1. Introduction
There is an increasing trend of using multiple-wavelength radars in an effort to accurately detect and retrieve cloud and precipitation microphysical properties. The dual-wavelength radar is, however, the most popular among these applications. It makes use of two wavelengths, one of which is such that Rayleigh scattering dominates for most precipitation-sized particles and the other short enough so that non-Rayleigh (or Mie) scattering occurs in the presence of large hydrometeors. For precipitation measurements, the combination of Ku (or X) and Ka band is found in many air/spaceborne radar applications. This choice is dictated not only by the desire for microphysical information but by practical considerations such as frequency allocation (for space-based radars), the advantages of matched radar beams, and the desire for good spatial resolution within the constraints of antenna size. Attenuation caused by rain and mixed-phase hydrometeors, among other factors, is also important in choosing frequencies not only because it adds difficulty in estimating precipitation but because it often prevents the radar signal from penetrating the entire storm.
The dual-wavelength precipitation radar (DPR), operating at Ku and Ka bands aboard the Global Precipitation Measurement (GPM) core satellite, is the first dual-wavelength radar in space (Hou et al. 2014). An important goal of the DPR is to derive rain rate and raindrop size distribution (DSD), the latter of which is often modeled by an analytical function (Ulbrich 1983; Gorgucci et al. 2000, 2002; Bringi et al. 2002; Feingold and Levin 1986). Several dual-wavelength radar retrieval techniques have been developed that are based on the standard dual-wavelength technique that uses the differential frequency ratio (DFR; i.e., the difference of radar reflectivities between two wavelengths) to first infer the DSD parameters, then correct attenuation, and then derive the rain-rate profile gate by gate either stepping forward or backward along the radar beam (Meneghini et al. 1992, 1997; Mardiana et al. 2004; Liao and Meneghini 2005; Rose and Chandrasekar 2005; Seto et al. 2013; Seto and Iguchi 2015; Liao et al. 2016). To improve the robustness of the dual-wavelength retrieval, several optimal estimation methods have been recently proposed, including the GPM DPR (version 5) algorithm in which one or more adjustment factors are used to modify nominal relationships between the parameters of the radar and the precipitation (such as k–Z and R–Dm relations, where k, Z, R, and Dm are specific attenuation, radar reflectivity, rain rate, and mass-weighted diameter) to account for variations in the DSD in space and time (Grecu et al. 2011; Seto and Iguchi 2015). Adjustment factors at each gate/profile are determined by optimizing predefined cost functions that constrain the values of the reflectivities and path integral attenuations (PIA).
In the standard DFR-based dual-wavelength technique, the two unknown DSD parameters, related to the characteristic size and number density of raindrops, are estimated at each range gate. However, this is successful only if there exist unique solutions of the DSD parameters to the dual-wavelength radar equations. For the gamma DSD model, ambiguities in the estimation of Dm often occur, particularly for light-to-moderate rain rates. As there is no additional information aiding in selection of the solution, the standard dual-wavelength technique becomes problematic for rain retrievals. Retrieval errors caused by improper selection of the solution worsen as retrieval proceeds deep into the storm as a result of error propagation. It has been revealed in our study from a large collection of the measured DSD data from various NASA-sponsored field campaigns that negative DFR values occur frequently. Since these negative DFR values lead to ambiguities in the estimate of Dm and therefore R, and further that these errors are propagated in range because of attenuation correction, the overall performance of the dual-frequency retrieval will be adversely affected.
To circumvent the double-value problem of Dm, a modified DFR, denoted by DFR*, is introduced in this study. Instead of weighting equally the Ku- and Ka-band reflectivities, the DFR* weights the Ka-band reflectivity less than the Ku-band reflectivity so that the possibility of double solutions diminishes. Removal of the double values of Dm with use of DFR* aids in finding Dm unambiguously from the Dm–DFR* relations; the trade-off, however, is that the DFR* is no longer exclusively a function of Dm but a function of the number concentration parameter as well.
In this study, our focus is on examining the potential of DFR* for the retrieval of the vertical rain and DSD profiles by employing measured DSD data. For understanding the uncertainties in rain-rate estimation and also in gaining insight into ways to improve the algorithms, a physical evaluation of the standard (DFR based) and modified (DFR* based) dual-wavelength techniques is conducted by applying them to simulated vertical rain profiles. The hydrometeor profiles for use in our evaluation are constructed by using the measured DSD acquired from various storm systems.
In this paper, we begin with briefly describing the principle and issues associated with the standard dual-wavelength technique in section 2, followed by the discussion of the modified dual-wavelength technique in section 3. Described in section 4 are the methodology related to the radar profiling algorithms and their procedures for the retrieval of rain and DSD profiles using DFR and DFR* while evaluation of the performance of the profiling algorithms is made in section 5. Summary and remarks are given in section 6.
2. Standard dual-wavelength radar technique
Computational results of (left) Dm vs DFR and (right) zKu-normalized rain rate vs DFR as the gamma DSD model is assumed with μ values of 0, 3, 6, and 10.
Citation: Journal of Applied Meteorology and Climatology 58, 1; 10.1175/JAMC-D-18-0037.1
The axes of the plots in Fig. 2 are the same as in Fig. 1 but in this case, the results, displayed as 2D probability density functions (PDF), are derived from measured DSDs acquired during the Iowa Flood Studies (IFloodS), the Integrated Precipitation and Hydrology Experiment (IPHEx), and data from the NASA Wallops Flight Facility in Wallops Island, Virginia. For the sake of comparisons with model results, theoretical computations from the gamma DSD model with a fixed μ of 3 are given by the white curves. It is worth mentioning that use of the zKu-normalized rain rate, as in Figs. 1 and 2, is a convenient and effective way to directly infer rain rate from radar parameters. In particular, given a (positive) DFR and an assumed μ, the R/zKu is obtained from the curves in the figure; multiplication of the result by zKu then provides the rain-rate estimate. The term Nw is obtained from either Ku- or Ka-band reflectivities along with the derived Dm. For the scattering computations, the raindrops are prescribed as horizontally oriented oblate spheroids with axis ratios that follow the shape–size relations reported by Thurai et al. (2007). Symmetrical axes of the raindrops are aligned in the vertical direction. The T-matrix method is used to compute the scattering properties of single particles (Mishchenko and Travis 1998). It is important to point out that vast majority of the DSD measurements (nearly 70%), as found in Fig. 2, fall into the range where the DFR is negative, which suggests a serious problem for operational retrieval algorithms if the DFR–Dm relations are used. For the GPM DPR whose minimum detectable reflectivities are approximately 12 and 17 dB at Ku- and Ka-wavelengths, respectively, the percentage of the DSD data with the DFR less than 0 is around 37%.
The 2D probability density functions, as derived from the DSD measurements taken during several NASA field campaigns, between (left) Dm and DFR and (right) zKu-normalized rain rate vs DFR. Theoretical computations from the gamma DSD model of μ = 3 are also plotted as references.
Citation: Journal of Applied Meteorology and Climatology 58, 1; 10.1175/JAMC-D-18-0037.1
3. Modified dual-wavelength radar technique
Figure 3 illustrates the DFR*–Dm relation, derived from the DSD data, as γ takes on different values. The thick black solid lines are the means of the data taken at each point along the horizontal axis. Theoretical computations (blue curves) derived from the gamma DSD model with a fixed μ of 3 are plotted for Nw varying from 10 to 105 mm−1 m−3. For reference, the contours of ZKu (dBZ; red curves) are also given in the plots. Shown in the top-left panel is the case for the single Ku band (i.e., γ = 0), while the results from the standard dual-wavelength technique (γ = 1) are shown in the bottom-right panel. The cases where γ is equal to 0.5 and 0.7 are displayed in the top-right and bottom-left panels, respectively. Several interesting features are revealed in Fig. 3; in particular, Dm has a one-to-one relation with DFR* for all the cases shown except for the case where γ = 1. As expected, the DFR*–Dm relations depend on Nw if γ is not 1, and the degree of the dependence depends on γ. The strongest dependence of Nw on these relations occurs at γ = 0 (Ku band only), as shown from both the DSD-derived data points and the model computations. Variability in the DFR*–Dm with respect to Nw tends to decrease for Dm greater than 1 mm as γ increases. The DFR* at γ = 1, or the standard DFR, is independent of Nw. It is worth reiterating that one of the crucial differences between the DFR* (γ ≠ 1) and the standard DFR is that the former gives a unique solution to Dm for given Nw and μ, while the latter results in double values of Dm for large amount of the radar measurements. Similar behavior is also found in the rain-rate estimates, as depicted in Fig. 4. Unique solutions to Dm and rain rate from the DFR* are perhaps the most desirable features from the perspective of radar retrievals, but it comes with the cost of adding the variable Nw to the equations and leading to an underconstrained retrieval. As one of the goals of the spaceborne radar is to obtain hydrometeor profiles along the radar beam, the rest of the paper is focused on how the modified dual-wavelength technique described above is implemented in the profiling algorithms and how its performance compares with algorithms that rely on the standard dual-wavelength technique.
Scatterplots of Dm vs modified DFR as derived from the DSD data for the cases of the weighting factor γ of the modified DFR equal to 0, 0.5, 0.7, and 1. Colors of the data points depict Nw values that are derived from the same DSD data. The mean values of the data are given by the thick dark curves. Theoretical model computations, represented by blue and red curves, are provided as the DSD is modeled as gamma distributions with a fixed μ of 3 and Nw (blue curves) ranging from 10 to 105 mm−1 m−3. Contours of ZKu at the values from 0 to 40 dB (red curves) are plotted as well for reference.
Citation: Journal of Applied Meteorology and Climatology 58, 1; 10.1175/JAMC-D-18-0037.1
Scatterplots of ZKu-normalized rain rate vs modified DFR as derived from the DSD data for the cases of the weighting factor γ of the modified DFR equal to 0, 0.5, 0.7, and 1. Colors of the data points depict Nw values derived from the same DSD data. The mean values of the data are given by the thick dark curves. Theoretical model computations, represented by blue and red curves, are provided as the DSD is modeled as gamma distributions with a fixed μ of 3 and Nw ranging from 10 to 105 mm−1 m−3. Contours of ZKu at the values from 0 to 40 dB (red curves) are plotted as well for reference.
Citation: Journal of Applied Meteorology and Climatology 58, 1; 10.1175/JAMC-D-18-0037.1
4. Hydrometeor profile retrieval
Although the modified dual-wavelength method or DFR* shows potential to improve accuracy of radar estimates over the standard DFR in terms of the uncertainties associated with double solutions, a physical evaluation of the hydrometeor profiling retrieval is required not only for understanding the uncertainties but also in gaining insight into ways to improve the retrievals. To achieve this, the measured DSD data acquired from a variety of storm systems during the NASA field campaigns are used to construct realistic hydrometeor profiles. Using measured DSD spectra, both radar and hydrometeor’s parameters, which include true and measured radar reflectivities, attenuation, rain, and DSD characteristic parameters (Dm and Nw), can be computed. These DSD-calculated parameters then serve as truth. An evaluation of algorithm performances is actually conducted by comparing the radar estimates with the truth. Before showing the results on the DFR* profiling algorithm performance, we will first describe the radar and hydrometeor profiles simulated by using measured DSD and then detail the procedures of the DFR* profile algorithms.
a. DSD profiles
For simplicity but without losing generality, two types of vertical DSD profiles are created: One type is the vertically uniform DSD profile where a single DSD spectrum is used for entire column, and the second type is the nonuniform profile in which the DSD changes gate by gate along the column. The DSD data used in this study are primarily from measurements made by Particle Size Velocity (PARSIVEL) disdrometers during the IFloodS field experiment from 1 May to 15 June 2013, IPHEx from 1 May to 15 June 2014, and the observations at the NASA Wallops Flight Facility in Wallops Island, Virginia, from 2013 to 2014. The PARSIVEL disdrometers are used to measure rain DSD and fall velocities of particle sizes from 0.3 to 20 mm.
Shown in Fig. 5 is an example of the simulated Ku-band radar reflectivities (top panels) and rain rates (bottom panels) from a segment of 200 DSD vertically uniform (left column) and nonuniform (right column) profiles. The range resolution is 0.125 km, which is comparable to the case for the GPM DPR, with the rain height set at 5 km. To construct realistic nonuniform hydrometeor profiles, the time series DSD data are converted to vertical profiles by assigning 1-min-integration-time DSD data to the gates from the top to the surface consecutively. The mean values of the correlations of rain, Dm, and log10Nw over all simulated nonuniform profiles between the range gates considered to the surface gate are plotted in Fig. 6 for the cases in which rain rate is less than 5 mm h−1 (left), greater than 5 mm h−1 (center), and greater than zero (right), which approximately represent the stratiform, convective, and stratiform–convective combined rain systems. These correlations are found to be comparable to the results reported by Adirosi et al. (2016) based on a combined study of Micro Rain Radar (MRR) and DSD measurements.
(top) Ku-band radar reflectivities and (bottom) rain rate simulated from a segment of the DSD profiles generated from measured DSD data. (left) Homogeneously vertical DSD profiles; (right) nonuniform DSD profiles.
Citation: Journal of Applied Meteorology and Climatology 58, 1; 10.1175/JAMC-D-18-0037.1
Averaged spatial correlations of rain rate, Dm, and log10Nw for the nonuniform DSD profiles between radar range heights and surface. Results from three rain ranges, i.e., R < 5, R ≥ 5, and R > 0 mm h−1 (all rain), are provided for approximately representing stratiform, convective, and their combined systems.
Citation: Journal of Applied Meteorology and Climatology 58, 1; 10.1175/JAMC-D-18-0037.1
b. Modified DFR profiling algorithms
As shown in Fig. 3, Dm can be obtained from DFR* if Nw is known. Likewise, rain rate can be derived if DFR* is used along with ZKu, as shown in Fig. 4. These procedures for the retrieval of Dm and rain rate constitute the basis for the DFR*-based profiling algorithms. On the other hand, the Ku- and Ka-band radar signals undergo losses caused by attenuation of the signals out to the gate at which the radar measurements are made. Accurate attenuation correction is therefore important for the algorithm’s overall performance. For down-looking air/spaceborne radar, forward and backward recursive procedures are often used for attenuation correction. In the forward approach, the retrieval starts from storm top and then proceeds into storm gate by gate until reaching the surface or final clutter-free gate. Conversely, the backward approach begins at the surface with a known PIA that is usually estimated by the surface reference technique (SRT) or other independent measurements (such as radiometers), with the correction moving upward until reaching the storm top. Benefits and demerits of the forward and backward approaches depend on the types of error sources involved in the algorithms. Nevertheless, it is clear that uncertainties associated with PIA only affect the performance of the backward recursive method, while errors associated with attenuation corrections prior to rain, such as attenuation resulting from cloud water and the melting region, are directly related to the performance of the forward recursive method.
To retrieve rain and DSD profiles using DFR*, one simple way is to assign the same Nw to each range gate of one vertical profile as an initialization process. With a known Nw, Dm can be uniquely derived from the DFR* if a fixed μ is assumed; rain rate and attenuation coefficients can then be derived by using DFR* along with ZKu. Attenuation corrections to the measured radar reflectivity profiles are carried out consecutively gate-by-gate along the radar beam either in the forward or backward direction. Since the derived hydrometeor profiles depend on the value of Nw assumed, finding the proper Nw that leads to accurate DSD and rain profiles is a key to the retrieval procedure.
More detail on the optimization described above is given by the flowchart in Fig. 7, which shows the gate-by-gate progression of the retrieval from the rain top to the surface (forward approach). The procedure starts with equally spaced values of log10(Ñw,k) (k = 1, 2, …, K, where K = 100) in the range from 0 to 6, from which the radar and hydrometeor profiles are estimated. The procedure is repeated K times, and the final Nw is chosen as the value that maximizes (12). Once Nw is selected, profiles of Dm and rain rate are found from lookup tables (LUT) that express Dm as a function of Nw and DFR*(γ). Accumulated Ku- and Ka-band attenuations (AKu and AKa; dB) are computed from the specific attenuations (kKu and kKa; dB km−1) of previous gates. The backward approach is done in the same fashion. Unlike the forward approach where the retrieval starts at the rain top where the attenuation is assumed to be negligible, the backward approach begins at the gate near surface where the SRT-derived PIAs are used to estimate the attenuation-corrected reflectivities from the measured radar reflectivities. The backward retrievals are therefore dependent on the accuracy of the PIA estimate.
Flowchart of DFR* for retrieval of DSD parameters and rain rate. Outer-loop variable k goes from 1 to K to provide equal-spaced values of log10(Ñw,k) at each step, while inner-loop variable n, running from 1 to N, represents radar range gates at which the retrievals are carried out.
Citation: Journal of Applied Meteorology and Climatology 58, 1; 10.1175/JAMC-D-18-0037.1
5. Evaluation of DFR* algorithms
Assessments of the modified DFR performance in estimates of rain and Dm are conducted by comparing the estimated results with the true values that are obtained from direct computations from the simulated DSD profiles. Figures 8 and 9 depict, in terms of 2D probability density functions, the comparisons of Dm and rain rates derived from the modified DFR with γ = 0 and 0.7 (left and center columns) and the standard DFR (right columns) to the corresponding true values of Dm and R. The forward recursive approach, as described in Fig. 7, is applied to the nonuniform DSD profiles that are prescribed earlier for these retrievals. To track the statistical errors along the profiles, the comparisons are made at the rain top and the surface that correspond to the beginning and end of the profiles. An unbiased δPIA model with the standard deviation of 0.8 dB, which is similar to what has been found from the GPM DPR measurements over ocean, is used to characterize the PIA errors. The σ1, σ2, and σ3 in (13)–(15) are 3.45 log10(mm−1 m−3), 1.6 dB (2 times the standard deviation of δPIA) and 2 dBZ, respectively. The choice of these values seems to lead to reasonable and robust estimates after extensive tests. For the standard DFR method, when negative DFR occurs, the larger of the two Dm solutions is chosen. To be consistent with the GPM DPR, the DSD data are used only if the DSD-computed Ku-band reflectivities are greater than 12 dBZ and the Ka-band reflectivities are greater than 17 dBZ (DPR sensitivity thresholds). Comparisons of the results derived from the modified and standard DFR methods clearly indicate that the modified DFR delivers far more accurate estimates than does the standard DFR for both rain rate and Dm. The results show generally good correlations between the modified-DFR estimates and the true values despite a slight degradation in accuracy as the retrieval progresses down to the surface. This degradation is the result of accumulating errors in the attenuation correction for ranges deeper into the storm. The rain estimates are more accurate than the Dm estimates because retrieval of Dm is more sensitive to the DSD model assumed, such as selection of μ for the gamma distribution, than is the rain retrieval. It is noticed that there are only slight differences of R and Dm estimates between the γ values of 0 and 0.7. This is because the nonuniform DSD profiles presented in this study do not actually result in significantly large variations of Nw in the profiles. For the case of relatively highly uncorrelated DSD profile, in which the differences in the estimates between γ = 0 and γ = 0.7 are clearly noticeable; that is, the estimates at γ = 0.7 appear less variable than those at γ = 0 (not shown).
The 2D probability density functions of Dm estimated by the modified DFR with γ of (left) 0 and (center) 0.7 as well as (right) the standard DFR (γ = 1) with the true Dm for the range gate at (top) the rain top and (bottom) the surface. The one-to-one lines are given by the black solid curves, while the means and 2 times the standard deviations of the data are given by the white solid curves and vertical bars, respectively. For both the modified and standard DFR methods, the forward recursive approach is used and applied to the nonuniform DSD profiles. An unbiased statistical δPIA model with the standard deviation of 0.8 dB is assumed.
Citation: Journal of Applied Meteorology and Climatology 58, 1; 10.1175/JAMC-D-18-0037.1
As in Fig. 8, but for rain rate.
Citation: Journal of Applied Meteorology and Climatology 58, 1; 10.1175/JAMC-D-18-0037.1
As expected, larger uncertainties in the estimates of rain and Dm using the standard DFR take place when the double values occur, as shown in the top-right panels of Figs. 8 and 9. These errors, however, tend to decrease as Dm or the rain rate becomes large; that is, when the DFR–Dm relationship is single valued. The ambiguities associated with the double solutions not only adversely affect the retrievals of rain rate and Dm but also introduce errors into the attenuation corrections. This effect can be seen in the bottom-right panels of Figs. 8 and 9 where errors associated with the Ku- and Ka-band attenuation corrections become severe enough at the surface to strongly degrade the retrieval estimates, particularly estimates of Dm. The modified and standard DFR comparisons also have been made at the intermediate gates (not shown). These profiles at intermediate gates behave similarly to those in Figs. 8 and 9 and are bounded by the results estimated at the beginning and end of the profiles.
Figures 10 and 11 show the same retrieval results as in Figs. 8 and 9 but for the backward approach. The general behavior of the results obtained from the backward approach largely remains the same as that obtained from the forward approach. Like the forward approach, the modified DFR is superior to the standard DFR method. Because of a reversal of direction of the solution, the last gate of the backward method terminates at the rain top at which the path attenuations from radar are generally small. Unlike the forward method, the accuracy of the PIAs plays a crucial role for the overall performance of the backward procedure.
The 2D probability density functions of Dm estimated by the modified DFR with γ of (left) 0 and (center) 0.7 as well as (right) the standard DFR (γ = 1) with the true Dm for a range gate at (top) the rain top and (bottom) the surface. The one-to-one lines are given by the black solid curves, while the means and 2 times the standard deviations of the data are given by the white solid curves and vertical bars, respectively. The backward recursive approach is applied in all cases to the nonuniform DSD profiles. Unbiased statistical PIA and δPIA models with standard deviations of 2 and 0.8 dB are assumed, respectively.
Citation: Journal of Applied Meteorology and Climatology 58, 1; 10.1175/JAMC-D-18-0037.1
As in Fig. 10, but for rain rate.
Citation: Journal of Applied Meteorology and Climatology 58, 1; 10.1175/JAMC-D-18-0037.1
For the results shown in Figs. 8–11, the DSDs along the vertical column are taken to be nonuniform, that is, partially correlated along the column. If the DSDs are assumed to be fully correlated instead of partially correlated, the rain rate and Dm estimates from the modified DFR tend to improve (not shown). This improvement is the result of the fact that fully correlated DSDs are consistent with the assumption that Nw is constant over the full profile. Changes in σ1, σ2, and σ3 values modify the details of the retrievals but not the general characteristics of the results. The functions p1, p2, and p3 of (13)–(15), serving as constraints on the solutions, play various roles in constraining the results, and their impact on the solutions change with rain intensity. For example, p1 constitutes a fairly strong constraint in finding the solutions throughout entire rain-rate range. Function p2 is a weak constraint for rain rates less than 1 mm h−1, a moderate constraint for rain rates between 1 and 10 mm h−1, and a strong constraint for rain rates greater than 10 mm h−1. Function p3 operates in a manner quite similar to p2 in the sense that it is a strong constraint at moderate and heavy rains, and is a slightly stronger constraint than p2 for light rain. Overall, the combination of p1, p2, and p3 provides stronger constraints to the retrieval at moderate–heavy rain rates than for light rain. This is because relative errors of δPIA [in p2 or (13)] become large for the cases where the PIAs are small, which are often associated with the light rain. The Ka-band radar does not provide much additional information if the rain rate is light. In these cases, the backscattering at both Ku and Ka band is primarily Rayleigh and the second frequency adds little additional information.
6. Summary and remarks
Development of a realistic and physically based framework to evaluate dual-wavelength radar techniques from space is important in assessing the accuracy of current and future retrieval algorithms for the dual-wavelength algorithms. In this study we employ measured DSD data from various NASA-sponsored field campaigns to construct vertical DSD profiles. With these profiles, the measured and attenuation-corrected radar reflectivity factors are simulated along with the corresponding rain rate and DSD parameters. The simulated radar reflectivities and hydrometeor parameters, because they are directly computed from the measured DSD, are considered as truth. Evaluation of the retrieval algorithms is conducted by comparing the hydrometeor parameters derived from the algorithms that use the DSD-simulated measured reflectivities as inputs with those directly computed from the DSD profiles. For the radar simulations the effect of radar random error and calibration bias are not taken into account.
The standard dual-wavelength technique, which relies on DFR–Dm relations, has long been considered as a possible means to derive rain and DSD parameters. However, for the Ku and Ka combination, negative values of DFR lead to ambiguities in the estimation of Dm that can result in large uncertainties in the rain and DSD estimates. If the forward approach is employed, these uncertainties grow larger as the retrieval proceeds deep into the storm as errors are accumulated along the retrieval direction. In an attempt to circumvent the ambiguities in the Dm estimation that the standard dual-wavelength method faces, we introduce a modified version of the standard technique, that is, the modified dual-wavelength technique, which partially weights the Ka-band measurements in the radar differential-frequency-ratio equations. The modified DFR or DFR* can be viewed as the generalized version of the DFR, from which the standard DFR and the single Ku-band equations are the extreme cases with weighting factors of 1 and 0, respectively.
It has been shown that the modified DFR has a unique relation with Dm if the weighting factor (or γ) is less than 0.8 for a given Nw. Unlike the standard DFR (where γ = 1), the DFR*–Dm relations are dependent on Nw. The degree to which the DFR* depends on Nw depends in turn on the value of γ. Generally speaking, dependence of the DFR*–Dm on Nw tends to become weaker as γ increases. From the perspective of radar retrieval, the strategy is to choose as large a γ as possible as long as a unique solution of Dm is achievable. However, a choice of γ near 1 leads to an abrupt transition (nonlinear behavior) in the DFR*–Dm relations in the range where Dm is less than 1, causing unstable solutions at small values of Dm. Appropriate choice of γ will also depend on some other factors, such as nonuniformity of hydrometeor profiles and model assumptions.
To apply the modified DFR for the retrieval of rain profiles, following the work of Seto et al. (2016), an optimal profiling retrieval algorithm is first described and then evaluated using the simulated DSD profiles. This is a recursive procedure that infers hydrometeor parameters and makes attenuation correction gate-by-gate along either forward (storm top, toward the surface) or backward (surface, toward the storm top) along the radar beam. As our primary focus is on examining the mechanics of rain retrieval techniques and assessing their retrieval errors, the uncertainties associated with cloud water and melting hydrometeors that often appear atop the rain as well as the issues related to nonuniform beamfilling (NUBF) and multiple scattering are excluded in our evaluation. As such, the attenuation at the first gate for the forward approach (rain top), the gate where the retrieval starts, is assumed to be zero. The PIA is used as the attenuation of the first gate for the backward approach, which is the range gate closest to the surface. The DFR*-based retrievals are carried out K times for a given DSD profile in correspondence with the K equally spaced initialized log10(Nw). The final Nw is then determined as that value that maximizes the product of three probability functions that minimize the differences between the radar measurements and the estimates. Once Nw is fixed, the range-profiled rain rate and Dm are derived.
It should be pointed out that constant Nw along the vertical profile is an approximation, which is true only if the vertical DSD is uniform. For nonuniform DSD profiles, such as the cases studied in the paper, Nw varies from gate to gate. The constant Nw approximation leads to errors, but these errors are limited by the fact that the natural vertical variations in the DSD are not too strong and the fact that the retrieved Nw will be close to a weighted mean of the actual Nw values along the path.
Comparisons of the estimates of the rain rate and Dm obtained from the modified and standard DFR to the true values indicate that the DFR*-based profiling retrievals are superior to those from the standard DFR. This is true for both forward and backward approaches. As mentioned earlier, attenuations caused by melting layer and cloud water are not included in our evaluation framework, and errors associated with these attenuations are not taken into account. These errors, if considered, will lead to performance degradation for the forward approach but not for the backward method. The backward approach, however, depends on the accuracy of PIA. Highly unreliable PIA will degrade its performance. It is necessary to point out that for the snow retrieval in which the DFR usually has a unique relation to Dm from the Ku- and Ka-band radar, the standard technique is often advantageous over other approaches because it fully accounts for the particle size distribution (PSD) variations in space and time. This is probably true for other frequency pairs that exhibit either a unique DFR–Dm relation or much-reduced region of DFR where the double solutions occur.
The benefits of introducing the DFR* are twofold: varying γ from 0 to 1 coincides with a transition from the single-frequency case to the full dual-frequency case; the double-valued solutions obtained when γ = 1 can be mitigated by decreasing γ. Although the DFR* with γ < 1 introduces dependency on Nw, it facilitates the use of optimal estimation methods similar to that used in the DPR operational retrieval method. Seeking the most reliable and accurate optimization method, though not the main purpose of this study, is important and will therefore be pursued in future studies.
Acknowledgments
This work is supported by Dr. R. Kakar of NASA Headquarters under NASA’s Precipitation Measurement Mission (PMM) Grant NNH15ZDA001N-PMM. The authors also wish to thank Dr. Ali Tokay from University of Maryland, Baltimore County, for providing DSD data.
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