The inverse problem of linear radiation transport theory

Abstract

where I(T, CL) is the radiation intensity at an optical depth T in the direction 8 (p=cos8), J.(T) is the albedo of single scattering, g(T) is the power of the radiation sources in the medium, and ~(7, CL, p') is the scatter indicatrix, -lGp<l, OdrC-. Equation (1) and its generalizations play a fundamental role in the theory of light scattering in the atmospheres of stars and planets, in neutron diffusion theory, and in other fields of physics /l-3/. Various aspects of direct problems connected with (1) have been studied, see /3-E/,primarily because of the needs of neutron physics and the theory of atomic reactors. In astrophysics and the physics of the atmospheres of planets, inverse problems for Eq.(l) are more important, where, given the output radiation, we wish to find the characteristics of the atmosphere. In practice, inverse problems have been solved by the method of inspection, i.e., by solving the direct problem for some model of the medium, and attempting, by choosing the parameters, to construct a model such that the boundary data are as close as possible to the measured data. Though direct solution of inverse problems has been attempted in recent years, see /g-11/, the attempts have been primarily of a formal type, considering the case when the medium characteristics are independent of T . Our present aim is to study inverse problems of radiation transport in a plane semi-infinite layer with characteristics dependent on T, and to justify the formal results of /12/. In Sect.1 we quote some results concerning the direct problem and give the formal apparatus for studying inverse problems. In Sect.2 we discuss in very general terms the question of the unique solvability of inverse problems. In Sect.3 we prove the Tikhonov correctness of the inverse problem in the case of isotropic scattering, and estimate rhe coditional stability of the solution.