Matrix Operator Theory of Radiative Transfer. 1: Rayleigh...

Matrix Operator Theory of Radiative Transfer.1: Rayleigh Scattering

Gilbert N. Plass, George W. Kattawar, and Frances E. Catchings

An entirely rigorous method for the solution of the equations for radiative transfer based on the matrixoperator theory is reviewed. The advantages of the present method are: (1) all orders of the reflectionand transmission matrices are calculated at once; (2) layers of any thickness may be combined, so that arealistic model of the atmosphere can be developed from any arbitrary number of layers, each with differ-ent properties and thicknesses; (3) calculations can readily be made for large optical depths and withhighly anisotropic phase functions; (4) results are obtained for any desired value of the surface albedo in-cluding the value unity and for a large number of polar and azimuthal angles including the polar angle 0= 0; (5) all fundamental equations can be interpreted immediately in terms of the physical interactionsappropriate to the problem; (6) both upward and downward radiance can be calculated at interior pointsfrom relatively simple expressions. Both the general theory and its history together with the method ofcalculation are discussed. As a first example of the method numerous curves are given for both the re-flected and transmitted radiance for Rayleigh scattering from a homogeneous layer for a range of opticalthicknesses from 0.0019 to 4096, surface albedo A = 0, 0.2, and 1, and cosine of solar zenith angle u = 1,0.5397, and 0.1882. It is shown that the matrix operator approach contains the doubling method as aspecial case.

1. Introduction

The radiation field arising from multiple scatteredphotons in a planetary atmosphere is of great practi-cal interest. Elegant and elaborate mathematicalsolutions have been developed for isotropic and Ray-leigh scattering by Chandrasekhar,' Sekera,2' 3 Kour-ganoff,4 and others. Numerical values for the vari-ous parameters of the diffuse radiation from a plane-tary atmosphere with Rayleigh scattering have beengiven in a set of tables published by Coulson et al.5

These tables are limited to optical thicknesses ofunity or less. Numerical results for Rayleigh scat-tering have also been presented by Herman andBrowning,6 Dave and Furukawa, 7 Kahle,8' 9 Howelland Jacobowitz,10 and Fymat and Abhyankar,"among others. Among these authors only Dave andFurukawa7 and Kahle9 give radiance values for anoptical thickness as large as 10. Most of the pub-lished results are for a surface albedo of zero.

More realistic problems for planetary atmospheresthat take account of the highly anisotropic scatteringfrom aerosols as well as the variations in the atmo-spheric parameters with height have not been suc-

The authors are with the Physics Department, Texas A&M Uni-versity, College Station, Texas 77843,

Received 24 May 1972.

cessfully solved in terms of standard mathematicalfunctions. Various numerical techniques have alsobeen proposed to calculate the radiance for aniso-tropic scattering. Some of these methods, such asthat of Herman et al.,12 appear to be applicable onlyfor small optical thicknesses. Important resultsusing matrix or doubling methods have been givenby Twomey et al.,13 Dave and Gazdag,14 Dave,15Hansen,' 6-' 9 and Hansen and Pollack.20 As inter-esting as they are, these results are limited by vari-ous factors. Those of Twomey et al.13 use a phasefunction that is averaged over a range of angles.Most of these authors use a doubling method that isapplicable to homogeneous layers but has not beenapplied as yet to the real atmosphere where variousparameters vary with height. Furthermore in mostof these methods, all orders of the reflection andtransmission matrices are calculated by an iterativeprocedure.

The matrix operator theory has several importantadvantages over these other methods. The historicaldevelopment of the method is quite interesting sincemany researchers in apparently diverse fields havecontributed to it over a period of many years. Nosingle individual can be credited as the sole origina-tor of the method as it presently exists. The earliestreference to scattering and transmission in multi-layered media dates back to Stokes2' in which heconsidered transmission and reflection through a

314 APPLIED OPTICS / Vol. 12, No. 2 / February 1973

stack of identical glass plates. By noting the factthat the resulting reflection and transmission for thestack were invariant with respect to the particularchoice of the combinations of plates, Stokes obtainedcertain commutativity relations. The continuousproblem was first considered by McClelland22 andlater by Schmidt.23 Schmidt's method is the ear-liest example of the invariant imbedding methodwhich has been actively pursued by Bellman andcolleagues (extensive references to their work aregiven by Bellman et al. 2 4 ). The first attempt toapply these ideas to radiative transfer theory wasmade by Ambarzumian2 5 where he obtained anequation for reflection from a semi-infinite atmo-sphere by noting that the reflection operator is leftunchanged upon addition of a new layer. Chandra-sekharl used elegant mathematical techniques to ex-pand and generalize Ambarzumian's analysis andthus to obtain reflection and transmission operatorsfor a finite layer.

During the period 1944-1949 Redheffer, who wasstudying microwave transmission problems, con-structed an algebra for the reflection and transmis-sion operators that contained the Stokes commuta-tivity relations, the Ambarzumian invariance princi-ple, and the general functional equations encompass-ing a variety of transfer procedures. The algebra ofthese operators is excellently summarized by Redhef-fer.26 Preisendorfer27 used the concepts of reflectionand transmission operators to make significant con-tributions to the area of radiative transfer. Van deHulst 2 8 was the first to formulate a set of doublingequations for reflection and transmission operatorswhich have proven to be very versatile. The matrixmethod used by Twomey et al. 2 9 is essentially thesame as that of Van de Hulst. More recently Grantand Hunt,30 utilizing the algebra developed byRedheffer and the radiative transfer methods devel-oped by Preisendorfer, have rederived the matrixtheory previously published by others but haveadded a term which represents the contribution frominternal sources. In particular, the fundamentalequation, which relates the transmission and reflec-tion operators for a combined layer to those of itsparts, Eq. (3.5) of Grant and Hunt,30 was originallyderived by Redheffer2 6 [see Eq. (20)] including theconcept of the star semigroup. The infinitesimalgenerators and differential equations for this prob-lem were derived by Redheffer2 6 [see Eqs. (40)-(53)]and are essentially the same as those repeated byGrant and Hunt30 [see Eqs. (4.1)-(4.12)]. Theirtreatment of internal fluxes is unnecessarily compli-cated [see Eqs. (5.1)-(5.13)]; a much simpler treat-ment is given by Bellman 3 l [see Eqs. (3)-(4) on p.348 and Eqs. (5)-(6) of this paper]. In a secondpaper Grant and Hunt3 2 show the nonnegativity ofthe reflection and transmission operators whichform a monotone sequence, a subject alreadycovered by Bellman3l (see pages 346-347 and Exer-cise 5).

The main results of matrix operator theory arepresented in Sec. II of this paper. We believe thatthe derivation given here is particularly'simple andcan be interpreted in meaningful physical terms ateach step. No claim to originality is made for thissection as the main results have all appeared else-where in the literature. However, these diverse re-sults have not previously been collected in one placeso that the external and internal radiances can bothbe calculated from simple expressions with directphysical meaning. The basic simplicity of the ma-trix operator theory has not been apparent frommost previous interpretations. The basic equationsare rederived in Sec. II [Eqs. (7)-(9)] in order todemonstrate the ease with which they can be ob-tained. The numerical techniques that are used inthe solution of these equations are discussed in Sec.III. An appendix shows that the doubling equationsare a special case of a more general algorithm. Afterpresentation of the general theory and the method ofnumerical calculation, results are given for Rayleighscattering from a homogeneous layer. Despite thenumerous articles that have been written on Ray-leigh scattering, there is much new data in our fig-ures, particularly for optical thicknesses greater thanunity and for surface albedos different from zero.

The particularly simple form of the matrix opera-tor theory that is presented here is especially de-signed for application to physical problems of radia-tive transfer. This is an entirely rigorous method forthe solution of the equations of radiative transfer.The theory as given here has the following impor-tant advantages (many of which apply equally wellto previously published versions of the theory as wehave already discussed): (1) all orders of the reflec-tion and transmission matrices are calculated atonce with a corresponding reduction in computertime over methods involving iterations; (2) layers ofany thickness may be combined, so that a realisticmodel of the atmosphere may be developed from anyarbitrary number of layers of any predeterminedthicknesses, each with different properties; (3) calcu-lations can readily be made for large optical depthsand with highly anisotropic phase functions; (4) re-sults are obtained for any desired value of the sur-face albedo including the value unity as well as forany polar angle that corresponds to one of the setgiven by the Lobatto integration scheme for thenumber of integration points chosen (the polar angle0 = 0 is always included in the set); (5) all funda-mental equations can be interpreted immediately interms of the physical interactions appropriate to theproblem; (6) both the upward and downward ra-diance can be calculated at interior points from rela-tively simple expressions.

I. Matrix Operator Theory

In a plane parallel medium all properties dependon a single spatial coordinate x. For conveniencethe position within the medium is denoted by the

February 1973 / Vol. 12, No. 2 / APPLIED OPTICS 315

I+(To), II = I (), 1+ = I+(Ti), J01+ = J+(TO, Ti),andJ1o = J (Ti, To)-

ThusI+ = toIo+ + r1oI, + Jo+

Io, = rI 0+ + tl- + J10-,(2a)

(2b)

where to, = t(To, T) is a diffuse transmission opera-tor and r = r(To, Ti) is a diffuse reflection operator.For homogeneous layers roi = r and toi = tio, butthese relations no longer hold in general for inhomo-geneous layers. The operators r and t are I x I ma-trices that multiply the column vectors I.

The problem treated in this section is the deriva-tion of the expression for the diffuse reflection andtransmission operators for a combined layer (from To

to T2) from the known operators for two separatelayers (from To to T, and from T1 to T2). The result-ing expression is valid for the combination of anytwo layers of finite size and different properties. Inthe next section the relation between these operatorsand the local physical properties (single scatteringfunction and single scattering albedo) are derived.

Equations similar to Eq. (2) are valid for theemerging radiance of the layer from T1 to T2

Fig. 1. Graphic representation of matrix operator vectors.

optical depth T, defined as the distance x within themedium divided by the attenuation length. Dividethe medium into any desired number of layers; theboundaries between layers are indicated in orderfrom the top of the medium by the value of the opti-cal depth at the boundary To, Ti, T2. The mediummay be inhomogeneous within any layer (see Fig. 1).

Let I+(T) be the specific radiance at the opticaldepth T for the radiation in the downward directionaccording to the usual definition. In matrix opera-tor theory I+ (T) is a column matrix

I+(TP2)| (T) = . ~~~~~~~~~(1)_I+(T,,U)_,

where I+(T,,Mi) is the downward radiance at the angleOi = cos--Li(O < Il < 1). The azimuthal or 0 de-pendence can be separated out of the equation as isshown in Sec. II, and its inclusion here is unneces-sary. Similarly let I-(T) be a column matrix thatrepresents the upward radiance.

Consider the radiance values of the radiationemerging from a layer whose boundaries are at TO

and Ti. These radiances, I+(T1) and I-(TO), dependlinearly on the incident radiances, I+(TO) and I-(Tj),and the contribution from the sources within thelayer, J+(To, T1) and J-(T1, TO) (the downward ra-diance at Ti and the upward radiance at TO due tosources within the layer). Except when it is neces-sary to emphasize the dependence on T, we use thefollowing concise notation: Io- = I(TO), Io+ =

I2 = tl2I + r21I2 + J12 ,

II = "12I+ + tI 2 J21-,and also for the combined layer from To to T2

(3a)

(3b)

I2 = t2IO + r20I2 + JoW, (4a)

I(- = rij+ + t2OI2 + J20, (4b)

Multiply Eq. (2a) by r 2 from the left, add thisequation to Eq. (3b), and multiply the resultingequation from the left by (E - r2ro) -, where E isthe identity matrix, to obtain

I = (E - r12r10) (t2I2_ + rI2to0I,+ + J21 - + r,,Jo+). (5)

This equation relates the upward radiance at any in-terior point Ti in the layer from TO to T2 with the in-cident upward and downward radiance, I2 and I0+,respectively, at the boundaries and the source func-tions for the layers. The physical interpretation ofthis type of equation is discussed later.

Multiply Eq. (3b) by r from the left, add thisequation to Eq. (2a), and multiply the resultingequation from the left by (E - riOr 2 ) -L to obtain

I+ = (E - ror,2) l(toIo+ + r,,t2II2 + JOI+ + rioJ21 ) (6)

This expression gives the downward radiance at anyinterior point T in the layer from To to T2.

Substitute the expression for Ii+ from Eq. (6) intoEq. (3a). The resulting equation expresses I2+ interms of Io+, I2-, and J 0 2+- A comparison of thecoefficients of these quantities with Eq. (4a) yieldsthe following expressions:

2 = t12(E - rjor,2) to, (7a)

?':N1 = 21 + t 2(E - rorI 2J r10t21,

J2+ = J2+ + t2(E - roro2)(JoI+ + rJ 21j).

(8a)

(9a)

Similarly the substitution of the expression for I,-


from Eq. (5) into Eq. (2b) and comparison of this re-sult with Eq. (4b) yields

t20= t1o(E - r2r 1o)f t21, (7b)

rO2 = ro, + t1o(E - r 2 r 0)' r2t0 l, (8b)

J20- = J + to(E - r2r10)(J21 + rlJoi). (9b)The physical meaning of these equations is found

if a formal expansion of the quantity ( - ror12)-iis made

(E - rri,) = (rwor.)0 (10)k-0

Substifution of this expression in Eq. (8a), for exam-ple, yields

r2 = r2 + t 2 Z(r,,r 12)0

r0Ot20. (11)0-0

According to Eq. (4a), r20 gives the contribution tothe downward radiance at the lower boundary, I2+,

from an incident upward radiance 12-. The contri-bution from the term = j in Eq. (11) correspondsto radiation that hs been diffusely reflected fromthe layer (To, Ti) j times. Equation (11) states thatthe radiation reflected from the layer (To, T2) is equalto the radiation reflected directly from-the layer (Ti,T2), r2l, to which is added the radiation that under-goes diffuse transmission through the lower layer,t 2 l, times the diffuse reflection from the upper layer,rio, times j diffuse reflections from the lower andthen the upper layer, (rlorl2)j, times the diffusetransmission through the lower layer, t1 2 .

The physical meaning of each of the other expres-sions for the diffuse transmittance, reflectance, andemittance is obtained in the same manner. One ad-vantage of the matrix operator method is that eachquantity in the final equations has a direct physicalmeaning. The key results are given by Eqs. (7)-(9),which give expressions for the diffuse transmittance,reflectance, and emittance for the combination ofany two atmospheric layers of arbitrary thickness.These equations were derived by Redheffer26 with-out the source term, which was added by Grant andHunt.30 The upward and downward diffuse, ra-diance at the boundary of the combined layer is ob-tained from Eq. (4).

The diffuse radiance at any point within a layer isgiven by Eqs. (5) and (6), an expression appreciablysimpler than that given by Grant and Hunt.3 0

These equations without the source term were givenby Bellman.3' The physical meaning of each termin these equations is obtained by the same methodas discussed in connection with Eq. (8a). For exam-ple, the upward diffuse radiance at the interior pointT, is given by Eq. (5) as: (1) the reflection to everyorder between the layers of contributions from theupward diffuse radiance at the lower boundary r2transmitted through the lower layer (1,2); (2) thedownward diffuse radiance at the upper boundary Totransmitted through the upper layer (0,1) and re-flected from the lower layer (1,2); (3) the upwarddiffuse emission of the layer (1,2); and (4) the down-ward diffuse emission from the upper layer (0,1) re-flected from the lower layer (1,2) as represented by

the first, second, third, and fourth terms of the right-hand bracket in Eq. (5), respectively.

Any reflecting properties of a lower boundary sur-face can easily be included in the formalism. Alayer extending from To to Ti can be combined with alower boundary surface if the diffuse transmissionand reflection operator can be defined. Consider thelower surface as extending from T to T2. Theboundary condition is that I2+ = 12- = 0, i.e., no ra-diation either leaves or is incident on the T2 bounda-ry of the ground. The appropriate transmission andreflection operators are t 2 = t 2 l = 0 and ri 2 = r2l=rG. Thus, it follows from Eq. (2) that

I,-= rl,+ (12)The matrix r is determined for any surface by

calculation of the diffuse radiance reflected at everyangle from a given downward radiance at a particu-lar angle of incidence. The angle of incidence isthen varied through all values considered in theproblem. The albedo of the surface is obtained fromthe ratio of the upward flux to the downward flux atthe surface. The elements of rG in a given columnare all equal for a Lambert surface.11. Numerical Techniques

The theory is extended in this section to includethe case where the radiance depends on both y and0. Fourier analysis is used to decouple the azimuthdependence in the equations. Consider the equationof transfer for the diffuse radiance I(T; Y, 0), i.e.,

( + 1)1(7;9'P = Fexp(-T1/o)W(r)p(T;, (P;,U, 0)

+ f fp(T;, ,P;',')I(T;',p')du'dO', (13)

where F is the incident solar flux, w is the singlescattering albedo, and -1 ' u 1 and 0 < (T) < 1.The phase function p(T; , ; o, o) is subject tothe following normalization condition:

f fp(r;u,P;ju',P')du'dP' = 1.-_ 0

(14)

Since the phase function depends only on the cosineof the angle between the directions (, ) and (g','),

p,;",'= P(HP';p',P)p(-,O; - 'Q') = P(Y,0;p',0') (15)pOyP,;- M',') = p(-os'; p', P') = P(,f- j, 1).

The standard Legendre polynomial expansiontechnique (Chandrasekharl) is used to decouple the1 dependence. Assume that the phase function con-sists of only a finite number of terms N, so that

Np(cosO) = Z p, (q')cosl(O - ').

This equation together with Eq. (14) yields

-Po (gty)du'= (27r)-_ 1

(16)

(17)

for all 0 < , ' 1. Assume that the radiance I has asimilar expansion, namely,

N

I(T, ,U¢) = ZI(T-,,)COSl(0-(P)1=0

(18)


Table 1. Upward Radiance

T = 16,384 T = 4096

1.000000 0.3507 0.3506 0.35040.995740 0.3503 0.3502 0.35010.985743 0.3495 0.3494 0.34930.970098 0.3482 0.3481 0.34800.948896 0.3465 0.3464 0.34630.922259 0.3443 0.3442 0.34400.890339 0.3416 0.3415 0.34130.853320 0.3384 0.3383 0.33810.811413 0.3347 0.3346 0.33440.764858 0.3304 0.3303 0.33020.713921 0.3256 0.3255 0.32540.658896 0.3203 0.3202 0.32010.600097 0.3143 0.3143 0.31410.537860 0.3076 0.3074 0.30740.472542 0.3001 0.3001 0.29990.404518 0.2917 0.2915 0.29140.334177 0.2822 0.2820 0.28190.261921 0.2713 0.2711 0.27110.188166 0.2586 0.2584 0.25830.113323 0.2432 0.2429 0.24290.037850 0.2229 0.2222 0.2221

1=21.D A=0.0

This expansion together with Eqs. (16) and (13) yields

t pd + 1)II(T, ) = W(T)Fexp(-T//o)p(A,Y0)

+ r(1 + 6O1) f pIL, ii )I,(Tg')d'-1

(19)

for I = 0, 1, 2, . . ., N. We have thus reduced Eq.(13) to a system of N + 1 independent equations.Rewriting Eq. (19) in terms of the upward (I-) anddownward (I+) radiance we have

d-+ l)I,+ (T. ) = .()Fexp(-/pO)p,(ppo)

+ w(T)'r(1 + 601) [ Pt (A' ,U) I ( .u )du0

+ fpo( -I')I,-(T,L')d#t']

( dT I)I / (TU) (T)Fexp-Tl )p,( rp)d

+ c, (T) 7rQ1 + 6 [fPI (-AU )Il (,u)du0

(20)

In matrix operator theory the integrals are discre-

J=1.0 R=0.2

+2 + + *4 * ++

xxxxxXXX

XXXXXXX~~~~~c~~c~-

+ -'* + + ++++ ++

A t & A I A A IAA~4

0.40 r----

iF -

IY

IXF

0.10 H

+E

I

D D ) D D ( (DCD u;

-- y --Ty-Y

Y Y Y Y X < X x K u

x*

x

X

x+ * * + + +44+ 4+++44 .

4

4t, 4 4, 4 0 t, 4 ¢, ,,

X

AIn

x

d+

XX X X X X XX X X

+ + +tAA

2E TRU=0.0019-D TRU=0.0039A TRU=0.0156

- + TRU=-0.0625X TRU=0.254,( TRU=1.0

I I I I.0

+ TRU=I.

X rU=IG. 0

Z TRU=64. 0Y TRU=256. 0X TRU=1024..0* TRU=4096.0

I I

0.2 0.4 0.6 0.8 1.0

(b)

Fig. 2. Upward radiance at the top of atmosphere for Rayleigh scattering for Ho = 1 and various values of the optical thickness T as afunction of the cosine of the nadir angle ,u. (a) A 0; (b) A = 0.2. The liimiiting curve for large -r is the same as the last plotted curve

within the width of the symbols.


x

X

,, +4', t

XX X

++

X

+-4.

-

I C1 _2,

LL0¢ r,_a:: 1o-2

Q:

0r :0

10I -3 --

I o-4 0

A

A

A

A

CD

ED (D cD

rr C

X X O00*

ED I , I .n In . E4

I I I II

0.2 0.4 0.6 0.8 1.0 ' 0

(a)

I+ f P,(-,,-

I

I I I I I I

� �

II

ILL

tized by some appropriate quadrature formula. Inall the calculations reported in this article Lobattointegration is used since it allows computations of theradiance at = 1. The quadrature is performedas follows:

ditions of the phase function [see Eq. (15)] that 1'++= F- - and F,+- = F-+. This symmetry leads toreciprocity in the equation of transfer. The opera-tors r and t for an infinitesimal layer can now be ex-pressed as follows (see Redheffer 2 6 ):

fg()du = Zg(yj)cj + R-,_1 j-0

(21)

where the error term Rn vanishes for a polynomial ofdegree 2n - 3. The abscissas yj and weights jwere taken from Stroud and Secrest.33

Let us adopt the following matrix notation for thematrix operator case.

+1

lJ+(T, i,Yj) =

P,+ (T;Pi,,U,) =- L

P Iml

+0I1 I .I .2 . . . IL_

Ito . .+ +

ImO ... Imm X

P112 P11 m

++ ++P 2 . . . P Imw ,

Pillt 112 . . .

P+ +- p+-

p.+ 1i,2o .. *-

Pooo exp(-T/pl) ... P+mexp(-T/M,,)]

Jl (T,pi,gj) =Fw(T) I

P Imjexp(-T/Aj) ;**Plmnlexp(-T/pM,)

P 111 exp(-T/ go) . pim exp(-Ti/g,)

J, (r,p,j) =Fw(r)

Pt, 1 mexp(-T//IA) . . . Pnexp(-T/a

C [ Cjjk]

M = [ojbjk ],

where gi > 0, m = n/2 n is the order of the quadra-ture in Eq. (21)], and each column in the radiancematrix corresponds to a different angle of incidencefor the source.

Equation (20) in this representation becomes

M d + I (T) = Jl (T) + W,(T)T(1 + bol)

X [p (-r)cll(T) + p,(T)cll (T)], (23)

-M d + 17 () = Jjo(T) + W(T)ir(i + bot)

X [PI -(T)CI,(T) + p 7(T)CIj-(T)].

With the equation of transfer expressed in thisform, the infinitesimal generators of the star semi-group are as follows (see Redheffer 2 6 and Grant andHunt3 0 ):

F,/ (r) = M l[E-(T)r(i + 50O)p1/ (T)C1,rl -(T)= M'W (T)T(1 + 6oj)p (T)C,

rl =(T) M w(T)r(1 + 6)plG()c, (24)rl7(T) = M [E - W(T)or( + 6OI)pl (r)C,

1 (T) =M J. (T),

Z, (T)= M J0 (T),

where E denotes the identity matrix.It should be noted that due to the symmetry con-

t= E - r++ ()A1 0 +o (Aio),

to = E -r (Q)Aio +o(Alo),rol= r+(Q)Alo + o(Alo),

r, = r,-+(o)A0 + o(Ai0),

(25)

where To < < T and Ai10 = T - To. 'The nota-tion f(x) = og(x)] is used to mean limf(x)/g(x) = 0,where the sense of the limit can be determined fromthe context. We also have the relations

2; = 0 + (t)oA1 + O(Aoo), (26),

01 = 21 ()A 0 + O(A00).

Since the operators for an infinitesimal layer havebeen defined, the star product algorithm [see Eqs.(7a)-(9b)] can be used to build up a layer of any de-sired thickness.

In order to insure that the matrices in Eq. (25) arenonnegative, the following condition must be satis-fied (see Grant and Hunt 30 )

(22)

(27)o < (T, -To)< minjyi/[1-/w(T)].pi

A value of 2 -15 for zAlo is chosen in order to insurethat the truncation error is small enough for the sin-gle scattering approximation to hold. Thus whenthe thickness of the layer is doubled each time, avalue of T = 1 is reached in fifteen steps.

A. Computational Aspects

The stability of the matrix operator algorithm forlarge optical depths depends on careful attention toseveral points. The first of these involves the nor-malization of the phase function [see Eq. (17)]. Atabular phase function is used to compute the coeffi-cient'p(Ati, Aj1 ) in Eq. (16) in the following way:

=1 10-0Po(yoyj) jVEp(igk0),

2 N-1Pi He 10= N Y_ P(ui,uj, k0)cos(lk0), 0,

N0k0(28)

where = 2r/N; linear interpolation is used to eval-uate the phase function at intermediate points. Foreach pii and lij the number of I terms needed is foundby successively adding terms in the series expansionuntil the original phase function has been approxi-mated to the desired accuracy. If pO(-i, Aj) comput-ed in this manner is substituted into Eq. (17), asmall error in the normalization is found in general.Furthermore the normalization constant is a func-tion of A, i.e.,

e, = Zp o(yjiAi)ci -(27T)0,j = 1,2,...,m,,1

(29)

for 0 < -j < 1. A correction matrix is computedfrom the errors Ej. This matrix is made symmetric


1 0=0.1882 R=0.0 0=900

I I I I I I I

lb X X~o*t x X

93 T W W A 6 2 i W"

A + +

I X -

+ X

- X

X

lo-'

10-2

I . I I 10

.0 0.2 0.4 0.6 0.8 1.0 C

.- hA

F

I I

XX

+ Xx

4 , <>

X

+X X

++

A + + +A

+

I -

-t 17XX

+ ++ + +4A

AA

CD

In

'AA

AA

CD

CD

In

in

InCD

(DIn

CD

A A A &A &A&

Winn j D C D )

V In I n w~~~~~~~~~~i [! E ] In~

1.0 0.2 0.14 0.6 0.8 1.0

AlI

(a) (b)

Fig.3. (a)Upwardradianceforyo = andA = (b)Ao = 0.1882,A =0,andP = 900. SeeFig.2forkey.

to preserve reciprocity. To insure that this correc-tion matrix is small a large number of quadraturepoints must be used in the case of a highly aniso-tropic scattering function.

The stability of the doubling equations was testedby calculating the total flux for a conservative scat-tering atmosphere for various optical depths up to T

= 4096. The flux was conserved in all instances to 1part in 105. It is particularly noteworthy that theentire calculation, which includes the computation ofthe reflected and transmitted radiance for opticaldepths up to 4096, twenty-one angles of incidence,and Lambert ground albedos of 0, 0.2, and 1.0 took'only 4 min on a CDC 7600.

B. Lambert Surface

By definition a Lambert surface of albedo A con-verts any radiance distribution impinging upon itinto a uniform distribution; if F is the downwardflux, AF+ is the upward flux. The matrix represen-tation of rG from Eq. (12) in matrix operator theoryis

A .r,;= X, I . . (30)

[Cuj l2Cl .. tM.Cw

The only remaining operator to be defined is XG-which is the source term for the Lambert surface dueto the unscattered incident flux. The matrix rep-resentation of this operator is given by

A Po exp(-r/u,) .* *. eXp(-T/pm) (= . ~~~~~~~~~~~~~~(31)

27rcI , io exp(-T/',).- . *,u exp(-T/P.)

The star product algorithm [Eqs. (7a)-(9b)] can nowbe used directly by treating the surface extendingfrom Tr to T2 as a degenerate slab, i.e., r 2 = r2 = rGand t 2 = t2l = 0.

It is shown in the Appendix that the matrix opera-tor approach contains the doubling method as a spe-cial case.

IV. Upward Radiance for Rayleigh Scattering

The matrix operator method described here hasbeen used to calculate the radiance for Rayleighscattering. A forty-two term Lobatto integrationwas used that is equivalent to fitting the functionwith an 81st degree polynomial. The results wereprinted for each combination of the twenty-onevalues of ,u and O as well as the seven 0 values andthree different values of the surface albedo A. Thesevalues have been checked against all published


0.40

LLJC--)7I0.30

C]CE I

CE3n

.2D

0.20 0

, . I l

PO=l.O R=l.O

I -

m

I I I I

values of the radiance. As an example, the upwardradiance is given in Table I as a function of the co-sine () of the nadir angle for the sun at the zenith(Ao = 1). The upward radiance for a semi-infiniteRayleigh atmosphere is given in the second column.These values were calculated from interpolation ofthe H functions given by Chandrasekhar.' Thevalues in the third and fourth columns were calcu-lated from our matrix operator code for opticaldepths of 16,384 and 4096, respectively.

A different kind of check was made by repeatingour calculation with only a fourteen term Lobatto in-tegration. These results agree at all optical thick-nesses to within 0.2%. For the conservative case fluxwas conserved to 1 part in 106 for optical depthsgreater than 4000. The incoming solar flux is nor-malized to unity in a direction perpendicular to thesolar beam in our calculations.

The upward radiance when the cosine of the solarzenith angle AO = 1 is shown in Fig. 2(a) for a surfacealbedo A = 0. There is limb brightening for smallvalues of the optical depth ; however, when T isgreater than a value near unity, limb darkening oc-curs. In all the figures given here for the upward ra-diance the curve for the largest T value in each case

uo -0. 1882 =O. 0 =0

I I

is a limiting curve that does not change further bymore than the width of the symbol as T increasesfurther.

In all cases the radiance is plotted as a function ofji instead of the zenith or nadir angle 0. The quanti-ty u seems more fundamental than 0 since: (1) theequations of radiative transfer involve only ,u and donot explicitly involve 0; (2) equal infinitesimal inter-vals of , contain the same area of the sky. Radianceplots as a function of A often show a rapid variationof the radiance near the zenith or nadir which is notobserved when the same results are plotted against 0.

The upward radiance when AO = 1 and A = 0.2 isgiven in Fig. 2(b). When T is very small, the upwardradiance is essentially constant, since the observedradiation consists almost entirely of photons thathave been reflected uniformly in all directions by theLambert surface and very few that have been scat-tered in the atmosphere. As T increases from asmall value, limb brightening changes to limb dark-ening when T 1.

The upward radiance when O = 1 and A = 1 isshown in Fig. 3(a). There is relatively little varia-tion with ; the curve changes from nearly a con-stant to one with limb darkening as T increases. For

= 1800

I I I I

x x

+

AA

AA

D A

[D

In (D

InIn

X X

+

A,

In

+

A,

CD

4, "t, , t

X

X XX X $ X +

Xx,X x + + iX XX

-4-

A

A Al

I +

, AI& I& 'am

Al

+ +

A AlA

AA

+

A

InIn0

In0an k ' I n 1

0.0 0.2 0.4 0.6 0.8 1.0 0.8 0.6 0.4

Al

Al& A ~In4

-4

In In-i

1

0.2 0.0

Fig. 4. Upward radiance for uo = 0.1882, A = O and p = 00 and 1800. See Fig. 2 for key.


X

+ X

+

o-'

10-IW

LLW

cc-a: - ;: (

M .

ccCc,

A

In

In

w [ x -In

-rI * , ,I

l

I

I

kb-0. 1882 R=. 2 =00

I I

4, (.) +

if 4,. + _

X 4 , +*

X 4, 14^~ 4,X 10 I.1 t

4,4,

X

x

X

+

4,>

* e

4,.

x

X X w x

t x X

X

+X

XX +

X

+

X ~ X +1A + X Xk X + ACD + ~~~~~X X + D+ ~~X XL A + + XX oX +++ A X 1 j

i m s ] i ~~+ ^ ̂ ^ X + 31 In A A + + + +++ A +

In CD A +~~+ ++A A CA A ~~++++.I ++A A (D n

I I 0. 0.L_ . I , I , I J__ 0,, 00j)0 0.2 0.4A 0.6 0.8 1.0 0.8 0.G 0.4A 0.2 0.0

Fig. 5. Upward radiance forgo = 0.1882, A = 0.2, and 0 = 00 and 1800. See Fig. 2 for key.

large T values the ratio of the upward radiance at thezenith to that near the horizon (last computed point, = 0.03785 or 0 = 87.83°) is 1.576.

The upward radiance when AO = 0.1882 and A = 0is shown in Fig. 4 for 0 = 0 (left-hand side) and 0 =1800 (right-hand side) and in Fig. 3(b) when 0 = 90°.The azimuthal angle 0 is measured from the incidentplane which contains the incident ray and the zenithdirection; the antisolar point occurs when = 180°.One particular curve represents the radiance thatwould be observed in the principal plane from thesolar horizon on the left of the figure to the nadirthrough the antisolar point and back down to theopposite or antisolar horizon. One of the most inter-esting features of these curves is the strong asymme-try near the nadir when T is small; in this region theradiance is always greater for = 180° than the cor-responding value for 0 = 0 and the same , value.

The limiting curve for large T shows a pronouncedlimb brightening at both horizons. The variationwith 0 is relatively small.

The upward radiance when o = 0.1882 and A =0.2 is given in Fig. 5 when = 0 and 180° and inFig. 6(a) when = 900. The radiation when T < 1is nearly independent of the nadir angle. On theother hand the limiting curve for large T is nearly thesame as when A = 0.

The upward radiance when o = 0.5379 or M =0.1882 and A = 1 is shown in Fig. 7 when ;b = 0 and180° and in Fig. 6(b) when = 900. When T < 1the upward radiance is independent of , since al-most all the observed photons have been reflectedfrom the Lambert surface. As T increases more vari-ation develops in the radiance curves for gO = 0.1882than for M = 0.5379. A slight limb brightening de-velops in the principal plane for Mo = 0.5379 as T in-creases, while a slight limb darkening arises at =90°. For = 0.1882 there is a more pronouncedlimb brightening in the principal plane as T increasestogether with an asymmetry in the curve around thenadir. The limb brightening is also present for q =900.

V. Downward Radiance for Rayleigh Scattering

The downward radiance when O = 1 and A =0,0.2, and 1 is shown in Figs. 8(a), 8(b), and 9(a), re-spectively. As T increases from a very small value,the downward radiance increases for a given ,u valueand reaches a maximum value when T 1 for A = 0and 0.2. Limb brightening also changes to limbdarkening near T = 1. When A = 1 the downwardradiance approaches a constant limiting value on thescale of these curves when T - 16; since there is no


0.20

x

I I I I

xw1

4

X

L+

X

X

0.10

LJ.JL)

cc

C:7

IcCc

0rCCCKn

0.01 r

+

+

0= I Bo

I I I

Ii

absorption either in the atmosphere or at the bound-ary surface, a uniform radiation field develops deepin such an atmosphere. A quantity of interest is theratio of the downward radiance at the zenith to thevalue near the horizon (last computed point =0.03785 or 0 = 87.83°) for large values of T; this ratiois 2.713, 2.115, and 1.000 when A = 0, 0.2, and 1, re-spectively. The ratio is independent of the value ofMto in the limit of large T.

The downward radiance in 'the principal planewhen O = 0.5379 is shown in Fig. 10. The samefeatures appear in these curves as in those for AO = 1already discussed. There is an asymmetry in thecurves around the zenith direction when T < 1.

The downward radiance when O = 0.1882 and A= 0 is shown in Fig. 11 when 0 = 00 and 1800 and inFig. 9(b) when = 90°. Because of the low solarangle, the maximum value of the downward radiancefor M values near the horizon now occurs for values ofthe optical thickness appreciably less than unity.

The downward radiance when O = 0.1882 and A= 0.2 is shown in Fig. 12(a) for = 900. Thesecurves as well as those at other 0 angles are verysimilar to the corresponding curves for A = 0. Thedownward radiance is not sensitive to the value ofthe surface albedo in this range, since a photon re-

PO=0.1882 R=0.2 =900

flected from the surface into an upward directionmust be scattered another time in order to join thedownward stream of radiation.

The downward radiance for gO = 0.1882 and A = 1is shown in Fig. 13 for = 0 and 1800 and in Fig.12(b) for = 900. The main difference betweenthese curves and those for solar angles nearer the ze-nith is that the downward radiance near the horizonfor M = 0.1882 is appreciable greater than its limit-ing value for large T over a range of T values from ap-proximately 0.02 to 1. For T values larger thanunity the radiance approaches a constant limitingvalue independent of M. The curves for T < 1 havethe same shape as those for A = 0 and 0.2, but theactual radiance values are greater.

VI. Conclusion

A simplified physical version of the matrix opera-tor theory of radiative transfer has been presentedhere. This theory has relatively simple expressionsfor the radiance at any depth in a plane parallel at-mosphere with each term representing a physical in-teraction appropriate to the problem. No iterationprocedures are used, but rather all orders of the re-flection and transmission matrices are calculated atonce. This theory can be used when the atmospher-

PeO0.5379 R=1.0 0=900

.-1--------T ��-1 7 '� I

4

X

X

'ZI ++

<>

4,

X4,

4,

+ + +4s

4,

XX

X

+I

++

A

+ +

XX

XX

0.20

0.10

X XX

+ + + +

i 0 I . . I I I 11.0 0.2 0.4 0. 6 0.8 1.0 0.03

0.

I l I I I I I

M In$~ WT WWTT

- "C=0.1882 R=1.0 O=900

+

- +§+~xA + X ~A + + +

A + X * ~ ~ ~ +XX0

I , I I I I

.0 0.2 0.4 0.6 0.8 1.0

A-I

(a) (b)

Fig. 6. (a) Upward radiance forgo = 0.1882, A = 0.2, and p = 900; (b) go = 0.5379 and 0.1882, A = 1, and o = 90°. See Fig. 2 for key.


4

A x t 0

- X+ ,X

10-1

L)

Cal

CE:M

ci

ci-cc:

m)

10-2C

+-A

ACD

In A

A(D A A

-I-- - -I- I -1 I

I

II

.

u=0.5379 R=I.0 =0

I I _ I _ F -X

*^^il*WW$WeI i

; IIM W Wa

--TV--- ' ~' t ~~~F- ~ -7

8~~ t t t E t + XX X XX X *T'+ X X+ + + ~ dj i dilb ± t In In In In

O=1800

o It, + + a ++6 A

+ i4,

4,

+ A

+ J,+AI

AA In'.

M [D M i~~~~~~~~.

0.0 0 i ,...1... _ I 1 . .1......t.-.. _..0.0 0.2 0.4 0.1

.8 1.0 0.8 ____ 0 .6 I0.8a 1.0 0.8a 0. 6 0. 4 0.2 0.0

AU

Fig. 7. Upward radiance forgo = 0.5379 and 0. 1882, A = 1, and 0 = 0° and 1800. See Fig. 2 for key.

ic parameters vary with height, since layers of anydesired thicknesses may be combined. The radiancemay readily be calculated for large optical depths,highly anisotropic phase functions, for all values ofthe surface albedo including A = 1 and for a numberof polar angles including = 1. As a first applica-tion of the method detailed results have been givenfor Rayleigh scattering from a homogeneous layer.The upward and downward radiance is given foroptical thicknesses T from 0.0019 to 4096, for surfacealbedo A = 0, 0.2, and 1, and for cosine of solar ze-nith angle M = 1, 0.5379, and 0. 1882.

This work was supported in part by grant NGR44-001-117 from the National Aeronautics and SpaceAdministration.

where the constant a involves the single scatteringalbedo and a constant depending on the Fourier de-composition that is irrelevant to this treatment.

Define the new operators S A and TA by

So _ p cA;

T, _ p++aA,

and

exp(-A/p1) 0.. .00 = exp(-A/j 2 ). .0 *

L0 0. .. exp(-A/p.)j

Equation (Al) canform:

(A2)

now be rewritten in the following

Appendix: Doubling Equations: A Special Case

In this Appendix the relationship between thedoubling equations of van de Hulst34 and the matrixoperator theory is investigated.

The infinitesimal operators ro, and to can be writ-ten from Eqs. (24) and (25) in matrix form as followsfor a layer of infinitesimal thickness

n = M-'SAc,

t,,, = F + M-'Tac, (A3)

where we have used the fact that E - M- 1 A is theexpansion to the first order in A of the matrix F .

Let us now use Eq. (8b) to couple two identical in-finitesimal layers. In terms of SA' and TA,

r,,2 = M-'SAc + (FA + M-'TAc)[E-(M' SAc)2 ]-'

r, = M-'p+-caA, X M-'Sc(F, + M-'Tc).

t,=E-M-'A + M-'p+caA, (Al) M-1, c, and F are diagonal matrices and com-


' I

* L

. +

II4

0.30

Lii

ccrC])

CEO. 10cc

cc¢1

CL

i-b0.1882 R=1.0 o=00

4,

4,

L + 4,

7

a]

+A A

in InMn+

(A4)

r

o-1800

if T T mm T ifIt, + +

t. 10 t. !t.10-0'01,$.Me

mute with one another; furthermore M-'Sc com-mutes with [E - (M-'S'c) 2 ]-'. Thus Eq. (A4) canbe rewritten after some matrix algebra as follows:r.2 = M-ISA + (Fj + TacM-1 )SA[E -(cM-SA)2 ]-l (Fi + M-cTjA c. (A5)

A comparison of Eqs. (A5) and (A3) suggests that amatrix operator S2A. be defined as follows:

S2g = S + (FA + TcM- t )SA[E-(cM-t S 2 ]- [FAM-cTj]. (A6)

Next let us consider the doubling method of vande Hulst 3 4 as employed by Hansen.1 6 The equationfor the scattering function S'(r, Mu, M.o) for a particu-lar Fourier component is given as follows:

S(2A;1, g,)) = S'(A; gg0) + exp(-T/)Y(T; u,p,)exp(-T/po)

+ exp(-T/,U0)J T'(A;y u, y)Zo (A;W' , dP

+ exp(-T/i) o(A;u')T'(A;u', Aj dyi

+ J T'(A;gg")f((;g",p)T'(4p ,Ao) ,, -d-, (A7)where multiplicative constants involving a particularFourier component have been dropped. Hansen de-fines lo(A; , M') as follows:

-G-lO. R0.0

100

10-1

LLJLU

cr io-2

cr:rac 1-3cc

z2 L

io-

0.2 O A 0. 6 0.3 iLU

;F( A;g,ugu) = 5: S'(A;,u,), n = 0, 1, 2,....i _2n +I

!$ * 4t .*

(A8)

where

Sl'(A;g,g0 ) = S'(A;g,gO)

and

Si'(A;,u,)= f S'(A;w')Si-I (A;,p o)dp'

If the integrals are discretized by some appropriatequadrature with weights c and abscissas i, Eqs.(A7) and (A8) can be rewritten in the following ma-trix form:

S'2,, = S', + (F, + T',AcM-') 2o(F, + M-cT ),

1 = SE +(CM-'SA)2 + (CM-'S,)4 +... ].

(A7a)

(A8a)

It should be noted that the term in brackets on theright-hand side of Eq. (A8a) is merely the power se-ries expansion of [E - (cM-1SA)2 ]-1. Thus Eq.(A7a) can be rewritten in the following concise form:

S'2= S' + (F, + T',cM-)[E-(cM-lS') 2]- (F + M-'cT'A). (A7b)

This is a simpler and more meaningful expression forthe doubling equation than the somewhat cumber-

i-=-.0 R0.2

100

10-1

10-4'

x

'I]

Y

I I I I I I I -.'1 4 oZ XX

X Xx x x

++++ + +

Z Z Z

'AA AAY

Y9Y0

C0I 0

Inj

3( X WKxg

I I ,

A

A

0

009 )K A I . I I I

0X

In

± :

4.W X X X yy'trAA A .

XX0 80cc

0.2 0.4 0.6 0.8 i1O

ll.

(a) (b)

Fig. 8. (a) Downward radiance at bottom of atmosphere for go = 1 and A = 0; (b) go = 1 and A = 0.2. See Fig. 2 for key.


I I I t I I

+1¢ + + ta

-^ *-X X X X Xx x

xx ++

-_ ^ ~ z Z Z + ± Z +z A

Y (D) -In A~~~~~~~I

- 2 ) X . * i. ! x xxXXXe

X

3K~~~~~) 3K 3K )K A W-, I

I I)K

- . -

- -

? X

z zg F ++"

R

M

I

j0=l.U HF=1.U

10-'

10-2

10-3

1o-4

010.2 014 O, 08 i1.W

jo=0.1882 R=0.0 0=900

- I I , I I ,

:+X x x+ X-At ,, ~b * 0 * ,&.

+ 3+ 3K 3 Kx

, + + ± , + + * 4 + +C *±+ ' + + +

_ye ~ ~ ~ ~ ~ ~ ~ ~

-n x x A,x A xxx Ax- In InCD0ZZ ZZZ

- z Z inCD 9 DInY InY CDYV3 I I

- x x X X X x x X X XxXO

* 3 X X *

* I

.0

(a)

0.2 0.4 0.6 0.8 1.0

(b)

Fig. 9. (a) Downward radiance foruo = 1 and A = 1; (b) go = 0.1882, A = O and k = 90°. See Fig. 2 for key.

mj=0.5379 R=0.0 O=00

lo-'

10-2

LL

ccM

CDcc: i0-3

CECDz2 ioI'

0.0

0=180

0.2 0.4 0.6 0.8 1.0 0.8 0.6 0.4 0.2 0.0

AF.

Fig. 10. Downward radiance forgo = 0.5379, A = 0, and tk = QO and 180'. See Fig. 2 for key.


100

10-'

LUJ(2

c

-l

clC jo-2

3

zM 1 0-

- I I I,

-,£ + x x x x x xx xx xxx

-+ X+

A~~~~A~~~~~0 AAAAi

In 0 0In ~0In 0 0

In 0000000. ̂ dA._~~~~~~~~C C) OD o

,, ln~ ,n In , l , l

. . . . . . . . .

I

I

I

E

1.

xbJ-O. 1882 R=0. 0 0.08

I I yIx IXX-4-

-- 7-- -----

+ XA ++~+ **++ 404 ,* + + + *.++ ++

+ * A A A A x x+ +x

In C (D AlCD A~~~x I C AA z ZZz

In I

z zz I s D C C s O 0y El , ! ) D(

Y

X

*

YrT.'Y Y

4,4 X X

.++: X

A A

Z z

CDCD

Iw

X X X XX*XX X X

X X

3K X

X X X X X X X

* A * X X A * X X X K X K

I L I I I I I

++ +~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

X XX X Xx Q+2 R + +0' 0

A -z

Q) 19~~~~~~~:

x x x A ~x x4

ID In

3K %XX X X -3

X_

I I ~ X

0.2 0.4 0.6 0.8 1.0 0.8 0.6 0.4 0.2 0.0

Fig. 11. Downward radiance foruo = 0.1882, A = O and =0 and 1800. See Fig. 2 for key.

some representation of Hansen.' 6 When the dou-bling equation (A7b) is compared with Eq. (A6) ob-tained from the matrix operator method, it is foundthat the two have the same form. This same con-struction can be used to complete the proof by in-duction for the doubling of any two homogeneouslayers of arbitrary optical thickness.

In order to complete the analysis, we must consid-er the operator to2 defined in Eq. (7a). When this iscombined with Eq. (Al), we have

to2 = (F, + M-'Tjc)[E(M- 1Sc)2] ' LF, + M-'TAc]. (A9)

Since

[E (M-'Sc)2l' = E +(M'Sc) 2[E (M-1SC)2 -,

and FA, M-1, and c commute with each other, Eq.(A9) can be rewritten after some matrix algebra as

to2 = F21 + FAM-TAc + M 'TicFA + (M TAC)2

+ (F, + M-lTc)(M-'Sac)2[E - (MSc)2j]l(F, + MTAc). (A9a)

By analogy with Eq. (A3) define to2 as

to2 = F2A + M-'T.ac. (A3a)

When Eq. (A3a) is substituted into Eq. (A9a), it isfound that T2A is

T2, = FjTA + TaFa + TjcM-T, + (F + TcM-')

X SAcM'SjE-(cM-'SA) 2 1'X IFA + M-'cTA]. (A10)

The doubling equation for the transmission opera-tor T2 ' obtained from Hansen'16 can be written as

T,,' = T'FA + FT,' + T'cM- 1 TA' (F, + T~cM-')2XjF, + cM-'Ta]

(All)

where

2;XA = Z Si' n = 1, 2, 3,....i=2n

(A112)

Now Eq. (A12) can be written in a more conciseform than that employed by Hansen if we note that

2;A = SAcM-'SJ'[E (cM -Sj,)2]-I. (Al2a)

We obtain by the substitution of Eq. (Al2a) into Eq.(All):

T21' = T'FA + FT1J + T,'cM-'T,'

+ (FA + TA'cM"')SA'cMS'[E - (cM-'T')2]-l

X (F + cM-'T'). (Alla)

This equation is identical in form to Eq. (A10) ob-tained from the matrix operator theory. Thus thematrix operator theory includes the doubling equa-tions as a special case.


X-A

0

zIK

KX

lo-'

10-2

LL

CD1 0 -3c,

Cc

cc 10-4cc

CDG

0.0

l , , -- 1 r----� -- i--S

l l l

l l

l

--- J

L - - i , l -I II-

)= 1800

I06C

PUo=U. ooc n--I. u

1 n-lT- T ~I I I-*I|

X X

E + x xx

L XX X .* + : + + r A + + ++ -

In A +xx x x~

x x 1 0

x , A -1 10-4-0.0 O.,zZ Z AK~~~ Z Z AAA

In CD CD ) !In CD DCDD

Ex , 1 0 -3L

1.0 0.2 0.11 0. 6 0.8 1.0 0.

(a)

X

'

I I

1r- -1 I I I I I

+ x x x I- x -E

+A

++ +

A+

AA

CD

ID

AA

AA

CD

ED

A

+-+ i

-I

In In I ID~ m

ut*L^

A

QDCD

QD

In CD~in 19 1

0 0 0 . 0 1 --- I 1 .

,0 0.2 0.4 O.ti 0.8 1.0

(b)

Fig. 12. (a) Downward radiance forgo = 0.1882, A = 0.2, and o = 900; (b) o =0.1882, A 1, and k = 0° and 180°. See Fig. 2 for key.

mc=0.1882 R=1.0 0=00

100

10-1

LU

2:CD)

cc 10o-cc:

cca-

n 103

I0-4 1

=180

0.2 0.4 0.6 0.8 1.0 0.8 0.6 0.4 0.2 0.0

Fig. 13. Downward radiance forgo = 0.1882, A = 1, and 0 00 and 180°. See Fig. 2 for key.


lo-,

1-10-2

LLJ

10-3

crcc-

CDcc 10 -4c

CDC

106-C

Mu=U. I bbd H-U. e Q=IJU-

- X X X x xx>�%+

4

In, this Appendix the term doubling equations re-fers to the algorithm proposed by Van de Hulst34

and applied to homogeneous layers, the only case de-scribed so far in the published literature. Thesedoubling equations are shown in the Appendix to bea special case of the matrix operator method. Ofcourse, in view of the equivalence shown here, theVan de Hulst doubling equations can be generalizedto an inhomogeneous atmosphere. If this were done,these new equations would be equivalent to the gen-eral matrix operator method.

References1. S. Chandrasekhar, Radiative Transfer (Oxford U. P., New

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William R. Hutchins, Federal Systems Grou'p of SandersAssociates, Inc.


Matrix Operator Theory of Radiative Transfer. 1: Rayleigh...

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