Quantitative Precipitation Estimation with Operational Polarimetric Radar Measurements in Southern China: A Differential Phase–Based Variational Approach

Hao Huang Key Laboratory for Mesoscale Severe Weather, Ministry of Education, and School of Atmospheric Sciences, Nanjing University, Nanjing, China
State Key Laboratory of Severe Weather, and Joint Center for Atmospheric Radar Research of China Meteorological Administration–Nanjing University, Chinese Academy of Meteorological Sciences, Beijing, China
School of Meteorology, and Advanced Radar Research Center, University of Oklahoma, Norman, Oklahoma

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Kun Zhao Key Laboratory for Mesoscale Severe Weather, Ministry of Education, and School of Atmospheric Sciences, Nanjing University, Nanjing, China
State Key Laboratory of Severe Weather, and Joint Center for Atmospheric Radar Research of China Meteorological Administration–Nanjing University, Chinese Academy of Meteorological Sciences, Beijing, China

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Guifu Zhang Key Laboratory for Mesoscale Severe Weather, Ministry of Education, and School of Atmospheric Sciences, Nanjing University, Nanjing, China
School of Meteorology, and Advanced Radar Research Center, University of Oklahoma, Norman, Oklahoma

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Qing Lin Guangdong Meteorological Observatory, Guangzhou, Guangdong, China
Key Laboratory for Mesoscale Severe Weather, Ministry of Education, and School of Atmospheric Sciences, Nanjing University, Nanjing, China

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Long Wen Key Laboratory for Mesoscale Severe Weather, Ministry of Education, and School of Atmospheric Sciences, Nanjing University, Nanjing, China

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Gang Chen Key Laboratory for Mesoscale Severe Weather, Ministry of Education, and School of Atmospheric Sciences, Nanjing University, Nanjing, China

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Zhengwei Yang Key Laboratory for Mesoscale Severe Weather, Ministry of Education, and School of Atmospheric Sciences, Nanjing University, Nanjing, China

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Mingjun Wang Key Laboratory for Mesoscale Severe Weather, Ministry of Education, and School of Atmospheric Sciences, Nanjing University, Nanjing, China

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Dongming Hu Guangzhou Meteorological Observatory, Guangzhou, China

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Abstract

Quantitative precipitation estimation (QPE) with polarimetric radar measurements suffers from different sources of uncertainty. The variational approach appears to be a promising way to optimize the radar QPE statistically. In this study a variational approach is developed to quantitatively estimate the rainfall rate (R) from the differential phase (ΦDP). A spline filter is utilized in the optimization procedures to eliminate the impact of the random errors in ΦDP, which can be a major source of error in the specific differential phase (KDP)-based QPE. In addition, R estimated from the horizontal reflectivity factor (ZH) is used in the a priori with the error covariance matrix statistically determined. The approach is evaluated by an idealized case and multiple real rainfall cases observed by an operational S-band polarimetric radar in southern China. The comparative results demonstrate that with a proper range filter, the proposed variational radar QPE with the a priori included agrees well with the rain gauge measurements and proves to have better performance than the other three approaches, that is, the proposed variational approach without the a priori included, the variational approach proposed by Hogan, and the conventional power-law estimator-based approach.

© 2018 American Meteorological Society. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

Corresponding author: Kun Zhao, zhaokun@nju.edu.cn; Guifu Zhang, guzhang1@ou.edu

Abstract

Quantitative precipitation estimation (QPE) with polarimetric radar measurements suffers from different sources of uncertainty. The variational approach appears to be a promising way to optimize the radar QPE statistically. In this study a variational approach is developed to quantitatively estimate the rainfall rate (R) from the differential phase (ΦDP). A spline filter is utilized in the optimization procedures to eliminate the impact of the random errors in ΦDP, which can be a major source of error in the specific differential phase (KDP)-based QPE. In addition, R estimated from the horizontal reflectivity factor (ZH) is used in the a priori with the error covariance matrix statistically determined. The approach is evaluated by an idealized case and multiple real rainfall cases observed by an operational S-band polarimetric radar in southern China. The comparative results demonstrate that with a proper range filter, the proposed variational radar QPE with the a priori included agrees well with the rain gauge measurements and proves to have better performance than the other three approaches, that is, the proposed variational approach without the a priori included, the variational approach proposed by Hogan, and the conventional power-law estimator-based approach.

© 2018 American Meteorological Society. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

Corresponding author: Kun Zhao, zhaokun@nju.edu.cn; Guifu Zhang, guzhang1@ou.edu

1. Introduction

In combination with the conventional reflectivity factor (ZH), polarimetric measurements, including the differential reflectivity (ZDR), the differential phase/specific differential phase shift (ΦDP/KDP), and the copolar correlation coefficient (ρhv), provide information to improve the accuracy of radar quantitative precipitation estimation (QPE). Various QPE algorithms mostly in power-law form have been proposed to estimate rainfall rate (R) from the polarimetric radar measurements, for example, R(ZH), R(ZH,ZDR), R(KDP), R(KDP,ZDR), and R(AH) (Brandes et al. 2002; Bringi and Chandrasekar 2001; Marshall and Palmer 1948; Ryzhkov et al. 2005a, 2014, 2005b; Seliga and Bringi 1976). Different rainfall estimators show different error patterns that are related to the precipitation microphysics and the quality of polarimetric radar data (Lee 2006). The radar data quality can be attributed to various reasons, including radar miscalibration, power attenuation, nonmeteorological target contamination, hail contamination, and random error (Berne and Krajewski 2013; Cifelli et al. 2011). To improve the accuracy of the rainfall estimation, different QPE algorithms are combined according to their error patterns or the precipitation classification, which are known as synthetic algorithms (Chen et al. 2017; Cifelli et al. 2011; Giangrande and Ryzhkov 2008; Ryzhkov et al. 2005b).

Among the polarimetric radar variables, ΦDP, which is defined as the difference in the propagation phase shift of the horizontal and vertical polarized signals, and KDP, which is the range derivative of ΦDP, are immune to radar miscalibration, attenuation, partial beam blockage, and contamination of solid-phase hydrometeors. QPE with KDP is less affected by raindrop size distribution (DSD) variability than QPE using a ZHR relation, since KDP is approximately the 4.2nd moment of DSDs (Huang et al. 2017) that is closer to the 3.67th moment of DSD (related to rain rate) than the 6th moment (ZH for Rayleigh scattering). Unlike ZH or ZDR, KDP is not a direct radar measurement, so a straightforward way is to estimate KDP from ΦDP before using it in QPE. A lot of methods have been proposed for ΦDP processing and KDP estimation in the literature (Hubbert and Bringi 1995; Hubbert et al. 1993; Wang and Chandrasekar 2009). Random error and backscattering phase in the observed ΦDP could result in negative KDP estimates that do not match the physical truth for rain measurements. To mitigate the impact of the abnormal KDP values on QPE, Ryzhkov et al. (2005a) proposed to include the sign of KDP in the estimator . However, including the sign could occasionally lead to negative accumulated rainfall. Recently, the linear programming method has been applied to KDP estimation of rainfall (Giangrande et al. 2013, hereafter G13). Nonnegative KDP constraints are imposed in the algorithms to reduce the impact of ΦDP errors and eventually lead to better QPE performance.

An alternative way for QPE with ΦDP is to use a forward approach (Rodgers 2000). In this method, R is regarded as a state variable, and a forward model is built to predict ΦDP values from R. Generally, optimum R estimates can be obtained when the difference between the predicted and the measured ΦDP is minimized. This method directly connects R with ΦDP measurements, so the two-step error propagation from ΦDP to KDP and then from KDP to R is avoided. Additional constraints, such as the nonnegativity of R and the self-consistency of polarimetric variables, can be incorporated into the algorithm. The essential idea of this method is to optimally retrieve R under the constraints posed by all the available information. The variational analysis is a common way to achieve this goal (Hogan 2007, hereafter H07). H07 proposed a variational QPE approach based on the ZR relation. The coefficient in the ZR relation is adjusted according to ZDR and ΦDP measurements. In addition, the forward operator is built from idealized DSDs (gamma distribution with fixed μ) in his work.

Based on the R–KDP relation derived from the real DSD measurements, we develop a QPE approach with the variational approach acting as the optimization framework in this work. Section 2 gives the description of the variational QPE approach. An idealized case is utilized to verify the algorithm in section 3. In section 4 the radar measurements of multiple precipitation cases in southern China are utilized to quantitatively evaluate the algorithm’s performance. Conclusions are given in section 5.

2. Description of the proposed variational QPE framework

a. DSD dataset used to construct the variational QPE approach

As the variational QPE approach is designed for S-band polarimetric radars, and the evaluation is on the basis of the data collected by a polarimetric radar and rain gauges in southern China, we use the DSD data collected in the same region to find the statistical relations. A two-dimensional video disdrometer (2DVD), which is a highly accurate instrument for measuring precipitation particles (Schönhuber et al. 2007), deployed in Guangdong Province, China, collected DSD data from May to July in 2016 and 2017. The location of the 2DVD is shown in Fig. 1. To build the connection between the polarimetric variables (ZH, ZDR, and KDP) and the rainfall rate, we calculate them from the DSD data following the formulas given by Cao et al. (2012). The scattering amplitudes are estimated with the T-matrix method (Mishchenko et al. 2000). The temperature is assumed to be 20°C; the wavelength is 10.4 cm. The statistical terminal velocity relation and the raindrop axis ratio relation are assumed to be the same as those given by Brandes et al. (2002).

Fig. 1.
Fig. 1.

The topography around Guangzhou S-POL (black triangle). The elevation of the terrain is indicated (shading). The 2DVD (black dot) is used to derive the statistical property of the precipitation in this region.

Citation: Journal of Atmospheric and Oceanic Technology 35, 6; 10.1175/JTECH-D-17-0142.1

b. The basic variational analysis

To estimate precipitation, the state vector for an n-gate ray in a radar scan is defined as follows:
e1
where R(i) is rainfall rate at the ith range gate and the superscript T means transpose. The ΦDP measurements can be denoted in vector form as follows:
e2
where y is called the observation vector and ΦDP(i) is the ΦDP measurement at the ith range gate. The system phase and the folding in ΦDP measurements should be preprocessed beforehand. By assuming Gaussian-distributed ΦDP errors, the cost function of the variational analysis, which is sum of the cost functions for the a priori () and the observations (), can be defined as
eq1
eq2
e3
where and are called the a priori and the corresponding error covariance matrix, respectively, which will be elaborated in section 2d; is the innovation vector; and is the error covariance matrix of ΦDP measurements (Kalnay 2003), which is usually determined by radar signal processing. In this study the measurement errors of ΦDP are assumed to be independent at different range gates. As a consequence, the diagonal entries of are set to the standard deviation of the ΦDP estimation errors and the off-diagonal elements are set to zero. The innovation vector , defined as , is the difference between and the vector of propagation differential phase (ϕDP) predicted from the rainfall rate , and is called the forward operator mapping the state variables to the measurements (Rodgers 2000). In this study the fitted power-law relation between R and KDP—that is, (a = 5.767 × 10−3, b = 1.274), which corresponds to shown in Fig. 2—is utilized to convert to the KDP vector. Then, the predicted ϕDP at jth range gate can be obtained with the accumulation, as follows:
e4
where is the predicted KDP at the ith range gate and is the range gate resolution.
Fig. 2.
Fig. 2.

The statistical QPE relations R(KDP) for S-band radars obtained from the regression of the observations (gray dots).

Citation: Journal of Atmospheric and Oceanic Technology 35, 6; 10.1175/JTECH-D-17-0142.1

The statistically optimum R for all the range gates is the state vector when Eq. (3) is minimized. In fact, the forward operator is not linear (); the minimization is actually a nonlinear optimization problem as a consequence. A common method for simplifying the optimization is to linearize the forward operator, converting Eq. (3) to a quadratic function, and then to minimize the cost function iteratively. This method is usually applied on the data assimilation of numerical weather prediction (Kalnay 2003). The estimate of the state vector at the iteration () is denoted as . Then, the innovation vector at the kth iteration can be linearized as
e5
where is the Jacobian matrix containing the partial derivative of the predicted observation with respect to each element of , expressed as
e6
and
e7
where R(i) is the elements of ; is a lower triangular matrix; and the entries of its main diagonal are also zero.
The linearized cost function and the corresponding gradient at the kth iteration are given by
e8
and
e9
where is the gradient of Jk with respect to xk, and ∇ represents the gradient. With the cost function, the corresponding gradient, and the constraints for the state variables (nonnegative rainfall rate), the problem can be iteratively solved using the constrained optimization algorithms (Sun and Yuan 2006), in which the Hessian of the cost function is approximately estimated automatically (Byrd et al. 1995). The optimization starts from a first guess, that is, , which can be a statistical state, the a priori, or the solution of the previous ray of the radar scan. In our study the a priori is utilized, that is, . Once the linearized cost function is minimized, the solution is utilized as the guess for the next iteration. Once the solution is converged satisfactorily, it is the statistically optimum estimate. A convergence test can be used here.

c. Range smoothing with the cubic B spline

The smoothing filter plays an important role in the variational approach. The forward model [see Eq. (4)] has a cumulative-sum form, and the state variable in a single range gate provides less information than the random errors. The rainfall rate will be overfitted to the errors if no additional constraint is introduced. Fortunately, the precipitation in nature measured by radars tends to be spatially continuous, which can be ensured by a smoothing mechanism in the variational analysis. Commonly, there are three ways to smooth the state vector in the minimization procedures of the cost function. The first is to introduce a smoothing filter for the state vector; the second is to add a penalty term, which is commonly a mean square of a Laplacian of the state vector or any other term with a similar function, to the cost function to impose additional smoothing constraints of the state vector; the third way is to use a nondiagonal a priori covariance matrix (H07; Maesaka et al. 2012).

In our study the cubic basis spline (B-spline) similar to that used by H07 is adopted. A low-resolution state vector, denoted as , is introduced. It is converted to the rainfall-rate estimates at the radar range gates with an matrix consisting of the weights of the cubic B-spline,
e10
Matrix is essentially an interpolation operator, and it constructs the high-resolution from the interpolation of . The smoothness of is mainly controlled by the smoothing factor c (. The details of matrix can be found in the appendix of H07’s study. With the matrix included, the cost function and the corresponding gradient become
e11
and
e12
Now the variable to be optimized is , and once convergence is achieved, is the estimated rainfall rates at the radar range gates. And the intermediate variables—for example, ϕDP and KDP—can be estimated from rainfall rates using the forward operator.

d. The a priori and the error covariance matrix

The a priori and the error covariance matrix are the information we know before the estimation and its uncertainty. If we use no additional information for QPE, then the diagonal entries of the a priori error covariance can be set to infinity. As a consequence only ΦDP is used in the variational analysis. In such a case, the approach is similar to the conventional R(KDP) method with the additional physical constraints included (G13; Huang et al. 2017). As we know, when the rainfall is light, the useful information provided by ΦDP measurements is contaminated by the errors in ΦDP. In the synthetic rainfall algorithms (Bringi et al. 2011, 2009; Cifelli et al. 2011), the ZH or R(ZH) is used as the threshold conditions for selecting the QPE algorithms. For example, R(ZH) is usually used for light rain because the rainfall estimated from ZDR or KDP has a relatively large error. Inspired by this idea, we incorporate R(ZH) from the statistical ZR relation into the variational analysis as the a priori term to provide more useful information in the variational analysis. Note that this is different from the conventional a priori term that usually uses a fixed value estimated from the climatic statistics, but we still call it the a priori because the adopted statistical ZR relation can represent the mean characteristics of the DSDs in the regime. The a priori error covariance matrix is utilized to characterize the error properties of the a priori. The diagonal entries of (denoted as ) describe the statistical standard deviation of the error of the a priori at each range gate, which consists of two parts, that is, the impact of the DSD variability and the measurement error of ZH (Lee 2006), and the off-diagonal part is the joint variability of the a priori at the range gates.

The main focus in this study is the QPE of the S-band polarimetric radars in southern China. In our study the statistical relation (see Fig. 3a) is utilized as the a priori for both the idealized experiment and the real cases. Meanwhile, the error covariance is also calculated statistically using the dataset elaborated on in section 2a. The is attributed to the DSD variability and the measurement errors of ZH. The uncertainty resulting from the DSD variability () can be derived as a function of rainfall rate, as shown by the dashed–dotted line in Fig. 3b. The uncertainty caused by the measurement noise () for the power-law relation is related to the measurement error of ZH (Bringi and Chandrasekar 2001), and is given as
e13
where represents the standard deviation of the linearized horizontal reflectivity factor (). If the error of the measurement noise of is about 1 dB, then is about 14% for our ZHR relation, and total uncertainty () is the collective effect of and (Lee 2006), which yields
e14
We denote the uncertainty () as a function of the rainfall rate f(R). Thus, in the variational analysis, the diagonal entries of the a priori error covariance matrix is estimated from the a priori, using . Since the ZH measurements are independent at the different range gates, the off-diagonal parts of the a priori error covariance are set to zero. The is set to 2 mm h−1 if the estimated value is less. In real applications, hail probably exists at the range gates where ZH values are greater than 53 dBZ. Then, the corresponding values are set to infinity.
Fig. 3.
Fig. 3.

(a) The statistical RZH relation (black solid line) for S-band radars regressed from the observations (gray dots). (b) The normalized standard deviation in R for the ZHR relation caused by DSD variability (; black dashed–dotted line) and measurement noise (; black dashed line). The total error (; black solid line), which is a combination of and .

Citation: Journal of Atmospheric and Oceanic Technology 35, 6; 10.1175/JTECH-D-17-0142.1

For radars of short wavelength, such as C or X band, the backscattering phase in ΦDP measurements and propagation path attenuation are major obstacles for variational QPE. In this situation both ΦDP and ZH provide less useful information. Then, the attenuation correction of ZH is necessary for the a priori (Carey et al. 2000), and the attenuation effect should also be included in the error covariance, which is left for future research. In this study we focus on QPE of the operational S-band radar in southern China.

3. Verification with an idealized case

a. Experiment design

In a sense the temporal evolution of rain properties is similar to the spatial variability, since the weather systems are moving. Thus, we simulate a range profile of radar measurements to test our QPE method based on the 2DVD observations. A rainfall event passed over the 2DVD site from 0830 to 1100 UTC 28 June 2016, in which the maximum 1-min rainfall rate exceeded 100 mm h−1. Since the 2DVD has a small sampling volume, the short-term rain properties can be significantly different from the climatic statistics as a result of the internal microscale variability of precipitation systems. Thus, a Bayesian method similar to that developed by Cao et al. (2010) is employed so that the simulation is closer to the statistics. First, the series of synthetic ZH and ZDR are calculated from the 2DVD data based on the scattering amplitudes used earlier. DSD parameters N0, μ, and Λ are calculated from the synthetic ZH and ZDR series using the Bayesian method based on the μ–Λ relation derived from the 2-yr summertime DSD observations (). The range resolution of the profiles is assumed to be 250 m. Considering that the storm moved more than 250 m in 1 min and our goal is to create simulated truth to test the retrieval method, the DSD parameters N0, μ, and Λ are interpolated and filtered with the five-gate moving median and mean filters to make the storm scale more realistic. Then, the range profiles of R, ZH, and KDP can be calculated from N0, μ, and Λ with the assumption of the gamma distribution. The ΦDP range profile is obtained from the integral of KDP following Eq. (4). To simulate the observations, unbiased Gaussian random errors should be added to ZH and ΦDP. The attenuation effect and the backscattering phase are neglected for this S-band experiment. Figure 4 shows an example of the simulated ZH and ΦDP observations. The standard deviations of simulated errors in ZH and ΦDP are 1 dB and 2°, respectively. The range profiles of the intrinsic R, R(ZH), and R(KDP) are shown in Fig. 4c. The heavy precipitation center is located about 38 km away from the radar. The rainfall rates estimated from ZH and KDP are not exactly same with the intrinsic R because of DSD variability. As a consequence, the uncertainty in the forward operator is included in the idealized case.

Fig. 4.
Fig. 4.

The range profiles of the polarimetric variables and the rainfall rate used in the idealized case. (a) The intrinsic ϕDP (red line) and the simulated ΦDP (blue line) measurements, and the intrinsic KDP (green line). (b) The intrinsic ZH (red line) and the simulated ZH (blue line) measurements. (c) The intrinsic rainfall rate (red line), and the rainfall rates estimated from the statistical RKDP relation (green line) and the ZH–R relation (blue line).

Citation: Journal of Atmospheric and Oceanic Technology 35, 6; 10.1175/JTECH-D-17-0142.1

In later subsections three tests are conducted to show the performance of the variational QPE approach. First, estimates of R using a different smoothing factor are intercompared so that a proper smoothing factor is decided for the idealized case and also the real application. Then, the performance of the variational QPE approach is compared with that of G13’s method. At last, the impact of the a priori statistics on the R estimates is investigated. To quantitatively evaluate the estimation, the correlation coefficient (CC), the root-mean-square error (RMSE), and the normalized absolute error (NE) are utilized, which are defined as
e15
e16
e17
Here, is the standard rainfall rate at the ith range gate, which represents the intrinsic values; is the rainfall-rate estimate at the ith range gate; and are the corresponding expected values; and represents the total number of range gates.

b. Impact of the smoothing filter

As stated earlier, the smoothness of the estimates is decided by c. A larger smoothing factor results in smoother estimates, but it will also suppress the physical variation of the precipitation. A proper smoothing filter, which keeps the continuity of the rainfall while retaining as much detail as possible, is one of the keys to high-performance QPE. To find a proper smoothing factor, we use a statistical method. Based on the random-generated Gaussian measurement errors, 1000 range profiles of the simulated ZH and ΦDP are generated from the simulated rainfall-rate series. The standard deviations of the simulated ZH and ΦDP errors in each range profile are assumed to be equal to 1 dB and 2°, respectively. In the variational analysis, the ΦDP errors used in Eq. (11) are assumed equal to 2°, with no error correlation assumed. The diagonal entries of the a priori error covariance matrix are set close to infinity (1020 mm2 h−2), so the a priori provides almost no information here. The impact of the a priori will be studied in section 3d. The smoothing factor c () is changed from 5 to 22. Then, the statistics of the CCs, RMSEs, and NEs corresponding to the 1000 estimated range profiles of R are utilized to represent the performance of the variational approach using a specific smoothing factor, as shown in Fig. 5.

Fig. 5.
Fig. 5.

The statistics of the (a) CCs, (b) RMSEs, and (c) NEs corresponding to the 1000 range profiles of R estimated with the variational approach using different smoothing factors. The mean (black solid line), the 95th percentile (red dashed line), and the 5th percentile (green dashed–dotted line) of the statistical variables are marked.

Citation: Journal of Atmospheric and Oceanic Technology 35, 6; 10.1175/JTECH-D-17-0142.1

With a smaller smoothing factor (less than 10), the mean CC is relatively lower and the mean RMSE/NE is relatively higher because of the impact of the residual low-frequency part of the random errors. When the smoothing factor increases, the impact of the random errors is weakened, the mean CC gets higher, and the mean RMSE/NE becomes lower at first, and then the mean CC decreases and the mean RMSE/NE increases again because of the excessive smoothing of the physical variation of rain. The performance of the variational approach achieves the best result when the weakening of the random errors and the smoothing of the physical variation generally balance each other. Overall, by using the smoothing factor between 11 and 15, the mean CC obtains the highest level and the mean RMSE/NE reaches the lowest level, indicating a satisfactory performance. To reduce the effects of the random errors while keeping the physical variation as much as possible, the smoothing factor is set to 11 in this study.

c. Comparisons with KDP estimates based on linear programming

The variational QPE approach is known as an optimization using the forward model based on ΦDP measurements. In contrast, the conventional QPE involves the inverse method, which estimates KDP from ΦDP first and then calculates R using the R–KDP relations. G13 proposed to incorporate the nonnegativity constraint when estimating KDP from ΦDP based on the linear programming, which proves to perform well. To investigate the performance of the variational QPE approach, we compare its statistical results with those based on G13’s method.

The comparisons are shown in Fig. 6, which is also based on 1000 range profiles of ZH and ΦDP with random errors. The configuration of the variational QPE approach is the same as that used in section 3b, except that we use 11 as the smoothing factor () here. For G13’s linear programming method, a nine-point derivative filter is utilized, which proves to be the best configuration for the idealized case (the procedure is not shown). Overall, uncertainty exists in the optimized ϕDP, the optimized KDP, and the estimated R from both algorithms as a result of the random errors in the simulated observations. The optimized ϕDP is highly consistent with the intrinsic value, with the mean CC close to 1, the mean RMSE less than 0.30°, and the mean NE less than 0.01 for both methods (not shown). For the KDP estimation, both methods give satisfactory results, with the mean values of CC greater than 0.96 and relatively low mean RMSE/NE. The median values of the KDP estimates (denoted with the blue solid line) are generally close to the intrinsic values (denoted with the red solid line) as shown in Figs. 6a and 6b. For the R estimation, a little bit of inaccuracy exists because of the uncertainty in the forward operator (the R–KDP relation). From the statistical perspective, the results from the variational approach are slightly better than those from G13’s method. The median values from the variational approach also coincide better with the intrinsic values. Note that there is a slight underestimation in R and KDP from G13’s method at the most severe part of the echo (intrinsic R greater than 100 mm h−1). This is because the derivative filter used is a bit too large. With a longer derivative filter, the underestimation becomes more evident. The uncertainty in the R estimates () caused by the random errors is generally comparable for the two methods. Within the range of 0 ~ 15 km, where the intrinsic R values are low, the variance in the R estimates is relatively small because of the nonnegative bounds utilized. This is one of the advantages of the variational QPE approach.

Fig. 6.
Fig. 6.

(a)–(d) The statistical comparisons of the R estimates and KDP estimates using the variational approach, and G13’s method based on 1000 range profiles of ZH and ΦDP with random errors. No a priori information is utilized in the variational estimation here. (a) Estimates of KDP based on the variational approach, (b) KDP estimates based on G13’s method, (c) R estimates based on the variational approach, and (d) R estimates based on G13’s method. The median values of the estimates are indicated (blue solid lines). The regions between the 5th and 95th percentiles are marked (gray shading). The intrinsic R and KDP are indicated (red solid lines). (e) The comparison of the standard errors in the R estimates using the variational approach (red solid line) and G13’s method (blue solid line).

Citation: Journal of Atmospheric and Oceanic Technology 35, 6; 10.1175/JTECH-D-17-0142.1

When the a priori statistics provide no information, the performance of the variational QPE approach is comparable with G13’s method, generally. Nevertheless, the advantage of the variational QPE approach is its flexibility. Additional information can be incorporated through the a priori or the upper and lower constraints easily. The impact of the a priori on the R estimation will be shown in section 2d.

d. Impact of the a priori

As shown in Fig. 7a, the a priori estimated from the simulated ZH in Fig. 4 shows some deviation from the intrinsic R because of both the DSD variability and the random noise. The corresponding error bars denote the uncertainty . With incorporating the information of ZH into the variational analysis as the a priori, the statistical performance of the estimated R is shown in Figs. 7b and 7c. Compared to the results where the a priori provides no information (see Fig. 6), the mean CC increases and both the mean RMSE and NE decrease. Another significant improvement is that the uncertainty () of the R estimation decreases (gray shading in Fig. 7b; red line in Fig. 7c). Although there is significant uncertainty in , we can obtain more confident R estimates with the combination of the a priori and the measurements (ΦDP). The accuracy of the R estimation is obviously better and its uncertainty () is less than the estimation based on G13’s method. There is also a shortcoming in the R estimates. Because of the sensitivity of the ZR relation to the DSD properties, the R(ZH) tends to underestimate the rainfall rate. This effect is also included in the variational analysis. However, the underestimation of the results of the variational analysis is not significant and is acceptable when we take the decease of the uncertainty () of the R estimation into consideration. We will investigate how the inclusion of the a priori affects the QPE performance in real cases in section 4.

Fig. 7.
Fig. 7.

(a) An example of the a priori (black solid line) utilized in the optimization and its corresponding errors (error bars). The intrinsic R is shown (red solid line as a reference). (b) As in Fig. 6c, but for the R estimates with the a priori used in the variational approach. (c) The comparison of the errors (standard deviations) in the R estimates with the a priori used in the variational approach (red solid line), the R estimates with no a priori used in the variational approach (red dashed line), and the R estimates based on G13’s method (blue dashed line). The mean standard deviation of the errors of the a priori is also shown (black solid line).

Citation: Journal of Atmospheric and Oceanic Technology 35, 6; 10.1175/JTECH-D-17-0142.1

4. Verification with the data of an S-band polarimetric radar

a. Data and the quality control

In Guangdong Province, an S-band polarimetric radar (called Guangzhou S-POL) is deployed at the city of Guangzhou. This radar is part of the Chinese operational radar network, whose main configuration is shown in Table 1. Guangzhou S-POL was a Doppler weather radar built several years ago, and it has been upgraded with polarimetric capability since 2016. In May and June 2016, about seven precipitation events occur within the coverage of Guangzhou S-POL. In this section QPE based on the aforementioned variational approach is verified using these events.

Table 1.

Settings and parameters of Guangzhou S-POL and its observations.

Table 1.

Guangzhou, one of China's most developed cities, is still undergoing rapid development, so the topographic condition around the radar site changes every year. The radar observations suffer from ground clutter contamination resulting from the tall buildings, the TV towers, and so on. Ground clutter contamination needs to be taken into consideration in QPE. Algorithms such as ground clutter detection and mitigation techniques are usually employed in radar digital signal processing (DSP) to mitigate the impact of ground clutter (Doviak and Zrnić 1993; Zhang 2016). Nevertheless, its impact on the radar measurements can seldom be fully eliminated, especially when the power of the ground clutter is dominant or the radial velocity of precipitation is close to zero. For polarimetric radars, the ground clutter mitigation can make the signals of the horizontal and the vertical channels incoherent/unbalanced. Thus, the accuracy of ZDR, ΦDP, and ρhv measurements, which are defined as the power ratio, the phase difference, and the correlation of the horizontal and the vertical signals, respectively, will probably decrease when the echo is contaminated by ground clutter. As we know, ΦDP is based on the propagation effect of electromagnetic wave. Besides, ZDR is significantly affected by ground clutter, especially when the signal of the weather echoes is weaker than the clutter. QPE with the variational approach using ΦDP and ZH proposed in this paper is considered to be less affected by the clutter than QPE with the algorithms with ZDR used.

Before being used in QPE, the quality of the data from Guangzhou S-POL is carefully controlled via the following four steps. As documented by Chen et al. (2014), the internal calibration system of Guangzhou S-POL using internally generated test signals and routine maintenance guarantee the ZH precision to be within 1 dB.

  1. Echoes with ρhv less than 0.85 are identified as potential nonmeteorological echoes. If the areas of the nonmeteorological echoes embedded inside the meteorological echoes are small with the number of clustered gates less than 20, they are marked back to precipitation. Some nonmeteorological echoes can have large ρhv greater than 0.85 as a result of the random measurement errors. Similarly, if the echo area has fewer than 20 clustered gates, they are removed utilizing a de-speckle filter.

  2. Term ZDR is calibrated using light rain observations. The averaged intrinsic ZDR value of the light drizzle echo with ZH between 10 ~ 20 dBZ is assumed to be very close to 0 dB (Giangrande and Ryzhkov 2005). The difference of the averaged value of the observed ZDR and the intrinsic value of the light rain region is considered as the ZDR bias.

  3. The system ΦDP is estimated from the statistics of the ΦDP values of the meteorological echo close to the radar site.

  4. For meteorological echoes, if the difference of the ΦDP values at two neighboring range gates is greater than 35°, then the ΦDP measurements at these two gates are considered to be contaminated by clutter and removed. After the unfolding, the data at these gates are refilled by linearly interpolating the ΦDP measurements at the surrounding gates.

A plane position indication (PPI) image of radar data on 10 May 2016 at the elevation angle of 1.5° is shown in Fig. 8. The echoes to the north of the radar site characterized with low ρhv are probably from nonprecipitation scatterers. In our processing, only the precipitation echoes are retained (see Fig. 8d). The precipitation echoes contaminated by ground clutter near the radar site are also processed with the clutter mitigation algorithms in the DSP. It causes inconsistency in the signals from the horizontal and vertical channels. The corresponding ρhv values decrease and the reliability of the radar measurements also decreases. Besides, the large backscattering phase from hail will also bring in uncertainty. Thus, the removal and the refill of the ΦDP measurements contaminated by ground clutter or hail (see step 4) will ameliorate the rain measurements.

Fig. 8.
Fig. 8.

The PPI images of a precipitation event collected by Guangzhou S-POL at 0030 UTC 10 May 2016 at the elevation angle of 1.5°. (a) ZH (dBZ), (b) ρhv, (c) ZH after the removal of nonprecipitation echoes, and (d) the mask utilized to separate the precipitation and nonprecipitation echoes.

Citation: Journal of Atmospheric and Oceanic Technology 35, 6; 10.1175/JTECH-D-17-0142.1

After quality control, the errors of ZH, ZDR, and ΦDP are assumed to be equal to 1 dB, 0.2 dB, and 2.0°, respectively, according to the configuration of the radar (Table 1). The rainfall rates for the radar scans are optimized based on the variational approach with R(ZH) as the a priori. The results for the same radar scan shown in Fig. 8 are shown in Fig. 9. The main precipitation core (southeast of the radar site) is characterized with large ZH and a sharp radial increase in ΦDP. The optimized rainfall rates (Fig. 9b) of this region are over 150 mm h−1. The intermediate variables ϕDP and KDP are also shown. The optimized ϕDP values (Fig. 9c) are consistent with the unfolded ΦDP measurements (Fig. 9a). Most of the random errors are removed and the ϕDP values are also consistent in the azimuthal direction. The optimized KDP values (Fig. 9d) of the main precipitation core are over 3° km−1. There is no negative value in KDP, which indicates that the optimization in our study follows the physical constraints. This is an evident advantage of this method as compared with the conventional least squares fit based KDP estimation and QPE (Huang et al. 2017).

Fig. 9.
Fig. 9.

The retrieval results for the radar scan shown in Fig. 8. (a) The unfolded ΦDP after the removal of the system phase, and (b) the optimized R from the proposed variational approach. (c) The optimized ϕDP and (d) the optimized KDP are shown.

Citation: Journal of Atmospheric and Oceanic Technology 35, 6; 10.1175/JTECH-D-17-0142.1

Besides the proposed variational approach with R(ZH) included as the a priori (notated with VWAP), the rainfall rates for the radar scans are also estimated with three other approaches. To show the impact of the a priori, the results for the variational approach with no additional information used in a priori (notated with VWOAP) are utilized as references. The variational approach proposed by H07 (notated with VH07) is also employed to show the impact of different forward operators. In VH07, the forward operator is based on the gamma distribution with a fixed μ = 3, and the statistical terminal velocity relation is the same as that used by Brandes et al. (2002). And last, the power-law estimator based on the least squares fitted KDP (notated with RKDP) is also included. The configuration of the least squares fitting is the same as that used in Huang et al. (2017) except that the fitting window for a gate with ZH below 40 dBZ Z is set to 9 km. The notation means the sign of KDP is included in the rainfall-rate calculation as proposed by Ryzhkov and Zrnić (1996). In later subsections, the QPEs based on these four different approaches are compared against the rain gauge observations to show their performance. The radar-estimated rainfall rates above the gauge sites are extracted and temporally integrated into the accumulated rainfall amounts (ARs) of different temporal resolutions, which facilitate the quantitative comparisons. To mitigate the impact of the ground clutters and the partial beam blockage on the radar QPEs, the scans at the 1.5° elevation are utilized. On the other hand, there are more than 600 rain gauges within a distance of 75 km from the radar site. To make a convincing and objective comparison, the quality of the gauge data is controlled in advance using a method of speckle filtering (Lee et al. 1999). The spatial continuity of the accumulated rainfall amounts is checked and the outliers are probably from the unreliable gauge sites. These sites are excluded from the QPE evaluation.

b. QPEs in one case

First, a precipitation event on 9–10 May 2016 is selected for the QPE evaluation. The event caused the maximum accumulated rainfall, over 150 mm, in southern China. The rain gauges’ derived total ARs of the whole event () are shown in Fig. 10. The relative differences between the ARs based on VWAP () and , defined as , are also included. The heavy rainfall band extended from northwest to southeast with the core located near the radar site. There is a slight underestimation in the radar-derived total ARs near the radar site (within the range of 20 km) and a slight overestimation outside the range of 30 km from the radar. The biases of the radar-derived ARs and the rain gauge–based ARs are caused by the mismatch of the sampling volumes, the vertical microphysical structure change or horizontal propagation when rain falls from the radar sampling volumes to the ground, the interstorm DSD variability, the quality of the radar data and the rain gauge data, the uncertainty in the QPE algorithms, and so on. Overall, the underestimation or overestimation of the total ARs is relatively slight.

Fig. 10.
Fig. 10.

The difference in the total accumulated rainfall estimated from the rain gauges and the radar-derived total accumulated rainfall (the elevation angle of 1.5°) based on the proposed variational approach from the precipitation event from 0000 UTC 9 May to 1200 UTC 10 May 2016. The places where the rain gauges were deployed (circles). The size of the circles denotes the total accumulated rainfall from the rain gauges, and the relative difference (shading). The radar site position (black triangle) is marked.

Citation: Journal of Atmospheric and Oceanic Technology 35, 6; 10.1175/JTECH-D-17-0142.1

The radar hourly rainfall estimates for all the gauge sites denoted in Fig. 10 on the basis of the four approaches are intercompared, as shown in Fig. 11. The performances of the different approaches vary. Generally, VWAP has the best performance among the four approaches. It shows a high correlation coefficient (CC = 0.94) and relative low errors (RMSE = 3.38 mm, NE = 0.34). When no information is provided as the a priori, VWOAP still shows satisfactory performance. The correlation coefficient decreases slightly (CC = 0.92) and the errors slightly increase (RMSE = 3.66 mm, NE = 0.44), which demonstrates the positive effect of the inclusion of R(ZH) as the a priori. For VH07, all three kinds of radar measurements (ZH, ZDR, and ΦDP) are utilized, while the performance of the rainfall estimation is roughly comparable with VWOAP, in which only ΦDP measurements are used for QPE. For the conventional RKDP method, even after the temporal accumulation, negative rainfall estimates may exist, which may introduce extra uncertainty (Huang et al. 2017). As a consequence, RKDP has the worst performance in the hourly rainfall estimation.

Fig. 11.
Fig. 11.

Scatterplot of the hourly rainfalls observed by all the rain gauges (shown in Fig. 10) vs the hourly rainfall estimates from the PPI scans at the elevation angle of 1.5° based on (a) VWAP, (b) VWOAP, (c) VH07, and (d) RKDP.

Citation: Journal of Atmospheric and Oceanic Technology 35, 6; 10.1175/JTECH-D-17-0142.1

Figure 12 shows the comparisons of the ARs derived from the four approaches and the rain gauge observations at each gauge site. Generally, the site-by-site comparisons show that the hourly ARs estimated with VWAP are in good agreement with those observed by the rain gauges, with the averaged CC being 0.95 and the averaged NE being 0.45. VH07 is also satisfactory in this situation, but it is not as good as VWAP. The results for RKDP not surprisingly show the worst performance. Especially at the gauge sites with lighter ARs (see Fig. 10), the results for RKDP have low CCs and high NEs.

Fig. 12.
Fig. 12.

Hourly rainfall comparisons at the rain gauge sites for (a) VWAP, (b) VWOAP, (c) VH07, and (d) RKDP. The places where the rain gauges were deployed (circles). The size of the circles represents the correlation coefficient between the time series of the radar-derived ARs and the time series of the gauge-derived ARs, and the the normalized error between them (shading).

Citation: Journal of Atmospheric and Oceanic Technology 35, 6; 10.1175/JTECH-D-17-0142.1

c. QPEs in all seven events

To illustrate the case dependence of different QPE approaches, the CCs, RMSEs, and NEs for the hourly rainfall comparisons for all seven rainfall events are shown in Table 2. The QPE accuracies vary in different rainfall events, probably because of the DSD variability and different durations. However, VWAP generally has the best performance in all the events and RKDP has the worst performance. VWOAP and VH07 have comparable performance according to Table 2. The radar QPE in the event at 0400 ~ 0900 UTC 7 June exhibits the worst accuracy compared with the other events, mainly because of the short duration (about 5 h) and the smaller sampling number (about 500).

Table 2.

Case-by-case comparisons of the hourly rainfall amounts estimated based on four approaches and those derived from the rain gauges.

Table 2.

Figure 13 shows the comparison between the radar-derived hourly rainfall amounts from all the cases compared and those observed by gauges. Similar conclusions can be drawn from the comparison. The total sampling number is more than 40 000, which guarantees the objectivity of the evaluation. For VWAP and VWOAP, the hourly rainfall amounts are generally symmetrically distributed along the identity line. But for VH07 and RKDP, the results are not as good as the results for VWAP and VWOAP. The hourly rainfall amounts estimated based on VWAP coincide best with the gauge-based results, with CC being 0.92, RMSE being 2.52 mm, and NE being 0.42. For VWOAP, the results get a little bit worse than the results for VWAP. The difference mainly exists when the gauge-derived hourly rainfall amounts are less than 10 mm. The results based on VH07 are comparable with VWOAP, with similar values of CC, RMSE, and NE. The conventional power-law estimator-based QPE (RKDP) exhibits the worst performance. The CC is only 0.83, RMSE is about 4.1 mm, and NE is very large (1.02). The negative hourly rainfall amounts for RKDP mainly occur when the precipitation is light. From the gauge observation, most of the hourly rainfall amounts are less than 10 mm. Thus, negative KDP values estimated from the least squares fitting method have a very large impact on QPE.

Fig. 13.
Fig. 13.

Integrated comparisons of the hourly rainfalls derived from the gauges vs the hourly rainfall estimates from the PPI scans at the elevation angle of 1.5° based on (a) VWAP, (b) VWOAP, (c) VH07, and (d) RKDP for all seven cases listed in Table 2. The probability densities of the samples (shading), for which the category sizes of the rainfall are 1 mm.

Citation: Journal of Atmospheric and Oceanic Technology 35, 6; 10.1175/JTECH-D-17-0142.1

The QPE evaluations are also separated into two groups as shown in Table 3. The evaluation for the gauge-derived hourly rainfall amounts less than 10 mm represents the result for light rain. VWAP still shows the best performance with the highest CC and the lowest RMSE/NE. Meanwhile, the performance of VWOAP worsens as a result of the lack of information from ZH measurements. For light rain, the useful information in ΦDP is mainly outweighed by measurement errors. Nevertheless, the variational analysis still results in better QPE than the conventional RKDP, since the least squares fitting method has no capability to constrain KDP within the physical limits. For VH07, the performance is between VWAP and VWOAP. The evaluation for the gauge-derived hourly rainfall amounts greater than 10 mm is for heavy rain. VWOAP and RKDP now have similar performance, mainly because the least squares fitted KDP gets more accurate as the rainfall gets heavier. The results based on VWAP are still better than VWOAP, which means the ZH measurements still positively contribute to QPE. VH07 now has the worst performance, mainly because it is not tuned for southern China and its forward operator cannot capture the DSD characteristics of these rainfall events well.

Table 3.

Comparisons of the hourly rainfall amounts based on four approaches and those derived from the rain gauges for all seven cases listed in Table 2 under two different conditions, i.e., RA(gauge) ≤ 10 mm and RA(gauge) > 10 mm.

Table 3.

5. Conclusions and discussion

In this study a variational QPE approach based on ΦDP measurements is proposed. With the physical constraints utilized in the variational approach, erroneous negative rainfall-rate and KDP estimates (which is an evident disadvantage for the conventional least squares fitting–based KDP estimation and QPE) no longer exist. The introduction of the range smoothing filter is essentially a utilization of the continuity of the precipitation system and the spatial coherence of the rainfall rate, which makes the rainfall-rate estimates suffer less from the random errors in ΦDP measurements. However, oversmoothing will oversuppress the physical variation in the rainfall rates. The best smoothing factor is determined through the comparison experiments. The QPE is further improved via incorporating the rainfall rates estimated from ZH into the variational analysis as the a priori information. The weighting of the information from the ΦDP measurements and the a priori is decided by their statistical errors. This has been demonstrated via an idealized case in section 3. With the a priori utilized, the results from the variational approach outperform those from the linear programming method.

Four QPE approaches are applied to the observations of Guangzhou S-POL in May and June 2016 to show the performance of our proposed variational approach. Generally, the VWAP results are the best rainfall estimates because of the integrated utilization of ZH and ΦDP measurements. VH07 is on the basis of the ZR relation and the gamma distribution with a fixed μ. It has better performance than VWOAP when the rainfall is light [RA(gauge) < 10 mm] and more information is available from the measurements (ZH, ZDR, and ΦDP). However, even when only ΦDP measurements are used in VWOAP, QPE results are still better than VH07 when the rainfall is heavier [RA(gauge) > 10 mm], probably because of the inaccurate forward operator of VH07. The QPE results based on the conventional RKDP do not coincide well with the gauge-based results, especially when the rainfall is light. Unphysical negative rainfall amounts can result from the negative least squares–fitted KDP values. For heavier rainfall, RKDP shows comparable performance to VWOAP, but it is still not as good as VWAP.

According to the case-by-case evaluation (Table 2), QPE performance based on VWAP and VWOAP varies mainly because of the uncertainties in the statistical R–KDP relation and the ZHR relation. Different weather systems usually have different DSD characteristics and cannot be accurately characterized with fixed relations. Of course, if the microphysics of the weather system is very different from the climatic statistics and the statistical R–KDP and ZHR relations are not appropriate for characterization of the system, then the performance of the variational approach will be very poor. Nevertheless, the advantage of the forward operator in the proposed approach is that KDP is approximately the 4.2nd moment of DSD, and the R–KDP relation should not change very significantly from the climatic statistics. The inclusion of ZDR may help to reduce the impact of DSD uncertainty in the calculation of the a priori. However, the measurement errors of ZDR may deteriorate the accuracy of the a priori. The 0.2-dB error in ZDR can result in about 20% error in rainfall-rate estimates (Lee 2006). The accuracy of ZDR measurement is affected by the radar hardware, the surrounding environment, attenuation, and many other factors. Yet, the calibration of ZDR is not an easy task. It is not easy to ensure the accuracy of ZDR and to characterize the error of the a priori if ZDR is included. We use only R(ZH) as the a priori, and the uncertainties resulting from DSD variability and measurement errors are also considered as in the a priori error covariance matrix. Comparative studies in section 4 have demonstrated that the combination of the forward operator and the a priori helps to improve radar QPE. The variational approach should result in satisfactory QPE unless the DSD properties of the events are extremely different from the climatic statistics, which will also be a problem in other QPE approaches.

As we know, all kinds of radar QPE approaches suffer from two sources of uncertainties and errors, that is, the uncertainties of estimation models and the measurement errors. Usually, the combination of polarimetric variables can result in better estimation models, while QPE may be affected more by the measurement errors. Composing different rainfall estimators according to the rain type classification (e.g., separation of stratiform and convective precipitation or identification of coalescence-dominant precipitation) and the error analysis is a promising way to minimize the impact of uncertainties and errors (Porcacchia et al. 2017; Steiner et al. 1995). It will be included in the current variational QPE framework and discussed in future work.

Acknowledgments

This work was primarily supported by the National Key Research and Development Program of China under Grant 2017YFC1501703, the National Fundamental Research 973 Program of China (2013CB430101), the National Natural Science Foundation of China (Grants 41475015 and 41322032), the Open Research Program of the State Key Laboratory of Severe Weather, and the Beijige Open Research Fund for Nanjing Joint Center of Atmospheric Research. The authors thank Dr. Bowen Zhou (NJU) for the careful reading and editing of the manuscript. The authors also thank the scientists and engineers of NJU-2DVD, Guangzhou S-POL, and other instruments. The data are archived on a specialized computer at Nanjing University and are managed by the project office (yang.zhengwei@nju.edu.cn). These data can be accessed upon request, but their usage may be restricted by relevant Chinese government rules and regulations that are beyond the control of the authors.

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  • Zhang, G., 2016: Weather Radar Polarimetry. CRC Press, 322 pp.

  • Fig. 1.

    The topography around Guangzhou S-POL (black triangle). The elevation of the terrain is indicated (shading). The 2DVD (black dot) is used to derive the statistical property of the precipitation in this region.

  • Fig. 2.

    The statistical QPE relations R(KDP) for S-band radars obtained from the regression of the observations (gray dots).

  • Fig. 3.

    (a) The statistical RZH relation (black solid line) for S-band radars regressed from the observations (gray dots). (b) The normalized standard deviation in R for the ZHR relation caused by DSD variability (; black dashed–dotted line) and measurement noise (; black dashed line). The total error (; black solid line), which is a combination of and .

  • Fig. 4.

    The range profiles of the polarimetric variables and the rainfall rate used in the idealized case. (a) The intrinsic ϕDP (red line) and the simulated ΦDP (blue line) measurements, and the intrinsic KDP (green line). (b) The intrinsic ZH (red line) and the simulated ZH (blue line) measurements. (c) The intrinsic rainfall rate (red line), and the rainfall rates estimated from the statistical RKDP relation (green line) and the ZH–R relation (blue line).

  • Fig. 5.

    The statistics of the (a) CCs, (b) RMSEs, and (c) NEs corresponding to the 1000 range profiles of R estimated with the variational approach using different smoothing factors. The mean (black solid line), the 95th percentile (red dashed line), and the 5th percentile (green dashed–dotted line) of the statistical variables are marked.

  • Fig. 6.

    (a)–(d) The statistical comparisons of the R estimates and KDP estimates using the variational approach, and G13’s method based on 1000 range profiles of ZH and ΦDP with random errors. No a priori information is utilized in the variational estimation here. (a) Estimates of KDP based on the variational approach, (b) KDP estimates based on G13’s method, (c) R estimates based on the variational approach, and (d) R estimates based on G13’s method. The median values of the estimates are indicated (blue solid lines). The regions between the 5th and 95th percentiles are marked (gray shading). The intrinsic R and KDP are indicated (red solid lines). (e) The comparison of the standard errors in the R estimates using the variational approach (red solid line) and G13’s method (blue solid line).

  • Fig. 7.

    (a) An example of the a priori (black solid line) utilized in the optimization and its corresponding errors (error bars). The intrinsic R is shown (red solid line as a reference). (b) As in Fig. 6c, but for the R estimates with the a priori used in the variational approach. (c) The comparison of the errors (standard deviations) in the R estimates with the a priori used in the variational approach (red solid line), the R estimates with no a priori used in the variational approach (red dashed line), and the R estimates based on G13’s method (blue dashed line). The mean standard deviation of the errors of the a priori is also shown (black solid line).

  • Fig. 8.

    The PPI images of a precipitation event collected by Guangzhou S-POL at 0030 UTC 10 May 2016 at the elevation angle of 1.5°. (a) ZH (dBZ), (b) ρhv, (c) ZH after the removal of nonprecipitation echoes, and (d) the mask utilized to separate the precipitation and nonprecipitation echoes.

  • Fig. 9.

    The retrieval results for the radar scan shown in Fig. 8. (a) The unfolded ΦDP after the removal of the system phase, and (b) the optimized R from the proposed variational approach. (c) The optimized ϕDP and (d) the optimized KDP are shown.

  • Fig. 10.

    The difference in the total accumulated rainfall estimated from the rain gauges and the radar-derived total accumulated rainfall (the elevation angle of 1.5°) based on the proposed variational approach from the precipitation event from 0000 UTC 9 May to 1200 UTC 10 May 2016. The places where the rain gauges were deployed (circles). The size of the circles denotes the total accumulated rainfall from the rain gauges, and the relative difference (shading). The radar site position (black triangle) is marked.

  • Fig. 11.

    Scatterplot of the hourly rainfalls observed by all the rain gauges (shown in Fig. 10) vs the hourly rainfall estimates from the PPI scans at the elevation angle of 1.5° based on (a) VWAP, (b) VWOAP, (c) VH07, and (d) RKDP.

  • Fig. 12.

    Hourly rainfall comparisons at the rain gauge sites for (a) VWAP, (b) VWOAP, (c) VH07, and (d) RKDP. The places where the rain gauges were deployed (circles). The size of the circles represents the correlation coefficient between the time series of the radar-derived ARs and the time series of the gauge-derived ARs, and the the normalized error between them (shading).

  • Fig. 13.

    Integrated comparisons of the hourly rainfalls derived from the gauges vs the hourly rainfall estimates from the PPI scans at the elevation angle of 1.5° based on (a) VWAP, (b) VWOAP, (c) VH07, and (d) RKDP for all seven cases listed in Table 2. The probability densities of the samples (shading), for which the category sizes of the rainfall are 1 mm.

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