Modified differential approximation for radiative transfer in a scattering medium with specularly reflecting boundaries

A modified differential approximation proposed by Olfe is extended to the treatment of a scattering medium with specularly reflecting boundaries. The extension admits of direction-dependent reflectivities at boundaries. The reflectivity and transmissivity, which results from either diffuse or collimated incident radiation on a slab, are presented as a function of refractive index, optical thickness and scattering albedo. The apparent properties are obtained from the unmodified differential approximation, modified differential approximation and integral equation approach. Comparison of the approximate results with the correct results obtained from the integral equation approach shows that, in general, the modified differential approximation is superior to the unmodified differential approximation.