Four-stream spherical harmonic expansion approximation for solar radiative transfer

This paper presents a four-stream extension of the δ-Eddington approximation by considering the higher-order spherical harmonic expansion in radiative intensity. By using the orthogonality relation of the spherical harmonic functions, the derivation of the solution is fairly straightforward. Calculations show that the δ-four-stream spherical harmonic expansion approximation can reduce the errors in reflection, transmission, and absorption substantially in comparison with the δ-Eddington approximation. For the conservative scattering case, the error of the new model is generally less than 1% for optical thicknesses greater than unity except for grazing incident solar beam. For nonconservative scattering cases (single scattering albedo ω = 0.9), the error is less than 5% for optical thicknesses greater than unity, in contrast to errors of up to 20% or more under the δ-Eddington approximation. This model can also predict the azimuthally averaged intensity to a good degree of accuracy. The computational time for this model is not as intensive as for the rigorous numerical methods, owing to the analytical form of the derived solution.