Convergence acceleration of radiative transfer equation solution at strongly anisotropic scattering

All the numerical methods of the radiative transfer equation (RTE) solution are based on the replacement of scattering integral by the finite sum. The main problem of such scattering integral representation is the presence of singularities in the radiance angular distribution (RAD): they cannot be included in any quadrature formula in essence (Krylov, 2006). To solve this problem various methods are used by different researchers: Wiscombe (1977), Thomas and Stamnes (2002), Rozanov et al. (2005) and others. The physical model of radiative transfer theory is the ray approximation. In this approximation any break in the boundary conditions spreads into the depth of the medium and generates singularities in the RAD. Scattering in the medium gives, along with the singularities, an anisotropic part in the light field. The difficulties in the calculation of the anisotropic part of the solution led to various truncation methods of the scattering anisotropy that are not completely true: in the general 3-D medium geometry the core of the problem is not the scattering, but the presence of singularities in the RAD. Krylov (2006) showed that the most effective and accurate way of including the singularities in the quadrature formula is their analytical, perhaps approximate, representation and an analytical integration - the method of singularities elimination.