A review of the matrix-exponential formalism in radiative transfer

- The matrix exponential description of radiative transfer is outlined.
- Several solution methods are derived using the matrix exponential formalism.
- The equivalence between the methods in computing the reflection matrix is shown.
- The Waterman's approximation is refined by including an additional term.

This paper outlines the matrix exponential description of radiative transfer. The eigendecomposition method which serves as a basis for computing the matrix exponential and for representing the solution in a discrete ordinate setting is considered. The mathematical equivalence of the discrete ordinate method, the matrix operator method, and the matrix Riccati equations method is proved rigorously by means of the matrix exponential formalism. For optically thin layers, approximate solution methods relying on the Padé and Taylor series approximations to the matrix exponential, as well as on the matrix Riccati equations, are presented. For optically thick layers, the asymptotic theory with higher-order corrections is derived, and parameterizations of the asymptotic functions and constants for a water-cloud model with a Gamma size distribution are obtained.