A Unified Formulation of Radiative Transfer in Plane-Parallel Atmospheres Based on General Decomposition of Radiance. Part I: The Theory

Abstract

Abstract A unified formulation of the general decomposition of radiance and the radiative transfer equation (RTE) of plane-parallel atmospheres and the general solutions of the generally decomposed RTE system (GD-RTES) are presented. It is shown that the eigenvalues of the coefficient matrix of the GD-RTES are real and the eigenvectors are independent of each other when the single-scattering albedo is less than unity; the general solution of the GD-RTES for a homogeneous layer can then be expressed in combinations of the exponential functions of the optical depth. The solution for nonhomogeneous atmosphere–surface systems can be obtained through either the matrix operator method or the linear system method. An outline of the general procedure is given, and a detailed example will be provided in Part II of this work. The formulation can serve as a meta-algorithm from which the prototypes of new algorithms can be developed and tested. Some of the conventional methods such as the discrete ordinate method and the spherical harmonic method can be considered as instances of the unified formulation. With the unified formulation, some interesting topics about the RTE and its solution can be discussed more generally, such as those about correction for energy conservation, treatment of boundary conditions, and transformation for numerical stability; some of the methods that have already proved useful to these issues in the well-established algorithms can be generalized to be available to the other methods that can be derived from the unified formulation.