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  • View in gallery

    Liquid portion of CSU-HIDRO algorithm (Cifelli et al. 2011) with Kdp in ° km−1, Zh in dBZ, and Zdr in dB.

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    Comparison between theoretically calculated rain rates (i.e., R) with min raindrop diameter (i.e., Dmin) of 0 and 0.25 mm (solid circles represent DSD samples, solid line indicates identical rain rates).

  • View in gallery

    Measured and modeled (a) average fall velocity and (b) mean axis ratio values as functions of equivalent raindrop diameters.

  • View in gallery

    Comparisons of rain-rate estimations from Case I and Case II for individual rain-rate estimators. Solid lines represent 0% error.

  • View in gallery

    Comparisons of rain-rate estimations from (a) Case I and (b) Case II for combined rainfall algorithm. (c) Rain-rate estimators used in (b) are specified using different symbols (see the legend). Solid lines represent 0% error.

  • View in gallery

    Error (%) in rain-rate estimation using combined fall velocity and axis ratio errors in the CSU-HIDRO radar rainfall retrieval algorithm; the solid lines represent 0% error.

  • View in gallery

    (left) Max values and (right) std dev of errors (%) in rain-rate estimations induced by raindrop fall velocity and axis ratio errors. For the ease of the reader, projections of the (a),(b) three-dimensional plots on the following planes are also provided: (c) ; (d) ; (e) (for values ranging from 0% to −10%); (f) (for values ranging from 0% to −10%); (g) (for values ranging from 0% to +10%); and (h) (for values ranging from 0% to +10%).

  • View in gallery

    Max values and std dev of errors (%) in rain-rate estimations induced by raindrop axis ratio errors and deviations of fall velocity (+10% for D ≤ 0.6 mm and −10% for D > 0.6 mm).

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Impacts of Raindrop Fall Velocity and Axis Ratio Errors on Dual-Polarization Radar Rainfall Estimation

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  • 1 Glenn Department of Civil Engineering, Clemson University, Clemson, South Carolina
  • | 2 Civil and Environmental Engineering, University of California, Los Angeles, Los Angeles, California
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Abstract

Motivated by the field observations of fall velocity and axis ratio deviations from predicted terminal velocity and equilibrium axis ratio values, the combined effects of raindrop fall velocity and axis ratio deviations on dual-polarization radar rainfall estimations were investigated. A radar rainfall retrieval algorithm [Colorado State University–Hydrometeor Identification Rainfall Optimization (CSU-HIDRO)] served as the test bed. Subsequent investigations determined that the available field measurements, which were very limited in scope, of the fall velocity and axis ratio deviations indicated rain-rate estimation errors of approximately 20%. Based on these findings, a sensitivity study was then performed using uncorrelated fall velocity and axis ratio deviations around the predicted values. Significant rain-rate estimation errors were observed for the realistic combinations of fall velocity and axis ratio deviations. It was shown that the maximum rain-rate estimation error can reach up to approximately 200% for combinations of fall velocity and axis ratio deviations (5000 drop size distribution samples were simulated for each combination) between −10% and +10% of the predicted values for each. The maximum standard deviation of errors was as great as 75% for the same combinations of fall velocity and axis ratio deviations. The authors found that use of dual-polarization radars to accurately estimate rainfall, during natural rain events, also requires a reanalysis of the parameterizations for raindrop fall velocity and axis ratio. These parameterizations should consider both the coupling between these two parameters and factors that may introduce any possible deviations of the predicted values of these parameters.

Corresponding author address: Dr. Firat Y. Testik, 110 Lowry Hall, Glenn Department of Civil Engineering, Clemson University, Clemson, SC 29634-0911. E-mail: ftestik@clemson.edu

Abstract

Motivated by the field observations of fall velocity and axis ratio deviations from predicted terminal velocity and equilibrium axis ratio values, the combined effects of raindrop fall velocity and axis ratio deviations on dual-polarization radar rainfall estimations were investigated. A radar rainfall retrieval algorithm [Colorado State University–Hydrometeor Identification Rainfall Optimization (CSU-HIDRO)] served as the test bed. Subsequent investigations determined that the available field measurements, which were very limited in scope, of the fall velocity and axis ratio deviations indicated rain-rate estimation errors of approximately 20%. Based on these findings, a sensitivity study was then performed using uncorrelated fall velocity and axis ratio deviations around the predicted values. Significant rain-rate estimation errors were observed for the realistic combinations of fall velocity and axis ratio deviations. It was shown that the maximum rain-rate estimation error can reach up to approximately 200% for combinations of fall velocity and axis ratio deviations (5000 drop size distribution samples were simulated for each combination) between −10% and +10% of the predicted values for each. The maximum standard deviation of errors was as great as 75% for the same combinations of fall velocity and axis ratio deviations. The authors found that use of dual-polarization radars to accurately estimate rainfall, during natural rain events, also requires a reanalysis of the parameterizations for raindrop fall velocity and axis ratio. These parameterizations should consider both the coupling between these two parameters and factors that may introduce any possible deviations of the predicted values of these parameters.

Corresponding author address: Dr. Firat Y. Testik, 110 Lowry Hall, Glenn Department of Civil Engineering, Clemson University, Clemson, SC 29634-0911. E-mail: ftestik@clemson.edu

1. Introduction

Accurate estimation of rainfall at high spatial–temporal resolutions over large geographic regions has been the subject of considerable research over the past few decades (Wilson and Brandes 1979; Austin 1987; Ryzhkov and Zrnić 1995; Bringi and Chandrasekar 2001; Krajewski and Smith 2002; Cifelli and Chandrasekar 2010), making weather radars indispensable. Accurate radar rainfall estimation is important for various hydrometeorological applications that rely on weather radars as their primary source of rainfall information, including water resources management, flood warning and control, flood and drought forecasting, urban hydrology studies, and climate change impact assessments (see, e.g., Sene 2009). The use of weather radars for estimating rainfall can be traced back to the late 1940s (Wexler and Swingle 1947; Marshall et al. 1947). With a single (horizontal) polarization, radar signals are transmitted and received in the same (horizontal) plane. A typical example of the single-polarization radar was the U.S. Weather Surveillance Radar-1988 Doppler (WSR-88D; Fulton et al. 1998) maintained by the National Weather Service (NWS). Using single polarization, rain rate R, defined as the amount of rainfall per unit time (e.g., mm h−1), is estimated as a function of the radar reflectivity factor. The dual-polarization radar was introduced by adding a polarimetric capability perpendicular to the single polarization of the conventional radar (Bringi and Chandrasekar 2001; Ryzhkov et al. 2005a,b). The NWS WSR-88D has recently been upgraded to dual-polarization capability (Saxion and Ice 2012). Compared with single polarization, dual polarization has significantly improved accuracy because of its capability to provide extensive information about the size, shape, orientation, and phase of the particles (Cifelli and Chandrasekar 2010). The three variables that are directly related to the dual-polarization radar rainfall estimation are the horizontal reflectivity factor , the differential reflectivity , and the specific differential phase . Different radar optimization algorithms (Ryzhkov et al. 2005b; Giangrande and Ryzhkov 2008; Cifelli et al. 2003, 2011) have been developed based on these three polarimetric radar variables.

In radar rainfall estimations, the rain rate is estimated using an empirical equation that was developed using a large set of simulated radar variables and the theoretically calculated rain rates. For parameterization development purposes, the rain rate in still air can be calculated theoretically as a function of the equivalent-volume drop diameter (defined as the diameter D of a sphere having the same volume as the raindrop), the drop size distribution (DSD), and the drop terminal velocity . Bringi and Chandrasekar (2001) discussed in great detail how to determine the radar observables and the corresponding rain rate. Briefly, the radar scattering amplitude and phase are first calculated using the T-matrix method (Mishchenko et al. 1996) based on the predicted oblate spheroidal shape (i.e., diameter and axis ratio) of the raindrop falling at terminal velocity and the probability density function of the drop orientation. The polarimetric variables are then computed by integrating the calculated scattering amplitude or phase over the DSD. Finally, radar rain-rate estimations are obtained using the empirical relationship between the polarimetric variables and the rain rate.

Extensive studies have been performed to model as a function of D (Atlas et al. 1973; Beard 1976; Atlas and Ulbrich 1977; Rogers 1989; Rogers and Yau 1989; Lhermitte 1990). At terminal velocity, the force balance among hydrostatic, surface tension, and aerodynamic forces was applied to derive equilibrium raindrop shapes (Green 1975; Beard and Chuang 1987). The relationships between the raindrop axis ratio α and the equivalent drop diameter were also developed using empirical equations fitted to either theoretical formulations or field and laboratory measurements (e.g., Pruppacher and Beard 1970; Green 1975; Beard and Chuang 1987; Andsager et al. 1999). In rain events, raindrop collisions and airflow turbulence may deform the possible equilibrium shape of a raindrop, causing them to oscillate (Beard et al. 1983; Beard 1984; Johnson and Beard 1984; Tokay and Beard 1996; Testik and Barros 2007). Drop oscillations with substantial axis ratio variations have been reported in a number of studies (Beard et al. 1989, 1991; Beard and Tokay 1991; Andsager et al. 1999; Tokay et al. 2000; Thurai and Bringi 2005). Based on field observations, Testik et al. (2006) reported nonlinear raindrop oscillations accompanied with lateral drifts (with drift speeds of around 20%–30% of the terminal velocities). A detailed summary of the existing laboratory, field, and model investigations of the raindrop shapes can be found in Testik and Barros (2007) and Beard et al. (2010).

Nevertheless, raindrops do not necessarily fall at terminal velocities (Testik et al. 2006; Montero-Martinez et al. 2009; Thurai et al. 2013). Indeed, raindrop fall velocity and shape are closely related. A nonterminal raindrop may undergo acceleration or deceleration, which will in turn deviate the drop shape from equilibrium. Similarly, a raindrop with a transient shape (e.g., an oscillating raindrop) may exhibit fall velocity deviations from terminal because of the unbalanced forces acting on the drop. Raindrop fall velocity and shape deviations from equilibrium characteristics become more pronounced under conditions such as high rain rates, horizontal wind, and in the presence of up-/downdraft. Accurate rain-rate calculations and estimations (as detailed in section 2), especially under such conditions, require consideration of this transient state.

Testik et al. (2006) observed a nonterminal oscillating raindrop using a high-speed camera, which was falling approximately 10% slower than the predicted terminal velocity of the same size (diameter of 1.9 mm). Montero-Martinez et al. (2009) reported that hydrodynamic instability or collision-induced drop breakup may induce small (~0.1 mm) raindrops to fall 10 times faster than the expected terminal velocities. It is difficult to provide a detailed microphysical discussion on the cause of these superterminal raindrops because Montero-Martinez et al. (2009) presented optical array spectrometer probe measurements of drops with diameters up to 3 mm. They observed that during intense rain events, the number of superterminal drops were 80%, 50%, and 20% of the total number of drops captured with diameters of around 0.24, 0.44, and 0.64 mm, respectively. During intense rain events, subterminal velocities, which were almost 10% lower than the predicted terminal velocities, were also reported for drops larger than 0.8 mm. In their recent analysis of two moderate-to-intense rain events, both of which had a 4-h rain accumulation of roughly 25 mm, Thurai et al. (2013) observed significantly lower raindrop fall velocities than terminal velocities [predicted by the equation of Atlas et al. (1973)] and a wide velocity distribution of the 3-mm drops. In an 80-m bridge experiment, Thurai and Bringi (2005) reported drop axis ratio distributions, in terms of Gaussian means and standard deviations, for different drop diameters. Moreover, their comparison of drop fall velocities with the Gunn and Kinzer (1949) data and the empirical equation of Brandes et al. (2002) for terminal velocity predictions determined that the mean fall velocities were lower than the predicted values for large drops.

In their study of the drop mean axis ratios from aircraft measurements, Chandrasekar et al. (1988) observed that the mean axis ratios had large standard deviations and that the mean axis ratio values were higher than the predicted values by the theoretical parameterization of Green (1975). The observed large standard deviations were primarily caused by drop oscillations. Andsager et al. (1999) developed an average axis ratio parameterization using field measurements of Chandrasekar et al. (1988) and laboratory observations of Beard et al. (1991) and Kubesh and Beard (1993). Keenan et al. (2001) conducted a sensitivity study to analyze the effects of different axis ratio formulations [including those given in Pruppacher and Beard (1970), Chuang and Beard (1990), Andsager et al. (1999), and others] on C-band radar polarimetric variables and rainfall estimations. They found that the C-band radar variables were rather sensitive to the axis ratio relations, and that the two-measurement rainfall estimator was more accurate and less sensitive to the axis ratio formulations than the single-measurement estimator .

The main goal of this study is to point out the potential error sources and their impacts for quantitative rainfall estimations by polarimetric radar systems. Given the recent upgrade of the WSR-88D to dual-polarization capability, the potential rainfall estimation errors should be corrected as the advantages of using radar systems make them an indispensable component in precipitation estimations. In this study, we analyzed the impacts of both raindrop fall velocity and axis ratio errors (i.e., unaccounted deviations from the raindrop terminal velocity and equilibrium axis ratio values from parameterizations used in retrieval algorithms) on S-band dual-polarization radar rainfall estimations. The impacts of these errors on rain rates estimated using empirical equations, which were developed for a radar rainfall algorithm, were the primary focus. The T-matrix method, the normalized Gamma DSD (Ulbrich 1983; Willis 1984), and the Beard and Chuang (1987) equilibrium raindrop shape were first used to reproduce the selected rainfall retrieval algorithm. A case study with the aforementioned field observations for raindrop fall velocity and axis ratio was then presented to elucidate the actual affect in natural rain events. The coupling between the raindrop fall velocity and axis ratio has not been elucidated yet. Therefore, different combinations of independent errors for fall velocity and axis ratio values were considered in our sensitivity analysis.

This article is organized as follows. In section 2, we describe our methodology of estimating rainfall using radar. We present and discuss the results of our simulation procedure and our sensitivity analysis in section 3 and summarize our conclusions in section 4.

2. Methodology

Dual-polarization techniques make it possible to estimate rain rates using an individual or a combination of empirically developed rain-rate estimators. The commonly used estimators, classified as single- and two-measurement estimators depending on the number of the polarimetric variables used, are , , , and (Cifelli and Chandrasekar 2010). The Colorado State University–Hydrometeor Identification Rainfall Optimization (CSU-HIDRO) radar rainfall retrieval algorithm uses a combination of the rain-rate estimators to maximize the performance of the polarimetric radar, the details of which are described in Cifelli and Chandrasekar (2010) and Cifelli et al. (2011). By comparing the rain-rate estimations using several radar rainfall retrieval algorithms with the rain gauge measurements, Cifelli et al. (2011) concluded that the CSU-HIDRO algorithm was superior to the other tested algorithms. Given its wide use and capabilities, we adopted the liquid portion of the CSU-HIDRO algorithm (see Fig. 1) as our test bed to analyze the combined effects of the raindrop fall velocity and axis ratio errors on dual-polarization radar rainfall estimations.

Fig. 1.
Fig. 1.

Liquid portion of CSU-HIDRO algorithm (Cifelli et al. 2011) with Kdp in ° km−1, Zh in dBZ, and Zdr in dB.

Citation: Journal of Hydrometeorology 15, 5; 10.1175/JHM-D-13-0201.1

The empirical formulations for the single- and two-measurement rain-rate estimators [e.g., in Fig. 1] were developed by nonlinear regressions between the simulated polarimetric variables and their corresponding rain rates (Bringi and Chandrasekar 2001). Bringi and Chandrasekar used the following equation to theoretically calculate the true rain-rate values (mm h−1):
e1
where is the raindrop fall velocity (m s−1) and , also known as the DSD, is the number of drops in a unit volume (m3) and a unit interval of the drop diameter D (mm). When estimating rain rates using the polarimetric radar, it has been assumed that raindrops fall at terminal velocities. Specifically, the parameterization [see Eq. (2)] developed by Atlas and Ulbrich (1977) for estimating the terminal velocity values (m s−1) is used in the CSU-HIDRO retrieval algorithm:
e2
where D is in millimeters. It should be noted that this empirical relationship deviates from the observations of Gunn and Kinzer (1949) at both ends of the raindrop size spectrum (Atlas and Ulbrich 1977). Nevertheless, this relationship is used in this study to be consistent with the CSU-HIDRO rainfall retrieval algorithm. The normalized Gamma DSD (mm−1 m−3; Ulbrich 1983; Willis 1984), which is commonly used in radar rainfall estimations, can be expressed as follows:
e3
where is the median volume diameter (mm), is the shape parameter of Gamma distribution, is the complete Gamma function, and is the normalized intercept parameter (mm−1 m−3) defined as a function of the water content (Bringi and Chandrasekar 2001). Substituting Eqs. (2) and (3) into Eq. (1), Eq. (1) can be re-expressed as
e4
where
e5
Bringi and Chandrasekar (2001) used the T-matrix method to perform a large number of simulations with different DSDs by varying the Gamma DSD parameters , , and in Eq. (3). In each of their simulations, the values of , , and were randomly sampled from uniform distributions over the range from 3 to 5 for Nw (mm−1 m−3), 0.5 to 2.5 mm, and −1 to 5, respectively, with R ≤ 300 mm h−1. Using these simulations and Eqs. (4) and (5), they created a large set of calculated radar parameter values and the corresponding true rain-rate values. The ranges of the distribution parameter values used by Bringi and Chandrasekar were narrower than those from disdrometer observations by Tokay et al. (2013). Nevertheless, we still adopted their parameter value ranges in our simulations in order to implement the CSU-HIDRO algorithm correctly. In our simulations, the maximum raindrop diameter was set as 8 mm (Bringi and Chandrasekar 2001; Testik and Barros 2007), and the minimum drop diameter Dmin was set as 0 mm because of the limitation of the T-matrix code (Leinonen 2014). Setting the minimum drop diameter as 0 mm had negligible impact on the results, which can be seen in Fig. 2. In this figure, the rain-rate calculations using Eq. (4) with minimum drop diameter set as 0 and 0.25 mm are compared. Here, the value of 0.25 mm for minimum raindrop size was selected because drops that have diameter less than 0.25 mm are classified as cloud drops (Testik and Barros 2007). The dielectric constant of water at 20°C (Ray 1972) was used for the S-band radar with a wavelength of 10 cm. The raindrops were assumed to follow the Beard and Chuang (1987) equilibrium shape with a canting angle of 7°. Beard and Jameson (1983) showed that more than 95% of the raindrops have canting angles less than 7°. In estimating the Beard and Chuang equilibrium axis ratios, a polynomial equation that was fitted to the Beard and Chuang equilibrium shape model (Bringi and Chandrasekar 2001), was used:
e6
Here, a and b are the semimajor and semiminor axis lengths, respectively, of the equilibrium raindrop shape. Finally, these simulated radar variables and the correspondingly calculated rain rates were used to construct the empirical formulas for the rain-rate estimators through nonlinear regressions. A list of the empirical formulas used in the S-band CSU-HIDRO radar rainfall algorithm (Fig. 1) that we used in this study is given below (Bringi and Chandrasekar 2001; Cifelli et al. 2011):
e7
e8
e9
e10
Fig. 2.
Fig. 2.

Comparison between theoretically calculated rain rates (i.e., R) with min raindrop diameter (i.e., Dmin) of 0 and 0.25 mm (solid circles represent DSD samples, solid line indicates identical rain rates).

Citation: Journal of Hydrometeorology 15, 5; 10.1175/JHM-D-13-0201.1

Following the procedure outlined above, we reproduced the results of the CSU-HIDRO retrieval algorithm by simulating 5000 rain events (Case I). The T-matrix calculations were conducted using a Python code (Leinonen 2014) that was developed from the FORTRAN code written by Mishchenko and Travis (1998). Our simulation results led to the same empirical formulations given in Eqs. (7)(9). Equation (10), which was borrowed from the single-polarization WSR-88D rainfall retrieval algorithm without conducting the same nonlinear regression analysis, indicated bias errors from our simulation results (see Figs. 4g,h, described in greater detail below) as expected. After establishing confidence in the implementation of the CSU-HIDRO algorithm, a case study using fall velocity and axis ratio measurements from observed rain events (Case II) and a sensitivity analysis for the raindrop fall velocity and axis ratio deviations (Case III) were performed.

3. Results and discussion

The 5000 simulations were first conducted with the fall velocity [Eq. (2)] and axis ratio [Eq. (6)] formulas replaced by observations in natural rain events from Montero-Martinez et al. (2009), Thurai et al. (2013), and Chandrasekar et al. (1988) (Case II). Though these observed data were either from one particular rain event or from events having similar rain rates, in this study, we applied them to all simulations to augment the lack of field measurements. For this reason, we also included the axis ratio data from laboratory experiments (Beard et al. 1991) to cover small raindrops. Thurai et al. (2013) did not discuss the underlying microphysics for their observations on the subterminal large drops in detail. Given the operation principle of the two-dimensional video disdrometers (2DVDs), their findings may include instrument artifacts. Tokay et al. (2013) analyzed a number of rain events from the same site as Thurai et al. (2013) using 2DVD and collocated Parsivel disdrometers. Tokay et al. reported that while raindrop fall speeds deviated from one event to the other for 2DVD measurements, measured raindrop fall speeds were indifferent from one event to the other using the Parsivel disdrometers. Tokay et al. attributed this observation to the 2DVD instrument artifacts. The findings of Thurai et al. (2013) are, however, in agreement with the findings of Montero-Martinez et al. (2009), in which subterminal large raindrops were reported [see Fig. 2 of Montero-Martinez et al. (2009) and Fig. 2 of Thurai et al. (2013)]. It is important to note that two different instruments, though neither of the instruments is ideal for studying raindrop fall speeds, were used in these two different studies [i.e., optical array spectrometer probe for Montero-Martinez et al. (2009) and 2DVD for Thurai et al. (2013)]. This can be considered as a support for the presence of nonterminal raindrops in rain events that was also observed by Testik et al. (2006) using a high-speed imaging technique.

The measured average fall velocity ( in Fig. 3a) and mean axis ratio ( in Fig. 3b) values, with respect to the equivalent drop diameters, are plotted in Fig. 3. For purposes of comparison, we also plotted Eqs. (2) and (6) in Fig. 3. For raindrops outside the data range, their fall velocities were assumed as identical to the closest boundary values (i.e., the first square and the last circle in Fig. 3a), while their axis ratios were taken from the Beard and Chuang (1987) polynomial fit. Since we do not know the accurate quantitative values for the deviations of the raindrop fall velocity and axis ratio from the predicted values, we undertook a sensitivity study to show the potential errors in dual-polarization radar rainfall estimations due to the combined effects of the raindrop fall velocity deviations and axis ratio deviations (Case III). A maximum deviation of ±10% (with 2% increments) around the original fall velocity and axis ratio values calculated from Eqs. (2) and (6) were considered. This led to a total of 121 fall velocity and axis ratio combinations in our analysis. Since previous studies indicated superterminal raindrops for small drop sizes and subterminal raindrops for large drop sizes, a subcase with more realistic combinations of fall velocity (+10% for D ≤ 0.6 mm and −10% for D > 0.6 mm) and axis ratio (from −10% to +10% with 2% increments) deviations was also included in Case III. The threshold D value of 0.6 mm was adopted per the findings of Montero-Martinez et al. (2009). The 10% limit was deemed as a reasonable value in that it was consistent with values observed previously (Testik et al. 2006; Montero-Martinez et al. 2009). For raindrops with diameters less than 1 mm, their axis ratios were also assumed to follow the Beard and Chuang (1987) polynomial fit throughout the study. Nevertheless, their axis ratio values were close to one because the surface tension dominates over the other forces acting upon the drops (Testik and Barros 2007; Szakall et al. 2010). To ensure consistency, Case III simulations were also performed for 5000 rain events for every combination.

Fig. 3.
Fig. 3.

Measured and modeled (a) average fall velocity and (b) mean axis ratio values as functions of equivalent raindrop diameters.

Citation: Journal of Hydrometeorology 15, 5; 10.1175/JHM-D-13-0201.1

Comparisons between the simulation results from Case I and Case II are shown in Fig. 4 for each of the individual rain-rate estimators (Figs. 4a–h) and in Fig. 5 for the combined CSU-HIDRO rainfall algorithm. It should be noted that Fig. 5 has more practical significance than Fig. 4 because dual-polarization radar rainfall estimation relies on a combined rainfall algorithm (see Fig. 1) rather than individual estimators for the entire range of rain rates. Rain-rate errors εR are defined in terms of percentage differences between the empirically [i.e., from Eqs. (7)(10)] and theoretically [i.e., from Eq. (4)] calculated rain rates . Case II, in which raindrop fall velocity and axis ratio deviations from the expected equilibrium characteristics are considered, indicated larger errors (both biases and uncertainties) compared to Case I for all four rain-rate estimators [i.e., , , , and ] and the combined rainfall algorithm. As can be seen from Fig. 4, the estimated rain rates in Case II are generally under-, over-, and underestimated for roughly an additional 20% as compared to the estimations in Case I using estimators , , and , respectively. For the estimator (see Figs. 4g,h), however, larger errors in Case II are observed only during light rain rates. It should be noted here that is insensitive to the vertical length scale of raindrops; hence, it is less sensitive to drop axis ratio deviations from predicted equilibrium values than other rain-rate estimators. Larger errors observed in small rain-rate events in Fig. 4h occur mainly because of large fall velocity deviations of small raindrops from the predicted terminal velocities (see superterminal drops in Fig. 3). Note that even though significant decreases of fall velocities for large drops are observed in Fig. 3, the rain rates are hardly controlled by the small number of large drops even during intense rain events (see Figs. 4g,h). It should also be noted that there are large rain-rate errors at light rainfall events. The significance of these errors depends on the specific hydrometeorological application. A comparison of the rain-rate estimations using the CSU-HIDRO radar rainfall retrieval algorithm for Case I and Case II is presented in Figs. 5a and 5b, and we specify in Fig. 5c the rain-rate estimator used to estimate given rain-rate data in Fig. 5b. Estimation errors using the radar algorithm are observed as a combination of the errors introduced from each of the above-mentioned rain-rate estimators (Figs. 4a–h). Note that the purpose of a radar rainfall retrieval algorithm is to maximize the performance of the radar rainfall estimations by minimizing the errors, which include both parametric and radar measurement errors (Bringi et al. 1996; Bringi and Chandrasekar 2001). In this study, we only considered the parametric errors and assumed the absence of measurement errors.

Fig. 4.
Fig. 4.

Comparisons of rain-rate estimations from Case I and Case II for individual rain-rate estimators. Solid lines represent 0% error.

Citation: Journal of Hydrometeorology 15, 5; 10.1175/JHM-D-13-0201.1

Fig. 5.
Fig. 5.

Comparisons of rain-rate estimations from (a) Case I and (b) Case II for combined rainfall algorithm. (c) Rain-rate estimators used in (b) are specified using different symbols (see the legend). Solid lines represent 0% error.

Citation: Journal of Hydrometeorology 15, 5; 10.1175/JHM-D-13-0201.1

For Case III, we determined the CSU-HIDRO rainfall rate estimation errors by conducting a large number of simulations with hypothetical deviations of the raindrop fall velocity and axis ratio values from the predicted equilibrium values. For both fall velocity and axis ratio values, deviations from −10% to +10% (with a 2% increment) were introduced to the predicted values using Eqs. (2) and (6), respectively. Although the correlation between the fall velocity and the axis ratio is realized, the quantitative relation between them is still unknown. So in this study, the deviations of fall velocity and axis ratio are considered independently. These introduced deviations, henceforth referred to as the fall velocity error and the axis ratio error , resulted in a total of 121 error combinations. The subcase with superterminal small drops and subterminal large drops resulted in 10 additional error combinations, which are discussed later in this section. The percentage errors for rain-rate estimations using CSU-HIDRO for 4 of these 121 error combinations are presented in Fig. 6. Here, the 10% fall velocity and −10% axis ratio error combination (Fig. 6a) and the 10% axis ratio and −10% fall velocity error combination (Fig. 6d) produce the lowest and highest maximum rain-rate estimation error and standard deviation, respectively, among the four selected cases. If remains the same (Figs. 6a,b or 6c,d), increases by more than 50% for majority of the rain-rate events, and more rain rates are likely to be underestimated when changes from −10% to +10%. If remains the same (Figs. 6a,c or 6b,d), decreases by nearly 50% for most of the simulated events when changes from −10% to +10%. The maximum absolute values and the standard deviations of the rain-rate estimation errors with respect to the combinations of the considered fall velocity and axis ratio errors are presented in Fig. 7. Typical and values are also tabulated in Table 1, along with the corresponding and values selected to cover the entire range of the error deviations. From Fig. 7 and Table 1, we can derive the following specific conclusions:

  1. The maximum is about 200% and the maximum is approximately 75% because of the combined fall velocity and axis ratio deviations. Both of these values occur when is −10% and is +10%. Note that this error combination (i.e., the negative percentage error for the fall velocity and the positive percentage error for the axis ratio) is the realistic case in that the field measurements indicate subterminal large drops with increased oblateness (i.e., larger axis ratio values than predicted; see Fig. 3).
  2. The continued increase in results in a corresponding increase in the data spread of both the (Figs. 7c,e,g) and the (Figs. 7d,f,h), which is caused by the variation of . That is to say, for a given , the deviation of introduces more errors in the radar rainfall estimation as increases. This observation clearly indicates that the rain-rate estimation errors are a function of combined fall velocity and axis ratio deviations.
  3. For a given , both (Figs. 7a,e,g) and (Figs. 7b,f,h) increase when increases either from 0% to −10% or 0% to +10%. Progressive increases (at an increased rate) are also observed for both and as increases, indicating a nonlinear effect of on the dual-polarization radar rainfall estimates.
  4. For a fixed value of , decreases as varies from −10% to +10% (see Figs. 7b,f,h for decreasing trends of same markers). The changing trends of , however, vary with different values. As can be clearly seen from Figs. 7e and 7g and Table 1, the changing trends of vary from increase to decrease as increases, with changing from either 0% to −10% or 0% to +10%. This quantitative analysis clearly indicates that the percentage error for estimating the rain rates is more sensitive to the axis ratio errors than to the fall velocity errors (if these two errors were uncorrelated).
The simulation results from the subcase of Case III are provided in Fig. 8, which presents the maximum rain-rate estimation errors and the standard deviation of the errors. This figure indicates that, for the same , the magnitudes of both and are close to their magnitudes in Fig. 7, with value of −10%. This finding indicates the dominance of the subterminal large drops (i.e., D > 0.6 mm) in radar rainfall estimations for this subcase.
Fig. 6.
Fig. 6.

Error (%) in rain-rate estimation using combined fall velocity and axis ratio errors in the CSU-HIDRO radar rainfall retrieval algorithm; the solid lines represent 0% error.

Citation: Journal of Hydrometeorology 15, 5; 10.1175/JHM-D-13-0201.1

Fig. 7.
Fig. 7.

(left) Max values and (right) std dev of errors (%) in rain-rate estimations induced by raindrop fall velocity and axis ratio errors. For the ease of the reader, projections of the (a),(b) three-dimensional plots on the following planes are also provided: (c) ; (d) ; (e) (for values ranging from 0% to −10%); (f) (for values ranging from 0% to −10%); (g) (for values ranging from 0% to +10%); and (h) (for values ranging from 0% to +10%).

Citation: Journal of Hydrometeorology 15, 5; 10.1175/JHM-D-13-0201.1

Table 1.

Max values and std dev of rain-rate estimation errors of 25 of the selected combinations of raindrop fall velocity and axis ratio errors.

Table 1.
Fig. 8.
Fig. 8.

Max values and std dev of errors (%) in rain-rate estimations induced by raindrop axis ratio errors and deviations of fall velocity (+10% for D ≤ 0.6 mm and −10% for D > 0.6 mm).

Citation: Journal of Hydrometeorology 15, 5; 10.1175/JHM-D-13-0201.1

4. Conclusions

In this study, the combined effects of the fall velocity and axis ratio deviations from the predicted terminal velocity and equilibrium axis ratio values, respectively, on dual-polarization radar rainfall estimations were investigated. A radar rainfall retrieval algorithm, CSU-HIDRO, was utilized for this investigation. The substantial effect of the fall velocity and the axis ratio errors on radar rainfall estimations were observed by first introducing errors based on measurements from natural rain events (Case II), then through a sensitivity study in which fall velocity and axis ratio errors were varied (Case III). The difference of percentage errors in rain-rate estimations between Case II and Case I, which is the base case without any error introduced, was generally about 20%. For Case III, the maximum rain-rate errors of up to 200% with maximum standard deviations of errors up to 75% were observed for different combinations of raindrop fall velocity and axis ratio errors, both of which were deviated between −10% and +10% with 2% increments.

These simulation results indicate the presence of significant error potential in polarimetric radar rainfall estimations because of deviations in the raindrop fall velocities and the axis ratios from equilibrium values that are predicted using available parameterizations. Such errors, if not corrected, may have the potential for notable negative impacts on various hydrometeorological applications that use rainfall inputs from polarimetric radar estimations. For example, large errors in radar rainfall rate estimations for heavy (light) rainfall events may cause flood (drought) related problems (see Fig. 6 for the combinations of rainfall rate and estimation error values from our simulations). Considering the recent update of the WSR-88D rainfall retrieval algorithm (Vasiloff 2012; Berkowitz et al. 2013) by adopting the rain-rate estimator from CSU-HIDRO, the error quantification conducted in this study has even more practical significance. The presented cases cover realistic scenarios and intend to make a contribution toward the awareness of the potential errors in polarimetric radar rainfall estimations and the need to correct these errors. The error sources highlighted in this study must be considered and minimized to improve the accuracy of radar rainfall estimations from polarimetric radars. This is of utmost practical significance in hydrological and water resource applications (e.g., forecasting of streamflow and river flooding and water resource management), where accurate estimation of rainfall is needed. We would like to emphasize the importance and need for careful and systematic investigations on the relevant rainfall microphysics. Consequently, parameterizations for predicting actual raindrop fall velocities and axis ratios of nonterminal raindrops must be the subject of future research.

Acknowledgments

This research was supported by the funds provided by the National Science Foundation under Grant AGS-1144846 to the second author (F.Y.T.). The first author is a graduate student under the guidance of F.Y.T.

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