## 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 *r*_{min} to *r*_{max}, *r*_{min} and *r*_{max} are the minimum and maximal values of radius *r*, *n*(*r*)d*r* denotes the fraction of the total number of particles per unit volume having radii between *r* and *r* + d*r*, *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 *ε*_{min} to *ε*_{max}, *ε*_{min} and *ε*_{max} 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.