0, we certainly\nhave the maximum value N of N(x,5) greater than the maximum\nvalue No of No (x,5). Then (6) implies:\nN(x,5) -\np\n(7)\n<\n1-p\nfor every x in X and E in E. The comparative merit of\n(7) over (6) consists in equation (7)'s ability to express\nthe error of truncation in terms of a relative error, that\nis the error relative to the prevailing magnitude N of the\nlight field. Hence for the medium at hand, carrying out the\nnatural solution to five terms results in a relative error\nof less than 1 percent.\nBefore closing we shall examine the inequalities (5)\nand (6) for some insight they may yield about the relative\nimportance of the various components of the decomposition of\nthe natural light field. For example, (5) shows that n-ary\nradiances are on the whole less by a factor of p than (n-1)\nary radiances. Thus if p = 1/2, say, then N1 (x,5) is on the\nwhole, about half the magnitude of No (x,5) and the magni-\ntude of N2 (x, 5), in turn, is about half that of N° (x,E), and\nso on. Thus the overall magnitude of n-ary radiances de-\ncrease exponentially with scattering order n. Inequality\n(6) also shows that for small p (near 0), a given n-ary radi-\nance varies directly as the nth power of p, whereas for large\np (near 1), the n-ary radiances vary essentially hyperbolically","SEC. 5.5\nTRUNCATED SOLUTIONS\n49\nwith 1 -p, i.e., as 1/(1-p). Similar observations can be\nmade using (6) or (7). We shall return to the matter of\ntruncated natural solutions in the following section and re-\nconsider them for transient light fields. The reader wishing\nradiance bounds in a slightly more general steady state case\nthan that considered in this section, may consult Sec. 22 of\nRef. [251].\n5.6 Optical Ringing Problem. One-Dimensional Case\nThe object of this section is to formulate the optical\nringing problem in the context of radiative transfer theory\nand to indicate how the natural mode of solution may be used\nto solve the problem. In order to explain the ideas behind\nthe optical ringing problem and its natural mode of solution\nwithout too many geometrical complications, we consider first\nthe one-dimensional case of the problem. The three-dimensional\ncase will be discussed in the following section.\nThe term, \"optical ringing\" has an analogous meaning to\nthe term \"reverberation\" as used in the theory of sound. In\nfact the well-known term \"reverberate\" applies in principle\nequally to optical and acoustical phenomena. However, until\nrecently, the relative difficulty of producing and recording\noptical reverberation because of the immeasurably short per-\niods of time involved has given the acoustical discipline\nalmost exclusive use of the term. We can use the popular\nacoustical meaning of the term \"reverberation\" to give the\nfollowing nontechnical definition of the phenomenon at hand:\nOptical ringing in an optical medium is the optical reverber-\nation of the medium set up by a narrow short pulse of mono-\nchromatic light. Hence the appropriate acoustical analogy to\noptical ringing would be the reverberation set up by a direc-\ntional, short clap of one-note thunder. In more technical\nparlance the optical ringing problem in a medium X is the\nproblem of determining at time t > 0, the time-dependent radi-\nance function over X which is the solution of the equation of\ntransfer, given a directional, spatial, and temporal Dirac-\ndelta function input of radiance to the medium at time t = 0.\nThis problem has applications to the description of time -\ndependent radiance fields set up by laser beams with their\ncharacteristic high power, narrow-beam, short-pulse shafts\nof monochromatic radiant flux. While interest in the optical\nringing problem has reawakened because of the advent of the\nlaser, it should be noted that the problem is a venerable one\nin radiative transfer theory and neutron transport theory,\nand was first studied purely for its intrinsic interest and\nas a fundamental block on which to build solutions with\narbitrary initial time-varying, inputs (see, e.g., [211],\n[235], [236]).\nGeometry of the Time-Dependent Light Field\nThe formulation of the time-dependent radiant flux\nproblem in an optical medium X will be facilitated by find-\ning an efficient means of depicting the space-time disposi-\ntion of the radiant flux throughout the optical medium. We\nshall now construct such a means. In the present discussion\nthe medium X is one-dimensional and is represented in Fig. 5.3 (a)","50\nNATURAL SOLUTIONS\nVOL. III\nline segment. We shall consider the medium to extend\nas\na\nindefinitely on either side of the origin point 0 of the\nmedium, with distance measured as positive toward the right.\n+\n(a)\nX\nt=0\no\n-uto\npulse\nX\n(b)\nt= to\nO\nuto\npulse\n(c)\nX\nt>to\nO\n+\nFIG. 5.3 Positions of a finite light pulse along a one-\ndimensional medium.\nNow suppose that point 0 becomes a source of radiant\nflux starting at time t = 0 and that 0 continues to emit flux\nin an arbitrary fashion in both directions about 0 until time\n= to, at which time the source at 0 is shut off. Let\nt\n\"No (0, t, +)\" and \"N o (0,t,-)\" denote these radiances of 0 at\ntime t in the positive and negative directions, respectively.\nFigure 5.3 (b) shows the position of the pulse emitted by 0\njust after time to. The pulse is speeding away from point 0\ninto the medium on either side of 0. Figure 5.3(c) shows\nthe position of the pulse some time later than to. Figures\n5. (a) through 5.3(c) are like three snapshots of the medium","SEC. 5.6\nRINGING PROBLEM, ONE-DIMENSIONAL\n51\nX at three separate instants of time subsequent to the emis-\nsion of the pulse. It would be quite instructive if instead\nof still shots of X at discrete time instants, , we could have\na moving picture of the pulse as it moves out into X from 0\nand generates the field of scattered light within X. Such a\nmeans of communication is obviously unfeasible for the pres-\nent work. However, an alternate and in some ways superior\nmeans of visualizing the time -dependent light field in X con-\nsists in a static space-time diagram of the pulse in X of\nthe kind depicted in Fig. 5.4.\nThe description of the pulse of radiant flux from point\n0. becomes relatively simple when given in terms Fig. 5.4.\nThe space-time - portrait of the pulse is given by the shaded\nV-shaped region in the space-time diagram. To find the in-\nstantaneous position of the pulse in X at time t', , first go\nalong the time axis erected perpendicular to X until time\nincreasing time\nA\nB\nC\npulse\nto\nX\no\nut'\nFIG. 5.4 A space-time portrait of the pulse in Fig. 5.3.\nThe world region of the pulse is shaded.","NATURAL SOLUTIONS\nVOL. III\n52\npoint t' is reached. Then draw a straight line through t'\nparallel to X. This line will intersect the shaded region in\ngenerally two segments A and B. The perpendicular projection\nof these segments down onto X will then give the location of\nthe pulse in X at time t' 0. The slope of any straight line\nsegments parallel to the boundaries of the shaded region of\nthe pulse are such that, as t' units of the time axis are\ntraversed, vt' units of the space axis are traversed, where\nV is the speed of light in X. We assume V to be constant\nover X. The shaded region of Fig. 5.4 is called the world\nregion of the pulse.\nIt follows from the axioms of special relativity that,\nrelative to the frame at 0, the space-time line traced out\nby a material particle in X cannot have an arbitrary slope,\nbut rather one which is bounded as follows. If (t) is the\ndistance of the particle from 0 at time t, then:\ndr(t)\n< V\ndt\nfor every t for which r(t) is defined in the frame anchored\nat 0. In particular, the slopes of the world lines (i.e.,\nspace-time trajectories) of the photons comprising the pulse\nof light from 0 are exactly of magnitude V, with respect to\nthe time axis. Thus on the one hand, the world line of a\nparticle stationary in X is a vertical line, and on the other\nhand, that of a photon is parallel to one of the boundary\nlines of the shaded region in Fig. 5.4. A11 naturally moving\nparticles in X must therefore have the tangents to their\nworld lines always between (or coincident) with these two\nextremes, with respect to the r, frame of reference at 0.\nThe space-time diagram also aids in visualizing the\nvarious possibilities of radiometric interactions between\npoints of X. Thus, points a, b, and C in Fig. 5.4 depict the\nthree possible dispositions of points in space time with re-\nspect to the pulse from 0. Point (=(r,t) is in the world\nregion of the pulse, and so represents a point of X at dis-\ntance r from 0 which at time t is being irradiated by radi-\nant flux comprising some of the pulse from 0. Points a and\nC on the other hand are not in the world region of the pulse.\nPoint a in particular represents a point in X after the pulse\nhas gone by it (to find the contemporaneous pulse to a, draw\na horizontal line through a, and the segment it determines\nwith the world region of the pulse is the requisite position\nof the pulse). Point C represents a point in X before the\npulse has gone by it. Points a and C thus have the property\nin common that they do not lie on the world region of the\npulse from 0; however, points a and C differ from one another\nin a fundamental sense. Indeed, the point in X corresponding\nto a may eventually feel the effects of the pulse through\nscattering of flux from the pulse; however, the point in X\ncorresponding to point C in the space time plane is \"forever\"\nimmune to the direct or indirect effects of the pulse. Here\nwe are implicitly adopting another empirical fact of macro-\nscopic physics : Effects of an event may propagate futureward\nin space-time but not pastward. When this fact is combined\nwith that about the limits on the slopes of the world lines","SEC. 5.6\nRINGING PROBLEM, ONE-DIMENSIONAL -\n53\nof particles mentioned above, we can readily delimit those\nparts of the space-time plane over (or through) which they\ncan effect or be effected by a given event (represented as a\npoint) in the plane. These regions are shown in Fig. 5.5 (a)\nfor an arbitrary point a. In general, for two points a and b\nin the space-time diagram associated with X, the common re-\ngion of possible interaction is the shaded intersection of\nthe futureward sector of b with the pastward sector of a, as\nshown, in Fig. 5. (b). If the intersection region is empty,\nthen the two points cannot interact.\nWith these preliminary observations in mind, we may now\nuse the the general space-time diagram to help in the study\nof the time-dependent - radiant flux problem on X. Starting\nfutureword\nof a\nelseward\nelseward\n(a)\nof a\nof a\npostword\nof G\nr\nr\na\n(b)\nb\nr\nFIG. 5. 5 Part (a) depicts those points of space-time\nabout point a which lie in a's future, past, and elsewhere\nfrom a. Part (b) shows the common region (shaded) shared\nby the future cone of b and the past cone of a. When this\nshaded region exists, then b can send a light signal to a.","54\nNATURAL SOLUTIONS\nVOL. III\nwith a fresh space-time diagram of the pulse emitted by point\n0 in X, as in Fig. 5.6, we see that the pulse effects at time t\nat some point a distance r from 0 in the medium arrive through\nthe pastward sector of the point (r,t). In particular, the\nregion of X contributing scattered flux of all orders to (r,t)\nis bounded by a (r,t), b(r,t), where we have written:\n\"a(r,t)\" for (r-vt)/2\n(1)\n\"b(r,t) for (r+vt)/2\n(2)\nFor example, if r = 0, then the interaction region of X at\neach time t is an interval on X of length vt centered on 0.\nThe route of radiant flux from 0 to point (r,t) may be quite\ndevious. Two sample routes from 0 to (r,t) are shown by the\na fourth order path\n(r,t)\na first order path\nto\nr\nE)\na(r,t)\n(0,0)\nb(r,t)\nut\nr\nut\nFIG. 5.6 Computing the scattered light reaching space-\ntime point (r,t) after starting from the origin (0.0).","SEC. 5.6\nRINGING PROBLEM, ONE-DIMENSIONAL\n55\ndashed lines in Fig. 5.6. In one of the cases the flux reach-\ning (r,t) is intended to be fourth order radiant flux. The\nspatial component of the path taken by this sample of radiant\nflux is obtained by projecting the space-time path onto X.\nObserve that in this particular example the only way radiant\nflux can reach (r,t) from 0 is by undergoing at least one\nback scattering operation.\nThe Equation of Transfer\nThe integral form of the equation of transfer for the\none-dimensional optical medium X defined above will now be\nderived. Before going into the details, however, it may be\nwell to reemphasize that the significance of a one-dimensional\noptical medium lies not so much in its power to represent an\nactual physical setting as it does in its ability to depict\nwith a minimum of geometric complication the essential alge-\nbraic structures of the associated three-dimensional problem.\nTherefore, the resultant equation of transfer derived below\nfor the present one-dimensional setting will, in all its\nalgebraic essentials, be representative of the full three-\ndimensional case, but will not be encumbered with details\narising from the latter's relatively complex geometrical\nstructure. These details will be faced in the following\nsection.\nUnder suitably adapted definitions of the radiance\nfunction and inherent optical properties for X, the equation\nof transfer for the one-dimensional optical medium X follows\nformally from the integral form of (4) of Sec. 3.15. In this\nway we extend the logical chain from the interaction principle\nof Chapter 3 to the present radiative transfer discussion.\nIn particular the present equation of transfer is obtained\nby postulating the characteristic form of the volume scatter-\ning function for one-dimensional media:\no(x;E';5,t) = p(x,t) S(E+E') + T(x,t) S(E'-5)\nwhere & is one of the two directions (=E) along the medium,\nand S is the well-known Dirac-delta function. The functions\np and T are, respectively backward and forward scattering\nfunctions for X. Furthermore, the values of the radiance\nfunction are now of the form N(x,t, +) or N(x,t,-), where \"+\"\nand \" \" \" denote flux in the direction + or respectively.\nThat is, we have written:\n\"N(x,E',t)\" for N(x,t,+) S(E'-5) + N(x,t,-) S(E'+E)\nSince the points X in X are located by one number only, namely\nthe signed distance r from 0 to X, we will write \"r\" in place\nof \"x\" throughout the one-dimensional setting. It now fol-\nlows from (8) of Sec. 3.14 with the adopted form of o and N\n(and assuming here only that S is idempotent, i.e. 2 = 8,\nat least formally) that the path function values N*(r,t,+)\nassociated with directions + are:\nN*(r,t,+) = N(r,t,+) Tr,t) + N(r,t, p(r,t)\n(3)","56\nNATURAL SOLUTIONS\nVOL. III\nN*(r,t-) = N(r,t,-) T (r,t) + N(r,t,+) p(r,t)\n(4)\nThe time-dependent integral form of the equation of transfer\nfor the lone-dimensional case therefore consists of the fol-\nlowing two equations (one for each direction (+,-):\n(5)\nb(r,t)\n(6)\nwhere u(r) = 1 if r > 0, and u(r) = 0 if r < 0. A11 terms\nexcept the transmittance terms in these two equations have\nbeen defined in the present section. The transmittances are\nrepresented as in (3) of Sec. 3.11; thus for the present case\nwe have:\n\"Ts-r\"\nfor\nexp\nin which matters are arranged so that r < S.\nOperator Form of the Equation of Transfer\nWe next cast the pair of transfer equations (5), (6)\ninto an operator form which at once suggests the appropriate\ninstance of the natural solution for the present case. Thus,\nwe agree to write:\n\"No(r,t)\" for u(r)N_(0,t-|r/v/,+) Tr\n\"N°(r,t)\" for u(~r)N_(0,t-|r/v,,-) Ir\nand further, we write:\nr\n\"T+\"\nfor","SEC. 5.6\nRINGING PROBLEM, ONE-DIMENSIONAL\n57\n\"R_\" for S. [ ]pTr-r' dr'\nr\n(r,t)\nb(r,t)\nS\n\"T\"\nfor\nb(r,t)\n\"R+\"\nfor\nJoTT'-r\ndr'\nWith these assignations, (5), (6) become:\nN(r,t,+) = N°(r,t) + NT +(r,t) + NR_(r,t)\n+ NR+(r,t)\nThe notation \"NT+(r,t)\", e.g., denotes the value of the func- -\ntion NT. at (r,t), and NT+ is the result of acting on N with\nthe operator T+. These equations may be made more compact\nand at the same time more algebraic in appearance by writing:\n\"N+\" for N(,,+)\n\"N_\" for N(,,-)\n\"NO\" for No(,)\n\"NO\" for N°(.,.)\nWith these abbreviations for the four radiance functions we\nthen can write (5) and (6) as :\n=\n+\n(7)\n(8)\nThis form of the equation of transfer now suggests that we\nwrite:","NATURAL SOLUTIONS\nVOL. III\n58\n(9)\n\"S\"\nfor\nR_ T.\nalong with:\n\"N\" for (N+,N_)\n(10)\nand\n\"No\" for (No , N°)\n(11)\nso that the system (7) and (8) written in vector notation\nbecomes :\n(N N ) = (No,No + (N+,N_)s\n(12)\nor, succinctly:\nN = N° + NS\n(13)\nIn this way we have reattained the basic structure of the\nintegral equation of transfer, now for the simple one-\ndimensional context (recall, e.g., the derivation of (4) of\nSec. 5.4). It follows that we may at once apply the natural\nsolution procedure to (13) and thereby compute directly the\nscattering order components of N to as great a degree of\naccuracy as desired. This will now be done.\nThe Natural Solution\nStarting with equation (13) , and treating N as if it\nwere an unknown in a simple linear algebraic equation we\nobtain:\n= N°(I-s) - 1\nwhere (I-S)-1 may be shown to be expandable into an infinite\nseries:\n(14)\nWe have encountered such a type of expansion several\ntimes before in the present work. For instance it was used\nin Example 15 of Sec. 2.11, and it occurred many times in\nthe examples of Chapter 3. Finally, closely related series\nwere encountered earlier in this chapter (see (2) of Sec. 5.4).\nHence the requisite solution of the time-dependent equation\nof transfer for the one-dimensional optical medium takes the\nform:","SEC. 5.6\nRINGING PROBLEM, ONE-DIMENSIONAL\n59\n(15)\nAn Example\nAs an illustration of the natural solution for the\npresent one-dimensional optical ringing problem suppose the\nmedium X is homogeneous and in the steady state, so that p\nand T are constant valued functions over space and time.\nSuppose further that No and No are each constant valued and\nover a time period from t = 0 to t = to > 0 (a slight simpli-\nfication occurs if these are of Dirac-delta temporal struc-\nture; however, a temporally finite pulse, is at present a\nmore useful and realistic input for X, and accordingly is\nadopted). Then, carrying out the expansion (15) to second\norder scattering, we have:\n(N+,N_) = (No,No) +\n(16)\nSince\n=\n= T2+R+R_ T.R++R.T\nR_T++T_R_ R_R++T2\nwe have from (16) for the first component N+ of the vector\n(N+N_):\n+ + (17)\nand for the second component N of the vector (N+,N_)\nN - = N° + [Nor++N°I]] + N°(I+R+ + (18)\nEquations (17) and (18) show how the natural solution (15)\ncan be constructed order by order for an evolution of (N+,N-).\nIf still another scattering order is needed, we include S3","60\nNATURAL SOLUTIONS\nVOL. III\nTHE\nT2+R_R_\nT.R++R+T\ns2s\n=\nR_R++T2\nR_T+ +T_R_\n+ T2R+ + RRR\n=\nR_T2+I_R_T R RRR + T2R R_T_R+ + T_R_R+ +\nTo show how the second order operators in (17) and (18)\nare applied in practice, let us assume explicitly that\nNo (0,t,- = 0 for all t, and that N is the constant value of\nthe radiance pulse No (0,t,+) of duration to, starting at\nt = 0, in the direction E, i.e., , of increasing r. The pres-\nent situation then constitutes an approximate model of the\nlight field generated by a laser-like beam pulse of duration\nto seconds in the immediate vicinity of the beam. The out-\ngoing field N+ evaluated at r = 0 for every t > 0 is then,\naccording to (17)\nN(0,t,+) = No(0,t,+) + N°T (0,t + N°(T2 +\n(19)\nThe incoming field N_ evaluated at r = 0 for every to is,\naccording to (18) :\nN (0,t,-) = NOR+(0,t + No(T+R+ + (T)(0,t)\n(20)\nIn each of these equations, we have No (0,t,+) = N for\nt to and No(0,t,+) = 0 for every other t.\n0\nLet us consider (20) in more detail. The first order\nscattering term, unraveled, becomes :\nb(r,t)\n,\nin which we are to set r = 0, and t' = t-|r A study\nof\npart (a) of Fig. 5.7, which depicts the present situation,\nand a study of the definitions No and No, shows that this\nintegral is best evaluated by establishing two cases: Case\n(i) pertains whenever t < to; Case (ii) pertains whenever\nt > to. The particular forms of (21) for these two cases\nare as follows. Case (i), ((0,t) in the pulse) :","SEC. 5.6\nRINGING PROBLEM, ONE-DIMENSIONAL\n61\n(o,t)\n(a)\nto\nuto\n2\nb(o,t)\na(o,t)\nut\n2\nspace-time integration\nspace-time integration\npath for N°R+\npath for N°T\n(r,t)\n+\npulse world\nregion\n(b)\nb(r,+)=(r+ut)\na(r,t)= (r-ut)\nFIG. 5.7 Space-time path integration details.\nvt/2\n-2ar' dr'\nNOR+ (0,t) = Np\n0\nNo (1 - e-avt)\n(22)\nCase (ii), ((0,t) after the pulse) :","62\nNATURAL SOLUTIONS\nVOL. III\nvt/2\nNOR+(0,t) Np dr'\n=\n(23)\nEquations (22) and (23) describe the first order scattered\nradiance flowing in the negative direction of X, at r = 0.\nFor the radiance at a general space-time (r,t), we\nonce again require two cases: Case (i) pertains when\n(t-to)v vt, the primary radi-\nance is clearly zero, as may be seen by reviewing the geom-\netry of the space-time plane discussed earlier. Furthermore,\nthis value is approached by (24) as (r,t) approaches the\nlower boundary of the pulse's world region, i.e., the line\ndefined by r = vt. Hence NoR+ is uniquely defined throughout\nthe whole space-time - diagram.\nWe turn next to illustrate the evaluation of the second\norder scattering terms in (20). We first consider NOTER\nThis is interpreted to be the result of the operation of R+","SEC. 5.6\nRINGING PROBLEM ONE-DIMENSIONAL\n63\non NOT+. The latter, in turn, gives the primary scattered\nradiance in the direction + for a general space-time point\n(r,t):\ndr'\nin which we are to set t' = t-r'/v. A study of Fig. 5.7\nshows that, for the present source condition, we have\nNt(r',t') = 0 for r' < 0 (no source radiant flux in the di-\nrection + at any time for points r' < 0). Hence the inte-\ngration may begin at r' = 0, instead of a (r,t) (=(r-vt)/2).\nFurthermore, T(r',t') is constant of fixed value T for all\nr' and t'. Hence, Case (i), ((r,t) in the pulse):\nVOT+(r,t) NT dr'\nHence:\n(26)\n=\nCase (ii), , ((r,t) after the pulse):\nNOT+(r,t) = (\n(27)\nEquations (26) and (27) give the primary scattered radiance\nin the direction + E at a general space-time point (r,t)\nfutureward of the origin (0,0).\nWe are now ready to evaluate the second order terms.\nThus we have, Case (i), ((r,t) in the pulse)\n(r+vt)/2\nVoT.(r',t') (r',t') Tr'-r dr'\n(r+vt)/2\n= Ntp\nr\n(r+vt)/2\nr'e-2ar'\n= = Ntp e\ndr'\nr\n#\n(28)\n=","64\nNATURAL SOLUTIONS\nVOL. III\nCase (ii), , ((r,t) after the pulse)\n(r+vt)\n(r',t') p(r',t') Tr'-r dr'\n=\nr' e-2ar' dr'\near\nNtp\n=\n=\n(29)\nThe final term in the second order expansion of N(0,t,-)\nas given in (20) is N+R+T_, that is, the result of operating\non NOR+ with T.. Once again it is convenient to consider two\ncases: Case (i), ((r,t) in the pulse) :\nb(r,t)\nNOR.T_(r,t) = S.\nMQR_(r',t') t(r',t') Tr'-r dr'\nS\nb(r,t)\nNpt\ndr'\n(r+vt)\n-e-avt + (r-vt) [r-vt]\n(30)\nCase (ii), , ((r,t) after the pulse):","65\nRINGING PROBLEM, ONE-DIMENSIONAL\nSEC. 5.6\nb(r,t)\nS.\nNOR+T_(r,t)\nT(r',t') Tr'-r dr'\n=\nr\nb(r,t)-\n1.\nTr','t') Tr'-r dr'+\n=\nr\nb(r,t)\n/\nT : (r', t') Tr'-r dr'\n+\n(31)\nThe integration in Case (ii) is shown split into two parts:\nthat part of the integration over the segment of the space-\ntime path after the pulse, and that over the segment of the\nspace-time path in the pulse. The result of an integration\nover the futureward region of the pulse is in general not\nzero for secondary and higher order scattering.\nThe first integral in (31) uses Case (ii) for NoR+\nevaluated in (25), , and the second integral uses Case (i)\nabove by replacing the lower limit in (30) by (r+vt/2) - (vto/2).\nThe requisite value N(0,t,-) is now obtained by setting r=0\nin the appropriate cases in (24), (25), (28), , (29), (30),\n(31) and adding the appropriate terms, in accordance with\n(20).\nConcluding Observations\nWe have carried the evaluation of N(0,t,-) far enough\nto show the essentials of the natural solution procedure for\nthe one-dimensional time-dependent problem. It should be\nparticularly noted how each step builds on the preceding step\nand--manipulative difficulties aside- - - how each step is in\nprinciple directly constructable in a finite number of oper-\nations using elementary calculus. With the advent of ever\nmore sophisticated symbolic manipulation programs for general\npurpose electronic computers, it should eventually be possi-\nble to have a program which would permit the symbolic term-\nby-term integration of the natural solution series (15). We\nhave carried the solution of the present problem far enough\nto show that only integrals of the type\nC\ndr\nb","66\nNATURAL SOLUTIONS\nVOL. III\nwill be encountered in the natural solution for one-dimensional\ntime-dependent radiative transfer problems on homogeneous\nspaces. With such general information a program should in\nprinciple be possible which combines simple algebraic and\ncalculus manipulations, and which will give the two components\nof the nth term of (15) mechanically and relatively quickly.\nBy having the machine run out several more terms than the\nsecond order, obtained so laboriously above, a trained human\nlooking at the emerging terms could perhaps discern a pattern\nin this (or subsequently more complex problems) and thereby\nprepare for an inductive leap to the general term of the se-\nries. The advantages of symbolic over numerical integration are\nobvious. The former is exact at each stage whereas the latter is\nplagued by cumulative round-off errors. Once a symbolic inte-\ngration has been performed, it may then be evaluated for the\nparticular numerical case of interest.\nOne final observation can be made about the natural\nsolution of one-dimensional time-dependent problems. This\nconcerns extension of the analogy between the class of acous-\ntical and optical reverberations, or as they are more common-\nly called, \"electrical circuit transients. By studying the\nLaplace transform techniques of solving the problems of tran-\nsients in electrical circuits (see, e.g., Chapter IX of Ref.\n[39]], one sees the possibility of interpreting certain terms\nin the final solution as analogous to the nth order scatter-\ning terms developed above. This suggests the possibility of\na thoroughgoing theory, built along natural-solution lines,\nwhich should underlie and unify the particular ringing prob-\n1ems in the fields of optics, acoustics, transmission-line\ntheory and electromagnetics. Mathematicians can view this as\nextensions of the Neumann series to space-time linear settings.\nAn approach to such a unification can be based on the formal-\nities developed in the present chapter since many of the op-\nerator equations appearing here are clearly interpretable in\nterms of the concepts of each of the preceding fields.\n5.7\nOptical Ringing Problem. Three-Dimensional Case\nWe examine next how the natural mode of solution of the\nequation of transfer can be applied to the problem of deter-\nmining the time-dependent radiance field in a natural optical\nmedium. The program to be followed here is that which sys-\ntematically generalizes the developments of Sec. 5.1 to the\ntime-dependent case; in particular the generalizations of the\nR and T operators will be the key steps in the present dis-\ncussion. We begin by introducing an important geometrical\nconcept connected with the time-dependent problem.\nThe Characteristic Ellipsoid\nLet x and y be two points in an extensive natural opti-\ncal medium X. Suppose that at time t = 0, a spherical pulse\nof light is emitted from X. This pulse expands about X as\ncenter and at time r/v passes point y, where r is the dis-\ntance from x to y. Here V is the speed of light in X, as-\nsumed independent of location and time throughout this dis-\ncussion. Just after the wave front of the pulse passes y, a","RINGING PROBLEM: THREE-DIMENSIONAL\n67\nSEC. 5.7\nmultiply-scattered radiant flux field is generally incident\non y from all directions about y. We now ask: What is the\nregion of points in X which can send radiant flux to y at an\narbitrary time t > r/v? It is easy to see that at exactly\nt = r/v, this region is the straight line segment between X\nand y. Any points X of X off this line segment could not\nsend scattered flux to y because the detour, however, slight,\nwould delay the scattered flux's arrival time at y. 0 For times\nt of arrival at y such that t > r/v, such detours are possi-\nble to some extent. The region in which the scattering de-\ntours are possible and which allow arrival at y at time t is\ngenerally an ellipsoid of revolution with X and y as foci.\nThis may be seen by studying Fig. 5.8, and recalling that\ndefinition of an ellipsoid which characterizes it as the locus\nof points Z such that the sum of distances d 1(x,z) + d(z,y) is\na constant.\nCHARACTERISTIC ELLIPSOID AT TIME t\nd(x,z)\nd(z,y)\nZ\n0\nX\ny\nn\nr(y,E,t)\nd\nD = ut\nFIG. 5.8 The characteristic ellipsoid relative to the\nsource at X and receiver at y at time t.","68\nNATURAL SOLUTIONS\nVOL. III\nFor the case at hand these distances are all initially con-\nsidered in terms of times of travel t (x,z) and t(z,y) across\nthe respective distances and we are interested in all those\npoints Z in X such that:\n(xx) + d(z,y) = vt\n(1)\nThis defines at each instant t > r/v an ellipsoid of revolu-\ntion in X, with foci X and y. From (1) we see that the major\naxis of the ellipsoid is of length vt. We call the ellipsoid\nso defined, the characteristic ellipsoid E(x,y;t) associated\nwith X and y at time t > r/v. A useful polar representation\nof E(x,y;t) with y as pole, is given by the equation:\nD2 - d2\nr(y,5,t) = 2(0-d cos 6)\n(2)\nwhere 0 is the angle between the unit vectors E and n, as\nin Fig. 5.8, and where we have written:\n\"D\"\nfor\nvt\n\"d\" for d(x,y)\nThe eccentricity E of the characteristic ellipsoid E(x,y;t)\nturns out to be d/D. At time t such that t = d(x,y)/v = r/v,\nwe have E = 1. As time increases indefinitely, E decreases\nto zero, so that- if the space is infinite in all directions\nabout y--the characteristic ellipsoid approaches a sphere\nwhich takes on very nearly the polar form:\nThe exact spherical form of E(x,y;t) occurs at finite times\nif X = y, i.e. whenever d = 0. In such a case, E (x,x;t)\nbecomes the characteristic spheroid S(x;t) with radius vt/2.\nTime-Dependent R and T Operators\nand the Natural Solution\nWith the necessary geometrical preliminaries out of the\nway we can now adapt the R and T operators of Sec. 5.1 to the\ntime-dependent case. We shall limit the present discussion\nto a homogeneous steady medium X with point source at a fixed\npoint 0 and such that the characteristic ellipsoid E(0,x;t)\nis contained in X for all t under discussion. We shall then\nwrite:\n\"R\" for\n[\n(1)\nand:","RINGING PROBLEM: THREE-DIMENSIONAL\n69\nSEC. 5.7\n(x,E,t)\nr\n(4)\n\"T\"\nfor\nComparing this pair of operators with their namesakes\nin Sec. 5.1, we see that the essential difference between the\ntwo pairs rests in the limit of integration for T. Now we\ncan limit the integration to the characteristic ellipsoid\nECO, s s t ) , whereas before (see Fig. 5.1) the limit of integra-\ntion for T was generally the distance from X to the boundary\nof X in the direction -E.\nIf we go on to write:\n\"s1\" for RT\n(5)\nand then :\n\"Nn+1,, for N° s Superscript(1)\nfor every n > 0, it follows that we can construct the time-\ndependent natural solution for the time-dependent equation\nof transfer (4) of Sec. 3.15, just as in 5.4. In particular\nthe solution verification may be repeated line for line and\nculminating as in (4) of Sec. 5.4, with the form:\nN(x,E,t) = N°(x,E,t) + N*(x,E,t)\n(5a)\nbut now each term has a time-dependent interpretation.\nTruncated Natural Solution\nJust as in the steady case in Sec. 5.5 we may now trun-\ncate the time-dependent natural solution and obtain an esti-\nmate of the accuracy of the truncated solution. It turns out\nthat the truncation estimates of the time-dependent solution\ncan be much sharper than their steady state counterparts, OW-\ning to the use of the characteristic ellipsoid in the time-\ndependent computations. In this discussion suppose the source\nstarts at t = 0 and emits in an arbitrary manner thereafter.\nThe light field sweeps out from 0 as center in the form of a\nspherical field, building up radiant flux of all scattering\norders within the sphere as time goes on.\nLet N° be the maximum (or supremum, if need be) of the\ninitial radiance function N° over the sphere of radius vt,\ncenter 0. See Fig. 5.9. Then observe that:\nN° (1-e-ar(max),\n(6)\nfor every E in (1) at X and time t, where p = s/a and where we\nhave written:","70\nNATURAL SOLUTIONS\nVOL. III\nX\nE(0,x,t)\no\nut\nFIG. 5.9 The spherical wave front of the pulse has radius\nvt. . The characteristic ellipsoid relative to 0 and X at time\nt defines those points of the medium which can send flux to\nx from 0 at time t.\n\"r(max)\" for max r(x,E,t)\nE E\nHence:\nr (max) = (D + d)/2\nD = vt\n,\nBy letting x vary over the spherical region of radius vt,\ncenter 0, (6) leads to:\nN° (x,5,t) = n°s1 (x,E,t) < N° (1-e-avt)\n(7)\n,","71\nSEC. 5.7\nRINGING PROBLEM: THREE-DIMENSIONAL\nfor every X in X and E in E. This may be compared with (3)\nof Sec. 5.5. Using (7) we can estimate the upper bound of\nprimary scalar irradiance and radiant energy over X in terms\nof that of residual scalar irradiance or radiant energy. Us-\ning the basic idea contained in (7), we can construct a chain\nof inequalities for n-ary radiances. For (7) yields an upper\nbound of primary radiance over the sphere of radius vt, cen-\nter 0, and this upper bound now can be turned around to play\nthe role of No in the estimate of the next scattering order,\nnamely, N (x,E,t). Thus in general, since:\nNn = Nn-151\nit readily follows that:\nN°(x,5,t)\n(8)\nfor every X in X, E in E, and integer n > 0. This inequality\nreduces to (5) of Sec. 5.5 in the steady state, i.e., when\nt 80. The inequality (8) shows that for X sufficiently close\nto 0 and for small times t,\nN°(x,E,t) 21 (svt)\"\n(9)\nwhere S is the total volume scattering function.\nNow, just as in the steady state case of Sec. 5.5, we\ncan estimate the error of truncation of the natural solution\nseries. Thus using (8), we have:\nHence:\n(10)\nN(x,5,t) - 1 - [p(1-e-avt),\nfor every x in X, and E in E at time t. For large times,\n(10) reduces to (6) of Sec. 5.5. The space and source condi-\ntions giving rise to this estimate are stated at the outset\nof this discussion.\nIt should now be a relatively simple matter to reduce\nthe preceding analysis to pulselike sources at 0, such as\nthat considered in Sec. 5.6. The general method of analysis\nand its results developed between (6) and (10), , of course\nremain the same for such sources, but sharper time-dependent\nestimates of N° are now possible. These truncation estimates\nare evidently capable of a large variety of treatments and","72\nNATURAL SOLUTIONS\nVOL. III\nwith the general mode of analysis now clear, each special\ncase is best left to individual treatment by the interested\ninvestigator.\n5.8\nTransport Equations for Residual, Directly Observable,\nand n-ary Radiant Energy\nIn this section we shall prepare the way for the exten-\nsion of the concept of the natural solution of the equation\nof transfer to the radiant energy field in an optical medium.\nWe shall derive from the time-dependent equations of transfer\nfor the n-ary radiances the corresponding time-dependent\ntransport equations for n-ary radiant energy. We shall even-\ntually find that the latter equations are completely solvable\nin terms of simple closed algebraic forms in all homogeneous\noptical media. This fact will allow an important insight\ninto the structure of the associated time-dependent radiance\nfield in the same medium, and thereby shed further light on\nthe difficult optical ringing problem in natural optical\nmedia, introduced in Secs. 5.6 and 5.7. We begin with a dis-\ncussion and solution of the transport equation for zero-order\nradiant energy (or alternatively, the residual radiant energy)\nin an optical medium with an arbitrary source. Then the\ntransport equations for nth order radiant energy will be de-\nrived along with the transport equations for directly observ-\nable radiant energy. Throughout this section the optical\nmedium will be homogeneous with arbitrary sources of radiant\nflux distributed throughout. The volume scattering function\nis to be arbitrary but of fixed directional dependence, and\nunless otherwise specified the scattering-attenuati ratio\np is also arbitrary but fixed, with 0 < p < 1.\nResidual Radiant Energy\nIn order to help fix the main ideas in the present dis-\ncussion, let the optical medium X under consideration be de-\npicted as in Fig. 5.10, that is, as an extensive region X\nwith a boundary Y on each point y of which is incident a\nradiance distribution N (y, .) which may be extended into\nX\nto obtain initial radiance distributions N° (x, ) at each\npoint x in X, after the manner of (1) of Sec. 5.1. In the\nterminology of Sec. 3.10 (see, e.g., (4) of Sec. 3.10) N°(x,5)\nis the transmitted (or residual) radiance at X in the direc-\ntion E. The alternative term \"residual radiance\" will be\nparticularly appropriate in the context of the present dis-\ncussion, and so is singled out for special use.\nSuppose now that sources of radiant flux are present\nwithin X. This is a relatively new condition since (except\nfor the brief discussion of example 3 of Sec. 3.9), no sys-\ntematic explicit use of internal sources was required. We\nhave now arrived at a point in our developments where the\nadvent of the special radiometric concept needed for the\ndescription of internal sources takes place naturally. We\ntherefore hypothesize the existence of an emission radiance\nfunction Nn, defined for each time t in some time period and\nat each point X in X, and direction E in E. The dimensions\nof Nn are precisely those of N (radiance per unit length)","SEC. 5.8\nTRANSPORT EQUATIONS\n73\nNo(y,)\ny\nO\nY\nX\nr>ut\nX'\nY'\nFIG. 5.10 Computing residual radiant energy in medium X.\nand the use of Nr n may be best understood by keeping this\nequality of dimensions in mind. Physically, Nn(x,E,t) is\nintended to describe the radiance emitted at x and time t per\nunit length in the direction E. . We envision Nn(x,E,t) to be\ngenerated by some radiant emission mechanism in X. This mech-\nanism generally takes two distinct forms, which may be in\noperation singly or simultaneously. These forms are described\nin Sec. 19 of Ref. [251] and therefore need not be repeated\nat length here. It suffices for our present purposes to ob -\nserve that the radiance Nn (x,E,t) arises generally either\nthrough scattering by change in frequency from an arbitrary\nfrequency to the one under consideration, or through the\nemission processes of conversion of nonradiant energy to\nradiant energy.\nWhen internal sources, characterized by means of an\nemission radiance function Nn, are present throughout a medi-\num X, the initial radiance function N° is defined throughout\nX as follows. We write:","NATURAL SOLUTIONS\nVOL. III\n74\n\"N°(z,E,t)\" for N o o(x,5,t-r/v)Ty(x,5\nr\n,(x',5,t')T (x',5) dr'\n(1)\nN\n+\n0\nThis definition takes place in the same general geo-\nmetrical setting of (2) of Sec. 3.10 and reduces to (2) of\nSec. 3.10 when X is source-free - and the light field is in the\nsteady state. Here as usual Z = X + Er , and t' = t-r'/v\nA slightly more general definition can be written if X itself\nhas changing inherent optical properties. Also, if scatter-\ning with change of frequency is to be explicitly taken into\naccount, we may replace Nn by the true emission function Ne.\nThe details of this more general definition of N° may be\nfound in Sec. 22 of Ref. [251]. Such generality will not be\nrequired in any of our discussions, and so in the interests\nof simplicity of exposition, the present definition will be\nretained. Immediately forthcoming from (1) is the equation\nof transfer for initial radiance in the presence of internal\nsources:\n1 V an° at + E N° = - aN° + N n\no\n(2)\nThis is obtained by taking the lagrangian derivative of the\ndefinitional identity which (1) implies. That is, while\nfollowing in imagination a photon packet along a natural path\nthrough X, we differentiate the right side of (1) by adapt-\ning the general procedure used to obtain equation (3) of Sec.\n3.15 from equation (1) of that section. Now, we use D/Dt\ninstead of d/dr, where D/Dt is defined in (5) of Sec. 3.15.\nEquation (2) is a direct generalization of (2) of Sec. 5.2.\nWe are now ready to define the notion of residual\nradiant energy and to establish its various analytical repre-\nsentations. By setting n = 0 in the definitions (16) and\n(17) of Sec. 5.1 we obtain the definitional identity:\nU°(x,t) = 17\nN (x,E,t) do(E)\ndV(X)\n(3)\nX\nE\nU o (X,t) is the residual (or reduced or unattenuated) radiant\nenergy in X at time t. When X is understood and fixed\nthroughout a discussion (as in the present one) its name may\nbe dropped from the notation and we will write \"U°(t)\" for\nthe residual radiant energy. The term \"residual\" is partic-\nularly well adapted to the photon interpretation of light.\nFor in that interpretation, U° (t) is simply the radiant","SEC. 5.8\nTRANSPORT EQUATIONS\n75\nenergy content of X at time t associated with photons which\nhave not been scattered or absorbed relative to the incident\nand emission sources of flux on X. Thus the photons making\nup U° (t) are those left over and in their original unscat-\ntered state after t units of time have elapsed since the\nexternal sources over X (represented by No) and the internal\nsources over X (represented by Nn) have been turned on.\nTransport Equation for Residual Radiant Energy\nThe transport equation for residual radiant energy can\nbe obtained directly from (2) by applying the integral oper-\nations occurring in (3) to each side of (2). Thus, inte-\ngrating (2) term by term, the time derivative term becomes :\nI.I.\nN°(x,5,t) dd(E) dV(x) = au°(t)\n1 a\n(4)\nat\nX\n(1)\nNext, we observe that the spatial derivative term may be\nwritten as :\n. (EN°) o\n,\nsince E is a variable independent of location on X. Then we\nobserve that the integral:\n1.\nE N° (x,E,t) dn(5)\n(1)\ndefines the residual radiance counterpart to the vector\nirradiance function H, as developed in Sec. 2.8. If we\nwrite \"HO (x, t)\" \" for the preceding integral, we can then go\non to perform the remaining integration, as required by (3),\nto obtain:\n|\nV . H (x,t) dV(x)\nX\nwhich by the divergence theorem may be written as a surface\nintegral of H over the boundary Y of X; thus:\nH°(x,t)\nH (x,t dV(x) = -\nV\nn(x) dA(x)\n(5)\n.\n,\nX\nY","NATURAL SOLUTIONS\nVOL. III\n76\nwhere n (x) is the unit inward normal to X at each X on Y, and\nA is the area measure of Y. Suppose we write:\n\"F°(t)\" or for\n(x,t)\nn(x)\ndV(x)\n(6)\nY\nThus p° (Y, t) is the net inward flux to X across the boundary\nY of X. Finally we write:\n1.1\n\"Pn(t)\" or \"Pn(x,t)\" for\nN\n(x,5,t)\ndo(E)\ndV(x)\nn\nX\n(1)\n(7)\nThus Pn (x, t) is the input radiant flux over X at time t.\nAssembling the results summarized in (4)-(7), - equation (2)\nbecomes:\n(8)\ndt = - +\nwhere we have written:\n1\n\"Ta\"\n(9)\nfor\nva\nEquation (8) is the requisite transport equation for residual\nradiant energy in medium X at time t.\nThe Attenuation Time Constant\nThe quantity T defined in (9) and which has the dimen-\nsion of time, is the attenuation time constant for X. The\nsignificance of a will become apparent as the discussions\nof this section proceed. However, a preliminary insight into\nits significance can be obtained as follows. Imagine all of\nE 3 to be an infinite homogeneous three-dimensional optical\n3\nmedium about the origin 0. Let the initial radiant energy\ncontent of E3 be zero. Let the sources in E 3 be confined to\na point source at 0 which is turned on at time t = 0 and\nwhich pours radiant flux out into X at a constant rate Pn\n(i.e., Pr (t) is independent of t, t > 0). At any finite time\nt > 0 the spherical wave front traveling outward from 0 is of\nradius vt. For every t > 0, let Y' be any given sphere of\nradius r(>vt), and let X' be the medium bounded by Y', as\nin Fig. 5.10.\nUnder these conditions we have in particular p° (t) = 0\nfor every t, 0 t < r/v, and (8) reduces to:","SEC. 5.8\nTRANSPORT EQUATIONS\n77\ndu°(t)\ndt = - U° + Pn\n(10)\nwith initial condition:\nU°(0) 0 .\n(11)\nThe solution of (10), subject to (11), is:\n(12)\n=\nover the time interval (0, r/v), and where we have written:\n\"U°(00\"\nfor\nPnTa\nThe significance of Ta now springs into view if we\nrecall a well-known result of elementary circuit analysis\nconcerning the charging of a simple capacitance-resistance -\nDC circuit such as that depicted in Fig. 5. 11. When switch\nS\nis closed at time t = 0, battery B of voltage V pumps\nelectrons along the circuit A which has resistance R, until\nthe capacitor of capacitance C (initially discharged) is\nfully charged. The amount q (t) of charge on the capacitor\nCIRCUIT A\nB\nC\nR\nS\nWW\nFIG. 5.11 The analogy between an electric circuit and\nan optical medium.","78\nNATURAL SOLUTIONS\nVOL. III\nat time t > 0 is given by the equation:\n= 9(00) (1-e -t/RC)\n(14)\nwhere we have written:\n\"9(00)\"\nfor\nCV\n(15)\nWith the strong structural resemblance between (12) and (14)\nin mind, we can make the following pairings between the radi-\native transfer concepts and the electrical circuit concepts:\nIn the Optical Medium\nIn the Electrical Circuit\nThe medium X\nThe circuit A\nThe Source Point 0\nThe battery B\nV/R\nP\nn\nU° (t)\nq (t)\n1/v\nC\n1/a\nR\nT (attenuation time\nRC (circuit time constant)\na\nconstant)\nHence the buildup of residual radiant energy in an ex-\ntensive homogeneous medium X is analogous to the charging of\na capacitor in a simple DC capacitor resistance circuit. The\ninternal source of radiant flux Pn is analogous to the basic\ncurrent associated with the battery voltage V and circuit re-\nsistance R. The capacitance of the circuit is, for given\ngeometry, dependent on the materials of the plates. Thus the\nsmaller the speed of propagation in the material, the larger\nthe capacitance, and the larger the steady state charge 9(00).\nAnalogously, the smaller the speed of propagation V in the\noptical medium, all other things being equal, the larger the\nsteady state stored energy UO (00). On this basis (which is\nnot, however, logically compelling) we pair 1/v with C. Fur-\nthermore, the less dense the conducting material of the cir-\ncuit, the smaller is the conductance 1/R; similarly, the less\ndense the material of the optical medium the smaller is a.\nOn this basis we pair 1/a with R. The standard circuit time\nconstant RC then pairs off with Ta. This pairing of time\nconstants is relatively strongly suggested by direct compari-\nson of (12) and (14), whereas the suggested pairings of 1/v\nwith C and 1/a with R are not as strong and, indeed, the pair-\nings may be switched without affecting the important pairing\nof Ta with RC, the pairing of principal interest at the mo-\nment. However, the indicated optical counterparts to R and\nC are quite interesting to contemplate, particularly when it\nappears that the analogy between the medium X and the circuit\nA can be extended quite far by establishing a link with the\nanalogies summarized in the closing paragraph of Sec. 5.6.\nApparently, if these analogies can be extended far enough,","SEC. 5.8\nTRANSPORT SOLUTIONS\n79\nthen with sufficient care and ingenuity, some of the time-\ndependent radiative transfer problems can possibly be solved\nby electrical (or even acoustical) analog methods in which\nthe time-dependent electrical (or reverberating acoustical)\nfield replaces the radiant field.\nJust as in the electrical case, the attenuation time\nconstant Ta is the time required for the residual radiant\nenergy to attain 63 percent of its steady state value. Below\nis given a table for the values of U° '(t)/U°(00) for various\nvalues of t in terms of multiples of Ta\nTABLE 1\nValues of i°(t)/U°(00) for various values of\nt in terms of multiples of Ta\nt nT\nU°(t)/U°(00)\na\nT a\n0.63\n2T\n0.86\na\n3T\n0.95\na\n4T\n0.98\na\n5T\n0.99\na\nGeneral Representation of\nResidual Radiant Energy\nThe solution (12) of the differential equation for\nresidual radiant energy is a special case of the more general\nsolution:\nt\n(t'-t)/Ta p.(t') dt'\n(16)\ne\n0\nwhere we have written:\n\"Po(t)\" for +\n(17)\nThe solution (15) represents the residual radiant energy in\na general homogeneous optical medium X with known combined\ninternal and external source flux function Po, as given by\n(17).","80\nNATURAL SOLUTIONS\nVOL. III\nTransport Equation for n-ary\nRadiant Energy\nWe derive next the transport equation for the second\nmain radiometric concept of this section, the n-ary radiant\nenergy Um (t). The definition of Un (t) was given in steady\nstate form in Sec. 5.1. Thus we have for every nonnegative\ninteger n,\nN°(x,5,t) d2(E)\ndV(x)\n(18)\nWe shall write \"Un(t)\" for un(x,t) whenever X is understood.\nStarting with the time-dependent radiance field in X\nwe apply to (5) of Sec. 5.7 the lagrangian derivative oper-\nator D/Dt in exactly the way d/dr was applied to (11) of Sec.\n5.1 to yield (1) of Sec. 5.2. We have, as a consequence, for\nevery integer n, n > 1:\nFNn-a . = -\n(19)\nwhich is the time-dependent equation of transfer for n-ary\nradiance Nn , and which is to be compared to (2) above and\n(1) of Sec. 5.2. Applying the integral operations in (18)\nto each member of each side of (19), we find that:\n1.1.\ndv(x) = 3U (t)\ndn(E)\n(20)\n=\n.\nX\nWe write:\n\"H\"(x,t)\"\nEN\"(x,5,t) d2(5)\nfor\n(21)\nE\nand\n\"pn(t)\" or \"Fn(Y,t)\" for\n(22)\nY\nwnere n(x) is defined as in (6). Finally we observe that:","SEC. 5.8\nTRANSPORT SOLUTIONS\n81\n1.\ndV(x) = S un-1(t)\nN°(x,E,t) ds(E)\n(23)\nWith the results (20) through (23) in mind, (19) yields\nup the following transport equation for n-ary radiant energy:\ndun(t) dt = - Un(t) +\n(24)\nS\nfor every integer n > 1. The main details of derivation of\n(24) thus proceed as in the case of the residual radiant\nenergy (8). Here we have written:\n\"Ts\" for 1/vs\n(25)\nIn equation (24), , (t) is the net inward radiant flux\nacross the boundary Y of X at time t. The radiant flux pn(t)\nhas scattering order n relative to that of P°(t). A term by\nterm interpretation of (24) is instructive: the time rate of\nchange of n-ary radiant energy in X at time t is the sum of\na growth term un-1(t)/Ts (which is the rate of conversion of\n(n-1)-ary scattered energy into n-ary scattered energy), a\ndecay term -Un(t)/To (which is the rate of conversion of\nn-ary energy into (n+1)ary energy and nonradiant energy),\nand\nfinally a general net rate of growth term giving the net\nbalance of influx and efflux of n-ary radiant energy across\nthe boundary of X. The quantity Ts is the scattering time\nconstant for the medium X. It is a concept which helps write\n(24) in a uniform manner in terms of the fundamental timelike\nquantities Ta and Ts.\nTransport Equation for Directly\nObservable Radiant Energy\nThe radiant energy U associated with directly observ-\nable radiance N, using a standard radiance meter is called\nthe directly observable radiant energy. This energy is to\nbe held both in conceptual and empirical contrast to the\nn-ary radiant energy Un, n > 1, which is not directly observ-\nable in practice. (The residual radiant energy is indirectly\nobservable using techniques alluded to in Sec. 3.10 and Sec.\n16 of Ref. [251].) We now derive the transport equation for\nU(t). We begin with the definitional identity:\nV 1.1\nU(x,t) =\nN (x,E,t) do(E)\ndV(x)\n(26)\nX\nE","82\nNATURAL SOLUTIONS\nVOL. III\nbased on (2) and (12) of Sec. 2.7. As usual we shall drop\nreference to X, when X is understood.\nStarting with the time-dependent radiance equation (4)\nof Sec. 3.15, we now apply the integral operations in (26) to\neach side of the transfer equation and obtain, in a manner\nanalogous to that culminating in (8) and (24) above, the re-\nsult:\ndU(t)\n(27)\n=\n+\n-\ndt\nThis is the transport equation for directly observable radi-\nant energy. In the equation we have written:\n1\n\"Ta\" for va\nand where a in turn is the value of the constant volume ab-\nsorption function in X. Furthermore, we have written:\n\"P(t)\" or \"F(Y,t)\" for\nH(x,t) n(x)dA(x)\n(28)\nY\nThe unit vector n(x) is defined as in Fig. 5.10, and\nso P(t) is the net inward radiant flux into X over the bound-\nary Y of X.\nThe Natural Solution for Directly\nObservable Radiant Energy\nIt is a relatively easy matter to verify (using (5a)\nof Sec. 5.7) that:\n(29)\nholds for every t > 0, where U(x,t) is defined as in (26)\nand the (X,t) are defined as in (18). Thus, once each\nUJ (x, t), j > 0, is known, U(X,t) is known and computable.\nEquation (29) represents the natural solution of the directly\nobservable radiant energy.\nIn the case of radiant energy the natural solution pro-\ncedure is not as vitally essential in the solution of U(t) as\nin the natural solution procedure for the case of radiance in\nSecs. 5.6 and 5.7. Indeed, the solution of (27) is written\ndown quite readily, assuming P(t) and Pn(t) given. Thus,\nwriting,\n\"P(t)\" for P(t) + Pn(t)\n(30)","83\nTRANSPORT SOLUTIONS\nSEC. 5.8\nwe have, analogously to (16) :\n(t'-t)/Ta p(t') )\ndt'\n(31)\nU(t) = U(0)e\ne\n0\nThe quantity T a is the absorption time constant for X and is\nrelated to Ta and Ts as follows:\n1-1-1\n(32)\n+\nT\nS\nThe natural solution procedure for radiant energy is, however,\nquite useful in throwing light on the inner workings of time-\ndependent light fields, for the solutions of the transport\nequations for Un are readily obtained in simple closed forms\nwhich are quite amenable to all manners of explicit, rear-\nrangements and manipulations. Some of the properties of time\ndependent radiant energy fields will be explored in the next\nfew sections.\nWe conclude this section with an important observation\nwhich will facilitate the studies below. This concerns the\nconnection between the net fluxes pn(t), , n > 0 occurring in\n(8) and (24), and the net flux P (t) occurring in (27). This\nconnection is established by means of the natural solution\nrepresentation of the directly observable radiant energy U(t)\nas given in (29). Thus, by summing over all n > 1 in (24) :\nn=1 = +\nand adding to these terms the corresponding terms of (8), we\nobtain:\n+\n+\ndt\nn=0\na\ncomparing this with (27) we conclude that:\n(33)\n.\nSolutions of the n-ary Radiant Energy Equations\n5.9\nWe shall now solve the transport equation for n-ary\nscattered radiant energy for every n > 1, and deduce from\nthe solutions several interesting properties of the scatter-\ning order decomposition of natural light fields. These prop-\nerties are both of intrinsic interest and of use in further-\ning the natural solution of the radiance field in optical","84\nNATURALS SOLUTIONS\nVOL. III\nmedia. They are also helpful in studying the light storage\nproblems in such media. These latter two applications will\nbe considered in Secs. 5.12 and 5.13. For the present we\nconcentrate on the immediate mathematical and physical fea-\ntures of the transport equations for Un. Throughout this\nsection, unless specifically noted otherwise, the optical\nmedium will be as in Sec. 5.8, that is homogeneous, with\narbitrary sources, arbitrary directional structure for o,\nand arbitrary fixed P, 0 < p < 1.\nNatural Integral Representations\nof n-ary Radiant Energy\nStarting with (24) of Sec. 5. 8, , we treat the indicated\ndifferential equation, for given n > 1, as an ordinary dif-\nferential equation with unknown function Un and known func-\ntions un-1 and pn , and with given parameters Ta, Ts. The\ninitial condition for Un is:\nUn(0) = 0\n(1)\n,\nfor every n > 0. The general solution under this condition\ncan therefore be patterned after (16) or (31) of Sec. 5.8\nwith the initial values set to zero. Specifically:\nt\ndt'\n(2)\n= e\n0\nNow, to simplify matters we shall assume that:\npn(t) =\n(3)\nfor every n > 0 over a given interval (0, t1) of time which\nis to include the time interval in which we shall be inter-\nested in the solutions of (24) of Sec. 5.8. Physically this\nmeans in effect that the collective expanding wave fronts of\nall sources in X are completely within the boundary Y of X\nover the time interval (0, t1). See Figure 5.10. With as-\nsumption (3) in force, (2) becomes:\nt\ndt'\ne\n0\nt\n= e t/Ta S\nt'/Ta un-1(t')) dt'\n(4)\ne\n0","SOLUTIONS OF EQUATIONS\n85\nSEC. 5.9\nwhich holds for n > 1 and 0 < t < t 1 . The form of (4) sug-\ngests a recursive construction of Un (t) starting with n = 1\nand using knowledge of U°(t) as given in (16) of Sec. 5.8.\nBy (3) , Po (t) in (16) of Sec. 5.8 uses only the internal\nsource function Pn. Hence Un(t) should be expressible in\nterms of UO (t) (or Pn (t)) along with Ts and Ta. Thus, start-\ning with (4) now applied to Un-1(t), n-1 1, we have:\ne\ndt\"\ndt'\n0\n0\n\"\n(t-t') et'/Iaun-2(t') dt'\n(5)\n=\n0\nThis process can be continued as long as the scatter-\ning order in the integrand is greater than zero. The pattern\nforming in (4) and (5) is clear. Applying (4) once again,\nnow to Un-2 the pattern is crystallized:\n(6)\n.\nThus, applying the representation (4) in all k times, OK\n< n-1, we have for Un(t):\n(t')\ndt'\n(7)\nIf in (7) we let k = n-1, then the desired integral repre-\nsentation of Un(t), t1 is obtained:\ndt'\n(8)\n0\nor, in terms of Pn:","86\nNATURAL SOLUTIONS\nVOL. III\ndt'\n(9)\nEquations (8) or (9) are the desired integral representations\nof Un(t). Observe that (8) holds for n > 1 and (9) holds for\nn >0.\nNatural Closed Form Representations\nof n-ary Radiant Energy\nThe formulas (8) or (9) are the requisite representa-\ntions of Un (t) under the given initial conditions (1), and\nthe conditions on the medium hypothesized in (3) and at the\noutset of this section. In order to evaluate the integrals\nwe must specify the nature of Un or Pn over the time inter-\nval (0, t1). We now illustrate the use of (9) by choosing\ntwo important instances of Pn. The first instance is where\nPr is the Dirac-delta function centered at t = 0 and with\nn\nradiant energy content Un. The second instance is where Pn\nis constant valued over (0, t1) with its constant magnitude\ndenoted by \"Pn\". In the first instance, we have:\n(10)\n=\nfor\nover the interval (0,t1) and for n > 0. We shall refer to\nthis case as the optical reverberation case (cf. the intro-\nduction to Sec. 5.6).\nThe second instance yields the representation:\n(11)\n=\nfor\nPn(t) = Pn.\n.\nover the interval (0, t1) and for n > 0. Here UO(00) is as\ndefined in (13) of Sec. 5.8. These two specific instances\nof (9) are verified by direct integration of (9) in each\ncase.","SEC. 5.9\nSOLUTIONS OF EQUATIONS\n87\nGeneral Integral Representations\nof n-ary Radiant Energy\nThe integral representation (9) of Un will now be gen-\neralized to the case for which the initial conditions on UJ,\nj < n, are arbitrary. That is, we now relax the conditions\n(1). However, we shall retain condition (3). The resultant\nrepresentation will permit the construction of relatively\ngeneral representations of the time-dependent n-ary radiant\nenergy in a homogeneous medium for which the wave fronts of\ninternal sources have not yet passed the boundaries. Thus,\nby successive applications of the type of solution displayed\nin (16) of Sec. 5.8, we eventually arrive at :\n= +\ne\ndt'\n+\nT5\nS\n0\n(12)\nThis is the desired generalization of (9), which holds for\nn 120.\nStandard Growth and Decay Formulas\nfor n-ary Radiant Energy\nof the infinite variety of possible time-dependent\nradiant energy fields attainable in principle via (12), two\ntypes stand out as particularly interesting. These are suf-\nficiently instructive to isolate and set un here as standards.\nThe first of these light fields is that given by (11) above.\nThis equation we shall call the standard growth formula for\nUn. Recall that in this case the initial values for the UJ.\nj < n, are all zero and that Pn is a positive constant over\nsome time interval (0, t1). Suppose we write:\nfor e -t/Ta\n\"Fn(t/Ta)\"\n(13)\nj=0\nThen we summarize the standard growth formula as follows: If\n(a) The optical medium is homogeneous.\n(b) un(0) = 0 and Pn(t) = Pn for t in (0,t1) and n > 0.","88\nNATURAL SOLUTIONS\nVOL. III\n(c) pn(t) = 0 for t in (0,t1) and n 0.\nThen:\n(14)\nfor every t in (0,t1) and n > 0.\nThe second standard case is that which describes the\ndecay of the n-ary light field from a given steady state\nlevel. Thus if an opaque curtain were suddenly drawn over\nthe ocean in which previously all internal radiant sources\nwere turned off, the following standard decay formula for\nwould describe very closely the decay of Un (t) for t > 0 for\nevery n > 0 in the ocean; thus: If\n(a) The optical medium is homogeneous.\n(b) and Pn = 0 for t in (0,t1) and\nn - 0 .\n(c) pn(t) = 0 for t in (0,t1) and n > 0.\nThen :\n(15)\n=\nfor every t in (0,t1) and n > 0.\nA few words about condition (b), the initial condition\nfor Un are in order. An examination of the general repre-\nsentation (11) of Un(t) shows that at steady state (i.e., the\nlimit of Un(t) as t 00) the various magnitudes Un (00) are not\narbitrary. Indeed, they generally depend on Pn and the ini-\ntial values Un (0), as explicitly shown in (12). Hence when\na steady state light field begins to decay after sources have\nbeen turned off, the initial values Un (0), n > 0 are general-\n1y not expected to be independent of each other. For example,\nif\nthe standard growth conditions are in effect, then (11)\nshows that:\nL\n=\n=\nfor every n > 0. Thus we see that the standard decay formula\nis intended to describe the decay of a light field which has\nbeen attained under standard growth conditions as given by\n(14) for t\n80.","SEC. 5.9\nSOLUTIONS OF EQUATIONS\n89\nWe can combine the standard growth and decay formulas\n(14) and (15) into a single standard formula as follows:\nIf\n(a) The optical medium is homogeneous.\n(b) Un (0), n > 0 is given as steady state value\nattained under a previous standard growth condi-\ntion and Pn(t) = Pn for t in (0,t1):\n(c) pn(t) = 0 for t in (0,t1) and\nThen:\nUn(t)==\"n\"(w) = +\n(16)\nand where Un(00) is determined by (14) for the present source\ncondition. As an interesting consistency check, observe that\nif the previous steady state condition (b) above is induced\nby Pn as given in (b), then un (t) in (16) is independent of\ntime, because Un (0) = Un (00).\nAs a final standard type of growth and decay formula,\nwe consider the case in which a standard growth begins at\nt = 0 and continues until time to, at which time the source\nis shut off and the existing light field decays from that\npoint on until some arbitrary time t1 under standard decay\nconditions. Equation (12) shows that the decay formula is:\n=\n+\n+\n(17)\nto and n > 0. For t to, Un (t) is given by\nfor\n(11). Formula (17) may be used to describe the transient\nradiant energy fields induced in large bodies of air or\nwater by radiant sources which are intermediate between the\nDirac-delta pulse and the steady source described in (10)\nand (11). Since any source output Pn over a time interval\n(0,to) can be approximated by a step function, we see that\nby superimposing fields of the type given by (17), we can\nrepresent n-ary radiant energy fields induced by finite non-\nconstant sources under the general conditions of this section.\n5.10 Properties of Time-Dependent n-ary Radiant Energy\nFields and Related Fields\nWe now turn to examine in detail some of the more in-\ntuitively interesting properties of time-dependent radiant\nenergy fields. In order to present the properties in their\nsimplest forms, we shall adopt for study throughout this sec-\ntion a light field evolving under either standard growth or\ndecay conditions or optical reverberation conditions in an","VOL. III\nNATURAL SOLUTIONS\n90\noptical medium X over a time interval (0,t 1) (Sec. 5.9). It\nwill be clear from the results stated below how analogous or\ncomplementary statements and properties can be formulated\nunder still more general conditions. We begin with a study\nof some of the fine-structure properties of n-ary radiant\nenergy fields and then go on to a formulation of the various\nrepresentations of related radiant energy quantities.\nSome Fine-Structure Properties\nof n-ary Radiant Energy\nProperty 1. Let t be a fixed time in (0,tq). Then the\nsequence UO(t), , U1 (t), Un(t), of n-ary radiant ener-\ngies at time t is a monotonic decreasing sequence with limit\n0. The proof of this property is based on (14) of Sec. 5.9.\nBy (13) of Sec. 5.9 we see that:\n(1)\nHence by noting that 0 < p < 1, we see that:\nlim\nso that\nfor t in (0,t1). As for the monotonicity of the sequence, it\nsuffices to note that:\n(2)\n1-Fn(t/Ia)\nand that F (t/Ta) increases monotonically, with n, to unity.\nThis may be seen by verifying that:\n-\nfor every n > 0 and every positive t. The limit part of\nproperty 1 follows also from (2) by using the ratio test for\nconvergent infinite series.\nProperty 2. Under standard growth conditions,\nfor every t in (0,t1). The proof is immediate. For example,\none may use (14) of Sec. 5.9 directly with the calculus, or\none may use algebra with the fact that dun(t)/dt is the","SEC. 5.10\nTIME-DEPENDENT - FIELDS\n91\ndifference given in (24) of Sec. 5. 8, with pn(t) = 0. Prop-\nerty 2 shows in particular that each n-ary radiant energy\ncomponent increases monotonically with time. Property 2 is\nto be compared with:\nProperty 3. Under standard decay conditions\n-\nfor every t in (0,t1). The proof is immediately obtainable\nfrom (15) of Sec. 5.9. Hence the rates of growth and decay\nof n-ary radiant energy under standard conditions are, to\nwithin a constant multiplicative factor, identical in struc-\nture within a given space.\nProperty 4. Under standard growth conditions,\nuntk(t)\nun(t)\nfor every t in (0,t1) and positive integers n, k. This fol-\nlows from property 2 and (24) of Sec. 5. 8 with pn(t) = 0.\nThe inequality is reversed under standard decay conditions.\nProperty 5. In the steady state of the standard growth\nprocess,\nfor every n > 0. Hence:\nuntk(a) . .\nUn(00)\nfor every pair n, K of nonnegative integers.\nProperty 6. In the optical reverberation case (equa-\ntion (10) of Sec. 5.9) we have the ratio:\nfor n > 1 and t in (0,tq). Thus, the ratio of successive\nn-ary radiant energy contents increases linearly with in- -\ncreasing time and decreases hyperbolically with increasing\nn.\nProperty 7. In the optical reverberation case with\npoint source (equation (10) of Sec. 5.9) Un(t), for a given\nscattering order, attains a maximum when the radius of the\nwave front is n times the attenuation length 1/a. Further,\nfor any given total volume scattering value s and time t in\n(0,t1), that component Un(t) is maximal whose order n makes\nthe absolute value of\n(vts) - 1 = (t/nT5) - 1","VOL. III\nNATURAL SOLUTIONS\n92\na minimum. The geometric content of properties 6 and 7 are\nsummarized in part (a) of Figure 5.12.\nProperty 8. In the optical reverberation case, the\ndirectly observable radiant energy U(t) is given by:\nU(t) = Un e-t/Ta\nThe proof rests on (10) of Sec. 5.9 and (29) of Sec. 5.8 and\nthe simple calculation:\nj\nt\nT\nU(t) = and j=0 U'(t) = Un e-t/Ta j=0\nS\n= -t/Ta. t/Ts = U e -t/Ta\nin which (32) of Sec. 5.8 was used. It follows immediately\nfrom property 8 that, in optical media with no absorption,\ni.e., for which a = 0, U(t) is independent of t in the rever-\nberation case. Part (b) of Figure 5.12 gives plots of Un (t)\nfor the first four scattering orders in the optical reverber-\nation case in which a = 0 and Un = 1. In the figure we have\nUn(t)\no\n2\n4\nt/Ta\no\n6\n2\n8\n3\nn\n10\n4\nFIG. 5.12 (a) The geometric version of property 7 of\nscattered radiant energy.","SEC. 5.10\nTIME - DEPENDENT FIELDS\n93\nOPTICAL REVERBERATION CASE\n(IO) of sec. 5.9\n0.6\n0.5\n0.4\nn= I\n0.3\nn=2\nn=3\nn=4\n0.2\n0.1\nI\n2\n3\n4\n5\n6\n7\n8\n9\n10\nT= 4/Ts\nFIG. 5.12(b) The geometric version of property 7 of\nscattered radiant energy. - - Concluded.\nwritten \"I\" for t/Ts. Thus the medium is a nonabsorbing me-\ndium (p = 0) with conserved directly observable energy. Note\nhow the scattering order components of U (t) well up one after\nanother, reaching their maxima, as described by property 7.\nFinally, according to property 8, the sum of the ordinates of\nall the curves at each T should add up to unity.\nScattered, Absorbed, and\nAttenuated Radiant Energies\nWe now round out the roster of the types of radiant\nenergy fields most commonly encountered in theoretical dis-\ncussions of time - dependent light fields. Until further\nnotice, source conditions are arbitrary and with P(t) = 0.\nSo far we have introduced the residual radiant energy\n( (3) of Sec. 5.8), , the n-ary radiant energy ( (19) of Sec.\n5.8), and the directly observable radiant energy ( (26) of","VOL. III\nNATURAL SOLUTIONS\n94\nSec. 5.8) with its natural representation ((29) of Sec. 5.8).\nBy writing:\n\"U*(t)\" for E. U3(t)\n(3)\nj=1\nwe define the scattered (or diffuse) radiant energy (in X)\nat time t. We then have from (29) of Sec. 5.8 the following\nradiant energy counterpart to the time-dependent integral\nequation of transfer (cf. (4) of Sec. 5.4):\nU(t) = U°(t)+U*(t)\n(4)\nUsing the emission radiant flux function Pn n and recalling\nthat we have set P(t) = 0 for t in (0, t1) let us write:\nI\nt\nU°(t)\n(5)\n\"U(t;a)\"\nfor\ndt'\n-\n0\nfor t in (0, t1). The meaning of this new radiant energy\nbecomes clear when it is recalled that U°(t) is the residual\n(i.e., the unattenuated) radiant energy. Therefore, since\nthe integral gives the total radiant energy input to the me-\ndium, the difference in (5) must be all the energy present at\ntime t that has undergone attenuation (absorption or at least\none scattering operation). We call U(t;a) the attenuated\nradiant energy (in the medium X) at time t. Only part of\nU(t;a) is detectable. In fact, the detectable part of U(t;a)\nis precisely U* (t). Thus let us write:\n\"U(t;a)\" for U(t;a) - U(t;s)\n(6)\nwhere, for uniformity of notation and heuristic purposes, we\nhave agreed momentarily to write\n\"U(t;s)\" for U*(t)\n(7)\nThen from (6) we have:\n(8)\nU(t;a) = U(t;a) + U(t;s)\na formula remarkably similar in structure to the basic rela-\ntion:\na = a + s\nderived from (4) of Sec. 4.2. We call U(t;a) the absorbed\nradiant energy (in X) at time t. The absorbed radiant energy\nis radiant energy that has disappeared from the present radi-\nometric scene via absorption processes.","SEC. 5.10\nTIME-DEPENDENT FIELDS\n95\nRepresentations of U(t;a),\nU(t;s), and U(t;a)\nThe transport equations for the three auxiliarly radi-\nant energies and their solutions are relatively easy to ob - -\ntain. We shall illustrate the power of the natural solution\nprocedure by basing the derivations of these equations and\nrepresentations directly on the knowledge of the n-ary radi-\nant energies developed so far.\nWe begin with the derivation of the differential equa- -\ntion for attenuated radiant energy U(t;a). From the defini-\ntion (5) we have\ndu°(t)\nFrom (8) of Sec. 5.8 we obtain:\ndu(tia)=UQ(E) =\n(9)\ndt\nrecalling that the condition pn (t) = 0 is in force for every\nn > 0 (hence po(t) = 0, , in particular, holds). This elegant\nformula for the growth rate of U(t;a) shows perhaps most\nclearly the reservoir source of U(t;a) (namely, U°(t)) and\nthe main line which taps the reservoir (namely, Ta, i.e.,\nattenuation). At standard steady state (9) shows that:\ndu(@;a)=Pn\n(10)\nThus in the steady state attained under standard growth con-\nditions the rate of increase of U(t;a) is precisely the in-\nput rate Pn, so that attenuated radiant energy in the medium\nincreases as fast as it is put into the medium by the source.\nNext we consider the scattered radiant energy U(t;s),\nor \"U* (t) 11 as we would call it ordinarily. The representa-\ntion (3) of U(t;s) gives rise to the associated differential\nequation for U(t;s) by computing (with the help of (24) of\nSec. 5.8) the following derivative:\ndU(t;s) =\nj=1\n=\n= 1-1-1 1 S (t;s) + U°(t)","96\nNATURAL SOLUTIONS\nVOL. III\nHence:\nU(t;s)\n(11)\ndt\nHere we begin to see some of the utility of the various\ntime constants Ta, Ts, Ta. They serve to remind one of the\ncorrect dimensions of each term in an equation or representa-\ntion, and they serve also to show the physical mechanism as -\nsociated with that term. Thus we see at a glance from (11)\nthat the rate of growth of U(t;s) - - the scattered radiant en-\nergy- - - is augmented by scattering of residual radiant energy\nUo(t) and decreased by absorption of scattered radiant ener-\ngy U(t;s). .\nThere is no need to solve (11) since we need only sum\nthe representations of the UJ(t) in (3) to obtain the desired\nrepresentation of U(t;s). Thus, under standard growth condi-\ntions ((14) of Sec. 5.9):\nUtis)t1-F(a)\nt/Ta\nHence:\nUts)1ta =\n(12)\nAn alternate representation of U(t;s) is obtained by\ndistributing (00) throughout the preceding representation\nThe result is:\nU(t;s) =\n(13)\nFrom this we obtain immediately the representation for the\ndirectly observable radiant energy. For, by (4) and (13),\nwe have:\nU(t) =\n(14)\nwhich is clearly a solution of (27) of Sec. 5.8 under stan-\ndard growth conditions.\nFinally the absorbed radiant energy is represented\nmost simply as :","SEC. 5.10\nTIME -DEPENDENT FIELDS\n97\nU(t;a) 0 = P n t - U (t) =\n(15)\nunder standard growth conditions. This representation fol-\nlows from (4), , (5), and (8). A representation under more\ngeneral growth conditions is obtained by retaining the inte-\ngral in (5). The differential equation for U(t;a) under\nstandard growth conditions is readily obtained:\n(t;a) = dU(t;a) - dU(t;s)\ndt\ndt\ndt\nUO (t)\nU° (t)\nU(t;s)\n+\n=\n-\nT\nT\nT\nS\na\na\nU(t;s) + UO(t)\n=\nT\na\nHence:\ndu(t;a) = U(t)\n(16)\ndt\nT\na\nWe have made a point of deriving the differential equation\nfor U(t;a) so as to make possible the comparison between it\nand (9). The comparison lends valuable insight into the\ngeneral roles of scattering and absorption in radiative\ntransfer phenomena. Thus, in the case of (16), the reservoir\nsource for U(t;a) is the directly observable radiant energy\nand the energy is tapped via the process of absorption.\n5.11 Dimensionless Forms of n-ary Radiant Energy Fields and\nRelated Fields\nWe shall now develop the dimensionless forms of the\nvarious equations and solutions for n-ary radiant energy,\nresidual radiant energy, directly observable radiant energy,\nand the related energy fields introduced in Sec. 5.10. We\nshall also explore the various possibilities for the defini-\ntion of time constants which are to characterize time-depend-\nent light fields in optical media. Before going on to the\ndetails of the discussion, some preliminary observations on\nphysical theories using dimensionless concepts are in order.\nWhen the analytical representation of a natural phenom-\nenon can be placed into such a form that the terms of the new\nrepresentation are dimensionless, this usually indicates that\nthe given phenomenon is a member of an inclusive class of\nphenomena whose members exhibit a common mathematical repre-\nsentation, but which ostensibly may have different external\nappearances. The mathematics used to represent the concepts\nof electrical network theory is a good example of this kind;\nfor the mathematical procedures employed in that theory are","NATURAL SOLUTIONS\nVOL. III\n98\noften equally applicable to problems in mechanical dynamics.\nAs a result of this common understructure, researchers in\neach of these fields have enriched the mathematical methods\nof the other by noting the applicability of the same set of\ntechniques in each field of study. (See Sec. 5.15.)\nSome of the discussions in this chapter have already\nindicated that the set of transient radiant energy phenomena\nmay be treated as a member of the class of natural phenomena\nwhich includes electrical network behavior ((14) of Sec. 5.8 ;\nsee also concluding comments of Sec. 5.6). We can alsopoint\nout that the natural mode of solution leads to radiant energy\nequations which have the same mathematical structure as the\nequations governing the growth and decay of families radio-\nactive substances. In this case, the counterparts to n-ary\nradiant energy Un are the population counts Pn of the nth\nspecies Sn of radioactive atoms which are the decay products\nof species Sn-1 and where Sn itself decays into species Sn+1\nStill other and ostensibly different natural phenomena share\nthe same mathematical substructure as the time-dependent\nradiant energy equations. For example, interacting biologi-\ncal species Sn often are arranged in a predatory hierarchy\nso that members of species Sn prey upon those in species\nSn-1 and are in turn preyed upon by those in species Sn+1\nThe time-dependent equations governing the population counts\nof the nth interacting species be they animal, vegetable, or\nmineral--often have a common fundamental mathematical core\nwhich is obtainable by stripping away the accidental topog-\nraphy of the equations associated with the particular case.\nThe advantages of attaining such dimensionless formulations\nlie in the resultant conceptual simplifications and economy\nof description of natural processes.\nThe casting into dimensionless form of the basic dif-\nferential equations of transient radiant energy and their\nassociated solutions has practical as well as conceptual ad-\nvantages. For example, dimensionless formulas allow the\ninclusion of a wide range of special cases in a single tabu-\nlation or graph, the specific case being recoverable after\nmultiplication by a suitable factor. The dimensionless forms\nthus compress a huge amount of particular numerical informa-\ntion into a relatively small space.\nWe turn now to the details of the discussion. For\nsimplicity we shall adopt throughout this section the stan-\ndard growth conditions in a homogeneous optical medium (re:\n(14) of Sec. 5.9). The developments of this section may\nserve as a pattern for generalizations to the nonstandard\ncases.\nConversion Rules for\nDimensionless Quantities\nAn examination of the various analytic representations\nof U°(t), U* (t), U(t), and related radiant energy concepts\nin Sec. 5.10, with an eye toward achieving dimensionless\nversions of these representations, brings to light the essen-\ntial observation that, without exception, each of the repre-\nsentations within the standard growth context obtains its","SEC. 5.11\nDIMENSIONLESS FORMS\n99\ndimension of energy from the presence of the product PnTa in\nthe form of UO(00). For example, (12) of Sec. 5.8 states that\nand (11) of Sec. 5.9 states that:\n=\nA perusal of U(t;a), U(t;s), (i.e., U*(t)) and U(t;a) in the\npreceding section will corroborate the observation still fur-\nther. This leads us to the following definition.\nDefinition of the Dimensionless form of U. Let \"U#\"\ndenote any of the following radiant energy expressions :\nUn(t), U(t;a), U(t;s), U(t;a), , U(t). Then we shall write:\n\"U#\" for\nand we call # the dimensionless form of U.\nThe next observation concerns the presence of terms of\nthe form t/Ta, t/Tc, t/Ta, Ta/Ta, Ts/Ta, and Ta/Ts in the\nvarious equations constructed so far. These expressions\nare already dimensionless The observation to make at pres-\nent is that these six terms, which involve four separate con-\ncepts, can be represented compactly by means of only two dis-\ntinct concepts, namely the ratio t/Ta and the scattering-\nattenuation ratio p(=s/a). To see this, let us write:\n(1)\n\"I\" for t/Ta\nWe call T the relative time. Its connection with steady\nstate concepts is very close and may be stated succinctly\nby first writing\n\"La\" for 1/a\n.\nWe call La the attenuation length associated with the opti-\ncal medium. Since Ta is 1/va, we see that:\n(2)\nLa=vT a\nso that:\n(3)\nT = t/Ta = vt/La\nFrom (3), T may be interpreted not only in a temporal sense\n(i.e., the number of attenuation times in a certain time t) ,\nbut in a spatial sense, too, namely the number of attenuation\nlengths in a certain path (traversed by light in real time t).\nThe representation of the six dimensionless terms displayed\nabove may be made in terms of p and T as follows:","100\nNATURAL SOLUTIONS\nVOL. III\nTABLE 2\nRepresentation of six dimensionless terms.\nt/T\nT\na\nt/T\npt\nS\nt/T a\n(1-p)\n, S\np\n(1-p)\nTa/T a\nTS/T, a\n(1-p)/p\nWe are now ready to state the conversion rules by which\none is guided to the dimensionless differential equations and\nassociated solutions for the various radiant energy fields.\nTowards this end, we note that the derivative:\ndU# (t)\ndt\nmay be written as:\ndU# (T)\ndt\n.\ndT\ndt\n,\nwhere:\ndt dt = 1/T a\nso that:\n(4)\na\nConversion rule 1. To convert du#(t)/dt to dimension-\nless form under standard growth conditions, multiply by\nTa/UO (00) and change all time ratios of the kind t/Tx and\nTx/T2 into their equivalent forms in terms of p and T, using\nTable 2.\nConversion rule 2. To convert U#(t) to dimensionless\nform under standard growth conditions, multiply by 1/U° (00)\nand change all time ratios of the kind t/Tx and Tx/Ty into\ntheir equivalent forms in terms of p and T, using Table 2.","SEC. 5.11\nDIMENSIONLESS FORMS\n101\nDimensionless Forms for U°(t)\nStarting with (8) of Sec. 5.8 under the standard growth\ncondition, we have\nTo apply conversion rules 1 and 2, we write this as :\n= +\nand then go on to obtain:\n(5)\nThe solution of (5) is:\n(T) = 1-e-T\n(6)\n=\nThe only dimensionless parameter in the representation\nof UO (T) is the relative time T. The absence of p from (5)\nand (6) indicates that the growth of residual radiant energy\nis basically independent of the medium in which it takes\nplace. At any rate U° (T) will be seen to differ from Un(T),\ne.g., the growth and decay of which depends critically on the\nparameter p.\nDimensionless Forms for un(t)\nStarting with (24) of Sec. 5.8 under the standard\ngrowth condition, we have:\nwhich we may write as :\n= - +\n,\nwhich by conversion rules 1 and 2 become:\n(7)","VOL. III\nNATURAL SOLUTIONS\n102\nwhich has the solution:\n(8)\n=\nwhere Fn is defined in (13) of Sec. 5.9. From (8) we have\nimmediately that:\n()\n(9)\nfor every n > 1, and a study of (7) shows that this relation\nholds also for n = 0.\nIt is interesting to note how (7), even though defined\nonly for n > 1, actually reduces to the correct relation when\nn = 0. A comparison of (5) and (7), suggests that we can\nidentify the term pün-1 (T) with 1 when n = 0, i.e. , we are\nencouraged to extend the meaning of UJ (T) to the case where\nj = -1. Thus let us write:\n\"-1(T)\" for 1/p\n(10)\nIn full dimensional form this means that we have the defini-\ntional identity:\n(11)\nWith this extension, we may use (7) as the basic n-ary\ndifferential equation which then includes (5) as a special\ncase.\nDimensionless Forms for U*(t)\nApplying the conversion rules to (11) of Sec. 5.10, we\nhave, under the standard growth condition:\ndÜ*(T) = - (1-p) *(T) + U°(T)\n(12)\nwith solution:\n(13)\nIt is interesting to see how (13) predicts the growth\nof scattered radiant energy in extreme media, i.e., media for\nwhich p = 0 and for which a = 1, e.g., in purely absorbing\nand scattering media, respectively. To see this, observe\nthat:","SEC. 5.11\nDIMENSIONLESS FORMS\n103\nThen we have from (13) :\n() = (t-1) = + e-T\n(14)\nThus in purely scattering media, at T = 0, U*(0) = 0, and for\nvery small relative times T:\nU*(T) =\n,\nso that *(T) commences growth parabolically from T = 0. For\nsômewhat larger T, U*(T) grows essentially linearly with T,\nas might be expected. In the case of the other extreme type\nof space, the purely absorbing space, i.e., one for which\np = 0, equation (13) predicts U*(T) = 0 for every T, as ex-\npected. In general for normal spaces, i.e., for spaces in\nwhich there is present both scattering and absorption, so\nthat 0 < p < 1, (13) predicts the steady state value of U*\nto be\n()\n(15)\nThis agrees with the natural solution computation based\non (9):\n(16)\nThe growth pattern of *(T) is relatively interesting\nbecause the rate of growth of U* (T) exhibits a maximum at a\ncertain finite time which depends on on p. Thus, from (13)\nwe have:\n(17)\n.\nFor normal spaces, i.e., , when 0 1, this rate of growth\nis zero for T = 0 and T = 8 and positive for all intermediate\nT. The T for maximum growth rate is obtained in the usual\nmanner using calculus, and is of the form Tmax, where we have\nwritten:\n\"Tmax\" for - ln(1-p)\n(18)\nWe shall have occasion to return to this relative time in\nthe discussion below on time constants.\nDimensionless Forms for U(t)\nApplying the conversion rules to (27) of Sec. 5.8, we\nhave, under the standard growth condition:","104\nNATURAL SOLUTIONS\nVOL. III\ndÜ(T) = - (1-p) (t) + 1\n(19)\nwhose solution is:\n(20)\nNote that for purely scattering media (p = 1) :\ndu(+) = 1\nwhich implies:\n(T) = T\nfor all T > 0. For purely absorbing media, U(T) = (T). In\nnormal spaces the steady state value of U(T) is:\n(21)\nDimensionless Forms for U(t;a), U(t;a)\nFrom (9) of Sec. 5.10 and the conversion rules we\nobtain:\ndu(t;a)= i°(+)\n(22)\ndT\nwhence, under standard growth conditions:\ne-r\nj(T;a) = (t-1)\n(23)\n+\nThis agrees with the special case (14) of the representation\nof U*(T) (alias U(T;s)), i.e., under the special case where\ns=a. Finally, from (16) of Sec. 5.10:\ndU(T;a)=(1-p) (((+)\n(24)\nwhence, under standard growth conditions:\n(25)","SEC. 5.11\nDIMENSIONLESS FORMS\n105\n1.0\nn=0\nn=1\n10-1\nn=2\n10-2\nn=3\nn=4\n10-3\n10-4\n() = pn[1-()\nP = 0.2\n10-5\no\nI\n2\n3\n4\n5\n6\n7\n8\n9\nIO\nFIG. 5.13 A plot of n (T) versus T for n = 0, 1, 2, 3, 4\nin an optical medium which has p = 0.2 (see (8) of Sec. 5.11).","106\nNATURAL SOLUTIONS\nVOL. III\n1.0\nn=0\nn=1\nn=2\n10-1\nn=3\nn=4\n10-2\n()= = n -\np=0.4\n=\n10-3\n10-\n4\n8\n9\n10\n2\n3\n4\n5\n6\n7\nO\nI\nT\nFIG. 5.14 A plot of n (T) versus T for n = 0, 1, 2, 3, 4\nin an optical medium which has a = 0.4 (see (8) of Sec. 5. 11).\nNote that the vertical spread of the curves is decreasing,\nand that the steady state values of un (T) crowd closer to-\ngether for higher p values.","SEC. 5.11\nDIMENSIONLESS FORMS\n107\n1.0\nn=0\nn=1\nn=2\nn=3\nn=4\n10-1\n10-2\n() pn n [1-()-\np=0.6 =\n10-3\n6\n7\n8\n9\n10\no\nI\n2\n3\n4\n5\nT\nFIG. 5.15 Continuation of Figures 5.13, 5.14.","108\nNATURAL SOLUTIONS\nVOL. III\nn=0\n1.0\nn=1\nn=2\nn=3\nn=4\n10-\np=0.8\n10-2\n10-\no\nI\n2\n3\n4\n5\n6\n7\n8\n9\n10\nII\nT\nFIG. 5.16 Continuation of Figs. 5.13 through 5.15.","SEC. 5.11\nDIMENSIONLESS FORMS\n109\n1.0\n=0\nn\n10-1\nn=2\n() = pn[1-()\np=1.0\nn=3\n10-2\nn=4\n10-3\no\n2\n3\n4\n5\n6\n7\n8\n9\nIO\nT\nFIG. 5.17 Conclusion of Figs. 5.13 through 5.16.\n10\n9\n8\n7\n6\n5\n4\n4\n3\n2\nI\no\nn\nFIG. 5.18 A plot of time constants for n (T),\nn = 0,1,2,3,4 in which C = 0.98. (See (27) of Sec. 5.11.)","VOL. III\nNATURAL SOLUTIONS\n110\n10\np=1.0\np=0.9\np=0.8\np=0.6\nI.O\np=0.4\np=0.2\n10-\np=0.1\n(T;s) = 110 -e-(1-p)T)-(1-e-t)\nP\n=\n10-2\n9\n10\n7\n8\n6\n4\n5\nI\n2\n3\no\nT\nFIG. 5.19 Plots of U(T;S) (=Ü*(T)) versus relative time\nEach curve represents a different scattering attenuation\nT.\nratio p. U(T;S) is the dimensionless form of U(t;s), and\nthis latter quantity is the total amount of scattered radiant\nenergy in the optical medium at time t after the steady source\nhas been turned on. U(t;s) is the sum of all n-ary radiant\nSome of the latter\nenergy components un(t), , n = 1,2,3\nquantities are plotted in Figs. 5.13 through 5.17, in dimen-\nsionless form. Each curve in the present figure, except for\np = 1, levels off to approach the asymptote p/(1-p). (See\n(15) of Sec. 5.11.)","SEC. 5.11\nDIMENSIONLESS FORMS\n111\n1.0\np=1.0\np=0.8\n10-\np=0.6\n2p 10-2\np=0.4\n10-3\np=0.2\nd(;s)\n= (ePT-1)e-t\ndt\np=0.1\n10-4\no\nI\n2\n3\n4\n5\n6\n7\n8\n9\n10\nT\nFIG. 5.20 Showing the evolution, , in time, , of the scat-\ntered radiant energy (see (17) of Sec. 5.11).","112\nNATURAL SOLUTIONS\nVOL. III\n1.0\n0.8\nn\nj\nU\n(T)\n0.6\nI\n3\nU(T;s').0.4\n2\n0.2\n2\n4\n6\nT\n8\n10\n00\nFIG. 5.21 A plot showing the relative magnitude of the\nsum of the first n scattering orders\nn\n{ U(T)\nj=1\nof radiant energy at time T as compared to the total amount\nU(T;s) of scattered radiant energy at the same time. The\nplot is for a space with scattering-attenuation - ratio p =\n0.8. Observe that for fixed n, the ratio is monotonic de-\ncreasing with time T. For fixed time T, the ratio increases\nwith increasing scattering order. As an example, , let n = 3,\nand T = 5. Then the ratio of UJ(T) to U(T;s) is 0.8; for\nT = 10, the ratio is 0.6; and in the limit, as T +00, the ra-\ntio is 0.48. Hence, at steady state the amount of radiant\nenergy having been scattered, once, twice, or three times is\n48 percent (= 1 - pn) of all that has been scattered in gen-\neral (see Fig. 5.22).","113\nDIMENSIONLESS FORMS\nSEC. 5.11\n1.0\np=0.1\np= 0.2\n0.9\np= 0.4\n0.8\np=0.6\n0.7\np=0.8\n0.6\n0.5\np=0.9\n0.4\n0.3\n0.2\n0.1\no\n8\n9\n10\n5\n6\n7\nI\n2\n3\n4\nn\nFIG. 5.22 The limiting values, for T = 00, of the ratios\nin Fig. 5.21.\n2.5\nTmax = In(1-p) P\n2.0\n1.5\n1.0\n0.8\n1.0\n0.6\n0.2\n0.4\nO\nP\nFIG. 5.23 The relative times for the occurrences of the\nmaxima in Fig. 5. 20, plotted as a function of p. For example,\nthe curve labeled \"p = 0.08\" in Fig. 5.20 has its maximum at\nabout T = 2.","114\nNATURAL SOLUTIONS\nVOL. III\n40\n30\n20\n10\no\n0.2\n0.4\n0.6\n0.8\nP\nFIG. 5.24 The time constant 0.98 as a function of\nscattering-attenuation ratio p. (See (26) of Sec. 5.11.)\nA Discussion of Time Constants\nTime-dependent natural phenomena may be broadly classed\ninto two main groups: those that are periodic and those that\nare not periodic over a given time interval. Periodic phe-\nnomena can in turn be characterized in part by means of their\nperiods, i.e. the smallest intervals of time over which they\nexhibit a basic cycle of behavior. Nonperiodic phenomena on\nthe other hand have very many ways of being nonperiodic, and\nthere is no simple single number which suggests itself as a\nsuitable measure of such general nonperiodicities. Of the\ngreat variety of nonperiodic phenomena, however, there are\nthose which appear to eventually tend with increasing time\ntoward a well-defined limit. These nonperiodic limiting\nphenomena can then be characterized in a manner analogous to\nthe periodic phenomena, i.e., , by means of single numbers which\nsuitably measure such simple nonperiodicities. One useful\nmeans is the concept of the time constant of such phenomena.\nThe time constant, broadly speaking, is that interval of time\nover which the nonperiodic limiting phenomenon evolves from\nsome standard initial state until it arrives just within a\nprescribed \"distance\" of its limit state.\nTime-dependent light fields in natural optical media\nare generally phenomena of the nonperiodic limiting type dis-\ncussed above. Therefore the notion of a time constant char-\nacterization of such phenomena seems worthwhile exploring.\nIn the discussion that follows we shall examine some possible\ncandidates for time constants of transient light fields in\nnatural optical media. One major fact that will emerge from","TIME CONSTANTS\n115\nSEC. 5.11\nthe discussion is that there is a large number of possible\ncandidates for time constants, each valuable in the context\nin which it is found and used. Thus it will turn out that,\nin the long run, no one single time constant will suffice\nfor the description of every instance in the great variety of\ntime-dependent radiant energy fields encountered in the vari-\nous natural media (oceans, lakes, atmosphere) The best\nchoice of time constant that can be made will vary jointly\nwith the type of radiometric concept used (radiance, irradi-\nance, or any of the variety of radiant energies discussed so\nfar) and the space in which the light field is evolving.\nTo illustrate the thesis just stated, consider once\nagain the residual radiant energy UO(t) discussed in Sec. 5.8,\nnow in comparison with the directly observable radiant energy\nU(t). We saw in Sec. 5.8 the exact analogy that held between\na simple resistance-capacitance DC circuit and an infinite\nhomogeneous optical medium in which UO (t) was evolving. This\nanalogy suggested that the candidate for the time constant\nassociated with U° in the medium was Ta. Comparing the form\nof Uo(t) with that of U(t) as given in (14) of Sec. 5.10, we\nsee that in the same medium, but now with reference to U(t),\nthe most obvious candidate for the time constant is Ta. Thus\nby switching from U° (t) to U(t) the appropriate choice for\ntime constant correspondingly goes from Ta to Ta.\nAs another illustration of the thesis of this discus-\nsion, consider the scattered radiant energy (t) (=U(t;s))\nas given in (12) of Sec. 5.10 and its dimensionless graphical\nrepresentation in Fig. 5.19. The steady state value of U* (T)\nis (1-p) in normal spaces, i.e., spaces in which 0 < p 1.\nFigure 5.19 shows how U* (00) approaches this value asymptot-\nically for selected values of p. For example, if p = 0.4 then\nU*(00) = 0.4/(1-0.4) = 0.67. This value has been attained (at\nleast visually, according to the graph) at about eight rela-\ntive time units. More generally, in a given space with\n0 p < 1, let C be any number such that 0 < C < 1. Then we\nrequire that value TC of T such that:\n1-p cp = *(Tc)\n1 - (1-e-tc)\n(26)\nFor every p, 0 < p < 1, the number T always exists since\nU*(T) is continuous and increases monotonically toward its\nlimit, and so eventually takes on the value co/ (1-p) for\n0 c 1. A graph of TC for C = 0.98 is given in Fig. 5.24\nas a function of p. For example, for p = 0.4, TC = 8, and\nso we return to the visual estimate given above. The graph\nof Fig. 5.24 shows generally that the greater the scattering\nattenuation ratio, the greater TO 98 this much could be\nguessed on intuitive grounds however, the exact quantitative\nmanner of the increase in TO 98 is interesting to observe.\nThe numbers TC, therefore, can serve as time constants for\nscattered radiant energy after a choice of C is made.","116\nNATURAL SOLUTION\nVOL. III\nThe time-dependent structure of the scattered radiant\nenergy U* (T) has an additional feature to that of asymptot-\nicity which may serve to be a workable basis for the defini-\ntion of a time constant. A study of the rate of growth of\nU*(T) in Sec. 5.11 showed that the derivative of the rate of\ngrowth starts out positive, becomes zero at relative time\n1-p)/p, and then remains negative for all subsequent rela-\ntive times in very given normal medium (cf. (18) of sec. 5.11).\nThis suggests that max' the relative time of the maximum\nrate of growth, is a possible candidate for a time constant\nfor a given medium, for it defines a distinguishable point\nof inflection on the growth curve of U*(T). Figure 5.23 de-\npicts Tmax as a function of p for a selected range of normal\nspaces. The point to observe here is that we need not always\nbase time constant definitions on the feature of asymptoticity\nof a nonperiodic phenomenon. Well-defined maxima or minima\nor points of inflection of growth curves may also serve as\nadequate bases for time constants.\nIt is interesting to observe how the notion of a time\nconstant can be extended to each of the n-ary radiant energy\nfields Un, n > 0. The best candidate for the time constant\nvaries with the scattering order n. Thus, suppose C is any\nnumber such that 0 < C < 1. Let (n) be that relative time\nfor which:\n= pn [1-Fn(Tc(n))]\nholds. That is we require c(n) such that:\n1-c = Fn (tc(n))\n(27)\n.\nAs in the case of (26), Tc (n) exists for every n > 1 and C\nsuch that 0 < 1. The basis for this conclusion is prop-\nerty 2 of un (t), stated in Sec. 5. 10, which implies that Un(T)\nincreases monotonically and continuously to its limit. Figure\n5.18 depicts a plot of Tc(n) for C = 0.98 and n = 0, 1, 2, 3, 4.\nStill one more variation in the concept of time çon-\nstant follows from the observation that the curves of Un(T)\nhave inflection points at relative times T = n. Thus setting:\n,\ndt\nimplies\nT=n\n(28)\nHence, as in the case of * (T), , we can use the inflection\npoints as identifiable characteristics of the growth curves\nof Un. Observe how the time constants suggested by (28) are\nindependent of p, and hence the medium, and depend only on\nn; yet the similar type of time constant for the sum U* of t\nthe n-ary fields in indeed depends on p.","SEC. 5.11\n117\nTIME CONSTANTS\nWith these illustrations we rest our case concerning\nthe nonexistence of a single universally applicable time con-\nstant for characterizing transient light fields in extensive\noptical media. Perhaps, if a single time constant were de-\nmanded which could be pressed into use more often than all\nthe other time constants discussed in the present chapter,\nthen we might tentatively suggest I a for consideration. For\nTa appears quite often in the energy context and most criti-\ncally in the radiance context of (10) of Sec. 5.7. Further-\nmore, Ta is based on the one inherent optical property (namely\na) of optical media which is the most thoroughly documented\nand which is the most readily measured member of the basic\ntrio a, o, a.\nFinally, we observe that all our preceding deliberations\nconcerned unbounded media - or very extensive media in which\ntheir boundaries played a negligible role. For a discussion\nof the theory of time constants in bounded media in which the\nsensitivity of radiometer instruments also plays a role the\nreader may consult the papers in parts IV, V of [236]. These\nreferences are part of a set of five reports in which the\nmain discussion centers on the study of the general metric\nproperties of time dependent light fields. The theory of the\ntime constant found in [236] is one of the several applica-\ntions of the general metric theory developed in the series.\n5.12 Global Approximations of General Radiance Fields\nIn this and the following section some of the theory of\ntime-dependent n-ary radiant energy fields will be applied to\ntwo general problems of radiative transfer theory. In the\npresent section attention will be directed to the problem of\nfinding relatively simple approximations of time dependent\nand steady state radiance fields in optical media. In par-\nticular it will be shown how the n-ary radiant energy fields\nmay be used to obtain approximations of the observable radi-\nance field such that the approximations are exact on a global\nlevel over the given medium.\nThe precise meaning of this phrase will become clear\nduring the course of the constructions of the approximations,\nto which we now turn. Unless specifically stated otherwise,\nall constructions will take place on a general optical medium\nX with arbitrary source conditions.\nWe begin with the observation that the operator formula\n,\nbased on the theory of Sec. 5.1, suggests the following simple\napproximation, where we write:\nN°\nn,\n(1)\n\"N\nfor\ng\nHere Un N > 1, is the n-ary radiant energy in X, and N1 is\nthe primary radiance function in X. N g is called the global\napproximation of Nn for n > 1.","118\nNATURAL SOLUTIONS\nVOL. III\nThe reason for such a name and structure of Nn lies in\ng\nthe following observations. Note first that Nn has scatter-\ning order \"dimensions\" of n-ary radiance. Next, observe that\nthe global approximation for Nn yields the estimate:\n(1)\nfor the radiant density function u in X. If we write \"un\"\nfor this function, then we see that:\n(2)\nfor n > 1. Finally:\nI\" X g u1(x,t)dV(x)\nu\n= un(t)\nThis shows that the approximation Nn to Nn has the property:\nun(t)\ndo(E)\ndV\n(x)\n(3)\n=\nX\n(1)\nIn other words, Nn yields the same radiant energy content\nof X at each time t as does Nn , the actual n-ary radiance\nfunction on X. Thus Nn yields an exact prediction of approx-\ng\nmation of Nn on an overall (or global) basis. The direction-\nal or local structure of Nn is approximated by that of N°,\na relatively easily computed function.\nThe global approximation of Nn may be used to obtain a\nglobal approximation of the directly observable radiance N\nby means of the natural solution representation of No, where\nwe have written:\n\"N\" \" for\n(4)","119\nSEC. 5.12\nGLOBAL APPROXIMATIONS\nFor, by the definition of the N. we have:\n(5)\nThe requisite global approximation of N is obtained by writ-\ning\n\"Ng\" for N° N N*\n(6)\nIt follows that:\nU(t) =\nN g(x,5,t)d2(5), dV(x)\n(7)\nX\nE\nso that N o g indeed endows X with the same radiant energy con-\ntent as N, the actual observable radiance function on X. The\nfunction N g may then be used to assign to each X in X, and &\nin E at time t the radiance:\n(8)\nwhere, in case standard growth conditions are in force in X,\nU*(t) (alias U(t;s)) and U1(t) are given by (14) of Sec. 5.9\nand (12) of Sec. 5.10. In the steady state attained under\nstandard growth conditions, (8) yields:\n(9)\nwhich is defined for 0 < P < 1.\nGlobal Approximations of Higher Order\nThe global approximation Nn in (1) above is but the\nlowest rung on an infinitely high ladder of global approxima- -\ntions of the radiance function in the medium X. We now formu-\nlate the global approximation to N of arbitrarily high order.\nThus let us for every n > 1, write:\n\"N\" for","120\nNATURAL SOLUTIONS\nVOL. III\nHere we choose to use the same name \"N\". for the approximat\ng\ning function, and we have now written, ad hoc:\n\"N(C)\",\"for\nfor\nj=1\nand\nk\n'U(k),\nuj\nfor\nNn° g is the global approximation of the kth order of Nn. It is\neasy to verify that Nn again is globally exact in the general\ng\nsense of (3). Defining Ng as in (6) and N* as in (4), now\nfor the kth order context, by stopping the sums in (4) and\n(6) at j = k, it follows that:\nN(k) (x,5,t) = i°(x,E,t)\n(10)\nwe call N (k) in (10) the global approximation of the kth order\nN. on (k) is globally exact in the sense of (7), i.e., ,\nof\nusing NSk) in (7) will yield U(k) (t). Observe that this ap-\nproximation also has the virtue of converging to N as k\n00.\nThat is:\n(11)\nThis follows from (10) and the facts that:\n1imk+ou(k)(t) = U*(t)\n(12)\nand that:\nlim N (k) = N*\n.\nk+00\nIn this way we see that the global approximations to N\nhave one additional property over the truncated solutions of\nSec. 5.5, namely the global exactness property. The steady\nstate limit version of (10) attained under standard growth\nconditions is:\n(13)\nand which is defined for k > 1, and 0 < p < 1. Under stan-\ndard growth or decay conditIons, one may use in (10) the","SEC. 5.12\nGLOBAL APPROXIMATIONS\n121\nexpressions for (t) and Un (t), developed in Sec. 5.11, to\ngenerate useful approximations to time-dependent radiance\nfields. First or second order global approximations should\nsuffice for many practical settings.\nWe note in passing that preliminary and informal numer-\nical studies seem to indicate that the shapes (the direction-\nal structure) of Nn appear to be spherical (or very nearlyso)\nwhen n is larger than some integer p which depends on the\nmedium X and p. If this conjecture can be proved in general,\n(probably by means of the set up in 10.5) then an enormous\nadvance in the practical utility of (13) can be made. This\nconjecture of the limiting shape of Nn as n 80, bears a\nstriking analog to the asymptotic radiance theorem studied\nelsewhere in this work (cf., e.g., Chapter 10). An important\napplication would be to diffusion theory (see (78) of Sec.\n6.6).\n5.13 Light Storage Phenomena in Natural Optical Media\nThe applications of the natural mode of solution of\nradiative transfer problems in optical media discussed in\nthis chapter will now be concluded with a definition and dis-\ncussion of the light-storage phenomena in such media.\nEveryday Examples of Light Storage\nThose who have looked out of a window of an airplane as\nit descended into a sunbathed cloud layer may recall the sud-\nden transition to a brilliant ambient field of light, and how\nthe sensation of brightness in every direction increased to\ndazzling proportions as the airplane descended further into\nthe upper regions of the cloud. This phenomenon is but one\nof many common examples of the storage of light by the mecha-\nnism of scattering. One can also see evidence of light stor-\nage on overcast nights on the outskirts of large cities: the\ncloud layer hovering low over the city is deeply and exten-\nsively illuminated from the street and building lights below.\nFlashes of lightning in storm clouds can light up an exten-\nsive cloud layer from horizon to horizon even though the ac-\ntual volume taken up by the network of electrical discharges\nis a minute fraction of the illuminated volume. Lighthouses\non densely fogged nights pour a well-defined beam of light\ninto a surrounding fog with the result that the beam and the\nlighthouse are imbedded in a field of scattered light which,\nunder suitable conditions, may be observed by approaching\nmariners far sooner than the light of the revolving beam. As\none descends into a lake or the ocean on a sunny day, there\nis a shallow region near the surface in which the radiance\nmeasurably increases with increasing depth for various hori-\nzontal and upward-looking lines of sight.\nThese examples illustrate the phenomenon of the storage\nof light in scattering media. The sense of the work 'storage\"\nis used in its everyday sense: the accumulation or building\nup of radiant energy in the scattering material that surrounds\nthe source of the energy. If one were to quickly extinguish\nthe light source, the stored light would not immediately dis-\nappear with the extinction of the source; rather the scattered","VOL. III\nNATURAL SOLUTIONS\n122\nlight stored in the earth's atmosphere would take on the order\nof a score of microseconds to be lost into space, or converted\ninto longer wavelengths of radiation and other forms of ener-\ngy. The decaying atmospheric light field is like the dimin-\nishing reverberation of organ notes in a spacious auditorium\nin which the acoustical energy is momentarily entrapped and\nredirected by the walls of the auditorium (cf. Sec. 5.6). In\nthe case of light, the walls of the auditorium are replaced\nby multitudes of tiny scattering centers comprising clouds,\nfogs, or parts of the entire atmosphere, and the hydrosphere\nof the earth: the light impinges on the scattering centers\nand is redirected again and again by scattering.\nThus, the energy of a pencil of photons, which ordi-\nnarily traverses a given volume of empty space in one micro-\nsecond, could, in principle, be cycled and recycled within\nthe confines of the volume for a period of several dozens of\nmicroseconds before it escapes or is transformed. Therefore,\nif a continuous steady beam of light is poured into such a\nvolume, the steady state density of scattered light stored\nwithin the volume could be tens of times greater than the\naverage density of the light ordinarily within the beam.\nDo all these phenomena have a common simple description?\nIs there a small set of properties of the medium and of the\nsource that, when isolated, can serve as the salient parame-\nters in an analytical description of the stored light field?\nThe answer is 'yes'; the natural mode of analysis of light\nfields plays an essential role in formulating the details of\nthe answer.\nIn this section we embark on a preliminary attempt to\ndescribe the phenomenon of light storage in precisely defined\nterms. Once we have decided on an exact radiometric defini-\ntion of the term \"stored light energy,\" we go on to formulate\na simple mathematical model of the light field in a scattering-\nabsorbing medium which can describe how the stored light\nenergy depends on the inherent optical properties of the\nmedium, the geometry of the medium, and the properties of the\nlight source.\nIt turns out that there are several ways in which we\nmay formulate the description of \"stored light energy.\" The\nform of the description depends on one's choice of the radi-\nometric quantity used in the description. For example, we\nfind that there is a description associated with the radiom-\netric concept of radiance, another description with irradi-\nance, another with radiant density, and still another with\nradiant energy.\nIn the present discussion we will limit our attention\nto the description of stored light energy exclusively by\nmeans of the concept of radiant energy. The resulting de-\nscription is by far the most natural of all the various\npossibilities; it is, by a happy coincidence, also the most\nsimple to deal with, and the easiest from which to draw\nexamples.\nIn the event that more detailed descriptions of storage\nphenomena than those developed in the present study are ever\nrequired, such as n-ary radiance Nn or radiance N, recall that","SEC. 5.13\nLIGHT STORAGE\n123\nwe have formulated the requisite time-dependent transport\nequations of these radiometric quantities in Sec. 5.2. There-\nfore, the work of this section should readily be extended to\nthe radiance case by interested researchers. The investiga-\ntion of the time-dependent radiant flux problem made in the\npreceding sections also supplements the results of the pres-\nent study by providing detailed numerical and graphical il-\nlustrations (Figs. 5.13-5.24) of the solutions of the n-ary\nradiant energy equations, and related radiometric concepts,\nwhich play an important role in the storage capacity concept.\nStorage Capacity\nLet \"U\" represent the directly observable steady state\nradiant energy attained in an arbitrary medium X under arbi-\ntrary growth conditions; let \"UO\" represent the amount of U\nconsisting of residual radiant energy from the source (asso-\nciated with photons which have not yet been scattered or\nabsorbed subsequent to entry into X) and finally, let \"U*\"\nrepresent the amount of U consisting of scattered radiant\nenergy within the medium (associated with photons which have\nundergone at least one scattrring operation). The ratio U*/U\nis then a measure of the relative amount of scattered radiant\nenergy in the medium X. It is a number which lies between\nzero and one and will be referred to as the storage capacity\nof the medium X.\nIn the case of an infinite homogeneous medium whose\nsteady state light field has been attained under standard\ngrowth conditions (Sec. 5.11), the storage capacity has a\nparticularly simple representation in terms of the total\nvolume scattering coefficient S, and the volume attenuation\ncoefficient a of the medium:\nstorage capacity =\n(1)\nwhere p is the scattering-attenuation ratio. In the case of\nnonhomogeneous or finite media, the storage capacity is a\nmore complicated function of p and the geometry of the medium.\n(Examples of more general storage capacity formulas will be\ngiven below in (5) and (6).) But even in the present simple\ncontext, we gain important insight into storage phenomena in\ngeneral: the storage capacity depends basically on the rela-\ntive magnitudes of S and a. Thus if we consider two media,\none in which S = 0.01/m, a = 0.02/m, and another in which\nS = 0.10/m, a = 0.20/m, we see that the former medium has an\nattentuation length of 1/a = 50 m while the latter while the\nlatter medium is an order of magnitude more optically dense\nwith an attenuation length of 1/a = 5 m. However, the\nscattering-attenuation ratio for each medium is p = 0.5.\nThus, despite the great disparity in optical density of these\nmedia, their storage capacities have a common value, namely\nU*/U = 0.5, indicating that in the steady state in each\nmedium, the stored radiant energy (in scattered form) is 50%\nof the total observable energy within each medium.","NATURAL SOLUTIONS\nVOL. III\n124\nMethods of Determining Storage Capacity\nThe problem of determining the storage capacity of an\ninfinite or very extensive optical medium (one in which the\nboundaries play a negligible role) is readily solved using\nthe results developed in the preceding sections on n-ary\nradiant energy. In particular, for homogeneous infinite\nmedia, the storage capacity reduces to a very simply obtained\nsingle number p, as shown above. The number p is readily\ndetermined in practice by a few local measurements. However,\nthe infinite settings are occasionally inadequate models of\nreal situations. In real media in terrestrial settings we\nusually dispense with computation programs and go directly to\nthe medium (clouds, lakes, oceans) to perform measurements\nin\nsitu over the given region. By following the definition of\nstorage capacity to the letter, we need only try to measure\nthe radiant energy U* and U by measuring scalar irradiance\nat each point throughout the medium and find the quotient\nU*/U. However, to probe the medium point by point is always\nlaborious and occasionally impossible. A practicable scheme\nfor measuring storage capacity of real media would be one in\nwhich all internal probings are obviated. We thus set up\nthe following problem for study: Is there some way of deter-\nmining U* (X) /U (X) for a medium X by limiting all radiometric\nmeasurements to the boundary of X? The answer is in the\naffirmative. We now present the details of a possible empir-\nical procedure leading to the storage capacity of a natural\noptical medium.\nThe discussion begins with the steady state version of\n(24) of Sec. 5.8 applied to a homogeneous, bounded region X\nof some real optical medium. The incident radiant flux on X\nis arbitrarily disposed over the boundary and X is assumed to\nhave no internal emission sources. Thus we begin with:\n0 = - aun(x) +\n(2)\nfor n > 1. Here pn (X) is the net inward radiant n-ary flux\nacross the boundary of X. The n-ary radiant flux is indexed\nrelative to the incident radiant flux on the boundary of the\noptical medium in which X is located. Thus if the optical\nmedium is the ocean and X is a cube 10 m on a side whose cen-\nter is located 100 m below the surface, then the n-ary radi-\nant flux in the cube is relative to the incident radiant flux\non the surface of the ocean. Summing each side of (2) over\nall n > 1:\n8\n8\n0 = - a n=1 (X) n=1 un-1 n=1\nUsing the natural solution properties this becomes:\n(3)\n0 = - au*(X) + sU(X) +","SEC. 5.13\nLIGHT STORAGE\n125\nwhere we have written:\n8\npn(x)\n\"P*(X)\"\nfor\n(4)\nn=1\nIn accordance with our preceding remarks, we are inter-\nested in estimating the quantity U*(X) with the ultimate goal\nin mind of estimating the ratio U*(X)/U (X) . But any such\nestimation must be couched in terms of observable or simply\ncalculable quantities. U* (X) is not directly observable; and\nU(X), , while observable, is not simply calculable. (It re-\nquires a determination of observable radiant density u(x) at\neach point X of X. ) In casting about for easily observable\nand simply calculable quantities, the observable net flux\nP(X), the residual net flux po (X) and the residual energy\nU°(X) immediately come to mind. If we can obtain an expres-\nsion for U* (X) /U (X) in terms of P(X), PO (X) and UO (X), we\nwill have obtained the best solution possible to the problem\nof empirically determining the storage capacity of a finite\nhomogeneous medium.\nIt turns out that the characterization of U*(X)/U(X)\nin terms of P(X), P°(X) and UO (X) is relatively easy to\nachieve. Starting with (3) and noting by (33) of Sec. 5.8\nthat we have:\nP(X) =\nwe can recast (3) into the form:\n= - aU* (X) + su°(X)\nWe can then represent the nonobservable U*(X) in terms of\nobservable and calculable quantities\n[P(X)\nHence\nU*(X)\na\n(5)\n=\nU(X)\n-\nEquation (5) gives the desired general formulation of the\nstorage capacity of a finite homogeneous medium X in terms\nof the directly observable net inward flux P (X) over the\nboundary of X, the calculable net inward residual flux PO(X)\nover the boundary of X, and the calculable residual energy\ncontent U° (X) of X. The volume absorption coefficient a and\nthe volume attenuation coefficient a are the inherent optical\nproperties of X which enter into the calculation and which\nare assumed known.","126\nNATURAL SOLUTIONS\nVOL. III\nIt should be remarked that equation (5) is an exact and\ncomputable formula for the storage capacity U* (X) U(X) when-\never X is any finite homogeneous medium with a > 0, irradiated\nby sources in an arbitrary manner and in which the resultant\nlight field is in steady state. If X is infinite in all di-\nrections or very extensive, then it may be that P(X) = po(X),\n,\nand (5) reduces to (1). The condition PO (X) = P(X) means that\n(X) = 0, i.e., that there is no net scattered flux across the\nboundaries of X. This could happen when the boundaries are\ninfinitely far removed, or when a small volume is deep inside\nan extensive medium.\nExample\nTo illustrate how (5) is used in particular contexts,\nconsider for example a horizontally extensive cloud stratum,\nor ocean layer with upper boundary on the surface, which is\nof finite geometric depth under a clear sunlit sky or clear\nmoonlit sky. To fix ideas, consider the ocean layer. We\nagree that the principal source of flux is to be the sun or\nmoon, as the case may be, with negligible auxiliary sources\nassociated with the sky and ground (or lower layers in the\ncase of the ocean). Suppose the sun cannot be seen through\nthe given layer as one is looking up from below. It may be\nchecked that the difference P(X) - PO(X) in (5) then reduces\nessentially to -P* *(X,+), where p* (X,+) is the total net out-\nward rate of flow of stored energy across the two boundaries\nof X. (The inward flow p* (X,-) is set to zero.) Suppose\nalso that the outward rate of low from X over its lower bound-\nary is small compared to that of its upper boundary (which is\ncompatible with the assumptions above) Then:\nN°SA\np°(x,-)\n=\n0\nva\nwhere N° is the radiance of the sun or moon at the upper\nboundary of X, 0 its angle from the zenith, So is its solid\nangle subtense, and A is the area of the upper boundary of\nthe cloud. The second equality follows from the definition\nof inward residual flux PO (x, over the upper boundary of X.\nHence (5) becomes\nU*(X)__p-R(X)\n(6)\nU(X) \" 1-R(X)\nwhere \"R(X)\" stands for P*(X,+)/P°(X,-), the reflectance of\nX at its upper boundary, a directly measurable quantity.\nAs a simple numerical illustration of (6), suppose that\nwe take the case of a part X of the ocean for which (6) holds\nand for which it is found that p = 0.4 and that R(X) = 0.02\nfor a given wavelength of light around the middle of the vis-\nible spectrum. Then the storage capacity U*/U is:\n0.4 - 0.02\n0.38\n= 0.39\n=\n1 - 0.02\n0.98","SEC. 5.13\nLIGHT STORAGE\n127\nIf some time later UO is known to be a certain amount over\nthe same layer, then, if \"C\" denotes the storage capcity,\nclearly:\nU = 1-C U°\n(7)\nand hence the directly observable radiant energy in the layer\nis estimable from UO and knowledge of C.\nEquations (6) and (7) illustrate but two of the many\npractical formulas which may be deduced--under - - various\nhypotheses - - from the exact formula (5). The preceding de-\nrivation will suffice to indicate the general outline of such\nprocedures, and we leave the exploration of other possibili-\nties to the interested reader.\n5.14 Operator-Theoretic - Basis for the Natural Solution\nProcedure\nWe close the present chapter with an overview of the\ntheoretical activities of the chapter. As in the earlier\ngeneral discussions of the canonical equations (Sec. 4.7)\nthe present discussion will perhaps not so much increase our\nability to solve specific problems of applied radiative trans-\nfer as it will deepen insight into the essential structure\nof the natural solution procedure, and therefore radiative\ntransfer theory. In particular the general results below\nwill show how radiative transfer theory, via the integral\nform of the equation of transfer, is connected to those parts\nof the main stream of mathematical physics which share with\nthe present field certain operator equations whose mode of\nsolution coincides, on the abstract level, with the natural\nmode of solution studied in this chapter. The discussion is\nintended to be intuitive, as far as the material will allow.\nLet L be a general (not necessarily linear) operator\ndefined on a domain D of functions such that Lf is in D when-\never f is in D Thus L maps elements of D into D. Next\nsuppose D has a \"distance function\" d defined on it such that\nif f and g are in D, then d(f,g) is a nonnegative real number\nwith the properties:\n(i) d(f,g) = 0 if and only if f = g\n(ii) d(f,g) = d(g,f)\n(iii) d(f,h) d(f,g) + d(g,h)\nThe function d is called a metric for D, and as can be\nseen, it has the three main properties of ordinary distance\nrelation of everyday life. We summarize all this by saying\nthat the pair (D d) is a metric space.\nNow the connection between (D d) and the radiative\ntransfer setting of this chapter is quite easily made. Let\nX\nbe an optical medium with initial radiance N° and let s Superscript(1) be\nthe operator in (5) of Sec. 5.7. Then write:","128\nNATURAL SOLUTIONS\nVOL. III\n\"L\" for N° + (.) 1\n(1)\nand we have an example of the operator L above, where D is\nnow the set of all radiance functions on X. Thus if N is a\nradiance function on X (i.e., N has the dimensions of radi-\nance) then certainly\nN° + NS 1\nis again radiance function on X. We are not asserting at the\nmoment that N is a solution of the equation of transfer, but\nmerely making an observation that the function displayed above\nhas the dimensions of radiance, and that is all at the moment\nthat is required for admission into D. Hence L as defined\nin (1) maps elements of D back into D.\nNext we show that there is a very natural counterpart\nin radiative transfer theory to the abstract metric d for\neach fixed time t and bounded optical medium X. Let us write\nto\n1\n\"d(f,g)\" for\n|f(x,5,t) g(x,E,t) dr(E)\ndV(x)\nv(x)\nX\nE\n(2)\nIt is easy to verify that if f = g, then d(f,g) = 0,\nand that if d(f,g) = 0, then f = g except on sets of direc-\ntions E and points X of zero measure. This exception can be\nsmoothed over by advanced technical devices, * and we hence-\nforth can assume condition (i) for a metric to be satisfied.\nNext one can verify conditions (ii) and (iii) with ease and\nthe verification is left to interested readers. We call the\nmetric function d as defined in (2), the radiometric. By\nvarious standard techniques (e.g., averaging) (2) can readily\nbe extended to unbounded media. An alternate choice of met-\nric can also be made by writing\nfor sup |f(x,5,t) - g(x,E, t)\n\"d(f,g)\"\nI\n(2a)\nx,5\nwhere\n\"sup h(x,5)\"\nx,5\n*In particular, this can be done by means of equivalence\nclasses of functions, an equivalence class being the set of\nall radiance functions on a domain Y which differ from one\nanother at most on subsets of Y of zero measure. Then we go\non to work with equivalence sets of functions rather than in-\ndividual functions. However, for the present wework directly\nwith the radiance functions, with no essential loss of rigor.","SEC. 5.14\nOPERATOR BASIS\n129\nmeans the supremum (the maximum) of the values of h(x,5) as\nx,5 vary over all permissible values in the domain of h. The\nfunction d in (2a) also satisfies all the properties (i) to\n(iii) of a metric. We shall call d in (2a) the supremum met-\nric.\nWe summarize what has been done so far: The operator\n(1) associated with the integral equation of transfer of\nclassical radiative transfer theory may be viewed as a spe-\ncial case of an abstract operator L on a metric space (D,d),\nthe particular classical form of the operator being given in\n(1), with D being the class of all radiance functions on X,\nand with d the radiometric as defined in (2) or the supremum\nmetric as given in (2a). In what follows we allow D to con-\ntain negative valued radiance functions as well as nonnega-\ntive valued radiance functions. Of course in physically\nmeaningful applications we shall always work with the latter;\nhowever, for mathematical purposes it is convenient also to\nhave the former.\nWe now come to a key property of the radiative transfer\noperator s1 which can be abstracted from the setting of the\npresent chapter and carried out far into the reaches of ab-\nstract operator theory, where its general utility can be more\neasily discerned. In Sec. 5.7 we showed that if N is an up -\nper bound (or supremum) of a radiance function, then (cf. (7)\nof Sec. 5.7):\n(NS1)(x,5,t) No(1-e-t/Ia)\nfor every X in X, & in (1) and t in (0,t), where \"Ta\" stands\nfor 1/va. From this we are led to deduce that for every pair\nf,g of radiance functions, and with the supremum metric (2a).\nd(Lf,Lg) C d(f,g)\n(3)\nwhere C is a number which depends only on t, p and Ta, i.e.,\nwhere we have written:\n\"C\" for p(1-e-t/Ia)\nIn all normal optical media (i.e., for which 0 < p < 1),\nwe have 0 < C < 1 whenever t > 0. The proof of (3) is imme-\ndiate, using the definitions (1) and (2a). Whenever an oper-\nator L on a general metric space (D,d) has property (3), we\nsay that L is a contraction mapping or that it has the con-\ntraction property. Hence our particular classical radiative\ntransfer operator L given in (1) is a contraction mapping,\nrelative to (2a). The reader may show that (3) also holds\nunder suitable conditions, relative to (2).\nTo summarize our findings so far: The operator L asso-\nciated with the time-dependent integral equation of transfer\nmay be viewed as a special case of a contraction mapping L\non a metric space (D,d).\nWe now have developed enough abstract machinery to il-\nlustrate the essential activity of the natural solution pro-\ncedure, on a very general level a level which is in contact","130\nNATURAL SOLUTIONS\nVOL. III\nwith the general representations of widely different natural\nphenomena in modern physics. Let us choose any function f()\nin D and write:\n\"f(1),, for Lf(0)\nThus we operate on f(o) in D with L to obtain f(1) in D. We\nrepeat this operation a finite number of times to obtain f(n)\nwhere we have written:\n\"f(n),, for Lf(n-1)\nIn this way we obtain a sequence\nof functions in D. As in the case of Sec. 5.1, we can de-\nfine iterates Ln of L so that (cf., e.g., (11) of Sec. 5.1):\nf(n) = Inf(0)\nBefore going on, the reader should verify that if we use L in\n(1), , and N° for f(1), then f(n) is simply\nn\nE Nj\n,\nj=0\ni.e., the sum of the n-ary radiances up to order n.\nSince L is a contraction mapping, we have, for m > n:\nc\"\n+\n(1 - c)\n(4)\nSince C is less than 1, cn is arbitrarily small for suffi-\nciently large n. Thus the sequence\n{ ( 0 ) , f(1) ....f(n)....)\nconstructed above is a Cauchy sequence (in the sense of\nmodern calculus). By establishing this feature of the se-\nquence we have reached the penultimate step in our general\ndiscussion of the natural solution procedure.","131\nOPERATOR BASIS\nSEC. 5.14\nThe significance of the Cauchy sequence feature of\n{ ( (0) , f (1)\nf (n)\n}\n,\n,\n,\nis this: In all physically meaningful settings for the met-\nric space (D d), it is possible to arrange matters so that,\nwhenever a sequence\n{ ((n)}\nof elements in L is a Cauchy sequence in the sense of (4),\nthen that sequence has a limit in D. In general, whenever a\nmetric space (D, d) has this property, we say that (D,d) is\ncomplete. It is easy to show that all physically meaningful\nradiative transfer settings always can be represented by com-\nplete metric spaces (D,d) Let us assume therefore for the\nremainder of the discussion that (D, ,d) is complete.\nTaking up the thread of the argument at (4) we now can\nassert the existence of a limit function f to the sequence\nconstructed above. Thus let us write:\n\"f\" for lim f(n)\n(5)\nWe now show that f has two very important properties:\nf satisfies the operator equation f = Lf\n(i)\n(ii) f is the only function in L for which (i) holds,\ni.e., if g = Lg and f = Lf, then f = g.\nProperty (i) follows readily by noting that, by definition,\n= Lf(n-1) Hence applying the limit operation to each\nside of this identity, the result follows by observing that\nL is a continous mapping* (so that the limit operation can\nbe pushed past L and made to act directly on f(n-1), Prop-\nerty (ii) follows from (i) and the contraction property of L :\nd(f,g) = d(Lf,Lg) < C d(f,g)\nFrom this (since C < 1) we must have d(f,g) = 0, so that\nf = g.\nLet us now make the final summary of what has been done\nso far in this section: The natural mode of solution in\nradiative transfer theory has been found to take its place as\na special case of a very general operator technique in modern\nfunctional analysis. This technique is based on the follow-\ning theorem (cf., , e.g., [140])\nTheorem (Principle of Contraction Mappings). Every\ncontraction mapping L on a complete metric space (D, gen-\nerates one and only one solution of the equation f = Lf.\n*A point which is readily established in functional\nanalysis texts (cf., e.g., [140]).","132\nNATURAL SOLUTIONS\nVOL. III\nThe classical radiative transfer setting entities are\npaired off with the abstract setting entities of the preced-\ning theorem as follows:\nIn Radiative Transfer Theory\nIn the Theorem\nD\na) Set D of all radiance functions\non an optical medium X\nb) The radiometric d, as in (2) or\nd\n(2a)\nc) The operator L, as in (1)\nL\nWe will make one final remark on the existence of the\nsolution f of the general operator equation f = Lf. This is\nthe observation that the solution f defined in (5) is inde-\npendent of the initial function f (o) starting the chain\nof\niterations Lnf(°). This fact becomes clear, at least logi-\ncally, by noting the uniqueness property (ii) above. For if\nf(0) and g (o) are two distinct initial functions, then con-\nstruction of their iteration sequences yields f and g such\nthat property (i) holds for each.\n5.15 Bibliographic Notes for Chapter 5\nThe natural mode of solution of the equation of trans-\nfer studied in this chapter, as noted in the introduction,\nplays a unique, fundamental role in radiative transfer theory.\nThe formal power of the method and its intuitive simplicity\ncannot be overemphasized. For some historical notes on the\nnatural mode of solution, see Secs. 26 and 42 of Ref. [251].\nFor recent modifications of the iterative concept of solu-\ntions of functional equations, especially for numerical pur-\nposes, see [171].\nThe development of the natural solution, as presented\nin Secs. 5.1 and 5.4, follows in the main that given in Ref.\n[251]. The canonical representation of primary radiance in\n(8) or (9) of Sec. 5.3 is occasionally referred to as \"See-\nliger's formula,\" and is to be conceptually distinguished\nfrom the more useful and accurate representation of N* given\nin (5) of Sec. 4.4. The only common feature of the two\nradiance representations is that they both fall within the\npurview of the basic canonical formula (4) of Sec. 4.7.\nThe discussion of the \"optical ringing problem\" in\nSecs. 5.7 and 5.8 is based on the matural-solution approach\nto the time-dependent radiative transfer problem, and is\ndesigned to be more precise than simple time-dependent clas-\nsical diffusion theory (Sec. 6.6). The approach outlined in\nthese sections is drawn from the results in Ref. [211]. A\nrelated approach to the optical ringing problem from the\npoint of view of temporal metric spaces was tentatively ex-\nplored in the series of reports [236]. Further approaches\nto time-dependent radiative transfer problem are possible\nvia the higher-order diffusion equations. See Table 1 of\nSec. 6.5. The truncated natural-solution inequalities in\nSec. 5.7 are based on [239]. Further inequalities in this\ncircle of ideas may be found in Ref. [67].","SEC. 5.15\nBIBLIOGRAPHIC NOTES\n133\nThe material of Secs. 5.8 to 5.12 is drawn, with minor\nrevisions, from Ref. [211]. The light storage discussions in\nSec. 5.13 are based on Ref. [237]. The abstract overview of\nthe natural mode of solution in Sec. 5.14 uses advanced con-\ncepts of functional analysis (in particular, the principle of\ncontraction mappings) which may, e.g., be studied in Ref.\n[140].\nIn the opening remarks of Sec. 5.11, it was emphasized\nthat the dimensionless forms of the equations describing\nn-ary radiant energy fields are shared by many natural proc-\nesses, some quite distinct conceptually from the time-depend- -\nent evolution of radiant energy in optical media. For a\nbrief exploration of such alternate processes governed by the\nsame equations, see Chapter 14 of Ref. [39] and the footnotes\nin that chapter.\nThe analogies between radiative transfer phenomena and\nother transport phenomena discussed in Sec. 5.11 also can be\npursued further, e.g., in [259] and [312].","CHAPTER 6\nCLASSICAL SOLUTIONS OF THE EQUATION OF TRANSFER\n6.0 Introduction\nIn this chapter we shall conduct an exposition of the two\nmost important classical modes of solution of the equation of\ntransfer used in practice besides the canonical and the natu-\nral modes discussed in the preceding two chapters. These\nclassical modes are the powerful spherical harmonic method,\nand the mathematically interesting diffusion method. The\nspherical harmonic method is classical in the sense that it\ndates back to Eddington and Jeans [120], two of the pioneers\nof radiative transfer theory. The spherical harmonic method\nrepresents radiance functions in terms of sums of products of\ntwo factors: one factor being purely spatial, the other\npurely directional, an intuitively natural representation for\nfunctions defined on the phase space X x E. On the other\nhand, there are two main theories of diffusion: the classi-\ncal and the exact theories. The classical diffusion method\nis based on Fick's law and views photons in optical media as\nswarms of particles diffusing with great speed, but generally\nin the manner of classical diffusion processes, such as heat\nconduction and Brownian motion. The exact diffusion method,\nwhich in its essential modern form dates back to the work of\nHopf [111], transcends in accuracy the classical diffusion\nmethod but is less general in applicability than the spheri-\ncal harmonic method, in that it applies strictly only to gen-\neral transport media whose volume scattering function values\no (x;5' ;5) are independent of the directions E' and 5. How-\never, the relatively great tractability of the equation of\ntransfer resulting from the introduction of this simplifica-\ntion has led to many interesting and fairly detailed exact\nsolutions of the transfer equation, some of which are quite\nvaluable in practice. For this reason we include in our\npresent discussions a brief exposition of the two main diffu-\nsion methods. Together, the spherical harmonic method and\nthe diffusion methods form useful adjuncts to the basic natu-\nral mode of solution and the canonical mode of solution dis-\ncussed earlier in this work.\nThe plan of the chapter is as follows: We begin with\nthe spherical harmonic method. To show the extraordinarily\nwide scope and power of the method and also its inherent\nsimplicity we derive it in much more general settings than is\ncustomary, and from an abstract algebraic point of view. This\nwill be done in Sec. 6.2, after a preliminary section devoted\n134","SEC. 6.0\nINTRODUCTION\n135\nto motivating the method. Then follows a specialized develop-\nment of the method using the functions which have given the\nmethod its name (Sec. 6.3) but which, in view of the exposi-\ntion of Sec. 6.2, need no longer exclusively be used. An il-\nlustrative example of the spherical harmonic method is given\nin Sec. 6.4 for plane-parallel media. The discussion of the\nalgebraic idea underlying the spherical harmonic method will\nbe taken up again as a matter of course in Chapter 7 wherein\nwe shall view the method from a more fundamental point of view,\nnamely from the viewpoint of the generalized invariant imbed-\nding relation (Sec. 7.10). In Sec. 6.5, we turn to the dif-\nfusion methods, developing them directly from the equation of\ntransfer by imposing the characteristic assumptions of each\ntheory into the equation. The solutions of some of the more\nfamous models in the classical diffusion method are discussed\nin Sec. 6. 6. In Sec. 6.7 the Milne model for infinite media\nwith point sources is discussed, followed by some relatively\nrecent results on a related problem on point source problems\nin semi-infinite media. The chapter is concluded in Sec. 6.8\nby a brief bibliographic survey of other classical methods of\nsolution comprising some of the stock in trade of current\nradiative transfer theory.\n6.1\nThe Bases of the Spherical Harmonic Method\nIn this section we shall describe the physical and\nmathematical bases of the spherical harmonic method. We be-\ngin with a brief discussion of the motivation for factoring\nthe radiance function values N(x,5) into a sum of products\nof the form: (x) g (5). We then go on to show how this intui-\ntively and physically natural decomposition is sanctioned and\ngiven a direct representation in terms of vector space theory.\nTo accomplish this program, the mathematical prerequisites\nwill entail no more than standard advanced calculus techniques.\nPhysical Motivations\nThe steady state radiance function is essentially a\nfunction of two variables: the spatial variable x and the\ndirectional variable E. When one examines the equation of\ntransfer, in either its integrodifferential or integral forms,\none is confronted with the complicating presence of the inte-\ngral term--which represents an integration over the direc-\ntional variable. If it weren't for that integral term, the\nequation of transfer would be a simple differential equation\nand the theory would long ago have been worked out and for-\ngotten by mathematicians! When an investigator, new to the\nfield of radiative transfer theory, encounters the equation\nof transfer, one of his more probable actions would be to see\nwhat would happen if the radiance function N is assumed to be\nthe product of two functions f and g, such that:\nN(x,5) f(x)g(5)\n(1)\nCould the radiance function in some optical media be repre-\nsented simply as such a product? It would be instructive to\nfollow the consequences of this query, as it is at once one","136\nCLASSICAL SOLUTIONS\nVOL. III\nof the most natural and fruitful of questions to investigate\nin the task of solving transfer problems.\nThe immediate effect of such an assumption as (1) would\nbe the reduction of the path function N* to the form:\nN*(x,5)\nN(x,5') o(x;E';5) do(E)\n(1)\nI\n(x)\ng(E') o(x;E';E) do(E')\n(2)\n.\n(1)\nIt looks as if the assumption (1) is ineffective unless\na similar assumption is made about the volume scattering func-\ntion. Thus, in the spirit of (1), another assumption is made,\nnow about o: We assume that two functions C and p exist and\nare such that:\n(3)\nUsing (3) in (2), the representation of N ( x , 5 ) becomes:\nI,\nN(x,s)=f(x)c(x)\nE\n=ff(x) 8#(5)\n(4)\nwhere f* and g* are defined in the obvious way. Therefore,\nunder the additional assumption (3), the path function N* may,\nlike N itself be represented as a product of two functions:\none of x alone, the other of E alone.\nThe next step in the explorations would be to see if\nthe equation of transfer becomes more tractable with (1) and\n(3) as starting points. Thus, starting with the equation of\ntransfer:\n= a(x,5) N(x,5) + N*(x,5) , (5)\nand using (1) and (3), the equation becomes:\ng(E) df(x) = = - a(x,5) f(x) g(5) + f*(x) g* (5)\n(6)\nHaving split apart the spatial and directional components of\no, as shown in (3), it is physically reasonable (but not","SEC. 6.1\nSPHERICAL HARMONIC METHOD\n137\nlogically necessary) to do likewise with a. Succumbing for\nthe moment to physical reasonability, so that the discussion\ncan proceed, we assume a(x,.) to be constant valued on (1) for\nevery x in X, and write simply \"a(x)\" for this common value\nat X. Then (6) can be rearranged into the form:\nB#(5)/g(5) - a(x)\n(7)\nTwo observations may now be made. First, the results\nof the accumulated assumptions, succinctly residing in (7),\nshow that f(x) is in principle determinable by a simple inte-\ngration of the differential equation (7) along a path of sight\nprovided the values of the parenthesized terms in (7) are\nknown. The second observation is that the values of the pa-\nrenthesized terms in (7) are known once the quotient\nis known. By an inspection of (7), , it is clear that this\nquotient must be some number independent of E. Hence we\nwrite:\ng*(5)/g(E) = A\n(8)\n,\nwhich then in turn requires the function g to satisfy the\nintegral equation of the form:\n(9)\n1g(5) =\n(1)\nThe net result of the assumptions (1) and (3) are to\nreduce the problem of the solution of (5) into subproblems:\nthe solution of an integral equation (9) for g, with an ap-\npropriate X; and a solution of the simple ordinary differ-\nential equation (7) for f, using the A obtained in process\nof finding g.\nIt appears therefore that up to this point a definite\nstep has been made in the solution of (5) by adopting the\nassumptions (1) and (3). It seems worthwhile to follow this\npromising start and to attempt to carry the solution of (9)\nto completion. If this can be done for all physically rea-\nsonable assumptions on p(5';5) in (3) then a general solu-\ntion of the equation of transfer will have been found. To-\nward this end we will adopt for p(5' ;5) the property of (weak)\nisotropy, i.e., the property that for every E' and E:\n1.\nI\n(5';5) S(E) =\np(5';5) d&(E')\n(1)\nSince either integral will be independent of E, or E', we\nshall set its fixed value equal to 1. This puts the burden\nof the correct magnitude of o on c(x) in (3). In fact we now\nsee that C (x) is none other than the volume total scattering","CLASSICAL SOLUTIONS\nVOL. III\n138\nvalue at x in X because, by (3) of Sec. 4.2:\n1.\n1.\ns(x;5')\n(x;5';5) do(E) = c(x)\np(5';5) dn(E)\nin\n=c(x)\nHence S (x;5') is independent of E', , and we write \"s (x)\" for\nthis common value at X. In this way we simultaneously nor-\nmalize p and give C a physical interpretation.\nA similar normalization can be made of g in (1) with\nthe corresponding effect of giving f a convenient physical\ninterpretation. Thus, requiring g to have the property:\n1.\ng(5) ds(E) = 1\nE\nit follows from (1) that:\n!\nh(*)\ng(E) ds(E) = f(x)\nN(x,5) SS(E) = f(x)\n=\n.\n(1)\nHence f is in this case simply the scalar irradiance function\nh.\nReturning now to the two reduced equations (7) and (9)\nwe have from (7) and (8) that:\ndh(x) = h x) (2s(x) - a(x)\n(10)\nFurthermore, from (9), by integrating each side over E, we\nfind:\n[1.\na 1/2 (5) do(E) = g(E')\np(5';;5) d2(E) do(E')\nwhence\n(11)\nA=1\nso that (10) reduces to:\ndh(x) = - a (x)h(x)\n(12)\n=\ndr","SEC. 6.1\nSPHERICAL HARMONIC METHOD\n139\nand (9) becomes:\n|\ng(5) =\ndo(E')\n(13)\nWe now have reduced the problem of determining the radi-\nance function N, under the assumptions (1) and (3), to the\nproblem of a simple integration of (12) along a path with re-\nspect to path length r, and the solution of (13). The solu-\ntion of equation (12) presents no difficulty, the general\nsolution being:\nr\n(x) = h x ) exp\n(x') dr'\n(14)\n0\nwhen the integration is taken along a straight path Pr(xo,5)\nof length r from point xo to X. The intermediate point\nx' is the form XO + r'E, 0 < r' <\nr.\nFinally, we turn to (13) and immediately observe that\nany constant function on E, whose value for every E in (1) is\nsome arbitrary fixed value go, is a solution. It follows\nthat, if g is any nonconstant solution of (13) then so will\ng + go be a solution of (13). This nonuniqueness of solu-\ntions of (13) is a most undesirable state of affairs for a\nphysical model of the light field. This means that, on phys-\nical grounds, we must generally reject the model constituted\nby equations (12) and (13). It follows further that we must\nreject either or both assumptions (1), (3) which gave rise to\n(12) and (13). Since (3) is quite tenable on physical\ngrounds, it follows that we must generally* reject (1).\nIn\nthis way we have shown that the initial attempt to factor N\ninto a product of a scalar irradiance function h and a di-\nrectional function g is untenable on physical grounds. By\nrepeating the essential steps of the arguments between (1)\nand (13) the same negative conclusion may be deduced for the\ncase where N is represented as a finite sum of terms of the\nform higi.\nThe intuitive concept of factoring N into spatial and\ndirectional components in general media has thus been shown\nto be unsupportable on practical physical grounds. However,\nthe factoring may be possible in certain geometrically and\nphysically ideal media. Indeed, as we saw in Sec. 4.4, plane-\nparallel media with uniform volume scattering functions per-\nmit such a factoring of N. According to (9) of Sec. 4.4, we\n*In particular, if a g can be found which satisfies (13),\nthen some approximate models may be found by adjusting go\nempirically in N(x,5) = h(x) (g(5) + go(s)).","140\nCLASSICAL SOLUTIONS\nVOL. III\nhave\n1\ng(E) 1 +(a)cos\n(15)\n0\nwhere K is now determined by the requirement that the normal-\nization property of g holds. Thus by adding two more assump-\ntions to (1) and (3), namely that h (x) varied exponentially\nwith a certain fixed exponential decay rate K, and that\no (x;5';5) is independent of E' and E, a very special factor-\nable radiance function is forthcoming.\nThe additional physical conditions of the required spe-\ncial exponential character of h and the uniform directional\nstructure of o are quite severe restrictions to impose on\ngeneral media in order to obtain a factoring of N. However,\nas we shall see later (40) of Sec. 6.6 and (3) of Sec. 7.10\nand Sec. 10.5], it is a property of certain extensive homoge-\nneous media that the radiance function N at great distances\nfrom the boundaries of such media comes arbitrarily close\n(for correspondingly great distances) to functions of the form\nhg, i.e., to factored form, in which there is a spatial fac-\ntor h and a directional factor g.\nThe conclusions of the various arguments presented above\nmay now be summarized.\n(i) In general media X for which (3) holds, the assump-\ntion that there exists a function g on (1) such that N(x,5) =\n(x) g(5) for every x in X and E in (1) is generally untenable\non physical grounds (the associated solutions are not unique).\nMore generally, finite representations of the form\nn < 00\nare also untenable.\n(ii) In some extensive, homogeneous media X, there\nexists a function g on E such that N(x,5) h(x)g(E) for\nevery E in E and x sufficiently far from the boundaries of X.\nBy\ncomparing the conclusions summarized in (i) and (ii), we\nsee from (i) that on the one hand the original intuitive guess\nas to the factorability of N into the form gh was generally\nincorrect; by conclusion (ii), on the other hand, there is a\nsmall solid core of truth inherent in the intuitive guess.\nFurthermore, while finite representations of N in the form\nn\nheigi\ni=0\nare generally incorrect, these representations may possibly\nbe so constructed that they increase in accuracy with an in-\ncrease in the number of terms of the sum. In particular it\nwould seem that by choosing sufficiently large numbers of\nterms for","141\nSEC. 6.1\nSPHERICAL HARMONIC METHOD\nn\n[ higi\n,\ni=0\nthese approximations to N may be improved at all points of a\nmedium X. Then at large distances from the boundaries of X\nthere will, by (ii), , be a single term higi of\nn\nE higi\ni=0\nwhich will dominate the others and which will essentially\nrepresent N in those regions.\nWith these observations we have reached the last stage\nof the physical motivation for the abstract harmonic repre-\nsentation method. We thereby are led to consider infinite\nseries of the form:\ni=0\nwhich, for given fixed X in X, represents the radiance dis-\ntribution values N(x,5) for every direction E in E.\nAn Algebraic Setting for Radiance Distributions\nThe preceding discussion has motivated the representa-\ntion of a radiance distribution N(x,) at a fixed point X in\nan optical medium X by means of an infinite series of func-\ntions, in the form:\n(16)\nThis constitutes the first step in constructing the\nabstract harmonic representation of N(x,.)).\nThe next step calls for the construction of an infinite\n} of functions, each with (1) as domain,\nfamily {00,\nand with the Following properties. First, the ii's are gen-\nerally allowed to be complex valued. This provides a great\ntheoretical convenience and in no way forces N to be complex\nvalued under specific physical conditions. Second, we re-\nquire that the family {00, P1, Q2, } be orthonormal, i.e.,\n(17)\n(1)\nwhere Sij is the Kronecker delta, i.e., Sij is zero whenever\ni + j, and one whenever i = j. This operation of integration","142\nCLASSICAL SOLUTIONS\nVOL. III\nand others similar to it will arise sufficiently often in the\nfollowing discussion that it will be convenient to abbreviate\nit in general by writing:\n() T(E)da(E)\n\"[0,4]\"\nfor\n(18)\nwhere and 4 are any two functions on E so that the integral\nof their product, as in (18), is defined. The bar over a\nfunction denotes complex conjugation. We call [0,4] the in-\nner (or scalar) product of and 4.\nThe reason for the terminology \"inner product\" stems\nfrom the deep similarity of this inner product with the clas-\nsical scalar product x y of two vectors x and y in euclidean\nthree space. The most striking similarities are paired off\nin the list below. Their proofs are immediate:\n(i)\n(i)\nIf {00, Q1, }\nIf a1, A2, a3 are\npairwise orthogonal\nis an orthonormal\nunit vectors of E3,\nfamily of functions\non E, then [di,4j]\nthen ai aj Sij\n= Sij\n(ii)\n(ii)\nIf, for a function\nIf, for a vector E\ng on E, there exist\nin (1) there exist\nn numbers Co, C1, C2,\nthree numbers C1,\n..., Cn, such that\nC2, C3 such that\nE C1a1 + C2a2\nn\n+ C3A3, then\nCi=E.ai\nthen Ci = [g,oi]\n(iii)\n(iii)\nx(y+z) = x°y + X°Z\n[f,g+h] = [f,g] + [f,h]\n[f+g,h] = [f,h] + [g,h]\n(x+y) -z = x°Z + y°Z\n(iv)\n(iv)\n[cf,g] =c[f,g]\n(cx).y = c(xy) = x°cy)\n[f,cg] = c[f,g]\nThe physical motivations discussed above have led us to\nconsider infinite series, so that the vector-spacelike prop-\nerty (ii) for inner product will be postulated to hold for\ninfinite series. The specific form of the infinite version\nof (ii) we shall adopt is as follows (the mathematical regu-\nlarity properties of integrability are omitted for simplicity\nof exposition)\nCompleteness property of {00, 01, Q2, ...}. If F is a\nfunction on E, and if for every j > 0 we write:\n\"fi\" for [F,qj]","143\nSEC. 6.1\nSPHERICAL HARMONIC METHOD\nthen:\n(19)\n=\nfor every E in E.\nThe algebraic setting for radiance distributions dis-\ncussed in example 15 of Sec. 2.11, now may be used once again.\nIn fact we can easily extend that setting for our present\npurposes. We therefore imagine all possible radiance distri-\nbutions at a fixed point x in X and imagine further all their\nnegatives and imaginaries (-N(x,) is the negative of N(x,.)),\niN(x,.) where i = V-I, is the imaginary of N(x, )) thrown in\nwith them. The totality n(x) of these and all possible sums\nof them form a vector space in the general sense: Sums of\nmembers of n(x) are again in n(x); and multiplication of mem-\nbers of n(x) by complex numbers are again in n(x). The addi-\ntional details of verification are simple and need not detain\nus here. The main fact to observe is that the set of all\nintegrable radiance distributions at a point x in X can be\nimbedded in a vector space of functions on (1) which includes\northonormal set {00, 01, Q2, } such that the complete-\nan\nness property holds for {00, 01, Q2 }. This is the alge-\nbraic setting for radiance distributions in which the ab -\nstract spherical harmonic method will be discussed.\nAbstract Spherical Harmonic Method\n6.2\nThe motivation and prerequisites of the abstract spher-\nical harmonic method having been dispatched in Sec. 6.1, we\nturn directly to the method itself, now applied to the gener-\nal time-dependent equation of transfer with source term ((14)\nof Sec. 3.15):\n1 ON\n(1)\n+ E . VN = - aN + N* + N n\n.\nat\nV\nwhere N is defined on a general optical medium X which may be\nfinite or infinite, generally inhomogeneous, but isotropic.\nWe assume furthermore that there exists an orthonormal family\n01, Q2, } of functions on E which has the complete-\n{00,\nness property.\nThe completeness property of {00, 01, O2, } applied\nto the radiance distribution N(x,.) at X in X yields:\n(2)\nN(x,E,t) = j=0 fj(x,t) pj (5)\nwhere we have written:\n(3)\n\"fj(x,t)\" for\nThus fj (x, t) is the scalar obtained by performing the integra-\ntion:","144\nCLASSICAL SOLUTIONS\nVOL. III\n(x,E,t)\n.\n(1)\nIn a similar manner we obtain:\n(E)\n(4)\nas the representation of the emission funcion Nn where we\nhave written:\n\"fn,j(x,t)\" for\n(5)\n.\nThe representation of the volume scattering function o\nis next. Since o uses two directional variables, we use the\ncompleteness property twice. First we obtain:\n(6)\n=\nwhere we have written:\n\"0j(x;5';t)\" for [0(x;5';-;t), ]\n(7)\nNext we obtain:\n(8)\nwhere we have written:\nfor [0(;*;t),\n(9)\nCombining these representations, we have:\n(x;5';5;t) = j=0 TK(E') (j)5)\n(10)\nThe reason for introducing the conjugates of the ok into (10)\nwill become clear shortly.\nNow the whole purpose of the spherical harmonic method,\nas we have seen in Sec. 6.1, is to effectively separate the\nspatial variables from the directional variables in the equa-\ntion of transfer so that the latter may be contained in a\nsystem of simple, directly integrable differential equations\ninvolving spatial variables only. We now apply the abstract\nharmonic representations of N, Nn, and o to the equation of\ntransfer (1), and effect such a separation of variables. On","SEC. 6.2\nABSTRACT METHOD\n145\nthe right side of (1) we have Nn already represented. Then\nfor the term N* (the summations all go from 0 to oo):\nN+(x,E,t) - = da(s)\ndr(E)\nda(E )\nE\n=\n(11)\nSince the medium X is assumed isotropic, the volume\nattenuation function values a(x;5) are independent of E, and\nso a need not be represented by a series of the complete\nfamily {00, 01, Q2, ...}. By means of (4), (10), and (11) we\ncan therefore represent the right side of (1) in the form:\nj=0 +\n(12)\nAttention is now directed to the left side of (1). The\ntime derivative term is directly treated to yield:\n(13)\n.","146\nCLASSICAL SOLUTIONS\nVOL. III\nThe spatial derivative term becomes:\n(VN(x,E,t)\n(14)\nCombining (12), (13), and (14) according to (1), we have:\n(15)\nIf it weren't for the spatial derivative term the contents of\nthe square bracket would have been free of the variable E,\nand a system of equations would have been obtained by setting\neach bracketed jth term to zero. At any rate we can elimi-\nnate the presence of E by an integration over E. The ortho-\nnormality property of {00, 01, O2, } is available for\nuse\nin this task. Thus multiplying each side of (15) by (5)\nand integrating over E, the orthonormality property immediate-\n1y yields\n(1)\n- a(x)fk(x,t) (x;t) + fn,k(x,t)\n(16)\n=\nIf we now write:\n\"Djk\"\nfor\n. do(E) ,\n(17)\nthen we obtain, at last, the spherical harmonic analysis of\n(1)\n=\n(18)\nk = 0, 1, 2,\nThis is the requisite abstract spherical harmonic system of\npartial differential equations for the family {fo, f1, , f2,\n}\nof functions, the abstract harmonic coefficient functions of\nthe radiance distribution N(x,.)). Knowledge of these fj","SEC. 6.2\nABSTRACT METHOD\n147\nallows construction of N(x,.) according to (2). The heart of\nthe abstract harmonic method of solving the equation of trans-\nfer thus resides in (18).\nFinite Forms of the Abstract\nHarmonic Equations\nAn inspection of the system (18) of abstract harmonic\nequations governing the harmonic coefficient functions fk\nshows two infinite series involved in the system. The pres-\nence of these infinite series could occasionally negate the\npractical utility of the system, for example in numerical so-\nlution work. It is interesting to observe, however, that\nthese infinite series may be rigorously removed and replaced\nby finite sums under the combined action of two very general\nconditions, one physical, the other mathematical. The mathe-\nmatical condition simplifies the differential operator series;\nthe physical condition simplifies the scattering term series.\nWe shall now briefly indicate the nature of these conditions.\nWe shall say that the family {00, Q2, } of func-\ntions on (1) has the finite recurrence property of degree v if\nfor every element E' in (1) and every in the\nfamily,\nthere\nexist V constants Ajk and V elements\ndav of {00,\n01, 2 ...} such that\n(5)\n(19)\nholds for every E in E. The motivation for this property\narises in an attempt to simplify the form of the operators\nDik and to reduce to a finite series the infinite series in-\nvolving them in (18). For example, in an orthogonal, three-\ndimensional coordinate frame in which X = (X1, X2, X3), we\nhave:\na\nWe use this form in (17) to obtain the representation\na\na\n(20)\n+ dx2 + C dx3\nwhere we have written:\n\"ajk\" for\ndo(E)\n(21)\n(22)\n\"bjk\"\nfor","148\nCLASSICAL SOLUTIONS\nVOL. III\nTK(5)\n\"Cjk\"\nfor\ndo(E)\n(23)\nE\nBy postulating a finite recurrence property of degree v for\n{00, 01, 2 2 }, if follows that ajk 0 whenever the in-\ndices k and j differ by a sufficiently large amount: indeed\n= 0 for all but at most V terms. Similarly with bjk and\najk\nCjk: This means that for fixed k Djk = 0 whenever j is suf-\nficiently large, and so the number of terms on the left of\n(18) become finite in the present case. It turns out that\nany orthonormal family obtained from suitable nth order\nordinary differential equations (a rich source of orthonormal\nfamilies by means of Sturm-Liouville theory) will possess a\nfinite recurrence property of degree V.\nFinally, the physical condition which simplifies the\nabstract harmonic equations is that of isotropy of the medium.\nIn the present case the isotropy reduces the general func-\ntional dependence of o on the independent variables E' and E\nto the special dependence of o on the scalar product 5' E\nof the directions. This simplified structure of o in turn\nmanifests itself in a simplification of the representation\n(10) to the form:\n(24)\nWe shall not go into the derivation details of this re-\nlation in the present abstract case. It suffices to note\nthat this form can be obtained when the members of the ortho-\nnormal family {00, 01, Q2, } obey a general type of addi-\ntion theorem often valid for functions arising in Sturm-Liou-\nville theory. Examples of addition theorems for such func-\ntions, are, e.g., in [318]. (See (12) and (15) of Sec. 6.3.)\nThe simplifying effect of (24) becomes evident when we\nrecalculate N+(x,E,t) after the manner of (11):\n= dn(E')\nN*(x,E,t)\n{{{1(x,t)\ndo(E')\ndr(E)","SEC. 6.2\nABSTRACT METHOD\n149\n() Sij\n(x;t)\n(25)\n=\nBy combining the preceding two conditions, the total\neffect on (18) is a complete finitization of each equation in\nthe system of equations, thereby rendering them more effec-\ntive for numerical computations. We may summarize these con-\nstructions as follows:\nLet X be an arbitrary isotropic, inhomogeneous optical\nmedium with internal emission radiance function Nn and gener-\nal time-dependent radiance field N as governed by the equa-\ntion of transfer (1). Let {00, 01, Q2, } be an orthonor-\nmal family of functions defined on the unit sphere (1) such\nthat: the family (a) possesses the completeness property (see\n(19) of Sec. 6.1); (b) possesses the finite recurrence prop-\nerty (19); (c) satisfies an addition theorem (24). Then each\nmember of the general abstract harmonic system of partial\ndifferential equations (18) reduces to the following finite\nform: For some positive integer V:\n1 af k\nv\n{\nk = 0,1,2,\nfjDj\n+\n+\nn,k\n=\nv st\njk\nj=0\n(26)\n6.3 Classical Spherical Harmonic Method: General Media\nThe general theory of the abstract harmonic method de-\nveloped in the preceding section will now be illustrated for\nthe classical case in which the orthonormal family is con-\nstructed from familes of associated Legendre functions of the\nfirst kind and circular (trigonometric) functions. The opti-\ncal medium X will be generally inhomogeneous and isotropic,\nwith time varying inherent optical properties, and given in-\nternal sources.\nThe Orthonormal Family\nWe begin by observing that the classical spherical har-\nmonic method customarily uses the ordered pair (u, of num-\nbers to specify a point E in E, where we have written \"u\" for\ncos 0, and where (0, ) are the two angles customarily used to\nspecify E in (1) (see Sec. 2.4 and also example 14 of Sec. 2.11\nfor an earlier use of H in conjunction with Legendre polynomi-\nals). The range of the variable u is thus the interval [-1,1],\nand the range of , [0, 2]. Every E in (1) determines a unique\n(0,0), that is a unique H in [-1,1] and a unique in [0,2].","150\nCLASSICAL SOLUTIONS\nVOL. III\nConversely, any pair (H,O) in [-1,1] x [0,2] determines a\nunique E in E.\nThe values of associated Legendre functions are usually\ndenoted by \"Pm(u)\". The integer n is nonnegative, i.e., n 0\nand the integer m satisfies the inequalities: -n m n. The\ngeneral relations in the theory of Legendre polynomiaIs we\nshall use below may be found fully developed, e.g., in [318],\n[289], and [119]. In particular we shall note that:\npnm =\n(1)\nand that:\nPO S P n\n(2)\nPM=0 for m > n\nwhere \"Pn\" denotes the Legendre function of the first kind\nand of degree n. For our present purposes, we note that the\nassociated Legendre function pm n is a real valued function\nwith domain [-1,1] and defined for all integers m,n such that\nn is nonnegative and /m/ < n. The associated Legendre func-\ntions include, by (2), the Legendre polynomials as special\ncases. Any functions pm n arising in the subsequent discussions\nfor which n < 0, are to be zero-valued functions. In view of\n(1) and (2) only pm with n+1 nonnegative indices m need be\ntabulated.\nThe orthogonality property of the family of associated\nLegendre functions takes the form:\n1\n0, ,\nwhenever n\nr\nP(m) du =\n2\n(n+m)\nwhenever n If r\n2n+I ( n-m)! '\no\n-1\n,\n(3)\nThe integral properties of the family of circular func-\ntions needed here are summarized by the equations:\n2\nsin mo do = 0\n0\n2\n{\n0 if m # 1 0\ncos mo do =\n(4)\n2 if m = 0\n0","Sec. 6.3\nGENERAL MEDIA\n151\nwhere m is confined to integral values. These properties can\nbe succinctly summarized by using complex variables. Thus,\nall three equations in (4) may be expressed by writing:\n12\n2\n(5)\nom\n0\nwhere Som is an instance of the general Knonecker delta Sij\nThe use of complex variables will considerably facilitate our\nwork in this section, and so they will be retained throughout.\nOne can always return to the real number setting by finding\nand considering separately the real and imaginary parts of\na\ncomplex term.\nThe details of the construction of the requisite ortho-\nnormal family on (1) are clearly indicated by considering (3)\nand (5). Thus to an arbitrary E in E, (to which corresponds a\nunique pair (1,0)) and to every pair of integers m, n, with\nn > 0, m < n we assign the complex number m (5) where we\nhave written:\nfor\n(6)\nwhere in turn we have written\n1/2\n\"Am\"\nfor\n(7)\nBy observing that:\nwe can limit tabulations of Am to nonnegative indices m.\nFurthermore, by recalling (1), the complex conjugate of ()\nmay be expressed as follows:\n)i((\n(8)\nThe orthonormality property of the family of functions\n&m over (1) may now be verified. For example:\n(1) dn(E) =\ndu\n= nr\n.\n8nr","152\nCLASSICAL SOLUTIONS\nVOL.\nIII\nThe remaining case where the upper indices of m may differ\nis straightforward using (5). Hence we have:\n(m(5)\ndo(E)\n(9)\ns\n=\nmb\nna\n(1)\n|m|\nfor every n, a 0 and b, m such that\n/b/\n<\n<\nn.\nAn exact one-to-one correspondence can be established\nbetween the abstract family {00, 01, Q2,\n}\nof Sec. 6.2 and\nthe spherical harmonic family presently under consideration.\nThus to oj of the earlier discussion we pair m, where =\nn2\nn. This correspondence arises when one contemplates\nFig. 6.1 in which each dot in the figure is paired with the\ninteger couple (m,n) n > 0, |m| < n, corresponding to the\nindices of . Then counting each row of dots by reading\nfrom left to right and counting rows from bottom to top, each\ndot is given a single index j. For example the dot in the\nfirst row, corresponding to (0,0) is given the index 0. The\ndot corresponding to (-1,1) is given the index 1, (0,1) the\nindex 2, (-3,4) the index 17, etc. In general:\nn\n(m,n)=(1,3)\nm\n-4 -3 -2 -1\n2\n3\n4\nFIG. 6.1 Scheme for establishing the correspondence be-\ntween the abstract and classical spherical harmonic method.","SEC. 6.3\nGENERAL MEDIA\n153\n(m,n) is paired with the index j = n2 + + n\n(10)\nand\n$ m is paired with j\n(11)\nObserve that the pairings are unique: given (m,n) there is\nprecisely one j > 0 corresponding to this pair; given j > 0,\nthere is precisely one pair (m,n) on the array corresponding\nto j and is readily obtained under the conditions on m,n de-\nscribed above.\nProperties of the Orthonormal Family\nWe shall now show that the family of spherical harmon-\nics\nn on (1) possesses the three main properties sufficient to\ninsure a reduction of the general abstract harmonic system\n(18) of Sec. 6.2 to its finite version (26) of Sec. 6.2.\n(The proof of the orthonormality of the family of spherical\nharmonics was outlined in the discussion leading to (9).)\nThe completeness property of the set of spherical har-\nmonics holds. However, the property depends on some rela-\ntively advanced arguments, and the interested reader is re-\nferred to Chapter 7 of [47] for the general theory of com-\npleteness of families of functions arising from nth order\ndifferential equations.\nThe addition theorem for Legendre functions holds and\ntakes the form (see, e.g., [119]):\n= cos m(-.')\n(12)\nwhere E and E' are any two directions in (1) and (H,O), (H',''')\nare their corresponding angular representations. Using (1),\n(2) , the evenness of cosine, and the oddness of sine, (12)\nmay be compactly written as :\nPe\n(13)\nThe argument of Pn in (13) is the scalar product of E'\nand E. This scalar product is reminiscent of the isotropy\ncondition for an optical medium. We now show how the isot-\nropy condition leads in the present case to the representa-\ntion of o in the form of (24) of Sec. 6.2. When isotropy","VOL. III\nCLASSICAL SOLUTIONS\n154\nholds, the value of o (for a fixed x and t) is known once\nE . E' is known, i.e., once a number u = E.E. . in the inter-\nval [-1,1] is specified. This value of o under isotropy con-\nditions will be denoted by \"o(x;55';t)\". Therefore, the\nfamily of Legendre polynomials Pn being complete (a fact also\nsupplied by the general theory in [47] cited above), we may\nexpress as follows:\n(14)\nwhere we have written:\n1\no (x;u;t) Pj(1) du\n(15)\n\"oj(x;t)\" for 2\n-1\nUsing (13) to represent P(EE') in (14), we have:\n=\n(16)\nThis is reducible to the form of (24) of Sec. 6.2 as may be\nseen by using the correspondence between oj and estab-\nlished above. (To show the correspondence in complete detail,\nlet o (x;t) be denoted ad hoc as \"of(x;t)\" and require it to\nhave value (x;t) for m in the range - < m < j.)\nIn this way we see how the addition theorem for the pm\nand the isotropy condition on scattering combine to form the\nextremely useful representation (16). The reader may now\nextend this idea to still other complete orthonormal families\nof functions defined on [-1,1] provided an addition theorem\nof the kind (13) is available for the family.\nNext, we observe that the orthonormal family of func-\ntions m satisfies the finite recurrence property of degree 2.\nThis observation is based on the following three well-known","SEC. 6.3\nGENERAL MEDIA\n155\nrecurrence properties of associated Legendre functions (see,\ne.g., [289], [119]):\nn-1 (u) + (n+1-m)\n(17)\n(u)\nsin\n0\n(18)\nsin 0 (n-m +2) (m-n-1) pm-1 n+1 (u) + (n+m-1) (n+m) pm-1 n-1 (u)\n(2n+1)\n(19)\nAs an example of how these recurrence relations give\nrise to instances of the general recurrence property (19) of\nSec. 6.2, consider (17). . Here we recall that \"H\" denotes\nE.K; k is the unit vector along the positive z-axis. Hence E'\nin (19) of Sec 6.2 is now k. Next, multiply each side of\n(17) by Am m eimo. Applying the general definition (6) and\nmaking some algebraic rearrangements the net result is:\nm-1 (5) + C(n+1,m) n+1 (5)\n(20)\nwhere we have written:\n\"C(n,m)\" for [\n1/2\n(21)\nHence in (19) of Sec. 6.2, we have v = 2, and the Ajk are now\nin the form of c(j,k), with j = n2 + m + n, and a1 = (n-1)2\n+ m + (n-1), a2 =_(n+1) + m + (n+1). The specific represen-\ntation of E . k $(((5) in (20) is now used in (20) of Sec. 6.2\nto effect an evaluation of the number cjk, and hence the sum:\n(22)\nj=0\nwhich forms part of the operation:\nv\nj=0 E fjb3\n(23)","CLASSICAL SOLUTIONS\nVOL. III\n156\nin (26) of Sec. 6.2. To see how (22) is evaluated, let us\nrepresent N(x,E,t) by means of the functions :\n(24)\nm = -n\nwhere we have written:\n\"Fm(x,t)\"\nN(x,E,t)\n(25)\nfor\n(1)\nThus Fm n in the present context corresponds to fi in the\nabstract context of Sec. 6.2, just as m corresponds to\nFurthermore, the correspondence of j in fi with the pair of\nindices (m,n) of Fm is once again that established above.\n(See Fig. 6.1 and (10), (11).)\nReturning to (22), we consider it in the context of (18)\nof Sec. 6.2, but now using the present family { }} of ortho-\nnormal functions. We threfore are to consider:\nm=-n\nE\nn=0\nm=-n\n(26)\n= C(a+1,b) +\nin which k = a2 + b + a.\nThus the infinite sum of z-derivatives in (18) of Sec.\n6.2 is reduced to a sum of two such derivatives.\nThe general procedure should now be clear: by placing\nthe recurrence relations (18) and (19) into their appropriate\ncounterparts of (20), , the numbers ajk and bjk in (21), (22)\nof Sec. 6.2 are readily evaluated. Then the sums :\n,\nj=0\nare evaluated analogously to the manner displayed in (26).\nThese details may be left to the reader.","157\nSEC. 6.3\nGENERAL MEDIA\nGeneral Equations for Spherical\nHarmonic Method\nThe net result of the reduction calculations on (26)\noutlined above may be written in the form:\n-B(a+1,-b+1)\n= [ +\na = 0,1,2,\n(27)\nwhere we have written:\n(28)\n\"B(a,b)\" for\nand where C(a,b) is defined generally in (21). Furthermore,\nwe have written:\n\"Fn,a(x,t)\"\n(29)\nfor\n(1)\nanalogously to (25), so that Nn has the representation:\n(30)\nm = -n\nThe set of equations (27) forms a coupled infinite sys-\ntem of equations in the unknown functions Fa, a = 0,1,2,\n1b1 < a. The functions Fb are generally complex valued,\naccording to their defined construction (25), , and such that\nN(x,E,t) is real valued, according to (24). The general ini-\ntialconditions for the system (27) are:","158\nCLASSICAL SOLUTIONS\nVOL. III\nN°(x,5,0) (C(E) do(E)\n(31)\n=\n,\nE\nfor every x in x, and where N° is the given initial radiance\nfunction on X x E at t = 0. For steady state versions of\n(27), the time derivative term is zero. The functions Fb\nthen have domain X and (31) is replaced by:\nFa(xo) =\ni°(x),5) (6(5) da(E)\n(32)\nfor X C over some appropriate subset of the boundary\nof X (cf. e.g., (26) of Sec. 6.4).\nThe system (27) is of sufficient generality to solve\nsuch problems as point source, beam source, and general in-\nternal source problems in the sea; natural light field prob-\n1ems in lakes, harbors, and the sea. Observe that the in-\nherent optical properties in the form of a and a may be\nquite general, and that the term FR a provides for internal\nsources of radiant flux, such as artificial light sources\n(laser beams, searchlights, submerged incandescent point\nsources, etc.) or natural light sources (phosphorescence,\nanimal sources, etc.). The general methods of solution of\n(27) and its manifold variants are well known and may be im-\nplemented by programmed machine procedures. If the model is\nsufficiently simple (as, e.g., in the illustration of Sec.\n6.4) the associated simplified form of system (27) may be\nsolved by hand and evaluated numerically or even used for\ngeneral theoretical reasoning.\n6.4\nClassical Spherical Harmonic Method: Plane-Parallel\nMedia\nThe classical spherical harmonic method developed in\nthe preceding section for general media will now be illus-\ntrated in a setting of primary importance in hydrologic (and\nmeteorologic) optics: the plane-parallel optical medium.\nThroughout this section, then, we shall assume that X is a\nplane-parallel medium of arbitrary (finite or infinite) depth.\nThe incident light field and the optical properties of X are\nassumed to be in the steady state and independent of the x\nand y coordinates throughout X, thus establishing a stratified\nmedium and stratified steady radiance field throughout X.\nUnder the present conditions on the medium X, the gen-\neral system of equations (27) of Sec. 6.3 reduces to:\nC(a,b) aFb -az a-1 + C(a+1,b) = (-a+oa) + Fb n,a\n(1)\n1b\n==0,1,2,\na\na","PLANE-PARALLEL MEDIA\n159\nSEC. 6.4\nHere we have adopted the terrestrially based coordinate\nsystem for hydrologic optics (Sec. 2.4) wherein depth Z is\nmeasured positive downwards from the air-water boundary. Thus\n\"-z\" in (1) now replaces \"X3\" in the general formula (27) of\nSec. 6.3, and \"x\" and \"y\" replace \"X1\" and \"X2\", respectively.\nThe functions a and a may vary with depth.\nThe first few equations of system (1), written out in\ngroups for each value of a, are:\n=\naFi\na = 1; b = -1:\nHF21\na = 1;b = 0: =\n+\na = 1;b = = 1:\na = 2; b = -2:\n+\na = 2; b = -1:\nC(2,- aF1 (-a+o2) F21 + F-1\na = 2; b = 0: =\nC(2,0)\na = 2; b = = 1:\nC(2,1)\na = 2; b = 2:\n(3.2)","160\nCLASSICAL SOLUTIONS\nVOL. III\nThus the group of equations for a = 0 consists of one equation;\nthe group for a = 1 consists of three equations; the group\nfor a = 2 consists of five equations. In general the group\nfor a = n consists of 2n + 1 equations. Some of the deriva-\ntive terms are missing in the displayed system above because\nof the conditions placed on the indices at the outset of the\ndiscussion. Thus Fb = 0 if a < 0 or a < 1bl. A convenient\nauxiliary rule to observe in this respect is that: whenever\na-b = 0 or a+b = 0, then C(a,b) = 0.\nA Formal Solution Procedure\nThe system (1), which represents the system of equations\nfor the spherical harmonic method in a plane-parallel setting,\ndisplays an interesting type of coupling among the functions\nFb. Observe how the upper index b is fixed in each equation\nof the system. We shall now show how this feature permits a\nsimplification of the general solution procedure of the sys-\ntem. The manner of simplification may be easily seen by means\nof the diagram in Fig. 6.2.\nEach dot in Fig. 6.2 represents an ordered pair (b,a)\nof indices corresponding to Fa. The effect of the rather\nweak coupling among the unknown functions Fb a of system (1) is\nsuch that we can partition the set of unknown functions into\nsubsets, corresponding to the vertical columns of dots, and\na\nautonomous columns\n4\n3\n2\nb\n-4 -3 -2 -1\n2\n3\n4\nb\nFIG. 6.2 A way of grouping the functions F a into\nautono-\nmous families, for solution purposes.","161\nSEC. 6.4\nPLANE-PARALLEL MEDIA\nsolve for the unknown functions associated solely with each\ncolumn. That is, , the unknowns Fb in the bth column can be\nobtained independently of the unknowns in the other columns of\nof the array. This observation can be put into a convenient\nmathematical form as follows. Let us write:\n\"Fb11 for Flb|+2'\n(2)\nand\n)\n\"ph\" for Fn,|b|+1'\n(3)\nThus, e.g.,\n= ...)\nF-1 =\n...)\n=\ny-2 = (F22, F22, F 2 2\n= ...)\n,\nand so on. With this notation, we see that the part of sys-\ntem (1) corresponding to an arbitrary fixed index b may be\nwritten succinctly in vector form as :\n(4)\nwhere we have written:\nc(Ib)=1,b)\nc(/b/+1,b)\n0\n0\n0\nc(/b/+2,b)\n0\n0\nc(|b/+2,b)\nC(lb++,b)\nc(|b/+3,\nC(l+++,3,b)\n0\n0\n\"Cb,,\nC(lb/+4,b)\n0\n0\n0\nc(|b/+5,b)\n0\n0\nfor\n0\n(5)","162\nCLASSICAL SOLUTIONS\nVOL. III\nand where we have written:\n0\n0\n0\n00\n0\n0\n+\n0\na\n01\n0\n0\na + O2\n\"a\" for\n(6)\nThe system (4) may be rearranged into the form:\n(7)\nb = 0, + 1, 2,\nwhere we have written:\n\"Bb,, for accb)-1\n(8)\n\"Gh\" for pheeb,\n(9)\nwhere \"(c),-\" denotes the formal inverse of eb.\nand\nThe formal solution procedure for (1) is now seen to be\nreduced to that associated with (7) and thereby becomes rela-\ntively straightfoward on either the numerical or manual levels.\nOf course, in practice, when numerical solutions are desired,\nthe system (7) must be truncated to a finite system along\nwith the number of components of Fb, and the formal inversion\nof eb must be reduced to a workable procedure. Before going\non to consider such truncations, we can place the system into\na standard form occasionally useful for formal theoretical\nconsiderations and which also shows the general overall struc-\nture of the system (1). Thus we first agree to write:\n\"p\" for (..., F-2, F-1, F0, p1,\n)\n\"Gn\" for (..., Gn2, G-1 Go, G1, G2,\n)\nand finally:","SEC. 6.4\nPLANE-PARALLEL MEDIA\n163\nB-2 B-1, B°, B1, B2\nB11\nfor diag (\n..)\nwhere \"diag\" denotes a diagonal block matrix with Bi as the\ni th block matrix along the diagonal. Then the system (7)\ntakes the generic form:\n(10)\nGn\nThis is the desired vectorial version of the system (1), show-\ning the overall linear form of the system, a form reminiscent\nof the equation of transfer without the path function term.\nThus we see from still another vantage point that the net\neffect of the spherical harmonic method is the removal of the\ncomplex directional dependence of the radiance field gener-\nated by the presence of the path function term N* in the gen-\neral equation of transfer.\nA Truncated Solution Procedure\nAs an illustration of the use of (7) in practice, we\nconsider the case of an arbitrarily stratified source-free\nplane-parallel medium. Thus in (7) we set:\nch=0\n(11)\nfor every integer b, /b/ > 0. This is a commonly occurring\nradiometric situation in most natural media in geophysical\noptics, so that the present illustration retains a wide range\nof applicability. The effect of condition (11) is rather far-\nreaching. To see this effect, observe that by the definition\nof C(a,b) we have:\nC(a,-b) = C(a,b)\nFrom this it follows that, formally\nC-b - eb and so B-b B Bb =\n(12)\nThus we need only consider:\nF B b , b = 0\n(13)\nNow the truncation procedure which we intend to apply\nto (13) may best be described by returning to the original\nsystem (1) and keeping in mind the diagram of Fig. 6.2. This\nreturn to (1) is also desirable, so as to bypass the formal\ninversion procedure leading to (13). It is clear from the\ndiagram in Fig. 6.2 that a truncation may take place at the\nth row, in the sense that no unknown functions Fb will be","164\nCLASSICAL SOLUTIONS\nVOL. III\nallowed in the system which have indices a > m. Then the\ntruncated autonomous system of equations associated with\nb = 0 is:\na = 0 :\na = 1:\nC(1,0)\nC(2,0)\na = 2:\n(14)\na = m-1: C(m-1,0) +\nC(m,0)\na = m :\nThe effect of the truncation becomes clear on inspec-\ntion of the equation corresponding to the case a = m. The\nderivative of Fm+1 is omitted from the equation for this\ncase. Thus in the system displayed above there are (m+1)\ndifferential equations and m+1 unknown functions: Fj, j = 0,\n1, , m.\nThe truncated autonomous system of equations associated\nwith b = 1 is:\na = 1 :\nC(2,1)\na = 2 :\nC(3,1)\na = 3:\n(15)\na = m-1: C (m-1,1)\na m :\nHere the system associated with b = 1 consists of m\ndifferential equations in m unknown functions: Fl, j = 1,\n..., m. In general the system associated with b, where b < m,\nconsists of m+1-b differential equations in the m+1-b unknown\nfunctions Fj, j = b, b+1, ..., m. Thus for the case b = m-1,","SEC. 6.4\nPLANE-PARALLEL MEDIA\n165\nwe have two equations:\nC(m,m-1) -\na = m - 1:\n(16)\nafm-1\na=m:\nFinally, for the case b = m, there is only one equation,\nnamely: a\n0 = ( ato m Fm\n(17)\na - m\nwhence Fm = 0, = provided (-a+om) t 0.\nOnce the = (m+1)2 functions Fb have been ob\ntained, where 0 m, and /b/ < a, the associated repre -\nsentation of N(x,5) Is, according to the general pattern (24)\nof Sec. 6.3:\n(18)\n=\nb = - a\nEquation (18) is the requisite order spherical har-\nmonic approximation to the radiance function N on a strati-\nfied plane-parallel source-free - optical medium in the steady\nstate.\nVector Form of the Truncated Solution\nIt is of interest to place the truncated system (14)\nto (17) into the compact form of (13). Thus let us write:\n\"F(b,m)\" for (Fb, Fb+1 ..... Fb)\n(19)\nF(b,m) is a function which assigns to each depth Z in the\nplane-parallel medium the (m+1) -component - vector F(b,m;a),\ni.e.,\nBy studying the general forms of (14) to (17), we see that\nthe truncated associate of Cb in (5) is the (m-b+1) x (m-b+1)\nmatrix:","VOL. III\n166\nCLASSICAL SOLUTIONS\n0\nC(b+1,b)\n0\n0\n0\nC(b+1,b)\nC(b+2,b)\n0\n0\n0\nC(b+2,b)\n0\n0\n0\n0\nC(b+3,b)\n0\n0\n0\n0\n0\n0\n0\n0\n0\n(20)\nC(m-1,b)\n0\n0\n0\n0\nc(m,b)\n0\n0\n0\n0\nC(m,b)\n0\n0\n0\n0\nwhich we shall denote by \"C(b,m)\". This matrix is invertible\nwhenever (m-b) is odd as we shall see below. Furthermore,\nif\nwe write:\n0\n0\n0\n-a+oo\n0\n0\n0\n0\n-a+01\n0\n0\n0\n0\n-a+o,\n(21)\n\"a(m)\" for\n0\n0\n0\n0\n-a+o\nm-1\n0\n0\n0\n0\n-a+o\nm\nthen the general representative of the systems of equations\n(14) to (17) is of the form:\n= F(b,m) a(m)\n(22)\nFinally, if we write:\n\"B(b,m)\" for a(m) C-1(b,m)\n(23)\nwe have:\n-\n(b,m) = (b,m) B(b,m)\n(24)\nb < m; m-b odd\nwhich is the desired vector form of the system (14) to (17)\nof m mth order spherical harmonic equations. We have now\nreached the stage where the system (1) is in a form amenable","SEC. 6.4\nPLANE-PARALLEL MEDIA\n167\nto solution by any of several well-known theoretical or nu-\nmerical techniques in the theory of ordinary differential\nequations (see, e.g., [23] or [47]). Of course (1) itself\ncan always be programmed directly for solution.\nThere is one instance of (24) whose solution can be\nwritten down immediately in \"closed form,\" namely the case\nwhere a and o are independent of depth; in other words, for\nthe case of an homogeneous medium X. Then, if we write:\nexp {B(b,m) }\n(25)\nfor\nj=0\nwhere Bj (b, m) is the jth power of the matrix B(b,m), and de-\nnote the value of F (b, m) at Z by \"F(b,m;z)\" then:\nF(b,m;z) = F(b,m;0) exp (B(b,m) 2 )\n(26)\n< m; m-b = 1, 3, 5,\nUsing the theories of [37], (26) may on the one hand be\nput into closed algebraic terms using the Jordan canonical\nforms of matrices; and on the other, (26) may be programmed\nfor direct evaluation on general-purpose electronic computers\nusing the techniques, for example, in [23].\nTo facilitate computations of F (b,m) using (26), we may\narrange matters so that the inverse of C(b,m) can be written\ndown by inspection whenever it exists. This may be done as\nfollows. First we verify the fact already noted, that e(b,m)\nhas an inverse whenever m-b is an odd integer. For example,\nwhen m-b = 1, and b > 0\nC(b+1,b)\n0\nC(b,m) =\nC(b+1,b)\n0\nthen\ndet C(b,m) = - c2(b+1,b) = # 0\nwhere \"det A\" denotes the determinant of a matrix A. Hence\nC(b,m) has an inverse. Again, when m-b = 2, and b > 0\nC(b+1,b)\n0\nC(b+2,0)\n0\nC(b,m) =\nC(b+1,b)\nC(b+2,b)\n0\n0\nthen","CLASSICAL SOLUTIONS\nVOL. III\n168\ndet e(b,m) = 0\nso that C(b,m) has no inverse in this case. Once more, for\nm - b = 3, > 0,\n0\nC(b+1,b)\nC(b+2,b)\n0\nC(b+1,b)\nC(b,m) =\nC(b+3,b)\nC(b+2,0)\n0\nC(b+3,)\n0\n0\n0\nand\ndet ((b,m) = c2(b+1,b) c2(b+3,b)\n(2b+5)(2b+7)*0\n=\nThe pattern forming should now be clear. By induction we\nhave, for integers b > 0, p 0 such that m - b = 2p + 1.\n(27)\ndet + 0\nWe next introduce the permutation matrix P which per-\nmutes the m-b+1 rows of C(b,m), where m -b = 2p + 1, p 0,\nin such a way as to near-diagonalize C(b,m), in the following\nsense. Return to C(b,m) above where m - b = 1 and note that\nwe can diagonalize it by interchanging its two rows. Simi-\nlarly, by interchanging the rows of C(b,m) where m - b = 3\nin pairs, starting with the first two rows, then going on to\ninterchange the second pair of rows (i.e., row three and\nfour) we obtain:\nC(b+2,b)\n0\nC(b+1,b)\n0\n0\nC(b+1,b)\n0\nPC(b,m) =\n0\nC(b+3,b)\n0\nC(b+3,b)\n0\nC(b+2,0)\n0\nwhere\n0\n1\n0\n0\n1\nP =\n1\n0\n0\n0\n0","SEC. 6.4\nPLANE-PARALLEL MEDIA\n169\nThe general structure of P for the case of an arbitrary m - b\n(= 2p +1) should now be evident: P is a 2 (p+1) x 2(p+1) m.a.\ntrix obtained from the identity matrix I of the same order by\ninterchanging the rows of I in pairs, as illustrated by the\nspecial case just considered. The utility of the permutation\nP rests in the fact that the inverse of P C(b,m) where m - b\n= 2p + 1, is readily written down by inspection. To see how\nthe inversion proceeds, consider once again the case of m -\nb\n3 = 2p + 1 (so that p = 1). To simplify the illustration,\n=\nwe shall write \"C\" for C(b+j,b), with \"b\" understood. In-\nspection of PC(b,m), with m-b = 3, shows that its inverse\nmust have the same overall structure as PC(b,m) itself and\nwhose main diagonal consists simply of elements of the form\n1/Cj. With this in mind, we may write:\n1\n0\n1\n0\nC1\n0\n0\n0\n0\n0\n[Pe(b,m)][Pe(b,m)]~1=\n0\n0\nC3\n0\n0\n0\n0\nC2\n0\n0\n0\nx2\n1\n0\n0\n0\n0\n0\n= I\n0\n0\n1\n0\n0\n0\n0\n1\nAs yet the entires X1 , X2 of the matrix are not known. How-\never, it is clear that X1, X2 satisfy the conditions:\nwhence\n2\nAs another example, let m - b = 5 = 2p + 1 (so that p = 2)\nOnce again the overall structure of [PC(b,m) 1-1 is the same","VOL. III\nCLASSICAL SOLUTIONS\n170\nas e(b,m) ; i.e. , near diagonal, where P is now the requisite\n6 row permutation matrix. To find [PC(b, m) 1 we write:\n[PC(b,m)] [PC(b,m)]\"\n0\n0\n0\n0\n0\n0\nx1\n0\nC1\n2\n0\n0\n0\n0\n0\n0\n0\n0\nC.\nC1\n1\n1\n0\n0\n0\n0\n0\nC3\n0\nC4\n0\n0\n0\n0\n0\n0\n0\nC\n0\nC2\n0\n3\n2\n3\n3\n1\n0\n0\n0\nC5\n0\n0\n0\n0\n0\n0\n0\nC5\n1\n0\n0\nC4\n0\n0\n0\n4\n5\n0\n0\n0\n0\n1\n0\n0\n0\n0\n0\n1\n0\n0\n0\n0\n0\n0\n1\n= I\n1\n0\n0\n0\n0\n0\n0\n1\n0\n0\n0\n0\n1\n0\n0\n0\nX4 are now readily determined\nThe remaining entries X1\nas in the case of p = 1. By direct inspection:\nthe\nThese two examples for the cases p = 1,2 clearly indicate the\nnature of [PC(b,m)] 1 with m-b = 2p + 1 for general integers\np > 0. The general rule may be phrased as follows: the main\ndiagonal of [PC(b,m) ]-1 consists of elements of the form\n1/C(b+(2j+1) b) arranged successively in pairs for j = 0, 1,\n, p. The nonzero off-diagonal elements in [PC(b,m)\noccur in exactly the same places as in PC(b,m) and each may\nbe obtained by dividing the corresponding entry Cj of PC(b,m)","SEC. 6.4\nPLANE-PARALLEL MEDIA\n171\nby (-1) times CKC&, where Ck and Cl are, respectively, the\nelements of PC(b,m) in the same row and column as C. The\nreader should now construct the [PC(b, m) ] for p = 3 to test\nthis rule. What is the rule's general form?\nFinally, we can rearrange (26) so as to specifically\ninclude within the formalism the preceding simple inversion\nprocedure. Returning to (22), we can write:\n- dz d\nWriting\n\"G(b,m)\" for F(b,m)p-1\n(28)\n\"D(b,m)\" for [Pa(m)] [PC(b,m)]-1\n(29)\nwe have\nG(b,m) = G(b,m) D(b,m)\n(30)\nm; m-b = 1, 3, 5,\nas the present counterpart to (24). The inverse [PC(b,m)]\"\nis the one whose simple rule of formation we have generated\nin the preceding discussion. Then, corresponding to (26), we\nhave:\nG(b,m;z) = G(b,m; 0) exp {-D(b,m)z}\n(31)\n0 b < m; m-b = 1, 3, 5,\nBecause of the autonomy of these equations with respect to b,\nwe can vary the truncation parameter m for each given b, so\nas always to have m-b odd, and therefore, to always have the\nalgorithm (31) at hand. Suppose, for example, we wish to\nfind all F6 with a < 4, as indicated by the diagram in Fig.\n6.2, and so as to have the representation of N(x,5) in (18)\nfor the case m = 4. Thus we are to find (4+1) 2 = 25 func-\ntions in all. In solving for the family {F} we accordingly\nmay truncate at F? (rather than F4) and solve for Fo, a = 0,\n1, 2, 3, 4, 5 using (31), taking advantage of the oddness of\nm-b = 5-0 = 5. In solving for the family {Fa}, we use (31)\ndirectly since now m-b = 4-1 = 3. A similar tactic is em-\nployed for extending by one additional member the family {F2},\nas in the case of Fa , and so on, to the end of the computa-\ntion procedure.\nEquations (26) and (31) are the final forms of the mth\norder spherical harmonic equations we shall study in this\nwork. Having deduced (26), (or its variant (31)) we reach","VOL. III\nCLASSICAL SOLUTIONS\n172\nthe threshold of the invariant imbedding domain of radiative\ntransfer theory. Thus the equation (26), say, may be viewed\non the one hand, as the logical culmination of the train of\ndeductions begun in Sec. 6.2 in the development of the clas-\nsical spherical harmonic method; and on the other hand (26)\nforms a bridge between the classical method of solution of\nthe equation of transfer and the invariant imbedding tech-\nniques for the solution of the equation of transfer. These\nlatter techniques will be considered in Sec. 7.10.\nSummary\nIn the preceding four Secs. 6.1 to 6.4 the spherical\nharmonic method is developed and applied after an appropriate\nmotivation of the method in Sec. 6.1. The main purpose of\nthe discussions is to make clear the fundamental ideas on\nwhich the method rests, in particular the general role of the\northonormal family of functions used to represent the radi-\nance function as a sum of products of purely spatial and di-\nrectional terms. This was done in Secs. 6.2 and 6.3. To\nshow the applicability of the method to the case of plane-\nparallel media, the setting of greatest utility in the study\nof hydrologic and meteorologic optics, the discussion of the\npresent section is added to the general remarks. In particu-\nlar, equation sets (14) to (17) above explicitly exhibit the\ntruncated forms of the spherical harmonic equations, where\nthe truncation arbitrarily sets to zero all functions Fb with\nindices a > m. The resultant system (24) can be used to solve\nfor the unknown complex valued functions Fb 0 < a, < m, |b\n< a. To solve (24) directly we must know No (in (31) or (32)\nof Sec. 6.3) from experiments. If N° is to be found theoreti-\ncally, we may use invariant imbedding methods which will give\nthe aerosol's or hydrosol's reflectance to incident light\n(Volume IV, et seq.).\n6.5 Three Approaches to Diffusion Theory\nThe term \"diffusion theory\" in the context of radiative\ntransfer theory denotes a discipline based on not any single\nequation, but rather a collection of more or less loosely in-\nterconnected theories each springing from some analytic ex-\npression which, in turn, is based on the fundamental equation\nof transfer. For our present purposes we may broadly classify\nthis collection of diffusion theories into two main groups:\nthe approximate and the exact theories. A diffusion theory is\napproximate to a greater or lesser degree depending on the\namount of modification undergone by the analytic structure of\nthe equation of transfer as the equation is subject to sim-\nplifying assumptions. In the present section our purpose is\nto approach this complex of diffusion theories from three\ndifferent directions so as to gain a useful overall perspec-\ntive of the sub-discipline of diffusion theory within general\nradiative transfer theory. In particular we shall approach\none of the more useful approximate diffusion theories (called\nclassical diffusion theory, for reasons which will eventually\nbecome clear) by starting from the equation of transfer and","SEC. 6.5\nDIFFUSION THEORY\n173\nproceeding to transform the equation by adopting the assump-\ntion of Fick's law for diffusing photons. Then we shall\nstart again, this time proceeding via spherical harmonic\ntheory which, depending on the order of terms retained in the\nbasic system (27) of Sec. 6.3, opens up a multitude of paths\ninto the domain of approximate diffusion theory. This ap-\nproach serves to show the extremely large number of diffusion-\ntype theories generally possible, and to throw light on the\nclassical diffusion theory by appropriately placing the lat-\nter in the hierarchy of approximate diffusion theories spring-\ning from the system of spherical harmonic equations of Sec.\n6.3. Finally, we start afresh once more from the equation of\ntransfer and develop the basic equation for an important\nexact diffusion theory which applies rigorously to optical\nmedia whose volume scattering functions o are independent of\nthe directions 5' and E.\nThe Approach via Fick's Law\nWe begin with the general time-dependent equation of\ntransfer (re (4) of Sec. 3.15) with source term in a general-\nly inhomogeneous optical medium X:\n1\n+ VN(x,E,t) = - a(x,t) N(x,E,t)\nat\nV\nNt(x,E,t)+Nn(x,E,t)\n(1)\nDiffusion theory is characteristically interested in\nthe description of the scalar irradiance ((x,t) rather than\nthe radiance N(x,5,t). That is, the density of the total\nflow at X in all directions is of interest rather than the\ndensity of the flow in each direction E at X. Thus we are\nled to integrate each term of (1) over direction space E.\nThe reduction of the resulting integrated form of (1) is\nfacilitated by recalling from (4) of Sec. 4.2 that:\na(x,t) = a(x,t) + s(x,t)\n(2)\nand from (2) of Sec. 2.8 that we write:\nN(x,E,t) Eds(5)\n(3)\n\"H(x,t)\" for\n,\n(1)\nwhere H(x,t) is the vector irradiance at x at time t.\nThe reduced integrated form of (1) is:\n1 ah(x,t)\n+ V H(x,t) = - a (x,t) h (x,t) + hn(x,t) (4)\nat\nV\nwhere we have written:","174\nCLASSICAL SOLUTIONS\nVOL. III\nhn(x,t)\" for\nn(x,5,t) S((E)\n(1)\nEquation (4) lacks utility in our present efforts to\ndescribe the scalar irradiance throughout X. The presence of\nthe divergence term for the vector irradiance blocks immedi-\nate usage of (4) in this respect: If, somehow, V H could be\nreplaced by a single function of h, then the resulting form\nof (4) would be a useful statement involving only scalar ir-\nradiance. It is at this point that the customary appeal to\nFick's law of diffusion is made. This law states that, for\nsome nonnegative valued function D, on X :\nH(x,t) = - D(x,t) vh(x,t)\n(5)\nfor every t in some time interval. In other words, at each\npoint x and time t, the vector H(x,t) has the direction of\nthe negative of the gradient of the scalar irradiance field\nh. In still other terms, H has the direction from the great-\nest to the smallest values of h in the neighborhood of a\npoint. The spatial and temporal variation of D is required\nto be quite mild, and for essentially all practical applica-\ntions D is assumed constant. The types of media for which\nFick's law is a reasonably good description of the state of\naffairs between H and h are those for which the scattering\nattenuation ratio p is large, say on the order of 0.6 and\nabove. A11 other things being equal the closer p is to 1\n(i.e., the larger the proportion of scattering compared to\nabsorption), the closer does Fick's law describe H in terms\nof h. Furthermore, Fick's law, all other things being equal,\nincreases in accuracy with distance from the boundaries and\nhighly directional or concentrated sources of the medium un-\ntil the effects of these boundaries and sources have dis-\nappeared. Any physical breakdown of a formula of the result-\nant theory is eventually traceable to a marked inapplicability\nof Fick's law. Using (5) in (4), we have:\n1 . ah(x,t) - V (D(x,t) vh(x,t)) = - a(x,t) (x,t) + (x,t)\nat\nV\n(6)\nEquation (6) is the desired scalar diffusion equation\nfor scalar irradiance h. D is the diffusion function (or\nconstant, as the case may be), a is the volume absorption\nfunction, and hn the emission or source term for the equation.\nThe diffusion theory based on (6) is the classical (scalar)\ndiffusion theory. When D is assumed constant over the space","SEC. 6.5\nDIFFUSION THEORY\n175\nX and a given time interval, an assumption which henceforth\nshall be in force, (6) may be written:\n1 at ah - D 2 h = - ah + h n\n(7)\nV\nEquation (7) has the Gestalt of the diffusion equation\nof classical heat conduction and other diffusion phenomena\nwith source term (hn) and annihilation term (-ah), , hence the\nmathematics of the diffusion of photons as governed by (7) is\nidentical to that of the diffusion of heat and other classi-\ncal diffusion phenomena, the theory of which is thoroughly\nunderstood. Therefore (7) may possibly be applied to such\nproblems as describing the transient light field set up by\npulsed sources. Equation (7) and related equations are\nstudied further in Table 1 below, and in Sec. 6.6.\nThe Approach via Spherical Harmonics\nThe next approach to diffusion theory we shall describe\nis that via the spherical harmonic theory developed in Sec.\n6.4. It will be seen that the approach can take place on\nseveral levels of generality and in an infinite number of di-\nrections on each level. We shall begin our discussion with\none of the simpler directions of approach on a very practical\nlevel, the goal being once again the classical scalar diffu-\nsion equation (7). However, now awaiting us at the goal is\nthe added bonus of a theoretical representation for the diffu-\nsion constant D and a formula describing the radiance function\nin a general diffusing medium in terms of the vector and\nscalar irradiances.\nIn our present approach to diffusion theory we shall be\nguided by the following two special principles concerning the\ncomponents Fa of the spherical harmonic representation of the\nradiance function:\n(i) A11 components Fb other than F:, Fi1, F 1 are set\nequal to zero in the system (27) of Sec. 6.3. A11 components\nof Fn, a other than Fb 0\nare zero.\n(ii) A11 time derivatives of the components Fb other\nthan Fo are set equal to zero in the system (27) of Sec. 6.3.\nThe reason for these two special principles stems ulti-\nmately from our intuitive conception of a diffusive flow of\nmaterial (or light) particles: (i) the amount of diffusive\nflow about a point varies mildly from direction to direction,\nand (ii) the overall directional structure of the flow itself\nvaries mildly from moment to moment. With this intuitive\nconception in mind, the rules of action stated in (i) and (ii)\nabove are arrived at by pairing Fo with h and by identifying\nthe components Fi1, Fi, F1 as the first three of an infinite\nset of components describing the overall directional flow of\nradiant energy at a point. The basis of this pairing of F o","CLASSICAL SOLUTIONS\nVOL. III\n176\nwith h is as follows. By (6) and (25) of Sec. 6.3 we have\nthe definitional identity:\nN(x,E,t)\n=\nN(x,E,t) PO(5) ds(5)\n(8)\n=\n.\nThe fact that the three components Fi1, Fi, F1 are associated\nwith the overall directional structures of the radiant flux\nis established by first noting that:\nH(x,t)\nN(x,E,t)\nEds(5)\n=\n(1)\nEds(5)\n(9)\n=\nn=0 m = - n o\nFurthermore, we have (cf. Fig. 2.4):\nE = sin 0 cos oi + sin 0 sin I + cos 0 k\n(10)\nIf we could now express the quantities sin 0 COS 0,\nsin 0 sin and cos as linear combinations of the then\nwe could directly evaluate the integral in (9) using the or-\nthonormality properties of the Toward this end we recall\nthat sin 0 = (1-cos2666 0) 1/2 = (1-12)1/2. Furthermore, an ex-\namination of any list of associated Legendre functions reveals\nthat:\n=\nThen:\nsin 0 (cos + i sin\n+1(5)/A1","SEC. 6.5\nDIFFUSION THEORY\n177\nSimilarly:\nsin 0 (cos - i sin ) ) = - 2PT1 e-id\n=\nFrom these expressions we deduce that:\n(11)\nsin 0 cos\n(12)\nsin 0 sin\nFinally, we observe that:\nCOS = u = =\n(13)\nUsing (11) to (13) in (10), we have the requisite representa-\ntion of E as a linear combination involving only members m n\nof the orthonormal family. The conjugates of om are obtained\nusing (8) of Sec. 6.3. As a result, (9) reduces immediately\nto:\nH(x,t)\n(14)\n-\nThis is the desired representation of the vector irradi-\nH(x,t) in terms of the spherical harmonic components Fb of the\nradiance function N. The representation reveals the role\nplayed by the three components Fi1, Fi, F} in the description\nof the overall directional structure of the light field (see\nalso (29) below).","CLASSICAL SOLUTIONS\nVOL. III\n178\nWith the basis for the two special principles (i) and\n(ii) now reasonably well established, we next apply these\nspecial principles to the system (27) of Sec. 6.3. According\nto principle (i) , we need consider only the cases a = 0, 1.\nAccording to principle (ii), all time derivatives, except\nthat of F:, vanish. The resultant set of four equations is:\n= +\n=\n(15)\n(1,1)\n(16)\n=\naFo\nF9\nC(1,0)\n+\n(a=1,b=0 in F =\n(17)\n= +\n(a=1,b=1 in Fo =\n(18)\nOur present goal is to obtain a single diffusion equa-\ntion for h(x,t) from the system (15) to (18). In view of the\nconnection between F0 and h stated in (8), we see that the\ngoal will be in sight if we use (16) to (18) to replace each\noccurrence of Fi1, Fi, F1 in (15) in terms of F: . Thus the\nterm:\nC(1,0)\nin (15), with the help of (17), becomes:\n(19)\nFurther the term:","SEC. 6.5\nDIFFUSION THEORY\n179\nB(1,1) Fil\nin (15), with the help of (16), becomes :\nIn a similar way the term:\nF12\nin (15), with the help of (18), becomes:\nCombining these terms in (15), the result is:\n(20)\nWe are now ready to pair off the terms in (20) with their\ncorrespondents in (7). Multiplying each side of (20) by\nand using (8), we can replace each occurrence of\n\"F:\" in (20) by \"h\". Next, by (15) of Sec. 6.3, we have:\n(;t)=\nPO(H)\ndu\n-1\ndo(E)\ns(x,t)\nIn other words, in (20) is the volume total scattering\ncoefficient. Hence:","CLASSICAL SOLUTIONS\nVOL. III\n180\nby virtue of (2). Finally, from (29) of Sec. 6.3 and the\ndefinition of hn in (4), , we have:\nFn,0 - hn\n.\nIn view of these observations, we may say that the structure\nof equation (20) is identical with that of (7). Therefore\nthe diffusion coefficient D in (7) is represented by the re-\nlation:\nD = 3(a-01)\n(21)\nwhere a is the volume attenuation coefficient and O1 is de-\nfined as in (15) of Sec. 6.3 (setting j = 1). This represen-\ntation of D rests on the basis of the spherical harmonic de- -\ncomposition of the equation of transfer subject to the special\nprinciples (i) and (ii) stated above which fix the level of\napproximation of the spherical harmonic decomposition. In\nsum, then, the left side of (21) arises when we approach dif-\nfusion theory via Fick's law; the right side arises when we\napproach diffusion theory via the spherical harmonic method.\nAt the point where the twain shall meet, we generate (21).\nThere are several alternate but equivalent forms of (21)\narising in practice. For example, if we write\n1\n\"H(x,t)\"\n(x;u;t) udu\n(22)\nfor\n-1\nThen, by (15) of Sec. 6.3, we have:\n1(x;t)(x,t)s(x,t)\n(23)\nThus we see that (x,t) is a mean value of the cosine = cos 0\n= E @ E' of the scattering angle 0. Another way of writing\n(22) to see this more clearly is to note that, when isotropy\nholds:\nL\n1\n(x;u;t) udu =\no(x;E';E;t) dn (5)\n(24)\n2\n-1\nHence (22) becomes:","SEC. 6.5\nDIFFUSION THEORY\n181\n(x,t) Eds(E)\n(25)\n/go(x;5';5;t) da(E)\nand from this the mean value property of (x,t) is quite\nclear; and by a mean value theorem of integral calculus,\n(26)\nFor optical media with large forward scattering values for o,\nthe values of u are near 1. For media with uniform scatter-\ning, i.e., o independent of E' and E, the value of u is 0.\nFor media with predominant backward scattering values, u has\nnegative values. Thus, in this sense, u is a measure of the\nrelative amount of the forward or backward scattering occur-\nring in a beam of flux within the medium. Returning now to\n(21) we use (23) to obtain:\nD=3(a-us)\n(27)\n3a(1-)\n=\n3(1-up)\nwhere p is the cattering-attenuation ratio and where \"La\"\ndenotes the attenuation length for the medium; that is, we\nhave written \"La\" for 1/a. Hence the diffusion coefficient\nhas the dimensions of length and in particular is equal to\nthe attenuation length of the medium divided by the factor\n3 (1-up).\nRadiance Distribution in\nDiffusion Theory\nWe conclude the discussion of the present approach by\nderiving the characteristic form of the radiance distribution\nN(x, . t) at a point x about which exists a diffusion process\nwith the properties (i) and (ii). Thus, the radiance N(x,E, t)\nat X at time t in the direction E is of the general form:\nN(x,E,t) = Fo(x,t) (1)(5)\nFq(x,t) (1)(5) + F1(x,t) 1(55)\n(28)","VOL. III\n182\nCLASSICAL SOLUTIONS\nThis form follows by using the present diffusion properties (i)\nand (ii) in (24) of Sec. 6.3. By evaluating each of the eight\nfactors in the four terms of (28), and simplifying, we obtain:\n(29)\n+\nEquation (29) displays the relatively mild structure of the\nradiance distribution associated with a classical diffusion\nprocess in an arbitrary optical medium. The greatest radi-\nance occurs in the direction of H(x,t). In directions E per-\npendicular to H(x,t) the radiance is simply h(x,t)/4m. Ob-\nserve that the overall graphical structure of N(x, . t) at a\npoint is simply that of a cardioid of revolution with axis\nalong the direction of H(x,t). Using (5) we may cast (29)\ninto radiometric terms involving h (x) only:\n(30)\nAs a representative indication of the details of the\nderivation of (29) from (28), observe that by (8) :\nFg(x,t) = (4r)-1/2 h(x,t)\n=\nand that:\n=\nHence:\nF0(x,t) h(x,t)/4\n(31)\n.\nFurthermore, by (16)\no\n=\nAlso:\nHence:\nF11 sin 0 (cos o - i sin\nIn a similar way it can be found that:\n1(x,t) D . sin 0 (cos 0 + i sin","SEC. 6.5\nDIFFUSION THEORY\n183\nFq(x,t) = - 3.1.D.COS ah(x,t)\n(33)\naz\nNote that the two expressions in (32) are complex conjugates;\nso that, upon addition, the imaginary terms cancel. On add-\ning together (31) to (33), equation (30) is obtained. Then\nusing (5), equation (29) is obtained.\nEquation (29) constitutes an effective means of verify-\ning empirically whether a given light field satisfies the\nconditions (i) and (ii) for a diffusion approximation. A11\nthree radiometric concepts, N, h, and H in (29) are readily\nmeasurable in practice. Hence if an empirical radiance dis-\ntribution comes to within an accepted interval of approxima-\ntion of a cardioid of revolution, then the classical diffu-\nsion equation may be used to describe such a light field. We\nnote a rather interesting near-confirmation of the steady\nstate form of (29) in the case of heavily overcast skies.\nEmpirical measurements reported in [186] show that the radi-\nance of the underside of a heavy cloud overcast has essen-\ntially the form of (29), i.e., the cardioidal form.\nApproaches via Higher Order\nApproximations\nWe pause in our description of the three main approaches\nto diffusion theory to place the discussion of the preceding\nparagraphs into perspective. We wish to show in particular\nhow the classical diffusion equation (20) (or its equivalent\nform (7)) takes its place somewhere near the bottom of an in-\nfinitely high ladder of successively more detailed diffusion-\ntype equations, each obtainable by following well-defined\nprinciples of modification, such as (i) and (ii) above, of\nthe basic system (27) of Sec. 6.3.\nIn order to facilitate the classification of the vari-\nous approaches possible via the system (27) of Sec. 6.3, let\nus write:\nF1, , Fa, Fa)\n\"Fa\" for\nThus, e.g., \"Fo\" denotes (F:), \"F1\" denotes (Fi1 Fi, F1),\nand so on. In other words Fa is a (2a+1) component vector\ncentered on the component Fo. When we say Fa is zero, we\nme an that each of its 2a+1 components is zero. Further, when\nwe write \"afa/dt\" we shall mean\nIn\na similar way we can define F\nNow the two principles (i) and (ii) used above to ar-\nrive at the classical diffusion equation (20) (or its equiva-\nlent (7)) may be recast into the following equivalent forms:\n(i) (if a > 1, then F a = 0) and (if a > 0, then Fn,a .\n(ii) if a > 0, then 0\n.","VOL. III\nCLASSICAL SOLUTIONS\n184\nThis relatively succinct way of describing the modification\nof the system (22) of Sec. 6.3 may form the basis of classi-\nfying various diffusion processes. Thus in the following\nlist, let the vectors Fa, Fn, a and their derivatives appear-\ning there be the only vectors not set equal to zero in the\nindicated approximation derived from (27) of Sec. 6.3. The\nsymbol in the \"process type\" column to the left of the non-\nzero vectors is a succinct way of denoting the numerical\nclassification of the approximation; some suggestive names\nfor the approximations are given to the right of the vectors.\nThus the approximation [1/0] is that giving rise to the clas-\nsical scalar diffusion equation derived earlier by setting to\nzero all terms in (27) of Sec. 6.3 except those of F , OF1 /at,\nF13 F n,0\nTABLE 1\nA short list of diffusion processes\nName of\nassociated\nProcess\nNonzero terms in (27) of Sec. 6.3\ndiffusion\ntype\nprocess\nEquilibrium\n[0/1]\nFo; n,\nF\nMonotonic\n[0/t]\nFo,\nScalar\n[1/0]\nFo,\nFo, dFo/at; F1; 2F1/at; n,1\nWave\n[1/t]\nFo, aFo/at; F1, 3F1/at;\nTensor\n[2/0]\nFo, do/dt; F1, dFi/dt; F2, dF2/dt; Fn,2\nWave-tensor\n[2/t]\nThe present classification of diffusion processes places two\ntheories below the scalar diffusion theory (\"below\" in the\nsense of \"less complex\"). The first of these, the equilib-\nrium diffusion theory, merely serves to describe the radio-\nmetric state of affairs in an equilibrium situation by means\nof the equation:\n+ F 0 = 0\nwhich may be written:\nh(x,t) = hn(x,t)\n(34)\nThus (34) holds for a uniform, steady light field in equilib-\nrium with its emission sources distributed throughout a medi-\num X. The term hn/a is reminiscent of Kirchhoff's law in\nradiometry, or of the equilibrium radiance N (see (2) of\nSec. 4.3). A slightly more detailed description is given by\nthe monotonic diffusion equation:","SEC. 6.5\nDIFFUSION THEORY\n185\n1 ah\n= -ah + h\n(35)\nat\nV\nn\nThus the diffusion process [0/t] described in (35) gives rise\nto a light field whose scalar irradiance h at a point gener-\nally grows or decays monotonically with time. The scalar\ndiffusion process [1/0] was discussed in detail above.\nWe next encounter the processes [1/t], which is one\nstep more accurate and complex than the classical diffusion\nprocess [1/0]. This new process is called the wave diffusion\nprocess by virtue of the fact that its associated equation\n(derived from (27) of Sec. 6.3 in the general manner illus-\ntrated for the case of [1/0]) is a wave equation of the form\n2\na h\ndh\n2\nA\nB\nD\nh =\nV\n-ah\nh\n(36)\n+\n-\n2\nat\nat\nwhere we have written:\n\"A\" for 3D/ 2\n\"B\" for (1 + 3Da)/ (37) (38)\n,\nComparing (36) with (7), we see that the process [1/t] adds\nthe next higher derivative term to the equation for the pro-\ncess [1/0], plus slightly modifying the coefficients of the\nderivatives of the latter's equation. The physical processes\ncorresponding to (36) and to (7) differ markedly: (36) de-\nscribes a general damped wave-like process which propagates\noutward from any epicenter at the finite speed v/v3. Indeed,\n(36) is the well-known telegrapher's equation, which describes\nin another context the propagation of wave signals through a\nresistive wave-conducting medium. Equation (7), on the other\nhand, is the classical diffusion equation which describes a\ngeneral monotonic decaying (or growing) diffusion process\n(with absorption and emission of the diffusing entities)\npropagating with infinite speed from a given epicenter. Equa-\ntion (7) may be essentially obtained from (36) by letting V\nbecome so large that the second-derivative term in (36) be-\ncomes negligible, i.e., so that A is small compared to B.\nThe next higher diffusion process beyond wave diffusion\nis the process [2/0]. A new entity enters the picture here\nwith F2. Whereas F1 describes the vectorial properties of\nthe radiant flux (see the description of the vector irradi-\nance H in terms of the components of F1, in (14)), F2 de-\nscribes the tensorial properties of the radiant flux, proper-\nties very much like those described by the stress tensor in\nfluid dynamics.\nOur present goal has essentially been reached; we have\nshown the place of the classical diffusion theory in the hi-\nerarchy of diffusion theories possible in radiative transfer\ntheory. It is seen that the classical diffusion equation (7)\nis neither the beginning nor the end of the possibilities of","VOL. III\nCLASSICAL SOLUTIONS\n186\ndescribing diffusive transport of photons in an optical medi-\num. However, equation (7) is on the borderline between those\ntheories which, on the one hand, are too crude to admit use-\nful descriptions, and those which, on the other hand, are\nmore accurate in their descriptive powers, but which are rel-\natively complex and intractable in the light of current math-\nematical techniques. It is because of this convenient mid-\ndling ground straddled by the diffusion equation (7) that it\nhas been so popular with researchers looking for easily han-\ndled, reasonably accurate quantitative accounts of natural\nlight fields. Some of the simple models arising from (7)\nwill be considered in Sec. 6.6.\nThe Approach via Isotropic Scattering\nThe third and final main approach to diffusion theory\nwe shall consider in this section is that via the assumption\nof the isotropic scattering property for an optical medium.\nThe nature of this assumption is quite different from those\nused in the preceding two approaches. The earlier approaches,\nvia Fick's law and via the spherical harmonic method, were\ngotten under way by first tampering with the directional\nstructure of the light field, i.e., by reducing its awesome\ndirectional complexity to some relatively innocuous, mildly\nvarying form (see, e.g., (29)) so that, for example, either\nFick's law or the [1/0] process defined in Table 1 above\ncould cope with the resultant weakened field. The nature of\nthe assumption we shall adopt the present discussion is\nsuch that it leaves inviolate the intricate geometric struc-\nture of the radiance field; but in order to inculcate a sem-\nblance of manageability into the field, it is to be hypothe\nsized that the volume scattering function o is independent\nof E' and E throughout the medium. The resultant light field\nbelonging to such a o is a relatively tame analytic object by\nnatural light field standards-- so tame, in fact, that some\nquite elegant mathematical analyses of the classical mold can\nbe employed to carry to completion the exact solution of the\nresulting equations for scalar irradiance. The associated\ntheory is called exact diffusion theory. The \"exactness\" of\nthe theory resides in its mathematical procedures, and not\nnecessarily in its fidelity as a physical theory.\nThe manner in which we shall approach exact diffusion\ntheory will be such as to show the necessity of the isotropic\nscattering assumption in the construction of the theory. By\nholding back the invocation of the isotropic scattering as-\nsumption until the last stage of the main analysis, it shall\nbecome quite clear that this is the essential physical con-\ncession made by an otherwise elegant, powerful theory which\nin principle is applicable to arbitrary (finite or infinite)\ninhomogeneous media with both internal and external sources.\nTo begin, let the optical medium X be of arbitrary spa-\ntial extent (in Fig. 6.3 it is shown as being finite), gener-\nally inhomogeneous, with arbitrary volume scattering function\no and volume scattering attenuation function a, and with ar-\nbitrary emission function Nn defined throughout X, and bound-\nary radiance distribution No. For simplicity of exposition,","SEC. 6.5\nDIFFUSION THEORY\n187\nIN\nXOMN\nx'\nFIG. 6.3 Setting up the exact diffusion theory.\nwe postulate a steady-state radiance field N through X X E.\nThe corresponding formulation for the time-dependent - field\nis obtained by simple modifications of the steady-state - case.\n(See, e.g., (12) of 7.14.) The present discussion will be\nfacilitated if at the outset we define certain integral oper-\nators. First, there is the path function operator R of Sec.\n3.17:\nR =\n[]o(x;) an(E')\n(1)\nThe path radiance operator T of Sec. 3.17 will also be needed:\nr(x,5)\nT =\n[ ] Tr-r'(x'.5) dr'\n0\nThe variables occurring in these operators are depicted in\nFig. 6.3. Further, we shall write:\n\"U\" for\ndd(s)\n(39)","VOL. III\nCLASSICAL SOLUTIONS\n188\nThis operator maps radiance distributions N(x,.) at a point x\ninto their associated scalar irradiances h (x), thus:\nI\n(40)\nN(x,5) dn(E)\nh (x) = NU(x) =\n(1)\nor simply:\nh = NU = vu\nfor short, where vu is an alternate form of h (Sec. 2.7) involv-\ning radiant density u, and the speed of light, V. We shall\nalso need the following two compositions of operators. First,\nthe scattering operator s Superscript(1) of Sec. 5.1:\nand the composition V, where we have written:\n(41)\n\"V\" for TU\nThe reader may verify directly from its definition that V has\nthe representation:\n(42)\nV =\nwhich is the iteration of the integral operators T and U,\nwhere for every x' and X in the medium we have written:\nfor Tr-r'(x',5) 17-11/20\n(43)\n\"Ka(x',x)\"\nand where E = (x-x')/|r-r'|; |r-r'| is the distance /x-x'|\nfrom point x' to point x as measured along the path of direc-\ntion E. (As usual, \"x\" denotes a point of E3, and as such is\nan ordered triple of real numbers. ) The integration in V is\nwith respect to the volume measure V. Thus dV (x) = r2drd((E),\nwhere X = X o + r E.\nWith all this machinery securely in place, we can go\non\nto obtain the requisite equations so as to keep easily in\nview at all times the essential physical and mathematical\nfeatures of the derivation.\nThe integral form of the equation of transfer ((2) of\nSec. 3.15) with emission function N is:\n*The notation \"NU(x)\" denotes the value at X of the\nfunction NU, and NU in turn is the result of operating on the\nfunction N with the operator U.","SEC. 6.5\nDIFFUSION THEORY\n189\nN(x,5) = = No + Nn) T(x,5) +\n(44)\nwhere* No is the initial radiance function within the medium\ndue to boundary radiances, i.e., where we have written:\n\"No(x,5)\" for No(x),5) ($(x-xg)\nand where No(xo,.) is the given incident radiance distribu-\ntion at an arbitrary point XO of X. By writing:\n\"No(x,5)\" for\n(44) becomes:\nN(x,5) = N°(x,5) + NS1(x,E)\nApplying U to each side, we have\nNU(x) = N°U(x) + NS1U(x)\nwhence:\n)=ho(x) + (NR)TU(x)\n=hq(x)+N,TU(x)\nHence\n(45)\nwhere we have written:\n\"hm(x)\" for Nou(x)\n(46)\nEquation (45) is but one step away from being an integral\nequation for scalar irradiance h. On first sight it might\nappear promising to use the operator U on N* to obtain the\nproduct of the volume total scattering function S (x) and\nscalar irradiance as follows:\nN.U(x)=s(x)h(x)\nToward this end, the N* term in (45) may have the identity\noperator I in the form of UU-1 slipped between N* and V, thus:\n*The notation: \"(N + N,)T(x,5)\" denotes the value at\n(x,5) of the function (Ro + Nn)T.","VOL. III\nCLASSICAL SOLUTIONS\n190\nN.uu-1v(x) sh(u-1v)(x)\nso that (45) could be written:\n+ sh(u-1v)(x)\nwhich is an operator equation in the unknown h. Unfortunately\nthe inverse U-1 to the operator U does not generally exist,\nfor the reason that there are many distinct radiance distri-\nbutions at a point X giving rise to the same scalar irradi-\nance h(x). This shows the necessity for assuming isotropic\nscattering for the medium if we are to obtain an integral\nequation for h. For then we have:\n(47)\nN.(x,5) =\nwhere we have assumed that:\n(48)\n(x;5';5) = s (x) / 41\nUsing N* (x,5) in (45) as given by (47) we have:\n(49)\n(hs) v(x)\nThis is the requisite general form of the basic equation of\nexact diffusion theory.\nThe natural solution of (49) is obtained by rearranging\nit as follows:\nv(x)\n(50)\nwhere we have written:\n(51)\n\"V*\"\nfor\nIt is easily shown that the inverse [I - V*]-1 of I - V* gen-\nerally exists, i.e., , that V, * has the contraction property\n(cf. Sec. 5.14). Hence (44) yields:\n(52)\n(x)\n=","SEC. 6.5\nDIFFUSION THEORY\n191\nwhere generally:\n(53)\n-\nHere V2 is VAVA, i.e., the operator V* followed by V*. In\ngeneral Vlis the operator vi followed in application by V*.\nThis solution procedure is quite general. The operator V*,\nwhich depends on the space X and its optical properties a\nand S, requires only the contraction property to be verified\nbefore it can be used in theory or practice.\nAn alternate form of (49), the form most often used in\nthe classicial solution procedures, is obtained by rewriting\n(45) as :\nso that:\nh (x) = h°(x) + V(x)\n(54)\nIn order to obtain an equation in h only (all other\nterms being given functions) it follows, for the same reasons\nas those leading to (49), , that the isotropic scattering as-\nsumption (48) must be adopted. In addition, if we are to re-\ntain the particular grouping of terms exhibited in (54), we\nmay (though it is not strictly necessary to do so) also as-\nsume that Nn is of uniform directional structure, i.e., we\nassume:\n(55)\nwhere hr n is defined in (4). Under these conditions, (54)\nreduces to:\n= + V(x)\n(56)\nIf the space X is infinite in all directions about x, and a\ngenerally is not zero, then h°(x) = 0, and (56) becomes:\n(57)\nwhich is the somewhat special but customary form of the inte-\ngral equation on which the exact diffusion theory is based.","VOL. III\n192\nCLASSICAL SOLUTIONS\nWe now sketch the customary method of solution of (57).\nThe medium is assumed homogeneous, so that (x) is indepen-\ndent of x and so that Ka(x',x) depends only on the difference\n|x-x' This assumption of homogeneity is necessary if the\nFourier transform method (the usual method used) is to be\napplied to (57). Thus, if 117\" denotes the three-dimensional\nspatial Fourier transform operator for functions on X (which\nis now all of euclidean three space) we have, applying 7 to\neach side of (57):\n(7h) (k) = 1 / 71(h, + hs) V] (k)\nwhere k is the spatial frequency variable associated with the\nspatial variable X. The value of 7[h] at k is written as\n\"F[h;k]\", \"(7h)(k)\", or \"h(k)\", similarly with the inverse\ntransform. Using the convolution theorem for Fourier trans-\nforms, (see, e.g., (6) of Sec. 7.14) this becomes:\nf(4) (hn(k) + sh(k)) ka(k)\n(58)\nwhere for brevity we also write:\n\"Ka(k)\" for F[Ka;k]\nThe carat over the letter \"h\" denotes, e.g., that h is the\nFourier transform of h. The beauty and power of the Fourier\ntransform method is now strikingly evident in (58) : the inte-\ngral operator equation (57) has been reduced to an algebraic\nequation in h (k) so that (58) may be directly solved for h(k):\nfn (k)\nTaking the inverse Fourier transform of each side, we have:\nh (x) 7-1\n(x)\n(59)\nwhich rivals the natural solution (52) in simplicity and\nelegance (but evidently not in power and scope). The solu-\ntions of (57) will be discussed in more detail in Sec. 6.7.\nThe present discussion is concluded with the observa-\ntion of how the radiance distribution N(x,.) is obtained from\nknowledge of scalar irradiance h (x) when using exact diffu-\nsion theory. Once the scalar irradiance field h has been ob-\ntained from either (52) or (59), we use the representation of\nN*, as given by (47), in the general relation (44) :\nN(x,5) = + T(x,5) + N.T(x,5)\nThus :","SEC. 6.5\nDIFFUSION THEORY\n193\nThus:\nN(x,5) =\n(60)\nIf the medium is source-free, so that No = 0, then\nN(x,5) =\n(61)\nIf the medium is in addition infinite, so that No = 0 at all\ninterior points of X then\n(62)\nN(x,5) =\nIf the medium is also homogeneous, then\nN(x,5) = (s/4) [hT(x,5)]\n(63)\n.\n6.6\nSolutions of the Classical Diffusion Equations\nIn this and the following section we shall exhibit some\nof the more useful general solutions of the classical and\nexact diffusion equations introduced in the preceding section.\nWe begin with the classical diffusion equation in its simplest\ncontext.\nPlane-Parallel Case\nConsider an homogeneous plane-parallel source-free op-\ntical medium with a steady, stratified light field generated\nby incident flux at its upper boundary. For example, natural\nlight fields in the seas, lakes, and harbors can supply such\ninstances. Further instances may be found in heavy fogbanks\nand thick cloud layers. Suppose that the conditions for the\ndiffusion equations hold in such media. What are the result-\nant forms of the light field--say the radiance distribution\nand associated scalar irradiance function - - that the classi-\ncal diffusion theory predicts for such media? We now seek\nthe answers to these questions.\nStarting with equation (7) of Sec. 6.5, and imposing\nthe source-free, steady light field condition, we have:\nD V 2 h - ah = 0\n(1)\nRecall that in a ree-dimensional Cartesian coordinate\nsystem:","194\nCLASSICAL SOLUTIONS\nVOL. III\nSince the light field is stratified, the x and y derivatives\nin 2h will be zero. Thus (1) reduces to:\nah = 0\n(2)\n-\n.\ndz\nTherefore, in its simplest guise, the classical diffu-\nsion equation (7) of Sec. 6.5 takes the form of a linear,\nsecond-order differential equation whose general solution for\na + 0 is of the form:\n= e KZ + c_ e-kz\n(3)\nwhere we have written:\na\n\"K\" for\n(4)\nWe call K, as defined in (4), the (classical) diffusion coef-\nficient. Recalling (27) of Sec. 6.5, we can express K alter-\nnatively as:\n3a(a - us)\nK =\n3a(a + (1-u)s)\n=\nThe diffusion coefficient K is the physical core of the\nsolution (3) and, indeed, of all of the solutions of the\nclassical diffusion equation. There may be variations in the\ngeometry of a medium--spherically symmetric, cylindrically\nsymmetric, plane parallel, as in the present case--and corre-\nsponding variations in the forms of solutions, as we shall\nsee, but running through these cases, and common to them all,\nis the notion of the diffusion coefficient K. Observe how K\ndepends jointly on the volume absorption coefficient a, the\ntotal volume scattering coefficient S, and on the me an cosine\nu, which is a measure of the anisotropic scattering property\nof the medium.\nAs a special solution of (3) let the plane-parallel me-\ndium be infinitely deep, so that on physical grounds C+ = 0\nin (3) (see (12)). Then (3) can be shown to reduce to:\n= h ( ) ) - KZ\n(6)\nThis is at once the most useful and representative example of\nthe analytic form of light fields in natural optical media.\nThe models for light fields in natural optical media come in\nall orders of complexity and power of representation, but in\nthe final analysis all exhibit, in greater or lesser degree,\nand with an accuracy that generally increases with increasing\ndepth, the overall exponential structure of natural light\nfields. The simplest of models of light fields in natural\nmedia--namely (2) - already exhibits this exponential structure","SEC. 6.6\nCLASSICAL DIFFUSION EQUATIONS\n195\nof the light fields. More sophisticated models will give\ncorrespondingly more detail on the structure of h(z) as a\nfunction of z; and still other models may sharpen the depen-\ndence of K on a and S. Yet for all its simplicity, (2) has\ncaptured the salient analytic property of the light in nat-\nural hydrosols: that of exponentiality.\nHow does the magnitude of the diffusion coefficient\ncompare with that of the volume attenuation coefficient a?\nWe note first of all that these quantities are indeed com-\nparable, both having dimensions of inverse length. From the\nrepresentation (5) of K we can build up the following chain\nof inequalities leading to a:\nK = (a + (1-)s) < /3a(a+s) < v3(a+s)(a+s) = /3 a (7)\nA more instructive inequality can be deduced provided\nthat some explicit relation between S and a is hypothesized.\nSuch a relation has already been observed in connection with\nthe validity of diffusion theory. In the remarks following\nFick's law (5) of Sec. 6.5, it was noted that the law holds\nwhen, among other things, the scattering-attenuation ratio p\nis at least 0.6. This condition on a in turn requires that:\nS > (10/4) a > 2a. It therefore seems reasonable to be able\nto use this inequality between S and a whenever diffusion\ntheory itself is being used. Therefore, starting the chain\nof inequalities in (7) once again, we are now led to:\nK = /3a(a + (1-)s) < /3a(a+s) < (a+s) (a+s) = a\n.\nHence we see that, whenever diffusion theory is applicable,\nwe must have:\n(9)\nK n. There is no question about the convergence\nof the infinite series in (45) since we have assumed that\neach Xj is embedded in a small but finite volume of given\nminimum size. Hence the points xi cannot all cluster in any\nfinite region of space. The exponential factors in (45) then\nassure convergence of the infinite series, since the distances\n1x-xjl increase regularly with j, in the limit.\nThe relation (45) has a deceptive amount of generality.\nWe could, if required, partition all of euclidean three space\n(except some arbitrarily small neighborhood of x) into cubes\nof varying sizes. if need be. Then each cube with center Xi\nis assigned an output Po(xj). Equation (45) then gives the\ntotal scalar irradiance at x generated by these discrete\nsources throughout space.\nAs an example of the preceding observation, suppose\nthat small, finite, contiguous volumes are used to similate\na thin cylindrical region with a straight-line segment in\nspace as axis and along which sources are distributed. Such\ncylinders may simulate narrow beams of radiant flux sent out\nby highly directional sources, for example laser sources. In\nthis case Po(xj) is generated by the scattering, within the\njth volume segment, of the residual flux of the beam reaching\nthe jth volume. Thus, suppose a laser source is at point xo\nand directed along the path Pr(x0,5) with initial point X-Superscript(c) and\ndirection E, as in Fig. 6.4. Partition the beam, which has\ninitial radiance No, into n parts, each a cylinder of length\nr/n and initial point Xj (= xo + (jr/n)). Finally, suppose\nthe volume scattering function o is independent of E', E, i.e.,\nthat isotropic scattering prevails throughout the medium.\nThen it is clear that:\nN e jra/n","CLASSICAL SOLUTIONS\nVOL.\nIII\n204\nX\n+\nr.\nr/1\nr\nn\nX\nNo\nO\nXo\nFIG. 6.4 Geometry for a narrow cylindrical beam source\nof radiant flux in diffusion theory.\nis the residual radiance reaching the initial point x of the\njth cylindrical part of the beam. From this and the defini-\ntion of path function it follows that:\nNo o jra/n (s/4)\nis the path function value at the initial point xj of the jth\ncylindrical part of the beam. Because scattering is isotropic,\nthis value is assigned to each direction about Xj. Since\npath function values have the dimension of intensity per unit\nvolume (e. g. , see note (h) for Table 3 in Sec. 2.12), we can\nmake the following assignation: To\nP(x)/4m \" (= J(xi))\nin (45), we assign:","SEC. 6.6\nCLASSICAL DIFFUSION EQUATIONS\n205\nNo e jra/n (sV(xj)/4m)\n,\nwhere V(xj) is the volume of the jth part of the beam, so\nthat (45) now becomes :\nh(x) j=1 n -jra/n e-k/x-xj V(xj)\n(46)\n/x-xjl\nThis shows how the discrete-source case can simulate\nimportant internal source problems in natural optical media,\nprovided, of course, that the basic diffusion point source\nmodel is valid for the given medium.\nThe radiance distribution associated with a discrete\nsource scalar irradiance field given by (45) is obtained by\nappeal to the interaction principle, so that by simply adding\ntogether terms of the form shown in (40), the desired radi-\nance distribution is obtained. An alternate representation\nof N(x,5) is obtainable as follows: From (39) and the inter-\naction principle it is clear that the vector irradiance gen-\nerated by the point sources at X1, X2, ...\nis:\nH(x) D (hj(x) (1+k/x-x;l)\n(47)\nIj\n/x-xjl\nj=1\nwhere we have written:\ne-k/x-xj\nP\no(xi)\n\"hj(x)\"\nfor\n4wD|x-x;\nand where rj is the unit vector directed from the observation\npoint x to the jth source point xj (see Fig. 6.4). Then us-\ning H (x) and h (x) as given, respectively, by (47) and (45),\nthe radiance N(x,5) at X in the direction E is given once\nagain by (29) of Sec. 6.5.\nContinuous Source Case\nWe now make the transition from the discrete source\ncase, just concluded, to the continuous source case. We be-\ngin with the finite version of (45) in which we have parti-\ntioned a subset Xn of the infinite medium into a set of n small\nvolumes Xj (\"small\" in the sense of less than one attenuation\nlength in diameter) each of which has a radiant flux output\nof PO (xj), where xj is a point of Xj. Hence the radiant flux\noutput per unit volume about xj is very nearly Po(x)/V(X),\nwhere V(xj) is the volume of X. We assume that the radiant\nflux output of Xj is uniform in all directions about Xj.\nThen the radiant intensity per unit volume:","VOL. III\nCLASSICAL SOLUTIONS\n206\nPo(xj)\nmay be represented by an emission radiance distribution\nNn(xj,5) which is independent of direction E. (Recall that\nNn has the same dimensions as path function N*, and that the\nlatter's dimensions may be characterized as radiant intensity\nper unit volume). Therefore, using the definition of hn in\n(4) of Sec. 6.5, we may write:\nPo(xj)\n(48)\n,\nso that:\nhn(xy)\n(49)\nWith this meaning of hn(xj), the finite version of (45)\nmay be rewritten as :\n(50)\nBy letting the partition of Xn become finer, so that in\nthe limit the associated Riemann integral over Xn is obtained,\n(50) becomes :\n(51)\ndV(x')\nh(x) =\nThis is the desired representation of the scalar irrad-\niance h (x) generated by isotropic point sources of strength\nhn(x) watts per unit volume, at points x' throughout a region\nXn of the medium X. In analogy to (43) of Sec. 6.5 we write:\n\"Kk(x',x)\" for 4TD|x-x' -k/x-x'\n(52)\nand\n(53)\n[] Kk(x',x)\ndV(x')\n\"W\"\nfor\nXn","SEC. 6.6\nCLASSICAL DIFFUSION EQUATIONS\n207\nso that (51) may be written:\nh(x) =\nkk(x',x) dV(x') = h,w(x)\n(54)\nX\nFinally, the vector irradiance H(x) in the continuous\nsource case can be obtained by starting with (47) and going\nto the Riemann integral counterpart of that sum. Thus, sup-\npose initially the sum is finite and that the sources are\nconfined to a part Xn of the medium. Then, as before the set\nXn is partitioned and (xj)\" introduced to denote the unit\nvolume output of the medium at point xj in Xn. Thus (47) be-\ncomes :\nK\n4/x-xj\nin which (52) is used. Observe that -r- is (x-xj)/|x-x- so\nthat as the partition of Xn is made suitably fine, the sum\nhas the limit:\nH(x) =\n(x-x') dV(x')\n(55)\nWhen h (x) and H(x), as given by (51) and (55), are used\nin (29) of Sec. 6.5, we obtain the appropriate radiance func-\ntion for the diffusing light field generated by a continuous\ndistribution of sources in Xn. The limitations of the point\nsource case are as considered above. Indeed, since the point source\ncase fails for points of observation too near the point\nsource, it follows that points of observation x in (51) and\n(55) should not be in Xn, and preferably at some distances\nfrom Xn. We must impose this limitation on all diffusion in-\ntegrals in practice. This problem of the proximity of the\nsources of the diffusing field will be examined in the follow-\ning paragraphs.\nPrimary Scattered Flux as Source Flux\nTime and again in the preceding illustrations of the\ndiffusion method, precautionary observations were required on\nthe use of the various derived equations because of possible\ninapplicability of Fick's law. For example, when an observa-\ntion point x is too near a point source point xo in an other-\nwise suitably diffusing medium, the radiance distribution","CLASSICAL SOLUTIONS\nVOL. III\n208\nabout x may depart too markedly from the cardioidal distribu-\ntion indigenous to classical diffusion theory. This depar-\nture is due principally to the highly directional residual\nradiance originating at xo and arriving at X. It would there-\nfore seem desirable to improve the radiometric conditions\nprior to applying the classical diffusion theory by first\ncomputing the primary scattered radiance field generated by\nthe given sources and using this radiance field as the source\nfield in the continuous diffusion case considered above. We\nshall explore this possibility and its generalization in this\nand the subsequent paragraph.\nIn order to correctly implement the present discussion\nit seems best to return directly to the basic equation of\ntransfer for scalar irradiance, (1) of Sec. 6.5. Our immedi-\nate task is to decompose the steady-state scalar irradiance\nh(x) into its residual component ho and its diffuse component\nh*, where the basis for these concepts were defined in (15)\nand (22) of Sec. 5.1. Thus, using the operator U in (39) of\nSec. 6.5, we write:\n\"h*(x)\" for N*(x,.) U\n(56)\nso that:\nh(x) = h o (x) + h*(x)\n(57)\nand\nh*(x) = [ h (x)\n(58)\n.\nIn other words, the scalar irradiance h(x) consists of the\nsum of all scalar irradiances hn(x) associated with n-ary\nradiance distributions Nn(x).) at X. Hence h* (x) consists\nof\nradiant flux having undergone one or more scattering opera-\ntions. Clearly, (57) may be obtained immediately from (4) of\nSec. 5.4 by applying the operator U (cf. (39) of Sec. 6.5).\nThat is, from\nN = N° + N*\n(59)\nwe obtain\nNU = (N° + U = N°U + N*U\nthat is:\nh = ho + h\n(60)\nWe now use this mode of decomposition of h in the steady\nstate version of (1) of Sec. 6.5. The details are as follows,\nstarting with:\nS\nE VN = - aN +\nNodn + N\nn","SEC. 6.6\nCLASSICAL DIFFUSION EQUATIONS\n209\nwe first decompose N as in (59) to obtain:\nE.V N*) = -a(N° + N*) +\n(N°\nN*)\nods\nNn\n+\n+\nn\nHence:\n|\nN°ods\nE.VN = - aN* +\nN*ods\n(61)\n+\n(1)\nwhere we have used the relation:\nE.D N° = -ano + Nn\nwhich follows from (2) of Sec. 5.8. Recalling the definition\nof N1 ( (2) of Sec. 5.1), (61) can be cast into the form:\nI\nN*ods + N1\nEN*= - aN* +\n(62)\n(1)\nThis is the equation of transfer which governs the dif-\nfuse radiance field N* consisting of primary and higher order\nscattered flux. An alternate derivation of (62) was performed\nin (7) of Sec. 5.2. The source for the field N* is the first\norder path function N1. Because the residual radiance N°\ncoming in from the boundaries of the medium, and emission ra-\ndiance Nn are now absent from N*, the directional structure\nof N* is considerably milder than that of N1 so that Fick's\nlaw is more likely to hold for N* than N.\nIt is to the scalar irradiance h* induced by N* that we\nnow direct attention and derive from (62) the required diffu-\nsion equation for h*. Thus, applying the operator U to (62)\nwe have:\nV.H* = -ah* + h1\n(63)\nwhere:\nh1 = hos\nand where we write:\n(64)\n\"H*\" for","210\nCLASSICAL SOLUTIONS\nVOL. III\nAssuming Fick's law to hold between H* and h * (cf. (5)\nof Sec. 6.5), i.e., assuming:\n(65)\n(63) becomes:\nVh* + ah*\n(66)\nThis is the requisite steady-state diffusion equation for h*\nin which the primary scattered scalar irradiance h1 serves as\nan auxiliary source to the basic emission sources hn in the\nmedium. The assumption of Fick's law for h* in (65) has a\nbetter chance of being valid than for h, since h has ho as a\ncomponent which can be associated with highly directional\nflows from boundaries and internal sources.\nThe theory of the continuous source developed above and\nsummarized in (51) and (55) may now be applied to the case\nwhere h in those equations is replaced by h1. The proof of\nthis procedure is based on the fact that the derivation of\n(51) and (55) ultimately rests on the steady-state version of\n(7) of Sec. 6.5; and this has just been shown to be identical\nwith (66) in which hn in the earlier equation is now replaced\nby h1.\nWe now illustrate the use of (66) by means of a simple\nexample. We consider an isotropic point source in an infi-\nnite homogeneous medium which scatters isotropically (i.e.,\nis independent of E' and 5) The source is at the origin and\nin reality constitutes a very small, essentially transparent\nsphere of radius r which has a uniform surface radiance No.\nThus the radiant emittance of the spherical surface is TN,\nand therefore the total flux output is 422 The average\nflux per unit volume of the spherical source is\n= 3No/ro. It is this output which would customarily be used\nin the estimate of hn in the continuous case (cf. (49)). How-\never, now the source is allowed first to generate a primary\nscattered flux field h1 in the space surrounding it. In prin-\nciple this primary scattered flux is generated at every point\nof the medium and may be estimated as follows at a point x' a\ndistance r' > ro from the center of the spherical source.\nFirst note that r' = /x'l. Then let (=sd(r')) be the\nmagnitude of the solid angle subtended by the sphere at van-\ntage point x' Then very nearly:\nN1(x',5)\nN°o(x';5';E) (5') = N°2(1')s/4n==\n=\n(1)\nfor every E. Hence:","SEC. 6.6\nCLASSICAL DIFFUSION EQUATIONS\n211\nh1(x) =\nN1(x',5)\ndo(E)\nNo e-ar'a(r')s\n(67)\nThis representation is not exact because the integration\nover the set of directions from the emitting sphere assumed\nthe distances from the point x' to the various points on the\nspherical surface were all equal to the fixed distance r'.\nHowever (67) should give excellent estimates of h1 (x') for\npoints x' when the sphere is viewed as a point source. We\nshall adopt (67) as a working basis in the present example.\nWe now use equation (51) with hn (x') in that equation\nreplaced by h1(x') as given in (67) Here r' is the distance\nfrom x' to the origin; hence r' = 1x'l. With these observa-\ntions (51) now lets us write:\n(68)\nh*(x) = Nos\n4wD|x-x'|\nX\nFinally, the residual scalar irradiance ho (x), was essentially\nevaluated in arriving at (67); that is, the scalar irradiance\ninduced by the small sphere is:\n(69)\nThe full scalar irradiance h (x) for the present problem\nis, according to (57), the sum of h°(x) and h* (x) as they are\ngiven in (68) and (69). A generalization of (68) is readily\neffected by letting N° vary in direction. A11 this means\nformally is that \"No\" goes under the integral sign in (68).\nIn this case, the approximation of h°(x) by Non(1xi) e-a/x|\nmust be examined. This will not be attempted here.\nHigher Order Scattered Flux\nas Source Flux\nThe preceding example of the use of primary scattered\nradiant flux as source flux in the classical diffusion equa-\ntion seems sufficiently useful to encourage carrying out the\nunderlying idea of the example to its logical conclusion.\nToward this end, suppose that it is possible to compute the\nfirst n+1 scattering orders for radiance: Nj, j = 0, 1,\nn. We then supplement this exact calculation by estimating\nthe radiance function","212\nCLASSICAL SOLUTIONS\nVOL. III\nj=n+1\nusing diffusion theory. Clearly this procedure includes that\nof the preceding discussion as a special case; in fact it is\nthe case n = 0.\nAs in the special investigation for the case n = 0, we\nbegin with the steady-state - equation of transfer:\nE.VN==.NN -\nNods + Nn\n+\nand now write N as :\nNj\n=N(n)+N(n,*)\n(70)\nwhere the definitions of the two terms N(n) and N(n,*) are\nimplicit in (70). Thus in particular N(0)= No and N(o,* =N*.\nUsing this decomposition in the equation of transfer,\nwe have:\nE.D(N\nods\n(71)\nNn\nNow, from (1) of Sec. 5.2 we have for every j > 1\n= + Nj-1 odd\n(72)\nand from (2) of Sec. 5.8:\n(73)\n= Nn\nBy adding equations (72) and (73) together from j = 1 up to\nj = n, we obtain:\n(74)\nE.VN +","SEC. 6.6\nCLASSICAL DIFFUSION EQUATIONS\n213\nThis equation is now used with (71) to reduce the lat-\nter to:\n(75)\nThis equation is the direct generalization of (62), the lat-\nter being obtained by setting n = 0 in (75).\nNext the operator U ((39) of Sec. 6.5) is applied to\neach side of (75) ; the result is:\n(76)\nThe final step is to hypothesize that Fick's law holds be-\ntween H n, *) and h (n, )):\n(77)\nso that (76) becomes:\n-D V h (n,*) +\n(78)\nThis is the requisite diffusion equation for h (n, *)\nIt is a direct generalization of (66) which is the case n = 0.\nThe source term for the flux h(n,*) is (n+1)-ary scattered\nflux, which should have relatively mild direction structure,\nso that (77) has a good chance of holding in practice. In\ngeneral, the greater the n, the more likely--on intuitive\ngrounds - (77) would seem to hold. (See the discussion follow-\ning (13) of Sec. 5.12.)\nOnce h(n,*) is obtained by solving (78) with the contin-\nuous source hn+1, using, e. g., (51) with hr replaced by hn+1,\nwe then find the complete scalar irradiance h by noting that\n(79)\n+\nwhere we write:\n(n),, for N(n) U\n(80)\nand:\n(n,*) for (n,*) U\n(81)\nFrom h (n,*) we can then find H (n, *) (x) using (77)\nand so, in turn, N N (x,5) using the diffusion equation","VOL. III\nCLASSICAL SOLUTIONS\n214\n(29) of Sec. 6.5 as a model. This diffusion-based estimate\nof N(n,*) (x,5) is then added to the known radiance N(n)(x,E).\nTime-Dependent Diffusion Problems\nTime-dependent radiative transfer problems arise, for\nexample, whenever extremely short pulses of radiant energy\nare released in scattering-absorbing media, and when the\nevolution of the subsequent scattered radiant energy of the\npulse is to be described or predicted in detail. We study\nnow a particularly simple and useful model of time-dependent\nlight fields based on classical diffusion theory, in particu-\nlar, equation (7) of Sec. 6.5.\nConsider an infinite homogeneous optical medium with a\nsingle point source at x' which at time t' emits a single\nDirac-delta - pulse of unit radiant energy. That is, we assume\nhn in (7) of Sec. 6.5 to have the form: hn (x,t) = Und(x-x').\ns (t-t') , where at present Un = 1, and Un in general has the\ndimensions of radiant energy.\nIt may be verified directly from (7) of Sec. 6.5 (by\nperforming the indicated differentiations and simplifying)\nthat the resultant scalar irradiance h (x, t) t > t', varies\nin space and time according as Kk(x', x;t', t), where we have\nwritten:\nV\n[4mvD(t-t\")]37Z exp\n\"K_(x',x;t',t)\" for\nexp\n-\n(82)\nThat is, for fixed x' and t', , the function Kk (x', t',\ndefined by (82) satisfies (7) of Sec. 6.5 at every space-time\npoint (x,t), such that x' + x and t > t'. The function\nKk(x', ; t', . ) first arose in the theory of transient heat\nconduction.\nIn general, with a continuous source distribution\nhn(x',t') defined throughout a part Xn of the medium for all\ntimes > t', we have, by means of the interaction principle,\nthe resultant scalar irradiance field given by:\nt\n(x',t') K K (x',x;t',t) dt dV(x')\nh\nh(x,t) =\n- 00\n(83)\nOf course, hn may be set equal to zero for all times t'\nearlier than some fiducial time to', so that hn(x',t') in\n(83) represents the general source condition (7) of Sec. 6.5.\nTherefore the resultant scalar irradiance field h defined by","SEC. 6.6\nCLASSICAL DIFFUSION EQUATIONS\n215\n(83) is the general solution of (7) of Sec. 6.5, as may be\nestablished by a direct appeal to (7) of Sec. 6.5.\nIt is of interest to connect (83) with two results ob-\ntained earlier in the present work. First we will show that\nif a steady point source condition subsists for all time,\ni.e., hn (x',t') is independent of time t' for all t' < and\nis zero for all points x other than a given point x' on the\nmedium, then:\nt\nKk(x',x)\nk(xx,\ndt'\n(84)\nso that (83) reduces to the steady state case (54). To see\nthis we note that Kk(x', x;t',t) has the general Gestalt of:\naditional\nwhere we have written, ad hoc:\nV\n\"a\" for [4mvD]3/2\n\"b\" for x-x'l2\nand:\n\"C\" for av\nand have replaced occurrences of \"(t-t')\" by \"t\". Then it\nis clear that on setting t = u2:\n[\nt\ndt'\ndu\n0\n- 00\n=\nThe second connection we can make is that between (83)\nand the earlier result which describes the behavior of radi-\nant energy under standard decay conditions, namely, property","CLASSICAL SOLUTIONS\nVOL. III\n216\n8 of Sec. 5.10. To establish this connection we now assume\nthat hn (x, t) = Un s (x) s (t). . This simulates the instantane-\n-\nous localized introduction of an amount U of radiant energy\ninto the medium. However, the actual manner of introduction\nis immaterial for the present discussion. With this condi-\ntion on hn, , (83) yields:\n,\nso that the radiant energy content of the medium at time t\nis:\nh(x,t) dV(x)\nU(t)\n=\nX\nU\nK (0,x;0,t) dV(x)\n=\nV\nK\nX\navt\nexp { - 1x/2 4vDt 2\nU\ne\nn\ndV(x)\n=\n3/2\n[4mvDt]\nX\nHence:\nU(t) = Une-avt =\n,\nwhich is precisely the analytic content of property 8 of Sec.\n5.10. This most interesting result shows that the classical\ndiffusion theory is globally exact and thereby may be used to\nhelp fill, in a consistent manner, the general gap in our\nknowledge about the local radiance distributions within a\ntime-dependent radiant field. That is, we may use (83) to\nsupplement the exact theory of the time-dependent radiant\nenergy field studied in Chapter 5, by giving approximate but\nuseful estimates of the radiant density throughout the medium.\nTo implement the program just outlined of supplementing\nthe exact radiant energy theory of Chapter 5 by diffusion\ntheory, we construct the basic diffusion equations for n-ary\nscalar irradiance from the time-dependent equation of trans-\nfer (19) of Sec. 5.8. Thus, by applying the operator U to\nthe equation of transfer for n-ary radiance, we have for\nn > 1:\n1 ah\" at + V . Hn = -ahn + shn-1\n(85)\nV","SEC. 6.6\nCLASSICAL DIFFUSION EQUATIONS\n217\nwhere for every n > 1 we have written:\n\"Hn\"\nfor\nand:\n(86)\n\"hn\" for\n(1)\nAssuming Fick's law holds between Hn and hn for every n,\nn >1, i.e., assuming:\n(87)\n,\nthen (85) yields the time-dependent diffusion equation for\nn-ary scalar irradiance,\n= - ahn + shn-1\n(88)\nOne immediate application of (88) is the direct general-\nization, to the time-dependent setting, of the results (68)\nand (79) of the continuous source cases with all the analytic\nadvantages of those results now transferred to the time-\ndependent context. In particular, we can replace hn(x', t;)\nin (83) by h1(x',t') which is computed exactly as in (67),\nbut with suitable time lag to account for the travel of the\ninitial pulse of the source from the source to x'. Then we\ncompute h*(x,t) as follows:\nL\nt\nh*(x,t)\nKx(x',x;t',t)\ndt'\ndV(x')\n=\n(89)\nso that:\nh(x,t) = h°(x,t) = + h ( x , t)\n(90)\nwhere ho (x,t) is the residual scalar irradiance computed from\nthe given source condition, which may be discrete or finite.\nThe theoretical basis for (89) is the time-dependent\ncounterpart to (66). This time-dependent - counterpart is ob-\ntained, e. .g., by adding up all equations in (88) for n= 1,2,","CLASSICAL SOLUTIONS\nVOL. III\n218\nThe result is:\n1 ah* - 2 h* = - ah* h1\n(91)\nat\nV\nObserve how the infinite number of Fick's laws in (87) imply\n(65). On the basis of (91), the representation (89) is\nestablished by simply repeating the arguments leading to (83).\nFinally, the generalization of (91) to the time-dependent\nversion of (78), and the derivation of the corresponding\nrepresentation of (79), , is readily made following the patterns\nof derivation established in that steady-state case.\nSolutions of the Exact Diffusion Equations\n6.7\nThe exact diffusion equation on which we base the dis-\ncussion of the present section is (57) of Sec. 6.5. In full\nnotation, this equation is of the form:\nh(x) = (h + sh) v(x)\ns(x') h(x')) K a (x',x) dV(x')\n1\n=\nX\n1 s(x') dV(x')\n(1)\n=\nX\nThe current settings in which this integral equation is\nto describe the scalar irradiance field h are infinite and\nsemi-infinite homogeneous media with arbitrary sources de-\nscribed by hn within X. Once a solution h is found for a\nspace X, the associated radiance distribution throughout X\nis obtained by means of (60) of Sec. 6.5. The first of our\ntwo main goals in this section is to solve (1) for a point\nsource in an infinite medium and arrange the solution in such\na manner as to be directly applicable to problems of finding\nradiance distributions associated with general source condi-\ntions in X. It will be seen that by judiciously tabulating\nthe point source solution of (1) all solutions of (1) corre-\nsponding to the possible source conditions within X, are ob -\ntainable in principle by relatively straightforward numerical\nprocedures based on the tabulated solution. The second main\ngoal is to discuss the solutions of (1) for semi-infinite\nmedia (infinitely deep, plane-parallel media) with arbitrary\ninternal sources.","SEC. 6.7\nEXACT DIFFUSION EQUATIONS\n219\nInfinite Medium with Point Source\nWe begin with (1) for the case of an infinite homogene-\nous medium X with a point source at the origin. The homoge-\nneity assumption frees a (x) and S (x) of dependence on x through-\nout X and lets us write:\n(2)\nIf\nwhere, as usual \"x\" denotes a point in X, and where |x-x'| is\nthe distance between points x and x'. The point source con-\ndition is represented by:\n(3)\nwhere P, is the quantity of radiant flux emitted steadily in\ntime and uniformly in all directions by the point source at\nthe origin. We may leave the nature of this source quite\narbitrary throughout the discussion. As a result, we shall\nbe able to adapt various solutions of (1) for the point source\ncase, by means of integration, and in such a manner that the\nactual nature of the source may vary from true emission pro-\ncesses, through transpectral scattering processes, on through\nelastic scattering processes. This will be illustrated later\nin the discussion. For the present we go on to investigate\nthe case of (1) with a single point source. The requisite\nform of (1) is:\ne\ndV(x') (4)\nThe theory of the solution of (4) is thoroughly under-\nstood; a representative detailed development of the solution\nof (4) may be found, e.g., in [40]. Therefore, beyond the\ngeneral observations leading from (39) to (59) of Sec. 6.5,\nwe shall not need to discuss the details of the solution pro-\ncedure of (4) in the present work. However, we wish to dis-\nplay the solution of (4) in such a manner that the results of\n[40] may be readily adapted to the radiative transfer context.\nSuch an adaptation requires the preliminary transition to a\ncertain class of dimensionless geometric parameters, which we\nnow define.\nThroughout this section we shall write:\n\"t(x,x') for I\nr\n(x\")\ndr\"\n(5)\n0","VOL. III\n220\nCLASSICAL SOLUTIONS\nwhere a is the volume attenuation function for the medium.\nThe integral is a line integral along a path @p(x,5) with\ninitial point X and terminal point x'. Since the medium X is\nisotropic and homogeneous, paths are straight-line - segments\nand\nt(x,x') = a/x-x'|\n(6)\nWhen no confusion will result, we will simply write:\n\"T\" for T(x,x')\n,\nwith x, and x' thereby being understood.\nThe quantity T assigned to the distance |x-x' I between\nx and x' is dimensionless, and by virtue of (6) may be viewed\nas the number of attenuation lengths La between x and x'.\nNext, for every subset Y of X we write:\n3(x)\ndV(x')\n(7)\n\"Va(Y)\"\nfor\nY\nThe quantity Va(Y) is dimensionless. Throughout this\nsection, both T(x,x') and V (Y) may be thought of and referred\nto as optical lengths and optical volumes, respectively, with-\nout fear of confusion with the classical notions of the same\nnames.\nWith definitions (5) and (7) in mind, (4) may be re-\nwritten as:\nP h(x') dV (x')\n(8)\nX\nwhere p is the cattering-attenuation ratio s/a. Equation\n(8) is the required dimensionless version of (4); and for\npurposes of a solution tabulation, we now impose the unit\nsource condition in the context of (8):\nP = 1\n(9)\nprovided that the Dirac-delta function S with dimensions L-3\n(to go with the volume measure V) is retained. Otherwise, if\na dimensionless Dirac-delta function S (to go with the opti-\ncal Va) is adopted, in (3) we write hn S (x') and the unit\nsource condition is\nn = 1\n(9a)\n.","SEC. 6.7\nEXACT DIFFUSION EQUATIONS\n221\nThe scalar irradiance field h governed by (8) is clear-\nly spherically symmetric about the point source so that h de- -\npends only on radial distance r or (now that the transition\nto dimensionless parameters has been made) on T. Let us de- -\nnote the solution of (8), , under the unit source condition\n(9a), by \"KE\". Then it can be shown (cf. [40]) that the\nscalar irradiance at optical distance T from the origin is\nKE(T), where:\n(10)\nand where, in turn we have written:\n(11)\n\"A(p,T)\" for\nand\n(12)\n\"B(p,T)\" for\nto point up the fact that KE (T) is simply a linear combina-\ntion of the dimensionless diffusion kernel KK(T) (cf. (52) of\nSec. 6.6) where now we write:\ne-kot\n(13)\nand the dimensionless beam transmittance kernel Ka(T) (cf.\n(43) of Sec. 6.5) where now we write:\n\"K_(1)\" for\n(14)\n.\nIt remains to specify the terms e(p,T), Ko, /, and\nDo. The latter term is simply aD, where D is the diffusion\nconstant (cf. (27) of Sec. 6.5) for the classical diffusion\ntheory. The remaining three terms form the heart of the\nexact solution and are tabulated in Tables 1 and 2 below for\nvarious values of p and T.\nThus from (10), we have\n(15)","222\nCLASSICAL SOLUTIONS\nVOL. III\nTABLE 1\nThe function E (p,T)\n0\np = 0.1\np = 0.2\np = 0.3\np = 0.4\np = 0.5\nT\np\n=\n0.0\n1.0000\n1.0000\n1.0000\n1.0000\n1.0000\n1.0000\n0.1\n1.0000\n1.0210\n1.0418\n1.0542\n1.0526\n1.0420\n0.2\n1.0000\n1.0382\n1.0773\n1.1000\n1.0962\n1.0756\n0.3\n1.0000\n1.0532\n1.1088\n1.1409\n1.1346\n1.1046\n0.4\n1.0000\n1.0667\n1.1375\n1.1781\n1.1692\n1.1301\n0.5\n1.0000\n1.0790\n1.1640\n1.2126\n1.2008\n1.1529\n0.6\n1.0000\n1.0904\n1.1888\n1.2448\n1.2300\n1.1736\n0.7\n1.0000\n1.1010\n1.2121\n1.2752\n1.2571\n1.1926\n0.8\n1.0000\n1.1109\n1.2342\n1.3038\n1.2826\n1.2100\n0.9\n1.0000\n1.1202\n1.2552\n1.3311\n1.3066\n1.2262\n1.0\n1.0000\n1.1291\n1.2753\n1.3571\n1.3293\n1.2412\n1.5\n1.0000\n1.1674\n1.3644\n1.4724\n1.4273\n1.3034\n2.0\n1.0000\n1.1990\n1.4402\n1.5699\n1.5068\n1.3504\n2.5\n1.0000\n1.2258\n1.5068\n1.6551\n1.5738\n1.3874\n3.0\n1.0000\n1.2494\n1.5667\n1.7311\n1.6314\n1.4171\n3.5\n1.0000\n1.2704\n1.6213\n1.8000\n1.6818\n1.4415\n4.0\n1.0000\n1.2895\n1.6718\n1.8630\n1.7265\n1.4617\n4.5\n1.0000\n1.3070\n1.7188\n1.9214\n1.7665\n1.4786\n5.0\n1.0000\n1.3231\n1.7630\n1.9757\n1.8026\n1.4928\n2.0745\n6.0\n1.0000\n1.3521\n1.8443\n1.8654\n1.5147\n7.0\n1.0000\n1.3779\n1.9182\n2.1630\n1.9182\n1.5304\n8.0\n1.0000\n1.4010\n1.9863\n2.2432\n1.9634\n1.5412\n9.0\n1.0000\n1.4222\n2.0497\n2.3169\n2.0024\n1.5486\n10.0\n1.0000\n1.4417\n2.1094\n2.3851\n2.0366\n1.5531\n11.0\n1.0000\n1.4599\n2.1659\n2.4499\n2.0667\n1.5554\n12.0\n1.0000\n1.4770\n2.2196\n2.5086\n2.0933\n1.5559\n13.0\n1.0000\n1.4931\n2.2710\n2.5652\n2.1172\n1.5550\n14.0\n1.0000\n1.5084\n2.3204\n2.6188\n2.1385\n1.5529\n15.0\n1.0000\n1.5230\n2.3682\n2.6700\n2.1578\n1.5498\n16.0\n1,0000\n1.5370\n2.4141\n2.7190\n2.1752\n1.5459\n17.0\n1.0000\n1.5503\n2.4586\n2.7658\n2.1910\n1.5413\n18.0\n1.0000\n1.5632\n2.5019\n2.8109\n2.2055\n1.5361\n19.0\n1.0000\n1.5757\n2.5439\n2.8543\n2.2186\n1.5304\n20.0\n1.0000\n1.5877\n2.5849\n2.8963\n2.2307\n1.5243\nNow that it is clear how KE (T) depends on the diffusion\nkernel K K (52) of Sec. 6.6) and the attenuation kernel Ka\n((43) of Sec. 6.5) we write (10) in its explicit form:","SEC. 6.7\nEXACT DIFFUSION EQUATIONS\n223\nTABLE .--Concluded\nThe function E (p,T).\np = 0.6\np = 0.7\np = 0.8\np = 0.9\np = 1.0\nT\n0.0\n1.0000\n1.0000\n1.0000\n1.0000\n1.0000\n0.1\n1.0269\n1.0099\n0.9921\n0.9745\n0.9564\n0.2\n1.0474\n1.0162\n0.9843\n0.9528\n0.9222\n0.3\n1.0643\n1.0206\n0.9767\n0.9341\n0.8934\n0.4\n1.0786\n1.0236\n0.9693\n0.9173\n0.8683\n0.5\n1.0909\n1.0257\n0.9621\n0.9019\n0.8460\n0.6\n1.1017\n1.0271\n0.9551\n0.8878\n0.8260\n0.7\n1.1113\n1.0279\n1.9483\n0.8747\n0.8077\n0.8\n1.1198\n1.0282\n0.9417\n0.8625\n0.7910\n0.9\n1.1275\n1.0282\n0.9353\n0.8510\n0.7755\n1.0\n1.1343\n1.0278\n1.9290\n0.8402\n0.7612\n1.5\n1.1601\n1.0229\n0.9002\n0.7936\n0.7019\n2.0\n1.1763\n1.0149\n0.8748\n0.7562\n0.6568\n2.5\n1.1866\n0.0054\n0.8519-\n0.7250\n0.6207\n3.0\n1.1929\n0.9952\n0.8313\n0.6982\n0.5908\n3.5\n1.1963\n0.9847\n0.8124\n0.6749\n0.5655\n4.0\n1.1978\n0.9742\n0.7951\n0.6543\n0.5437\n4.5\n1.1976\n0.9637\n0.7791\n0.6358\n0.5246\n5.0\n1.1963\n0.9534\n0.7643\n0.6191\n0.5076\n6.0\n1.1912\n0.9334\n0.7374\n0.5901\n0.4788\n7.0\n1.1838\n0.9144\n0.7137\n0.5654\n0.4550\n8.0\n1.1749\n0.8964\n0.6926\n0.5440\n0.4349\n9.0\n1.1651\n1.8793\n0.6734\n0.5253\n0.4175\n10.0\n1.1547\n0.8631\n0.6560\n0.5086\n0.4024\n11.0\n1.1438\n0.8477\n0.6400\n0.4936\n0.3890\n12.0\n1.1327\n0.8330\n0.6252\n0.4800\n0.3769\n13.0\n1.1215\n0.8190\n0.6114\n0.4676\n0.3661\n14.0\n1.1102\n0.8055\n0.5985\n0.4562\n0.3562\n15.0\n1.0989\n0.7926\n0.5864\n0.4456\n0.3471\n16.0\n1.0876\n0.7802\n0.5750\n0.4357\n0.3387\n17.0\n1.0764\n0.7683\n0.5643\n0.4265\n0.3310\n18.0\n1.0653\n0.7568\n0.5540\n0.4178\n0.3238\n19.0\n1.0542\n0.7457\n0.5443\n0.4096\n0.3170\n20.0\n1.0433\n0.7349\n0.5349\n0.4019\n0.3107","224\nCLASSICAL SOLUTIONS\nVOL. III\nTABLE 2\nThe functions K 0 and dk6/dp 2\ndk 0 2 dp\np\nK 0\n0.0\n1.000000\n0.000000\n0.1\n1.000000\n0.164892 (-5) *\n0.2\n0.999909\n0.009094\n0.3\n0.997414\n0.116201\n0.4\n0.985624\n0.373272\n0.5\n0.957504\n0.731896\n0.6\n0.907332\n1.145954\n0.7\n0.828635\n1.590033\n0.8\n0.710412\n2.051119\n0.9\n0.525430\n2.522370\n0.92\n0.474002\n2.617473\n0.94\n0.413976\n2.712805\n0.96\n0.340829\n2.808348\n0.98\n0.242983\n2.904085\n0.99\n0.172511\n2.952020\n1.00\n0.000000\n3.000000\n*Note: \"(-5)\" means \"multiply by 10 5 \"\nIn this way we can see that, for computation purposes, the\nscalar irradiance KE (T) at optical distance T from the origin\nconsists of two terms, one which may be attributed to resid-\nual flux (the first term) and the other which may be attri-\nbuted to scattered flux. This type of partitioning of the\nexact representation of h(x) into a residual part (ho) and a\nscattered part (h*) was already encountered in the classical\ndiffusion theory, e.g., in (7) of Sec. 1.5, in (57) of Sec.\n6.6, and more generally in (79) of Sec. 6.6. Also, in the\ntime-dependent case, this partition was encountered in (90)\nof Sec. 6.6.\nA tabulation of 4 T 2 K (T) is given in Table 3 for two\ncases of p and for a range of T from 0 to 10 units. These\nchoices of p are representative orders of magnitude for p in\nthe case of the ocean (p = 0.3) and the atmosphere (p = 0.9)\nfor wavelengths around 500 mu, for the middle of the visible\nspectrum. For the determination of KE(T) for values of p\nother than p = 0.3, 0.9, Tables 1 and 2 may be used. It must\nbe kept in mind that these tabulations are for the unit source\ncondition (9a).","SEC. 6.7\nEXACT DIFFUSION EQUATIONS\n225\nTABLE 3\nThe function 4 T2 KE (T)\np = 0.3\np = 0.9\nT\n0.0\n1.0000\n1.0000\n0.1\n0.9644\n1.1211\n0.2\n0.9196\n1.2343\n0.3\n0.8710\n1.3384\n0.4\n0.8209\n1.4326\n0.5\n0.7708\n1.5168\n0.6\n0.7215\n1.5914\n0.7\n0.6737\n1.6567\n0.8\n0.6277\n1.7130\n0.9\n0.5838\n1.7607\n1.0\n0.5421\n1.8006\n1.5\n0.3675\n1.8974\n2.0\n0.2441\n1.8660\n2.5\n0.1599\n1.7547\n3.0\n0.1037\n1.5992\n3.5\n0.0668\n1.4239\n4.0\n0.0427\n1.2454\n4.5\n0.0272\n1.0742\n5.0\n0.0173\n0.9158\n6.0\n0.0069\n0.6483\n7.0\n0.0028\n0.4467\n8.0\n0.0011\n0.3018\n9.0\n0.0004\n0.2007\n10.0\n0.0002\n0.1318\nInfinite Medium with\nArbitrary Sources\nWe now develop a procedure whereby Table 3, and more\ngenerally (15), , may be used to compute scalar irradiance\nfields generated by arbitrary sources. Suppose the source\nterm hn (x) is given throughout an infinite medium X; hn (x)\nmay be associated with plane sources, finite volume sources\nof flux, etc., and may be of quite arbitrary spatial depen-\ndence throughout X. It is clear either intuitively or for-\nmally (from the interaction principle using the theorems of\nSec. 3.16) that the scalar irradiance h (x) associated with\nhn(x) is given by:","VOL. III\nCLASSICAL SOLUTIONS\n226\n1\n(16)\nh n (x') KE(x',x) dVa(x')\n(x)\n=\n-\na\nX\nwhere we have written:\n(17)\n\"KE(x',x)\" for KE (t(x,x'))\nThe reason for the presence of \"a\" in (16) may be found by\ntracing back through the unit source condition (9a) and ulti-\nmately to (3) and (4). If h, is given in watts per cubic\nmeter, and a in per meter, then h is given in units of watts\nper square meter.\nA practical computation scheme for (x) may be based on\nthe following procedure: given hn (x) throughout a subset Xn\nof X, divide Xn into n small cubes C(xi) (or any other con-\nveniently shaped regions) over each of which both T(x,x') and\nhn(x) vary only slightly. Thus each cube C(x) is represen-\ntative of the radiometric properties of X around xi, where\nxi is the cube's centerpoint. Then (16) may be replaced by\nthe approximating finite sum:\n(18)\nh(x) = i=1 h (xi) Kg (xi,x) va(c(xi))\nThe evaluation of (x) using (18) is facilitated by us-\ning Table 3 for optical distances T(x,x') up to 10. More\ngenerally, (15) would be used with Tables 1 and 2.\nAs a specific example of a setting in which (18) may be\napplied, consider the problem of determining the irradiance\nfield generated in an infinite homogeneous medium by a beam-\ntype source, such as that associated with powerful search\nlights or laser beams. The geometrical relations of the\npresent example are summarized in Fig. 6.5. The source may\nbe represented as a small sphere of radius ro with surface\nradiance No and which is allowed to emit uniformly over a\nconical set E0 of directions with central direction 50. Thus\nE0 may be all directions E such that E . 50 > cos 00 where 0 o\nis the half angle opening of Eo. By varying 00, the cone can\nrepresent everything from narrow beams (small eo) to uniform\npoint sources (00 = II).\nWith these geometrical preliminaries fixed, we now\nreturn to the discussion in Sec. 6.6 which developed the\ntheory of primary scattered flux as source flux and which cul-\nminated in the formulas (67) through (69) of Sec. 6.6. We can","SEC. 6.7\nEXACT DIFFUSION EQUATIONS\n227\nif\no\nx'\n52(r)\n80\nr\no\nXo\n+ X\nFIG. 6.5 Geometry for a nonisotropic point source of\nradiant flux in diffusion theory.\nimmediately adopt for our present purposes the formula (67)\nof Sec. 6.6 which describes the primary scalar irradiance\nh1(x') in terms of the inherent radiance No, the total scat-\ntering coefficient s, the beam transmittance e-ar', and the\nsolid angle so(r') subtended by the point source at point x'\n(See Fig. 6.5.) Now h1(x') replaces hn(x') in (16) or hn (xi)\nin (18). Thus (16) becomes:\ne-\nh(x) = pN\n(x'\nx)\ndVa(x')\n(19)\no\nX\nand (18) becomes:","VOL. III\nCLASSICAL SOLUTIONS\n228\nn\n(x) = pN o { sc(xxl) K (xj,x) v_(c(xi))\n(20)\ni=1\nIn (19) the integration may be limited to the subset X of X\ndefined by the cone E0 of directions. Thus point x' is in X\nif and only if x'//x' is in Eo. In (20) the sum is over all\ncells C(xi) which partition X0. Because of the exponentials\nand the solid angles So ( x i / ) in (20), the sums (for a given\nNo) need not be extended over very many attenuation lengths\nwithin X before good estimates of h (x) can be made.\nSemi-Infinite Medium with\nBoundary Point Source\nThe exact diffusion solution (16) holds for media which\nextend indefinitely far in all directions about the point\nsource. Such a situation will hold more or less in natural\nwaters when the source and observer are at relatively great\ndepths (several attenuation lengths, say). However, if the\nsource is relatively near the surface, the reflectance prop-\nerties of the remaining thin layer of medium above the source\nwould differ noticeably from that of an infinitely deep layer\nabove the source, so that the scalar irradiance (T) at shal-\nlow depths in a light field induced by a point source near\nthe boundary would differ markedly from that predicted by (16).\nSimilar observations may be made for fogs and cloud banks in\nthe atmosphere. In the present example, we summarize some\nresults of exact diffusion theory which can predict h(T) for\nrelatively shallow depths in natural waters (or for points\nnear flat cloud or fog boundaries) when the point source is\non the boundary. The reflection effects of the air-water\nsurface are not included in the present analysis and must be\naccounted for separately. In the second example below the\nresults will be extended to the case of internal point sources.\nBoth examples are based on the results by Elliott given in\nRef. [88]. A generalization of the equations developed below\nand their appropriate place in the general theory of radia-\ntive transfer in media with internal sources, will be given\nin Sec. 7.13.\nThe starting point for the present discussion is equa-\ntion (8) in which the medium X is now an infinitely deep ho-\nmogeneous plane-parallel medium exhibiting isotropic scatter-\ning and with a point source of small positive radius r o at\ndepth X = C > 0. We shall use the terrestrially based refer-\nence system for natural hydrosols (cf. Sec. 2.4). Further-\nmore we use the unit source condition (9a) in (8).\nThus we start with (8), now in the form:\n(x,x')\n1\ne\n(x')\n(21)\nph(x'))\ndV\n(8(x'-xo)\n+\na\n(x,x')\nX+","SEC. 6.7\nEXACT DIFFUSION EQUATIONS\n229\nwhere x+ is the set of all x (=(x1,X2, X3)) in the terrestrial\ncoordinate frame such that X3 = z > 0. The Dirac-delta func-\ntion E in (21) is dimensionless, and is centered on the point\nxo (=(0,0,c)), C > 0. Furthermore, it is to be explicitly\nnoted that for the remainder of this section all coordinates\nX1, X2, X3 (hence all distances, areas, and volumes) are to be\nmeasured in units of optical length (cf. (5), (7)).\nNow the procedure in Ref. [88] is to take the Fourier\ntransform of (21) with respect to the variables X1 and X2 over\nan arbitrary horizontal plane at depth X3 (=z). Thus let W1\nand W2 be the spatial frequencies along the X1 and X2 direc-\ntions and let us write:*\nh(x) ei(x101 + x2w2) dA(x)\n\"fo(z;wy,w2)\"\nfor\nX2\n(22)\nwhere X2 is the horizontal plane at depth z, and A is the\narea measure over Xz. Thus fo is the Fourier transform of h\nover X, and fo has the same dimensions as h. Therefore,\napplying the operator:\nla\nx 2\nto each side of (21), we obtain:\nfo(z;w) I(|z-z'/,w) dz'\n0\n(23)\nwhere we have written:\n8\n\"I(| 2-2'/,w)\" for /z-z'|\n(24)\n*In the present exposition, we retain the Fourier trans-\nform conventions used in [88] in order to facilitate the study\nof the results therein.","VOL. III\nCLASSICAL SOLUTIONS\n230\nwhere Jo is a zero-order Bessel function, and where, for\nbrevity, we have written:\n(25)\n\"fo(z;w)\" for fo(z,w1,w2)\nThe next step in the solution procedure is the observa-\ntion that (23) can be solved using the Wiener-Hopf technique\nprovided that C = 0, i.e., , that the source is at the boundary.\nThis solution procedure is quite intricate and beyond the im-\nmediate interests of the present work; therefore the inter-\nested reader is referred to Ref. [88] for details and further\nreferences. The main results of the present example may be\nunderstood without recourse to the solution details. We need\nonly observe that the required scalar irradiance is obtained\nfrom the solution fo(.;w) of (23) by means of the following\nintegration which is the inverse Fourier transformation to\nthat in (22):\n#\n(26)\nfo(2,w) w To(wr)\ndw\n0\nin which:\nx = (x1,x2,2)\nand:\nw2=w2+w2\n(27)\nr2=x2+x2\n(28)\nSince h (x) depends only on depth Z and the radial distance r,\nwe agree to write:\n(29)\n\"h(z,r)\" for h(x)\nFigure 6.6 depicts the geometrical details of the case\nwhere the point source is at the boundary. Observe that the\nmedium is divided into region A (shaded) and conical region B\n(unshaded). It is found that h (z,r) for points X = (X1,X2,Z)\nin region A is approximated by the relation:\n(30)\n(Valid in region A, Fig. 6.6.)\nwhere in turn 41(z) is evaluated in [172] and is tabulated in\nTable 4, and Ko is given in Table 2. Table 4 may be extended,","SEC. 6.7\nEXACT DIFFUSION EQUATIONS\n231\npoint\nZo\nsource\n45.9\n0\nB\nZ\nFIG.\n6.6 Domains of validity of approximate solutions\n(30) and (31).\nif necessary, using the eddingtonian approximation to U1:\n2\n+\nThe functions En(z) are the exponential integrals\ndu\nand are tabulated. The farther the point x (= (x1 , X2, X3)) in\nregion A is from the dashed dividing lines between regions A\nand B, the better the approximation (30).","VOL. III\nCLASSICAL SOLUTIONS\n232\nTABLE 4\nEvaluation of 41(2)\n41 (z)\nZ\nZ Z o\n0.7104\n0.5773\n0\n0.7204\n0.5982\n0.01\n0.02\n0.7304\n0.6154\n0.7404\n0.6312\n0.03\n0.05\n0.7604\n0.6607\n0.7279\n0.1\n0.8104\n0.8495\n0.2\n0.9104\n0.9633\n0.3\n1.0104\n1.0731\n0.4\n1.1104\n1.1803\n0.5\n1.2104\n1.2858\n0.6\n1.3104\n0.7\n1.4104\n1.3901\n0.8\n1.5104\n1.4935\n0.9\n1.6104\n1.5963\n1.0\n1.7104\n1.6985\n1.2\n1.9104\n1.9019\n1.5\n2.2104\n2.2051\n2.0\n2.7104\n2.7079\n2.5\n3.2104\n3.2092\n3.0\n3.7104\n3.7098\n3.5\n4.2104\n4.2101\n4.0\n4.7104\n4.7102\nThe error of the approximation by (30) is of the order\nof magnitude of and (30) is applicable when p is 0.6\nor more.\nFurthermore, it is found that h(z,r) for points\nx\n(=(x1,x2,z)) in region B is approximated by the relation:\n3\n0\ncos\ne-kod(1+k d)\n(31)\n=\ne\n2\n(Valid in region B, Fig. 6.6. )\nwhere we have written:\n\"d\" for r 2 + (z+zo) 2\n(32)","SEC. 6.7\nEXACT DIFFUSION EQUATIONS\n233\nand where:\ntan 0 27/20\n(33)\nand:\n20==0.7104\n(34)\nThis approximation improves with the distance of X\n(=x1,X2,Z) ) in region B from the dashed dividing lines be-\ntween regions A and B. The error of approximation by (31)\nis of the order of magnitude of |1/d5 I and (31) is applicable\nwhen p is 0.6 or more.\nA study of (30) and (31) readily shows the effect on\nh (x) of the presence of the boundary at depth Z = 0. Suppose\nfor the moment that Ko = 0 (no absorption case). Then in\nregion A of Fig. 6.6, and for fixed z, the scalar irradiance\nfalls off as the inverse cube of the distance r from the sym-\nmetry axis of the field, whereas in region B, which is rela-\ntively farther removed from the boundary than region A, the\nscalar irradiance falls off only as the inverse square of the\ndistance d. The fixed number 20 (known as the \"extrapolation\nlength\") in (34) arises in the correct adjustment of boundary\nconditions of the present problem.\nSemi-Infinite Medium with Internal\nPoint Source\nThe results of the preceding example will now be ex-\ntended to the case of a semi-infinite homogeneous medium with\na point source at X = (0,0,c), C > 0, i.e., , with a point\nsource in the interior of the medium rather than on the bound-\nary. Let us denote the solution of (21) for this case by\n\"fc(z;w)\". Hence, when C = 0 we are to have fo (z;w) of (23)\nback once again, and fc is to be a proper generalization of\nfo. Now assume a general point source condition hn/o (cf.\n(9a)). Then the functional relations connecting fc and fo,\nas derived by Elliott [88] are of the form:\nZ\nS\nf\n(t,w) fo(t+c-z,w) dt, ,\n(35)\nZ < C\n0\nC\nf\nc(z,w) =fo(1z-c/,w) f\n(t,w) fo(t-c+z,w) dt\n(36)\n2>C\n,\n0","VOL. III\nCLASSICAL SOLUTIONS\n234\nZo\npoint source\n45°\nA\n0\nB\nZ\nr\nDomains of validity of approximate solutions\nFIG.\n6.7\n(38) through (40).\nOnce (z,w) is obtained using (35) or (36), h(z,r) can be\nobtained by means of the inversion formula:\n8\nh(z,r) - 1/\n(37)\nf c(z,w) w J o (wr) dw\n0\nwhich is simply (26) now with fc in place of fo. A few ob- -\nservations on these functional relations will be made below,\nbut for the present we go on to their immediate consequences.\nFigure 6.7 depicts the semi infinite medium with point source\nat (0,0,c). The medium is divided into two regions with the\nshaded region A and the conical region B, exactly analogously\nto the partition depicted in Fig. 6.6. Corresponding to (30)\nwe now have the approximate solution:","SEC. 6.7\nEXACT DIFFUSION EQUATIONS\n235\nYo\nYt\nYc\nZZC\nYz\nYo\nYt\nYz\nC\nz\nYc\nFIG. 6.8 Relative placement of source (c) and observation\n(z) levels in (35) and (36).\ndt e-Kor(1+kgr)\nh\n(38)\nfor C\nC\ndt -Kor(1+kgr)\n,\n(39)\nfor z z c\n(Valid in region A, Fig. 6.7.)","VOL. III\nCLASSICAL SOLUTIONS\n236\nA11 the terms occurring in (38) and (39) were defined\nin (30). The ranges of integration may be visualized with\nthe help of Fig. 6.8. Observe how (39) reduces to (30) when\nC = 0. The errors of approximation are on the order of /c3/r5/\nfor (38) and for (39). The approximations (38), (39)\nare applicable for media with p = 0.6 or more.\nCorresponding to (31) we now have:\n/3h\nhtt, C (3) cos 0 -Kod (1+kod)\n(40)\n(Valid in region B, Fig. 6.7.)\nObserve in this instance, also, how (40) reduces to its limit-\ning case (31) for C = 0, where now in (40) we have written:\n\"d\" for /22+(2+20-c)2\n(41)\nand also where\ntan 0 2420-C\n(42)\nThe approximation (40) holds for large /z-cl and has an error\non the order of magnitude of |c/d3 for media with p = 0.6 or\nmore.\nObservations on the Functional\nRelations for fc and f o\nThe various solutions displayed above for h(z,r) in a\nsemi-infinite medium are of great interest for two reasons.\nThe first reason is clear enough: They supply additional\ninformation on the behavior of (x) in deep plane-parallel\nmedia in which there are point sources near the boundaries.\nThe second reason for interest in these solutions does not\nexist so much on a practical level as on a theoretical or\nconceptual level. This interest centers on the form of the\nfunctional relations (35) and (36) which seem to hold consid-\nerable importance for radiative transfer theory. These two\nremarkable relations show how to connect the point source\nsolution for the case C = 0 with that for the case C > 0. The\ngeneral form of the functional relations (35) and (36) are\nthose of the relations usually found by the techniques of\ninvariant imbedding, the techniques growing out of the clas-\nsical invariance principles of Chandrasekhar. It will be\nshown in Sec. 7.13 how the general counterparts of (35) and\n(36) for radiance fields may be deduced from the invariant\nimbedding relations (cf. also examples 2, 3, 5 of Sec. 3.9).\nAs a result of the derivations in Sec. 7.13, there will be a","SEC. 6.7\nEXACT DIFFUSION EQUATIONS\n237\nunified set of analytical techniques for solving internal-\nsource problems in general optical media.\n6.8\nBibliographic Notes for Chapter 6\nThe discussions of Sec. 6.1 leading to (36) of that\nsection are based on some elementary properties of complete\northonormal families of functions, which in turn find their\nrightful place in Hilbert space theory, or general vector\nspace theory. For an exposition of these ideas, see, e.g.\n[104]. The isolation of the two properties, namely: the\nfinite recurrence property of the orthonormal family and the\nisotropy property of the medium led to the finite forms (26)\nof the abstract harmonic equations in Sec. 6.2. This expli-\ncit delineation of the necessary properties to be held jointly\nby orthonormal families and optical media, which lead to the\nabstract harmonic equations (26) of Sec. 6.2, appears to be\nnew.\nThe exposition of the classical spherical harmonic\nmethod in Sec. 6.3 is based on that of Refs. [175] and [314].\nThe solution procedures of the classical spherical harmonic\nequations for plane-parallel media in Sec. 6.4 are based on\nmodern algebraic methods in differential equation theory,\nsuch as those in [47]. Some innovations in numerical pro-\ncedures in the spherical harmonic method may be found in [323]\nand [325]. The manner of approach to diffusion theory in\nSec. 6.5 is dictated by the specific needs and outlook of\ngeophysical radiative transfer theory. The classification\nof diffusion processes in Sec. 6.5 is of course only partial-\nly complete; a systematic investigation of such classified\nprocesses appears to be of some interest to radiative trans-\nfer theory, and offers interesting physically based problems\nin partial differential equation theory.\nThe general solutions of the classical diffusion equa-\ntions in the opening paragraphs of Sec. 6.6 are widely known,\nuseful formulas for scalar irradiance. The various primary\nscattered flux source methods and those based on higher or-\ndered scattered flux sources in the latter part of Sec. 6.6\noffer some novelty in the otherwise quite thoroughly formed\nclassical method of treatment of the diffusion of light\nthrough scattering media. Furthermore, the particular needs\nof hydrologic optics and meteorologic optics has caused some\nemphasis to be placed on the representation of the radiance\ndistribution N (x, ) throughout diffusing media. This resulted\nin derivations of formulas for N(x,5) in general diffusion\ncontexts, such as (29) of Sec. 6.5; and (14) and (40) of Sec.\n6.6, which do not appear to be too widely known.\nThe solutions of the exact diffusion equations in Sec.\n6.7 for the case of infinite media are based on the work in\n[40]. This work also contains many useful tables and graphs\nof associated solutions. The theory of semi-infinite media\nwith point sources is relatively unexplored. However, refer-\nence [88] forms a definitive beginning of such a theory, and\nthe latter half of the discussions in Sec. 6.7 are based on\nthe results of [88].","CLASSICAL SOLUTIONS\nVOL. III\n238\nFurther References\nFurther references beyond those mentioned above and\nwhich contain contributions to the classical theory of trans-\nport phenomena may be briefly mentioned here. First of all\nthere is the early definitive work by Hopf [111] on mathemat-\nical problems of radiative transfer in media which are in\nthermodynamic equilibrium. This work contains the germ of the\nmodern operator theoretical approach to transfer problems\nwhich is continued in [37] and [143], and more recently in\n[251]. Another early definitive work on classical radiative\ntransfer theory is that of Chandrasekhar [43] which develops\na minor variant of the spherical harmonic method of the kind\nformulated by Wick in [319]. Applications of the Chandra-\nsekhar theory are made by Lenoble in [108], [155], [156].\nBy far the most significant contribution in [43] is that of\nthe principles of invariance, which were discussed in general\nin Chapter 3 above and which will be considered further in\nChapter 7 below. The reference [62] also contains much use-\nful mathematical information which is applicable to practical\nradiative transfer contexts. A relatively recent survey of\nradiative transfer theory and classical and exact diffusion\ntheory may be found in [288].\nSome tabulated solutions of the equation of transfer\nare given in [53], [91], and [11]. Diffusion theory from the\npoint of view of Monte Carlo techniques is explored in [41]\nand [176]. Some recent numerical solutions for light fields\nin homogeneous slabs (with isotropic scattering) which blend\nthe spherical harmonic method and the technique of invariant\nimbedding are given in [15] and [16].","239\nBIBLIOGRAPHY\nBIBLIOGRAPHY FOR VOLUME III\n5. Armed Forces NRC Vision Committee, Minutes and Proceed-\nings, 23rd Meeting, pp. 123-126 (March 1949).\n11. Beach, L. A., , et al., Comparison of Solutions to the One-\nVelocity Neutron Diffusion Problem (Naval Res. Lab.\nReport 5052, December 23, 1957).\n15. Bellman, R. E., Kalaba, R. E., and Prestrud, M. C.,\nInvariant Imbedding and Radiative Transfer on Slabs\nof Finite Thickness (American Elsevier Pub. Co.,\nNew York, 1963).\n16. Bellman, R. E., Kagiwada, H. H., Kalaba, R. E., and\nPrestrud, M. C., Invariant Imbedding and Time-\nDependent Transport Processes (American Elsevier Pub.\nCo., New York, 1964).\n23. Birkhoff, G., , and Rota, G., Ordinary Differential Equa-\ntions (Ginn and Co., New York, 1962).\n28. Bouguer, P. , Optical Treatise on the Gradation of Light\n(Paris, 1760, trans. by W. E. K. Middleton, Univ. of\nToronto Press, Toronto, 1961).\n37. Busbridge, I. W., The Mathematics of Radiative Transfer\n(Cambridge Univ. Press, London, 1960). .\n39. Carslaw, H. S. and Jaeger, J. C., Operational Methods in\nApplied Mathematics (Dover Pub., Inc., New York, 1963).\n40. Case, K. M., de Hoffman, F., and Placzek, G., Introduction\nto the Theory of Neutron Diffusion (Los Alamos Sci-\nentific Laboratory, Los Alamos, New Mexico, June 1953),\nvol. I.\n41. Cashwell, E. D. and Everett, C. J., (A Practical Manual\non the) Monte Carlo Method for Random Walk Problems\n(Pergamon Press, New York, 1959). .\n43. Chandrasekhar, S., Radiative Transfer (Oxford, 1950).\n47. Coddington, E. N., and Levinson, N., The Theory of Ordi-\nnary Differential Equations (McGraw-Hill, New York,\n1955).\n53. Coulson, K. L., Dave, J. V., and Sekera, Z., Tables Re-\nlated to Radiation Emerging from a Planetary Atmo-\nsphere with Rayleigh Scattering (Univ. of Calif.\nPress, Berkeley, 1960).\n62. Davison, B. Neutron Transport Theory (Clarendon Press,\nOxford, 1957). .","VOL. III\n240\nCLASSICAL SOLUTIONS\n67. Dresner, L., \"Isoperimetric and other inequalities in the\ntheory of neutron transport,\" J. Math. Phys., 2, 829\n(1961).\n71. Duntley, S. Q., \"The reduction of apparent contrast by\nthe atmosphere, J. Opt. Soc. Am., 38, 179 (1948)\n82. Duntley, S. Q., and Preisendorfer, R. W., The Visibility\nof Submerged Objects (Final Report N5ori-07864, Visi-\nbility Laboratory, Massachusetts Institute of Tech-\nnology, 31 August 1952).\n88. Elliott, J. P., \"Milne's problem with a point source,\"\nProc. Roy. Soc., A, 228, 424 (1955).\n91. Feigelson, E. M., et al., Calculation of the Brightness\nof Light in the Case of Anisotropic Scattering, Part 1\n(translated from Transactions (Trudy) of the Inst. of\nAtm. Phys. No. 1, Consultants Bureau, Inc., , New York,\n1960)\n104. Halmos, P. R., Introduction to Hilbert Space (Chelsea Pub.\nCo., New York, 1951).\n108. Herman, R., , and Lenoble, J., \"Etude du Regime Asymptotique\ndans une Milieu Diffusant et absorbant,\" Revue d'\nOptique, 43, 555 (1964).\n111. Hopf, E. Mathematical Problems of Radiative Equilibrium\n(Cambridge Univ. Press, 1934).\n117. Ivanoff, A., and Waterman, T. H., \"Elliptical polariza-\ntion of submarine illumination,\" J. Mar. Res., 16, 255\n(1958).\n118. Ivanoff, A. , and Waterman, T. H., \"Factors, mainly depth\nand wavelength, affecting the degree of underwater\npolarization, \" J. Mar. Res., 16, 283 (1958).\n119. Jahnke, E., and Emde, F., Tables of Functions with For-\nmulae and Curves (Dover Publications, Inc., New York,\n1945), 4th ed.\n120. Jeans, J. H., \"The equations of radiative transfer of\nenergy,\" Monthly Not. Roy. Astron. Soc., 78, 28 (1917).\n140. Kolmogorov, A. N., and Fomin S. V., Elements of the Theory\nof Functions and Functional Analysis (Graylock Press,\nRochester, New York, 1957), vol. 1.\n143. Kourganoff, V., with the collaboration of I. W. Busbridge,\nBasic Methods in Transfer Problems (Oxford Univ. Press,\nLondon, 1952).\n155. Lenoble, J., \"Application de la methode de Chandrasekhar\na l'etude,\" Rev. d'Optique, 35, 1 (1956).","BIBLIOGRAPHY\n241\n156. Lenoble, J., \"Etude theorique de la pénétration du\nrayonnement dans les milieux diffusants naturels,\"\nOptica Acta, 4, 1 (1957).\n157. Lenoble, J., \"Theoretical study of transfer of radiation\nin the sea and verification on a reduced model,\" Sym-\nposium on Radiant Energy in the Sea, International\nUnion of Geodesy and Geophysics, Helsinki Meeting,\nAugust 1960 (L'Institute Géographique National, Mono-\ngraph No. 10, Paris, 1961).\n171. Luchka, A. Y., The Method of Averaging Functional Correc-\ntions: Theory and Applications (Academic Press, New\nYork, 1965).\n172. Mark, C., \"The neutron density near a plane surface,\"\nPhys. Rev., 72, 558 (1947).\n175. Meghreblian, R. V., and Holmes, D. K., Reactor Analysis\nMcGraw-Hill, New York, 1960).\n176. Meyer, H. A. (Symposium on) Monte Carlo Methods (Statis-\ntical Lab., University of Florida, March 1954).\n177. Middleton, W. E. K., Vision Through the Atmosphere (Univ.\nof Toronto Press, 1952).\n186. Moon, P., and Spencer, D. E., \"Illumination from a non-\nuniform sky,\" Illum. Eng., 37, 707 (1942).\n211. Preisendorfer, R. W., A Preliminary Investigation of the\nTransient Radiant Flux Problem (Lecture Notes, vol. II,\nVisibility Laboratory, Scripps Inst. of Ocean., Univ-\nversity of California, San Diego, February 1954).\n212. Preisendorfer, R. W., \"Apparent radiance of submerged ob-\njects,\" J. Opt. Soc. Am., 45, 404 (1955).\n235. Preisendorfer, R. W., \"Time-dependent principles of in-\nvariance,\" Proc. Nat. Acad. Sci., 44, 328 (1958).\n236. Preisendorfer, R. W., Temporal Metric Spaces in Radiative\nTransfer Theory (five papers: I. Temporal Semimetrics;\nII. Epoch Times and Characteristic Functions; III.\nCharacteristic Spheroids and Ellipsoids; IV. Temporal\nDiameters; V. Local Temporal Diameters) (Scripps Inst.\nof Ocean. Refs.: I, 59-2; II, 59-7; III, 59-10; IV,\n59-17; V, 59-18; University of California, San Diego,\n1959).\n237. Preisendorfer, R. W., A Study of Light Storage Phenomena\nin Scattering Media (Scripps Inst. of Ocean. Ref.\n59-12, University of California, San Diego, 1959)\n239. Preisendorfer, R. W., Radiance Bounds (Scripps Inst. of\nOcean. Ref. 59-20, University of California, San Diego,\n1959).","242\nCLASSICAL SOLUTIONS\nVOL. III\n250. Preisendorfer, R. W., \"A model for radiance distributions,\"\nin Physical Aspects of Light in the Sea (Univ. of\nHawaii Press, 1964), J. E. Tyler, ed.\n251. Preisendorfer, R. W., Radiative Transfer on Discrete Spaces\n(Pergamon Press, New York, 1965).\n259. Redheffer, R., \"On the relation of transmission-line theory\nto scattering and transfer,\" J. Math. and Phys., 41, 1\n(1962).\n279. Schuster, A., \"Radiation through a foggy atmosphere,\" As-\ntrophys. J., 21, 1 (1905).\n281. Schwarzschild, K., \"Ueber das gleichgewicht das sonnen-\natmosphäre,\" Gesell. Wiss. Gottingen, Nachr. Math.-phys.\nKlasse, p. 41, (1906)\n282. Schwarzschild, K., \"Uber diffusion und absorption in der\nsonnenatmosphäre, \" Sitzungsberichte der Königlich\nPreussischen Akad. der Wiss., p. 1183 (Jan.-Jun. 1914).\n288. Sobolev, V., A Treatise on Radiative Transfer (Van Nos-\ntrand, New York, 1963).\n289. Sommerfeld, A., Partial Differential Equations in Physics\n(Academic Press, New York, 1949).\n298. Tyler, J. E. , \"Radiance distribution as a function of\ndepth in an underwater environment,\" Bull. Scripps\nInst. Ocean., 7, 363 (1960).\n312. Want, Alan Ping-I, Scattering Processes (Ph D Math. Thesis,\nUniversity of California, Loa Angeles, 1966).\n314. Weinberg, A. M., and Wigner, E. P., The Physical Theory of\nNeutron Chain Reactors (Univ. of Chicago Press, 1958).\n318. Whittaker, E. T., and Watson, G. N., A Course of Modern\nAnalysis (Cambridge Univ. Press, 1952), 4th ed.\n319. Wick, G. C., 'Uber ebene diffusionsprobleme,\" Z. f. Phys.,\n121, 702 (1943)\n323. Wilf, H. S. , \"Numerical integration of the transport\nequation with no angular truncation,\" J. Math. Phys.,\n1, 225 (1960)\n325. Yabushita, S. , \"Tschebyscheff polynomial approximation\nmethod of the neutron transport equation,\" J. Math.\nPhys., 2, 543 (1961).","INDEX\n243\nAbsorbed radiant energy, 94\nContraction mapping, 129;\nAbsorption time constant, 83\nprinciple of, 131\nAbstract spherical harmonic\nContraction property, 129\nmethod, 143\nCosine (mean value II), 180\nAddition theorem for spheri-\ncal harmonics, 148, 153\nDecomposed radiometric\nAlbedo for single scatter-\nfunctions, 36\ning (see scattering-\nDiffuse radiometric func-\nattenuation ratio)\ntions, 36; stored energy,\nAlgebraic spherical harmonics,\n123\nDiffusion coefficient (K), 194\n141\nApparent radiance, canonical\nDiffusion equation, scalar,\nrepresentation, 16\n174; wave, 184; tensor,\nAttenuated radiant energy, 94\netc., 184\nAttenuation length, 99, 196\nDiffusion function (D), 174,\nAttenuation time constant, 76\n180, 181\nDiffusion length, 196\nBouguer's work, 1\nDiffusion processes, a short\nBounds, on radiance, 47\nlist, 184\nDiffusion theory, three ap-\nCanonical equations, sense of\nproaches, 172\nthe term, 1; classical, 9;\nDimensionless forms of radi-\nexperimental verification,\nant energy fields, 97\n13; general media, 15; strat- Directly observable, radiant\nified media, 18; polarized\nenergy, equation of trans-\nradiance, 21; abstract ver-\nfer, 81\nDiscrete sources in diffusion\nsions, 24\nCanonical representation of\ntheory, 202\napparent radiance, 16; of\nabstract functions, 27\nE (epsilon) function, 222\nCauchy sequence, 130\nElectric circuit analogy\nCharacteristic ellipsoid, 66;\n(with an optical medium),\nspheroid, 68\n77, 123\nClassical diffusion theory,\nElsewhere (in space-time),\n134; basic diffusion equa-\n53\ntion, 175; approaches via\nEquation of transfer, n-ary\nhigher order approximations,\nradiance, 36; unscattered\nradiance, 37; diffuse radi-\n183; hierarchy of processes,\nance, 37; path function, 38;\n184; plane-parallel solu-\ntions, 193; spherical (point)\nnatural solution, 43, 127;\nsolutions, 200; discrete\nfor optical ringing, 56;\nsolved symbolically, 65;\n(extended) solutions, 203;\ncontinuous (extended) solu-\nresidual radiant energy, 76;\nn-ary radiant energy, 80;\ntions, 206; primary sources,\ndirectly observable radiant\n207; for higher order scat-\ntered scalar irradiance, 213;\nenergy, 81; dimensionless\ntime dependent, 214\n(for radiant energy), 97;\nClassical spherical harmonic\nscalar irradiance (diffu-\nsion equation), 175; scat-\nmethod, plane-parallel\ntered radiance, 209; scat-\nmedia, 158\nComplete metric space, 131\ntered scalar irradiance,\nCompleteness property of\n210, 213\nspherical harmonics, 142,153 Equilibrium radiance, 6\nCone (in space time), 53\nEquivalence classes of func-\nContinuous sources in diffu-\ntions, 128\nsion theory, 206","INDEX\nVOL. III\n244\nn-ary radiometric concepts,\nExact diffusion theory, 134;\n31; radiance, 33; scalar\nbasic equation, 190, 192, 218\nirradiance, 34; radiant\ninfinite medium with point\nenergy, 34; general, 35;\nsource, 219; infinite medium\ncanonical equations for\nwith arbitrary sources, 225;\nnatural closed forms for\nscalar irradiance, 226; semi-\nradiant energy, 86; time\ninfinite medium with boundary\ndependent properties, 89;\npoint source, with internal\ndimensionless forms, 97\npoint source, 228, 233; on\nNatural solution, for radi-\nthe Elliot functional rela-\nance, 42; truncated, 45;\ntions, 236\ntime-dependent, 58; sym-\nExponential property of dif-\nbolic integration, 65;\nfusion field (plane-parallel\nfor directly observable\ncase), 194\nradiant energy, 82; time\ndependent properties, 90;\nFick's law, 174\ndimensionless forms, 97;\nFinite recurrence property,\noperator-theoretic basis,\n147, 154\n127; for scalar irradiance,\nFirst order scattered radi-\nance, equation for, 41\n191\nNormal space (0 < p < 1), 103\nFourier transform, of exact\ndiffusion equation, 192\nOperator-theoretic basis for\nFunctional relations for\nnatural solution, 127\nfc, fo, in exact diffusion\nOperators, R (path function),\ntheory, 236\n32; T (path radiance), 32;\nFuture (in space-time), 53\nS (radiance) 33; time\ndependent, 68; contraction,\nGlobal approximations for\n129; U (scalar irradiance)\nradiance, higher order, 117,\n188; V (= TU), 188\n119\nOptical length, 220\nOptical medium, transparent,\nInelastic scatter, 5\n3; absorbing, 3; fundamental\nInequality for K,a, 195\n5; electric circuit analogy,\nIntegral equation for scalar\n77; as a metric space, 132\nirradiance, 189\nOptical reverberation case, 86\nIrradiance, vector, via spher-\nOptical ringing problem, one-\nspherical harmonics, 177;\ndimensional, 49; three-\nscalar, via higher order\ndimensional, 66\nscattering, 213 (see also\nOptical volume, 220\nscalar irradiance, vector\nirradiance)\nPast (in space-time), 53\nPath function, equation of\nK (kappa) for classical dif-\ntransfer, 38\nfusion theory, 194;\nPath radiance, first order\nKo dimensionless form, 221\nform, 11\nK-function, general, 15\nPoint source case, in clas-\nKa, 188; Kk, 214; KE, 221\nsical theory, 198; in exact\nKoschmieder's equation, 5\ndiffusion theory, 219\nPolarized radiance, canonical\nLight field, time dependent,\nrepresentation, 19\n49\nPrimary radiance, equation\nLight storage phenomena in\nfor, 41\nnatural optical media, 121\nPrimary scattered flux as\nsource flux, 207\nMetric, supremum, 129;\nPurely absorbing medium, 31\nradio-, 128\nMetric space, complete, 127,\n131","INDEX\n245\nRadiance, in transparent media, Scatter processes, inelastic\n2; in absorbing media, 3;\nor transpectral, 5; single,\nequilibrium, 6; maximum\n10\nnatural waters), 12; trans-\nScatter time constant, 81\nmittance, 14; polarized, 19;\nScattered flux, higher order,\nresidual (reduced, unscat-\n211\ntered), 31; n-ary, 33; nat-\nScattered radiance, equation\nural solution for, 42; bounds,\nof transfer, 209\n47; global approximations,\nScattered radiant energy, 94\n117, 119; distribution in-\nScattered scalar irradiance,\ndiffusion theory, 181, 197,\nequation of transfer, 210\n201\nScattering-attenuation ratio,\nRadiant energy, n-ary, 34;\n10, 47\nresidual representation, 79;\nScattering-order - decomposition,\nequation of transfer for\n30\nn-ary, 81; natural closed\n\"Seeliger's formula\", 132\nform representation, 86;\nSimple model, for polarized\noptical reverberation case,\nlight fields, 21\n86; standard growth and\nSpace-time diagrams, 51, etseq.\ndecay case, 87; time depen-\nSpherical (point source) dif-\ndent properties, 89; scat-\nfusion field, 200\ntered, absorbed, attenuated,\nSpherical harmonic method, 134;\n93; stored, 123; time de-\nbases, 135; motivating argu-\npendent (check), 216\nment, summarized, 140; alge-\nRadiant flux, net inward, 76;\nbraic setting, 141; complete-\nsource, 76; net n-ary, 80\nness property, 142, 153; ab -\nRadiative transfer analogues,\nstract, 143; finite abstract\n77, 133\nforms, 147, 149; classical\nRadiative transfer theory, on\nmethod, general media, 149;\na metric space, 132\nfinite recurrence property,\nRadiometric (as a metric), 128\n154; general differential\nRadiometric functions, general\nequations, 157; classical\nn-ary, 35; diffuse, decom-\nmethod, plane-parallel media,\nposed, 36\n158; truncated solution pro-\nReflectance, in diffusion\ncedure, 163\ntheory, 198, 202\nStandard growth and decay case,\nRelative error in radiance\n(for n-ary radiant energy),\ncomputations, 48\n87\nRelative time, 99\nStorage capacity (of an opti-\nResidual, radiance, 31; trans-\ncal medium), 123\nfer equation, 74; radiant\nStratified media, canonical\nenergy, 79\nequation for radiance, 18\nReverberation, optical, 49\nSupremum metric, 129\nSymbolic integration (term by\nScalar diffusion equation, 174;\nterm for natural solution),\nhigher order form, 213\n65\nScalar irradiance, exponential\nform, 11; n-ary, 34, 217;\nTelegrapher's equation, 185\nequation of transfer (diffu-\nTensor diffusion equation,\nsion equation), 175; inte-\n184\ngral equation, 189; scat-\nTime constant, attenuation,\ntered (equation formula)\n76; scattering, 81; ab-\n210, 218; higher order (equa-\nsorption, 82; dimensionless\ntion formula), 213; integral\nforms, 100; for n-ary radi-\nform, 214; time dependent\nant energy, 109; general\nn-ary (diffusion equation),\ndiscusion, 114\n216; exact diffusion theory,\nTime dependent light field,\n226\n49","VOL. III\nINDEX\n246\nUnit source condition, 220\nTime dependent n-ary radiant\nUnscattered radiance, 31;\nenergy field, properties,\nequation of transfer, 37\n89\nTime dependent operators, 68\nVector irradiance, via\nspherical harmonics, 177;\nTransmittance, radiance, 14,\nin classical diffusion\n17\ntheory, 198, 201, 207;\nspectral scatter, 5\n1\nscattered form, 210; n-ary\nT1 'sport (transfer) equations,\nesidual radiant energy, 76;\n217\nVolume absorption function,\nn-ary radiant energy, 80;\ndirectly observable radiant\n4\nVolume total scattering\nenergy, 81\nfunction, 4\nTrue absorption, 5\nTruncated natural solution, 45;\nWave diffusion equation,\ntime dependent, 69\n185\nTruncated spherical harmonic\nWorld region, 52\nmethod, 163\nU.S. GOVERNMENT PRINTING OFFICE: 1976-677-883/ 38 REGION NO. 8"]}