A comparative study of the delta-Eddington and Galerkin quadrature methods for highly forward scattering of photons in random media

Abstract

A versatile and accurate treatment for the highly forward-peaked phase function in the three-dimensional (3D) radiative transfer equation (RTE) based on the discrete ordinates method (DOM) is crucial for biomedical optics. Our first objective was to compare the delta-Eddington (dE) and Galerkin quadrature (GQ) methods. The dE method decomposes the phase function into a purely forward-peaked component and the other component, and expands the other component by Legendre polynomials as well as the finite order Legendre expansion (FL) method does. The GQ method conducts the weighting procedure in addition to the Legendre expansion. Although it was reported that both methods can provide the accurate results for calculations of the RTE, the versatility of both methods is still unclear. The second objective was to examine a possibility of a conjunction of the GQ method with the dE method, called as the GQ-dE method, which has the advantages of both methods. We examined numerical errors in the moment conditions of the phase function using the FL, dE, GQ, and GQ-dE methods at various types and orders of the quadrature sets, mainly in the region of the errors induced by the angular discretization using the DOM. The errors were reduced by the dE method from those by the FL method, however the error reduction depended on the types and orders of the quadrature sets. Meanwhile, the errors were significantly reduced by the GQ and GQ-dE methods, regardless of the quadrature sets. We also verified the numerical calculations of the time-dependent 3D RTE by the analytical solution of the RTE for homogeneous media in the region of the scattering length scale, where the highly forward-peaked phase function strongly influences the RTE-results. The errors in the RTE-results were similar to those in the moment conditions. Our results suggest the higher versatility and accuracy of the GQ and GQ-dE methods than those of the FL and dE methods.