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A. F. Cheng (Applied Phys Lab Johns Hopkins Univ)
New, accurate numerical solutions of the radiative transfer equation are compared with the Hapke (1981) analytic approximation which is widely used to analyze planetary observations. The numerical solutions use the Ambartsumian invariance principle as do the well-known Chandrasekhar H-function solutions, but the invariance principle has been re-expressed in a form which allows high order-accurate numerical integrations without any required interpolations. The new numerical solutions reproduce the Chandrasekhar H-function solutions for Legendre phase functions to within O(10-4), but the new solutions allow single scattering phase functions of arbitrary form. For example, the numerical solution and the Hapke solution are compared for the Henyey-Greenstein phase function with albedo 0.4, width parameter b = 0.2, and forward scattering parameter c = 0.7. In this case, the error of the Hapke solution generally increases as incidence angles increase, i.e., lowest errors occur near normal incidence. This contrasts with the dependence on emission angles, where the greatest errors occur at intermediate emission angle. The Hapke solution for bidirectional reflectance I/F is systematically as much as 4 errors in the Hapke solution depend in general on the form of the scattering phase function, and are typically greater for backscattering phase functions where the errors can exceed 20% Results from an extensive survey will be reported.