In radar polarimetry, the differential phase consists of the propagation differential phase and the backscatter differential phase . While is commonly used for attenuation correction (i.e., estimation of the specific attenuation A and specific differential phase ), recent studies have demonstrated that can provide information concerning the dominant size of raindrops. However, the estimation of and is not straightforward given their coupled nature and the noisy behavior of , especially over short paths. In this work, the impacts of estimating 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 and A. In addition, an improved technique is proposed to compute from and 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 and A showed that more accurate and stable α and A are obtained if is estimated at range resolution, which is not possible by conventional range filtering techniques. Accurate 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 .
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 (°) is independent of attenuation, miscalibration, and partial beam blockage (PBB) effects (Doviak and Zrnić 1993). However, measurements can include phase shifts in the backward direction as a result of Mie scattering, the so-called backscatter differential phase (°), and random fluctuations ε (°) on the order of few degrees. In general, a range profile is modeled as
where (°) 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 are the specific differential phase (° 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 (or ) from when is significant is given by Hubbert and Bringi (1995), and several attempts have been proposed to improve 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 has been used to correct measurements of reflectivity Z (dBZ) affected by radar calibration and PBB (Giangrande and Ryzhkov 2005). In addition, 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 , comprehensive research on is still needed because it is a challenge to provide accurate from noisy measurements of .
Existing methods to estimate A in rain assume that A = , where α is a constant for a given frequency (Bringi et al. 1990). Testud et al. (2000) also used the relation between A and in their rain profiling ZPHI technique, to express A in terms of the difference of and measurements of Z, avoiding 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 and α (Bringi et al. 2001; Ryzhkov and Zrnić 2005).
In contrast to and A, limited research has been conducted on the applications of . For example, can be a suitable candidate to mitigate uncertainties related to the differential reflectivity (dB) because and offer a correlated behavior (Scarchilli et al. 1993; Testud et al. 2000) and because is independent of attenuation and radar calibration; see Eq. (1). These aspects of could be useful to establish relations between and the median drop diameter (mm) (Trömel et al. 2013) because is often expressed in terms of (Matrosov et al. 2005; Kim et al. 2010). Moreover, Otto and Russchenberg (2010) included estimates to retrieve DSD parameters. Hubbert and Bringi (1995), Otto and Russchenberg (2011), and Trömel et al. (2013) estimated by subtracting from , while Schneebeli and Berne (2012) included a Kalman filter approach. The effectiveness of estimating at high resolution is rather complicated because of the cumulative and noisy nature of and possible remaining fluctuations on .
The purpose of this work is to 1) explore the role and impact of estimated 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 in rain while keeping the spatial variability of drop sizes. For such purpose, two (or ) 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 algorithm is introduced, which integrates estimates of 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 technique is evaluated. Section 6 focuses on the statistics of α, A, Z, and to conduct further assessments of the presented methods. Finally, section 7 draws conclusions of this article.
2. Estimation techniques for -based variables
a. Estimation of
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 (km) are filtered from a profile. Fluctuations at range scales larger than (i.e., “bumps”) are removed by applying the same filter multiple times to new generated profiles by combining a previous filtered and original profile. In this manner the corresponding profile is obtained and is given by taking a range derivative of . For the generation of , a predetermined threshold τ (°) is required, which is on the order of 1–2 times the standard deviation of , hereafter (°). One of the limitations of this technique is that accurate estimates of and at scales are hardly achieved (Grazioli et al. 2014).
An adaptive approach that estimates at high spatial resolution while controlling its standard deviation (° 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 . Besides , attenuation-corrected Z and profiles are also required, as well as a predefined pathlength interval  (km). For gate i, located at range , a set of samples are obtained from  using a theoretical expression of . The pathlength that minimizes the set is selected and denoted as . Assuming the correlated behavior between and , samples in the range [; ] that do not satisfy the condition are filtered to avoid contamination from . The standard deviation of the profile is denoted as . The spatial variability of at scales is captured by downscaling each remaining sample from to scale. A downscaling parameter [0, 1] is derived from Z and in the same interval [; ], and is estimated as
where M represents the number of samples with negligible . The actual is calculated using the terms inside the sum operation in Eq. (2). The and profiles are obtained by repeating the same procedure over the remaining gates, while the corresponding profile is calculated by simply integrating in range. In addition, a profile of the normalized standard error (NSE) of is given by the ratio between actual and . This approach was demonstrated for rain particles at X-band frequencies, and therefore any undetected Z and echoes from hydrometeors other than rain can lead to inaccurate estimates. The two 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 profiles are represented as and , respectively, where z (dBZ) and (dB) represent the attenuated reflectivity and the attenuated differential reflectivity, respectively; and PIA (dB) indicates the two-way path-integrated attenuation in reflectivity and PIA (dB) in differential reflectivity.
Bringi et al. (1990) introduced the differential phase (DP) approach such that = and PIA, 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 , 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 in a path interval , where . First, is expressed as a function of two known variables, and , and one unknown, . Then, is obtained using and the empirical relation PIA = , where PIA = PIA − PIA and = . In this way, is estimated at scales, reducing errors related to . Although  can be freely selected; could be inaccurate at short path intervals and/or be contaminated by and . In addition, if includes localized observations of hail or mixtures of rain and hail in , then might be biased over the entire path interval.
Using a constant α may lead to limited approximations of and PIA 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 , and is estimated according to the ZPHI method. The estimated is integrated over  to build a differential phase profile denoted as . 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 , which implies the need for a proper way to filter noise and components from 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 . The inputs and outputs associated with the three presented attenuation correction methods are summarized in Fig. 1.
To determine PIA, integrate the specific differential attenuation (dB km−1) that is given by . 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, will be given by = (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), dB (°)−1; Kim et al. (2010), dB (°)−1; and Snyder et al. (2010) , dB (°)−1; while Ryzhkov et al. (2014) estimated γ equal to 0.14 for tropical rain (i.e., low and high ) and 0.19 for continental rain (i.e., high and low ). 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 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
A approach is presented to identify and separate Mie scattering signatures from noise and random fluctuations embedded in . A flowchart of the algorithm is illustrated in Fig. 2. Three inputs are required: a 2D field measured in rain, the corresponding field obtained from the AHR approach, and the A field estimated by the CZPHI method. Given these inputs, the resulting field is based on the following five steps:
Design and apply a filter to smooth strong outliers from a profile, taking into account. Correct each smoothed profile for system phase offset by subtracting the mean of over the first 5% of measured gates.
Obtain by integrating profiles of A, if they are associated with a minimum error E, otherwise by integrating profiles. Next, subtract from , profile by profile, as a first attempt to estimate the corresponding field. The next steps are related to 2D processing.
- Remove unusual values larger than 12° from the field. According to Testud et al. (2000), Trömel et al. (2013), and Schneebeli et al. (2014), the simulated values at X-band frequencies rarely reach 12°. The remaining noise in is reduced by assuming that similar values of are collocated with similar values of as follows. Set as the minimum of and as , where (° km−1) is given by Eq. (4). Define S as a set of samples, whose gates are collocated with values in the interval . Reject samples from S that are outside the interval [; ], where and indicate the arithmetic mean and the standard deviation of the samples in S, respectively; υ is a predefined threshold in the interval  and a value of 1 is chosen. This process is iterated by shifting  toward high values in small steps such that = and = until is equal to the maximum of . To obtain sufficient samples in S, is given as values are less frequent than small values (e.g., see the fields in Figs. 3, 8, and 11).
Apply a 2D interpolation method to fill empty gaps on 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 that are linked to < 0.4° km−1 (i.e., weak rain echoes) by a representative value. This value is chosen as the mean of samples constrained by < 0.4° km−1 and , where indicates the mean of samples obtained in a similar manner as in step 3 but using after step 4. The value of 0.4° km−1 is found to match the 30-dBZ level used in this work for storm cell identification.
3. Evaluation of 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 (dB) are used, such that range gates with larger than −18 dB are discarded from , z, and fields. Further preprocessing includes suppressing isolated segments of a profile smaller than 0.25 km and rejecting a profile if the percentage of gates with measurements is less than 5%. Because a 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 profile is determined based on , which represents the average of multiple samples by running a five-gate window along the profile. If 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 values are less than , starting at the last measured gate and moving toward the radar. The ending range is used to limit the corresponding extent of z and profiles. After this, is calculated again to estimate by the conventional technique.
b. Comparison between and A
Next, (C) and (AHR) will be compared against A(ZPHI) using the empirical relation A = , 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 techniques and their impact on Z.
To estimate (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. Such a filter design is found suitable for 0.03 km. For the estimation of (AHR), values of L on the order of 3 km are associated with theoretical values of 0.5° km−1 for = 0.03 km (Reinoso-Rondinel et al. 2018) and therefore  is predefined as  km. The z and 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 profiles. To estimate a five-gate window is run along a given profile. For the calculation of A(ZPHI), is derived from (C) instead of (AHR) to evaluate (AHR) in an independent manner. A path interval  is defined by the first and last data points, in the downrange direction, of a (C) profile. In cases where 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 field shows a rapid increment in range on the north side of the storm, whereas rarely increases on the south side. Note that the 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-dBZ contour is obtained from the attenuation-corrected Z using the ZPHI method [i.e., after calculating A(ZPHI) as explained previously]. Comparing (C) and (AHR), the (AHR) field is able to maintain the spatial variability of the storm down to range resolution scale, eliminating areas of smaller than −1° km−1, which are present in (C). However, the coverage of the (AHR) field is smaller than that of (C). This is because in the AHR approach, it is not always possible to obtain samples with negligible ; that is, in Eq. (2). Note that isolated segments smaller than 2 km were removed from both fields in order to avoid estimates of that could be associated with noisy areas and/or low accuracy.
The scatterplots (C)–A(ZPHI) and (AHR)–A(ZPHI) resulting from the same event, E1, are compared in Fig. 4. In Fig. 4a, it can be seen that the (AHR)–A(ZPHI) scatterplot (14 783 data points) is more consistent than that of (C)–A(ZPHI) (15 490 data points) with respect to the empirical relation A = 0.34. In a quantified comparison, the correlation coefficient between (C) and A(ZPHI) is equal to 0.65, whereas for (AHR) and A(ZPHI) it is 0.96. Their corresponding standard deviations with respect to the empirical relation are 1.20 and 0.41° km−1, respectively. To compare the impact of both 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 dBZ. The errors quantified by , , 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 (AHR) and A(ZPHI), determined by two independent methods, show a strong agreement to the empirical relation , leading to equivalent Z(DP,AHR) and Z(ZPHI,C) results. In the contrary, the agreement between (C) and A(ZPHI) is less evident, and although (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 processing on the CZPHI method
In this section, the ability to estimate by both 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 = , with i = , where represents the minimum error at range . As such, the arithmetic mean and standard deviation of , (°) and (°), respectively, will be used as quality measures.
At X-band frequencies,  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 should be at least 3 km and that be larger than 10°. In addition, if the (C) profile is used in Eq. (3), then the percentage of gates with 0° km−1 should be at least 50%, whereas if the (AHR) profile is used, the percentage of gates with 0.5° km−1 and NSE < 20% should be larger than 80%. The percentage threshold for (C) is less than for (AHR) because the conventional method rarely avoids negative 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 (C) is used is much larger than when (AHR) is used and their corresponding optimal values for α are α–(C) = 0.24 and α–(AHR) = 0.34 dB (°)−1. The reason why the two α values are different can be explained by observing the measured and the estimated (C) and (AHR) profiles shown in Figs. 5b and 5c, respectively. First, note that might include (i) a bump in the range  km and (ii) oscillations in the range  km. Second, the bump is more noticeable in (C) than in (AHR). In consequence, the matching between (C) and (CZPHI) shown in Fig. 5b is not as good as the one observed in Fig. 5c. Note that (CZPHI) represents in Eq. (3). The extent of the (AHR) profile is less than that of (C) because M in Eq. (2) appears to be 0 at the beginning and ending ranges of . However, this limited extent of (AHR) avoids the oscillations seen at the ending ranges of .
The selected α–(C) and α–(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–(AHR) values are close to 0.34 dB (°)−1, whereas those related to (C) are mostly smaller than 0.34 dB (°)−1 and sometime equal to . An optimal α that equals or could be associated with an inadequate matching between the input and the obtained (CZPHI), which can lead to incorrect α. The resulting values associated with (C) and (AHR) are 2.16° and 0.20°, respectively, and their corresponding values are 0.75° and 0.08°. These results come from the azimuthal sector , which covers approximately the north side of the storm shown in Fig. 3. Outside this sector, the constant α was selected, associated with either (C) or (AHR), because the stated conditions were not met.
2) Performance analysis
The impact of the optimal selection of α–(C) and α–(AHR) on the estimation of A(CZPHI) is measured using (AHR) as a reference because of the consistency between (AHR) and A(ZPHI) demonstrated in section 3b and 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)–(AHR) and A(CZPHI, AHR)–(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–(C) values. The correlation coefficient from A(CZPHI, C)–(AHR) is equal to 0.78, while from A(CZPHI, AHR)–(AHR) it is 0.98. Their corresponding standard deviations with respect to A = are 0.28 and 0.05 dB km−1, respectively, where α values are given by α–(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 (AHR) and α–(AHR). Their root-mean-square errors (RMSE) are equal to 1.67 and 0.10 dB, respectively, for Z(DP, AHR) ≥ 35 dBZ. 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 (C) as a reference instead of (AHR) and the results are summarized next. The correlation coefficient between A(CZPHI, C) and (C) is equal to 0.59 and smaller than those shown in Fig. 6a. This is because of the limited accuracy associated with (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–(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 and attenuation-corrected Z(CZPHI, AHR) and 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 (see Fig. 3). A similar situation is observed by comparing the fields of and , where enhanced areas of correspond to oblate raindrops. From the 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 fields may not represent calibrated measurements. Furthermore, the radial pattern presented in the and 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 , they do not seem to affect estimates of 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–(AHR) instead of α–(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 approaches.
The resulting Z(CZPHI, AHR) and fields at 1450 UTC, associated with PIA (PIADP) 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 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 arc signature as a result of possible size sorting processes (Kumjian and Ryzhkov 2008). The fields of (C) and (AHR) are also illustrated in Fig. 8. It is seen that the (AHR) field retains the spatial variability of the storm better than the (C) field while reducing negative estimates. Note that both fields show enhanced values along the inflow edge of the storm with values as high as 12° km−1 collocated with the arc. Estimates of over the echo appendage, in both fields, are not possible because of its narrow width of less than 1 km.
The selected values for α–(C) and α–(AHR) are given in Fig. 9 as a function of azimuth. Observe that the optimization of α using (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 (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 as shown in Table 3. On the other hand, the optimization of α using (AHR) occurred in multiple azimuthal profiles, resulting in values mostly larger than 0.34 dB (°)−1 in contrast to those resulting from (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 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), , (C), and (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 and values, located in the center of the white circles. It can be observed that (AHR) preserves the storm structure better than (C) because the AHR approach avoids a segmented texture and negative values, which are observed in the (C) field. Maximum values of PIA and PIADP reached 18 and 3 dB, respectively, while fully attenuated areas (south side) occurred behind strong rain echoes associated with values on the order of 10° km−1.
The resulting values of α–(C) and α–(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 values are smaller than 10°. Optimal values of α–(AHR) are predominantly found between 0.34 and 0.50 dB (°)−1. The absence of 0.50 dB (°)−1, in contrast to event E2, may indicate the lack of big drops present at this time. Selected values of α–(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 and , together with , , and RMSE are given in Table 3, showing that (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 dBZ 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), , (C), and (AHR) at 0558 UTC are shown in Fig. 11. Comparing the fields of Z and , the (AHR) field maintains the spatial structure of the storm better than (C). It can be seen that the magnitudes of (C) and (AHR) are frequently smaller than 4° km−1, implying a slow incremental behavior of estimated profiles. As such, only the azimuthal sectors [75°; 150°] (east side) and [250°; 280°] (west side) were associated with 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–(C) were smaller than 0.34 dB (°)−1, while values of α–(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 (C) and (AHR) are indicated in Table 3, showing a decreased performance of the CZPHI method compared to the results of E1–E3.
5. Evaluation of estimates
For each storm event, the preprocessed (section 3a), the obtained (AHR) fields, and A(CZPHI, AHR) fields were set as inputs to the algorithm for its evaluation. As part of the approach (step 1), a low-pass FIR filter specified by a 32-filter order and 1-km cutoff range scale was applied to the field.
The estimated 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 that are given by a uniform value correspond to the areas of Z smaller than the 30-dBZ level, which defines the shape of the described storm cells. Moreover, a spatial correlation between the fields and their corresponding fields is observed, which confirms the correlation nature between and (e.g., compare Figs. 12a and 7d). Such a spatial correlation is not exclusive to and because a similar correlation is also observed between the fields of , Z, and , exemplifying the self-consistency relation (Scarchilli et al. 1996) between , Z, and in a comparable manner.
The ability of the algorithm to capture the spatial variability of is substantial. For example, in Fig. 12a, significant 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