2020 年 98 巻 6 号 p. 1129-1146
Dual-frequency Ku/Ka-band radar retrievals of snow parameters such as liquid-equivalent snowfall rate (R) and mass-weighted diameter (Dm) have two principal errors, namely, the differences between the assumed particle size distribution (PSD) model from the actual PSD and inadequacies in characterizing the single-scattering properties of snowflakes. Regarding the first issue, this study, based on radar simulations from a large amount of observed PSD data, shows that there exist relatively high correlations between the estimated snow parameters and their true values derived directly from the measured PSD. For PSD data with R greater than 0.1 mm h−1, a gamma PSD model with a fixed shape factor (µ) equal to 0 (or exponential distribution) provides the best estimates of R and Dm. This is despite negative biases of up to −15 % in R and underestimates and overestimates in Dm for small and large Dm, respectively. The µ = 0 assumption, however, produces relatively poor estimates of normalized intercepts of a gamma PSD (Nw), whereas the best estimates are obtained when µ is considered either 3 or 6. However, the use of an inappropriate scattering table increases the errors in snow retrieval. Simple evaluations are made for cases where the scattering databases used for the algorithm input differ from that used for retrieval. The mismatched scattering databases alone could cause at least 30–50 % changes in the estimates of snow water content (SWC) and R and could affect the retrievals of Dm and Nw and their dependence on µ.
Dual-frequency radars have been increasingly used for detecting and retrieving cloud and precipitation, such as the Dual-frequency Precipitation Radar (DPR) aboard the Global Precipitation Measurement (GPM) core satellite (Hou et al. 2014; Skofronick-Jackson et al. 2018; Radhakrishna et al. 2020; Yamaji et al. 2020), the NASA High-altitude Imaging Wind and Rain Airborne Profiler (Li et al. 2016), and the NASA JPL Airborne Second/Third Generation Precipitation Radar (APR-2/APR-3) (Im 2003; Sadowy et al. 2003). The general principle is that the use of two wavelengths provides information on the characteristic size of the hydrometeor distribution. One wavelength is such that Rayleigh–Gans or Rayleigh scattering (hereafter, both are referred to as Rayleigh scattering for convenience) dominates for most snow precipitatingsized particles, and the other has a wavelength short enough so that non-Rayleigh (or Mie) scattering occurs in the presence of large hydrometeors.
Many studies have been conducted to explore the capabilities of dual-frequency radar in the retrieval of precipitating snow parameters from the ground (Matrosov 1998; Szyrmer and Zawadzki 2014a, b) and air/spaceborne radar measurements (Meneghini et al. 1992, 1997; Meneghini and Kumagai 1994; Liao and Meneghini 2005; Liao et al. 2008, 2016; Heymsfield et al. 2005; Wang et al. 2005; Matrosov et al. 2005; Grecu et al. 2018). This study evaluates the performance of the standard technique, which uses the differential frequency ratio (DFR), defined as the difference of radar reflectivity between two frequencies to estimate snow microphysical properties and the associated bulk parameters for the GPM DPR operating at the Ku- and Ka-bands. Although the DFR-based technique is effective in obtaining snow properties, its retrieval accuracy depends on model assumptions, including the parameterization of the particle size distribution (PSD), empirical mass–size relation linking the observed geometrical size of a particle to its mass, and radar scattering model. The complex nature of snow-flakes, such as shape and structure, and the inability of the modeled PSD to represent actual snow spectra lead to errors in the estimates of snow parameters. Additionally, uncertainties associated with the scattering computations of snowflakes also affect the accuracy of the DPR snow retrieval. Therefore, understanding the uncertainties of snow precipitation estimation that depend on PSD parameterizations and scattering models of individual particles is critical in evaluating the overall performance of dual-frequency retrieval techniques. Furthermore, separating the uncertainties associated with the PSD models and scattering models/databases and their respective contributions to overall uncertainties are useful for gaining insight into how to improve the retrieval methods.
Snow PSD is usually modeled as a gamma distribution with two or three free parameters depending on whether its shape factor (µ) is fixed or a function of the mass-weighted diameter (Dm). In this study, we assess the uncertainties in snow estimates arising from PSD parameterization and mass-size relation by employing the measured PSD data. The snow mass spectra, which can be converted from the measured PSD using an empirical density–size relation, are used to obtain the PSD parameters, such as the mass-weighted diameter and the normalized intercept of a gamma PSD (Nw), and snow bulk parameters, such as the snow water content (SWC) and liquid-equivalent snowfall rate (R) if the measured fall velocity is used. By coupling the measured PSD with the particle scattering model, the measured radar parameters can be computed, which are subsequently used as inputs to the standard dual-frequency algorithm. The retrieval accuracy is evaluated by comparing the radar estimates of Dm, Nw, SWC, and R with the same quantities directly computed from the PSD spectra (or truth).
To achieve the goal of this study, a large amount of PSD data will be used, including measurements of snow PSD and fall velocity acquired from the Snow Video Imager/Particle Image Probe (SVI/PIP) (Newman et al. 2009) during the winter of 2014 at the NASA Wallops Flight Facility site in Wallops Island, Virginia. The scattering databases of snow particles developed at the NASA Goddard Space Flight Center (GSFC) (Kuo et al. 2016) and Florida State University (Liu 2008; Nowell et al. 2013) are used to characterize radar scattering properties.
This article is divided into four sections. Section 2 describes the method and its procedures for the assessment of dual-wavelength snow retrieval, and an analysis of the uncertainties associated with the PSD model assumption and scattering databases are provided in Section 3, followed by remarks in Section 4.
Figure 1 schematically illustrates the procedures for assessing dual-frequency snow retrieval. The left-hand block (enclosed by a red dashed line) depicts forward computations simulating radar reflectivity factors at both frequencies and snow characteristic parameters such as Dm, Nw, SWC, and R using PSD measurement data coupling with an empirical mass–size relation. Snow parameters directly derived from the PSD measurements are considered as the true values. The right-hand block displays retrieval procedures that start with an assumed PSD model. It then uses the pre-computed retrieval look-up tables that relate radar measurement reflectivity factors to snow parameters. The singlescattering database (shown in the top-middle) used for producing both the measured radar parameters and retrieval look-up tables provides the backscattering and extinction cross-sections of individual snow particles. The snow parameters, retrieved by taking the PSD-simulated radar parameters as input, are compared with the same quantities computed directly from the same PSD data generating the radar parameters.
Flowchart illustrating the retrieval error assessment procedures. The left column enclosed by a dashed-line rectangle represents the forward computations. The computations start from the measured PSD when coupling with a mass–size (m–D) relation to obtaining true snow liquid-equivalent parameters (Dm, Nw, SWC, and R) and radar parameters such as Z(Ku), Z(Ka), and DFR when coupling with the single-scattering database. The dashed line-enclosed right column shows the retrieval steps taking on the radar reflectivity computed by the forward model (left column) as the input to the integral retrieval look-up table (algorithm) calculated from an assumed PSD model and the same single-scattering database as used in the forward model. Comparisons between the estimated and true snow parameters are made for the assessment of retrieval errors.
The degree to which the retrieved and true snow parameters (directly computed from the measured PSD) agree offers a measure of retrieval accuracy resulting from the PSD model assumed in the algorithm. Notably, the principal difference in the PSDs between the left- and right-hand blocks is that the former is the observed PSD or actual particle size spectrum whereas the latter is the PSD model assumed in the retrieval equations. Because discrepancies between the actual and assumed PSDs would lead to errors in the snow estimates, selecting an appropriate PSD model is one of the primary ways to improve overall retrieval accuracy. The following subsections are devoted to discussions on the single-scattering database, measured PSD-based forward computations, and retrieval method.
2.1 Single-scattering databaseThe scattering properties of individual snowflakes at radar frequencies provide the basis for estimating snow parameters. The computation of the single scattering of snowflakes depends on their detailed structure, shape, and orientations relative to the incident electromagnetic field. Affected by atmospheric conditions and storm system types, snow appears complex in both shape and structure. Therefore, the accurate modeling of snowflakes is challenging for computing snow-scattering characteristics. Using recently developed high-performance computing systems, the single-scattering database from an enormous collection of crystals and aggregates generated using a 3D growth model has recently become available (Kuo et al. 2016). The computations are based on the discrete dipole approximation (DDA) numerical method to retrieve the scattered field (Draine and Flatau 1994). The basic elements comprising snow aggregates are pristine ice crystals. This database was developed at the NASA/GSFC, hereafter called the GSFC scattering database or table. Using the same numerical approach, i.e., DDA, the scattering properties from the aggregates formed by a collection of bullet rosettes have been created and stored at Florida State University (FSU) (Liu 2008; Nowell et al. 2013), hereafter called the FSU database. In both the GSFC and FSU databases, the particles are assumed to be randomly oriented in space.
Figure 2 shows the results of the backscattering (top row) and extinction (bottom row) coefficients, defined as backscattering and extinction cross-sections normalized by liquid-equivalent sphere cross-sections, respectively. These are obtained from the GSFC (blue dots) and FSU (red filled-circles) databases at the Ku- (left column) and Ka-bands (right column) as a function of the liquid-equivalent diameter. The heavy blue lines represent the mean values in 0.02-mm size bins from the GSFC database, and the vertical blue bars represent the ±1 standard deviation from the mean. For reference, Fig. 2 also gives the results from the simple randomly oriented spheroidal model with an aspect ratio of 0.6 and constant mass densities of 0.05, 0.1, and 0.2 g cm−3 (Liao et al. 2013), computed using the T-matrix method (Mishchenko and Travis 1998). Notably, the masses of the particles from the models (GSFC, FSU, and spheroid) are preserved for a given liquid-equivalent diameter, but for a fixed mass, the individual particle shape, structure, and density will be different. As anticipated, less variation in the back-scattering and extinction coefficients is found at the Ku-band from the different particle models because the scattering from these particles is approximately in the Rayleigh regime, where the scattering properties depend primarily on the particle mass rather than the detailed structure and shape. By contrast, the scattering results at the Ka-band show a greater degree of variation, especially at liquid-equivalent diameters greater than 1 mm. This occurs because the scattering from relatively large particles enters the non-Rayleigh (or Mie) regime, where the shapes and particle structure matter. Compared with the extinction results, the backscattering coefficients reveal relatively large fluctuations not only from model to model but also within the GSFC database, which includes aggregates of various types of pristine crystals. Attenuation due to snow, as computed from the extinction cross-sections, is small at the Ku- and Ka-bands, and therefore, it is regularly ignored.
Backscattering (top row) and extinction (bottom row) coefficients of snow obtained from the GSFC (green dot) and FSU (red filled-circle) scattering databases as a function of the liquid-equivalent diameter at the Ku- (left column) and Ka-bands (right column). Also plotted are the mean (solid blue curve) and two-time standard deviation (vertical blue bar) of the GSFC database. For reference, the scattering results computed from the randomly oriented oblate-spheroidal models with an aspect ratio of 0.6 are provided for densities of 0.05, 0.1, and 0.2 g cm−3, denoted by dotted, dashed, and dotted-dashed curves, respectively.
Notably, because of the significant increase in computing power and memory required and convergence issues for electrically large particles, the largest liquide-quivalent diameters of the scattering data from both the GSFC and FSU databases are limited to be approximately 3 mm. Although this particle size range covers most snow events, it could lead to truncation errors in estimates of heavy snowstorms in which particle sizes regularly exceed the database limits. However, scattering computations from the spheroidal model using the T-matrix method (Mishchenko and Travis 1998) are efficient and apply to all particle sizes found in nature. Figure 2 shows that the FSU database follows well the spheroidal particle results with a density of 0.2 g cm−3. For the GSFC data, high variability in the backscattering results is found at the Ka-band, which could be due to the different pristine crystals that are used to construct the database. Despite this, the results are bounded between 0.05 and 0.2 g cm−3, suggesting a way to extend the mentioned databases to larger particle sizes.
2.2 Forward computations from PSD measurementsAmong the instruments used to measure PSD are the Snow Video Imager/Particle Image Probe (SVI/PIP), Two-Dimensional Video Disdrometer (2DVD), and Autonomous Parsivel Unit, all providing measurements of particle size spectra as a function of particle geometrical sizes. The measured PSD, denoted by Nm (DL) in mm−1 m−3, is given as a function of particle geometrical sizes, which are typically the largest diameters (DL in mm) of spheres circumscribing non-spherical snowflakes. To compute radar reflectivity and liquid-equivalent snow parameters, a conversion from particle size to mass (m) in grams is required. To obtain it an empirical density–size (or ρ–DL) relation is used. Most ρ–DL relations available in the literature use a power law, as in Eq. (1):
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(ρw is the mass density of water (g cm−3) and D (mm) is the liquid-equivalent diameter), so we have
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(mm2) is the backscattering cross-section as a function of DL. |Kw|2 is the dielectric factor, and by convention, it takes on a value of 0.93. In the scattering tables described in Section 2.1, the backscattering cross-section of snow is given as a function of the liquid-equivalent diameter D, which is referred to as σb (D). Note that either DL or D, which are related by (3), can be used to characterize the size/mass of a snow particle. We find (DL) in (8) by first obtaining D from DL using (3) and then search in the scattering database for σb corresponding to D. Therefore, (DL) is equal to σb (D) if DL and D are related by (3).
The PSD-simulated Ze (λ), taken as the radar data, is the input to the retrieval algorithm. Notably, snow attenuation, though correctable, is typically negligibly small for most Ku- and Ka-band spaceborne radar measurements, and therefore, it is not considered in this study. The PSD-derived snow parameters from (4)–(7) are regarded as the true values or truth serving as the benchmark in calculating retrieval accuracy.
2.3 Dual-frequency radar retrieval techniqueFrom the perspective of snow retrieval, describing the particle spectrum requires a PSD model. The three-parameter gamma distribution, however, is the most frequently adopted to represent the hydrometeor size distribution (Gorgucci et al. 2000, 2002; Bringi et al. 2002). The form of this distribution is expressed as
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Snow retrieval look-up tables in which SWC (top-left), R (top-right), and Nw (bottom-right) are normalized by the Ku-band radar reflectivity ZKu, and Dm (bottom-left) is expressed as a function of DFR (= 10 log10(ZKu/ZKa)). For these computations, the gamma PSD model with a fixed µ of 0 is assumed. The results from the GSFC and FSU scattering databases are denoted by the heavy blue and red solid curves, respectively. Also provided are the results from the spheroidal model with the snow densities ranging from 0.05 to 0.5 g cm−3 (thin black curves) for reference.
Snow retrieval look-up tables in which SWC (top-left), R (top-right), and Nw (bottom-right) are normalized by the Ku-band radar reflectivity ZKu, and Dm (bottom-left) is expressed as a function of DFR (= 10 log10(ZKu/ZKa)) computed from the GSFC (blue) and FSU (red) scattering databases for µ values of 0, 3, and 6.
Using (9)–(16) and applying the scattering databases prescribed earlier, the relations between the snow parameters and radar measurements can be simulated. In Fig. 3, the various normalized quantities are shown under the assumption of a gamma PSD with µ = 0. In these plots, the DFR is given along the abscissa, whereas SWC (top-left), R (top-right), and Nw (bottom-right) are normalized by Ze (Ku) or ZKu. The bottom-left panel shows the plots of Dm versus DFR. The results corresponding to the GSFC and FSU databases are depicted by the blue and red heavy-solid curves, respectively. Also shown are the results derived from the fixed-density spheroidal particle model with mass densities varying from 0.05 to 0.5 g cm−3 (thin black curves). The analysis of these plots reveals that the GSFC-based results lie between those corresponding to a randomly oriented spheroidal particle model with snow densities of 0.1 g cm−3 and 0.2 g cm−3, whereas the FSU results are close to those derived from spheroids with a density of 0.2 g cm−3. Note that the largest Dm from the GSFC and FSU results is limited to 1.5 mm. This is less than half of the largest diameters defined in these scattering databases to avoid possible truncation errors. Specifically, for the fixed-density spheroidal model, Dm can be as large as 4 mm, whereas the largest D used in the integral computations is set to 8 mm.
The results linking the snow parameters to dual-frequency radar measurements, as shown in Fig. 3, offer a means to infer snow properties. The basic retrieval procedures are described as follows: Starting with the measured Ku- and Ka-band radar reflectivity factors (ZKu and ZKa), we have the DFR (DFR = ZKu – ZKa), from which we find the corresponding values of SWC/ZKu, R/ZKu, and Nw/ZKu based on Fig. 3. Multiplying these ratios by ZKu yields values for SWC, R, and Nw. The estimates of Dm can be obtained directly from the DFR data.
The relationships between the radar measurements and snow parameters, as shown in Fig. 3, comprise retrieval look-up tables (LUTs), which can be repeated using different scattering databases or scattering models. Because the GSFC and FSU scattering databases are built on physically realistic crystal growth models, our study focuses on retrieval from these scattering databases. Note that compared to the simple spheroidal model, these scattering databases lack scattering results from larger particle sizes, limiting retrieval only to light-to-moderate snowfall rates. Figure 3 shows that the differences in the LUTs between the GSFC and FSU results tend to increase with an increase in DFR because the scattering characteristics of large particles depend more on particle shape and detailed structure than those of smaller particles. This is specifically true for the Ka-band. Therefore, snow estimates rely strongly on the choice of LUT.
In addition to the scattering database, the PSD model, specifically the µ value in the gamma PSD, affects the retrieval LUT. Figure 4 shows comparisons of the LUTs for three µ values of 0 (solid), 3 (dotted), and 6 (dashed). The blue curves depict the results from the GSFC scattering database, whereas the red curves are from FSU. Differences between these two sets of results are evident.
Retrieval uncertainties in the dual-frequency radar algorithm described above can be evaluated using the measured PSD data. In this study, we use the measurements of snow PSD taken from the SVI/PIP during the winter of 2014 at the NASA Wallops Flight Facility, located in Wallops Island in Virginia. The data set encompasses eight snow events, with the details listed in Table 1, including the start and end times of the snowfall event, mean temperature, and total accumulation of each event. Although the PIP measures the dimensions or sizes of the snowflakes and their fall velocities, it does not provide measurements of particle mass, prohibiting one from obtaining the liquid-equivalent PSD parameters and snow bulk properties directly from the measurements.
One way to circumvent this is to adopt a density–size empirical relation to derive particle mass from size, implying that the retrieval accuracy will depend on the density–size relation used. Given the variations in the ρ–DL relations published in the literature, this dependence complicates the assessment of retrieval. Therefore, it is necessary to examine the impact of the ρ–DL relation on retrieval. This is presented in detail in Section 3.1. As mentioned earlier, one of our primary goals is to evaluate the uncertainties associated with the PSD models to identify the most appropriate PSD for snow retrievals. To this end, retrieval uncertainties associated with the PSD model will be investigated in Section 3.2. Because of the highly complex nature of snowflakes, variations in the scattering properties are to be expected, which is reflected by the differences in the scattering properties between the GSFC and FSU scattering databases, as discussed earlier. To investigate the influence of the scattering databases on retrieval accuracy, a comparative study will be conducted in Section 3.3 to evaluate the retrieval accuracy from the GSFC and FSU scattering databases. For the studies mentioned in Sections 3.1–3.3, the errors arising from the single-scattering tables are not considered because the scattering database used for simulating the radar measurements from the observed PSD is the same as that used for constructing the retrieval LUTs. One expects that an imperfect scattering database would incur additional errors. Rigorously examining the scattering databases, though beyond the scope of this study, is a critical task in the overall improvement of snow estimates. Validating the scattering databases requires collocated measurements of the Ku- and Kaband radar and particle mass spectra. In this study, the retrievals will be conducted in Section 3.4 for the cases in which one scattering database (GSFC) is used for generating the radar measurements, whereas the other (FSU) is used for retrieval and vice versa. Such tests help in disclosing the extent of the uncertainties resulting from inaccurate scattering databases.
3.1 Impact of ρ–DL relations on retrieval accuracyFigure 5 shows comparisons of the snow parameters retrieved from the dual-frequency algorithm using the same quantities derived from the measured PSD spectra or truth when applying the density–size relation documented by Heymsfield et al. (2010) to derive particle mass spectra from the PSD measurements. The snow parameters involved in the comparisons include SWC, R, Dm, and Nw. The gamma PSD with a fixed µ = 0 is assumed for retrieval, and the GSFC scattering database is applied for retrieval. For comparison, the two-dimensional probability density functions (PDFs) from the data points of the estimated and true values are displayed (top row), whereas the bin-averaged mean (blue solid lines) and standard deviation (vertical bars extending one standard deviation above and below the mean) of the data are shown (bottom row). The results show relatively high correlations between the estimates and truth. An exception to this is Nw, showing more scatter and larger standard deviations than the others. To see how the ρ–DL relations alter the retrieval results, algorithm performance is evaluated from the three well-known density–size relations, i.e., Fabry and Szyrmer (1999), Brandes et al. (2007), and Heymsfield et al. (2010), hereafter called the Fabry, Brandes, and Heymsfield relations, respectively. The values (a, b) in (1) for the Fabry, Brandes, and Heymsfield relations are (0.15, −1), (0.178, −0.922), and (0.104, −0.95), respectively. The quantity, known as the relative error of the estimate, is used as a benchmark to quantify the uncertainties in the retrieval. Let xi and yi (i = 1, 2, …, n) stand for the i-th pair of data points representing the true and estimated quantities, respectively, where n is the total number of data points. The relative error is defined by
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Comparisons of the two-dimensional probability density functions (PDFs) and the means (blue curves) (respectively shown in the top and bottom panels) from the estimated and true snow parameters, including SWC (left column), R (second column from the left), Dm (third column from the left), and Nw (right column). The two-time error standard deviations (blue vertical bars) are also provided in the bottom panel. Correlations denoted by corr are given in the top panel, and one-to-one relations (black curves) are given in the bottom panel. The fixed µ = 0 gamma PSD model is assumed, and the GSFC scattering database is adopted for retrieval.
Relative errors of SWC (top-left), R (top-right), Dm (bottom-left), and Nw (bottom-right) estimated from three density–size relations as the fixed µ = 0 gamma PSD model is assumed, and the GSFC scattering database is adopted for retrieval.
A fixed µ gamma PSD is assumed in the dual-frequency retrieval algorithm, as described earlier. To answer the question of how the µ value is selected to achieve the best estimates, the uncertainties associated with the gamma PSD model from three µ values, in the form of fractional errors, are compared in Fig. 7. Shown in the figure are relative errors as large as 20–70 % at µ = 0 in the SWC estimates when the SWC is less than 0.03 g m−3 and 0–70 % relative errors at µ = 0 in the R estimates when R is less than 0.1 mm h−1. However, these errors become relatively small (between −30 % and 10 % when R < 0.1 mm h−1) when µ is set to either 3 or 6. Where SWC and R exceed 0.03 g m−3 and 0.1 mm h−1, respectively, the gamma PSD with µ = 0 produces the best overall estimate of SWC and R, with most of the errors less than 15 %. This is to be compared with values of µ set to 3 and 6 that produce errors greater than 20 %. Notably, the measured PSD data with SWC greater than 0.03 g m−3 roughly correspond to the data with R greater than 0.1 mm h−1. Given the GPM DPR detectability, where only precipitation rates of 0.1 mm h−1 or higher can be measured, the only relevant data are those with R over 0.1 mm h−1. Therefore, the gamma PSD model with a fixed µ = 0 yields the best estimates of SWC and R from the perspective of DPR retrieval.
Relative errors of SWC (top-left), R (top-right), Dm (bottom-left), and Nw (bottom-right) estimated from three different µ values as the Heymsfield ρ–DL is adopted, and the GSFC scattering database is used for retrieval.
The behavior of Dm and its dependence on µ differ from those of SWC and R. Specifically, for Dm greater than 0.8 mm, the choice of µ = 0 leads to more accurate estimates than when µ = 3 or 6. By contrast, Nw derived from µ = 0 is the worst among the three µ values. The results of Nw from µ = 3 and 6 are close to each other, with overall biases less than ±200 %. Note that the Heymsfield density–size relation is applied to the error analysis shown in Fig. 6. Other ρ–DL relations, such as Brandes and Fabry relations, lead to similar results (not shown) because the ρ–DL relations have a little impact on retrieval accuracy, as concluded in Section 3.1.
3.3 Dependence of retrieval accuracy on scattering databasesThe assessments shown in Figs. 4–7 were conducted using the GSFC scattering database. To evaluate how the retrieval accuracy changes if different scattering tables are used, we repeat the same procedure that produced the results in Figs. 4–7 by substituting the GSFC database with the FSU database. It is found that most of the results are similar to those derived from the GSFC database. This includes the weak dependence on the ρ–DL relations and similar behavior of the estimated R, Dm, and Nw, as shown in Fig. 8, regarding the µ values. For example, the smallest bias occurs when µ = 0 in the estimates of R and Dm if R is greater than 0.1 mm h−1 and Dm exceeds 0.8 mm. The Nw estimates are also consistent with those from the GSFC database, yielding the best agreement with the true values when µ = 3 and 6. The SWC estimates from the FSU database, however, are most accurate at µ = 3 rather than at µ = 0, as obtained from the GSFC results. Careful comparisons between Fig. 7 and Fig. 8 reveal that a change in the scattering database from the GSFC to the FSU positively shifts the relative errors in SWC, whereas changes in the estimates of R and Dm are small. For instance, the error in SWC estimated at µ = 0 jumps from ∼20 % to ∼50 % when the FSU scattering database replaces the GSFC table. The error in Nw increases going from the GSFC to the FSU database at µ = 0, but the errors remain nearly unchanged at other µ values.
Relative errors of SWC (top-left), R (top-right), Dm (bottom-left), and Nw (bottom-right) estimated from three different µ values as the Heymsfield ρ–DL is adopted, and the FSU scattering database is used for retrieval.
Although fully addressing the retrieval uncertainties regarding the accuracy of the scattering database is beyond the scope of this study, it is still worthwhile evaluating how and to what degree the retrieval accuracy is affected if an inappropriate retrieval LUT is used. Up to now, the retrieval errors have been assessed by excluding an error caused by an inappropriate retrieval LUT because the single-scattering table used for simulating the measurements of radar reflectivity is the same as that used for generating the retrieval LUT. In other words, the two scattering databases, one for the simulation of radar measurements (serving as inputs to the algorithm) and the other for the retrieval procedure, are matched. Errors in retrieval should increase if different scattering databases are applied for simulation and retrieval.
To understand the error caused by using unmatched scattering tables, a test is conducted where the GSFC database is used for generating measurements of the radar reflectivity and the FSU scattering database is used for retrieval and vice versa. Analogous to Fig. 5, Fig. 9 shows the PDFs (top panel) and means (bottom panel) of the data from the estimated and true snow parameters for the case in which the GSFC scattering database is used to generate radar measurements and the FSU scattering table is used for retrieval. For retrieval, a gamma PSD with µ = 0 is assumed. In contrast to the results shown in Fig. 5, where matched scattering tables are used, the results in Fig. 9 show the decreasing trend in correlations for the case of unmatched databases. The relative errors illustrated in Fig. 10 show biases of the estimates as a function of µ. Despite better estimates of SWC at µ = 0 than the matched scattering databases, possibly for the wrong reason, the SWC estimates at µ = 3 and 6 tend to depart more from the truth. Similar behavior can be seen in the estimates of R, Dm, and Nw with all three µ values evaluated. Roughly, the relative biases are within ±50 % in the estimates of SWC, R, and Dm. The biases of Nw are found to be much higher than the biases in the other quantities.
Comparisons of the two-dimensional probability density functions (PDF) and the means (blue curves) (respectively shown in the top and bottom panels) from the estimated and true snow parameters, including SWC (left column), R (second column from the left), Dm (third column from the left), and Nw (right column) for the case where the scattering database is inappropriately used for retrieval. The two-time error standard deviations (blue vertical bars) are also provided in the bottom panel. Correlations denoted by corr are given in the top panel, and one-to-one relations (black curves) are given in the bottom panel. The fixed µ = 0 gamma PSD model is assumed, the GSFC scattering database is used for producing the measurements of radar reflectivity (inputs to the algorithm), and the FSU database is used for the retrieval algorithm.
Relative errors of the estimates of SWC (top-left), R (top-right), Dm (bottom-left), and Nw (bottom-right) for three µ values as the GSFC scattering database is used to simulate the Ku- and Ka-band radar reflectivity, the inputs to the retrieval algorithm, and the FSU database is used for generating the retrieval look-up tables.
By switching the roles of the GSFC and FSU databases in the assessment to use the FSU database for creating the measured radar reflectivity and the GSFC database for retrieval, Fig. 11 provides the results of the relative errors of the snow estimates. Using the results in Fig. 10 as a benchmark, the errors in SWC and R move upward so that the SWC and R have the smallest errors at µ = 3 and µ = 6, respectively. However, the SWC and R in Fig. 10 have the smallest error at µ = 0. Furthermore, the shapes of the relative error curves in Dm and Nw are altered, compared to Fig. 10, and µ equal to either 3 or 6 yields the best estimate.
Relative errors of the estimates of SWC (top-left), R (top-right), Dm (bottom-left), and Nw (bottom-right) for three µ values as the FSU scattering database is used to simulate the Ku- and Ka-band radar reflectivity, the inputs to the retrieval algorithm, and the GSFC database is used for generating the retrieval look-up tables.
The tests conducted in Figs. 9–11 show that there are at least 30–50 % changes in the estimates of SWC and R resulting exclusively from unmatched scattering databases. The retrieval errors depend not only on µ but also on the scattering tables selected for the forward simulation and retrieval procedure. Inaccurate scattering tables also affect the retrievals of Dm and Nw and their dependence on the choice of µ. Regardless of how the scattering databases are assigned in the tests, the gamma PSD with µ = 6 yields the best estimate of Nw. This is true for all results presented, including those from the retrievals using the matched scattering tables.
The complex and widely varying nature of snowflakes poses a challenge for the dual-frequency radar retrieval of the microphysical and bulk properties of snow. Inaccuracies in the modeling of the PSD and challenges in characterizing the single-scattering characteristics of snow constitute two principal error sources in radar retrieval. The studies presented in this paper address the extent of errors associated with the PSD model assumed in the retrieval and scattering database used. To assess the uncertainties in the estimates of snow parameters (SWC, R, Dm, and Nw), the PSD data taken from the PIP measurements are used to simulate the radar measurements that serve as the input to the dual-frequency radar retrieval algorithm. The snow parameters inferred from the algorithm are then compared with the same quantities derived directly from the measured PSD data. The degree to which the estimates agree with those obtained from the measured PSD provides a measure of retrieval accuracy. Although the primary focus of this study is to assess the retrieval uncertainties associated with the gamma PSD model with various fixed-µ values for two scattering databases, the impact on retrieval when an inappropriate scattering table is applied is investigated.
Two scattering databases, the GSFC and FSU databases, and the spheroid fixed-density model are compared in the space of the single-scattering parameters (backscattering and extinction coefficients vs. particle sizes) and for generating the relationships between snow parameters and dual-frequency radar measurements (retrieval LUTs). At the Ku-band, the GSFC and FSU databases and spheroidal model yield similar scattering parameters because Rayleigh scattering dominates at this wavelength. Under the Rayleigh scattering approximation, the scattering properties primarily depend on the particle mass, and therefore, they are independent of particle shape and structure. This, however, is not the case at the Ka-band, where the GSFC and FSU databases show differences when the liquid-equivalent diameter exceeds 1 mm. This is also true for the spheroidal model, where deviations between the scattering properties between the simple and more accurate simulated particles become evident in the non-Rayleigh scattering regime. The disparities between the GSFC and FSU databases are also reflected in the retrieval LUTs, which are more pronounced when DFR or Dm increases.
The PIP provides measurements of the particle size spectrum but not mass. To obtain particle mass, an empirical density–size relation is required. Recognizing the variability in the ρ–DL relations documented in the literature, the conversion from particle size to mass is not unique. To evaluate how various ρ–DL relations affect retrieval, three well-known ρ–DL relations are used. It is found that the retrieval accuracy is almost independent of the ρ–DL relation chosen largely because both simulated measurements of radar reflectivity (input to the algorithm) and retrieval LUTs are affected similarly by the ρ–DL relation. As such, the influence of ρ–DL variability is mostly canceled in the processes of forward simulation and inverse retrieval. The insensitivity of the density–size relations to retrieval accuracy ensures the validity of the retrieval assessment presented in this study because the ρ–DL relations are merely used for converting particle size to its mass. Notably, if the PSD measurements could provide both the particle geometrical size and liquid-equivalent diameter distribution, no ρ–DL relation would be needed. Moreover, if collocated radar measurements were available, it would provide a better assessment of retrieval errors and enable an assessment of the scattering databases.
An assessment of uncertainties in estimates of snow properties from a fixed µ gamma PSD model was first conducted for the case in which the same scattering database was used for both the forward simulation and retrieval procedure. Except for Nw, showing high standard deviations and biases in the estimates, high correlations between the estimates and true values for SWC, R, and Dm exist. For the comparisons of the data when R > 0.1 mm h−1 and the case where the same scattering table is used for both generating the measured reflectivity and forming integral LUTs (the case of matched scattering tables), a gamma PSD model with a fixed µ = 0 (or exponential distribution) provides the best estimates of R. It shows slightly negative biases up to −15 % in the data range from 0.1 to ∼20 mm h−1, but with overestimates in SWC from the GSFC and FSU scattering tables of ∼20 % and ∼50 %, respectively. The gamma PSD with µ = 0 also moderately overestimates/underestimates Dm when Dm is larger/smaller than ∼0.8 mm, whereas the PSD with µ = 3 and 6 mostly overestimates Dm to some extent despite the small Dm (Dm < 0.5 mm). By contrast, the best estimates of Nw are obtained when µ = 3 and 6. The estimates of Nw at µ = 0 are significantly worse than those from higher values of µ. Overall, the gamma PSD model with µ = 0 provides the best estimates of R and Dm but not necessarily SWC and Nw. The retrieval with µ = 3 and 6 shows a clear advantage in estimating Nw.
Errors in snow retrieval become larger if an inappropriate scattering table is used in the retrieval procedure. This study conducted simple tests for cases where the scattering databases used for the algorithm input differ from those used in retrieval. The mismatched scattering databases alone could cause at least 30–50 % changes in the estimates of SWC and R, affecting the retrievals of Dm and Nw and their dependence on µ. However, the gamma PSD with µ = 3 and 6 yields the best estimate of Nw.
Considering all of the uncertainties associated with the PSD models (µ from 0 to 6) and scattering tables (matched and unmatched), the relative errors are in the ranges of −40 % to 70 %, −50 % to 40 %, and −30 % to 50 % in the estimates of SWC, R, and Dm, respectively. The Nw estimates are mostly overestimated, with errors as large as 1000 % at µ = 0, reducing to a range of −100 % to 200 % for µ = 3 or 6. Though the PSD model assumed in the retrieval procedures is significant, the accuracy of the scattering computations of snow is crucial for the development and improvement of algorithm techniques from the perspective of radar dual-wavelength retrieval. Validation and evaluation of existing scattering databases and scattering models require collocated measurements from dual-wavelength radar and PSD and liquid-equivalent snow parameters such as SWC, R, or Dm. These measurements, though extremely useful, are still unavailable.
This work was supported by Dr. G. D. Skofronick-Jackson of NASA Headquarters under NASA's Precipitation Measurement Mission (PMM) Grant NNH 18ZDA001N-PMMST. The authors also wish to thank NASA PMM ground validation team for providing and processing SVI/PIP data and Dr. K-S. Kuo of University of Maryland and Dr. G. Liu for providing the scattering databases.
