Anisotropic scattering in inhomogeneous media—1. A new representation formula for the solution of the auxiliary integral equation for the source function

A new representation formula for the solution of the auxiliary integral equation for the source function in inhomogeneous, anisotropically scattering media is presented. It involves two new functions Φ and ψ of two variables instead of the original five variables. This generalizes earlier results of Kagiwada et al. (1969) and Sobolev (1972) applicable to homogeneous atmospheres. The corresponding Bellman-Krein formula for the resolvent kernel is also derived. The present representation for the solution of Fredholm integral equations of second kind with unsymmetric kernels provides a new approach to radiative transfer in anisotropic inhomogeneous media.