Radar observations focused on snowfall from platforms outside the established weather surveillance radar networks have become ubiquitous over the last 2 decades, largely due to increased interest in the role of snowfall in mid- and high-latitude microphysics, hydrology, and climate. This research accelerated with the advent of satellite-borne radars flown by missions to quantify global hydrometeor and precipitation properties. These satellite-borne radars (specifically the CloudSat mission's Cloud Profiling Radar (CPR) and the Global Precipitation Measurement (GPM) mission's Dual-frequency Precipitation Radar (DPR) , with two others anticipated to launch in the coming decade) are capable solely of measuring vertical profiles of radar reflectivity factor (hereafter, reflectivity) along with path-integrated attenuation under certain conditions. To understand the capabilities of these satellite-borne radars for quantifying snowfall, we must know how well radar reflectivity observations constrain snowfall properties.

To these ends, CloudSat and GPM have contributed to multiple field experiments involving ground-based radars and designed to provide, in part, ground validation data for the radar remote sensing of snowfall: the Canadian CloudSat-CALIPSO Validation Project (C3VP) , the Global Precipitation Measurement (GPM) Cold-season Precipitation Experiment (GCPEx) , the Light Precipitation Validation Experiment (LPVEx) , the International Collaborative Experiment during the PyeongChang 2018 Olympics and Paralympics (ICE-POP) , and the Olympic Mountains Experiment (OLYMPEx) . These and a number of smaller, more focused field campaigns have made extensive use of small K-band profiling radars, e.g., METEK's Micro Rain Radar MRR,, but several experiments, including C3VP, GCPEx, and ICE-POP, have deployed ground-based, W-band scanning, or profiling radars. Although these ground-based radars may provide advanced capabilities such as Doppler velocity measurement, their reflectivity measurements in snowfall are a valuable resource for examining the capabilities of the satellite-borne radars .

The ability of radar reflectivity to constrain snowfall properties, however, has not been well-evaluated. Snowfall exhibits a wide range of microphysical characteristics that influence radar reflectivity and snowfall rate. Most notable to casual observers are variations in particle habits: pristine dendrites, needles, columns, plates, and bullets; aggregates of the same; pellets and graupel for example. Underlying these differences in habit are variations in mass, and, given a particular mass, variations in how mass is distributed within the particle. Unlike longer-wavelength radars for which radar backscattering properties of snow particles are sensitive primarily to particle mass, at millimeter wavelengths those properties are additionally sensitive to particle shape. Investigations of particle mass and area (an aspect of shape) have painstakingly determined the broad extent of these variations. Along with differences in single-particle properties, populations of falling snow particles vary substantially in their concentrations with size (i.e., the spectral particle size distribution, PSD) based on measurements from the ground e.g., and, more recently, with the advent of imaging particle probes, from aircraft e.g.,and references therein. The observed particle concentrations vary over several orders of magnitude.

In radar-based remote sensing scenarios when these properties are not known, these variations produce uncertainty in the relationship between radar reflectivity factor (hereafter, reflectivity) and associated water content and snowfall rate. A common approach to estimating this uncertainty has been to evaluate modeled reflectivities, water contents, and snowfall rates using a range of assumed particle models and PSDs. The results are often expressed using relationships between reflectivity and snowfall rate (“Z–S” relationships) . This approach allows the uncertainty in a retrieved snowfall rate to be estimated, but the existing studies have not provided insight into the dominant sources of uncertainty nor into the ability of observed radar reflectivity to constrain various properties controlling the snowfall rate. examined uncertainties and information content for radar observations of mixed- and ice-phase regions of a convective storm but targeted radar systems with more advanced capabilities. used CloudSat and aircraft observations to assess how uncertainties in the ice particle mass-dimension relationship contribute to radar reflectivity forward model uncertainties but used known, observed particle size distributions and did not examine the influence of the mass-dimension uncertainties on snowfall retrieval performance.

In this work, we provide uncertainty and information content analyses for retrieving snowfall from observations of radar reflectivity at millimeter wavelengths, focusing on W-band (94 GHz). The results are representative of the general problem of estimating snowfall from such remote radar reflectivity observations without supplementary collocated observations of snow particle mass-dimension relationships, fall speeds, and particle size distributions. The results apply particularly to observations by the CPR and by the DPR's Ka-band radar, but also to reflectivity measurements from ground-based radars such as the MRR and the Department of Energy Atmospheric Radiation Measurement (ARM) program's Millimeter Wavelength Cloud Radar and Ka-band ARM Zenith Radar (KAZR) . The retrieval method used here is the foundation for the retrieval used for CloudSat's 2C-SNOW-PROFILE product; that application is the subject of a future companion paper. Our objectives here are to identify the snowfall properties that are best constrained by such observations and the most significant sources of uncertainty in the radar retrieval of snowfall. The results establish a performance baseline for reflectivity-only observations of snowfall, indicate where uncertainty reduction efforts should be focused, and suggest what improvements to radar-observing systems could be most beneficial.

The analyses use the optimal estimation (OE) retrieval technique , which inherently diagnoses information content and uncertainties in retrieved quantities subject to specified uncertainties in measurements, forward models, and a priori knowledge of the quantities to be retrieved . The retrieval produces best estimates of snow size distribution parameters by using the radar reflectivity observations to refine a priori estimates of those parameters (Sect. ). The information content metrics provided by OE require all sources of uncertainties in the retrieval process to be specified. These are discussed in Sect. . Ground-based radar and precipitation observations allow the retrieval to be tested, showing that size distribution width is best constrained by the retrieval and that uncertainties in retrieved size distribution parameters (but not uncertainties due to the assumed exponential form of the PSD itself) are the strongest contributors to uncertainties in estimated snowfall rates (Sect. ). The results suggest that the retrieved size distribution widths could be useful for diagnosing changes in PSD resulting from microphysical processes and that improved observational constraints on size distribution parameters, as might be provided by dual-wavelength radar observations , would likely enhance snowfall retrieval performance (Sect. ).

The retrieval uses measurements of reflectivity to estimate snow microphysical properties and to quantify water content and snowfall rate. At the wavelengths characteristic of cloud radars such as CloudSat and shorter-wavelength precipitation radars, scattering by precipitation-sized particles does not follow the Rayleigh approximation, and both attenuation and multiple scattering may affect the radar signal. At these wavelengths, snow particle scattering and extinction properties depend not only on mass, but on shape as well. With even simple parameterized expressions for particle mass, shape, and size distribution, single-frequency observations of radar reflectivity alone are insufficient to reasonably constrain the resulting set of parameters.

To address this insufficiency, retrievals must incorporate a priori information about particle microphysical and scattering properties. This is accomplished here using OE , a Bayesian technique that allows a priori information to be included explicitly. The input for this retrieval is the Ze observed by the radar for a range gate identified as containing snow. For notational consistency with other work, we show this as a vector: 1y=Ze,1obs. A forward model F(x,b̃) relates y to x, a state vector of unknown properties to be retrieved, as 2y=F(x,b̃)+ϵ, where b̃ are parameters not being retrieved but which influence the forward model results. The forward model approximates the true physical relation between x and y, and there are uncertainties associated with both the observations y and the forward model parameters b̃. ϵ represents the total uncertainty due to all sources. OE attempts to find x^, an estimate of the state which maximizes the posterior conditional probability density function (PDF) P(x|y), subject also to prior knowledge about the values of x. This prior knowledge is described by expected values xa and their covariances Sa. Assuming Gaussian statistics for the model-measurement errors and the a priori state, minimizing the cost function 3Φ(x,y,xa)=y-F(x,b̃)TSϵ-1y-F(x,b̃)+x-xaTSa-1x-xa with respect to x gives this PDF, where Sϵ is the covariance matrix representing the uncertainties ϵ. The Gaussian assumption is reasonable if the expected values and covariance matrices are known for the model-measurement uncertainties and the a priori state, but other details are lacking. In that case, the Gaussian form maximizes the entropy of a PDF . Assuming an alternate form would introduce constraints on the retrieval that are not justified based on the limited knowledge of the PDF.

Provided the forward model is not excessively nonlinear, Newtonian iteration 4x^i+1=x^i+Sa-1+KiTSϵ-1Ki-1KiTSϵ-1y-F(x^i,b̃)-Sa-1(x^i-xa) leads to x^, where K is the Jacobian of the forward model with respect to x, and Ki=K(x^i). Iteration continues until the squared difference in successive x^i normalized by the current estimate of the a posteriori covariance S^x is much smaller than the number of state vector elements . At convergence, this covariance of x^ is obtained as 5S^x=K^TSϵ-1K^+Sa-1-1, where K^=K(x^). As a diagnostic test of the results, a χ2 statistic is calculated using the retrieved state vector in Eq. (). A value near the number of observations suggests correct convergence . Several metrics, determined from the retrieved state and based on information theory, provide insight into the retrieval performance; these metrics are presented in Sect. .

Applying the exponential distribution in Eq. (), the singly-scattered non-attenuated reflectivity Zess is 14ZessR=Λ4Kw2π5∫DM,minDM,maxN0exp⁡-λDMσbk(DM,b̃)dDM. The backscatter cross section σbk has been written to show its dependence on a vector of parameters b̃ as well as on DM. The vector b̃ includes the parameters for the mass- and area-dimension relations α, β, γ, and σ which were used to construct the particle models from which the scattering properties were calculated. The tilde indicates that these parameters are approximations of the true values and a source of uncertainty.

During C3VP, a vertically pointing W-band radar (the Jet Propulsion Laboratory's Airborne Cloud Radar, ACR) was deployed on the ground at CARE. In all, about 28 h of ACR radar profiles of snowfall were recorded at approximately 2.8 s intervals. These observations represent 17 distinct snow events that occurred over 18 d between 3 November 2006 and 2 March 2007; however, most of the accumulations were concentrated during nine of the events (Table ). These observations include portions of three of the cases that were used to develop the snow particle microphysical models (cases SYN1, LES1, and LES2, ). Of the nearly 36 000 ACR profiles in these observations, approximately 7300 are from cases SYN1, LES1, and LES2. Further, as described in , ACR reflectivities from 12 of the events between 2 December 2006 and 26 February 2007 were used to constrain the snow particle models' scattering properties to give unbiased reflectivities. This overlap should be kept in mind when evaluating the retrieved snowfall rates and estimated accumulation, but it should not substantially affect the assessments of retrieval uncertainties, uncertainty sources, and information content metrics that follow.

The retrieval was applied to the ACR reflectivities observed in the single range bin nearest the surface, at 197 m above ground level (AGL). Temperatures and pressures needed by the retrieval to perform snow detection, calculate fall speeds, and establish the a priori states were obtained from nearby surface meteorology observations. Because of the short distance to the target range bin, attenuation along the path was neglected. The retrieved snowfall rates produce a Z–S relationship that is most similar to that developed by for an aggregate particle model denoted as the aggregate (HA) (Fig. ). For warmer temperatures and mid-range reflectivities, the Z–S relationship becomes more similar to that of and the LR3 relationship of Kulie and Bennartz.

For comparisons, snowfall rate observations were obtained at 1 min intervals from the Vaisala FD12P and scaled to provide unbiased accumulations relative to the nearby Dual Fence Intercomparison Reference, or DFIR . The retrieved ACR snowfall rates, PACR, were matched to the nearest-in-time observed snowfall rate, PFD12P.

Time series of PACR and PFD12P show a high degree of agreement over most of the observing period (Fig. a). This is not extraordinary given the dependence of the retrieval's particle microphysical and scattering properties on portions of the C3VP data. Two notable exceptions occur near time indices 25 000 and 32 500, however, when the FD12P recorded snowfall rates above 1 mm LWE h-1, while the retrieved values are substantially smaller. Examining the time series of ACR reflectivities shows that the ACR did not observe high reflectivities during these periods (Fig. b). The first of these anomalies occurred on 22 February 2007 from 11:20 to 12:05 UTC, while the second occurred on 1 March 2007 between 22:15 and 22:50 UTC. For both, the ACR operator made note of the heavy snowfall, suggesting that both the FD12P and the ACR observed similar snowfall rates. Based on soundings, Environment Canada forecasts, and ACR operator observations, these anomalies appear to correspond to melting aloft, ice pellets, and freezing rain . These conditions could also have been favorable for formation of large, heavy aggregates. It seems likely that the conditions produced snowfall whose properties were strongly inconsistent with the particle properties assumed in the retrieval, although reflectivities did not change substantially.

Accumulations were calculated from both PACR and PFD12P with and without the two anomalies described above (Fig. ). Accumulations agree substantially during the first 16 h but diverge somewhat beyond that, again noting the dependence of the retrieval's assumed microphysical and scattering properties on portions of the C3VP data. With the anomalies included the final difference between the accumulations is 2 mm. With the anomalies removed that difference is reduced to 0.7 mm. For individual events, absolute fractional differences between the ACR and FD12P accumulations can range to 50 % and upwards (Table ), but these large values are associated mainly with events with small accumulations. For events with larger accumulations, the absolute fractional differences are mostly below 30 %. At seasonal timescales, the random components in event-total accumulations are likely uncorrelated, leading to offsetting errors when calculating seasonal accumulations. The time series of absolute fractional differences between the ACR-derived and FD12P accumulations begins with large fractional differences. Within 5 h and over the initial three events, the fractional differences reduce to less than 5 % and then remain below 20 % for the remainder of the season.

While millimeter-wavelength, single-frequency radar reflectivity observations alone would seem to have limited utility for retrieving snowfall properties, the results herein demonstrate capabilities for quantifying snowfall rate, accumulation, and aspects of the snow PSD. The results were obtained by applying the radar observations to constrain a priori information appropriate to a broad range of snowfall regimes. The results indicate that the approach would provide useful information when applied to observations such as those from satellite-borne radars, which observe a range of snowfall regimes and for which radar observables are limited to reflectivity.

The results demonstrate the ability of the retrieval to produce reliable estimates of snow accumulation, particularly over timescales involving multiple events and more than several hours of snowfall duration, in spite of large uncertainties in retrieved instantaneous snowfall rates. For the C3VP season, the retrieval reproduced the observed accumulation within 13 % at the end of the season. These results were achieved by omitting two particular time periods during which the retrieval's particle property assumptions were likely very inconsistent with the observed snowfall. Without this adjustment, the end-of-season absolute difference was 18.9 %, illustrating the need for adequate discrimination of the precipitation phase in the retrieval process. Keeping in mind that certain a priori assumptions of the retrieval were also sourced from the C3VP observations, these results are probably best viewed as indicating proper function of the retrieval. The time series of seasonal accumulation shows that while the initial fractional differences reach almost 80 %, the differences diminish with time and increasing accumulation, reaching values of less than 5 % within 5 h. These results are partly due to offsetting errors between events; however, for individual events, best agreement between the observed and retrieved snow accumulations were achieved for events that were longer in duration and produced more substantial accumulations. The observed accumulations for these events were mostly near or within the tighter uncertainty bounds produced by a decorrelating error model applied to the retrieved accumulation. The modest decorrelation used in the model produces uncertainties in the retrieved event accumulations of only 1 % to 20 %. Thus, despite large instantaneous snowfall rate uncertainties for these single-frequency, millimeter-wavelength retrievals, retrieved rates can be expected to prove of value for quantifying accumulations over events, months, seasons, and longer.

Uncertainties in instantaneous retrieval-estimated snowfall rate are dominated by uncertainties in the retrieved state (the uncertainties in the estimated PSD), followed by uncertainties in particle model parameters and, to a lesser extent, the uncertainties in the fall-speed model. Uncertainties due to the assumption of an exponential PSD form are negligible. There is a degree of ambiguity here. The uncertainties in the particle model parameters contribute to the uncertainties in the estimated snowfall rate due to the appearance of the mass term in Eq. () but also contribute to uncertainties in the retrieved state. We treat these as independent contributions to the uncertainty. There is likely some covariance that could reduce total uncertainties, but this is not addressed in the treatment of snowfall rate uncertainty presented here.

Retrieval performance, quantified in terms of information content metrics, is determined by the sensitivity of the observations to the desired state vector, the uncertainties assessed for the forward models and measurements, and the explicit assumptions about the uncertainty in the a priori knowledge of the state. For W-band modeled reflectivities in dB, the magnitudes of sensitivities for log⁡λ are 3–4 times those for log⁡N0, and sensitivities are opposite in sign. This contributes to log⁡λ being better constrained by the retrieval than is log⁡N0. The consequences of these sensitivities are described more fully in Appendix . To the extent that process information can be gleaned from changes in the slope parameter over time or space, the retrieval may be useful for process analyses when more direct observations of PSD are not available.

Model-measurement uncertainties are dominated by uncertainties in the particle model parameters (e.g., the coefficients and exponents of the mass- and area-dimension relationships, Table ), and it is the uncertainties in mass parameters that are the most substantial contributor . For these near-surface observations, contributions to uncertainties in W-band radar reflectivity from shape, the assumption of an exponential form for the PSD, and the discrete-truncated form of the integrations over size distribution were not significant. For longer wavelength radars that might be used in similar applications (e.g., the MRR or KAZR), shape uncertainty will likely be even smaller due to less prevalent non-Rayleigh scattering.

These baseline results suggest several avenues for improving such single-frequency, radar reflectivity-based snowfall retrievals. Improved constraints on snow PSD parameters, through either reduced a priori uncertainties or better observational constraints, are paramount. For ground- or aircraft-based observations, ancillary measurements of snow PSDs can improve the a priori constraints. For retrievals from satellite-borne radar where such measurements are not available, the a priori state is given by more broadly applicable relationships for PSD parameters like those presented here. To the extent that a priori states for specific snowfall regimes might have smaller uncertainties, knowledge of regime-specific PDFs for snow PSD parameters would improve retrieval results provided the correct regime can be diagnosed by the retrieval. Coincident dual-frequency radar observations may also provide improved constraints on the snow PSD parameters but among current satellite-borne instruments, the CPR is single-frequency, and while the GPM DPR provides dual-frequency observations, the DPR sensitivities limit observations to heavier snowfall and implementation of dual-frequency snowfall retrieval has proven difficult . Finally, model-measurement uncertainties can be reduced by reducing uncertainties in particle mass estimates. This may require a more synergistic approach in which improved PSD information is coupled with additional observations such as Doppler velocity to better constrain the assumed particle model used in the retrieval, e.g., moving toward the approach used by with ground-based observations. The methods presented here, easily adaptable to other observing systems providing multiple frequency or collocated Doppler velocity observations, provide the basis from which such improvements can be tested and evaluated.