Fast Radiative Transfer Solution Using the Discrete Ordinate Radiative Transfer Method and the Small-Angle Approximation

By dividing the phase function and radiance into the forward and the diffuse components, the original radiative transfer equation can be decomposed into two equations: one only including the forward component, named the forward equation, and one including both the forward and diffuse components, named the diffuse equation. The forward equation can be approximately solved by using the small-angle approximation (SAA). The SAA is an analytical approach associated with the Gaussian distribution with coefficients given by the forward phase function. The diffuse equation can be numerically solved by using the discrete ordinate radiative transfer (DISORT) method. The solar source term from the forward component can be approximated in terms of the Dirac delta function. The source term scattered from other scattering directions is rearranged so that the effective phase functions for single-scattering and multiple-scattering are different. Subsequently, the homogeneous and the particular solutions in DISORT are different to the conventional DISORT solutions. The general solution is given by summing the homogeneous and particular solutions subject to appropriate boundary conditions. The total solution of the original function is given by the summation of the forward solution based on the SAA and the diffuse solution based on the DISORT. For a phase function with a strong diffraction peak, the advantage of the present method is that the DISORT computation can be efficient because the forward peak is considered by using the SAA.