## Abstract

In a two-part paper, radar rain-rate retrievals using specific attenuation *A* suggested by Ryzhkov et al. are thoroughly investigated. Continuous time series of overlapping measurements from two twin polarimetric X-band weather radars in Germany during the summers of 2011–13 are used to analyze various aspects of rain-rate retrieval, including miscalibration correction, mitigation of ground clutter contamination and partial beam blockage (PBB), sensitivity to precipitation characteristics, and the temperature assumptions of the *R*(*A*) technique. In this paper, the relations inherent to the *R*(*A*) method are used to estimate radar reflectivity *Z* from *A* and compare it to the measured *Z* in order to estimate PBB and calibration offsets for both radars. The fields of *Z* estimated from *A* for both radars are consistent, and the differences between *Z*(*A*) and measured *Z* are in good agreement with the ones calculated using either consistency relations between reflectivity at horizontal polarization *Z*_{H}, differential reflectivity *Z*_{DR}, and specific differential phase *K*_{DP} in rain or a digital elevation model in the presence of PBB. In the analysis, the dependence of *A* on temperature appears to have minimal effects on the overall performance of the method. As expected, the difference between *Z*(*A*) and attenuation-corrected measured *Z* observations varies with rain type and exhibits a weak systematic dependency on rainfall intensity; thus, averaging over several rain events is required to obtain reliable estimates of the *Z* biases caused by radar miscalibration and PBB.

## 1. Introduction

The potential utilization of specific attenuation *A* for rainfall estimation, mitigation of partial beam blockage (PBB), and radar networking was discussed in the paper by Ryzhkov et al. (2014, hereafter RAL14). Like Atlas and Ulbrich (1977) and Matrosov (2005), they showed that, theoretically, the relation between precipitation intensity *R* and *A* depends less on drop size distribution (DSD) than traditional rainfall algorithms based on radar reflectivity *Z*, differential reflectivity *Z*_{DR}, and specific differential phase *K*_{DP} in a wide range of rain intensity. The estimation of specific attenuation is, however, subject to further uncertainties, particularly without polarimetric measurements (Berne and Uijlenhoet 2006; Delrieu et al. 1999, 2000; Trömel et al. 2014b). RAL14 estimate *A* with the so-called ZPHI method from the radial profile of measured *Z* and the total span of differential phase ΔΦ_{DP} following Bringi et al. (1990) and Testud et al. (2000). Since the *A* estimate based on differential phase shift is unaffected by reflectivity biases caused by radar miscalibration, attenuation, PBB, and wet radomes (Kurri and Huuskonen 2008), the same holds for rain retrievals based on *R*(*A*) and for radar reflectivity retrievals based on *Z*(*A*) relations, which may possibly be exploited to estimate these error sources quantitatively. RAL14 tested the *R*(*A*) method on data collected by two closely located X-band radars in Germany and polarimetrically upgraded WSR-88D S-band radars in the United States. They demonstrated for one rain event that the two adjacent X-band radars, one of which was strongly miscalibrated and the other affected by PBB, produce almost indistinguishable fields of rain rate when using *R*(*A*). It was also shown that the *R*(*A*) method yields robust estimates of rain rates and rain totals at S band, where specific attenuation is vanishingly small.

While the methodology described in RAL14 shows great promise, a more thorough validation study is necessary to quantify the ensuing quantitative precipitation estimation (QPE) improvement and to determine the sensitivity of the method to potential uncertainties associated with the variability of temperature and the ratio *α* = *A*/*K*_{DP}, which is one of the key parameters in the ZPHI method. The operational implementation of the *R*(*A*) method also raises some technical issues that need to be settled, such as a minimal required span of total differential phase ΔΦ_{DP} along the propagation path in rain, the difference in the algorithm performance in light and heavy rain, and the mitigation of possible ground clutter contamination. Moreover, *K*_{DP}- and *A*-based estimators usually benefit from combinations with *Z*-based estimators at low rain rates, low ΔΦ_{DP}, or otherwise compromised *K*_{DP} or ΔΦ_{DP} measurements (Figueras i Ventura et al. 2012; Vulpiani et al. 2012; Park et al. 2005). Thus, correcting *Z* measurements for PBB and calibration errors will remain advantageous for QPE, in addition to the need of unbiased reflectivity measurements when analyzing the vertical structure of precipitation (Berne et al. 2005) or hydrometeor classification (Al-Sakka et al. 2013). Because DSDs change in different rain regimes (Bringi et al. 2003), it also remains to be seen if the *R*(*A*) and *A*(*Z*) relations derived by RAL14 from a disdrometer dataset in Oklahoma are transferable to our observation area and to what extent natural DSD variability affects the stability of these relations. The use of *Z*(*A*) may yield an operationally efficient way to monitor calibration and PBB without the need of additional measurements apart from already observed *Z* and Φ_{DP}.

This paper is the first part of a series of two papers. In the first part, we introduce the radars used for the study and compare the performance of the attenuation-based technique for the correction of *Z* for miscalibration and PBB with alternative methods. In the second part of the series (Diederich et al. 2015, hereafter Part II), the rain totals derived from different methods including the new *R*(*A*) algorithm are compared with measurements of precipitation from networks of rain gauges. Section 2 of this first paper describes the observations and the digital elevation model (DEM)-based PBB estimation. In section 3, we derive PBB maps and radar calibration biases from the difference between *Z*(*A*) and attenuation-corrected *Z* measurements and consistency relations between *K*_{DP}, *Z*_{DR}, and *Z* in rain. Section 4 analyzes the dependency of *Z*(*A*) relations on temperature and precipitation characteristics, while section 5 demonstrates the quantitative consistency between *Z*(*A*) from both X-band radars and an overlapping C-band radar. Section 6 summarizes and discusses the results and expected implications for QPE.

## 2. Observations and DEM-based PBB estimation

Two polarimetric X-band Doppler radars, BoXPol and JuXPol (see Table 1 for details), were installed recently in western Germany at a distance of 48.5 km from each other (Fig. 1). Both radars have been measuring precipitation with 100-km maximum range since 2010. Thus, a large area of overlapping radar observations exists. Observations from both radars are considerably affected by ground clutter. While JuXPol suffers from notable miscalibration during the investigated period, several sectors of BoXPol observations are strongly affected by PBB. Thus, the twin system is ideally suited to evaluate algorithms for clutter reduction, beam blockage correction, calibration, and compositing. The German weather service [Deutscher Wetterdienst (DWD)], in the ongoing process to upgrade its complete radar network, began operation of a polarimetric C-band weather radar in Offenthal (Table 1, Fig. 1) in 2011 at approximately 130-km distance from Bonn, for which data were kindly made available for comparison in the study.

The BoXPol radar of the Transregional Collaborative Research Centre 32 (TR32; www.tr32.de) is installed on a 30-m tall building next to the Meteorological Institute of the University of Bonn (Table 1) at 99.9 m MSL. At 1.5° elevation (see Figs. 2, 3b; see Table 2) two hills block the beam between 128° and 175° azimuth, with a minimum visible elevation of 2° seen by the radar at 161° azimuth. Other notable partial beam blockers are the Siebengebirge east of the radar at approximately 6–10-km distance, some rather flat hills to the southwest, two industrial chimneys to the northeast, the so-called “Post Tower” building between the radar and the Siebengebirge, the city administration building at 65° azimuth, and another tallish building at 212°.

A DEM combined with theodolite observations were used to calculate the PBB for BoXPol independently of the *R*(*A*) or consistency methods discussed in the next section. The Shuttle Radar Topography Mission (Farr et al. 2007) data at 3-arc-s resolution (SRTM3; approximately 93 m × 59 m spatial resolution at Bonn) was used to derive the elevation angle at which the radar beam should hit the terrain, assuming standard atmospheric refraction. The lowest visible elevation for each radar bin was calculated with a resolution of 0.1° in azimuth and 50 m in range using nearest neighbor interpolation. Theodolite measurements made directly at the radar antenna, together with slant ranges obtained from satellite images available from Google Maps, were included in the maps of minimum visible elevation to account for blockages caused by the most important structures not captured by the SRTM3 data.

The normalized two-way antenna beam pattern was calculated for a 1° at half maximum antenna beamwidth with 0.1° azimuth and 0.005° elevation resolution, assuming a perfect Gaussian shape for an angle range between −1.5° and +1.5° from the beam center. Additional horizontal “smearing” of the antenna beam pattern due to azimuthal rotation was taken into account by simulating an effective antenna pattern in horizontal direction. The polar-gridded maps of minimum unobstructed elevation are then combined with the effective beam pattern to calculate the PBB percentage by dividing the remaining beam power *P*_{rem} by the original beam power *P*_{orig}. This PBB-related reflectivity bias is converted into a correction factor given in logarithmic units (see Fig. 3b for elevation 1.5°) by

The JuXPol radar of the Terrestrial Environmental Observatories (TERENO) program (http://teodoor.icg.kfa-juelich.de/) of the Helmholtz Association is installed on the Sophienhöhe, a hill created from open-pit mining, roughly 200 m above the surrounding terrain. While JuXPol is practically unblocked at elevations above 0.5° because of its position on the hilltop, its observations are severely influenced by ground clutter, especially from the numerous windmill power generators, which are “seen” by unblocked side lobes of the antenna beam pattern. The Offenthal polarimetric C-band radar of DWD is partially blocked at 0.5° elevation by the Taunus mountain range approximately 40 km to the northwest of its location.

Weather station measurements of temperature taken approximately 80 m from BoXPol were extrapolated using a 6.5 K km^{−1} vertical gradient to provide rough estimates of temperature at radar beam heights. While numerical weather model or radiosonde temperature data may be accurate in certain situations, this simple extrapolation had the best cost–benefit ratio considering the time frames and sensitivities involved. Because the *R*(*A*) method is intended for liquid phase only, these temperatures were used to avoid melting layer contamination by removing measurements in areas colder than 6°C. They were also used in the temperature-dependent relations defining *α*, *R*(*A*), and *Z*(*A*) shown in Tables 3–5. The relations listed in these tables were derived from T-matrix simulations and a large DSD dataset collected in Oklahoma (adapted from RAL14). Figure 1 also includes locations of rain gauge measurements used in Part II.

## 3. Deriving beam blockage maps and radar calibration from *Z*_{H}(*A*_{H}) and from the *Z*_{H}–*Z*_{DR}–*K*_{DP} consistency relation

*Z*

*A*

*Z*

*Z*

*K*

Quantities derived from differential phase are in principle immune to PBB and radar miscalibration, which can be important systematic error sources in radar reflectivity measurements. With the *R*(*A*) method from RAL14, we were also provided with a tool for estimating rain rate and radar reflectivity independently of radar calibration and PBB, using total differential phase shift. The difference between attenuation-corrected *Z* and the *Z* estimate from specific attenuation therefore quantifies PBB and miscalibration in terms of power. To this goal, we estimated specific attenuation *A* with the ZPHI method from the radial profile of measured *Z* and ΔΦ_{DP} following Bringi et al. (1990) and Testud et al. (2000) as described in RAL14. The radial segments uninterrupted by PBB can be determined via the DEM, manually, or by detection of surface clutter that is usually associated with the onset of PBB. With *Z*_{H,U} as the observed reflectivity at horizontal polarization before ground clutter filtering, *Z*_{H} as the reflectivity at horizontal polarization after ground clutter filtering, and *ρ*_{H,V} as the cross-correlation coefficient, we used the condition *Z*_{H,U} > 35 dB*Z* and *Z*_{H,U} − *Z*_{H} > 5 dB*Z* (or *ρ*_{H,V} < 0.87) to identify potential beam blockers. This method led to good results with the X-band radars when comparing with the DEM-based and manual methods, but it may be unsuitable in regions where the beam is affected by many ground targets with high reflectivity but weak PBB, causing the beam to become excessively segmented.

The path-integrated two-way attenuation along the radial path *s* for an arbitrary segment (*r*_{1}, *r*_{2}) is estimated as

with the net ratio *α* = *A*/*K*_{DP} assumed constant or temperature dependent; we approximated this dependency by linear interpolation between the given values for rain at four different temperatures in Table 3. To avoid contamination from the melting layer, segments were ended if the temperature at the top of the beam dropped below 6°C. In Eq. (2), *T* is the temperature, which may vary along the beam path (*r*_{1}, *r*_{2}). The integrals

and

were calculated for each beam segment using an average exponent *b*(*T*) in the *A*_{H} = *a*(*Z*_{H})^{b} relation linearly interpolated from the values given in Table 4. Specific attenuation is then calculated using the simplified ZPHI formula (RAL14):

By inverting the *A*_{H}(*Z*_{H}) relations in Table 4, the estimate of *Z*_{H}(*A*_{H}) can now be calculated from specific attenuation at horizontal polarization from *A*_{H}. Because *A*_{H} is immune to radar miscalibration, PBB, and the impact of wet radome, the estimate *Z*_{H}(*A*_{H}) is also not affected by these factors as well as path-integrated attenuation. However, the estimate of *Z*_{H} from *A*_{H} is sensitive to the DSD variability and temperature. This dictates the need for averaging or summation over a sufficiently large spatial–temporal domain to mitigate the effect of the DSD variability. It is suggested to estimate the bias of *Z* attributed to miscalibration and PBB using the following relation:

where *Z*_{H} + *IA*_{H} is the observed reflectivity corrected for attenuation by adding

and the *α* values are determined via linear interpolation of the results in Table 2. The extension of the spatial–temporal domain over which summation in Eq. (6) is performed will be discussed later. The name *B*_{A} stands for “*Z* bias determined from *A*” expressed in decibels. In the blocked beam segments, *B*_{A} is a cumulative bias caused by PBB and radar miscalibration while *B*_{A} in the unobstructed beam segments is attributed to miscalibration. Thus, we accumulated these estimates in all unobstructed beam segments to first obtain the radar miscalibration bias and in obstructed beam segments to obtain the PBB bias after subtracting the prior. The accumulations can be done for individual pixels (Fig. 3c) or for beam segments over which PBB can be assumed to be constant (Fig. 3d).

Close to the radar, short segments interrupted by PBB show unrealistic high biases prior to the location where PBB should take place (Fig. 3d). This is most likely a result of clutter contamination of the Φ_{DP} estimates through backscatter differential phase shift, in combination with sampling effects of a “ΔΦ_{DP} greater than” condition when accepting or rejecting ZPHI segments into the accumulations. Both lead to an overestimation of ΔΦ_{DP}. Applying a median filter instead of a running mean to Φ_{DP} for smoothing, filtering values within 200 m of identified clutter pixels, and an azimuthal smoothness criterion on *Z*_{H}(*A*_{H}) instead of a “ΔΦ_{DP} greater than” condition for *Z*_{H}(*A*_{H}) value acceptance reduced the effect, but provided fewer nonrejected *Z*(*A*) segments.

The *Z*_{H} biases caused by PBB and radar miscalibration can also be estimated using the consistency relation between *Z*_{H}, *Z*_{DR} (corrected for integrated attenuation *IA*_{H} and *IA*_{DR}), and *K*_{DP} in rain (Vivekanandan et al. 2003; Giangrande and Ryzhkov 2005; Borowska et al. 2011):

where the sums include only measurements with 45 > *Z*_{H} > 20 dB*Z* and *ρ*_{H,V} > 0.97 in each pixel (Fig. 3e) or these pixels integrated along uninterrupted segments (Fig. 3f). Eliminating *Z*_{DR} biases is required, however, in order to exploit this relation, which can be problematic because of possible changes of *Z*_{DR} due to a system bias drift with time, in blocked sectors, or through radome effects (Figueras i Ventura et al. 2012). Locations affected by clutter show anomalies again, but they are much smaller than for *Z*_{H}(*A*_{H}) because of the stricter filtering of *ρ*_{H,V} > 0.97 performed prior to the application of the consistency relation.

As a third bias estimator, the consistency between *Z*_{H} and *K*_{DP} was used following the study of Zhang et al. (2013) for S band. However, we did not assume that attenuation is negligible, and instead of deriving the mean factor *a* in the *Z*_{H} = *aK*_{DP}^{b} relation for every scan, we used *Z*_{H}(*A*_{H}) = *Z*_{H}(*αK*_{DP}), where *A*_{H} was calculated as the product of the measured *K*_{DP} and the factor *α* rather than using the ZPHI procedure. This makes it possible to calculate the bias, termed *B*_{K}, as

while using the same filter criteria (45 > *Z*_{H}> 20 dB*Z* and *ρ*_{H,V} > 0.97) as in Eq. (8). Our approach also differs from Zhang et al. (2013) by not using the whole integral of *K*_{DP} (i.e., ΔΦ_{DP}) along an uninterrupted segment. This way of calculating *B*_{K}, which in some respects is very similar to *B*_{A}, differs slightly because of the removal of weak signals (where *K*_{DP} and ΔΦ_{DP} are more noisy) and heavy rain [where Φ_{DP} and *K*_{DP} can be contaminated with backscatter differential phase shift (see, e.g., Matrosov et al. 1999, 2002; Otto and Russchenberg 2011; Schneebeli and Berne 2012; Trömel et al. 2013, 2014a) and where *Z*_{H} may be affected by Mie scattering]. The resulting biases, now for both BoXPol and JuXPol, estimated for each individual pixel accumulated from May to September 2011 are shown in Fig. 4 (bottom). The result is nearly identical with the ZPHI-based *B*_{A} (Fig. 4, top) but has a slightly higher spatial variability, which originates from the noisiness of the *K*_{DP} measurements. All methods suggest a calibration bias in unobstructed beams of 2–3 dB for BoXPol and 8 dB for JuXPol.

The outermost uninterrupted beam segments usually cover the largest area of the radar scan. These segments therefore also have more samples to calculate biases with, the smallest random error (if summation for *B*_{A}, *B*_{C}, and *B*_{K} is done over the entire segment), and the largest contribution to measured rainfall throughout the scan. The PBB biases for BoXPol calculated through *Z*_{H}(*A*_{H}), *Z*_{H}–*Z*_{DR}–*K*_{DP} consistency, and *Z*_{H}(*K*_{DP}) consistency for outer segments also show an overall good agreement with the DEM estimates (Fig. 5a). However, the PBB of the buildings at azimuths 65°, 207°, and 212° shows up stronger in the radar-derived PBBs than in the DEM-derived ones. When the DEM-based PBB correction is calculated using an antenna width of 1.2° instead of 1° and an elevation of 1.4° instead of 1.5°, they match more closely (not shown). This may be a result of imperfect and slightly varying antenna positioning (e.g., by wind) as BoXPol does not have a radome. Despite these differences, all three methods of estimating PBB give very consistent results.

To evaluate how much accumulation time is required to produce robust results, *B*_{A}, *B*_{C}, and *B*_{K} were calculated in unblocked beams of the BoXPol radar coverage area with accumulation times of 1, 9, and 19 days of rainfall (not counting the nonrainy days in between). Their temporal variability during the period of May–October 2011 is illustrated in Fig. 5b. It is apparent that in most cases averaging over 1-day periods is not sufficient to produce a reliable estimate of the *Z* bias related to radar miscalibration (dots in Fig. 5b). Averaging over 9- and 19-day periods of rain yields much more stable results (thin and boldface lines in Fig. 5b, respectively). All three bias estimates are within a 2-dB interval. It can be concluded that, for the 6-month period, the radar reflectivity measured by BoXPol was underestimated by 2 ± 1 dB.

As a measure of how well the PBB-related bias is estimated using these accumulation times, the estimated radar miscalibration bias is first subtracted, and the root-mean-square (RMS) deviation of the residual biases toward the DEM-derived biases is calculated (Fig. 5c). Again, averaging over 1-day periods results in too high uncertainties (dot symbols), whereas the RMS errors in the PBB-related bias estimates are generally within 1 dB for the 19-day period. The *B*_{C} estimate (using the *Z*_{H}–*Z*_{DR}–*K*_{DP} consistency) appears to be more stable at single-day accumulation times (blue dots in Fig. 5c) while being very sensitive to miscalibration of *Z*_{DR}. For the analysis shown here, the temporal drift in the *Z*_{DR} calibration had to be compensated by manual postprocessing. The *Z*_{DR} calibration changes were strong enough to change *B*_{C} over 1 dB within the investigated time frame.

## 4. Dependency of *Z*(*A*) bias on temperature and rainfall characteristics

*Z*

*A*

Through the dependence of *α* on temperature, different assumptions on temperature along the beam will manifest themselves through varying biases of *Z*_{H,V}(*A*_{H,V}) or *R*(*A*_{H,V}). To evaluate the consequences of different temperature-dependency assumptions—next to other effects like DSD variability calibration changes—*B*_{A} and *B*_{C} [Eqs. (6) and (8)] were calculated for all unblocked azimuths of individual BoXPol scans from April to October for the years 2011–13 (Figs. 6a,b). The required sums in Eqs. (6) and (8) covered 100 consecutive scans comprising a minimum amount of rain for each scan (at least 5% of the scan). In Fig. 6a, the black curve indicates the temporal evolution of *B*_{A} during a 3-yr period provided that the temperature dependence of the factor *α* and the parameters of the *Z*_{H}(*A*_{H}) power-law relation along the radar beam is taken into account assuming a temperature lapse rate of 6.5 K km^{−1} and the surface temperature equal to the one actually measured at the radar. The blue curve corresponds to the assumption of the same lapse rate and a constant surface temperature equal to 15°C. Green and red curves are obtained assuming that temperature along the beam is constant and equal to 15° and 20°C, respectively. Note that the four curves are almost indistinguishable, with no noticeable seasonal cycle in the offset. Obviously, the impact of temperature assumptions on the bias estimates is much less important than the DSD variability, and the use of a fixed value, or a typical annual temperature cycle extrapolated vertically with a 6.5 K km^{−1} lapse rate, gives satisfactory results. The correlation between *B*_{A} and observed temperature is below 0.15 for both radars regardless of which temperature assumptions are used.

The *B*_{C} obtained from the *Z*_{H}–*K*_{DP}–*Z*_{DR} consistency [Eq. (8)] still agree fairly well with those obtained from *Z*_{H}(*A*_{H}) for both BoXPol and JuXPol (Figs. 6b,c). Note that the more negative the overall calibration bias is, the more negative the *B*_{A} also is compared to *B*_{C}. This could be explained as follows: if the bias is strongly negative and, consequently, the radar sensitivity is lower, then many reflectivity measurements fall below the noise cutoff threshold. Attenuation in these invisible areas of rain will be attributed to the remaining areas of visible reflectivity, which then appear to cause more attenuation/phase shift than they really did. Thus, *Z*_{H}(*A*_{H}) in these areas will be higher, making the original *Z*_{H} measurement erroneously appear more negatively biased than it really was.

When comparing *Z*_{H}(*A*_{H}) to attenuation-corrected *Z*_{H} observations, it is commonly observed that *B*_{A} is larger during light rain than during heavy rain. In Fig. 7a, *B*_{A} is plotted against . The two separate clusters of points in Fig. 7a correspond to very different *B*_{A} during the years 2011–12 and 2013 (Fig. 6). Whether the correlation between bias and rain intensity is linked to the noise-threshold effect mentioned above, sampling effects linked to the ΔΦ_{DP} measurement errors and the impossibility of negative attenuation in the algorithm, inadequate parameters in the used relations in Tables 3–5, artificial enhancement of *IA*_{H} (and therefore also *Z*_{H} + *IA*_{H}) with backscatter differential phase shift in heavy rain, or real changes in the *Z*_{H}(*A*_{H}) relation occurring because of the DSD variability could not be established with absolute certainty. We assume a combination of all. The effect that is evident for both BoXPol and JuXPol appears to be a dominant source of uncertainty of *B*_{A} and is probably what leads to a correlation coefficient of 0.58 (0.75 if smoothed over five consecutive scans) of *B*_{A} calculated in the largely overlapping regions for BoXPol and JuXPol seen in Fig. 7b. The effect is mitigated if accumulations are restricted to measurements where *Z*_{H} + *IA*_{H} and *Z*_{H}(*A*_{H}) are between 15 and 45 dB*Z* (similarly as in *B*_{K}), or when the median of *Z*_{H} + *IA*_{H} − *Z*_{H}(*A*_{H}) is calculated instead of the sums in Eq. (6), which is an indication that ΔΦ_{DP} sampling and backscatter differential phase shift plays a significant role in it.

Despite these remaining uncertainties, the estimated calibration offsets of BoXPol and JuXPol are very consistent when comparing long time series of collocated *Z*_{H}(*A*_{H}) estimates from the two radar systems in Figs. 3–7. Comparisons of collocated measurements of BoXPol and JuXPol suggest that radar reflectivity factors retrieved from specific attenuation for both radars are in good agreement (Fig. 7c, red) and can be efficiently used to make a seamless composite map of *Z*. The slight 1-dB overestimation of *Z*_{H}(*A*_{H}) by JuXPol compared to BoXPol in the direct comparison is most probably linked to the same effect of attenuation by invisible precipitation being attributed to the still-visible signals that was mentioned in relation to Fig. 6. The random error of *B*_{A} derived from individual scans shown in Fig. 7 is obviously larger than 3 dB in many cases, which makes it unsuitable to reliably correct all of the *Z*_{H} observations in a scan for calibration on a scan-to-scan basis. Thus, temporal accumulation over preferably several events/days remains necessary for the purpose of calibrating *Z*_{H} to acceptable levels. Using measured *Z* after correction for attenuation (i.e., *Z*_{H} + *IA*_{H}) would result in about 6 dB discrepancy between BoXPol and JuXPol reflectivities due to the difference in their calibration biases (Fig. 7c, black).

## 5. Consistency of *Z*(*A*)-derived corrections for PBB and miscalibration from two X-band radars and one C-band radar

*Z*

*A*

We finalize our analysis by testing the accuracy of *Z*(*A*) and *B*_{A} during an extensive rain event between 0900 and 1700 UTC 22 June 2011. In addition to the two X-band radars, we use data from the polarimetrically upgraded C-band radar in Offenthal (Fig. 1, Table 1). Figure 8 shows *B*_{A} for all radars, accumulated over the event and within beam segments uninterrupted by the automated blocker identification described in section 3. Both BoXPol and Offenthal show significant PBB in their overlapping areas. The segments between Offenthal and the PBB-causing Taunus mountains obviously contained too few rain observations to make a reliable *B*_{A} estimate. All radars show slight azimuthal variability of *B*_{A} even in unblocked sectors, caused by the dependency of *B*_{A} on rain characteristics as described in sections 3 and 4, and insufficient sample size. Figure 9 shows a reflectivity composite generated by choosing the radar observation closest to the ground at every location. The circular structure seen in the *Z*_{H} + *IA*_{H} composite (Fig. 9a) around BoXPol is the result of the strongly miscalibrated 0.5° elevation scan of JuXPol PPI intersecting the cone of the 1.5° BoXPol scan. Similar disparities become visible at bordering JuXPol and Offenthal measurements (Fig. 9a, top) and in sectors affected by PBB (Fig. 9a, center). The use of *Z*_{H}(*A*_{H}) mitigates these offsets but exhibits compromised estimates in segments containing only light rain and thus low ΔΦ_{DP} (Fig. 9b, left). The automated blocker detection also removed values from the Offenthal scan over the Taunus mountains (Fig. 9b, bottom right). If *Z*_{H} + *IA*_{H} is corrected with the cumulative *B*_{A} displayed in Fig. 8, the performance in light rain is greatly improved, giving a more complete coverage of the event (Fig. 9c). The only remaining discrepancies following correction through *B*_{A} are the measurements removed by noise cutoff at different intensities by radars affected by miscalibration (JuXPol) and PBB (BoXPol) and the region between Offenthal and Taunus mountains, which did not contain enough *Z*_{H}(*A*_{H}) samples.

In Fig. 10, we compare collocated estimates of each reflectivity estimator between radars for the entire event in scatterplots (correlation coefficient and mean bias given above the plots). The uncorrected PBB leads to lowered correlations and slight biases in *Z*_{H} + *IA*_{H} comparisons (Fig. 9a) containing Offenthal and BoXPol measurements, while miscalibration of JuXPol adds a strong negative bias. The *Z*_{H}(*A*_{H}) has overall higher correlation coefficients between the radars because of corrected PBB and less bias mostly due to calibration corrections, but more scattering for reflectivity values below 20 dB*Z* (Fig. 10b). The increased scatter is the consequence of larger relative random errors in ZPHI segments with ΔΦ_{DP} < 6°. The *B*_{A} correction in *Z*_{H} + *IA*_{H} + *B*_{A} compensates calibration errors and PBB well and gives more precise low reflectivity estimates than *Z*_{H}(*A*_{H}) (cf. Figs. 10b,c). However, very high reflectivity values add more scatter in the plots for estimators containing *Z*_{H} + *IA*_{H} compared to *Z*_{H}(*A*_{H}), which is most likely a consequence of inadequate attenuation correction *IA*_{H} and contamination by backscatter differential phase shift. Note that some random scatter will always remain because the collocated measurements were made up to 150 s apart from each other, and the compared observations may have different radar volume sizes because of beam broadening. If segments with ΔΦ_{DP} < 6° are removed from the comparison, the correlations of *Z*_{H}(*A*_{H}) become comparable to those of *Z*_{H} + *IA*_{H} + *B*_{A}, by the cost of removing most observations below 20 dB*Z* (not shown).

Although the *B*_{A} corrections derived during one extensive event were still less precise than those calculated using larger accumulations, they still substantially improve reflectivity estimates in this case. For less extensive rain events, accumulations over more than one event/day remain necessary to obtain useable *B*_{A}.

## 6. Summary and conclusions

The *Z*_{H} bias associated with radar miscalibration and PBB can be deduced and corrected using the *Z*_{H}(*A*_{H}) relations in Tables 3 and 4 with estimates of *A*_{H} obtained from the ZPHI formula in rain of sufficient intensity, if ΔΦ_{DP} in a segment is large compared to the error of the Φ_{DP} measurement (typically 4°–6°). The overall accuracy of calibration- and PBB-related biases derived from *Z*_{H}(*A*_{H}) is very good when accumulating over long time intervals; we conclude this from comparisons with *K*_{DP} and *Z*_{DR} consistency relations, with PBB calculated from DEM, and between radars.

Our analysis demonstrates that the accuracy of the *Z*_{H} bias retrieval in the blocked regions using the *Z*_{H}(*A*_{H}) relations is comparable to, or more accurate than, the one obtained from techniques based on the use of a DEM. The use of self-consistency relations between *Z*_{H}, *K*_{DP}, and *Z*_{DR} gave similar results, but can be problematic in cases of *Z*_{DR} calibration drift or radome effects or in blocked areas where an independent estimate of the *Z*_{DR} bias should be made. We found that in long-term accumulations, contamination of the Φ_{DP} measurement by ground clutter can cause significant biases in neighboring beam segments. We were able to mitigate these effects by an adapted Φ_{DP} processing and a stricter filtering, which excluded potentially clutter-contaminated measurements.

When varying the temperature assumptions inherent to *A*_{H}(*Z*_{H}) and *R*(*A*_{H}), we found that the use of constant *α* values for typical temperatures (*α* = 0.3 for 15°C and 0.27 for 20°C) together with their *A*_{H}(*Z*_{H}) and *R*(*A*_{H}) counterparts from Tables 4 and 5 did not compromise the technique significantly. A difference in the behavior of the *Z*_{H} − *Z*_{H}(*A*_{H}) bias in heavy or light rain became evident, but it could not be established with certainty whether this was solely an artifact caused by the technical limitations of the method (mainly the precision of the Φ_{DP} estimate) or also the consequence of DSD variability and associated changes in the relation between *A*_{H} and *Z*_{H}. If the latter is the case, this weakness in the *Z*_{H}(*A*_{H}) calculation will likely become an asset of the *R*(*A*_{H}) method over *R*(*Z*_{H}), because rain rate is more closely correlated to *A* and *K*_{DP} than to *Z*.

When radar calibration errors led to biases of *Z*_{H} below −7 dB (JuXPol 2011–12 and BoXPol 2013), the *Z*_{H}(*A*_{H})-based correction tended to slightly exaggerate the bias by approximately 1 dB compared to *Z*_{H}–*Z*_{DR}–*K*_{DP} consistency and compared with a less severely miscalibrated radar. The effect may be associated with areas of rainfall below the minimum detectable signal, and attenuation caused by invisible rain is attributed to visible signals. The same is likely to happen when strong attenuation or PBB hides weaker precipitation behind the obstacle. This is an undesirable effect when calculating true reflectivity biases, but when accumulating rain from *R*(*A*_{H}) [and also of *R*(*Z* + *B*_{A})], the invisible rain is indirectly added to visible raining areas, which in return leads to less biased total rainfall accumulations over longer time periods.

Since the *Z*_{H} bias estimators *B*_{A} [Eq. (6)], *B*_{C} [Eq. (8)], and *B*_{K} [Eq. (9)] have different strengths and weaknesses with regard to *Z*_{DR} calibration, ground clutter, and contamination through backscatter differential phase shift, it is suggested to consult or combine all of them rather than choosing one over the others. We recommend accumulating and storing the necessary sums to calculate the biases *B*_{A}, *B*_{C}, and *B*_{K} operationally on a daily basis for every scan bin. The resulting bias maps can be used to visually identify blocked and unblocked azimuths so that a radar operator can easily flag and join uninterrupted segments (for example, using a simple point and click interface and automated smoothness criteria). The sums can then be accumulated over any number of days and over entire unblocked or partially blocked regions, which should remove remaining artifacts in very short beam segments close to the radar. The resulting corrections for PBB and miscalibration through *B*_{A} can thereby directly be deduced from observed *Z*_{H} and Φ_{DP}, which represents an efficient alternative to techniques that require additional measurements. Although *Z*_{H}(*A*_{H}) should be calculated in ZPHI segments containing pure rain and sufficient differential phase shift, the resulting average *B*_{A} can later be transferred to all measurements. Thus, the use of *Z*_{H}(*A*_{H}) for radar reflectivity bias estimation provides a tool that can efficiently improve not only estimates of rain rate but also of radar reflectivity, which also finds use in applications beyond quantitative precipitation estimation.

## Acknowledgments

This research was carried out at the Hans Ertel Centre for Weather Research, a network of universities, research institutes, and the Deutscher Wetterdienst, funded by the BMVBS (Federal Ministry of Transport, Building and Urban Development). The work was also supported by the D5 project of TR32 “Patterns in Soil–Vegetation–Atmosphere Systems,” funded by the Deutsche Forschungsgemeinschaft (DFG). X-band weather radar data were provided by the ABC/J Geoverbund, TR32, and the TERENO project of the Helmholtz Association. The C-band weather radar of Offenthal was provided by the Deutscher Wetterdienst (DWD). The topography map that the DEM-derived PBB calculations are based on was obtained from NASA and the Shuttle Radar Topography Mission. Special thanks to Kai Mühlbauer, Martin Lennefer, Normen Hermes, and Heye Bogena for providing the X-band data and to Kathrin Wapler and Patrick Tracksdorf for providing the C-band radar data. Funding for A. Ryzhkov and P. Zhang was provided by NOAA/Office of Oceanic and Atmospheric Research under NOAA–OU Cooperative Agreement NA11OAR4320072, U.S. Department of Commerce, and by the U.S. National Weather Service, Federal Aviation Administration, and U.S. Department of Defense program for modernization of NEXRAD radars.

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