On Inverse Boundary-Condition Problems

in Radiative Transfer

L. B. Barichello a, R. D. M. Garcia b and C. E. Siewert c

a Instituto de Matem?atica, Universidade Federal do Rio Grande do Sul, 91509-900 Porto Alegre, RS, Brazil
b Centro T?ecnico Aeroespacial, Instituto de Estudos Avan?cados,
12231{970 S~ao Jos?e dos Campos, SP, Brazil
c Center for Research in Scientific Computation, Mathematics Department, North Carolina State University, Raleigh, NC 27695{8205, USA

Abstract

Some elementary computations are reported to suggest that a certain type of inverse boundary-condition problem in radiative transfer can, in some cases, be solved quite simply.

1 Introduction

In a recent paper [1] concerning inverse-source problems in radiative transfer, it was mentioned that inverse boundary-condition problems could be solved in a manner similar to the one used to solve the source problems. Here we report some computations to support that suggestion.

We consider the equation of transfer [2] for the radiation intensity I(o; ?), written as

? @

@o I(o; ?) + I(o; ?) = $

2

LX

l=0
lPl(?)
Z 1

?1 Pl(?0)I(o; ?0)d?0 (1)

where o 2 (0; o0) is the optical variable, ? 2 [?1; 1] is the cosine of the polar angle (as measured from the positive o axis) used to describe the direction of propagation of the radiation and $ is the albedo for single scattering. In addition, the l are the coefficients in a Legendre polynomial expansion of the scattering law. For direct problems in radiative transfer, we normally

Preprint submitted to Elsevier Science 4 March 1996