A developed solution of the stochastic Milne problem using probabilistic transformations

Abstract

This paper considers the solution of the Milne problem of radiative transfer with isotropic scattering in a continuous stochastic medium. Properties of the medium are assumed to be continuous random functions of the spatial dimensions. The available solutions – in literature – for this stochastic integro-differential equation (SIDE) are represented only by the ensemble average of the radiant energy density. In this paper, a developed algorithm, based on the implementation of the random variable transformation technique together with an integral transformation to the stochastic properties, is introduced. A complete stochastic solution represented by the probability-density function (p.d.f) of the radiant energy density is obtained. Using the closed form of the p.d.f, the nth moment of the stochastic solution is evaluated. In realization of this work, Exponential and Gaussian statistics for the medium properties are assumed. Results are physically acceptable and found to be compatible with those in the literature.