The Impact of Reflectivity Gradients on the Performance of Range-Oversampling Processing

Christopher D. Curtis aCooperative Institute for Mesoscale Meteorological Studies, University of Oklahoma, Norman, Oklahoma
bNOAA/OAR/National Severe Storms Laboratory, Norman, Oklahoma

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Sebastián M. Torres aCooperative Institute for Mesoscale Meteorological Studies, University of Oklahoma, Norman, Oklahoma
bNOAA/OAR/National Severe Storms Laboratory, Norman, Oklahoma

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Abstract

Range-oversampling processing is a technique that can be used to lower the variance of radar-variable estimates, reduce radar update times, or a mixture of both. There are two main assumptions for using range-oversampling processing: accurate knowledge of the range correlation and uniform reflectivity in the radar resolution volume. The first assumption has been addressed in previous research; this work focuses on the uniform reflectivity assumption. Earlier research shows that significant reflectivity gradients can occur in storms; we utilized those results to develop realistic simulations of radar returns that include effects of reflectivity gradients in range. An important consideration when using range-oversampling processing is the resulting change in the range weighting function. The range weighting function can change for different types of range-oversampling processing, and some techniques, such as adaptive pseudowhitening, can lead to different range weighting functions at each range gate. To quantify the possible effects of differing range weighting functions in the presence of reflectivity gradients, we developed simulations to examine varying types of range-oversampling processing with two receiver filters: a matched receiver filter and a wider-bandwidth receiver filter (as recommended for use with range oversampling). Simulation results show that differences in range weighting functions are the only contributor to differences in radar reflectivity measurements. Results from real weather data demonstrate that the reflectivity gradients that occur in typical severe storms do not cause significant changes in reflectivity measurements and that the benefits from range-oversampling processing outweigh the possible isolated effects from large reflectivity gradients.

© 2021 American Meteorological Society. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

Corresponding author: Christopher Curtis, chris.curtis@noaa.gov

Abstract

Range-oversampling processing is a technique that can be used to lower the variance of radar-variable estimates, reduce radar update times, or a mixture of both. There are two main assumptions for using range-oversampling processing: accurate knowledge of the range correlation and uniform reflectivity in the radar resolution volume. The first assumption has been addressed in previous research; this work focuses on the uniform reflectivity assumption. Earlier research shows that significant reflectivity gradients can occur in storms; we utilized those results to develop realistic simulations of radar returns that include effects of reflectivity gradients in range. An important consideration when using range-oversampling processing is the resulting change in the range weighting function. The range weighting function can change for different types of range-oversampling processing, and some techniques, such as adaptive pseudowhitening, can lead to different range weighting functions at each range gate. To quantify the possible effects of differing range weighting functions in the presence of reflectivity gradients, we developed simulations to examine varying types of range-oversampling processing with two receiver filters: a matched receiver filter and a wider-bandwidth receiver filter (as recommended for use with range oversampling). Simulation results show that differences in range weighting functions are the only contributor to differences in radar reflectivity measurements. Results from real weather data demonstrate that the reflectivity gradients that occur in typical severe storms do not cause significant changes in reflectivity measurements and that the benefits from range-oversampling processing outweigh the possible isolated effects from large reflectivity gradients.

© 2021 American Meteorological Society. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

Corresponding author: Christopher Curtis, chris.curtis@noaa.gov

1. Introduction

On weather radars, range-oversampling processing can be used to provide radar data with faster updates and/or reduced variance (Torres and Zrnić 2003). For example, it has been used to achieve faster updates on the National Weather Radar Testbed phased-array radar (Curtis and Torres 2011) and to improve estimates of the polarimetric variables using data from the National Severe Storms Laboratory’s KOUN research radar (Curtis and Torres 2014). In a nutshell, oversampling in range at a rate that is L times faster than the inverse of the transmitter pulse width results in received baseband complex samples [commonly referred to as the in-phase and quadrature-phase (IQ) samples] that are correlated in range. Traditionally, a wider-bandwidth receiver filter (L times wider than a conventional matched filter) has been used, but a conventional matched filter can also be utilized effectively for range-oversampling processing. Details about both approaches can be found in Torres and Curtis (2020). After the receiver filter is applied, a given set of L samples in range can be decorrelated using a linear transformation; the resulting transformed IQ samples are then used to estimate sample-time correlations for each of the L oversampled gates. Next, these L correlation estimates can be averaged, and the result is used to obtain radar variables with reduced variance. At high signal-to-noise ratios (SNR), the variance is reduced by a factor of L relative to conventional sampling and processing (Torres and Zrnić 2003). It is important to note that collecting L times more samples via range oversampling does not result in longer dwell times. That is, relative to conventional sampling and processing, the use of range-oversampling processing with the same dwell times results in radar data with reduced variance. Alternatively, shorter dwell times can be used to obtain radar data with the same variance. Thus, by changing the dwell times, a trade-off can be found between faster updates and reduced variance of radar-variable estimates.

Range-oversampling processing can be implemented adaptively to ensure that the variance of a given radar-variable estimate is minimized for particular weather signal characteristics (Curtis and Torres 2011, 2014). This is important because the application of the linear transformation may increase the noise power; thus, the adaptive implementation finds the best trade-off between noise enhancement and variance reduction. Thus, at low SNRs, the linear transformation obtained by the adaptive algorithm approaches a conventional matched filter and results in little-to-no variance reduction. At high SNRs, the linear transformation approaches a whitening transformation (complete decorrelation) and results in a reduction of variance by a factor of L. There are two main assumptions for using range-oversampling processing: accurate knowledge of the range correlation of IQ samples and uniform reflectivity in the radar resolution volume. We have previously addressed the importance of accurately measuring (or knowing) the range correlation of IQ samples and quantified the impacts of using a mismatched range correlation in the performance of range-oversampling processing (Torres and Curtis 2012). The main drawbacks of adaptive range-oversampling processing are increased computational complexity and some loss in range resolution. The trade-offs between variance reduction and range resolution have been explored in greater detail in Torres and Curtis (2015). In this paper, we explore the effects of using range-oversampling processing when the reflectivity is not uniform in the radar resolution volume. In other words, we study the effects of reflectivity gradients on the performance of range-oversampling processing.

Reflectivity is the radar term for the backscattering cross section per unit volume (Doviak and Zrnić 1993); it depends on the type, size, shape, aspect, and number of hydrometeors in the radar resolution volume. In general, the hydrometeor characteristics change from one location to another, so the reflectivity also changes in all radar spatial dimensions: azimuth, elevation, and range. A change of reflectivity in one or more dimensions creates a nonzero reflectivity gradient. In this work, we focus on reflectivity gradients in the range dimension because they have the potential to affect the implementation and performance of range-oversampling processing. For simplicity, we use the term “reflectivity gradient” to refer to the magnitude of the projection of the reflectivity gradient vector in the range dimension.

In the early years of weather radar, reflectivity gradients were studied to better understand biases in reflectivity estimates from using different types of radar receivers. We can take advantage of that research to help focus our analyses on the type and range of reflectivity gradients that may be realistically present when sampling different storm types. For example, Rogers (1971) stated that “reflectivity gradients exceeding 20 dB km−1 are not uncommon in echoes from stratiform or convective rain”; however, they did not discuss the frequency of occurrence of these “not uncommon” gradients. One study from Mueller (1977) based on severe thunderstorms found that “gradients as large as 50 dB km−1 occur,” but the study also showed that “over 80% of all observations occurred with the absolute magnitude of the gradient less than 12 dB km−1” and that “some 50–60% occurred with gradients less than 6 dB km−1.” This tells us that whereas larger reflectivity gradients do occur, the vast majority are below 20 dB km−1. Another study from a summer hailstorm found “gradients from 0–30 dB km−1 with a few around 35 dB km−1” (Scarchilli et al. 1986). As far as the nature of the gradients, they also reported that “reflectivity expressed in dBZ varies linearly in space.” In more recent research, Kurdzo et al. (2014) show an observation of a “convective cell with a strong reflectivity gradient” of 45 dB km−1. For this study, we chose to examine linear dBZ gradients from 0 to 50 dB km−1.

The presence of significant reflectivity gradients reported in the literature confirms that, in practice, the uniform reflectivity assumption for range-oversampling processing is often violated. Thus, it is important to quantify the impacts of reflectivity gradients on range-oversampling processing performance. In section 2, we examine the effects of reflectivity gradients on radar reflectivity measurements. This leads to the development of simulations that are used to systematically study the impacts of reflectivity gradients on range-oversampling processing in section 3. Section 4 looks at the impacts on real data in light of simulation results, and section 5 summarizes our results and conclusions.

2. Effects of reflectivity gradients on radar reflectivity measurements

To better understand the effects of reflectivity gradients, it is useful to look at the contributions to the received radar signals from individual hydrometeors. The two-way effective antenna radiation pattern weights contributions from individual hydrometeors based on their azimuth and elevation position with respect to the direction of the radar beam. Our focus is on the range dimension, where the range weighting function (RWF) weights contributions from individual hydrometeors based on their range location with respect to the center of the resolution volume. The radar resolution volume is conventionally determined by contours corresponding to the 6-dB width of the two-way effective antenna radiation pattern and the 6-dB width of the RWF (Doviak and Zrnić 1993).

In simplified versions of the weather radar equation, the reflectivity is often assumed to be constant and is removed from the integrals that represent the spatial weighting. An example of this type of simplified radar equation is
P¯(r0,θ0,ϕ0)=Cη(r0,θ0,ϕ0)r020r2|WS(r)|2dr×02π02πf4(θ,ϕ)sinθdθdϕ.
In this equation, P¯ is the average received signal power corresponding to the radar resolution volume centered at r0, θ0, and ϕ0 in spherical coordinates; C is a system constant that includes the transmit peak power, the antenna gain, the radar wavelength, and system and propagation losses; η(r0, θ0, ϕ0) is the radar reflectivity at the center of the resolution volume; WS is the RWF centered at r0; and f4 is the two-way effective antenna radiation pattern centered at (θ0, ϕ0). In simpler versions of the radar equation like Eq. (1), we cannot account for the impacts that the RWF and the antenna radiation pattern can have on radar measurements in the presence of reflectivity gradients. A more accurate version of the radar equation is given by
P¯(r0,θ0,ϕ0)=Cr020r202π02πη(r,θ,ϕ)f4(θ,ϕ)|WS(r)|2sinθdθdϕdr,
where the reflectivity is inside the integrals because it varies as a function of r, θ, and ϕ. This formulation of the radar equation allows us to consider the effects of reflectivity gradients in all dimensions. In this work, we focus on the effects of gradients in the range dimension and seek to understand how different RWFs can result in different radar measurements when the reflectivity is not constant.

Even with conventional sampling (i.e., no range oversampling and a receiver matched filter) and processing, the presence of reflectivity gradients can impact the interpretation of radar data, where the common assumption is that each observation (i.e., each radar-variable estimate for a given range bin) relates to average properties of all the hydrometeors in the radar resolution volume whose location in range coincides with the peak of the RWF (Johnston et al. 2002). When the reflectivity is not constant, data may be assigned to the incorrect range, and radar observations may reflect the properties of only a fraction of the hydrometeors in the resolution volume. This problem gets compounded when the RWF is not the same for all range bins.

In general, the RWF is determined by the shape of the transmitted pulse, the receiver impulse response, and the effective linear transformation from range-oversampling processing. Since the transmitted pulse and receiver impulse response are normally fixed, our focus will be on the effects from range-oversampling processing, which produces different RWFs when compared with conventional matched-filter processing (Torres and Curtis 2012). Moreover, for adaptive pseudowhitening (Curtis and Torres 2011, 2014, 2017; Torres and Curtis 2020), the RWF can potentially change at every range location. This is because adaptive pseudowhitening uses signal characteristics to find a single parameter, p, that determines the appropriate linear transformation. This linear transformation is calculated to minimize the variance of radar-variable estimates and also determines the RWFs (Torres and Curtis 2012). As p varies from 0 to 1, the pseudowhitening linear transformation varies from matched-filter-like to a whitening transformation. Because different linear transformations result in different RWFs, as p changes, the RWF also changes. Figure 1 shows examples of different RWFs for a few values of p using the Weather Surveillance Radar-1988 Doppler (WSR-88D) transmit pulse with a receiver bandwidth that is L times wider than the bandwidth of the pulse (typically τ−1). The RWF corresponding to p = 0 is the closest to the RWF corresponding to conventional sampling and processing using a matched receiver filter. A matched receiver filter minimizes the variance at low SNRs, but as the SNR increases, we can achieve a lower variance of estimates by decorrelating the samples in range. In general, as p increases, the decorrelation increases, which also leads to an increase in the 6-dB width of the RWF. Hence, the whitening transformation (p = 1) results in the widest RWF.

Fig. 1.
Fig. 1.

Examples of area-normalized RWFs for conventional processing using a matched receiver filter (CP) and range-oversampling processing using a wide-bandwidth receiver filter (ROP) and select linear transformations corresponding to values of p from 0 (matched filter-like transformation) to 1 (whitening transformation). The range is relative to the center of the radar resolution volume.

Citation: Journal of Atmospheric and Oceanic Technology 38, 8; 10.1175/JTECH-D-20-0201.1

In the presence of reflectivity gradients, radar measurements can change in different ways for different RWFs. The next sections will explore by how much radar measurements change in the presence of reflectivity gradients when using range-oversampling processing in comparison with conventional processing.

3. Simulations and analysis

In this section, we use simulations to systematically study the impact of reflectivity gradients on the performance of range-oversampling processing with two different receiver filters. In modern radar receivers, received complex signals are downconverted (first stage) and digitized at the radar’s intermediate frequency (IF), and the receiver filter and second downconversion stage (to baseband) are implemented digitally. Thus, to include the effects of the receiver filter, our simulations start with a set of radially aligned independent scattering centers with range spacing Δr given by the IF. That is, Δr = c(2fIF)−1, where c is the speed of light and fIF is the IF. In this simulation, each scattering center represents the combined contributions of all scatterers in a “slab” of space with dimensions given by the extent of the radar resolution volume in azimuth and elevation and the range sampling spacing in range. A flow diagram is shown in Fig. 2.

Fig. 2.
Fig. 2.

Simulation flowchart. Inputs to each processing block are listed to the left of the block. The notation f() is loosely used to make the dependency of inputs on basic simulation parameters explicit. The dimensionality of nonscalar inputs is indicated in parentheses. The dimensionalities of outputs are listed to the right of each processing block. The total number of scattering centers is G = (L − 1)D + F + Np − 1, where Np is the length of the transmit pulse, F is the length of the receiver impulse response, D is the decimation factor, and L is the range oversampling factor; M is the number of samples in the dwell. An asterisk next to an operation is used to indicate that the same operation is performed on each row of the input matrix. Separate processing branches are needed for conventional processing (right branch) and ROP (left branch). Each simulation run produces a single signal-power estimate. For this work, the simulation is repeated for 20 000 realizations.

Citation: Journal of Atmospheric and Oceanic Technology 38, 8; 10.1175/JTECH-D-20-0201.1

For each scattering center, we use the procedure in Curtis (2018) to simulate M samples of baseband IQ signals with sample-time spacing given by the pulse repetition time (PRT) and with arbitrary signal characteristics defined by the mean signal power S, the mean Doppler velocity υ, the spectrum width συ, and the SNR. For this work, and without loss of generality, we adopted the following signal characteristics: S = 1, υ = 0 m s−1, συ = 2 m s−1, and SNR = 30 dB. We also adopted the acquisition parameters of the lowest elevation angle of the WSR-88D volume coverage pattern 12. That is, the maximum unambiguous velocity is υa = 8.5 m s−1, and the number of samples per dwell is M = 15. The IF is selected so that, after decimation to baseband, the resulting range-sampling spacing is 50 m, which corresponds to a range-oversampling factor of L = 5 for the conventional WSR-88D radar-data range spacing of 250 m. To match a typical configuration of the digital receiver in the WSR-88D, we adopted a base-band decimation factor D = 32, which results in fIF = DLc/(500 m) = 95.92 MHz and Δr ≈ 1.56 m.

Next, the scattering-center IQ signals are manipulated along the range dimension to impose the desired reflectivity gradients and to simulate the effects of the transmitted pulse, the receiver filter, and the range-time signal processing. To obtain a single set of reflectivity estimates after signal processing, we simulate G = (L − 1)D + F + Np − 1 scattering centers, where F is the length of the finite-impulse-response (FIR) receiver filter, and Np is the number of samples of the transmitted pulse when sampled at the IF. To simplify the notation and without loss of generality, we assume that G is odd. First, range weights w are applied to the G scattering-center IQ signals to simulate gradients of reflectivity that are linear in decibel units. That is, w(lΔr) = 10[l−(G+1)/2]Δr|∇Z|/20, where l = 0, 1, …, G − 1, and |∇Z| is the magnitude of the reflectivity gradient (dB km−1). These range weights are chosen such that the one for the scattering center in the middle of the set is always 1 (or 0 dB). Second, we convolve the resulting signals with the transmitted pulse ptx, which in our case approximates the WSR-88D transmitted pulse sampled at the IF (Np = 181), and this result is convolved with the receiver-filter impulse response h. The receiver filter is a digital FIR filter that is created using the window method (as on the WSR-88D) and is designed to have either a matched bandwidth (B6 = τ−1) or an L-times-wider bandwidth (B6 = −1) (Torres and Curtis 2020). In both cases, it has F = 201 taps. With these settings, G, the number of scattering centers, is 509. After the two convolution operations, we obtain (L − 1)D + 1 sets of samples along the range dimension. We emphasize that all operations are performed along the range dimension, so the same signal manipulations are repeated at all M sample times. Next, we decimate the signals to baseband by a factor of D to obtain L sets of M-sample IQ signals; these are the range-oversampled inputs to the signal processor V, where V is an L-by-M matrix with complex (IQ) entries. Last, using conventional or range-oversampling processing, we obtain a reflectivity estimate.

With conventional sampling and processing, we use a receiver filter with a matched 6-dB bandwidth (i.e., B6τ = 1) hmf, and the range-oversampled signals are decimated by a factor of L (i.e., we retain only the first row of V) prior to estimating the signal power in the conventional manner. With range-oversampling processing, we use either a receiver matched filter (B6τ = 1) as recommended in Torres and Curtis (2020) for minimizing receiver changes or one with L-times-wider bandwidth (B6τ = 5) as was originally recommended to accommodate range oversampled signals. For each receiver filter, we apply two processing transformations T: a matched-filter-like transformation (with transformation parameter p = 0) and a whitening transformation (p = 1). These correspond to the extreme cases for adaptive pseudowhitening (Curtis and Torres 2011, 2014) and define the range of performance that can be expected when p is chosen adaptively in the interval from 0 to 1. The generation of linear transformations for range-oversampling processing requires a priori knowledge of the range correlation matrix of the weather signals CV. We can obtain the required range correlation by computing the autocorrelation of the modified pulse, which is obtained as the convolution of the known transmitted pulse and receiver-filter impulse response. The processing entails applying the transformation to the input range-oversampled signals as X = TV, computing mean signal powers for each of the L sets of IQ samples, averaging each set of L signal powers, and using these to estimate the signal power in the conventional manner. For more details about range-oversampling processing, the reader is referred to Torres and Curtis (2020).

The RWF depends on the transmitted pulse, the receiver impulse response, and, as mentioned in the previous section, the range-oversampling processing transformation (defined by p for adaptive pseudowhitening). Figure 3 shows the RWF for conventional processing (CP), and for range-oversampling processing (ROP) with four combinations using one of the two receiver filters (B6τ = 1 or 5) with one of the two processing transformations (p = 0 or 1). It can be seen that, for the same processing transformation, the use of a receiver matched filter results in wider RWFs relative to the wide-bandwidth receiver filter. Similarly, for the same receiver filter, the use of a whitening transformation results in wider RWFs relative to using a matched-filter-like transformation. In general, the width of the RWF is proportional to the degree of decorrelation needed, which is a function of the correlation of the range oversampled data before and after the linear transformation. The correlation before the linear transformation is dictated by the bandwidth of the receiver filter, with narrower receiver filters resulting in more correlation. The correlation after the linear transformation is a function of p, with higher values of p resulting in more decorrelation. Intuitively, the results in Fig. 3 make sense because achieving higher degrees of decorrelation requires the use of more “information” or, equivalently, a wider RWF.

Fig. 3.
Fig. 3.

Normalized RWFs for CP and for ROP with the four combinations of receiver filter (B6τ = 1 or 5) and processing transformation (p = 0 or 1). The range is relative to the center of the radar resolution volume.

Citation: Journal of Atmospheric and Oceanic Technology 38, 8; 10.1175/JTECH-D-20-0201.1

Figure 4 shows the change in the mean of reflectivity estimates when using range-oversampling processing with respect to using conventional processing as a function of reflectivity gradients for the four combinations of receiver filter and transformation described above. The solid curves were obtained from simulations, which include all sampling and processing effects: the transmit pulse, the receiver filter, the baseband downconversion, and the range-oversampling processing. In this case, the results were obtained as averages of Monte Carlo simulations, by which the simulation process described above (and shown in Fig. 2) was repeated for R = 20 000 realizations (a new realization of IQ signals corresponding to the scattering centers in the first step of the simulation are produced on each iteration). The circle markers were obtained by summing the product of the RWFs (corresponding to the four combinations of receiver filter and transformation) with the range profiles of scattering-center signal powers (corresponding to all reflectivity gradients from 0 to 50 dB km−1 in steps of 2 dB km−1). This process produces the expected signal-power measurements for each case (i.e., each combination of receiver filter and transformation). We use the term “expected” here because this computation only takes into account the transmit pulse and the receiver filter, effectively ignoring any artifacts that might be introduced by baseband downconversion and range-oversampling processing (e.g., due to mismatches between the true and measured signal and noise range correlations).

Fig. 4.
Fig. 4.

Change in the mean of reflectivity estimates when using range-oversampling processing with respect to using conventional processing as a function of reflectivity gradients. The solid curves were obtained from simulations and correspond to different combinations of receiver filter (B6τ = 1 or 5) and processing transformation (p = 0 or 1). The circle markers correspond to the “expected” changes (based only on the associated RWF) of the mean of reflectivity estimates for the same cases.

Citation: Journal of Atmospheric and Oceanic Technology 38, 8; 10.1175/JTECH-D-20-0201.1

Because the expected results capture only RWF effects and the simulation results capture all processing stages, including filtering, downconversion, and range-oversampling processing, the remarkable agreement between the expected and simulated results indicates that the measurement differences observed in the presence of reflectivity gradients come exclusively from the RWFs. This is important because it implies, for example, that the range correlation is not affected by reflectivity gradients. In other words, we should expect no biases in radar reflectivity measurements from correlation-mismatch effects if the range correlation matrix is known (or measured accurately) for the no-gradient case. As expected, in the presence of reflectivity gradients, range-oversampling processing leads to different measurements of reflectivity when compared with conventional processing. This is because range-oversampling processing results in a different RWF, which is a function of the normalized receiver-filter bandwidth B6τ and the range-oversampling transformation p. The predicted measurement differences are larger for larger reflectivity gradients and also when the input IQ data are more correlated in range (e.g., when using a receiver matched filter). Figure 5 shows the change in standard deviation of reflectivity estimates for the same cases as Fig. 4 using simulations. The results are not surprising: the standard deviation of reflectivity estimates obtained with range-oversampling processing and a whitening transformation is significantly lower than that of estimates obtained with conventional processing.

Fig. 5.
Fig. 5.

Improvement in the standard deviation of reflectivity estimates when using range-oversampling processing with respect to use of conventional processing as a function of reflectivity gradients. The curves were obtained from simulations and correspond to the same combinations of receiver filters and processing transformations as in Figs. 3 and 4.

Citation: Journal of Atmospheric and Oceanic Technology 38, 8; 10.1175/JTECH-D-20-0201.1

Taking into consideration their frequency of occurrence, we can infer that, in practice, reflectivity gradients should not significantly affect the performance of range-oversampling processing. According to Mueller (1977), reflectivity gradients of 12 dB km−1 or less make up over 80% of measured gradients, with most of them being less than 6 dB km−1. For gradients of 12 dB km−1, the reflectivity measurement differences between conventional and range-oversampling processing are between 0.03 and 0.2 dB for the wide-bandwidth receiver filter and between 0.06 and 0.5 dB for the matched receiver filter. Very few cases of gradients up to 35 dB km−1 have been reported (Sirmans and Doviak 1973; Scarchilli et al. 1986). For these, the measurement differences are between 0.2 and 1.4 dB for the wide-bandwidth receiver filter and between 0.4 and 3.2 dB for the matched receiver filter. In these extreme situations, the large measurement differences experienced with the use of a whitening transformation are not negligible. However, the impact of significant measurement differences is isolated, and these unlikely circumstances would need to be weighed against the wide-ranging improvements in standard deviation that can be achieved when using range-oversampling processing. It is difficult to know for sure how a small number of extreme gradients could affect meteorologists’ interpretation of the data, but an initial approach is to use real data to explore the trade-off between standard deviation improvements and reflectivity gradient impacts.

4. Real data analysis

We have seen in section 3 that, for large reflectivity gradients, the change in the measured radar reflectivity can be significant when using range-oversampling processing with a whitening transformation. We also know from previous studies that the percentage of radar bins with large reflectivity gradients is relatively small. The purpose of this section is to explore the impacts of reflectivity gradients when using range-oversampling processing with real data.

We selected a data case from 8 April 2012 collected with the S-band KOUN radar in Norman, Oklahoma, at approximately 2336 UTC. The data were collected using an oversampling factor of L = 5 and a wide-bandwidth receiver filter with B6τ = 5. In addition, M = 17 pulses were collected with a PRT = 3.1 ms (υa = 8.9 m s−1) at an elevation of 0.5° and with azimuthal sampling spacing of 1°. The case includes severe convective storms with large reflectivity gradients in range (Fig. 6a). Three different types of processing were applied to the collected data: conventional processing using a digital matched filter to emulate the matched receiver filter from section 3, range-oversampling processing after the application of that same digital matched filter, and range-oversampling processing after the application of a wide-bandwidth receiver filter. To emulate the B6τ = 1 receiver filter, the digital matched filter was applied to the range oversampled data. Adaptive pseudowhitening was utilized for both types of range-oversampling processing. For the B6τ = 5 receiver filter, the range correlation was computed from the data as in Curtis and Torres (2013), and the noise was assumed to be white based on the wider receiver bandwidth. For the B6τ = 1 case, we attempted to use the same method to compute the range correlation after the digital matched filter was applied but determined that the obtained reflectivity estimates were unexpectedly biased when using that approach. Further analysis revealed that, when using a matched received filter, range-oversampling processing is sensitive to even the smallest range-correlation mismatches that can arise from measuring the range correlation directly from the data. To mitigate this issue, we computed the range correlation by theoretically combining the effects of the digital matched filter with the range correlation measurement from the wide-bandwidth receiver filter. Because of the higher sensitivity to small errors in range correlation measurements, we recommend that another method is used to accurately measure the range autocorrelation when using a matched receiver filter. Other possible methods include directly measuring the modified pulse from either a strong point target or by directly injecting a delayed version of the transmitted pulse into the radar receiver. For the B6τ = 1 receiver filter, we also need the noise correlation, which was computed from the digital matched filter and was utilized as described in Torres and Curtis (2020). The noise power was measured using the radial-by-radial noise estimator described in Ivić et al. (2013).

Fig. 6.
Fig. 6.

(a) Reference reflectivity field and zoomed-in reflectivity fields corresponding to (b) conventional processing and (c) adaptive-pseudowhitening processing using a matched receiver filter and (d) a wide-bandwidth receiver filter. The range-oversampled data were collected with the KOUN radar on 2336 UTC 8 Apr 2012 at an elevation of 0.5° and with azimuthal sampling spacing of 1°. The black contour in (a) encloses the data used in Fig. 7; Fig. 9 includes data from three additional cases. The black dotted lines in (b)–(d) correspond to the radial in Fig. 8.

Citation: Journal of Atmospheric and Oceanic Technology 38, 8; 10.1175/JTECH-D-20-0201.1

Figure 6 shows the results for all three receiver-filter and processing combinations using the real data. Figure 6b shows the conventional processing with a zoom level clearly showing the reflectivity gradients in range for the convective storm. Figure 6c illustrates the effects of applying adaptive pseudowhitening to the digitally matched-filtered data. Figure 6d shows the effects of using adaptive pseudowhitening on the data as originally collected (i.e., with the wide-bandwidth receiver filter). Based on the simulations, the B6τ = 1 receiver filter with range-oversampling processing (Fig. 6c) should lead to the largest changes in radar reflectivity measurements (relative to conventional processing) in the presence of strong reflectivity gradients. Qualitatively, the data appear to be unbiased, and the effects of gradients do not seem to be significant. The measured reflectivity values appear to be smoother, which is consistent with the reduced variance from range-oversampling processing. Recall that, when using adaptive pseudowhitening, range bins with low SNR near the fringes of storms lead to values of p close to 0, and range bins with high SNR lead to values closer to 1. Even in the high-SNR areas with strong gradients, the radar reflectivity measurements with range-oversampling processing do not depart significantly from those obtained with conventional processing. For the B6τ = 5 receiver filter (Fig. 6d), these data are very similar to the data from Fig. 6c. As expected, the reflectivity estimates are smoother as a function of range than when using conventional processing (Fig. 6a) and appear to be unbiased.

Figure 7 shows histograms of measured radar reflectivity differences between the range-oversampling-processing data and the conventional-processing data from range gates at the same locations focused on a sector with weather returns (azimuths 120°–300° and ranges 40–465 km) and bins with SNR > 10 dB. The histograms capture the variability in these gate-to-gate differences when using different combinations of receiver filters and processing. The histograms are nearly the same for both cases of adaptive pseudowhitening processing; this supports our assertion that reflectivity gradients do not have a significant additional effect on reflectivity measurements when using range-oversampling processing. Based on the simulations from section 3, if reflectivity gradients were a major factor, we would have expected the differences when using the B6τ = 1 receiver filter to be greater than for the B6τ = 5 one. The mean values for the differences are 0.153 dB for the B6τ = 1 case and 0.045 dB for the B6τ = 5 case; this could be an effect of the reflectivity gradients since the means are positive and larger for the matched-filter-receiver case, but given the large number of range gates (over 20 000) and the fact that reflectivity gradients are not ubiquitous, these differences could also be due to a small mismatch in the range correlation and/or statistical variability in the data. At least in this case, the benefits of range-oversampling processing in terms of reducing the variance of estimates outweigh any minor reflectivity differences.

Fig. 7.
Fig. 7.

Histograms of reflectivity differences between the adaptive-pseudowhitening processing (APTB) data and conventional processing data (shown in Fig. 6) from range gates at the same locations. Only range bins between 120° and 300° in azimuth and 40 and 465 km in range with SNR at or above 10 dB are included; these 14 500 bins correspond to the most significant portions of the storms where the largest reflectivity gradients are observed.

Citation: Journal of Atmospheric and Oceanic Technology 38, 8; 10.1175/JTECH-D-20-0201.1

Figure 8 illustrates the differences in processing for a particular radial (azimuth 248°) that has significant reflectivity gradients. The measured radar reflectivities are shown for conventional processing and both cases of adaptive pseudowhitening processing. For both cases of adaptive pseudowhitening processing, the results are similar and have less variability than the conventional processing case, as expected. Adaptive pseudowhitening does not seem to cause substantial increases in the reflectivity values, even in the case of strong reflectivity gradients. This is consistent with the results shown in Fig. 6. It also seems that any increases in reflectivity from range-oversampling processing would be small relative to the spatial variability of the data obtained with conventional processing, which has larger variance. Some of these differences in spatial variability are quantified and discussed in more detail in Curtis and Torres (2014).

Fig. 8.
Fig. 8.

Range profile of reflectivity estimates for the data in Fig. 6 at the 248° azimuth using conventional processing and both cases of adaptive-pseudowhitening processing.

Citation: Journal of Atmospheric and Oceanic Technology 38, 8; 10.1175/JTECH-D-20-0201.1

Figure 9 shows average differences in measured radar reflectivities between both cases of range-oversampling processing and conventional processing as a function of reflectivity gradient for the data in Fig. 7 plus three additional cases to increase the range of gradients. The reflectivity gradients in the data under analysis span a wide range from 0 dB km−1 to values close to 50 dB km−1 with a total of about 65 000 data points. Despite the limited number of weather scenarios, this large number of points with a wide diversity of gradients are sufficient to validate the simulation results. The reflectivity gradients were estimated from the data that were computed using adaptive pseudowhitening processing and the wide-bandwidth receiver filter, which represent the best compromise between precision and range resolution. We excluded gates with weak returns (SNR < 10 dB) as in Fig. 7. The reflectivity differences were binned according to the measured gradient in 5 dB km−1 intervals (i.e., 0–5, 5–10, …, 45–50 dB km−1), and the average difference for each bin was then obtained. We calculated the average difference to minimize the statistical variability from the estimates. From these averages, we subtracted the corresponding systematic bias (similar to the bias identified in Fig. 7 but including all datasets). The black dotted curve shows the number of gates in each 5 dB km−1 gradient bin. The simulation results in Fig. 4 are also included as a reference, where the shaded areas correspond to the full range of p values for each receiver filter. Because the reflectivity estimates obtained with adaptive pseudowhitening processing may correspond to p values between 0 and 1, the curves from Fig. 4 represent lower and upper bounds for the curves obtained from the real data. It is important to note that the curves obtained from the real data may not perfectly agree with our simulation results due to 1) the statistical variability in the measured reflectivity data, 2) the effects of the RWF on the true reflectivity gradient, 3) the finite number of gates in each 5 dB km−1 gradient bin (which drastically decreases as the gradients increase), 4) the binning process, and 5) the variability of p (because the SNR of the data under analysis is always 10 dB or more, most of the values of p should be close to 1). Nevertheless, the curves from the real data exhibit the expected behavior: larger measured reflectivity changes from range-oversampling processing for larger reflectivity gradients and larger reflectivity changes from range-oversampling processing when using a matched receiver filter. Also, the measurements fall within their predicted bounds (i.e., the respective shaded areas obtained from simulations). Overall, the results from this real data case support the conclusion that range-oversampling processing reduces the variability of the radar data without causing significant differences in radar reflectivity measurements in the presence of the types of reflectivity gradients found in practice.

Fig. 9.
Fig. 9.

Average of change in measured radar reflectivities between range-oversampling processing and conventional processing as a function of reflectivity gradient from data collected with the KOUN radar. The blue curve corresponds to a matched receiver filter, and the red curve corresponds to a wide-bandwidth receiver filter (left axis). The black dotted curve shows the number of range gates (logarithmic units) in each 5 dB km−1 gradient bin (right axis). The shaded areas correspond to the reflectivity differences with respect to conventional processing obtained from simulations (see Fig. 4) using a matched receiver filter (blue) and a wide-bandwidth receiver filter (red) for all values of p between 0 and 1.

Citation: Journal of Atmospheric and Oceanic Technology 38, 8; 10.1175/JTECH-D-20-0201.1

5. Conclusions

Range oversampling processing leads to increased computational complexity and some loss of range resolution. Earlier work has shown that this loss in range resolution can be offset by a decrease in the variance of radar-variable estimates (Torres and Curtis 2015). The focus of this paper is to examine the effects of reflectivity gradients when using range-oversampling processing. This is important because range-oversampling processing techniques such as adaptive pseudowhitening were developed using a uniform reflectivity assumption. However, we know from earlier studies that significant reflectivity gradients occur in storms with some gradients above 30 dB km−1 and possibly even higher. At the same time, the research of reflectivity gradients in storms also shows that the vast majority of gradients are less than 12 dB km−1. After looking at possible effects from reflectivity gradients, we found that the only significant effect comes from changes to the range weighting function. These changes, which can occur on a gate-by-gate basis, cause differences in reflectivity measurements in the presence of gradients.

To quantify these effects, we developed simulations that used receiver filters similar to those implemented currently on the WSR-88D. We studied gradients from 0 to 50 dB km−1 that were linear in the reflectivity domain, which is consistent with earlier research. These simulations showed that the differences in reflectivity measurements increase as the gradients increase in magnitude. The differences are larger for whitening than for matched-filter processing. The differences are also larger for narrower matched-filter-like receiver filters than for wider receiver filters. We also found that reflectivity gradients can also cause biases when measuring the range correlation matrix for matched-filter-like receiver filters using our previously developed measuring technique (Curtis and Torres 2013). Because these measurement biases can lead to significant reflectivity biases, we recommend using an L-times-wider receiver filter when using this real-time measurement technique. If a narrow, matched filter is utilized, we recommend a different approach for measuring the range correlation.

In addition to the theoretical simulations, we also examined the effects on real data. As mentioned previously, the vast majority of gradients are less than 12 dB km−1, and the qualitative comparisons between conventional processing and adaptive pseudowhitening processing did not show any significant differences. We also analyzed real data with significant reflectivity gradients in range, and we concluded that the measurement differences from reflectivity gradients tend to be small relative to the estimate variability for conventional processing. Overall, we conclude that the widespread improvement in estimate standard deviation from range-oversampling processing outweighs the possible measurement differences in the presence of strong gradients. These large gradients occur relatively infrequently and should not significantly affect the performance of range-oversampling processing.

Acknowledgments

The authors thank Igor Ivić and three anonymous reviewers for providing comments to improve the paper. Funding was provided by the NOAA/Office of Oceanic and Atmospheric Research under NOAA–University of Oklahoma Cooperative Agreement NA11OAR4320072, U.S. Department of Commerce.

REFERENCES

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    • Crossref
    • Search Google Scholar
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  • Curtis, C., and S. Torres, 2011: Adaptive range oversampling to achieve faster scanning on the National Weather Radar Testbed phased-array radar. J. Atmos. Oceanic Technol., 28, 15811597, https://doi.org/10.1175/JTECH-D-10-05042.1.

    • Crossref
    • Search Google Scholar
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  • Curtis, C., and S. Torres, 2013: Real-time measurement of the range correlation for range oversampling processing. J. Atmos. Oceanic Technol., 30, 28852895, https://doi.org/10.1175/JTECH-D-13-00090.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Curtis, C., and S. Torres, 2014: Adaptive range oversampling to improve estimates of polarimetric variables on weather radars. J. Atmos. Oceanic Technol., 31, 18531866, https://doi.org/10.1175/JTECH-D-13-00216.1.

    • Crossref
    • Search Google Scholar
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  • Curtis, C., and S. Torres, 2017: Adaptive range oversampling processing for nontraditional radar-variable estimators. J. Atmos. Oceanic Technol., 34, 16071623, https://doi.org/10.1175/JTECH-D-16-0051.1.

    • Crossref
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  • Ivić, I. R., C. Curtis, and S. M. Torres, 2013: Radial-based noise power estimation for weather radars. J. Atmos. Oceanic Technol., 30, 27372753, https://doi.org/10.1175/JTECH-D-13-00008.1.

    • Crossref
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    • Crossref
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  • Rogers, R. R., 1971: The effect of variable target reflectivity on weather radar measurements. Quart. J. Roy. Meteor. Soc., 97, 154167, https://doi.org/10.1002/qj.49709741203.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Scarchilli, G., E. Gorgucci, and R. Leonardi, 1986: Theory and optimization of the excess bias measurement. J. Atmos. Oceanic Technol., 3, 217229, https://doi.org/10.1175/1520-0426(1986)003<0217:TAOOTE>2.0.CO;2.

    • Crossref
    • Search Google Scholar
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  • Sirmans, D., and R. J. Doviak, 1973: Meteorological radar signal intensity estimation. NOAA Tech. Memo. ERL NSSL-64, 92 pp.

  • Torres, S., and D. Zrnić, 2003: Whitening in range to improve weather radar spectral moment estimates. Part I: Formulation and simulation. J. Atmos. Oceanic Technol., 20, 14331448, https://doi.org/10.1175/1520-0426(2003)020<1433:WIRTIW>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Torres, S., and C. Curtis, 2012: The impact of signal processing on the range-weighting function for weather radars. J. Atmos. Oceanic Technol., 29, 796806, https://doi.org/10.1175/JTECH-D-11-00135.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Torres, S., and C. Curtis, 2015: The impact of range-oversampling processing on tornado velocity signatures obtained from WSR-88D superresolution data. J. Atmos. Oceanic Technol., 32, 15811592, https://doi.org/10.1175/JTECH-D-14-00196.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Torres, S., and C. Curtis, 2020: Revisiting the optimum receiver filter bandwidth for range-oversampling processing. J. Atmos. Oceanic Technol., 37, 507515, https://doi.org/10.1175/JTECH-D-19-0057.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
Save
  • Curtis, C., 2018: Weather radar time series simulation: Improving accuracy and performance. J. Atmos. Oceanic Technol., 35, 21692187, https://doi.org/10.1175/JTECH-D-17-0215.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Curtis, C., and S. Torres, 2011: Adaptive range oversampling to achieve faster scanning on the National Weather Radar Testbed phased-array radar. J. Atmos. Oceanic Technol., 28, 15811597, https://doi.org/10.1175/JTECH-D-10-05042.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Curtis, C., and S. Torres, 2013: Real-time measurement of the range correlation for range oversampling processing. J. Atmos. Oceanic Technol., 30, 28852895, https://doi.org/10.1175/JTECH-D-13-00090.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Curtis, C., and S. Torres, 2014: Adaptive range oversampling to improve estimates of polarimetric variables on weather radars. J. Atmos. Oceanic Technol., 31, 18531866, https://doi.org/10.1175/JTECH-D-13-00216.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Curtis, C., and S. Torres, 2017: Adaptive range oversampling processing for nontraditional radar-variable estimators. J. Atmos. Oceanic Technol., 34, 16071623, https://doi.org/10.1175/JTECH-D-16-0051.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Doviak, R., and D. Zrnić, 1993: Doppler Radar and Weather Observations. 2nd ed. Academic Press, 562 pp.

  • Ivić, I. R., C. Curtis, and S. M. Torres, 2013: Radial-based noise power estimation for weather radars. J. Atmos. Oceanic Technol., 30, 27372753, https://doi.org/10.1175/JTECH-D-13-00008.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Johnston, P. E., L. M. Hartten, C. H. Love, D. A. Carter, and K. S. Gage, 2002: Range errors in wind profiling caused by strong reflectivity gradients. J. Atmos. Oceanic Technol., 19, 934953, https://doi.org/10.1175/1520-0426(2002)019<0934:REIWPC>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Kurdzo, J. M., B. L. Cheong, R. D. Palmer, G. Zhang, and J. B. Meier, 2014: A pulse compression waveform for improved-sensitivity weather radar observations. J. Atmos. Oceanic Technol., 31, 27132731, https://doi.org/10.1175/JTECH-D-13-00021.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Mueller, E. A., 1977: Statistics of high radar reflectivity gradients. J. Appl. Meteor., 16, 511513, https://doi.org/10.1175/1520-0450(1977)016<0511:SOHRRG>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Rogers, R. R., 1971: The effect of variable target reflectivity on weather radar measurements. Quart. J. Roy. Meteor. Soc., 97, 154167, https://doi.org/10.1002/qj.49709741203.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Scarchilli, G., E. Gorgucci, and R. Leonardi, 1986: Theory and optimization of the excess bias measurement. J. Atmos. Oceanic Technol., 3, 217229, https://doi.org/10.1175/1520-0426(1986)003<0217:TAOOTE>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Sirmans, D., and R. J. Doviak, 1973: Meteorological radar signal intensity estimation. NOAA Tech. Memo. ERL NSSL-64, 92 pp.

  • Torres, S., and D. Zrnić, 2003: Whitening in range to improve weather radar spectral moment estimates. Part I: Formulation and simulation. J. Atmos. Oceanic Technol., 20, 14331448, https://doi.org/10.1175/1520-0426(2003)020<1433:WIRTIW>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Torres, S., and C. Curtis, 2012: The impact of signal processing on the range-weighting function for weather radars. J. Atmos. Oceanic Technol., 29, 796806, https://doi.org/10.1175/JTECH-D-11-00135.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Torres, S., and C. Curtis, 2015: The impact of range-oversampling processing on tornado velocity signatures obtained from WSR-88D superresolution data. J. Atmos. Oceanic Technol., 32, 15811592, https://doi.org/10.1175/JTECH-D-14-00196.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Torres, S., and C. Curtis, 2020: Revisiting the optimum receiver filter bandwidth for range-oversampling processing. J. Atmos. Oceanic Technol., 37, 507515, https://doi.org/10.1175/JTECH-D-19-0057.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Fig. 1.

    Examples of area-normalized RWFs for conventional processing using a matched receiver filter (CP) and range-oversampling processing using a wide-bandwidth receiver filter (ROP) and select linear transformations corresponding to values of p from 0 (matched filter-like transformation) to 1 (whitening transformation). The range is relative to the center of the radar resolution volume.

  • Fig. 2.

    Simulation flowchart. Inputs to each processing block are listed to the left of the block. The notation f() is loosely used to make the dependency of inputs on basic simulation parameters explicit. The dimensionality of nonscalar inputs is indicated in parentheses. The dimensionalities of outputs are listed to the right of each processing block. The total number of scattering centers is G = (L − 1)D + F + Np − 1, where Np is the length of the transmit pulse, F is the length of the receiver impulse response, D is the decimation factor, and L is the range oversampling factor; M is the number of samples in the dwell. An asterisk next to an operation is used to indicate that the same operation is performed on each row of the input matrix. Separate processing branches are needed for conventional processing (right branch) and ROP (left branch). Each simulation run produces a single signal-power estimate. For this work, the simulation is repeated for 20 000 realizations.

  • Fig. 3.

    Normalized RWFs for CP and for ROP with the four combinations of receiver filter (B6τ = 1 or 5) and processing transformation (p = 0 or 1). The range is relative to the center of the radar resolution volume.

  • Fig. 4.

    Change in the mean of reflectivity estimates when using range-oversampling processing with respect to using conventional processing as a function of reflectivity gradients. The solid curves were obtained from simulations and correspond to different combinations of receiver filter (B6τ = 1 or 5) and processing transformation (p = 0 or 1). The circle markers correspond to the “expected” changes (based only on the associated RWF) of the mean of reflectivity estimates for the same cases.

  • Fig. 5.

    Improvement in the standard deviation of reflectivity estimates when using range-oversampling processing with respect to use of conventional processing as a function of reflectivity gradients. The curves were obtained from simulations and correspond to the same combinations of receiver filters and processing transformations as in Figs. 3 and 4.

  • Fig. 6.

    (a) Reference reflectivity field and zoomed-in reflectivity fields corresponding to (b) conventional processing and (c) adaptive-pseudowhitening processing using a matched receiver filter and (d) a wide-bandwidth receiver filter. The range-oversampled data were collected with the KOUN radar on 2336 UTC 8 Apr 2012 at an elevation of 0.5° and with azimuthal sampling spacing of 1°. The black contour in (a) encloses the data used in Fig. 7; Fig. 9 includes data from three additional cases. The black dotted lines in (b)–(d) correspond to the radial in Fig. 8.

  • Fig. 7.

    Histograms of reflectivity differences between the adaptive-pseudowhitening processing (APTB) data and conventional processing data (shown in Fig. 6) from range gates at the same locations. Only range bins between 120° and 300° in azimuth and 40 and 465 km in range with SNR at or above 10 dB are included; these 14 500 bins correspond to the most significant portions of the storms where the largest reflectivity gradients are observed.

  • Fig. 8.

    Range profile of reflectivity estimates for the data in Fig. 6 at the 248° azimuth using conventional processing and both cases of adaptive-pseudowhitening processing.

  • Fig. 9.

    Average of change in measured radar reflectivities between range-oversampling processing and conventional processing as a function of reflectivity gradient from data collected with the KOUN radar. The blue curve corresponds to a matched receiver filter, and the red curve corresponds to a wide-bandwidth receiver filter (left axis). The black dotted curve shows the number of range gates (logarithmic units) in each 5 dB km−1 gradient bin (right axis). The shaded areas correspond to the reflectivity differences with respect to conventional processing obtained from simulations (see Fig. 4) using a matched receiver filter (blue) and a wide-bandwidth receiver filter (red) for all values of p between 0 and 1.

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