Modeling of scattering properties of mineral aerosols using modified beta function

Authors

  • Hong Tang,

    1. College of Metrology and Measurement Engineering, China Jiliang University, Hangzhou, China
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  • Jian-Zhong Lin

    Corresponding author
    1. College of Metrology and Measurement Engineering, China Jiliang University, Hangzhou, China
    2. State Key Laboratory of Fluid Power Transmission and Control, Zhejiang University, Hangzhou, China
    • Corresponding author: J-Z. Lin, College of Metrology and Measurement Engineering, China Jiliang University, Hangzhou 310018, China. (linjz@cjlu.edu.cn)

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Abstract

[1] In the combined experiment from Amsterdam and modeling study, the spheroid is applied to represent the real mineral aerosols, and the single scattering properties of spheroids are computed using the T matrix combined with the improved geometric optical method in the wider size range and aspect ratio range. After that a modeling method for the scattering matrices of the feldspar, quartz, and red clay is proposed. In this approach, the ensemble-average scattering matrix is calculated by using the number density distribution deduced from the fit of the measured normalized projected surface area distribution with the modified beta function and the shape distribution obtained by the least squares fit of all the six measured scattering elements. On the other hand, the models of scattering matrices using the lognormal function to fit the measured normalized number distribution and fit the measured normalized projected surface area distribution are also conducted as comparison. Furthermore, the asymmetry parameter is calculated in order to further verify the reliability of the proposed method. Simulation experiments indicate that the simulated scattering matrices with the proposed method are rather close to the measured scattering matrices for the three mineral aerosols, and this proposed numerical model can provide a simple, reliable, and efficient method to reproduce the scattering properties of the mineral aerosols based on the Amsterdam database.

1 Introduction

[2] Mineral aerosol particles are important constituents of the atmospheres of Earth, and they influence the Earth's radiation budget, air quality, and cloud formation [Nousiainen and Vermeulen, 2003; Muñoz et al., 2007]. Recently, scattering of electromagnetic waves on the mineral aerosol particles becomes of growing importance in the remote sensing of the Earth's atmosphere [Schmidt et al., 2009]. A thorough understanding and explanation of the impact of aerosols on the global radiative forcing and atmospheric visibility degradation phenomena require knowledge of the aerosol optical properties [Kaskaoutis and Kambezidis, 2006].

[3] The scattering properties of mineral aerosol particles are an important diagnostic tool for the remote detection and the retrieval of information on the physical properties of the grains [Mishchenko et al., 2003]. But accounting for light scattering of mineral aerosols has been known to be a source of significant uncertainties, and it is also a very difficult problem to solve accurately [Yang et al., 2007; Volten et al., 2001; Kahnert, 2004]. This is partly because the aerosol optical properties that are fundamentally determined by the complicated morphologies, compositions, and complex refractive indices of aerosol particles are not well understood, and the mineral particles seem to be with a broad range of irregular shapes and distributed in the size from submicron to millimeter [Yang et al., 2007; Muñoz et al., 2001]. In fact, a large mass fraction of the mineral aerosols is composed of the irregular particles, and the nonsphericity of the aerosol particles strongly affects their scattering properties, including the polarization and their spectral behavior [Volten et al., 2001]. So many efforts have been focused on the research on the light scattering of the nonspherical mineral aerosol particles [Vilaplana et al., 2004; Kahnert et al., 2002a; Mishchenko and Travis, 1994; Meland et al., 2012; Dubovik et al., 2006].

[4] In recent years, a large amount of experimental light scattering data with the light scattering facility in Amsterdam have become available and have wide applications [Volten et al., 2005; Mishchenko et al., 2000; Hovenier et al., 2003]. The Amsterdam light scattering database contains the measured scattering matrices as functions of the scattering angles and the size distributions as well as some shape information. However, the laboratory measurements do not provide sufficiently complete information about the light scattering and certainly do not include all types of aerosols occurring in the atmosphere [Volten et al., 2001]. Therefore, it is desirable to compute the scattering properties numerically and find a simple and yet sufficiently accurate model to reproduce the scattering properties of real mineral aerosols. Moreover, the numerical calculations may help us to analyze the measured results adequately. In general, the combined experimental data and a powerful numerical model is the ideal method to analyze the scattering properties of real mineral aerosols [Muñoz and Hovenier, 2011; Laan et al., 2009; Nousiainen et al., 2011b].

[5] Due to the complicated shapes of mineral aerosols, most case computations of light scattering have to be replaced by some simple particle models [Muñoz et al., 2011]. Model calculations for spheroid, polyhedral prism, cylinder, and Gaussian random sphere have been carried out and compared with various experimental data [Nousiainen et al., 2006, 2011a; Kahnert et al., 2002b; Veihelmann et al., 2006]. Mishchenko et al. [1997] modeled the phase function for the dustlike aerosols using a mixture of spheroids with random orientations. They concluded that, although the natural dust particles were typically mixtures of irregular shapes, their cumulative phase function could be adequately modeled using a wide aspect ratio distribution of prolate and oblate spheroidal particles [Mishchenko et al., 1997]. From the modeling standpoint, it is highly desirable to use the simple and symmetrical particles in order to keep the computation efficiency [Kahnert, 2004]. Because the simplest nonspherical particle is the spheroid, understanding the scattering by the spheroids is very helpful to determine the light scattering of nonspherical particles. Furthermore, the spheroid has a simple axially symmetric shape, and one can use it to span shapes ranging from flat oblate disks to spheres to elongated prolate needles by varying just the aspect ratio [Mishchenko et al., 1997; Kahnert et al., 2005]. It is already clear that many mineral aerosols present the nonspherical shapes and have broad distribution ranges in the size and shape. When using an ensemble of nonspherical particles to fit the measured scattering matrix, the average over the particle size, orientation, and shape may reduce the information contained in the scattering data, which allows for the use of the simple model particles for reproducing the scattering matrix of an ensemble of particles with more complex shapes [Kahnert, 2004]. Thus, in this paper, we consider the spheroid to reproduce the measured scattering matrices of three well-characterized mineral aerosol samples in the Amsterdam light scattering database, including the feldspar, quartz, and red clay.

[6] In order to simulate the ensemble-average scattering matrix, the single scattering properties of spheroids must be calculated beforehand. The single scattering properties of spheroids have been studied for a long time. Existing numerical methods such as the discrete dipole approximation (DDA) and finite difference time domain technique (FDTD) are more flexible but are very time consuming and in practice can be applied only to particle size comparable to or smaller than the incident wavelength [Zhou et al., 2003]. The geometric optical method (GOM) is an approximation method that can calculate the scattering properties of spherical and nonspherical particles at larger size region [Zhou et al., 2003]. To achieve a definite accuracy, the conventional GOM should be applied for particles whose size parameters are large enough. Whereas the minimum limitation of particle size can be decreased greatly when the improved geometric optical method (IGOM) developed by Yang and Liou is applied to calculate the scattering properties of particles [Yang et al., 2005; Yang and Liou, 1996]. On the other hand, the fastest and most powerful numerical tool for rigorously computing spheroid light scattering is the T matrix method. Some researchers have used the T matrix method to study the scattering matrices of mineral particles [Nousiainen and Vermeulen, 2003; Schmidt et al., 2009; Kahnert, 2004]. However, these scattering matrices are determined only within moderate sizes and moderate aspect ratios. The maximal aspect ratio for the prolate spheroid and the reciprocal of minimum aspect ratio for the oblate spheroid decrease with the increasing size parameter when the T matrix convergence is achieved [Mishchenko and Travis, 1998].

[7] Since the real mineral aerosols have broad ranges of shape distributions and size distributions, the research of the light scattering for particles with larger size parameters is also necessary [Mishchenko et al., 1997]. In fact, there is no single computational program that can cover the entire size parameter ranging from the Rayleigh regime to the geometric optics regime. So we decide to combine the T matrix with the IGOM to calculate the single scattering properties of spheroids. Through this combination, the single scattering properties of spheroids can be calculated in the wider size range and aspect ratio range.

[8] Actually, it needs two distributions, the size distribution n(r) and shape distribution p(ε) (r is the radius of the projected surface area–equivalent sphere in micrometers, r is in the size range from rmin to rmax, rmin and rmax are the minimum and maximal values of radius r, n(r)dr denotes the fraction of the total number of particles per unit volume having radii between r and r + dr, ε is the aspect ratio of a spheroid, for a prolate spheroid ε > 1, for an oblate spheroid ε < 1, ε is in the aspect ratio range from εmin to εmax, εmin and εmax are the minimum and maximal values of ε, p(ε) denotes the probability distribution of ε), to describe the whole distribution of polydisperse spheroids [Kahnert et al., 2002b]. Many efforts have been focused on the determination of appropriate spheroid distributions to simulate the scattering matrices of real mineral particles [Nousiainen and Vermeulen, 2003; Kahnert, 2004; Nousiainen et al., 2006; Kahnert et al., 2005]. In these researches, the lognormal function is commonly used to obtain the size distribution of spheroids whose characteristic parameters are calculated by the fit of the measured distribution data from the Amsterdam database first, and then the shape distribution of spheroids is obtained through several manners to reproduce the scattering properties of real mineral aerosols. Usually, the equiprobable distribution is used as the shape distribution. In addition, obtaining the shape distribution by the fit of the measured scattering matrix is also applied. Actually, not only the shape distribution of spheroids but also the size distribution is of crucial importance. Meanwhile, it is very necessary to know how differences in the size distribution of spheroids can affect their scattering matrices. Different particle systems may have different size distribution forms. The lognormal function, however, has a very limited fitting capacity, and it may not always adequately describe the actual size distributions of mineral aerosols [Yu and Standish, 1990; Popplewell et al., 1988; Peleg and Normand, 1986].

[9] Motivated by this, the present study focuses on the size distribution of spheroids, and we decide to use a more precise and universal function, instead of the lognormal function, to describe the size distribution of spheroids. Meanwhile, the effects of the size range and aspect ratio range on the scattering matrices of spheroids are also considered after using the T matrix combined with the IGOM to calculate the single scattering properties of spheroids. In doing so, we will obtain the size distribution of spheroids accurately and then reproduce the scattering properties of real mineral aerosol particles from the Amsterdam light scattering database successfully.

2 Methods and Computer Simulations

2.1 Fit of Particle Distributions

[10] According to the electromagnetic scattering theory, if the light is scattered by an ensemble of randomly oriented particles and time reciprocity applies, the Stokes vectors of the incident light and the scattered light are related by a 4 × 4 scattering matrix [van de Hulst, 1981; Mishchenko et al., 2002]. When the ensemble of randomly oriented particles and their mirror particles are presented in equal numbers, the scattering matrix F has the block diagonal form [van de Hulst, 1981; Mishchenko et al., 2002]:

display math(1)

where F is called the scattering matrix, θ is the scattering angle, Fij are the elements of the scattering matrix, i, j = 1–4, the element Fij contains information about the size, shape, and complex refractive index of the scatter [van de Hulst, 1981; Mishchenko et al., 2002; Hovenier et al., 2003].

[11] In the Amsterdam light scattering database, the scattering matrices by ensembles of some natural irregular particles can be used easily. All elements, except F11(θ), are given relative to F11(θ), and F11(θ) is normalized to unity for θ = 30°. Moreover, the scattering matrix elements are measured at scattering angles from θ = 5° to 170° with 5° angular resolution and from 170° to 173° with 1° angular resolution [Volten et al., 2006a; Volten et al., 2006b]. The measurement data are available both at wavelengths 442 and 633 nm. Since the size parameters of particles at 442 nm are about 30% larger than those at 633 nm, the modeling of the scattering properties is performed at 633 nm in this paper [Nousiainen and Vermeulen, 2003; Nousiainen et al., 2006].

[12] The laboratory measurements, however, do not provide the scattering properties of all types of aerosols, and they are not suitable to be used at an arbitrary wavelength in the visible and infrared regimes. Thus, it is necessary to compute the scattering properties with some models numerically [Muñoz et al., 2007]. In addition, numerical simulations may help us to analyze the measured results and then be studied independently.

[13] In this paper, we decide to use an ensemble of spheroids to model the mineral aerosols based on the Amsterdam database. When using the spheroid with random orientation to calculate its scattering matrix and scattering cross section, the exact solution covering all sizes and shapes does not exist. Currently, the fastest and most powerful numerical tool for rigorously computing spheroid light scattering is the T matrix method. In the reference [Mishchenko and Travis, 1998], the authors demonstrated the converged size parameters were varied with different input conditions. It is clear that the maximal convergent size parameter greatly depends on the complex refractive index and asphericity, and it can be significantly decreased when the particle becomes more aspherical. On the other hand, it is necessary to calculate the single scattering properties of larger spheroids in view of the wide size range of real mineral aerosols. Therefore, due to the computational constraints of T matrix method, we use the T matrix to calculate the scattering matrix and scattering cross section of the spheroid in the small size range up to the convergence point. From that point onward, we continue calculation using the IGOM method. Only in this way, we can obtain the single scattering properties of spheroids with good accuracy in the wider size range and aspect ratio range.

[14] For the samples of feldspar, quartz, and red clay in the database, the normalized projected surface area distribution S(logr) has been measured using the laser diffraction method (S(logr) describes the distribution of projected surface area of the projected surface area–equivalent sphere, inline image). Meanwhile, the normalized number distribution N(logr) is also given in the database because the number distribution is often required for the numerical calculation (N(logr)dlogr is the relative number fraction of projected surface area–equivalent sphere per unit volume in the size range from logr to logr + dlogr, inline image) [Volten et al., 2006a, 2006b]. Actually, N(logr) is deduced from S(logr), S(logr) = c3r2N(logr), c3 is determined by the equation inline image. These data are listed in tabular form, and they are easily accessible from the Amsterdam database.

[15] Figure 1 shows the normalized projected surface area distributions and the normalized number distributions of three samples. Here since all these data are from the Amsterdam database, we also call them the measured normalized projected surface area distribution and the measured normalized number distribution, respectively. According to Figure 1, it is clear to see that the distribution forms of three samples are different, and the three samples have different size ranges. Moreover, these distributions are broad and partly overlap. On the other hand, it is suggested that in any function, when used to describe the distribution of a real monomodal particle system, it is reasonably expected that there exists a minimum size and a maximum size. For r = rmin and r = rmax, its distribution values are always zero [Yu and Standish, 1990]. So we can see that the measured normalized projected surface area distribution of the three samples, with a bell shape, is more appropriate to this condition.

Figure 1.

Distributions of three mineral aerosol samples.

[16] In the numerical modeling of scattering properties from the Amsterdam database, we need to know information about the number density distribution n(r) (here the number density distribution is also the size distribution of spheroids, the relation between n(r) and N(logr) is listed in equation (2)) and the shape distribution to obtain the ensemble-average scattering matrix (see equation (3)). In doing this work, most researches choose the lognormal function to fit the measured normalized number distribution in order to obtain the number density distribution first and then adjust the shape distribution to reproduce the scattering matrices of mineral aerosols [Nousiainen et al., 2006; Kahnert, 2004]. Actually, the real particle system with a finite size range may not have a perfectly smooth size distribution, and the lognormal function may not always adequately describe the actual particle distribution regulation. Especially, the lognormal function will fail to describe the particle system whose size distribution is skewed to the left [Popplewell et al., 1988]. To overcome these difficulties, a search for an alternative function to fit the measured distribution data is made.

display math(2)

where c1 is a constant; it can be determined by making the normalization of n(r) over the entire size range to be one [Volten et al., 2005].

display math(3)

where Fij(θ, λ) is the ensemble-average scattering matrix element, ε is in the range from ε1 to εM with M different values; inline image, Csca is the scattering cross section with a equivalent radius r, wavelength λ, and aspect ratio εk; inline image is the scattering matrix element of a spheroid with a given scattering angle θ, radius r, wavelength λ and aspect ratio εk [Meland et al., 2012; Nousiainen et al., 2006].

[17] It has been generally accepted that there is no unique function which can represent all the particle size distributions encountered in reality, but there are still several versatile functions to represent the most commonly used particle size distributions. Yu suggested that the Johnson's SB function be used as a general function for the monomodal particle size distribution. The Johnson's SB function is explicitly given by [Yu, 1994]

display math(4)

where M1 and σ2 are the characteristic parameters of Johnson's SB function.

[18] Popplewell et al. proposed the modified beta function to represent the monomodal particle size distribution which can be described by [Popplewell et al., 1988]

display math(5)

where α and m1 are the characteristic parameters of modified beta function.

[19] Figure 2 shows the fitting results of the Feldspar sample. In Figure 2a, “○ measured” denotes the measured normalized projected surface area distribution and “beta,” “SB,” “L-N” denote the modified beta, Johnson's SB, and lognormal functions are used to fit the measured normalized projected surface area distribution, respectively. According to these fitting results, we deduce the corresponding number density distributions (see equation (6)), and the deduced number density distributions are presented in Figure 2b. In Figure 2b, “○ measured” denotes the feldspar number density distribution is directly deduced from the measured normalized number distribution (here we also call “○ measured” in Figure 2b the measured number density distribution). In this study, the least squares is used as the fitting method. Since the real particle populations have a finite size range determined by physical considerations. As a result, it is suggested that any function, when used to describe a particle size distribution, should satisfy, besides the general conditions for a frequency distribution, the following boundary condition: it is safe to assume rmin = 0 and rmax can be determined beforehand by the physical considerations. So there are two unknown parameters to describe the modified beta and Johnson's SB functions.

Figure 2.

Fitting results of feldspar sample.

display math(6)

where c2 is a constant; it can be determined by making the normalization of n(r) over the entire size range to be one [Volten et al., 2005].

[20] In Figure 2, it appears that the fit of the measured normalized projected surface area distribution for the Feldspar sample with the modified beta function is better than that with the lognormal function and the Johnson's SB function. Moreover, the number density distribution deduced from the fit of the measured normalized projected surface area distribution with the modified beta function is obviously excellent. The slight difference with the modified beta function relative to the measured number density distribution is perhaps unavoidable, because it is impossible that the fit is exactly the same as the measured number density distribution. In fact, the particle size distribution follows the law of probability. Johnson presented a general investigation of the mathematical properties of the Johnson's SB distribution function and its role in the theory of statistical distribution functions [Johnson, 1949]. Meanwhile, Popplewell et al. have proved that the modified beta function has the ability to describe the symmetric distributions as well as asymmetric distributions skewed to either the right or the left because of the independent characteristic parameters of the modified beta function [Popplewell et al., 1988]. However, the Johnson's SB function has weaker ability to describe the distributions skewed to the left. On the other hand, the lognormal function may be regarded as an extreme case of the Johnson's SB function; it is likely that a similar property may also be found for the Johnson's SB function and lognormal function [Yu, 1994]. Thus, the modified beta function is more suitable for the description of the feldspar distribution in the Amsterdam database.

[21] Figure 3 presents the fitting results of the Feldspar number density distributions. In Figure 3, “L-N number” and “beta number” denote that we use the lognormal function and modified beta function to fit the measured normalized number distribution and then deduce the number density distributions according to equation (2). Meanwhile, “L-N area” and “beta area” denote the number density distributions are deduced from the fit of the measured normalized projected surface area distribution with the lognormal function and modified beta function, respectively.

Figure 3.

Fitting results of number density distribution for feldspar sample.

[22] It can be observed from Figure 3 that there is reasonably good agreement between the measured number density distribution and the number density distribution deduced by fitting the measured normalized projected surface area distribution with the modified beta function, especially for particles with small radii. However, there is some obvious difference between the measured number density distribution and the number density distribution with the “L-N number” at the two ends of the total radius range. Moreover, the number density distribution with the “L-N area” and “beta number” are all not very satisfactory.

[23] Figures 4 and 5 give the fitting results of the red clay and quartz, respectively. In Figures 4a and 5a, “□measured” denotes the measured normalized projected surface area distribution, and “L-N,” “beta” denote we use the lognormal and modified beta functions to fit the measured normalized projected surface area distribution, respectively. In Figures 4b and 5b, the meanings of “□ measured,” “L-N number” and “beta area” are the same as those in Figure 3. According to these fitting results, it is evident that the number density distribution deduced from the fit of the measured normalized projected surface area distribution with the modified beta function is better than that deduced from the fit of the measured normalized number distribution with the lognormal function. There is poor agreement in the number density distribution between “□ measured” and “L-N number,” especially when r is greater than 10 µm. It should be noted that, because of the small percentage of very large size particles over the total number density distribution, the impact on the difference of the number density distribution between “□ measured” and “beta area” for much larger particles is not very significant. So the number density distributions of red clay and quartz samples can be evaluated by means of the fit of the measured normalized projected surface area distribution with the modified beta function quite successfully. The main advantages of the modified beta function are that it has the ability to describe symmetric distributions as well as asymmetric distributions skewed to either the right or the left.

Figure 4.

Fitting results of red clay sample.

Figure 5.

Fitting results of quartz sample.

[24] According to the above analysis, we can use the modified beta function to fit the measured normalized projected surface area distribution and then deduce the corresponding number density distribution. In doing so, the number density distribution can be reproduced very well for the feldspar, quartz, and red clay in the Amsterdam database. The number density distribution with the “beta area” is clearly superior to that with the “L-N number.”

2.2 Modeling of Scattering Matrices

[25] After obtaining the number density distribution deduced from the fit of the measured normalized projected surface area distribution with the modified beta function, we can reproduce the scattering matrix by assuming a known shape distribution or by varying the shape distribution of the spheroids until a good fit of the measured scattering matrix is obtained [Muñoz et al., 2006].

[26] Figure 6 shows the comparison of the measured and simulated scattering matrices for the feldspar sample. It should be noted that all the legends in the subfigures are the same, and only one of them is displayed. The horizontal ordinate is the scattering angle, and the vertical coordinate is the scattering matrix element Fij. “○ measured” denotes the measured scattering matrix elements from the Amsterdam database, and the measured errors are denoted by error bars. In this test, “discrete” denotes we use the measured number density distribution as n(r). That is to say the distribution data at the corresponding radii for this n(r) are all given indirectly according to the database, and we cannot change these values and the discrete intervals. Meanwhile, the equiprobable shape distribution and the shape distribution obtained by least squares fit of all the six measured scattering elements are used, respectively. The aspect ratio of the spheroid is varied from 1/2.6 to 2.6 with M = 17, and we choose the following values: 1/2.6, 1/2.4, 1/2.2, 1/2, 1/1.8, 1/1.6, 1/1.4, 1/1.2, 1, 1.2, 1.4, 1.6, 1.8, 2, 2.2, 2.4, 2.6. And then the integration is performed over the measured number density distribution according to equation (3). Since there are 31 discrete points in the overall radius range from 0.07 to 13 µm for the feldspar in the database, the trapezoidal integration rule is used, and the number of integration interval is 30 (for the measured number density distribution of feldspar, what we have are only the 31 sets of data including 31 distribution data and 31 corresponding radii). In this paper, the least squares minimization routine is used to produce the scattering matrix numerically. And the specific execution approach can refer to the reference [Kahnert, 2004]. Since the exact value of complex refractive index for the feldspar is unknown, m = 1.5 + 0.001i is used according to the information provided in the database.

Figure 6.

Comparison of measured and simulated scattering matrices for feldspar (1/2.6 ≤ ε ≤ 2.6).

[27] In Figure 6, it is clear to see that the simulated scattering matrix follows the general trend of the measured scattering matrix. However, this comparison shows relatively significant differences between the measured and simulated scattering matrices for −F12/F11, F22/F11, and −F34/F11. Thus, the simulated scattering matrix cannot adequately reproduce the measured scattering matrix by using the measured number density distribution as n(r), even in the case that the shape distribution of spheroids is obtained by least squares fit of all the six measured scattering elements.

[28] In Figure 7, the simulated scattering matrices are calculated using the lognormal and modified beta functions. That is to say the integration in equation (3) can be performed over the above two distribution functions with more size bins. Actually, after obtaining the characteristic parameters of the lognormal and modified beta functions by the fit of the measured normalized projected surface area distribution/the measured normalized number distribution, the distribution values of the lognormal and modified beta functions at any radius point are known. Here “L-N number” denotes the number density distribution is deduced from the fit of the measured normalized number distribution with the lognormal function. “L-N area” and “beta area” denote the number density distribution is deduced from the fit of the measured normalized projected surface area distribution with the lognormal and modified beta functions, respectively. Since the size range of the feldspar sample is about 0.07 µm < r < 13 µm, and the computation is very time consuming for the scattering properties of particles at larger particle sizes, especially in the case that more size bins are employed. We try to truncate the particles at the larger sizes. Here particles larger than 2.5μm are excluded from this simulation [Nousiainen et al., 2006; Nousiainen and Vermeulen, 2003; Nousiainen et al., 2011b]. In the reference [Nousiainen and Vermeulen, 2003], the effect of the truncation had been tested. Due to truncation, about 7% of the total cross-sectional surface area of the sample particles was excluded from the simulations, and the truncation at 2.5 µm had shown a negligible impact on the ensemble-average scattering matrix. Furthermore, the integration in equation (3) is performed from 0.07 to 2.5 µm in step of 0.02 µm. The aspect ratio of spheroids is varied from 1/2.6 to 2.6 with M = 17, and the shape distribution is obtained by least squares fit of all the six measured scattering elements. In Figure 7, the simulated scattering matrices using the lognormal function or modified beta function to fit the measured normalized number distribution or the measured normalized projected surface area distribution yield a clear improvement over those shown in Figure 6. Figure 8 gives the similar comparison as that in Figure 7, but with different aspect ratio range. Here 1/3 ≤ ε ≤ 3 with M = 21. As shown in Figure 8, the agreement is surprisingly good for F11, F22/F11, F33/F11, and F44/F11 when using the shape distribution by the least squares fit of all the six measured scattering elements and the number density distribution deduced from the fit of the measured normalized projected surface area distribution with the modified beta function to compute the ensemble-average scattering matrix.

Figure 7.

Comparison of measured and simulated scattering matrices for feldspar (1/2.6 ≤ ε ≤ 2.6, 0.07 < r < 2.5).

Figure 8.

Comparison of measured and simulated scattering matrices for feldspar (1/3 ≤ ε ≤ 3, 0.07 < r < 2.5). In each subfigure, circle denotes the measured scattering matrix element, solid line denotes n(r) is obtained by the “beta area” and p(ε) is obtained by the “least squares fit for Fij,” dotted line denotes n(r) is obtained by the “L-N number” and p(ε) is obtained by the “least squares fit for Fij,” dash-dotted with light gray color denotes n(r) is obtained by the “L-N number” and p(ε) is assumed to be the equiprobable distribution.

[29] Meanwhile, we also can notice in Figure 8 that the F11 obtained by the “beta area” model is remarkably well in the forward and intermediate scattering angles. However, the differences compared with the measured data for F33/F11 and F44/F11 when using the “L-N number” model are very obvious. In the scattering matrix, −F12/F11 describes the degree of linear polarization of the single scattering light, and it has a characteristic bell shape at intermediate scattering angle. It is clear to see that the simulated −F12/F11 with the “beta area” model follows the general trend like the measured −F12/F11, and it also has positive polarization values at scattering angles from about 90° to 150°. In addition, we also see that the use of equiprobable shape distribution of spheroids cannot produce a significant improvement in the reproduction of the scattering matrix corresponding to the least squares fit shape distribution. What is more important is the introduction of larger variation range in the aspect ratio can greatly improve the agreement with the measured scattering matrix.

[30] Figure 9 describes the shape distribution obtained by the least squares fit of all the six measured scattering elements. The input conditions are the same as those in Figure 8, and the number density distribution uses the “beta area” model. The horizontal ordinate is the aspect ratio ε, and the vertical coordinate is the shape distribution p(ε). Obviously, the distribution of weights for the shape distribution by least squares fit of all the six measured scattering elements is not equiprobable, and it has large weights at the two edges of the aspect ratio range. Figure 10 gives the relative errors of the simulated scattering matrices for feldspar. The input conditions are the same as those in Figure 8. The calculated equations for the relative errors for Fij are(F11) = abs((F11,mes − F11,sim)/F11,mes).100, Err(Fij/F11) = abs(Fij,mea/F11,mea − Fij,sim/F11,sim.100).

Figure 9.

Least squares fit shape distribution for feldspar. (1/3 ≤ ε ≤ 3, M = 21, 0.07 < r < 2.5, n(r) is obtained by the “beta area” and p(ε) satisfiesinline image).

Figure 10.

Relative errors of the simulated scattering matrices for feldspar (1/3 ≤ ε ≤ 3, 0.07 < r < 2.5). In each subfigure, solid line denotes n(r) is obtained by the “beta area” and p(ε) is obtained by the “least squares fit for Fij,” dash-dotted with light gray color denotes n(r) is obtained by the “L-N number” and p(ε) is obtained by the “least squares fit for Fij,” dotted line denotes n(r) is obtained by the “L-N number” and p(ε) is assumed to be the equiprobable distribution.

[31] Figure 11 shows the comparison of the measured and simulated scattering matrices for the feldspar. In this simulation, the integration in equation (3) is performed from 0.07 to 13 µm in step of 0.07 µm. The aspect ratio of spheroids is varied from 1/2.6 to 2.6 with M = 17. Clearly, the increase of the size range cannot greatly improve the agreement with the measured scattering matrix when the range of aspect ratio is not sufficiently wide, even if in the case that the modified beta function is applied to the fit of the measured normalized projected surface area distribution and the shape distribution is obtained by least squares fit of all the six measured scattering elements.

Figure 11.

Comparison of measured and simulated scattering matrices for feldspar (1/2.6 ≤ ε ≤ 2.6, 0.07 < r < 13). In each subfigure, circle denotes the measured scattering matrix element, solid line denotes n(r) is obtained by the “beta area” and p(ε) is obtained by the “least squares fit for Fij,” dotted line denotes n(r) is obtained by the “L-N number” and p(ε) is obtained by the “least squares fit for Fij,” dash-dotted line with light gray color denotes n(r) is obtained by the “L-N number” and p(ε) is assumed to be the equiprobable distribution.

[32] From Figures 6 to 11, it is seen that the model, in which the number density distribution of feldspar sample is deduced from the fit of the measured normalized projected surface area distribution with the modified beta function and the shape distribution is obtained by least squares minimization fit of all the six measured scattering elements, can yield equally good reproduction of the scattering matrix compared with that using the number density distribution deduced from the fit of the measured normalized number distribution with the lognormal function. The simulated scattering matrix with the least squares fit shape distribution is better than that with the equiprobable shape distribution when other input conditions are the same. It should be noted that the variation in the aspect ratio can result in a larger variation in the scattering matrix. So the range of the aspect ratio should be as wide as possible. That is why we apply the T matrix combined with the IGOM to calculate the single scattering properties of spheroids.

[33] Figure 12 shows the comparison of the measured and simulated scattering matrices for the quartz sample. There are 39 discrete points in the total radius range from 0.07 to 53 µm in the Amsterdam database. That is to say the distribution values (S(logr) and N(logr)) are known at the corresponding 39 discrete radius points. In Figure 12, we perform the truncation for particles larger than 2.5 µm. The aspect ratio of spheroids is varied from 1/3 to 3 with M = 21, and the shape distribution is obtained by the least squares fit of all the six measured scattering elements. For the quartz sample, we use m = 1.54 given in the database. Here “discrete” denotes the measured number density distribution in the radius range from 0.07 to 2.5 µm is used as n(r). That is to say this number density distribution is deduced from the measured normalized number distribution, and the distribution data at the corresponding radii for this n(r) are given indirectly according to the database. Since there are 21 discrete points in the radius range from 0.07 to 2.5 µm for the quartz in the database, the trapezoidal integration rule is used and the number of integration interval is 20 in the “discrete” model. In the two models of “L-N number” and “beta area,” the integration in equation (3) is performed from 0.07 to 2.5 µm in step of 0.02 µm, respectively. Figure 13 depicts the similar comparison as that in Figure 12 but with different size range. In Figure 13, we also perform the truncation for particles larger than 13μm.

Figure 12.

Comparison of measured and simulated scattering matrices for quartz (1/3 ≤ ε ≤ 3, 0.07 < r < 2.5). In each subfigure, ○ denotes the measured scattering matrix element, solid line denotes n(r) is obtained by the “beta area” and p(ε) is obtained by the “least squares fit for Fij,” dotted line denotes n(r) is obtained by the “L-N number” and p(ε) is obtained by the “least squares fit for Fij,” dash-dotted with light gray color denotes n(r) is obtained by the “discrete” and p(ε) is obtained by the “least squares fit for Fij.”

Figure 13.

Comparison of measured and simulated scattering matrices for quartz (1/3 ≤ ε ≤ 3, 0.07 < r < 13). In each subfigure, circle denotes the measured scattering matrix element, solid line denotes n(r) is obtained by the “beta area” and p(ε) is obtained by the “least squares fit for Fij,” dotted line denotes n(r) is obtained by the “L-N number” and p(ε) is obtained by the “least squares fit for Fij,” dash-dotted with light gray color denotes n(r) is obtained by the “L-N area” and p(ε) is obtained by the “least squares fit for Fij.”

[34] According to the comparison in Figures 12 and 13, it is evident that the simulated scattering matrix with 0.07 µm < r < 13 µm yields a clear improvement over that with 0.07 µm < r < 2.5 µm. From Figure 1, it is clear to see that the quartz has a wider distribution range and larger particle size. Especially, the measured normalized projected surface area distribution of quartz occurs broadening phenomena at logr larger than 0.5. So the light scattering of quartz at larger particles has an obvious effect on the ensemble-average scattering matrix. Thus, the simulated scattering matrix with truncation at 13 µm is closer to the measured scattering matrix. Poor reproduction of the measured scattering matrix is achieved with 0.07 µm < r < 2.5 µm even if the aspect ratio is varied from 1/3 to 3. That is to say, we should not make the truncation at 2.5 µm if we want to obtain good reproduction of the measured scattering matrix for the quartz sample. In Figure 13, the shape of −F34/F11 shows an almost perfect agreement between the measurement and simulation with “least squares fit for Fij, beta area” model. The −F12/F11 with the “least squares fit for Fij, beta area” model also shows obvious good fit compared with other simulated models. Meanwhile, the F11, F33/F11, and F44/F11 using this model present qualitative agreement with the measured scattering elements, and F11 is much flatter at scattering angles ranging from 120° to 160°. The F22/F11 with “least squares fit for Fij, beta area” model is almost 1 when the angles close to the forward direction and increases again at backscattering angles.

[35] Figure 14 shows the comparison of the measured and simulated scattering matrices for the red clay. For the red clay sample, there are 37 discrete points in the total radius range from 0.07 to 38 µm. That is to say the radii and their distribution values (S(logr) and N(logr)) are known at the 37 discrete points. In this test, we also perform the truncation for particles larger than 2.5 µm, and the measured number density distribution in the radius range from 0.07 to 2.5 µm is used as n(r). Since there are 21 discrete points in the radius range from 0.07 to 2.5 µm for the red clay in the database, the trapezoidal integration rule is used and the number of integration interval is 20 in the “discrete” model. The aspect ratio of spheroids is varied from 1/3 to 3 with M = 21, and the shape distribution is obtained by least squares fit of all the six measured scattering elements. Figure 15 describes the similar comparison as that in Figure 14 but with equiprobable shape distribution.

Figure 14.

Comparison of measured and simulated scattering matrices for red clay (1/3 ≤ ε ≤ 3, n(r) is obtained by the “discrete” and p(ε) is obtained by the “least squares fit for Fij”).

Figure 15.

Comparison of measured and simulated scattering matrices for red clay (1/3 ≤ ε ≤ 3, n(r) is obtained by the “discrete” and p(ε) is assumed to be the equiprobable distribution).

[36] The exact value of complex refractive index for the red clay is unknown. As shown in Figures 14 and 15, there are no significant differences in the scattering elements, expect −F12/F11 and −F34/F11, when the complex refractive index of red clay is varied. That is to say −F12/F11 and −F34/F11 are very sensitive to the change of the complex refractive index of red clay. In addition, we find very a little effect on the scattering matrix when the real part of the complex refractive index is varied from 1.5 to 1.7. In general, we decide to use m = 1.7 + 0.0001i to simulate the scattering matrix of red clay.

[37] Figure 16 gives the comparison of the measured and simulated scattering matrices for the red clay. In this test, the integration of n(r) in equation (3) is performed from 0.07μm to 13μm in step of 0.07μm. The aspect ratio of spheroid particles is varied from 1/3 to 3 with M = 21. Figure 17 presents the similar comparison as that in Figure 16 but with different radius range (the integration of n(r) in equation (3) is performed from 0.07 to 2.5 µm in step of 0.02 µm).

Figure 16.

Comparison of measured and simulated scattering matrices for red clay (1/3 ≤ ε ≤ 3, 0.07 < r < 13). In each subfigure, circle denotes the measured scattering matrix element, solid line denotes n(r) is obtained by the “beta area” and p(ε) is obtained by the “least squares fit for Fij,” dotted line denotes n(r) is obtained by the “L-N number” and p(ε) is obtained by the “least squares fit for Fij,” dash-dotted with light dray color denotes n(r) is obtained by the “beta area” and p(ε) is assumed to be the equi-probable distribution.

Figure 17.

Comparison of measured and simulated scattering matrices for red clay (1/3 ≤ ε ≤ 3, 0.07 < r < 2.5). In each subfigure, circle denotes the measured scattering matrix element, solid line denotes n(r) is obtained by the “beta area” and p(ε) is obtained by the “least squares fit for Fij,” dotted line denotes n(r) is obtained by the “L-N number” and p(ε) is obtained by the “least squares fit for Fij,” dash-dotted with light dray color denotes n(r) is obtained by the “L-N number” and p(ε) is assumed to be the equiprobable distribution.

[38] According to the comparison in Figures 16, 17, it is clearly seen that the simulated scattering elements with 0.07 µm < r < 2.5 µm show relatively large deviations corresponding to those with 0.07 µm < r < 13 µm, even if the aspect ratio is varied from 1/3 to 3. That is because the red clay has a wider distribution at large size region relative to the feldspar, and the light scattering of red clay at larger particles has an obvious effect on the ensemble-average scattering matrix. So the truncation size of size distribution for the red clay should be as large as possible, and at the same time the determination of the truncation point should consider the computation resource. In Figure 16, we can obtain reasonably good reproduction of the red clay scattering matrix by assuming the number density distribution is deduced from the fit of the measured normalized projected surface area distribution with modified beta function, and the shape distribution is obtained by least squares fit of all the six measured scattering elements in the wider size range and aspect ratio range.

[39] According to the analysis from Figures 6 to 17, we propose a modeling method of the scattering matrix based on the Amsterdam database. In this method, the number density distribution of an ensemble of spheroids is deduced from the fit of the measured normalized projected surface area distribution with the modified beta function, and then the shape distribution is obtained by the least squares fit of all the six measured scattering elements. It should be noted that the size range and aspect ratio range of spheroids need to be chosen carefully according to the distribution characteristics of real samples. In doing so, the numerical calculation of scattering matrix gives very good agreement with the measured data for the feldspar, quartz, and red clay from the Amsterdam database.

[40] Through the T matrix combined with the IGOM, the single scattering properties of spheroids can be obtained in the wider size range and aspect ratio range. The calculation method of the single scattering properties with this combination is only the necessarily premise foundation. On that basis, excellent reproduction results can be achieved by using the “least squares fit for Fij, beta area” model within the wider size range and aspect ratio range. However, the simulated scattering matrices with the “least squares fit for Fij, L-N number” and “least squares fit for Fij, L-N area” models are poorly reproduced even if the single scattering properties of spheroids are calculated with T matrix combined with the IGOM in the wider size range and aspect ratio range.

[41] By just using the modified beta function to fit the measured normalized projected surface area distribution, rather than the measured normalized number distribution, the number density distribution can be determined extremely well. This fitting approach is very simple and yet highly effective. It sufficiently utilizes the distribution information provided in the database and is well applicable to the feldspar, quartz, and red clay. Meanwhile, perfect reproduction of the measured scattering matrix can be obtained with only two kinds of scattering inputs including the scattering cross section and the scattering matrix with this proposed method.

2.3 Calculation of Asymmetry Parameter

[42] In order to further verify the reliability of the proposed method above mentioned, we also use different models to compute the asymmetry parameter. The asymmetry parameter is defined as [Veihelmann et al., 2004]:

display math(7)

where P11 is the phase function, it can be obtained by F11, and the relationship between P11 and F11 is described in the reference [Volten et al., 2006b].

[43] Once obtaining the phase function in the full angles from 0° to 180°, the asymmetry parameter can be calculated with equation (7). Owing to the lack of the measured scattering element F11 for the scattering angles in the ranges 0°-5° and 173°-180°, we cannot use this F11 to calculate the asymmetry parameter [Nousiainen et al., 2006; Liu et al., 2003; Laan et al., 2009]. Some extrapolation methods have been used for obtaining the synthetically measured F11 (that is, the measured F11 combined with the extrapolated F11) in the full range from 0° to 180° [Volten et al., 2006b; Liu et al., 2003]. Here we use the extrapolation method developed by Liu et al. in which the theoretical Lorenz-Mie computation for the projected area–equivalent sphere is applied in the forward scattering direction, and a cubic spline extrapolation is used in the backscattering direction [Liu et al., 2003]. In doing so, we can determine the synthetically measured F11 in the full angles from 0° to 180° and then obtain the measured asymmetry parameter (that is, the asymmetry parameter is calculated by the synthetically measured F11). It should be noted that it need no extrapolation for the asymmetry parameter obtained by the simulated scattering matrix.

[44] Tables 1-3 list the asymmetry parameters for the feldspar, quartz, and red clay. In these tables, the asymmetry parameters are obtained through the phase function calculated by different numerical models and the phase function calculated by the synthetically measured F11, respectively. In general, the asymmetry parameter with the “least squares fit for Fij, beta area” model is the closest to the measured asymmetry parameter.

Table 1. Asymmetry Parameters Calculated From Measured and Simulated Scattering Element F11 for Feldspar (1/3 ≤ ε ≤ 3, 0.07 < r < 13)
Asymmetry Parameter
MeasuredL-N Number, Least Squares Fit for FijBeta Area, Least Squares Fit for FijL-N Number, Equiprobable
0.73480.72000.72120.7078
Table 2. Asymmetry Parameters Calculated From Measured and Simulated Scattering Element F11 for Quartz (1/3 ≤ ε ≤ 3, 0.07 < r < 13)
Asymmetry Parameter
MeasuredL-N Number, Least Squares Fit for FijBeta Area, Least Squares Fit for FijL-N Area, Equiprobable
0.68500.71260.70870.6733
Table 3. Asymmetry Parameters Calculated From Measured and Simulated Scattering Element F11 for Red Clay (1/3 ≤ ε ≤ 3, 0.07 < r < 13)
Asymmetry Parameter
MeasuredL-N Number, Least Squares Fit for FijBeta Area, Least Squares Fit for FijBeta Area, Equiprobable
0.75130.62860.67320.6266

3 Conclusions

[45] In this paper, the scattering properties of three mineral aerosols from the Amsterdam database are studied, and a numerical modeling method based on the spheroid shape for the reproduction of the measured scattering matrix is proposed. In order to reproduce the measured scattering matrix, the T matrix combined with the IGOM is used first. Through this combination, we can calculate the single scattering properties of spheroids in the wider size range and aspect ratio range. After that we pay more attention to the size distribution of spheroids and propose a modeling method on the ensemble-average scattering matrix. That is the number density distribution is deduced from the fit of the measured normalized projected surface area distribution with the modified beta function, and the shape distribution is obtained by least squares fit of the measured scattering matrix. It should be noted that the variation in the aspect ratio can result in a larger variation in the scattering matrix. So the ensemble of the spheroids should have as wide aspect ratio range as possible. Meanwhile, the determination of the truncation size should be considered carefully according to the distribution characteristics of real mineral aerosols. In addition, the modeling using the lognormal to fit the measured normalized number distribution and fit the measured normalized projected surface area distribution to reproduce the scattering matrix is also performed. Furthermore, the asymmetry parameter is calculated to verify the goodness of the proposed method. Simulation results indicate that the modeling using the “beta area” number density distribution combined with the least squares fit shape distribution can yield equally good reproduction of the measured scattering matrices for the feldspar, quartz, and red clay. Through fitting the measured normalized projected surface area distribution with the modified beta function, the number density distribution of spheroids shows relatively little deviation corresponding to that deduced from the fit of the measured normalized number distribution with the lognormal function, and then we can achieve better reproduction of the measured scattering matrix from the Amsterdam database. This proposed method appears to be attractive because of its simplicity and efficiency. Moreover, the scattering matrices of the mineral aerosols having larger size parameters from the database can be well established with the ensemble of spheroids based on the proposed method.

Acknowledgments

[46] This work was supported by the National Natural Science Foundation of China (11132008 and 11202202) and Zhejiang Province Natural Science Funds (Y6110147). The authors are grateful to M. I. Mishchenko from NASA, Goddard Institute for Space Science, for providing the T matrix code and Prof. Ping Yang from Texas A&M University for the IGOM code and H. Volten and O. Muñoz et al. for making the measurement data available. The authors are also grateful to the anonymous referee for comments that improved the manuscript.