Approximations of phase function in calculating the spectral albedo of snow surface with multiple scattering

Four types of approximations of the Mie phase function were studied in calculating multiple scattering by snow particles with the doubling method. These involve the two renormalizations of Hansen and Grant, the delta-M method and direct truncation. These four approximations were compared for snow surface albedo with effective grain radii of 50, 200 and 1000μ m in a wavelength region from 0.3 to 3.0μ m with the delta-Eddington approximation as a reference. In the Hansen's renormalization, the maximum albedo error exceeds 0.1 for snow with an effective radius of 1000 μ m at small solar zenith angles. The delta-M method overestimates snow albedos at all solar zenith angles in a wavelength region smaller than 1.4 μ m for snow with effective radius of 1000 μ m. This is due to insufficient angle resolution (0.1° in a scattering angle region less than 2°) in the forward peak region of the look-up table of the Mie phase function. It has been shown that even with ten times higher resolution in the scattering angle region less than 10° a sufficiet accuracy could not be obtained for an effective radius of 1000μ m in a wavelength region smaller than 0.6μ m. Reasonable results were obtained by the Grant's renormalization and direct truncation approximation for all cases of effective grain radii studied. It was also found that these methods save computation time and memory because sufficient accuracy is obtained even with a low angle resolution of 0.1° in the forward peak region of phase function. In direct truncation, the result was not sensitive to the choice of a truncation angle between 5° and 20°.