Matrix Riccati equation formulation for radiative transfer in a plane-parallel geometry

In this paper, we formulate the radiative transfer problem as an initial value problem via a pair of nonlinear matrix differential equations (matrix Riccati equations or MREs) which describe the reflection (R) and transmission (T) matrices of the specific intensities in a plane-parallel geometry. One first computes R and T matrices of some small but finite layer thickness (equivalent optical thickness tau similar to 0.01) and then repetitively applies the doubling method to the reflection and transmission matrices R(tau) and T(tau) until reaching the desired layer thickness. The initial matrices R(tau(0)) and T(tau(0)) can be computed quite accurately by either of the following methods: multiple-order, multiple-scattering approximation, iterative method or fourth-order Runge-Kutta techniques. In addition, the reflection coefficient matrix of a semi-infinite medium satisfies an algebraic matrix equation which can be solved repetitively by a matrix method. MREs offer an alternative way of solving plane-parallel radiative transport problems. This method requires only elementary matrix operations (addition, multiplication and inversion). For vector and/or beam-wave radiative transfer problems, large matrices are required to describe the physics adequately, and the MRE method provides a significant reduction in computer memory and computation time.