Statistical analysis for stochastic radiation transfer in finite participating media with Rayleigh scattering

The stochastic radiation transfer problem is studied in a finite participating planar continuously random medium. The problem is considered for specular-reflecting boundaries with Rayleigh scattering. The importance Rayleigh scattering arises in the atmospheric applications in the form of radiative transfer through clouds or in neutron transport with a quadratic formula of scattering. Such cluttered media should be analyzed in a statistical sense. As a result, the random variable transformation (RVT) technique is used to obtain the average of the solution functions that are represented by the probability-density function (PDF) of the solution process. The transport equation is solved deterministically to obtain a closed form of the solution as a function of optical depth x and optical thickness L. The solution is used to obtain the PDF of the solution functions by applying the RVT technique among the input random variable (L) and the output process (the solution functions). The obtained averages of the solution functions are used to obtain the complete analytical averages for some interesting physical quantities, namely, reflectivity and transmissivity, at the medium boundaries. The numerical results are calculated and represented graphically for different probability distribution functions.