## Abstract

In radar polarimetry, the differential phase $\Psi DP$ consists of the propagation differential phase $\Phi DP$ and the backscatter differential phase $\delta hv$. While $\Phi DP$ is commonly used for attenuation correction (i.e., estimation of the specific attenuation *A* and specific differential phase $KDP$), recent studies have demonstrated that $\delta hv$ can provide information concerning the dominant size of raindrops. However, the estimation of $\Phi DP$ and $\delta hv$ is not straightforward given their coupled nature and the noisy behavior of $\Psi DP$, especially over short paths. In this work, the impacts of estimating $\Phi DP$ on the estimation of *A* over short paths, using the extended version of the ZPHI method, are examined. Special attention is given to the optimization of the parameter *α* that connects $KDP$ and *A*. In addition, an improved technique is proposed to compute $\delta hv$ from $\Psi DP$ and $\Phi DP$ in rain. For these purposes, diverse storm events observed by a polarimetric X-band radar in the Netherlands are used. Statistical analysis based on the minimum errors associated with the optimization of *α* and the consistency between $KDP$ and *A* showed that more accurate and stable *α* and *A* are obtained if $\Phi DP$ is estimated at range resolution, which is not possible by conventional range filtering techniques. Accurate $\delta hv$ estimates were able to depict the spatial variability of dominant raindrop size in the observed storms. By following the presented study, the ZPHI method and its variations can be employed without the need for considering long paths, leading to localized and accurate estimation of *A* and $\delta hv$.

## 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).

One of the advantages of polarimetric radars is given by the measurements of differential phase between the horizontally and vertically polarized signals caused by the delay of one with respect to the other as both signals propagate through hydrometeors. In this way, the differential phase $\Psi DP$ (°) is independent of attenuation, miscalibration, and partial beam blockage (PBB) effects (Doviak and Zrnić 1993). However, $\Psi DP$ measurements can include phase shifts in the backward direction as a result of Mie scattering, the so-called backscatter differential phase $\delta hv$ (°), and random fluctuations *ε* (°) on the order of few degrees. In general, a $\Psi DP$ range profile is modeled as

where $\Phi DP\u2061(r)$ (°) represents the differential phase in the forward direction and *r* (km) indicates the distance from the radar. Two useful variables that can be estimated from $\Phi DP$ are the specific differential phase $KDP$ (° km^{−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 $KDP$ (or $\Phi DP$) from $\Psi DP$ when $\delta hv$ is significant is given by Hubbert and Bringi (1995), and several attempts have been proposed to improve $KDP$ estimates at X-band frequencies (Wang and Chandrasekar 2009; Giangrande et al. 2013; Schneebeli et al. 2014; Huang et al. 2017). The specific differential phase $KDP$ has been used to correct measurements of reflectivity *Z* (dB*Z*) affected by radar calibration and PBB (Giangrande and Ryzhkov 2005). In addition, $KDP$ has led to improved estimation of rainfall rate, mostly in heavy rain or mix rain, because of its quasi-linear relation to liquid water content (Lim et al. 2013). Although radar measurements seem to benefit from using $KDP$, comprehensive research on $KDP$ is still needed because it is a challenge to provide accurate $KDP$ from noisy measurements of $\Psi DP$.

Existing methods to estimate *A* in rain assume that *A* = $\alpha KDP$, where *α* is a constant for a given frequency (Bringi et al. 1990). Testud et al. (2000) also used the relation between *A* and $KDP$ in their rain profiling ZPHI technique, to express *A* in terms of the difference of $\Phi DP$ and measurements of *Z*, avoiding $KDP$ calculation. However, it is known that *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 $\Phi DP$ and *α* (Bringi et al. 2001; Ryzhkov and Zrnić 2005).

In contrast to $KDP$ and *A*, limited research has been conducted on the applications of $\delta hv$. For example, $\delta hv$ can be a suitable candidate to mitigate uncertainties related to the differential reflectivity $ZDR$ (dB) because $\delta hv$ and $ZDR$ offer a correlated behavior (Scarchilli et al. 1993; Testud et al. 2000) and because $\delta hv$ is independent of attenuation and radar calibration; see Eq. (1). These aspects of $\delta hv$ could be useful to establish relations between $\delta hv$ and the median drop diameter $D0$ (mm) (Trömel et al. 2013) because $D0$ is often expressed in terms of $ZDR$ (Matrosov et al. 2005; Kim et al. 2010). Moreover, Otto and Russchenberg (2010) included $\delta hv$ estimates to retrieve DSD parameters. Hubbert and Bringi (1995), Otto and Russchenberg (2011), and Trömel et al. (2013) estimated $\delta hv$ by subtracting $\Phi DP$ from $\Psi DP$, while Schneebeli and Berne (2012) included a Kalman filter approach. The effectiveness of estimating $\delta hv$ at high resolution is rather complicated because of the cumulative and noisy nature of $\Psi DP$ and possible remaining fluctuations on $\Phi DP$.

The purpose of this work is to 1) explore the role and impact of estimated $\Phi DP$ profiles on the performance of the extended ZPHI method at X-band frequencies to improve estimates of *α* and *A* over short paths and 2) develop a technique to compute $\delta hv$ in rain while keeping the spatial variability of drop sizes. For such purpose, two $KDP$ (or $\Phi DP$) methods, by Hubbert and Bringi (1995) and Reinoso-Rondinel et al. (2018), are reviewed in section 2 as well as three attenuation correction approaches, by Bringi et al. (1990), Testud et al. (2000), and Bringi et al. (2001). In addition, the $\delta hv$ algorithm is introduced, which integrates estimates of $KDP$ and *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 $\delta hv$ technique is evaluated. Section 6 focuses on the statistics of *α*, *A*, *Z*, and $\delta hv$ to conduct further assessments of the presented methods. Finally, section 7 draws conclusions of this article.

## 2. Estimation techniques for $\Psi DP$-based variables

### a. Estimation of $KDP$

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 $\Delta r$ (km) are filtered from a $\Psi DP\u2061(r)$ profile. Fluctuations at range scales larger than $\Delta r$ (i.e., $\delta hv$ “bumps”) are removed by applying the same filter multiple times to new generated $\Psi DPg$ profiles by combining a previous filtered and original $\Psi DP$ profile. In this manner the corresponding $\Phi DP$ profile is obtained and $KDP$ is given by taking a range derivative of $\Phi DP$. For the generation of $\Psi DPg$, a predetermined threshold *τ* (°) is required, which is on the order of 1–2 times the standard deviation of $\Psi DP$, hereafter $\sigma P$ (°). One of the limitations of this technique is that accurate estimates of $\Phi DP$ and $KDP$ at $\Delta r$ scales are hardly achieved (Grazioli et al. 2014).

An adaptive approach that estimates $KDP$ at high spatial resolution while controlling its standard deviation $\sigma K$ (° km^{−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 $\Delta V$. Besides $\Psi DP$, attenuation-corrected *Z* and $ZDR$ profiles are also required, as well as a predefined pathlength interval [$Lmin;Lmax$] (km). For gate *i*, located at range $ri$, a set of $\sigma K$ samples are obtained from [$Lmin;Lmax$] using a theoretical expression of $\sigma K$. The pathlength that minimizes the $\sigma K$ set is selected and denoted as $L\u2061(i)$. Assuming the correlated behavior between $ZDR$ and $\delta hv$, $\Delta \Psi DP$ samples in the range [$ri\u2212L\u2061(i)$; $ri+L\u2061(i)$] that do not satisfy the condition $|\Delta ZDR|<\sigma ZDR$ are filtered to avoid contamination from $\Delta \delta hv$. The standard deviation of the $ZDR$ profile is denoted as $\sigma ZDR$. The spatial variability of $\Psi DP$ at $\Delta r$ scales is captured by downscaling each remaining $\Delta \Psi DP$ sample from $L\u2061(i)$ to $\Delta r$ scale. A downscaling parameter $w\u2061(i)$$\u2208$ [0, 1] is derived from *Z* and $ZDR$ in the same interval [$ri\u2212L\u2061(i)$; $ri+L\u2061(i)$], and $KDP\u2061(i)$ is estimated as

where *M* represents the number of $\Delta \Psi DP$ samples with negligible $\Delta \delta hv$. The actual $\sigma K\u2061(i)$ is calculated using the terms inside the sum operation in Eq. (2). The $KDP$ and $\sigma K$ profiles are obtained by repeating the same procedure over the remaining gates, while the corresponding $\Phi DP$ profile is calculated by simply integrating $KDP$ in range. In addition, a profile of the normalized standard error (NSE) of $KDP$ is given by the ratio between actual $\sigma K$ and $KDP$. This approach was demonstrated for rain particles at X-band frequencies, and therefore any undetected *Z* and $ZDR$ echoes from hydrometeors other than rain can lead to inaccurate $KDP$ estimates. The two $KDP$ methods will be referred to as the conventional (C) and the adaptive high-resolution (AHR) approaches, respectively. A diagram is presented in Fig. 1 to briefly indicate the inputs and outputs of each method.

### b. Estimation of A

For attenuation correction purposes, *Z* and $ZDR$ profiles are represented as $Z\u2061(r)=z\u2061(r)+PIA\u2061(r)$ and $ZDR\u2061(r)=zdr\u2061(r)+PIADP\u2061(r)$, respectively, where *z* (dB*Z*) and $zdr$ (dB) represent the attenuated reflectivity and the attenuated differential reflectivity, respectively; and PIA$\u2061(r)$ (dB) indicates the two-way path-integrated attenuation in reflectivity and PIA$\u2061(r)DP$ (dB) in differential reflectivity.

Bringi et al. (1990) introduced the differential phase (DP) approach such that $A\u2061(r)$ = $\alpha KDP\u2061(r)$ and PIA$\u2061(r)=\alpha \Phi DP\u2061(r)$, where *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 $KDP$, while estimates of PIA lead to *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 $A\u2061(r)$ in a path interval [$rp;rq$], where $rq>rp$. First, $A\u2061(r)$ is expressed as a function of two known variables, $z\u2061(r)$ and $z\u2061(rq)$, and one unknown, $A\u2061(rq)$. Then, $A\u2061(rq)$ is obtained using $z\u2061(rq)$ and the empirical relation $\Delta $PIA = $\alpha \Delta \Phi DP$, where $\Delta $PIA = PIA$\u2061(rq)$ − PIA$\u2061(rp)$ and $\Delta \Phi DP$ = $\Phi DP\u2061(rq)\u2212\Phi DP\u2061(rp)$. In this way, $A\u2061(r)$ is estimated at $\Delta r$ scales, reducing errors related to $KDP\u2061(r)$. Although [$rp;rq$] can be freely selected; $\Delta \Phi DP$ could be inaccurate at short path intervals and/or be contaminated by $\delta hv\u2061(rp)$ and $\delta hv\u2061(rq)$. In addition, if $z\u2061(r)$ includes localized observations of hail or mixtures of rain and hail in [$rp;rq$], then $A\u2061(r)$ might be biased over the entire path interval.

Using a constant *α* may lead to limited approximations of $A\u2061(r)$ and PIA$\u2061(r)$ because *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 [$\alpha min;\alpha max$], and $A\u2061(r)$ is estimated according to the ZPHI method. The estimated $A\u2061(r)$ is integrated over [$rp;rq$] to build a differential phase profile denoted as $\Phi DP\u2061(r,\alpha )$. Repeating this procedure for the remaining values of *α*, the optimal *α* is the one that minimizes the error *E* (°) given by

Note that the optimization process requires the estimation of $\Phi DP$, which implies the need for a proper way to filter noise and $\delta hv$ components from $\Psi DP$ while maintaining its spatial variability. However, meeting such requirements is not straightforward; therefore, the reliability of an “optimal” *α* to estimate *A* and PIA depends on the performance of the chosen approach to estimate $\Phi DP$. The inputs and outputs associated with the three presented attenuation correction methods are summarized in Fig. 1.

To determine PIA$\u2061(r)DP$, integrate the specific differential attenuation $ADP\u2061(r)$ (dB km^{−1}) that is given by $ADP=\gamma A$. The DP and ZPHI methods assume *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, $ADP$ will be given by $ADP$ = $\gamma A$(CZPHI), where *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), $\alpha =0.315$ dB (°)^{−1}; Kim et al. (2010), $\alpha =0.35$ dB (°)^{−1}; and Snyder et al. (2010) ,$\alpha =0.313$ dB (°)^{−1}; while Ryzhkov et al. (2014) estimated *γ* equal to 0.14 for tropical rain (i.e., low $ZDR$ and high $KDP$) and 0.19 for continental rain (i.e., high $ZDR$ and low $KDP$). It is important to note that other authors have suggested smaller average values for *α*. For example, Bringi and Chandrasekar (2001) simulated polarimetric variables in rain and indicated that $\alpha =0.23$ dB (°)^{−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 $\delta hv$

A $\delta hv$ approach is presented to identify and separate Mie scattering signatures from noise and random fluctuations embedded in $\Psi DP$. A flowchart of the $\delta hv$ algorithm is illustrated in Fig. 2. Three inputs are required: a 2D $\Psi DP$ field measured in rain, the corresponding $KDP$ field obtained from the AHR approach, and the *A* field estimated by the CZPHI method. Given these inputs, the resulting $\delta hv$ field is based on the following five steps:

Design and apply a filter to smooth strong outliers from a $\Psi DP$ profile, taking $\Delta r$ into account. Correct each smoothed $\Psi DP\u2032$ profile for system phase offset by subtracting the mean of $\Psi DP\u2032$ over the first 5% of measured gates.

Obtain $\Phi DP$ by integrating profiles of

*A*, if they are associated with a minimum error*E*, otherwise by integrating $KDP$ profiles. Next, subtract $\Phi DP$ from $\Psi DP\u2032$, profile by profile, as a first attempt to estimate the corresponding $\delta hv$ field. The next steps are related to 2D processing.- Remove unusual $\delta hv$ values larger than 12° from the $\delta hv$ field. According to Testud et al. (2000), Trömel et al. (2013), and Schneebeli et al. (2014), the simulated $\delta hv$ values at X-band frequencies rarely reach 12°. The remaining noise in $\delta hv$ is reduced by assuming that similar values of $\delta hv$ are collocated with similar values of $KDP$ as follows. Set $Kmin$ as the minimum of $KDP$ and $Kmax$ as $Kmin+\Delta K$, where $\Delta K$ (° km
^{−1}) is given by Eq. (4). Define*S*as a set of $\delta hv$ samples, whose gates are collocated with $KDP$ values in the interval [$Kmin;Kmax$]. Reject $\delta hv$ samples from*S*that are outside the interval [$\delta \xafhv\u2212\upsilon \sigma \delta hv$; $\delta \xafhv+\upsilon \sigma \delta hv$], where $\delta \xafhv$ and $\sigma \delta hv$ 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 [$Kmin;Kmax$] toward high values in small steps such that $Kmin$ = $Kmax$ and $Kmax$ = $Kmin+\Delta K$ until $Kmax$ is equal to the maximum of $KDP$. To obtain sufficient samples in*S*, $\Delta K$ is given asbecause high $KDP$ values are less frequent than small $KDP$ values (e.g., see the $KDP$ fields in Figs. 3, 8, and 11). Apply a 2D interpolation method to fill empty gaps on $\delta hv$ 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 $\delta hv$ that are linked to $|KDP|$ < 0.4° km

^{−1}(i.e., weak rain echoes) by a representative value. This value is chosen as the mean of $\delta hv$ samples constrained by $|KDP|$ < 0.4° km^{−1}and $|\delta hv|<\sigma \xaf\delta hv$, where $\sigma \xaf\delta hv$ indicates the mean of $\sigma \delta hv$ samples obtained in a similar manner as in step 3 but using $\delta hv$ 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 $KDP$ 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.

To remove areas that include particles other than rain and/or areas with low signal-to-noise ratio (SNR), measurements of linear depolarization ratio $LDR$ (dB) are used, such that range gates with $LDR$ larger than −18 dB are discarded from $\Psi DP$, *z*, and $zdr$ fields. Further preprocessing includes suppressing isolated segments of a $\Psi DP$ profile smaller than 0.25 km and rejecting a $\Psi DP$ profile if the percentage of gates with measurements is less than 5%. Because a $\Psi DP$ profile could be noisy at ranges behind strong reflectivity echoes associated with low SNR and fully attenuated signals, its range extent needs to be determined. The ending range of a $\Psi DP$ profile is determined based on $\sigma \xafP$, which represents the average of multiple $\sigma P$ samples by running a five-gate window along the $\Psi DP$ profile. If $\sigma \xafP$ is less than 1.5°, then the ending range is given by the last measured gate in the downrange direction. Otherwise, the ending range is set by the middle gate of the second consecutive window whose $\sigma P$ values are less than $\sigma \xafP$, starting at the last measured gate and moving toward the radar. The ending range is used to limit the corresponding extent of *z* and $zdr$ profiles. After this, $\sigma \xafP$ is calculated again to estimate $KDP$ by the conventional technique.

### b. Comparison between $KDP$ and A

Next, $KDP$(C) and $KDP$(AHR) will be compared against *A*(ZPHI) using the empirical relation *A* = $\alpha KDP$, where *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 $KDP$ techniques and their impact on *Z*.

To estimate $KDP$(C), a finite impulse response (FIR) filter is used such that the order of the filter is 36 and the cutoff range scale is 1 km, including a Hann window. The required threshold *τ* is set to 1.5$\sigma \xafP$. Such a filter design is found suitable for $\Delta r=$ 0.03 km. For the estimation of $KDP$(AHR), values of *L* on the order of 3 km are associated with theoretical values of $\sigma K<$ 0.5° km^{−1} for $\Delta r$ = 0.03 km (Reinoso-Rondinel et al. 2018) and therefore [$Lmin;Lmax$] is predefined as [$2;5$] km. The *z* and $zdr$ inputs are corrected for attenuation and differential attenuation, respectively, according to the DP method, in which a linear regression fit of 1 km is applied to $\Psi DP$ profiles. To estimate $\sigma ZDR$ a five-gate window is run along a given $ZDR$ profile. For the calculation of *A*(ZPHI), $\Delta \Phi DP$ is derived from $\Phi DP$(C) instead of $\Phi DP$(AHR) to evaluate $KDP$(AHR) in an independent manner. A path interval [$rp;rq$] is defined by the first and last data points, in the downrange direction, of a $\Phi DP$(C) profile. In cases where $\Delta \Phi DP<$ 0° as a result of a reduced SNR profile, the estimation of *A*(ZPHI) is avoided.

Results from the storm event E1 at 1216 UTC are shown in Fig. 3. The $\Psi DP$ field shows a rapid increment in range on the north side of the storm, whereas $\Psi DP$ rarely increases on the south side. Note that the $\Psi DP$ field is not adjusted for phase offset. The attenuated *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 $KDP$(C) and $KDP$(AHR), the $KDP$(AHR) field is able to maintain the spatial variability of the storm down to range resolution scale, eliminating areas of $KDP$ smaller than −1° km^{−1}, which are present in $KDP$(C). However, the coverage of the $KDP$(AHR) field is smaller than that of $KDP$(C). This is because in the AHR approach, it is not always possible to obtain $\Delta \Psi DP$ samples with negligible $\Delta \delta hv$; that is, $M=0$ in Eq. (2). Note that isolated $KDP$ segments smaller than 2 km were removed from both $KDP$ fields in order to avoid estimates of $KDP$ that could be associated with noisy areas and/or low accuracy.

The scatterplots $KDP$(C)–*A*(ZPHI) and $KDP$(AHR)–*A*(ZPHI) resulting from the same event, E1, are compared in Fig. 4. In Fig. 4a, it can be seen that the $KDP$(AHR)–*A*(ZPHI) scatterplot (14 783 data points) is more consistent than that of $KDP$(C)–*A*(ZPHI) (15 490 data points) with respect to the empirical relation *A* = 0.34$KDP$. In a quantified comparison, the correlation coefficient $\rho KA$ between $KDP$(C) and *A*(ZPHI) is equal to 0.65, whereas for $KDP$(AHR) and *A*(ZPHI) it is 0.96. Their corresponding standard deviations $\sigma KA$ with respect to the empirical relation are 1.20 and 0.41° km^{−1}, respectively. To compare the impact of both $KDP$ techniques on the DP method, *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 $\rho KA$, $\sigma KA$, and bias *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.

From the previous analysis, the following can be highlighted. The values of $KDP$(AHR) and *A*(ZPHI), determined by two independent methods, show a strong agreement to the empirical relation $A=\alpha KDP$, leading to equivalent *Z*(DP,AHR) and *Z*(ZPHI,C) results. In the contrary, the agreement between $KDP$(C) and *A*(ZPHI) is less evident, and although $KDP$(C) barely includes substantial errors on attenuation-corrected *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 $KDP$ processing on the CZPHI method

In this section, the ability to estimate $\Phi DP$ by both $KDP$ approaches is studied and their impact on the performance of finding optimal *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* = $\u2211ei$, with *i* = $p,\u2026,q$, where $ei$ represents the minimum error at range $ri$. As such, the arithmetic mean and standard deviation of $ei$, $e\xafmin$ (°) and $\sigma emin$ (°), respectively, will be used as quality measures.

At X-band frequencies, [$\alpha min;\alpha max$] is predefined as [0.1;0.6] dB (°)^{−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 $rq\u2212rp$ should be at least 3 km and that $\Delta \Phi DP$ be larger than 10°. In addition, if the $\Phi DP$(C) profile is used in Eq. (3), then the percentage of gates with $KDP>$ 0° km^{−1} should be at least 50%, whereas if the $\Phi DP$(AHR) profile is used, the percentage of gates with $KDP>$ 0.5° km^{−1} and NSE < 20% should be larger than 80%. The percentage threshold for $\Phi DP$(C) is less than for $\Phi DP$(AHR) because the conventional method rarely avoids negative $KDP$ values. If these conditions are met, *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 $\Phi DP$(C) is used is much larger than when $\Phi DP$(AHR) is used and their corresponding optimal values for *α* are *α*–$\Phi DP$(C) = 0.24 and *α*–$\Phi DP$(AHR) = 0.34 dB (°)^{−1}. The reason why the two *α* values are different can be explained by observing the measured $\Psi DP$ and the estimated $\Phi DP$(C) and $\Phi DP$(AHR) profiles shown in Figs. 5b and 5c, respectively. First, note that $\Psi DP$ might include (i) a $\delta hv$ bump in the range [$3.5;5.5$] km and (ii) oscillations in the range [$6.5;8.5$] km. Second, the $\delta hv$ bump is more noticeable in $\Phi DP$(C) than in $\Phi DP$(AHR). In consequence, the matching between $\Phi DP$(C) and $\Phi DP$(CZPHI) shown in Fig. 5b is not as good as the one observed in Fig. 5c. Note that $\Phi DP$(CZPHI) represents $\Phi DP\u2061(ri,\alpha )$ in Eq. (3). The extent of the $\Phi DP$(AHR) profile is less than that of $\Phi DP$(C) because *M* in Eq. (2) appears to be 0 at the beginning and ending ranges of $\Psi DP$. However, this limited extent of $\Phi DP$(AHR) avoids the oscillations seen at the ending ranges of $\Psi DP$.

The selected *α*–$\Phi DP$(C) and *α*–$\Phi DP$(AHR) values as a function of azimuth for the same storm are depicted in Fig. 5d. Values for *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*–$\Phi DP$(AHR) values are close to 0.34 dB (°)^{−1}, whereas those related to $\Phi DP$(C) are mostly smaller than 0.34 dB (°)^{−1} and sometime equal to $\alpha min$. An optimal *α* that equals $\alpha min$ or $\alpha max$ could be associated with an inadequate matching between the input $\Phi DP$ and the obtained $\Phi DP$(CZPHI), which can lead to incorrect *α*. The resulting $e\xafmin$ values associated with $\Phi DP$(C) and $\Phi DP$(AHR) are 2.16° and 0.20°, respectively, and their corresponding $\sigma emin$ values are 0.75° and 0.08°. These results come from the azimuthal sector [$280\xb0;310\xb0$], which covers approximately the north side of the storm shown in Fig. 3. Outside this sector, the constant *α* was selected, associated with either $\Phi DP$(C) or $\Phi DP$(AHR), because the stated conditions were not met.

#### 2) Performance analysis

The impact of the optimal selection of *α*–$\Phi DP$(C) and *α*–$\Phi DP$(AHR) on the estimation of *A*(CZPHI) is measured using $KDP$(AHR) as a reference because of $1)$ the consistency between $KDP$(AHR) and *A*(ZPHI) demonstrated in section 3b and $2)$ the fact that the presented data were collected from one radar. Hence, the following analysis is based on internal polarimetry consistency.

The scatterplots *A*(CZPHI, C)–$KDP$(AHR) and *A*(CZPHI, AHR)–$KDP$(AHR) resulting from event E1 are shown in Fig. 6a. Observe that multiple *A*(CZPHI, C) estimates are smaller than those from *A*(CZPHI, AHR) as a consequence of selecting “small optimal” *α*–$\Phi DP$(C) values. The correlation coefficient $\rho AK$ from *A*(CZPHI, C)–$KDP$(AHR) is equal to 0.78, while from *A*(CZPHI, AHR)–$KDP$(AHR) it is 0.98. Their corresponding standard deviations $\sigma AK$ with respect to *A* = $\alpha KDP$ are 0.28 and 0.05 dB km^{−1}, respectively, where *α* values are given by *α*–$\Phi DP$(AHR). In Fig. 6b, attenuation-corrected *Z*(CZPHI, C) and *Z*(CZPHI, AHR) are compared against *Z*(DP, AHR), where *Z*(DP, AHR) is obtained from $KDP$(AHR) and *α*–$\Phi DP$(AHR). Their root-mean-square errors (RMSE) are equal to 1.67 and 0.10 dB, respectively, for *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.

A similar analysis of *A*(CZPHI) is performed using $KDP$(C) as a reference instead of $KDP$(AHR) and the results are summarized next. The correlation coefficient between *A*(CZPHI, C) and $KDP$(C) is equal to 0.59 and smaller than those shown in Fig. 6a. This is because of the limited accuracy associated with $KDP$(C). The resulting RMSE between *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*–$\Phi DP$(C) values, leading to similar attenuation correction results. Nonetheless, even if *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 $zdr$ and attenuation-corrected *Z*(CZPHI, AHR) and $ZDR$ fields from event E1 are displayed in Fig. 7. The *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 $\Psi DP$ (see Fig. 3). A similar situation is observed by comparing the fields of $zdr$ and $ZDR$, where enhanced areas of $ZDR$ correspond to oblate raindrops. From the $ZDR$ field, it seems that its lower bound is between −2 and −1 dB, which could be due to radar miscalibration rather than prolate-shaped particles, and therefore *Z* and $ZDR$ fields may not represent calibrated measurements. Furthermore, the radial pattern presented in the $zdr$ and $ZDR$ fields may be associated with an azimuthal modulation as result of a metallic fence near the radar causing PBB effects (Giangrande and Ryzhkov 2005). Although such error sources may cause uncertainties on *Z* and $ZDR$, they do not seem to affect estimates of $KDP$ 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*–$\Phi DP$(AHR) instead of *α*–$\Phi DP$(C). Nonetheless, event E2 shows specific signatures associated with the spatial distribution of raindrop size that can be used to illustrate the ability of selecting proper *α* values using the outcome of both $KDP$ approaches.

The resulting *Z*(CZPHI, AHR) and $ZDR$ fields at 1450 UTC, associated with PIA (PIA_{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 $ZDR$ field shows an area of significantly enhanced values along the inflow edge (arrow 3). This feature, commonly seen in supercell storms, is referred to as the $ZDR$ arc signature as a result of possible size sorting processes (Kumjian and Ryzhkov 2008). The fields of $KDP$(C) and $KDP$(AHR) are also illustrated in Fig. 8. It is seen that the $KDP$(AHR) field retains the spatial variability of the storm better than the $KDP$(C) field while reducing negative $KDP$ estimates. Note that both $KDP$ fields show enhanced values along the inflow edge of the storm with values as high as 12° km^{−1} collocated with the $ZDR$ arc. Estimates of $KDP$ over the echo appendage, in both $KDP$ fields, are not possible because of its narrow width of less than 1 km.

The selected values for *α*–$\Phi DP$(C) and *α*–$\Phi DP$(AHR) are given in Fig. 9 as a function of azimuth. Observe that the optimization of *α* using $\Phi DP$(C) was possible only in three azimuthal profiles of the mini-supercell. This is because in multiple azimuthal profiles, the percentage of gates per profile with $KDP$(C) > 0° km^{−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 $\rho AK$ as shown in Table 3. On the other hand, the optimization of *α* using $\Phi DP$(AHR) occurred in multiple azimuthal profiles, resulting in values mostly larger than 0.34 dB (°)^{−1} in contrast to those resulting from $\Phi DP$(C). According to Ryzhkov and Zrnić (1995) and Carey et al. (2000), such large values are expected in areas of big raindrops, which is consistent with the $ZDR$ arc signature.

### 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), $ZDR$, $KDP$(C), and $KDP$(AHR) fields at 1955 UTC are shown in Fig. 10. The *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 $ZDR$ and $KDP$ values, located in the center of the white circles. It can be observed that $KDP$(AHR) preserves the storm structure better than $KDP$(C) because the AHR approach avoids a segmented $KDP$ texture and negative $KDP$ values, which are observed in the $KDP$(C) field. Maximum values of PIA and PIA_{DP} reached 18 and 3 dB, respectively, while fully attenuated areas (south side) occurred behind strong rain echoes associated with $KDP$ values on the order of 10° km^{−1}.

The resulting values of *α*–$\Phi DP$(C) and *α*–$\Phi DP$(AHR) as a function of the azimuthal sector [0°; 360°], not shown here, indicate that for most azimuthal profiles, *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 $\Delta \Phi DP$ values are smaller than 10°. Optimal values of *α*–$\Phi DP$(AHR) are predominantly found between 0.34 and 0.50 dB (°)^{−1}. The absence of $\alpha >$ 0.50 dB (°)^{−1}, in contrast to event E2, may indicate the lack of big drops present at this time. Selected values of *α*–$\Phi DP$(C) are frequently smaller than or equal to 0.34 dB (°)^{−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 $e\xafmin$ and $\sigma emin$, together with $\rho AK$, $\sigma AK$, and RMSE are given in Table 3, showing that $\Phi DP$(AHR) profiles lead to more reliable values of *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), $ZDR$, $KDP$(C), and $KDP$(AHR) at 0558 UTC are shown in Fig. 11. Comparing the fields of *Z* and $KDP$, the $KDP$(AHR) field maintains the spatial structure of the storm better than $KDP$(C). It can be seen that the magnitudes of $KDP$(C) and $KDP$(AHR) are frequently smaller than 4° km^{−1}, implying a slow incremental behavior of estimated $\Phi DP$ profiles. As such, only the azimuthal sectors [75°; 150°] (east side) and [250°; 280°] (west side) were associated with $\Delta \Phi DP>$ 10°. In both sectors, the optimization process was characterized by an inadequate performance because, in multiple azimuthal profiles, repetitive values equal to 0.1 dB (°)^{−1} were selected and the associated errors were larger than those found in E1–E3. In the remaining profiles, values of *α*–$\Phi DP$(C) were smaller than 0.34 dB (°)^{−1}, while values of *α*–$\Phi DP$(AHR) were comparable to 0.34 dB (°)^{−1}, indicating the absence of raindrops of considerable size. The results associated with the selection of *α* using $\Phi DP$(C) and $\Phi DP$(AHR) are indicated in Table 3, showing a decreased performance of the CZPHI method compared to the results of E1–E3.

## 5. Evaluation of $\delta hv$ estimates

For each storm event, the preprocessed $\Psi DP$ (section 3a), the obtained $KDP$(AHR) fields, and *A*(CZPHI, AHR) fields were set as inputs to the $\delta hv$ algorithm for its evaluation. As part of the $\delta hv$ approach (step 1), a low-pass FIR filter specified by a 32-filter order and 1-km cutoff range scale was applied to the $\Psi DP$ field.

The estimated $\delta hv$ fields resulting from storm events E1–E4 at 1216, 1450, 1955, and 0558 UTC, respectively, are shown in Fig. 12. In all events, it can be seen that the areas of $\delta hv$ that are given by a uniform value correspond to the areas of *Z* smaller than the 30-dB*Z* level, which defines the shape of the described storm cells. Moreover, a spatial correlation between the $\delta hv$ fields and their corresponding $ZDR$ fields is observed, which confirms the correlation nature between $\delta hv$ and $ZDR$ (e.g., compare Figs. 12a and 7d). Such a spatial correlation is not exclusive to $\delta hv$ and $ZDR$ because a similar correlation is also observed between the fields of $\delta hv$, *Z*, and $KDP$, exemplifying the self-consistency relation (Scarchilli et al. 1996) between $ZDR$, *Z*, and $KDP$ in a comparable manner.

The ability of the algorithm to capture the spatial variability of $\delta hv$ is substantial. For example, in Fig. 12a, significant $\delta hv$ values are more visible on the north side than on the south side of the storm cell, indicating the presence of Mie scattering. Another example of the spatial variability and consistency aspects of $\delta hv$