A new snowflake aggregation model is developed to study single-scattering properties of aggregate snowflakes. Snowflakes are generated by random aggregation of six-bullet rosettes constrained by size-density relationships derived from previous field observations. Due to random generation, aggregates may have the same size or mass, yet different morphology allowing for a study into how shape influences their scattering. Single-scattering properties of the aggregates were investigated using discrete dipole approximation (DDA) at 10 frequencies. Results were compared to those of Mie theory for solid and soft spheres (density 10% that of solid ice) and to T-matrix results for solid and soft spheroidal cases with aspect ratios of 0.8. Above size parameter 0.75, neither the solid nor the soft sphere and spheroidal approximations accurately represented the DDA results for aggregates. Asymmetry and the normalized scattering and backscattering cross sections of the aggregates fell between the soft and solid spherical and spheroidal approximations. This implies that evaluating snow scattering properties using realistic shapes, such as the aggregates created in this study, is of paramount importance. Concerning the morphology of the aggregate snowflakes created in this study, the dependence of their single-scattering properties on each aggregate's detailed structure seemed of secondary importance. Using normalized standard deviation as a measure of relative uncertainty, it is found that the relative uncertainty in backscattering arising from the different morphologies caused by random aggregation is typically ~15% for individual particles and ~18% when integrated over size distributions. Relative uncertainties for other single-scattering parameters are less.
 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  used discrete dipole approximation (DDA) to compute the asymmetry factor (g), scattering, and absorption cross sections (Csca and Cabs) 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 = 2πreff/λ, where reff is the radius of an equal-mass, solid ice sphere, and λ is the wavelength of the incoming radiation. Wood  also noted that at 94 GHz, the nonspherical nature of ice crystals can cause deviations from Rayleigh scattering even at small sizes.
 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 , Bruggeman  and Sihvola . 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  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, Csca, and Cabs. Kulie et al.  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.  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  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.  and Botta et al.  reached the same conclusion as the solid and soft spherical cases mentioned above.
 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.
Maruyama and Fujiyoshi  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.  with two-dimensional dendrites as constituent particles.
 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  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.
 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.
2 Generation of Aggregate Flakes
2.1 Constituent Particles
 In the present study, six-bullet rosettes are used as the constituent particles of the aggregate flakes due to their common presence in the upper level cloud deck. Many field campaigns have shown the existence of bullet rosettes in cirrus clouds [e.g., Arnott et al., 1994; Heymsfield and Iaquinta, 2000; Heymsfield et al., 2002; Field and Heymsfield, 2003]. Heymsfield et al.  conducted a field campaign at the Atmospheric Radiation Measurement site in Oklahoma during March 2000. Observed particles above 100 µm were frequently bullet rosettes or rosette aggregates, and the average number of bullets in these rosettes was 5.8 ± 0.6, or approximately 6, supporting the use of six-bullet rosettes as aggregate constituent crystals.
 Similarly, using images from a Particle Measuring System 2D-C imaging probe, Woods et al.  found during the Improvement of Microphysical Parameterization through Observation Verification Experiment that bullet rosettes form below −40°C and sideplanes, sectors, bullets, and their assemblages and aggregates form below −18°C. In addition, data sets collected by M. Kajikawa at Mt. Hachimontai Observatory of Akita University showed that out of 45 different particle types observed in cirrus clouds, 39 of them were classifiable as bullet rosettes [Heymsfield et al., 2002]. It must be noted, however, that rosettes are mostly found at the top of clouds at fairly cold temperatures.
 Furthermore, most large snowflakes are made of dendritic crystals with aggregation maxima at 0 and −15°C [Pruppacher and Klett, 1997]. Constituent crystals for aggregate flakes can vary as temperature and water vapor concentration levels change throughout a cloud layer leading to different crystal formations. Aggregate growth by collision with other ice crystals is a function of radii of the particles and their fall speeds [e.g., Westbrook et al., 2004]. This would mean that aggregates of purely rosette crystals would be extremely rare as collisions occur when an aggregate and/or crystal descends through a cloud layer and interacts with crystal types formed by the differing environment. Our aggregates of six-bullet rosette crystals are a first attempt for generating a more complex aggregate shape.
 In cirrus clouds, bullet rosettes tend to have maximum dimensions between 200 and 800 µm [Pruppacher and Klett, 1997] and follow a range of size-density relationships. In this study, the “bullets” in the rosettes are assumed to be columns made of equal-sized cubes with orthogonally oriented columns that share the same single cube in the center (Figure 1). The effective density of any rosette with n number of blocks per column, assuming the blocks are an odd number thereby representing a symmetric rosette, can therefore be calculated by:
where ρi is the density for solid ice (0.916 g cm−3). The density of rosettes with n = 3, 5, 7, and 9 are 0.336 g cm−3, 0.162 g cm−3, 0.0913 g cm−3, and 0.058 g cm−3, respectively. Figure 2 is a plot of these densities along with a size-density relationship for rosettes captured in the field campaign in Heymsfield et al. :
where D is the maximum projected dimension and ρe is the rosette's density. As seen in Figure 2 and from the density calculation in equation (2), rosettes 200 µm and 400 µm in length (hereafter referred to as 200 µm or 400 µm rosettes) approximately satisfy both the theoretical density of a rosette in this study and the density determined from Heymsfield et al. . The 200 µm rosette corresponds to each column being five cubes long while the closest fit for a 400 µm rosette is seven cubes long. For the 200 µm and 400 µm rosettes, this corresponds to cube lengths of 40.0 µm and 57.1 µm. In the following section, we use these two types of six-bullet rosettes to form aggregate snowflakes.
 Other studies [e.g., Maruyama and Fujiyoshi, 2005; Ishimoto, 2008; Petty and Huang, 2010; Botta et al., 2011; Tyynelä et al., 2011] have constructed aggregates using different constituent crystal types. Maruyama and Fujiyoshi  assumed that the constituent elements used for aggregation were spheres with densities ranging from 0.1 to 0.5 g cm−3. In the Ishimoto  study, the constituent elements do not take a shape, but are assumed to be 100 µm in maximum dimension and occupy 50% of the grid space; the decision to use 50% fractional occupancy is rather arbitrary without a plausible justification. In Petty and Huang , the realistic snowflakes were an aggregate of needles and three different dendrite aggregates. These, however, were not constrained by size-mass/density relationships.
 The Tyynelä et al.  generated aggregates were comprised of either stellar or fernlike dendrites created by an aggregation model based on one by Westbrook et al. . The crystals followed the 2D growth algorithm put forward by Reiter  and crystal thickness relations from Magono and Lee . Particle mass is determined by the gamma function, αDmaxβ, where α and β are determined experimentally [e.g., Pruppacher and Klett, 1997], with α = 0.0036 and β = 1.57 for fernlike aggregates and α = 0.0086 and β = 1.48 for stellar aggregates. Previous studies from Heymsfield et al.  (aggregates), Schmitt and Heymsfield  (aggregates), and unrimed and rimed dendritic aggregates seen by Locatelli and Hobbs  have indicated values for α = 0.006, 0.0068, 0.0073, and 0.0037 and β = 2.05, 2.22, 1.4, and 1.9, respectively. The generated Tyynelä et al.  aggregate flakes are fairly similar to these predetermined size-mass relations. A spheroidal model they examined underestimated their aggregates' backscatter cross sections by factors of 10 and 50–100 at the Ka (35.6 GHz) and W (94 GHz) bands, respectively.
Botta et al.  built their aggregates using the Distributed Hydrodynamic Aerosol-Radiation-Microphysics Application Cloud Resolving Model (CRM), which modeled results from one particular case study. Each flake consisted of many stellar-type crystals, constructed out of ice spheres, with random sizes and orientations that adhered to the mass-dimensional relationships of the constituent crystal type. The aggregates also followed the mass-dimensional relationships defined in the CRM and were allowed to freely vary in aspect ratio. Backscatter properties of their aggregates were calculated by the Generalized Multiparticle Method [Xu, 1995] and determined by averaging over a size bin. These results were compared against two spherical, two spheroidal cases (one matching the aggregates aspect ratio and one set to 0.6) and the in situ case study values. They concluded that in order to match observed radar reflectivity histograms collected during the case study, the ice hydrometers must have a “refined” representation that yields realistic looking aggregates as the spherical and spheroidal cases did not adequately model the observed reflectivities.
2.2 Aggregate Formation
 The model developed in this study generates snowflakes by aggregating the six-bullet rosettes discussed in section 2.1. Starting in a 3D grid space, rosette crystals of either 200 µm or 400 µm in maximum dimension are generated, depending on the type of model run. The model runs can be homogenous, meaning that they are either comprised of all 200 or 400 µm rosettes, or they can be a combination of both the 200 and 400 µm rosette crystals. A 400 µm rosette is shown in Figure 3a and two 400 µm rosettes in Figure 3b. Each rosette does not share point locations, but rather has the outer faces of the crystals touching each other. Therefore, our model does not allow rosettes to generate if they share lattice points with a previously existing rosette (flowchart in Figure 4). These flakes were allowed to grow randomly in all directions and resulted in fairly spherical flakes. Figure 5 is an example of 400 µm rosette aggregate flakes with maximum dimensions of (a) 0.95 mm, (b) 5.05 mm, (c) 6.59 mm, and (d) 10.45 mm.
 Since heavy snowfalls and large flakes are typically aggregates [e.g., Matrosov, 1992; Hanesch, 1999; Westbrook et al., 2006; Petty and Huang, 2010], several studies have been conducted to determine the size-density relationship of aggregate snowflakes [e.g., Magono and Nakamura. 1965; Holroyd, 1971; Muramoto et al., 1995; Fabry and Szyrmer, 1999; Heymsfield et al., 2004; Brandes et al., 2007]. A summary of size-density relationships from several of the above aggregate studies is given in Brandes et al.  along with their own relation, as shown in Figure 6. The differences among these relations are believed to be partially due to how the maximum dimension of the snowflake is defined and what type of flake is included in different studies (i.e., dry snowflakes versus studies that included heavily rimed snowflakes). Particularly, Magono and Nakamura  and Holroyd  used the geometric mean of the particle's major and minor axes as seen from above, while Muramoto et al.  determined the maximum dimension based on the maximum horizontal dimension. The Fabry and Szyrmer  and Brandes et al.  relations are based on the equivalent volume diameter while, Heymsfield et al.  used the diameter of the minimum circumscribed circle [Brandes et al., 2007].
 As can be seen from Figure 6, density decreases with increasing particle size. By controlling the growth parameters in our aggregation model (Figure 4) that control the size and density of our aggregates, the generated flakes closely follow the Brandes et al.  diameter-density relationship to within 20%. Note that the definition of maximum diameter used in this study is not the same as Brandes et al. . The maximum diameter of the flake is similar to the Heymsfield et al.  definition. The Brandes et al.  relation was chosen as it appears to represent a consensus between the different size-density relationships. Also, deviations of up to 20% from the Brandes et al.  relation were allowed to account for the different size-density relationships. Additionally, this study's aggregate flakes tend to be more spherical in nature resulting in maximum dimensions close to those used by Magono and Nakamura , Holroyd , and Muramoto et al. . The flakes generated in this study are also plotted in Figure 6, with the aggregates made of 200 µm rosettes shown as red circles, made of 400 µm rosettes as green circles, and the combination flakes as blue circles. Using the aggregation method developed in this study, 557 aggregate snowflakes are generated with maximum dimensions ranging from 765 to 12,585 µm with reff between 180 and 1460 µm.
2.3 Aspect Ratio
 While there are numerous studies investigating the aspect ratio of raindrops, few exist for aggregate snowflakes. One of the first studies which involved falling snow was done by Magono and Nakamura  who took pictures of falling snowflakes and measured the maximum vertical length directly from the photographs. The horizontal dimensions were measured by tracing the outline of the flake on dyed filter paper. They found that for flakes below 10 mm, the vertical and horizontal dimensions were nearly identical indicating that the flakes were mostly spheroidal. Above 10 mm, the horizontal dimension became larger than the vertical leading to aspect ratios (maximum vertical dimension divided by maximum horizontal dimension), ar, of close to 0.9.
Brandes et al.  studied snowflakes associated with winter storms in Colorado using a two-dimensional video distrometer that examined the shape, size, and terminal velocities of snowflakes above 0.4 mm. Flakes smaller than 0.4 mm were regarded as “suspect” due to wind effects and other measurement issues. Defining ar the same way as Magono and Nakamura , Brandes et al.  found that at 2 mm, ar values ranged from 0.5 to 5, while at 8 mm, the range was narrower from 0.5 to 1.5. Taking the modal value of 0.2 mm bins, they calculated the following equation to describe ar with respect to maximum diameter, Dmax, in mm:
 Equation (3) indicates that the flakes are nearly spheroidal (~0.9 to 1.0) in agreement with Magono and Nakamura .
 A study conducted by Korolev and Isaac  examined ar values of aggregate crystals in ice clouds. Images of aggregates were collected during snow events over the Canadian and U.S. Arctic in addition to the Great Lakes. The ar values were calculated differently from the previously mentioned studies. First, Dmax, which does not need to be the true horizontal or vertical dimension, was calculated. The maximum width of the flake perpendicular to Dmax (Dw) was determined allowing for ar values to be calculated as:
 The Korolev and Isaac  study found that the ar values of ice particles were typically 0.6 for large particles above a Dmax of 80 µm.
 In this study, ar values are calculated using the Korolev and Isaac  method, with two axes (XZ and YZ) used to simulate a 2D flake image. Results are summarized in Table 1 by flake type. While the range is large between minimum and maximum ar values, the modal value is around 0.8. These aggregates are closer to the near-spherical flakes observed by Magono and Nakamura  and Brandes et al.  and somewhat greater than those measured by Korolev and Isaac .
Aspect ratios, ar, for the different flake types for XZ and YZ axes. Included are the minimum and maximum ar values along with the standard deviation.
0.8317 ± 0.0904
0.8463 ± 0.0892
0.8192 ± 0.1077
0.8015 ± 0.1328
200 and 400 µm
0.8044 ± 0.1098
0.7894 ± 0.1025
2.4 Fractal Dimension
 Fractal dimension (df) is a measure of the complexity of an object determined by calculating how fast the length, area, volume, etc., change with respect to smaller and smaller scales [Petigen et al., 1992]. Researchers such as Muramoto et al. , Westbrook et al. [2004, 2006], Maruyama and Fujiyoshi , and Ishimoto  studied the df of snowflakes captured during field campaigns against their generated aggregates as a way to validate their modeled flakes. Unlike some other quantities, df is only slightly affected by the angle of the camera and the particle's area [Maruyama and Fujiyoshi, 2005]. df is related to a flake's mass and diameter/linear span by:
where M is the mass of the aggregate and r is the radius [Westbrook et al., 2004]. In this equation, a is a nonuniversal constant that is determined by information about the pristine crystal particles comprising the aggregate. When equation (5) is applied to the studies of Heymsfield et al. , Locatelli and Hobbs , and Mitchell , df values were 2.04, 1.9, and 2.1, respectively, for the aggregates containing bullet rosettes.
Muramoto et al.  studied images of falling snowflakes using the divider method, in which only the outline of the flake images was used in determining df. They examined four different aggregates using two different views: one from the side and one from above. The calculated dimensions were between 1.10 and 1.19, with less than 0.04 variation in any of the four cases.
Maruyama and Fujiyoshi  studied their aggregates in a similar fashion, but used the box-counting method. In this method, the image of interest, in this case an outline of a 2D flake image, is overlaid by a mesh grid of a given size δ. Next, the number of the mesh boxes that contain part of the object (N(δ)) is tallied. A graph of log(δ) on the x axis and log(N(δ)) on the y axis is generated with the df equaling the slope of the best fit line between all data points [Falconer, 1990; Petigen et al., 1992]. Different methods can result in different df values, yet they are usually related to one another [Petigen et al., 1992]. In their calculations, Maruyama and Fujiyoshi  further found that fractal dimension increased with increasing size of the flake and recorded dimensions of 1.15 to 1.24 for aggregates between 3 and 5 mm.
 We chose to calculate df using the box-counting method from a generated 2D outline image using the software package ImageJ along with the plugin FracLac [Rasband, 1997–2008; Karperien, 1999–2007]. The dfs for the flakes generated in this study are given in Table 2 and are broken down by flake type. Our flakes have an average fractal dimension of 1.19 – 1.20, depending on flake type, with a standard deviation of a tenth or less. While this is on the upper end of the average determined by previous studies, our sample size is comprised of many more flakes. Looking at the minimum and maximum values, we observed a similar size range to that observed by Muramoto et al.  with larger particles having dfs of over 1.25, mimicking the trend seen in Maruyama and Fujiyoshi .
The minimum, maximum, and average fractal dimensions for the 200 µm, 400 µm, and combination flakes.
1.1925 ± 0.0285
1.2001 ± 0.0310
200 and 400 µm
1.1897 ± 0.1346
3 Single-Scattering Properties of Aggregate Snowflakes
 Aggregate snowflakes are generated using the method described in section 2. These aggregates not only follow the size-density relations derived previously from in situ microphysical measurements for both bullet rosettes and aggregates, but also have aspect ratios and fractal dimensions similar to previous studies. In this section, we study the single-scattering properties of these aggregate snowflakes using DDA simulations. In particular, the DDA model, DDSCAT, developed by Draine and Flatau , is used in this study.
3.1 DDA Results
 In DDA, the target object is approximated as an array of polarizable points that are on a cubic lattice. The polarizable points acquire dipole moments via their response to the local electric field and the electric fields of surrounding points. This method requires that the spacing between dipoles must be sufficiently small compared to the interacting electromagnetic wavelength determined from the relationship |m|kd < 1 [Draine and Flatau, 2009], where m is the complex index of refraction, k is the angular wavenumber, and d is the dipole spacing. In this study, we made sure that |m|kd values are well below the acceptable threshold of 1. DDA modeling is performed for the following frequencies: 10.65, 13.6, 18.7, 23.8, 35.6, 36.5, 89.0, 94.0, 165.5, and 183.31 GHz, which are the frequencies to be used in the CloudSat CPR, GPM radar, and radiometers. Since the temperature dependence of the refractive index of ice is very weak [Mätzler, 2006], DDA modeling is performed only for −10°C. Results for the flakes are given in Figure 7 for all frequencies mentioned above. In this figure, asymmetry parameter (g), normalized scattering (Qsca), backscattering (Qbsc), and absorption (Qabs) cross sections are plotted against the size parameter. The cross sections are normalized by the sectional area of the equal-mass, solid ice spheres. For example, Qsca is the unitless normalized scattering cross section given by , where Csca is the scattering cross section. For reference, the above parameters are computed for equivalent mass solid ice spheres and soft spheres along with solid and soft spheroids with an ar = 0.8 and shown in Figure 7. The soft cases have a density 10% that of solid ice (i.e., ρ = 0.0916 g cm−3) and use the Maxwell Garnett  mixing rule for deriving their effective refractive indices.
 Results for the aggregate flakes are plotted as open circles, squares, and triangles of differing colors to show the influence of frequency on the scattering results. Solid spheres and spheroids are shown as solid blue and red curves with the soft spheres and spheroids given as dashed blue and red curves, respectively. The scattering properties for the spheres are determined using Mie theory at a frequency of 183.31 GHz. The spheroids are calculated using the T-matrix method [Mishchenko et al., 1996; Mishchenko and Travis, 1998] also at a frequency of 183.31 GHz. (Note that this particular method does not easily allow for the calculation of the asymmetry parameter and is therefore not shown). All calculations are done assuming the particles are randomly oriented. Results for spheres and spheroids at other frequencies are not shown in the figure. Since the dielectric properties of ice are not very sensitive to frequency in the microwave spectrum, their results at other frequencies are very similar to those of 183.31 GHz when plotted against size parameter.
 The values of single-scattering properties for solid spheres and spheroids are very similar. As seen in Figure 7, these solid particle approximations are not a sufficient substitute for aggregate flakes as their results do not replicate the single-scattering properties especially at large size parameters outside the Rayleigh regime. Unlike Kim  who noted scattering differences between spheres and simple aggregates occurring at x = 2.5, results of our more complex aggregates start to deviate noticeably from those of spheres and spheroids at x = 1 for Qsca (Figure 7b) and Qabs (Figure 7a). Above x = 1, Qsca results from DDA increase at a steady rate, eventually surpassing the values predicted by solid spheres and spheroids at x = 3. The aggregates do not have the variations in Qsca as seen in both solid sphere and spheroidal cases, most likely due to the lack of internal reflections causing interference patterns in the results. Above x = 1, Qabs for aggregates is always less than that for equivalent solid spheres/spheroids. Of additional note, Qabs results have noticeable frequency dependence due to different values in the imaginary part of the refractive indices for different frequencies.
 The soft sphere and spheroidal approximations follow the trend observed in the DDA scattering and absorption results of aggregate flakes better than those of solid spheres and spheroids. Up until x ~ 3.5, Qsca results for the aggregates fall between those results for solid and soft cases. Asymmetry parameter (Figure 7d) for the aggregates does not appear to be adequately modeled for solid spheres even at x < 1 as it consistently underestimates g regardless of size parameter unlike Kim  where the deviations occurred at x = 2.5 for the simple aggregates. At larger size parameters, g converges to approximately 0.9. While soft spheres follow the trend in g more accurately than solid spheres, soft sphere approximation tends to overestimate. The aggregate values of g fall between those of soft and solid spheres. This is true as well for Qsca at x < 3.5. This trend was also observed in Liu  with simpler, nonaggregate pristine crystal types.
 Due to the difference of magnitude between sphere/spheroid and aggregate values for Qbsc, a separate figure of x vs. Qbsc is shown in Figure 8. This figure is the same as Figure 7c, but with Qbsc ranging from 0 to 1. Examining Figure 8, Qbsc follows the values for solid spheres until x ~ 0.75. Between x = 0.75 and x = 1.5, the aggregates' backscatter is less than that for the solid spheres. But unlike Qsca above, the aggregates do mimic the peak in values for solid ice spheres between size parameters 0 and 1.5. At larger size parameters, however, Qbsc for the aggregates does not follow the patterns observed for solid spheres just like Qabs, Qsca, and g. While solid spheres overestimate Qbsc for aggregates, soft spheres tend to underestimate with the values for aggregates falling in between the two approximations. The solid spheroidal case does follow Qbsc for some of the flakes, especially for the higher frequencies, until x ~ 1.75. After this, it overestimates just like the solid sphere case.
Qsca and Qbsc both have increasing spreads in values for larger size parameters as seen in Figures 7b and 8. This scatter of values suggests that the shape of individual aggregates has some influence on the intensity of scattering, in addition to the particle's mass/radius. Qbsc has a relatively larger scatter indicating that this single-scattering property is more influenced by particle shape. This dependence makes parameterization of the backscatter by snow aggregates difficult. Qabs and g appear to have a smaller shape influence, instead being more dependent on mass/radius.
3.2 Spread of Single-Scattering Properties
 To quantify the amount of spread in the single-scattering properties between particles of similar size parameters with differing shapes, we calculated a relative uncertainty statistic at only one of the frequencies used in the above computations. All the results of the aggregates at only 183.31 GHz are divided into different size bins based on reff. The bins are 100 µm in size starting at 150 µm with each bin containing 7 to 29 flakes. After calculating the mean value and standard deviation of each of the four single-scattering properties for a given bin, the relative uncertainty is represented by the standard deviation normalized by the mean. The results are given in Figure 9.
 The relative uncertainty overall tends to decrease with increasing size parameter, except for x ~ 1.5. Qbsc is the most scattered of the single-scattering properties with relative uncertainty ranging from 9% at x ~ 1.2 to 29% at x ~ 1.5 with a typical value of 15% when x > 2. The other three parameters generally have lower relative uncertainty. The statistic for Qsca ranges from 2% at x ~ 5.3 to 27% at x ~ 1.5, with a typical value of between 2 and 4% at larger x values. Qabs follows a similar pattern to Qsca with a minimum value of 1.7% at x ≥ 5, maximum value of 13% at x = 1.5, with most values at or below 2% for x > 2. The statistic values for g are typically 1% or less for x > 2, with the largest value being 11% when x ~ 0.75.
3.3 Size Distribution Averaged Values
 As discussed above, particle shape influences, to some degree, the single-scattering properties of individual particles. In an effort to see how greatly the scattering properties are influenced by shape for a given atmospheric volume in which particles with various sizes and shapes coexist, we calculated the volume scattering and absorption coefficients by taking into account particle size distributions. For the aggregate flakes, we used the Sekhon and Srivastava  exponential distribution, which is given by , where N0s and λs are functions of the liquid equivalent snowfall rate (R) and Dmelt is melted diameter. The volume scattering, backscattering, and absorption coefficients (σsca, σbsc, and σabs) are calculated by:
where t represents either sca for scattering, bsc for backscattering, or abs for absorption. The volume averaged asymmetry parameter is determined by:
 To populate the size distribution, all flakes with reff sizes ranging from 180 to 1460 µm are divided into size bins of 200 µm. Within each size bin, one flake is randomly chosen for each integration run, then the volume coefficients and are determined for R values from 0.1 to 2.0 mm h−1. A total of 100 runs are conducted for each R value at a single frequency of 183.31 GHz to discount any impacts different frequencies would have on the scattering values. The mean and standard deviation of σsca, σbsc, σabs, and from the 100 runs are given in Figure 10. Since the particles are selected randomly in each bin for each integration run, the standard deviation gives the indication of how the particle shape influenced the volume integrated scattering properties.
 From Figure 10, as R increases, the mean value and standard deviation also increase for σsca, σbsc, and σsca while the standard deviation decreases for as it approaches 1. This increase in mean value is due to the effects of R dependent variables N0s and λs. To further examine the uncertainties caused by different particle shapes, the normalized (by mean) standard deviations are calculated as an indicator of relative random error, and the results are shown in Figure 11. Besides for the very first R value of 0.1 mm h−1, the relative uncertainty values for σsca, σbsc, and σabs are fairly consistent and increase with increasing R with a range between 6% and 22% for R values of 0.25 and 2.0 mm h−1, respectively. The σbsc tends to have a slightly larger relative uncertainty than σsca by several percentage points at most R values. The relative uncertainty in σabs is consistently a few percent lower than either σsca or σbsc. The relative uncertainty values for g are much less, typically 1% or less.
 In summary, while the single-scattering properties of aggregate snowflakes are primarily determined by a particle's size or mass, the detailed shape of individual particles also influences these properties, especially for backscattering. For individual snow particles created in this study, the relative uncertainty as defined by standard deviation divided by the mean in Qbsc is typically 15%, and for Qsca, Qabs, and g, it is typically less than 5%. When integrating over size distributions, the relative uncertainty is typically 15% for backscattering, scattering, and absorption coefficients, and a few percent for asymmetry parameter. This will lead to uncertainty errors in radar reflectivity values caused by particle shape of ~15% as radar reflectivity is proportional to the integrated σbsc values.
 This study aims to understand the single-scattering properties of aggregate snowflakes. A new snowflake aggregation model was developed to generate realistic flakes that could be observed in nature. The model uses six-bullet crystal rosettes, with size and density consistent with previous field studies, as building blocks. Two different sized rosettes, 200 and 400 µm, were used for the three snowflake types: 200 µm only, 400 µm only, and a combination of 200 and 400 µm. These rosettes were allowed to randomly grow together, but the aggregation was constrained by a size-density relationship for aggregates derived from previous observations. Therefore, by design, the size-density relation of the generated aggregates follows what has been observed. For a particle with same size or mass, its shape or morphology can be different each time flakes are generated by the model because they are generated at random. By doing this, we may examine how the detailed shape or morphology influences the particles' scattering and absorption properties even if they have the same mass or size. Several other statistical parameters were also used to further assess the validity of the flakes, including aspect ratio and fractal dimension. All flake types have aspect ratio values around an average of 0.8 in line with estimates of Magono and Nakamura  and Brandes et al. . The values of fractal dimension are consistent with those observed during field studies of actual falling flakes and mimicked the trend of increasing fractal dimension with increasing size.
 The single-scattering properties of the generated particles were computed using DDA. As pointed by previous investigators, absorption, scattering, and backscattering cross sections as well as the asymmetry parameter of the snowflakes all differed greatly from those predicted by Mie theory of equivalent solid and soft spheres, especially at larger size parameter values outside the Rayleigh regime. The T-matrix solutions for solid and soft spheroids differed as well. Below a size parameter of approximately 0.75, results of a solid sphere can be used to somewhat approximate the snowflakes' scattering and absorption properties.
 Above size parameter x ~ 0.75, the snowflakes' scattering properties start to diverge from those of solid spheres and spheroids. In particular, over the size parameter range examined in this study (up to 6), the scattering properties of snowflakes computed by DDA have much smoother (less wavy) variation with size parameter than those for solid ice spheres, giving the possibility of parameterizing the scattering properties by size parameter as done by Liu . While soft sphere (with a density 10% of solid ice) approximation does tend to mimic the patterns of asymmetry, scattering, and absorption cross sections more accurately, asymmetry and scattering cross sections are, respectively, over- and underestimated. The soft spheroid case (also with a density of 10% of solid ice) performs even worse for scatter cross-section results than its soft sphere counterpart. The backscatter cross section is not well duplicated by the soft or solid spherical/spheroid approximations with aggregate results being in between the soft and solid results.
 In order to examine how the detailed shape of a snowflake influences its scattering properties, we estimated the relative uncertainty as defined by the normalized standard deviation of their values given a similar size parameter. The results show that the relative uncertainty for Qbsc ranges from 9% to 29% with a typical value of 15% for size parameters larger than 2. Qsca, Qabs, and g had less variation with particle shape with a typical relative uncertainty of only a few percent. This indicates that backscattering cross sections, to some extent, are more sensitive to the actual shape of the flake while the other properties are mainly tied to the flake's mass or effective radius when proper size-density relationships are used as constraints in snowflake generation.
 To assess the particle shape variation-induced relative uncertainty over an atmospheric volume, we performed integration over a size distribution of particles for 100 times at a given snowfall rate. In each integration, we randomly selected a particle in each size bin and then integrated scattering properties over the size distribution. Then, the relative uncertainty of the integrated quantities is assessed by the standard deviation of the 100 runs. The results show that relative uncertainty is typically 15% for scattering, backscattering, and absorption coefficients, and only a few percent for asymmetry parameter over the snowfall rate range of 0.1 to 1.5 mm h−1.
 This study is in agreement with numerous others that indicate spheres and spheroids are not an adequate model for aggregate flakes as they do not adequately mimic the single-scattering properties. This indicates a need for development of an algorithm or a database where more reliable estimates of aggregate single-scattering properties can be obtained for use in retrieval algorithms. The single-scattering results of the aggregates flakes used in this study are available in a database at cirrus.met.fsu.edu.
 These aggregate flakes comprised of six-bullet rosette crystals are just a first approximation. Ultimately, it would be desirable to have an aggregate flake that is made up of many different crystal types to more accurately model what is seen in nature and provide that as an inclusion to the database. Also, more in situ data studies into the crystal-type distributions present in the layers of a cloud during a snow-producing event would provide a solid basis on which better aggregate flakes can be constructed. Field campaigns where the scattering properties of aggregate flakes are measured by aircraft also provide valuable knowledge against which we can ultimately compare the results of the modeled flakes.
 This research has been supported by NASA grants NNX10AG76G and NNX10AM30G and NSF grant AGS-1037936. The authors are grateful to B. T. Draine and P. J. Flatau for providing their DDA model for our simulations, to Benjamin Johnson for helpful suggestions and discussions, and the three anonymous reviewers for their very helpful comments.