Initial Value Problem for Exit Function for Multiple Light Scattering

Abstract

To alleviate the nonuniqueness problem of the solution of the integral equation derived by Hovenier for the exit function, a generating function for reflection and transmission functions for homogeneous, plane-parallel media, we have derived an integrodifferential equation, so that the function can be obtained in the form of the initial value problem. The numerical stability of our equation was then tested against the problem of multiple scattering in layers of a wide range of optical thickness, characterized by the Henyey-Greenstein phase function of varying degrees of anisotropy. The main features of the present technique are (1) an application of the fast invariant embedding method of Sato et al. and (2) an effective interpolation scheme to estimate numerically the limiting values of the singular terms involved in the equation. The results were compared with those of the doubling method. It was found that the maximum relative errors of the reflection function, ranging from 0.005% to 0.03%, generated from the exit function was 1 to 2 orders of magnitude smaller than those of the transmission function. Nevertheless, even the transmission functions rarely show errors larger than 2%, unless the optical thickness is substantially less than 0.1. Our results indicate that the equation presented in this work has a high degree of numerical stability and that it can be potentially useful for multiple scattering computations.