# Investigation of Weather Radar Quantitative Precipitation Estimation Methodologies in Complex Orography

^{*}

## Abstract

**:**

_{dp}), single polarization reflectivity factor (Z

_{hh}), and differential reflectivity (Z

_{dr}), once accurately processed, in most cases performs better among those tested and those that make use of Z

_{hh}alone, K

_{dp}alone, and Z

_{hh}, and Z

_{dr}. Improvements greater than 25% are found for the total rain accumulations in terms of normalized bias when the VPR extrapolation is applied.

## 1. Introduction

_{hh}), the differential reflectivity (Z

_{dr}), and the specific differential phase shift (K

_{dp}) as well as the estimates of near-surface rain rate (R), which is usually obtained from the radar quantities measured aloft. In particular, in complex orography environments, when the radar antenna tilts have to be set higher than usual to limit beam blocking (say above 3 or 4 deg with respect to the horizon) or the radar is positioned at a high altitude (e.g., above 1500 m), the differences in the sampled radar quantities aloft with respect to those that would be observed close to the surface can be significant. Some past studies have focused on rain precipitation estimation in complex orography [19,20] while others have extensively addressed the issue of the vertical extrapolation of the reflectivity factor, Z

_{hh}, down to the surface level to estimate the rain rate, R, taking into account, in particular, the distinct signatures of the vertical profile of reflectivity in stratiform and convective rain regimes [21,22,23]. However, as far as the authors have been able to ascertain, none of the works referenced above have addressed the issue of vertical extrapolation when dealing with weather radar dual polarization rain estimation formulas, although there is growing interest in the characterization of the signature of the vertical profile of K

_{dp}and Z

_{dr}[24].

_{hh}), R(K

_{dp}), R(Z

_{hh},Z

_{dr}) relationships, as well as those based on one of their combinations; (ii) show the impact of the use of quasi-vertical profiles of rain and K

_{dp}filtering settings on QPE algorithms. In this work, the acronym VPR, often ambiguously referred to vertical profile of reflectivity, indicates the vertical profile of rain. The use of quasi-VPR, when dealing with dual polarization radar variables, is a key aspect of this study. The term “quasi-VPR” actually refers to the fact that we are not looking at vertical radar observations but instead we derive vertical information by exploiting slanted paths [25]. In this study, quasi-VPRs are used to overcome the complication that would result if we attempted to perform QPEs using dual polarization estimators (i.e., using Z

_{hh}, Z

_{dr}, and K

_{dp}or a subset of them at the same time). Indeed, various QPE estimators exist based on several combinations of Z

_{hh}, Z

_{dr}, and K

_{dp}, with estimation coefficients adapted as a function of rain regimens observed by radar (e.g., [26,27]). Those QPE estimators might benefit from the use of quasi-VPR as defined in this study because it allows near surface extrapolations after rain rate retrievals instead of performing them before the retrieval step.

## 2. Methodology for Quantitative Precipitation Estimation

#### 2.1. QPE from Radar Observables

_{hh}, Z

_{dr}, and K

_{dp}, correctly processed. They are listed below for the reader’s convenience:

_{hh}and Z

_{dr}are in linear units, respectively, whereas K

_{dp}is in (deg·km

^{−1}). All the variables are defined at longitude (x), latitude (y), altitude (h), and position

**p**= (x,y,h) with respect to the radar location. The conversion coefficients a

_{q}, b

_{q}, and c

_{q}are defined so that the resulting rain rate R

_{q}is in (mm·h

^{−1}), where the subscript q indicates one of the three estimation formulas through the labels “Z”, “K”, and “DR”. Finally, Equation (4) (with label q = KZ), adapted from the formulation proposed in [28], is a weighted combination of R

_{K}and R

_{Z}estimates from Equations (2) to (3) proposed by [29,30], respectively. In this case, the value of the weight (w) is set up so that the rain estimation that comes out using K

_{dp}, i.e., R

_{K}, weighs more for higher precipitation rates than that when making use of Z

_{hh}and Z

_{dr}only, i.e., R

_{DR}. The values of filtered K

_{dp}are used to discriminate high rain precipitation regimes from lower ones. In particular we fixed w = 1 for K

_{dp}≥ 0.5, w = 0 for K

_{dp}≤ 0.25, and w = 4·K

_{dp}− 1 for 0.25 < K

_{dp}< 0.5. The coefficients a

_{q}, b

_{q}, and c

_{q}used are listed in Table 1. On the other hand, the estimation coefficients applied to implement the various estimation formulas can vary significantly depending on the way they are calculated and the assumptions made. We decided to fix those estimation coefficients using the results shown in [31] at C band (5.6 GHz), where the relations between the rain rate and the other radar variables are driven by experimental observations of drop size distribution obtained from a 2DVD disdrometer collected for two months in a field campaign in the Rome area, coupled with electromagnetic simulations of the propagating radar signal (i.e., using T-matrix routines). In these simulations an environmental temperature of 20 °C, the drop shape model of Beard and Chuang [32] and a Gaussian variation of the drop canting angle with a zero mean and a standard deviation of 7.5° are assumed. The choice of using the set of estimation coefficients derived by the same population of drop size distribution for the same geographical area for all the estimation formulas should help avoid inconsistencies that might arise when using parameterizations weighted by different approaches and measurement techniques.

#### 2.2. Extrapolation of QPE at the Ground Using Quasi-Vertical Profiles

_{q}, performed using Equations (1)–(4) are in principle produced at any of the sampling positions,

**p**, within the radar volume. Of course, R

_{q}(

**p**) is meaningful for

**p**within areas of rain precipitation. This is likely if

**p**describes positions below the bottom of the freezing band. However, this is not a condition that can be always met and depends on the radar position with respect to the surrounding orography (the terrain altitude variations). In such complex orography conditions,

**p**can point to ice or mixed regions, making the values of estimated R

_{q}agree less with the ground reality. Even when the radar antenna scan strategy and the surrounding orography allow for a sampling of rain below the bottom of the freezing band, i.e., in rain layers, modifications of the drop size distribution caused, for example, by spontaneous drop breakup, collision-induced breakup, coalescence, and/or evaporation redistribute the water mass contained in the raindrops across the size spectra. These modifications should lead to changes of the apparent trend of R

_{q}with altitude. As a consequence, theoretically, the analysis of the vertical variations of R

_{q}can be important for enhancing the QPE performance close to the ground level and may represent a smart way to deal with operational environments. To this goal, in this work we want to quantify the benefit, if any, of the use of quasi-vertical profiles of R

_{q}(VPR

_{q}

^{obs}) extracted routinely by each of the radar volumes to perform rain rate extrapolations at the ground. To do that, a VPR

_{q}

^{obs}is primarily defined as an average “< · >” of R

_{q}(

**p**) within altitude intervals (∆h) centered at fixed altitudes h

_{j}(x,y) = h

_{r}(x,y) + j·∆h/2, where j is an index within [0, j

_{max}] and h

_{r}is the lowest altitude detected by the radar for each horizontal position (x,y):

_{q}

^{obs}. However, the spatial average in Equation (5) is not performed in the entire radar domain but the grid points (x,y) are selected using the criteria that comply with the following rule: Z

_{hh}> 0 dBZ and Z

_{dr}> −0.5 dB and |∆Z

_{hh}/∆h| > 25 dBZ. This rule should avoid the inclusion of spikes and measurements too close to the signal-to-noise ratio level and guarantee good stability of the extracted profiles.

_{j}would not be regularly spaced due to the irregular radar spatial sampling along with the vertical direction. This should not be an issue when considering the VPR for QPE extrapolations, but in the case of visualization of the time series of VPR, azimuthal averages need to be interpolated in some way before displaying them.

_{q}

^{obs}we used it to correct R

_{q}for the altitude effects by extrapolating the rain estimates at positions

**p**= (x,y,h) up to the ground level (h

_{GRD}); i.e., at positions

**p**

_{GRD}= (x,y,h

_{GRD}). The extrapolation is performed by applying a regression on the values of VPR

_{q}

^{obs}(h

_{j}) to obtain a modelled VPR

_{q}

^{mod}(h). Differently from past work, where it is a common practice to describe VPR

_{q}

^{obs}by stepwise linear models, in this study we propose the use of a non-linear regression model to keep the implementation as simple as possible:

_{1}and p

_{2}are regression coefficients in mm·h

^{−1}·km

^{−1}and mm·h

^{−1}, respectively and the subscript q can assume one of the labels “Z”, “K”, “DR”, and “KZ”, indicating the rain estimator used. It is worth noting that the regression model used in Equation (6) is unlikely to catch strong vertical variations of R

_{q}but it is probably enough to correct for most of the bias in situations of particularly complex orography. Of course, Equation (6) can be updated with more accurate regression models driven by experimental evidence. The final correction on QPE estimates in Equations (1)–(4) is achieved by adding a correction quantity as formulated below:

_{m}, and where

**p**

_{GRD}is a position at the ground level. The value of h

_{m}can be set to h

_{r}, the minimum altitude detected by the radar, or set to zero or set to the value below which R

_{q}is constant. In the following sections we will apply the methodology just formalized to case studies that occurred in the inland area of Rome close to the Apennine chain acquired by the P55C radar. Then we will compare the outcomes of Equations (1)–(4) with those obtained by implementing Equation (7). It is worth remarking that VPR

^{obs}and, as a consequence, VPR

^{mod}, can be quite variable in space and time and a mean profile over a large area in space and time could be not representative of the vertical variation of precipitation at each position of the radar grid. One aspect we want to examine in this study is the level of improvement obtained when considering average vertical profiles, as in Equations (5) and (6), in estimating rain precipitation in complex orography using the dual polarization formulations in Equations (1)–(4) and (7).

## 3. Experimental Setup and Case Studies

#### 3.1. Description of the Case Studies

_{hh}, Z

_{dr}, and K

_{dp}and their temporal average, calculated using Equation (5), respectively. It follows a synthetic summary of the three events considered.

**12 October 2012**. This event, associated with a secondary trough over the central Mediterranean Sea, carried heavy precipitation mainly over the Lazio and Umbria regions on 11–12 October 2012. The storm started over the Tyrrhenian Sea, where several mesoscale convective systems developed during the night of 11 October. Convective systems reached central Italy in the early morning of 12 October, with heavy precipitation over the northern part of the Lazio and Umbria regions. The rain was not persistent, but the maximum cumulative rainfall registered exceeded 150 mm in 6 h (Figure 1a). The P55C radar was able to observe only part of the convective systems, mainly beyond 80 km from the radar site. In the last part of the event, the system moved to the south and farther away from the radar (100 km). The main activity developed between 03:15 UTC and 08:35 UTC, as shown in Figure 2, top left, where the time series of the quasi-vertical profile of Z

_{hh}is shown. In terms of temporal averages of quasi-vertical profiles, Figure 3, top panels, shows the mixed nature of this event, with a stratiform rain regime at the beginning (labeled Time 1 in Figure 2 and Figure 3) with the typical pronounced peak of Z

_{hh}just below the freezing level, lower values of Z

_{dr}near zero close to the ground, and accompanied by low values of K

_{dp}. Within the main phase (Time 2), the quasi-vertical profile of Z

_{hh}moved to higher values with nearly constant behavior below the freezing level.

**15 October 2012**. This event took place on 14 and 15 October 2012 across a vast area in the western Mediterranean. The regional weather service issued a severe weather warning for central Italy and particularly for the city of Rome. The precipitation event was associated with a trough that deepened significantly over the northern Tyrrhenian Sea, pushed a frontal system eastward, and advected moist air from the Tyrrhenian Sea inland to fuel deep convection on the coast west of Rome. The synoptic scale system moved rapidly to the east and triggered a line of convection that crossed the western half of central Italy from north to south in a few hours. Most of the precipitation produced by this system occurred in the late afternoon of 15 October across the western part of central Italy and on the western slopes of the Apennines chain (Figure 1b). The main core of the event developed around 17:30 UTC, when a line of convective cells evolved quickly, then started to weaken after 19:00 UTC (Figure 2, middle left panel). The total rainfall amount recorded in Rome was less than had been predicted. The highest daily amount recorded within the historic area of Rome was modest and equal to 35 mm by operational rain gauge. This event is quite characteristic because it clearly shows a Z

_{dr}tower during the core phase (Figure 3, middle center panel, “Time 2” curve). The time average profiles in Figure 3, middle panels, show the convective origin of this event and confirm the values of Z

_{dr}larger than zero well above the freezing level.

**14 October 2015**. During October 14 and 15, 2015, strong thunderstorms accompanied by strong winds and heavy rain swept up several regions along central and southern Italy as well as the western and northwestern Balkans, causing floods and landslides across the affected areas. In central Italy, in the Lazio and Abruzzo regions, within the P55C radar coverage, authorities reported landslides and casualties caused by river overflow. In those regions, the extreme weather left three people dead and residents were evacuated for safety on 14 October 2015. A woman died in the town of Paliano near Frosinone after the car she was in fell into a crater that opened up due to flooding after torrential rainfall in the area. Closer to Rome, the Aniene River overflowed its banks at Tivoli, flooding homes and basements and driving some residents onto their roofs for safety. Official sources said 100 people were evacuated from the town of Canistro in the province of L’Aquila in the hills surrounding Rome after the Rio Spalto overflowed. The city of Rome was spared most of the bad weather, while surrounding towns and cities were mostly impacted. The single railway line connecting Rome and Cassino was taken out of action by a landslide. Local rain gauges registered peaks larger than 170 mm within 24 h (Figure 1c). This event was characterized by a severe convection phase with Z

_{hh}maximum values up to 60 dBZ and vertical maximum extension up to 10–12 km, Z

_{dr}values reaching 2 dB close to the surface and, at some times, extending with positive values over the zero temperature level and K

_{dp}values around 2 deg/km (Figure 2 and Figure 3, bottom panels).

_{dr}and K

_{dp}, which start sharply increasing just above the freezing level and below it, with different strengths probably depending on the degree of convection of the specific event. This characteristic shape can modify in the presence of Z

_{dr}towers or ice crystal aggregation processes.

#### 3.2. Radar System Specifications

_{dr}), whose quality is expected to be higher than in the systems making use of simultaneous transmissions as the power transmitted is fully transferred to the two orthogonal polarization states.

_{hh}, Z

_{dr}, differential phase shift (φ

_{dp}), and the Doppler velocity (V).

#### 3.3. Available Measurements and Settings for the Analyzed Case of Study

_{dr}calibration issues. Three sequences of RHI were performed. Sequence 1 includes RHIs at 13:11, 13:17, and 13:21 UTC for the 66 deg azimuth; Sequence 2 includes RHIs at 13:30 and 13:31 UTC for the 190 deg azimuth, and sequence 3 includes RHIs at 13:34 and 13:36 UTC for the 90 deg azimuth. Useful vertical scans were available from 08:10 to 08:14 UTC, for a total of 2160 vertical profiles.

#### 3.4. Definition of Complex Orography

_{TA}). However, G

_{TA}alone does not take into account the radar perspective. For example, a radar system positioned on the top of a mountain or in a valley nearby will likely have different QPE performance although the surrounding environment does not change (i.e., G

_{TA}is the same within the radar coverage domain). For this reason, we consider an additional term, which is the gradient of the radar beam altitude (G

_{BA}). In this way we can define the gradient of the radar orography G

_{RO}as follows:

**p**the operator ∆

_{r}h

_{X}is the difference of the altitudes h

_{X}along with the radial direction, and where δ

_{r}is the radar radial resolution. The symbol “X” stands for “TA” or “BA”, indicating the terrain altitudes and sampled radar beams altitudes with respect to sea level. Note that in flat terrain and when considering PPI (plan position indicator) acquisitions, G

_{BA}coincides with the local tangent of the antenna elevation angle. In that special case, sampling at higher elevations produces higher G

_{RO}. In general, the term G

_{BA}increases G

_{RO}as a function of the radar position and the sampling strategy adopted, (e.g., PPI or LBM: lowest elevation beam or VMI: vertical maximum intensity, etc.). A practical example that will be discussed later on will help to clarify this aspect. So far, Equation (8) is not general enough for practical use in different contexts. In order to constrain the dynamic of G

_{RO}to the interval [0, 1] we use an exponential compression formula that allows defining the ROI as follows:

_{RO}reaches its maximum value, ROI = 1 (b = 1/max(G

_{RO})). However, to avoid having the few peaks of G

_{RO}corrupt the ROI, we considered the 90th percentile (prc

_{90}) of G

_{RO}instead of its maximum (b = 1/prc

_{90}(G

_{RO})).

_{TA}is considered instead of G

_{RO}. These three curves highlight the sensitivity of ROI to the altitude of the radar position with respect to the surrounding terrain orography. In addition, the separation between ROI curves and TAI highlights the contribution of G

_{RO}with respect to G

_{TA}. In the result sections below, the mask in Figure 4b is used to evaluate the performance of QPEs in different portions of the P55C coverage domain.

## 4. Results

_{q}, using single and dual polarization measurements sampled at the lowest elevation beam (LBM), i.e., at positions

**p**=

**p**

_{LBM}. In addition, Equation (7) was coupled with the four rain estimations in Equations (1)–(4) to carry out the adjustments related to the vertical variability of the rain precipitation and produce estimations at the positions of the ground levels,

**p**=

**p**

_{GRD}. Equation (7) was used in combination with Equation (6) where h

_{m}= 0 and the coefficients p

_{1}and p

_{2}are dynamically calculated at each time step. In this respect, Figure 5a,d,g, show an example of quasi-vertical profiles of rain estimates and its regression model VPR

^{mod}as formulated in Equations (5) and (6), respectively. Figure 5b,c,e,f,h,i, show the statistical distribution of the coefficients p

_{1}and p

_{2}, respectively, for the entire event. It is worth noting that despite the variability of the events, the coefficients p

_{1}and p

_{2}are quite stable in terms of average and standard deviation. As a first approximation, the average values of p

_{1}and p

_{2}can be used as constant parameters in Equation (6) to adjust the rain estimates aloft and produce estimates closer to those registered at the surface.

_{q}(

**p**

_{LBM}) are shown, we can see R

_{Z}(

**p**

_{LBM}) behaves most poorly, with large underestimations (Figure 6i). A large improvement occurs when using R

_{K}(

**p**

_{LBM}), which seems to correctly represent the area of heavy precipitation (Figure 6g). When introducing Z

_{dr}; i.e., when using R

_{DR}(

**p**

_{LBM}), (Figure 6e), we have a small improvement with respect to R

_{Z}(

**p**

_{LBM}) but we still register an appreciable underestimation compared to R

_{K}(

**p**

_{LBM}). Finally, when combining all R

_{K}and R

_{DR}into a single rain estimator, R

_{KZ}(

**p**

_{LBM}), (Figure 6b), we have something that is more consistent with the rain gauges with respect to using R

_{K}(

**p**

_{LBM}) alone. On the other hand, when applying the quasi-vertical profile adjustment (panels in the right columns of Figure 6) we have an increase of rain precipitation values for all the implemented estimates and a more qualitative agreement with rain gauges, especially in the tendency to represent the maxima of accumulated precipitation (compare, for example, Figure 6g,h with the reference Figure 6a). A confirmation of this is given in the example shown in Figure 7, where a zoomed version of R

_{KZ}(

**p**

_{LBM}) and R

_{KZ}(

**p**

_{GRD}) is displayed for the case of 14 October 2015. This figure can be particularly informative in the light of combined radar and rain gauge products. Indeed, in these merging approaches and in particular for those that make use of the Kriging methodology, the spatial structure of the rain field from radar, usually described in terms of spatial covariance or variogram, is used to train the interpolation algorithms [35]. As is evident in Figure 7, the spatial structure of rain precipitation fields changes depending on the effect of the spatial adjustment performed (quasi-vertical profile adjustment in our case), and this effect can cause variations in the final results of the combined radar vs. rain gauge products.

_{dp}is a key factor when estimating rain in complex orography environments, whereas the use of quasi-vertical profile adjustment is able to partially compensate for the overall bias, although the differences in this case can be subtle. For example, note the distribution of sample pairs around the reference 1:1 line and the error scores discussed below. On the other hand, the estimated precision (the variability of the estimates around the 1:1 line) seems to not get a strong benefit from the application of quasi-vertical profile adjustments.

_{KZ}(

**p**

_{GRD}) (red dashed line, which shows the results closest to the blue curve of the rain gauge total accumulation). On the contrary, for the case on 15 October 2012 (Figure 9b) we have a different outcome, with quite a large overestimation of R

_{DR}(

**p**

_{GRD}) and R

_{KZ}(

**p**

_{GRD}) and the best performance obtained when considering R

_{Z}(

**p**

_{GRD}) or R

_{K}(

**p**

_{GRD}). Finally, it is worth noting that the performance of the K

_{dp}-based rain estimator depends to some extent on how it is retrieved. In particular, Figure 9d, shows the performance of R

_{KZ}(

**p**

_{LBM}) as a function of four thresholds used (K

_{dp_}

_{min}) for the retrieval of K

_{dp}as specified in the figure legend for the case study on 14 October 2015. See Appendix A and Figure A3 for more details on the filtering of the differential phase shift (φ

_{dp}) and K

_{dp}estimation. For the analyzed cases we used K

_{dp_}

_{min}= −1.5 deg/km, which seems to be a reasonable choice in terms of φ

_{dp}filtering.

_{q}(

**p**

_{LBM}) since many operational QPE algorithms use a gauge-bias correction procedure to fold all of the error sources that compromise the QPEs. Basically, AF is an average ratio of rain gauge accumulations with respect to those from radar estimates as a function of the distance from the radar. We described AF using an exponential law and derived its regression parameters for each of the R

_{q}(

**p**

_{LBM}) algorithms in Equations (1)–(4) by considering the three case studies together. In addition, the right column in each table lists the average relative improvement between R

_{q}(

**p**

_{LBM}) and R

_{q}(

**p**

_{GRD}) when they are evaluated for two distinct parts of the radar domain: the domain where ROI ≤ 0.6 (lower degree of complex orography) and ROI > 0.6 (higher degree of complex orography).

_{Z}estimator (i.e., when moving from R

_{Z}(

**p**

_{LBM}) to R

_{Z}(

**p**

_{GRD})), although the increment was as low as 4% in the case on 15 October 2012 (Table 3). The biggest improvement, as noted previously, is made when using R

_{K}. In this case, the improvement from R

_{Z}(

**p**

_{GRD}) to R

_{K}(

**p**

_{GRD}) is between 18% and 30%, with the exception of 15 October 2012, where the improvement reaches only 7%. The combined rain estimator, R

_{KZ}, gives a further small improvement between 6% and 10% with respect to R

_{K}(

**p**

_{GRD}) and between 30% and 38% compared to R

_{Z}(

**p**

_{GRD}). Even in this case, the only exception is given by the performance obtained on 15 October 2012 that register a deterioration of R

_{KZ}(

**p**

_{GRD}) with respect to the other estimators when quasi-vertical profiles are used. Actually, the strongest improvement brought by the use of quasi-vertical profile adjustment seems to be in terms of normalized bias, with results significantly reduced by a factor larger than 25% when moving from R

_{Z}(

**p**

_{LBM}) to R

_{Z}(

**p**

_{GRD}) and from R

_{KZ}(

**p**

_{LBM}) to R

_{KZ}(

**p**

_{GRD}) on 12 and 15 October 2012.

## 5. Discussion

_{KZ}with the use of quasi-vertical profile adjustment seems to perform better than the others, although an exception exists. Recent literature on QPE studies seems to be more oriented toward a combination of various QPE algorithms whose choice can guided by information related to the type of precipitation [22,26,27]. In this respect, the concept of adjusting rain estimates using quasi-vertical profiles of rain, VPR, as introduced in this work can add to the existing techniques. The advantage of using quasi-VPR is that it allows adjusting rain estimates from its vertical variability in a direct and self-consistent way. However, complications in using quasi-VPR can arise because its variability reflects that of the QPE algorithm used. In other words, the shape of the VPRs depends on the specific QPE algorithm used, and a characterization of the VPR for each QPE algorithm would be needed. For example, we expect that a VPR obtained considering R

_{K}will have a lower negative slope than R

_{Z}in the ice part given the typical values of Z

_{hh}and K

_{dp}in the ice region. Then, an ideal procedure that used VPRs and a combination of several QPE algorithms should address this issue. On top of this, the co-existence of different types of precipitation in different radar domains adds complexity to the procedure because a single VPR is unlikely to represent the entire radar scene. Thus, a more sophisticated VPR retrieval would require the separation of the observed radar scene in areas of different precipitation types. Then, for each of those areas we should identify a characteristic VPR and apply the adjustment separately. This lack of further sophistication is probably responsible for the larger errors found when applying the VPR adjustment for the event on 15 October 2012. In that case, another aspect to highlight is the fact that the QPE algorithms we are using do not take into account peculiar situations such as, for example, Z

_{dr}columns. For this reason, in general, caution is needed when dealing with QPE algorithms that are based on Z

_{dr}. A last issue that would need to be addressed when dealing with quasi-vertical profiles is that they are “apparent” (i.e., they are the result of the antenna pattern convolution with the actual unknown measured quantity) and obtained using slant observations. This consideration is open to the possibility of further refinement, which should include the identification of the actual VPR using deconvolution techniques [36] and estimation theory e.g., [37].

## 6. Conclusions

_{dp}, the single polarization reflectivity factor Z

_{hh}, and the differential reflectivity Z

_{dr}, once accurately post-processed, performs better in most of the cases tested than those that make use of Z

_{hh}alone, K

_{dp}alone, and Z

_{hh}and Z

_{dr}. An improvement larger than 25% is found in terms of normalized bias when applying VPR extrapolations to Z

_{hh}-based QPE and to the combined QPE algorithm. Many limited improvements are registered in terms of root mean square errors because the error standard deviation probably accounts for most of the discrepancies registered between radar QPE and reference rain gauge.

_{dp}estimates, through its filtering, has an impact on the performance of the QPE algorithms, although this impact is less significant when the combined algorithm is considered. In addition, in one case analyzed, where we observed Z

_{dr}columns we found that the VPR adjustment produced quite large overestimations, leading us to conclude that methods based on Z

_{dr}can worsen the final results if we do not apply any special care in the preprocessing step as well as in the QPE algorithms.

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Appendix A. Processing of the Radar Measurements

#### Appendix A.1. Signal-to-Noise Ratio Thresholding

_{hh}(r)·r

^{−2}), we can rewrite it in terms of the minimum detectable reflectivity and the minimum detectable power signal abbreviated as MDZ and MDS, respectively. Thus, the radar equation in terms of these minimum quantities is MDZ = C

^{−1}·r

^{2}MDS. The filtering of the values affected by noise is accomplished by discarding all values of Z

_{hh}lower than MDZ at each range gate r. To do that, only MDS needs to be known (the constant C cancels out with MDS, as will be seen later). This can be done by exploiting the raw data itself. Indeed, we can reasonably expect that in clear air conditions the values of P

_{r}detected at the farthest distances r

_{max}are a good approximation of the minimum level detectable (P

_{min}≅ MDS). Such clear air conditions are also met for relatively high elevations at far distances even during a precipitation event. Thus, MDS = C·Z

_{hh}(r

_{max})·r

_{max}

^{−2}, and when regrouping all terms into MDZ we have MDZ = (r/r

_{max})

^{2}Z

_{hh}(r

_{max}).Then a stable measure of Z

_{hh}(r

_{max}) allows estimating MDZ and then filtering out Z

_{hh}< MDZ.

#### Appendix A.2. Clutter Identification and Its Suppression

_{hh}, Z

_{dr}, and the unfiltered differential phase shift plus an experimental clutter polar volume obtained by considering time averages of Z

_{hh}calculated separately offline. All the above-mentioned information is then weighted using the fuzzy logic scheme as in [20] and well described in [38]. The output quantity of the fuzzy tool we used is a quality index that describes the strength of membership degree of a given grid point within the radar grid to the class of ground clutter echoes with respect to the rest of the possible target typology. Note that since we make use of texture features of input radar variables, their calibration is not strictly required at this step of the processing chain. An interesting behavior that we verified is that the fuzzy approach we used also tends to suppress reasonably well the radio frequency interferences, although this is achieved at the cost of filtering out some of the meteorological features at the border of the rain cells in the clutter class. An example of the clutter filtering outcome is given in Figure A1 where raw Z

_{hh}(Figure A1a) is compared with its filtered version (Figure A1b) obtained applying the signal-to-noise ratio thresholding and the fuzzy logic filtering steps. As can be seen from the unfiltered raw data in Figure A1a, the environment is made particularly challenging by the presence of a lot of radio frequency interferences (look at the radial signatures departing from the radar position at the center of the map) that add to the ground clutter more densely present in the eastern part of the map. In Figure A1b, after applying the clutter filtering, all the apparent contaminations due to clutter and radio frequency interferences seem to be reasonably suppressed, although some radar resolution volumes of precipitation might be erroneously labelled as clutter and then put to void values. An additional example of clutter suppression is given in Figure A2b, in terms of an ad hoc RHI acquisition of Z

_{hh}, where at the hill peaks, labelled as H1 and H2, the radar signal results were correctly suppressed with respect to raw Z

_{hh}in Figure A2a. The same figure also shows the appearance of Z

_{hh}when other processing steps are applied, as will be discussed in the following sections of this Appendix.

**Figure A1.**Example of the outcome of the clutter filtering procedure on 14 October 2015 at 12:10 UTC for a Plan Position Indicator (PPI) at 1.59 deg radar antenna elevation. (

**a**) Raw reflectivity factor, Z

_{hh}, (dBZ); (

**b**) filtered version of Z

_{hh}after the clutter filtering. Contours of terrain altitudes at 500 and 1500 m asl. (grey contours) and constant range circles at 65 and 120 km from the radar are overlaid.

**Figure A2.**Example of the outcome of the radar processing chain in terms of specifically arranged Range Height Indicator (RHI), acquired by P55C radar on 14 October 2015, 13:17 UTC at 66 deg azimuth (

**a**); after performing the Z

_{hh}(dB) bias compensation and clutter suppression (

**b**); partial beam blocking compensation (

**c**); and single polarization path attenuation compensation (

**d**).

#### Appendix A.3. Differential Phase Shift Filtering and K_{dp} Estimation

_{dp}estimation. Past developments on this topic include algorithms based on least-square fitting methods [39], FIR filters [40], wavelet analysis [41], linear programming constraints [42], and, more recently, Kalman filter approaches, which are promising under lower signal-to-noise ratio conditions [43]. All these methods aim at reducing the effects of the statistical errors affecting the measured differential phase (φ

_{dp}), including those error sources originated by statistical/sampling error, ground clutter, side lobes, second-trip echoes, mixed-phase hydrometeors (large melting aggregates and hailstones), non-uniform beam filling, and to some extent compensating for the effects of the differential backscattering phase (δ

_{hv}). The final result is a filtered differential propagation phase (φ

_{dp}) that should be monotonically increasing and not excessively smoothed in order to avoid overly processed K

_{dp}estimates that lose fine-scale precipitation features.

_{dp}and for its inverse estimation through temporary estimates of K

_{dp}. It has been demonstrated that this approach combines an easy practical implementation with a reasonable degree of estimation accuracy and the representativeness of the final K

_{dp}product in terms of spatial resolution features while using a filter length on the order of 1 km. In addition, this technique allows choosing the level of negative values of the output K

_{dp}(K

_{dp_}

_{min}) consistently with the filtered φ

_{dp}, thus modulating the inclusion of the contributions of δ

_{hv}, non-uniform beam filling, and other statistical errors, which usually cause negative values of K

_{dp}. Figure A3a, gives an example of raw φ

_{dp}and some of its filtered versions using various thresholds of K

_{dp_}

_{min}, as specified in the panel’s legend. The resulting filtered K

_{dp}are shown in Figure A3b. Note that when selecting K

_{dp_}

_{min}= −0.5 deg/km, φ

_{dp}is forced to be quasi-monotonically increasing for any negative range variation of φ

_{dp}, leading in that case to an overestimation of raw φ

_{dp}. However, this does not affect the peaks of estimated K

_{dp}but, instead, its values are closer to zero (Figure A3b). A more conservative choice is given when K

_{dp_}

_{min}= −1.5 deg/km, which is the value used in the main text.

**Figure A3.**Example of differential phase shift (φ

_{dp}) and estimated K

_{dp}on 14 October 2015 taken at 6.2 deg elevation angle and 107 deg azimuth. (

**a**) Raw φ

_{dp}(black line) and four φ

_{dp}filtered versions (colored lines) using several K

_{dp}minimum thresholds (K

_{dp_}

_{min}) as specified in the panel’s legend; (

**b**) Estimates of K

_{dp}for the values of K

_{dp_}

_{min}specified in the panel’s legend.

#### Appendix A.4. Partial Beam Blocking Compensation

_{hh}. In this work, we calculated the partial beam blockage levels (PBBL) representing, at each position and for a specific antenna elevation, the occultation degree varying from 0 (free of blockage) to 1 (totally blocked). This is accomplished by resorting to the simplified obstruction function proposed in [44] and applied to the difference between the terrain altitude and that of the expected radar ray paths. A model for the terrain altitudes is required to run this procedure. In in this work, the terrain altitudes are obtained from the SRTM 90 m digital elevation model database [45]. Then the compensation of Z

_{hh}for partial blocking is carried out by multiplying Z

_{hh}(in linear units) for a compensation multiplicative factor equal to 1/(1 − PBBL) for PBBL within [0.1, 0.7] (compensating beam obstructions from 10% to 70%). This means applying a multiplicative compensation factor of 1.111 and 3.333, which implies adding factors of 0.4571 dB and 5.2284 dB, respectively.

#### Appendix A.5. Calibration of Z_{dr}

_{dr}calibration is verified through vertical pointing of the radar antenna that were available during the analyzed dataset. The measured average value of Z

_{dr}(in dB) gives an estimate of its miscalibration being the expected signature of Z

_{dr}for rain drops observed vertically equal to 0 dB [46]. For the P55C, for the case studied here we compensated for a miscalibration factor of 2.69 dB with respect to an arbitrary reference factor value, which was verified using vertical observation in rain.

#### Appendix A.6. Path Attenuation Compensation

_{hh}and Z

_{dr}is accomplished by exploiting the quasi-linear relation between specific attenuations A

_{hh}, A

_{dr}in (dB/km), and K

_{dp}(deg/km), respectively, which is A

_{hh}= α

_{hh}·K

_{dp}

^{βhh}and A

_{dr}= α

_{dr}·K

_{dp}

^{βdr}. The coefficients α

_{hh}and α

_{dr}can vary in time and space depending on raindrop size, shape, and temperature. In this work we used the coefficients of α

_{hh}, β

_{hh}, α

_{dr}, and β

_{dr}from [47] at C band, which are 0.073, 0.99, 0.013, and 1.23, respectively. Those coefficients are valid in a rain regime only and for this reason we limited their application to in-rain layers only. To do that, we adopted an altitude threshold to identify the bottom of the freezing band. The nearest radiosounding is Pratica di Mare, Italy (Station identifier: 16245 LIRE, lat.: 41.65, long.:12.43, about 28 km far from the radar site), and is used for this purpose. All radar samples below that threshold were compensated for rain path attenuation. Everything above was maintained at the level of the last compensation applied in rain. The value of the zero thermal altitude registered by radiosoundings in Pratica di Mare for the analyzed event on 15 October 2015, was 3218 m and 2895 m asl, at 00 and 12 UTC, respectively.

#### Appendix A.7. Calibration of Z_{hh}

_{hh}is carried out following the methodology described in [48], where a third-degree polynomial formula relating the ratio K

_{dp}/Z

_{hh}as a function of Z

_{dr}is proposed to formulate the self-consistency approach proposed in [10]. Thus, we derived the bias that needs to be considered for the calibration of Z

_{hh}by minimizing the difference between the measured total differential phase shift for some selected radar samples in light rain and that reconstructed using the third polynomial formula as a function of variable possible values of the bias of Z

_{hh}. The total differential phase shift is calculated by adding selected values of K

_{dp}·2 δr along radial paths, where δr is the radar radial resolution in (km). However, we verified that in the above-mentioned procedure the selection of the samples in light rain requires particular attention. To this aim, we excluded the radar grid points that comply with the following rule: K

_{dp}≤ K

_{dp_up}or h

_{bin}> h

_{fl_bt}or Z

_{hh}> Z

_{hh_up}or S

_{dr}≤ S

_{dr_dw}or S

_{dr}> S

_{dr_up}or r < r

_{dw}, where K

_{dp}, h

_{bin}, Z

_{hh}, S

_{dr}and r is the filtered differential phase shift in (deg/km), the altitude asl. of the radar bin (km), the attenuation corrected radar reflectivity factor (mm

^{6}/m

^{3}), the standard deviation of the attenuation corrected differential reflectivity, and the distance from the radar (km). The thresholds used are set up as K

_{dp_up}= 0.5; h

_{fl_bt}= 3; Z

_{hh_up}= 10

^{(60 dBZ/10)}; S

_{dr_dw}= 0; S

_{dr_up}= 0.05; r

_{dw}= 5.

## Appendix B. Error Score Definitions

_{q}) and that from the rain gauge (R

_{g}). The subscript q indicates one of the rain estimators used in the main text in Equations (1)–(4). The operator <·> indicates the arithmetic average over the numbers of paired radar and rain gauge rain values. The symbols µ

_{x}and σ

_{x}

^{2}stand for average and variance of x, respectively.

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**Figure 1.**From left to right: 24 h rain accumulations for the case of studies on 12 October 2012, 15 October 2012 and 14 October 2015 registered by the Italian rain gauge network. See main text for details.

**Figure 2.**From left to right: time series of quasi vertical profiles of Z

_{hh}(dBZ), Z

_{dr}(dB), and K

_{dp}(deg·km

^{−1}), respectively, calculated using Equation (5). From top to bottom case studies on 12 October 2012, 15 October 2012 and 14 October 2015, respectively. The horizontal dotted lines indicate the value of zero thermal altitude as registered by the local radiosoundings in Pratica di Mare (id station 16245 LIRE) at 00 UTC and 12 UTC.

**Figure 3.**From left to right: average quasi vertical profiles of Z

_{hh}(dBZ), Z

_{dr}(dB), and K

_{dp}(deg·km

^{−1}), extracted from three time windows as specified in each panel’s legend and indicated in the left panels of Figure 2. The bi-dimensional histogram of the occurrences of each radar variable is overplayed as well. The horizontal dotted lines indicate the value of zero thermal altitude as registered by the local radiosoundings in Pratica di Mare (id station 16245 LIRE) at 00 UTC and 12 UTC. From top to bottom: case studies on 12 October 2012, 15 October 2012, and 14 October 2015, respectively.

**Figure 4.**(

**a**) Map of the Radar Orography Index (ROI) in the P55C radar domain when lowest beams are considered; (

**b**) radar mask where position having ROI > 0.6 are equal to 0 and 1 otherwise; (

**c**) distribution of the ROI values in (

**a**); (

**d**) average ROI as a function of the distance from the radar for radar altitudes at 200 m and 102 m and the curve of the Terrain Altitude Index TAI (Equation (9) using G

_{TA}instead of G

_{RO}).

**Figure 5.**(

**a**,

**d**,

**g**) Examples of quasi Vertical Profiles of rain rate intensity (VPR) as a function of altitude (

**h**) at the time specified in the panel’s legend for the case studies on 12 October 2012, 15 October 2012 and 14 October 2015, respectively. In this figure R

_{KZ}is used to calculate VPR. The dotted straight lines refer to linear regressions in the form VPR

^{m}

^{od}(dB) = p

_{1}·h(km) + p

_{2}; (

**b**,

**c**,

**e**,

**f**,

**h**,

**i**): statistical distribution of the parameters p

_{1}and p

_{2}, respectively, extracted from the P55C in each of the three case studies considered. The values of average and standard deviation for p

_{1}and p

_{2}are also shown in the panel’s legend.

**Figure 6.**Qualitative comparison between rain gauge accumulations within 24 h (top left) and rain P55C-radar accumulations using various rain estimators as specified in each of the panel’s title. Panels in the middle column refer to the implementation of Equations (1)–(4) whereas panels on the right side refer to the use of Equation (7). The rain accumulations are obtained summing those registered on 12 October 2012, 15 October 2012, and 14 October 2015 at each gauge position. To improve the qualitative comparison among the various estimators implemented, in each panel a spatial interpolation is applied starting from the rain accumulations extracted at the positions of the rain gauges.

**Figure 7.**Zoomed rain accumulation map within 24 h from Polar 55C radar using R

_{KZ}(

**p**

_{LBM}) (

**a**); and R

_{KZ}(

**p**

_{GRD}) (

**b**) on 14 October 2015.

**Figure 8.**Scatterplots of total rain accumulation over the three case studies on 12 October 2012, 15 October 2015 and 14 October 2015 of rain gauges and radar estimates as specified in each panel’s title. Left column: implementation of Equations (1)–(4) in the main text. Right panels: the same as in the left panel after implementing Equation (7) in the main text.

**Figure 9.**Spatial average of time accumulations of rain precipitation from rain gauges measurements and the analyzed radar estimates sampled at the rain gauge positions. (

**a**–

**c**) Radar and rain gauge comparison for the case studies specified in each panel’s title when the K

_{dp}minimum threshold filtering parameter (K

_{dp_}

_{min}) is set to −1.5 deg/km for all rain estimators as in the panel’s legend; (

**d**) radar and rain gauge comparison in terms of R

_{KZ}(

**p**

_{LBM}) radar estimator as a function K

_{dp_}

_{min}.

Coefficient Symbols | Coefficients Values | Estimation Formulas | Reference Literature |
---|---|---|---|

a_{Z}; b_{Z} | 0.0140; 0.728 | ${R}_{Z}={a}_{Z}\xb7{Z}_{hh}^{bz}$ | Adirosi et al., 2015 [31] |

a_{K}; b_{K} | 22.398; 0.813 | ${R}_{K}={a}_{K}\xb7|{K}_{dp}{|}^{{a}_{K}}\xb7sgn\left({K}_{dp}\right)$ | Adirosi et al., 2015 [31] |

a_{DR}; b_{DR}; c_{DR} | 6.96 × 10^{−3}; 0.934; −4.051 | ${R}_{DR}={a}_{DR}\xb7{Z}_{hh}^{{b}_{DR}}\xb7{Z}_{dr}^{{c}_{DR}}$ | Adirosi et al., 2015 [31] |

w | $\{\begin{array}{c}=1for{K}_{dp}\ge 0.5\\ =0for{K}_{dp}\le 0.25\\ =4\xb7{K}_{dp}-1for0.25{K}_{dp}0.5\end{array}$ | ${R}_{KZ}=\left(1-w\right)\xb7{R}_{DR}+w\xb7{R}_{K}$ | Tested in this work |

_{hh}, Z

_{dr}are in linear units. K

_{dp}is in (deg·km

^{−1}). The label “sgn” indicates the sign function.

**Table 2.**Scores of the radar vs. rain gauge daily accumulations on 12 October 2012. K

_{dp}_

_{min}= −1.5 deg/km. The best value in each row is in boldface. The error scores are defined in Appendix B *.

Score s, Alg. q | R_{q}^{LBM} = R(p_{LBM}) | R_{q}^{GRD} = R(p_{GRD}) | R_{q}^{AF} = AF×R_{q}^{LBM} | <100×[s^{GRD} − s^{LBM}]/s^{LBM}> | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

R_{Z} | R_{K} | R_{DR} | R_{KZ} | R_{Z} | R_{K} | R_{DR} | R_{KZ} | R_{Z} | R_{K} | R_{DR} | R_{KZ} | ROI ≤ 0.6 | ROI > 0.6 | |

RMSE | 18.32 | 16.31 | 18.30 | 12.55 | 15.82 | 13.41 | 15.83 | 12.08 | 27.90 | 17.89 | 23.06 | 15.12 | −9.85 | −21.07 |

NSE | 0.82 | 0.73 | 0.82 | 0.56 | 0.71 | 0.60 | 0.71 | 0.52 | 1.25 | 0.80 | 1.03 | 0.68 | −8.48 | −21.06 |

NB | −0.37 | −0.40 | −0.34 | −0.03 | 0.03 | −0.12 | 0.03 | 0.04 | 0.57 | −0.01 | 0.42 | 0.19 | −146.26 | −52.50 |

CC | 0.65 | 0.77 | 0.65 | 0.81 | 0.69 | 0.79 | 0.69 | 0.79 | 0.80 | 0.78 | 0.79 | 0.84 | +1.80 | +1.60 |

**p**

_{GRD}) with respect to R(

**p**

_{LBM}) when the radar scene is partitioned using the radar orography index (ROI). The label “s” indicates one of the score index in the first column.

**Table 3.**As in Table 2 but for 15 October 2012.

Score s, Alg. q | R_{q}^{LBM} = R(p_{LBM}) | R_{q}^{GRD} = R(p_{GRD}) | R_{q}^{AF} = AF×R_{q}^{LBM} | <100×[s^{GRD} − s^{LBM}]/s^{LBM}> | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

R_{Z} | R_{K} | R_{DR} | R_{KZ} | R_{Z} | R_{K} | R_{DR} | R_{KZ} | R_{Z} | R_{K} | R_{DR} | R_{KZ} | ROI ≤ 0.6 | ROI > 0.6 | |

RMSE | 16.24 | 14.67 | 16.23 | 14.57 | 15.65 | 14.63 | 15.91 | 18.84 | 17.39 | 15.81 | 16.47 | 15.04 | 1.82 | −14.33 |

NSE | 0.87 | 0.78 | 0.87 | 0.78 | 0.83 | 0.78 | 0.85 | 1.01 | 0.93 | 0.84 | 0.88 | 0.80 | 2.48 | −14.18 |

NB | −0.52 | −0.32 | −0.52 | −0.15 | −0.13 | 0.03 | 0.07 | 0.39 | 0.02 | 0.03 | −0.06 | −0.05 | −124.00 | −42.61 |

CC | 0.31 | 0.32 | 0.31 | 0.30 | 0.26 | 0.32 | 0.27 | 0.29 | 0.34 | 0.32 | 0.34 | 0.31 | −7.0 | −14.01 |

**Table 4.**As in Table 2 but for 14 October 2015.

Score s, Alg. q | R_{q}^{LBM} = R(p_{LBM}) | R_{q}^{GRD} = R(p_{GRD}) | R_{q}^{AF} = AF×R_{q}^{LBM} | <100×[s^{GRD} − s^{LBM}]/s^{LBM}> | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

R_{Z} | R_{K} | R_{DR} | R_{KZ} | R_{Z} | R_{K} | R_{DR} | R_{KZ} | R_{Z} | R_{K} | R_{DR} | R_{KZ} | ROI ≤ 0.6 | ROI > 0.6 | |

RMSE | 28.07 | 19.07 | 22.11 | 17.60 | 23.79 | 18.26 | 19.37 | 17.22 | 22.45 | 19.31 | 23.02 | 18.26 | −9.00 | −12.20 |

NSE | 1.12 | 0.76 | 0.88 | 0.70 | 0.95 | 0.73 | 0.77 | 0.68 | 0.89 | 0.77 | 0.92 | 0.73 | −8.90 | −11.40 |

NB | −0.69 | −0.32 | −0.49 | −0.23 | −0.52 | −0.19 | −0.30 | −0.04 | −0.45 | −0.04 | −0.49 | −0.31 | −57.18 | −14.92 |

CC | 0.73 | 0.81 | 0.79 | 0.84 | 0.77 | 0.82 | 0.80 | 0.84 | 0.76 | 0.82 | 0.76 | 0.83 | +1.85 | +3.65 |

© 2017 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license ( http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Montopoli, M.; Roberto, N.; Adirosi, E.; Gorgucci, E.; Baldini, L.
Investigation of Weather Radar Quantitative Precipitation Estimation Methodologies in Complex Orography. *Atmosphere* **2017**, *8*, 34.
https://doi.org/10.3390/atmos8020034

**AMA Style**

Montopoli M, Roberto N, Adirosi E, Gorgucci E, Baldini L.
Investigation of Weather Radar Quantitative Precipitation Estimation Methodologies in Complex Orography. *Atmosphere*. 2017; 8(2):34.
https://doi.org/10.3390/atmos8020034

**Chicago/Turabian Style**

Montopoli, Mario, Nicoletta Roberto, Elisa Adirosi, Eugenio Gorgucci, and Luca Baldini.
2017. "Investigation of Weather Radar Quantitative Precipitation Estimation Methodologies in Complex Orography" *Atmosphere* 8, no. 2: 34.
https://doi.org/10.3390/atmos8020034