## 1 Introduction

[2] Many satellite sensors, such as CloudSat's Cloud Profiling Radar (CPR) [*Stephens et al*., 2002; *Tanelli et al*., 2008], and the upcoming Global Precipitation Measurement's (GPM) Dual-Frequency Precipitation Radar and GPM Microwave Imager [*Smith et al*., 2007], have the capability to detect ice and snow particles in the atmosphere. A precipitation retrieval algorithm for these satellite sensors can be developed when the single-scattering properties of snowflakes are accurately calculated for use in radar equations and microwave radiative transfer models. Snowflake shapes are often modeled as spheres or oblate spheroids to ease the complexity of calculations, despite the fact that snowflakes are often observed to be aggregates of crystals [e.g., *Kajikawa and Heymsfield*, 1989; *Westbrook et al*., 2004]. Some of the first and most commonly used snowflake models are those of equivalent mass spheres of solid ice or a soft sphere mixture of ice, air, and occasionally water [e.g., *Marshall and Gunn*, 1952; *Smith*, 1984; *Matrosov*, 1992; *Bennartz and Petty*, 2001; *Liao et al*., 2008]. These spherical approximations allow for application of Mie theory to determine the scattering and absorption properties of the flakes [*Liu and Curry*, 2000; *Evans et al*., 2002]. While the scattering and absorption properties of solid spheres are easier to compute than irregularly shaped aggregate flakes, they are not always an adequate substitution. For example, *Kim* [2006] used discrete dipole approximation (DDA) to compute the asymmetry factor (*g*), scattering, and absorption cross sections (*C _{sca}* and

*C*) of three different simple aggregates made of cylinders from 0.06 to 5 mm in maximum dimension and found that Mie theory did not adequately predict the single-scattering properties for these simple aggregate flakes when the size parameter is greater than 2.5. Here, size parameter is defined as x = 2πr

_{abs}_{eff}/λ, where

*r*is the radius of an equal-mass, solid ice sphere, and

_{eff}*λ*is the wavelength of the incoming radiation.

*Wood*[2011] also noted that at 94 GHz, the nonspherical nature of ice crystals can cause deviations from Rayleigh scattering even at small sizes.

[3] An alternative approach to solid sphere approximation is that of a soft or fluffy sphere. In this case, the equal-mass sphere is made of a mixture of ice and air with an effective dielectric constant determined by mixing formulas such as those of *Maxwell Garnett* [1904], *Bruggeman* [1935] and *Sihvola* [1989]. Several investigators [e.g., *Liu*, 2004; *Kulie et al*., 2010; *Petty and Huang*, 2010] have studied scattering and absorption results of more realistic particle shapes. *Liu* [2004] compared the DDA results of sector-like and dendrite snowflakes to soft and solid sphere approximations. He determined that the aggregate results fell between those of the soft and solid spherical approximations for *g*, *C _{sca}*, and

*C*.

_{abs}*Kulie et al*. [2010] concluded that radiative transfer modeling results using the assumption of soft spheres did not match results from satellite microwave observations. In addition, they examined how the results from studies of simple crystal type particles performed in the radiative transfer model, including sector and dendrite flakes [

*Liu*, 2008], aggregates and droxtals [

*Hong*, 2007], hexagonal plates [

*Hong*, 2007;

*Liu*, 2008], hexagonal columns and bullet rosettes [

*Hong*, 2007;

*Kim et al*., 2007;

*Liu*, 2008].

*Kulie et al*. [2010] concluded that these simple crystal types produced results that were more realistic and closer to data from satellite microwave sensors than those based on soft spheres for most events.

*Petty and Huang*[2010] concluded that single-scattering properties with the soft sphere approximation did not adequately model those of more realistic aggregate snowflake shapes. There are also several studies that have examined aggregates and compared those to spherical and/or spheroidal cases. Studies done by both

*Tyynelä et al*. [2011] and

*Botta et al*. [2011] reached the same conclusion as the solid and soft spherical cases mentioned above.

[4] The use of simplistic crystals may be reasonably valid for studies dealing with visible and infrared radiations [e.g., *Grenfell and Warren*, 1999; *Yang et al*., 2005], which, when viewed from satellites, can only sense the top portion of the cloud where ice particles are commonly crystal like [e.g., *Field and Heymsfield*, 2003]. Microwave radars or radiometers can generally sense the entire layer of a cloud and snowfall below the cloud with the exception of sounding channels. Ice particles at the lower portion of the cloud, snowflakes in particular, are aggregations of tens to tens of thousands “constituent crystals” [*Houze*, 1993], which themselves can be rosettes with many arms [*Heymsfield et al*., 2002]. The structures of aggregate snowflakes, while being complicated, are not totally without regularity. In situ measurements have indicated that snowflakes commonly exhibit fractal properties with certain fractal dimensions [*Muramoto et al*., 1993; *Maruyama and Fujiyoshi*, 2005; *Ishimoto*, 2008] and the density of a snowflake aggregate decreases with its dimension following a power-law relationship [*Magono and Nakamura*, 1965; *Holroyd*, 1971; *Muramoto et al*., 1993; *Fabry and Szyrmer*, 1999; *Heymsfield et al*., 2004; *Brandes et al*., 2007]. This regularity provides us a guide to design a model for aggregate snowflakes. Note that due to the irregular shape of ice particles, the radius/diameter for an ice particle is customarily defined as the minimum radius or diameter of a sphere that would encapsulate the particle. The effective density is then defined as the ratio of the mass of the particle to the void space in the above sphere. This definition of density will be used in this study.

[5] *Maruyama and Fujiyoshi* [2005] concluded that when looking at detailed microphysics of the collision-coalescence process for aggregate creation, the resultant flakes could be modeled as fractals. Since they were concerned with generating a snowflake and a particle size distribution, their particular model for snowflake generation combined an aggregation model with the Monte Carlo method in which all constituent particles were assumed to be spheres with the same low density. Their aggregation model factored in variables such as vertical velocity differences among particles, coalescence efficiency, and rotational speed. A similar aggregation model was used by *Evans et al*. [2002] with two-dimensional dendrites as constituent particles.

[6] While the Maruyama and Fujiyoshi method has the intention of generating particles based on the physics of collision and coalescence processes, there are two shortcomings in such generated particles. First, the use of low-density spheres as constituent particles is unphysical and difficult to handle when the aggregate's scattering properties are calculated. Second, it cannot be guaranteed that the size-density relation of the resultant particles follows those observed by previous investigators. *Ishimoto* [2008] pursued a different approach and created snowflakes based on their observed fractal properties. While this method does produce realistic looking snowflakes, it does not have constraints in place to ensure that these flakes follow size-density relationships as derived from previous case studies.

[7] With the goal of generating realistic aggregates to represent snowflakes and studying the scattering and absorption properties of such snowflakes, we propose a new approach to generating aggregate flakes. These snowflakes are comprised of a realistic crystal type, the bullet rosette, and adhere to size-density relationships of both the bullet-rosette crystals and aggregate flakes. Then, the single-scattering properties of the so-generated aggregates are computed using DDA simulation. The DDA results are examined to see how closely aggregates can be approximated by the sphere and spheroidal cases and what particle properties might influence the single-scattering results.