Advances in Meteorology

Volume 2017, Article ID 1835169, 10 pages

https://doi.org/10.1155/2017/1835169

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

^{1}Key Laboratory of Meteorological Disaster, Ministry of Education (KLME)/Joint International Research Laboratory of Climate and Environment Change (ILCEC)/Collaborative Innovation Center on Forecast and Evaluation of Meteorological Disaster (CICFEMD), Nanjing University of Information Science and Technology, Nanjing 210044, China^{2}State Key Laboratory of Severe Weather, Chinese Academy of Meteorological Sciences, Beijing 100081, China^{3}Shenzhen National Climate Observatory, Shenzhen 518040, China^{4}State Key Laboratory of Numerical Modeling for Atmospheric Sciences and Geophysical Fluid Dynamics, Institute of Atmospheric Physics, Chinese Academy of Sciences, Beijing 100029, China

Correspondence should be addressed to Jian-Qi Zhao; nc.ca.pai.liam@iqnaijoahz

Received 26 May 2017; Accepted 30 July 2017; Published 18 September 2017

Academic Editor: Jia Yue

Copyright © 2017 Feng Zhang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### 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.

#### 1. Introduction

The dynamics and transmission of the atmosphere rely on the distribution and magnitude of the net radiative heating of the atmosphere system. In the stratosphere, the net radiative heating depends solely on the imbalance between infrared radiative loss and local absorption of solar UV radiation [1]. The distribution of the radiative sources and sinks exerts a zero order control on the large-scale seasonally varying zonal wind fields and mean temperature in the stratosphere [2]. It is known that the large-scale circulation in the stratosphere is different from that in the troposphere, and eddies are as elementary to the circulation as the differential solar radiative heating. Radiative processes play a key role in driving global climate change and establishing temperature structure of the atmosphere [3]. Phase function always displays very intricate structures, as the peak value of the forward scattering could be several orders of magnitude larger than that of the back scattering. Currently, Legendre polynomial expansion is widely used in representing the scattering phase function and it is sensitive to the forward scattering peak of phase function. Thus, the Legendre polynomial series converge very slowly; it could take literally thousands of Legendre polynomial terms to reconstruct the original phase function.

In order to improve the parameterization of phase function, several techniques have been developed such as the - method [4], the - method [5], 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. Even if the strong forward scattering peak has been removed by the above techniques, the phase function itself still needs to be parameterized with limited terms of Legendre polynomial expansion. How to represent the scattering phase function accurately and efficiently is the goal of this study. In the following, various phase functions are expanded by Legendre polynomial and the second kind of Chebyshev polynomial. In Section 3, the accuracies of the scattering phase functions reconstructed by Chebyshev and Legendre polynomial expansions are discussed. A short summary is given in Section 4.

#### 2. Theoretical Background

The scattering phase function using Legendre polynomial expansion can be written aswhere and are the Legendre function and the scattering angle, respectively. is the number of expansion terms. , determined from the orthogonal property of Legendre polynomial, can be written aswhere , , and is the asymmetry factor.

Using the second kind of Chebyshev function, the phase function can be expanded aswhere is the scattering angle and is the second kind of Chebyshev function. , . The second kind of Chebyshev polynomial of degree isThere is a recurrence formula for

The second kind of Chebyshev series is orthogonal polynomial with respect to the weighting function whereis the Kronecker delta.

Similarly, can be written from the orthogonal property of the second kind of Chebyshev polynomial in the form

#### 3. Comparison of Phase Function

In this section, the phase function expansions by the Legendre and the second kind of Chebyshev polynomials are compared for accuracy and efficiency. The samples will be taken from Henyey-Greenstein (HG) phase function and the phase functions of spherical and nonspherical particles.

##### 3.1. Henyey-Greenstein Phase Function

The HG function [11] has a remarkable analytical property aswhere is the asymmetry factor. The phase functions reconstructed in various terms of Chebyshev and Legendre polynomials against the benchmark results of HG phase functions are shown in Figure 1. In Figure 1(a), the phase functions reconstructed by Chebyshev and Legendre polynomials are shown for 8 terms (top panel), 16 terms (middle panel), and 24 terms (bottom panel); in Figure 1(b), the relative errors of phase function are shown.