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
.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.