The probabilistic method for problems of radiative transfer: Markov properties

摘要

On the basis of the stochastic model of multiple scattering of photons, we consider the diffuse reflection and transmission of a parallel beam of radiation by a finite, plane-parallel, nonemitting and homogeneous atmosphere with conservative and isotropic scattering. We assume that the stochastic process under consideration represents a homogeneous stationary evolution in a Markovian manner with respect to the optical depth.First we derive the forward and the backward integro-differential equations for the emission probability distributions from the Chapman–Kolmogoroff equations. Then, starting with the Laplace transform of these equations, we obtain the S- and T-functions of S. Chandrasekhar for monodirectional illumination of the upper and the lower boundaries, depending on the optical depths τ0andτ1 (0⩽τ<τ1). The results obtained with the aid of the forward equations reduce to those derived from the backward equations, because of the homogeneous optical properties of the medium. Some new functional equations for the source functions of the auxiliary equations are given.