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

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
thiswork.