Fast and accurate radiance calculations using truncation approximation for anisotropic scattering phase functions

The phase function for solar light scattering by large particles such as cloud droplets is strongly anisotropic due to very strong peaking in the forward direction. This creates numerical difficulties when attempting to calculate accurate reflected and transmitted radiances, which are important for remote sensing of atmospheric and surface properties. A popular approach uses the delta function to approximate the forward-scattering peak in a fraction of energy and a limited number of polynomial terms or a geometrically truncated function for the remaining fraction (so-called truncation approximations). This article compares and discusses several methods for fast and accurate calculations using truncation approximations. When using a single truncation approximation for all scattering orders, large biases appear in directions near the solar and anti-solar points. As shown here, high accuracy can be obtained using different truncation approximations depending on the order of scattering. Of particular importance is the use of phase functions close to the exact phase functions for the first few orders of scattering. Applying the method in combination with the Monte Carlo (MC) method, in which the truncation fraction for a scattering order depends on the scattering angle at the previous scattering event, obtains accurate radiance calculations under almost all geometrical and optical conditions, including in directions near the solar point. Because the method also reduces computational noise due to the MC sampling of radiance, it is useful for fast and accurate radiance calculations for cloudy atmospheres.