Solution of the Radiative Transfer Equation for Vertically Inhomogeneous Media by Numerical Integration Solvers: Comparative Analysis

Abstract

An efficiency of the singular value decomposition (SVD) method and ordinary differential equation (ODE) solvers in finding the reflection matrix are compared. A reflection matrix can be found by solving the one-dimensional radiative transfer equation. The latter’s solution based on the discrete ordinate method leads to the singular value decomposition (SVD) method. Alternative approach consists in transforming the original problem into a matrix Riccati equation written specifically for the reflection matrix. The matrix Riccati equation is solved using numerical integration techniques for ordinary differential equations (ODEs). It is found that for a single layer case, the SVD approach is faster than the ODE solvers by an order of magnitude. Yet as the number of layers increases, the ODE solvers become more efficient than the SVD approach. In addition, they outperform the SVD method when a solution for a set of optical thicknesses of the medium should be found or when retrieval of optical thickness should be performed. The comparison between different ODE solvers is performed, as well.