Generalization of Marshak boundary condition for non-Lambert reflection

Abstract

In this paper we generalize the Marshak lower boundary condition in the method of spherical harmonics (MSH) for an arbitrary law of surface reflection. The obtained analytical solution is based on the bilinear interpolation of surface bidirectional reflection distribution function (BRDF). Recurrent formulas presented here allow to avoid numeric integration and calculate matrices of boundary conditions for arbitrary orders of azimuthal and Legendgre polynomial expansions of MSH. For practical applications, we have studied the effect of discretization of the BRDF angular grid on the accuracy of calculation of reflected diffuse radiance. Simulations conducted for a large number of vegetative covers, soils and sand, show that an accuracy of 0.2% can be achieved with M=11 azimuthal harmonics at N=21 grid knots in cosine of zenith angle. Calculations over a ruffled water surface are much more demanding. For wind speed v≥10 m/ s , an accuracy of 0.2% requires discretization of M≥40, N≥51.