Intensity of emergent radiation for finite homogeneous slab which absorbs radiation and scatters it isotropicall

Bellman, R. E.

Kagiwada, H. H.

Kalaba, R. E.

English

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NASr - 21( 0 7 )  
. 
MEMORANDUM 
RM-4637-NASA 
JUNE 1965 
ESTIMATION OF 
INTERNAL SOURCE DISTRIBUTIONS USING 
EXTERNAL FIELD MEASUREMENTS IN 
RADIATIVE TRANSFER 
R. E. Bellman, H. H. Kagiwada and R. E. Kalaba 
This research is sponsored by the National Aeronautics and Space Administration under 
Contract No. NASr-21. This report does not necessarily represent the views of the 
National Aeronautics and Space Administration. 
. 
iii 
PREFACE 
This Memorandum was prepared as part of RAND'S continuing study 
of Satellite Meteorology for the National Aeronautics and Space 
Administration under contract number NASr-21(07). It should be useful 
in estimating source distributions in stellar and planetary atmos- 
pheres based on observations of emergent intensity patterns. 
V 
SUMMARY 
A finite homogeneous slab which absorbs radiation and scatters 
it isotropically possesses internal isotropic sources of radiation. 
The authors first show how to determine the intensity of the emergent 
radiation, making use of invariant imbedding techniques, and then 
show how to determine the distribution of the internal sources that 
best accounts for an observed emergent radiation pattern. This 
inverse problem is viewed as a nonlinear two-point boundary value 
problem which can be resolved numerically using quasilinearization. 
vi i 
CONTENTS 
PREFACE ...................................................... iii 
SUMMARY ........................................................ v 
Section 
I- INTRODUCTION ............................................ 1 
11. DERIVATION OF THE BASIC EQUATIONS ....................... 2 
111. APPROXIMATE SYSTEM OF ORDINARY DIFFERENTIAL 
EQUATIONS ............................................. 7 
IV.  INVERSE PROBLEM AND QUASILINEARIZATION .................. 9 
V. DISCUSSION ............................................. 12 
REFERENCES .................................................... 13 
1 
I. INTRODUCTION 
A basic problem in science and engineering is inferring causes 
on the basis of observed effects. In the construction of model 
stellar and planetary atmospheres and in the study of planetary 
entry phenomena, this problem is particularly significant. 
Memo.z;ax&m w e  shall-discuss some aspects of this problem in relation 
- -  
to the ability of the modern computer t o  integrate systems of a 
thousand or more ordinary nonlinear differential equations. 
-- --. .. -._^.- ._“..-e- 
._ 
The physical situation is as follows: A slab of finite optical 
thickness absorbs radiation and scatters it isotropically. It 
possesses internal sources of isotropic radiation. First, we dis- 
cuss the direct problem of determining the intensity of the radia- 
\
tion emerging from the slab, with particular emphasis on its angular 
distribution. We use standard ideas from the theory of invariant 
imbedding; (lY2) the X and Y functions of Chandrasekhar play a 
fundamental role. (3) Then we shift to the inverse problem. 
We assume that measurements of the emergent radiation field are 
made, and it is desired to determine the distribution of internal 
sources which best explains the measurements. This is viewed 
mathematically as a nonlinear two-point boundary value problem 
which is solved numerically using quasilinearization. (4y5) We con- 
clude with a discussion of some of the extensions of the model which 
are possible. 
The importance of such problems was brought out in conversations 
with Dr. Ralph Zirkind of the Advanced Research Projects Agency. 
2 
Consider a homogeneous s l ab  bounded by two p lanes  separated by 
an  o p t i c a l  d i s t a n c e  x The s l ab  both absorbs r a d i a t i o n  and scatters 
i t  i s o t r o p i c a l l y ,  t h e  albedo for  s i n g l e  s c a t t e r i n g  i s  A .  Within 
t h e  s l a b  are i s o t r o p i c  sources  of r a d i a t i o n .  
t h e  bottom s u r f a c e ,  t h e  s t r e n g t h  o f  t h e  source i s  B(y) p e r  u n i t  
volume pe r  u n i t  s o l i d  angle  p e r  u n i t  t i m e .  
0' 
A t  each p o i n t  y above 
We f i r s t  d e r i v e  computationally u s e f u l  equa t ions  f o r  t h e  
i n t e n s i t y  o f  t h e  r a d i a t i o n  emerging from t h e  top  s u r f a c e .  To do t h i s ,  
w e  cons ide r  t h e  s l a b  extending from t h e  bottom t o  t h e  o p t i c a l  a l t i -  
tude X ,  add a l a y e r  of t h i ckness  A ,  and no te  t h e  changes t h a t  occur  
i n  t h e  i n t e n s i t y  of t h e  r a d i a t i o n  emerging from t h e  top of t h e  s l ab .  
L e t  
T(v,x)dv = t h e  r a t e  of emission o f  energy p e r  u n i t  h o r i z o n t a l  
area through the upper s u r f a c e  of a s l a b  of thick-  
nes s  x ,  having a d i r e c t i o n  c o s i n e ,  w i t h  r e s p e c t  (1) 
t o  t h e  upward-directed normal, between v and 
v + dv. 
I n  a d d i t i o n  l e t  
p(v,x)dv = t h e  f r a c t i o n  of t h e  energy which i s  i s o t r o p i c a l l y  
emit ted a t  t he  top of t h e  s l a b  of t h i c k n e s s  x and 
which u l t i m a t e l y  emerges from t h e  top w i t h  a d i -  
(2) 
r e c t i o n  cos ine  between v and v + dv. 
3 
Then we may write 
T(v,x+A)dv = (1- i)T(v,x)dv + 
1 
T(v' ,x)dv'  7 A A, 
V 
0 
+ o(A). 
The f i r s t  term on t h e  right-hand s i d e  accounts f o r  t h e  l o s s e s  en- 
countered i n  pas s ing  through t h e  s l a b  o f  t h i ckness  A .  The f i r s t  
f a c t o r  i n  t h e  second term i s  t h e  t o t a l  ra te  of product ion of s c a t t e r e d  
r a d i a t i o n  i n  a c y l i n d e r  w i th  u n i t  base  a r e a  and a l t i t u d e  A ,  l oca t ed  
a t  t h e  top of t h e  s l a b  of thickness  x. The second f a c t o r  i s  t h e  
f r a c t i o n  of such r a d i a t i o n  which u l t i m a t e l y  emerges from t h e  top w i t h  
a d i r e c t i o n  cos ine  v.  The t h i r d  term, o(A), accounts f o r  a l l  
h i g h e r  o r d e r  processes  and c o n s i s t s  o f  terms i n  A of powers h i g h e r  
t han  the  f i r s t .  The l i m i t i n g  form, as  A tends t o  z e r o ,  i s  t h e  p a r t i a l  
d i f  f e r e n t i a l - i n t e g r a l  equat ion 
1 
0 
The s l a b  of t h i ckness  0 i s  e a s i l y  analyzed.  
s u r f a c e  i s  a p e r f e c t  abso rbe r  which l e a d s  t o  t h e  c o n d i t i o n  
W e  assume t h e  bottom 
T(v,O) = 0. ( 5) 
S i t u a t i o n s  i n  which t h e  bottom su r face  i s  an  emitter and a r e f l e c t o r  
a re  r e a d i l y  t r e a t e d ,  though w e  do n o t  d i s c u s s  them he re .  
4 
I n  o rde r  t o  i n t e g r a t e  Eq. ( 4 )  i t  i s  necessary  t o  d e r i v e  an 
equat ion  f o r  t h e  func t ion  p(v ,x) .  L e t  u s  i n t roduce  t h e  func t ion  
q(v,x)dv = t h e  f r a c t i o n  of t h e  energy i s o t r o p i c a l l y  e m i t t e d  
a t  t he  top o f  t h e  s l a b  o f  t h i ckness  x which 
u l t i m a t e l y  emerges from t h e  bottom w i t h  a d i r e c -  ( 6 )  
t i o n  cos ine ,  w i t h  r e s p e c t  t o  t h e  downward-directed 
normal, between v and v + dv.  
I n  Eq. ( 7 )  we  have cons idered  t h e  s l a b  of t h i ckness  x + A t o  be  made 
by adding a s l a b  o f  t h i ckness  A t o  t h e  bottom of  a s l a b  o f  t h i ck -  
nes s  x.  I n  a d d i t i o n ,  w e  have 
1 
( 8 )  
A 
V I  q(v,x+A) = q(v ,x )  ( 1  - 4) + p(v ,x)h  q(v ' ,x)dv ' -  + o ( A ) .  
0 
The l i m i t i n g  f o r m  of t h e s e  equa t ions ,  as A t ends  t o  z e r o ,  a r e  
and 
1 
1 dv ' 
qx(v,x) = - - V q(v,x)  + h p(v,x)  j q(v ' ,x)  7. 
0 
Thus 
5 
For t h i n  s l a b s  of thickness  yp- ha~.re 
277 dv 
4n p(v,A)dv = -+ o(A) 
1 q(v,A)dv = 5 dv + O(A). 
1 
P(VY0) = 'z, 
4(VYO) = 'z - 1 
Our b a s i c  equat ions a r e  
1 
dv 
T = - T + 477 p B + A p T(v',x) 7 , T(v,O) = 0. X V 
0 
Experience w i t h  many s i m i l a r  systems of n o n l i n e a r  d i f f e r e n t i a l - i n t e g r a l  
equa t ions  l eads  u s  t o  b e l i e v e  tha t  t h i s  system can be r e a d i l y  i n t e g r a t e d  
numerically by approximating t h e  i n t e g r a l s  through t h e  u s e  of f i n i t e  
sums w i t h  Gaussian quadra tu re  fornulas .  (1Y2) 
Before w e  d i s c u s s  t h i s ,  we w i l l  make s e v e r a l  s u b s t i t u t i o n s  t o  
p u t  t h e  system of equa t ions  (15)-(17) i n t o  a more convenient form. 
6 
We w r i t e  
dv ' X(V,O) = 1, 
0 
Thus X and Y a r e  t h e  s tandard  X and Y func t ions  of r a d i a t i v e  t r ans -  
f e r .  ( 3 J 6 y 7 )  The i n t e n s i t y  o f  t h e  r a d i a t i o n  emerging from t h e  upper 
s u r f a c e  o f  t he  s l a b  of  t h i ckness  x w i t h  d i r e c t i o n  cos ine  v i s  denoted 
t ( v ,  x) and i s  given by t h e  formula 
21l v t ( v , x )  = T(v,x) .  (22) 
With t h i s  s u b s t i t u t i o n ,  E q .  (17 )  becomes 
1 
= - 2 t + B  X +hX 5 t (v ' ,x )dv ' ,  t (v ,O)  = 0 .  (23) tx V V 2v 
0 
7 
111. APPROXIMATE SYSTEM O F  ORDINARY DIFFERENTIAL EQUATIONS 
Our b a s i c  system o f  equat ions c o n t a i n s  i n t e g r a l s  on t h e  i n t e r v a l  
( 0 , l ) .  
Gaussian quadra tu re  formula of  order  N,  
W e  approximate such an i n t e g r a l  o f  a f u n c t i o n  g(v) u s ing  a 
1 N 
where w i=lY2, . . . ,N,  are t h e  C h r i s t o f f e l  numbers, and v i=l,2, . . . ,NY 
i’ i’ 
are t h e  a b s c i s s a s  a t  which t h e  integrand i s  t o  b e  evaluated.  These 
are t a b u l a t e d  i n  Ref. 2. 
expect  t h a t  N=7 w i l l  y i e l d  high accuracy, L e t  u s  then i n t r o d u c e  t h e  
f u n c t i o n s  x .  (x) ,  yi(x) and t. (x) , 0 5 x s x 
system of o rd ina ry  d i f f e r e n t i a l  equat ions 
From experience i n  s i m i l a r  problems, (7,8) w e  
as s o l u t i o n s  o f  t h e  
1 1 0’ 
N . 
ti(0) = 0 ,  (27 )  1 t i + ? .  +- h = - -  V i 2vi x i  T.; t j  w j  2 
i i ti 
i=1,2, .  . . ,N. 
8 
This  i s  a non l inea r  system o f  3 N  o rd ina ry  d i f f e r e n t i a l  equa t ions  wi th  
a complete system of  i n i t i a l  cond i t ions .  I t s  i n t e g r a t i o n  on a modern 
computer i s  r o u t i n e  f o r  0 5 x I x 0' 
of  t h e  emergent i n t e n s i t y  a t  t h e  top,  t (vi ,xO),  i = l , Z , . . . , N ,  f o r  a 
given i n t e r n a l  source  d i s t r i b u t i o n  B(y),  0 5 y I x 
I n  t h i s  way w e  produce t h e  v a l u e s  
0' 
9 
I V .  INVERSE PROBLEM AND QUASILINEARIZATION 
Now l e t  u s  t u r n  t o  t h e  problem of determining the  source d i s t r i -  
b u t i o n  f u n c t i o n  B(y) which would b e s t  e x p l a i n  an observed emergent 
i n t e n s i t y  p a t t e r n .  F i r s t  w e  w i l l  cons ide r  t h e  case i n  which 
(28) 
2 
B(y) = a + b y  + cy , 
where a ,  b ,  and c a r e  t h e  t h r e e  cons t an t s  t o  be determined. W e  
suppose t h a t  t h e  observed i n t e n s i t y  of t h e  emergent r a d i a t i o n  with 
d i r e c t i o n  cos ine  v i s  b * i . e . ,  i i’ 
t . ( x  ) 2 b i = l , 2 , .  . . ,N. 1 0  i ’  
W e  wish t o  minimize t h e  sum of the squa res  o f  t h e  d e v i a t i o n s ,  
N ,. 
s =  
j=l  
through an a p p r o p r i a t e  cho ice  o f  t h e  c o n s t a n t s  a,  b ,  c ,  which e n t e r  
i n t o  E q s .  (25), (26), and (27) through B. 
This problem i s  r e a d i l y  solved numerical ly  u s i n g  t h e  q u a s i l i n e a r i -  
z a t i o n  technique. ( 4 ’ 9 )  W e  now p resen t  t h e  b a s i c  formalism. A 
system of M equa t ions  i s  w r i t t e n  i n  v e c t o r  form 
x = f(x,cy) , x(0)  = c ,  
where cy i s  an R-dimensional v e c t o r  cons t an t .  We wish t o  determine 
t h e  v e c t o r  cy so as t o  minimize t h e  form 
10 
where Q i s  a p o s i t i v e  d e f i n i t e  square m a t r i x  of o r d e r  M, and 
given v e c t o r .  
d i f f e r e n t i a l  equat ion 
i s  a 
We cons ide r  cy t o  b e  a func t ion  of t i m e  s a t i s f y i n g  t h e  
cy = 0. ( 3 2 )  
Now t h e  i n i t i a l  va lue  of cy is  unknown. Thus, ou r  o r i g i n a l  problem 
i s  e q u i v a l e n t  t o  one i n  which 
and some of  t h e  components of w ,  s ay  the  f i r s t  L ,  a r e  f r e e ,  and t h e  re- 
maining ones a r e  s p e c i f i e d .  W e  wish t o  minimize a q u a d r a t i c  form i n  
y(T),  t h e  f i n a l  va lue  of y .  This can  be done by s t a r t i n g  wi th  an 
i n i t i a l  approximation, w = w(O) 
t o  produce y( ' ) ( t ) ,  0 S t S T .  
f i n d i n g  y ( l ) ( t )  so  t h a t  
and i n t e g r a t i n g  the  system, Eq. ( 3 3 )  
The next  approximation i s  produced by 
where F i s  minimized; .J(y(')) i s  t h e  Jacobian ma t r ix  
Th i s  i s  done by numerical ly  producing an a p p r o p r i a t e  s y s t e m  of  L 
independent s o l u t i o n s  of t h e  homogeneous equa t ion  corresponding t o  
Eq. ( 3 4 )  and a l s o  a p a r t i c u l a r  s o l u t i o n .  
p o s s i b l e  s o l u t i o n s  of Eq. ( 3 4 )  are 
I n  an obvious n o t a t i o n ,  t he  
11 
i= 1
where c i = l Y 2 , . . . , N ,  are a r b i t r a r y  cons t an t s .  The minimizing va lues  
i’ 
of c 
mined from E q .  (36)) i n t o  F (from Eq. (31)).  A l i n e a r  a l g e b r a i c  
i = 1 , 2 , .  . . ,L, are determined by s u b s t i t u t i n g  y(’) (T) (as  de t e r -  
i’ 
system of equat ions  f o r  t h e  optimal choices  of c i = 1 , 2 , . . . , L y  
i’ 
r e s u l t s  from 
bF - = 0 , i=lY2,. . . ,L.  
i a, 
(37) 
Many previous  c a l c u l a t i o n s  a t tes t  t o  t h e  e f f i c a c y  of  t he  procedure.  
However, a t  t i m e s ,  s p e c i a l  devices  a r e  r equ i r ed  t o  c a r r y  through a l l  
t h e  s t eps .  (4210) The procedure i s  then repea ted  t o  produce y(2)  (t) , 
and so on. When t h e  i n i t i a l  es t imate  i s  s u f f i c i e n t l y  good, t h e  
d i g i t s  i n  each approximant doubles a sympto t i ca l ly .  
The c a s e  o f  a layered s l a b  i s  d i scussed  i n  Ref. 8. 
I f  B(y) i s  known only  as t h e  s o l u t i o n  o f  a d i f f e r e n t i a l  equat ion  
invo lv ing  some unknown c o n s t a n t  vec to r  13 
e s s e n t i a l l y  the  same procedure i s  used.  Equat ion (38)  i s  ad jo ined  to  
the  s y s t e m  of E q s .  (25)-(27)’ and t h e  c a l c u l a t i o n  proceeds a s  before .  (11) 
12 
V. DISCUSSION 
I n  the  previous paragraphs w e  have i n d i c a t e d  t h a t  a j ud ic ious  
use  o f  concepts from the  t h e o r i e s  o f  i n v a r i a n t  imbedding and quasi-  
l i n e a r i z a t i o n ,  t o g e t h e r  w i t h  modern d i g i t a l  computers,  could l e a d  t o  
t h e  determinat ion of  i n t e r n a l  source  d i s t r i b u t i o n s  us ing  e x t e r n a l  
f i e l d  measurements. It i s  p o s s i b l e  t o  examine t h e  e f f e c t s  o f  e r r o r s  
i n  t h e  measurements on t h e  accuracy o f  t h e  source  d i s t r i b u t i o n  
est imat ion.  This  might be  of  v a l u e  i n  p l ann ing  experiments.  I n  
a d d i t i o n ,  t h e  r e s u l t s  of t h e  c a l c u l a t i o n s  might  s e r v e  a s  a s t imulus  
t o  t h e  development of new t h e o r i e s  concerning t h e  sources .  
It would probably be necessary  t o  s t a r t  a computat ional  program 
wi th  the  s i m p l e  phys i ca l  model ske tched .  The b a s i c  i n v a r i a n t  imbedding 
equat ions f o r  v a r i o u s  g e n e r a l i z a t i o n s  a r e  a v a i l a b l e .  These inc lude  
sphe r i ca l  and c y l i n d r i c a l  geometry, t i m e  dependence, (2)  an i so t ropy  
of  t h e  s c a t t e r i n g  l a w ,  inhomogeneity o f  t h e  medium, (12) and energy 
dependence. (13) 
13 
REFERENCES 
1. Bellman, R. E., R. E. Kalaba, and M. C. Prestrud, Invariant Imbed- 
ing and Radiative Transfer in Slabs of Finite Thickness, American 
Elsevier, New York, 1963. 
2. Bellman, R. E., H. H. Kagiwada, R. E. Kalaba, and M. C. Prestrud, 
Invariant Imbedding and Time-dependent Transport Processes, 
American Elsevier, New York, 1964. 
3. Chandrasekhar, S., Radiative Transfer, Dover Publications, New 
York, 1960. 
4. Bellman, R. E., and R. E. Kalaba, Quasilinearization and Nonlinear 
Boundary Value Problems, American Elsevier, New York, 1965. 
5. Bellman, R. E., H. H. Kagiwada, and R. E. Kalaba, "Orbit Determina- 
tion as a Multi-point Boundary Value Problem and Quasilineariza- 
tion," Proc. Nat. Acad. Sci. USA, Vol. 48, 1962, pp. 1327-1329. 
6. Bellman, R. E., H. H. Kagiwada, R. E. Kalaba, and S. Ueno, A New 
Derivation of the Integro-differential Equations for Chandrasekhar's 
X and Y Functions, The RAND Corporation, RM-4349-ARPAY December 
1964. 
7. Bellman, R. E., H. H. Kagiwada, R. E. Kalaba, and S. Ueno, Numerical 
Results for Chandrasekhar's X and Y Functions of Radiative Trans- - fer, The RAND Corporation, RM-4531-PRY April 1965. 
8. Bellman, R. E., H. H. Kagiwada, R. E. Kalaba, and S. Ueno, Inverse 
Problems in Radiative Transfer: Layered Media, The RAND Corpora- 
tion, RM-4281-ARPAY December 1964. 
9. Kagiwada, H. H., "Computational Aspects of Inverse Problems in 
Analytical Dynamics, Transport Theory, and Wave Propagation," 
unpublished Ph. D. thesis, University of Kyoto, 1965. 
10. Bellman, R. E., H. H. Kagiwada, and R. E. Kalaba,"Wengert's 
Numerical Method for Partial Derivatives, Orbit Determination and 
Quasilinearization," Cow. ACM, Vol. 8, 1965, pp. 231-232. 
11. Bellman, R. E., H. H. Kagiwada, and R. E. Kalaba, On the Identifica- 
tion of Systems and the Unscrambling of Data - I: Hidden 
Periodicities, The RAND Corporation, RM-4285-PRY September 1964. 
12. Bellman, R. E., H. H. Kagiwada, R. E. Kalaba, and S. Ueno, Analytical 
and Computational Techniques for Multiple Scattering in In- 
hosoaeneous Slabs, The RAND Corporation, RM-4438-PRY January 1965. 
13. Bellman, R. E., and R. E. Kalaba, Nuclear Reactor Theory, "Trans- 
port Theory and Invariant Imbedding," G .  Birkhoff and E. Wigner 
(eds.), American Mathematical Society, Providence, 1961, pp. 206-218. 


Estimation of internal source distributions using external field measurements in radiative transfer

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