## 1. Introduction

Millimeter-wavelength (e.g., Ka and W band) radars have use for the remote sensing of precipitation, especially from space-based platforms owing to their sensitivity and relatively compact form (Mead et al. 1994). *CloudSat* (Stephens et al. 2008), for example, operates a nadir-viewing W-band profiling radar that observes latitudes from 82°N to 82°S, making it an important platform for sensing high-latitude precipitation. Scattering by precipitation-sized ice particles at these wavelengths, however, is sensitive to particle shape as well as mass, and accounting for the variation of snow microphysical properties and shapes is a substantial difficulty for retrieving snowfall with these instruments. Such retrieval problems require a priori assumptions about the microphysical and scattering properties of observed snowfall. To adequately characterize uncertainties in snowfall retrievals, the a priori information must characterize the variability of these properties as well as their expected values.

For snowfall at the surface, microphysical properties have typically been evaluated using time-consuming analyses of individual particles (e.g., Mitchell et al. 1990; Heymsfield and Westbrook 2010). The limited sampling in such an approach makes it difficult to characterize the expected values and variability of these properties. As an alternative, several recent field campaigns, including ground validation studies associated with satellite missions (Hudak et al. 2006; L’Ecuyer et al. 2010; Hudak et al. 2012) as well as a stand-alone study (Löhnert et al. 2011), have made intensive observations of snowfall properties using nearly collocated radars, disdrometers, and precipitation gauges. Although the locations sampled by these campaigns are currently limited, such observations offer the potential to characterize snowfall properties in varied locations and meteorological regimes.

Wood et al. (2014, hereinafter W14) introduced an optimal estimation (OE; Rodgers 2000) retrieval for analyzing such observations. The retrieval produces Gaussian probability density functions (PDFs) representing the expected values and uncertainties for parameters describing the variation of particle mass *m* and horizontally projected area

This work characterizes snow microphysical properties via the W14 retrieval, and then applies those properties to investigate the modeling of radar scattering properties of snow particles. Observations of snowfall rate, snow particle size distribution, size-resolved fall speeds, and 9.35-GHz (X band) radar reflectivity are used to estimate the parameters of power laws describing particle mass and *CloudSat*/*CALIPSO* Validation Project (C3VP; Hudak et al. 2006). The retrieved PDFs of snow microphysical parameters are then applied to constrain particle models used to evaluate W-band radar scattering properties via the discrete dipole approximation (Draine and Flatau 1994), and the performance of these models is evaluated using ground validation observations.

## 2. C3VP events

The snow events used in this work were observed at the Meteorological Service of Canada’s Centre for Atmospheric Research Experiments (CARE) at Egbert, Ontario, Canada, approximately 80 km north of Toronto. Observations used as inputs to the retrieval consist of 9.35-GHz radar reflectivities from the McGill University vertically pointing X-band radar (VertiX; Fabry and Zawadzki 1995), snowfall rates from a Vaisala, Inc., FD12P (Vaisala Oyj 2002) scaled to match snow accumulations from a double fence intercomparison reference (DFIR; Goodison et al. 1998), size-resolved fall speeds from Colorado State University’s 2D video disdrometer (2DVD; Kruger and Krajewski 2002; Huang et al. 2010), and size distributions from the National Aeronautics and Space Administration’s Snowflake Video Imager (SVI; Newman et al. 2009). The characteristics of these observations are described more completely in W14.

Four snowfall events were selected because of completeness of the required observations and because they represent a modest range of snowfall conditions. Observations by personnel on the ground at CARE (R. T. Austin et al. 2007, unpublished manuscript) along with daily operations logs (CIRA 2013) from the 10th Cloud Layer Experiment (CLEX-10), which operated jointly with C3VP, provide characteristics of three of the events. Synoptic event 1 (SYN1; 6 December 2006) involved a weak low passing northeastward over Ontario that produced snowfall at CARE mainly between about 1200 and 1530 UTC. Aircraft observations near CARE showed liquid phase near cloud top with mixed phase and ice below. Snowfall at CARE was described as light and dry early in the day transitioning to moderate wet snow later. VertiX echo-top heights were about 4 km above ground level (AGL) during the precipitation and SVI size distributions showed tails extending to 4–8 mm (Fig. 1). Temperatures during the most significant snowfall were near freezing. The 24-h accumulation for the event was 3.2 mm liquid water equivalent (LWE).

Lake-effect event 1 (LE1; 7 December 2006) consisted of lake-effect snow squalls that resulted from the cold air mass and northwesterly winds that followed the system of the previous day. CARE received snowfall over most of the day, with a 24-h accumulation of 10.2 mm LWE. Temperatures were near freezing early in the day and decreased with time, reaching 255 K at the day’s end (Fig. 2). VertiX echo-top heights were shallower than the previous day, varying from 1 to 3 km AGL. SVI size distributions were similar to those of the previous day but more variable over time. A period after 2100 UTC showed high concentrations of particles smaller than 2 mm and was associated with the coldest temperatures of the day.

As in Fig. 1, but for event LE1.

Citation: Journal of Applied Meteorology and Climatology 54, 4; 10.1175/JAMC-D-14-0137.1

As in Fig. 1, but for event LE1.

Citation: Journal of Applied Meteorology and Climatology 54, 4; 10.1175/JAMC-D-14-0137.1

As in Fig. 1, but for event LE1.

Citation: Journal of Applied Meteorology and Climatology 54, 4; 10.1175/JAMC-D-14-0137.1

Lake-effect event 2 (LE2; 27–28 January 2007) was a second lake-effect snow event that resulted as a warm front near CARE shifted to the south during the evening of 27 January and cold northwesterly winds entered the area. Snow fell mainly between 0100 and 0400 UTC, at temperatures between 267 and 270 K. Snowfall rates at CARE were initially light, but increased rapidly as a heavy snowband lingered over the site (Fig. 3). Large snowflakes, near 10 mm in diameter, fell during periods of heavy snow and visibility was near zero. SVI size distributions showed particles with sizes up to 10 mm early in the event. Total accumulations for the day were 4.6 mm LWE.

As in Fig. 1, but for event LE2.

Citation: Journal of Applied Meteorology and Climatology 54, 4; 10.1175/JAMC-D-14-0137.1

As in Fig. 1, but for event LE2.

Citation: Journal of Applied Meteorology and Climatology 54, 4; 10.1175/JAMC-D-14-0137.1

As in Fig. 1, but for event LE2.

Citation: Journal of Applied Meteorology and Climatology 54, 4; 10.1175/JAMC-D-14-0137.1

Synoptic event 2 (SYN2; 14 February 2007) occurred between intensive observing periods. Although C3VP observer reports of the conditions on the surface and aloft are lacking, the event has been recognized as a massive synoptic winter storm, producing extensive snowfall with substantial accumulations and societal impacts over northeastern United States and southeastern Canada (Grumm and Stuart 2007). At CARE, this system was significantly deeper than the other three events, with VertiX echo-top heights extending to about 6 km AGL (Fig. 4). Observations from precipitation gauges show that snowfall occurred throughout most of the day and produced accumulations of 8.3 mm LWE. This event was also the coldest of the four, with temperatures ranging from 256 to 261 K during the snowfall and with SVI size distributions that were narrow in comparison with those of the earlier events. Characteristics of the events are summarized in Table 1.

As in Fig. 1, but for event SYN2.

Citation: Journal of Applied Meteorology and Climatology 54, 4; 10.1175/JAMC-D-14-0137.1

As in Fig. 1, but for event SYN2.

Citation: Journal of Applied Meteorology and Climatology 54, 4; 10.1175/JAMC-D-14-0137.1

As in Fig. 1, but for event SYN2.

Citation: Journal of Applied Meteorology and Climatology 54, 4; 10.1175/JAMC-D-14-0137.1

Properties of C3VP events. Fraction of day snowing is the fraction of time for which the surface observations of precipitation rate are nonzero. Time is approximate time over which retrievals were performed. The max rate and day accumulation were evaluated using 1-min FD12P rates rescaled to match DFIR accumulations. Max *Z*_{e} was taken from the 10-s VertiX data for range bin 13 at 488 m AGL.

The causes for the large DFIR/FD12P accumulation ratios for events LE2 and SYN2 (Table 1) are not clear. The FD12P uses both an optical sensor and a heated capacitive sensor to estimate precipitation rates for snow and the capacitive sensor is subject to undercatch (Vaisala Oyj 2002). For event LE2, however, winds observed at the CARE meteorological tower were light at 0.5–2.5 m s^{−1} during the snowfall, suggesting that undercatch should not have been significant. Winds were stronger for event SYN2, generally below 5 m s^{−1}, but ranging as high as 6 m s^{−1} for short periods. Additionally, the size distributions for event SYN2 suggest high concentrations of small particles were present (Fig. 4). For this event, undercatch by the capacitive sensor may have been significant and biased the unscaled FD12P precipitation rates low.

Observations were averaged over independent 5-min samples and uncertainties estimated as described in Wood et al. (2013) and W14. Ground clutter caused the nearest usable VertiX reflectivities to be 488 m above the ground. From the reflectivities at 488 m, reflectivities at the surface and the corresponding uncertainties were estimated by considering vertical reflectivity gradients and the likely time separation between precipitation features observed aloft and their appearance at the surface, as described by W14. The resulting samples were then screened to ensure temperatures less than 273 K and wind speeds less than 5 m s^{−1} at the surface, giving 375 distinct samples from the four events: 33 from event SYN1, 173 from LE1, 43 from LE2, and 126 from SYN2.

The VertiX observations are susceptible to attenuation by wet snow accumulations on the radar’s conically shaped radome. Wet snowfall may have occurred for event SYN1 from roughly 1200 to 1500 UTC, and for event LE1 from 0000 to 0400 UTC, during periods of near-freezing temperatures. The VertiX was monitored and cleared of snow periodically, but the times at which this was done are not known. Comparisons of time series of observations over the CARE site by Environment Canada’s King City radar against coincident VertiX observations suggest the VertiX experienced about 2 dB of attenuation by the end of the SYN1 period. For the LE1 event, the time series comparisons suggest about 4 dB of attenuation, but it appears the radome was cleared after 0400 UTC, prior to the remaining 20 h of LE1 observations. Bias corrections for the VertiX observations were determined using these King City radar comparisons, as described in W14, and provide partial compensation for this attenuation.

## 3. The snow microphysics retrieval

*m*and horizontally projected area

Estimates of the a priori state for mass– and area–dimension parameters for use in the C3VP microphysics retrieval.

Selecting Gaussian distributions acknowledges the limited information available about the actual forms of these distributions, and also leads to a reasonably tractable form for the retrieval. For distributions with specified widths, Gaussian distributions maximize the entropy of the distribution (Rodgers 2000; Shannon and Weaver 1949) and impose the least constraint on the retrieval. Selecting other PSD forms without evidence of their appropriateness would introduce unjustified constraints.

Together,

## 4. Snow microphysics retrieval results

Considering the mass parameters, retrieved values of

Retrieved mass parameters for events (a) SYN1, (b) LE1, (c) LE2, and (d) SYN2. Points are sized and color coded to indicate the retrieval input values for reflectivity and temperature, respectively. Gray error bars show the a priori expected values and standard deviations; blue bars show them for the retrieved state. Black triangles in (a) show values from Mitchell (1996) for larger and irregularly shaped particles.

Citation: Journal of Applied Meteorology and Climatology 54, 4; 10.1175/JAMC-D-14-0137.1

Retrieved mass parameters for events (a) SYN1, (b) LE1, (c) LE2, and (d) SYN2. Points are sized and color coded to indicate the retrieval input values for reflectivity and temperature, respectively. Gray error bars show the a priori expected values and standard deviations; blue bars show them for the retrieved state. Black triangles in (a) show values from Mitchell (1996) for larger and irregularly shaped particles.

Citation: Journal of Applied Meteorology and Climatology 54, 4; 10.1175/JAMC-D-14-0137.1

Retrieved mass parameters for events (a) SYN1, (b) LE1, (c) LE2, and (d) SYN2. Points are sized and color coded to indicate the retrieval input values for reflectivity and temperature, respectively. Gray error bars show the a priori expected values and standard deviations; blue bars show them for the retrieved state. Black triangles in (a) show values from Mitchell (1996) for larger and irregularly shaped particles.

Citation: Journal of Applied Meteorology and Climatology 54, 4; 10.1175/JAMC-D-14-0137.1

Mass and area parameters from Mitchell (1996) used in Figs. 5 and 6. Corresponding habit codes are from Magono and Lee (1966). S1a indicates aggregates of side planes. Particle dimensions are in micrometers, and the parameters are in cgs units.

Differences between events are apparent in the mass parameters, with values for the two synoptic events (SYN1 and SYN2) larger than those for the lake-effect events (LE1 and LE2). The strongest contrasts are between events SYN1 and LE2. The mass parameter pooled means for these two events differ by almost a full standard deviation in both *D* = 1 to 5 mm for event SYN2. In contrast to the other events, event LE1 has a more mixed range of temperatures and reflectivities, and has retrieved parameters falling between those of the two synoptic events SYN1 and SYN2 and the other lake-effect snow event LE2.

Unlike the mass parameters, the area parameters

As in Fig. 5, but for area parameters.

Citation: Journal of Applied Meteorology and Climatology 54, 4; 10.1175/JAMC-D-14-0137.1

As in Fig. 5, but for area parameters.

Citation: Journal of Applied Meteorology and Climatology 54, 4; 10.1175/JAMC-D-14-0137.1

As in Fig. 5, but for area parameters.

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For

(left) Retrieved shape parameters ϕ vs size distribution slope λ fitted to the SVI-observed size distributions. Point sizes and colors for points and error bars are as in Fig. 5. (right) A priori (vertical gray bar) and event-pooled means and standard deviations (blue points and vertical bars).

Citation: Journal of Applied Meteorology and Climatology 54, 4; 10.1175/JAMC-D-14-0137.1

(left) Retrieved shape parameters ϕ vs size distribution slope λ fitted to the SVI-observed size distributions. Point sizes and colors for points and error bars are as in Fig. 5. (right) A priori (vertical gray bar) and event-pooled means and standard deviations (blue points and vertical bars).

Citation: Journal of Applied Meteorology and Climatology 54, 4; 10.1175/JAMC-D-14-0137.1

(left) Retrieved shape parameters ϕ vs size distribution slope λ fitted to the SVI-observed size distributions. Point sizes and colors for points and error bars are as in Fig. 5. (right) A priori (vertical gray bar) and event-pooled means and standard deviations (blue points and vertical bars).

Citation: Journal of Applied Meteorology and Climatology 54, 4; 10.1175/JAMC-D-14-0137.1

As anticipated from Figs. 5–7, the variances and covariances associated with the mass parameters have decreased compared to the a priori, while those associated with the area parameters and

Comparison of prior and posterior PDFs for the microphysics retrieval. “Exp” is expected value, “Var” is variance, and “Cov” is covariance.

One of the key questions to be answered by this analysis is whether covariances exist that were not present in the a priori. From (9), it can be seen that covariances that mix mass and area parameters {e.g.,

Correlations between microphysical parameters.

Citation: Journal of Applied Meteorology and Climatology 54, 4; 10.1175/JAMC-D-14-0137.1

Correlations between microphysical parameters.

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Correlations between microphysical parameters.

Citation: Journal of Applied Meteorology and Climatology 54, 4; 10.1175/JAMC-D-14-0137.1

While the expected values of

Several other studies have suggested values of

## 5. Application to radar retrieval of snowfall

A primary goal of the preceding analysis is to provide observational constraints on mass and shape that can be used for modeling snow particle scattering properties and their uncertainties for use in millimeter-wavelength radar retrievals of snowfall rate. Using these observational constraints helps insure a physically consistent relationship between the particle microphysical and scattering properties that influence the relationships between radar reflectivity and snowfall rate that are assumed implicitly or explicitly within such retrievals.

Because of shape sensitivity, the Rayleigh or Mie approaches commonly used to model scattering properties at lower frequencies are not applicable for other than small particles (Schneider and Stephens 1995; Liu and Illingworth 1997). Scattering properties of larger snow particles may be simulated with Mie theory using spheres composed of a mixture of air and ice (the soft sphere approximation); however, comparisons against less approximate methods have shown the inability of such models to reproduce backscattering properties across multiple frequencies (Petty and Huang 2010). Several more complex techniques may be used at millimeter wavelengths (see, e.g., Bohren and Singham 1991 for an overview). The discrete dipole approximation (DDA; Draine 1988; Draine and Flatau 1994), the method used for this work, allows for arbitrary geometry by replacing the continuous particle with an array of discrete dipoles on a spatial lattice.

The relationships for particle mass and

### a. Shape assumptions

Using constrained DDA particle models (appendix), scattering calculations were made for a variety of shapes intended to represent the features of more pristine, planar particles and of larger, irregular aggregates. For planar particles, a branched platelike shape with six branches, designated as shape SPp, was used (Fig. 9, upper left). With this planar shape, the horizontally projected area may be altered by changing the width of the branches. For narrow widths, the shape is much like a stellar crystal (P1d; Magono and Lee 1966) and has a small horizontally projected area. As the branch width increases, the shape approaches that of a crystal with broad branches (P1c) and at the limit of maximum branch width, is a hexagonal plate (P1a). For purposes of comparison, calculations were also done for hexagonal plates (shape HPp) that met the mass constraints from the snow microphysics retrieval but not the constraints on horizontally projected area.

Examples of shapes for dipole arrays: (top left) SPp, (top center) B6pf, (top right) Ep, (bottom left) B8pr-30, and (bottom right) B8pr-45.

Citation: Journal of Applied Meteorology and Climatology 54, 4; 10.1175/JAMC-D-14-0137.1

Examples of shapes for dipole arrays: (top left) SPp, (top center) B6pf, (top right) Ep, (bottom left) B8pr-30, and (bottom right) B8pr-45.

Citation: Journal of Applied Meteorology and Climatology 54, 4; 10.1175/JAMC-D-14-0137.1

Examples of shapes for dipole arrays: (top left) SPp, (top center) B6pf, (top right) Ep, (bottom left) B8pr-30, and (bottom right) B8pr-45.

Citation: Journal of Applied Meteorology and Climatology 54, 4; 10.1175/JAMC-D-14-0137.1

Spatial particles were represented with clusters of thick hexagonal branches and with scalene ellipsoids (Fig. 9). These spatial shapes are considered to be simplistic, somewhat abstract representations of aggregate particles. Three different configurations were examined: B6pf, with six branches all lying in the horizontal plane (Fig. 9, upper middle); B8pr-30, with eight branches, six of which intersected the horizontal plane at angles of 30° (Fig. 9, lower left); and B8pf-45 with eight branches, six of which intersected the horizontal plane at angles of 45° (Fig. 9, lower right). The orientation of the branches controls the aspect ratio of the particle and for a given aspect ratio, the horizontally projected area may be altered by changing the branch thickness. For shape B6pf, both aspect ratio and horizontally projected area get smaller as the branch thickness is reduced. Shape B8pr-30 has an aspect ratio of about 0.5, and B8pr-45 has an aspect ratio near 0.70.

The scalene ellipsoid Ep (Fig. 9, upper right) is a simple shape that also has the ability to meet the constraints on horizontally projected area and aspect ratio, but whose shape is fundamentally different than the branched particles. Given a desired size

### b. DDA results

^{−3}. For simulating radar reflectivities, the backscatter cross section is calculated from

Figure 10 shows the resulting

Backscatter cross sections for planar particle models compared with those for Rayleigh and Mie solid ice spheres.

Citation: Journal of Applied Meteorology and Climatology 54, 4; 10.1175/JAMC-D-14-0137.1

Backscatter cross sections for planar particle models compared with those for Rayleigh and Mie solid ice spheres.

Citation: Journal of Applied Meteorology and Climatology 54, 4; 10.1175/JAMC-D-14-0137.1

Backscatter cross sections for planar particle models compared with those for Rayleigh and Mie solid ice spheres.

Citation: Journal of Applied Meteorology and Climatology 54, 4; 10.1175/JAMC-D-14-0137.1

Figure 11 shows

As in Fig. 10, but for spatial particle models.

Citation: Journal of Applied Meteorology and Climatology 54, 4; 10.1175/JAMC-D-14-0137.1

As in Fig. 10, but for spatial particle models.

Citation: Journal of Applied Meteorology and Climatology 54, 4; 10.1175/JAMC-D-14-0137.1

As in Fig. 10, but for spatial particle models.

Citation: Journal of Applied Meteorology and Climatology 54, 4; 10.1175/JAMC-D-14-0137.1

Over most of the shown size range,

### c. Assessments

The modeled scattering properties were assessed using observed PSDs and collocated W-band radar reflectivities. PSDs were obtained from the SVI, which operated nearly continuously at CARE during the 2006/07 C3VP observing season. Observations at 1-min resolution were averaged using distinct 5-min samples as was done for the snow microphysics retrievals. These distributions, based on the Feret diameter, were converted to distributions on

The observed radar reflectivities were provided by the airborne cloud radar (ACR) (Sadowy 1999), a 95-GHz profiling radar deployed on the ground at CARE during C3VP. The ACR pointed vertically, and the range bin nearest the surface was centered at 197 m AGL. Comparisons of the reflectivities in this bin versus reflectivities in the adjacent bin above suggest this lowest bin was not substantially affected by ground clutter for reflectivities above about −15 dB*Z*_{e}, so observations from this lowest bin were used and assessments were limited to cases with reflectivity greater than −15 dB*Z*_{e}. Reflectivities in linear units at about 2.8-s resolution were averaged in time using 5-min samples, consistent with the treatment of the SVI observations. Although a formal calibration of the ACR was not performed immediately prior to C3VP, a previous intercomparison between the ACR and the University of Massachusetts Cloud Profiling Radar System showed average differences of 0.3 dB*Z*e (Sekelsky et al. 1999). Here, calibration errors are neglected, and it is noted that any biases in the ACR calibration could affect the results presented below.

#### 1) Reflectivity comparisons

Of the five shapes considered, the B8pr-30 shape provided the best agreement with the observed reflectivities, with a bias of −0.03 dB and root-mean-square (RMS) difference of 5.4 dB over all of the PSDs that were well sampled by the disdrometer (Fig. 12). The more compact shapes B6pf and SPp substantially overestimated reflectivities over most of the observed reflectivity range and produced considerably greater scatter versus the ACR observations. The less compact B8pr-45 and Ep underestimated reflectivities, with no improvement of RMS differences versus that for the B8pr-30 shape. Obviously, the use of the SPp shape over the full size range observed by the SVI represents an unrealistic and severe extrapolation. The largest observed

Comparisons of observed W-band reflectivities from the ACR vs those simulated using DDA models for various particle shapes. The blue points indicate SVI size distributions with <100 particles in the sample and with those particles distributed across five or fewer size bins. Reflectivity biases and RMS differences (in parentheses), computed over all the well-sampled (black) points, are shown above each panel.

Citation: Journal of Applied Meteorology and Climatology 54, 4; 10.1175/JAMC-D-14-0137.1

Comparisons of observed W-band reflectivities from the ACR vs those simulated using DDA models for various particle shapes. The blue points indicate SVI size distributions with <100 particles in the sample and with those particles distributed across five or fewer size bins. Reflectivity biases and RMS differences (in parentheses), computed over all the well-sampled (black) points, are shown above each panel.

Citation: Journal of Applied Meteorology and Climatology 54, 4; 10.1175/JAMC-D-14-0137.1

Comparisons of observed W-band reflectivities from the ACR vs those simulated using DDA models for various particle shapes. The blue points indicate SVI size distributions with <100 particles in the sample and with those particles distributed across five or fewer size bins. Reflectivity biases and RMS differences (in parentheses), computed over all the well-sampled (black) points, are shown above each panel.

Citation: Journal of Applied Meteorology and Climatology 54, 4; 10.1175/JAMC-D-14-0137.1

The difference in bias between the B8pr-30 and the B8pr-45 shapes corresponds with differences in the vertical aspect ratios: the vertical aspect ratio for the B8pr-30 particle is about 0.5 while that for the B8pr-45 is near 0.7, giving a particle that is more extended along the direction of radar beam propagation than is the B8pr-30 shape. Additionally, for a given particle size, the branches of the B8pr-45 particle are likely somewhat wider than those of the B8pr-30 particle. This increase in width is necessary for the different shapes to have equal

Clearly, combinations of the shapes used in this study could be found that give biases similar to that of the best-fit shape. For example, one can imagine a combination of the highly reflective SPp shape at small sizes with the less reflective B8pr-45 shape at large sizes. Since these shapes are constructed using the same

The comparisons shown in Fig. 12 also highlight the possible effects of the limited sample volume of the SVI compared to the radar. The blue points are cases for which the SVI detected fewer than 100 particles over the 5-min sample with those particles distributed in five or fewer size bins, suggesting the size distributions may have been poorly sampled. An examination of the ACR operator’s log (R. T. Austin et al. 2007, unpublished manuscript) showed that many of these cases were associated with the initiation or termination of snowfall at the surface, or with low-level stratocumulus lacking precipitation.

#### 2) Uncertainties in modeled reflectivity

Random placement of dipoles on the lattice (appendix) causes small variations in the scattering properties. To evaluate these variations, four distinct realizations of the random dipole arrays for the B8pr-30 shape were constructed and used to calculate

The variations in

Citation: Journal of Applied Meteorology and Climatology 54, 4; 10.1175/JAMC-D-14-0137.1

The variations in

Citation: Journal of Applied Meteorology and Climatology 54, 4; 10.1175/JAMC-D-14-0137.1

The variations in

Citation: Journal of Applied Meteorology and Climatology 54, 4; 10.1175/JAMC-D-14-0137.1

To evaluate the influence of random dipole locations on modeled reflectivities, the replicate realizations of the B8pr-30 scattering properties were used along with the SVI PSDs to calculate radar reflectivity per (15). Evaluations were limited to cases for which the SVI size distribution was well sampled (more than 100 particles observed or particles distributed over more than five size bins). As expected, since these random variations are uncorrelated over the range of particle sizes, the resulting uncertainties in the modeled radar reflectivity are negligible (Fig. 14), with typical fractional uncertainties in *Z*_{e} of 0.02.

Histogram of radar reflectivity uncertainties for shape B8pr-30 due to the random placement of dipoles.

Citation: Journal of Applied Meteorology and Climatology 54, 4; 10.1175/JAMC-D-14-0137.1

Histogram of radar reflectivity uncertainties for shape B8pr-30 due to the random placement of dipoles.

Citation: Journal of Applied Meteorology and Climatology 54, 4; 10.1175/JAMC-D-14-0137.1

Histogram of radar reflectivity uncertainties for shape B8pr-30 due to the random placement of dipoles.

Citation: Journal of Applied Meteorology and Climatology 54, 4; 10.1175/JAMC-D-14-0137.1

Derivatives of reflectivity (dB) with respect to the microphysical parameters α, β, γ, and σ. Derivatives for α and γ are taken with respect to the natural logarithms of these parameters. Values are plotted vs the slopes λ of exponential size distributions fitted to the SVI PSDs.

Citation: Journal of Applied Meteorology and Climatology 54, 4; 10.1175/JAMC-D-14-0137.1

Derivatives of reflectivity (dB) with respect to the microphysical parameters α, β, γ, and σ. Derivatives for α and γ are taken with respect to the natural logarithms of these parameters. Values are plotted vs the slopes λ of exponential size distributions fitted to the SVI PSDs.

Citation: Journal of Applied Meteorology and Climatology 54, 4; 10.1175/JAMC-D-14-0137.1

Derivatives of reflectivity (dB) with respect to the microphysical parameters α, β, γ, and σ. Derivatives for α and γ are taken with respect to the natural logarithms of these parameters. Values are plotted vs the slopes λ of exponential size distributions fitted to the SVI PSDs.

Citation: Journal of Applied Meteorology and Climatology 54, 4; 10.1175/JAMC-D-14-0137.1

Uncertainties in modeled reflectivities range from 5 to 15 dB and also depend markedly on the slope of the size distribution, with narrow distributions having much smaller uncertainties than broad distributions (Fig. 16), consistent with the behavior of the Jacobian for

Uncertainties in modeled reflectivities due to uncertainties in microphysical parameters, shown vs size distribution slope λ.

Citation: Journal of Applied Meteorology and Climatology 54, 4; 10.1175/JAMC-D-14-0137.1

Uncertainties in modeled reflectivities due to uncertainties in microphysical parameters, shown vs size distribution slope λ.

Citation: Journal of Applied Meteorology and Climatology 54, 4; 10.1175/JAMC-D-14-0137.1

Uncertainties in modeled reflectivities due to uncertainties in microphysical parameters, shown vs size distribution slope λ.

Citation: Journal of Applied Meteorology and Climatology 54, 4; 10.1175/JAMC-D-14-0137.1

Estimates of the contributions of uncertainties in α, β, γ, and σ to uncertainties in reflectivity (dB).

## 6. Conclusions

Descriptions of snow microphysical properties, consisting of expected values, uncertainties, and correlations of microphysical parameters, are essential constraints for the remote sensing of snowfall. Although estimates of parameters for mass– and area–dimension relationships for various habits are available from a number of prior studies, these studies largely lacked information needed to estimate the distributions of these parameters. Additionally, in many cases estimates of both mass and area parameters were not available from the same sample of particles, thus there was little guarantee of consistency between the mass and area representations. To address these issues, a retrieval algorithm was developed that estimates the PDFs for these microphysical properties using multisensor observations of snowfall, designed around the observations available from a highly instrumented ground site used for a snowfall remote sensing field validation campaign: Rayleigh-regime radar reflectivity, snowfall rate, particle fall speeds, and size distributions.

The algorithm was applied to observations from four snowfall events involving both synoptic front and lake-effect processes. The measurements were found to primarily determine

The results from these analyses provide information essential to developing physically consistent representations of snow particle microphysical and scattering properties needed by snowfall retrievals using millimeter-wavelength observations. Given the complexity of snow crystals and the highly variable shape of aggregates, the goal for such representations is to be sufficiently realistic for retrieval purposes. Particle models were constructed using the retrieved microphysical properties, and so should be consistent with observed X-band reflectivities, fall speeds, and snowfall rates. While the models did not incorporate explicit representations of pristine shapes, models with reasonable aspect ratios and inhomogeneous structure were able to reproduce observed W-band reflectivities.

Characterizing the microphysical properties in the form of PDFs allows such retrievals to better quantify forward model uncertainties, which in turn allows the uncertainties in retrieved snowfall rates to be better quantified. Using particle models constructed with the retrieved microphysical properties, we found that the uncertainties in the retrieved microphysical parameters, especially those associated with particle mass, cause substantial uncertainties in modeled reflectivities. The estimated uncertainties due to uncertainties in

The approach used to estimate these uncertainties assumes a linear error-propagation model and is likely a crude approximation for large uncertainties. Even so, in the applied context of retrievals that minimize differences between modeled and observed millimeter-wavelength radar reflectivities, these reflectivity model uncertainties are likely significant. As demonstrated by W14, however, improvements to ground-based observing systems may better constrain microphysical properties and reduce uncertainties. Additionally, extending these analyses to a more diverse range of snowfall regimes may reveal regime-dependent variations in microphysical properties and suggest approaches to further reduce their uncertainties. Several recent field experiments, including the Light Precipitation Validation Experiment (LPVEx) and GPM Cold Season Precipitation Experiment (GCPEx), have examined snowfall in different meteorological regimes and/or used enhanced suites of instrumentation. Although the OE method developed here was targeted specifically to observations from C3VP, it is readily adaptable to observations from these and similar future experiments.

## Acknowledgments

Parts of this research by NBW and TSL were performed at the University of Wisconsin–Madison and at Colorado State University for the Jet Propulsion Laboratory, California Institute of Technology, sponsored by the National Aeronautics and Space Administration. AJH acknowledges support from the JPL *CloudSat* Office and NASA Global Precipitation Measurement program Contract NN-X13AH73G. Computing resources for discrete dipole modeling were provided by the Community Computing Facility of the National Center for Atmospheric Research. Thanks are given to G.-J. Huang of Colorado State University, F. Fabry of McGill University, and L. Bliven of NASA Goddard Space Flight Center for making their C3VP datasets available and sharing their expertise. We appreciate the efforts of three anonymous reviewers who provided helpful feedback on the paper.

## APPENDIX

### Constrained Discrete Dipole Modeling Method

*m*and horizontally projected area

*d*is defined inside the instance (Fig. A1, top). Possible values for

*d*are obtained by dividing the maximum dimension

*n*is the complex refractive index of the dipole material and

*d*were limited to no more than 73 μm, suitable for solid ice dipoles at 250 K and frequencies up to 183 GHz based on the ice refractive index data of Warren (1984).

Cartoon illustrating construction of a dipole array for DDA calculations. Given a desired shape that meets the specified maximum dimension and horizontally projected area, (top) first a cubic lattice is constructed within the shape. (middle) Next, dipoles are placed so that the horizontally projected area is completely occupied by dipoles. (bottom) Last, the remaining dipoles, sufficient in number to meet the specified mass, are placed randomly on the lattice.

Citation: Journal of Applied Meteorology and Climatology 54, 4; 10.1175/JAMC-D-14-0137.1

Cartoon illustrating construction of a dipole array for DDA calculations. Given a desired shape that meets the specified maximum dimension and horizontally projected area, (top) first a cubic lattice is constructed within the shape. (middle) Next, dipoles are placed so that the horizontally projected area is completely occupied by dipoles. (bottom) Last, the remaining dipoles, sufficient in number to meet the specified mass, are placed randomly on the lattice.

Citation: Journal of Applied Meteorology and Climatology 54, 4; 10.1175/JAMC-D-14-0137.1

Cartoon illustrating construction of a dipole array for DDA calculations. Given a desired shape that meets the specified maximum dimension and horizontally projected area, (top) first a cubic lattice is constructed within the shape. (middle) Next, dipoles are placed so that the horizontally projected area is completely occupied by dipoles. (bottom) Last, the remaining dipoles, sufficient in number to meet the specified mass, are placed randomly on the lattice.

Citation: Journal of Applied Meteorology and Climatology 54, 4; 10.1175/JAMC-D-14-0137.1

*d*, the mass of a single dipole is

*m*, the required number of dipoles is

If ^{4}, a value again based on accuracy criteria (Draine and Flatau 1994), and if the lattice has enough nodes on which to place the dipoles, the lattice spacing is acceptable. In practice, a range of *d* will produce acceptable lattices, so a *d* is chosen that makes the number of dipoles as small as possible without falling below 10^{4}.

Dipoles are placed randomly on the lattice nodes in such a way that *m*. The random placement of dipoles means that the modeled scattering properties will have some amount of random variation. Scattering properties at millimeter wavelengths, however, should not be strongly sensitive to the fine elements of particle structure (Matrosov 2007). Additionally, since radar reflectivity is obtained by integrating over the particle size spectrum, the effects of random variations in scattering properties on radar reflectivity should be further reduced.

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