## 1. Introduction

Traditional radar methods of measuring precipitation are based on relations between water equivalent reflectivity factor (hereafter “reflectivity”), *Z _{e}*, and precipitation rate,

*R*. Even for a simpler case of rainfall, there are significant variations in

*Z*–

_{e}*R*relations due to variability of the drop size distributions. Estimating snowfall parameters from radar measurements is more challenging. Variability in snowflake shapes, densities, and fall velocities results in greater uncertainties in

*Z*–

_{e}*R*relations for snowfall. Notwithstanding these uncertainties, radar gives an opportunity to obtain near-real-time estimates of snowfall rate and accumulation, providing information that is not readily available with other remote sensors.

As for rainfall, snowfall radar estimates are typically obtained with longer wavelength (centimeter) radars such as those that operate at S band (10–11-cm wavelength, *λ*), C band (*λ* ∼ 5–6 cm), or X band (*λ* ∼ 3 cm). Snowfall rate as inferred from radar measurements is usually expressed in melted liquid equivalent, *S*, rather than in solid snow rate. Two different approaches are used to derive *Z _{e}*–

*S*relations. In one approach both

*Z*and

_{e}*S*are calculated from theoretical or experimental snowflake size distributions (SSD) and they are related in a statistical manner (e.g., Sekhon and Srivastava 1970; Matrosov 1992). In the second approach, radar measurements of reflectivity are related to the snowfall rate as observed by nearby snow gauges (e.g., Super and Holroyd 1998; Hunter and Holroyd 2002). Rasmussen et al. (2003) provides a review of some

*Z*–

_{e}*S*relations suggested for use with longer-wavelength radars. For such radars, snowflakes are typically much smaller than radar wavelength and scattering does not deviate significantly from the Rayleigh law.

At millimeter radar wavelengths such as those at K_{a} band (*λ* ∼ 8 mm) and, especially, W band (*λ* ∼ 3 mm), scattering by typical snowflakes is generally non-Rayleigh. Radars operating at these wavelengths have been primarily intended for applications in cloud remote sensing and they have typically not been used for snowfall measurements. It has been shown, however, that the use of such radars in pair with centimeter-wavelength radars can potentially improve the accuracy of snowfall parameter retrievals (Matrosov 1998; Liao et al. 2005; Wang et al. 2005) because the dual wavelength reflectivity ratio provides independent information on characteristic size of snowflakes. Knowing size information, one can better infer snowfall rate from single reflectivity measurements.

Currently, a number of vertically pointing millimeter-wavelength radars operate on a continuing basis in many locations including those that regularly or sometimes get snowfalls. Examples of such radars are vertically pointing K_{a}-band (34.6 GHz) millimeter-wavelength cloud radars (MMCR) operated by the U.S. Department of Energy’s Atmospheric Radiation Measurement Program (DOE ARM) in Oklahoma and the North Slope of Alaska (NSA). Another MMCR-type radar has been recently deployed in the Canadian Arctic. The W-band (94 GHz) nadir-looking CloudSat radar will provide global information on reflectivity profiles of hydrometeors (Stephens et al. 2002). Although the main objective of all these radars is cloud remote sensing, their measurements of snowfall can also be interpreted quantitatively, thus providing valuable information on snowfall accumulation and the vertical structure of snowfall rate, which can vary significantly (Mitchell 1988).

Since for most of the radars mentioned above there are no collocated centimeter-wavelength radars for application of dual-wavelength ratio techniques, *Z _{e}*–

*S*relations specifically tailored for millimeter wavelengths are needed for quantitative interpretation of snowfall measurements. This article presents results of modeling

*Z*–

_{e}*S*relations for the common millimeter-wavelength frequencies of 34.6 and 94 GHz based on available information on snowflake-size distributions and densities. The first of the aforementioned approaches for deriving

*Z*–

_{e}*S*relations is employed here because there are no reliable datasets available that combine collocated millimeter-wavelength radar and snow gauge measurements. This modeling is concerned with so called “dry” snow, the term usually used for describing snow with negligible amounts of liquid water/riming. “Dry” snowfalls usually contribute most significantly to snow accumulation at temperatures less than about −8°C.

## 2. Snowflake model

### a. Snowflake shape

*λ*for a snow particle with a major dimension

*D*can be formulated as follows:

*m*is the complex refractive index of snow. The great majority of derivations of

_{s}*Z*–

_{e}*S*relations in snowfall assumed that for snowflakes that satisfy these conditions the radar backscatter would be the same as that of a sphere of solid ice having the same mass. While this assumption precludes modeling polarimetric properties of backscatter cross sections, it provides a reasonably good approximation of

*Z*for low-density dry snow at centimeter wavelengths. The spherical model, however, becomes progressively less suitable as the size parameter

_{e}*x*increases beyond the bounds of the Rayleigh conditions. Larger nonspherical ice particles exhibit backscatter dependence on particle orientation (e.g., Matrosov 1993; Aydin and Singh 2004) and generally greater reflectivities (especially outside the Rayleigh scattering regime) compared to those of spherical particles of the same mass (Matrosov et al. 2005a).

The simplest geometrical shape that allows describing nonspherical hydrometeors is that of spheroid. In their classical observational in situ study, Magono and Nakamura (1965) used tracing outlines of snowflakes that revealed a generally spheroidal shape. The oblate spheroidal model is used widely in the radar community to model raindrops and it was also shown to satisfactorily describe and explain depolarization observations of snowflakes at K_{a} band (Matrosov et al. 1996, 2001). Note also that, at radar frequencies, the small details in a particle structure usually do not significantly affect the backscatter properties that depend largely on the overall shape (Dungey and Bohren 1993), which, in the case of spheroid, is determined by the spheroid aspect ratio, *r*. This is especially true for smaller snowflake size parameters *x* = *πDλ*^{−1} (Schneider and Stephens 1995). The applicability of the spheroid model for larger snowflakes with small values of *r* (e.g., dendrites that are larger than several millimeters) at W band is not so obvious. This model, however, satisfactorily explains experimental 94-GHz aggregate snow reflectivities (Matrosov et al. 2005a).

Based on the arguments given above, and also due to the fact that very large dendrites are not very common, the oblate spheroidal model was adopted here for snowflake modeling. Another reason to use this model is the lack of detailed shape information on aggregates, which makes the input information for the computational models that are able to account for fine shape details (e.g., the discrete dipole approximation; Lemke and Quante 1999) not very well defined. In the spheroid model, low aspect ratio values of about 0.1–0.3 are suitable for modeling pristine dendrite snow crystals, while values of *r* ≈ 0.5–0.7 are usually observed for larger aggregate crystals (Magono and Nakamura 1965; Korolev and Isaac 2003). The value around *r* ≈ 0.6 also best explains experimental dual-wavelength ratios from the airborne radars operating at 9.6 and 94 GHz (Matrosov et al. 2005a).

### b. Snowflake orientation

Magono and Nakamura (1965) noted that, due to aerodynamic forcing, snowflakes typically fall with their major dimensions oriented horizontally. Polarimetric radar depolarization measurements at different polarimetric bases and at different elevation angles also reveal that, in the absence of strong horizontal winds (and wind shear), single shape snowflakes are oriented preferably horizontally with a standard deviation (SD) of about 9° (Matrosov et al. 2005b). Thus it was assumed during modeling that symmetry axes of oblate spheroids that approximate snowflakes are oriented with the 0° mean and SD = 9° with respect to vertical. While there is no polarization dependence of radar measurables for vertically pointing radar for such snowflake orientations, snowflake shape and, to a lesser extent, the value of SD influence the magnitude of *Z _{e}* at millimeter-wavelength frequencies.

### c. Snowflake mass

*ρ*, of falling dry aggregate snowflakes that are larger than 0.2 cm diminishes with their size from approximately 0.03 g cm

^{−3}at

*D*= 0.2 cm to about 0.01 g cm

^{−3}at

*D*= 1.5 cm. While

*D*≈ 0.2 cm is the smallest size presented by Magono and Nakamura (1965), there are a number of microphysical studies in which ice aggregates with sizes less than 0.2 cm were studied (e.g., Brown and Francis 1995; Mitchell 1996; Heymsfield et al. 2004). Most of these studies indicate that the mass,

*m*, of an aggregate particle typically increases as

*D*where

^{b},*b*≈ 2. The following formulas provide a continuous mass–size relation for snowflakes for a large range of possible sizes:

*D*= 0.01 cm was chosen in (3a) because for smaller sizes this formula can provide results that correspond to unrealistic densities exceeding those of solid ice. While (3b) closely approximates data from Magono and Nakamura (1965), (3a) provides particle masses that are within the range of empirical relations of the aforementioned studies for smaller aggregate particles. Note that (3a) is very close to the relation for aggregates S3 (according to the Magono and Lee 1966 classification) suggested by Mitchell (1996):

Equation (3c) reflects the fact that the density of very large snowflakes does not exhibit any significant dependence on size.

*Z*–

_{e}*S*relations. According to Heymsfield (1972) density–size relations for dendrites (P1e according to the Magono and Lee classification) and stellars with broad arms (P1d) are practically identical. For such crystals Mitchell (1996) suggests the following relation:

### d. Snowflake fall velocity

*υ*, are determined by their mass, their area projected to the flow, their size, and such air properties as density

_{t}*ρ*, and dynamic viscosity

*η*. In a recent study, Mitchell and Heymsfield (2005) critically addressed existing fall velocity–size relations for aggregate ice crystals. They suggested a refined relation that accounts for the correspondence between the Reynolds and the Best or Davies numbers that is appropriate for aggregates. The data presented by Mitchell and Heymsfield (2005, their Fig. 5) indicate an approximately linear relation between

*υ*and the logarithm of aggregate size. This linear relation can be expressed as

_{t}*D*is in microns. For 0.16 cm <

*D*< 1 cm, expression (6) also approximates well the measurement-derived fall velocity relation for large unrimed aggregates of dendritic crystals suggested by Locatelli and Hobbs (1974).

Many theoretical (e.g., Mitchell and Heymsfield 2005) and experimental (e.g., Magono and Nakamura 1965) studies indicate that terminal fall velocities of larger snowflakes practically do not depend upon the size of the flake. Taking this into consideration, it was assumed that (6) describes aggregate snowflake fall velocities in the range 0.01 cm < *D* < 1 cm and the constant value of *υ _{t}* was then assumed for

*D*≥ 1 cm.

*h*can be calculated from those at the surface using the expression (Pruppacher and Klett 1997)

*η*and

*ρ*are the air dynamic viscosity and density [

*ρ*(0) = 1.23 kg m

^{−3},

*η*(0) = 1.65 × 10

^{−5}Pa s], respectively, and

*α*= 0.57 for snowflakes. It should be noted that the fall velocity approximations considered above are applicable to snowflakes with a negligible degree of riming. Rimed and wet snowflakes and also graupel generally exhibit higher fall velocities. Snowfalls consisting of such particles are usually observed at temperatures near freezing and when significant amounts of supercooled water drops are present in the atmosphere. This study focuses on “dry” snow.

## 3. Backscatter cross sections of individual snowflakes

*P*) and air (

_{i}*P*) are computed. The complex refractive index of a bulk snowflake mixture,

_{a}*m*

_{s}, is then calculated using the approach that is often used in modeling dielectric properties of dry snow at radar frequencies (e.g., Battan 1973; Oguchi 1983):

*m*is the complex refractive index of solid ice adopted from Warren (1984).

_{i}Figure 1 shows backscatter cross sections, *σ _{b}*, of horizontally oriented aggregate snowflakes with aspect ratios that are in the range 0.6 ≤

*r*≤ 1 when viewed at vertical incidence. The mass of snowflakes was modeled according to (3) and was preserved as

*r*was changing. This means that effective bulk densities are different for snowflakes with the same size but different aspect ratios. Though Mie-type backscatter oscillations are present for all considered aspect ratios, as can be seen from Fig. 1, backscatter cross sections of nonspherical snowflakes already begin to deviate noticeably from the spherical model already at major dimensions of about

*D*= 0.15 cm for

*ν*= 34.6 GHz, and

*D*= 0.05 cm for

*ν*= 94 GHz. This indicates that, for at least vertical-viewing geometry and millimeter radar wavelengths, the assumption that the backscatter cross section of an arbitrary snowflake would be the same as that of a sphere of ice having the same mass is not generally valid. This assumption, which is often used when calculating radar properties of snowflakes is, however, valid for lower radar frequencies in the Rayleigh scattering regime.

While, as mentioned above, *r* ≈ 0.6 is expected to adequately describe natural snowflake aggregates, it is also instructive to examine backscatter cross sections for particles with smaller aspect ratios. Figure 2 presents results for 0.2 ≤ *r* ≤ 0.5. Note that the T-matrix calculation routine often becomes unstable for particles with aspect ratios smaller than about 0.2, so no results are presented for *r* < 0.2. It can be seen from Fig. 2 that the dependence of backscatter cross sections *σ _{b}* on

*r*becomes progressively less pronounced as

*r*decreases. This is true especially for smaller particles.

Figure 3 shows backscatter cross sections for single pristine dendritic crystals (P1d) as a function of their size. Though the calculations presented in this figure were made for *r* = 0.2 they are expected also to be representative for smaller values of *r* because of the lesser sensitivity of *σ _{b}* to

*r*at smaller values of aspect ratios and also because pristine crystals usually are smaller than aggregates and they are not expected to exceed a few millimeters in size. Note also that because the effective bulk density of snowflakes changes with their size, the results presented in Figs. 1 –3 for one frequency cannot be obtained exactly from the results for the other frequency using simple frequency scaling.

## 4. Reflectivity of ensembles of snowflakes

### a. Snowflake size distributions

One of the most complete experimental studies of snowflake size spectra in terms of snowflake real sizes was presented by Braham (1990) who reported data that contained 49 experimental SSDs. Figure 4 shows these SSDs approximated by the exponential function with parameters *N*_{o} and Λ from Braham (1990). The majority of snowflakes in Braham’s study appeared to be aggregate snowflakes with some amount of single spatial dendrites. Heavier snowfalls are characterized by smaller values of the parameter Λ. Later experimental studies (e.g., Harimaya et al. 2000) showed experimental SSD features similar to those reported by Braham (1990).

### b. Z_{e}–S relations at millimeter wavelengths

*m*is the complex refractive index of water at a given wavelength,

*λ*, and the angular brackets in (11) indicate additional integrating over the snowflake orientations, which were assumed to be Gaussian with a 0° mean and SD = 9° as discussed in section 2b. Snowflake orientations in the azimuthal direction were assumed to be random, so for vertical incidence,

*Z*does not depend on the polarization state of radar signals.

_{e}*S*, can be expressed as

*ρ*is the density of water, and the mass–size and fall velocity–size relations were discussed in sections 2c and 2d. The values of

_{w}*D*

_{min}and

*D*

_{max}in (10) and (11) were assumed to be 0.01 and 1 cm. This choice of

*D*

_{min}and

*D*

_{max}proved to be not very sensitive for the considered experimental SSDs. Modest variations of

*D*

_{min}and

*D*

_{max}(about 2%–20%) in the integral (12) did not result in significant changes in either

*Z*or

_{e}*S*.

Figure 5 shows the scatterplots of radar reflectivity versus liquid equivalent snowfall rate for frequencies 34.6 and 94 GHz calculated for the experimental SSDs with parameters from Fig. 4. Aggregate snowflake aspect ratio was assumed to be 0.6 and the mass and terminal fall velocities were calculated using (3) and (6). The best power-law fits drawn with *Z _{e}* as an independent variable for the aggregate snowflake assumption are

*Z*= 56

_{e}*S*

^{1.20}(34.6 GHz) and

*Z*= 10

_{e}*S*

^{0.8}(94 GHz), where

*Z*in mm

_{e}^{6}m

^{−3}and

*S*in mm h

^{−1}. Note that first the

*S*–

*Z*relations were developed considering

_{e}*Z*as an independent variable and then these relations were converted to the conventional

_{e}*Z*–

_{e}*S*form. The data scatter around the best-fit lines is rather modest. Due to stronger non-Rayleigh scattering effects, reflectivities at 94 GHz are significantly smaller than those at 34.6 GHz for the same values of snowfall rate

*S*. The millimeter wavelength

*Z*–

_{e}*S*relations are quite different from those obtained by various researches for longer wavelength radars (Rasmussen et al. 2003).

While it is expected that aggregate snowflakes are the dominant hydrometeor type in appreciable snowfalls, for comparison reasons, calculations were also performed using the dendritic crystal model and the same experimental SSDs. The mass–size and fall velocity–size relations in this case are given by (5) and (7). The results for which dendrite reflectivities were greater than 0 dB*Z* are also shown in Fig. 5. For 34.6 GHz the aggregate best fit also rather well describes the dendrite data. It is not quite so for the 94-GHz data, although deviations of the dendrite data from the best power-law fit for aggregates are still not very significant. Reflectivities that are less than about 0 dBZ are more characteristic of nonprecipitating clouds and were not considered here.

### c. Variability of Z_{e}–S relations due to assumptions

The main assumptions affecting radar reflectivity values, *Z _{e}*, include those about the particle mass and shape. While in the Rayleigh scattering regime, backscatter cross sections of individual particles are proportional to the square of particle masses, the resonance effects diminish the backscatter–mass dependence. The shape effects, which are not very pronounced for small particles, however, become progressively more important (especially for vertical incidence at horizontally oriented scatterers) as particles grow larger and transition to the sizes that are outside the Rayleigh scattering regime. The assumption about particle mass also affects the liquid equivalent snowfall rates,

*S*. The values of

*S*are also affected by the assumption about the terminal fall velocity–size relation.

The aggregate particle mass–size relation given by (3) and the terminal fall velocity–size relation given by (6) and an assumption about the *r* = 0.6 aspect ratio were used to produce the data presented in Fig. 5. Table 1 shows how the best fit power-law *Z _{e}*–

*S*relations would vary if one of the main assumptions is changed. To assess this variability the particle mass and fall velocity were changed by 20% compared to values from Eqs. (3) and (6), respectively, and the particle aspect ratio was changed from 0.6 to 0.8. A more random orientation of snowflakes will be manifested similarly as an increase in

*r*. These changes in the main assumptions may represent the reasonable uncertainties of corresponding assumptions. Indeed, the fall velocity approach of Mitchell and Heymsfield (2005), for example, provides fall velocities of 1 cm aggregates (which is a typical aggregated snowflake size) of about 1 m s

^{−1}. The Khvorostyanov and Curry (2002) approach provides 1.4 m s

^{−1}for the same particle size. This is about ±20% from their mean value of 1.2 m s

^{−1}. Two different mass–size relations for aggregates from Mitchell (1996) provide similar percentage differences if applied to larger aggregate sizes.

It can be seen from Table 1 that there is an appreciable sensitivity of the *Z _{e}*–

*S*relations to main assumptions. Assuming the independence of the assumption uncertainties, one can expect variability of at least a factor of 2 in the coefficients of the

*Z*–

_{e}*S*relations. Variability in the exponents of these relations is more modest, though it also contributes to the overall uncertainty in the correspondence between

*Z*and

_{e}*S.*It should be noted also that, compared to the low observational angles, the vertical radar beam observations (which is the most common configuration for K

_{a}- and W-band radars) and the preferable orientation of snowflakes with their major dimension in the horizontal plane result in larger variability of backscatter due to changes in the particle aspect ratio.

## 5. Attenuation of radar signals in snowfall

_{a}- and W-band signals in snowfall. The specific attenuation coefficient,

*a*, can be obtained by integrating imaginary parts of the forward scattering amplitudes

*f:*

*f*relates to the dimensionless amplitude matrix element

*S*

_{o}, which customarily is used in optics as

Figure 6 shows scatterplots of the attenuation coefficients at −5°C versus liquid-equivalent snowfall rates at K_{a}- and W-band radar frequencies calculated for the assumed snowflake aspect ratio *r* = 0.6 and the same experimental SSDs that were used in reflectivity modeling. Complex refractive indices of solid ice corresponding to a temperature of −5°C are about 1.78 −*i*0.0024 (34.6 GHz) and 1.78 −*i*0.0043 (94 GHz).

The best-fit power law approximations for the data in Fig. 6 are *a* = 0.011*S*^{1.1} and *a* = 0.12*S*^{1.1} for 34.6 and 94 GHz, respectively. It can be seen that the dry snow attenuation at K_{a} band is fairly small and can be neglected for most practical cases. The attenuation at W band, however, is about one order of magnitude higher than that at K_{a} band and, probably, needs to be accounted for when measuring snowfall with higher precipitation rates in thick snow layers. The total attenuation in such cases can be appreciable and reach or even exceed 1 dB, which is on the order of a typical absolute calibration uncertainty of radars. For wet and/or rimed snow, attenuation can be significantly larger.

It should be noted also that the attenuation in the atmospheric gases (i.e., water vapor and oxygen) generally should be accounted for when interpreting measurements at W band at longer ranges (Stephens et al. 2002). Gases attenuation at K_{a} band at vertical incidence that typically amounts to fractions of 1 dB often can be neglected especially in dry cold atmospheres.

## 6. Summary and conclusions

Millimeter-wavelength radars operating at K_{a} and W bands are being increasingly used in the atmospheric remote sensing. These radars are typically employed either in a vertical-pointing mode (for ground based radars) or in a nadir-pointing mode (for airborne and spaceborne radars). A number of ground-based millimeter wavelength cloud profiling radars have been deployed in the colder climates and spaceborne radars are usually designed to operate at polar orbits, thus providing frequent coverage of high latitudes where snowfall is common. Although these radars’ primary use is remote sensing of nonprecipitating clouds, they can also potentially be used for snowfall measurements. Adding quantitative snowfall information derived from radar measurements will be useful in many climate and cloud physics applications.

In this study, modeling of reflectivity–liquid-equivalent snowfall rate was performed for the common frequencies employed at K_{a} band (∼35 GHz) and W band (∼94 GHz). “Dry” snowfall, which is characterized by negligible amounts of liquid water in snow and snowflake riming, was studied. Nonspherical aggregate and single crystal dendritic snowflake models were used for calculations. These models included realistic mass–size and terminal fall velocity–size relations and assumed some flutter around the preferred orientation of snowflakes with their maximum dimensions in the horizontal plane. Experimental size distributions describing real (not melted) snowflake sizes were employed in this study.

Most common aggregate snowflake sizes are already outside the Rayleigh scattering regime at K_{a}- and especially at W-band frequencies. For vertical/nadir-pointing geometry, backscatter properties of snowflakes noticeably depend on snowflake overall shape (i.e., aspect ratio). The typical radar reflectivity–snowfall rate relations for aggregate snowflakes calculated for a set of basic assumptions (see Table 1) are *Z _{e}* = 56

^{1.20}(34.6 GHz) and

*Z*= 10

_{e}*S*

^{0.8}(94 GHz). These relations are generally valid also for snowfall consisting of single dendrite crystals (P1d), though snowfall rates and thus accumulations of such snowfalls are usually significantly less compared to the snowfalls consisting of aggregates. Though these relations were developed for “dry” snowfall, they are expected to be applicable for snowfall with some light degree of riming. Dry snow typically forms at colder temperatures (usually less than about −12°C) and is characterized by mean vertical Doppler velocities that are less than about 1.2–1.4 m s

^{−1}. A significant increase in snowflake riming and/or wetness is likely to cause changes in

*Z*–

_{e}*S*relations.

The strong non-Rayleigh behavior of backscatter at millimeter wavelengths in K_{a} and W bands results in reflectivity being proportional to the lower moment of the snowflake size distribution. As a result the exponents in the *Z _{e}*–

*S*relations are closer to unity compared to the longer wavelength radars. The proportionality of

*Z*and

_{e}*S*to similar moments of the size distribution can, in part, explain small data scatter in Fig. 5. For precipitation radar frequencies (3–10 GHz), both coefficients and exponents in

*Z*–

_{e}*S*relations are larger. Typically for such frequencies, coefficients vary from 50 to 400 while the exponents vary between 1.5 and 2 (e.g., Rasmussen et al. 2003).

Uncertainties in the coefficients of *Z _{e}*–

*S*relations caused by variability in the snowflake shape, and in the parameters of mass–size and terminal fall velocity–size relations are rather substantial. These coefficient uncertainties can easily be as high as a factor of ∼2; thus, the errors in the snowfall rate and accumulation estimations for radar reflectivity measurements could be appreciable. These errors can potentially be reduced if the coefficient tuning is performed based on collocated radar and snow gauge measurements. Such a tuning approach is often used in rainfall measurements when parameters of standard

*Z*–

_{e}*R*(

*R*here is rainfall rate) relations are adjusted based on the rain gauge measurements that are deployed in the radar coverage area. There are, however, no high quality collocated datasets for millimeter-wavelength radars and snow gauge data available at the moment, so this direction for refinement of

*Z*–

_{e}*S*relations should be left for further studies.

## Acknowledgments

This research was supported by the Office of Science (Biological and Environmental Research), U.S. Department of Energy, Grant DE-FG02-05ER63954, and the CloudSat project.

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As in Fig. 1 but for aspect ratios 0.2, 0.3, and 0.5.

Citation: Journal of the Atmospheric Sciences 64, 5; 10.1175/JAS3904.1

As in Fig. 1 but for aspect ratios 0.2, 0.3, and 0.5.

Citation: Journal of the Atmospheric Sciences 64, 5; 10.1175/JAS3904.1

As in Fig. 1 but for aspect ratios 0.2, 0.3, and 0.5.

Citation: Journal of the Atmospheric Sciences 64, 5; 10.1175/JAS3904.1

Backscatter cross sections of horizontally oriented fordendritic crystals (P1d) as a function of crystal major dimension.

Citation: Journal of the Atmospheric Sciences 64, 5; 10.1175/JAS3904.1

Backscatter cross sections of horizontally oriented fordendritic crystals (P1d) as a function of crystal major dimension.

Citation: Journal of the Atmospheric Sciences 64, 5; 10.1175/JAS3904.1

Backscatter cross sections of horizontally oriented fordendritic crystals (P1d) as a function of crystal major dimension.

Citation: Journal of the Atmospheric Sciences 64, 5; 10.1175/JAS3904.1

Scatterplot of the intercept and slope parameters in experimental snowflake size distributions.

Citation: Journal of the Atmospheric Sciences 64, 5; 10.1175/JAS3904.1

Scatterplot of the intercept and slope parameters in experimental snowflake size distributions.

Citation: Journal of the Atmospheric Sciences 64, 5; 10.1175/JAS3904.1

Scatterplot of the intercept and slope parameters in experimental snowflake size distributions.

Citation: Journal of the Atmospheric Sciences 64, 5; 10.1175/JAS3904.1

Scatterplots of radar reflectivity at vertical incidence and snowfall rate at frequencies (a) 34.6 and (b) 94 GHz for experimental snowflake size distributions.

Citation: Journal of the Atmospheric Sciences 64, 5; 10.1175/JAS3904.1

Scatterplots of radar reflectivity at vertical incidence and snowfall rate at frequencies (a) 34.6 and (b) 94 GHz for experimental snowflake size distributions.

Citation: Journal of the Atmospheric Sciences 64, 5; 10.1175/JAS3904.1

Scatterplots of radar reflectivity at vertical incidence and snowfall rate at frequencies (a) 34.6 and (b) 94 GHz for experimental snowflake size distributions.

Citation: Journal of the Atmospheric Sciences 64, 5; 10.1175/JAS3904.1

Scatterplots of the attenuation coefficient at vertical incidence and snowfall rate for experimental snowflake size distributions.

Citation: Journal of the Atmospheric Sciences 64, 5; 10.1175/JAS3904.1

Scatterplots of the attenuation coefficient at vertical incidence and snowfall rate for experimental snowflake size distributions.

Citation: Journal of the Atmospheric Sciences 64, 5; 10.1175/JAS3904.1

Scatterplots of the attenuation coefficient at vertical incidence and snowfall rate for experimental snowflake size distributions.

Citation: Journal of the Atmospheric Sciences 64, 5; 10.1175/JAS3904.1

Variability of *Z _{e}*–

*S*(

*Z*in mm

_{e}^{6}m

^{−3},

*S*in mm h

^{−1}) relations due to main assumptions about snowflake mass

*m*, terminal fall velocity

*υ*, and aspect ratio

_{t}*r*.