## Abstract

The sensitivity of radar backscattering cross sections on different snowflake shapes is studied at C, Ku, Ka, and W bands. Snowflakes are simulated using two complex shape models, namely, fractal and aggregate, and a soft spheroid model. The models are tuned to emulate physical properties of real snowflakes, that is, the mass–size relation and aspect ratio. It is found that for particle sizes up to 5 mm and for frequencies from 5 to 35 GHz, there is a good agreement in the backscattering cross section for all models. For larger snowflakes at the Ka band, it is found that the spheroid model underestimates the backscattering cross sections by a factor of 10, and at W band by a factor of 50–100. Furthermore, there is a noticeable difference between spheroid and complex shape models in the linear depolarization ratios for all frequencies and particle sizes.

## 1. Introduction

Currently, there are a number of planned or active satellite missions that utilize radar observations for cloud and precipitation studies. The National Aeronautics and Space Administration (NASA) Global Precipitation Measurement (GPM) mission planned for launch in 2013 will carry a dual-frequency precipitation radar that uses Ku and Ka bands (13.6 and 35.6 GHz, respectively). The currently flying NASA *CloudSat* and planned European Space Agency (ESA) Earth Clouds, Aerosols and Radiation Explorer (EarthCARE) missions utilize W-band (94.0 GHz) cloud radars to profile clouds and precipitation from space. Ground-based radars that operate at lower frequencies are used as ground validation for satellite algorithms. It is therefore important to have a good understanding of how physical properties of the cloud and precipitation particles are linked to the radar observables at those frequencies.

In a number of recent studies it was shown that interpretation of radar observations, especially at high-frequency bands, that is, Ka and/or W bands, depends on an assumption about a particle shape. Westbrook et al. (2006) compared scattering from aggregate snowflake models to a soft sphere model, and found that as the size of the snowflakes was increased, the differences in the dual-frequency ratio (Ka versus W) between the models also increased mainly because of the differences in the shapes. Ishimoto (2008) used fractal shapes to model backscattering at X, Ku, and W bands. He also found large differences in the backscattering cross sections between the fractals and equal volume hexagonal columns. Petty and Huang (2010) used an aggregate shape model for snowflakes and found that the dual-frequency backscatter ratio (Ku versus Ka) is very sensitive to particle shape for larger snowflake sizes, and may not be suitable for snowflake mass estimation. Botta and Aydin (2010) showed similar results.

The main challenge in such studies is to ensure that the complex particle shape models represent realistic snow particles. In the present manuscript, we compare and model snowflakes of various sizes and shapes using fractal, aggregate, and soft spheroid shape models. To ensure that the modeled particles resemble real snowflakes, we adjust the model parameters such that the key physical properties, namely, the mass–size relation and aspect ratio, are realistic.

The scattering computations are carried out at four different radar frequencies (C, Ku, Ka, and W bands). The discrete dipole approximation (DDA) is used for complex shapes, while the T-matrix method (TMM) together with the Maxwell–Garnett effective medium approximation (EMA) is used for the soft spheroids. We are particularly interested in possible differences in scattering between the two complex shape models, and how they compare with the simple soft spheroid model.

## 2. Shape modeling

For particles smaller than the wavelength, that is, in the Rayleigh region, the internal field of the particle can be considered homogeneous and therefore does not produce shape-dependent effects in scattering. However, the different dimensions of the particle respond to the incident field differently, especially for particles in fixed or preferential orientations. These differences can be seen in the polarimetric observables. For particles comparable in size to the wavelength, the internal field is no longer homogeneous and various interference phenomena produce distinct features in the scattered field, which vary as a function of size, shape, refractive index, and orientation of the particle. We thus compare different shape models for snowflakes at different radar frequencies and assess their applicability.

### a. Aggregate model

The aggregate model is based on the model by Westbrook (2004). The model simulates the aggregation process in a physically realistic way using an iterative and stochastic method. First, a cloud of single ice crystals in random orientations is generated. During each iteration step, two aggregates or single ice crystals are chosen randomly from the generated cloud and placed in random positions within the cross sections defined by their radii of gyration. The probability of collision is determined by the fall velocity difference of the particles and their radii of gyration. If there is a collision, then the particles are connected from their nearest points projected to the horizontal plane. The generated aggregate is returned to the cloud. If there is no collision, a new pair is chosen randomly.

In our model, we reorient the new aggregates based on their maximum moments of inertia. This increases their radii of gyration, which are computed with respect to the vertical axis. According to, for example, Cho et al. (1981), ice crystals tend to fall with their major dimensions oriented horizontally. For single ice crystals, we generate stellar and fernlike dendrites by using the 2D crystal growth algorithm of Reiter (2005). For the thickness, we use the relation by Magono and Lee (1966). For their sizes, we use the gamma distribution function (e.g., Mitchell 1991) fitted to the size distribution of unrimed dendrites by Hariyama (1975) with the maximum dimensions between 0.3 and 2.0 mm for stellar dendrites, and between 1.0 and 5.0 mm for fernlike dendrites. The model does not take into account snowflake breakup during aggregation. This might make the generated particles fluffier than real snowflakes.

In the present study, we generate a total of 100 random samples for both stellar and fernlike aggregates. The number of single ice crystals is varied between 10 and 100 in increments of 10 with 10 random samples for each case.

### b. Fractal model

The fractal model is based on the model by Ishimoto (2008). In the model, the snowflake is generated by an iterative procedure, starting with a single ice dipole at the center of the grid. At each iteration step, the dipole grid is doubled in each dimension, and a number of new ice dipoles are positioned at random lattice sites adjacent to the existing dipoles. The total number of dipoles occupied after each iteration step is controlled by the fractal dimension *fd*. It is chosen so that the mass–diameter relation is as close as possible to the relation computed for the modeled aggregates. Here we use *fd* = 2.05 for stellar and *fd* = 1.88 for fernlike dendrites. Because of the similar mass–diameter relation between fractals and aggregates, it is possible to study the influence of different shape models on scattering.

We fix the dipole spacing *d* for a given particle type to ensure that it is small enough to correctly represent a solid part of the snowflake, and that all the generated fractals and aggregates obey the same validity criteria for DDA. In our computations, we have chosen *d* = 19 *μ*m for stellar and *d* = 47 *μ*m for fernlike dendrites. The mass of single dipoles is *m _{d}* =

*ρ*

_{ice}

*d*

^{3}(

*ρ*

_{ice}= 0.917 g cm

^{−3}is the density of ice) for both fractals and aggregates.

There are a few disadvantages of using the fractal model to represent real snowflakes. One is that, especially for a small fractal dimension (*fd* < 2.0), some parts of the fractal can be separated and therefore would not be aggregates in a strict sense. Another disadvantage is that in order to model very large snowflakes, the number of maximum iterations must be so large (*N*_{iter} > 8) that the available computer memory will become a limiting factor. This can be circumvented to some degree by increasing the dipole spacing.

In the present study, we generate a total of 200 random samples for both *fd* = 2.05 and *fd* = 1.88 cases. Values of 5, 6, 7, and 8 are used for *N*_{iter}, with 50 random samples per *N*_{iter}. In Fig. 1, we show sample shapes of the generated aggregate, fractal, and soft spheroid models with different sizes.

### c. Fitting particle models to measurements

The derived mass of snowflakes is a power-law function of the measured maximum diameter *D*_{max} (e.g., Pruppacher and Klett 1997); , where *α* and *β* are usually experimentally determined parameters. The mass and diameter are in centimeter gram seconds. Based on field measurements, Heymsfield et al. (2004) report *α* = 0.006, *β* = 2.05, and Schmitt and Heymsfield (2010) report *α* = 0.0068, *β* = 2.22 for aggregates. Schmitt and Heymsfield (2010) also noted that *β* is a function of snowflake type, ranging from 1.4 (aggregates of unrimed side planes) to 3.0 (lump graupel), with values between 1.8 and 2.3 being the most common. Locatelli and Hobbs (1974) report *α* = 0.0073, *β* = 1.4 for unrimed aggregates of dendrites, and *α* = 0.0037, *β* = 1.9 for rimed aggregates of dendrites. We obtain *α* = 0.0036, *β* = 1.57 for the fernlike aggregates, and *α* = 0.0086, *β* = 1.48 for stellar aggregates.

The maximum diameters are sometimes determined from the shadowed area of an equivalent ellipse/sphere. In this study, we use the maximum horizontal extent of the particle in the *x* and *y* directions as the maximum diameter *D*_{max}, mimicking the imaging geometry of 2D video disdrometers.

In the fractal shape model, *fd* has been adjusted to fit to the aggregate shapes so that the mass–diameter relation is similar between the complex shape models. This is also used for the spheroids to compute the effective refractive indices. It is noted that when fixing both *fd* and *N*_{iter}, the physical properties of fractals, such as mass and diameter, are also largely fixed. This can be seen in Fig. 2, which shows the mass of both the modeled aggregates and fractals as a function of *D*_{max}. We also show the empirical mass–diameter relations of Locatelli and Hobbs (1974), Heymsfield et al. (2004), and Schmitt and Heymsfield (2010), as well as the fitted curves for the samples using simple linear regression. As can be seen, the masses of the modeled particles agree reasonably well with the empirical formulas. If considering both stellar and fernlike aggregates (black triangles and circles, respectively), the slope parameter *β* for the empirical relations is reproduced well, with *α* being underestimated by a factor of 2. This could be due to the absence of breakup, which tends to produce denser snowflakes.

Figure 3 shows the density of the modeled snowflakes and several empirical density–diameter relations from various authors. The volume of the particles was chosen to be that of the area-equivalent sphere, where the area is determined by the projected image of the particles in the *y*–*z* plane. This mimics the way snowflakes are imaged in the field using, for example, disdrometers. The volume of snowflakes can also be determined by the maximum equivalent sphere, spheroid, or ellipsoid. Again, the modeled particles agree well with the empirical formulas.

In Fig. 4, we show the aspect ratios of the modeled particles, and the average measured value by Korolev and Isaac (2003). The aspect ratios are defined as the ratio between the maximum vertical extent in the *z* direction to the maximum horizontal extent in the *x* and *y* directions. Because the snowflakes are reoriented based on their maximum moment of inertia, the average aspect ratio is about 0.65 for aggregates and 0.68–0.7 for fractals. The value of 0.6 by Korolev and Isaac (2003) is based on cloud-particle imager (CPI) measurements from aircraft. The aspect ratio is estimated from the images by first finding the direction of the maximum extent of the particle, and then by measuring the maximum extent in direction orthogonal to it. The aspect ratio is the ratio of these two maximum extents. One advantage of this definition as compared to ours is that it should be independent of particle orientation effects, such as turbulence, which is common in both CPI and disdrometer measurements. Because of the reorientation of the modeled particles, however, our average aspect ratios are close to the value obtained by Korolev and Isaac (2003).

Figure 5 shows the roundness factor for the modeled particles. Again, the average values for different size ranges as measured by Korolev and Isaac (2003) are shown for reference. The roundness *R* is defined as the ratio of the average projected area of the particle in the *x*, *y*, and *z* directions to the cross section of maximum diameter–equivalent sphere

This definition is also used by Ishimoto (2008). For CPI images, the projected area can be estimated only in one direction.

As can be seen, the roundness decreases as the diameter of the particle increases. At the maximum measured size (1 mm), the value is close to 0.4, which is what Ishimoto (2008) obtained for the fractal snowflakes. We obtain a smaller value, about 0.32 on average, for the aggregates.

## 3. Scattering modeling

We model the falling snowflakes using three different snowflake shape models: the fractal model, aggregate model, and soft spheroid model. Scattering computations are conducted using the Amsterdam DDA code (ADDA) by Yurkin et al. (2007) and the TMM code by Mishchenko (2000). ADDA is an implementation of DDA, a numerical method in which the scatterer is approximated by a set of dipoles. It can be applied to any shape, and can be considered a semiexact method. The accuracy of DDA is determined by the number of dipoles, dipole spacing, and refractive index of each dipole, as discussed, for example, by Yurkin et al. (2006), Draine (2008), and Zubko et al. (2010). For improved accuracy, we use filtered coupled dipoles in ADDA (Yurkin et al. 2010), which changes the form of the Green’s tensor and polarizability of dipoles. TMM is an exact and efficient method, but the implementation used in this study can only be applied to rotationally symmetric and homogeneous particles. In TMM computations, the snowflakes are assumed to be oblate spheroids, and the interior of the particle is approximated by EMAs. We use the Maxwell–Garnett EMA with ice as an inclusion in an air matrix. The refractive indices of ice are from Mätzler (2006). The aspect ratio of the spheroids is 0.65, corresponding to the average aspect ratio of the aggregates. We use two different EMA solutions, using the two mass–diameter relations from the smaller stellar and larger fernlike aggregates, to demonstrate that differences in mass–diameter relations can produce large differences in radar cross sections for the soft spheroid model.

For the orientations of spheroids, we assume a Gaussian distribution with a 20° standard deviation from the horizontal plane. This assumption is a compromise between the values reported by others in the literature, for example, Russchenberg and Ligthart (1996) and Matrosov (1991). For the fractals and aggregates, we use a fixed orientation, such that the maximum moment of inertia is horizontally aligned.

We use four different radar frequencies: C (5.6 GHz), Ku (13.6 GHz), Ka (35.6 GHz), and W (94.0 GHz). For the C band, we consider a horizontal incidence similar to ground-based radars, while for the higher frequencies a vertical incidence of satellite radars is adopted.

## 4. Results and discussion

In Figs. 6 and 7, we plot the horizontal backscattering cross section ( is an element of the amplitude scattering matrix, and *k* is the wavenumber) and the linear depolarization ratio (LDR) *σ*_{vh}/*σ*_{hh} as a function of *D*_{max}. We also show the maximum diameter sphere size parameter.

Figure 6 shows *σ*_{hh} for the modeled particles. At the C band (top left panel), both fractals and aggregates agree well. The two TMM solutions also agree well with their corresponding aggregate and fractal results, but show large mutual differences at small sizes. This shows that, because of different ice crystal habits, the mass–diameter relation will be different, and therefore will produce very different cross sections for different sizes when using TMM and EMA for soft spheroids.

As the radar frequency is increased, the dispersion in the cross sections increases because of the increased influence of shape. At the Ku band (top right panel), the presence of the first non-Rayleigh-type interference feature decreases the cross sections for spheroids at large *D*_{max}. At the Ka band (bottom left panel), there are already several interference minima present, and the TMM computations seem to underestimate the cross sections by a factor of 10 at large *D*_{max}. When moving to the W band (bottom right panel), the aggregates and fractals still largely coincide, albeit with a considerable dispersion, but the TMM computations seem to underestimate the cross sections as much as by a factor of 50–100. There are also many interference minima present for spheroids. Fractals and aggregates, on the other hand, do not show such clear interference features. This may be due to the more regular shapes of the spheroids, giving rise to more regular internal fields.

Figure 7 shows *σ*_{vh}/*σ*_{hh} (LDR) as a function of *D*_{max}. Overall, there is a great deal of variation (60 dB) for both the fractals and the aggregates for all sizes and wavelengths, indicating that LDR is very sensitive to shape. The aggregates produce higher values than the fractals. This could be because of the difference in mass for a given diameter. The mass–diameter relations are not exactly the same for these shape models. As the radar frequency increases, LDR increases. At Ka band, the LDR values tend to be larger than at the W band for aggregates, which is surprising. We do not have a satisfying explanation for this yet. For the spheroids, the interference features are again clearly visible, while fractals and aggregates do not show such features. TMM computations underestimate LDR by about 30 dB at all frequencies.

It should be noted that the detection limit for LDR in modern weather radars is about −30 dB, which means that only the largest snowflakes generated in this study could in principle be detected at Ka and W bands.

## 5. Conclusions

Dual-polarization radar backscattering by three different snowflake models at four different frequencies has been studied. Two of the models, fractals and aggregates, have realistic shapes and have been simulated using DDA. The third model, soft spheroids, employ EMAs to account for the air–ice mixture, and have been simulated using TMM. The models were tuned to represent physical properties, such as the mass, density, and aspect ratio, of real snowflakes.

The radar backscattering cross sections of aggregates and fractals agree well at all the studied frequencies, while spheroids agree fairly well only at the C and Ku bands, but can underestimate the cross sections by a factor of 10 at the Ka band and a factor of 50–100 at the W band. For LDR, there are consistent differences between the complex shapes and spheroids at all studied frequencies. The differences are about 30 dB at all frequencies for larger sizes (*D*_{max} > 5 mm). There are also some differences between fractal and aggregate models, but these can be explained by different mass–diameter relations.

It should be noted that the aspect ratios of aggregates and fractals vary, whereas they are fixed for the spheroids. Some of the differences between the models may be related to this. It would be interesting to study how an aspect ratio distribution would affect the backscattering cross sections and LDR by spheroids.

## Acknowledgments

We thank the Finnish Center for Scientific Computing (CSC) for the permission to use their superclusters. This work was supported by the Academy of Finland (Contracts 125180 and 128328).