Improved Attenuation-Based Radar Precipitation Estimation Considering the Azimuthal Variabilities of Microphysical Properties

Hao Huang Key Laboratory for Mesoscale Severe Weather/MOE, and School of Atmospheric Sciences, Nanjing University, Nanjing, Jiangsu, China
State Key Laboratory of Severe Weather, and Joint Center for Atmospheric Radar Research, China Meteorological Administration, and Nanjing University, Beijing, China

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Kun Zhao Key Laboratory for Mesoscale Severe Weather/MOE, and School of Atmospheric Sciences, Nanjing University, Nanjing, Jiangsu, China
State Key Laboratory of Severe Weather, and Joint Center for Atmospheric Radar Research, China Meteorological Administration, and Nanjing University, Beijing, China

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Haonan Chen Cooperative Institute for Research in the Atmosphere, Fort Collins, Colorado
NOAA/Earth System Research Laboratory, Boulder, Colorado

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Dongming Hu Guangdong Meteorological Observatory, Guangzhou, Guangdong, China

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Peiling Fu Guangzhou Meteorological Observatory, Guangzhou, Guangdong, China

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Qing Lin Guangdong Meteorological Observatory, Guangzhou, Guangdong, China

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Zhengwei Yang Key Laboratory for Mesoscale Severe Weather/MOE, and School of Atmospheric Sciences, Nanjing University, Nanjing, Jiangsu, China
State Key Laboratory of Severe Weather, and Joint Center for Atmospheric Radar Research, China Meteorological Administration, and Nanjing University, Beijing, China

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Abstract

The attenuation-based rainfall estimator is less sensitive to the variability of raindrop size distributions (DSDs) than conventional radar rainfall estimators. For the attenuation-based quantitative precipitation estimation (QPE), the key is to accurately estimate the horizontal specific attenuation AH, which requires a good estimate of the ray-averaged ratio between AH and specific differential phase KDP, also known as the coefficient α. In this study, a variational approach is proposed to optimize the coefficient α for better estimates of AH and rainfall. The performance of the variational approach is illustrated using observations from an S-band operational weather radar with rigorous quality control in south China, by comparing against the α optimization approach using a slope of differential reflectivity ZDR dependence on horizontal reflectivity factor ZH. Similar to the ZDR-slope approach, the variational approach can obtain the optimized α consistent with the DSD properties of precipitation on a sweep-to-sweep basis. The attenuation-based hourly rainfall estimates using the sweep-averaged α values from these two approaches show comparable accuracy when verified against the gauge measurements. One advantage of the variational approach is its feasibility to optimize α for each radar ray, which mitigates the impact of the azimuthal DSD variabilities on rainfall estimation. It is found that, based on the optimized α for radar rays, the hourly rainfall amounts derived from the variational approach are consistent with gauge measurements, showing lower bias (1.0%), higher correlation coefficient (0.92), and lower root-mean-square error (2.35 mm) than the results based on the sweep-averaged α.

Corresponding author: Kun Zhao, zhaokun@nju.edu.cn

Abstract

The attenuation-based rainfall estimator is less sensitive to the variability of raindrop size distributions (DSDs) than conventional radar rainfall estimators. For the attenuation-based quantitative precipitation estimation (QPE), the key is to accurately estimate the horizontal specific attenuation AH, which requires a good estimate of the ray-averaged ratio between AH and specific differential phase KDP, also known as the coefficient α. In this study, a variational approach is proposed to optimize the coefficient α for better estimates of AH and rainfall. The performance of the variational approach is illustrated using observations from an S-band operational weather radar with rigorous quality control in south China, by comparing against the α optimization approach using a slope of differential reflectivity ZDR dependence on horizontal reflectivity factor ZH. Similar to the ZDR-slope approach, the variational approach can obtain the optimized α consistent with the DSD properties of precipitation on a sweep-to-sweep basis. The attenuation-based hourly rainfall estimates using the sweep-averaged α values from these two approaches show comparable accuracy when verified against the gauge measurements. One advantage of the variational approach is its feasibility to optimize α for each radar ray, which mitigates the impact of the azimuthal DSD variabilities on rainfall estimation. It is found that, based on the optimized α for radar rays, the hourly rainfall amounts derived from the variational approach are consistent with gauge measurements, showing lower bias (1.0%), higher correlation coefficient (0.92), and lower root-mean-square error (2.35 mm) than the results based on the sweep-averaged α.

Corresponding author: Kun Zhao, zhaokun@nju.edu.cn

1. Introduction

Quantitative precipitation estimation (QPE) is one of the major missions in radar meteorology (Bringi and Chandrasekar 2001; Zhang et al. 2016). The uncertainty in the raindrop size distributions (DSDs) of precipitation and radar measurement errors are two major error sources for radar QPE (Lee 2006). For conventional weather radars without polarimetric capability, QPE is achieved by assuming power-law relations between rainfall rate R and reflectivity factor Z, i.e., ZR relations (Marshall and Palmer 1948), which may suffer a lot from the DSD variabilities. After the polarimetric upgrade, weather radars can measure differential reflectivity factor ZDR and specific differential phase shift KDP/differential phase shift ΦDP in addition to the horizontal reflectivity factor ZH. By including ZDR and KDP in the power-law estimators, e.g., R(ZH, ZDR), R(KDP), and R(KDP, ZDR), the rainfall estimation will be less sensitive to the DSD uncertainty (Bringi et al. 2011; Chen et al. 2017; Ryzhkov et al. 2005a,b). By combining the polarimetric rainfall estimators and conventional R(Z) estimators according to their error structures in terms of DSD variabilities and measurement errors, the QPE performance can be further improved (Ryzhkov et al. 2005a,b).

Recently, the relation between R and specific attenuation A is utilized for QPE (Diederich et al. 2015; Ryzhkov et al. 2014; Wang et al. 2014). As documented in Ryzhkov et al. (2014), the R(A) estimator is less sensitive to DSD variabilities than R(Z) and R(KDP) estimators at S, C, and X bands. For attenuation-based QPE, the key factor is to estimate A accurately. In their work, A is estimated using the rain profiling algorithm called “ZPHI” from the attenuated Z and the given total path integrated attenuation (PIA) (Testud et al. 2000). One of the ZPHI algorithm’s advantages is its immunity to the miscalibration of Z. However, the accuracy of the A estimated by the ZPHI algorithm heavily depends on the accuracy of the PIA. In the ZPHI algorithm, the PIA is calculated from ΔΦDP multiplied by a coefficient α, where ΔΦDP is the change of differential phase shift over the propagation path and the coefficient α is the mean ratio of A to KDP. In the ZPHI attenuation estimation algorithm (Ryzhkov et al. 2014; Testud et al. 2000), a fixed α is utilized. However, α is sensitive to the DSD variabilities and environmental temperature, which limits the accuracy of the ZPHI algorithm. The ZPHI algorithm is extended by adjusting α through minimizing the difference between the measured ΦDP and reconstructed ΦDP from retrieved A (Bringi et al. 2001). However, since the coefficient α is sensitive to DSD variabilities, the constructed ΦDP may also lack precision (Gou et al. 2019; Gu et al. 2011; Ryzhkov et al. 2014). Gu et al. (2011) and Carey et al. (2000) tried to correct attenuation using the hot spot (HS) technique by adjusting the α values for the precipitation within the hot spot regions which usually have larger ZDR values. This technique can mitigate the impact of DSD variabilities on attenuation correction but still suffer from the uncertainties to some degree. For example, the identification of the HS regions is not easy especially for radar rays with multiple HS regions. In addition, only two discrete α values are used for precipitation within and beyond the HS regions, but the actual α values usually have a higher spatial variance.

The DSD characteristics significantly vary for different rain types (e.g., tropical and continental rainfall). Thus, a slope of the ZDR dependence on ZH in rain, which is an indicator of rain type, was used to optimize α on a sweep-to-sweep basis for attenuation-based QPE (Cocks et al. 2019; Ryzhkov and Zrnić 2019; Wang et al. 2019). This method is generally immune to the system biases in ZH and ZDR. And the optimized α values result in better rainfall estimates than a fixed α in their assessment (Cocks et al. 2019; Wang et al. 2019), making it promising for operational applications. Nevertheless, limitations exist in this method too, according to Wang et al. (2019). First, this method derives a single α value for a whole radar sweep. However, the DSD properties, especially the mean raindrop size, vary in different regions of a radar sweep. Using a single α value may introduce underestimation (overestimation) of rainfall for the echoes with small (large) mean raindrop size. On the other hand, to optimize an α value from the ZDR slope, this method requires a large number of valid rainfall samples; the threshold is 30 000 in Wang et al. (2019). If a radar sweep does not have a sufficient number of valid samples, a default or recently updated α value is used until a sufficient number of samples is accumulated. This strategy mitigates the impact of measurement errors and the skewed distribution of the ZHZDR pairs; however, this may enlarge the impact of the DSD variability for the sweeps without enough samples.

Considering the potential differences in DSD characteristics in different regions of a radar sweep, one may improve the performance of attenuation-based QPE if a radar sweep is segmented into subregions with different α values used for them. In our paper, this possibility is investigated using a variational algorithm. In this algorithm, the propagation effect (attenuation and phase shift) is considered in the forward operator and optimized when minimizing a cost function; the horizontal specific attenuation AH, differential attenuation ADP, and KDP are parameterized using ZH and ZDR. After estimating attenuation using the variational approach, the optimized coefficient α are derived for QPE based on R(A) estimators, in which the azimuthal variabilities of α are considered. The paper is organized as follows. The dataset used for this study and the relevant approaches for attenuation estimation and QPE are introduced in section 2. In section 3, the measurements of an operational radar and surrounding gauges in south China are used to evaluate the proposed algorithm. Conclusions and discussions are given in section 4.

2. Data and methods

a. Dataset

Influenced by the East Asian summer monsoon, heavy rainfall usually occurs in south China from May to June, causing enormous calamity (Luo et al. 2017). Since 2016, the Chinese operational radar network in south China has been gradually upgraded with polarimetric capability. As part of the operational radar network, the S-band polarimetric radar in Guangzhou City, i.e., GZ SPOL, has been proved to have good performance and flexibility for scientific studies (Huang et al. 2018; Wu et al. 2018). Therefore, GZ SPOL is used in this paper as the prototype for algorithm validation. Accordingly, all the retrieval models in the paper are constructed based on the DSD data collected from a two-dimensional video disdrometer (2DVD) in south China in May–July in 2016 and 2017. The location of GZSPOL and the 2DVD can be found in Fig. 1 of Huang et al. (2018). In the construction of the algorithms, polarimetric variables and R are calculated from the DSD data according to formulas in Cao et al. (2012), with the scattering amplitudes estimated using the T-matrix method (Mishchenko et al. 2000). The axis ratios and terminal velocities of raindrops are parameterized from raindrop diameters following Brandes et al. (2002). In the QPE evaluation, the algorithms are mainly applied to the GZ SPOL measurements in precipitation events during May and June 2016; the radar scans at 1.5° elevation angle are used. These radar-derived rainfall rates are temporally integrated and compared against the accumulated rainfall measured by gauges within the distance of 75 km from the radar, in which the gauge data are quality controlled using a speckle filter described in Huang et al. (2018).

In the variational approach, the quality of polarimetric radar data is crucial for its success. The quality control procedures are similar to those in Huang et al. (2018) and are listed below. For GZ SPOL, the reflectivity is calibrated by an internal system using generated test signals and the system is annually maintained, which guarantees the ZH precision to be within 1 dB (Chen et al. 2014). Then, to mitigate the impact of the nonmeteorological echoes on the retrieval, the data with the copolar correlation coefficient ρhv lower than 0.85 are removed. The remaining echoes having less than 20 consecutive range gates are also removed. The differential reflectivity ZDR is calibrated by monitoring the echo with ZH between 10 and 20 dBZ, for which the averaged intrinsic ZDR value is 0.14 dB according to DSD statistics. In the processing of ΦDP, the system bias is estimated from the statistics of the ΦDP values of the meteorological echo close to the radar site using the method in Maesaka et al. (2012). Then, the ΦDP measurements with the differences from the values at the neighboring range gates greater than 35° are removed and refilled by linear interpolation after unfolding. After the quality control, the random errors of ZH, ZDR, and ΦDP are assumed to be 1 dBZ, 0.2 dB, and 2.0°, respectively, according to the specifications of GZ SPOL (Huang et al. 2018).

b. Review of attenuation-based QPE

In Ryzhkov et al. (2014), attenuation is estimated using the ZPHI algorithm before QPE. The ZPHI algorithm was originally designed for spaceborne rain radar of the Tropical Rainfall Measuring Mission (TRMM). It was then applied to ground-based polarimetric radars by Testud et al. (2000) and has been utilized in many studies (Bringi et al. 2001; Gou et al. 2019; Gu et al. 2011; Ryzhkov et al. 2014). To derive the ZPHI algorithm, AH is assumed to be related to the horizontal reflectivity factor by

AH(r)=a(r)[Zh(r)]b,

where Zh is the horizontal reflectivity factor (mm6 m−3). Parameter b is nearly constant for GZ SPOL (b = 0.65) according to the scattering simulation using the DSD data. With a(r) assumed to be constant along each radar beam, the AH profile for the path between r0 and r1 can be estimated using the measured (i.e., attenuated) reflectivity factor Za and the corresponding total PIA, following

AH(r)=[Za(r)]b(e0.23bPIA1)I(r0,r1)+I(r,r1)(e0.23bPIA1),

where

I(r0,r1)=0.46br0r1[Za(s)]bds,

and

I(r,r1)=0.46brr1[Za(s)]bds.

Note that the uncertainty on attenuation estimation and QPE caused by the assumption of constant a(r) in (1) is less than 20% as shown in Ryzhkov et al. (2014). In (2), the only unknown for the AH estimation is the PIA between path (r0, r1). For polarimetric radars, PIA is usually calculated from path integrated differential phase (ΔΦDP) over the path (r0, r1) (Bringi et al. 2001; Ryzhkov et al. 2014; Testud et al. 2000) by

PIA(r0,r1)=α[ΦDP(r1)ΦDP(r0)]=αΔΦDP,

where α is the ray-averaged ratio between AH and KDP.

After estimating AH, the rainfall rate can be calculated from R(AH) estimators constructed for GZ SPOL. As documented in Ryzhkov et al. (2014), the relation between R and AH is less sensitive to DSD variabilities than R(ZH) and R(KDP) especially for S band, but it is sensitive to radar wavelength and temperature. To minimize the impact of these two factors, the R(AH) estimators are fitted at 0°–30°C with a 1°C interval for GZ SPOL using a nonlinear least squares approach. Taking the regression result at 20°C (R=2311.7AH0.95) as an example (Fig. 1), R can be well parameterized by AH. The correlation coefficient (CC) between the R values calculated from DSDs and those predicted from R(AH) is close to 1 and the root-mean-square error (RMSE) is low (1.99 mm h−1). The definition of CC, RMSE, normalized error (NE), and relative bias (RB) can be found in Huang et al. (2018, 2020). The dependence of R(AH) on temperature is also shown in Table 1. When temperature changes from 0° to 30°C, the exponent keeps nearly constant (0.95) and the intercept raises from 1361 to 2923. In practical applications, the R(AH) estimator for a specific radar range gate is selected from prefitted ones according to the temperature from the interpolation of the sounding measurements. The sounding station at Qingyuan (station number 59280) is within the coverage of GZ SPOL, and it provides measurements of temperature profiles twice a day.

Fig. 1.
Fig. 1.

Scatterplot of R directly calculated from the DSDs vs those calculated from R(AH) at 20°C for GZ SPOL simulated from the measurements of the 2DVD.

Citation: Journal of Hydrometeorology 21, 7; 10.1175/JHM-D-19-0265.1

Table 1.

The R(AH) estimators and α values at different temperatures (0°, 10°, 20°, and 30°C) for GZ SPOL.

Table 1.

The key for the attenuation-based QPE is the accuracy of the α used in (5). In the initial implementation of the ZPHI algorithm (Testud et al. 2000) and the attenuation-based QPE (Ryzhkov et al. 2014), a fixed α is used. However, as revealed by the simulation using DSD data (Fig. 2), the ratio of AH to KDP (α) changes in terms of mean raindrop size (denoted by mass-weighted mean diameter, Dm) as well as temperature. For a specific temperature, α generally decreases when Dm increases (Ryzhkov et al. 2014). As a result, the mean α can vary sweep by sweep or even ray by ray due to the spatial variation of precipitation. Two methods, i.e., the method based on the ZDR slope and the variational approach, for α optimization are introduced below.

Fig. 2.
Fig. 2.

Scatterplot of the mass-weighted mean diameter Dm vs AH/KDP at 0°C (red), 10°C (blue), 20°C (green), and 30°C (yellow) for GZ SPOL simulated from the measurements of the 2DVD.

Citation: Journal of Hydrometeorology 21, 7; 10.1175/JHM-D-19-0265.1

c. Estimating α using ZDR slope

As DSD properties can be indicated by the rainfall type, which can be distinguished from the slope of ZDR dependence on ZH, Wang et al. (2019) proposed to optimize α from a ZDR slope on a sweep-to-sweep basis. For a radar sweep, a ZDR slope K is calculated from the ZH and ZDR measurements. Then, a relation between α and K, which is constructed from DSD simulations, is adopted to calculate the α for the sweep. In our paper, for the verification of the variational approach detailed in the next subsection, we use the QPE results based on the ZDR slope method similar to that in Wang et al. (2019) as a benchmark.

In Wang et al. (2019), the bilinear αK relation is obtained from the DSD measurements in Oklahoma. The DSD data with simulated ZH values between 20 and 50 dBZ are separated into bins with 2-dBZ width. Within each bin, the simulated ZDR values are sorted and DSD data are separated into five groups with equal sizes according to the sorted ZDR values. For each group, the α value is calculated by AH/KDP, where summation is performed over all DSDs; the K value is the difference of the median ZDR values for two ZH bins centered at 20 and 50 dBZ divided by 30. Then, the bilinear αK relation is obtained from the five αK pairs. In our paper, this method is applied to DSD measurements in south China, as shown in Fig. 3. Since the rainfall in Guangzhou is more tropical than in Oklahoma, the αK pairs in Fig. 3 indicates that a larger α exists for a specific K than that in Oklahoma. It is also found that five points (Fig. 3) are not enough for a robust relation fitting. Thus, we manually adjust the relation from Oklahoma (the red dashed line) for south China (the blue dashed line), as follows:

α=0.0550.75K,ifK<0.0467,α=0.02,ifK0.0467.
Fig. 3.
Fig. 3.

The relation between factor α (dB per degree) and ZDR slope as simulated from DSD observations collected by a 2DVD in south China. Black triangles represent each simulated variable value for the 0%–20%, 20%–40%, 40%–60%, 60%–80%, 80%–100% percentile groups of ZDR. The red dashed line represents the bilinear fits from the DSDs in Oklahoma (Wang et al. 2019). The blue dashed line is the adjusted relation for south China.

Citation: Journal of Hydrometeorology 21, 7; 10.1175/JHM-D-19-0265.1

With (6), we can now determine the α from the ZHZDR pairs, for which the radar sweeps at 1.5° elevation are used. The ZHZDR pairs are selected by the criteria: 1) data below 4 km to mitigate potential contamination from ice particles; 2) ρhv > 0.98; 3) −4 < ZDR < 4 dB; and 4) 20 < ZH < 50 dBZ. The ZHZDR pairs are separated into 16 ZH bins with 2-dBZ width from 20 to 50 dBZ; within each bin, a median ZDR value is calculated if the sample size within the bin is larger than 200. Then, the ZDR slope K is obtained from the linear regression fitting of the median ZDR values in terms of the ZH bin centers. Afterward, the optimized α is calculated from K using the bilinear αK relation, i.e., (6). Note that if the sample size of the valid ZHZDR pairs in a radar sweep is smaller than 30 000, the recently updated α is used until enough samples are accumulated.

d. Estimating α using a variational approach

To get the α values on a ray-to-ray basis, a variational approach is proposed in this paper. In the variational approach, the forward operators are based on the relations among the polarimetric variables. According to Gorgucci et al. (1992), ZH (dBZ), ZDR (dB), and KDP (° km−1) triplets reside within a limited three-dimensional space for rainfall, and KDP can be well related to ZH and ZDR, known as a self-consistent relation KDP(ZH, ZDR). Similarly, AH (dB km−1) and ADP (dB km−1) can also be parameterized using ZH and ZDR, i.e., AH (ZH, ZDR) and ADP(ZH, ZDR). These relations can be sensitive to temperature. Thus, we fit these three relations at 0°–30°C with a 1°C interval using the 2DVD data for GZ SPOL. To parameterize KDP, AH, and ADP from ZH and ZDR, the relations are also selected from the prefitted ones according to the temperature from the sounding measurements. The examples of fitted KDP(ZH, ZDR), AH (ZH, ZDR), and ADP(ZH, ZDR) at 20°C are shown in Fig. 4, which can be formulated as

KDP=3.52×107×Zh1.00×(ZDR310.7ZDR2+45.1ZDR90.4),
AH=2.52×108×Zh1.07×(ZDR38.9ZDR2+26.7ZDR30.0),
ADP=1.03×1010×Zh0.99×(ZDR3+36.2ZDR2183.9ZDR+616.6),

where Zh (mm6 m−3) equals to 100.1ZH. It can be found all the KDP, AH, and ADP can be well parameterized by ZH and ZDR, with most of the samples located near the one-to-one lines, which means these relations are not very sensitive to DSD variabilities.

Fig. 4.
Fig. 4.

Scatterplots of the regression results of the parameterizations of (a) KDP, (b) AH, and (c) ADP in terms of ZH and ZDR at 20°C for GZ SPOL simulated from the measurements of the 2DVD. The text at the left top corner of the subplots indicates the fitted relations.

Citation: Journal of Hydrometeorology 21, 7; 10.1175/JHM-D-19-0265.1

With the above relations, the attenuation of a radar sweep can be estimated ray by ray. We first assume a state vector x containing the unattenuated ZH and ZDR for the n-gate radar ray,

x=[ZH(1),ZH(2),,ZH(n),ZDR(1),ZDR(2),,ZDR(n)]T.

The measurement vector y containing the measured reflectivity ZHm, measured differential reflectivity ZDRm, and measured differential phase shift ΦDPm at each ray can be expressed as

yZH=[ZHm(1),ZHm(2),,ZHm(n)]T,yZDR=[ZDRm(1),ZDRm(2),,ZDRm(n)]T,yΦDP=[ΦDPm(1),ΦDPm(2),,ΦDPm(n)]T,y=[yZH;yZDR;yΦDP].

Since KDP, AH, and ADP can be parameterized from the unattenuated ZH and ZDR using (7)(9), the polarimetric measurements can be predicted from x with the propagation effects (attenuation and phase accumulation) considered in the integration terms. This is known as the forward operator H(x). Then, the difference between the predicted and measured polarimetric measurements normalized by the error covariance matrix R is known as the cost function J,

J=Jb+Jy,Jb(x)=12(xxb)TB1(xxb),Jy(x)=12[H(x)y]TR1[H(x)y].

As the conventional cost function used in data assimilation, xb and B are the a priori and its error covariance matrix, and Jb is the corresponding cost function. In this study, we do not use any a priori constraint for x and Jb is set to 0 hereafter. In the error covariance matrix R for y, we assume the errors of ZH, ZDR, and ΦDP are generally independent from each other and are independent from gate to gate according to Hogan (2007) and Cao et al. (2013). Thus, R is composed of diagonal matrices RZH, RZDR, and RΦDP:

R=[RZHORZDRORΦDP]

with O representing the zero matrices.

To obtain the optimized polarimetric radar data, the key process is to minimize the cost function J which contains the nonlinear forward operator H(x), in which the limited-memory Broyden–Fletcher–Goldfarb–Shanno algorithm for bound constrained large-scale nonlinear optimization (LBFGS-B) is utilized (Byrd et al. 1995). Since the raindrops are statistically oblate (Brandes et al. 2002), nonnegative constraints are used for unattenuated ZDR in the optimization. The minimization of a radar ray starts from a first guess, we use 30 dBZ and 0.5 dB for all the ZH and ZDR of the ray. In later iterations of the minimization, the value of the cost function and its gradient in terms of x is required. The gradient term with the a priori term omitted can be written as

g(x)=HTR1[H(x)y],

where H is the Jacobian matrix of the forward operator. In the minimization, it is obtained using automatic differentiation (Hogan 2014).

To further increase the accuracy of the result, the continuity of the precipitation is further introduced to the variational analysis by using radial B-spline filters in a similar way to Huang et al. (2018). The B-spline filter can enforce the smoothness of the state variables, which could mitigate the impact of random measurement errors on the state variables. The formulas are similar to those used in Huang et al. (2018) and not repeated here.

After the minimization of the cost function, the unattenuated ZH and ZDR in the state vector are considered to be optimized for the ray. Then, the calculation is continued for the next radar ray in the same sweep until finished. Afterward, AH and KDP can be calculated from the optimized state variables using (7) and (8). Then, the α is estimated for each radar ray by AH/KDP where summation is performed for all valid gates of the ray. In addition, to better compare against the method based on the ZDR slope, we further calculate a single α for the whole sweep by AH/KDP where summation is performed for all valid gates of the sweep.

e. Different configurations for attenuation-based QPE

As documented in section 2b, the accuracy of attenuation-based rainfall estimation is closely related to the accuracy of α used for AH estimation. Six different configurations for PIA calculation in (5) and the R(A) estimation are considered for further comparison. The first one is to use α values fitted from the DSD measurements (AH = αKDP). To consider the sensitivity of α to temperature, the α values are fitted at 0°–30°C with a 1°C interval, with the examples shown in Table 1. In this configuration (denoted as CONV), the coefficient α in (5) is chosen from the fitted values according to the mean temperature of the radar sweep. In (5), another important issue is the way to determine ΔΦDP over the path (r0, r1). For CONV, the ΦDP measurements are filtered by a nine-gate running mean filter and the values of ΦDP(r0) and ΦDP(r1) are the median of the ΦDP values at the seven gates near the gates r0 and r1. The second configuration (denoted by SLP) uses the same way to determine ΔΦDP as for CONV, and its α is determined from the ZDR slope based on the αK relation adjusted for south China, i.e., (6).

The other four configurations involve the results from the variational analysis described in section 2d. In these four configurations, ΔΦDP is obtained from the difference of ϕDP between the gates r0 and r1, in which ϕDP is predicted from the optimized state variable of the variational retrieval. After the optimization, the ϕDP is close to ΦDP but with the random errors filtered out. In such a way, the ΔΦDP is expected to result in better attenuation and rainfall rate than the one used in CONV and SLP. For the selection of α, the third (fourth) configurations, denoted as CP (SP), uses the same strategy as in the CONV (SLP). For the fifth configuration, we use the sweep-averaged α calculated from the variational approach. This configuration is denoted as the VU (the variational approach with a uniform α). For the last configuration, the α estimated for each ray by the variational approach is used. In such a way, the impact of DSD variabilities on α is considered ray by ray, which is denoted as the VAZ (the variational approach with α values optimized for azimuths).

3. QPE evaluation

a. An example of attenuation estimation

On 28 May 2016, a mesoscale convective system passed over GZ SPOL. The plan position indicator (PPI) images of the radar scan at 1.5° elevation at 0600 UTC are shown in Fig. 5. The measured ZH, ZDR, and ΦDP are exhibited by Figs. 5a–c. Multiple quasi-linear convections are distributed to the southeast of the radar site, and large areas of stratiform precipitation exist to the northwest of the convections. In the stratiform regions, the ZDR values are generally low and the ΦDP values increase slowly; while in the convective regions where rainfall rate may be higher, the ZDR values are larger and the ΦDP values increase faster. The PIA optimized by the variational analysis from the radar site to each range gate is shown in Fig. 5d. Due to the generally higher AH associated with larger KDP, it can be found that the increase of PIA is larger in the heavy rainfall regions than the weak echo regions.

Fig. 5.
Fig. 5.

The PPI images of a precipitation event collected by GZ SPOL at 0600 UTC 28 May 2016 at the elevation angle of 1.5°. (a)–(c) The radar measurements of ZH, ZDR, and ΦDP, respectively; (d) the PIA calculated from the optimized state variables of the variational analysis.

Citation: Journal of Hydrometeorology 21, 7; 10.1175/JHM-D-19-0265.1

Figure 6 exhibits the further comparison of the α and total PIA based on different configurations of QPE for each ray of the scan in Fig. 5. To demonstrate the differences of DSD properties for different ray, we define a variable: the KDP-weighted ZDR (denoted as ZDRw), which is also shown in terms of azimuth in Fig. 6. The formula of ZDRw for an n-gate ray is

ZDRw=i=1nZDR(i)×KDP(i)i=1nKDP(i),

where ZDR(i) and KDP(i) are the ZDR and KDP at the ith gate calculated from the optimized state vector of the variational approach. In the definition of ZDRw, ZDR is an indicator of mean raindrop size, which is a monotonically increasing function of Dm if the gamma distribution with a fixed shape parameter is assumed; KDP has a quasi-linear relation with rainfall rate (Bringi and Chandrasekar 2001). Thus, ZDRw can be used as a proxy for rainfall rate-weighted mean raindrop size, similar to that defined in Ryzhkov et al. (2005a). In Fig. 6a, it is found that the ray-averaged α values from the variational approach for the VAZ (the green solid line) show large azimuthal variances. The maximum value (~0.47 dB per degree) is approximately twice the minimum value (~0.23 dB per degree). Despite the impact of measurement errors, most of the variances are caused by the differences in the DSD properties of different rays. The ray-averaged α values show clear relations with the ZDRw; the α values for the VAZ generally decreases when the ZDRw increases in the azimuth. This is consistent with decreasing in α with the increase of Dm in Fig. 2, which may indicate that the impact of azimuthal DSD variabilities can be accounted for by the ray-averaged α values from the variational approach. For the azimuthal angles between 135° and 190° where most convective lines with larger ZDR exist, it can be found that the ray-averaged α values are generally low compared to the other regions. For the CONV, CP, SLP, SP, and VU, constant α values are used for different azimuths in the PIA calculation. The α determined from the ZDR slope of the radar sweep for the SLP and SP (red line) is about 0.033 dB per degree, and it is close to the sweep-averaged value from the variational approach for the VU (blue line). The α determined from DSD statistics for the CONV and CP (black line) is generally too low compared to the optimized values from the other two methods.

Fig. 6.
Fig. 6.

(a) The α used for the CONV/CP, SLP/SP, VU, and VAZ and (b) the corresponding max PIA as functions of azimuth angle corresponding to the radar scan in Fig. 5. (c) The ZDRw for each ray is also shown.

Citation: Journal of Hydrometeorology 21, 7; 10.1175/JHM-D-19-0265.1

In Fig. 6b, due to the differences in the α values, the maximum PIA values for different QPE configurations are not the same. The PIA values from the SP and VU are close because of the close α values. For the azimuthal angles where most convective lines with larger ZDR exist (135°–190°), the PIA values estimated for the VAZ are generally smaller than the ones for the SP and VU. This may indicate the potential overestimation of PIA and the rainfall rate in the convective regions, which is also pointed out by Wang et al. (2019).

b. QPE evaluation in seven cases

The comparison of radar-derived hourly rainfall against the gauge measurements in seven cases in May–June 2016 is shown in Table 2. The performance of the attenuation-based QPE approaches varies in different cases. With no DSD uncertainty considered, the performance of the CONV is generally the worst, with generally the lowest CC, highest RMSE, highest NE, and highest RB in each case. For the SLP, the α values are optimized by ZDR slopes for radar sweeps, and the SLP results in better rainfall estimates than the CONV. Compared to the CONV and SLP, the ΔΦDP obtained from the forward operator of the variational approach is used for the calculation of PIA in (5) by the CP and SP. As a result, increases in CC values and decreases in RMSE values can be found for the results of the SP (CP) compared to those for the SLP (CONV). Similar to the SLP and SP, the α values are optimized for each sweep for the VU with the variational approach applied. As a result, the performance of the VU is also close to the SP. The accuracy of the VAZ is the best among the six approaches, with the highest CC, the lowest RMSE, and the lowest NE, while the RB values for the VAZ are close to those for the VU. The good performance of the VAZ may be attributed to the consideration of azimuthal DSD variabilities in the ray-averaged α.

Table 2.

The comparisons of the hourly rainfall estimated based on four configurations and those derived from the rain gauges in seven precipitation events in May–June 2016, including the correlation coefficient (CC), root-mean-squared error (RMSE), normalized error (NE), and relative bias (RB).

Table 2.

Since the performance of the SP is similar to that of the VU, we compare the α values for the two configurations in Fig. 7. These two configurations aim to considered the DSD properties in the optimization of α by using the αK relation (the SP) and the variational analysis (the UV). It can be found that the general trends of the α derived from the ZDR slope and the sweep-averaged α from the variational analysis are similar. For example, in the precipitation event between 0000 UTC 27 May and 1200 UTC 28 May, the α values for the SP and UV are very close. But differences still exist in the sweep-averaged α values from these two configurations. This is mainly because of their different retrieval basis. The ZDR-slope algorithm mainly involves an assumption of the relation between the mean raindrop size and the ZDR slope K, while the variational approach uses the relations among polarimetric variables which has accounted for most DSD variabilities. Besides, in the α estimation from ZDR slope, the α estimates only update when enough valid samples exist or accumulate, which results in the steps in the corresponding α curves. This strategy mitigates the impact of measurement errors while gains the impact of DSD variabilities for the sweeps without enough samples for α estimation. On the other hand, one sweep-averaged α value is determined for one radar sweep in the variational method. For the radar sweep with a small sample size, larger α uncertainty may be caused by the measurement errors. This could be the reason for the larger sweep-to-sweep α variabilities between 0000 and 1200 UTC 9 May. However, considering the performance of the UV, this uncertainty caused by measurement errors is acceptable for the accuracy of QPE and is not further processed here. In Fig. 7, the α values (about 0.023 dB per degree in all the cases) from DSD statistics are also shown for comparison, which results in the relatively worse performance of CONV and CP.

Fig. 7.
Fig. 7.

The time series of sweep-averaged α values (dB per degree) (blue solid lines) obtained by the variational approach (i.e., the ones used in the UV) for the precipitation events shown in Table 2. The black (red) solid lines represent the α values used in the CP and CONV (SLP and SP).

Citation: Journal of Hydrometeorology 21, 7; 10.1175/JHM-D-19-0265.1

The performance of the VAZ is the best compared to the other configurations denoted by Table 2. This is mainly because the α value is optimized for each radar ray for the VAZ, in which the DSD properties, e.g., smaller or larger mean raindrop size, in the radar ray are considered, which has been revealed by Fig. 6. To further illustrate this, it is useful to show the azimuthal and temporal variabilities in the optimized α values and the ZDRw, which is an indicator of the mean raindrop size for rays. The result for the precipitation event over the period 0000 UTC 27 May–1200 UTC 28 May 2016 is shown in Fig. 8. Besides the temporal variabilities which have been shown by the sweep-averaged α values used by the VU (Fig. 7), the ray-optimized α values (Fig. 8a) also show clear azimuthal variabilities for the whole event. Comparing the α values optimized for the radar rays (Fig. 8a) with the ZDRw values (Fig. 8b), it can be found that the larger α values are generally corresponding to the lower ZDRw, which indicates lower mean raindrop sizes. This is consistent with the change of α in terms of Dm in Fig. 2. If a uniform α is used for a whole sweep, the rainfall rate tends to be overestimated (underestimated) for the rays with larger (smaller) mean raindrop sizes. By considering the azimuthal DSD variabilities (indicated by the values of ZDRw) in the optimization of α values, the VAZ results more accurate rainfall estimation than the other configurations with higher CC, lower RMSE and lower NE values (Table 2).

Fig. 8.
Fig. 8.

The ray-averaged (a) α and(b) ZDRw for radar rays as functions of time and azimuth in the precipitation event occurred from 0000 UTC 27 May to 1200 UTC 28 May 2016.

Citation: Journal of Hydrometeorology 21, 7; 10.1175/JHM-D-19-0265.1

To show the overall performance of rainfall estimation in May–June 2016, Fig. 9 exhibits the comparisons of radar-derived hourly rainfall using the above six QPE configurations against the gauge measurements in all the seven cases, in which the total sample size exceeds 40 000. Similar to the results shown in Table 2, the hourly rainfall estimated by the CONV (Fig. 9a) has the worst performance due to the utilization of fixed α values derived from the statistics of the DSD observations; clear underestimation (RB = −10.4%) can also be found along with the lowest CC (0.90), largest RMSE (2.75 mm), and largest NE (0.44). The results of the CP (Fig. 9c) are better due to the utilization of ΔΦDP from the variational analysis, but underestimation still exists. Compared to the CONV (Fig. 9a), the SLP (Fig. 9b) uses the α values optimized by the ZDR slope, which results in the slightly lower RMSE (2.70 m), NE (0.43), and RB (−4.5%). The improvement over the results from the SLP (Fig. 9a) can also be found for those from the SP (Fig. 9d) for which the ΔΦDP has been optimized. This improvement is even larger than the improvement of the SLP over the CONV in our cases, especially according to the decreases of the RMSE and NE. For the VU which also uses sweep-averaged α values but from the variational analysis (Fig. 9e), the performance is similar to that of the SP (Fig. 9d), with similar CC, RMSE, and NE. In the VAZ, due to the consideration of azimuthal DSD variabilities (see Fig. 8a), the rainfall estimates have the highest CC (0.92), lowest RMSE (2.35 mm), lowest NE, and lowest RB in the comparison against the gauge measurements, which proves that the VAZ is the best configuration for attenuation-based QPE.

Fig. 9.
Fig. 9.

The joint distributions (shading) of the hourly rainfall amounts from the gauge measurements vs those derived from the PPI scans of GZ SPOL at the elevation angle of 1.5° for all seven cases listed in Table 2. The category sizes of the hourly rainfall amounts are 1 mm. (a)–(f) The results for the attenuation-based rainfall estimation using the six different configurations: CONV, SLP, CP, SP, VU, and VAZ, respectively.

Citation: Journal of Hydrometeorology 21, 7; 10.1175/JHM-D-19-0265.1

At last, the verification of radar-derived total accumulated rainfalls against the gauge measurements in the cases in Table 2 is shown in Fig. 10. Since the CONV and SLP have worse performance for the estimation of hourly rainfalls than the other configurations according to Fig. 9, their results for the total accumulated rainfalls are not shown here. In Fig. 10, it is found that all the three configurations (SP, VU, and VAZ) with α values optimized can generally result in good total accumulated rainfalls in May–June 2016, except some rainfall underestimation far from the radar site. At these range gates farther away from the radar site, the sampling altitudes are higher. Due to the dominance of the collision–coalescence process, the rainfall rate of south China generally increases toward the ground (Liu and Zipser 2013; Wu et al. 2018), which probably is the reason for the underestimation. Besides, the terrain is also higher in these regions far away from the radar site, which could cause larger uncertainty. The improvement of results based on the VAZ over those from the VU and the SP is not very significant, mainly because the impact of DSD uncertainty in different azimuths becomes lower in the long-term accumulation of rainfall. In contrast to the results of the SP, the VU, and the VAZ, the results based on the CP show more significant underestimation. This is because the α values only from DSD statistics are too low for these cases.

Fig. 10.
Fig. 10.

Comparisons of the total accumulated rainfalls derived from the gauges and those from the PPI scans at the elevation angle of 1.5° based on (a) the CP, (b) the SP, (c) the VU, and (d) the VAZ in all the cases listed in Table 2. The positions of the circles show the gauge locations. The size of the circles denotes the total accumulated rainfall from the rain gauges, and the color shading shows the relative bias of the radar-derived rainfalls. The gray shading represents the terrain height. The radar site position is marked as black triangles.

Citation: Journal of Hydrometeorology 21, 7; 10.1175/JHM-D-19-0265.1

4. Conclusions and discussion

In consideration of the good relation between R and AH, the AH values estimated from the ZPHI algorithm are used for rainfall estimation for GZ SPOL in south China. A variational approach based on the relations among polarimetric variables is proposed to optimize the coefficient α in the ZPHI algorithm, which is compared against an approach for α optimization based on a slope of ZDR dependence on ZH. Six configurations for attenuation-based rainfall estimation are designed to show the performance of these approaches.

Generally, the hourly rainfall based on the CONV has the worst performance with a significant underestimation, mainly because the α values are derived from the climatic measurements of 2DVD without consideration of the DSD variabilities in specific cases. The SLP with α optimized based on the ZDR slope for each radar sweep shows better performance (a lower system bias) than the CONV. Besides the accuracy of α, the random errors in ΦDP also have an impact on the accuracy of the estimation of PIA, which further affects the accuracy of attenuation and rainfall estimation. Once the ϕDP optimized by the variational approach is utilized for the calculation of the ΔΦDP in the CP (SP) configuration, the CC between the radar-derived hourly rainfall against the gauge measurements increases and the RMSE/NE decreases compared to the results of the CONV (SLP). However, the system bias generally has a minor change according to the RB values. Similar to the SP, sweep-averaged α values are derived from the variational approach, which is used for QPE in the VU. The VU has a similar performance to the SP for rainfall estimation (CC = 0.92, RMSE = 2.56 mm, and NE = 0.39). The advantage of the variational approach is that an averaged α value can be calculated for each radar ray instead of the whole radar sweep. In such a way, the azimuthal DSD variability can be mitigated in the α optimization and the corresponding QPE. For the VAZ which uses the ray-averaged α, the accuracy of the estimated hourly rainfall is further improved than the VU and SP. The RMSE decreases about 8% of that for the VU (from 2.56 to 2.35 mm). According to the above results, the VAZ proves to be the best configuration for attenuation-based QPE. The results from the VAZ can be also compared against those obtained by the variational approach based on the RKDP relation used in Huang et al. (2018). It is revealed that the results from the VAZ have better performance with larger CC and lower RMSE values, which is mainly because the RAH relation suffers less from the DSD variabilities.

Although the variational approach has a good performance in the optimization of α and the rainfall estimation shown by the previous comparison, there are also limitations. First, even the relations among the polarimetric variables in Fig. 4 are generally solid, some outliers still exist especially in the parameterization of AH (Fig. 4b). Besides, in an individual case, the DSD variabilities can still cause different levels of uncertainty in the forward operator based on the parameterization of KDP, AH, and ADP in terms of ZH and ZDR, which will further affect the QPE performance. This is probably why the QPE performance in different rainfall events (Table 2) varies.

Since the forward operator involves ZH and ZDR, the variational approach requires a good data quality of polarimetric variables. In our study, the quality control procedures are used to guarantee the accuracy of the radar variables for the variational analysis. However, currently, good data quality cannot always be expected from all operational radars due to the common calibration issues (Zittel et al. 2015), which may limit the applicability of the variational approach. As shown by Wang et al. (2019), the approach for α optimization based on the ZDR slope is less affected by miscalibration, even though the azimuthal changes of the system biases of ZH and ZDR can still impact its accuracy. Currently, the α optimization approach based on the ZDR slope is more broadly applicable than the variational approach considering the data quality issues. However, the results of the VAZ demonstrate the potential room for improvement by considering the azimuthal variability of α. It is also possible for the variational approach to segment the rainfall into finer subregions, for example, not only in azimuth but also in different ranges. However, in the meantime, a finer segmentation can reduce the sample size for α optimization and the impact of the radar measurement errors can increase. This topic is left for the future studies.

It is also worth mentioning that fixed standard deviations of the measurement errors of ZH (1 dB), ZDR (0.2 dB), and ΦDP (2°) are used for the variational analysis according to the radar specification. This may not be realistic since the measurement errors of ZH, ZDR, and ΦDP can increase when the signal-to-noise ratio or ρhv decreases. It is not easy to accurately account for this impact; however, this impact on retrieval is expected to be low because the measurement errors of ZH, ZDR, and ΦDP are in the denominator terms of the cost function and can partially cancel with each other when they increase simultaneously.

Acknowledgments

The authors would also like to thank the scientists and engineers working on GZ SPOL, 2DVD, and the other instruments used in this study. These data used in this study 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. This work is primarily supported by the National Key Research and Development Program of China (Grant 2018YFC1506404 and Grant 2017YFC1501703) and the National Natural Science Foundation of China (Grant 41905019 and 41875053), the Open Research Program of the State Key Laboratory of Severe Weather, the Operational Technology Experiment and Demonstration Project for Hail Suppression and Precipitation Enhancement in the East Side of Taihang Mountain, China (HBRUWCSY_2017_04), the 5th “333 High-level Personnel Training Project” of Jiangsu Province (BRA2019037), the Guangzhou Science and Technology Project (201804020038), and the Science and Technology Project of Zhejiang Province (Grant 2017C03035).

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  • Brandes, E. A., G. Zhang, and J. Vivekanandan, 2002: Experiments in rainfall estimation with a polarimetric radar in a subtropical environment. J. Appl. Meteor., 41, 674685, https://doi.org/10.1175/1520-0450(2002)041<0674:EIREWA>2.0.CO;2.

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  • Fig. 1.

    Scatterplot of R directly calculated from the DSDs vs those calculated from R(AH) at 20°C for GZ SPOL simulated from the measurements of the 2DVD.

  • Fig. 2.

    Scatterplot of the mass-weighted mean diameter Dm vs AH/KDP at 0°C (red), 10°C (blue), 20°C (green), and 30°C (yellow) for GZ SPOL simulated from the measurements of the 2DVD.

  • Fig. 3.

    The relation between factor α (dB per degree) and ZDR slope as simulated from DSD observations collected by a 2DVD in south China. Black triangles represent each simulated variable value for the 0%–20%, 20%–40%, 40%–60%, 60%–80%, 80%–100% percentile groups of ZDR. The red dashed line represents the bilinear fits from the DSDs in Oklahoma (Wang et al. 2019). The blue dashed line is the adjusted relation for south China.

  • Fig. 4.

    Scatterplots of the regression results of the parameterizations of (a) KDP, (b) AH, and (c) ADP in terms of ZH and ZDR at 20°C for GZ SPOL simulated from the measurements of the 2DVD. The text at the left top corner of the subplots indicates the fitted relations.

  • Fig. 5.

    The PPI images of a precipitation event collected by GZ SPOL at 0600 UTC 28 May 2016 at the elevation angle of 1.5°. (a)–(c) The radar measurements of ZH, ZDR, and ΦDP, respectively; (d) the PIA calculated from the optimized state variables of the variational analysis.

  • Fig. 6.

    (a) The α used for the CONV/CP, SLP/SP, VU, and VAZ and (b) the corresponding max PIA as functions of azimuth angle corresponding to the radar scan in Fig. 5. (c) The ZDRw for each ray is also shown.

  • Fig. 7.

    The time series of sweep-averaged α values (dB per degree) (blue solid lines) obtained by the variational approach (i.e., the ones used in the UV) for the precipitation events shown in Table 2. The black (red) solid lines represent the α values used in the CP and CONV (SLP and SP).

  • Fig. 8.

    The ray-averaged (a) α and(b) ZDRw for radar rays as functions of time and azimuth in the precipitation event occurred from 0000 UTC 27 May to 1200 UTC 28 May 2016.

  • Fig. 9.

    The joint distributions (shading) of the hourly rainfall amounts from the gauge measurements vs those derived from the PPI scans of GZ SPOL at the elevation angle of 1.5° for all seven cases listed in Table 2. The category sizes of the hourly rainfall amounts are 1 mm. (a)–(f) The results for the attenuation-based rainfall estimation using the six different configurations: CONV, SLP, CP, SP, VU, and VAZ, respectively.

  • Fig. 10.

    Comparisons of the total accumulated rainfalls derived from the gauges and those from the PPI scans at the elevation angle of 1.5° based on (a) the CP, (b) the SP, (c) the VU, and (d) the VAZ in all the cases listed in Table 2. The positions of the circles show the gauge locations. The size of the circles denotes the total accumulated rainfall from the rain gauges, and the color shading shows the relative bias of the radar-derived rainfalls. The gray shading represents the terrain height. The radar site position is marked as black triangles.

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