A complete probabilistic solution for a stochastic Milne problem of radiative transfer using KLE-RVT technique

Abstract

A novel technique, named (KLE-RVT), is constructed to find a full probabilistic solution of a finite Milne problem of radiative transfer in spatially stochastic atmosphere. This technique is a combination between the random variable transformation (RVT) technique and the Karhunen–Loève expansion (KLE) of the input stochastic process (the total cross section of the medium) to find the probability density function of the solution stochastic process. The RVT technique is applicable only if the probability density function of the input random variable (process) is known in a closed form. To overcome this obstacle, KLE is applied to represent the spatially continuous random cross section, defined only by its mean and covariance function, in terms of a finite number of uncorrelated random variables with known probability density functions. By this technique, the probability density function of the solution process is evaluated dealing with the input process itself instead the integral transformation of it. This solution is general and valid for any input second order stochastic process. Numerical results of our findings are presented to realize this work.