Similarity of radiative transfer equation: Error analysis of phase function truncation techniques

Abstract

Accurate radiance computations with highly peaked phase functions is a challenging problem. The developed truncation methods replace the peak of phase function using different approximations in the cone of forward scattering. The main goal of this paper is to employ a new integral form of similarity conditions to the error analysis of truncation techniques. This analysis emphasizes two main error sources of these methods from (1) truncation of Legendre series, and (2) truncation of the forward cone for peaked phase functions. The first error has an oscillating pattern and is effectively suppressed by the single scattering correction. The second, often overlooked, error manifests itself as a bias which weakly depends on the number of Legendre terms used in the solution unless it becomes comparable to the total order of Legendre expansion series. This paper presents a comparative theoretical and numerical error analysis of the Delta function method [15], Delta-fit method [7], and Delta-M method [21]. The Delta-M method, combined with the single scattering correction, is shown to provide the best overall accuracy for the intensity computations.