Radiative transfer models for light scattering from planetary surfaces

New, accurate numerical solutions of the radiative transfer equation are co mpared with the Hapke [1981, 1984, 1986] analytic approximation, which is w idely used in planetary data analyses. The numerical solutions use the Amba rtsumian invariance principle as do the well-known Chandrasekhar [1960] H f unction solutions. The invariance principle has been reexpressed in a form which allows high order-accurate numerical integrations without any require d interpolations. The new numerical solutions reproduce the Chandrasekhar H function solutions for Legendre phase functions but also allow single-scat tering phase functions of arbitrary form. Accurate numerical solutions of t he radiative transfer equation for Henyey-Greenstein phase functions reveal that the errors in the Hapke model for estimating bidirectional reflectanc e values range from <2% rms for dark surfaces like the Moon to <7% rms for bright surfaces such as Europa. Further comparisons demonstrate that Hapke model fitting procedures estimate single particle scattering albedo values to within <3% for both dark and bright surfaces.