Comparison of Chebyshev and Legendre Polynomial Expansion of Phase Function of Cloud and Aerosol Particles

Feng Zhang,Quan Yang,Kun Wu,Kun Liu,Jian-Qi Zhao

doi:10.1155/2017/1835169

Abstract

Chebyshev and Legendre polynomial expansion is used to reconstruct the Henyey-Greenstein phase function and the phase functions of spherical and nonspherical particles. The result of Legendre polynomial expansion is better than that of Chebyshev polynomial for around 0-degree forward angle, while Chebyshev polynomial expansion produces more accurate results in most regions of the phase function. For large particles like ice crystals, the relative errors of Chebyshev polynomial can be two orders of magnitude less than those of Legendre polynomial.

Highlights

The dynamics and transmission of the atmosphere rely on the distribution and magnitude of the net radiative heating of the atmosphere system
In order to improve the parameterization of phase function, several techniques have been developed such as the δ-M method [4], the δ-fit method [5], GT approximation in geometrical truncation [6], MRTD scattering model [7, 8], Q-space analysis [9], and invariant imbedding T-matrix method [10]
It can be seen that the patterns of phase function and relative errors are similar to the results of pure water cloud

Summary

Introduction

The dynamics and transmission of the atmosphere rely on the distribution and magnitude of the net radiative heating of the atmosphere system. In order to improve the parameterization of phase function, several techniques have been developed such as the δ-M method [4], the δ-fit method [5], GT approximation in geometrical truncation [6], MRTD (multiresolution time domain) scattering model [7, 8], Q-space analysis [9], and invariant imbedding T-matrix method [10]. These techniques tend to remove the strong forward scattering peak instead of seeking a fast convergence expression of phase function.

Theoretical Background

Comparison of Phase Function

Summary