## 1. Introduction

When considering tornado warnings, National Weather Service (NWS) forecasters routinely assess the strength of Doppler radar tornadic vortex signatures (TVSs) or tornado signatures (TSs) by evaluating their rotational velocity through the azimuthal gate-to-gate Doppler velocity difference (e.g., Brown et al. 1978; Mitchell et al. 1998). Unfortunately, the detection of smaller, weaker, and/or more distant tornadoes using radar is more challenging because of the curvature of the earth and the fact that the radar beam broadens as the range increases. Brown et al. (2002) described the impact of the effective antenna beamwidth and the azimuthal sampling interval on the capabilities of the Weather Surveillance Radar-1988 Doppler (WSR-88D) to resolve tornado cyclones: TVSs get weaker as the ratio between the circulation diameter and the effective beamwidth decreases or as the azimuthal sampling gets coarser. With these findings, it became evident that tornado detection on the WSR-88D could be improved by reducing the effective antenna beamwidth and/or using finer azimuthal sampling (Brown et al. 2005). Thus, in 2008 the NWS upgraded all WSR-88D radars to produce *superresolution data* at lower elevation angles, which are routinely used by forecasters as major inputs into the tornado warning decision process.

The implementation of superresolution data on the WSR-88D entails producing reflectivity fields with finer sampling in range (250 m instead of 1 km) and all radar variables with finer sampling in azimuth (0.5° instead of 1°). It also entails improving the azimuthal resolution for all radar variables by reducing the effective antenna beamwidth from ~1.4° to ~1° (Torres and Curtis 2007). Finer azimuthal sampling is achieved by processing overlapped 1° radials that share the first half of the samples with the previous radial and the second half with the next radial. The reduction in the effective beamwidth with respect to legacy-resolution data is achieved by applying a tapered data window (von Hann) to the samples in each radial at the expense of an increase in the variance of all radar-variable estimates. Because superresolution data do not meet the system’s requirements for data quality, they are not used by any radar product generator (RPG) algorithms. Instead, the RPG algorithms recombine the superresolution data stream to form a *recombined data* stream with legacylike resolution and quality (Torres and Curtis 2007). In contrast, forecasters routinely use superresolution data in their warning decision process, so they would be in a position to benefit from improvements in the superresolution data quality.

Range-oversampling processing is a promising technique that has been used on pulsed Doppler weather radars to reduce the variance of radar-variable estimates. Range oversampling by a factor of *L* involves collecting samples at a rate of *L**τ*^{−1} (i.e., *L* times faster than with conventional sampling) and processing sets of *L* samples with a decorrelating matrix transformation, after which sample-time autocovariances can be estimated and incoherently averaged to reduce their uncertainty (Torres and Zrnić 2003a,b; Torres et al. 2004; Curtis and Torres 2011, 2014). Thus, range oversampling can be used to reduce the variance of superresolution data on the WSR-88D. In the short term, producing more precise superresolution radar data would improve forecaster confidence, which is critical for the warning decision process. In the future, algorithms could be modified to benefit from finer sampling and improved azimuthal resolution. However, the benefits of range-oversampling processing come at the price of additional computational complexity and a broader range weighting function; that is, data produced with range-oversampling processing can be more precise, but there is a slight degradation in range resolution (Torres and Curtis 2013). For example, the typical range resolution (defined as the 6-dB width of the range weighting function) of the WSR-88D with conventional processing is 246 m, whereas data produced with 5-times range oversampling followed by a whitening transformation can have a fivefold reduction in variance but a degraded range resolution of 455 m.

The goal of this work is to quantify the effects of range-oversampling processing on velocity signatures from tornadoes when using WSR-88D superresolution data. In particular, the variance reduction and degraded range resolution will be quantified by analyzing the low-altitude azimuthal velocity difference (Δ*V*) from simulated idealized Doppler velocity fields for tornadolike vortices corresponding to different radar sampling and processing schemes. Based on the simulation framework of Wood and Brown (1997), Doppler spectra and velocity fields are produced with different range-oversampling processing options (corresponding to different range weighting functions) and for legacy-resolution and superresolution sampling schemes (corresponding to different azimuthal sampling intervals and different effective antenna patterns). Thus, it is possible to assess how different spatial samplings, antenna patterns, and/or range weighting functions affect Δ*V*. The rest of the paper is organized as follows. Section 2 describes the radar simulator, section 3 contains the simulation results, and section 4 presents an example using tornado data collected with the National Weather Radar Testbed (NWRT) in Norman, Oklahoma. The paper ends with a discussion of the trade-offs involved in the use of range-oversampling processing to produce Doppler velocity data, a key element of the tornado warning decision process.

## 2. Simulation of Doppler velocity data from tornado-like vortices

This section describes the simulator used to quantify the effects of range-oversampling processing on the velocity signature of tornadoes when using WSR-88D superresolution data. The purpose of this simulator is to produce synthetic mean Doppler velocity fields (and associated Doppler spectra) that reasonably represent how tornado vortices are sampled by the radar, a task that can be achieved with different approaches and different degrees of realism and complexity. For example, Capsoni and D’Amico (1998) proposed a realistic but highly complex physically based radar simulator that combines the returns from individual hydrometeors advected by an underlying wind field. Cheong et al. (2008) used a similar approach based on fields from a high-resolution numerical weather model. May et al. (2007) developed a less complex method based on 3D gridded fields (instead of individual hydrometeors) that describe the state of the atmosphere. With the same goal of examining the effects of radar sampling on the measurement of Δ*V*, Wood and Brown (1997) and Brown et al. (2002) developed a simpler simulator based on idealized 2D vortex models. While not as realistic as physically based simulators, this approach is computationally simpler and, more importantly, sufficient for testing the impact of range-oversampling processing on Δ*V*.

The basic structure of our simulator closely matches the simulation model described in Wood and Brown (1997). Velocities of equivalent scattering centers on a fixed 2D grid are combined in range using the range weighting function and in azimuth using the effective antenna pattern. Since the model is two-dimensional and the velocity fields are constant, we make the following simplifying assumptions: 1) the fields are stationary over a few dwell times, 2) the fields are uniform with height, and 3) the beam axis is horizontal. Figure 1 is a diagram showing the basic geometry of the simulator, including an example of a scattering center. The gray swath is the area of interest for the simulator and can contain multiple radar resolution volumes. Additional scattering centers beyond the swath are included to accommodate the weightings in both range and azimuth. The swath is extended in range by *c* is the speed of light, and *θ*_{p}, which is half of the support of the effective antenna pattern (including sidelobes). The scattering-center grid has azimuthal spacing Δ*θ*, which corresponds to the rotation of the antenna during the pulse repetition time (*T*_{s}), and range spacing Δ*r*, which corresponds to the spacing of the oversampled gates. The velocities of the scattering centers are calculated based on an idealized 2D vortex model, and the data that are used to compute Δ*V* come from the gray swath.

*V*, so we simulate several different vortex positions inside of a radar resolution volume and average the results. There are also several tornado vortex models from which to choose. Our simulator uses the Burgers–Rott vortex model (Burgers 1948; Rott 1958) that is discussed (along with other models) in Davies-Jones (1986). This model was suggested by Wood and Brown (2011) when studying one-celled tornadoes. The Burgers–Rott model is expressed aswhere

*V*

_{t}is the tangential velocity at a given point,

*V*

_{x}is the maximum tangential velocity,

*R*is the distance from the center of the vortex, and

*R*

_{x}is the core radius (the distance from the center of the vortex to the points with maximum tangential velocity). The Burgers–Rott model is used to simulate both velocity and reflectivity, except that the reflectivity model uses a core radius that is twice as large as the one used for velocity to account for the centrifuging of hydrometeors and debris (Brown and Wood 2012). The last major difference in the simulators is the method for computing the Doppler velocity (and its variance). This computation and further details of the simulator are discussed next.

In the Wood and Brown (1997) simulator, the velocity value for the scattering center is computed directly from the tornado vortex model. This is equivalent to sampling the velocity field at a single point in the middle of the scattering-center grid cell. Because of the steep velocity gradients present in the tornado vortex model (especially for small tornadoes), we define a finer grid in each scattering-center grid cell. The tangential velocity and the corresponding power are computed from the tornado vortex model at each of the finer grid points. These tangential velocities are then projected onto a vector connecting the radar and the fine grid point to find the corresponding Doppler velocities. Finally, the powers and Doppler velocities in the fine grid are used to compute the Doppler spectrum (i.e., the power-weighted distribution of radial velocities in the scattering-center grid cell), the total power (i.e., the sum of all powers in the scattering-center grid cell), and the Doppler velocity (i.e., the power-weighted average of radial velocities in the scattering-center grid cell) for each scattering center.

*S*) for a radar resolution volume centered at a particular range

*r*

_{0}and a particular azimuth

*θ*

_{0}is an extension of (A.4) from Wood and Brown (1997):where

*k*is the spectral index (0 ≤

*k*<

*M*),

*S*

_{0}is the Doppler spectrum for a scattering center,

*θ*and Δ

*r*values are the scattering-center grid spacings in azimuth and range, respectively; Δ

*υ*is the spectral resolution defined as 2

*υ*

_{a}/

*M*, where

*υ*

_{a}is the maximum unambiguous velocity and

*M*is the number of samples per dwell. The centers of both

*I*+ 1 points in azimuth and 2

*J*+ 1 points in range. Doppler velocities (

*υ*) are computed in a similar way aswhere

*P*

_{0}and

*υ*

_{0}are the powers and Doppler velocities for the scattering centers, respectively. Unlike Wood and Brown’s (1997) equation, (3) includes the weather signal power, which is not assumed as uniform in the model, and uses the effective two-way antenna pattern, which combines the intrinsic two-way antenna pattern with the data window. The reader should note that whereas the Doppler spectrum in (2) is aliased in the Nyquist cointerval, Doppler velocities in (3) are not aliased. This is equivalent to assuming perfect velocity dealiasing for the computation of Δ

*V*.

*J*

_{2}is the Bessel function of second order, and

*θ*

_{1}as

*λ*is the transmit wavelength. The pattern is sampled every Δ

*θ*to match the spacing of the scattering-center grid. The effective two-way pattern is then computed as the convolution of the two-way antenna pattern and the square of the data window as in Torres and Curtis (2006):where

*f*

^{4}is the two-way pattern calculated from (4),

*d*is the

*M*-sample data window, and

*T*

_{s}is the pulse repetition time. The effective two-way pattern is zero outside of the interval

*θ*

_{p}is half the support of the two-way effective pattern. In the simulation, the data window is either rectangular or von Hann. The von Hann window is used to generate a narrower effective beamwidth and is part of the process for producing superresolution data on the WSR-88D.

_{i,j}designates the element in the

*i*th row and

*j*th column of matrix

_{j,j}is an element along the diagonal. Matrix

*r*increments to match the spacing of the oversampled range gates. Examples of the three range weighting functions used in the simulation (digital matched filtering, pseudowhitening, and whitening) are shown in Fig. 3.

As described before, Doppler spectra and mean Doppler velocities can be computed for any radar volume inside the gray swath area in Fig. 1. For this study, radar volumes are centered on a radar sampling grid with 0.5° or 1° spacing in azimuth (corresponding to superresolution or legacy-resolution data, respectively) and 250 m in range (i.e., the nonoversampled range gate spacing). The extent of the swath in azimuth is set to capture the full vortex (it decreases with range because the azimuthal extent of the vortex also decreases with range) and accounts for the random positioning within the central radar resolution volume. The extent of the swath in range is also set to capture the full vortex, including random positioning, but stays constant because the range gate spacing does not change as a function of range. After the Doppler spectra and mean Doppler velocities are calculated for all the radar volumes in the swath, the mean and variance of Δ*V* estimates are computed as described next.

*V*is obtained as the largest constant-range maximum velocity difference in the simulation swath; that is, maximum (

*V*estimates can be expressed asit depends on the variance of the maximum and minimum Doppler velocity estimates and on their correlation coefficient

*ρ*) is obtained by normalizing the inverse Fourier transform of the Doppler spectrum, and a high signal-to-noise ratio (SNR) is assumed; that is,We impose an upper bound of

_{∞}as defined by Torres et al. 2004) and computed aswhere

_{m}are the same as in (6). The correlation coefficient in (7) is given by the amount of overlap between the radar volumes corresponding to

This simulation is used next to quantify the effects of range-oversampling processing on the mean and variance of Δ*V* estimates.

## 3. Simulation results

The simulator described in the previous section is used to produce Doppler spectra and velocity radar fields corresponding to different radar sampling and signal processing schemes. We consider the four tornado models in Brown et al. (2002); Table 1 lists the peak rotational velocities (*V*_{x}) and core radii (*R*_{x}) for each of them. The radar resolution volume containing the tornado vortex center is placed at ranges of 5–100 km in steps of 5 km. Figure 2 shows Doppler velocity fields for the four tornado models at selected ranges. A radar resolution volume of 250 m by 1° is overlaid on top of the fields to help visualize the relative scale and strength of the vortices. For each range, the simulator places the vortex center at 500 random locations within the radar resolution volume. We use the acquisition parameters of the 0.5° elevation scan of volume coverage pattern (VCP) 12, which is typically used to observe severe convective storms with the operational WSR-88D. The number of samples in the dwell time is 50, and the maximum unambiguous velocity is 35 m s^{−1}, corresponding to a pulse repetition time of 0.78 ms. The range-oversampling factor is 5, as proposed for a future operational implementation of range oversampling on the WSR-88D. Thus, the range sampling interval is 50 m, and data are produced with 250-m spacing after processing, which consists of conventional matched filtering, range oversampling with a pseudowhitening transformation, or range oversampling with a whitening transformation. In this case, the pseudowhitening transformation is based on a sharpening filter with fixed parameter *p*, where *p* controls the degree of decorrelation: as *p* ranges from 0 to 1, the resulting pseudowhitening transformation ranges from a matched filter to a whitening transformation (Torres et al. 2004). For this study, we chose *p* = 0.8 to illustrate the trade-off between variance reduction and range resolution.

Peak rotational velocities and core radii for the four tornado models.

Figure 3 shows the range weighting functions for the modified pulse measured on the KOUN radar (Norman) and the three processing options. As anticipated, the range weighting function gets broader as the range-oversampling transformation changes from a digital matched filter to a whitening transformation. The computed range resolution for these cases is 246 m for the digital matched filter, 411 m for the pseudowhitening transformation, and 455 m for the whitening transformation. The azimuthal sampling is either that of legacy-resolution data (i.e., azimuthal increments of 1° and samples in the dwell time weighted with the rectangular window) or superresolution data (i.e., azimuthal increments of 0.5° and samples weighted with the von Hann window). Figure 4 shows the two-way effective antenna patterns for legacy-resolution and superresolution azimuthal samplings. These are obtained by convolving the intrinsic two-way antenna pattern of the WSR-88D (

The mean and variance of Δ*V* are computed for each tornado model, each vortex position, and each sampling and processing mode as described in the previous section. Figure 5 shows the mean value of Δ*V* as a function of tornado distance from the radar for different sampling and processing modes and for each of the four tornado models. Note that each point represents the average of the values obtained for the 500 random placements at each range. As expected, the mean of Δ*V* gets smaller for all cases as the range from the radar to the tornado increases (Brown et al. 1978). As shown by Brown et al. (2002), superresolution data produced with a matched filter leads to larger Δ*V* values compared to legacy-resolution data. Also evident in these figures is the expected impact of the range-resolution degradation due to the broadening of the range weighting function from range oversampling using pseudowhitening or whitening transformations. For all cases under analysis, it is seen that the largest impact of the range-resolution degradation occurs for smaller tornadoes at close ranges.

Figure 6 quantifies the degradation in the mean of Δ*V* for superresolution data produced with either pseudowhitening or whitening transformations (subscripts “SR&PW” and “SR&W,” respectively) with respect to superresolution data produced with a matched filter (subscript “SR”). It can be seen that the degradation in Δ*V* from range-oversampling processing is no more than 14% for any tornado model under consideration; it is less than 8% for stronger and larger tornadoes C and D, and also less than 8% for weaker and smaller tornadoes A and B beyond ~50 km. Compared to a whitening transformation, a pseudowhitening transformation produces less degradation in Δ*V*. For a pseudowhitening transformation with *p* = 0.8, Δ*V* is no more than 10% for any tornado model; it is less than 6% for stronger and larger tornadoes C and D, and also less than 6% for weaker and smaller tornadoes A and B beyond ~50 km.

Figure 7 is analogous to Fig. 6, except that it quantifies the improvement in the variance of Δ*V*. In general, the variance of Δ*V* estimates depends on the number of samples, the maximum unambiguous velocity, the type of signal processing, and the resulting Doppler spectrum after radar sampling. As mentioned before, different processing schemes lead to different variance reduction factors (VRF_{∞} = 5 for whitening and ~3.4 for pseudowhitening with *p* = 0.8). This explains the general trend of the curves at ranges beyond ~30 km. The departures at close ranges and across tornado models can be attributed to differences in the Doppler spectra corresponding to the radar volumes for *V* variance reduction factor from range-oversampling processing with a whitening transformation is no less than ~1.8 for any tornado model, and it is more than 5 for all tornadoes beyond ~50 km. The use of a pseudowhitening transformation produces less degradation in Δ*V* but leads to smaller variance reduction factors. For a pseudowhitening transformation with *p* = 0.8, the variance of Δ*V* is reduced more than ~1.6 times for any tornado model, and it is larger than ~3.5 for all tornadoes beyond ~50 km.

In summary, this analysis quantifies the fundamental trade-off associated with the computation of Δ*V* using superresolution data with range-oversampling processing: higher-precision data are obtained at the price of a slight degradation in range resolution, which leads to a reduction in the magnitude of the measured Δ*V*.

## 4. Real data

Unfortunately, there are no tornado time series data cases collected with the current WSR-88D radars that can be used for this study because range oversampling has not yet been implemented operationally. There is a small amount of range-oversampled data collected with the research WSR-88D radar in Norman (KOUN), but no tornado cases were collected. Thus, we utilize range-oversampled tornado data collected with the NWRT phased-array radar (PAR) in Norman (Zrnić et al. 2007). Unlike the WSR-88D, the NWRT PAR antenna is stationary, which avoids the beam smearing associated with rotating antennas. However, the beamwidth of the NWRT PAR increases as the beam is electronically steered away from broadside. Even with these differences, the NWRT PAR data can still be used to quantify the relative degradation of Δ*V* from range-oversampling processing compared to conventional matched-filtering processing. This is because the performance of range-oversampling processing depends on the modified pulse and the processing transformation but not on the antenna pattern.

Figure 8 shows data collected at ~2320 UTC 19 May 2013 at an elevation of 0.5° using two different PRTs. The time series data that were used to produce reflectivity estimates were collected with *M* = 12 samples and *T*_{s} = 3.1 ms. The time series data that were used to produce velocity and spectrum width estimates were collected with *M* = 25 samples and *T*_{s} = 0.8 ms (the Nyquist velocity is *υ*_{a} = 29.3 m s^{−1}). The data in the top row were processed using an adaptive pseudowhitening algorithm whereby the pseudowhitening parameter *p* adapts to the received signal characteristics (Curtis and Torres 2011). At high SNR (as in this case of a strong tornado close to the radar), the adaptive pseudowhitening algorithm leads to values of *p* that are very close to 1. Thus, the data in the top row of Fig. 8 are very similar to data processed with a whitening transformation. In contrast, the data in the bottom row were processed using a digital matched filter. The scanning strategy utilized superresolution azimuth spacing, but the effective antenna beamwidth matches the beamwidth of the phased-array antenna (~1.5° at broadside) because the beam is stationary during the dwell time.

The reflectivity and storm-relative velocity data obtained using adaptive pseudowhitening exhibit lower variance as expected; the fields are smoother and less noisy than the matched-filter fields. For this case, the Doppler velocity couplet is apparent in both of the velocity fields. The value of Δ*V* is calculated with the same method used in the simulation. The Δ*V* for adaptive-pseudowhitening processing is 75.4 compared to 78.3 m s^{−1} for matched-filter processing. There is some degradation due to the change in range resolution, but the effect of range-oversampling processing in this particular case is relatively small (~4%). For this strong tornado close to the radar, the increase in data quality from range-oversampling processing is probably more significant than the slight degradation in the magnitude of Δ*V*.

## 5. Conclusions

Superresolution data enhances the strength of the TVS, which is measured by the azimuthal differential velocity (Δ*V*). However, superresolution data also leads to radar-variable estimates with higher variance. Range-oversampling processing can be used to reduce the variance of superresolution-data estimates, but it also degrades the range resolution. In principle, this degradation could affect the depiction of small-scale features and reduce the value of Δ*V*. Through simulations, we quantified the impact of range-oversampling processing on the mean and variance of Δ*V* for tornado-like vortices. For all four studied tornado models, the impact was shown to be larger at closer ranges, with smaller tornadoes being affected more. In the worst case, a whitening transformation leads to a mean Δ*V* degradation of at most 14% with a variance reduction factor of ~1.8. However, beyond 50 km, the impact reduces to a mean Δ*V* degradation of less than 8% with a variance reduction factor above 5. For adaptive pseudowhitening, the performance is expected to be between matched filtering (*p* = 0) and range oversampling with a whitening transformation (*p* = 1). For the pseudowhitening transformation illustrated here (*p* = 0.8), the mean Δ*V* degradation is less than 6% beyond 50 km at the price of less variance reduction (a factor around 3.5 beyond 50 km). These simulation results are supported by the analysis of a tornadic data case collected with the NWRT phased-array radar and processed using both a conventional matched filter and adaptive pseudowhitening. In this case, the Δ*V* degradation from range-oversampling processing was measured at ~4%, while the variance reduction from adaptive pseudowhitening was apparent in the smoother texture of the fields.

An operational implementation of range-oversampling processing on the WSR-88D must consider both operational benefits (e.g., reduced variance of estimates) and detrimental effects (e.g., degraded range resolution). However, the precise impact of such a trade-off on the warning decision process can only be assessed with a comprehensive study that involves NWS forecasters and a wide variety of weather scenarios. For this paper, we chose to focus on tornadic signatures because they are critical for tornado warnings and are likely to be affected by changes in the spatial resolution of the radar. In addition, tornado warnings involve risk to life and property and are some of the most visible products issued by the NWS. The NWS Warning Decision Training Division (WDTD) defines an “operator identified tornadic shear” as “an intense, roughly azimuthal shear associated with tornadic-scale rotation” (NOAA 2015a), which is assessed by a forecaster by interrogating superresolution Doppler velocity fields. As described in WDTD’s training course, three criteria are used by forecasters to identify a TVS: 1) minimal shear, 2) vertical extent, and 3) persistence. There is no hard lower threshold for the shear; an expert forecaster assesses the strength of the vortex by considering its range, its apparent size, the near-storm environment, and the forecaster’s experience. The vertical extent of the TVS should be at least 1500 m, and the signature must persist for 5 min or longer (NOAA 2015b).

Although a comprehensive assessment of the effects of range-oversampling processing on the tornado warning decision process is beyond the scope of this work, focusing on the three criteria mentioned above can give insight into how range-oversampling processing affects an important aspect of this process. To address the first criterion, minimal shear, we quantified the effects of range-oversampling processing on Δ*V* for tornadic vortices and found that the processing will have a small impact on the shear strength measured by the radar. The simulation study shows that the degradation in Δ*V* decreases for tornadoes at far ranges, making it less of an issue where superresolution data are more critical to the tornado warning decision process. Since forecasters do not use a hard lower threshold on Δ*V* for determining minimal shear, the predicted small degradation in Δ*V* should have a minor impact on this particular aspect of identifying tornadic shear. However, the expected variance reduction of Δ*V* (and of the entire Doppler velocity field) will lead to a more accurate assessment of the spatial and temporal continuity of the radar data. This will significantly contribute to forecaster confidence when assessing the other two criteria: vertical extent and persistence of the TVS. Overall, the benefits of range-oversampling processing in terms of improved data quality should outweigh the relatively small degradation to Δ*V* and thus contribute to the tornado warning decision process by improving forecaster confidence in the radar data.

The authors thank R. Brown, D. Zrnić, and three anonymous reviewers for providing comments that improved the manuscript. Funding was provided by NOAA/Office of Oceanic and Atmospheric Research under NOAA–University of Oklahoma Cooperative Agreement NA11OAR4320072, U.S. Department of Commerce.

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