Radiative Transfer through Arbitrarily Shaped Optical Media. Part I: A General Method of Solution

A general transform method is presented for studying problems of radiative transfer through absorbing, emitting and anisotropically scattering media exposed to arbitrary radiation conditions on its boundaries. The method permits quite arbitrary horizontal and vertical variability in the scattering and extinction properties of the medium bounded by a surface whose albedo and bidirectional reflection function varies from point to point. The technique developed incorporates a two-dimensional Fourier transform of the radiative transfer equation and a full Fourier expansion in azimuth. The general solution is based on the use of invariant imbedding principles in the form of doubling and adding algorithms. In developing these algorithms the principles of invariance are derived for three-dimensional geometry. Differences and similarities to the one-dimensional transfer problem are highlighted throughout. The method is applied to two special problems, namely the reflection by an atmosphere overlying or surface possessing an albedo step function and the transfer through an inhomogeneous Gaussian shaped medium.