## Abstract

The authors evaluate rainfall estimates from the new polarimetric X-band radar at Bonn, Germany, for a period between mid-November and the end of December 2009 by comparison with rain gauges. The emphasis is on slightly more than 1-month accumulations over areas minimally affected by beam blockage. The rain regime was characterized by reflectivities mainly below 45 dB*Z*, maximum observed rain rates of 47 mm h^{−1}, a mean rain rate of 0.1 mm h^{−1}, and brightband altitudes between 0.6 and 2.4 km above the ground. Both the reflectivity factor and the specific differential phase are used to obtain the rain rates. The accuracy of rain total estimates is evaluated from the statistics of the differences between radar and rain gauge measurements. Polarimetry provides improvement in the statistics of reflectivity-based measurements by reducing the bias and RMS errors from −25% to 7% and from 33% to 17%, respectively. Essential to this improvement is separation of the data into those attributed to pure rain, those from the bright band, and those due to nonmeteorological scatterers. A type-specific (rain or wet snow) relation is applied to obtain the rain rate by matching on the average the contribution by wet snow to the radar-measured rainfall below the bright band. The measurement of rain using specific differential phase is the most robust and can be applied to the very low rain rates and still produce credible accumulation estimates characterized with a standard deviation of 11% but a bias of −25%. A composite estimator is also tested and discussed.

## 1. Introduction

As polarimetric radar technology is maturing, there has been a significant resurgence of interest in rainfall measurement with the 3-cm (X band) wavelength polarimetric radars (Matrosov et al. 2002, 2005; Anagnostou et al. 2004, 2010; Park et al. 2005; Maki et al. 2008; Chandrasekar et al. 2009; Matrosov 2010; Wang and Chandrasekar 2010). Attractive attributes of these instruments compared to the 5-cm wavelength radars used by most national weather services in Europe and the 10-cm wavelength radars are lower cost, smaller size, better performance in ground clutter environment, and higher sensitivity of the differential phase to the variations in rain rate. These advantages come with a price; attenuation of the 3-cm wavelength radiation by rain is much larger than attenuation at longer wavelengths. Thus, such radars are suitable for measurements of light and moderated rain to short distances.

Small gap-filling X-band radars could be used in mountain valleys or small areas/watersheds where measurements near ground are needed. Indeed, there have been several studies indicating the value of X-band radar measurements in complex terrain in California (Matrosov 2010), on the Mediterranean islands (Anagnostou et al. 2009), Italian Alps (Anagnostou et al. 2010), French countryside near Paris (Moreau et al. 2009), and mountainous and flat metropolitan areas in Japan (Park et al. 2005). A network of X-band radars is under investigation for complementing long-range, high-power weather radars (Brotzge et al. 2006). Several mobile X-band radars have been used in scientific studies (Wulfmeyer et al. 2008) and four stationary radars are operating in a network configuration (Wang and Chandrasekar 2010).

Errors in rainfall measurements by polarimetric radars at the 10-cm wavelength have been documented in a study by Ryzhkov et al. (2005b), who compare hourly rain totals derived from radars and gauges. Polarimetry offers definitive improvement at all rain rates yet such an improvement is relatively modest for light rain. The “synthetic” algorithm by Ryzhkov et al. (2005b) yields the fractional RMS error of 68% for hourly accumulations less than 5 mm.

A similar problem exists at the 3-cm wavelength although specific differential phase, *K*_{DP}, has lower statistical fluctuations compared to its mean value because the phase shift is inversely proportional to the wavelength. Radar reflectivity factor *Z* and differential reflectivity *Z*_{DR} are much more affected by attenuation at the 3-cm wavelength than at the 10-cm wavelength (Matrosov et al. 2002; Anagnostou et al. 2004; Park et al. 2005; Matrosov 2010), but polarimetry offers ways to account for attenuation by relating specific attenuation and specific differential attenuation to *K*_{DP} (Bringi et al. 1990). In light–moderate rain, these relations are stable and have been successfully applied (e.g., Park et al. 2005; Matrosov 2010).

Several relations between rain rate and polarimetric variables at the 3-cm wavelength have been utilized in previous studies. These include single-parameter relation *R*(*K*_{DP}) and multiparameter relations like *R*(*Z*, *Z*_{DR}) and *R*(*Z*, *Z*_{DR}, *K*_{DP}) (e.g., Matrosov 2010). A different approach is described by Testud et al. (2000), who extend the methodology developed by Marzoug and Amayenc (1991) for the Tropical Rainfall Measuring Mission by using the differential phase as an external constraint (as opposed to attenuation) to estimate the rain profile along the radar beam from the reflectivity profile. In addition, composite estimators (Park et al. 2005; Maki et al. 2008; Chandrasekar et al. 2009; Matrosov 2010) have been suggested to take advantage of the corrected reflectivity factor to estimate low rain rates (for *Z* < 35 dB*Z*) from *R*(*Z*) and higher rain rates from *R*(*K*_{DP}). In recent validation studies, Park et al. (2005), Moreau et al. (2009), and Matrosov (2010) report biases in hourly rain totals within 3% and the corresponding fractional RMS errors between 15% and 25% for polarimetric radar measurements at X band.

One perennial issue confronting radars is quantitative precipitation estimation (QPE) if beams intercept or exceed the bright band. Attempts to solve this problem through reconstruction of Vertical Profiles of Reflectivity (VPR) (Koistinen 1991; Kitchen et al. 1994; Seo et al. 2000; Matrosov et al. 2007; Anagnostou et al. 2010) are ongoing. It has been demonstrated that the application of polarimetry can help to identify rain, melting layers, and snow regions that are useful for correcting VPRs (Matrosov et al. 2007; Anagnostou et al. 2010). Classification exasperates the issue of how to deal with the fact that either instrument accumulates rain during times that the other is censored. In the text and appendix we quantify this discrepancy.

The main objective of our work is to examine the performance of the X-band polarimetric radar continuously operating in the cool season, which is characterized by low rain rates and low freezing levels. The emphasis is on continuous monitoring and estimation of light rain accumulation over a period of about a month. Thus we evaluate conventional and polarimetric rainfall accumulations obtained from 21 November to 31 December 2009 (containing 17 rainy days) with the polarimetric X-band radar in Bonn, Germany. The observed precipitation events were mainly stratiform with occasional embedded weak convection. Ground validation is made with the nearby network of rain gauges covering the distance between 17 and 61 km from the radar; the average rain rate was 0.1 mm h^{−1} and the maximum 47 mm h^{−1}. A notable feature of this dataset is the low freezing level (between 0.6 and 2.4 km), so that most of the time the radar beam at elevation 1.5° (and occasionally at 0.5°) intercepted the melting layer, thus affecting rainfall measurements. At short ranges (<30 km) and elevation of 0.5° contamination from ground clutter was often present.

Section 2 of our paper describes the characteristics of the observing radar, its data, gauge data, and experimental setup. Accumulation estimates and comparison with gauges are in section 3. Matched relations for wet snow (in the bright band) based on classification and comparison with gauges are in section 4. The appendix describes the censoring of radar data.

## 2. Radar and gauge data

In 2009, the Meteorological Institute of the University of Bonn upgraded its 3.2-cm Doppler weather radar by adding polarimetric capabilities (BoXPol). BoXPol operationally uses the simultaneous transmission and reception of horizontally (H) and vertically (V) polarized signals (so-called SHV mode) as well as the transmission of the H-polarized signal and reception of H- and V-polarized signals [so-called LDR (linear depolarization ratio) mode]. The radar has been in continuous operation since summer 2009. Recently a radar of exactly the same configuration has been sited about 40 km west on the Sophienhöhe close to Jülich, Germany (JüXPol). Both radars were installed to achieve an unprecedented monitoring of the Ruhr valley. In this paper we utilize only the BoXPol observations.

### a. Radar and its data

The main radar characteristics are in Table 1. Radar data collected in the SHV mode are analyzed herein. These are reflectivity factors at horizontal and vertical polarizations *Z _{h}* and

*Z*, differential reflectivity

_{υ}*Z*

_{DR}, differential phase Φ

_{DP}, and the correlation coefficient between horizontally and vertically polarized returns

*ρ*. During data collection, the radar samples in range were spaced at 100-m intervals up to the maximum range of 100 km and typically 90 time samples were used to compute radar variables.

_{hυ}The recorded *Z*_{DR} and *ρ _{hυ}* were processed to eliminate bias by noise. The differential phase Φ

_{DP}was filtered with a 40-point running average filter. The

*K*

_{DP}was obtained from a least squares fit of the radial slope of Φ

_{DP}over 20 points (2 km). The

*Z*and

_{h}*Z*

_{DR}have been corrected for attenuation using the formulas

and

according to Park et al. (2005). In (1), attenuation-related biases Δ*Z* and Δ*Z*_{DR} are expressed in dB, whereas Φ_{DP} is in degrees; these are consistent with Matrosov’s (2010) corrections.

Self-consistency between *Z _{h}*,

*Z*

_{DR}, and

*K*

_{DP}in rain as well as comparison with observations by the operational radars from the German Weather Service (DWD) were used to check the absolute calibration of the radar reflectivity factor. The self-consistency equation for X band was obtained using simulations based on the measured raindrop size distributions (DSDs) in Oklahoma (Ryzhkov et al. 2005a):

where *Z* is expressed in dB*Z*, *K*_{DP} is in ° km^{−1}, and *Z*_{DR} is in dB and (2) is valid for 20 < *Z* < 45 dB*Z*. Note that the drop size distributions in Oklahoma could be considerably different from the ones in Bonn. Moreover, the uncertainty in attenuation correction of *Z*_{DR} might propagate the error into the final estimation of the *Z* offset.

Using (2) and following the approach of Ryzhkov et al. (2005a) and Marks et al. (2009), the *Z* bias is determined by matching the integrals

and

where summation is performed over the rain areas where 20 < *Z* < 45 dB*Z* and *ρ _{hυ}* > 0.97. Then the

*Z*bias is

From data classified as rain at 1.5° elevation (satisfying *ρ _{hυ}* > 0.97; see section 3) the bias was determined to be 6.4 dB

*Z*. Comparing attenuation-corrected reflectivity to data from the German Weather Service radars (5-cm wavelength) yields a bias of between 6 and 7 dB

*Z*bracketing the self-consistent

*Z.*The

*Z*

_{DR}was calibrated using measurements at vertical incidence.

We checked the stability of absolute calibration of *Z* and *Z*_{DR} by monitoring the reflection of stable ground scatterers (Silberstein et al. 2008) and found the variations of *Z* and *Z*_{DR} within 0.5 dB*Z* and 0.2 dB, respectively.

Available for our analysis are volumetric data from elevation scans at 0.5°, 1.5°, 2.4°, 3.4°, 4.4°, 5.7°, and 8.1° in 5-min intervals; however, only data from the three lowest elevations are examined.

### b. Rain gauge data

The rain gauges are of a weighing bucket type and belong to the Erftverband and the Aggerverband water authorities (named after the Rhine tributaries, which they manage) in Germany. The Erftverband gauges cover the sector from 188° to 331° azimuth; the Aggerverband gauges are located between 32° and 60°. In total, 42 gauges are operating within 61 km of the radar (Fig. 1) and 37 of these are within 50 km. Every 60 s, 34 gauges report measurements in millimeters per hour while the remaining 8 gauges provide accumulations every 60 s. The amount of beam blockage was determined at each gauge from terrain profiles and the radar beam position. Twenty locations (between 299° and 60° azimuth) had small (0.07–0.98 dB) but correctable power loss due to blockage. These were chosen for further analysis.

## 3. Rainfall estimators and comparison with gauges

We quantify the accuracy of radar rainfall estimates by comparison with rain gauges using such standard metrics as fractional bias, fractional standard deviation (SD), and correlation (see, e.g., Ryzhkov et al. 2005b for definitions).

### a. Rainfall estimators

We start with the following estimators of rain rate:

and

The *R*(*Z*) in (6) is the standard relation (i.e., *Z =* 197*R*^{1.5}, very close to the Marshal–Palmer formula) on the German Weather Service’s 5-cm wavelength radars. It is fully applicable to the 3-cm wavelength radar for measurement in light rain where Rayleigh scattering is exclusively present. Relation (7) without the sign multiplication was proposed by Matrosov (2010), who derived it by assuming a polynomial-shape size model of Brandes et al. (2002) and used it for *K*_{DP} > 0.1° km^{−1}; otherwise he applied the *R*(*Z*) = 0.073*Z*^{0.57} relation (i.e., *Z* = 100*R*^{1.76}). We modify Matrosov’s *R*(*K*_{DP}) relation by introducing the sign of *K*_{DP} to mimic a similar equation applied at the 10-cm wavelength (Ryzhkov et al. 2005a). Its purpose is to eliminate biases that would otherwise occur if negative *K*_{DP} values (caused by statistical uncertainty) were eliminated (Ryzhkov and Zrnić 1996). The relation (7) does not need frequency-dependent adjustment because the Bonn radar (9.337 GHz) has almost the same frequency as the National Oceanic and Atmospheric Administration (NOAA) radar used by Matrosov (9.41 GHz); the adjustment for different frequencies is often overlooked.

The radar estimates of rain rate at a particular gauge location were obtained by averaging *R*(*Z*) and *R*(*K*_{DP}) over the domain centered on the gauge and containing 20 successive range gates (2-km distance) and two adjacent azimuths (1° apart).

### b. Methodological issues

Estimators (6) and (7) have been applied indiscriminately with no special effort to calibrate the reflectivity except to compensate the effects of attenuation using (1). It became immediately obvious that the measured radar reflectivity was negatively biased and application of the *R*(*Z*) relation (6) resulted in large underestimates of rain prompting recalibration as described in section 2a. Adding 6.4 dB to the measured *Z* calibrates the values at 1.5° and 2.4° elevations. At 0.5° the correction was adjusted to account for partial beam blockage at each gauge location.

The median values of differential phase Φ_{DP} at gauge locations over the whole period of observations were between 6° and 8° and maxima reached 30°. Therefore, according to Eq. (1), the median value of the attenuation-related bias of *Z* is 1.5–2.0 dB*Z* and maximal *Z* bias approaches 7.5 dB*Z*.

Because of physical reasons (disparity in resolution volumes, advection, and observations aloft versus on the ground), measurements by radar and gauges may not be perfectly matched at any given moment; that is, either one indicates rain while the other does not. To partially account for fall speed and/or advection, we added 2 min to the gauge time for comparisons with radar data. However, what counts at the end is the total accumulation by either radar or gauge. To distinguish all possible binary combinations we use set notation such that the intersection *R* ∩ *G* means both radar and gauge measure *R* > 0, the union indicates either one or both measure *R* > 0, and or means that the one with the overbar indicates zero rain rate while the other does not. This notation (in addition to verbal description) is found throughout the paper and in the tables to alert readers about which comparison is being made.

To reduce false radar-derived rain accumulations associated with radar noise and nonmeteorological echo, it was assumed initially that radar detects rain (i.e., *R* > 0) if the signal-to-noise ratio (SNR) is larger than 4 dB and correlation coefficient *ρ _{hυ}* is larger than 0.8. Because of the uncertainty of SNR estimates and to avoid contamination by ground clutter, we subsequently applied a threshold based on the reflectivity values (see the appendix).

### c. Radar–gauge comparisons

The scatterplots of total rain accumulation over the whole period of observations (i.e., 17 rainy days within 40 successive days) derived from the radar measurements (6) and (7) versus gauge accumulations (in the *R* ∪ *G* sense) are in Fig. 2. The two data entries (Fig. 2a) marked with triangles are outliers attributed to ground clutter contamination. The distance of the corresponding gauges is relatively close to the radar (17 km for the smaller and 34 km for the larger outlier). Gradually increasing the *ρ _{hυ}* threshold from 0.85 to 0.97 reduces these two outliers, but also eliminates valid data. This fact indicates that the sole use of

*ρ*is not sufficient to get rid of ground clutter because occasionally ground scatterers may be characterized by very high

_{hυ}*ρ*. (Zrnić et al. 2006). Inclusion of the Φ

_{hυ}_{DP}and

*Z*

_{DR}textures as an additional discriminator (Gourley et al. 2007) would likely be beneficial. An obvious problem of the

*R*(

*Z*) estimator besides the two outliers is underestimation of higher accumulations.

A similar examination of the *R*(*K*_{DP}) totals indicates negative bias (Fig. 2b). The outliers associated with gross overestimation by the *R*(*Z*) relation are now manifested as large underestimates (triangles). There is one more outlier corresponding to the gauge with a rain total of 50 mm for which the *R*(*K*_{DP}) relation yields almost zero rain total (diamonds). In contrast, the *R*(*Z*) relation produces very satisfactory rainfall estimates at this particular gauge location. The two *R*(*K*_{DP}) outliers with almost 0 rain totals correspond to the closest gauges (17 and 17.9 km from the radar) and disappear in the measurements at elevation 1.5°.

Another data quality problem was uncovered for the day of maximum rain accumulation (23 November 2009) when radar reflectivity at elevation 0.5° was badly corrupted. Furthermore, the median *ρ _{hυ}* sustained a drop from >0.85 to about 0.77 and hence was not suitable for classification. The daily areal rain total obtained from the

*R*(

*Z*) estimator was only 2% of the actual rain total measured by gauges (16.8-mm-average gauge rainfall). It is noteworthy that this unknown artifact affects the performance of the polarimetric estimator

*R*(

*K*

_{DP}) to a much lesser extent so that it produced credible daily rain total (17.8-mm-average rainfall). Apparently the phase measurement is very robust and little affected by interference and/or wild signal fluctuations. We observe substantial improvement in the performance of both radar relations (6) and (7) after three gauges with contaminated data and the whole day, 23 November 2009, were excluded from considerations (Fig. 3). The point

*R*(

*G*) = 45 mm,

*R*(

*K*

_{DP}) = 80 mm (Fig. 2b) is also flawed; the gauge accumulation over time was consistently lower by the same relative amount. This negative gauge bias is attributed to very tall nearby trees, which for this reason have been cut in February 2010. Recording on one gauge was at 15-min intervals rather than every minute. Therefore we have eliminated five gauges from quantitative (tabular) comparisons in all cases except when considering matched relations (section 4c) for which we eliminated two more gauges that experienced timing problems (these do not affect the total accumulations but affect partitioning into bright band part and rain part).

Radar accumulation estimates from the data collected at 1.5° elevation have much less scatter than from 0.5° elevation and no contamination by ground clutter (Fig. 4). The radar accumulations are significantly higher. Further analysis (section 4) indicates that contribution by the melting snow in the bright band is the principal cause. Indeed, the minimum height of the bright band for 2 days of measurement was between 1.9 and 2.2 km, for 5 days it was between 0.6 and 1.0 km, and for 16 days it was between 1.1 and 1.5 km. Therefore at the 0.5° elevation, contribution by the bright band was primarily at ranges larger than 50 km (6 gauges), while at 1.5° the contributions could be at ranges larger than 34 km where the majority of gauges reside. Note that little or no bias is at locations closest to the radar (17, 18, and 23 km) where the beam center for the 1.5° elevation is at less than 0.6 km above the ground.

Statistics of the differences between radar rain accumulations, *R*(*K*_{DP}) and *R*(*Z*), and gauge accumulations at 0.5° and 1.5° are quantified in Table 2 (15 gauges). Evidently the performance of the *R*(*K*_{DP}) estimator at the 0.5° elevation is excellent while *R*(*Z*) produces inferior but acceptable results. At the 1.5° elevation, both estimators have huge bias attributed to the melting snow. The correlation coefficient between radar and gauge accumulations is >0.9 at 1.5°, but for *R*(*Z*) drops to 0.78 at 0.5°.

## 4. Matched estimators and *R*(*Z*) relation

Motivations for this section are (i) to see how the single-parameter estimators (applied at 0.5°) perform if the dominant type of scatterers (rain or wet snow) in the radar resolution volume is identified and the relation appropriate for that scatterer type is applied as suggested by Giangrande and Ryzhkov (2008); (ii) to see how well rain can be measured at 1.5° when the bright band dominates the returns, with obvious application to locations where the lowest scans are blocked; (iii) to evaluate the quality of a composite matched relation; and (iv) to quantify the value of polarimetric information to the *R*(*Z*) estimator.

### a. Classification

The sophisticated classification scheme presented by Giangrande and Ryzhkov (2008) and further advanced by Park et al. (2009) is applicable to the 10-cm wavelength but does not transfer directly to the 3-cm wavelength. Matrosov et al. (2007) and Anagnostou et al. (2010) used a simpler approach to identify the melting layer at X band. We distinguish rain, melting snow (bright band), and nonmeteorological scatterers (including ground clutter) using only the magnitude of the correlation coefficient *ρ _{hυ}*. Rain is classified if

*ρ*> 0.97, wet snow if 0.8 <

_{hυ}*ρ*< 0.97, and nonmeteorological echo if

_{hυ}*ρ*< 0.8. A borderline of

_{hυ}*ρ*= 0.97 between rain and wet snow is consistent with the choice by Giangrande and Ryzhkov (2008), Park et al. (2009), Matrosov et al. (2007), and Anagnostou et al. (2009). The choice of the

_{hυ}*ρ*threshold required scrutiny because a small fraction of meteorological echo for which

_{hυ}*ρ*is below the threshold is lost.

_{hυ}We tested the amounts of rainfall that would be lost as a function of the lowest *ρ _{hυ}* threshold (Table 3; i.e., the condition ). The chosen value is a subjective compromise between loss of rainfall and contamination by artifacts. The loss is largest at the lowest elevation (Table 3) and unequivocally decreases with elevation suggesting that ground and other clutter is the prevailing contributor lowering

*ρ*. With the

_{hυ}*ρ*threshold at 0.8, data from the 1.5° elevation exhibit a loss of 1% or 0.8 mm per gauge out of the 71.8-mm total accumulation. The loss reduces to 0.4% (0.3 mm) if the

_{hυ}*ρ*threshold is decreased to 0.75, but this choice would lead to some increase in the errors of estimates.

_{hυ}To test our classification, scattergrams of *Z*_{DR} versus *Z* (over 15 gauge locations at 5-min intervals) are plotted in Fig. 5 for the three elevation angles and two conditions on *ρ _{hυ}*. If the classification is correct there should be a clear difference between median values of

*Z*

_{DR}(at a given

*Z*) for the two conditions on

*ρ*. The scattergrams from supposedly pure rain (

_{hυ}*ρ*> 0.97; Fig. 5, left panels) appear very consistent and similar for all three elevations. The median values of

_{hυ}*Z*

_{DR}are 0.62, 0.52, and 0.43 dB at the 0.5°, 1.5°, and 2.4° elevations, indicating a decrease in median drop size with elevation. This is primarily due to the effects of evaporation on the drop size distributions (Kumjian and Ryzhkov 2010), although at 2.4° the contribution from dry aggregated snow above the bright band might be a factor as it is characterized by very low

*Z*

_{DR}. Such snow has a correlation coefficient larger than 0.97 and is misclassified as rain.

The values of *Z*_{DR} associated with wet snow within the bright band (0.8 < *ρ _{hυ}* < 0.97; Fig. 5, right panels) are definitely higher than in rain. Median values are 1.18, 0.96, and 0.81 dB at the elevations of 0.5°, 1.5°, and 2.4°.

Using this classification scheme, we can identify the dominant type of scatterers for each radar–gauge pair at all three antenna elevations and estimate the relative contribution of the bright band to the radar-derived rain total. Our analysis indicates that such a contribution is 24% (16.6 mm), 66.1% (46 mm), and 47.3% (33 mm) at the 0.5°, 1.5°, and 2.4° elevations, respectively.

### b. Relations for the bright band

We derive special *R*_{ws}(*Z*) and *R*_{ws}(*K*_{DP}) relations for the estimation of rain rate from returns identified as bright band. To find these relations, we took radar data classified as rain at 0.5° and data classified as bright band at the elevation of 1.5°. We did this above each gauge location (total of 15) and computed the accumulations using the relations valid for rain. Then we found the median of the ratios *R*(1.5°)/*R*(0.5°) over the 15 locations, where *R* stands for rain accumulation at a given elevation and above a gauge location; it can be either from *R*(*Z*) or *R*(*K*_{DP}). The median is the factor by which the corresponding relation, *R*(*Z*) or *R*(*K*_{DP}), should be divided to produce an equivalent rain rate.

We have found that the division of *R*(*Z*) by 2.09 and *R*(*K*_{DP}) by 2.9 transforms measurements in the bright band to rain rates. We have applied these transformations to the bright band at 0.5° and at 1.5° for comparisons with gauges.

Model studies indicate the *R*(*K*_{DP}) in the bright band is about 1.6 times larger than in the rain below if dry snow above the bright band is rimed; it is 2.25 times larger if the snow is not rimed (Ryzhkov et al. 2008). Well-pronounced bright bands, like the ones in our data, are formed from snow with little riming. Thus our factor 2.9 for the *R*(*K*_{DP}) estimator is consistent with the data although larger than the model prediction. This could be due to enhancement of the *K*_{DP} in the bright band due to nonuniform beam filling (Ryzhkov and Zrnić 1998). The values of *Z* and *K*_{DP} associated with the bright band depend on the position and thickness of the bright band and on scattering properties of the dry snow, wet snow, and rain within the radar resolution volume. In our data at elevation 0.5° only wet snow and rain could have been contributors. At 1.5° elevation, there could have been some contribution from dry snow above the bright band at two gauge locations when its height was below 1 km (on 5 days).

The divisor 2.09 for the *R*(*Z*) relation is smaller than the typical value (~4 at average *Z* = 21 dB*Z*) for the ratio of peak reflectivity in the bright band to the reflectivity below observed with the vertically pointing antenna by Fabry and Zawadzki (1995). This is likely due to smoothing by the beam of the peak *Z* at the low-elevation angles.

### c. Comparison of accumulations

Having determined the appropriate *R*_{ws}(*Z*) and *R*_{ws}(*K*_{DP}) relations, we combined these with (6) and (7), respectively, according to the designation from the values of *ρ _{hυ}*. That is, we have matched the relation to the type of precipitation. Motivated by Matrosov’s (2010) algorithm we have also tested a “matched composite” algorithm whereby the matched (6) is used if

*K*

_{DP}< 0.1° km

^{−1}but the matched (7) is used otherwise. In the quantitative comparisons that follow we have excluded seven gauges: three because of significant ground clutter at 0.5° elevation, one because of known underestimation by the gauge, one because sampling in time was 15 min, and two because they were faulty.

As the final product of the matched algorithms we present the statistics of total accumulation (*R* ∩ *G*) in Table 4. The unexpected result concerning the 0.5° elevation scan is the increase in the negative difference between the *R*(*K*_{DP}) accumulations and gauges and decreased bias in *R*(*Z*) accumulations compared with the nonmatched relations (Table 2) favoring this *R*(*Z*) estimator. The composite matched estimator is a close second in RMS value but has the smallest SD. It is noteworthy that *R*(*K*_{DP}) accumulations have smaller standard deviation than accumulations from *R*(*Z*). The high correlation between gauge and *R*(*K*_{DP}) accumulations and lower standard deviation implies that it should be possible to construct an unbiased *R*(*K*_{DP}) relation that would be better than a corresponding unbiased *R*(*Z*); simple multiplying factors can unbias both relations. Nonetheless, such an artifice, without revealing the underlying causes of the biases, is unsatisfactory and likely would be valid only for this dataset.

For the *R*(*Z*) and composite algorithm the results from the 1.5° elevation are the same in SD and correlation, both better than *R*(*K*_{DP}), but the latter has the smallest bias. These findings require further explanation and we turn to accumulations as a function of the distances from the radar.

In the case of segregated precipitation categories (rain and the bright band), comparisons can be made only on data for which both instruments indicate precipitation (i.e., *R* ∩ *G*; see Table A3). Plotted in Figs. 6a,b are the biases *R*(*K*_{DP}) − *R*(*G*) in accumulations versus range; circles stand for the portion of the total classified as rain while stars are for the portion classified as wet snow. At 0.5° elevation (Fig. 6a) the points designated as rain are almost independent of range and definitely biased low indicating a deficiency of the algorithm (7) for light rain. The wet snow part is consistent with the rain part although its bias is smaller (Table A3). The contribution by rain sensed at the 1.5° elevation (Fig. 6b) has no range dependence and exhibits a much smaller bias. The variability of the wet snow contribution is considerably larger and the bias becomes more negative with range.

Whereas the matched *R*(*K*_{DP}) improved the accuracy of accumulations obtained from the 1.5° observation, it degraded the accuracy at 0.5°. The good performance of the relation [(7), not matched] at 0.5° is achieved because the *R*(*K*_{DP}) from wet snow is significantly larger than from rain, thus fortuitously balancing the pure rain part of underaccumulation. The measurement designated as rain at 1.5° accounts for 44% of rain at the ground, whereas the measurement at 0.5° corresponds to 75%.

The increase in negative bias [i.e., *R*(*K*_{DP}) − *R*(*G*)] at 0.5° compared to the bias at 1.5° is likely caused by the transient properties in drop size distribution (Prat and Barros 2009). Note that the *Z*_{DR} at lower elevation is larger (Fig. 5) but corresponds to the smaller rain rate. This observation is consistent with Kumjian and Ryzhkov’s (2010) finding that in a low-intensity rain regime (<10 mm h^{−1}) evaporation on the way to the ground reduces the concentration of the smallest drops. This reduction causes a slight increase in *Z*_{DR} but more significant decrease in *K*_{DP} (Kumjian and Ryzhkov 2010, their Fig. 9) as observed here. We have applied the model of Kumjian and Ryzhkov with median conditions in Bonn—humidity on the ground of 87%, temperature of 7.6°C, and 1-km change in height—to a gamma DSD (shape factor *μ* = 5, rain rate < 3 mm h^{−1}). It produced a 22% relative decrease in *K*_{DP} over the 1-km height interval. The change in average *K*_{DP} in our measurements from pure rain between 1.5° and 0.5° was 20%, suggesting that the influence of evaporation on the drop size distribution might have caused the decrease.

The other related issue is the inability of relation (7) at 0.5° elevation to produce unbiased rain accumulations. The reason that it works better on data from 1.5° is because larger *K*_{DP} maps into a higher rain rate. Note that the relation (7) at *K*_{DP} < 0.1° km^{−1} produces larger rain rates than other documented *R*(*K*_{DP}) relations (i.e., Park et al. 2005; Anagnostou et al. 2010; Wang and Chandrasekar 2010; or Testud et al. 2000), hence a reduction in bias is not possible by applying any one of these relations.

The examination of biases in accumulation *R*(*Z*) − *R*(*G*) from 0.5° elevation (Fig. 7a) reveals that the matched procedure evenly balances the biases for either of the two contributors (rain or wet snow). There is no perceptible trend with range implying that the nearly Marshal–Palmer *R*(*Z*) in use by the German Weather Service suits the cool-season, light-rain regime, but variability in accumulation from rain is significantly larger than from wet snow.

The results from the 1.5° (Fig. 7b) accumulations indicate some overestimation for the rain part at closer range. The wet snow contribution is also biased. The bias decreases with range, possibly because of averaging by the radar beam. At more distant ranges on days with a low bright band, the beam encounters a considerable amount of dry snow (or is totally filled with it if the bright band is below 900 m). Precise determination of the brightband position and thickness with respect to the beam is needed to better quantify the amount of precipitation it produces on the ground.

The *R*(*K*_{DP}) accumulations from pure rain have smaller standard deviations than corresponding *R*(*Z*) accumulations whereas the reverse is true for accumulations from the bright band (Table A3). The decrease in *ρ _{hυ}* and nonuniform beam filling associated with the bright band cause larger variations in estimates of

*K*

_{DP}than in power estimates.

Finally, for comparison and to highlight the value of attenuation correction we evaluate the *R*(*Z*) relation. Thus, the *Z* data have not been corrected for attenuation by rain but have been adjusted for miscalibration and partial beam blockage. No doubt the bias of −17.4 mm (−25%) in accumulations (at 0.5° elevation; Table 5) is induced by attenuation. At the 1.5° elevation, attenuation (by rain and/or wet snow) is overwhelmed by the increase in reflectivity from the wet snow and therefore the bias is positive (22.7 mm, or 33%).

## 5. Summary and conclusions

We explored quantitative rainfall measurements with the polarimetric 3-cm wavelength radar in Bonn, Germany, during November/December 2009 when 17 days had rainfall (about 2800 5-min accumulations). The rain regime was characterized by predominantly light rain and a low bright band, a situation often encountered in Europe but not sufficiently explored with polarimetric radar. The total accumulations have been compared to estimates from a network of rain gauges.

About 1.7 mm of accumulation (2.4%) out of the average total (71.8 mm per gauge) are not registered by radar, partly due to the censoring of data (by the reflectivity *Z* and correlation coefficient *ρ _{hυ}* thresholds) and likely also due to the effects of advection (i.e., by the time rain reaches the gauge the precipitation aloft has adverted downstream). A larger portion of average accumulation (9.7 mm or 13.5%) is sensed by the radar but not present at the gauges. The likely cause is the enormous difference in the size of the resolution volumes; the gradients of rain field can be such that there is no rain at a gauge location while the average over the radar measurement cell indicates rain. For that reason, we favor accepting

*R*> 0 classified as precipitation even if the gauge registers no rain. Independently measured total accumulations by both instruments (i.e.,

*R*∪

*G*) matter as these are used to quantify rainfall. But for a quantitative assessment of the gauges versus radar a prudent approach is the conditional comparison of accumulations from times when both indicate rainfall (i.e., comparison is within the subset

*R*∩

*G*). In our data, the statistics from either approach are comparable.

Accumulations made with the regional *R*(*Z*) (similar to Marshal–Palmer) and with an *R*(*K*_{DP}) were compared to gauge accumulations. The *R*(*K*_{DP}) provided almost unbiased estimates of accumulation (point and areal) and the lowest variability (SD of 14%, correlation of 0.95) from observations at 0.5° elevation. Accumulations from the *R*(*Z*) had a bias of 22%, SD of 23%, and correlation of 0.78. Estimates from the 1.5° elevation of both accumulations had a huge bias (≥100%) and large SDs (≥42%), pointing to the significant influence of the bright band.

We applied classification based on *ρ _{hυ}* to separate contributions of nonmeteorological scatterers, wet snow, and rain. It is meant for the cold season wherein the bright band is close to the ground and within the beam (at elevations ≤ 1.5°) but dry snow is not a contributor. Then we used a vertical profile methodology on

*Z*and

*K*

_{DP}to obtain matched relations

*R*

_{ws}(

*Z*) and

*R*

_{ws}(

*K*

_{DP}) for wet snow. It entailed comparing measurements from the bright band at 1.5° elevation with corresponding values in pure rain at 0.5° and matching the former to the latter. This improved accumulation estimates from both relations at the 1.5° elevation. The bias in

*R*(

*K*

_{DP}) accumulation shrunk from 100% to −2% and in

*R*(

*Z*) from 121% to 39%; less dramatic but significant reductions in SDs were from 45% to 23% and from 42% to 17%, respectively. The bright band (at 1.5° elevation) contributed 66% of average accumulation per gauge. This and the good result suggest that the procedure could be applied to similar conditions elsewhere and improve QPE. A “matched” composite relation consisting of the matched

*R*(

*Z*) for

*K*

_{DP}< 0.1° km

^{−1}and the matched

*R*(

*K*

_{DP}) otherwise was also applied and its performance on 1.5° elevation data was extremely close to the matched

*R*(

*Z*).

The matched relations were applied to 0.5° and the matched composite relation had the smallest SD and highest correlation but the matched *R*(*Z*) had the lowest bias and RMS error. The matched *R*(*K*_{DP}) produced smaller SD in accumulations than *R*(*Z*) and almost the same correlation as the “composite matched,” but it had the largest bias.

The points concerning *R*(*K*_{DP}) are as follows: (i) the vertical profile of *R*(*K*_{DP}) can be used to convert the measurements in the bright band to precipitation on the ground. (ii) There is an apparent influence of evaporation on the DSD, thus reducing the *R*(*K*_{DP}) on the way to the ground. However, we could not exclude as a factor the excess of negative values generated by the estimator. (iii) Existing *R*(*K*_{DP}) relations (admitting negative rain) are biased low in very light-rain regimes; finding one based on physical principles is in order. (iv) The very good performance of the matched *R*(*Z*) estimator was predicated on a very careful calibration and elimination from comparison of gauge locations and a full day of observations that had data problems. The *R*(*K*_{DP}) was essentially immune to these problems.

As confirmed here and elsewhere, polarimetry can transform the 3-cm wavelength radar into a reliable, quantitative, rain-measuring instrument. It restores attenuated reflectivities and is essential for converting returns from the bright band into realistic rainfall on the ground. This adds value in environments whereby the radar beam is blocked at the lowest elevations. The accumulations from the conventional *R*(*Z*) relation (without polarimetric adjustment) at 0.5° elevation underestimate by 18% the gauge values (SD = 24%), undoubtedly because of attenuation. The accumulations derived from the data at 1.5° elevation are overestimated by 33% (SD = 18%) because of the contribution by the bright band.

Resilience of the *R*(*K*_{DP}) estimator for long-term average rainfall accumulation clearly stands out. It is hardly affected by thresholds (or lack thereof) and noise; both increase the random error in estimates but add no bias. Even in adverse conditions caused by strong interference, the daily accumulation agreed remarkably well with gauges, in contrast to total loss of the *R*(*Z*) accumulations. The processing scheme and long-time averaging eliminate random errors and stabilize the *R*(*K*_{DP}) cumulative estimates; herein the SD of the difference (matched *R* − *G*) was 11% of the total (for 0.5° elevation). The correlation of 0.96 implies that the bias could be eliminated without sacrificing the SD with an *R*(*K*_{DP}) suited for very light-rain regimes.

## Acknowledgments

We thank the engineers Dr. Andreas Schneider, Martin Lennefer, and Kai Mühlbauer for the maintenance and operation of the Bonn X-band radar. The radar is a recently established facility operated by the Meteorological Institute of the University of Bonn and was manufactured by Electronic Enterprise Corporation (EEC) and Gamic. We thank Dr. Küll for software consulting. Funding for the CIMMS authors was provided by NOAA/Office of Oceanic and Atmospheric Research under NOAA/University of Oklahoma Cooperative Agreement NA17RJ1227, U.S. Department of Commerce. Dr. Lesya Borowska was funded by a grant from the Deutsche Forschungsgemeinschaft (DFG) in the framework of Transregional Research Center on “Patterns in Landsurface-Vegetation-Atmosphere Interactions” (TR32). We also acknowledge additional funding of Dr. Borowska’s work by a grant from the National Research Council (NRC), NOAA N 0940760. We thank DWD for providing their radar data for calibration and the Aggerverband and Erftverband for providing their rain gauge data. Mattew Kumjian provided ratios of *K*_{DP} values from his evaporation model. We thank three anonymous reviewers for criticisms, especially one suggesting the vertical profile of *K*_{DP} for deriving the relation in bright band, and Dr. Barros for pointing out issues with drop size distributions.

### APPENDIX

#### Quantitative Evaluation of Threshold Effects

A concern of practical significance is the manner in which valid data are separated from noise. Having applied an SNR threshold of 4 dB we soon found out that even small clutter residue can add enough accumulations to erode the accuracy of measurement. Therefore we have examined the minimum reflectivity values corresponding to noise and/or contamination by clutter at each gauge location when there was no rain. The minima were between −2.6 and 7.4 dB*Z* at an elevation of 0.5°. At higher elevations the largest minimum was 0.9 dB*Z* and the rest were all smaller than −1.1 dB*Z*. We attribute the increase in *Z*_{min} at 0.5° elevation to ground clutter. The reflectivity thresholds were set to 0.5 dB above the minima for each gauge and at each elevation. Computations of rainfall were also made with no threshold whatsoever, with a threshold applied to *ρ _{hυ}*, and with a threshold applied to both

*ρ*and

_{hυ}*Z*. The potential for missing rain by the gauges at the

*ρ*> 0.8 and

_{hυ}*Z*

_{min}threshold combinations (i.e., ) is listed in Table A1. As can be seen the loss is rather small and there is a negligible increase if thresholds on both variables are applied. At 1.5° elevation the censoring of

*Z*removes a slightly smaller amount, likely because there are more contributions from the bright band.

We tested the accumulation (*R* ∪ *G*) obtained with the relations (6) and (7) for four different censoring schemes (see Table A2). The good quality of the *R*(*K*_{DP}) measurement for the 0.5° elevation might seem surprising in that the bias between its accumulation and the gauges is almost zero. Moreover, all the results (rows) are almost the same, thus independent of the censoring scheme. Further analysis suggests that the relation for rain underestimates the amounts while the contribution by the bright band (at 0.5°) overestimates the amounts so that the two fortuitously cancel.

The values from *R*(*Z*) are inferior both in terms of bias and standard deviation. This is partly due to contamination by the bright band. Also, artifacts at 0.5° affect power measurements (from which *Z* is derived) much more than measurement of phase (i.e., *K*_{DP}).

The results for the 1.5° elevation of both radar estimators are dismal, independent of the censoring scheme. Clearly the bright band causes a significantly larger increase in the differential phase and the reflectivity than rain. Thus the relations grossly overestimate the rain at the ground and require appropriate adjustment. Measurements at 2.4° (not shown) fare better than the ones at 1.5° because the propagation path through the bright band is on average shorter and there is a substantial path through the snow region above the bright band.

Statistics of rain accumulations for the matched algorithm segregated according to pure rain, bright band, and matched are in Table A3. This comparison is over the field *R* ∩ *G* for which both indicate rain (total gauge accumulations for each row in Table A3 are in the last column); segregation by type is not feasible in the *R* ∪ *G* field. At 0.5° matching produces a good result for the *R*(*Z*) estimator but increases the bias in the *R*(*K*_{DP}) because the relation for the bright band is matched to the deficient relation (7) for rain. At the 1.5° elevation the *R*(*K*_{DP}) for the rain part has a very small bias. Close to cloud base drops are larger than near ground and hence produce larger *K*_{DP} values resulting in a closer agreement with gauges on the ground. Last we computed the accumulation sensed by the radar but not registered by the gauges (i.e., the subset ). The average value per gauge was 9.7 mm (13.5% of 71.8 mm).

## REFERENCES

## Footnotes

^{*}

Present affiliation: NOAA/National Severe Storms Laboratory, Norman, Oklahoma.