Invariant imbedding and perturbation techniques applied to diffuse reflection from spherical shell

Bellman, R. E.

Kagiwada, H. H.

Kalaba, R. E.

English

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MEMORANDUM 
RM-4730-NASA 
AUGUST 1985 
NASr- 21( 0 7) 
INVARIANT IMBEDDING AND 
PERTURBATION TECHNIQUES APPLIED TO 
DIFFUSE REFLECTION FROM 
SPHERICAL SHELLS 
R. E. Bellman, H. H. Kagiwada and R. E, U b a  
This research is sponsored by the National Aeronautics and Space Administration under 
Contract No. NASr-21. This r e p t  does not necessarily represent the views of the 
National Aeronautics and Space Administration. 
I 7& R4 ll D w  
I 1700 M A I N  ST. S A N l A  M O N I C A  - C A L I C O P N I A  * e0104  
iii 
PREFACE 
This Memorandum w a s  prepared as part of RAND'S continuing study 
of Satellite Meteorology for the National Aeronautics and Space 
Administration under contract number NASr-21(07). 
diffuse reflection of light fran a spherical shell is of central 
importance in meteorology and astrophysics. 
a method f o r  assessing quantitatively the effects of sphericity on 
diffuse reflection patterns. 
for further work devoted to inverse problems for spherical shell atmos- 
phere. 
ological satellite experiments. 
The problem of 
This Memorandum presents 
This study is a necessary prerequisite 
This should aid in the planning and interpretation of meteor- 
V 
This  Memorandum presents a method f o r  in tegra t ing  numerically 
the invariant  imbedding r e l a t ion  for  the d i f fuse  r e f l ec t ion  coeff ic-  
i e n t  of a spherical  she l l .  The re f lec t ion  coef f ic ien t  of a s h e l l  is 
re l a t ed  t o  tha t  of a s l ab  v i a  a perturbation technique. 
The r e s u l t s  of some t e s t  calculat ions are presented; they com- 
pare well  with results obtained i n  an earlier Memorandum(1) using a 
d i f f e ren t  method. In te res t ing ly  enough, the per turbat ion technique 
proves useful f o r  qui te  large values of the per turbat ion parameter. 
v i i  
CONTENTS 
PREFACE ......................................................... iii 
SUMMARY ......................................................... v 
S e c t i o n  
I. INTRODUCTION ............................................ 1 
11. PERTURBATION TECHNIQUE .................................. 3 
111. COMmrrATIOW RESULTS ................................... 5 
REFERENCES ...................................................... 8 
1 
The equation f o r  the sca t te r ing  funct ion of a spher ica l  s h e l l  is  
a nonl inear  p a r t i a l  d i f f e r e n t i a l  i n t eg ra l  equation. 
tional experience is ava i lab le  in the  nlnnerical so lu t ion  of such cam- 
plex funct ional  equations. 
approach based upon the  numerical est imat ion of p a r t i a l  der iva t ives .  
The r e s u l t s  obtained seemed sa t i s f ac to ry  from both the physical and 
computational viewpoints. Nevertheless, the importance of the equation 
and the newness of the  ca lcu la t ion  made i t  seem advisable t o  approach 
the problem from a second and independent po in t  of view. 
t o  use  a per turbat ion technique based upon an expansion in inverse 
powers of the inner radius  of the s h e l l .  
L i t t l e  computa- 
In an earlier paper,") we described an 
W e  e lec ted  
Results of t h i s  numerical experiment confirmed our earlier cal- 
cu la t ion  t o  severa l  s i g n i f i c a n t  f igures .  Remarkably enough, this was 
achieved using only the f i r s t  and second terms in the  per turbat ion 
expansion. 
L e t  conical flux of n e t  i n t ens i ty  TT per  uni t  area be uniformly 
incident  upon a hollow s h e l l  of inner radius  a and outer  rad ius  z. 
The intensity of d i f fuse ly  re f lec ted  r ad ia t ion  in the  d i r ec t ion  
arc cos v, due t o  incident  flux w i t h  d i rec t ion  arc cos u ,  is 
where the S funct ion satisfies the r a the r  
equation 
( 1) 
formidable funct ional  
2 
1 1  (1-v 2 ) -s 2 +(;+--) s-(-)s U 
zu v 
z + zv sv + zu 
1 1 
= h [l + 1 S(z ,v ' ,u )  51 [l + 4 J S(z ,v ,u ' )  q y 
U t  2 o  0 
f o r  z 2 a ,  with the i n i t i a l  condition S(a,v,u) = 0 .  (2'3) 
the  albedo f o r  s ing le  sca t t e r ing .  We suppose t h a t  the core i s  a 
pe r fec t  absorber. 
which can be used e i t h e r  t o  test  the accuracy of the numerical so-  
l u t ion ,  o r  t o  reduce the s i z e  of the problem. 
Here h i s  
The function S i s  symmetric i n  v and u ,  a f a c t  
In  a previous paper,") we obtained a numerical so lu t ion  of t h i s  
equation by means of an estimation of S 
the  values of S a t  su i tab ly  chosen p o i n t s .  
Method I. 
1 - and compare the r e s u l t s  of the two ca lcu la t ions .  a 
exce l len t .  
a s  a l i n e a r  combination of 
This w i l l  be ca l led  
U 
In t h i s  no te ,  we use a per turbat ion expansion i n  powers of 
The agreement i s  
. 3 
11. PERTURBATION TECHNIQUE 
L e t  us introduce the t h i c b e s s  of the  s h e l l ,  x = z-a, assume that 
5 << 1, and s u b s t i t u t e  the formal expansion a 
s1 s2 s = so + - +  2 + ... a a (3) 
in Eq. (2), with the quantities Soy S1, S2, ..., independent of a. 
turns  out t o  b e  the S-function fo r  a p l a n e  p a r a l l e l  s l ab ,  a func- 
t ion  which can be eas i ly  and accurately calculated.  (1s3)  
coe f f i c i en t s  of powers of - we obtain the following equations 
Equating 
1 
a' 
and 
2 2  - (S ) + (+)(so)v + ( - ) ( s o ) u  + (; 1 1  + s1 - -u +v 2 2 so u v  l x  
i j  i j  i j  (x) = So(x,vi,v.) ,  S1 (x) = S,(x,viyvj), U 0 J L e t  us  wr i te  S 
V (x) = (So)v, where the vi  a r e  the roots  of the  sh i f t ed  Legendre 
polynomials. Using quadrature techniques as described i n  Refs. 4 and 
5, we reduce Eq. (4) t o  a system of ordinary d i f f e r e n t i a l  equations 
(x) = (So)u, 
i j  
4 
involving the functions introduced above. To solve these,  we require  
the functions Vi’ and Vi’. These pa r t i a l  der iva t ives  are produced in 
two d i f f e r e n t  ways. 
formula of Ref. 1. We s h a l l  r e f e r  t o  t h i s  as Method IIa. Secondly, 
we can turn t o  the equation f o r  So and obtain,  by m e a n s  of p a r t i a l  
d i f f e r e n t i a t i o n ,  corresponding equations f o r  (So)u and (So)v. 
are adjoined t o  Eq. (4) t o  provide a complete set of equations f o r  the 
four  functions S o ,  (So)u, (So)v, S1. Quadrature techniques applied t o  
these produce the desired ordinary d i f f e r e n t i a l  equations. This pro- 
cess involves the  numerical solut ion of N(3N+1) simultaneous equations.  
We s h a l l  r e f e r  t o  t h i s  as Method IIb.  
F i r s t  of a l l ,  we can use the numerical estimation 
These 
5 
Calculation of the ref lect ion function r(x,v,u) was carried out 
with the two variat ions of the perturbation technique. Sane checks 
consist of varying the order of the quadrature formula, varying the 
step length of integration, and increasing the inner radius a so as 
to  approach the limiting value a = m. O t h e r  checks consis t  of com- 
parisons among the three computational schemes (Methods I, IIa, and 
I=). W e  found the resu l t s  t o  be satisfactory.  
In Fig. 1, we present three sets of curves f o r  the ref lect ion 
function r,  f o r  the case in which the albedo h = 1, the inner radius 
a = 50, and the thickness x = 3. 
angle of incidence, arc cosine u = 17.6 degrees, 60.0 degrees, and 
Each set corresponds to  a d i f fe ren t  
87.3 degrees. 
Four curves are plotted f o r  each incident direct ion,  although these 
Ihe abscissa is arc cosine v ,  the angle of reflection. 
curves l ie on top of one another in certain instances. 
are labelled "Slab,*' " I , i 1  "ILa," and "IIb.** The curves labelled 
These curves 
t*Slabll correspond to  the r function in the l i m i t  as a + -. 
**I1* are produced by a d i r ec t  application of the estimation of deriva- 
Curves  
t i v e  formula, the method described in Ref. 1. Curves *rIIall are pro- 
duced by the perturbaticm method with the use of the derivat ive esti- 
mation formula, and curves **IIb" are produced by the perturbation 
method alone. 
with programs written in the FOR!lWUl IV source language. 
Ihe calculations a re  performed on an IBH 7044 machine 
We used 
N = 5, an integration step size of 0.005, and an Adams-Moulton inte- 
gration formula. Running times a re  comparable f o r  Methods I I a  and' Ifb, 
6 
0.E 
0.7 
0.6 
0.5 
r 
0.4 
0.3 
0.2 
0.1 
 [_a. angle 
Slab 17.6 
1 
~ I I  a 
I1 b 
0 
Angle of reflection (deg) 
Fig.1 -Some reflected intensity patterns for a shell 
with albedo X = l  I inner radius a =50, and thickness 
x = 3 ,  for various angles of incidence 
. 7 
No d i f f i c u l t i e s  are encountered with the  use of the per turbat ion 
technique. 
Ref. 1 is qu i t e  s a t i s f ac to ry .  The present  technique has the  advantage 
of producing r e f l e c t i o n  functions f o r  a v a r i e w  of shell inner r a d i i  
in a s ing le  ca lcu la t ion .  Moreover, it appears t o  be s t a b l e ,  even 
h e n  is f a i r l y  large.  This r a t i o  i s  - = 0.06 in the case discussed 
above. The technique is su i t ab le  f o r  many other  types of 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  equations, and f u r t h e r  r e s u l t s  w i l l  be presented 
subsequently. 
The comparison of r e s u l t s  with the d i r e c t  ca lcu la t ion  of 
50 
8 . 
REF E RENCE S 
1. Kagiwada, H. H., R. E. Kalaba, and R. E. Bellman, Numerical E s t i -  
mation of Derivatives with an Application t o  Radiative Transfer 
i n  Spherical She l l s ,  The RAND Corporation, RM-4617-NASA, June 
1965. 
2. Bailey, P. B . ,  "A Rigorous Derivation of Some Invariant  Imbedding 
Equations of Transport Theory," J. Math. Anal. App l., Vol. 8, 
1964, pp. 144-169. 
3. Bellman, R. E . ,  and R. E. Kalaba, "On the  P r inc ip l e  of Invariant  
Imbedding and Diffuse Reflection from Cylindrical  Regions ,I1 
Proc. N a t .  Acad. Sc i .  USA, Vol. 43, No. 6 ,  June 1957, 
pp. 514-517. 
4. Bellman, R. E., H. H. Kagiwada, R. E. Kalaba, and M. C. Prestrud, 
Invariant Imbedding and Time-Dependent Transport Processes, 
American Elsevier  Publishing Company, New York, 1964. 
5. Bellman, R. E.,  R. E. Kalaba, and M. C. Prestrud, Invariant  I m -  
bedding and Radiative Transfer i n  Slabs of F i n i t e  Thickness, 
American Elsevier Publishing Company, New York, 1963. 


Invariant imbedding and perturbation techniques applied to diffuse reflection from spherical shells

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