ANALYSIS\n69\noperators can be reduced, formally at least, to just those in\n2 (a,b). This power of representation of the members of\nT2 (a,b) will be used again in the present section to derive\nthe requisite differential equations for all invariant im-\nbedding operators from those for m (x,y). From these differ-\nential equations, in turn, the operators may be systematically\nconstructed by various solution procedures using the inherent\noptical properties of the appropriate media.\nThroughout this section, unless otherwise stated, we\nshall work with an arbitrary source-free plane-parallel medi-\num X(a,b), as b, with arbitrary incident radiance distribu-\ntions N_ (a) and N+ (b) over the upper boundary Xa and the\nlower boundary Xb, respectively. As in the case of Sec. 7.4,\nthe present results are readily generalized to wider settings,\nnamely general one-parameter settings and general unparameter-\nized optical media. Also, as in the case of Sec. 7.4, the\nexposition is primarily heuristic, with rigorous developments\nleft for future study.\nDifferential Equations for m(x,y)\nStarting with the basic equation concerning the opera-\ntor M(x,y); namely:\nN(y) N (x) m(x,y)\nintroduced and studied in Ex. 7 of Sec. 3.7, we apply the\ndifferential operator d/dy to each side of this equation,\nwhere the differential operator occurs in (1) of Sec. 7.1.\nThus:\nday = ay\n.\nFor the reader unfamiliar with analytic (i.e., differential,\nintegral, and general limit) operations on operators, we may\nnote here that the rules governing these operations are the\nsame in all essential respects as those for the everyday type\nof function encountered in the domain of elementary calculus.\nHence for the purposes of the present discussion, the reader\nwill require no more advanced techniques than those encounter-\ned in such a domain. Needless to add, however, the physical\ncontent of the ensuing statements are far from trivial and\nare worthy of further analysis and application.\nContinuing now with the derivation, we use (9) of Sec.\n7.1 to reduce the preceding result to:\ndN(Y) = N(y) K(y) = N(x) dm(x,y) =\ndy\ndy\nUsing the basic equation for m(x,y) once again, we obtain:\nN(x) dm(x,Y)\nN(y)K(y) =","INVARIANT IMBEDDING TECHNIQUES\nVOL. IV\n70\nUnder the present lighting conditions, N(x) is arbitrary so\nthat:\nam(x,y) =\n(1)\nay\nwhich is our first main result and which holds for arbitrary\nx,y in X(a, b) This differential equation for m(x,y) harbors\nthe four differential equations for its four components, which\nare operator-valued functions. For future reference, these\nare:\ndM++(x,y)\n= M++(x,y)t(y) + M+_(x,y)p(y)\n(2)\ndy\ndM-+(x,y)\nM-+(x,y)t(y) + m (x,y)p(y)\n(3)\n=\ndy\nam+-(x,y)\n= M++(x,y)p(y) + M+_(x,y)t(y)\n(4)\ndy\nam__x,y)\n= M_+(x,y)p(y) + (x,y)t(y)\n(5)\nay\nObserve how (2),(4) and (3), (5) are autonomous and indeed are\ncopies of (5), (6) in Sec. 7.1. Hence (M++,M+-) ++' pair with\n(N+,N_) as do also (M-+,M--). The initial value of\n(M++,M+ is (I,0), while that of (m 1-4,M--) is (0,1). .\nA companion equation to (1) (its adjoint) is obtained\nwhen the differentiation is performed with respect to x rather\nthan y. To obtain the companion equation observe that:\nam(y,x) m(x,y) + M(y,x) am(x,y)\n.\nHence:\nam(x,y) = -m(x,y) am(y,x) m(x,y)\n(6)\nax\nax\nApplying (1) to the derivative on the right in (6) we have\nax\nwhence:\nam(x,y) -K(x) m(x,y)\n(7)\nax","SEC. 7.5\nINVARIANT IMBEDDING ANALYSIS\n71\nEquations (1) and (7) are reducible to nxn matrix equations\nusing angular discretization techniques such as those to be\ndescribed in Secs 7.7, 7.9 and 7.10. The initial values are\nof course in each case M(x,x) = I, the identity matrix. The\nfour functions taken in the above pairs comprising m, are\ncalled the fundamental solutions of the equation of transfer.\nAs we have already seen in 7.4, by judicious linear combina-\ntions of these solutions, we can obtain all the useful scat-\ntering properties (e.g., R, T, R, J) of an optical medium.\nThe tactic used above to find (6) is a special case of\nthe general procedure for finding the derivative of the in-\nverse A-1 of an operator, knowing the derivative of A. Thus,\nif A is differentiable and depends on X :\n0 = = + A dA-1\n,\ndx\ndx\ndx\nwhence:\ndA-1 = - A 1 da A-1\n(8)\ndx\nThis formula is based on the standing assumption that the in-\nverse of A exists so that the product AA-1 is defined, and\nthat A and A-1 are in some sense differentiable. Equation\n(8) is a general form of the formula:\n(1/y) - dy = -y-1 y-1\ny2\ndx\nencountered in elementary calculus for the derivative of the\nnumerical valued function 1/y in terms of that of y. Now,\nhowever, we generally are not permitted to join together the\ntwo inverses A-1 in (8) since operator multiplication is gen-\nerally not commutative.\nDifferential Equations for M(x,y,z)\nBy means of the representation of M(x,y,z) in terms\nof Mly, z) and M(y,x), as given in (42) of Sec. 7.4, we can\nfind the differential equations governing M(x,y,z). There\nare generally three such equations, one arising from differen-\ntiation of M(x,y,z) with respect to each of the three dis-\ntinct depth variables x,y, Z within [a,b]. Throughout the de-\nrivations, then, x,y, Z will be distinct variables, unless\nspecifically noted otherwise.\nThus, differentiating each side of:\nM(x,y,z) = [m(y,2)C+ + m(y,x)c_]-\n(9)\nwith respect to X and using (8) and (1):","INVARIANT IMBEDDING TECHNIQUES\nVOL. IV\n72\nam(x,y,z) = -M(x,y,z -M(y,x)C_]M(x,y,2)\ndx\n= (x,y,z)m(y,x)K(x)C_m(x,y,z)\nNow, the general relation:\nM(x,v,z) = M(x,y,2)M(y,v)\n(10)\nfollows from the definition of m(y,v) and (44) of Sec. 7.4\nalong with the general relation (84) of Sec. 3.7. Compare\n(10) with (39) of Sec. 7.4. Using (10), we have, in particu-\nlar:\nM(x,y,z)M(y,x) = M(x,x,z)\n,\nwhich allows us to write:\nM(x,y,2) = M(x,x,z) K(x)C_ M(x,y,z)\n(11)\nax\nWe defer discussion of (11) until its two companions have\nbeen derived. Toward this end, differentiating each side of\n(9) with respect to y and this time using (8) along with (7):\nam(x,y,z) = -M(x,y,z)\nay\n=\n=\nHence:\nM(x,y,z) = M(x,y,z) K(y)\n(12)\nay\nFinally, differentiating each side of (9) with respect to Z :\naz\nwhich may be simplified, using (10), to:\nM(x,y,z) =\n(13)\naz","SEC. 7.5\nINVARIANT IMBEDDING ANALYSIS\n73\nNow for a brief discussion of (11), (12), and (13).\nAll three equations show how to construct M(x,y,z) given the\nrelatively simpler operators M(x,z,z), K (z), C+ (in the case\nof (13) or M(x,x, z) K (x) , C_ (in the case of (11)) For\nexample, in the case of (11), part (a) of Fig. 7.9 shows that\nby starting with the basic slab X(y,z) (shaded) and building\nit up to level x as shown, we can compute m(x,y,z) for every\nx, such that < 2. A11 that is needed to start the calcu-\nlation is information on M(x,y,z) for the special case x=y.\nThis information, in view of (44)-(47) of Sec. 3.7, is tanta-\nmount to knowledge of the standard operators R(y,z) and\nT(y,2 z). of course, one must know in addition the local trans-\nmittance and reflectance operators for the required range of\nx above the level y. A similar observation holds for (13)\nwhose geometric significance is depicted in part (c) of Fig.\n7.9. Finally, Eq. (12) strikes the middle road between (11)\nand (13) and shows how M(x,y,z) can be obtained by working\ninward from either boundary of X(x,z) and initially knowing\nM(x,x,z) or M(x,z,z), , as the case may be. The former of\nX\ny\n(a)\nZ\nX\n(b)\ny\nZ\nX\n(c)\ny\nZ\nFIG. 7.9 Three ways in which to generate invariant im-\nbedding operator M(x,y,z).","INVARIANT IMBEDDING TECHNIQUES\nVOL. IV\n74\nthese cases is shown in (b) of Fig. 7.9.\nIt is instructive to unravel some of the information\ncontained in these equations. We begin with (12) which yields\nthe following four differential equations for the complete re-\nflectance and transmittance operators:\nJJ(2,y,x) = T(z,y,x)t(y) + R(z,y,x)p(y)\n(14)\nay\nR(x,y,z) = R(x,y,z)t(y) + T(x,y,2)p(y)\n(15)\nay\nIR(2,y,x) = T(z,y,x)p(y) + R(z,y,x)t(y)\n(16)\nay\nJJ(x,y,2) = R(x,y,z)p(y) + T(x,y,z)t(y)\n(17)\ndy\nObserve how (14), (16) are fundamentally similar to (5), (6)\nof Sec. 7.1, while (15), (17) are likewise autonomous and sim-\nilar. Recall also the discussions of (1), , (7) above. These\nearlier equations are no more fundamental than the present\nequations. Indeed, (14)-(17) - may be used as the basis for\nall two-point boundary value problems by adopting the set of\n(two-point) fundamental solutions defined in (38)-(40) - below.\nNext, since the situations depicted in parts (a) and\n(c) of Fig. 7.9 are basically alike, we shall give only the\ndetails of unravelling Eq. (11), which goes with part (a) of\nthe figure. The result is readily obtained by first noting\nthat:\nT(z,x,x) Q(z,x,x) -T(x)\np(x)\n0\n0\n-M(x,x,z)X(x)C_ = -\nr(x,x,z)\n-p(x)\nT(x)\nin\np(x)\nT(z,x)\n= -\nT(z,x)p(x)\n0\n=\n0\nIn view of this, (11) reduces to\nT(z,x)p(x) T(z,y,x) R(z,y,x)\nM(x,y,z) [:\nR(x, z) p (x)+t(x) R(x,y,z) T(x,y,z)\nax\nwhence:","SEC. 7.5\nINVARIANT IMBEDDING ANALYSIS\n75\naJ(2,y,x)\nR(x,y,z)\n=\n(18)\ndx\n2((2,y,x)\n=\n(19)\ndx\nR(x,y,z) = (R(x, z) ( p x (x)),(R(x,y,z)\n(20)\ndx\nT(x,y,z) = (R(x, z)p(x) + t(x))J(x,y,y)\n(21)\nax\nThe various physical interpretations of these equations\nare instructive and the reader may gain understanding of the\ndynamics of scattering problems by translating each of the\npreceding equations into words or appropriate mental images.\nFor example, (18) describes how steadily flowing upward radi-\nance (imagined incident at level z) changes at level y in\nX(x, z) when material is added to X(x,z) at its upper boundary\nXx. Thus (refer to part (a) of Fig. 7.9) when a thin layer\nis added to X(x,z) at level x, the normally transmitted radi-\nance (represented by T(z,x)) is now locally reflected in the\nnew layer (represented by (x)) and then globally reflected\n(as represented by R(x,y,z)) down to layer y in X(x,z).\nFurther elucidation of the dynamics of scattering-ab-\nsorbing media is forthcoming from the present differential\nequations for M(x,y,z) by observing how the differential\nequations for R and T, as derived in Sec. 7.1, may be derived\nanew in the present setting. As an example, consider Eq. (18)\nof Sec. 7.1. That equation describes, in essence, how re-\nflected radiance at level a in X(a,b) changes when an incre-\nmental layer is added to X(a,b) at level a. In terms of the\npresent equations such a change in R(a,b) is the sum of the\nchanges in R(a,y,b) when a and y are simultaneously varied\nfor the special instance when a = y, i.e., , when the deriva-\ntives of R(a,y,b) with respect to y and a are added together\nfor the case a y. Thus, from (15) and (20):\n- dx =\ndy\n= R(x,y,z)t(y) + T(x,y,z)p(y) + (R(x,z)p(x)+t(x))R(x,y,z)\n(22)\nLetting y approach x, the right side of this equation becomes,\nafter rearrangements:\np(x) (x)R(x,z) + R(x,z)t(x) + R(x,z)p(x)R(x,z) .\nFurthermore, the left side of (22) is related to R(x,z) by\nthe equation:","INVARIANT IMBEDDING TECHNIQUES\nVOL. IV\n76\n+ R(x,x,y,z)\nRR(x,2) =\n(23)\nlim\ndx\nax\ndy\ny+x\nwhich follows from:\nR(x,z) = lim R(x,y,z)\n.\nThese limit equalities devolve on (45) of Sec. 3.7 and the\nusually available continuity of Q(x,y,z) and its derivative.\nCombining these results, (18) of Sec. 7.1 is obtained from\n(22) but now as seen in the light of a superposition of\nchanges of the complete reflectance function R(x,y,z).\nThe remaining three equations of Sec. 7.1 may also be\nviewed from the new vantage point of the invariant imbedding\nrelation. For instance, Eq. (27) of Sec. 7.1 may be obtained,\nin essence, from (14) and (18) via the observation that:\n]\n+ JJ(2,y,x)\ndT(z,x)\n(24)\nlim\n=\ndx\ndy\ndx\ny+x\nwhich follows from:\nT(z,x) = lim T(z,y,x)\n.\ny+x\nThese limit equalities devolve on (44) of Sec. 3.7 and the\nusually available continuity of T(z,y,x) and its derivatives.\nOn the other hand, Eq. (28) of Sec. 7.1 is obtained directly\nfrom (19) after passing to the limit y+z and suitable rear-\nrangement of coordinate variables. Finally, (29) of Sec. 7.1\nfollows directly from (21) in a similar way. The reason for\nthe direct derivations in the latter two cases stems from the\nobservation that:\nR(z,x) = lim R(z,y,x)\nand\nT(x,z) = lim T(x,y,z)\ny+z\nand that the derivatives in (19) and (21) are with respect to\nX.\nDifferential Equations for M(v,x;u,w)\nOur starting point for the present derivations may be\neither (50), (51) or (52) of Sec. 7.4. We choose the repre-\nsentation (51) of M(v,x;u,w) so as to build directly on the\nresults (11)-(13) just obtained and to gain some practice in\nthe semigroup properties of the M-operators. In the present\nnotation, (51) becomes:","SEC. 7.5\nINVARIANT IMBEDDING ANALYSIS\n77\nM(v,x;u,w) = m(v,y,u)m-1(x,y,W)\n(25)\nWe generally expect four distinct differential equations to\ngovern each member of 4 (a,b). Thus, assuming u,v,w,x,y,2 to\nbe pairwise distinct variables, we have first of all:\n.\nav\nBy (11)\nam(v,y,u)=-m(v,v,u)((v)c_m(v,y,u)\n.\nav\nHence:\nam(v,x;u,w) = (v,v,u)K(v)C_m(v,x;u,w)\n(26)\nav\nThis is the first of the requisite differential equations.\nNext, from (25):\nBut from (8) and (11):\nam-1(x,y,w) = -M-1 (x,y,w)[-m(x,x,w)k(x)cm(x,y,w)]m-1 (x,y,w)\n.\ndx\nHence the second requisite differential equation is:\nam(v,x;u,w) = M(V,x;u,w)m(x,x,w)K(x)C\n(27)\ndx\nThis equation may be simplified by recalling (76) of Sec. 3.7\nwhich states that:\nm(x,x,w)m(x,x;w,x)\nand also the semigroup property (84) of Sec. 3.7:\nM(v,x;u,w) M(x,x;w,x) = M(v,x;u,x)\n= M(v,x,u)\nso that (27) becomes:\nam(v,x;u,w) = M(v,x,u)K(x)C_ = am(v,x,u)\nC_\n(28)\n.\ndx","78\nINVARIANT IMBEDDING TECHNIQUES\nVOL. IV\nof the two preceding differential equations, (27) is the nat-\nural form of the requisite differential equation (the opera-\ntor sought occurs explicitly on both sides of the equation).\nFrom a conceptual and computational point of view, (28) shows\nthat, as far as dependence on the variable X is concerned,\nm 1(v,x;u,w) behaves essentially like the members of 2 (a,b)\nor T3(a,b) (cf. (1) and (12)).\nNext, from (25):\ndu\ndu\nBy (13):\n=\ndu\nso that:\nam(v,x;u,w) = -m(v,u,u)k(u)C+m(v,x;u,w)\n(29)\ndu\nFinally, from (25) once again:\ndw\nFrom (8) and (13):\nam-1 (x,y,w) = -m-1 (x,y,w) (-m(x,w,w)k(w)c.m(x,y,w))m-1(x,y,w)\naw\nIt then follows that:\na((M(v,x;u,w) = M(v,x;u,w)?\n(30)\ndw\nThis equation, as (27), can be simplified slightly if we use\nthe fact that:\nM(v,x;u,w)m(x,w,W) = m(v,w;u,w)\n= M(v,w,u)\nHence (30) becomes:\nam(v,x;u,w) = (M(v,w,u) X (w)C.\ndw\n= am(v,w,w) c+\n(31)\naw","SEC. 7.5\nINVARIANT IMBEDDING ANALYSIS\n79\nThis equation and (28) show that, as far as the response var-\niables x, W are concerned, M(v,x;u,w) behaves essentially\nlike the members of 2 (a,b) or rs(a,b) (see (1) and (12)).\nThese observations could have been obtained directly using\n(52) of Sec. 7.4; however, the plausibility of (28) and (31)\nhas now been reinforced by taking the preceding route.\nDifferential Equations for M(x,y) and Y(s,y)\nIt is interesting to derive the differential equations\nfor the simplest of operators M(x,y) and the most complex of\noperators 4(s,y) encountered so far in our studies. Simpli-\ncity and complexity are measured here in terms of the osten-\nsible algebraic structure of the components of M(x,y) and\n4(s,y) As far as the simplicity and complexity of their dif-\nferential equations are concerned, matters are reversed, as\nwe shall now see. Thus for Y(s,y) we use (56) of Sec. 7.4\nand find that:\n24(s,y) =\nay\n= + (s(s)\n(32)\n,\n*\nwhence:\n34(s,Y) = v(s,y)k(y)\n(33)\ns y y\ndy\nThis result shows that the dependence of (s,y) on y is essen-\ntially that of M(x,y) on y. The integration of (33) starts\nfrom the initial given operator ((s,s). The derivation of\nthe differential equation showing how (s,y) varies with S is\nsomewhat more complex and left to the reader. The differen-\ntial equations for the Y-operators will be considered again in\nSec. 7.12 wherein they will be represented in terms of com-\nplete reflectance and transmittance operators.\nTurning now to the derivation of the differential equa-\ntion for M(x,y), we use as a base the representation given by\n(10) of Sec. 7.4. From this we see that it is necessary to\nfind:\na\naz\n= -\nin which we have used (8). Hence, with the aid of (7) and\n(11):\nFor all y we add I_8(s-y) where I_ = (o-fi) by (56) of Sec.\n7.4.","80\nINVARIANT IMBEDDING TECHNIQUES\nVOL. IV\naM(x,z) x\naz\nx\n[c_M(x,z) - c+]\n.\nHence\n= - C+M(x,z)]. (34)\naz\nReplacing M(z,x) in (34) by either of its representations,\n(11) or (12) of Sec. 7.4, the desired differential equation\nis obtained. The details are left to the interested reader.\nAnalysis of the Differential Equation for R(y,b)\nThe differential equation for R(a,b), as given in (18)\nof Sec. 7.1, was shown in the discussion of that section to\nbe of central importance in evaluating the reflectance and\ntransmittance operators associated with a plane-parallel med-\nium X(a,b). In view of this importance, it is desirable to\ngain as much insight as possible into the structure of the\ndifferential equation governing R(a,b). We now analyze the\nequation for R(a,b), in two different manners, into a rela-\ntively simple pair of linear operator equations using the in-\nvariant imbedding relation. The result will perhaps shed\nsome light on the methods of determining radiance fields with-\nin natural optical media.\nWe begin with the semigroup relation (53) of Sec. 3.7:\nR(a,z,b) = T(a,y,b)@ly,z,b)\nBy setting y = Z in this relation, R(y,z,b) becomes R(y,b),\nso that:\nR(y,b) =J-1(a,y,b)R(a,y,b)\n(35)\nThis is the key representation for the reflectance operator\nR(y,b) in X(a,b) using the complete reflectance and transmit-\ntance operators R(a,y,b) and T (a,y,b). The inverse of the\noperator T (a,y, b) usually exists in most natural media, and\nso we shall proceed on the assumption of its availability, in\norder to see where it leads. Now, it follows from (15) and\n(17) that:","SEC. 7.5\nINVARIANT IMBEDDING ANALYSIS\n81\na(R(a,y,b) = Q(a,y,b)t(y) + J(a,y,b)p(y)\n(36)\n-\ndy\nIT(a,y,b) = R(a,y,b)p(y) + J(a,y,b)t(y) o\n(37)\ndy\nTherefore, on differentiating each side of (35) with respect\nto y:\nJR(y,b),\ndy\n= JJ-1(a,y,b) Q(a,y,b) + J-1(a,y,b)\nany\ndy\n= -J-1 (a,y,b) J(a,y,b)t(y)]J-1(a,y,b)R(a,y,b)\n+ J-1 (a,y,b) [-R(a,y,b)t(y) J(a,y,b)p(y)]\nUsing (35) again, this may be simplified to\nJR(y,b) = -R(y,b)p(y)R(y,b) - T(y)R(y,b)\ndy\nR(y,b)t(y)- p(y)\n.\nOn rearranging the preceding equation, we obtain (18)\nof Sec. 7.1. Equations (35), (36), (37) therefore constitute\nthe required analysis of (18) of Sec. 7.1. We may summarize\nthis finding alternatively as follows: The system (36), (37)\nof Zinear differential equations for the complete Rand J\noperators together with (35) uniquely determines R(y,b) in\nX(a,b). The system (36), (37) may be represented succinctly\nby:\nda(y) = a(y) ((y)\n(38)\ndy\nwhere we have written:\n\"a(y)\" for (Q(a,y,b),J(a,y,b))\nTherefore the construction of R(a,y,b), a 0, to\nappear in J it is necessary that there exist components of\nN(y) such that they can have scattering orders of at most fi-\nnite order ri. No component of N(y) has this property, so\nthat the Ji do not occur in J. Hence the Jordan canonical\nform of K must then be such that S = 0, i.e., g consists\nonly of Jo. The resultant form of (25) is then quite simple:\nN(y) = N(0) exp {P Jop-1y}\n= N(0)P exp {Joy}p\nIt is easy to see that:\n1y\ne\n12Y\nexp 'Joy) =\ne\n12my\ne\nWe define the characteristic radiance vector by writing:\n\"N(y)\" for N(y)P\nThen (25) can be written:\n11y\ne\nl2y\n0\n(27)\n0\nA 2my\ne\nEquation (27) is the requisite equation for the characteristic\nradiance vector N(y). Observe that if N (y) is the jth com-\nponent of N(y), then we have:\nAjy\nNj(y) = e\n(28)\nHence each component (y) of the characteristic radiance vec-\ntor has a specific rate of growth (if Aj > 0) or decay (if\n1j < 0). In infinitely deep media such as X(0,00), wherein\nthere are no internal sources and the only incident radiance\nis at the upper boundary, the components Nj (0) associated with\nthe positive valued eigenvalues are set to zero. The eigen-\nvalues and components can be renumbered so that Nj (0) = 0 for\nV 2m. To see the effect of this on the physical radiance\nvectors N(y), let the elements of P be of the form aij, and\nthose of p-1 be of the form bij, then from (27):","126\nINVARIANT IMBEDDING TECHNIQUES\nVOL. IV\nN(y)\n(N1(0)e\n=\nN(y)P\nHence:\nA1y\nN(y) If (N1(0)e\nTherefore:\nwhich holds for j = 1,...,2\nWe observe from the definition\nof N(0) that:\n2m\nHence:\n2m\n(29)\nj=1,...,2m\nThis is the desired characteristic representation of N(y).\nEach 1i is non positive, i.e., 0 for i = 1, ...,M. Ob -\nserve that each of the 2m quantities Nj(y) is completely de-\nterminable, knowing the 2m quantities NK(0), the entries aki\nand bij of the matrices P and p-1, and of course the m eigen-\nvalues 1i. By retracing the steps leading to (29) and assum-\ning X(0,00) to be replaced by a finitely deep homogeneous med-\nium X(0,d), d< 00, we see that (29) changes only slightly:\nthe upper limit of the i-sum becomes 2m and the non negative\neigenvalues 1i, m+1si V 2m can enter the representation.\nEquation (29) or its counterpart for X(0,d) is representative\nof the general form of Chandrasekhar's equations in his clas-\nsical work [43]. The salient difference between them rests\nin the manner of representing N and o over E, thereby fixing\nthe associated values of aki, bij and 1j. Chandrasekhar uses\nGauss' method of representing the N and o functions by Legendre\npolynomials, whereas the present method appeals directly to the\nobservable partition of the radiance function as given in (1),\n(2) of Sec. 7.7 and (23), (24). In this way we have arrived\nat the first goal of the present discussion, namely, the illus-\ntration of the place of Chandrasekhar's mode of solution of\nthe equation of transfer in the general scheme of radiative\ntransfer theory, as seen from the invariant imbedding point of\nview.","SEC. 7.10\nMETHOD OF GROUPS\n127\nAsymptotic Property of N(y)\nThe final topic for discussion in this section is the\nmatter of the asymptotic property of radiance distributions\nin deep homogeneous media. The property states that the shape\nof the radiance distribution N(y,.) approaches a limit as y-00\nin X(0,00), , and that this limit is determined solely by the\nstructure of the volume scattering function o on X(0,00) and\nso, in particular, this limiting form of N(y,.) is independent\nof the radiance distriubtions at the surface of X(0,00). We\nshall discuss this matter in detail in Chapter 13. However,\nthere exists a simple instructive proof of the asymptotic ra-\ndiance property using the general system of equations (29),\ni.e., the characteristic representation of N(y), and while the\nmomentum of the present discussion is still high, we shall\ngive a demonstration of the asymptotic radiance property using\n(29) as a base.\nOur present goal, therefore, is to show that the 2m-com-\nponent vector N(y), whose jth component is given by (29), ap-\nproaches a 2m-component vector N(00) as a limit, that N(00) is\ndetermined only by o, and that N(00) is independent of N(0).\nNow, the first thing to notice is that the m numbers 1i are,\nin real media, all negative, so that N(y) generally goes to\nthe zero vector 0 (i.e., the 2m-component vector with all com-\nponents zero). This, of course is not the vector N(00) we are\nseeking. The decrease in size of N(y) as y-00 is distracting\nas one seeks its asymptotic shape, and this decrease can be\nerased by normalizing N(y) with respect to some factor which\ndecreases to zero with y at the same rate as N(y). The graph-\nical interpretation of this normalization is quite simple:\nthe radiance distribution at each depth y is magnified in size\nso that one of the radiance components, say that representing\nvertically downward radiance, is of unit magnitude. Then all\nother components arrange themselves in size relative to this\nunit component. If N(y), so plotted, approaches a fixed vec-\ntor, as y+00, then we say that the limit N(00) exists.\nIn the present case the 'normalization factor' may con-\nveniently be chosen as eky where k is the smallest of the num-\nbers -1j, i = 1 m. Specifically, we write; ad hoc:\n\"k\" for min {-11,-> -Am }\n.\nThis implies that -ky goes to zero with the least speed of\nall the factors sii In particular e (Xi+k)y goes to 0 as y\ngoes to 8 for every i, except for when 1i = -k. To be specific\nsuppose 1& = -k. Armed with this factor, we multiply each side\nof (29) by e ky and let y+00:\n2m\nlim y-00 N j (y) e ky = (0) b lj\n(30)\nk=1\nfor j = 1,\n2m.\n,","128\nINVARIANT IMBEDDING TECHNIQUES\nVOL. IV\nLet us write \"a \" for the th column of P, and \"bl\" for the\nl\n1th row of p- 1\nThen the system of 2m limits (29) can be\nwritten:\n= (N(0).aq)bl\n(31)\nwhere in turn we have written:\n\"N(oo)\"\n(N1(00),\nN\nfor\n(00)\n)\n2 m\nand:\nky\n\"Nj(a)\"\nfor\nlim\n(y)\nNj\nEquation (31) shows clearly that the directional structure of\nN(00) is simply that of the lth row of p-1 The lth row of\np-1 is determined solely by the matrices p and T which are\nmanufactured from o. Observe that the directional structure\nof N(0) is wiped out by the taking of the dot product of N(0)\nand the 1th column of P. Hence the asymptotic directional\nstructure of N(y) can be so determined solely by computing\np-1, and this is independent of N(0). Looking back on the\ntrail we have travelled, we recall that P is the matrix which\nmaps K into its Jordan canonical form. Thus we can find\nN(00) by purely algebraic operations on K which, as we have\nseen, is the infinitesimal generator of the group 2 (0, oo) of\ninvariant imbedding operators associated with the medium\nX(0,00).\nAsymptotic Properties of Polarized Radiance Fields\nWe conclude the discussion of the characteristic form of\nthe radiance solution by noting that the techniques just used\nfor the unpolarized context can equally well be applied to po-\nlarized radiance distributions. This means, in particular,\nthat the theoretical questions of the asymptotic properties of\npolarized radiance fields raised in Sec. 4.6 and still earlier\nin Chapter 1 can be fully resolved using the preceding tech-\nnique. Equation (31), as it stands, has the gestalt of the\ncorresponding equation for polarized radiance, differing from\nthe polarized version only in the dimensions of the vectors\nand matrices involved. This difference is precisely deter-\nminable: all vectors in the unpolarized context go over into\nthe polarized context with a four-fold increase in components,\nand all matrices go over with a corresponding four-fold in-\ncrease in their linear dimensions. However, beyond these\nquantitative differences, the two theories of polarized and\nunpolarized radiance distributions are algebraically alike.\n(See, e.g., Section 114 of Ref. [251].) Some experimental\nwork on the asymptotic polarized light field has been done by\nHerman and Lenoble [107]. Otherwise, there exists at present\nvery little experimental study of the asymptotic polarized\nlight field.","SEC. 7.11\nGENERAL METHOD OF GROUPS\n129\n7.11 Method of Groups for General Optical Media\nThe various methods of solution of the equation of trans-\nfer, such as the method of modules (Sec. 7.8), the method of\nsemigroups (Sec. 7.9), and the method of groups in the preced-\ning section hold within them a common core which, if extracted,\ncan guide the construction of a method of solution of the ra-\ndiance field in arbitrary optical media. This section is de-\nvoted to the isolation of the common conceptual kernel of\nthose methods and to a brief exposition of the general method\nof solution it suggests.\nAnalysis of the Group Method: Initial Data\nWe begin with a recapitulation of the ground-forms for\nthe two basic methods. The semigroup method rests on the\nsemigroup relation (12) of Sec. 7.8 for the complete transmit-\ntance operator T (r) (i.e., T(x,z,00) where r = z-x|) The\nfundamental equations for the light field in this method are\ngiven by the system (10) of Sec. 7.9 or the system (14) of\nSec. 7.8, depending on whether the continuous variable y or\nthe discrete variable y = jd is used. The method of groups\nrests on (1) of Sec. 7.10 for the invariant imbedding operator\n(r) (i.e., M(x,z) where r = z-x), which holds for all real\nnumbers r and S. The equations (9) or (25) of Sec. 7.10 may\nbe used to find the light field at any depth y in X(0,00).\nWhat are the basic data needed in the computational ap-\nplications of each method? The data needed are: (a) a,o\nthroughout X(0,00) and either (b) N(0), the complete radiance\ndistribution at level 0; or (c) (0) and Roo, i.e., the down-\nward incident radiance (0) at level 0, and the reflectance\noperator Roo for X(0,00). Thus, the inherent optical properties\na and o are indispensable in finding N(y) using either method.\nHowever, we clearly have an option on the initial radiance\ndata. Alternative (b) requires the full radiance distribution\nat level 0. Alternative (c) requires only the downward inci-\ndent radiance on level 0, but along with the reflectance oper-\nator for X(0,00) Alternative (b) is possible when preliminary\nempirical estimates of N(0) are available. As a result of\nhaving both N.(0) and N+ (0) available, we then obviate the\nneed of Roo. However, in theoretical studies only N_ (0) is gen-\nerally available for use. The remaining part of N(0), namely,\nN+ (0), is simply some more unknown data to be sought along\nwith N(y), y>0. Clearly, for deep homogeneous media X(0,00),\nhaving to find (0) is tantamount to finding Roo for X(0,00).\nWe thus come to the first conclusion in our analysis of the\ngroup and semigroup methods: Each method requires as given\ndata either alternatives (a) and (b); or (a) and (c). The\nfirst alternative is the empirical alternative; the second,\nthe theoretical alternative. In discussion the extension of\nthe method of groups to more general media, the theoretical al-\nternative demands more attention than the empirical alterna-\ntive. Hence when the extension is made below, it will be made\nwith an eye to the adoption of the theoretical alternative,\nthereby resulting in a more powerful method of solution in the\nsense that it does not depend on basically superfluous prelim-\ninary empirical measurements.","INVARIANT IMBEDDING TECHNIQUES\nVOL. IV\n130\nAnalysis of the Group Method:\nLimitations of the Equation of Transfer\nHaving settled the matter of what kind of initial data\nshall be required in the general method, we seek the theoreti-\ncal equations which may be the basis of the new method. Now,\nboth equations (12) of Sec. 7.8 and (1) of Sec. 7.10 use one-\ndimensional parameters, namely the depths r and S in X(0,00).\nPhysically, the interpretation of these equations is that,\ngiven the operators J (r) and (s) (or m (r) and n(s)) for\ntwo contiguous segments of a vertical path in X(0,00), one\nknows how to find the operator T (r+s) (or m(r+s)) associated\nwith the union of the two path segments. What is the analo-\ngous case in general media? To fix ideas, suppose we still\nhave X(0,00 but that X(0,00) is no longer homogeneous, nor\neven stratified: X(0,00) is a natural chaos of variations in\na,o and initial incident radiances over the upper boundary X\nIt is now clear that the light field can vary markedly over\nplanes Xy at depth y in X(0,y) . Hence it is no longer suffi-\ncient to simply give the depth in X(0,00) in the description\nof the light field in X(0,00); a full specification of the\npoint in question must be given.\nIn this more general setting what then is the counter-\npart to the simple vertical path used in the stratified plane-\nparallel case? Figure 7.16 depicts a possible candidate in\nthe form of a general path P with initial point X O at the\nboundary X of X(0, and terminal point x in X(0,00). Here\n\"xo\" and \"x\" denote ordered triples of real numbers giving\nthe coordinates of XO and X with respect to some terrestrial\nframe of reference. It should be noted that XO need not be\non X for what follows. We have simply placed it there to\nfix ideas. In the homogeneous stratified case, P can be\nvertical and, given N(0) at X0, we can find N(y) at any dis-\ntance y along P using (9) of Sec. 7.10 (in the method of\ngroups) or using (10) of Sec. 7.9 (using the method of semi-\ngroups). Alternatively, we can integrate directly along P\nusing (26) of Sec. 7.10 or (38) of Sec. 7.5 to find N(y).\nThis then suggests that we merely need to specify P as in Fig.\n7.16 and, with the initial radiance N(0) given at X O integrate\nmethodically along P. But what of the curvilinear structure\nof P? This appears to present no obstacles, at least in\nprinciple. For, let \"t\" denote the unit vector to P at x,\nand let E be given a fixed partition as in Fig. 7.15 (see\nalso (1), (2) of Sec. 7.7). Thus no matter where X is in\nX(0,00), E has the given fixed partition. Then for E in E the\nequation of transfer at x may be written:\ndN(x,5) x(x)N(x,E) + N*(x,E)\n(1)\ndr\nwhere r is distance measured along P at X in the direction of\nthe tangent t. If the partition of (1) is now introduced, an\napproximating system to (1) can be formed using the techniques\nexplained, e.g., in Sec. 7.7 or in (25) of Sec. 7.10. As a\nresult, at each point X of P a system of ordinary differen-\ntial equations just like (26) of Sec. 7.10 describes the light","SEC. 7.11\nGENERAL METHOD OF GROUPS\n131\nk\n=\n+\nXO\nX0\nE\ny\nx\nor\nFIG. 7.16 Analysis of the group method; limitations of\nthe equation of transfer.\nfield at X. . Thus, if we write:\n\"N;(r)\" for N(x,5j)\n\"N*()\" for N*(x,5i)\n\"a(r)\" for a(x)\nwhere is a fixed representative in the partition of (1) and\nX is at a distance r from X, then (1) becomes:\n-a(r)N;(r) +\nN*i(r),\n(2)\np where p = m+n, and we no longer explicitly dis-\ni = 1\n,\n.\ntinguish between the members Ai, Bj of the partition. It","INVARIANT IMBEDDING TECHNIQUES\nVOL. IV\n132\nappears that by knowing all p components Ni(r) at x, we can\ncompute dNi (r) dr for each i from the right side of (2), and\nthen use this derivative value to estimate each of the p val-\nues Ni(r+Ar) for some reasonable incremental distance Ar along\nthe path in the direction t. In this way we can perhaps com-\nputationally inch our way along P and find N(x, 5i), at least\nin principle, at any point X in X(0,00) and for any of the p\ndirections Ei!\nEncouraged by the seemingly successful generalization\nof the homogeneous stratified case to the nonhomogeneous case\nas outlined above, we go on to see whether the preceding com-\nputational scheme can be phrased succinctly in group-theoretic\nterms. Granted the system (2) can be integrated along a given\npath P starting with the initial radiance distribution N(xo),\nwe can then find N(x1) at X1, a point a distance r along P\nfrom xo. Then for the same reasons we may go on to find N(x2)\nat point X2 of P. Suppose we summarize the construction ac-\ntivity over the segment between XO and X1 by means of an oper-\nator n (xo,x1), and similarly let n(x1,X2) map N(x1) into\nN(x2) Then we could say,\nn(x0,x2) = n(xo,x1) n(x1,X2)\n(3)\nin analogy to the group relation which holds for the operator\nm(x,y) of (0,00). We thus appear to have arrived at the\nrequisite group-theoretic relations in the form of (3) for the\ngeneral case.\nBefore going any further and before we develop specific\nnumerical schemes on (3) as a base, it would be well to test\nthe validity of that scheme on some easily visualized case\nfor which we know the answers. Toward this end, we suppose\nX(0,00) is homogeneous once again. Now, however, we assume the\nincident radiance distributions on the upper boundary X of\nX(0,00) to vary with location on X0. To fix ideas, let N. (0)\nbe vertical collimated radiance and let it undulate sinusoidal-\nly in magnitude along the direction from left to right with\nperiod ro, as in Fig. 7.17, and be constant along directions\nnormal to the Figure. Thus the light field in X(0,00) is quite\nclearly not stratified, even though the inherent optical prop-\nerties of X(0,00) are about as innocuous as can be without be-\ning trivial. Since the argumentation leading to (3) was for\nan arbitrary path P in X(0,00), let us now choose P to be\na\nhorizontal infinitely long path going from left to right just\nbelow X0, as in Fig. 7.17. With this arrangement fixed we\nnow turn to the system (2) and observe that the associated\nmatrix operator K is, by the homogeneity of X(0,00), independ-\nent of distance r along P. It follows that, as far as the\nintrinsic structure of (2) is concerned, we have reverted to\nthe full group-theoretic context of Sec. 7.10. In particular,\nthe asymptotic radiance theorem states that N(r) should\nhave a limit as r+00, i.e., there should be a fixed radiance\ndistribution toward which the p-component vector N(r) goes.\nNow, under the conditions just defined this conclusion is pat-\nently false! Quite obviously the radiance distributions along\nP vary sinusoidally in dependence with distance r along P.\nObserve next that while we do not know exactly what N(r) is at\nr from xo, we do know that N(r) = N(r+ro), i.e., that the","SEC. 7.11\nGENERAL METHOD OF GROUPS\n133\nk\nro\nX\nO\nk\nA\nXo\nX\ny\nr\nFIG. 7.17 The equation of transfer cannot directly\nrelate radiances along parallel paths.\nlight field varies periodically with the same period ro as the\nincident radiance field on X We have thus come to a contra-\ndiction, and the next task is to understand just where the\nreasoning leading to the general integration scheme along P\nwas fallacious.\nTo detect the fallacy at hand, we quickly can dismiss\nequation (2) itself as the epicenter of difficulty; similarly,\nthe asymptotic radiance theorem based on the characteristic\nrepresentation (28) of Sec. 7.10 cannot be the trouble center.\nWe therefore descend on the remaining possibility; namely, the\nequation of transfer itself, and this, understandably, is done\nwith a measure of trepidation. The trouble appears to stem\nfrom the use of the equation of transfer in the setting de- -\npicted in Fig. 7.16. We therefore review with care the mean.\ning of the terms in (1). The only strange aspect of (1) is\nthe derivative term. But this has been correctly translated","INVARIANT IMBEDDING TECHNIQUES\nVOL. IV\n134\nfrom the general directional derivative term E.VN(x,E) that\ncustomarily appears in the equation of transfer. Indeed, let\nS be distance measured along direction E, as in Fig. 7.16.\nThen:\nE.VN(x,5) dN(x,5) = E.t dN(x,5)\ndr\nby virtue of the simple geometric fact that:\nE.tAs = Ar\nwhere As and Ar are two arbitrary distances along directions\nE and t but related as shown in Fig. 7.16. Therefore, what-\never the source of difficulty, it is not one born of simple\nalgebraic errors. It remains, therefore, to consider concep-\ntual errors of application of the equation of transfer.\nIn going from (2) to the integration scheme over path\nsome error of interpretation of (1) was committed. The error\nmust center on the intended meaning, i.e., the intended phys -\nical interpretation of the derivative term of (1). It was\nhoped that knowledge of the value of E.tdN(x, E) /dr would per-\nmit an estimate of N(y,5) at the neighboring point y = x+tAr\non P. When phrased in this way (rather than in the abbrevi-\nated notation of (2)) the difficulty starts to resolve: the\nderivative term E.VN(x, E) of the equation of transfer is in-\ntended to give the rate of change of N(x,5) at x in the direc-\ntion E and in no other direction. Hence an attempt at extrap-\nolating the value N(x,E) at x in some direction E', using the\nequation of transfer, is permissible only when E' = E. There-\nfore the integration scheme of (2) along P holds only when P\nis a straight line with direction E.\nThe preceding italicized observation stands out in bold\nrelief when, now forewarned, we consider the following quite\nsimple test situation. Let X(0,00) be a purely absorbing me-\ndium. Thus o = 0 throughout X(0,00) Let X(0, 00) be irradiated\nby vertical collimated light as in Fig. 7.16, but now the spa-\ntial dependence over the upper boundary need not even be peri-\nodic, but simply some arbitrary given form. The equation of\ntransfer can readily predict N(x, -k), the downward radiance at\nany point X along any vertical path P. However, given\nN(x, -k), we cannot use the equation to predict N(y, -k) where y\nis a point the same depth from X and just next to X. From\nthis we infer that in any optical medium the equation of trans-\nfer is generally powerless to describe or interrelate directly\nthe radiance flow at two neighboring points x and y which are\ndirected along parallel paths containing x and y.\nIn the simple case of a purely absorbing medium, it is\nclear that to know N(x,.) at each point of plane Xy in x(0,00),\nit is necessary to know N(0,.) at each point of plane X for\nthe directions E in E.. It seems reasonable that this is in-\ndeed the case also for media with arbitrary scattering mecha-\nnisms extant within them. It shall turn out that this is so.\nThus, in the counterexample of Fig. 7.17, the initial data at\npoint XO is generally inadequate to predict radiance parallel\nto P at points above and below P. What is needed is initial","SEC. 7.11\nGENERAL METHOD OF GROUPS\n135\ndata over a whole vertical plane A (seen dashed, end on)\nthroughout X(0,00) which then permits a methodical computation-\nal march away from the data plane in the direction of its nor-\nmal t, a march which eventually can in principle sweep through\nall of X(0,00).\nWe have deliberately travelled the route just taken,\ni.e., from (1) to the preceding fallacy, and then to the reso-\nlution of the fallacy just above, principally to uncover the\nresultant insight into the nature of the equation of transfer\nenunciated above. Thus, while it is quite obvious from fol-\nlowing any of the classical derivations of the equation of\ntransfer, just what the equation can do in a given optical\nmedium, it does not seem to have been emphasized what the\nequation of transfer cannot do by way of direct interrelation\nof the light fields at two neighboring points in the medium.\nAnalysis of the Group Method: Summarized\nThese observations on the limitations of the equation of\ntransfer complete our analysis of the semigroup and group meth-\nods by showing the minimum number of necessary steps that must\nbe taken in generalizing the methods to arbitrary optical me-\ndia. In particular, we have learned that the general opera-\ntors M(x,z) of the simple homogeneous case, which worked well\nin predicting N(z) knowing N(x) along a vertical path on a\nstratified plane-parallel X(0,00) must now be replaced by open\nators m(x,z) which relate the radiance distributions over all\nof plane Xx to the radiance distributions over all of plane\nXz. In short, we must develop the generalization of the group\nT2 (0,00) for X(0,00 in the context of a one-parameter represen-\ntation of the space X(0,00), i.e., a representation of X(0,00)\nwhich conceives of X(0,00) as a full three-dimensional body\ncomprised of a one-parameter set of parallel planes. In this\nway we return to the general concept of a one-parameter opti-\ncal medium as given in Example 2 of Sec. 3.9. Furthermore,\nthe discussions following (93) of Sec 7.4 may now be restudied\nwith profit. In the light of the preceding analysis, that dis-\ncussion now takes on a deeper meaning which can be developed\nin concrete terms as follows.\nThe General Method of Groups\nThe geometric setting for the general method of groups\nis a rectangular parallelepiped X(a,b,c) of dimensions a,b,c,\nand which is oriented and defined with the help of Fig. 7.18.\nThe unit vectors i,j,k of the usual right-hand cartesian co-\nordinate frame are shown. The standard hydrologic optics co-\nordinate system measures depth Z as positive, increasing in\nthe direction -k. Analogously, the X and y measurements are\npositive, increasing in the directions -i,-j, respectively.\nThis measuring convention is simply a logical extension of the\nuseful plane-parallel convention of measuring Z positive along\n-k. If desired, the i and j unit vectors may be reversed to\nobtain a right-hand coordinate system of more familiar appear-\nance. Also the k unit vector may be reversed with i and j","INVARIANT IMBEDDING TECHNIQUES\nVOL. IV\n136\nX(a,b,c) or X(o,c)\nk\nX(o)\ny\nb\na\nFIG. 7.18 . The parallelepiped within which an arbitrary\nradiative transfer process can evolve and be studied.\nsuitably adjusted. We shall not adopt this latter reversal,\nas it will necessitate a massive revision of all direction\nconventions developed so far, and will cause difficulties in\ntreating the unified planetary radiative transfer problems in\nwhich the atmosphere above the top plane boundary X(0) of\nX(a,b,c) is allowed to interact with X (a,b,c). . We call\nX(a,b,c) a monobloc: it is the general version of a plane-\nparallel medium. The latter type of medium is the special\ncase, of X (a, b,c) for which a = b = 8. We assume that\nin-\nan\ncident radiance distribution is defined over (0) and that\nthere are no further sources on or within X(a,b,c). We assume\nalso that a and o are specified throughout X(a,b,c). A one-\nparameter representation of the monobloc is fixed by writing:\n(4)\nX(a,b,c) =\nU X(z)\n)","SEC. 7.11\nGENERAL METHOD OF GROUPS\n137\nwhere x(z) is the plane section of X(a,b,c) normal to the\ndirection k and at depth Z below the plane X(0). Hence each\nX(2) is a plane of fixed dimensions a by b. Having fixed the\nparametrization direction of X(a,b,c) as being parallel to k,\nwe can suppress the dimensions a and b of X(a,b,c) and write\nsimply:\n\"X(0,c)\" for X(a,b,c)\n(5)\nMore generally, an arbitrary subslab of X(0,c) between levels\nX and Z is denoted by \"X(x,z)\" in complete analogy to the\nplane-parallel context discussed earlier in this chapter.\nWe note in passing that the choice of the direction of\nparametrization need not be along k. It can, for example, be\nalong i or j, i.e., we could slice up X(a,b,c) by planes nor-\nmal to i (in which case X(a,b,c) is denoted by \"X(0,a)\" in\nanalogy to (5)) or normal to j (so that X(a,b,c) is denoted\nby \"X(0,b)\"). For the general theory developed below we could\neven slice up X(a,b,c) by parallel planes cocked at some out-\nlandish angle, and being normal to an arbitrary direction 50.\nFinally, the parametrization could even be accomplished with\nnon-plane surfaces. However, as we shall presently see, the\napparently special monobloc X(a,b,c), the orthodox-looking\nparametrization (4), and the special lighting conditions are\nof sufficient generality to subsume all cases encountered in\npractice.\nThe basic equation of the group method is the operator\nform of the equation of transfer:\ndN(Y) = N(y) K(y)\n(6)\ndy\nas developed in Sec. 7.1. Under the present lighting condi-\ntions we have, from principle of invariance I of Example 2 in\nSec. 3.9:\nN(y)=N_(y)R(y,c) =\n(7)\nThis was obtained by setting Z = C in principle I and using\nthe fact that (c) = 0. Finally, for convenience, we repeat\n(18) of Sec. 7.1 here (now adapted to X(y,c)):\n- JR(y,c) = p(y) + T(y)R(y,c) + R(y,c)t(y) + R(y,c)p(y)R(y,c).\ndy\n(8)\nIt is clear that the derivation of (8), originally performed\nin a stratified plane-parallel setting, holds also for the\npresent monobloc setting. Indeed, as shown in Equation I' of\nSec. 25 of Ref. [251], the gestalt of (8) persists in arbi-\ntrary optical media in euclidean three space. Equations (6),\n(7) and (8) are the basic equations of the general method of\ngroups, and are used in numerical procedures as follows.\nStage One. Discretize the directional variables of equations\n(6), (7), (8) by partitioning (1) after the manner of Secs. 7.9,","VOL. IV\nINVARIANT IMBEDDING TECHNIQUES\n138\n7.10. The end results are matricial versions of the three\nbasic equations. The matrix version of each term will be\nwritten below in boldface type. Thus the function N(y) be-\ncomes the vector N(y) and the functional components of N(y),\nnamely N+ (y) and N_ (y) become vectors N+ (y), N_ (y) as illus-\ntrated earlier in this chapter.\nStage Two. Solve (8) for all reflectance matrices R(y,c) ,\nOsysc given the initial condition R(c,c) = 0 (the zero ma-\ntrix). Integration proceeds from R(c,c) through R(y,c) to\nR(0,c). Thus, we in effect build up X(0,c) layer by layer\nand compute R(y,c) at each intermediate stage X(y, c) of con-\nstruction.\nStage Three. Solve (6) for all radiance vectors N_ (y), Osysc\ngiven the initial radiance N_ (0). Toward this end, use (6) in\nexpanded form:\ndN_(y)\n= N_(y)t(y) + N.(y)p(y)\ndy\nin which (7) has been substituted in the equation for (y):\ndN_(y)\n(9)\n= N _(y)[[(y) + R(y,c)p(y)]\n.\ndy\nEquation (9) is solved, starting at level y = 0, and is\nused to work down through X(0,c to level C. At each level y,\nOsysc, R(y, c) is used, as indicated, and is taken from the\nresult of Stage Two. At each level y, N+ (y) is obtained from\nthe matricial version of (7). Equation (9) governs N (y); the\nlatter is an m-component vector (cf. (23), (24) of Sec. 7.10)\nEquations (7) and (9) are therefore used to burrow methodically\nfrom one layer in X(0,c) down to the next, gathering up new\nvalues of N+ (y) along the way. Observe how knowledge of (y)\nover all of level y permits the derivatives of the components\nof N. (y) to be computed, from which estimates of the components\nN. (y+Ay) are obtained. Equation (7) then yields N+(y+Ay).\nObservations on the Method of Groups\nA comparison of the preceding three stages of computa-\ntion, especially the last two, shows that we are generalizing\nthe method of semigroups as summarized in the system (13) of\nSec. 7.8. In the present case Roo is replaced by R(y,c and\nthe infinitesimal generator A (as in (12) of Sec. 7.9) now uses\ndepth-variable operators P, T, and R.\nA further examination of the three stages outlined above\nshows that in any radiative transfer problem, of all the glob-\nal properties of an extended medium, only its standard reflec-\ntance is really indispensable along with the complete trans-\nmittance operator J(0,y,c):\nN_(y) = (0)J(0,y,c)","SEC. 7.11\nGENERAL METHOD OF GROUPS\n139\nwhich implies (9) upon differentiation of each side with re-\nspect to y. (Use (17) of Sec. 7.5 and (52), (53) of Sec.\n3.7.) These two operators and their governing laws are exhi-\nbited in general in (52) and (53) of Sec. 3.7. These were\nused in Sec. 7.8 to develop the semigroup method in the homo-\ngeneous plane-parallel setting. The group-theoretic struc-\nture residing just below the surface activity of Stages One,\nTwo, and Three is latent in (14) of Sec. 7.10 and may be sum-\nmarized as follows. Write:\n\"a(x,z)\" for R(x,z,c),J(x,2,c))\n(10)\nwhere R(x,z,c) and T(x,z,c) are the complete reflectance\nand transmittance operators associated with X(0,c) and x,z\nare arbitrary levels in X(0,c) such that XSZ. Observe that\n2(x,z) operates on N-(x) to yield (z) ,N-(2)). For any\ntwo such operator pairs as a(x,y) and a(y,z), write:\nfor (J(x,y,c)R(y,z,c),J(x,y,c)J(y,z,c))\n(11)\nBy (52) and (53) of Sec. 3.7 we see immediately that:\nA(x,y)*A(y,2) = a(x,z)\n(12)\nFurthermore, the binary operation * defined in (11) is asso-\nciative and a(x,x) for every X clearly serves as the identity\nelement in the sense that:\nA(x,x)*A(x,y) = a(x,y)\n(13)\nBy extending the meaning of T(x,y,c) and Q(x,y,c) to the\ncase where X and y are not restricted to the relation x sy,\nthe set {a (x,y) 0 x x , ysc} becomes a partial group. This\nextension can be made by following the suggestions given\naround (44) of Sec. 7.4. The partial group (a(x,y): 0sx,\nysc}, which we denote by \"A2(0,c)\", is clearly isomorphic to\n2 (0,c) introduced in Example 7 of Sec. 3.7.\nIt may be well to also make some observations of a\npractical nature concerning the integration of equations (8)\nand (9). Consider (9) first. The ith component of the ra-\ndiance vector function N_ (z) at level Z is of the form\nN(x,y,z,5i) where i is in the ith partition element Ai of\nE. Here x,y,z now are the three coordinates of a point in\nthe monobloc (see Fig. 7.18). Let us write \"Ni(x,y,2)\" or\n\"Ni(p)\" for this ith component of (2), , where \"p\" stands for\n\"(x,y,z)\". Equation (9) gives the rate of change of Ni (p) at\npoint p from which we may estimate i(p+EAr) where Ar is an\nincrement of path length along the direction Si. Our detailed\nanalysis of the equation of transfer earlier in this section\nshows that this is the only type of extrapolation that the\nequation permits. However, now that Ni (p) is known for every\nm, and for every p over the plane X(z), this limited\ni = 1,\nmode of extrapolation is clearly adequate to propagate the","INVARIANT IMBEDDING TECHNIQUES\nVOL. IV\n140\n4\nk\n(x,y,z)=p\nX(z)\nar\nX(z+az)\np+Ejar\nAZ\nFIG. 7.19 How to propagate the radiance computations\nfrom one parameter surface to the next.\nradiance field from plane X(z) to plane X(z+Az) for suitably\nchosen Az. Fig. 7.19 helps show how the function N_ (z+Az)\nover X(z+Az) is obtained from N_ (z) known over X(z). In par-\nticular we have for each p (= (x,y,z)) in X(z) and i in E.:\n(14)\nwhere:\n(15)\nOnce N (z+Az) has been obtained, we use (7) to find N+(z+Az)\nby means of the formula:\n(16)","SEC. 7.11\nGENERAL METHOD OF GROUPS\n141\nNow that (14) and (15) have been displayed, and the\nmode of propagation of N (z) to N (z+Dz) has been made clear,\nit may be well to observe that there is no royal road to the\nsolution of the radiative transfer problem in a general mono-\nbloc such as X(a,b,c). The sheer number of dimensions of\nX(a,b,c) and E_ must always combine to dampen the enthusiasm\nof the most intrepid computer. At least now that the invar-\niant imbedding techniques have shown the fundamental struc-\nture of the present type of transfer problem (as outlined in\nStages One to Three above), we can rest in the knowledge that\nthe theory has progressed as far as it can go on the phenomeno-\nlogical level, and that what remains is the development of\nmore adequate numerical procedures to use on (8) and (9) for\nthe general monobloc X(a,b,c). of course this is not meant\nto discount the use of other procedures such as those based on\nthe classical techniques of Chapter 6, or on the natural mode\nof solution (Chapter 5), or the canonical mode (Chapter 4), or\ntheir equivalents. As far as Eq. (8) is concerned, one such\nattempt has been made using invariant imbedding techniques in\nRef. [251] wherein the solution of (8) is carried out on a\nmonobloc using the approach of discrete-space theory developed\nin that reference (see, in particular, Chapter X).\nThe Method of Groups and the Inner Structure\nof Natural Light Fields\nWe now round out our discussion of the method of groups\nand also bring to a close some matters raised in Example 7 of\nSec. 3.7 by outlining a proof of the general group-theoretic\nstructure of light fields in natural optical media.\nLet X be an arbitrary connected source-free subset of\neuclidean three-space. Let a,o be given throughout X and let\nX be irradiated arbitrarily on its boundary. A parametriza-\ntion of X is introduced so that:\n(17)\nX =\nX(z)\nU\naszsb\nThis decomposition of X into a family of two-dimensional sur-\nfaces X(z) is illustrated in (a) of Fig. 7.20. In this way X\nbecomes a one-parameter optical medium.\nTo each subslab X(x,z) of X, shaded in (a) of Fig. 7.20,\nwe can assign reflectance and transmittance operators after\nthe manner explained in Examples 2, 4, and 5 of Sec. 3.9 so\nthat the invariant imbedding relation holds for X. Hence\nequations (6), (7), (8) can be suitably extended to the set-\nting in X so that the general counterpart to (12) holds, and a\npartial group (a,b) can be assigned to X. In particular a\ncomputation procedure for (y) can be initiated and sustained\nthat will propagate N (z) across each parameter surface X(z)\nwithin X in a manner completely analogous to that based on\n(14) - (16).\nThe parametrization (17), being quite general, leads to\nan instructive mode of description of the inner structure of\nthe light field. As an interesting special case of (17),","INVARIANT IMBEDDING TECHNIQUES\nVOL. IV\n142\nX(y)\nX(z)\nX(x)\na\nb\nN_(z)\n(a)\nX\nX(r)\nX(s)\nX(x)\np+\n(b)\nq\nX\nFIG. 7.20 There are many ways in which an optical medi-\num can be made over into a one-parameter medium.\nconsider the parametrization of X by spherical shaped surfaces\nX(r) within X of radius r about an internal point p of X. We\ndefine X(r) as the intersection of a sphere of radius r with\nX. See (b) of Fig. 7.20. Suppose the light field is given\non arbitrarily small spherical surface X(r). Then using the\ngeneral one-parameter versions of (12), , the radiance field\ncan be computed at any point q in X, where q lies on X(s) for\nsome radius S. Conversely, knowledge of the light field on\nsome sphere about q as center could lead in principle to the\ndetermination of the light field at p after re-parametrization\nof X about 9. This then is the most general description of\nthe inner structure of natural light fields in an arbitrary\noptical medium X as defined above.","SEC. 7.12\nHOMOGENEITY, ISOTROPY, ETC.\n143\n7.12 Homogeneity, Isotropy and Related Properties of Optical\nMedia\nIn this section we collect together some special know\nledge that has been gathering during the development of this\nand earlier chapters, knowledge concerning the properties of\nhomogeneity, isotropy, polarity, and related concepts associ-\nated with optical media. This accumulation of facts is time-\nly in that it will play an important role in rounding out the\ntheory of internal-source generated light fields in natural\noptical media to be considered in the following section, and\nin Example 10 of Sec. 8.7.\nAs we shall see, the problem of internal sources in op-\ntical media requires for its solution no new concepts beyond\nthose presented in Example 3 of Sec. 3.9. However, this bat-\ntery of concepts gives rise to some relatively complex (but\nhighly instructive) operations with the standard reflectance\nand transmittance operators for optical media. Any insights\ninto the reduction of the number of the participating opera-\ntors and their assemblies in the final formulations will cor-\nrespondingly reduce the amount of labor required to effect\nspecific numerical or theoretical answers to the source prob-\nlems.\nOne of the classical means of simplifying radiative\ntransfer formulations is the use of \"symmetry principles\",\nchief among which are various reciprocity principles governing\nthe R and T functions. It is one of the purposes of the pres-\nent section to define and discuss these symmetry properties,\noutline their extensions to general media, and to indicate\nwhen the extensions are or are not helpful. Perhaps the most\nimportant outcome of this discussion, at least from a practi-\ncal point of view, is the unpleasant fact that most of the\n\"symmetry principles\" of the classical theory no longer hold\nin the general settings of arbitrary optical media. In other\nwords, many of the \"symmetries\" that arose in the classical\nsettings arose because the settings themselves were symmetri-\ncal and generally quite idealized, and not because there sub-\nsisted some inherent invariant character of the symmetry.\nFor example, by graduating from the use of irradiance\nor from scalar irradiance (or radiant density) within infinite\nor semi-infinite homogeneous isotropic media, to the use of\nradiance in such media, at least one important reciprocity\ntheorem falls by the wayside. By making the space inhomogen-\neous, but still isotropic, an important symmetry property van-\nishes into the void. By making the space finite, inhomogen-\neous and irregular in geometric structure essentially all but\none of the classical symmetry properties (reciprocity for ra-\ndiant density) leave the investigator with handfuls of func-\ntional equations whose associated analytic difficulties must\nbe squarely faced without any essential help forthcoming from\nthe lone surviving symmetry principle. In short, the moment\none steps from the nice one-dimensional spaces with their nice\none-dimensional radiometric concepts and enters the represent-\ner of the real world, namely euclidean three-space, and at -\ntempts to describe radiant flux in that setting in terms of","144\nINVARIANT IMBEDDING TECHNIQUES\nVOL. IV\nradiance rather than radiant density, then, except in the most\nsingular cases, the classical symmetries no longer subsist and\nhence are no longer available to facilitate numerical and the-\noretical activity.\nDespite the predominantly negative features of the fol-\nlowing discussion it will still be instructive for the reader\nto have consolidated and clarified some of the more frequently\nused \"symmetries\" and \"uniformities\" in the processes of con-\nstructing models of natural optical media. To this task we\nnow turn.\nThroughout this section let X be an optical medium in\neuclidean three-space E 3 to which is associated a volume atten-\nuation function a, a volume scattering function o and an in-\ndex of refraction n. The domain of n is X, the domain of a is\nX x E, that of o is X x E x E, the values a (x,5 and o(x;5';5)\nare non negative real numbers. It seems best to proceed by\nmaking formal definitions and following them with appropriate\ncomments that illustrate their physical meanings and interre-\nlate them. We begin with the local concepts, i.e., concepts\nassociated with the points of X. Following this the global\nconcepts are introduced (those associated with subsets of X)\nand an attempt will be made to define the global concepts anal -\nogously to the local concepts whenever possible. One of the\nmain problems of the present area of radiative transfer theory\nis to determine whether a valid local concept in a given medi-\num X carries over to the global context. We shall indicate,\nby theorem and example, some instances of this problem as the\ndiscussion proceeds.\nLocal Concepts\nDefinition 1. X is said to be homogeneous if the values n(x),\na(x,) and o(x;5';E) are independent of X for every in E.\nSince a depends generally on X and E, the values (x,5)\nin a homogeneous space, while independent by definition of x,\nmay possibly depend on E. Thus, e.g., while a(x,5) = a(x', 5)\nfor every x, x' in X, this common value may depend on E. Per-\nhaps this is an academic point in the sense that homogeneity\nis rarely found in such a general form in nature. Be that as\nit may, the present definition, being necessarily framed with\nn, a and o as the basic concepts at hand, some decision must\nbe made as to the E-dependence of n, a and o in the homogen-\neous case. The decision adopted above imposes the least re-\nstrictions on the functions while capturing the basic idea\nbehind homogeneity: the uniformity in the spatial domain of\nthe values of n, and o.\nHomogeneity helps simplify the equations of radiative\ntransfer in many ways. The most immediate effect is in the\nstructure of the beam transmittance function. In general, for\na path Pr(xo,5) we have:\nr\nT r (x0,5) = [n2(x)/n2(xo)] exp dr' } ,\n(1)","SEC. 7.12\nHOMOGENEITY, ISOTROPY, ETC.\n145\nn(x') is the index of refraction at x' = xo + r'E, a distance\nr' along Pr(xo,5) from the initial point xo. When X is homo-\ngeneous, the index of refraction function n is independent of\nlocation in X so that in particular Pr(x0,5) is a straight\nline segment with direction E. Further, n(x) = n(xo), along\nwith a(x',5) = a(xo,5). Thus under homogeneity, Tr(xo,5) be-\ncomes:\n-a(xo,5)r\nTr(x),5)==\n(2)\nwhere a (x0,5) is the fixed value of a in X associated with\nthe direction E, and r is the length of Pr(xo,5).\nIt may be possible to have the index of refraction es-\nsentially constant on X without having a or o independent of\nlocation. When this is the case we have restricted inhomoge-\nneity of X. Such inhomogeneity is ideal for the theorist in\nradiative transfer: he has the opportunity of studying the\nmain problems of radiative transfer without the annoying and\ndistracting possibility of curved or broken paths Pr(x,5),\nand of varying radiance values in an otherwise clear medium\n(cf. Sec. 21 of Ref. [251]), or in the beam transmittance\nfunction (cf. (1)). Therefore, throughout this section, when\nwe consider X to be inhomogeneous it will be understood to be\na restricted inhomogeneity of X.\nWe conclude this discussion of homogeneity by rephrasing\nthe definition in terms of the notion of a displacement trans-\nformation on X. A function D on X with values in X is a dis-\nplacement transformation if, and only if, there exists a fixed\npoint y such that:\nD(x)\nHere we are using the fact that the points of E3 (and hence\nthose of X) are ordered triples of numbers, as in analytic\ngeometry, so that there is an algebraic basis for adding them\ntogether. The main part of Definition 1 may now be phrased\nanalytically as follows. \"X is homogeneous\" means:\nwhenever D(x) is in X, then n (D(x)) = n(x)\nand a (D(x),5) = a(x,5) and o (D(x);5';5) o(x;5';5)\n.\nA less restrictive notion than homogeneity but one that\nstill permits all the analytic blessings of homogeneity to\nbe\nenjoyed by the theorist is the notion of separability of X :\nDefinition 2. X is said to be separable if the index of re-\nfraction function is constant and a(x,5) is independent of E\nand if is independent of x for every E', E in E.\nThe reason for the name \"separable\" becomes clear on\nfixing x in X and writing:\n\"p(x;5';5)\" for\n(3)\n.","VOL. IV\nINVARIANT IMBEDDING TECHNIQUES\n146\nThe function p on X x (1) x E so defined is called the phase\nfunction in astrophysical optics (cf. Ref. [43]] and by means\nof it o(x;5';5) may be written:\n(4)\nHence in separable media may be written as the prod-\nuct of two functions: one which is free of x and the other\nwhich depends on X--SO that the spatial dependence is uncou-\npled or separated from the main directional dependence. In\nparticular, in a separable medium the spatial dependence of\no is carried by a, while the directional dependence of o is\ncarried by p. The utility of the separability assumption be-\ncomes clear on examining, e.g., the definitions of the ma-\ntrices r. (a) , t (a), etc., occurring in (9) of Sec. 7.7. If\nthe medium were assumed separable, then r_ (a), t-(a), etc.\nwould be independent of a, while still allowing a measure of\ninhomogeneity of the medium to be present.\nIn separable optical media, the natural measure of dis-\ntance is not geometric distance but optical distance, in the\nfollowing sense: If (x,E) is a path in a separable medium,\nthe\nin-\nthen its optical length is the number\ntegration being taken along the path. (ra(x') dr' This\n,\nnum-\nber is usually designated by \"T (r)\" and enters into the theory\nvia the equation of transfer when a transition from r to Tr\nis made. Thus the equation:\ndN = -aN + N*\ndr\nbecomes:\n1 dN\n(N*/a)\n-N\n+\ndr\na\nSince:\ndt = a\ndr\nwe have:\ndN =\n(5)\ndt\nBeam transmittance in separable media becomes:\n(6)\nIf the dependence of T on r is suppressed and T is made the\nbasic measure of distance, then the medium X is homogeneous,\nin the sense of Definition 1, with respect to the distance\nmeasure T. Furthermore, the volume attenuation function in\nsuch a separable medium with optical distance T is replace-\nable by a unit-valued function at all points of X. In other\nwords, in a separable optical medium one can normalize the\nvolume attenuation function and effectively remove it from","SEC. 7.12\nHOMOGENEITY, ISOTROPY, ETC.\n147\nthe scene, and the volume scattering function is replaceable\nby the phase function.\nDefinition 3. An optical medium X is said to be isotropic at\nif the values a(x,5) are independent of E and the values\no(x;5';5) depend only on the scalar product 5'.5 of is and E.\nX is isotropic if it is isotropic at every point.\nFrom this we see, first of all, that while homogeneity\nof X is constancy of a and o on X, isotropy of X is a con-\nstancy of a on (1) along with a certain special constancy of o\non 11) x E. Specifically, \"X is isotropic at x\" means\nfor every E',E in E, a(x,5') = a(x,E), and\n(7)\nfor every 51,52,53,54 in E, if 51.E2 = E3.E4, then\n= 0(x;53;54)\n(8)\nIsotropy of X can be characterized by means of rotation trans-\nformations of E3 Let T be a rotation of E, at X. Then the\npreceding isotropy conditions may be rendered as:\n((x,T(E)) a(x,5)\nand\n(x,T(E),T(E')) = T(x;5';5)\nfor every E', E in (1) and every rotation T at X.\nDefinition 4. A scattering process (or a) is said to be iso-\ntropic at X in X if o(x;5';5) is independent of E' E in E.\nAn attenuation process (or a) is said to be isotropic at x in\nX if a(x,5) is independent of E in E. A scattering or atten-\nuation process is isotropic if it is isotropic at every X in X.\nThe distinction between the medium X being isotropic and\nthe scattering process on X being isotropic is thus clear.\nThe connections between the two ideas are as follows: If a\nand o are isotropic, then X is isotropic. On the other hand,\nif X is isotropic then a is isotropic, but o need not be iso-\ntropic. This anomaly of symmetry in the isotropy properties\nstems from the fact that o has two spatial variables while a\nhas only one. Hence nailing down isotropy of X fixes that of\na but leaves o a margin of variability, a margin, incidentally,\nwhich has been found most useful in the classical theory.\nObserve that if o is isotropic at x, then:\no(x;5';5) s(x)/4t\n(9)\nwhere S (x) is the value of the volume total scattering func-\ntion of X. Furthermore, if X is separable and o is isotropic,\n(4) and (9) combine to yield:\n(x) If a(x)p(E';5)\nso that:\np(5';5) s(x)/a(x)\n(10)","VOL. IV\nINVARIANT IMBEDDING TECHNIQUES\n148\nFrom this we see that the phase function value p(5';5) is in-\ndependent of E' and E and is a real dimensionless number be-\ntween 0 and 1, a number which we have called the scattering-\nattenuation ratio.\nDefinition 5. A scattering process (or o) is said to be re-\nversible at x in X if the following property of o holds: for\nevery E',E in E, o(x;5';5) = o(x;-5;-5'). An attenuation pro-\ncess (or a) is said to be reversible at x in X if\na(x,5) = a(x,-E) for every E in E. A scattering ( or atten-\nuation) process is reversible if it is reversible at every x\nin X.\nIt is clear that if a medium is isotropic at a point x,\nthen o is reversible at x, for, indeed, since 5'.E = (-5)(-5')\nthe reversibility follows from (8). However, the converse\nneed not be true: reversibility of o at X does not logically\nimply isotropy of X at x, and the reader may devise theoretical\nexamples which show this.\nWe summarize the four main local properties of an opti-\ncal medium in Fig. 7.21 which shows the class of all optical\nmedia in E3 grouped into families which are homogeneous, sep-\narable, isotropic, and reversible. Observe how the class of\nhomogeneous spaces is included in the class of separable\nspaces, and of how the class of reversible spaces (i.e.,\nspaces with reversible o) includes the isotropic spaces as\nspecial cases. The classes partially overlap in the Figure,\nshowing that generally a space may have several, one, or none\nof the four general uniformities.\nGlobal Concepts\nWe shall now show that the local concepts of homogeneity\nand isotropy can be carried over, after suitable modifications,\nto the global description of the scattering properties of ex-\ntended media. To keep the introduction to these ideas simple\nand intuitively meaningful we shall at first consider only\nstratified plane-parallel media, i.e., media whose a and o\nare independent of location on planes parallel to the bounda-\nries. Later in the discussion more general media will be\nbriefly discussed.\nNow the counterpart to o in the global context is the\nreflectance function R(a,b;5';5) and the transmittance function\nT(a,b;E';5) associated with a plane-parallel medium X(a,b).\nThese pairings are intuitive and not to be taken in a formal\nsense. They suggest various analogous properties of the global\nfunctions that one may seek. For example, the analogous global\nproperty to homogeneity is the condition that R(x,z;5';5) and\nT(x,z;5';5) depend only on the difference z-x, where, e.g., E'\nis in E., and E is in E+, as the case may be. It is easy to\nsee that, if and only if X(a,b) is homogeneous or separable,\nthen this property holds for R and T, either directly, or\nafter shifting over to the optical length parameter.\nThe next concept which may be profitably extended to the\nglobal setting is that of isotropy of the medium X at a point X.","SEC. 7.12\nHOMOGENEITY, ISOTROPY, ETC.\n149\n[\nPolar\nnon separable or\nnon reciprocal\nSeparable\nIsotropic\nHomogeneous\nReversible\nSet of all optical media in euclidean space\nFIG. 7.21 The four principal categories for local prop-\nerties of optical media and their general logical interdepen-\ndence.\nInstead of a point we now have a general subslab X(x,z) in\nX (a,b), , and instead of the condition that 5'.E be fixed in\nmagnitude, we require that the directions be related by means\nof a reflection in a plane parallel to Xa thus:\nDefinition 6. A stratified plane-paralle1 medium X(a,b) is\nsaid to be symmetric if the following properties hold for\nevery subslab X(x,z) of X(a,b):\nR(x,z;5';E) R(2,x;M(E');M(E))\n(11)\nand:\nT(x, 2;5';5) = T(2,x;M(E');M(E))\n(12)\nfor every reflection transformation M of (1) in a plane parallel\nto X","150\nINVARIANT IMBEDDING TECHNIQUES\nVOL. IV\nXa\nE'\nX\nE.\n+\nE\nZ\nP\nX\nE.\n+\nZ\nM(E')\nM(E)\nFIG. 7.22 The reflection of directions E,E' in plane P,\nas used in describing polarity of an optical medium.\nThe opposite notion to that of symmetry in the present\ncontext is polarity. Thus X(a,b) is polar or exhibits polar-\nity if it is not symmetric; and this, by Definition 6, means\nthat there exists a subslab X(x,z) of X(a,b) such that either\nR(2,x;M(E');M(E)\nor:\nT(x,z;5';5) T(2,x;M(E');M(E))\nfor some reflection transformation M of (1) in a plane P paral-\n1el to Xa (see Fig. 7.22 for the case of reflectance). The\nmain theorem about polarity is the following:\nPolarity Theorem: Let X(a,b) be a stratified plane-parallel\nmedium. (a) If X(a,b) is separable and isotropic, then X(a,b)\nis symmetric; (b) If X(a,b) is non separable and isotropic,\nthen X(a,b) is polar.","SEC. 7.12\nHOMOGENEITY, ISOTROPY, ETC.\n151\nThe proof of the theorem may be made to devolve on the differ-\nential equations for R(a,b) and T(a,b) in Sec. 7.3, but will\nbe omitted here. The main point of the theorem is that sym-\nmetry of X(a,b) may be lost by the presence of essential in-\nhomogeneities in X(a,b) by \"essential inhomogeneity\" is\nmeant that the medium is not just separable, but rather such\nthat 0(2;5';5) x(z) depends on depth Z in X(a,b). A proof of\nthe polarity theorem along with examples for discrete spaces\nis given in some detail in Sec. 57 of Ref. [251]\nGoing on now to the global counterpart of reversibility,\nwe have:\nDefinition 7. A stratified plane-parallel medium X(a,b) is\nsaid to be reciprocal if the following properties hold for\nevery subslab X(x,z) of X(a,b):\nR(x,z;E';5) = R(x, z;-E;-5')\n(13)\nand:\nT(x,z;E';5) = T(z,x;-E;-E')\n(14)\nand:\nR(z,x;-5;-5')\n(15)\nR(z,x;5';5)\n=\nand:\nT(z,x;5';5) = T(x,z;-5;-5')\n(16)\nExamples can be given which show that symmetry and re-\nciprocity of X(a,b) are generally independent notions. Thus\nX(a,b) may be symmetric but not reciprocal; and conversely,\nX(a,b) may be reciprocal but exhibit polarity. That this is\nplausible may be seen without too much preliminary work by\nletting X(x,z) approach zero thickness so that symmetry of\nX(x,z) becomes a manifestation of isotropy of o; and recipro-\ncity of X(x, z) reduces nearly to reversibility of o. Since\nreversibility and isotropy of o are partially independent,\nthis independence can be inherited at least by very thin slabs\nX(x,z). The main theorem on reciprocity is the following:\nReciprocity Theorem. Let X(a,b) be a stratified plane-paral.\nlel medium. If o on X(a,b) is reversible, then X(a,b) is re-\nciprocal.\nObserve that this theorem, which can be proved using the\ndifferential equations for R(a,b) and T(a,b) in Sec. 7.3, holds\nin particular for nonseparable media. The theorem was first\nstated and proved for separable plane-parallel media X(a,b) by\nChandrasekhar in Ref. [43]. A proof of the reciprocity theorem\nfor general isotropic media is sketched in Ref. [40].\nSummary\nTo summarize the main results of this section so far we\nmay say that in going from the local to the global level in\nstratified plane-parallel media one generally can carry over\nthe concept of reciprocity but not symmetry. More precisely,\nand in terms of the defined concepts above, a locally rever-\nsible medium is always reciprocal, but a locally isotropic\nmedium may exhibit polarity.","INVARIANT IMBEDDING TECHNIQUES\nVOL. IV\n152\nThe loss of symmetry (where we use the term in the\nsense of Definition 6) is a phenomenon that arises because of\nthe adoption of radiance as the basic radiometric concept ra-\nther than irradiance or alternatively, radiant density. Had\nwe used the latter concept, then symmetry would hold (in the\nscalar irradiance context) for inhomogeneous isotropic plane-\nparallel media. Symmetry would be lost in such a context only\nwhen isotropy was lost. By adopting radiance over irradiance\nwe reap the benefits of a more detailed description of the\nlight field at the expense of the classical symmetries pos-\nsessed by irradiance. Furthermore, the reflectance and trans-\nmittance functions R and T in the scalar irradiance context,\nbeing scalars, commute; i.e., symbolically, RT = TR. By adopt-\ning radiance, R and T become integral operators or matrices,\nand these objects are notoriously noncommutative, thus block-\ning still further the passage of certain symmetries of the\nscalar formulations to the field of operator formulations.\nConclusion\nIn conclusion, then, the elevation of the local notions\nof homogeneity, separability, isotropy, and reversibility to\nthe global settings in plane-parallel media is quite possible.\nHowever, only the local concept of reversibility is generally\ninherited by the space on the global level (in the form of re-\nciprocity). But this inheritance is precarious and can con-\nceivably vanish on graduation to arbitrarily shaped anisotrop-\nic media in which the radiometric concept used is radiance\nrather than irradiance or scalar irradiance. Thus all the\nclassical symmetries are in principle left behind in the\nsearch for general invariant properties of scattering-absorb-\ning media. The general principles of invariance, the invariant\nimbedding relations and their various semigroup properties are\nimportant examples of general properties of optical media\nwhich are invariant under the transition from local to global\nformulations within those media. This has been shown in de-\ntail in Chapter VI of Ref. [251], for general discrete spaces.\nFurther study of the problem of the extension of local\nsymmetries to the global level are best handled by means of\nthe standard - operator & (X;a,b). A detailed study of such\nextensions has yet to be made. It would be of interest to\nformulate the appropriate counterparts to homogeneity, and\nisotropy for general media using & (X;a,b), and then to find\ntheorems, if possible, which are the appropriate generaliza-\ntion of the Polarity and Reciprocity theorems.\n7.13 Functional Relations for Media with Internal Sources\nIn this section we return to the problem of internal\nsources in optical media introduced in Example 3 of Sec. 3.9\nand reconsidered in Sec. 6.7 From a theoretical point of\nview the problem was completely solved in Sec. 3.9 and, in\nview of the methods of determination of the R and T operator\ndiscussed in Sec. 7.7, we may say that the practical numerical\nmeans of solving the internal-source problem are also well in","SEC. 7.13\nINTERNAL SOURCES\n153\nhand. However, there remain several most interesting ques-\ntions on the conceptual level, questions that arise when one\nexamines the functional relations (35) and (36) of Sec. 6.7\nwith an eye toward the intuitive meaning of the equations and\nof their connection with the invariant imbedding relations\nthat may be written down for the same medium. Specifically,\nwe are confronted with two equations (35), (36) of Sec. 6.7,\nderived directly from the equation of transfer, and which are\nostensibly statements of a certain type of invariance for sca-\nlar irradiance h(x). Their physical meanings as given by\nElliott [88], are, however, occluded by the fact that their\nmain terms fo and fc are Fourier transforms of the scalar ir-\nradiance function h (x) (cf. (23) of Sec. 6.7) rather than\nh (x) itself. Therefore, one of the principal goals in this\nsection is the development of a systematic method of deriva-\ntion of the counterparts to (35), (36) of Sec. 6.7 for the\ncase of radiance in a general one-parameter optical medium\nX(a,b) with an arbitrary set of sources on various levels\nwithin X(a,b), using only the concepts inherent in the invar-\niant imbedding relation for the medium. We therebye shall\nestablish intuitively meaningful generalizations of the\nElliott equations and also extend their domain of validity.\nAn additional dividend is accrued throughout in the form of\nfurther insight into the interconnections among the Y-operators\nand the invariant imbedding operators. These connections\narise as a matter of course during the derivations. Through-\nout this section, let \"X(a,b)\" denote a one-parameter optical\nmedium with artibrary a,a. In particular X(a,b) will not be\nassumed isotropic, so that there are generally four local op-\nerators P + (t), T+ (t) (cf. Sec. 7.1). Throughout this section\nsources shall be confined, for simplicity and without any\nserious loss of generality, to single depths S within the slab\nX(a,b), asssb. For sources at several discrete levels, su-\nperposition of the results below will yield the desired field\nexpression. By passing to the limit of numbers of discrete\nsources, the theoretical way to continuously distributed\nsources is opened. These generalizations are left to the\nreader. For helpful hints in this direction see (36) of Sec.\n3.9 and its discussion. Also see the paragraph on Two-D\nModels for Internal Sources in Sec. 8.5, and consult Example\n10 of Sec. 8.7.\nPreliminary Relations\nOne important dividend of the present efforts is a col-\nlection of auxiliary functional relations between the a, J\noperators and the Y -operator of Sec. 3.9. These equations\nplace the interrelations of the Y -operator into a deeper per-\nspective than is available from (31)-(34) - of Sec. 3.9. of\nparticular interest at present are the connections between\nR (a,s,b), J(a,s,b), and 4(s,s), where asssb. It follows\nfrom (20) - (23) of Sec. 3.9 and (40) - (43) - of Sec. 3.7 that:\n(1)\nR(a,s,b) = T(a,s) Y (s,s:a,b)\n(2)\nR(b,s,a) = T(b,s) Y (s,s:a,b)","154\nINVARIANT IMBEDDING TECHNIQUES\nVOL. IV\nT(a,s,b) = T(a,s) (I + Y (s,s:a,b))\n(3)\nT(b,s,a) = T(b,s) (I + Y (s,s:a,b))\n(4)\nwhere we have written; for asssb, asysb:\n\"Y(s,y:a,b)\" for Y(s,y)\n(5)\nto point up specifically the fact that ( a,y) belongs to\nX(a,b). (See the remarks following the applications of the\ninteraction method in Example 3 of Sec. 3.9.)\nEquations (1) - (4) show clearly the interrelation between\nthe complete reflectance operators and the local - operator\nfor X(a,b). . These relations will be helpful in constructing\na dual class of a or J operators needed subsequently.\nAnother set of functional relations, needed in the der-\nivations below, is the following, which again is based on\n(20)-(23) of Sec. 3.9:\n4++(s,s:a,b) = 4+ (s,s:a,b)R(s,b)\n(6)\nR(s,a) 4.+(s,s:a,b)\n(7)\n(s,s:a,b) =[I-R(s,a)R(s,b)] =\n(8)\nY (s,s:a,b) = (s,s:a,b)R(s,a)\n(9)\n(10)\nR(s,b)\n+ Y (s,s:a,b) = [I - R(s,b)R(s,a)]\n(11)\n(12)\n[I\"++(s,s:a,b)]R(s,a)\n(13)\n+(s,s:a,b) R(s,b)[I\"++(s,s:a,b)]\n(14)\n[IY__(s,s:a,b)]R(s,b)\n(15)\nIntegral Representations of the Local Y -Operators\nWe are now ready for the derivations of the representa-\ntions of the local Y-operators in terms of the simpler R\nand J operators of the invariant imbedding relation. We fix\nattention at first on the setting within X(a,b) depicted in\nFig. 7.23.\nAn internal source is at level S in X(a,b). The source\nmay be a point, or some arbitrary discrete or continuous set\nof points on level s, , and of arbitrary directional structure\nat each point of the set. We consider first the upward","SEC. 7.13\nINTERNAL SOURCES\n155\na\nN°(s)\nS\nNo(s)\nb\nFIG. 7.23 An internal source situation in medium X(a,b).\nThe first main case: source level (s) below observation level\n(t).\ncomponents NO (s) of the source N° (s). The resultant radiance\nfield at level S generated throughout X (a, b) by this source\ncomponent is, as explained in Sec. 3.9, given by the local\nY -operator components Y (s, s:a,b) and Y (s,s:a,b). The\nfirst of these gives the resultant upward field, the second\ngives the resultant downward field. Now consider Y ++ (s,s:a,b).\nWe wish to study the dependence of Y (s,s:a,b) on\nholding\na,\nS and b fixed. It will turn out that knowledge of this de-\npendence will lead directly to the requisite integral repre-\nsentation of Y (s,s:a,b), in terms of a family of invariant\nimbedding operators for X(a,b).\nThus, imagine a family {X(t,b) a stsb} of optical me-\nin which the original medium X(a,b) is imbedded (i.e.,\ndia\nX(a, b) is a member of the family). The associated family of\nlocal Y -operator is (s,s:t,b): astsb}. The effect\nthe\nin Y (s,s:t,b) can be determined by taking the\nof\nvarying\nt\nderivative of Y ++ (s,s:t,b) in the following way:\nY (s,s:t,b) d y + (s,s:t,b)\nR(s,b)\n=\nat\nat\nwhich is suggested by (6) Furthermore, (13) suggests that we\nwrite:","INVARIANT IMBEDDING TECHNIQUES\nVOL. IV\n156\n(s,s:t,b) (s,s:t,b)]R(s,t)]\nat\nat\nR(s,t)\n-[I+W++(s,s:t,b)]T(s,t)p+(t)T(t,s)\nThe last equality comes from (28) of Sec. 7.1, applied to\nX(t,s). . The minus sign comes from the fact that depth is\nmeasured positive from a to b. Returning to the original\nequation, we have:\nR(s,t)T(s,b)\nat\nat\nThe derivative term may therefore be solved for and found to\nbe of the form:\nIN IY++(s,s:t,b)]T(s,t)p+(t)T(t,s)R(s,b) [I - R(s,t)R(s,b)] -1\nUpon examination, this seemingly complex representation col-\nlapses into the composition of three highly intuitive forms.\nOne of these is R(t,s,b), since, by (42) of Sec. 3.7:\nR(t,s,b)=T(t,s)R(s,b)[I If - R(s,t)R(s,b)]-1\n.\nThe next term to consider is:\n[I + +++ (s,s:t,b)]T(s,t)\n(16)\nThis is an unfamiliar combination of operators. It has\nnot arisen in our work as yet. However, there is a tantaliz-\ning asymmetry between it and (3) above. When the variables\nt,s,b are placed in (3) we have:\nT(t,s,b) = T(t,s)[Iy__(s,s:t,b)]","SEC. 7.13\nINTERNAL SOURCES\n157\nThe interpretation of T(t,s,b) is at this stage of our\nstudies well understood: downward incident radiance at level\nt generates a light field in X(t,b) and T(t,s,b) gives the\ndownward component of that field of level S in X(t, b) (See\nFigure 7.23) The term (16) seems to give a dual interpreta-\ntion to that of T(t,s,b). Thus, it says that upward source\nradiance at level S generates a light field within X(t,b) and\n(16) gives the upward component of that light field at level\nt\nin X(t,b). Hence (16) acts like a complete transmittance\noperator, but one whose input level (s) and output level (t)\nare exactly reversed from their customary relative orientations\nwithin X(t,b) With these interpretations and the dual kin-\nship of (16) and J (t,s,b) in mind, let us write \"J+ (s,t,b)\n11\nfor (16). Then the differential equation for Y +(s,s:t,b)\nbecomes:\na\na Y (s,s:t,b)\nt\not(s,t,b)p+\n(t)\nR(t,s,b)\n(17)\n=\nS\nat\nb\nIntegrating each side of (17) over the interval [a,s], and\nusing the fact that:\n(s,s:s,b) = 0\n,\nwe have:\nY (s,s:a,b) = (s)+(s,t,b)p+(t) R(t,s,b) dt\na\nt\nS\na\nastsseb\nb\n(18)\nThe simple physical interpretation of (18) should not\nescape notice. Consider X(t,b). Imagine the source No(s) at\nlevel S giving rise to the upward emergent radiance at level\nt in the space X(t,b) Then imagine a thin incremental layer\nadded to X(t,b) at level t. This thin layer reflects some of\nthe emergent flux via (p+ (t)) back down into X(t,b). This re-\nflected flux sets up a light field in X(t,b), the upward com-\nponent of which at level S being given by R(t,s,b). . By let-\nting X(t,b) grow another thin layer at level t, still another\nincremental light field is added to that at level S. By adding\nup all such increments, starting from level S and working up\nto level a, we obtain the total field at level S induced by\nthe upward source component NO (s) . The analytical representa-\ntion (18) summarizes all this compactly, as shown. The little\nideograph next to (17) and (18) serves to depict the relative\npositions of the depth variables in X(a,b).\nThe representation (18) is one of a pair of representa-\ntions for Y (s,s:a,b), the other arising when we imbed X(a,b)\nin the family {X(a,t): astsb of spaces. (See Figure 7.24)\nThen we consider:","158\nINVARIANT IMBEDDING TECHNIQUES\nVOL. IV\na\nN° (s)\nS\nNo(s)\nb\nFIG. 7.24 An internal source situation in medium X(a,b).\nThe second main case: source level (s) above observation\nlevel (t)\n(s,s:a,t) = R(s,a) (s,s:a,t)\nat\nusing (7). This is analyzed further using (14), which yields:\nar =\n(s,s:a,t)\n+\nat\nat\n+\nOnce again a complex group of terms can be collapsed into a\ncomposition of three physically meaningful groups of terms.\nThe operator P (t) separates the two remaining terms. The\nlast group of terms is simply T(t,s,a) (cf. (43) of Sec. 3.7).\nThe first group of terms is one of those tantalizing duals to\nthe invariant imbedding operators. This time, by studying\n(41) of Sec. 3.7 and (1) above, we see that we should write\n(26) below) so that the preceding dif-\n\"Rt(s,t,a)\" (see\nferential equation for Y\n(s,s:a,t) becomes:\n++ (s,s:a,t)\na\nat(s,t,a)p_(t)J(t,s,a)\n(19)\nS\n=\nat\nt\nb","SEC. 7.13\nINTERNAL SOURCES\n159\nIntegrating (19) over the interval [s, b] and using the fact\nthat:\n(s,s:a,s) = 0\n,\nwe have:\nb\na\nS\nY (s,s:a,b) (t)J(t,s,a) dt\n(20)\nt\nb\nassstsb\nThe physical interpretation of (20) is as follows:\nconsider the medium X(a,t) with upward source No (s) at level\nS (Fig. 7.24). The light field in X(a,t) generated by this\nsource has a downward component at level t given by Rt(s,t,a).\n(If source flux is upward directed at s, then, since at is a\nreflector, response flux is downward directed at t.) Adding a\nthin increment to X(a,t) at level t causes a corresponding in-\ncrement of reflected radiance (via p- (t)) to re-enter X(a,t)\nand to be completely transmitted by J(t,s,a) to level S. By\nadding all such incremental layers on X(a,t) from t = S to\nt = b, the representation of Y (s,s:a,b) is obtained.\nThe pattern emerging in these derivations should now be\nclear.\nOn the basis of this emerging pattern the differen-\ntial equations for Y (s,s:t,b) and Y (s,s:a,t) and other\ncorresponding integrals can be written down directly without\nany further detailed derivation. However, the interested\nreader should verify the formulas so obtained:\nay (s,s:t,b)\na\nR+(s,t,b)p.(t)J(t,s,b)\n(21)\nt\n=\n-\nat\nS\nb\na\nt\n(s,s:a,b) = (t)J(t,s,b) dt\nY\n(22)\nS\nb\nastsssb\nThese equations are companions to (17) and (18) and go with\nFig. 7.23. The following equations are companions to (19) and\n(20) and go with Fig. 7.24.\na\nS\nay (s,s:a,t)\nt\nr+(s,t,a)p_(t)R(t,s,a)\n(23)\nb\nat\n(s,s:a,b) = (\"++(s,t,a)p_ t)R(t,s,a) dt\na\n(24)\nY\nS\nt\nas, say. This being so, our attention turns im-\nmediately to 4(s,y:s,b) and 4(s,y:y,b) and we ask: under what\nconditions on the geometry of X(a,b) and its inherent optical\nproperties do we have:\n4(s,y:s,b) = 4(y,s:s,b)\n(88)\n- S\n- y\n- b\nas a valid equation?\nThe diagrammatic insert under the equation shows the geo-\nmetric context in which the question is asked. It should be\nobserved that this is equivalent to the equation:\n4(s,y:s,b) = Y(s,y:y,b)\n(89)\n- S\n- y\n- y\n- S\n- b\n- b\nin the geometric context where names of the levels S and y are\ninterchanged in the same medium as shown by the diagrams below\n(89).","INVARIANT IMBEDDING TECHNIQUES\nVOL. IV\n180\nTo see the conditions under which (88) holds, it is suf-\nficient to examine each of the four operator equations within\n(88). Thus, consider, for example, the equation arising from\n\"++\" components of (88):\nthe\n(90)\n4++(s,y:s,b) If (y,s:s,b)\nFrom this we see at once that (88) cannot generally hold\non the operator level since the left side is always zero,\nwhile the right is generally not zero. This establishes the\nfact that the simple scalar condition (87) has no exact coun-\nterpart in the general operational transport formulations we\nare now considering. However, we still may inquire as to the\nother pairs of components in (88). Those pairs that are not\nzero--are they ever equal? Or: are the sums of the compo-\nnents of the left side equal to the sums of the components on\nthe right side of (88)? The latter question is prompted by\nenergy conservation considerations. The latter question will\nbe considered subsequently in Chapter 8 in a setting where\nthe question makes physical sense (Example 10, Sec. 8.7). For\nthe present we examine the former question out of simple cur-\niosity.\nThe diagram below (88) suggests that if we are to find a\ncorresponding pair of nonzero components in (88), it would be\nthose with the signature \"-+\". (Cf. (63), (69).) Consider\nthen, for possible validity, the statement:\n4-+(s,y:s,b) =\nwhich is equivalent to:\nR(s,y,b) = Q+(y,s,b)\nBy (1) and (25) this is equivalent to:\nR(y,b)[I-R(y,s)R(y,b)]-1T(y,s) = T(s,y) [I-R(y,b)R(y,s)]-1R(y,b)\nwhich in turn is equivalent to:\n(I-R(y,s)R(y,b)]T(s,y)R(y,b) = R(y,b)T(y,s) [I-R(y,s)R(y,b)]\n.\nFor this to be valid, it is sufficient to have commutation\nfreely possible between R(y,s), R(y,b) and T(s,y), T(y,s) along\nwith\n(91)\nT(s,y) = T(y,s)\nand, among other things:\n(92)\nR(y,s)R(y,b)T(s,y) = T(y,s)R(y,s)R(y,b)\nAt this point, our studies of Sec. 7.12 may be used to\nhelp clear the air of present question. The polarity theorem\nasserts that a plane-parallel medium X(a,b) must be isotropic\nand separable in order that (91) hold. This is not too strin-\ngent a requirement on the medium and its inherent optical prop-\nerties. However, if X(a,b) is not plane-parallel, it is gen-\nerally the case that (91) no longer holds, no matter how","SEC. 7.13\nINTERNAL SOURCES\n181\nregular its inherent optical properties. That commutativity\nand condition (92) are also to hold--i.e., to have a recipro-\ncity condition is hopeless in general. One exception occurs\nin the scalar context, i.e., when the R(y,b), R(y,s) and\nT(s,y) are real valued functions of s, y, b and not matrices\nor integral operators (as in the present discussion).\nIn this way we see that (35) and (36) of Sec. 6.7 can-\nnot be directly generalized to the operator level without loss\nof the rather special reciprocity condition (87). This is a\nsmall loss in view of the fact that (85) and (86) are capable,\nas they stand, of solving in principle the most general point\nsource problems on continuous one-parameter optical media.\nTheir complementary counterparts associated with cases 1 and 3\nin stage 3 of Fig. 7.25 are also capable of performing this\nservice. The derivation of the associated equations are left\nto the reader as an important exercise (cf., (108)-(111) below.\nA Royal Road to the Internal-Source Functional Relations\nIt was perhaps somewhat of an anticlimax for the atten-\ntive reader to see the four operator equations of case 2,\nstage 3 (in Fig. 7.25), so hard-won through the early portions\nof this section, unceremoniously collapsed into the simple op-\nerator equation (85). Still another such revelation may have\noccurred when (86) was reached. Be that as it may, the rela-\ntive simplicity of (85) and (86), compared with the system of\ntheir progenitors, attests to the correctness of the deductions\nand to the power of the invariant imbedding approach which gave\nus the general Y-operator concept. But yet the very simplicity\nof these results invites an attempt of a correspondingly simple\nderivation of (85) and (86). We shall now indicate the out-\nlines of such a derivation. We shall be very careful not to\nadd all the rigorous details or else we shall simply retrace\nthe work of this section. Thus we shall embark on a 'royal\nroad' to (85) and (86), in the sense that it is ostensibly\nwell-paved with no long steep grades, and along which the ana-\nlytic and algebraic pitfalls have been filled and smoothed with\nrhetoric.\nWe choose as a setting case 2 of stage 3 in Fig. 7.25.\nThe present derivation begins with a partition of X(a,b) by\nthe internal surface Xt, astsy. The only source on or in\nX(a,b) is at level s, y~~