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

Authors: Feng Zhang; Kun Liu; Quan Yang; Kun Wu; Jian-Qi Zhao
Journal: Advances in Meteorology
 (2017)

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## 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.


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## Body

## 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 theδ-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. 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 as(1)Pcos⁡Θ=∑n=0NωnPncos⁡Θ,where Pn and Θ are the Legendre function and the scattering angle, respectively. N is the number of expansion terms. ωn, determined from the orthogonal property of Legendre polynomial, can be written as(2)ωn=2n+12∫-11Pcos⁡ΘPncos⁡Θdcos⁡Θ,where ω0=1, ω1=3g, and g is the asymmetry factor.Using the second kind of Chebyshev function, the phase function can be expanded as(3)Pcos⁡Θ=∑n=0NχnUncos⁡Θ,where Θ is the scattering angle and Un is the second kind of Chebyshev function. U0=1,U1=2x, U2=4x2-1. The second kind of Chebyshev polynomial Un of degree n is(4)Unx=∑k=0n/2-1kn-k!k!n-2k!2xn-2k.There is a recurrence formula for Un(5)Un+1x+Un-1x=2xUnx.The second kind of Chebyshev series is orthogonal polynomial with respect to the weighting function1-x2(6)∫-111-x2UmxUnxdx=12πδnm,where(7)δn,m=1n=m0n≠m,is the Kronecker delta.Similarly,χn can be written from the orthogonal property of the second kind of Chebyshev polynomial in the form(8)χn=2π∫-111-cos2ΘPcos⁡ΘUncos⁡Θdcos⁡Θ.


## 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 as(9)PHGcos⁡Θ=1-g21-2gcos⁡Θ+g23/2,where g 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.Figure 1
HG phase function (a) withg=0.8 and relative error of phase function (b) in 8, 16, and 24 terms.
(a)
(b)It can be seen in Figure1 that the reconstructed phase function becomes more accurate with the increasing of terms. On the whole, the results of Chebyshev polynomial expansion are more accurate compared to the Legendre polynomial expansion between 12 degrees and 168 degrees. In addition, Chebyshev polynomial provides better results at large scattering angles. On the contrary, Legendre polynomial has shown good results at forward (0-degree) and back (180-degree) angles.


### 3.2. Phase Function of Spherical Particles

For particles of equivalent spheres, Mie theory is primarily used to calculate the single-scattering properties. In the following, the phase functions of soot and water cloud at a wavelength ofλ=0.55 μm are shown.


#### 3.2.1. Soot

The radiative impact of soot is size distribution dependent. The soot size distribution is used to be described by the log-normal distribution(10)nr=1rσg2πexp⁡-ln2⁡r/rg2σg2,where rg=52nm and σg=0.42, which are obtained by fitting measurements of diesel soot emissions [12]. The phase function for the ensemble system is given by(11)Pcos⁡Θ=λ2/2π2∫i1Θ,r+i2Θ,rnrdr∫Qscarr2nrdr,where Qsca is the scattering efficiency, i1 is the square of vertical scattering amplitudes, and i2 is the square of horizontal scattering amplitudes. The asymmetry factor of soot g is 0.53 at wavelength λ=0.55μm.Figure2 shows the results of soot. The accuracy of the Chebyshev polynomial expansion is slightly better than that of the Legendre polynomial expansion in a 60–120-degree scattering angle. Because of the small size of soot particles, the phase functions reconstructed by 8 terms of Chebyshev and Legendre polynomials are both very close to the benchmark value of phase function.Figure 2
Phase function of soot (a) and relative error of phase function (b) in 8 terms.
(a)
(b)


#### 3.2.2. Water Cloud

The size distribution of water cloud droplets tends to closely resemble gamma distributions [13] as(12)nr=Arαe-βr,where r is the radius of the water cloud and A, α, and β are constants. The effective radius and effective variance are re=(α+3)/β and ve=1/(α+3), respectively. We adopted ve=0.172 as a constant value in cloud optical property parameterizations [14]. The effective radius is set as 5.89μm, with asymmetry factor g being 0.85.Figure3 shows the phase function for water cloud. The relative errors of Legendre polynomial can be up to 10,000% between the scattering angles of 60 degrees and 180 degrees, and while the relative errors of Chebyshev polynomial are about one order of magnitude less than those of Legendre polynomial in 24 terms, however, the Legendre polynomial gives better results at small scattering angle.Figure 3
Phase function of water cloud (a) and relative error of phase function (b) in 8, 16, and 24 terms.
(a)
(b)Figure4 shows the results of aerosol internally mixed with water cloud. In this case, the water cloud particle is the same as that in Figure 3 and the concentration of black carbon aerosol has been considered as 5000 ppb (parts per billion). It can be seen that the patterns of phase function and relative errors are similar to the results of pure water cloud.Figure 4
Phase function of black carbon internally mixed with water cloud (a) and relative error of phase function (b) in 8, 16, and 24 terms.
(a)
(b)


### 3.3. Phase Function of Nonspherical Particle

The shapes of dust and ice cloud particles are exclusively nonspherical. It is well known that the single-scattering properties of these particles are vastly different from those spherical particles. The scattering for nonspherical particles can be calculated by finite-difference time domain method [15], bridging technique [16], T-matrix method [17], and the improved geometric optics method [18].


#### 3.3.1. Dust Aerosol Phase Function

For dust aerosols, the radiative forcing depends strongly on their size distribution and the log-normal distribution is usually used(13)nr=1rN2πln⁡σgexp⁡-12ln⁡r-ln⁡rgln⁡rg2,where r and N are the radius of the equivalent sphere, with the same volume as a spheroid, and the number density of dust aerosols, respectively. σg and rg are the standard deviation and mode radius, respectively. The effective radius and the effective variance are re=rgexp⁡5ln2⁡σg/2 and ve=exp⁡(ln2⁡σg)-1, respectively. Dust particles are assumed to be spheroids [19, 20] with an aspect ratio of 1.7 [21, 22]. σg=2.15 and rg=1.90μm are set for dust coarse mode. The single-scattering properties of dust aerosols are from a dust database [23], which is calculated by the combination of the T-matrix and improved geometric optics method (IGOM) [24, 25]. The asymmetry factor g is 0.81 for coarse mode of dust at the wavelength of λ=0.55μm.Figure5 shows the results of phase function for coarse mode of dust. In the cases of 16- and 24-term reconstruction, both Chebyshev and Legendre polynomials can hardly fit the forward scattering peak well. In comparison with Legendre polynomial expansion, the results of Chebyshev polynomial expansion are much more accurate except for the forward scattering phase function, as the relative errors are reduced by one order of magnitude.Figure 5
Phase function of coarse mode of dust (a) and relative error of phase function (b) in 8, 16, and 24 terms.
(a)
(b)


#### 3.3.2. Ice Cloud Phase Function

Ice crystals with a hexagonal prism are selected for our study. In order to characterize the size distribution of ice cloud, a mean effective size as the ratio of ensemble particle volume and the particle projected area can be written as [26, 27](14)De=32∫VLnLdL∫ALnLdL,where V(L) and A(L) are the volume and projected area, respectively, L is a maximum dimension of an ice crystal, and nLdL denotes the number density of ice crystals in the size interval (L,L+dL). The single-scattering properties are from a database of ice crystals [28], which is calculated based on a combination of the T-matrix method, the improved geometric optics method (IGOM), and the Amsterdam discrete dipole approximation (ADDA). The asymmetry factor of ice cloud is g=0.786 at the wavelength of λ=0.55μm.Figure6 shows the phase function of ice cloud with De=50μm. Because of the strong forward scattering peak, the relative errors by Legendre polynomial expansion are up to 10,000%, while the accuracy of phase function is dramatically improved by using Chebyshev polynomial expansion which has been two orders of magnitude improvement in relative errors except for the small scattering angle.Figure 6
Phase function of ice cloud (a) and relative error of phase function (b) in 8, 16, and 24 terms.
(a)
(b)


## 3.1. Henyey-Greenstein Phase Function

The HG function [11] has a remarkable analytical property as(9)PHGcos⁡Θ=1-g21-2gcos⁡Θ+g23/2,where g 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.Figure 1
HG phase function (a) withg=0.8 and relative error of phase function (b) in 8, 16, and 24 terms.
(a)
(b)It can be seen in Figure1 that the reconstructed phase function becomes more accurate with the increasing of terms. On the whole, the results of Chebyshev polynomial expansion are more accurate compared to the Legendre polynomial expansion between 12 degrees and 168 degrees. In addition, Chebyshev polynomial provides better results at large scattering angles. On the contrary, Legendre polynomial has shown good results at forward (0-degree) and back (180-degree) angles.


## 3.2. Phase Function of Spherical Particles

For particles of equivalent spheres, Mie theory is primarily used to calculate the single-scattering properties. In the following, the phase functions of soot and water cloud at a wavelength ofλ=0.55 μm are shown.


### 3.2.1. Soot

The radiative impact of soot is size distribution dependent. The soot size distribution is used to be described by the log-normal distribution(10)nr=1rσg2πexp⁡-ln2⁡r/rg2σg2,where rg=52nm and σg=0.42, which are obtained by fitting measurements of diesel soot emissions [12]. The phase function for the ensemble system is given by(11)Pcos⁡Θ=λ2/2π2∫i1Θ,r+i2Θ,rnrdr∫Qscarr2nrdr,where Qsca is the scattering efficiency, i1 is the square of vertical scattering amplitudes, and i2 is the square of horizontal scattering amplitudes. The asymmetry factor of soot g is 0.53 at wavelength λ=0.55μm.Figure2 shows the results of soot. The accuracy of the Chebyshev polynomial expansion is slightly better than that of the Legendre polynomial expansion in a 60–120-degree scattering angle. Because of the small size of soot particles, the phase functions reconstructed by 8 terms of Chebyshev and Legendre polynomials are both very close to the benchmark value of phase function.Figure 2
Phase function of soot (a) and relative error of phase function (b) in 8 terms.
(a)
(b)


### 3.2.2. Water Cloud

The size distribution of water cloud droplets tends to closely resemble gamma distributions [13] as(12)nr=Arαe-βr,where r is the radius of the water cloud and A, α, and β are constants. The effective radius and effective variance are re=(α+3)/β and ve=1/(α+3), respectively. We adopted ve=0.172 as a constant value in cloud optical property parameterizations [14]. The effective radius is set as 5.89μm, with asymmetry factor g being 0.85.Figure3 shows the phase function for water cloud. The relative errors of Legendre polynomial can be up to 10,000% between the scattering angles of 60 degrees and 180 degrees, and while the relative errors of Chebyshev polynomial are about one order of magnitude less than those of Legendre polynomial in 24 terms, however, the Legendre polynomial gives better results at small scattering angle.Figure 3
Phase function of water cloud (a) and relative error of phase function (b) in 8, 16, and 24 terms.
(a)
(b)Figure4 shows the results of aerosol internally mixed with water cloud. In this case, the water cloud particle is the same as that in Figure 3 and the concentration of black carbon aerosol has been considered as 5000 ppb (parts per billion). It can be seen that the patterns of phase function and relative errors are similar to the results of pure water cloud.Figure 4
Phase function of black carbon internally mixed with water cloud (a) and relative error of phase function (b) in 8, 16, and 24 terms.
(a)
(b)


## 3.2.1. Soot

The radiative impact of soot is size distribution dependent. The soot size distribution is used to be described by the log-normal distribution(10)nr=1rσg2πexp⁡-ln2⁡r/rg2σg2,where rg=52nm and σg=0.42, which are obtained by fitting measurements of diesel soot emissions [12]. The phase function for the ensemble system is given by(11)Pcos⁡Θ=λ2/2π2∫i1Θ,r+i2Θ,rnrdr∫Qscarr2nrdr,where Qsca is the scattering efficiency, i1 is the square of vertical scattering amplitudes, and i2 is the square of horizontal scattering amplitudes. The asymmetry factor of soot g is 0.53 at wavelength λ=0.55μm.Figure2 shows the results of soot. The accuracy of the Chebyshev polynomial expansion is slightly better than that of the Legendre polynomial expansion in a 60–120-degree scattering angle. Because of the small size of soot particles, the phase functions reconstructed by 8 terms of Chebyshev and Legendre polynomials are both very close to the benchmark value of phase function.Figure 2
Phase function of soot (a) and relative error of phase function (b) in 8 terms.
(a)
(b)


## 3.2.2. Water Cloud

The size distribution of water cloud droplets tends to closely resemble gamma distributions [13] as(12)nr=Arαe-βr,where r is the radius of the water cloud and A, α, and β are constants. The effective radius and effective variance are re=(α+3)/β and ve=1/(α+3), respectively. We adopted ve=0.172 as a constant value in cloud optical property parameterizations [14]. The effective radius is set as 5.89μm, with asymmetry factor g being 0.85.Figure3 shows the phase function for water cloud. The relative errors of Legendre polynomial can be up to 10,000% between the scattering angles of 60 degrees and 180 degrees, and while the relative errors of Chebyshev polynomial are about one order of magnitude less than those of Legendre polynomial in 24 terms, however, the Legendre polynomial gives better results at small scattering angle.Figure 3
Phase function of water cloud (a) and relative error of phase function (b) in 8, 16, and 24 terms.
(a)
(b)Figure4 shows the results of aerosol internally mixed with water cloud. In this case, the water cloud particle is the same as that in Figure 3 and the concentration of black carbon aerosol has been considered as 5000 ppb (parts per billion). It can be seen that the patterns of phase function and relative errors are similar to the results of pure water cloud.Figure 4
Phase function of black carbon internally mixed with water cloud (a) and relative error of phase function (b) in 8, 16, and 24 terms.
(a)
(b)


## 3.3. Phase Function of Nonspherical Particle

The shapes of dust and ice cloud particles are exclusively nonspherical. It is well known that the single-scattering properties of these particles are vastly different from those spherical particles. The scattering for nonspherical particles can be calculated by finite-difference time domain method [15], bridging technique [16], T-matrix method [17], and the improved geometric optics method [18].


### 3.3.1. Dust Aerosol Phase Function

For dust aerosols, the radiative forcing depends strongly on their size distribution and the log-normal distribution is usually used(13)nr=1rN2πln⁡σgexp⁡-12ln⁡r-ln⁡rgln⁡rg2,where r and N are the radius of the equivalent sphere, with the same volume as a spheroid, and the number density of dust aerosols, respectively. σg and rg are the standard deviation and mode radius, respectively. The effective radius and the effective variance are re=rgexp⁡5ln2⁡σg/2 and ve=exp⁡(ln2⁡σg)-1, respectively. Dust particles are assumed to be spheroids [19, 20] with an aspect ratio of 1.7 [21, 22]. σg=2.15 and rg=1.90μm are set for dust coarse mode. The single-scattering properties of dust aerosols are from a dust database [23], which is calculated by the combination of the T-matrix and improved geometric optics method (IGOM) [24, 25]. The asymmetry factor g is 0.81 for coarse mode of dust at the wavelength of λ=0.55μm.Figure5 shows the results of phase function for coarse mode of dust. In the cases of 16- and 24-term reconstruction, both Chebyshev and Legendre polynomials can hardly fit the forward scattering peak well. In comparison with Legendre polynomial expansion, the results of Chebyshev polynomial expansion are much more accurate except for the forward scattering phase function, as the relative errors are reduced by one order of magnitude.Figure 5
Phase function of coarse mode of dust (a) and relative error of phase function (b) in 8, 16, and 24 terms.
(a)
(b)


### 3.3.2. Ice Cloud Phase Function

Ice crystals with a hexagonal prism are selected for our study. In order to characterize the size distribution of ice cloud, a mean effective size as the ratio of ensemble particle volume and the particle projected area can be written as [26, 27](14)De=32∫VLnLdL∫ALnLdL,where V(L) and A(L) are the volume and projected area, respectively, L is a maximum dimension of an ice crystal, and nLdL denotes the number density of ice crystals in the size interval (L,L+dL). The single-scattering properties are from a database of ice crystals [28], which is calculated based on a combination of the T-matrix method, the improved geometric optics method (IGOM), and the Amsterdam discrete dipole approximation (ADDA). The asymmetry factor of ice cloud is g=0.786 at the wavelength of λ=0.55μm.Figure6 shows the phase function of ice cloud with De=50μm. Because of the strong forward scattering peak, the relative errors by Legendre polynomial expansion are up to 10,000%, while the accuracy of phase function is dramatically improved by using Chebyshev polynomial expansion which has been two orders of magnitude improvement in relative errors except for the small scattering angle.Figure 6
Phase function of ice cloud (a) and relative error of phase function (b) in 8, 16, and 24 terms.
(a)
(b)


## 3.3.1. Dust Aerosol Phase Function

For dust aerosols, the radiative forcing depends strongly on their size distribution and the log-normal distribution is usually used(13)nr=1rN2πln⁡σgexp⁡-12ln⁡r-ln⁡rgln⁡rg2,where r and N are the radius of the equivalent sphere, with the same volume as a spheroid, and the number density of dust aerosols, respectively. σg and rg are the standard deviation and mode radius, respectively. The effective radius and the effective variance are re=rgexp⁡5ln2⁡σg/2 and ve=exp⁡(ln2⁡σg)-1, respectively. Dust particles are assumed to be spheroids [19, 20] with an aspect ratio of 1.7 [21, 22]. σg=2.15 and rg=1.90μm are set for dust coarse mode. The single-scattering properties of dust aerosols are from a dust database [23], which is calculated by the combination of the T-matrix and improved geometric optics method (IGOM) [24, 25]. The asymmetry factor g is 0.81 for coarse mode of dust at the wavelength of λ=0.55μm.Figure5 shows the results of phase function for coarse mode of dust. In the cases of 16- and 24-term reconstruction, both Chebyshev and Legendre polynomials can hardly fit the forward scattering peak well. In comparison with Legendre polynomial expansion, the results of Chebyshev polynomial expansion are much more accurate except for the forward scattering phase function, as the relative errors are reduced by one order of magnitude.Figure 5
Phase function of coarse mode of dust (a) and relative error of phase function (b) in 8, 16, and 24 terms.
(a)
(b)


## 3.3.2. Ice Cloud Phase Function

Ice crystals with a hexagonal prism are selected for our study. In order to characterize the size distribution of ice cloud, a mean effective size as the ratio of ensemble particle volume and the particle projected area can be written as [26, 27](14)De=32∫VLnLdL∫ALnLdL,where V(L) and A(L) are the volume and projected area, respectively, L is a maximum dimension of an ice crystal, and nLdL denotes the number density of ice crystals in the size interval (L,L+dL). The single-scattering properties are from a database of ice crystals [28], which is calculated based on a combination of the T-matrix method, the improved geometric optics method (IGOM), and the Amsterdam discrete dipole approximation (ADDA). The asymmetry factor of ice cloud is g=0.786 at the wavelength of λ=0.55μm.Figure6 shows the phase function of ice cloud with De=50μm. Because of the strong forward scattering peak, the relative errors by Legendre polynomial expansion are up to 10,000%, while the accuracy of phase function is dramatically improved by using Chebyshev polynomial expansion which has been two orders of magnitude improvement in relative errors except for the small scattering angle.Figure 6
Phase function of ice cloud (a) and relative error of phase function (b) in 8, 16, and 24 terms.
(a)
(b)


## 4. Summary

In scattering and radiative transfer calculations, the complicated phase functions have to be represented by certain polynomial expansions. The second kind of Chebyshev polynomial is proposed to approximate the scattering phase functions. Compared with Legendre polynomial expansion in the scattering cases of HG, spherical and nonspherical particles, Chebyshev polynomial expansion shows more accurate results at a wide range of scattering angles, especially for large particles like ice crystals, where the relative errors are reduced by two orders of magnitude, while in the forward peak of the phase function, the Legendre polynomial produces better results than Chebyshev polynomial. It is concluded in this short note that Chebyshev polynomial is a better choice at a wide range of scattering angles. Besides, the Legendre polynomial always has better results in the forward scattering angles. In further research,δ-fit or δ-M [4, 5] should be used in the calculation to produce more accurate results.



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