The truncation problem

Abstract The truncation procedure is an approximation commonly used in plane-parallel radiative transfer codes, which consists in removing the forward scattering peak observed in the phase function of large particles (few microns) or cloud droplets, due to diffraction. This approximation allows faster calculations but introduces biases in the radiances modelled at the ground-based level and also at the top of the atmosphere in narrow angular intervals (cloud bows and glory). In this study, we recalled the principle of the truncation and present a new method to correct its flaws. In comparison with previous studies, we present here a full comprehensive correction and analysis of the truncation biases for the downward and upward radiances. For ground-level measurements, we add to truncated calculations an approximate expression of the successive scatterings in the truncated forward peak to restore the solar aureole. In case of satellite measurements, we reduce the biases found for narrow angular signatures simply by changing the expression of the primary scattering. After correction, maximal errors do not exceed 0.001 for the degree of linear polarization for optical thickness smaller than 2.0, which is a sufficient accuracy for most applications based on polarimetric measurements. This correction is available for both total and polarized radiances and is now implemented in successive order of scattering code used in the Generalized Retrieval of Aerosol and Surface Properties (GRASP) algorithm. The analysis of the problem is based on the method of successive orders, but the suggested corrections are applicable for any other resolution method (e.g. adding-doubling).