## 1. Introduction

The Global Precipitation Measurement (GPM) mission core satellite planned for launch in 2013 will utilize the combination of a Dual-Frequency Precipitation Radar (DPR) and the multifrequency GPM Microwave Imager (GMI) to estimate rainfall and snowfall over a large fraction of the earth’s surface (Smith et al. 2007). Physical modeling and retrieval algorithm development efforts for GPM will build on experience with earlier satellite missions using active and/or passive microwave instruments, including the Tropical Rainfall Measuring Mission (TRMM; Kummerow et al. 2000), the Advanced Microwave Scanning Microwave Radiometer (Kawanishi et al. 2003), and CloudSat (Stephens et al. 2002; Grecu and Olson 2008).

GPM will depend on state-of-the-art cloud–radiative transfer models and multisensor techniques to infer surface precipitation rates and other cloud properties under a wide variety of conditions. The more observables that are simultaneously exploited in a retrieval scheme—that is, the higher dimensional the space in which observations are compared with model calculations, the greater the vulnerability of that scheme to errors in the computed radiative properties of atmospheric scatterers, including spectral dependence, polarization, and the relationship between microwave emission or attenuation and radar backscatter. With this in mind, and in view of the near-absence of corroborating in situ measurements of these properties for snow, a reevaluation of common approximations is in order.

For most of the twentieth century, calculations of electromagnetic scattering and absorption by atmospheric particles of intermediate size—neither much smaller nor much greater than the wavelength—were practical only for spherical or nearly spherical particles owing to the availability of an exact theory for that special case (Mie 1908). Highly nonspherical ice and snow crystals and snow aggregates have therefore commonly been modeled for microwave radiative transfer purposes, including radar backscatter, as spheres.

In particular, a snowflake (i.e., aggregate of numerous smaller vapor-grown crystals) has often been modeled as an equal-mass sphere consisting of a blend of ice and air (Petty 2001; Bennartz and Petty 2001; Liu 2004). The dielectric properties of the mixture are then calculated using an effective medium approximation, such as that of Maxwell Garnett (1904) or Bruggeman (1935). This general approach is informally known as the “soft sphere” (or sometimes “fluffy sphere”) approximation.

Advances in computing power have greatly increased the range of particle shapes for which scattering properties may be computed with some accuracy. A comprehensive survey of the current state of the art is given by Mishchenko et al. (2000). For complex particles, the two most common methods are the finite-difference time domain (FDTD) method (Taflove and Hagness 2000) and the coupled dipole approximation (CDA; also commonly known as the discrete dipole approximation, or DDA). The latter method, which we employ in this paper, was first demonstrated by Purcell and Pennypacker (1973) and subsequently refined by a number of authors (Dungey and Bohren 1991; Draine and Flatau 1994; Rahmani et al. 2004).

CDA results for fairly simple ice particle shapes have been presented by Evans and Stephens (1995), Liu (2004), Kim et al. (2007), Hong (2007a,b), and Grecu and Olson (2008). Hong (2007b) also included a very compact aggregate of simple hexagonal prisms based on Yang and Liou (1998). These results generally differed significantly from those obtained for soft spheres, although the choice of mappings between particle properties and sphere parameters was not fully explored and may not be unique. For example, Liu (2004) considered only equal-mass spheres as candidate proxies for the nonspherical particles.

Even when CDA or similar numerical methods have been applied to nonspherical particles, the results have sometimes been recast in terms of equivalent soft spheres, usually with the requirement that the spheres be of equal mass, though the degree of “softness” may be a function of frequency (Liu 2004).

Few numerical results have been published for complex aggregates of dendritic snow crystals of a type resembling those found in real snowfall. The first application of CDA to a model of a snow aggregate appears to have been by Osharin (1994), who computed extinction and absorption efficiencies from 10 to 130 GHz for a stochastically generated aggregate consisting of five dendrite-like “crystals” and containing a total of 554 dipoles. For this structure, reasonable agreement was found with the extinction resulting from an appropriately defined equivalent sphere, but agreement for the absorption efficiencies was less satisfactory.

Ishimoto (2008) used the FDTD method applied to fractally generated snow aggregates to compute radar backscatter at 9.8, 35, and 95 GHz. He found that backscatter for larger fractal particles was up to an order of magnitude lower than that computed for volume-equivalent spheres.

One historical impediment to the systematic investigation of scattering and extinction properties of highly complex particles has been computational: both the FDTD method and the most commonly used CDA code, called DDSCAT (Draine 1988), requires computer memory in proportion to the bounding rectangular domain, even if most of the lattice points are unoccupied by dielectric material.

In the case of DDSCAT, the memory requirement is further multiplied by that method’s utilization of efficient numerical solution schemes. Thus, while DDSCAT is the most convenient CDA package for those particles that can be accommodated within the limits of the available machine memory, it is easy to construct sparse structures for which the memory demands are problematic even though the total mass of the particle falls well within the range of what is tractable for a more compact particle.

In this study, we therefore utilize our own CDA code that has been optimized for sparse structures. Only processor time, not physical memory, limits the size and complexity of particles that can be considered.

The theoretical calculation of microwave extinction and scattering by snow has two distinct and equally essential parts: (i) the statistical description of the distribution of sizes, shapes, and densities of snow particles under varying conditions and (ii) the accurate modeling of the electromagnetic properties of the individual particles. This paper examines selected aspects of only the second of these problems. In particular, we wish not only to highlight the feasibility of accurately simulating microwave scattering and extinction by quasi-realistic snow aggregates but also to demonstrate the clear inadequacy of the soft sphere approximation for these aggregates, especially when spectral characteristics are considered.

Secondary objectives include a brief examination of the validity of the Rayleigh–Gans approximation (RGA) for radar backscatter from snowflakes as well as a preliminary parameterization of the mass extinction coefficient for snowfall as a function of particle mass. The latter results, especially the ratios of extinction for pairs of frequencies, lend themselves to direct validation via a ground-based microwave attenuation link.

It is not the purpose of the present paper to undertake a definitive parameterization of all important scattering and extinction properties, as such an effort will require a broader range of shape models as well as field measurements to help constrain and/or validate the choice of models. We therefore focus here mainly on radar backscatter at DPR frequencies and microwave extinction at selected GMI frequencies.

In the next section, we review the theory behind the CDA and some key developments in the recent literature. In section 3, we outline the specific assumptions and properties of our sparse coupled dipole scattering program (SCDScat). In section 4 we compare SCDScat results with Mie results for “soft” dielectric spheres. Comparative memory and CPU requirements for the DDScat and SCDScat are discussed in section 5.

Finally, we present sample results for complex, quasi-realistic snow aggregates. We show that there is no “equivalent sphere” that simultaneously reproduces all relevant scattering properties for these particles. We also show that the Rayleigh–Gans approximation gives significant errors for these particles.

## 2. Theory

### a. The coupled dipole approximation

The coupled dipole approximation was first demonstrated by Purcell and Pennypacker (1973), who calculated approximate radiative cross sections of simple nonspherical interstellar dust grains. According to this method, an arbitrary particle is represented by an array of *N* pointlike polarizable particles located on a cubic lattice. To approximate a homogeneous medium, the spacing *d* of the particles, or dipoles, must be small relative to the wavelength; specifically, |*m*|*d* ≪ *λ*, where *λ* is the wavelength in the external medium (e.g., vacuum or air) and *m* is the (complex) relative index of refraction of the material comprising the particle.

**p**of the constituent dipoles. The polarization is proportional to the local field, so thatwhere

**is a complex 3 × 3 polarizability tensor for the dipole and**

*α***E**is the electric field experienced by the dipole. For a medium such as ice that may be approximated as electrically isotropic, the polarizability reduces to a scalar.

*N*dipoles, the total electric field experienced by the

*j*th dipole is given bywhere

**E**

_{0,j}is the incident electric field at that node.

_{j,k}is the (complex) interaction matrix describing the coupling between the

*j*th and

*k*th dipoles. See Tsang et al. (2001) and Markel (1993) for additional background.

Equation (3) is a complex linear system of dimension 3*N* that can be solved by direct matrix inversion or via iterative methods. Inversion yields the most complete solution but requires considerable memory; iteration requires far less memory but yields a solution for only one incident wave direction and polarization at a time. Once the dipole polarizations have been obtained, all particle radiative properties for a particular direction and polarization of incidence can be readily computed.

### b. Dipole polarizability and particle properties

*α*of the individual dipole, the material volume

*d*

^{3}represented by the dipole, and the complex index of refraction

*m*. The Clausius–Mossotti relation (CMR; Jackson 1999) is given byQuestions have been raised about the validity of the CMR for dipole radiation in view of the fact that, when applying the optical theorem (Jackson 1999) in the Rayleigh–Gans limit, it fails to predict extinction by scattering with pure real

*m*(Draine and Flatau 1994; Tsang et al. 2001). This observation has led to a series of modifications of the CMR to correct this perceived deficiency to account for the radiation reaction effect (Chaumet et al. 2004).

*σ*

_{ext}(Draine and Flatau 1994) to include a compensation for energy loss by radiation reaction, given by (⅔)

*k*

^{3}

**p**

_{j}·

**p***

_{j}. With this modification, which allows us to use the original CMR, our extinction, absorption, scattering, and parallel (radar) backscatter cross sections arewhere

**p**

*is the polarization for dipole*

_{j}*j*and

**e**

_{0}is the unit vector describing the polarization of the incident electromagnetic wave

**E**

_{0}. One may further obtain the single-scatter albedo as

Also of interest in radiative transfer calculations are the scattering phase matrix 𝗣(Θ), where Θ is the scattering angle, and the asymmetry parameter *g*, which is the first moment of the phase function with respect to cosΘ. While these quantities are readily computed from the polarizations, we do not discuss them further in this paper.

## 3. Algorithm description

Our simulations entail the following steps: (i) specify the (*x*, *y*, *z*) positions of the *N* dipoles constituting a target particle shape (the size of the particle may be varied, within limits, by changing the dipole spacing *d*), (ii) specify the polarizabilities *α* for each dipole, (iii) iteratively solve (3) to obtain the polarizations **p**, and (iv) evaluate the optical cross sections and other properties using (5)–(8).

Because dipole positions are specified in the form of a simple list of coordinates rather than by initializing a rectangular grid encompassing the extended particle, no memory is wasted on empty space, and there is no limit on the allowable maximum distance *D*_{max} between any pair of dipoles or on the volume of the bounding rectangular domain. Extremely sparse structures may therefore be easily accommodated; the total memory requirement is determined solely by *N*, irrespective of geometric arrangement.

Moreover, while we require contiguous dipoles to fall on a regular Cartesian lattice in order to ensure the validity of the CMR, we do not require more distant elements of the structure to follow the same lattice arrangement. Thus, we can construct aggregates of simpler particles, each of which has its own regular internal lattice arrangement, but each elementary “crystal” may be randomly oriented with respect to the others. Where two such lattices come into contact with each other, we delete any dipoles that are closer than *d* to any other dipole since this would imply a nonphysical “overlapping” of mass as well as locally invalidating the CMR.

*r*

_{veq}is the radius of the volume-equivalent homogeneous sphere. The mass of the particle iswhere

*ρ*

_{ice}is the density of ice, taken here to be 917 kg m

^{−3}. It is also common to express the mass or volume of an irregular ice particle in terms of a liquid-equivalent radiuswhere

*ρ*

_{liq}is the standard density of liquid water, 1000 kg m

^{−3}.

Because of their significance for current and future sensors, we present simulations for three passive microwave frequencies (18.7, 36.5, and 89.0 GHz) and two radar frequencies (13.4 and 35.6 GHz). The complex index refraction *m* of ice is based on the data of Warren (1984), assuming a temperature of −5°C (Table 1).^{1}

To solve (3) for the polarizations **p**, we employ a direct complex conjugate gradient method (Sarkar et al. 1988). While DDSCAT makes use of computational methods that are more efficient, it is also much more demanding of physical memory. DDSCAT also requires polarizabilities to be specified on a regular lattice, a requirement that is violated by the model aggregates discussed below. In section 5, we give examples of the trade-off between memory and CPU time for the two methods.

## 4. Validation

### a. Solid spheres

For homogeneous spherical particles with arbitrary refractive index *m*, Mie theory gives exact results. To validate our CDA code, we therefore begin by testing it on solid ice spheres for various values of *d* and *N* and comparing the results with those obtained from the BHMIE routine of Bohren and Huffman (1983).

Figure 1 depicts errors for fixed *d* = 20 *μ*m and varying *N*, assuming a value of *m* appropriate for ice at 35.6 GHz. The error is zero for a single dipole and approaches zero again for large *N*. For small values of *N* > 1, we attribute the appreciable error to the fact that a cluster of only a few dipoles cannot reasonably approximate a sphere.

Figure 2 depicts errors for fixed *N* = 14 147 and variable |*m*|*kd*. We find that satisfactory accuracy for the *σ*_{ext} and *ω*_{0} is obtained when |*m*|*kd* < 0.30 is satisfied. For accurate *σ*_{back}, however, *d* must be smaller still by at least a factor of 2. For the range of frequencies considered in this paper, we therefore require *d* ≤ 100 *μ*m.

### b. Soft spheres

#### 1) Effective medium approximations

*ϵ*is the dielectric constant and the subscripts 1, 0, and eff denote inclusion, matrix, and effective dielectric constant, respectively. In the case of a snowflake, the computational results strongly depend on whether the particle is modeled as matrix of air with embedded inclusions of ice or vice versa.

*ν*is an arbitrary positive number;

*ν*= 2 yields the Bruggeman formula while

*ν*= 0 yields Maxwell Garnett.

In the next subsection, we compare CDA calculations for stochastically generated soft spheres with calculations based on a combination of Mie theory and the above dielectric mixing formulas. We thereby not only validate the CDA for a nonhomogeneous sphere but also assess the appropriateness of alternative mixing formulas. We will show that the best overall agreement is obtained neither for Bruggeman nor for Maxwell Garnett, but rather for the generalized mixing formula with *ν* = 0.85.

#### 2) CDA simulations and results

*f*

_{targ}. Since the final

*N*yielded by this procedure fluctuated slightly, the true fraction

*f*iswhere

*N*

_{max}is the total number of lattice points within the sphere. By varying

*f*and

*d*, one obtains any desired combination of density and particle mass, subject only to the requirement that |

*m*|

*kd*< 0.1 for the shortest wavelength considered. Examples are shown in Fig. 3.

A commonly stated requirement for the validity of dielectric mixing rules is that the size of the inclusions be much smaller than the incident wavelength (Sihvola 1989). Because of the stochastic structure of the dipole representations of our soft spheres, this condition is not enforced; large clusters or chains of contiguous dipoles and/or voids are inevitable.

We performed CDA calculations for soft spheres with liquid-equivalent radius ranging from 0.1 to 1.2 mm for 13.4 and 35.6 GHz. For illustrative purpose, we present the extinction cross section per unit mass, backscatter cross section per unit mass, and single-scatter albedo for *r*_{liq} = 1.0 mm in Fig. 4. For all fractions, the optical properties yielded by CDA are bounded by the Maxwell Garnett (ice as inclusion) and Bruggeman rules. However, neither rule fits the CDA results except for very large or small fractions. Interestingly, Maxwell Garnett (ice as matrix) gives far worse results (higher extinction) than the other two cases for all fractions except very close to unity.

On the other hand, we find that a nearly exact fit to *all* CDA-computed properties is obtained using the Sihvola mixing rule (14) with air as the matrix and *ν* = 0.85. This good fit persists over the full range of particle masses and densities and microwave frequencies tested. Our best-fit value of *ν* also approximates the values obtained in a recent numerical study of 2D lossless material using the finite-difference time domain method (Karkkainen et al. 2000).

In summary, our CDA simulations suggest that when applying Mie theory to soft spheres with completely random internal structure, the generalized dielectric mixing rule with *ν* = 0.85 is superior to either the Maxwell Garnett or Bruggeman methods for almost all ice fractions. We will therefore use this method as the benchmark for evaluating the applicability of the soft sphere approximation to more complex particles.

## 5. Performance issues

We tested both DDSCAT and our own CDA code on an Intel Core2 1.8-GHz Linux system with 4 GB of physical memory and using the Portland Group f90 compiler. Processor time comparisons are given in Fig. 5. SCDScat is optimized for targets with low overall volume fraction. When the ice fraction is close to unity, the Fourier transform technique utilized by DDSCAT is more efficient, requiring ∼3% of the CPU time required by our code.

But when the mean particle density is sharply reduced (for fixed total particle mass), (i) the electromagnetic interaction between dipoles is weakened, allowing our code to converge more quickly, and (ii) the memory requirement for DDSCAT (and the associated CPU time) grows substantially while remaining constant for our code.

For ice fractions of only few percent, our code requires CPU times comparable to or even less than DDSCAT. Indeed, in the limit of zero density, our code would require only a single iteration, since each dipole would “see” only the incident EM field (i.e., the Rayleigh–Gans approximation). For *f* ≪ 1, the total CPU time for our code is roughly proportional to *N* ^{2}.

The memory requirement for version 7.0 of DDSCAT, single precision mode, is about 35 + 0.0010*N*_{bound} megabytes, where *N*_{bound} = *N _{x}N_{y}N_{z}* is the number of lattice points in the bounding rectangular volume (Draine and Flatau 2008). Thus, when the target object has a very low volume fraction or the maximum length in each dimension is large, the memory requirement can be prohibitive, especially on 32-bit machines that cannot address more the 4 GB of virtual memory. For SCDScat, the total memory requirement is only 0.33

*N*kilobytes where

*N*is the number of constituent dipoles, irrespective of their physical arrangement.

For the DA3 aggregate model employed in this study (see below), the DDSCAT virtual memory requirement would have been approximately 10 GB. Using our own code, the requirement was only 20 MB, a roughly 500-fold reduction.

## 6. Application to complex particles

Having validated our CDA code against both solid and soft spheres, we are now equipped to address the important question of whether “equivalent soft spheres” can be found that adequately approximate the scattering and extinction properties of complex structures *resembling* real snow aggregates. It is not our purpose here to obtain “correct” results for actual snowfall since we do not yet know how to accurately describe the shape of any single real snowflake, let alone the wide distribution of sizes and shapes found in a volume of falling snow. Nevertheless, if the soft sphere approximation is found to fail for the shapes considered here, there is little reason to expect it succeed for other aggregate shapes.

In any atmospheric volume containing falling or suspended snow particles, there exist few unambiguous physical or statistical descriptors of that snowfall. The single least ambiguous property is the combined mass of the ice within the volume. All others—particle densities, diameters, shapes, even number—are highly sensitive to one’s definition and may not be uniquely mappable from one particle structure (e.g., a real aggregate) to another (e.g., a soft sphere or an ensemble of smaller solid spheres).

*per unit mass*, radar backscatter

*per unit mass*, etc.:with analogous relationships defining

*κ*

_{abs},

*κ*

_{sca}, and

*κ*

_{back}.

### a. Shape generation

To date, CDA computations using DDSCAT have been published for relatively simple nonspherical ice structures (Evans and Stephens 1995; Kim et al. 2007; Liu 2004; Hong 2007a,b). Those structures were chosen to resemble a number of basic pristine ice crystal habits commonly found in nature, such as needles, plates, prisms, and simple spatial dendrites. Slightly more complex randomly aggregated structures have been modeled by Weinman and Kim (2007), but the constituent elements were simple cylinders. Casella et al. (2008) undertook CDA calculations for stochastically generated clumps of ice resembling graupel (snow pellets).

Here we examine the scattering properties of complex, quasi-realistic aggregates of elemental structures. First, each elemental structure is expressed as a regular array of dipoles. For example, a thin needle may be defined as a 2 × 2 × *n* prism. For dendritic crystals, we digitized selected photographs of real snowflakes published by Libbrecht (2007). An example is depicted in Fig. 6. We then use a stochastic algorithm to assemble an arbitrary number of randomly rotated copies of the elemental structures into aggregates. The resulting structures used in this study are depicted in Fig. 7. These include a needle aggregate (NA) and three dendrite aggregates (DA1, DA2, and DA3).

### b. Size and density characterization

The effective average density (or volume fraction) and the effective geometric radius of any particle are important parameters with respect to the particle’s radiative properties. While these parameters are well defined for spheres, they are not well defined for more complex structures. It has been common to define the effective geometric radius by direct or indirect reference to the bounding sphere. In particular, Liu (2004) defines an empirical “softness parameter” that describes where the radius of a radiatively equivalent soft sphere falls relative to the radius of the equal-mass solid sphere versus that of the bounding sphere.

For snow aggregates, the geometric bounding sphere is an inherently nonrobust particle property, as it is determined entirely by the extrema (e.g., random protruding “whiskers;” see for example Fig. 7a) that may or may not be radiatively significant. Thus, two nearly equivalent realizations of a stochastically generated aggregate might have significantly different bounding spheres.

*r*

_{eff}based on the root-mean-square distance of the particle mass from the center of gravity 〈

**r**〉 of the dipole array. Specifically,where the factor

*r*

_{eff}equals the actual radius when the particle is a homogeneous sphere. Our

*r*

_{eff}is therefore similar to, but 29% larger than, the “radius of gyration” employed by Osharin (1994) and Westbrook et al. (2006). The effective volume fraction is then

The above properties are given for all four aggregate models in Table 2 for the minimum and maximum values of *d* employed. Note that the fraction *f* depends only on the shape and does not depend on the scaling of the particle (this is true for any definition based on the bounding sphere as well).

## 7. Soft sphere approximation

In section 2, we showed that our CDA computations accurately reproduce Mie results for soft spheres when the latter are based on the exponential dielectric mixing rule with the empirically determined coefficient *ν* = 0.85. We can therefore now perform CDA calculations on more complex particle shapes and determine whether there exists an equivalent soft sphere with similar radiative properties.

### a. Criteria for comparison

There are two adjustable parameters in the soft sphere model: density (or volume fraction) and mass (or volume-equivalent radius). For a soft sphere of a given mass and density to be considered radiatively similar to a given nonspherical aggregate, it must simultaneously reproduce a number of key radiative properties, including their spectral dependence. For the purposes of the Global Precipitation Mission, the most important properties to consider, because of their relevance to combined active–passive retrieval strategies, include

- the microwave mass extinction coefficient
*κ*_{ext}at the GMI frequencies of 18.7, 36.5, and 89.0 GHz and, especially, the ratio of extinction for pairs of these frequencies; - the microwave single-scatter albedo
*ω*_{0}at the above frequencies; and - the radar backscatter cross section per unit mass
*κ*_{back}at the DPR frequencies of 13.4 and 35.6 GHz, and especially the backscatter ratio at these frequencies.

As already noted, we are primarily concerned with extinction and backscatter *per unit mass*. Thus, we allow that an equivalent soft sphere might have a different mass than the aggregate it is intended to approximate. In particular, an ensemble of snow aggregates might be best represented by a larger or smaller number of soft spheres having the same total mass. Thus, our criteria for matching soft sphere properties to those of a given aggregate are considerably less constrained than if we required equal masses per particle.

### b. Soft sphere properties

For our comparisons, we consider soft spheres with masses ranging from 0.004 to 16 mg, corresponding to liquid equivalent radii ranging from 10 *μ*m to 1.6 mm. The volume fraction *f* of ice is varied from 1% to 100%. Figure 8 depicts the relationships between backscatter at the two DPR frequencies (Fig. 8a) and extinction at selected pairs of GMI frequencies (Figs. 8b,c) for all combinations of density and mass.

These figures provide the framework for the subsequent interpretation of the CDA results for aggregates, so it is important to understand our novel graphical depiction of the soft sphere results in isolation first. Solid curves represent contours of constant mass *M* for varying volume fraction *f*. Dashed curves represent contours of constant *f* for varying *M*. Collectively, these contours define an envelope of two-frequency results for all possible combinations of mass and density. The intersection between any dashed and solid contour represents the soft sphere results for that particular combination of *M* and *f*.

Conversely, the actual extinction and backscattering properties of any arbitrary particle can be plotted as a point on each of these figures. If the point falls within the envelope of curves defined by the soft sphere calculations, then the mass and density of a radiatively equivalent (for that pair of properties) soft sphere can be determined. Ideally, we would wish for the so-determined mass and density to be the same for all possible pairs of frequencies and radiative properties, in which case we might claim to have found a truly equivalent soft sphere.

For most combinations of radiative properties (considered in pairs), there exists either a *unique* mass-density combination or else *no* equivalent soft sphere. But for DPR backscatter ratios less than about unity, more than one matching one mass-density combination exists—that is, the equivalent soft sphere is *non-unique*. This nonuniqueness occurs for particles with large physical radius (large mass and low density). Small particles (small mass and high density) are associated with the constant backscatter ratio predicted by the Rayleigh limit, that is, the fourth power of the ratio of the two DPR frequencies, or about 50 (i.e., 17 dB).

### c. Aggregate properties

CDA calculations were performed for each of the aggregate models ND, DA1, DA2, and DA3 for interdipole spacing *d* varying from 10 to 100 *μ*m. The effective ice fraction, as defined by (18), is constant for each model, but the total mass of the aggregate varies between the limits given in Table 2. Because radiative properties, especially backscatter, are strong functions of orientation for these highly asymmetric particles, results were averaged over 125 uniformly distributed orientations. Figure 9 presents results for each model plotted as dots overlaid on the envelope of curves previously obtained for soft spheres.

#### 1) Radar backscatter

For the smallest particles, the DPR backscatter (Fig. 9, top row) is nearly independent of the density of a soft sphere. The results for all aggregates are consistent with a more or less fixed density at these small sizes, though the effective soft sphere density is somewhat higher for ND and DA1 than the aggregate densities computed from (18).

For larger sizes, there is no consistent mapping between aggregate properties and those of radiatively equivalent soft sphere, neither for single aggregate of varying mass nor for different aggregates of the same mass.

#### 2) Extinction

Mass-normalized extinction cross sections for frequency pairs 18.7–36.5 GHz and 36.5–89.0 GHz are presented in the middle and bottom rows, respectively, of Fig. 9. Generally speaking, when the mass of the aggregate increases, the mass-normalized extinction cross sections increase as well, but at different rates. For each frequency, the CDA-calculated cross sections are generally greater than those of the soft sphere with equal mass and density except in the case of extremely small particles.

Overall, it can be seen that the extinction cross section pairs for each model deviate from that of a soft sphere with the same mass and density. In particular, no soft sphere of the same mass as that of the aggregate yields the correct radiative cross sections for any density. Also, at higher frequencies, aggregates more nearly resemble an ensemble of smaller-mass, higher-density soft spheres. For example, for the frequency pair 36.5–89.0 GHz, the DA2 model with a mass of 15.6 mg is equivalent to a 4.0-mg soft sphere with a density of 0.2, whereas for 18.7–36.5 GHz it corresponds to an 8.0-mg soft sphere with density of 0.1.

Finally, it is significant that the sphere-equivalent density of a given aggregate is not even approximately constant but rather varies both with frequency and with geometric scaling (varying mass) of the particle. Thus, a key failing of the soft sphere approximation is that there is no purely *geometric* basis for specifying the equivalent sphere’s effective density.

## 8. General extinction and scattering behavior

The results from the previous section suggest that the soft sphere approximation is essentially useless for parameterizing the multispectral extinction and backscattering properties of complex aggregates of snow crystals and therefore should probably be abandoned. This does not mean, however, that no parameterization is possible.

Figure 10 depicts the mass extinction coefficient *κ*_{ext} and single-scatter albedo *ω*_{0} for all four aggregate models, plotted as a function of total aggregate mass *M*, which in turn is proportional to *d*^{3} for any specific aggregate. When presented in this way, it is apparent that differences between the aggregate models are less important than the generic dependence on total particle mass *M*.

*A*and

*b*and the lower limit

*M*

_{min}of the fit are given in Table 3.

The *b* coefficients are very similar for all three frequencies, implying that the ratios of attenuation for two frequencies are only very weakly dependent on particle mass. In particular, we find that, for fixed *M* = 1 mg, (19) gives *κ*_{ext} as proportional (within 3%–5%) to frequency raised to the power 2.56; for *M* = 10 mg, the power is 2.47. In the absence of prior information about mean particle mass and in view of the variations among aggregate models, a power of 2.5 is suggested as an approximate model for the frequency dependence of microwave extinction in dry snow. This is remarkably similar to the power of 2.53 implied by the analytic fits of Casella et al. (2008) to CDA calculations for soft spheres.

Multisensor and multispectral retrieval methods are especially sensitive to the ratios of backscatter and extinction at multiple frequencies. These ratios are depicted in Fig. 11. Figure 11a shows the DPR backscatter ratio decreasing in a predictable manner with *M* for *M* < 3 mg, beginning with the Rayleigh limit of 17 dB. But the ratio becomes extremely sensitive to the choice of aggregate model for *M* > 3 mg, ranging from 0 to 10 dB.

Extinction ratios at GMI frequencies (Figs. 11b,c) are more variable for any given mass but nevertheless remain confined within fairly well-defined ranges for most larger values of *M* and show a weak tendency to decrease with *M* for masses greater than approximately 0.3 mg. In particular, if these calculations are representative of real snow, then any reasonable distribution of aggregate masses should yield extinction ratios of between 4 and 6.5 for 18.7 and 36.5 GHz and between 7 and 10 for 36.5 and 89.0 GHz.

## 9. Rayleigh–Gans approximation

For a first-order treatment of snowflake scattering that does not depend on an equivalent sphere assumption, one sometimes resorts to the Rayleigh–Gans approximation (van de Hulst 1957). The RGA is based on the assumption that the combined scattering properties of the particle arise from a superposition of scattered waves from each subelement responding only to the incident wave. In other words, (a) there is no electromagnetic coupling between subelements, and (b) there is effectively no significant net phase shift (or retardation) of the incident wave internal to the particle. Formally, the conditions |*m* − 1| ≪ 1 and |*m* − 1|*kr* ≪ 1 must be satisfied (Bohren and Huffman 1983). These conditions require that the target be small and not a strong dielectric.

Radar backscatter by snowflakes would appear to be a promising application of RGA on account of (i) the relatively low intrinsic refractive index of ice in the microwave band, (ii) the sparse structure of snowflakes, leading to further reduced internal coupling (Berry and Percival 1986), and (iii) the relatively large ratio of wavelength to snowflake diameter. Recent studies obtain the radar backscattering cross section by applying the RGA approximation for particles with low density and open structure (Matrosov 1992; Westbrook et al. 2006). In particular, Westbrook et al. (2006) found that the form factor for a sphere with the same radius of gyration *r* was consistent with RGA calculations for a complex aggregate as long as 2*kr* < 2. However, the RGA results themselves were not validated against more accurate methods.

Computationally, the RGA is exactly equivalent to running our CDA simulations for a single iteration, so that each dipole responds only to the incident field and not to the subsequently scattered waves from other dipoles. We may evaluate the validity of the RGA for two of our model particles, NA and DA1, by comparing results from the first iteration with those obtained after convergence.

Figure 12 shows the mass-normalized backscattering cross section derived from the CDA and RGA method for the two cases, plotted as the backscatter at 35.6 GHz versus that at 13.4 GHz. Very similar results are found for the other two aggregate models, DA2 and DA3 (not shown). In general, the RGA gives nearly correct backscatter *ratios* for most particle sizes, although with a tendency for the RGA to underestimate the ratio slightly (typically by no more than 1–2 dB) for larger particles. However, the absolute backscatter cross section is substantially underestimated for all particle sizes, typically by at least 2 dB for small sizes and as much as 7 dB for larger sizes.

## 10. Conclusions

Coupled-dipole calculations of microwave extinction and backscatter have been demonstrated for structures that we believe more nearly resemble real snow aggregates than have previously been investigated. For this purpose, we wrote a CDA code that is designed for sparse dipole structures defined in unbounded space, so that processor time, not computer memory, is the factor that practically limits the maximum dimensions of the particles that can be considered. For example, our DA3 aggregate would have required over 10 GB of memory using the well-established DDSCAT program; our code required only 20 MB for the same structure. Notwithstanding the sharp reduction in memory required, the computation time is still well within an order of magnitude of that required by DDSCAT for sparse particles of similar total mass and density.

Our computational method was first validated against Mie theory using dipole structures representing solid ice spheres. A simple modification of the traditional CDA expression for particle extinction cross section was shown to yield correct results not only for large ensembles of dipoles but also in the single-dipole limit, which has not always been the case for some formulations that neglect the so-called radiation reaction effect.

The CDA method was further applied to soft ice spheres consisting of variable fractions of ice and air, and the results were compared with the predictions of Mie theory in combination with the Bruggeman and Maxwell Garnett dielectric mixing formulas. We found that none of these widely used formulas adequately reproduced the CDA results over the full range of fractions from 1% to 100%; rather, the best fit was consistently achieved using the generalized dielectric mixing rule of Sihvola (1989) with *ν* = 0.85.

Using the above dielectric mixing rule and Mie theory as a reference, we sought to determine whether soft spheres existed that were radiatively equivalent to our snow aggregate models, taking into account the spectral dependence of backscatter and/or extinction per unit mass at key DPR and GMI frequencies. We found that even when spheres of nonequal mass are considered, there is no single combination of density and particle mass that simultaneously captures the multifrequency properties of these aggregates on a per-mass basis. We therefore find no persuasive basis for retaining any kind of soft sphere, equal mass or otherwise, as a model for the microwave properties of snowflakes.

Despite the failure of the soft sphere approximation, we found that all four aggregate models exhibited similar behavior in certain respects. In particular, the mass extinction coefficient at each of the key GMI frequencies of 18.7, 36.5, and 89.0 is a simple power-law function of total aggregate mass *M*. Moreover, the exponent of the fit is similar for all three frequencies, meaning that dual-frequency extinction ratios are only very weakly dependent on particle masses. Even in the absence of knowledge of the distribution of *M*, our results predict microwave extinction by snow aggregates to be roughly proportional to the frequency raised to 2.5.

It would be both straightforward and valuable to compare the extinction ratios predicted herein with direct field measurements of these ratios in dry snowfall using a sensitive and well-calibrated multifrequency microwave attenuation link. If similar extinction ratios are observed in real snow, it would lend credence to the present (and similar) models as a basis for parameterizing microwave scattering and extinction. On the other hand, it is known from field observations of snow particles (Locatelli and Hobbs 1974) that real aggregates, unlike our model aggregates, have densities that tend to depend on aggregate mass, and this behavior should be accounted for in any definitive parameterization of extinction by snow.

The computed backscatter ratio at DPR frequencies is a well-defined function of particle mass for *M* < 3 mg. Above this size, however, the ratio is extremely sensitive to aggregate structure. Because large particles can contribute disproportionately to radar backscatter from a given atmospheric volume, we must consider the possibility that a few large aggregates in a volume might compromise the utility of dual-frequency backscatter ratio measurements of the type envisaged for GPM. This potential concern deserves further investigation.

As a straightforward extension of the original objectives of this study, we examined whether the Rayleigh–Gans approximation yielded useful results for our model aggregates. While the dual-frequency backscatter ratio was reasonably well reproduced (to within 2 dB) by the RGA for the frequencies of the DPR, the absolute magnitude of the backscatter cross section was too low by as much as 7 dB. Recent results for radar scattering by aggregates obtained by Westbrook et al. (2006) using Rayleigh–Gans theory should be viewed with this possible bias in mind.

Notwithstanding the availability of increasingly sophisticated physical models of snow crystals and snow aggregates, it will be some time before we can have confidence in our ability to capture the *statistical* properties of real snowfall via detailed electrodynamic simulations of single particles. Thus, the single most important near-term contribution to be made in this area may be the systematic *measurement* of multifrequency microwave extinction and backscatter in falling snow, using suitable combinations of low-power radars and well-calibrated microwave attenuation links covering horizontal distances of one to several kilometers.

## Acknowledgments

Dipole structures were graphically rendered using the Raster3D package by Merritt and Bacon (1997). Three anonymous reviewers made suggestions that resulted in significant improvements of this paper. This work was supported by NASA Grant NNX08AD36G.

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Assumed refractive index for ice at −5°C for selected AMSR and DPR frequencies.

Physical properties of model aggregates.

^{1}

Warren and Brandt (2008) have published an updated compilation of *m* for ice. At the frequencies considered, the imaginary part of *m* is roughly half as large as that originally published, and the corresponding absorption cross sections are proportionally smaller as well. The extinction cross sections are dominated by scattering except in the small particle limit; therefore, the change is unimportant for most sizes of interest here.