Single-scattering solution for radiative transfer through a turbid atmosphere

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Abstract
A solution is presented to the radiative transfer of the solar irradiation through a turbid atmosphere, bounded by a Lambert surface, based on the single-scattering approximation, i.e., an assumption that a photon that underwent scattering either leaves the top of the atmosphere or strikes the surface. The solution depends on idealization of the scattering phase function of the aerosols. The equations developed here are subsequently applied to analyze quantitatively (1) the enhancement of the surface irradiation and (2) the enhancement of the scattered radiant emittance as seen from above the atmosphere, caused by the surface reflectance and subsequent atmospheric scattering. An order-of-magnitude error analysis is presented.