## 1. Introduction

The S-band weather radars in the United States are primarily utilized for the observations of severe storms. The major reason for this is that shorter-wavelength radars may experience significant attenuation in heavy precipitation. A long-standing problem of attenuation correction at shorter radar wavelengths can be efficiently resolved if the radar has dual-polarization capability.

According to the National Weather Service plans, single-polarization Weather Surveillance Radars-1988 Doppler (WSR-88D) will be retrofitted in the next few years by adding polarimetric capability and, in the long run, may be complemented by C-band and X-band polarimetric radars for better areal coverage. Note that all weather radars utilized by the Federal Aviation Administration in the terminal areas of airports operate at C band. Television companies also use C-band Doppler radars, some of which already have polarimetric diversity. Hence, it is important to investigate and demonstrate abilities of such radars to quantitatively assess characteristics of severe storms in the presence of strong attenuation.

Polarimetric methods for attenuation correction of radar reflectivity *Z* and differential reflectivity *Z*_{DR} utilize measurements of differential phase Φ_{DP}, which is immune to attenuation (Bringi and Chandrasekar 2001). Simplified versions of the attenuation correction techniques assume that the coefficients of proportionality *α* and *β* between the *Z* and *Z*_{DR} biases and Φ_{DP} do not vary much (Bringi et al. 1990). The correction factors *α* and *β* are equal to the ratios *A _{h}*/

*K*

_{DP}and

*A*

_{DP}/

*K*

_{DP}, respectively, where

*A*is specific attenuation of microwave radiation at horizontal polarization,

_{h}*A*

_{DP}is specific differential attenuation, and

*K*

_{DP}is specific differential phase [see Bringi and Chandrasekar (2001) for definitions]. However, at C band, these ratios are highly variable in convective cells containing large raindrops and hail because of effects of resonance scattering (Carey et al. 2000; Keenan et al. 2001; Bringi et al. 2001; Ryzhkov et al. 2006, 2007; Gourley et al. 2006; Vulpiani et al. 2008; Tabary et al. 2008, 2009; Borowska et al. 2009, 2011).

In rain, the ratio *α* = *A _{h}*/

*K*

_{DP}at C band usually varies between 0.05 and 0.18 dB per degree as reported by Bringi et al. (1990), Carey et al. (2000), Gourley et al. (2006), Ryzhkov et al. (2007), and Keranen and Yllasjarvi (2008). The ratio

*β*=

*A*

_{DP}/

*K*

_{DP}varies over an interval from 0.008 to 0.1 dB per degree and exhibits a strong correlation with the maximum value of

*Z*

_{DR}in an attenuating rain cell (Carey et al. 2000; Keenan et al. 2001; Ryzhkov et al. 2007; Tabary et al. 2009). Borowska et al. (2009, 2011) estimated much higher local values of

*α*and

*β*within “hot spots” containing large raindrops and melting hail.

More-sophisticated polarimetric schemes for attenuation correction attempt to take into account the variability of *α* and *β* (Carey et al. 2000; Bringi et al. 2001; Ryzhkov et al. 2007; Vulpiani et al. 2008). Bringi et al. (2001) suggested the self-consistent method with constraints to optimize the coefficients *α* and *β* by examining the radial profile of Φ_{DP} and imposing constraints on the corrected value of *Z*_{DR} at the far side of an attenuating rain cell. This method implies that these coefficients change from ray to ray but remain constant along each particular ray. Vulpiani et al. (2008) allowed for variability of the coefficients *α* and *β* along the propagation path by identifying the prevailing rain regime in each range gate, but a single correction factor weighted by *K*_{DP} for any given path is used for attenuation correction along the path. Carey et al. (2000) took into account that the parameters *α* and *β* can also vary along the radial and assigned different fixed values of them in hot spots or “big drop zones” and the rest of the propagation path. The idea of Carey et al. (2000) was further advanced by Ryzhkov et al. (2006, 2007) who proposed a procedure for a more objective estimation of *α* and *β* within hot spots. The latter method is described and explored in this paper.

The proposed method for attenuation correction is tested for two heavy-rain events that were observed with two different C-band polarimetric radars in central Oklahoma and in the Chicago, Illinois, metropolitan area. Both radars were built by the Enterprise Electronics Corporation. One of them belongs to Valparaiso University, in Valparaiso, Indiana, and another one belongs to the University of Oklahoma.

Results of attenuation correction are validated using self-consistency between radar polarimetric variables and comparisons with the measurements from a nearby polarimetric prototype of the WSR-88D (KOUN) in Oklahoma and from a single-polarization WSR-88D (KLOT) in the Chicago area that did not experience much attenuation in the storms.

The paper is organized as follows. The description of the hot-spot attenuation correction procedure and its comparison with the self-consistent method of Bringi et al. (2001) are presented in section 2. Section 3 contains a validation of the results of the attenuation correction at C band through direct comparison with the measurements from the nearby KOUN radar in Oklahoma. Section 4 is devoted to the analysis of the Chicago storm and statistics of the parameters *α* and *β* derived from C-band polarimetric measurements, and section 5 includes a summary of the results.

## 2. Algorithm for attenuation correction

### a. Brief review of polarimetric techniques for attenuation correction at C band

*Z*and

*Z*

_{DR}was suggested by Bringi et al. (1990). According to this method, the biases of

*Z*and

*Z*

_{DR}(Δ

*Z*and Δ

*Z*

_{DR}, respectively) are estimated from simple formulas:

*α*and

*β*are supposed to be constant. The coefficient

*α*is the ratio of specific attenuation

*A*and specific differential phase

_{h}*K*

_{DP}, whereas the coefficient

*β*is the ratio of specific differential attenuation

*A*

_{DP}and

*K*

_{DP}. Testud et al. (2000) proposed another correction algorithm for

*Z*(the “ZPHI” rain-profiling algorithm) that also assumes a fixed coefficient

*α*.

*r*

_{0},

*r*

_{m}) that contains radar echo so that

*A*(

_{h}*r*) is estimated using attenuated radar reflectivity

*Z*and ΔΦ

_{a}_{DP}through the formula

*b*is an exponent in the relation

*A*=

_{h}*aZ*and the radar reflectivity factor in (5)–(7) is expressed in linear units.

^{b}*α*for each radial of data (a self-consistent method with constraints). The self-consistent method does not require a fixed a priori value for

*α*, but rather searches for an optimal

*α*value within a predetermined range (

*α*

_{min},

*α*

_{max}) for each particular radial. For each

*α*value in the predetermined range, a model profile of differential phase

_{DP}(

*r*) over the entire range from

*r*

_{0}to

*r*through an attenuating rain cell. An optimal

_{m}*α*

^{(opt)}minimizes the difference Δ between the measured and calculated profiles:

*i*denotes the range gate index from

*r*

_{0}to

*r*.

_{m}*A*

_{DP}(

*r*) is prescribed to be proportional to the optimized

*A*[

_{h}*r*,

*α*

^{(opt)}]:

*γ*determined from

*Z*′

_{DR}(

*r*) are measured (biased by differential attenuation) and expected differential reflectivity, respectively, at range

_{m}*r*for corrected radar reflectivity

_{m}*Z*(

*r*). In other words, the corrected value of

_{m}*Z*

_{DR}at range

*r*has to be equal to what is expected (in rain) at reflectivity

_{m}*Z*(

*r*). This principle of

_{m}*Z*

_{DR}correction was first suggested by Smyth and Illingworth (1998) (with

*Z*′

_{DR}equal to 0 dB in the shadow of an attenuating cell) and later modified by Bringi et al. (2001) and Tabary et al. (2009), who assumed that the value of

*Z*′

_{DR}is a function of corrected radar reflectivity factor there.

### b. Attenuation and differential attenuation in hot spots

High variability of the correction factors *α* and *β* in strong convective cells is attributed to strong resonance scattering effects at C band that impact *A _{h}*,

*A*

_{DP}, and

*K*

_{DP}for raindrop sizes exceeding 5 mm (Carey et al. 2000; Zrnić et al. 2000; Keenan et al. 2001; Ryzhkov and Zrnić 2005). The effects of resonance scattering at C band are illustrated in Figs. 1 –3, where results of simulations of different radar variables from the measured drop size distributions (DSD) in central Oklahoma are presented. Computations were made assuming that temperature of raindrops is 20°C, their shape depends on equivolume diameter as prescribed by Brandes et al. (2002), the width of the canting angle distribution is 10°, and maximal raindrop diameter is 8 mm. Similar to analogous simulations by Keenan et al. (2001),

*D*

_{max}= 8 mm was chosen to accentuate the “large drop” resonance effects.

Differential reflectivity *Z*_{DR} exhibits extreme variability for *Z* > 45 dB*Z* (Fig. 1). Very high *Z*_{DR} can be associated with relatively moderate values of *Z*. The scatterplots of *A _{h}* and

*A*

_{DP}versus

*K*

_{DP}are shown in Fig. 2. The degree of scatter is substantially reduced for

*Z*

_{DR}< 3 dB and the ratios

*A*/

_{h}*K*

_{DP}and

*A*

_{DP}/

*K*

_{DP}are much more stable for lower

*Z*

_{DR}. In general, both parameters

*α*=

*A*/

_{h}*K*

_{DP}and

*β*=

*A*

_{DP}/

*K*

_{DP}tend to increase with increasing

*Z*

_{DR}(e.g., Carey et al. 2000; Keenan et al. 2001). This tendency is especially well pronounced for the ratio

*A*

_{DP}/

*K*

_{DP}(Fig. 3). Such a strong dependence of

*β*on the magnitude of

*Z*

_{DR}was recently confirmed by observations reported by Tabary et al. (2009). Very similar simulation results based on DSDs measured in tropical rain in Australia and reported by Carey et al. (2000) and Keenan et al. (2001) indicate the universal character of resonance effects attributed to large raindrops in different climate regions.

The major conclusion from the simulations and observations is that both *α* and *β* become extremely unstable for *Z* > 45 dB*Z* and *Z*_{DR} > 3 dB. Enhanced variability of *α* and *β* within hot spots was first noticed in the study of Carey et al. (2000), who suggested identifying hot spots and treating them separately from the rest of the ray. Carey et al. (2000) recommended using different pairs of *α* and *β* values inside and outside hot spots.

### c. Hot-spot method for attenuation correction

As mentioned in the introduction, Ryzhkov et al. (2006, 2007) utilized the idea of Carey et al. (2000) and proposed a procedure for automatic determination of the parameters *α* and *β* within hot spots. In this study, we adopt such an approach after its slight modification.

*α*

_{0}and

*β*

_{0}are constant outside hot spots for a given radar sweep. These background values can be set equal to their average climatological values or can be estimated from the data, as will be described later.

Identification of hot spots is a crucial component of the algorithm. For a given radar sweep, the rays potentially containing hot spots are identified using a simple reflectivity threshold *Z*^{(th)} (usually between 45 and 50 dB*Z*), after *Z* is preliminarily corrected using (1) with *α* = 0.06 dB per degree (which is considered to be an average climatological value). If the maximal *Z* associated with weather echo (where cross-correlation coefficient *ρ*_{hv} is higher than 0.7) does not exceed *Z*^{(th)} anywhere along the radial, then this radial does not contain hot spots and is qualified as a “non–hot spot” (NHS) radial. Carey et al. (2000) utilized the *ρ*_{hv} threshold of 0.97 to detect big-drop zones, which are similar to what are referred to as hot spots in this study. We refrain from using this threshold in our analysis because many areas with much lower *ρ*_{hv} were found in the observed storms that are obviously not associated with hot spots and might be affected, for example, by nonuniform beam filling (Ryzhkov 2007).

*α*

_{0}and

*β*

_{0}generally depend on temperature as well as the prevalent type of DSD (e.g., Jameson 1992) and can be roughly estimated using the data from the NHS radials by examining minimal values of measured

*Z*

_{DR}drops as low as −1 dB because of differential attenuation. Then,

*β*

_{0}is estimated as a median value of ratios

*Z*′

_{DR}is the expected value of

*Z*

_{DR}(not biased by differential attenuation) in the

*Z*

_{DR}minimum estimated from

*Z*after reflectivity is corrected for attenuation using (1) with

*α*= 0.06 dB per degree:

*β*

_{0}strongly depends on the prevalent

*Z*

_{DR}.

The background factor *α*_{0} depends on temperature (similarly to *β*_{0}), but it is much less sensitive to *Z*_{DR}. The scatterplots of *β*_{0} versus *α*_{0} simulated from disdrometer measurements in Oklahoma for two different temperatures, 10° and 20°C, are displayed in Fig. 5. Simulations are made for *Z*_{DR} < 3 dB to avoid contamination from hot spots and for *Z*_{DR} > 0.5 dB. At lower *Z*_{DR}, the ratio *A _{h}*/

*K*

_{DP}can increase dramatically as a result of lower

*K*

_{DP}associated with near-spherical drops. Nevertheless, our analysis shows that in Oklahoma the contribution of drops with

*Z*

_{DR}less than 0.5 dB does not exceed 10% of total

*A*integrated over the average DSD. Figure 5 shows that

_{h}*α*

_{0}almost linearly depends on

*β*

_{0}at a given temperature except for very low

*β*

_{0}and that the range of

*α*

_{0}variability is significantly smaller than that of

*β*

_{0}in a relative sense. Note that the slopes of the

*α*

_{0}–

*β*

_{0}dependences simulated from disdrometer data are lower than the factor of 3.33 suggested by Vulpiani et al. (2008) (dashed line in Fig. 5).

Once the background value *β*_{0} is determined for a given sweep, the corresponding value of *α*_{0} can be determined from the gray curves in Fig. 5 if the average temperature along the propagation path is known. Note that temperature uncertainty of 10°C results in about a 10% uncertainty in *α*_{0} for larger *β*_{0} and may be tolerated in a first approximation. According to the suggested method, the background values *α*_{0} and *β*_{0} may vary from sweep to sweep as functions of elevation and time.

*α*

_{0}and

*β*

_{0}are determined, a preliminary attenuation correction of

*Z*and

*Z*

_{DR}is performed for all radials in the sweep using equations

These preliminarily corrected *Z* and *Z*_{DR} are used to identify hot spots using the following criteria: 1) *Z* > *Z*^{(th)} and *ρ*_{hv} > 0.7 everywhere in the hot spot, 2) the maximal value of *Z*_{DR} within the hot spot exceeds _{DP} within the hot spot exceeds *α* and Δ*β* in (13) and (14) are assumed to be the same in all hot spots for a particular radial. However, these parameters are different for different radials.

_{DP}(HS) stands for the Φ

_{DP}increase within hot spots. Equation (18) stipulates that in the basic equation in (5) for the traditional ZPHI method the term

*α*ΔΦ

_{DP}(

*r*

_{0};

*r*

_{m}) should be replaced with the term

*α*

_{0}ΔΦ

_{DP}(

*r*

_{0};

*r*

_{m}) + Δ

*α*ΔΦ

_{DP}(HS):

_{DP}(

*r*

_{0};

*r*

_{m}) and ΔΦ

_{DP}(HS), are used for constraining the procedure instead of one. As a result, the radial profile of

*A*estimated from (5) becomes dependent on the value of Δ

_{h}*α*. The appropriate factor Δ

*α*should be defined from the iterative process of incrementing Δ

*α*until a certain condition is satisfied. Following Ryzhkov et al. (2006, 2007), it is required that

*A*is equal to

_{h}*α*

_{0}

*K*

_{DP}outside hot spots and that the corresponding integrals should also be equal.

*Z*and

*Z*are in reflectivity decibels (dB

_{a}*Z*) and

*A*[

_{h}*s*, Δ

*α*

^{(opt)}] is the profile of specific attenuation determined from (19) with the parameter Δ

*α*

^{(opt)}satisfying condition (21).

The conception of the method is illustrated in Figs. 4 and 6. Figure 4 shows simple model profiles of *Z* and Φ_{DP}. It is assumed that *Z* is equal to 45 and 53 dB*Z* outside and inside the hot spot, respectively, and that the parameter *α* is equal to 0.10 dB per degree within the hot-spot area and to 0.06 dB per degree outside it. The corresponding profile of *A _{h}* is computed using the relation

*A*= (2.98 × 10

_{h}^{−5})

*Z*

^{0.8}from Le Bouar et al. (2001) and is indicated by a solid line in the three panels of Fig. 6 for three possible locations of the hot spot along the propagation path. In these model examples,

*r*

_{0}= 0 km,

*r*

_{m}= 25 km, and the radial extension of the hot spot area is 5 km. Applying relations (19) and (20) with a varying parameter Δ

*α*in (13) results in the different retrieved profiles of

*A*shown in Fig. 6. Note that

_{h}*A*changes not only within the hot spot but also outside it. In Fig. 6,

_{h}*L*/

*R*means the ratio of the left-hand and right-hand sides of (21).

The retrieved *A _{h}* is lower than its true value if Δ

*α*= 0.0 dB per degree (dashed lines) and the retrieved radar reflectivity factor is underestimated. This is equivalent to utilizing the unmodified ZPHI equation in (5) with

*α*=

*α*

_{0}= 0.06 dB per degree everywhere along the propagation path. In this case, the ratio

*L*/

*R*is less than 1 regardless of hot-spot location along the ray. If Δ

*α*is too high (0.08 dB per degree), then

*L*/

*R*> 1 and retrieved

*A*is overestimated (dotted lines). The retrieved and true profiles of

_{h}*A*match precisely (solid lines) only if Δ

_{h}*α*= 0.04 dB per degree and

*L*/

*R*= 1, that is, if condition (21) is satisfied.

It is important that the method yields an unbiased estimate of Δ*α*^{(opt)} even in the situation in which there is no radar echo (or valid data) behind the hot spot or in the rear side of the convective cell with respect to the radar (*r*_{2} = *r*_{m}). This happens very often when the radar signal is totally extinct because of attenuation within a hot spot or differential phase becomes very noisy as a result of the drop of *ρ*_{hv} (Tabary et al. 2008). However, the sensitivity of the algorithm is somewhat diminished in such a situation. Indeed, the change of the ratio *L*/*R* is within 0.72–1.30 if Δ*α* varies between 0.0 and 0.08 dB per degree and the hot spot is at the near end of the propagation path (*r*_{1} is close to *r*_{0} as in Fig. 6b). The corresponding span of *L*/*R* is smaller (between 0.86 and 1.10) for *r*_{2} = *r*_{m} (Fig. 6c) and the algorithm is less robust.

*Z*

_{DR}attenuation correction is based on the original idea of Smyth and Illingworth (1998), according to which the measured value of

*Z*

_{DR}behind the attenuating cell is compared with what is expected in light rain in the shadow of this cell. This idea was later modified by Bringi et al. (2001) and Tabary et al. (2008, 2009), as described in section 2a. In our algorithm, the attenuation-related bias in differential reflectivity is determined as

*β*

_{0}and Δ

*β*are defined by (14) and [

*r*

_{1},

*r*

_{2}] is the range interval containing the hot spot (Fig. 4). The parameter Δ

*β*is estimated from

_{DP}(

*r*

_{0}) = 0, path-integrated attenuation Δ

*Z*(

*r*) and differential attenuation Δ

_{m}*Z*

_{DR}(

*r*) can be expressed as

_{m}## 3. Attenuation correction in heavy rain in Oklahoma

The performance of the HS algorithm for attenuation correction has been tested in the case of heavy rain that occurred on 10 March 2009 in Oklahoma. Multiple precipitation bands have been characterized by very high *Z* (exceeding 60 dB*Z*). According to the National Oceanic and Atmospheric Administration (NOAA) publication *Storm Data*, no hail was reported on the ground, but melting hail aloft should not be excluded. It is possible that most of the hail completely melted before reaching the surface because the freezing level was very high (at 3.4 km) on that day.

This event has been observed with nearly collocated C-band and S-band polarimetric radars. The C-band University of Oklahoma Polarimetric Radar for Innovations in Meteorology and Engineering (OU PRIME) has a 1-MW transmitted power and a ½° antenna beam that in combination with 0.125-km gate spacing provides very high spatial resolution of polarimetric data. The data collected by the polarimetric prototype of the S-band WSR-88D (KOUN) have been used for comparison and validation of the procedure for attenuation/differential attenuation correction. The S-band radar is at a distance of 6.86 km and an azimuth of 337.3° with respect to the C-band radar. The KOUN radar has a 1° beam, and the radial resolution of the data collected during the storm was 0.25 km.

Absolute calibration of *Z* for both radars was checked using comparisons with the nearby operational WSR-88D near Oklahoma City, Oklahoma (KTLX). The consistency between *Z* and *K*_{DP} in rain was also utilized to evaluate the absolute calibration of *Z*. The general principles of the polarimetric consistency checks are described by Gorgucci et al. (1992), Goddard et al. (1994), and Ryzhkov et al. (2005), among others. It is expected that *Z* and *K*_{DP} are consistent for moderate to heavy rain within the radar reflectivity interval between 40 and 50 dB*Z*.

Examination of *Z*_{DR} in dry aggregated snow or dry graupel above the melting layer and analysis of the *Z*–*Z*_{DR} scatterplots in rain for *Z* < 40 dB*Z* (in the areas where attenuation is insignificant) were utilized to remove the bias in the measurements of *Z*_{DR} at both C and S bands. According to Ryzhkov et al. (2005), intrinsic *Z*_{DR} in dry aggregated snow should be within the range 0.1–0.2 dB. The observed scatterplot of *Z*_{DR} versus *Z* in rain has to be in agreement with that obtained from theoretical simulations (see Fig. 1). We believe that the combination of both methods allows us to reduce the *Z*_{DR} calibration bias to 0.1–0.2 dB.

The scanning strategies of the two radars were not synchronized, and their antenna elevations were slightly different, which makes it difficult to achieve a good match between the two datasets. However, it was possible to select volume scans of data with a time difference less than 30 s for which the fields of C-band and S-band data exhibit very good resemblance. An example of such data is presented in Fig. 7, where the fields of the measured C-band and S-band *Z*, *Z*_{DR}, and Φ_{DP} taken at elevations of 0.41° (C band) and 0.48° (S band) and around 0309 UTC are displayed.

First, it is obvious that the better spatial resolution of the OU PRIME data is a great benefit, as comparison with KOUN data reveals. The impact of differential attenuation on the *Z*_{DR} measurements at C band is clearly visible in the sector with enhanced differential phase where *Z*_{DR} drops as low as −8.5 dB and the corresponding values of S-band *Z*_{DR} are mainly within the range between 0.5 and 1.5 dB. Negative bias in C-band *Z* is apparent in the area just behind the first precipitation line (marked as B) and within the second line of precipitation (marked as A), which is farther away from the radar.

The comparison of composite RHIs taken at azimuth = 285.5° at C band and azimuth = 280° at S band provides additional evidence of substantial differences between the C-band and S-band data caused by attenuation and effects of resonance scattering at C band (Fig. 8). Indeed, the second precipitation line is disconnected from the first one in the C-band plots and is associated with a reflectivity factor that is about 20 dB lower than that measured at S band. Differential reflectivity at C band is higher than at S band in the updraft area at the leading edge of the squall line (because of resonance scattering on large raindrops), but then it decreases rapidly along the propagation path deeper into the storm. Notable is a significantly lower cross-correlation coefficient *ρ*_{hv} at C band, which is also attributed to resonance scattering.

Estimation of the parameter *β*_{0} using the data from the radials void of hot spots on the radar scan shown in Fig. 7 yields a value of 0.03 dB per degree if the method described in section 2c is used. This is a typical background value in continental storms (Tabary et al. 2009) and is 2–3 times as high as in tropical rain dominated by smaller drops (Bringi et al. 2006; Ryzhkov et al. 2007). The corresponding value of *α*_{0} obtained from Fig. 5 at temperature *T* = 10°C is about 0.1 dB per degree.

In qualitative terms, the hot-spot algorithm does a good job in reducing attenuation-related biases, as Fig. 9 shows. The corrected fields of *Z* and *Z*_{DR} at C band agree very well with the corresponding S-band fields except for a narrow azimuthal sector marked by a dashed line where the OU PRIME beam is significantly blocked by the nearby building. It is apparent that negative *Z* bias in the areas A and B in Fig. 7 is almost eliminated after the attenuation correction procedure is applied. Intrinsic values of *Z*_{DR} at C band are noticeably higher than the ones at S band within the squall line, which is attributed to the resonance scattering. A spot of very high *Z*_{DR} combined with low *Z* at about *X* = −73 km and *Y* = 18 km in C-band panels is not an artifact but a real signature produced by size sorting in the updraft of a small growing convective cell, as a more detailed analysis of RHI plots indicates (not shown).

A more quantitative validation of the attenuation correction scheme was performed by converting C-band and S-band radar data from a polargrid to a Cartesian grid with 1 km × 1 km resolution and comparing the gridded *Z* and *Z*_{DR} data before and after correction for attenuation. Such a comparison was made only in the areas affected by noticeable attenuation where the measured (uncorrected) *Z*_{DR} at C band is below −1 dB. The scatterplots of the differences *Z*(S band) − *Z*(C band) vs *Z*_{DR}(S band) − *Z*_{DR}(C band) before and after correction using the hot-spot algorithm are displayed in Fig. 10. The scatter is very significant in both panels of Fig. 10 because of the spatial/temporal mismatch of the C-band and S-band radar fields and because of the differences in the intrinsic values of *Z* and *Z*_{DR} at the two radar wavelengths due to effects of resonance scattering. Nevertheless, the positive effect of attenuation correction is obvious: median values of the *Z* and *Z*_{DR} differences are very close to zero after the correction is performed.

A notable feature of the examined case of continental rain is significant attenuation/differential attenuation (over 20/7 dB) associated with very modest values of differential phase, which does not exceed 120°. If the background values *α*_{0} = 0.1 dB per degree and *β*_{0} = 0.03 dB per degree were used for attenuation correction utilizing (16) and (17) everywhere in the PPI, it would result in significant underestimation of path-integrated attenuation and differential attenuation along the radials containing hot spots, and the estimated *Z* and *Z*_{DR} biases would not exceed 12 and 3.6 dB, respectively. In fact, these are at least 2 times as high, as the top panel in Fig. 10 shows. The terms Δ*α*ΔΦ_{DP}(HS) and Δ*β*ΔΦ_{DP}(HS) in (26) and (27) compensate for this underestimation.

## 4. Attenuation correction for the Chicago storm

The second storm for which polarimetric attenuation correction at C band was tested was observed in the Chicago metropolitan area. This very severe thunderstorm hit the area at about 0000 UTC 5 August 2008, producing damaging winds and torrential rain. Thousands of travelers at Chicago O’Hare International Airport and fans attending a baseball game at Wrigley Field were evacuated. Many homes and businesses were damaged as a result of the storm. Wind gusts sped up to over 90 mi h^{−1} (≃40 m s^{−1}), and one fatality was reported in northwestern Indiana as a result of a falling tree. No hail was reported on the ground during this storm.

The storm was in the coverage area of the C-band “Sidpol” radar (owned by Valparaiso University) for at least 2 h before the leading edge of the squall line passed over the radar site at approximately 0149 UTC 5 August 2008 and radar data recording was interrupted because of lightning strikes. The storm position between the Sidpol radar and the Chicago WSR-88D (KLOT), which are approximately 90 km apart, provided an opportunity (although not as good a one as in Oklahoma) to compare radar reflectivities at C and S bands and assess the impact of attenuation at C band.

Sidpol radar data were available with a radial resolution of 0.125 km and an azimuthal resolution of about 0.83° within the range of 180 km from the radar. Extremely high values of Φ_{DP} have been measured in this storm, as opposed to the storm observed in Oklahoma. An example of the radial profile of measured Φ_{DP} is shown in Fig. 11a. The recorded differential phase exhibits double aliasing and needs to be dealiased before the estimation of its radial derivative, specific differential phase *K*_{DP}, can be performed. A three-step procedure was used for the Φ_{DP} dealiasing and processing. This implies downward shifting of differential phase, as shown in Fig. 11b; elimination of the Φ_{DP} jump caused by aliasing; and editing and smoothing of Φ_{DP} using the measurements of *ρ*_{hv} (Fig. 11c).

Attenuation correction of *Z* and *Z*_{DR} was performed for all radar scans every 6 min during the 2-h period of observations after data-quality issues were addressed. The estimated background parameter *β*_{0} in the attenuation correction scheme was significantly lower than in the Oklahoma storm and varied mostly between 0.008 and 0.012 dB per degree from scan to scan. Hence, the corresponding background value *α*_{0} obtained from Fig. 5 was about 0.06 dB per degree at an average temperature of 20°C.

The degree of attenuation at C band was striking in this storm, as can be seen from Fig. 12, where the fields of the measured *Z* (before correction for attenuation), corrected *Z*, differential phase Φ_{DP}, and *Z* obtained from the S-band KLOT radar at 0149 UTC are displayed. The difference between measured and corrected *Z* at C band approaches 30–40 dB over extended areas of the storm. This is not surprising given the fact that Φ_{DP} exceeds 300° in large azimuthal sectors west and north of Sidpol.

The WSR-88D provides good reference for validating attenuation correction at C band because the S-band signal experiences much lower attenuation and the WSR-88D is located behind the squall line at 0149 UTC; hence, the propagation path of the S-band microwave radiation through heavy rain is relatively short. It is evident that the corrected *Z* at C band agrees well with that measured by the WSR-88D within the squall line. However, if the attenuated C-band signal drops below the noise level, as in the remote areas in the northern and western azimuthal sectors, then attenuation correction is not possible. The difference between corrected C-band *Z* and S-band *Z* in the stratiform part of the storm is caused either by the height mismatch of the radar sampling volumes of the two radars at elevation 0.5° in this area or by possible error in the reading of antenna elevation by the Sidpol radar (i.e., its actual elevation might be higher than 0.5°). In the latter case, the Sidpol radar samples a good part of the melting layer, whereas the corresponding radar resolution volume of theWSR-88D is below the melting layer and, therefore, S-band *Z* is lower than C-band *Z* there.

Because the C-band and S-band radars were so far away from each other, it was hard to perform quantitative verification of the attenuation correction scheme by direct comparison of reflectivities measured by both radars as was done for the Oklahoma case. Instead, we checked the consistency of corrected *Z* and *Z*_{DR} with *K*_{DP} in rain by examining the scatterplot of the difference 10 log(*K*_{DP}) − *Z _{h}* versus

*Z*

_{DR}and comparing it with theoretical dependencies at C band (Fig. 13). The solid line in Fig. 13b corresponds to the simulations based on measured DSD in Oklahoma, whereas the dashed line depicts theoretical results of Gourley et al. (2006). Median values of

*Z*before correction for attenuation (shown by asterisks) are 5–12 dB below what is expected for a given

*K*

_{DP}and

*Z*

_{DR}(corrected for attenuation). After the attenuation correction is made, median values of

*Z*(shown by diamonds) are within 1 dB with respect to their model values as dictated by consistency, which attests to the good quality of the attenuation correction.

The largest recoverable attenuation bias of about 40 dB is estimated along the radial at an azimuth of 257.2° (line in Fig. 12), where Φ_{DP} as high as 602° has been measured (Fig. 14a). To the best of our knowledge, this is the highest value of differential phase ever reported. It is much higher than anything measured in the previous C-band studies in Europe and Australia. An amazing result is that the polarimetric algorithm for attenuation correction is capable of reliably restoring the radar reflectivity in the situation in which 99.99% of signal power is lost (Fig. 14b). Note that the Hitschfeld–Bordan attenuation correction scheme for a single-polarization radar experiences serious problems if attenuation barely reaches 10 dB or is even lower.

Total differential attenuation along the same ray reaches 7 dB, as Fig. 14c shows. The minimal reliably measured *Z*_{DR} measured at the end of this ray is about −6 dB, with the corresponding Φ_{DP} exceeding 600°. Thus, the net value of *β* averaged over the path is 0.01 dB per degree. Note that in the Oklahoma case *Z*_{DR} drops to lower values at the radials with maximal differential phase of only 120°, which is ⅕ of the maximal Φ_{DP} measured along the propagation path in Fig. 11. We speculate that the Oklahoma storm contained a higher concentration of large raindrops with resonance size than did the Chicago storm. This is consistent with the facts that maximal *Z*_{DR} measured in the Oklahoma event is 1–2 dB higher than in the Chicago case and that there is a strong correlation between maximal *Z*_{DR} and parameter *β* (Carey et al. 2000; Keenan et al. 2001; Tabary et al. 2009; Borowska et al. 2011).

*Z*and

*K*

_{DP}using the relations

*K*

_{DP}is expressed in degrees per kilogram and

*Z*is in reflectivity decibels. The

*R*(

*K*

_{DP}) estimate does not depend on attenuation. Figure 14d confirms that the

*R*(

*Z*) and

*R*(

*K*

_{DP}) profiles along the radial at azimuth = 257.2° are in a good agreement if the rain rate is less than 100 mm h

^{−1}.

The quality of the *Z*_{DR} correction for differential attenuation is illustrated in Fig. 15, where the composite PPI plot of *Z*, Φ_{DP}, *ρ*_{hv}, and three fields of *Z*_{DR} are presented for the radar scan at 0124 UTC. Uncorrected differential reflectivity exhibits strong differential attenuation in the sectors of high Φ_{DP} where measured *Z*_{DR} drops below −5 dB. The blank azimuthal sector in the northwestern direction is caused by total attenuation of the radar signal. A simplistic correction procedure for differential attenuation based on the use of (17) with *β*_{0} = 0.01 dB per degree significantly improves the *Z*_{DR} estimate but falls short of eliminating the relatively large areas of negative *Z*_{DR} where differential attenuation is especially severe (Fig. 15c, marked as linear correction). This means that the parameter *β* should be increased significantly in certain azimuthal directions to fix the problem. The hot-spot or “adaptive” technique that automatically determines an appropriate coefficient *β* apparently does a much better job (Fig. 15d, marked as adaptive correction) and ensures positive and more-realistic- looking *Z*_{DR}. There is no apparent artificial drop of *Z*_{DR} next to the blank sector with severe attenuation.

Because the reference S-band KLOT radar lacks polarimetric capability, it cannot be used for validation of the differential attenuation correction in the Chicago case. The algorithm robustness can be assessed by taking into account the absence of negatively corrected *Z*_{DR}, its general consistency with *Z*, and the spatial/temporal continuity of the fields of corrected *Z*_{DR}. Detailed analysis of the images of corrected *Z*_{DR} for 2-h periods of observation indicates that the suggested algorithm for differential attenuation correction is robust and reliable. Occasional “bad radials” of corrected *Z*_{DR} take place but they are relatively rare and can be eliminated by using considerations of azimuthal continuity.

Every radial of radar data containing hot spots is characterized by particular values of *α* and *β* that turn out to be highly variable. Scatterplots of parameters *α* and *β* versus the maximal value of *Z*_{DR} in hot spots are shown in Figs. 16a and 16b. These scatterplots summarize results for all radar scans at elevation 0.5° and indicate large variability of *α* and *β* in hot spots at C band. Most values of *α* are within the range of 0.05 and 0.20 dB per degree, whereas *β* varies mainly between 0.01 and 0.04 dB per degree. The estimates of *α* and *β* in this study are generally consistent with the previous findings of the authors for C-band observations in Alabama and Canada (Ryzhkov et al. 2007), the estimates by Tabary et al. (2008, 2009) in France, and the results of Keranen and Yllasjarvi (2008) in Finland. Ryzhkov et al. (2007) reported median values of *α* between 0.08 and 0.22 dB per degree in rain and rain/hail mixture, whereas Keranen and Yllasjarvi (2008) found most of these to be between 0.06 and 0.18 dB per degree. Tabary et al. (2008, 2009) and Keranen and Yllasjarvi (2008) claimed median values of *β* of 0.025 and 0.035 dB per degree, respectively, in their investigations. Much higher local values of *α* and *β* have been reported in melting hail in the recent study of Borowska et al. (2011).

Figures 16a and 16b show that both *α* and *β* tend to increase with increasing *Z*_{DR} in hot spots but that such a tendency is clouded by large scatter. The scatterplot of *β* versus *α* indicates a certain degree of correlation between the two parameters, as is to be expected in rain (Fig. 16c). Vulpiani et al. (2008) assumed that the ratio *β*/*α* is approximately constant and is equal to 0.3 in rain. Although the average slope of the *β*–*α* scatterplot in Fig. 16c is close to 0.3, the excessive scatter testifies to the fact that the coefficients *α* and *β* are only loosely connected. We do not exclude, however, the possibility that at least part of the excessive scatter might be attributed to the uncertainty in estimating the radial extension of hot spots and, therefore, the corresponding change in differential phase ΔΦ_{DP}(HS). Possible underestimation of hot-spot extension (due to radar miscalibration or utilization of undercorrected *Z* and *Z*_{DR}) may be associated with lower ΔΦ_{DP}(HS) and artificially high Δ*α* and Δ*β*. However, overestimation of Δ*α* and Δ*β* does not necessarily mean overestimation in path-integrated attenuation/differential attenuation because it is determined by the products Δ*α*ΔΦ_{DP}(HS) and Δ*β*ΔΦ_{DP}(HS) [see (26) and (27)], which are much more stable, as our analysis shows.

Unaccounted-for attenuation may severely restrict the capability of single-polarization radar to quantify precipitation at C band. The extent of this problem is illustrated in Fig. 17, where the fields of rain rates estimated from the measured and corrected C-band *Z* and *K*_{DP} and from *Z* measured by WSR-88D are displayed for the radar scan at 0149 UTC. Heavy underestimation of the rain rate retrieved from uncorrected *Z* is obvious practically everywhere within the storm. In the areas west of the Sidpol radar where rain rates estimated from corrected C-band *Z*, *K*_{DP}, and S band exceed 100 mm h^{−1}, the corresponding rain rates retrieved from the measured (uncorrected) C-band *Z* are less than 1 mm h^{−1}.

## 5. Conclusions

This investigation confirms the conclusions of several previous studies that the ratios *α* = *A _{h}*/

*K*

_{DP}and

*β*=

*A*

_{DP}/

*K*

_{DP}at C band can be anomalously high in hot spots and, therefore, that the hot spots should be treated separately from the rest of the storm for attenuation correction. The hot-spot method for attenuation correction originally suggested by Ryzhkov et al. (2006, 2007) has been modified and applied for two heavy-rain events (in Oklahoma and in the Chicago area) for which radar reflectivity factor exceeded 60 dB

*Z*but no hail was reported on the ground.

The proposed technique demonstrated good overall skill in correcting radar reflectivity *Z* and differential reflectivity *Z*_{DR}, as was testified to by direct comparisons with measurements by S-band radars; by the consistency of corrected *Z*, *Z*_{DR}, and *K*_{DP} in rain; by the absence of negative *Z*_{DR} in the corrected fields of differential reflectivity; and by the spatial/temporal continuity of the corrected fields of *Z* and *Z*_{DR}. For the first time, the results of attenuation correction at C band were validated through direct comparison with simultaneously collected data obtained with a nearly collocated S-band polarimetric radar in Oklahoma. In the Chicago case, the measured differential phase Φ_{DP}, which is proportional to path-integrated attenuation, exceeded 600° in some azimuthal directions and the radar reflectivity *Z* has been successfully recovered after the signal was attenuated by about 40 dB!

The values of estimated correction factors *α* and *β* in hot spots exhibit high variability and are consistent with those previously reported in the literature. We hypothesize that the high variability of the parameters *α* and *β* in hot spots at C band can be attributed to the effects of resonance scattering by large raindrops that may or may not be associated with hail. In other words, anomalous attenuation and differential attenuation may happen in pure rain as well. This is in agreement with previous findings by Carey et al. (2000) and Keenan et al. (2001), who reported anomalously high attenuation at C band in the absence of hail on the ground. However, the presence of hail aloft usually increases the supply of large drops that originate from melting hail. Recent theoretical studies of melting hail by Ryzhkov et al. (2009) indicate that shedding of water from melting hailstones leads to enhancement in the concentration of very large drops with size of about 8 mm. In other words, dry hailstones with very different sizes melt into giant raindrops with approximately the same size. Such an enhancement in the number of very large drops may not be offset by their breakup if there is plenty of melting hail in the storm.

Although the hot-spot method for attenuation correction demonstrated reasonably good performance for the two cases of heavy rain investigated in this study, further refinement of the algorithm and more validation studies are needed. There are indications that the methods for attenuation correction based on differential phase Φ_{DP} may fail if Φ_{DP} becomes excessively erratic in the shadow of attenuating cells as a result of a loss of correlation between orthogonally polarized signals and the effects of nonuniform beam filling (Ryzhkov 2007). In this situation, which can be very common in the presence of melting hail (Borowska et al. 2011), differential phase may not be usable for attenuation correction at all at shorter wavelengths and approaches using different principles may have to be explored.

## Acknowledgments

This research was supported by the National Research Foundation of Korea (NRF) through a grant provided by the Korean Ministry of Education, Science and Technology(MEST) in 2010 (Grant K20607010000). Authors A. Ryzhkov and P. Zhang are supported by the NOAA/Office of Oceanic and Atmospheric Research under NOAA–University of Oklahoma Cooperative Agreement NA17RJ1227, U.S. Department of Commerce. We are very grateful to Prof. R. Palmer, Dr. B. Cheong, and R. Kelly from the Atmospheric Radar Research Center at the University of Oklahoma for providing C-band polarimetric data from the recently established OU PRIME radar that was manufactured by Enterprise Electronics Corporation. Also, Drs. D. Zrnić and L. Borowska reviewed the manuscript and helped to clarify several aspects of our analysis. The authors also appreciate very constructive comments and suggestions by the anonymous reviewers.

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Scatterplots of (top) *A _{H}* and (bottom)

*A*

_{DP}vs

*K*

_{DP}in pure rain at C band for (a),(c) all

*Z*

_{DR}and (b),(d)

*Z*

_{DR}< 3dB. Radar variables are computed from 25 920 DSDs measured in Oklahoma.

Citation: Journal of Applied Meteorology and Climatology 50, 1; 10.1175/2010JAMC2258.1

Scatterplots of (top) *A _{H}* and (bottom)

*A*

_{DP}vs

*K*

_{DP}in pure rain at C band for (a),(c) all

*Z*

_{DR}and (b),(d)

*Z*

_{DR}< 3dB. Radar variables are computed from 25 920 DSDs measured in Oklahoma.

Citation: Journal of Applied Meteorology and Climatology 50, 1; 10.1175/2010JAMC2258.1

Scatterplots of (top) *A _{H}* and (bottom)

*A*

_{DP}vs

*K*

_{DP}in pure rain at C band for (a),(c) all

*Z*

_{DR}and (b),(d)

*Z*

_{DR}< 3dB. Radar variables are computed from 25 920 DSDs measured in Oklahoma.

Citation: Journal of Applied Meteorology and Climatology 50, 1; 10.1175/2010JAMC2258.1

Scatterplot of the ratio *A*_{DP}/*K*_{DP} vs *Z*_{DR} in pure rain at C band. Radar variables are computed from 25 920 DSD measured in Oklahoma.

Citation: Journal of Applied Meteorology and Climatology 50, 1; 10.1175/2010JAMC2258.1

Scatterplot of the ratio *A*_{DP}/*K*_{DP} vs *Z*_{DR} in pure rain at C band. Radar variables are computed from 25 920 DSD measured in Oklahoma.

Citation: Journal of Applied Meteorology and Climatology 50, 1; 10.1175/2010JAMC2258.1

Scatterplot of the ratio *A*_{DP}/*K*_{DP} vs *Z*_{DR} in pure rain at C band. Radar variables are computed from 25 920 DSD measured in Oklahoma.

Citation: Journal of Applied Meteorology and Climatology 50, 1; 10.1175/2010JAMC2258.1

Conceptual plot illustrating the hot spot between ranges *r*_{1} and *r*_{2} in the radial profiles of *Z* and Φ_{DP}.

Citation: Journal of Applied Meteorology and Climatology 50, 1; 10.1175/2010JAMC2258.1

Conceptual plot illustrating the hot spot between ranges *r*_{1} and *r*_{2} in the radial profiles of *Z* and Φ_{DP}.

Citation: Journal of Applied Meteorology and Climatology 50, 1; 10.1175/2010JAMC2258.1

Conceptual plot illustrating the hot spot between ranges *r*_{1} and *r*_{2} in the radial profiles of *Z* and Φ_{DP}.

Citation: Journal of Applied Meteorology and Climatology 50, 1; 10.1175/2010JAMC2258.1

Scatterplots of *α*_{0} vs *β*_{0} simulated from disdrometer data in Oklahoma for *T* = 10° and 20°C for *Z*_{DR} varying between 0.5 and 3.0 dB. Dashed line depicts the dependence *α*_{0} = 3.33*β*_{0} from Vulpiani et al. (2008).

Citation: Journal of Applied Meteorology and Climatology 50, 1; 10.1175/2010JAMC2258.1

Scatterplots of *α*_{0} vs *β*_{0} simulated from disdrometer data in Oklahoma for *T* = 10° and 20°C for *Z*_{DR} varying between 0.5 and 3.0 dB. Dashed line depicts the dependence *α*_{0} = 3.33*β*_{0} from Vulpiani et al. (2008).

Citation: Journal of Applied Meteorology and Climatology 50, 1; 10.1175/2010JAMC2258.1

Scatterplots of *α*_{0} vs *β*_{0} simulated from disdrometer data in Oklahoma for *T* = 10° and 20°C for *Z*_{DR} varying between 0.5 and 3.0 dB. Dashed line depicts the dependence *α*_{0} = 3.33*β*_{0} from Vulpiani et al. (2008).

Citation: Journal of Applied Meteorology and Climatology 50, 1; 10.1175/2010JAMC2258.1

Retrieved radial profiles of specific attenuation *A _{h}* for

*α*

_{0}= 0.06 dB per degree, different values of Δ

*α*in a hot spot, and different locations of a hot spot along the propagation path. The true value of Δ

*α*in a hot spot is equal to 0.04 dB per degree. Here,

*L*/

*R*is the ratio of the left and right sides of Eq. (21). The retrieved profile of

*A*for Δ

_{h}*α*= 0.04 dB per degree coincides with the “true” profile of

*A*.

_{h}Citation: Journal of Applied Meteorology and Climatology 50, 1; 10.1175/2010JAMC2258.1

Retrieved radial profiles of specific attenuation *A _{h}* for

*α*

_{0}= 0.06 dB per degree, different values of Δ

*α*in a hot spot, and different locations of a hot spot along the propagation path. The true value of Δ

*α*in a hot spot is equal to 0.04 dB per degree. Here,

*L*/

*R*is the ratio of the left and right sides of Eq. (21). The retrieved profile of

*A*for Δ

_{h}*α*= 0.04 dB per degree coincides with the “true” profile of

*A*.

_{h}Citation: Journal of Applied Meteorology and Climatology 50, 1; 10.1175/2010JAMC2258.1

Retrieved radial profiles of specific attenuation *A _{h}* for

*α*

_{0}= 0.06 dB per degree, different values of Δ

*α*in a hot spot, and different locations of a hot spot along the propagation path. The true value of Δ

*α*in a hot spot is equal to 0.04 dB per degree. Here,

*L*/

*R*is the ratio of the left and right sides of Eq. (21). The retrieved profile of

*A*for Δ

_{h}*α*= 0.04 dB per degree coincides with the “true” profile of

*A*.

_{h}Citation: Journal of Applied Meteorology and Climatology 50, 1; 10.1175/2010JAMC2258.1

Fields of measured (top) *Z*, (middle) *Z*_{DR}, and (bottom) Φ_{DP} at (left) C and (right) S bands for the storm at 0309 UTC 10 Mar 2009. Here, the elevation (C band) = 0.41° and the elevation (S band) = 0.48°. The C-band radar is at *X* = 0, *Y* = 0. The areas of visible negative bias of *Z* caused by attenuation at C band are marked as A and B in the top-left panel. In the left panels, a line indicates the azimuthal direction for which the RHI plot in Fig. 8 is displayed.

Citation: Journal of Applied Meteorology and Climatology 50, 1; 10.1175/2010JAMC2258.1

Fields of measured (top) *Z*, (middle) *Z*_{DR}, and (bottom) Φ_{DP} at (left) C and (right) S bands for the storm at 0309 UTC 10 Mar 2009. Here, the elevation (C band) = 0.41° and the elevation (S band) = 0.48°. The C-band radar is at *X* = 0, *Y* = 0. The areas of visible negative bias of *Z* caused by attenuation at C band are marked as A and B in the top-left panel. In the left panels, a line indicates the azimuthal direction for which the RHI plot in Fig. 8 is displayed.

Citation: Journal of Applied Meteorology and Climatology 50, 1; 10.1175/2010JAMC2258.1

Fields of measured (top) *Z*, (middle) *Z*_{DR}, and (bottom) Φ_{DP} at (left) C and (right) S bands for the storm at 0309 UTC 10 Mar 2009. Here, the elevation (C band) = 0.41° and the elevation (S band) = 0.48°. The C-band radar is at *X* = 0, *Y* = 0. The areas of visible negative bias of *Z* caused by attenuation at C band are marked as A and B in the top-left panel. In the left panels, a line indicates the azimuthal direction for which the RHI plot in Fig. 8 is displayed.

Citation: Journal of Applied Meteorology and Climatology 50, 1; 10.1175/2010JAMC2258.1

Composite RHI plot of (top) *Z*, (middle top) *Z*_{DR}, (middle bottom) Φ_{DP}, and (bottom) *ρ*_{hv} at (left) C and (right) S bands for the storm at 0309 UTC 10 Mar 2009. Here, azimuth (C band) = 285.5° and azimuth (S band) = 280°. The azimuthal direction of the vertical cross section is shown in Fig. 7.

Citation: Journal of Applied Meteorology and Climatology 50, 1; 10.1175/2010JAMC2258.1

Composite RHI plot of (top) *Z*, (middle top) *Z*_{DR}, (middle bottom) Φ_{DP}, and (bottom) *ρ*_{hv} at (left) C and (right) S bands for the storm at 0309 UTC 10 Mar 2009. Here, azimuth (C band) = 285.5° and azimuth (S band) = 280°. The azimuthal direction of the vertical cross section is shown in Fig. 7.

Citation: Journal of Applied Meteorology and Climatology 50, 1; 10.1175/2010JAMC2258.1

Composite RHI plot of (top) *Z*, (middle top) *Z*_{DR}, (middle bottom) Φ_{DP}, and (bottom) *ρ*_{hv} at (left) C and (right) S bands for the storm at 0309 UTC 10 Mar 2009. Here, azimuth (C band) = 285.5° and azimuth (S band) = 280°. The azimuthal direction of the vertical cross section is shown in Fig. 7.

Citation: Journal of Applied Meteorology and Climatology 50, 1; 10.1175/2010JAMC2258.1

Fields of (left) corrected C-band *Z* and *Z*_{DR} and (right) measured S-band *Z* and *Z*_{DR} for the PPI in Fig. 7. The elevation (C band) = 0.41° and the elevation (S band) = 0.48°. The dashed line in the left panels indicates the azimuthal direction with strong partial blockage of the OU PRIME beam.

Citation: Journal of Applied Meteorology and Climatology 50, 1; 10.1175/2010JAMC2258.1

Fields of (left) corrected C-band *Z* and *Z*_{DR} and (right) measured S-band *Z* and *Z*_{DR} for the PPI in Fig. 7. The elevation (C band) = 0.41° and the elevation (S band) = 0.48°. The dashed line in the left panels indicates the azimuthal direction with strong partial blockage of the OU PRIME beam.

Citation: Journal of Applied Meteorology and Climatology 50, 1; 10.1175/2010JAMC2258.1

Fields of (left) corrected C-band *Z* and *Z*_{DR} and (right) measured S-band *Z* and *Z*_{DR} for the PPI in Fig. 7. The elevation (C band) = 0.41° and the elevation (S band) = 0.48°. The dashed line in the left panels indicates the azimuthal direction with strong partial blockage of the OU PRIME beam.

Citation: Journal of Applied Meteorology and Climatology 50, 1; 10.1175/2010JAMC2258.1

Scatterplots of the differences *Z*(S band) − *Z*(C band) vs *Z*_{DR}(S band) − *Z*_{DR}(C band) (top) before and (bottom) after attenuation correction for the part of the radar scan in Figs. 7 and 9 for which measured (uncorrected) *Z*_{DR} is lower than −1 dB.

Citation: Journal of Applied Meteorology and Climatology 50, 1; 10.1175/2010JAMC2258.1

Scatterplots of the differences *Z*(S band) − *Z*(C band) vs *Z*_{DR}(S band) − *Z*_{DR}(C band) (top) before and (bottom) after attenuation correction for the part of the radar scan in Figs. 7 and 9 for which measured (uncorrected) *Z*_{DR} is lower than −1 dB.

Citation: Journal of Applied Meteorology and Climatology 50, 1; 10.1175/2010JAMC2258.1

Scatterplots of the differences *Z*(S band) − *Z*(C band) vs *Z*_{DR}(S band) − *Z*_{DR}(C band) (top) before and (bottom) after attenuation correction for the part of the radar scan in Figs. 7 and 9 for which measured (uncorrected) *Z*_{DR} is lower than −1 dB.

Citation: Journal of Applied Meteorology and Climatology 50, 1; 10.1175/2010JAMC2258.1

Radial profiles of the differential phase (top) that was measured by the Sidpol radar, (middle) after downward phase shift, and (bottom) after editing, unfolding, and smoothing at 0149 UTC 5 Aug 2008 at azimuth = 257.2° and elevation = 0.5°.

Citation: Journal of Applied Meteorology and Climatology 50, 1; 10.1175/2010JAMC2258.1

Radial profiles of the differential phase (top) that was measured by the Sidpol radar, (middle) after downward phase shift, and (bottom) after editing, unfolding, and smoothing at 0149 UTC 5 Aug 2008 at azimuth = 257.2° and elevation = 0.5°.

Citation: Journal of Applied Meteorology and Climatology 50, 1; 10.1175/2010JAMC2258.1

Radial profiles of the differential phase (top) that was measured by the Sidpol radar, (middle) after downward phase shift, and (bottom) after editing, unfolding, and smoothing at 0149 UTC 5 Aug 2008 at azimuth = 257.2° and elevation = 0.5°.

Citation: Journal of Applied Meteorology and Climatology 50, 1; 10.1175/2010JAMC2258.1

Fields of (a) measured and (c) corrected *Z* at C band, (b) Φ_{DP} at C band, and (d) *Z* measured by S-band radar (KLOT WSR-88D) at 0149 UTC 5 Aug 2008. The antenna elevation is 0.5°. The Sidpol radar is situated at *X* = 0, *Y* = 0 km. The star marks the location of the WSR-88D. The straight line indicates azimuth 257.2° (see Figs. 11 and 14).

Citation: Journal of Applied Meteorology and Climatology 50, 1; 10.1175/2010JAMC2258.1

Fields of (a) measured and (c) corrected *Z* at C band, (b) Φ_{DP} at C band, and (d) *Z* measured by S-band radar (KLOT WSR-88D) at 0149 UTC 5 Aug 2008. The antenna elevation is 0.5°. The Sidpol radar is situated at *X* = 0, *Y* = 0 km. The star marks the location of the WSR-88D. The straight line indicates azimuth 257.2° (see Figs. 11 and 14).

Citation: Journal of Applied Meteorology and Climatology 50, 1; 10.1175/2010JAMC2258.1

Fields of (a) measured and (c) corrected *Z* at C band, (b) Φ_{DP} at C band, and (d) *Z* measured by S-band radar (KLOT WSR-88D) at 0149 UTC 5 Aug 2008. The antenna elevation is 0.5°. The Sidpol radar is situated at *X* = 0, *Y* = 0 km. The star marks the location of the WSR-88D. The straight line indicates azimuth 257.2° (see Figs. 11 and 14).

Citation: Journal of Applied Meteorology and Climatology 50, 1; 10.1175/2010JAMC2258.1

Validation of *Z* and *Z*_{DR} attenuation correction for the radar scan in Fig. 12 using self-consistency among *Z*, *Z*_{DR}, and *K*_{DP} in rain. (a) Scatterplot of 10 log(*K*_{DP}) − *Z _{h}* vs

*Z*

_{DR}after

*Z*and

*Z*

_{DR}are corrected for attenuation using the hot spot method. (b) Median values of the estimated difference between 10 log(

*K*

_{DP}) and

*Z*as functions of

_{h}*Z*

_{DR}for uncorrected

*Z*(asterisks) and corrected

_{h}*Z*(diamonds) as compared with the corresponding theoretical curves based on simulations at C band from disdrometer data in Oklahoma (solid line) and from Gourley et al. (2006) (dashed line).

_{h}Citation: Journal of Applied Meteorology and Climatology 50, 1; 10.1175/2010JAMC2258.1

Validation of *Z* and *Z*_{DR} attenuation correction for the radar scan in Fig. 12 using self-consistency among *Z*, *Z*_{DR}, and *K*_{DP} in rain. (a) Scatterplot of 10 log(*K*_{DP}) − *Z _{h}* vs

*Z*

_{DR}after

*Z*and

*Z*

_{DR}are corrected for attenuation using the hot spot method. (b) Median values of the estimated difference between 10 log(

*K*

_{DP}) and

*Z*as functions of

_{h}*Z*

_{DR}for uncorrected

*Z*(asterisks) and corrected

_{h}*Z*(diamonds) as compared with the corresponding theoretical curves based on simulations at C band from disdrometer data in Oklahoma (solid line) and from Gourley et al. (2006) (dashed line).

_{h}Citation: Journal of Applied Meteorology and Climatology 50, 1; 10.1175/2010JAMC2258.1

Validation of *Z* and *Z*_{DR} attenuation correction for the radar scan in Fig. 12 using self-consistency among *Z*, *Z*_{DR}, and *K*_{DP} in rain. (a) Scatterplot of 10 log(*K*_{DP}) − *Z _{h}* vs

*Z*

_{DR}after

*Z*and

*Z*

_{DR}are corrected for attenuation using the hot spot method. (b) Median values of the estimated difference between 10 log(

*K*

_{DP}) and

*Z*as functions of

_{h}*Z*

_{DR}for uncorrected

*Z*(asterisks) and corrected

_{h}*Z*(diamonds) as compared with the corresponding theoretical curves based on simulations at C band from disdrometer data in Oklahoma (solid line) and from Gourley et al. (2006) (dashed line).

_{h}Citation: Journal of Applied Meteorology and Climatology 50, 1; 10.1175/2010JAMC2258.1

Radial profiles of (a) Φ_{DP} (thick line) and *ρ*_{hv} (thin line), (b) measured (thin line) and corrected (thick line) *Z*, (c) measured (thin line) and corrected (thick line) *Z*_{DR}, and (d) *R*(*Z*) (thin line) after *Z* is corrected and *R*(*K*_{DP}) (thick line) at azimuth 257.2° at 0149 UTC 5 Aug 2008. Attenuation correction of *Z* is performed using the hot-spot algorithm with constraint condition (21).

Citation: Journal of Applied Meteorology and Climatology 50, 1; 10.1175/2010JAMC2258.1

Radial profiles of (a) Φ_{DP} (thick line) and *ρ*_{hv} (thin line), (b) measured (thin line) and corrected (thick line) *Z*, (c) measured (thin line) and corrected (thick line) *Z*_{DR}, and (d) *R*(*Z*) (thin line) after *Z* is corrected and *R*(*K*_{DP}) (thick line) at azimuth 257.2° at 0149 UTC 5 Aug 2008. Attenuation correction of *Z* is performed using the hot-spot algorithm with constraint condition (21).

Citation: Journal of Applied Meteorology and Climatology 50, 1; 10.1175/2010JAMC2258.1

Radial profiles of (a) Φ_{DP} (thick line) and *ρ*_{hv} (thin line), (b) measured (thin line) and corrected (thick line) *Z*, (c) measured (thin line) and corrected (thick line) *Z*_{DR}, and (d) *R*(*Z*) (thin line) after *Z* is corrected and *R*(*K*_{DP}) (thick line) at azimuth 257.2° at 0149 UTC 5 Aug 2008. Attenuation correction of *Z* is performed using the hot-spot algorithm with constraint condition (21).

Citation: Journal of Applied Meteorology and Climatology 50, 1; 10.1175/2010JAMC2258.1

Fields of (a) corrected *Z*, (b) measured and (c),(d) corrected *Z*_{DR}, (e) Φ_{DP}, and (f) *ρ*_{hv} at 0124 UTC 5 Aug 2008. The antenna elevation is 0.5°.

Citation: Journal of Applied Meteorology and Climatology 50, 1; 10.1175/2010JAMC2258.1

Fields of (a) corrected *Z*, (b) measured and (c),(d) corrected *Z*_{DR}, (e) Φ_{DP}, and (f) *ρ*_{hv} at 0124 UTC 5 Aug 2008. The antenna elevation is 0.5°.

Citation: Journal of Applied Meteorology and Climatology 50, 1; 10.1175/2010JAMC2258.1

Fields of (a) corrected *Z*, (b) measured and (c),(d) corrected *Z*_{DR}, (e) Φ_{DP}, and (f) *ρ*_{hv} at 0124 UTC 5 Aug 2008. The antenna elevation is 0.5°.

Citation: Journal of Applied Meteorology and Climatology 50, 1; 10.1175/2010JAMC2258.1

Scatterplots of the measured parameters (a) *α* and (b) *β* vs maximal *Z*_{DR} in the hot spotat elevation 0.5° and (c) scatterplot of *β* vs *α* in the hot spots.

Citation: Journal of Applied Meteorology and Climatology 50, 1; 10.1175/2010JAMC2258.1

Scatterplots of the measured parameters (a) *α* and (b) *β* vs maximal *Z*_{DR} in the hot spotat elevation 0.5° and (c) scatterplot of *β* vs *α* in the hot spots.

Citation: Journal of Applied Meteorology and Climatology 50, 1; 10.1175/2010JAMC2258.1

Scatterplots of the measured parameters (a) *α* and (b) *β* vs maximal *Z*_{DR} in the hot spotat elevation 0.5° and (c) scatterplot of *β* vs *α* in the hot spots.

Citation: Journal of Applied Meteorology and Climatology 50, 1; 10.1175/2010JAMC2258.1

Fields of rain rates obtained from (a) measured and (b) corrected *Z* at C band, (c) *K*_{DP}, and (d) S-band *Z* at 0149 UTC 5 Aug 2008. A star indicates the location of the WSR-88D.

Citation: Journal of Applied Meteorology and Climatology 50, 1; 10.1175/2010JAMC2258.1

Fields of rain rates obtained from (a) measured and (b) corrected *Z* at C band, (c) *K*_{DP}, and (d) S-band *Z* at 0149 UTC 5 Aug 2008. A star indicates the location of the WSR-88D.

Citation: Journal of Applied Meteorology and Climatology 50, 1; 10.1175/2010JAMC2258.1

Fields of rain rates obtained from (a) measured and (b) corrected *Z* at C band, (c) *K*_{DP}, and (d) S-band *Z* at 0149 UTC 5 Aug 2008. A star indicates the location of the WSR-88D.

Citation: Journal of Applied Meteorology and Climatology 50, 1; 10.1175/2010JAMC2258.1