SOLUTIONS 0? THE EQUATIONS OF'' RADIATIVE TRANSFER

BY AN INVARIANT IMBEDDING APPROACH

APPROVED:

ssor

? , 'tzh.-Minor Professor

Director of the DcoiBrEn tment of Physics

Dep.n of the Graduate School

SOLUTION'S OF' THE FQUAl'TOi-fS Of RADTATIV5 TEffiPER

BY AM INVARIANT TdBBUDTMCr APPROACH

rnr "u»c1 T Q i i UiO lb

Presentee: to the Graduate Council of the

North Texas State Urn vorslty in Partial

Fulfillment of the Requirements

For the Uo^ree of

f-T a orni.-o AT? C r\ ~r i rr. rp

By

Char 1 er 2-1. *\6 ars, S,

* Tj9 YI r, q-" ^

J anus rv, 19 69

TABLE OF CCm'T ivPS

Pase

LIST 0?' TABLES . . iv

LIST OF ILLUSTRATIONS . v

Chapter

I. INTRODUCTION 1

II. REDUCTION OP TNF XATRIX FQUATIGN Gr TRANSFER . if

III. COMPUTATIONAL hETNODS 9

IV. DISCUSSION . . . . . 17

APPENDIX 21

B13L10 G Pi A P IIY 38

13 i

LIST O*1 TABLES

Table Pa, e

I, Corrjsri son of Comm> fcational Results for X- and Y~Functions for Conservative Isotropic Scattering with jx -• 0.50 . 11

IT. The Integral Functions of Ch^ndr&sekhar for |L = 0.500 and u ) 0 = 3 .00 13

III. Co-inarison of Fluxes for Conservative Baylel^h Scattering . . . . 15

IV. Percent Errors for the Vector and Scalar Reflected Intensities . . . . . . . 18

V. Percent Errors for the Vector and Scalar Transmitted Intensities . . , 20

iv

LIST OF ILLUSTRATION

Figure

1 .

3.

5.

6.

7.

8.

10.

11,

12.

13.

Ik.

15.

Reflected Intens ity for - 0.1933i (/ -- 0.0, s:aA - 0 . . . . . . . .

Reflected Intensity for- u 0.5000, 0 = 0.0, and A - 0 . . . . . . . .

Reflected Intensity for u0 =• 0.306?, f = 0.0, and A - 0 . . . . . . . .

k. Reflected Intensity for 0 -- 0.0, and A - 0

0.9-3^3

Reflected Intensity for u.c -- 0.1933-0 = 1.5?, and A = 0 . '

Reflected Intensity for U6 - 0.5000, 0 = 1.57, and A - 0 . . . . . . .

Reflected Intensity for - 0.806?, 0 = 1.57, and A - 0

Reflected intensity for U0 -- 0,98kl 0 = 1.5?, and. A - 0 . . . . .

Transmitted Intensity for U = 0 0 ~ 0.0, and A = 0

Transmitted Intensity for U0 = 0 0 = 0.0, and A - 0 . . . . . .

Transmitted Intensity for ua =. 0 0 =r 0.0, and A - 0 . . . . . .

Transmitted Intensity for p,„ = 0 0 =. 0.0, and A - 0 . .

Transmitted Intensity for 0 - 1.57 ? and A --- 0

Transmitted Intensity for .a = 0 0 = 1. 5'7 , and 0

Transmitted - Intensity for u 0 = 1.57, and A - 0 ' . . .

9

9 •

1 q q ? - ? ^ i

6000

80 6?,

98-11,

1933,

5000,

O.Q £9 I 3

]p £1 £T pi

22

. . . 2 3

. . . 2k-

. . . 25

. . . 26

. . . 27

. . . 28

. . . 29

. . . 30

31

32

33

3k

35

36

L I S T 0 ? TT,U731*EA ' £ 1 0 i f S ( € 0 2 : ? . ^

F i - n a r o

1 6 . I V s n s r i i t f e e c ' I r ) t e n s ' ; t; v f o :

? •- 1 . 5 7 i O r ' j P-rjo. u J J - o = 0 . 9 3 M

37

v i

CiiAVT;7: I

T!'-; ,llrj:")T'\i f f j r

Tiie r;o~.-ier in the i?r;nr.i wn!" Inbedc in'? rmoroaoVi is that

it replaces a system of 1 ineay in he :;ro ifferential equations

hnvX'o?, tr.'O — "noifit boundary con&itJors nlth a system of non-

linoai" in t e f*ro ~d i f f e r cu t i al ecu? tiorjs 'ooesessln:?; a set of

initial conci itions. It in the letter sot of equations which

are ideally suited for numerical calculations. With the

conventiora3 anoroach etr.r;!ovine the X- and Y-functions,

t>roble;ri3 of unicity of the solutions are prevalant as was

shovxi by Chandrase'khar (?) and Hullikin (k) . Bellman et al.

(1) successfully used the invariant imbedd .ing technique for

the solution of problems in radiative transfer in slabs of

finite thickness T'rith inotropic scattering. Ka^ivrada and

Kalaba (3) ha;/e numerically solved the scalar equation of

transfer for an atmosphere vrith a scatter in,a- uhase

function of the forti p(jx) = 1 + j.i» This solution vras

accoT-Dlished by s Fourier expansion of the single scattering

phase function in terms of associated Leprendre polynomials.

Similar expansions for the scattering srtd transrrission

functions yield Fouii er coeffioionts nith no explicit

azimuth dependence, The resulting inte~ro-differcntial

equations have initial conditions v.hich render then! easily

m o l m ' S l a 'rrj ..is-.inj h i •' s e e u r a t e m a n o r i a l

soi'ie'-i-;? ir- ' Jo ' f f ] .^ pruo.lr , \o?i «»jr1 t h e e t i c . A ? ri i i r*

tec-.m i c , r a T i l l on •:_a-?rl f o r s e a t h v ; r i n r a c c o r d i n a t o a

ria vl €i < • • > • . . n s e f u n c t i o n £•;<? H ^ y l e i a h r h a n e m s - t v i r e s u l M n * ? :

i n pp inor-: a s e i n t h e ; ; r - t e r or e c a a t i o n a and t h a i r

c o a r o l c r i t y . I t n h o u l o - a l so b e n o t e d c h a t when t h e s o l u t i o n

co r re r roon- l In - t o a o a r t a i n c r i t i c a l d e w t h i s o b t o i n o c l , t h e

s o L i u / s o ' i s a t a l l vi+ 'e-r-sdn a . te o : o t i c ° l d e p t h s c o n s i s t e n t w i t h

t h e s t e n c r a e u s e d i n t h e i n t e a r a t . i a n a r a a j a o o b t e i r . e ^ ,

Tno r a a u l t i - i a c a l en1! a t i o n a for b o t h t h o s c a l a r and

m a t r i ? ; e a s e i l l u s t r a t e t h e n e c e s s i t y f o r a c c o u n t i n ^ f o ^

p o l a r i z a t i o n e f f e c t s i n t h e t h e o r e t i c a l c a l c u l a . t i o n s of

r e s u} 1; I r. -"c i ^ t e n 8 i t i e b «

CHAPTr.B °<I BLIOG ji:i?r:Y

1 , 1 v.'n-i, H. B . , P . 2 , K a l a , pnd M. C. P r e s b r u d , I t •/•?.;riant Tnbe.id V"- *•"•(•* " . t ' v e T^ane "qf i-p Slr ibn of '• r1 Tbio1\fia'*a^ i-aa/ /o rb . , A' .aorican If,]_5)0viar C o a.a a n y 5 I r c . , 10 • '1.

2 , ChmnlraseV.i».? , 3 . , b?d ia? t i ye ' i ' rerun e r , Hew York , Dover P u b l i c a t i o n a , 19 60 . ~~

3 , K a a i a a b a , a , , and R, 3 , Ks l r> ;a , '' Es t i a a t i o n of L o c a l fir.I.aot^onl.c S c a t t e r i n g P r e o e r t : : or us iv r r i • o a s u r e a e n t s of b r l t i n l v S c a t co red R a d i a t i o n , t ! Tbe j_ourn£ l of °,yp.n ^ it-a t i ?e S ' o e n t r ' o e c o & n d Pa^ in t i vp Trvar'^-Ter. V I I "( 0 c t o b e r 71967")™, 2 9 5 - 3 0 ~ "

k . H a i l l a i n , 'I', W., !,A b on 1 i n e a r I n t e r r e d i f f e r e n t i a l E q u a t i o n i n R a d i a t i v e T r a n s f e r , " The J o u r n e ] of t h e S o c i o in, f o r I n c a s t r i a l Kathe ipe11 c s , ~XI T f~(Jvne7196"5^ 3 8 8 - M O , "

dT (€- \' p V r 2. n

V- = 1 ^ . 0 -

RSDUCTX05 0? TH3 r ATRIAC

KQUATTC:-! 0? T?A>;S?ViH.

The equation of transfer for scattering according to

a phase ma trix can be vrc.it tern in the foil loving form(1) :

— Ht.LL.pl ~ ™ \ - 'i ' "* » J . . o 5

*:here tr? is the incident flur on the upper surface of the

atmosphere in the direction (- e, ) where ©a- cos-t

(04^6il) an d ad e f i nee t h s an A'l e t ha t t h e pi an e containi n

the z-arAs and the incident besu? naives with an arbitrary

x-axis. The x-axis is taken to be in the direction of the

incoains beam so that f s. 0. The matrix ls t h e

single scattering phase matrix for the atmosphere. In the

case of Rayleigh scattering, the single scattering matrix

has the forF.

where

1 1 1

v 3 r ( a, - ~ w 1 ' 1 4- Y-l

1 1 \ o o

ll

5

A -

4 - ^ . C

0

-4.

r ? "/ ... . t- »JSi ^ +f-~- I

^ D 7

\' «*.< f r\ — .•/-. "\ V -fi S**J I

[ c> / /

D

0

0

7. 0

7. cos (. c'o- <p>)

and

C6S ZLkr fa \ -y)£>^ / •-! 4,

u w y; J

p? cosZ[\j>~£•) y?p.sirJ2Lf-$) \

C.&* d C y ~ - pk-susS ?, L o-.'i

• ? > \

Chatidr©sekhar (1) arid Sekera (^) have introduced the

properties of the Fourier components of the r>hase matrix

which aid in the factorization of the safcrix equations.

For the ca.se of the Eaylci^h phaae nnfcrix a Bet of scalar

eqnations was obtained. The resultIns equations and

their reduction may be shorn as follows.

The diffusely reflected intensity T , C K ^ i , OA 0 TC,

is obtained- from the Rcatterinm na';rir by the fol 1 owing

equation:

I ^ V - ^ ~ *u §>A'V)V'?$ ' \ v ^ ~ • Similarly, the diffusely transmitted intensity Ttl? - u ^

^ i) 1 }

CiAp.41 6^^1TC, is obtained from the transmission matrix

as folio /is: «—«• f l/j ~™~«»

= %- M-tUu-3 s /W I

i t:' It, (b) lr | 9 ' i o

where *C is the total ontical thio-'ness of the atmosphere.

6

"Z l'o s t cein ".j and 1.rans;11ssIon r;q triceo In p.

for'L. O. i r CO oriao O liTircle S C 9,1/16 T' 1 n nHsn^1 X

results in the equations

«*•*.»<»«» m £ wWWPw«t0l' V» ^ J W t # i ¥ * W * v 11,^ . where

p C.V0

and

~T~CW , - /r\^

i ^ u w r , \ ^ -- x V - i m o v - i , ! .

i:io soJ uij.CLI of the ocjviyt Ion of tronsfer mav read i 1 y be

obtained by nuriericaly calculations of the resulting scalar

equations for and , 1:=1,2.

The procedure necessary for the explicit equations of

the scalar functions % C , ^ and "\p;y.,^ for lc=l,2 and the

n.-i o r i 7. Gcj ^ j_ oiis for t n o & z iiju t-licil iiid6D6iid €/i t in/iti*! cg s Is

•*-: ~ d.' bCtlj_ D J o€.: 116 1' V, ( 2 j » "i1 fl8 T8S111.t121 -rC 6q UH fc 10'H B f OT

the above functions are:

^vl+ %.* - u)Lt^tlA Se'icj S 1.-V 1St£ j pyp."J

for k-1,2,

. UJ *

ind

H y l i + ^ V

The result I nc equat4 on?; for the azir. uthal independent

coaoonents of anr? 1 tc',y.,0j ° r t

I V * ^ * %A - % u3ltr) I \±l\X.) + 4A^S\t* / " v ' . J f x \ \ / i - » »

* t Ml a) + '-l \ \*\ i ylh S L13; V-", \l 1 f/u"! I ^ ^Vv^s. | A4*4—' / | / | fc> J

and

.. ., V _ N / ^ U . . -%L A/ e5--!-^ ,.M .M-i y ^ j , ^ - % ^ { Y \ r , - % +

W M ^ ' V + V a M c ^ T ^

vrhere

f ^ -lliL-f)

HCp-N' \ L O

rJ>

M(|i) is the transpose of the matrix K(yJ ,

The addition of a Lambert surfrc- at the lower surface

of the atmosphere and the resulting change in reflected and

transrai tted. intensities emerging frori the atit!ost>here ciay be

easily calculated from the previously cor.ou.tec1 functions (1),

CfiAPT£R B1 6LI0G"-l?:?!IY

1 . Chpndras^ch t i r , S . , Rnd 1 p t l v c Tr . - rnnfer . Hew York, Dover Pub l i c ca t i onr;, 19^0 .

2 . Seko.ra, Z, , " Redrc t i of t h e Squa t i o n s of R e l a t i v e T r a n s f e r f o r a P l a n e - P ^ r - a l l o l , P l a n e t a r y Atsrion^here: Pp. r ts I & I I , " The RAND Coi-co r a t i o n (BM-4952-PH,BM~ 50 56-PR) , S a n t a Mon i c a , Cal i f o r n i a .

CWp-'i'KH T1T

co"!iru,.'?.VL,ro'*MJ ;

The goto! ed nor.l.lnj-'-r h'tc yro--dif^erent^ pi enactions

obtained by reduction of the equation of transfer were solved

•usIn?; a co/ahination of numerical Fieans to obtain the greatest

noss ibi e accurao" in caloulat j sy the resultijp; intensities.

The final coo en rere written us Viz double precision arithmetic

on the International Business '.'"e chine 360/50 computer. The

codes are can able of con on tin--"1; results for ! ?rge optical

denths in a relatively short ti^c for an1/ ground albedo ] ess

thai one and err eater than- or ocu.al to zero end any single

scattering slbedo which soy vary Kith optical depth in the

atmosphere, MulliVin (6) has shown that singular solutions

erisb near the desired solutions and that all numerical

integration must be done with hi'hi precision to avoid then.

The integrals with respect to jx ore anororirna.ted by

Gauss Quadrature noraali ?,eo to the interval from zero to one.

A change in the order from seven to nine in the Gaussian

integration results in a chanae of at most one part in

300,000 in the total intensity for a y. and p0 value of 0.500

for all optical depth cor.sic cred. Fine! values obtained

for lar^e optical dxrothc un to a deoth of five were done

with a Gauss 0uadaaturc of order nine so that cosine values

near those c crura ted by Coulson e>t el. ( 5) could be obtained

10

for co-,io1-oo, Thr; irite at ipri vri th recioct to the optl cal

•rlcot-" \p. rs-rrroT'-eo b;' itIm-v Hi-Ilutta inte':ratjon as 9.

st-xi'tinT tn'occdnre and ther r-rlfcchin:-- to the fifth-order

Actar„.s-Bc.shforth rredictor counled with the fifth-order Wanfi-

Koin ten corrector. As s. ter; 1: of th = r.ijn'erieal rrocedrre, the

inte To-dif ferenti?! eauo.t ion.s of Che ndrasekhar ® s X- and Y-

f'unctions for an isotropic phase fur;otion vrere solved.

These results are shovm vn Table I with the numerical results

of Carl?.tedt and MulliVjti (2) and Bellnan et al. (]). The

results of Bellman et al. were calculated. by a sinilar

technique usins invariant imbedding. As can be seen from

this table,the agreement is excellent.

T/J-.T.-Vi T

COKPABTSON OF COiiPUTATIOr'-iL RESULTS FOB X- AI'U) Y-FUMC'l'IOi Jo FOR CONSERVATIVE ISOTROPIC SCATl'EHIMG V/XTlT jlc- 0.50

X P r e s e n t Scheme C a r l s t e 0. t --1 !u] „ 1 i k i n B e l l m a n

0 . 2 V 1 . 2 4 4 7 5 1 . 2 4 4 8 0 1 . 2 4 4 5 6 Y 3.9355':- --1 8 . Q 3 5 3 2 - 1 8 . 9 8 2 1 4 - 1

0 . 6 X 1 . 4 5 9 9 9 1 . 4 6 0 0 0 1 . 4 6 0 0 3 Y 6 . 5 7 0 2,5 - 1 6 , 5 7 0 32 - 1 6 . 5 7 0 9 5 - 1

1 . 0 v 1 . 5 7 4 0 3 1 . 57404 1 . 5740 3 Y 5 . 0 0 0 5 4 - 1 5 . 0 0 0 4 5 - 1 5 . 0 0 0 3 2 - 1

1 . 4 X 1 . 6 4 5 7 3 1 . 6 4 5 7 8 1 . 6 4 5 7 3 Y 4 . 0 0 6 2 7 - 1 4 . 0 0 6 2 1 - 1 4 . 0 0 6 2 6 - 1

1,6 X 1.67272. 1 , 6 7 2 7 2 1 . 6 7 2 7 2 Y 3 . 6 4 7 1 1 - 1 3 . 6 4 ? 0 7 - 1 3 . 6 ^ 7 1 5 - 1

2 . 0 V /V I . 6 9 5 6 I 1 . 6 9 561 1 . 6 9 5 6 1 Y 3 . 3 5 1 7 7 - 1 3 . 3 5 1 7 4 -3 3 . 3 5181 - 1

2 . 4 X 1 . 7 4 7 8 3 1 . 7 4 7 8 3 1 . 7 4 7 8 3 Y 2 , 7 1 9 2 1 - 1 2.719-10 - 1 2 . 7 1 9 2 1 - 1

2 . 8 X 1 - 7 7 3 5 8 1 . 7 7 3 5 3 1 . 7 7 3 5 3 V 2 . 4 2 9 1 5 - 1 2 . 4 2 9 1 3 - 1 2 . 4 2 9 1 4 - 1

3 . 0 X 1 . 7 8 4 5 9 1 , 7 8 4 5 9 1 . 7 8 4 5 9 Y 2 . 3 0 9 0 3 - 1 2 , 3 0 9 0 7 - 1 2 . 3 0 9 0 8 - 1

3 . 5 X 1 . 3 0 8 0 3 1 . 5 0 8 0 3 1 , 8 0 8 0 3 Y 2 . 0 6 0 0 9 - 1 2 . 0 6 0 0 8 - 1 2 . 0 6 0 0 9 - 1

1 1

1 2

A c h e o H o ~ r e s u l t s : > * o n ; t r i e C o / l c u l a t i o . n s o f t h e S t o k e * s

v 6 o t o t * i r i " t h ^ m a t r i x C c ' . s o T - m n a , l s n r - c c o : T l n l c h p - f ] # C h a n d r ? s e V . h a r

( 1 ) h a s s h o : - m fchftt S ' t V . u , v O - k J T i f . ; > i . t g n u ^ t b e e x p r e s s i b l e V V«» » \ i \ ° V ' A ^ ' |

% v a , I n t - e i T i ^ o f f u n c t i o n s o f ^ 9 ; ! ' 1 ^ i 0 . T i e s e f u n c t i o n s a r e

" X ' a O , W > » « T u v o , ( T i v O . & L i O . l f i \ U 5 9 - n d V ' l t O , w h e r

- . \ I \ v » 1 \ 1 I ' 1

* y ? + v - 4 L ^ %> • »

0 \ | a . \ - i ~ j x 7 , + • / i S o ( i - S ^ i . y M u / \ ° ^ ' Y ' ^

X c u A - L * V z X L u ' z y . , u . ) - v \ \ ' \ 1 ^ t / u '

[ ^ 1 • 1 r * ' I 1 '

0

- - y a X a - v ^ s ' % , p

? 1 L \ = \£e~vt + V z S ^ L ^ T y i ^ v . ' ) ,

- ( . 1 - | J ? ) e V + l / i S 6 t i - u ; 1 ) \ l ^ n ' l 5

5 - U O - 4 - Y z S L | / T ^ , u ' > d f / / y . J

1

B l u . \ - V i ^ L i - u ^ b f , i

1 0 I 1 1 I \

A ^ t t ^ f I c A

V ^ > - v a ^ L - i j . - v \ r l i ^ u s i Y ,

t r i v 0 - + X ' V ^ n Y )

\ \ y i . ) = + e 1

• Y ru \ j O = ^ W y . + • e . - y * ,

A u "

THE TPT3C-3AT, Fl'^CTTOL'TS 0 ? CKAFljPAS^KHAB FOR u. -:Q . 500 Mil) v V ^ . 0 0

• f c - 0 . 0 2 —

1 - - 0 „£55 t - 1 . 0 0

ADAMS ELEEftT P. ELBERT A DAH3 ELBERT

5 0 , 0 1 9 0 9 0 . 0 1 9 4 0 , 29 6 0 3 0 . 2 9 3 1 0 . 4 4 6 9 0 0 . 4 4 0 1

s» 0 , ?<%i/•. 0 . 2 6 1 7 6 0 . 1 0 3 B 0 . 4 1 1 3 2 0 . 4 -9804 0 . 4 9 8 6 8 '

f n . 7 9 3 56 0 . 7 9 3 0 2 1 . 0 8 30 5 1 . 0 8 0 1 0 1 . 1 6 6 2 1 1 . 1 5 6 4 6

X 1 , 0 3 5 5 5 1 . 0 3 5 1 ? 1 . 3 2 2 7 1 , 3 2 2 9 ? 1 . Z13395 1 . 4 3 4 4 3

? 0 . 0 0 1 9 6 0 . 0 0 1 9 ' ? 0 . 0 50 29 0 , 0 '3030 0 , 0 3 5 5 ? 0 , 0 8 5 1 2

? 0 . 2 5 2 0 0 0 . 2 5 1 9 2 0 . 2 3 1 1 ? 0 . 2 2 9 8 0 0 . 2 1 4 8 1 0 . 2 0 5 6 3

•n 0 . 7 6 3 6 5 0 . 7 6 7 4 0 0 . 5 2 9 7 8 0 . 5 3 6 0 7 0 . 8 2 3 3 0 0 . 3 5 5 8 0 '

<r 0 . 9 9 6 0 1 0 . 9 9 5 * 2 0 . 6 3 1 2 1 0 . 6 3 3 9 7 0 . 4 1 1.40 0 . 4 2 1 6 0

6 0 . 0 0 1 9 6 0 . O O I 0 7 0 , 0 4 6 1 8 0 . 0 4 4 9 9 0 . 0 6 7 9 8 0 . 0 6 2 8 2

V ' o . 9 - 0 36 0 . Q 3 0 39 0 . 6 6 3 7 s A c c q 0 c UUO CJ 0 . 4 9 6 9 5 0 . 5 0 7 3 4 -

YU> V 0 . 9 8 0 3 7 0 . 9 3 0 3 9 0 . 6 0 6 9 3 0 . 6 6 8 0 3 0 . 5 0 4 6 8 0 . 5 0 7 0 8

1 3

I >4

The fractions of ^ ? f:; — slightly different

than those of Chandrasehhrt (1) to the difference of

for7i for the reflected and transmitted intensities with no

a ivti'Ghal dependence. Shovm else- in Tsble IT arc the

re cult in calculations obtained by Chandrasekhar and Elbert

(/_;,) The agreement between correspond ing functions is very

-ood for an optical depth of 0.C2, while noticeable

differences occur for optical depths of 0.5 ana 1.0, In

the discussion of their results, Chandrasek-au and Elbert

note that for increasing optical depths their values laclcod

stability and were subject to a -neater error.

* ^q 1 ci eclr of tl'i0 numeri C9± j'osiij. ts t/jus o ou&.irico fox*

the diffusely reflected and trannr.itted fluz normal to the

utroer end lower boundaries in the scalar and matrix cases,

— . « x.v, ,,v-Tl,OT*,3 ^il1v ^ j:° flux * aud the Tne sun 01 u le upi^acu 1 — -- --i- 4 >

^ .5p * r, t<~ n ivi the ab^euce ciireci: soiar 1 iu:l i\- u « 1 r xo ^ fo-T - —-

of absorotion in the a trio c where. With this rotation. (B

+w% J/y^WRst .0. The resulting calculations are shovm in Table

ITT and agreement is auite ood.. To.e close a' re eiuent o.i tne

upward, and downward flures fo- both the matrix and scalar

a-oproach should be noted. In both cases for the optical,

depths shown., the relative stability of the two cases for

inchessinootical d.eoth nay be notcn, L^e l.ar;'ws !..• erior

occurs for snail values of y., which is to be expected;

roundoff error does not appear to be significant m either

c o Q

rr i\ 'o r r? 1* T T 1f\ X. f J L i- f-

COilPAJlISOii 0? Fl/JXLS FOR COrSS.lVATIVE RAYT.KIG1] 8GATT S x i l 0

V"° CASS "P - U

1 .00 0 .01592 SCALAR 0,0376 2 0 . 0 1 2 4 1 1 .000 58

MATRIX ,03769 .01234 1 .00041

0 .5000c SCALAR .75331 . 57447 1 .00004

y a - ri-D r v ,73416 . 57410 1 .00004

0 .98408 SCALAR 1 .06376 .90882 1 .00003

MATRIX 1.0628,6 .9096°' 1 ,00001

5 .00 0 .01592 3CAI.AR 0 .04548 .00A 59 1 ,00021

MATRIX .04548 .00454 1 .00011

o .50000 SCALAR 1 .28741 .28342 1 .00006

MATRIX 1 .28827 . 28251 1 .00004

0 .93403 SCAIA R 2 .288 50 ,78412 1 .00008

MATRIX 2. 28583 0 .78660 1 .00004

1 ^

CYAP^Yi BT.BLICC^AH-IY

1 . '-iel V:!?n, H. , B, E, K ^ a b a , and H, C. P r e s t r u d , Tr. vc' •"i an t Inbed r1 ina a~ id "lac 1^ hive T'- nvins f c r 1 n SI abs or f - T - i f . Yen York, Ansric-vn E l s e v i e r Co a na;i;/, In c , , 1 ° 6 3 .

2. C a r l s t e o t , J . L . , and 'T. V I . K u l l i k i t i , "The X- and Y-f u n c t i o n s of C h a r c l r a n e k h a r , P r j t r o r h v s l c g I J on -rn a l Sunnleins^t, XII ( June ,19^6) , Wi-9-501.

3 . C h a n d r 0 s r , S . , R a o l a t l v e T r a n s f e r , N e w York, Dover P n b l i c a fcions 5 1960.

i!-, Chandrasekhar , S. , and D. D. E l b e r t , "The I l l r r i i n a t i o r i and P o l a r i z a t i o n o f t he Sn.nlit Sky on Raylei 'ch S c a t t e r i n g , 1 1 The Tygnr.act 1 c:~is o'p t ^ e can P h i l o s o p h i c a l Society,"XLIV (December, 195*0, 6^3-729.

5. Cou 1 son, K, L . , J . V, Dave, and Z. Sekera , Tab] es RoJ ati ' '1 n to P a d i n t i o n Yriernitv fro~i a Plawetotry Atnos Yisre n'i th 3.3 ' ' l o i rh So 3 t te :H na , C a l i f o r n i a , U h i v e r c i t " of C a l i f o r n i a ?re?-s, 19-SO,

6. Y u l l i k i n , T. W., "AMon l inca r I n t c ~ r o - D i f f c r c n t i s l Equat ion in Pad ' i a t ive ^rani-;Cp."o," ^h 0 ,Tou.""na1. of t ^ Soci g tv f o r I n d u s t r i a l and A~.roi i ed Ya thens t i c s , XI I I (June ,1965) , 3-38 ~>0 6 , ~

16

DTSCU8SI0M

In Figures 1-8, ths reflectec Intensities for both the

scalar and vector solutions are plotted as a function of

for ^ —0 and $-1 , 5? with a ground albedo of zero for some

selected values of |l0 end tan, Since it is difficult to read

accurate values from ^ranhs, Table IV was developed to show

the percentage error bet?ieen the two calculations. The

percentage error is defined as (Tv -Is)*100/lv where I is

the intensity conputed from the correct vector field ap-oroach

and I & is the intensity commited fron the scalar theory. As

can be seen from this table the percentage error can be

quite significant, reaetonp; a maxiirtiru value of approximately

twelve per cent in absolute value for the largest and ^4with

0=0 and tau of one. For fixed ^ and y.0 the error as a

function of tau tends to reoc'n a mariTium and decrease

monotoni cally to some asymptotic value for , This

behavior is entirely expected since as the atmosphere becomes

very optically thick the very-hia;h-order collisions give

smaller contributions to the total. For corresponding: values

of and tJ the percentage error is almost alt-rays smaller

in absolute value for 0-1.57 than for fi=0

1?

IV

PEBCE' V T R S L A ? I 7 ^ Sn .R0H3 B;3TWK2K THIS VECTOR AND SCALAR R E r L E C T E D I S T E ; - . r 3 T T E S S

1 1 ^ 0 . 1-933, £ - 0 , A= 0 \ i .= ° - I 9 3 3 , 0 = ^ / 2 , A=0

Y X - o . 50 x = l . 0 0 t - •5.00 t - o . 5 0 X = i . 0 0 5 . 0 0

0 . 01 ?? 4 . 3 2 ^•.66 4 . 6 3 0 . 0 7 0 . 1 1 - 0 . 0 1

0 . 5ooo 2 . 7 7 3 . 8 0 2 . 6 7 - • 1 . 8 1 - 1 . 6 8 - 1 , 5 2

0 , 98'M - 1 1 . 3 0 - 1 1 . 7 0 - 8 . 1 1 - 9. '77 - 9 . 3 0 - 7 . 1 8

\ 1 \1-* = 0 . 5000, # = o , A= -0 u , - 0, , 5000 , p - f y 2 , A=0

v- t = 0 . 5 0 t = 1 . 0 0 < = = 5 . 0 0 t---0. 50 t = i . 0 0 t - •5.00

0. ,0159 3 . 3 7 4 . 6 2 4 . 51 - 1 . 7 9 - 1 . 9 0 - 1 . 8 7

0. ,5000 - 2 . 6 3 - 2 . 1 1 - 1 . 1 3 - 2 . 3 7 - 2 . 4 1 - 1 . 8 5

0. . 9 8 - 1 - 7 - 0 2 - 8 . 6 3 ~ 5 . 8 1 ~ 4 . 0 6 - 4 . 9 0 - 3 . 'i-O

\ I h = ° ' ,8067) p=0 , A= =0 0 . 8 0 6 7 , f - , A=0

1*" t = o . 5 0 %r~-l .00 t = = 5 . 0 0 x - 0 . 5 0 t - 1 . 0 0 x * = 5 - 0 0

0 . 0 1 5 9 - 1 . 2 0 - 0 . 8 6 _ 0 , 5 0 - 5 . ?4 - 5 . 7 4 - 4 . 7 6

0 , 5 0 0 0 - 1 0 . 1 0 - 1 0 . 5 0 - 6 . 7 1 - 3 . 3 3 - 3 . 8 1 2 . 7 2

0 . 9 8 4 1 2 . 2 3 0 . 6 5 - 0 . 54 3 . 8 2 3 . 4 0 1 . 6 ?

W= o . 0 8 4 1 , / t o , A= =0 ? • - 0 .Q841, fr-1?/2, , A--0

f < = 0 . 5 0 <=1 .00 - 5 . 0 0 ^ = 0 . 5 0 X - i .00 X--= 5 . 0 0

0 .0159 - 8 . 2 7 - 8 . 5 2 - 6.-I-7 - 8 . 7 5 - 9 . 0 4 - 6 . 8 7

0 . 5000 - 7 . 0 2 - 8 . 6 0 - 5 . 8 1 - 4 . 0 6 - 4 . 9 0 - 3 . 4 0

0 .9841 7 . 5 0 7 . 4 4 4 . 0 3 8 . 1 6 8 . 1 6 4 . 7 4

V*

19

In Ft fares Q--1 6 the correar^ndVnz transmitted intensities

are rlottecl , Table V >,JU'0e the coca- ^ondi n'< percenta^e errors

in trie trarsr.lttea inter.cit'.*. It rhov"1 ? bo noted that errors

in c:nce.eM of se^entesn rer cert can be encountered by us ins;

the incorrect sc ior the or?.'. As th<* ootic?1 thlcVnness of the

at:ao3ohere increases, the rerc^nto<;e error wil 1 decrease since

the nhotonG reaching the bottom Fill have un<3er^one thonssnds

of collisions and oolarination effects are virtually destroyed.

TA3LE V

PESC^T ERRORS 5ETWZE* THE VECTOR AMD SGALAR TEA 1-4SHITTED I¥TZHSITTES

|AP= 0 . 1 9 3 3 , 0 - 0 , A=0 y.& - 0 . 1 9 3 3 , f - ^ / 2 , A - 0

t - o . 5 0 t = i . o o t - 5 . 0 0 t = c . 5 0 t = 1 . 0 0 t = 5 . 0 0

0 . 0 1 5 9

o . 5 0 0 0

0 . 9 3 4 1

1 3 . 3 0 1 7 . 3 0 - 0 . 1 2

7 . 8 6 1 1 . 2 0 - 0 . 9 1 '

- 7 . 7 3 - 8 . 1 ? 1 . 0 1

- 1 . 8 7 - 4 . 6 1 - 3 . 7 4

- 2 . 1 8 - 2 . 6 3 - 2 . 2 4

- 1 0 . 4 0 - 1 1 . 1 0 - 1 . 4 5

0 . 5 0 0 0 . '/--:0, A - 0 y.o = 0 , 5 0 0 0 , j M V ? , A--0

t ^ O . 5 0 t - 1 . 0 0 ^ = 5 . 0 0 t f = 0 . 5 0 t ^ l . O O t - 5 . 0 0

0 . 0 1 5 9

0 . 5 0 0 0

0 . 9 8 4 1

6 . 6 0 9 . 6 0 0 . 4 2

8 . 4 5 1 1 . 3 0 1 . 8 5

- 0 . 5 8 - 0 , 9 6 0 . 1 1

- 3 . 0 0 - 4 . 1 0 - 2 . 7 9

- 2 . 5 5 - 2 . 8 5 - 1 . 4 2

- 4 . 2 1 - 5 . 2 3 - 0 . 6 8

r

i« — 0 A—n |^0 — U » W ^ v . J - - ~ 5 -

y . a - 0 . 8 0 6 7 , f - ^ / 2 , A - 0

r t - o . 50 t--=l . 0 0 t - 5 . 0 0 t - . -o . 50 1 5 = 1 . 0 0 t = 5 . o o

0 . 0 1 5 9

0 . 5 0 0 0

0 . 9 8 4 1

- 0 . 7 2 0 , 2 9 - 0 . 1 5

6 . 2 6 7 . 9 0 1 . 8 9

6 . 2 1 6 . 4 5 1 . 4 6

- 6 , 9 4 - 7 . 4 3 - 2 . 0 0

- 3 . 4 7 - 4 . 1 4 - 0 . 8 5

3 . 8 7 3 . 4 0 0 . 5 6

p a = 0 . 9 8 4 1 , 0 = 0 , A - 0 0 . 9 8 4 1 , 0 = ^ / 2 , A=0

^ = 0 . 5 0 t = i . o o 1 ^ 5 . 0 0 t ^ 0 . 5 0 t - 1 . 0 0 t - 5 . 0 0

0 . 0 1 5 9

0 . 5 0 0 0

0 . 9 8 4 1

- 9 . 5 5 - 9 . 6 5 - 1 . 5 0

- 0 . 5 3 - 0 . 9 6 o . l l

8 . 9 0 9 - 3 3 2 . 0 5

- 1 0 , 5 0 - 1 0 , 7 0 - 1 . 7 1

- i t . 2 1 - 5 . 2 3 - 0 . 6 8

8 . 2 6 8 . 4 5 1 . 7 1

20

APPENDIX

21

.0

T — — — r "*t~ —-p

H' 0.1933 0 = 0.0 A = 0 .0

MATRIX

SCALAR

t - 3.0

X =0.5

J 1 4 L. J 1 L. i_ .0 .2 .3 .4 .5 .6 .7 .8 .9 1.0

P Pi;-x, 1 --Reflected intensity for ^=0.1 933, 0*>O5O, and A=0.

.5

t 4 co s

z LiJ I -2

O - 3

UJ H O Ld _J Li_ UJ 9 or.

1 T- "i 1 r r " I — r

H = 0.5000 0 - 0.0 A = 0.0

MATRIX SCALAR

t = 5.0

X = 2.0

^ t = 1 .0

i I

. 0 J . 2 . 3 . 4 . 5 . 6 . 7 . 8 . 9 1.0

F F i g . 2 - - R e f l e c t e d i n t e n s i t y f o r | i a -0 . 5 0 0 0 , g f=0 . 0 , a n d A = 0

?!

0 . 8 0 6 7 0 = 0.0 A = 0 . 0

> t CO

2 Ll! h-2!

O LU

!— O LU

_J

LL

LU

CC

.5 h

X = 5.0

X = 3.0

X = I.O

MATRIX

SCALAR X - 0.5

.0 .! .2 .3 .4 .5 .6 .7 .8 .9 1.0

H

Pi5.3—Reflected in tens i ty fo r ^lo=0.8067, ^=0.0, and A=0

2h

8

0.9841 0 =

o .4 Ixi I -O LxJ

I Lu LjJ "2 ££

MATRIX SCALAR

X = 1.0

2 i i i i i—>—i 1 1 1 1 .0 .1 .2 .3 .4 .5 .6 J . .8 .9 1.0

P Fi.r.iL~~RGf3. ected intensity for ^,-0.98^1, 0=0,0, and A=0

25

0 = !.57 = 0.1933 A = 0 .0

MATRIX

SCALAR

.6 .7 .8 .9 1.0

P -V

Fls. 5 — R e f l e c t e d Intensity for ^ 0 . 1 9 3 3 , 0 = 1.5?, and A.=0,

26

p< = 0 . 5 0 0 0

.5 h

0= 1.57 A = 0.0

MATRIX

X= 5.0 SCALAR

4

.3 >-

t CO 2: LxJ I— Z

Q LU h-O LiJ _J U-UJ . cr .1

x-1.0

.1 .2 .3 .4 .6 .7 .8 .9 1.0

F

Fio;. 6—Reflected i n tens i t y fo r |t0-0.500, . 57, arid A=0

27

.8

.7

>-

t .6 CO Z LLJ

z

Q LD I— o 3 u _j u_ UJ 2 cr

.!

.0

, , •

P< = 0 . 8 0 6 7 0 = 1.57 A = 0 .0

MATRIX SCALAR

t =5.0

t = 1.0

x = 0.5

I I

P 0 .! .2 .3 .4 .5 .6 .7 .8 .9 1.0

Fisr,?-~Reflected int-ensit y i or |J.a --0 . P067, fr.,1. 5?, and. A=0 .

pq

o .4 Ui h-

LxJ .O _J LL. LtJ OC .2

.1

P-= 0.984! 0=1.57 A = 0.0

MATRIX SCALAR

t = 5.0

1 = 0.5

.0 .1 .2 .3 .4 .5 .6 .7 .8 .9 1.0

p —

Pis-,8--Reflected intensity for jio~0,98^-1, $-1.57, and A=<

29

Ue= 0.1933 0 = 0 .0 A = 0 .0

MATRIX

SCALAR

Z = 0.5

"C = 1.0

X =3.0

T=5 .0

.2 .3 .4 .5 .8 .9 1.0

P F i j r . 9 - - T r a n s m i t t e d i n t e n s i t y f o r . 1Q33, $ - - 1 . 5 7 , am d A=0

30

.40

> -

t oo z U.J I— 2

Q US I— t 5 01 21 < o: H

3 0

.20

.10

.00

"i r " ~ "T r i r

P ' ,= 0 . 5 0 0 0 0 - 0 . 0 A = 0 . 0

- M A T R I X - S C A L A R

x =0.5

.0

% =5.0

.2 .3 .4 .5 .6 -.7 .8 .9 1.0

F Flo;. 10~-~Trans:nitted i n tens i t y fo: -0 , 5000 , $-0 .0 , and A=0

31

.50

.40

.00

|.it= 0.806 7 0 - 0.0 A = 0.0

MATRIX

-SCALAR

t = I.O

w .20

t = 5,0

J L -.1 I I I I J_ .0 .! .2 .3 .4 .5 .6 .7 .8 .9 1.0

p _

Pie;. 11—Transmitted intensity for jao=0 .806?, 0=0.0, and A=0.

32

.00

F° = = 0 . 9 8 4 I 0 = 0 . 0 A = 0 . 0

MATRIX

SCALAR

X = 6.0

X = 1.0

2; .30

t *0.5 K .20

J _J L J I I I L .0 .1 .2 .3 .4 .5 .6 .7 .8 .9 i.O

P

F i . 12==Trans m .111 ed intensi ty for |io=0.9S4l, 0-0,0, and A-0 „

33

0 = 1.5 = 0 . 1 9 3 3 A = 0 . 0

MATRIX

CALAR

55 .1?-

t = 0.5

H .Ob

< .04 t =3.0

% =5.0

Fio:. 13~~Transmitted i n t e n s i t y f o r ^6=0.1933> and. A=0

34

0 = 1.57 = 0 . 5 0 0 0 A = 0 . 0

MATRIX

- - SCALAR

X = 1.0

w .15

lu .10 t =3.0

t: - 5.0

oc .05

.0 .1 .2 . 3 .4 .5 .6 .7 .8 .9 i.O

Fi.9c.l4-—Transmitted i n t e n s i t y f o r ^ = 0 . 5 0 0 0 , ^ = 1 . 5 7 , and A=0

35

> b-CO -3 2: LJ H 2!

Q ? ui • H-b 2 CO z < . CC •'

.0

(J0= 0 . 8 0 6 7 0 = 1 . 5 7 A - 0 . 0

MATRIX

SCALAR

t =1.0

J L 1 1 ! I I 1 L .0 .2 .3 .4 .5 .6 .7 .8 .9 1.0

P

F i« : .15 - -T ransn i t ted i n t e n s i t y f o r ^ = 0 . 8 0 6 7 , $=1.5?, and A-0,

36

.0

"T" ^ r --—-p—. ~ — r

p. 0.984! 0 = 1.57 A = 0.0

MATRiX

SCALAR

X = 1.0 <S .3

—.—JL JL_ J \ i - - 1 1 t I .0 .1 .2 .3 .4 .5 .6 .7 .8 .9 1.0

P X Fip;. l6~-TransTuitted i n t e n s i t y f o r |Aa = 0 ,98^-1 > P = 1 . 5 ? i

a n (i A = 0

37

BI!T.T0GR4P"'

Books

Bell mar., H, E. , R. E. ^laba, and C. Prestrud, Invariant I ibedd in ~ and Radiative Transfer in Slabs of Finite Thickness, 11 e.-r York, American Elsevier Company, Inc., 19^3.

Chandrasekbar, S., Radiative Transfer, New York, Dover Publica't ion s, 1960 .

Coulson, K. L., J. V. Dave, and Z. Selrera. Taoles Relating to Radiation Bner<?1.nr; from a planetary Atmosphere vrith Ra~rfel <xh Scattering, California, University of California Press, 19-0.

Articles

Carl steel t, J. L., and T. 'A. Kullikin, "The X- and Y~ Funct ions of Chandrasekbar,,} P. strongs leal ,Toumal Sun clement, XII (June ,1966), - 50 i .

Chandrasekhar, S., and D. D, Elbert, "The Illumination and Polarization of the Sunlit Sky or Rayleigh Scattering," The Transactions of the American Philosophical Society, XLIV (December, 195^) ,~ -':3-?29.

Kagiwada, H., and R. E. Kala.ba, •'Estimation of Local Anisotropic Scattering Properties using Measurements of Hultinly Scattered. Radiation,w The Journal of Quantitative Spectroscopy and Red l^nyr- Transfer. VII (October,1967), 295-303.

Hullikin, T, , "A Nonlinear Interredifferential Equation in Radiative Transfer,'5 The Journal of the Society for Industrial and Applied Mathenatl cs , XI11 (June ,1965), 388-410.

Sekera, Z., "Reduction of the Zauations of Radiative Transfer for a Plane-Pa.r all el, Planetary At no sphere : Parts I & II," The RAITD Corporation ( R.k-'I-952~PR,R:I~50 56-PR) , Santa I'ionica, Californla.