## 1. Introduction

In recent satellite missions, spaceborne radar observations, sometimes in combination with passive microwave radiometer measurements, are being used to quantify the precipitation rates of liquid, ice-phase, and mixed-phase hydrometeors. The Tropical Rainfall Measuring Mission (TRMM; see Kummerow et al. 1998) satellite observatory featured a 13.8-GHz (Ku band) radar and a microwave radiometer with nine channels ranging from 10 to 85 GHz. TRMM was recently succeeded by the Global Precipitation Measurement mission (GPM; see Hou et al. 2014), in which a dual-wavelength radar operating at 13.6 and 35.5 GHz (Ku and Ka bands) and a microwave radiometer with 13 channels from 10 to 183 GHz were deployed on the GPM *Core Observatory* satellite. Also, as part of the Afternoon Train (A-Train; see Stephens et al. 2002) constellation of sensors, a 94-GHz radar and a microwave radiometer with 14 channels from 7 to 89 GHz were deployed on two satellites having the same suborbital track. Physically based precipitation estimation on the basis of these radar/radiometer observations requires that a consistent model for the absorption and scattering properties of precipitation particles, spanning the range of active and passive channel frequencies, be utilized (Haddad et al. 1997; L’Ecuyer and Stephens 2002; Grecu and Olson 2008; Grecu et al. 2011).

In particular, remote sensing of falling snow presents a significant physical modeling challenge because of the diverse habits and vertical distributions of ice particles (Heymsfield et al. 2002; Heymsfield 2003). There are three main parts to the modeling effort: first, realistic ice particle geometries must be specified, since it has been demonstrated that the radiative extinction by particles is geometry dependent for particles with equivalent size parameters (defined as *x*_{eq} = *πD*_{eq}/*λ*, where *D*_{eq} is the liquid equivalent particle diameter and *λ* is the wavelength of radiation) that are greater than ~0.5 (Grecu and Olson 2008; Liao et al. 2013). This implies that particle geometry is important for computing extinction by particles with liquid equivalent diameters greater than ~1.3 and ~0.5 mm at 35 and 89 GHz, respectively. The asymmetry parameter is geometry dependent for even smaller effective size parameters (Grecu and Olson 2008). Second, the radiative single-scattering parameters of each particle need to be calculated for the specified particle geometries and assumed dielectric properties of ice. Third, the relationship between particle number concentrations and gross particle size (or mass) must be specified for the purpose of computing the bulk scattering properties of snow particle polydispersions.

In previous studies of the scattering properties of ice particles, particle geometries that are based upon the habits of individual pristine crystals (e.g., Tang and Aydin 1995; Liu 2004; Kim 2006; Hong 2007; Weinman and Kim 2007; Botta et al. 2013; Lu et al. 2014) and spheroid approximations (e.g., Liu 2004; Matrosov 2007; Tyynelä et al. 2011; Johnson et al. 2012; Liao et al. 2013; Ori et al. 2014) have been utilized, and aggregates have been simulated by combining spheres (Maruyama and Fujiyoshi 2005) or pristine crystals (e.g., Hong 2007; Petty and Huang 2010; Botta et al. 2010, 2011; Tyynelä et al. 2011; Nowell et al. 2013; Bi and Yang 2014; Ori et al. 2014) or by using statistical representations (e.g., Ishimoto 2008; Tyynelä et al. 2011). In this study, the objective is to describe the shapes of particles down to a relatively small scale (~50 *μ*m). Physical structures at this scale and larger, in individual crystals, are appreciable relative to the shortest radiation wavelengths of interest (~1 mm), and so these structures should be represented realistically so as to perform valid scattering calculations. Instead of using idealized habits to represent the geometries of pristine crystals, a 3D physical model is used to “grow” individual crystals. The pristine crystals of various sizes are then aggregated to create a set of composite particles that is diverse and extensive relative to those generated in previous studies. The ultimate objective is to simulate a set of particles that can span the range of aggregate ice particle shapes and sizes that may occur naturally.

In previous studies, the single-scattering properties of structured ice particles have been carried out using the discrete dipole approximation (DDA; Purcell and Pennypacker 1973; Draine and Flatau 1994), the finite-difference time-domain method (Kunz and Luebbers 1993), the generalized multiparticle Mie solution (Xu 1995; Xu and Gustafson 2001), the conjugate gradient–fast Fourier transform method (Sarkar et al. 1986; Liao and Sassen 1994), the pseudospectral time domain method (Liu et al. 2012), the Rayleigh–Gans approximation (Tyynelä et al. 2013; Leinonen et al. 2013; Hogan and Westbrook 2014) and modifications thereof (Lu et al. 2013), and extensions of the

The simulation of pristine crystals and aggregates is described in section 2. Single-scattering parameters associated with individual particles and polydispersions of particles are presented in section 3 for a selection of radar and microwave radiometer wavelengths that are relevant to TRMM and the GPM mission. Bulk reflectivity and radiance simulations for an idealized snow layer are also described in this section. In section 4, the results of the study are summarized, and ideas for extensions of the current work are discussed.

## 2. Ice-phase particle modeling

### a. Pristine crystals

The simulation of individual pristine ice crystals is based upon a method pioneered by Gravner and Griffeath (2009, hereinafter GG) called “snowfake.” In brief, snowfake is a computational algorithm that describes the diffusion of vapor and freezing (or attachment) of water substance at the boundary of an ice crystal, using a 3D planar triangular lattice of hexagonal prisms to discretize the moisture budget. By varying the initial crystal seed, the supersaturation level of the environment, the parameters that control the phase of moisture in crystal boundary prisms, and the thresholds for the attachment of boundary prisms to the crystal, a large variety of crystal habits that closely resemble crystals found in nature can be “grown” (GG). Note, however, that, even though the processes of crystal growth are represented in snowfake, the algorithm incorporates tunable parameterizations of these processes and the dimensions of the crystals are somewhat arbitrary. In this study, the nominal size of the prisms in the 3D lattice is assumed to be 17 *μ*m, and the crystals are interpolated to a 17-*μ*m Cartesian grid for subsequent processing.

The snowfake algorithm is parallelized and is run using virtually the same input parameters that GG used to create archetypal crystals. Shown in Fig. 1 are illustrations of the evolution of a fern dendrite, needle, and “sandwich” plate with broad branches (see Figs. 13, 29, and 43, respectively, of GG for reference). Note that, although the basic habits of the crystals are maintained during growth, subtle changes of crystal structure occur over time as well. In all, nine different crystal habits based upon the inputs from GG for crystals illustrated in their Figs. 13, 14, 4, 16, 29, 19, 44, 46, and 43 are grown; see the top row of Fig. 2 for thumbnail sketches of the nine crystal types. They include, in order, the fern dendrite, classic dendrite, dendrite with facets, simple star-shaped dendrite, needle, three dendrites with broadening arms, and a sandwich plate, as described in GG. Note that the crystals grown in the current study may be slightly different from those produced by GG because of small differences in input parameters and coding details, but they are a close approximation to the habits shown in Fig. 2. These crystals are sampled at regular time intervals of the crystal growth simulation to create pools of crystals of each habit.

The nine crystal types generated with snowfake represent crystal habits that are typically observed in warmer growth regimes, that is, at temperatures greater than −20°C; see, for example, Heymsfield (2003). Aggregates of these crystal types were collected in snow events at Sapporo, Japan, by Fujiyoshi and Wakahama (1985), and were photographed in situ using short-exposure cameras by Garrett and Yuter (2014) in the Wasatch Mountains of Utah. Although crystals with other habits (bullet rosettes, capped columns, etc.), rimed crystals, and aggregates of these crystals are also commonly observed, the crystals generated in this study are limited to unrimed crystals with simpler habits that were also simulated by GG. The simulation of crystals with alternative habits and rimed particles is left for future investigations.

### b. Aggregate snow particles

In situ microphysics probe observations indicate that the habits of particles with sizes greater than ~500 *μ*m are predominantly aggregates of crystals (Field 2000; Heymsfield et al. 2002; Heymsfield and Miloshevich 2003; Schmitt and Heymsfield 2014). Therefore, to produce full distributions of snow particles representing the diversity of observed habits, simulations of crystal self-collection are performed to create aggregate snow particles. In the current study, each aggregate is composed of only pristine crystals from the same snowfake growth simulation; that is, only crystals having one of the nine habits are used to create any aggregate. The crystals that are aggregated come from different growth stages of the crystal growth simulation described in section 2a, however. As a result, the component crystals in a given aggregate will be of different sizes but will have the same basic geometric structure.

The self-collection algorithm proceeds as follows. First, a “collector” particle is randomly selected from the growth history of a given pristine crystal. The collector is rotated such that its axis of maximum moment of inertia *I*_{max} is oriented parallel to the vertical direction, and then the particle is rotated about this axis by a random angle. Then, a “component” crystal from the same crystal growth history is selected at random. The *I*_{max} of the component crystal is also oriented vertically, and the crystal is rotated about this axis by a random angle, but the component particle undergoes another rotation about an axis perpendicular to the vertical axis (i.e., a random relative tilt angle). The relative tilt angle is assumed to be normally distributed with zero mean and a standard deviation of 60°, in an attempt to simulate the fall orientations of the collector and component particles in a turbulent environment. The angular deflection is not applied to the collector particle, because it is the relative difference in particle tilt that is important. Note that this assumed distribution of relative tilt angles is consistent with a particle median canting angle of roughly 28° if the collector and component particle canting are assumed to sum to the relative tilt and with a median canting angle of 40° if canting of only one of the two particles contributes to the relative tilt. Although lower canting angles are expected from theoretical considerations (see Klett 1995), these canting angles are in the range reported by Garrett et al. (2015), who used short-exposure photography to derive the distributions of snow particle canting from snow events in the Wasatch Mountains of Utah.

To simulate the joining of the collector and component particles, both particles are first projected onto a horizontal plane, and the centroids of the projections are determined. The mean distances *D*_{p} of the projection elements to each centroid are also obtained. Now, considering the collector particle, a random point in the horizontal projection of the particle is determined, with the distance between this random point and the projection centroid described by a normal distribution with zero mean and a standard deviation proportional to *D*_{p}. A corresponding point is similarly found in the horizontal projection of the component particle. These two points are aligned vertically, and the two particles are then brought into contact in this alignment and joined. The joined particle becomes the new collector particle. A new component particle is then selected, and the process is repeated iteratively until the aggregate reaches a liquid equivalent diameter of 3.1 mm or the number of component particles reaches a prescribed, shape-dependent limit.

Note that the locations of ice (and air) in the original pristine crystals are described on a 3D grid, but the rotation and tilting of the particles in the simulated collection process require transformations of the original ice coordinates to new coordinates. In this process, the transformed coordinates of ice in the rotated/tilted particles are retained in computer memory at high precision. Only the ice structures of the final aggregate particles are interpolated to a 17-*μ*m grid. This procedure leads to the simulation of aggregates with ice structures that are well preserved, as seen in Fig. 2.

In the self-collection procedure, it was first assumed that the collector and component crystals had a probability of occurrence that was an exponentially decreasing function of particle size. This led to an initial distribution of aggregate sizes that did not contain many particles of less than ~0.5-mm liquid equivalent diameter. Therefore, the initial set of aggregates is augmented by aggregates of size-limited (<100 *μ*m liquid equivalent) component crystals that are assumed to occur with equal probability, and this broadens the combined distribution of aggregates to include smaller particles.

Using the methods just described, a database of 6646 particles, including single pristine crystals and aggregate snow particles, is created. The relative proportions of crystals/aggregates derived from the nine pristine crystal types are given in Table 1. Particle properties that are stored in the crystal/aggregate database are listed in Table 2. In the database, the minimum particle liquid equivalent diameter is 164 *μ*m (maximum dimension of 260 *μ*m), and the maximum liquid equivalent diameter is 3.1 mm (maximum dimension of 14.26 mm). The largest particle size of the database is not dictated by the aggregation method but rather by the memory limitation for computing the single-scattering properties of the particles, as described in section 3a. A sampling of snow particles composed of the different pristine crystals is illustrated in the columns of Fig. 2. Although the self-collection method is heuristic, the properties of the resulting snow particles are consistent with observations. For example, mass versus the maximum particle dimension of the snow particles is plotted in Fig. 3. Superimposed on the particle simulations are relations of mass versus maximum dimension derived from airborne in situ probe observations by Heymsfield et al. (2010), categorized by cloud type. Note that the distribution of the plotted data clusters around the observed relationships; the aggregates of different habits tend to exhibit different relations of mass versus maximum diameter, however. For example, aggregates of needles can have relatively low masses for a given particle size, whereas aggregates of sandwich plates are generally more massive.

Habits of the nine simulated pristine crystals and the proportion of crystals/aggregates in the database derived from those pristine crystals. Note that the naming of crystals follows GG, and the figure numbers in GG that illustrate the crystal types are given in parentheses.

Listing of particle properties stored in the crystal/aggregate database.

The fractal dimension of each particle is determined using the box-counting method (Falconer 2003). The mean fractal dimension of the simulated snow (2.0) is also in the range of fractal dimensions derived from in situ observations (2.0–2.3) (Schmitt and Heymsfield 2010).

## 3. Particle microwave scattering properties

### a. Single particles

The single-scattering properties of the simulated snow particles are calculated using DDA as coded in DDSCAT 7.1 (Draine and Flatau 1994). In the DDA method, the complex geometry of each particle is represented, electromagnetically, by a distribution of polarizable points (dipoles), and the method yields accurate single-scattering properties for the specified distribution. The assignment of polarizabilities follows Gutkowicz-Krusin and Draine’s (2004) correction of Draine and Goodman’s (1993) method. The 3D simulation grid of the aggregate snow particles has a resolution of 17 *μ*m, but, since the minimum wavelength of radiation (1.6 mm) considered in the current study is much larger, the simulated particle’s dipole distribution can be represented on a coarser-resolution grid while still satisfying the maximum interdipole distance criterion of DDA. Therefore, the aggregate particles on the original simulation grids are resampled on a 50-*μ*m-resolution dipole grid using 3 × 3 × 3 averaging and thresholding of the particle ice distributions. Using the lower-resolution dipole grid, the computational burden of DDA single-scattering calculations is much reduced, and there is only a slight degradation of the structures of the simulated particles. The averaging of particle structures could be eliminated if computer memory limits increase or if DDA methods that take advantage of the sparse ice structures of snow particles are employed (Petty and Huang 2010).

It is also assumed that the snow particles are falling with random orientations, and so the single-scattering parameters of each particle are averaged over 900 different particle orientations to produce representative parameters. The single-scattering parameters computed for each particle are the extinction cross section, the scattering cross section, the asymmetry parameter, and the backscatter cross section (Table 2). Note also that the scattering properties are computed for particles up to a maximum size of 14.26-mm maximum dimension. This upper limit is due to the memory capacity of the computational nodes utilized to perform the DDA calculations; see section 4 for further discussion of computational limits.

^{−3}), as well as particles having ice–air densities that are a function of size, are considered here. The spherical particles with size-dependent densities are assumed to have the observed relation for mass versus maximum dimension that is described by Heymsfield et al. (2010):where

*m*is the particle mass in grams and

*D*is the particle maximum dimension in centimeters. The Bruggeman (1935) method for computing the dielectric properties of homogeneous mixtures is selected to compute the refractive indices of the spherical particles. Alternative dielectric mixing formulas have been intercompared in several studies (e.g., Meneghini and Liao 1996; Petty and Huang 2010; Johnson et al. 2012). The dielectric properties of mixtures involving liquid water are particularly sensitive to the choice of mixing formula. The dieletric properties of ice–air mixtures are less sensitive to, but still dependent on, the formula chosen, and the Bruggeman (1935) formula leads to scattering properties that are close to those derived from rigorous scattering calculations for spheres composed of random distributions of ice and air (Petty and Huang 2010).

Plotted in Fig. 4 are the single-scattering properties of both the nonspherical crystal/aggregate snow particles and the spherical, homogeneous ice–air particles. The single-scattering properties illustrated in Fig. 4 are calculated at selected dual-frequency precipitation radar (DPR; 13.6 and 35.5 GHz) and GPM Microwave Imager (GMI; 89 and 165.5 GHz) channel frequencies, which are similar to the channel frequencies of the High-Altitude Imaging Wind and Rain Airborne Profiler (HIWRAP; 13.9 and 33.7 GHz) and the Conical Scanning Millimeter-Wave Imaging Radiometer (CoSMIR; 89 and 165.5 GHz) (G. Heymsfield et al. 2013; Wang et al. 2013). Shown in the left two columns of Fig. 4 are the extinction and backscatter efficiencies at the radar frequencies; these parameters are used to define the radar attenuation and reflectivity of the precipitation. The efficiencies plotted in Fig. 4 are with respect to the geometric cross sections of liquid equivalent spheres.

Note that the extinction and backscatter efficiencies of the larger nonspherical particles substantially exceed the efficiencies computed for the spherical ice–air particles with observed size-dependent densities. As seen in Fig. 4, the single-scattering properties of the 0.1 g cm^{−3} constant-density particles are in better agreement with the properties of the nonspherical aggregate particles, but the interference nodes seen in the backscatter efficiencies of the spherical particles lead to discrepancies at the larger particle sizes at 35.5 GHz. The extinction and backscatter efficiencies of the 0.3 g cm^{−3} spherical particles exceed those of the nonspherical particles at 35.5 GHz, a result that differs from that of Liao et al. (2013), who found good agreement between spherical and nonspherical particle models if a 0.3 g cm^{−3} density was assumed for the spheres. These authors assumed that all nonspherical particles were aggregates composed of two types of bullet rosette crystals, and the bullet rosettes were aggregated symmetrically such that the resulting particles were roughly spherical in profile.

The computed extinction efficiencies and asymmetry parameters at 89 and 165.5 GHz are shown in the two right columns of Fig. 4. These frequencies correspond to the two higher-frequency window channels of the GMI radiometer. The single-scattering albedos of the nonspherical and constant-density spherical ice particles at these frequencies (not shown) have nearly the same magnitudes, and are close to 1 for particles with greater than ~0.5-mm liquid equivalent diameter. Therefore, extinction and scattering efficiencies are almost synonymous for the larger particles. Note that the extinction efficiencies of the larger nonspherical particles are substantially greater than those of the variable-density spheres, and they generally lie between the efficiencies of the 0.1 and 0.3 g cm^{−3} constant-density spheres. On the other hand, the asymmetry parameters of the larger nonspherical particles are mostly lower than those of the spherical ice–air particles with either variable or constant density. Moreover, the asymmetry parameters of the nonspherical particles at 165.5 GHz are lower than those simulated by Liao et al. (2013) at a similar frequency using aggregates of bullet rosettes; see Figs. 5 and 6 in their paper for comparison.

As seen in Fig. 4, higher-density (0.2 or 0.3 g cm^{−3}) spherical ice–air particles may better approximate the extinction by the nonspherical snow particles simulated in this study at higher frequencies, but a single choice of spherical particle density does not lead to a good approximation of the asymmetry parameters and backscatter efficiencies of the nonspherical particles over the full span of DPR and GMI frequencies—a result supported by the previous work of Liu (2004). For this reason, and because the scattering properties of nonspherical snow particles can be efficiently computed using parallel-processing techniques, the focus of the current work is on the utilization and testing of the nonspherical particle properties for remote sensing applications rather than on the adjustment of simpler models that approximate nonspherical particle calculations.

### b. Bulk scattering properties of snow particle polydispersions

*n*is the spectral concentration of particles (PSD) as a function of the liquid equivalent diameter

*D*

_{le}. In addition,

*N*

_{w}is the intercept,

*D*

_{o-le}is the liquid equivalent median volume diameter, and

*μ*is the shape factor of the normalized distribution. One advantage of the normalized version of the gamma distribution is that, for a given water content and

*D*

_{o-le}value,

*N*

_{w}is determined independent of

*μ*, whereas the traditional gamma distribution intercept varies strongly with

*μ*, or the shape of the distribution.

The simulated crystals/aggregates are a discrete distribution of particles, and so this discrete distribution must be accommodated to integrate the single-scattering properties of the particles over specified PSDs. The discrete distribution is accommodated by first computing the ensemble mean scattering properties of particles within contiguous 50-*μ*m intervals of liquid equivalent diameter. The mean scattering properties are then slightly smoothed by fitting a linear curve to the mean values of three consecutive 50-*μ*m intervals and replacing the mean value of the central interval by the regressed value. The resulting regression fits are plotted as red lines in Fig. 4.

Also note that the smallest particle in the crystal/aggregate database has a 164-*μ*m liquid equivalent diameter (260-*μ*m maximum dimension), placing it roughly at the upper end of the Rayleigh-scattering regime for the highest microwave frequencies considered in this study. Since the scattering properties of smaller particles are essentially functions of their masses only, 20 spherical ice particles with liquid equivalent diameters between 10 and 200 *μ*m are generated, and their scattering properties are computed using Mie theory. These particles and their scattering properties are added to the crystal/aggregate database to extend it to smaller particle sizes. The largest particle in the crystal/aggregate database has a 3.10-mm liquid equivalent diameter (14.26-mm maximum dimension).

The smoothed, ensemble-mean values of particle extinction, scattering, and backscattering cross sections are multiplied by the normalized gamma PSDs and numerically integrated over particle size to compute bulk extinction, scattering, and backscattering coefficients of the crystal/aggregate particle polydispersions. Radar reflectivity is derived from the backscattering coefficient, as in Johnson et al. (2012). Asymmetry parameters are weighted by the corresponding particle scattering cross sections and convolved by the PSDs; they are then normalized by the bulk scattering coefficients to obtain the bulk asymmetry parameters of the polydispersions. The numerical integrations of nonspherical particle scattering properties over particle size distributions are limited to the range from 10-*μ*m to 3.10-mm liquid equivalent diameter. Sensitivity tests indicate only small biases introduced by the truncation of the integrals over particle size for *D*_{o-le} ≤ 1.5 mm. All bulk scattering properties of spherical and nonspherical particle polydispersions in this study are computed as just described, although in Olson et al. (2016; hereinafter Part II) a filtering of the simulated nonspherical particles using a specified mass–size relation is performed in one sensitivity test.

As an illustration of simulated particle bulk scattering properties that are relevant to radar observations from the GPM mission, in Fig. 5 the difference of the Ku-band (13.6 GHz) reflectivity and the Ka-band (35.5 GHz) reflectivity is plotted against the Ku-band reflectivity for three values of *N*_{w} (Marshall–Palmer value *N*_{w-MP} and that value divided and multiplied by a factor of 10) and a range of *D*_{o-le} values, with *μ* fixed at a value of zero. The simulations are performed successively for three spherical snow particle models with constant densities of 0.1, 0.2, and 0.3 g cm^{−3} and for the nonspherical crystal/aggregate particle model. The Ku–Ka reflectivity difference is independent of *N*_{w}, and, with *μ* fixed, the Ku–Ka difference is an increasing function of *D*_{o-le}, as seen in Fig. 5 [see also Liao et al. (2005)]. From a remote sensing perspective, *D*_{o-le} could be estimated from the Ku–Ka reflectivity difference, and then *N*_{w} could be estimated from the Ku reflectivity.

Note, however, that the interpretation of the reflectivities is dependent on the ice density and shape. As the assumed density of the spherical ice particles increases, the *D*_{o-le} associated with a given Ku–Ka difference increases while the *N*_{w} associated with a given Ku–Ka difference and Ku reflectivity decreases. If one neglects attenuation, the estimated snow water content decreases as the assumed spherical particle density increases, for a given set of Ku- and Ka-band reflectivity measurements.

The nonspherical crystal/aggregate particle curves exhibit unique behavior, however. For the largest value of *N*_{w}, the curve of Ku–Ka difference versus Ku reflectivity is similar to that of the 0.1 g cm^{−3} density spherical particles. For the lowest value of *N*_{w}, the reflectivities of the nonspherical particles and the 0.1 g cm^{−3} spherical particles are similar at low Ku reflectivities, but the Ku–Ka differences of the nonspherical particles decrease in relation to the 0.1 g cm^{−3} spherical particle differences as the Ku reflectivity increases. The reason for this behavior can be traced to the backscatter efficiency curves of the 0.1 g cm^{−3} density spherical particles in Fig. 4. At 13.6 GHz the backscatter efficiency curves of the 0.1 g cm^{−3} spherical particles and nonspherical particles are nearly the same, whereas at 35.5 GHz the spherical particle curve exhibits a minimum “node” of efficiency for particles with diameters near 2.5-mm liquid equivalent while no such node appears in the nonspherical particle curve. The higher *D*_{o-le} calculations emphasize the efficiency contributions of larger particles, and so it follows that the nonspherical particles have higher 35.5-GHz reflectivities in a relative sense, and the Ku–Ka reflectivity differences of the nonspherical particles are reduced. Indeed, the Ku–Ka reflectivity differences of the 0.2 g cm^{−3} spherical particles at moderate Ku reflectivities are closer to those of the nonspherical particles, but the Ku–Ka differences of the 0.2 g cm^{−3} spheres are lower than those of the nonspherical particles at low Ku reflectivities (Figs. 5b,d). The general conclusion of this analysis is that there is no unique assumption about the density of spherical particles that can be made in such a way that they faithfully represent the reflectivity behavior associated with the nonspherical crystal/aggregate particles. However, 0.1–0.2 g cm^{−3} spheres can be used to roughly approximate the reflectivities of the nonspherical particles.

In addition to *N*_{w} and *D*_{o-le}, the PSD shape factor *μ* can potentially have an impact on bulk scattering properties. Plotted in Fig. 6 are the Ku–Ka reflectivity differences versus Ku reflectivities for *μ* values of −2 (Figs. 6a,b) and +2 (Figs. 6c,d), which represent low and high values of the shape factor within the observed range (A. Heymsfield et al. 2013). The reflectivities are plotted for both spherical particles with 0.1 g cm^{−3} density (Figs. 6a,c) and nonspherical crystal/aggregate particles (Figs. 6b,d), which can be compared with the corresponding *μ* = 0 calculations in Figs. 5a and 5d, respectively. With regard to the spherical particles, increasing *μ* tends to lower the Ku–Ka reflectivity differences and Ku reflectivities, for a given *N*_{w}/*D*_{o-le} combination. The trend of decreasing Ku–Ka reflectivity difference with increasing *μ* is not followed by the nonspherical particles, which show more-complex behaviors depending on the values of *N*_{w} and *D*_{o-le}.

As a prelude to a discussion of radiance calculations, simulated 89- and 165.5-GHz bulk extinction coefficients and asymmetry parameters of snow particle polydispersions are shown in Figs. 7 and 8, respectively, for both 0.1 g cm^{−3} spherical particles and nonspherical crystal/aggregate particles. Single-scattering albedos for these distributions of particles (not shown) are close to a value of 1 for water contents greater than ~0.1 g m^{−3}.

Note that the extinction coefficients of the spherical and nonspherical snow particles at 89 GHz are roughly in the same range. Extinction coefficients approximately double as the particle density increases from 0.1 to 0.3 g cm^{−3} (not shown). At 165.5 GHz, there is a divergence of the spherical and nonspherical particle extinction curves for the higher values of *N*_{w}, with the 0.1 g cm^{−3} spherical particles producing roughly one-half of the extinction of the nonspherical particles for the highest *N*_{w} value.

The asymmetry parameter is only a function of the angular distribution of scattered radiation, and therefore the simulations of asymmetry parameter depend on the shape of the assumed PSD only. Since *μ* is fixed, the asymmetry parameter is a function of *D*_{o-le} only, and this can be seen in the plots presented in Figs. 7 and 8. Note that, in general, the asymmetry parameters of the nonspherical particle polydispersions are significantly lower than those of the spherical particles. Spheres with a density of 0.3 g cm^{−3} produce somewhat lower asymmetry parameters (not shown) but not as low as those of the nonspherical particles. This implies that the polydispersions of nonspherical crystal/aggregate particles are not as forward scattering as spherical particles—an inference that is also supported by Fig. 4. Similar calculations are performed for *μ* values of −2 and +2 (not shown), but these calculations indicate that the choice of *μ* has a relatively minor impact on bulk extinction coefficients and asymmetry parameters.

The impact of particle shape on upwelling microwave radiances at the higher microwave frequencies is now examined. To isolate the effect of scattering, simulations of upwelling radiances at 89 and 165.5 GHz are performed for a homogeneous, 1-km-thick snow layer with a temperature of 273.15 K, overlying a blackbody surface that also has a temperature of 273.15 K. Downwelling 2.7-K cosmic background blackbody radiation impinges on the top of the snow layer from above. The snow water content in the layer is varied by changing *N*_{w} and *D*_{o-le} in the PSD, and alternative snow particle models are utilized, as in the previous examples. For simplicity, no gaseous absorption is assumed in the snow layer. Upwelling radiances are computed using Eddington’s second-approximation radiative transfer solution with delta scaling (Weinman and Davies 1978; Joseph et al. 1976), which is appropriate for a nadir-viewing instrument looking down at the top of the snow layer. The Eddington radiative transfer solution is currently used in the GPM combined radar–radiometer precipitation estimation algorithm, since it is computationally efficient and reasonably accurate. The radiance calculations, as they have been described here, are designed to represent the radiances upwelling from a snow layer atop an optically thick liquid precipitation layer with a phase transition at the freezing level, but in an idealized way that makes the interpretation of results more straightforward.

Computed upwelling radiances at 89 and 165.5 GHz are presented in Figs. 9 and 10, respectively. As expected, the upwelling radiances all decrease with snow water path because of the dilution of 273-K radiances upwelling from the base of the snow layer by backscattered 2.7-K cosmic background radiances in the layer [see also Roberti et al. (1994) and Olson et al. (2001)]. These scattering “depressions” are also greater in magnitude at 165.5 GHz relative to those at 89 GHz, given that the extinction (and scattering, since the single-scattering albedo is nearly 1) is greater at 165.5 GHz. Note also that layers of the more dense spherical particles produce greater scattering depressions than do layers of less dense particles.

Figure 10 indicates substantially lower 165.5-GHz radiances (because of scattering) based upon the nonspherical crystal/aggregate particle model, relative to radiances simulated using the 0.1 g cm^{−3} spheres. The greater scattering depressions are associated with the higher extinction (and scattering) coefficients and lower asymmetry parameters of the nonspherical snow particles (see Fig. 8), although the asymmetry parameters have a substantially greater influence on scattering depressions. To demonstrate this effect, radiance simulations that are based on nonspherical particle layers, but with the asymmetry parameters of 0.1 g cm^{−3} spheres substituted for the asymmetry parameters of the nonspherical particles, are plotted as dashed lines in Fig. 10d. It is clear that the lower asymmetry parameters of the nonspherical particles are required to produce the simulated scattering depressions of the nonspherical particle snow layers. Since the nonspherical particles are less forward scattering than the spheres, there is generally less upward-propagating scatter of the 273-K blackbody radiation that originates from below the snow layer, as well as greater backscatter of low-intensity cosmic background radiances. Therefore, the upwelling radiances at the top of the snow layer are reduced. The asymmetry parameters of the nonspherical particles at 89 GHz are also lower than those of the 0.1 g cm^{−3} spheres, but the impact on radiances is less dramatic (Fig. 9), owing to the lower bulk scattering by snow particles at 89 GHz.

Relative to the 0.1 g cm^{−3} spheres, spherical particles with 0.2–0.3 g cm^{−3} (0.4–0.5 g cm^{−3}) densities produce greater scattering depressions that are comparable to those derived from nonspherical particles at 89 GHz (165.5 GHz), as a result of their higher bulk scatter and lower asymmetry parameters (0.4–0.5 g cm^{−3} simulations not shown). It is evident, however, that a single choice of spherical particle density does not lead to consistent spherical and nonspherical particle radiance simulations across the microwave spectrum. The different scattering properties of spherical and nonspherical snow particles will have important consequences for the simultaneous fitting of airborne radar and radiometer observations of stratiform precipitation, to be described in Part II of this series.

Note from Figs. 9d and 10d that the radiances upwelling from the nonspherical particle snow layers show a dependence on liquid water path only. This fortuitous result is linked to the interplay of extinction and asymmetry of scatter of the snow particles. Both the spherical and nonspherical particles are almost purely scattering. In the limit of conservative scattering (single-scattering albedo = 1), the Eddington’s second-approximation radiative transfer solution with delta scaling is dependent on single-scattering properties through two parameters: the scaled extinction coefficient *k*′ = *k*(1 − *g*^{2}), where *g* is the asymmetry parameter, and a second parameter that depends on both *k* and *g* and that will be called *β* here: *β* ≡ *k*(1 − *g*). The parameters *k*′ and *β* are similar, and for the nonspherical particles they are essentially functions of water content only, whereas for spherical particles they show some additional sensitivity to *N*_{w}. More rigorous and extensive multistream or Monte Carlo radiative transfer calculations will be required to determine whether the radiance dependencies described here still hold, and under what conditions.

## 4. Summary and concluding remarks

A model that can be used to represent the nonspherical geometries of falling snow particles, including those of individual pristine crystals and aggregates of crystals, is developed in the current study. The simulated crystals and aggregates have mass-versus-size and fractal properties that are consistent with field observations. The single-scattering properties of the simulated snow particles are calculated using the discrete dipole approximation in a computationally efficient way that makes use of parallel-processing techniques. Thereby, a large set of snow particles and their radiative properties is generated, and, by averaging the radiative properties of the particle ensembles in mass bins and convolving the mean properties by assumed normalized gamma particle size distributions, the bulk single-scattering properties of snow particle polydispersions are also calculated. Using the bulk single-scattering properties of the particles, radar reflectivities are computed, and upwelling microwave radiances are simulated for different snow layers over an idealized “liquid precipitation” background. The single-scattering properties, reflectivities, and upwelling radiances simulated using the nonspherical particles are compared with the same quantities computed on the basis of spherical, homogeneous ice–air spheres with different assumed densities. The frequencies selected for the simulations are consistent with the operating frequencies of the GPM DPR and the snow-sensitive, higher-frequency channels of the GMI.

A general conclusion of this work is that spherical particle models cannot be used to approximate the single-scattering properties of the nonspherical snow particles generated in this study, across the range of channel frequencies spanned by the DPR and GMI instruments. Although the single-particle extinction (and scattering) efficiencies of 0.1 g cm^{−3} spheres and the nonspherical particles are similar at the 13.6- and 35.5-GHz frequencies of the DPR, the backscatter efficiencies of the spherical particles greater than 2.25-mm liquid equivalent diameter at 35.5 GHz are lower than those of the nonspherical particles with the same mass. The 0.1 g cm^{−3} density spherical particles also produce deficient extinction (and scattering) at the 89- and 165.5-GHz frequencies of the GMI, and the asymmetry parameters are much higher than those of nonspherical particles with the same mass at those two frequencies.

In contrast, Liao et al. (2013) found that the microwave backscattering in the DPR frequency range and the asymmetry of scatter at the 89- and 183.31-GHz GMI channel frequencies due to aggregates composed of bullet rosettes could be approximated reasonably well by spheres or spheroids with densities of 0.3 g cm^{−3}. The scattering properties of the bullet rosette aggregates used in the Liao et al. (2013) study are evidently different from those of the crystals/aggregates generated here, and more work will be required to determine what aspects of the respective ice particle geometries lead to differences in scattering properties. This study and previous work by Liu (2004, 2008), Kim (2006), Petty and Huang (2010), Kulie et al. (2010, 2014), Tyynelä et al. 2011, Leinonen et al. (2012), and others suggest that it may not always be possible to represent the scattering properties of snow particles using spheres, and it has been demonstrated here that the properties of a large number of diverse nonspherical particles can be computed feasibly. Although the calculations of the scattering properties of nonspherical particles in this study are limited to a maximum *x*_{eq} ≈ 5, recent calculations for particles up to *x*_{eq} = 30 have been performed with DDA using the “domain decomposition” message-passing-interface method described in Numrich et al. (2013). This implies that in future work the scattering properties of snow polydispersions can be calculated for relatively large median volume diameters without truncation errors.

A consequence of the differences between the single-scattering properties of the spherical and nonspherical snow particles is that the simulated bulk reflectivities and upwelling microwave radiances associated with polydispersions of the particles are also different. Therefore, the interpretation of observed reflectivities and radiances using such simulations will obviously be different. For example, a Ku–Ka reflectivity difference of 5 dB yields an estimated *D*_{o-le} of 1.1 mm for nonspherical particles (see Fig. 5), whereas for spherical particles *D*_{o-le} values of 1.0, 1.2, and 1.4 mm are estimated if particle densities of 0.1, 0.2, and 0.3 g cm^{−3}, respectively, are assumed. If in addition, the Ku reflectivity is 25 dB*Z*, then the corresponding estimated liquid equivalent water content is 0.31 g m^{−3} for nonspherical particles, and water contents of 0.42, 0.19, and 0.12 g m^{−3} are estimated if spherical particles with densities of 0.1, 0.2, and 0.3 g cm^{−3}, respectively, are assumed.

A question still to be addressed is, then, What particle models are simultaneously consistent with actual radar and radiometer observations at the DPR and GMI frequencies? Here, it is only *presumed* that the nonspherical particles have more realistic geometries and radiative properties. In Part II of this series, simultaneous observations of radar reflectivities from the HIWRAP and upwelling microwave radiances from the CoSMIR, as well as particle polydispersion properties derived from microphysics probe observations, will be utilized to discriminate between the particle models.

An unexpected result, illustrated by Figs. 9d and 10d, is that the simulated microwave scattering depressions at 89 and 165.5 GHz due to layers of nonspherical snow particles are very close to being unique functions of the equivalent liquid water path of the layers. The scattering depressions associated with spherical snow particles tend to show some additional sensitivity to particle size distribution parameters such as *N*_{w}. Although the radiance simulations are highly idealized, this result suggests that measured radiances could provide a useful constraint on the liquid water path of snow layers in radiometer-based or combined radar–radiometer estimation of precipitation. Choosing alternative values of the particle size distribution shape factor *μ*, incidence angle, and layer temperature structure leads to some additional radiance sensitivity that is ~10 K, at high incidence angles in particular (not shown). Differences of this order may be within the general uncertainties of the simulations that are due to details of the snow particle modeling. Another consideration that is not addressed here is the effect of snow particle orientation on the polarization of scattered radiation—an effect that could also lead to greater variation of upwelling radiances. In any event, more realistic multistream, polarimetric radiance simulations using liquid layer backgrounds or different Earth surfaces should be studied to see if these radiance sensitivities continue to hold.

One simplification of the current work is that the full database of simulated nonspherical particles is used to represent their radiative characteristics. As described in section 2, the database of snow particle simulations is composed of a variety of pristine crystals and aggregate particles representing different particle habits and mass-versus-size relations. Alternatively, the full database of particle simulations can be subdivided by component particle habit or filtered by prescribed mass–size relations. The impact of mass–size filtering on snow particle properties and reflectivity/radiance simulations will be explored in Part II of this series. Last, the focus of the current work is on snow particle habits typical of stratiform precipitation regions, and so the simulation of denser particles, such as rimed aggregates, graupel, and hail, will require further study.

The authors thank Janko Gravner and David Griffeath for helpful discussions that led to the development of the simulations of ice crystal growth in this study. The bulk of this investigation was supported by the NASA PMM and RST programs. To be specific, NASA Grants NNX10AI49G, NNX13AG87G, NNX11AR53G, NNX11AR55G, NNX13AG48G, NNX10AH67G, and NNX13AH73G provided support. Kwo-Sen Kuo also acknowledges the generous support of Ziad Haddad and Simone Tanelli.

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