## 1. Introduction

Conventional S- and C-band weather radars have been used for several decades to monitor the evolution of precipitation. In recent years the technology of those conventional radars has been upgraded to polarimetric technology in order to further improve weather radar measurements (Doviak et al. 2000). Severe weather can produce rapid and localized surface damage associated with, for example, heavy rain and tornadoes. In this context, a network of small polarimetric X-band weather radars may be suitable to obtain observations of fast-developing storms at close range and at resolutions higher than those from conventional radars (McLaughlin et al. 2009; Chandrasekar et al. 2018).

*ε*(°) on the order of few degrees. In general, a

*r*(km) indicates the distance from the radar. Two useful variables that can be estimated from

^{−1}) and the specific attenuation

*A*(dB km

^{−1}), which are commonly used for the estimation of rainfall rate and attenuation correction (Bringi and Chandrasekar 2001).

The traditional method to estimate *Z* (dB*Z*) affected by radar calibration and PBB (Giangrande and Ryzhkov 2005). In addition,

Existing methods to estimate *A* in rain assume that *A* = *α* is a constant for a given frequency (Bringi et al. 1990). Testud et al. (2000) also used the relation between *A* and *A* in terms of the difference of *Z*, avoiding *α* is sensitive to temperature, drop size distribution (DSD), and drop size variabilities; therefore, Bringi et al. (2001) extended the ZPHI technique to avoid a priori value for *α*. These methods have been adapted to address attenuation problems at X-band frequencies (Matrosov et al. 2002; Park et al. 2005a; Gorgucci et al. 2006; Lim and Chandrasekar 2016). Moreover, Ryzhkov et al. (2014), Wang et al. (2014), and Diederich et al. (2015) modified the extended ZPHI method to improve rainfall-rate estimation and to demonstrate that *A* can be used to reduce issues related to radar calibration and PBB. Despite these promising benefits, the potential of using *A* might be limited depending on the approach to obtain *α* (Bringi et al. 2001; Ryzhkov and Zrnić 2005).

In contrast to *A*, limited research has been conducted on the applications of

The purpose of this work is to 1) explore the role and impact of estimated *α* and *A* over short paths and 2) develop a technique to compute *A*. In section 3, the performances of the attenuation correction methods that assume a constant *α* are compared using four storm events. This comparison is extended in section 4 to examine the selection of *α* profile by profile and its impact on *A* and *Z*. In section 5, the *α*, *A*, *Z*, and

## 2. Estimation techniques for ${\mathrm{\Psi}}_{\text{DP}}$ -based variables

### a. Estimation of ${K}_{\text{DP}}$

In the conventional technique given by Hubbert and Bringi (1995), a low-pass filter is designed such that gate-to-gate fluctuations at scales of the range resolution *τ* (°) is required, which is on the order of 1–2 times the standard deviation of

^{−1}) is given by Reinoso-Rondinel et al. (2018). For notation purposes, the difference of a radar variable

*V*over a given pathlength is expressed as

*Z*and

*i*, located at range

*Z*and

*M*represents the number of

*Z*and

### b. Estimation of A

For attenuation correction purposes, *Z* and *z* (dB*Z*) and

Bringi et al. (1990) introduced the differential phase (DP) approach such that *α* [dB (°)^{−1}] is assumed to be a constant coefficient. Gorgucci and Chandrasekar (2005) studied the accuracy of this method using simulated radar variables at X-band frequencies and showed that estimates of *A* are very sensitive to inaccurate estimates of *Z* values associated with only a slight degradation of the average error for attenuation correction, ±1.5 dB.

To improve the DP method, Testud et al. (2000) introduced the ZPHI method that estimates

*α*may lead to limited approximations of

*α*is sensitive to DSD, drop shape, and temperature variabilities (Jameson 1992). To take into account the sensitivity of

*α*, Bringi et al. (2001) extended the ZPHI method to search for optimal

*α*values at C-band frequencies, called the CZPHI method. An initial value for

*α*is selected from a predefined interval [

*α*, the optimal

*α*is the one that minimizes the error

*E*(°) given by

*α*to estimate

*A*and PIA depends on the performance of the chosen approach to estimate

To determine PIA^{−1}) that is given by *γ* to be constant, whereas the CZPHI technique searches for an optimal *γ*, addressing its sensitivity to DSD variability (i.e., rain type). However, such sensitivity of *γ* is less at X-band frequencies than at C- and S-band frequencies (Ryzhkov et al. 2014). In this work, *A*(CZPHI) represents the specific attenuation determined by the CZPHI approach and *γ* is assumed a constant.

Representative values for *α* and *γ* at X-band frequencies can be given by the mean fit of simulated polarimetric relations using a large set of DSDs and different drop shapes and temperatures. For example, Kim et al. (2010) and Ryzhkov et al. (2014) demonstrated that *α* values vary in the interval [0.1; 0.6] dB (°)^{−1}, and Otto and Russchenberg (2011) obtained an average value of 0.34 dB (°)^{−1} for *α* and for *γ* a value of 0.1618. Similar results were suggested by Testud et al. (2000), ^{−1}; Kim et al. (2010), ^{−1}; and Snyder et al. (2010) ^{−1}; while Ryzhkov et al. (2014) estimated *γ* equal to 0.14 for tropical rain (i.e., low *α*. For example, Bringi and Chandrasekar (2001) simulated polarimetric variables in rain and indicated that ^{−1}. Matrosov et al. (2014) avoided simulations by using observations resulting from collocated X- and S-band radars and found *α* in the range of 0.20–0.31 dB (°)^{−1}. Thus, a representative value for *α* can vary depending on models and assumptions used to simulate polarimetric variables, on the type of observed storms and their geographical locations, and on the accuracy of measurements.

### c. Estimation technique for ${\delta}_{\text{hv}}$

A *A* field estimated by the CZPHI method. Given these inputs, the resulting

- Design and apply a filter to smooth strong outliers from a
${\mathrm{\Psi}}_{\text{DP}}$ profile, taking$\mathrm{\Delta}r$ into account. Correct each smoothed${\mathrm{\Psi}}_{\text{DP}}^{\prime}$ profile for system phase offset by subtracting the mean of${\mathrm{\Psi}}_{\text{DP}}^{\prime}$ over the first 5% of measured gates. - Obtain
${\mathrm{\Phi}}_{\text{DP}}$ by integrating profiles of*A*, if they are associated with a minimum error*E*, otherwise by integrating${K}_{\text{DP}}$ profiles. Next, subtract${\mathrm{\Phi}}_{\text{DP}}$ from${\mathrm{\Psi}}_{\text{DP}}^{\prime}$ , profile by profile, as a first attempt to estimate the corresponding${\delta}_{\text{hv}}$ field. The next steps are related to 2D processing. - Remove unusual
${\delta}_{\text{hv}}$ values larger than 12° from the${\delta}_{\text{hv}}$ field. According to Testud et al. (2000), Trömel et al. (2013), and Schneebeli et al. (2014), the simulated${\delta}_{\text{hv}}$ values at X-band frequencies rarely reach 12°. The remaining noise in${\delta}_{\text{hv}}$ is reduced by assuming that similar values of${\delta}_{\text{hv}}$ are collocated with similar values of${K}_{\text{DP}}$ as follows. Set${K}_{\mathrm{min}}$ as the minimum of${K}_{\text{DP}}$ and${K}_{\mathrm{max}}$ as${K}_{\mathrm{min}}+{\mathrm{\Delta}}_{K}$ , where${\mathrm{\Delta}}_{K}$ (° km^{−1}) is given by Eq. (4). Define*S*as a set of${\delta}_{\text{hv}}$ samples, whose gates are collocated with${K}_{\text{DP}}$ values in the interval [${K}_{\mathrm{min}};{K}_{\mathrm{max}}$ ]. Reject${\delta}_{\text{hv}}$ samples from*S*that are outside the interval [${\overline{\delta}}_{\text{hv}}-\upsilon {\sigma}_{{\delta}_{\text{hv}}}$ ;${\overline{\delta}}_{\text{hv}}+\upsilon {\sigma}_{{\delta}_{\text{hv}}}$ ], where${\overline{\delta}}_{\text{hv}}$ and${\sigma}_{{\delta}_{\text{hv}}}$ indicate the arithmetic mean and the standard deviation of the samples in*S*, respectively;*υ*is a predefined threshold in the interval [$1;2$ ] and a value of 1 is chosen. This process is iterated by shifting [${K}_{\mathrm{min}};{K}_{\mathrm{max}}$ ] toward high values in small steps such that${K}_{\mathrm{min}}$ =${K}_{\mathrm{max}}$ and${K}_{\mathrm{max}}$ =${K}_{\mathrm{min}}+{\mathrm{\Delta}}_{K}$ until${K}_{\mathrm{max}}$ is equal to the maximum of${K}_{\text{DP}}$ . To obtain sufficient samples in*S*,${\mathrm{\Delta}}_{K}$ is given as${\mathrm{\Delta}}_{K}=\{\begin{array}{cc}0.2& {K}_{\mathrm{min}}\le 2.5\xb0\mathrm{k}{\mathrm{m}}^{-1},\\ 0.5& 2.5\xb0<{K}_{\mathrm{min}}<8\xb0\mathrm{k}{\mathrm{m}}^{-1},\\ 1.0& {K}_{\mathrm{min}}\ge 8\xb0\mathrm{k}{\mathrm{m}}^{-1},\end{array}$because high${K}_{\text{DP}}$ values are less frequent than small${K}_{\text{DP}}$ values (e.g., see the${K}_{\text{DP}}$ fields in Figs. 3, 8, and 11). - Apply a 2D interpolation method to fill empty gaps on
${\delta}_{\text{hv}}$ caused by step 3. For this task, the inpainting (or image fill-in) algorithm (Bertalmio et al. 2003; Criminisi et al. 2004; Elad et al. 2005) is selected because it is one of the image processing algorithms commonly used to smoothly interpolate 2D images. The essential idea is to formulate a partial differential equation (PDE) for the “hole” (interior unknowns) and to use the perimeter of the hole to obtain boundary values. The solution for the interior unknowns involves the discretization of PDEs on the unknowns’ points into a system of linear equations. D’Errico (2006) implemented an inpainting code for 2D arrays that is freely available and used for this step. The code offers multiple methods to formulate a PDE, and the method referred to as the spring method is selected because it provides a reasonable compromise between accuracy and computational time. - (optional) To better distinguish storm cells from their background (i.e., for radar displaying purposes), it is recommended to replace areas of
${\delta}_{\text{hv}}$ that are linked to$\left|{K}_{\text{DP}}\right|$ < 0.4° km^{−1}(i.e., weak rain echoes) by a representative value. This value is chosen as the mean of${\delta}_{\text{hv}}$ samples constrained by$\left|{K}_{\text{DP}}\right|$ < 0.4° km^{−1}and$\left|{\delta}_{\text{hv}}\right|<{\overline{\sigma}}_{{\delta}_{\text{hv}}}$ , where${\overline{\sigma}}_{{\delta}_{\text{hv}}}$ indicates the mean of${\sigma}_{{\delta}_{\text{hv}}}$ samples obtained in a similar manner as in step 3 but using${\delta}_{\text{hv}}$ after step 4. The value of 0.4° km^{−1}is found to match the 30-dB*Z*level used in this work for storm cell identification.

## 3. Evaluation of ${K}_{\text{DP}}$ processing by the ZPHI method

### a. Datasettings and preprocessing

The polarimetric X-band International Research Center for Telecommunications and Radar (IRCTR) Drizzle Radar (IDRA; Figueras i Ventura 2009) is located at the Cabauw Experimental Site for Atmospheric Research (CESAR) observatory in the Netherlands (NL) at a height of 213 m from ground level (Leijnse et al. 2010). Its operational range and range resolution are equal to 15.3 and 0.03 km, respectively, while the antenna rotates over 360° in 1 min. Four storm events, E1–E4, that occurred in the Netherlands during the year 2011 will be used for demonstration and analysis purposes. A description of these events is summarized in Table 1.

Description of four storm events E1–E4 observed in the Netherlands.

To remove areas that include particles other than rain and/or areas with low signal-to-noise ratio (SNR), measurements of linear depolarization ratio *z*, and *z* and

### b. Comparison between ${K}_{\text{DP}}$ and A

Next, *A*(ZPHI) using the empirical relation *A* = *α* is 0.34 dB (°)^{−1}, as suggested by Otto and Russchenberg (2011). In this scheme, *A*(ZPHI) is used as a reference to evaluate both *Z*.

To estimate *τ* is set to 1.5*L* on the order of 3 km are associated with theoretical values of ^{−1} for *z* and *A*(ZPHI), *A*(ZPHI) is avoided.

Results from the storm event E1 at 1216 UTC are shown in Fig. 3. The *z* field represents a relatively small cell of a nonuniform structure in close proximity to the radar. The 30-dB*Z* contour is obtained from the attenuation-corrected *Z* using the ZPHI method [i.e., after calculating *A*(ZPHI) as explained previously]. Comparing ^{−1}, which are present in

The scatterplots *A*(ZPHI) and *A*(ZPHI) resulting from the same event, E1, are compared in Fig. 4. In Fig. 4a, it can be seen that the *A*(ZPHI) scatterplot (14 783 data points) is more consistent than that of *A*(ZPHI) (15 490 data points) with respect to the empirical relation *A* = 0.34*A*(ZPHI) is equal to 0.65, whereas for *A*(ZPHI) it is 0.96. Their corresponding standard deviations ^{−1}, respectively. To compare the impact of both *z* values are corrected for attenuation using the DP and ZPHI correction methods, and are denoted as *Z*(DP, C), *Z*(DP, AHR), and *Z*(ZPHI, C); see Fig. 1. The scatterplots *Z*(DP, C)–*Z*(ZPHI, C) and *Z*(DP, AHR)–*Z*(ZPHI, C) are compared in Fig. 4b such that *Z*(ZPHI, C) estimates are used as reference. It is observed that for relatively high values of *Z*(ZPHI, C), *Z*(DP, C) values are slightly overcorrected, which agrees with Gorgucci and Chandrasekar (2005) and Snyder et al. (2010). In contrast, *Z*(DP,AHR) values are found significantly consistent with *Z*(ZPHI, C) estimates. The mean biases associated with *Z*(DP,C) and *Z*(DP, AHR) are equal to 0.95 and −0.21 dB, respectively, for *Z*(ZPHI, C) ≥35 dB*Z*. The errors quantified by *Z* are summarized in Table 2. The remaining events, E2–E4, at 1450, 1955, and 0558 UTC, respectively, were also analyzed in a similar manner and the corresponding quantified errors are indicated in Table 2.

Comparison results between *A*(ZPHI) resulting from the ZPHI method for four storm events. Data points in each event are given: event E1 (~14 000), E2 (~13 000), E3 (~40 000), and E4 (~30 000).

From the previous analysis, the following can be highlighted. The values of *A*(ZPHI), determined by two independent methods, show a strong agreement to the empirical relation *Z*(DP,AHR) and *Z*(ZPHI,C) results. In the contrary, the agreement between *A*(ZPHI) is less evident, and although *Z*(DP,C), it can significantly impact estimates of *A* by the DP method. Similar findings at X-band frequencies were reported by Gorgucci and Chandrasekar (2005) but using simulated data.

## 4. Impact of ${K}_{\text{DP}}$ processing on the CZPHI method

In this section, the ability to estimate *α* values for the estimation of *A* and the correction of *Z* by the CZPHI method is measured. For analysis purposes, the minimum *E* obtained from Eq. (3) is expressed as *E* = *i* =

At X-band frequencies, [^{−1} with steps of 0.02 dB (°)^{−1}, as suggested by Park et al. (2005b) and Ryzhkov et al. (2014). For a correct optimization process, it is recommended that ^{−1} should be at least 50%, whereas if the ^{−1} and NSE < 20% should be larger than 80%. The percentage threshold for *α* is selected by minimizing *E*, considering only range gates that satisfy the stated conditions; otherwise *α* is equal to 0.34 dB (°)^{−1}.

### a. Event E1: Single cell

#### 1) Optimization analysis

Results involved in the optimization process along azimuth 288.1° for storm event E1 at 1216 UTC are shown in Figs. 5a–c. In Fig. 5a, it is seen that the minimum *E* when *α* are *α*–*α*–^{−1}. The reason why the two *α* values are different can be explained by observing the measured *M* in Eq. (2) appears to be 0 at the beginning and ending ranges of

The selected *α*–*α*–*α* that are related to a minimum *E* (i.e., optimal *α* values) are encircled by black edges, while those that are nonrelated to a minimum *E* are represented without edges. Note that optimal *α*–^{−1}, whereas those related to ^{−1} and sometime equal to *α* that equals *α*. The resulting *α* was selected, associated with either

#### 2) Performance analysis

The impact of the optimal selection of *α*–*α*–*A*(CZPHI) is measured using *A*(ZPHI) demonstrated in section 3b and

The scatterplots *A*(CZPHI, C)–*A*(CZPHI, AHR)–*A*(CZPHI, C) estimates are smaller than those from *A*(CZPHI, AHR) as a consequence of selecting “small optimal” *α*–*A*(CZPHI, C)–*A*(CZPHI, AHR)–*A* = ^{−1}, respectively, where *α* values are given by *α*–*Z*(CZPHI, C) and *Z*(CZPHI, AHR) are compared against *Z*(DP, AHR), where *Z*(DP, AHR) is obtained from *α*–*Z*(DP, AHR) ≥ 35 dB*Z*. This means that the attenuation-correctioned CZPHI method can lead to lower performance than the ZPHI method, comparing Fig. 6b with Fig. 4b. In this analysis, the RMSE was used instead of the mean bias to take into account the standard deviation of *Z*(CZPHI) estimates associated with the variability of *α*. The quantified errors used to evaluate the CZPHI method are summarized in Table 3.

Comparison results between *A*(CZPHI, C) and *A*(CZPHI, AHR) using

A similar analysis of *A*(CZPHI) is performed using *A*(CZPHI, C) and *Z*(CZPHI, C) and *Z*(DP, C) is equal to 0.82 and smaller than the case when *Z*(DP, AHR) is used as a reference. This is because *Z*(CZPHI, C) and *Z*(DP, C) are obtained from the same *α*–*Z*(DP, C) is set as a reference, their resulting RMSE is still larger than the one from *Z*(CZPHI, AHR)–*Z*(DP, AHR).

Attenuated *z* and *Z*(CZPHI, AHR) and *Z*(CZPHI, AHR) field restored attenuated *z* areas with PIA values up to 14 dB mostly on the north side of the storm cell, which is associated with rapid increments of *Z* and *Z* and *A* by neither of the discussed methods and they do not influence the results of the presented analysis.

### b. Event E2: Mini-supercell

The performance of the CZPHI method from event E2 at 1450 UTC is analyzed in a similar manner as for event E1 and the quantified errors are summarized in Table 3. The results show again that the CZPHI method performs better when *α* is given by *α*–*α*–*α* values using the outcome of both

The resulting *Z*(CZPHI, AHR) and _{DP}) values up to 10 dB (1.6 dB), are shown in Fig. 8. In the *Z*(CZPHI, AHR) field, a significant gradient can be seen along the inflow edge of the storm (arrow 1), as well as a narrow echo appendage (arrow 2). An echo appendage typically curves in the presence of a mesocyclone process; however, this feature was not seen during the considered period. The ^{−1} collocated with the

The selected values for *α*–*α*–*α* using ^{−1} is less than 50%, which led to the selection of the constant *α*, avoiding suboptimal *α* values. This means that in those profiles, *A* is given by the ZPHI method, leading to a reasonable correlation *α* using ^{−1} in contrast to those resulting from

### c. Event E3: Tornadic cell

This event was associated with a bow apex feature along the leading edge of the storm. According to Funk et al. (1999), cyclonic circulations can occur along or near the leading bow apex, which can produce tornadoes of F0–F3 intensity. For a detailed observation of event E3, only the southeast side of the *Z*(CZPHI, AHR), *Z* field shows a strong gradient along the leading edge (arrow 4), indicating a region of strong convergence and low-level inflow (white arrows). A bow apex attribute resulting possibly from a descending rear inflow jet (Weisman and Trapp 2003) is also noticeable (arrow 5). This feature seems to be associated with a rotation pattern in the form of a hook or weak-echo hole (Bluestein et al. 2007) (extended arrow 6) that caused wind and tornado damage as indicated in Table 1. It is also observed that the core of the weak-echo hole, whose inner diameter is approximately 0.75 km, is related to bounded weak _{DP} reached 18 and 3 dB, respectively, while fully attenuated areas (south side) occurred behind strong rain echoes associated with ^{−1}.

The resulting values of *α*–*α*–*α* values are associated with a minimum error *E*, except in the azimuthal sector of [40°; 120°], where estimates of *A* were determined by the ZPHI method. This sector was related to light and uniform rain profiles, where *α*–^{−1}. The absence of ^{−1}, in contrast to event E2, may indicate the lack of big drops present at this time. Selected values of *α*–^{−1} but in a few profiles they are equal to 0.1 or 0.6 dB (°)^{−1}, possibly as a result of an inadequate optimization process. The resulting *α* and better estimates of *A* and *Z*.

### d. Event E4: Irregular-shaped cell

In contrast to events E1–E3, E4 is mainly related to light rain with a few spots of moderate rain and it is not associated with any known reflectivity signatures. In addition, multiple radial paths with reflectivity echoes larger than 30 dB*Z* are mostly smaller than 5 km, in which PIA reached values of 2.5 dB, and only in few profiles it increased to 14 dB. The fields of *Z*(CZPHI, AHR), *Z* and ^{−1}, implying a slow incremental behavior of estimated ^{−1} were selected and the associated errors were larger than those found in E1–E3. In the remaining profiles, values of *α*–^{−1}, while values of *α*–^{−1}, indicating the absence of raindrops of considerable size. The results associated with the selection of *α* using

## 5. Evaluation of ${\delta}_{\text{hv}}$ estimates

For each storm event, the preprocessed *A*(CZPHI, AHR) fields were set as inputs to the

The estimated *Z* smaller than the 30-dB*Z* level, which defines the shape of the described storm cells. Moreover, a spatial correlation between the *Z*, and *Z*, and

The ability of the algorithm to capture the spatial variability of *Z*, *α* given in Fig. 9.

In event E3, estimates of

During the estimation of ^{−1} (i.e., areas of light rain) were replaced by a uniform value (step 5). The percentages *I* (%) of *U* (°) are 0.59°, 0.65°, 0.87°, and 0.13°, respectively, and they are summarized in Table 4.

Comparison results between

The resulting

## 6. Assessment on *A* and ${\delta}_{\text{hv}}$

### a. Performance of the CZPHI method

To further evaluate the CZPHI method, the same quality measures introduced in section 4 and the storm events E1–E4 are used but during time periods, as given in Table 1. For a representative and concise evaluation, only the results from event E2 will be discussed in detail. During the first 20 min, this event consisted of an ordinary storm cell of a small size, ^{2}. When this cell was exiting the “view” of the radar, around 1420 UTC, another storm cell entered the scope of the radar. This storm manifested the characteristics of a mini-supercell during the period 1430–1500 UTC and that of a decaying storm after 1500 UTC. The quality measures resulting from event E2 are shown in Fig. 14.

Figure 14a illustrates the time series of the mean and standard deviation of the errors related to the optimization of the parameter *α*. From these results, it can be inferred that the degree of similarity between the *α*–*α* did not occur because the storm scenario was given by weak rain echoes and rain paths of less than 3 km, which are not sufficient to meet the established conditions. The increments of

The impact of the optimization of *α* on the estimation of *A* is quantified by comparing *A*(CZPHI, C) and *A*(CZPHI, AHR) against *A* is given by *A*(CZPHI, AHR) instead of *A*(CZPHI, C). The two *A* and *α* are suboptimal, provided that the *α* values are alike. In contrast, the results of *α* because *A*(CZPHI) and *A*(DP, AHR) estimates. The decreasing behavior of *A*(ZPHI, C) values resulting from noisy estimates of *A*(CZPHI) on attenuation-corrected *Z*. It can be said that between *A*(CZPHI, C) and *A*(CZPHI, AHR), the estimates of *A*(CZPHI, C) can impact negatively on the accuracy of *Z*. Furthermore, the RMSE magnitudes for both cases tend to increase from a scenario given by an ordinary cell, before 1420 UTC, to a complex mini-supercell, after 1420 UTC. This tendency is due to the spatial structure of the storm cells that can pose a more or less challenging task to capture the sensitivity of *α* to DSD and drop size variabilities. Such a challenging level can be depicted from the *E*. The discontinuity of RMSE observed around 1420 UTC is because of the lack of *Z* samples ≥ 35 dB*Z* to compute RMSE. The quality measures resulting from the events E1, E3, and E4 presented similar results to those calculated from event E2. For example, in events E1–E4, *Z*(CZPHI, C) were found in the range of 1–2 dB, while for *Z*(CZPHI, AHR) they were seen between 0 and 0.5 dB.

To analyze the distribution of the optimal values for *α* associated with a minimum *E*, the histograms of *α*–*α*–^{−1}, while the sum of the bin heights is equal to 1. In events E1–E3, a frequent selection of *α*–^{−1} is observed, as a result of a recurrent mismatch between the measured *α*–*α* in the vicinity of 0.34 dB (°)^{−1} is more evident in the case of *α*–*α*–*α* that is obtained from simulations and fitting procedures. Nonetheless, the histogram of *α*–*α* larger than the empirical one. The reason for such a contribution is because of the increased size of raindrops associated with the mini-supercell structure as shown in Figs. 8 and 9 and an inadequate optimization process during the decaying period. In event E4, a repetitive selection of *α* equal to 0.1 and 0.6 dB (°)^{−1} is noted, indicating an unstable behavior of the optimization process, which agrees with the increasing behavior of the

### b. Performance of the ${\delta}_{\text{hv}}$ algorithm

To further assess the *I*. Recall that MAE and MSD were calculated using the empirical relation between

In terms of MAE, the improvement observed from ^{−1}, and a temporal increase of *α* values because the selection of suboptimal *α*–

In contrast to MAE, MSD time series depict an evident improvement obtained from *I* resulting from the estimation of *A*(CZPHI, AHR) or directly from *A*(CZPHI, C) or

## 7. Summary and conclusions

In weather radar polarimetry at X-band frequencies, the differential phase *A* has improved radar measurements affected by, for example, attenuation, miscalibration, and partial beam blockage. Another variable of interest is *A*, and *A* and thereby on the attenuation correction of *Z* using the extended version of the ZPHI method, the CZPHI method. Special attention was given to the optimization of the parameter *α* that relates *A* in rain. Also, a technique to improve the calculation of *α*) and the CZPHI technique (with a variable *α*) were adapted at X-band frequencies to estimate *A*, denoted as *A*(ZPHI) and *A*(CZPHI), respectively. Moreover, the results obtained from the AHR and CZPHI methods were included in the estimation of

In the analysis associated with a constant *α*, *A*(ZPHI) magnitudes show a strong consistency, leading to a correlation coefficient of *A*(ZPHI) present a low agreement; nonetheless, *Z*, with a slight degradation related to *Z*, but it can negatively impact the estimation of *A*. These findings confirm the conclusions of similar studies that when *A* (Gorgucci and Chandrasekar 2005).

In the study related to a variable *α*, the CZPHI method was tested using *α*, *α* values associated with *α* on the estimation of *A* was measured in terms of *α* is associated with *A*(CZPHI) on the correction of *Z* but measured in terms of the RMSE. For this analysis, *A*(ZPHI) and the fact that data were obtained from one radar (i.e., without independent measurements at the same time of the storm events). Nonetheless, such a methodology allows a volume-to-volume comparison between estimates of *A* and *α* is connected to *α* because the errors related to the optimization process are on the order of 0°–0.5°, in contrast to 1°–2.5° for *α*–*α* is consistent with empirical values of *α*. However, in scenarios dominated by light rain, the optimization can lead to the selection of erroneous *α* values. Nonetheless, the presented analysis shows the potential of combining the AHR and CZPHI approaches for a better estimation of *A* and correction of *Z* in rain recurring to the optimization of the parameter *α* over short range paths.

The proposed *Z* and *Z* and ^{−1}. The results of the time series analyses, which correspond to the evolution of the presented storms, show a significant agreement between estimated

Even though it was shown that the presented work can provide improved estimates of *α*, *A*, and *Z*, and *Z* and *A*, and the proposed algorithm to estimate

A careful *α* related to the CZPHI method and enhanced estimation of *α*, *Z*, and

Gratitudes to 4TUDatacentrum for its support of maintaining IDRA data, an open access dataset (Russchenberg et al. 2010). Also, the authors thank Dr. Jacopo Grazioli and Dr. Yadong Wang for their constructive discussion. This work was supported by RainGain through INTERREG IV B.

## REFERENCES

Bertalmio,M.,L. Vese,G. Spario, andS. Osher,2003:Simultaneous structure and texture image inpainting.

,*IEEE Trans. Image Process.***12**,882–889, https://doi.org/10.1109/TIP.2003.815261.Bluestein,H.,M. French,R. Tanamachi,S. Frasier,K. Hardwick,F. Junyent, andA. Pazmany,2007:Close-range observations of tornadoes in supercells made with a dual-polarization, X-band, mobile Doppler radar.

,*Mon. Wea. Rev.***135**,1522–1543, https://doi.org/10.1175/MWR3349.1.Bringi,V. N., andV. Chandrasekar,2001:

*Polarimetric Doppler Weather Radar Principles and Applications.*Cambridge University Press, 636 pp.Bringi,V. N.,V. Chandrasekar,N. Balakrishman, andD. Zrnić,1990:An examination of propagation effects in rainfall on radar measurements at microwave frequencies.

,*J. Atmos. Oceanic Technol.***7**,829–840, https://doi.org/10.1175/1520-0426(1990)007<0829:AEOPEI>2.0.CO;2.Bringi,V. N.,T. Kennan, andV. Chandrasekar,2001:Correcting C-band reflectivity and differential reflectivity data for rain attenuation: A self-consistent method with constraints.

,*IEEE Trans. Geosci. Remote Sens.***39**,1906–1915, https://doi.org/10.1109/36.951081.Carey,L.,S. Rutledge,D. Ahijevych, andT. Keenan,2000:Correcting propagation effects in C-band polarimetric radar observations of tropical convection using differential propagation phase.

,*J. Appl. Meteor.***39**,1405–1433, https://doi.org/10.1175/1520-0450(2000)039<1405:CPEICB>2.0.CO;2.Chandrasekar,V.,H. Chen, andB. Philips,2018:Principles of high-resolution radar network for hazard mitigation and disaster management in an urban environment.

,*J. Meteor. Soc. Japan***96A**,119–139, https://doi.org/10.2151/jmsj.2018-015.Criminisi,A.,P. Perez, andK. Toyama,2004:Region filling and object removal by exemplar-based image inpainting.

,*IEEE Trans. Image Process.***13**,1200–1212, https://doi.org/10.1109/TIP.2004.833105.D’Errico,J.,2006: Data interpolation by image inpaiting. Version 1.1.0.0, MATLAB Central File Exchange, http://www.mathworks.com/matlabcentral/fileexchange/4551.

Diederich,M.,A. Ryzhkov,C. Simmer,P. Zhang, andS. Trömel,2015:Use of specific attenuation for rainfall measurement at X-band radar wavelengths. Part I: Radar calibration and partial beam blockage estimation.

,*J. Hydrometeor.***16**,487–502, https://doi.org/10.1175/JHM-D-14-0066.1.Doviak,R. J., andD. S. Zrnić,1993:

*Doppler Radar and Weather Observations.*2nd ed. Academic, 562 pp.Doviak,R. J.,V. Bringi,A. Ryshkov,A. Zahrai, andD. Zrnic,2000:Considerations for polarimetric upgrades to operational WSR-88D radars.

,*J. Atmos. Oceanic Technol.***17**,257–278, https://doi.org/10.1175/1520-0426(2000)017<0257:CFPUTO>2.0.CO;2.Elad,M.,J. Starck,P. Querre, andD. Donoho,2005:Simultaneous cartoon and texture image inpainting using morphological component analysis.

,*Appl. Comput. Harmonic Anal.***19**,340–358, https://doi.org/10.1016/j.acha.2005.03.005.Figueras i Ventura,J.,2009: Design of a high resolution X-band Doppler polarimetric radar. Ph.D. thesis, Delft University of Technology, 162 pp.

Funk,T. W.,K. E. Darmofal,J. D. Kirkpatrick,V. L. Dewald,R. W. Przybylinski,G. K. Schmocker, andY.-J. Lin,1999:Storm reflectivity and mesocyclone evolution associated with the 15 April 1994 squall line over Kentucky and southern Indiana.

,*Wea. Forecasting***14**,976–993, https://doi.org/10.1175/1520-0434(1999)014<0976:SRAMEA>2.0.CO;2.Giangrande,S. E., andA. V. Ryzhkov,2005:Calibration of dual-polarization radar in the presence of partial beam blockage.

,*J. Atmos. Oceanic Technol.***22**,1156–1166, https://doi.org/10.1175/JTECH1766.1.Giangrande,S. E.,R. McGraw, andL. Lei,2013:An application of linear programming to polarimetric radar differential phase processing.

,*J. Atmos. Oceanic Technol.***30**,1716–1729, https://doi.org/10.1175/JTECH-D-12-00147.1.Gorgucci,E., andV. Chandrasekar,2005:Evaluation of attenuation correction methodology for dual-polarization radars: Application to X-band systems.

,*J. Atmos. Oceanic Technol.***22**,1195–1206, https://doi.org/10.1175/JTECH1763.1.Gorgucci,E.,V. Chandrasekar, andL. Baldini,2006:Correction of X-band radar observations for propagation effects based on the self-consistency principle.

,*J. Atmos. Oceanic Technol.***23**,1668–1681, https://doi.org/10.1175/JTECH1950.1.Gourley,J. J.,P. Tabary, andJ. P. D. Chatelet,2006:Data quality of the Meteo-France C-band polarimetric radar.

,*J. Atmos. Oceanic Technol.***23**,1340–1356, https://doi.org/10.1175/JTECH1912.1.Grazioli,J.,M. Schneebeli, andA. Berne,2014:Accuracy of phase-based algorithm for the estimation of the specific differential phase shift using simulated polarimetric weather radar data.

,*IEEE Geosci. Remote Sens. Lett.***11**,763–767, https://doi.org/10.1109/LGRS.2013.2278620.Huang,H.,G. Zhang,K. Zhao, andS. E. Giangrande,2017:A hybrid method to estimate specific differential phase and rainfall with linear programming and physics constraints.

,*IEEE Trans. Geosci. Remote Sens.***55**,96–111, https://doi.org/10.1109/TGRS.2016.2596295.Hubbert,J., andV. N. Bringi,1995:An iterative filtering technique for the analysis of copolar differential phase and dual-frequency radar measurements.

,*J. Atmos. Oceanic Technol.***12**,643–648, https://doi.org/10.1175/1520-0426(1995)012<0643:AIFTFT>2.0.CO;2.Jameson,A. R.,1992:The effect of temperature on attenuation correction schemes in rain using polarization propagation differential phase shift.

,*J. Appl. Meteor.***31**,1106–1118, https://doi.org/10.1175/1520-0450(1992)031<1106:TEOTOA>2.0.CO;2.Kim,D.-S.,M. Maki, andD.-I. Lee,2010:Retrieval of three-dimensional raindrop size distribution using X-band polarimetric radar data.

,*J. Atmos. Oceanic Technol.***27**,1265–1285, https://doi.org/10.1175/2010JTECHA1407.1.Kumjian,M. R., andA. V. Ryzhkov,2008:Polarimetric signatures in supercell thunderstorms.

,*J. Appl. Meteor. Climatol.***47**,1940–1961, https://doi.org/10.1175/2007JAMC1874.1.Leijnse,H., andCoauthors,2010:Precipitation measurement at CESAR, the Netherlands.

,*J. Hydrometeor.***11**,1322–1329, https://doi.org/10.1175/2010JHM1245.1.Lim,S., andV. Chandrasekar,2016:A robust attenuation correction system for reflectivity and differential reflectivity in weather radars.

,*IEEE Trans. Geosci. Remote Sens.***54**,1727–1737, https://doi.org/10.1109/TGRS.2015.2487984.Lim,S.,R. Cifelli,V. Chandrasekar, andS. Y. Matrosov,2013:Precipitation classification and quantification using X-band dual-polarization weather radar: Application in the hydrometeorology testbed.

,*J. Atmos. Oceanic Technol.***30**,2108–2120, https://doi.org/10.1175/JTECH-D-12-00123.1.Matrosov,S. Y.,K. Clark,B. Martner, andA. Tokay,2002:X-band polarimetric radar measurements of rainfall.

,*J. Appl. Meteor.***41**,941–952, https://doi.org/10.1175/1520-0450(2002)041<0941:XBPRMO>2.0.CO;2.Matrosov,S. Y.,D. E. Kingsmill,B. E. Martner, andF. M. Ralph,2005:The utility of X-band polarimetric radar for quantitative estimates of rainfall parameters.

,*J. Hydrometeor.***6**,248–262, https://doi.org/10.1175/JHM424.1.Matrosov,S. Y.,P. C. Kennedy, andR. Cifelli,2014:Experimentally based estimates of relations between X-band radar signal attenuation characteristics and differential phase in rain.

,*J. Atmos. Oceanic Technol.***31**,2442–2450, https://doi.org/10.1175/JTECH-D-13-00231.1.McLaughlin,D., andCoauthors,2009:Short-wavelength technology and the potential for distributed networks of small radar systems.

,*Bull. Amer. Meteor. Soc.***90**,1797–1817, https://doi.org/10.1175/2009BAMS2507.1.Otto,T., andH. W. J. Russchenberg,2010: Estimation of the raindrop-size distribution at X-band using specific differential phase and differential backscatter phase.

*Proc. Sixth European Conf. on Radar in Meteorology and Hydrology (ERAD 2010)*, Sibiu, Romania, Meteor Romania, 6 pp., https://www.erad2010.com/pdf/oral/thursday/xband/07_ERAD2010_0109.pdf.Otto,T., andH. W. J. Russchenberg,2011:Estimation of specific differential phase and differential backscatter phase from polarimetric weather radar measurements of rain.

,*IEEE Geosci. Remote Sens. Lett.***8**,988–992, https://doi.org/10.1109/LGRS.2011.2145354.Park,S.-G.,V. N. Bringi,V. Chandrasekar,M. Maki, andK. Iwanami,2005a:Correction of radar reflectivity and differential reflectivity for rain attenuation at X band. Part I: Theoretical and empirical basis.

,*J. Atmos. Oceanic Technol.***22**,1621–1632, https://doi.org/10.1175/JTECH1803.1.Park,S.-G.,M. Maki,K. Iwanami,V. N. Bringi, andV. Chandrasekar,2005b:Correction of radar reflectivity and differential reflectivity for rain attenuation at X band. Part II: Evaluation and application.

,*J. Atmos. Oceanic Technol.***22**,1633–1655, https://doi.org/10.1175/JTECH1804.1.Reinoso-Rondinel,R.,C. Unal, andH. Russchenberg,2018:Adaptive and high-resolution estimation of specific differential phase for polarimetric X-band weather radars.

,*J. Atmos. Oceanic Technol.***35**,555–573, https://doi.org/10.1175/JTECH-D-17-0105.1.Russchenberg,H.,T. Otto,R. Reinoso-Rondinel,C. Unal, andJ. Yin,2010: IDRA weather radar measurements—All data. 4TU Centre for Research Data, Delft University of Technology. Subset used: IDRA processed data with standard range, accessed 12 October 2012, https://doi.org/10.4121/uuid:5f3bcaa2-a456-4a66-a67b-1eec928cae6d.

Ryzhkov,A., andD. S. Zrnić,1995:Precipitation and attenuation measurements at 10-cm wavelength.

,*J. Appl. Meteor.***34**,2121–2134, https://doi.org/10.1175/1520-0450(1995)034<2120:PAAMAA>2.0.CO;2.Ryzhkov,A., andD. S. Zrnić,2005: Radar polarimetry at S, C, and X bands: Comparative analysis and operational implications.

*32nd Conf. on Radar Meteor.*, Albuquerque, NM, Amer. Meteor. Soc., 9R3, https://ams.confex.com/ams/32Rad11Meso/webprogram/Paper95684.html.Ryzhkov,A.,M. Diederich,P. Zhang, andC. Simmer,2014:Potential utilization of specific attenuation for rainfall estimation, mitigation of partial beam blockage, and radar networking.

,*J. Atmos. Oceanic Technol.***31**,599–619, https://doi.org/10.1175/JTECH-D-13-00038.1.Scarchilli,G.,E. Gorgucci,V. Chandrasekar, andT. A. Seliga,1993:Rainfall estimation using polarimetric techniques at C-band frequencies.

,*J. Appl. Meteor.***32**,1150–1160, https://doi.org/10.1175/1520-0450(1993)032<1150:REUPTA>2.0.CO;2.Scarchilli,G.,E. Gorgucci,V. Chandrasekar, andA. Dobaie,1996:Self-consistency of polarization diversity measurement of rainfall.

,*IEEE Trans. Geosci. Remote Sens.***34**,22–26, https://doi.org/10.1109/36.481887.Schneebeli,M., andA. Berne,2012:An extended Kalman filter framework for polarimetric X-band weather radar data processing.

,*J. Atmos. Oceanic Technol.***29**,711–730, https://doi.org/10.1175/JTECH-D-10-05053.1.Schneebeli,M.,J. Grazioli, andA. Berne,2014:Improved estimation of the specific differential phase shift using a compilation of Kalman filter ensembles.

,*IEEE Trans. Geosci. Remote Sens.***52**,5137–5149, https://doi.org/10.1109/TGRS.2013.2287017.Snyder,J.,H. Bluestein, andG. Zhang,2010:Attenuation correction and hydrometeor classification of high-resolution, X-band, dual-polarized mobile radar measurements in severe convective storms.

,*J. Atmos. Oceanic Technol.***27**,1979–2001, https://doi.org/10.1175/2010JTECHA1356.1.Testud,J.,E. L. Bouar,E. Obligis, andM. Ali-Mehenni,2000:The rain profiling algorithm applied to polarimetric weather radar.

,*J. Atmos. Oceanic Technol.***17**,332–356, https://doi.org/10.1175/1520-0426(2000)017<0332:TRPAAT>2.0.CO;2.Trömel,S.,M. R. Kumjian,A. V. Ryzhkov,C. Simmer, andM. Diederich,2013:Backscatter differential phase—Estimation and variability.

,*J. Appl. Meteor. Climatol.***52**,2529–2548, https://doi.org/10.1175/JAMC-D-13-0124.1.Wang,Y., andV. Chandrasekar,2009:Algorithm for estimation of the specific differential phase.

,*J. Atmos. Oceanic Technol.***26**,2565–2578, https://doi.org/10.1175/2009JTECHA1358.1.Wang,Y.,P. Zhang,A. V. Ryzhkov,J. Zhang, andP.-L. Chang,2014:Utilization of specific attenuation for tropical rainfall estimation in complex terrain.

,*J. Hydrometeor.***15**,2250–2266, https://doi.org/10.1175/JHM-D-14-0003.1.Weisman,M., andR. Trapp,2003:Low-level mesovortices within squall lines and bow echoes. Part I: Overview and dependence on environment shear.

,*Mon. Wea. Rev.***131**,2779–2803, https://doi.org/10.1175/1520-0493(2003)131<2779:LMWSLA>2.0.CO;2.