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Research article 16 Mar 2022
Research article  16 Mar 2022
Snow microphysical retrieval from the NASA D3R radar during ICEPOP 2018
 ^{1}Mesoscale Atmospheric Processes Laboratory, NASA Goddard Space Flight Center, Greenbelt, MD, USA
 ^{2}Universities Space Research Association, Columbia, MD, USA
 ^{3}Joint Center for Earth Systems Technology, University of Maryland, Baltimore County, Catonsville, MD, USA
 ^{a}now at: The Tomorrow Companies, Inc., Boston, MA, USA
 ^{1}Mesoscale Atmospheric Processes Laboratory, NASA Goddard Space Flight Center, Greenbelt, MD, USA
 ^{2}Universities Space Research Association, Columbia, MD, USA
 ^{3}Joint Center for Earth Systems Technology, University of Maryland, Baltimore County, Catonsville, MD, USA
 ^{a}now at: The Tomorrow Companies, Inc., Boston, MA, USA
Correspondence: Robert S. Schrom (robert.s.schrom@nasa.gov)
Hide author detailsCorrespondence: Robert S. Schrom (robert.s.schrom@nasa.gov)
A method is developed to use both polarimetric and dualfrequency radar measurements to retrieve microphysical properties of falling snow. It is applied to the Ku and Kaband measurements of the NASA dualpolarization, dualfrequency Doppler radar (D3R) obtained during the International Collaborative Experiments for PyeongChang 2018 Olympic and Paralympic winter games (ICEPOP 2018) field campaign and incorporates the Atmospheric Radiative Transfer Simulator (ARTS) microwave singlescattering property database for oriented particles. The retrieval uses optimal estimation to solve for several parameters that describe the particle size distribution (PSD), relative contribution of pristine, aggregate, and rimed ice species, and the orientation distribution along an entire radial simultaneously. Examination of Jacobian matrices and averaging kernels shows that the dualwavelength ratio (DWR) measurements provide information regarding the characteristic particle size, and to a lesser extent, the rime fraction and shape parameter of the size distribution, whereas the polarimetric measurements provide information regarding the mass fraction of pristine particles and their characteristic size and orientation distribution. Thus, by combining the dualfrequency and polarimetric measurements, some ambiguities can be resolved that should allow a better determination of the PSD and bulk microphysical properties (e.g., snowfall rate) than can be retrieved from singlefrequency polarimetric measurements or dualfrequency, singlepolarization measurements.
The D3R ICEPOP retrievals were validated using Precipitation Imaging Package (PIP) and Pluvio weighing gauge measurements taken nearby at the May Hills ground site. The PIP measures the snow PSD directly, and its measurements can be used to derived the snowfall rate (volumetric and water equivalent), mean volumeweighted particle size, and effective density, as well as particle aspect ratio and orientation. Four retrieval experiments were performed to evaluate the utility of different measurement combinations: Kuonly, DWRonly, Kupol, and Allobs. In terms of correlation, the volumetric snowfall rate (r=0.95) and snow water equivalent rate (r=0.92) were best retrieved by the Kupol method, while the DWRonly method had the lowest magnitude bias for these parameters (−31 % and −8 %, respectively). The methods that incorporated DWR also had the best correlation to particle size (r=0.74 and r=0.71 for DWRonly and Allobs, respectively), although none of the methods retrieved density particularly well (r=0.43 for Allobs). The ability of the measurements to retrieve mean aspect ratio was also inconclusive, although the polarimetric methods (Kupol and Allobs) had reduced biases and mean absolute error (MAE) relative to the Kuonly and DWRonly methods. The significant biases in particle size and snowfall rate appeared to be related to biases in the measured DWR, emphasizing the need for accurate DWR measurements and frequent calibration in future D3R deployments.
Estimation of snowfall rates and other properties from weather radar is made difficult by many of the same challenges that exist for rainfall estimation (primarily, the discrepancy between the sixthmoment dependence of radar reflectivity factor Z and the third to fourthmoment dependence of precipitation rate R), but additional factors further confound radar retrievals of snow. Whereas the shape and scattering properties of a raindrop depend only on its mass and temperature (e.g., Beard et al., 2010; Ekelund et al., 2020), there is tremendous diversity in ice crystals of a given mass, resulting from the infinite complexity of particle trajectories through differing thermodynamic environments resulting in growth by vapor deposition (e.g., Kuroda and Lacmann, 1982; Chen and Lamb, 1994; Fukuta and Takahashi, 1999), aggregation (e.g., Hosler et al., 1957; Hobbs et al., 1974; Connolly et al., 2012), and riming (e.g., Mitchell et al., 1990; Jensen and Harrington, 2015; Moisseev et al., 2017), as well as ablation by sublimation (e.g., Smith et al., 2009) and melting (e.g., Matsuo and Sasyo, 1981; Leinonen and von Lerber, 2018). All of these processes influence the scattering and aerodynamic properties of these ice particles in ways that can influence the interpretation of radar data (e.g., Hall et al., 1984; Vivekanandan et al., 1994; Bechini et al., 2013; Botta et al., 2013; Thompson et al., 2014). Despite these challenges, since the introduction of weather radar, these instruments have been important tools in gathering information about icephase precipitation. Early efforts focused on using Z to estimate the intensity of snow precipitation measurements with assumed ice particle size distribution (PSD) forms (e.g., Marshall and Gunn, 1952). These efforts relied on in situ ground measurements to derive empirical relations to the measured Z, yielding a variety of Z–S relations, depending on the climatological properties of snow at the measurement location.
Improvements upon these situational Z–S relationships can be made if multiparameter radar observations are available. For snow, these have historically proceeded along two pathways following advances in multifrequency/Doppler and dualpolarization radar technologies. Multifrequency methods essentially rely upon deviations from Rayleigh scattering to infer a characteristic particle size (e.g., Matrosov et al., 2005; Liao et al., 2016) and, with three frequencies (typically X or Ku, Ka, and W bands), density can also be inferred (Kneifel et al., 2015). For vertically pointing radars, Doppler velocities can also be used to refine the density estimate (Oue et al., 2015; Mason et al., 2018), since fall velocity of a snow particle depends (to first order) on its size and density (Heymsfield and Westbrook, 2010). These methods have been employed primarily towards data collected at a few wellequipped snow observatories such as the Hyytiälä Forestry Research Station in Hyytiälä, Finland (Hari and Kulmala, 2005), the Jülich Observatory for Cloud Evolution (JOYCE) in Germany (Löhnert et al., 2015), and the Department of Energy – Atmospheric Radiation Measurement Program facility in Alaska (de Boer et al., 2018) and applied to airborne and spaceborne radar datasets (Leinonen et al., 2018; Tridon et al., 2019; Chase et al., 2021).
Polarimetric radar measurements have shown value in inferring ongoing ice growth processes, due to the dependence of these measurements on the distributions of particle shapes, orientations, and sizes. In particular, enhancements in the differential reflectivity (Z_{DR}) and specific differential phase (K_{dp}) have been linked to the planar crystal growth near −15 ^{∘}C (e.g., Ryzhkov and Zrnić, 1998; Kennedy and Rutledge, 2011; Andrić et al., 2013; Schrom et al., 2015; Moisseev et al., 2015). Assessing changes in vertical profiles of the polarimetric radar variables also provides information on ice growth processes. Decreases in Z_{DR} and K_{dp} towards the ground have been observed with increases in reflectivity towards the ground, indicating growing ice particles becoming more chaotically oriented and more spherical, a result of some combination of aggregation and intense riming (e.g., Bechini et al., 2013; Oue et al., 2015; Ryzhkov et al., 2016; Schrom and Kumjian, 2016; Kumjian and Lombardo, 2017). Some recent efforts have been made to gain quantitative information about the ice particle properties (and thus associated microphysical processes and snowfall rates) from polarimetric radar measurements using empirically determined algorithms (e.g., Bukovčić et al., 2020) and microphysicalmodel informed parameter estimation (e.g., Schrom et al., 2021). However, there has been limited evaluation of these methods using additional remote sensing (e.g., multifrequency radar measurements) and in situ observations.
From the extensive literature on multifrequency and polarimetric radar studies of snow, it is evident that complementary information is contained in these measurements. However, relatively few studies have been performed to assess the information content of multifrequency, dualpolarization radar measurements of snow, partially because of a lack of radar platforms with these capabilities deployed in locations subject to frequent and highaccumulation snowfall events. The NASA dualpolarization, dualfrequency Doppler radar (D3R) is a premier radar for making such measurements. The D3R was built to provide ground validation measurements for the NASA Global Precipitation Measurement (GPM) mission's dualfrequency precipitation radar (DPR; Chandrasekar et al., 2010; Vega et al., 2014) and operates at Ku and Ka bands using novel solidstate transmitters. The D3R was first deployed in the GPM Coldseason Precipitation Experiment (GCPex; SkofronickJackson et al., 2015) and in subsequent GPM ground validation field campaigns. Upgrades to improve sensitivity and range resolution have been implemented (Kumar et al., 2017) since these early campaigns.
Recognizing the potential utility of D3R measurements to provide unique information about microphysics, dynamics, and quantitative precipitation estimation (QPE) in snowstorms, the Korean Meteorological Administration (KMA), organizers of the International Collaborative Experiments for PyeongChang 2018 Olympic and Paralympic winter games (ICEPOP 2018) cooperated with NASA to deploy the D3R radar in Daegwallyeong, South Korea, from November 2017–March 2018. The D3R was part of an extensive network of groundbased remote sensing and in situ instrumentation deployed during ICEPOP 2018 and formed a central observation point for measurements aligned perpendicular to the coastal mountain ranges of eastern South Korea. This measurement strategy was devised to examine the distribution of precipitation from the coast to the mountains in different winter synoptic weather situations and evaluate highresolution numerical weather prediction in this complex topographic region (Lim et al., 2020).
The objective of this study is to use the data collected by the D3R during ICEPOP 2018 develop a snow retrieval algorithm using realistic scattering models of pristine, aggregate, and rimed snow particles to further our understanding of the complementary nature of the dualfrequency and polarimetric radar measurements and their utility regarding snow microphysical characterization and QPE. The output of this algorithm is intended to aid in identifying microphysical processes during ICEPOP events and provide snow QPE during the deployment. The extensive network of ground instrumentation is leveraged to validate the algorithm output. The manuscript is organized as follows: the observational datasets are listed in Sect. 2, the particle scattering properties we use and the construction of lookup tables from these databases is described in Sect. 3, algorithm mechanics and information content are analyzed in Sect. 4, validation for selected cases is presented in Sect. 5, and the conclusions are given in Sect. 6.
2.1 D3R
The NASA D3R radar is a polarimetric Doppler weather radar operating at Ku (13.91 GHz) and Ka (35.56 GHz) bands, which utilizes novel design features including aligned antennas, solidstate transceivers, and a digital waveform generator to enable deployment in a wide range of environmental conditions on a mobile trailer platform (Chandrasekar et al., 2010). At both frequencies, the following parameters are measured: reflectivity (Z), differential reflectivity (Z_{dr}), differential propagation phase (ϕ_{dp}), copolar correlation coefficient (ρ_{hv}), radial velocity (V), and spectrum width (W).
During ICEPOP 2018, the D3R was located on the roof of the Daegwallyeong (DGW) regional weather office (36.677^{∘} N, 128.719^{∘} E; altitude 789 m m.s.l.). The D3R was configured to measure 150 m range gates out to a maximum range of 39.75 km, combining a short pulse for ranges <3.3 km with a medium pulse for the remaining range gates. This gives a Kuband sensitivity ranging approximately from −30 to −5 dBZ over the short pulse and from −15 to 5 dBZ over the long pulse (Kumar et al., 2017). The primary scan schedule during snow events conducted a Plan Position Indicator (PPI) scan at 5^{∘} elevation followed by Range Height Indicator (RHI) scans at 51, 231, and 330^{∘} azimuths, with each set of scans taking 5 min. For this study, we focus on the 231^{∘} RHI scans aimed towards the May Hills Supersite (MHS) 2 km downrange, which contained a wealth of ground instrumentation.
We use the D3R data available from the NASA ICEPOP data archive held at the Global Hydrology Resource Center (Petersen et al., 2018). Several snowfall events were observed by D3R during the ICEPOP 2018 campaign, but we selected only three events to analyze in this manuscript, listed in Table 1, for their diversity in synoptic forcing, environmental profiles, and microphysics. Time–height profiles from the Kuband polarimetric RHIs near the Precipitation Imaging Package (PIP) are shown for these in Fig. 1. The 9 January case was relatively shallow and had increases in reflectivity at the lowest altitudes of the radar beam close to the ground (Fig. 1a). The 28 February and 7–8 March cases were deeper, with reflectivity >20 dBZ extending above 2 km. These cases both had larger enhancements in K_{dp}, with the 28 February having the largest K_{dp} and Z_{H} values of the three cases (Fig. 1b and h). We disregarded events that had mixedphase precipitation, as the modeling of their scattering properties is less mature than ice particles for our needs, as well as events that did not have both the D3R and data from ground instruments at MHS available.
The D3R Z_{dr} and ϕ_{dp} were calibrated on an event basis, and the absolution calibration of Ku and Ka Z was established in the early period of the deployment (Chandrasekar et al., 2018). Notwithstanding these calibration efforts, in order for the retrieval algorithm to perform optimally, we examined the data for selfconsistency of Z_{dr} and the dualwavelength ratio (DWR), defined as Z_{Ku}−Z_{Ka}. The expectation is that, at nearzenith angles, Z_{dr} should be close to zero, and, in the limit of small particles, the DWR should equal the difference in attenuation, which should be small, but positive, depending on the pathintegrated attenuation from water vapor, cloud liquid water, and hydrometeors.
We examined time series of the PDFs of these quantities for the events listed in Table 1. In Fig. 2, the top two rows show the time series of the PDF of Ku and Ka Z_{dr} at elevation angles $>\mathrm{80}{}^{\circ}$ and altitudes above 1.5 km m.s.l. There is clearly some nonstationary behavior in these PDFs, so we applied additional calibration offsets (independently at each frequency) to the Z_{dr} for each RHI such that the average Z_{dr} at elevation angles $>\mathrm{80}{}^{\circ}$ is equal to zero. The time series of the calibrated Ku and Ka Z_{dr} are shown in the third and fourth rows of Fig. 2. In addition to the modification to Z_{dr}, we also noticed some unusual behavior in the smallparticle DWR PDF (fifth row of Fig. 2) during the 28 February case. When Z_{Ku}<10 dBZ, SkofronickJackson et al. (2019) found that Ku–Ka DWR is typically close to zero, but during this case, there is large increase in the DWR, peaking around 06:00 UTC, that is difficult to explain entirely by particle size or attenuation. Accumulation of wet snow on the radome is a possible explanation for this behavior (Venkatachalam Chandrasekhar, personal communication, 2020), but for this study we have chosen to avoid correcting the DWR because it is difficult to have a continuous independent estimate of the Ku–Ka relative calibration from a ground radar.
2.2 PIP
The primary source of validation for the microphysical retrievals in this study is PIP (Pettersen et al., 2020), which takes measurements that can be used to derive quantities including snowfall rate and effective density. The PIP is a video disdrometer made up of a single highspeed camera, continuously recording at 380 frames per second, and a halogen lamp, which is used to backlight the precipitation particles. The camera and lamp are separated by 2 m and the focal plane is located 1.33 m from the camera lens and uses an open sampling volume (i.e., the sampling volume is not enclosed within a box). The images produced are 640×480 pixels with a resolution of 0.1 mm×0.1 mm. The field of view (FOV), including the edge effects, is then 64−D_{eq} mm $\times \mathrm{48}{D}_{\mathrm{eq}}$ mm, where D_{eq} is the equivalent diameter in millimeters. The depth of field (DOF) also varies with particle size and is expressed as $\mathrm{117}/{D}_{\mathrm{eq}}$ (in mm). The sampling volume is a multiplication of the FOV, DOF, and the number of frames over a given time period. Considering 100 particles with uniform size of D_{eq}=1 mm, the sampling volume is 790 m^{3} for a 1 min observation period. As PIP only uses a single camera, the precipitation particle images are of a projected view of the particle and do not contain any information on the particle dimension along the viewing direction.
PIP determines the characteristics of the precipitation particles using an algorithm written using the National Instruments IMage AQuisition (IMAQ) software package. This algorithm determines the shape of the precipitation particle (i.e., the long and the short dimensions) by fitting an ellipse to the PIPimaged particle. The IMAQ software package defines the fitted ellipse as the ellipse having both the same area and the same perimeter as the PIPimaged particle. During our preliminary analysis of the data, we found that the IMAQfitted ellipses tended to overestimate the long dimension of the particle and underestimate the short dimension, resulting in an underestimate of the aspect ratio. As such, we have reprocessed the PIP data using an alternative, custombuilt ellipsefitting strategy (Helms et al., 2022). This strategy uses the method implemented in the fit_ellipse program in the Coyote Interactive Data Language (IDL) library (http://www.idlcoyote.com/, last access: 25 February 2022) to perform the actual fit. The fitting is performed on images of particles taken from videos that PIP records for troubleshooting purposes. These videos contain the first 2000 frames that contain precipitation particles within each 10 min period. For periods with fewer than 2000 frames containing precipitation particles, the videos will be shorter than 2000 frames.
The PIPdetermined orientation angle is defined as the counterclockwise angle from the positive horizontal axis, where positive is to the right, to the longest dimension of the particle. This results in orientation angles ranging from 0 to 180^{∘}. In order to combine the ellipse aspect ratio (minor axis length divided by major axis length) and orientation angle information, we have used the natural logarithm of the aspect ratio of the bounding box of the particle. The bounding box is defined as the smallest rectangle that is able to contain the particle and whose edges are either horizontally or vertically oriented.
2.3 Soundings
Radiosonde launches were performed every 3 h during precipitation events from the DGW Regional Weather Office, supplementing the normal 12hourly observations, and used Meteomodem M10 radiosondes (Meteomodem, 2021). The profiles of temperature and humidity from the nearestintime radiosonde were used as input to the D3R algorithm to calculate attenuation from atmospheric gases. These profiles are also used to qualitatively evaluate the retrieval output with respect to locations of wellknown thermodynamic importance (e.g., the −15 ^{∘}C dendritic growth maximum and layers that are supersaturated with respect to ice).
Gaining useful information from polarimetric radar measurements requires scattering properties of hydrometeors with preferred mean orientations. We incorporate such scattering properties into the retrieval algorithm described herein, by way of lookup tables (LUTs) that are derived by integrating these scattering properties over a prescribed PSD. Specifically, we use scattering properties for a pristine plate, an aggregate of plates, and graupel (all at a wide range of sizes) from the Atmospheric Radiative Transfer Simulator (ARTS; Eriksson et al., 2018; Brath et al., 2020; Ekelund et al., 2020) database. Scattering properties for each of these particles are available for a discrete set of frequencies (for this study, we use the calculations at 13.4 and 35.6 GHz), temperatures (190, 230, and 270 K), incident angles of the transmitted radiation, scattering angles of the scattered radiation, and, except for graupel (which is assumed to have total random orientation) zenithrelative orientation angles (we refer to this angle hereafter as β). Table 2 lists values of these properties that correspond to the scattering calculations.
The ARTS database aims to provide scattering properties for a set of particles over a large range of frequencies so that applications using both active and passive remote sensing measurements are consistent. The derivation of polarimetric radar variables from this database is described in Appendix A. For each particle type listed in Table 2, these variables are integrated over a PSD using a modified gamma form (e.g., Petty and Huang, 2011):
where D is the diameter of an equivalentvolume solid ice sphere in mm, D_{m} is the massweighted mean equivalentvolume diameter, and μ is the shape parameter. While this definition of D (and D_{m}) complicates the comparison to in situ measurements (where maximum dimension is typically used to describe size), it is directly related to particle mass, and, in the Rayleigh limit, radar reflectivity. To reduce the dimensionality of the LUTs while preserving the variables that control the shape of the PSD, only D_{m} and μ are varied in the construction of the LUT, and N_{0}(D_{m},μ) is a normalized concentration factor such that the ice water content of all PSDs stored in the LUT is 1 g m^{−3}:
where ρ_{i} is the density of solid ice (0.917 g cm^{−3}). The distribution is later scaled by a retrieved concentration factor to match the observed reflectivity. For the particle types with preferred orientation, these PSDs are further integrated over the range of β angles to account for the orientation distribution. We choose the von Mises distribution to represent the PDF of beta angles (Table 2). The von Mises distribution is a continuous function that represents the dispersion of variables in circular coordinates, and has been used to represent hydrometeor orientation retrieved from polarimetric radar (Bringi and Chandrasekar, 2001; Melnikov and Straka, 2013). For a zeromean canting angle, this distribution simplifies to
where κ is a dispersion parameter and G(κ) is a normalization factor such that the sum of probabilities is equal to 1. When κ=0, the distribution is uniform; as κ increases, the distribution becomes narrower and can be approximated by a normal distribution with standard deviation $\sqrt{\mathrm{1}/\mathit{\kappa}}$. In the construction of the LUTs, we make the simplifying assumption that κ is independent of particle size. While this is almost certainly an oversimplification (observations and Reynolds number analysis suggest that smaller particles of a given aspect ratio should have more randomly distributed canting angles; Klett, 1995), in the retrieval, this is dealt with by allowing the combination of pristine and aggregate PSDs of different κ values, as will be further described in Sect. 4.
The LUTs are constructed for each particle species. The LUT dimensions and ranges of the LUT indices are given in Table 3. The variables stored within the LUTs encompass three categories: radar variables, physical variables, and simulated PIP measurements. The radar variables are the singleparticle scattering properties necessary to construct the polarimetric radar measurements: the reflectivity factor Z at both horizontal and vertical polarizations, the extinction coefficient k at both polarizations, the specific differential phase K_{dp}, and the real and imaginary parts of the copolar conjugate product of scattering amplitudes (${C}_{\mathrm{hv}}={S}_{\mathrm{hh}}^{*}{S}_{\mathrm{vv}}$). The formulas used to forward model the radar measurements from these quantities are given in Appendix B.
In addition to the radar variables, we simulate several variables (denoted with subscript P) that represent PIPmeasured quantities. Because the PIP measures 2D projections of 3D particles, assumptions must be made to convert the PIP measurements to physical variables. While several formulas have been proposed to achieve this purpose (e.g., Jiang et al., 2017), we opted instead to use 2D projections of particles derived from the shapefiles used in the ARTS database to simulate the PIP measurements over the range of D_{m}, μ, and κ in the LUTs, as this process is less ambiguous than the alternative of attempting to derive the 3D properties from the 2D PIP measurements. Moreover, this process ensures an internally consistent comparison of the radar retrieval output with PIP measurements. Most of these quantities depend in some way on the equivalent diameter D_{eq} measured by the PIP. The simulated PIP measurements include the snowfall water equivalent rate (S), volumetric snowfall rate (S_{V}), effective density (ρ_{P}), normalized intercept ${N}_{\mathrm{P}}^{*}$, areaweighted mean particle volume diameter D_{P}, mean box aspect ratio ($\stackrel{\mathrm{\u203e}}{{a}_{\mathrm{box}}}$), and mean ellipse aspect ratio ($\stackrel{\mathrm{\u203e}}{{a}_{\mathrm{ell}}}$). The final two quantities are weighted by the projected area. The snowfall water equivalent rate is calculated as
where m(D) is the mass of the particle in grams, V_{t}(D,β) is the terminal velocity in m s^{−1}, and S is in units of mm h^{−1}. The terminal velocity is calculated following the process given in Heymsfield and Westbrook (2010), where the area ratio is simulated from the ARTS shapefiles (and thus depends on both D and β), and using values for the density and viscosity of air typical at the MHS location during ICEPOP 2018 snow events (−5 ^{∘}C, 90 %RH, 925 hPa). The volumetric snowfall rate is calculated using the same terminal velocity but, instead of mass, using the volume of the particle derived from the PIPmeasured diameter:
The PIPderived density (ρ_{P}) is defined as the ratio of the liquidequivalent snow rate to the volumetric snowfall rate multiplied by the density of liquid water ρ_{w} (Tiira et al., 2016):
The normalized intercept ${N}_{\mathrm{P}}^{*}$ is adapted from the definition given for ${N}_{\mathrm{0},\mathrm{23}}^{*}$ in Field et al. (2005):
where d_{max2D}(D,β) is the average 2D projected maximum particle dimension. This parameter is useful for providing temperaturedependent constraints on the PSD as will be shown in Sect. 4.
D_{P} is the ratio of the fourth to third moments of the PSD in terms of D_{eq}:
In order to evaluate the capability of the D3R polarimetric measurements to retrieve a bulk measurement of the aspect ratio, we defined the areaweighted mean ellipse aspect ratio ($\stackrel{\mathrm{\u203e}}{{a}_{\mathrm{ell}}}$):
where $\frac{b}{a}(D,\mathit{\beta})$ is the ratio of the short to long axis of the ellipse fitted to the simulated PIP image. In order to compare our retrievals to a variable that is influenced by the canting angle distribution parameter κ, we defined the areaweighted mean box aspect ratio ($\stackrel{\mathrm{\u203e}}{{a}_{\mathrm{box}}}$):
where $\frac{y}{x}(D,\mathit{\beta})$ is the ratio of the vertical to horizontal dimensions of the bounding box of the simulated PIP image.
While it is not feasible to illustrate every dimension of the LUTs, a few examples are provided to preview the expected sensitivity of the D3R measurements to microphysical parameters. In Fig. 3, the Ku–Ka dualwavelength ratio is plotted as a function of both the volumeequivalent D_{m} and PIPmeasured D_{P} for each of the three species. The DWR is nearly identical for the pristine plates and plate aggregates at D_{m}<1 mm and D_{P}<3 mm but reaches an upper limit of about 5 dBZ for the single plates while increasing to nearly 10 dBZ for the aggregates. Large Ku–Ka DWRs, sometimes exceeding 10 dBZ, have been observed coincident with large aggregates (20 mm projected diameter) in dendritic growth regimes during OLYMPEX (Chase et al., 2018), so it is important that the LUTs capture this DWR range. The DWR of graupel is lower than the unrimed plates and aggregates for a given volumeequivalent size but is similar or greater when viewed with respect to PIPmeasured size based on the projected area. This suggests that DWR alone is not sufficient to determine the PSD mean particle size and density. Such information can be provided by the polarimetric measurements. For this study, we assume graupel to be randomly oriented and thus have zero contribution to Z_{dr} and K_{dp}, which is a reasonable assumption for dry graupel (Kumjian, 2013). Meanwhile, the contributions to Z_{dr} and K_{dp} from plates and aggregates primarily depend on D_{m} and κ (Fig. 4). As expected, both Z_{dr} and K_{dp} increase with κ as the orientation distribution becomes less dispersive. The Z_{dr} rapidly decreases with aggregate size, as does K_{dp} (though less rapidly). Meanwhile, the Z_{dr} of the individual plates does not depend on D_{m}, since these plates have a fixed aspect ratio at sizes larger than about 0.2 mm (Brath et al., 2020). The K_{dp} of the individual plates also does not depend much on size, although some resonance effects lead to a small decrease in K_{dp} at larger size. From Figs. 3 and 4, it is clear that there is complementary information in the dualfrequency and polarimetric variables to discern particle size and species, which will be demonstrated in the next section.
Optimal estimation (OE; Rodgers, 2000) is a form of Bayesian inversion that assumes Gaussian error statistics and accommodates moderately nonlinear forward models. OE has been used for singlefrequency (L'Ecuyer and Stephens, 2002; Munchak and Kummerow, 2011) and multifrequency (Grecu et al., 2011; Mason et al., 2018) radar precipitation retrievals. It is applied here to the multifrequency, polarimetric observations provided by the D3R. In this section, the OE components (state vector, observation vector, and covariance matrices) are defined and the approach is illustrated with an example retrieval along a single radar ray.
4.1 Optimal estimation setup
The cost function that is minimized by optimal estimation is
where Y_{obs} is the observation vector and Y_{sim}(X) its forwardmodeled counterpart, S_{y} is the measurement and forward model error covariance matrix, X is the state vector and X_{a} its prior, and S_{a} is the state covariance matrix. For the D3R retrieval of snow microphysical properties, we define Y_{obs} to contain one or more of the following series of dualfrequency or dualpolarization observations at each range gate along a radial:
where each type of observation is filtered for ground clutter and only considered if the signaltonoise ratio exceeds 1. Thus, the number of observations of each type may differ. Note that ϕ_{dp} instead of K_{dp} was chosen because ϕ_{dp} is the more direct measurement, and additional assumptions (with potentially nonGaussian errors) are required to derive K_{dp} from noisy ϕ_{dp} measurements.
The state vector consists of quantities that describe the PSD, relative contribution of each species, as well as other quantities known to affect the Ku and Kaband polarimetric radar measurements:
where each quantity is defined at nodes (indicated by the superscript in Eq. 13) which may be arbitrarily placed (for this study, nodes are spaced 600 m horizontally and 250 m vertically). Each of these quantities, their priors, and ranges are defined in Table 5. Following Grecu et al. (2011), the measured Kuband Z_{hh} (which is not in Y_{obs}) is used as direct input to the forward model, and from these measurements and the quantities in the state vector X, the measurements in Y_{obs} are simulated. A detailed description of this forward model is provided in Appendix B.
Field et al. (2005)The observation error covariance matrix S_{y} must accurately describe the error characteristics of the measurements and forward model. Values that are unrealistically low can lead to overfitting, whereas overly conservative (large) error estimates can lead to underutilization of the information contained within the measurements. In this study, we consider the diagonal elements of S_{y} to be the sum of the measurement error and forward model error components, which are given in Table 6 for each measurement in S_{y}. The measurement errors are obtained from the gatetogate variance over many homogeneous scenes at low (<10^{∘}) elevation angles, where the assumption is that the true change in the measured quantity is small compared to the measurement noise. The measurement error is assumed to be uncorrelated in space and between variables. The forward model error is quantified by assessing the variance in the measurements for alternate aggregate particles of the same size as the ARTS aggregates. These alternate models include dendrites and columns, with different assumptions about the aggregation process (Schrom et al., 2021). Although these forward model errors are correlated between variables, analysis of the covariance matrices showed the correlation between DWR and Z_{dr} error to be insignificant. While Z_{dr}−K_{dp} error correlation is larger, the ϕ_{dp} error is dominated by upstream propagation error which is assumed to be uncorrelated to the Z_{dr} forward model error at a given range gate. Therefore, for this study, the offdiagonal elements of S_{y} are set to zero, although this is a choice that could be refined in future implementations of the retrieval.
4.2 Example ray
The optimal estimation process along a ray of radar data is illustrated in Fig. 5, which shows the observed (Y_{obs}) and simulated (Y_{sim}) measurements, and Fig. 6, which shows the various retrieval parameters in X as well as derived quantities of ice water content and D_{m} for each iteration until convergence. This ray is characterized by DWR peaks at 0–5 and 23–28 km, which reach 6 dB, dropping to 1–3 dB elsewhere. The Z_{dr} has several peaks, the most significant of which reach over 2 dB at 20 and 30 km range, bracketing the DWR peak. The ϕ_{dp} increases most rapidly in the 10–20 km range with smaller rates of increase elsewhere, reaching 15^{∘} at the Ku band and 50^{∘} at the Ka band. The initial profiles of the retrieval parameters ${N}_{\mathrm{P}}^{*}$, f_{r}, f_{p}, κ_{p}, κ_{a}, μ_{p}, μ_{a}, and the derived ice and cloud liquid water content and D_{m} of the aggregate and pristine PSDs are shown in the lightest shaded lines in Fig. 6. The iterative adjustments guided by the Jacobian matrix respond to the initial differences between Y_{obs} and Y_{sim}:

${N}_{\mathrm{P}}^{*}$ decreases slightly near the DWR peaks and increases elsewhere.

f_{p} decreases below 50 % in the DWR peak region but increases above 60 % in the 10–20 km range and again near 30 km, corresponding to the Z_{dr} peaks and steepest increase in ϕ_{dp}.

f_{r} is generally below 40 %, with minima near the Z_{dr} peaks.

κ_{p} is generally lower than κ_{a} but exhibits peaks corresponding to the Z_{dr} peaks, whereas κ_{a} does not vary as much with range.

μ_{a} and μ_{p} increase slightly from their initial values in most regions, although μ_{a} dips below zero near the DWR peaks (lower μ corresponds to a longer tail of the PSD at large sizes and can result in higher DWRs, all else being equal).

No significant cloud liquid water is detected or needed to explain the DWR observations. In fact, the nearzero DWR at the far ranges implies that there is little differential attenuation, and the observed nearfield DWR can be attributed to particle size effects.

D_{m} of the pristine population (controlled by the retrieval parameter D_{p}) generally increases as a proportion of the D_{m} of the aggregates in order to balance pristine particle contributions to Z_{dr} and ϕ_{dp}.
The most significant impact of these parameter changes is to increase D_{m} and reduce ice water content in the DWR peak regions, with the opposite changes elsewhere. The final retrieved state is in good agreement with the Z_{dr} and ϕ_{dp} observations at both frequencies. There is a residual high bias in the DWR, especially at the range beyond 30 km. Examining the DWR histogram for this case in Fig. 2 reveals a mean near or below zero, which may indicate a low bias in the relative calibration of the Ku to Ka reflectivity, and possible high bias in the ice water content retrieval.
As a consistency check, we show that the retrieval algorithm effectively reproduces the spatial distribution of the radar variables for an RHI (Fig. 7). The retrieval that uses only the DWR as input is able to reproduce the reflectivity and dualwavelength ratio measurements but overestimates the Φ_{dp} at both wavelengths and poorly represents the spatial structure of the Z_{DR}. In contrast, the retrieval using only the Kuband polarimetric measurements produces simulated Φ_{dp} and Z_{DR} that correspond well to the measurements but dualwavelength ratio simulations that fail to capture the observed dualwavelength ratio structure in the RHI. The simulated radar variables associated with the retrieval that incorporates all of the radar observations shows the best consistency with the observations, suggesting that the polarimetric and DWR measurements contain independent and complementary information.
4.3 Information content analysis
Some further insight into the adjustments made to the parameters can be gained by examining the Jacobian matrix K of partial derivatives of each element of Y_{sim} with respect to each element of X.
While the Jacobian is state dependent, the sign and relative magnitude of each element from the example ray shown in the previous section are illustrative of the general sensitivity of the forward model to the retrieval parameters. The Jacobian matrix is composed of blocks that are either diagonal or triangular, depending on whether the parameter has a significant downrange propagation effect on an observation. The effect of modifying the parameters in X can be explained by considering the change to the PSD at a fixed Kuband reflectivity:

Increasing ${N}_{\mathrm{P}}^{*}$ results in a smaller mean particle size and higher ice water content. This decreases the DWR and increases ϕ_{dp} downrange, while the effect on Z_{dr} is relatively small and state dependent.

Increasing f_{p} increases Z_{dr} and ϕ_{dp} downrange as a result of the increasing contribution of the pristine particles to the PSD, with little effect on DWR.

Increasing D_{p} also increases Z_{dr} (and decreases DWR) as the pristine particles become larger and contributes more to reflectivity but decreases ϕ_{dp} downrange as the ice water content is reduced.

Increasing κ_{a} and κ_{p} increases the Z_{dr} and downrange ϕ_{dp}, with the κ_{p} having a much larger magnitude effect.

Increasing μ_{a} reduces the DWR as the largeparticle tail of the PSD is truncated. There is almost no impact of changing μ_{p} on the simulated observations.

Increasing cloud liquid results in an increase in the downrange DWR due to increased differential attenuation but has no impact on the polarimetric measurements.

Increasing f_{r} reduces the DWR and all of the polarimetric measurements, as the rimed particles are assumed to have random orientation.
It is noteworthy that the sign of the observation response to perturbations varies in different ways for the different parameters in X. This is an indication that this optimal estimation retrieval, as we have formulated it, is well determined and is also a requirement for reducing ambiguity, or cross talk, in the retrieved state. The information content and cross talk among parameters can also be evaluated by examining the averaging kernel matrix of the retrieved state. The averaging kernel provides a measure of influence of the observations on the retrieved state and is defined as
Values close to 1 indicate strong influence of measurements, and values close to 0 indicate that the retrieval is heavily influenced by the prior. In Fig. 9, the median, 10 %, and 90 % quantiles of the averaging kernel are plotted. These statistics were derived from many retrieved states spanning the cases we examined in Table 1. The parameter with the highest information content is the normalized intercept (${N}_{\mathrm{P}}^{*}$), which both the DWR and ϕ_{dp} are highly sensitive to (Fig. 8). The two parameters that describe the pristine component of the PSD (f_{p} and D_{p}) also have a similar median and 90 % quantile values to ${N}_{\mathrm{P}}^{*}$, due to their sensitivity to the polarimetric parameters, but a lower 10 % quantile, likely originating from situation where the f_{r} is high and there is little sensitivity of the polarimetric variables to f_{p} and D_{p}. Similarly, κ_{p} has a large range between the 10 % and 90 % quantiles, indicating that the information content of this parameter is state dependent, and high values of f_{p} and D_{p} are required to maximize the sensitivity of the polarimetric observations to this variable. Another parameter with a wide range of averaging kernel diagonal values is f_{r}, which requires low f_{p} and D_{p} values for it to be the primary driver of the polarimetric variables. Some of this state dependence of the information content is also reflected in the offdiagonal values, which indicate significant cross talk, i.e., a strong correlation in the posterior state vector, between the following groups of variables: ${N}_{\mathrm{P}}^{*}$, μ_{a}, f_{r}; and f_{p}, D_{p}, κ_{p}. These groups have similar Jacobians making it difficult to determine them independently. Finally, it is notable that a few of the variables (κ_{a}, μ_{p}, and cloud liquid) have very low averaging kernel values, indicating that the observations are not particularly sensitive to them. This is not surprising, since the aggregates do not show much Z_{dr} or K_{dp} sensitivity to κ except at the smallest sizes (Fig. 4), and the shape parameter (μ) of the pristine PSD is not going to influence the reflectivity observations (DWR and Z_{dr}) because the shape of the largeparticle tail is only important to these observations if D_{p} is close to its upper limit of 1. The sensitivity of Z_{dr} and K_{dp} to κ does depend on the aggregate shapes. The ARTS large plate aggregate used herein may not represent cases of more horizontally exaggerated aggregates where the assumed orientation will have a larger impact on the simulated polarimetric variables. The low averaging kernel values for cloud liquid are reflective of the result that it was rarely retrieved in significant quantities, and since it is treated logarithmically, only large values will induce large changes to the DWR. However, there were several RHI scans, particularly on 28 February, where some cloud liquid was necessary to explain high DWR values that could not be achieved by differential scattering alone.
The primary tool used for validation of the D3R retrievals is the PIP located at the MHS location. The D3R retrievals were matched to the PIP by averaging the retrieved quantities in a 600 m wide by 500 m tall box centered above MHS. The lower altitude limit of this box was placed 250 m above the surface to avoid ground clutter contamination along the radials used for averaging. To evaluate the impact of the dualfrequency and dualpolarization measurements, four retrieval experiments were conducted:

Kuonly: a singlefrequency retrieval, equivalent to prescribing a temperaturedependent Z−S relationship;

DWRonly: a dualfrequency retrieval without polarimetric information; similar information content to the GPM dualfrequency precipitation radar (DPR);

Kupol: a singlefrequency polarimetric retrieval using Z_{dr} and ϕ_{dp} at the Ku band only;

Allobs: a dualfrequency polarimetric retrieval using DWR and the Z_{dr} and ϕ_{dp} at both frequencies.
To assess the quality of the PIP–D3R matchups, the Kuband reflectivity was calculated directly from the PIPderived PSDs using Mie theory (spherical particles) with the PIPderived particle densities (Tokay et al., 2022). Although there will be some departures from Mie theory for nonspherical particles, at the Ku band these are relatively small (Kuo et al., 2016). The larger contribution to error is the various assumptions required to derive particle density from the PIP size and fall speed (Tokay et al., 2022). Using an ensemble approach to these assumptions, a range of reflectivities was obtained and compared to the D3Robserved reflectivity in the averaging box (top row of Fig. 10).
Some different tendencies are noted for each event. The 9 January case had a consistently higher PIPderived reflectivity than the D3R measurement, especially during the middle hours of the event. This was a lowechotop cold low case, and D3Rderived time–height profiles (Fig. 1) indicate significant echo growth (perhaps due to aggregation) at low levels, which may have continued in the clutter region. The 28 February case exhibits a similar bias between the PIPderived and D3Rmeasured reflectivity at the Ku band in the first 6 h, after which the D3R reflectivity comes within the lower range of PIP estimates. This was a deeper, warm low case, and the D3R profiles do not indicate any substantial reflectivity increase towards the surface. Instead, it is suspected that excess attenuation from wet snow (the wet bulb temperature was above 0 ^{∘}C until 04:00 UTC and the air temperature was above 0 ^{∘}C until 08:00 UTC) accumulated on the radome is responsible for these differences (note also the sharp increase in DWR during the same time period attributed to this factor in Sect. 2). The 7–8 March case exhibited the best agreement between PIPderived and D3Rmeasured reflectivity throughout the event. This was also a warm low case with deep echo tops but colder wet bulb temperatures (between −4 and −2 ^{∘}C during the event).
The validation statistics presented in this section are derived from matchups of the D3Rderived and PIPmeasured quantities listed in Table 4. Because the PIP measurements are taken every minute, whereas the D3R RHI scans were conducted every 5–6 min, an optimal lag was found by maximizing the correlation between the lagged D3Rmeasured and PIPderived Kuband reflectivity time series. This lag time was between 2–7 min, depending on the event, which is consistent with a fall speed slightly greater than 1 m s^{−1} to cover the distance from the center of the averaging box to the surface. Only retrievals where the D3Rmeasured Ku band reflectivity was within the range of PIP estimates were considered in the calculation of statistics to ensure that the PIP measurements are reasonably representative of the D3R observations.
5.1 Snowfall rate and water equivalent
The time series of snow water equivalent rate (S) for each event are shown in Fig. 11. The D3R substantially underestimates snowfall throughout the 9 January and 28 February events, which is not surprising since the PIPderived reflectivity significantly exceeds the D3R measurement for reasons discussed previously in this section. It is worth noting, however, that the DWRonly and Allobs retrievals are in good agreement with the Pluviomeasured S accumulation on 9 January, which would be consistent with aggregation processes in the lowest levels that increase reflectivity but do not increase S. The 7–8 March event shows better agreement with the PIP measurements. In this case, which had the best reflectivity match, the DWRonly retrieval overestimates S, whereas the Kuonly and Kupol retrievals underestimate S relative to the PIP. The retrieval using all of the observations is the closest match and the accumulated S falls within the range of PIP estimates for most of the event. We note from the DWR time series for this event in Fig. 10 that the DWR is just above 0 dB for much of this event which implies a smaller mean particle size and higher S for a given reflectivity. Meanwhile, the Kupol retrieval gives a slight reduction in S from the Kuonly retrieval, which is already biased low for this event. Compared to the Pluvio, the Allobs retrieval is biased high; however, after correcting the Pluvio data for wind (Milewska et al., 2019), this retrieval is in better agreement. However, these same wind corrections bias the 9 January event higher than the retrievals and in better agreement with the PIP.
Error statistics for volumetric snowfall rate (S_{V}) and S from all events, filtered for times when the D3Rmeasured reflectivity was within the PIPestimated range, are presented in Table 7. The bias is the overall fraction difference between the accumulated PIP and D3Rderived amounts. All methods underestimate the amounts, with the DWRonly method providing the closest match for both S_{V} and S. However, this appears to be the result of compensating biases on the 28 February and 7–8 March events, and the mean absolute error (MAE) is highest for this method. The Kupol method provides the best correlation for S_{V} and S even though it has the largest magnitude bias of all the experiments. The high correlation coefficients, particularly for the multiparameter methods, are indicative of a response of the radar measurements to the microphysical properties that determine snowfall rate, but the large biases indicate either a calibration bias in the observations, biased forward model (i.e., unrepresentative scattering properties), or both.
5.2 Mean particle size and density
The D3R retrieval provides an estimate of the PIPmeasured areaweighted mean particle volume diameter (D_{P}) that is consistent with the particle shapes used to generate the LUTs and can be compared directly to the PIP measurement. The time series of this parameter is shown for each event in the top row of Fig. 12. On 9 January 2018, all of the radar retrievals were biased low with respect to the PIP, which is consistent with the reflectivity bias we noted on this day. On 28 February, the PIP measured a rapid increase of D_{P} to a maximum at 06:00 UTC, followed by a decrease to a minimum around 09:00 UTC and another maximum at 15:00 UTC. None of the retrievals did a particularly good job of capturing the peak at 06:00 UTC, but all methods except the Kuonly retrieval captured the increase toward the second maximum, although again, the peak values were underestimated. On 7–8 March, both methods that used the DWR (DWRonly and Allobs) captured the temporal variability of D_{P} quite well but were biased low; this is consistent with a suspected low bias in the DWR for this event (mean cloudtop DWRs were slightly below zero; see Fig. 2).
The statistics of the D_{P} retrievals are presented in Table 8 and, as with the snowfall rate statistics, only consider observations where the D3Robserved reflectivity was within the range of calculated PIP values. All of the retrievals are biased low with respect to the PIP, with the Kupol retrieval coming the closest. This is counterintuitive, since this retrieval does not consider the DWR, which is the measurement most sensitive to D_{P}. However, the suspected low bias of the DWR on 7–8 March, which dominates the statistics due to its close match to the observed Ku reflectivity, contributes heavily here. The MAE is only slightly lower for the Kupol method than the Allobs method, despite the much more significant bias. Meanwhile, the correlation coefficient is largest for the two methods that incorporate the DWR, suggesting that the DWR is indeed informative regarding the particle size; this underscores the need for DWR to be carefully calibrated to avoid significant bias. Combined use of polarimetric and DWR information seems to at least partially alleviate these biases.
The retrieved effective particle density ρ_{P}, defined as the snow water equivalent rate divided by the volumetric snowfall rate, can also be directly compared to the PIP measurement (note that this is different from the effective density defined by Pettersen et al., 2020, which is the ratio of fall speeds of an observed snow particle and raindrop of the same equivalent diameter). To first order, ρ_{P} is inversely proportional to D_{P} because the unrimed aggregates' density decreases with size. However, changes in f_{r} and f_{p} also affect ρ_{P}, so it is worthwhile to evaluate these retrievals as well. The bottom row of Fig. 12 shows the time series of ρ_{P} for each event. In general, we find that when the retrieved D_{P} is biased high with respect to the PIP observation, ρ_{P} is biased low, and vice versa. It is interesting that on 28 February, all of the D3R methods are in tight agreement, whereas on 7–8 March, the DWRonly and Allobs retrievals are higher than the Kuonly and Kupol retrievals, due to the shift toward smaller (denser particles) inferred from the low DWR on this event. In this case, the Kuonly and Kupol methods are less biased, but the variability is better represented by the DWRusing retrievals. This is again borne out in the statistics (Table 8), where the Kuonly and Kupol methods have the lowest magnitude bias and MAE, and unlike the D_{P}, the Kuonly method has the best correlation (although the Allobs retrieval is the next best). This suggests that the size–density relationship in the scattering models we chose may not be representative of the particles observed during ICEPOP, or that more information (e.g., Wband reflectivity) is needed to constrain the density (e.g., Kneifel et al., 2015).
5.3 Bulk particle orientation and aspect ratio
The use of polarimetric measurements in the Kupol and Allobs retrievals has the most impact on the retrieved pristine fraction, ratio of pristine to aggregate mean particle size, and pristine population orientation distribution. All of these parameters combine to influence the areaweighted ellipse ($\stackrel{\mathrm{\u203e}}{{a}_{\mathrm{ell}}}$) and box ($\stackrel{\mathrm{\u203e}}{{a}_{\mathrm{box}}}$) aspect ratios. In Fig. 13, the influence of the Z_{dr} and K_{dp} measurements (Fig. 10) can be observed in these two retrievals, whereas the Kuonly and DWRonly experiments did not appreciably change these parameters. The high Z_{dr} values before 16:00 UTC on 9 January result in low aspect ratios at that time. There is not much of a trend in Z_{dr} on 28 February, due to the constant presence of large aggregates which dominate the Z_{dr} measurement, but there is a notable peak in K_{dp} around 09:00 UTC which corresponds to the minimum aspect ratio for this event. The 7–8 March event had the least variable polarimetric signatures, but a short peak in Z_{dr} around 18:00 UTC on 7 March corresponds to a drop in aspect ratio at that time.
The comparison of the retrieved aspect ratios to those derived from the PIP is inconsistent from event to event. There appears to be little variability in the PIP time series on 9 January for either measure of aspect ratio. There is a steady increase in $\stackrel{\mathrm{\u203e}}{{a}_{\mathrm{ell}}}$ between 09:00 and 12:00 UTC on 28 February, matching the retrieved behavior, although the PIP range is considerably smaller than the retrieved range. The 7–8 March event does not show any significant trend in the PIPmeasured aspect ratios, consistent with the low Z_{dr} variability during this event. The aspect ratio error statistics are given in Table 9. The Kupol and Allobs methods provide the lowest MAE and least bias; however, the best correlation comes from the Kuonly method (for $\stackrel{\mathrm{\u203e}}{{a}_{\mathrm{ell}}}$) and the DWRonly method (for $\stackrel{\mathrm{\u203e}}{{a}_{\mathrm{box}}}$). This is a surprising result, especially in light of the very small range of these retrieved values in Fig. 13. In these methods, the very limited aspect ratio variability is entirely driven by changes in PSD, with smaller mean particle sizes being associated with smaller aspect ratios. The polarimetric measurements add significant variation to the retrieved aspect ratio, but the correlation statistics suggest that the relationship between these measurement and PIPderived aspect ratio and orientation is tenuous at best. Helms et al. (2022) provide further information on the algorithms used to derive aspect ratio and orientation from the PIP; noting that motion blurring and compression lead to artificially high aspect ratios, particularly for small particles, and quantization artifacts that lead to a maximum aspect ratio of 0.6 for particles less than 0.5 mm in diameter. In future deployments, we expect to measure these properties more precisely as these algorithms improve, and colocated highresolution cameras (such as the MultiAngle Snowflake Camera instrument; Garrett et al., 2012) provide complementary information about selected individual particles, while the PIP, with its wider field of view, provides more information about the PSD and bulk snowfall properties.
This study describes an algorithm that makes use of both polarimetric and dualfrequency radar measurements to retrieve microphysical properties of falling snow, including snowfall rate (volumetric and water equivalent); ice water content; particle size distribution; the relative contribution of pristine, aggregate, and rimed species; and particle orientation distribution. The algorithm is flexible in that it can use as many or as few measurements as available. In this study, it is applied to the Ku and Kaband measurements of the NASA D3R radar obtained during the ICEPOP 2018 field campaign but can be applied to additional or different frequencies. This is possible because it makes use of the ARTS microwave singlescattering property database for oriented particles (Brath et al., 2020), which encompasses ADDA (Yurkin and Hoekstra, 2011) scattering calculations over a wide range of frequencies. This differentiates it from methods that use Tmatrix or Rayleigh–Gans approximations but constrains it to use the particle geometries that are available (at this time, only hexagonal plates and aggregates composed of these plates). More geometries are available for randomly oriented particles, but these cannot make use of the polarimetric information (although they are used in this study to represent the rimed particles).
The retrieval uses optimal estimation to solve for several parameters that describe the PSD, relative contribution of each species, and the orientation distribution along an entire radial simultaneously. This is necessary (versus a gatebygate approach) to account for the measurements sensitive to propagation effects (e.g., DWR and ϕ_{dp}). Examination of Jacobian matrices and averaging kernels shows that the DWR provides information regarding the characteristic particle size, and to a lesser extent, the rime fraction and shape parameter of the size distribution. The Z_{dr} measurements provide information regarding the mass fraction of pristine particles and their characteristic size and orientation distribution. Meanwhile, the ϕ_{dp} measurements are sensitive to most of the same measurements as Z_{dr} but are also sensitive to the overall particle concentration. Thus, by combining the dualfrequency and polarimetric measurements, some ambiguities can be resolved that should allow a better determination of the particle size distribution parameters and integrated quantities (e.g., ice water content, snowfall rate) than can be retrieved from singlefrequency polarimetric measurements or dualfrequency, singlepolarization measurements.
The D3R ICEPOP retrievals were validated using PIP and Pluvio measurements taken nearby at the May Hills ground site. The PIP measures the snow PSD directly (Pettersen et al., 2020) and several useful parameters can be derived directly from its measurements or indirectly with additional assumptions. These include the snowfall rate (volumetric and water equivalent), mean volumeweighted particle size, and effective density (Tokay et al., 2022), as well as parameters describing the mean aspect ratio and orientation distribution. We validated the retrieval during three events representing both warm and cold snow regimes (Kim et al., 2021). These events were chosen based upon availability of both PIP and D3R data, significant accumulation at the MHS location, and absence of any mixedphase precipitation which the algorithm does not account for. Four retrieval experiments were performed to evaluate the utility of different measurement combinations: Kuonly, DWRonly, Kupol, and Allobs. In terms of mean absolute error and correlation, the volumetric snowfall rate was best retrieved (r=0.95), followed closely by the snow water equivalent rate (r=0.92). The Kupol method had the highest correlation to these parameters, while the DWRonly and Allobs methods had the lowest magnitude bias. These methods that incorporated DWR also had the best correlation to particle size (r=0.74), although none of the methods retrieved density particularly well (r=0.46). The ability of the measurements to retrieve mean aspect ratio was also inconclusive, although the polarimetric methods (Kupol and Allobs) had reduced biases and MAE relative to the Kuonly and DWRonly methods. The significant biases in particle size and snowfall rate appeared to be related to biases in the measured DWR (positive on 28 February and negative on 7–8 March), emphasizing the need for accurate DWR measurements and frequent calibration (e.g., colocated measurements at a nonattenuating frequency such as S or C band). Notwithstanding these calibration biases, during the most wellbehaved event (7–8 March), where the PIPderived reflectivity was closest to the D3R measurement, the Allobs method provided the best snowfall accumulation and closely approximated the observed time series of snowfall rate and particle size.
The D3R is scheduled to be deployed in Storrs, Connecticut, USA, during the 2021–2022 and 2022–2023 winters as part of the NASAsponsored Investigation of Microphysics and Precipitation for Atlantic CoastThreatening Snowstorms (IMPACTS) field experiment. A similar deployment setup is planned with nearby PIP measurements. Additionally, the airborne Ku and Kaband HIWRAP radar will be available to evaluate the D3R calibration, and other lessons learned (e.g., video monitoring to observe snow accumulation on the radome, siting to avoid ground clutter) will be applied. Ongoing improvements to the PIP processing algorithms, particularly regarding the estimation of particle aspect ratio, will also be advantageous to further refine the algorithm described in this work. Availability of scattering databases for oriented particles with different geometries will facilitate running these retrievals as an ensemble to provide more robust posterior distributions of the retrieved parameters. Finally, the methodology can be expanded to accommodate liquid and melting particles, although scattering databases for the latter, particularly with the polarimetric parameters from oriented melting particles, are not yet mature enough for this application.
Faithfully simulating the observables from polarimetric radar requires considering the incident and scattered Stokes vectors; these vectors are related via (Adams and Bettenhausen, 2012)
where I, Q, U, and V are the elements of the Stokes vector, r is the distance from the sensor to the particle, Z_{lm} are the elements of the scattering or phase matrix, and the i and s subscripts indicate incidence and scattering, respectively.
To generate tables of the polarimetric, singlescattering properties, we transform the Stokes matrix elements to singlescattering properties more commonly used in radar meteorology, such as backscatter cross section at horizontal and vertical polarizations (${\mathit{\sigma}}_{\mathrm{hh}}^{\mathrm{b}}$ and ${\mathit{\sigma}}_{\mathrm{vv}}^{\mathrm{b}}$, respectively). Additionally, we include the complex, copolar conjugate product C_{hv} between the scattering amplitude matrix elements at horizontal and vertical polarization (S_{hh} and S_{vv}, respectively) that allow for the copolar correlation coefficient ρ_{hv} to be calculated. These variables are defined in terms of the phase matrix elements as (Ekelund et al., 2020)
All the expressions above define the phase matrix elements as in the backscatter direction, or opposite the incident direction, and the phase matrix elements have units of mm^{2}.
Expressions are given below for the radar reflectivity factor
in units of mm^{6} m^{−3} and copolar correlation coefficient
The propagation variables, K_{dp}, and the extinction cross sections (${\mathit{\sigma}}_{\mathrm{h}}^{\mathrm{e}}$ and ${\mathit{\sigma}}_{\mathrm{v}}^{\mathrm{e}}$) can be calculated from the corresponding extinction matrix of the particle. The oriented, azimuthally random uniform particles we use from the ARTS database have only three independent extinction matrix elements: K_{11}, K_{12}, and K_{34}. The propagation variables are thus defined as (Ekelund et al., 2020)
The extinction matrix elements have units of mm^{2} and K_{dp} has units of degree km^{−1}.
The D3R measurements that comprise the observation vector Y (Eq. 12) are forward modeled from the measured Kuband vertically polarized reflectivity ${Z}_{\mathrm{v},\mathrm{Ku}}^{\mathrm{m}}$^{1} and the state vector X. This combination of frequency and polarization was chosen because it is least prone to error in the attenuation correction. To simulate the D3R measurements, we must obtain both the backscattering properties of the ice particles as well as the propagation scattering properties (i.e., the propagation phase shifts and attenuation at each polarization) from the LUTs defined in Sect. 3. The components of Y are defined by the following equations that include the propagation effects along a radial:
where K_{dp}(f) is the specific differential phase at frequency f, ϕ_{sys} is the system differential phase upon transmission, and ${Z}_{\mathrm{p},\mathrm{f}}^{\mathrm{m}}$ is the attenuated reflectivity at polarization p and frequency f defined as
Here, ${Z}_{\mathrm{p},\mathrm{f}}^{\mathrm{e}}$ is the intrinsic effective reflectivity factor and k_{ext}(p,f) is the specific attenuation, which includes contributions from gases, cloud liquid water, and hydrometeors. The symbols p and f indicate the polarization and frequency, respectively.
DWR, Z_{dr}, and ϕ_{dp} are forward simulated from Eqs. (B1)–(B4) by integrating numerically along each radial, following a conceptually similar process to that of Grecu et al. (2011). First, at each range gate, the attenuation contributions from gases (determined by the closestintime sounding from DGW) and cloud liquid (part of the retrieval state vector) are calculated. Next, ${Z}_{\mathrm{v},\mathrm{Ku}}^{\mathrm{e}}$ is calculated by inserting the observed ${Z}_{\mathrm{v},\mathrm{Ku}}^{\mathrm{m}}$ and the calculated twoway pathintegrated attenuation from previous range gates into Eq. (B4). Then, ${Z}_{\mathrm{p},\mathrm{f}}^{\mathrm{e}}$, k_{ext}(p,f), and K_{dp}(f), along with the simulated PIP measurements in Table 4, are calculated from interpolated LUTs that have been combined for all the species according to the retrieval parameters and scaled to the observed ${Z}_{\mathrm{v},\mathrm{Ku}}^{\mathrm{m}}$. These quantities are then used to forward model the measurements that comprise Y following Eqs. (B1)–(B4).
The process of simulating the radar measurements from combined LUTs can be described by considering a LUT for a variable V selected from Table 4. First, the LUT dimensionality for each species is reduced by linear interpolation over the following parameters: the radial zenith angle θ, temperature at the altitude of the range gate, and speciesspecific μ and κ values interpolated from the nodes for these parameters. At this stage, the remaining LUT dimensions are D_{m} and frequency (for the radar variables). Next, the unrimed aggregate (V_{u}) and rimed (V_{r}) LUTs are combined into a merged “aggregate” LUT (V_{a}) by weighting each species according to the rime mass fraction interpolated to each range gate r:
Recall from Sect. 3 that the LUTs are normalized to contain a constant 1 g m^{−3} ice water content; there exists a corresponding ${N}_{\mathrm{P}\mathrm{0}}^{*}\left({D}_{\mathrm{m}}\right)$ value consistent with that water content. Because ${N}_{\mathrm{P}}^{*}$ is prescribed at each range gate from the interpolated retrieval parameter $\mathit{\delta}{N}_{\mathrm{P}}^{*}\left(r\right)$ (see definition in Table 5), the LUT for each species s is rescaled element wise to be consistent with this prescribed ${N}_{P}^{*}$:
The pristine and aggregate LUTs are then merged – individually for the scaled and unscaled LUTs – according to the interpolated retrieval parameters f_{p} (pristine mass fraction) and D_{p} (ratio of the pristine to aggregate D_{m}). To perform this merging, first the mean massweighted diameter dimension of the combined LUT (D_{m,c}) is calculated from the prescribed f_{p}, D_{p}, and aggregate D_{m,a}:
Next, the variables in the pristine LUTs (V_{p} and ${V}_{\mathrm{p}}^{x}$) are interpolated to this new set of D_{m,c} values and merged with the aggregate LUTs (V_{a} and ${V}_{\mathrm{a}}^{x}$) according to the prescribed mass fraction to create a combined LUT:
Now there exists a onedimensional LUT for each parameter (and frequency for the radar variables) that is indexed by D_{m} and is consistent with the prescribed PSD characteristics from the retrieval state vector. The retrieved value of D_{m}(r) is that which produces the attenuationcorrected (from Eq. B4 over prior range gates) ${Z}_{\mathrm{v},\mathrm{Ku}}^{\mathrm{c}}\left(r\right)$ from the ${N}_{\mathrm{P}}^{*}$scaled Z_{v,Ku} LUT. From this retrieved D_{m}(r), the ice water content W(r) (in g m^{−3}) is simply the scaling factor that is required to produce this ${Z}_{\mathrm{v},\mathrm{Ku}}^{\mathrm{c}}\left(r\right)$ from the unscaled LUT. Once D_{m}(r) and W(r) are determined, all of the remaining LUT parameters, including those needed to simulate the measurements in Y as well as the simulated PIP measurements used for validation, can be readily calculated by interpolating the LUTs to D_{m}(r) and scaling by W(r). These parameters are saved to the output data structure, and the propagation equations are iteratively integrated to repeat the process at the next range gate.
The D3R data used in this study (Chandrasekar, 2019) can be accessed at https://doi.org/10.5067/GPMGV/ICEPOP/D3R/DATA101. The PIP data (Bliven, 2020) can be accessed at https://doi.org/10.5067/GPMGV/ICEPOP/PIP/DATA101.
SJM led this research by developing the lookup tables and optimal estimation methodology, processing the radar data, and calculating validation statistics. RSS assisted with the generation of lookup tables, polarimetric components of the radar forward model, and assessment of forward model error. AT processed the PIP and Pluvio measurements that were used for validation. CNH performed additional processing of the PIP data to derive the bulk aspect ratio and orientation parameters.
At least one of the (co)authors is a member of the editorial board of Atmospheric Measurement Techniques. The peerreview process was guided by an independent editor, and the authors also have no other competing interests to declare.
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This article is part of the special issue “Winter weather research in complex terrain during ICEPOP 2018 (International Collaborative Experiments for PyeongChang 2018 Olympic and Paralympic winter games) (ACP/AMT/GMD interjournal SI)”. It is not associated with a conference.
Robert S. Schrom's and Charles N. Helms' work was supported by an appointment to the NASA Postdoctoral Program at NASA Goddard Space Flight Center, administered by Universities Space Research Association under contract with NASA. S. Joseph Munchak and Ali Tokay were supported by the Global Precipitation Measurement (GPM) Ground Validation program. The authors would also like to acknowledge many insightful discussions about the D3R data quality and calibration procedures with Venkatachalam Chandrasekhar at Colorado State University. The authors greatly appreciate the participants in the World Weather Research Programme Research Development Project and Forecast Demonstration Project, International Collaborative Experiments for PyeongChang 2018 Olympic and Paralympic winter games (ICEPOP 2018), hosted by Korea Meteorological Administration (KMA), and would like to acknowledge GyuWon Lee of Kyungpook National University, Daegu, South Korea, for stimulating ongoing scientific research from datasets collected during ICEPOP 2018.
This research has been supported by the National Aeronautics and Space Administration (Internal Scientist Funding Model Work Package: GPM Ground Validation).
This paper was edited by GyuWon Lee and reviewed by two anonymous referees.
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Reflectivity is defined in linear units in this appendix unless otherwise noted.
 Abstract
 Introduction
 Datasets
 Particle scattering properties
 Algorithm description
 Validation
 Conclusions
 Appendix A: Derivation of singlescattering properties
 Appendix B: Radar forward model
 Data availability
 Author contributions
 Competing interests
 Disclaimer
 Special issue statement
 Acknowledgements
 Financial support
 Review statement
 References
 Abstract
 Introduction
 Datasets
 Particle scattering properties
 Algorithm description
 Validation
 Conclusions
 Appendix A: Derivation of singlescattering properties
 Appendix B: Radar forward model
 Data availability
 Author contributions
 Competing interests
 Disclaimer
 Special issue statement
 Acknowledgements
 Financial support
 Review statement
 References