## 1. Introduction

Both the mesocyclone and tornado detection algorithms currently used with the Weather Surveillance Radar 1988-Doppler (WSR-88D) system involve some measure of the azimuthal shear (e.g., Mitchell et al. 1998). Traditionally, the azimuthal shear of a circulation pattern has been estimated by finding the difference between the peak inbound and outbound velocity values at constant range and nearby azimuths, and dividing this difference by the distance between the two peak velocities. Since this simple azimuthal shear calculation only utilizes velocity data at two points, it is highly influenced by errors in the velocity field due to aliasing or noise. In addition, this “peak to peak” azimuthal shear calculation is highly affected by radar beam placement with regard to the circulation (Wood and Brown 1997).

The need for a radar-derived shear quantity that is more immune to noisy data and radar sampling issues prompted the development of the local, linear least squares derivatives (LLSD) method for calculating azimuthal shear (Smith and Elmore 2004). This method involves estimating the azimuthal derivative of the radial velocity by fitting the velocity field to a model with a low degree of freedom. Since the LLSD shear calculation is based on a low-order model, the resulting shear field is more noise tolerant than the traditionally used shear calculation (Smith and Elmore 2004). Several velocity values in the neighborhood of a particular point are used in the LLSD shear calculation, reducing the dependence of the shear calculation on any particular velocity value. Consequently, LLSD shear values of mesocyclone-scale circulations are not strongly affected by radar beam location or noise (Smith and Elmore 2004).

The LLSD shear calculation depends on the Doppler velocity field, which degrades with range as the radar beam widens and the number of scattering particles in the sampling volume increases; since more scatterers are included in the Doppler velocity calculation, high velocity values associated with circulations are averaged out at far ranges. Similarly, LLSD shear values of circulations degrade with range, as the values depend on Doppler velocity measurements (Smith and Elmore 2004). Thus, a weak circulation that is close to the radar could be associated with larger LLSD shear values than a strong circulation far from the radar. Ideally, the same shear threshold could be used to detect significant circulations regardless of their range from the radar; however, this is not possible if shear is range dependent.

To mitigate this issue, two range-correction methods, a linear regression model and an artificial neural network (ANN), were developed to correct the LLSD shear signatures of simulated circulations. By examining the range dependence of circulations of known strength and size, the true shear of the simulated circulations, calculated using their user-specified diameters and peak velocities, was related to radar-measurable parameters. Both range-correction methods were then applied to areas of tornadic and nontornadic shear in real radar data. The goal of the range correction is to produce shear values that depend more strongly on the strength and size of the circulation, as opposed to the range of the circulation from the radar.

## 2. Background

### a. Circulation detection algorithms

The tornado detection algorithm (TDA) that is currently used with the WSR-88D system employs the difference between velocity values at constant range and adjacent azimuths (the “gate-to-gate velocity difference”) to detect tornadic vortex signatures (TVSs); if this velocity difference exceeds a particular threshold, the gate pair is identified as a potentially tornadic region (Mitchell et al. 1998). Liu et al. (2007) discuss several issues related to using the gate-to-gate velocity difference to detect tornadoes. The actual value of this difference can be affected by the azimuthal offset of the radar beam center from the vortex (e.g., Wood and Brown 1997), as well as noisy data and velocity aliasing. In addition, as the radar beam widens with range, more scattering targets are used to estimate the Doppler velocity, leading to greater volume averaging and less accurate velocity measurements at far ranges. To mitigate these issues, Liu et al. (2007) propose using a wavelet analysis technique to measure the average velocity difference between regions of different scales. This technique utilizes velocity data at several different points, so it is less sensitive to noisy or inaccurate data.

As another alternative to the gate-to-gate method, Yu et al. (2007) suggest the use of Doppler spectra to identify tornadic regions in the radial velocity field. Unlike the TVS, Yu et al. (2007) argue that characteristics of a tornadic Doppler spectrum do not degrade significantly with range, making them ideal for tornado detection. Following the results of Yu et al. (2007), Wang et al. (2008) developed a TDA that uses spectral signatures and a fuzzy logic system to make detections.

### b. LLSD shear

*V*is the radial velocity,

_{r}*s*is the coordinate in the azimuthal direction,

*s*is the arc length from the center point of the calculation to the point (

_{ij}*i*,

*j*),

*V*is the radial velocity at (

_{ij}*i*,

*j*), and

*w*is a positive weight function that differs according to the current point (

_{ij}*i*,

*j*).

When the LLSD method is applied to radar data, the Doppler velocity data are first passed through a 3 × 3 median filter to reduce speckle noise. Next, Eq. (1) is applied to the filtered velocity data to estimate the azimuthal shear. The LLSD calculation is evaluated over a kernel of velocity data of a particular width and depth to estimate the LLSD shear of the center point of the kernel (Fig. 1). The radial and azimuthal dimensions of the kernel used in the calculation are held constant, such that fewer radials are used in the calculation at far ranges. Typical sizes for the radial and azimuthal kernels are 750 and 2500 m, respectively; these conventions were followed for this work.

### c. Advantages and disadvantages of the LLSD method

The robustness of the LLSD shear method was demonstrated by Smith and Elmore (2004). In their work, circulations were simulated with varying strengths and diameters at ranges from 20 to 200 km from a hypothetical radar using a Rankine combined vortex simulation (Wood and Brown 1997). Noise was added to the velocity data and random azimuthal offsets from the beam centerline were added to the circulation location to simulate real sampling limitations. The vortex was then “sampled” by a theoretical WSR-88D radar, with an azimuthal sampling interval of 1° and an effective beamwidth of 1.39° [i.e., a WSR-88D radar operating in legacy resolution mode; Brown et al. (2002)]. For each range, 1000 circulations were simulated with different Gaussian-distributed noise values and azimuthal offsets, and mean and 95% confidence intervals were calculated for the maximum LLSD and peak-to-peak shear values in the circulation. Calculations for a 5-km-diameter vortex using a 2.5-km azimuthal kernel indicated that the LLSD shear value is within 20% of its true value out to a range of about 140 km, with a much smaller variance than peak-to-peak shear (Smith and Elmore 2004).

However, the LLSD technique also has some disadvantages. Although the median filter used with the LLSD technique preserves areas with large velocity gradients, it also tends to smooth out peaks in the velocity field [see Mather (2004) for a more complete description of issues associated with median filters]. The smoothing of peaks can be beneficial if the peaks are spurious and correspond to noisy data. However, the median filter could effectively decrease the magnitude of the peak velocity values in a tornadic circulation. Thus, LLSD shear values may underestimate the true azimuthal shear of a circulation, particularly if the circulation is small (Mitchell and Elmore 1998).

Finally, in the LLSD technique, there is a trade-off between spatial resolution and noise resistance. Typical LLSD shear calculations use a kernel that is 2.5 km in azimuthal width. Smaller kernels tend to produce shear values that vary substantially when noise is incorporated, whereas larger kernels tend to underestimate the true shear values of circulations. Since the spatial scale of the calculation is approximately on the meso-*γ* scale (e.g., Markowski and Richardson 2010), the LLSD method will be most useful for detecting mesocyclone-scale circulations and large tornadoes. Thus, although the LLSD method cannot be used to detect small tornadoes, it will be more resistant to noisy data, which tend to occur in radar data on small spatial scales (e.g., Liu et al. 2007).

## 3. Range correction for LLSD shear

Like most other radar-dependent parameters, such as the peak velocity values of circulations, LLSD shear signatures degrade with range as a result of beam widening (e.g., Smith and Elmore 2004). In this work, a range correction was developed for the LLSD shear signatures of circulations. Rankine vortices were simulated and used to develop regression equations relating measured circulation parameters to the true azimuthal shear values of the simulated circulations. A linear regression model and ANNs were evaluated as possible range-correction methods.

### a. Description of Rankine vortex simulation

Various mesocyclone-scale circulations were simulated using the technique described by Wood and Brown (1997). This simulation creates modified Rankine (1901) vortices with user-specified peak velocities and core diameters, and then samples the vortices with a synthetic radar. For this work, simulated vortex core diameters ranged from 1 to 4 km in increments of 0.25 km, and peak velocities ranged from 20 to 50 m s^{−1} in increments of 2 m s^{−1}. Gaussian-distributed noise with a mean value of 0 m s^{−1} and a standard deviation of 2 m s^{−1} was added to the mean velocity values to simulate radar noise. The Gaussian distribution is often used to model radar noise in the velocity field, as the Doppler spectrum itself generally resembles a Gaussian distribution (Doviak and Zrnić 1993). In addition, random azimuthal offsets from the beam centerline were added to the circulation center location, as in Smith and Elmore (2004). For each circulation, 1000 simulations were produced with different distributions of random noise patterns and azimuthal offsets at ranges from 5 to 140 km in increments of 5 km.

A radar with a 0.5° azimuthal sampling interval and a 1.02° effective beamwidth was used to sample the vortices [i.e., a WSR-88D operating in superresolution mode; Brown et al. (2002)]. The simulated velocity field was then used to calculate the LLSD shear. Finally, the maximum velocity value, maximum LLSD shear value, and approximate diameter of each circulation were obtained from the simulated velocity data. The circulation diameter was estimated by finding the distance between peak positive and negative velocity values. Mean values of these parameters were calculated from all 1000 iterations for each circulation and used to develop the range correction. The target of the range correction is the “true” azimuthal shear, calculated from the true peak velocity and diameter values of the simulated circulations.

### b. Development of linear regression equations

#### 1) Selecting diameters for regression equations

To determine which circulations should be included in the range correction, shear error, defined as the difference between the true and measured LLSD shear values, was plotted for simulated circulations of different core diameters, peak velocities, and ranges (not shown). Since LLSD shear values are generally on the order of 0.01 s^{−1} for mesocyclone-scale circulations measured with a WSR-88D (Smith and Elmore 2004), a measured shear value that differed by more than 0.01 s^{−1} from the true shear value was flagged for a range correction. These simulations demonstrated that circulations with diameters greater than or equal to 3 km have LLSD shear values that are within 0.01 s^{−1} of their true shear values through a range of 140 km. Thus, only circulations with diameters from 1 to 2.75 km were used to develop the regression equations. In addition, the circulations were only simulated out to a range of 140 km from the radar; as a result, only fairly well-resolved circulations were included in the regression.

#### 2) Resolution limitations

Clearly, a circulation with a 1-km diameter will not be resolved at a range of 140 km. The minimum resolvable circulation diameter at a particular range is limited by the beamwidth of the radar, which widens with range. Since the range correction relies upon accurate circulation measurements, a range correction can only be applied to circulations that are well resolved.

Bluestein et al. (2010) classify well-resolved circulations as circulations with a horizontal scale that is at least 10 times larger than the spatial resolution of the radar beam. For their work involving a mobile X-band radar, this was a reasonable restriction, since the radar was always relatively close to the circulations being sampled. However, if this resolution restriction was applied to a WSR-88D, a 2.5-km vortex would only be “well resolved” at ranges out to about 15 km. Most circulations within the range of a WSR-88D will not meet the resolution restrictions used by Bluestein et al. (2010), so a different method was employed to determine which circulations were resolvable.

Using the Wood and Brown (1997) simulation, the mean estimated diameter at each range was calculated for circulations with peak velocities ranging from 20 to 50 m s^{−1} in increments of 2 m s^{−1} and true diameter values ranging from 1 to 2.75 km. If the estimated diameter differed from the true diameter by more than 10% for six or more assigned peak velocity values for a circulation at a particular range, the circulation was declared unresolvable at that range. (The “six or more” threshold was somewhat subjectively selected based on inspection of the simulated circulation variables.) Unresolvable circulations were not included in the regression equation development. The 10% rule was used to allow for changes in diameter due to noise, beam placement, and azimuthal resolution. Since the value of the estimated circulation diameter must be equivalent to integer multiples of the azimuthal resolution at each range, it is unlikely that the true diameter of a vortex will be measured.

#### 3) Regression procedure

A multiple linear regression technique was initially used to develop the range correction. Predictors for the regression were selected based on knowledge of the effects of different variables on azimuthal shear. For example, traditionally, azimuthal shear is calculated using the diameter and velocity difference of a circulation, so these parameters were candidates for predictors. The range of the circulation from the radar and the peak LLSD shear value of the circulation were also considered as predictors. All predictors varied approximately linearly with the true shear of the circulation, except for circulation diameter; the inverse of the diameter varied linearly with the true shear.

To examine the range degradation of LLSD shear signatures in greater detail, the ratio of measured LLSD shear to the true shear of different simulated circulations was plotted versus range (Fig. 2). Of the eight circulation sizes shown, the four smallest circulations have shear signatures that appear to decrease rapidly with range within the first 30 km of the radar. As a result of this rapid shear decrease, beyond 30 km, the smallest circulations have shear values that have degraded much more significantly in comparison to the shear values of the larger circulations. Because of this discrepancy, a piecewise regression was initially chosen for the range correction.

The smallest circulations (diameters from 1 to 1.75 km) were included in one regression procedure and the largest circulations (diameters from 2 to 2.75 km) were included in another procedure. Based on the resolvable ranges of the circulations, the small-diameter equation was only developed for circulations less than 85 km from the radar and the large-diameter equation was developed for circulations less than 135 km from the radar.

Predictors for both equations were screened using a forward selection technique (e.g., Wilks 2006); the results of the procedure are shown in Tables 1 and 2. A cross-validation procedure was used to examine the results of the regression on independent testing sets. The regression equation was developed using 90% of the data and tested on the remaining 10% of the data; each regression procedure was repeated 10 times using different training and testing sets to obtain mean and standard deviation values for the goodness-of-fit parameters.

Results from the forward-selection procedure for the small-diameter equation. Rows indicated in boldface depict the predictor that was added to the linear regression at each step. Mean values of parameters from 10 cross-validation procedures are shown, with standard deviations shown in parentheses. Table modeled after Wilks (2006).

As in Table 1, but for the large-diameter equation.

To select the number of predictors to include in each regression procedure, the mean squared error (MSE) and predictor correlations of the regression equations were examined. Figure 3 shows the MSE of the regression equations as a function of the number of predictors included in the regression. For the small-diameter equation, the change in the mean and standard deviation of the MSE as a result of the inclusion of the fourth predictor was negligible, suggesting that a good fit could be achieved with only the first three predictors. In contrast, the MSE of the large-diameter equation decreased steadily as each predictor was added to the regression procedure.

As three of the four predictors (shear, diameter, and peak velocity) are related to the true circulation strength, it is possible that two or more of these predictors are correlated. To investigate possible predictor correlations, the correlation matrices for the predictors for both the small- and large-diameter equations were calculated (Table 3). Based on the Pearson correlations, it is evident that the peak velocity, the LLSD shear, and the inverse of the estimated diameter are inversely related to the range of the circulation; this indicates, not surprisingly, that the estimated diameters of circulations increase with range and the shear and estimated peak velocities decrease with range.

Correlation matrices for predictors in small- and large-diameter equations. The Pearson correlation is shown for each unique pair of predictors. Table modeled after Wilks (2006).

The LLSD shear is directly related to the peak velocity for both the small-diameter (correlation = 0.671) and large-diameter (correlation = 0.933) equations. These correlations are significantly high when considering a regression procedure that assumes uncorrelated predictors. Therefore, only the first three predictors were used in the small- and large-diameter equations. By eliminating the LLSD shear predictor from the small-diameter equation and the peak velocity predictor from the large-diameter equation, all predictor correlations remained fairly low (<0.5).

#### 4) Resulting equations

*s** is the corrected shear,

*D*is the diameter (km),

*V*is the maximum radar-estimated velocity (m s

^{−1}), and

*r*is the range (km). The small-diameter equation can be applied at ranges out to 85 km for diameters between 1 and 1.75 km with the following restrictions:

- if the range ≥50 km, diameter must be ≥1.25 km;
- if the range ≥65 km, diameter must be ≥1.5 km; and
- if the range ≥75 km, diameter must be ≥1.75 km.

*s*is the maximum LLSD shear (s

^{−1}). The large-diameter equation can be applied at ranges out to 135 km for diameters between 2 and 2.75 km with the following restrictions:

- if the range ≥100 km, diameter must be ≥2.25 km;
- if the range ≥110 km, diameter must be ≥2.5 km; and
- if the range ≥120 km, diameter must be ≥2.75 km

The results for the small- and large-diameter equations are shown in Fig. 4. There are several corrected shear estimates for every true shear value because corrected shear values for all ranges are shown; different corrected shear estimates correspond to different ranges. Corrected shear results are shown for circulations of all core diameters and peak velocities that were included in the regression.

The correlation coefficient (*R*^{2}) values for both the small- and large-diameter equations are fairly high, 0.963 and 0.902, respectively, for the training sets (values for the testing sets were similar). For both equations, the regression appeared to show the most skill for circulations of intermediate diameters and velocities.

Figure 5 shows the residual distribution for the regression equations, defined as the difference between the true shear values and the corrected shear values. Residuals are plotted as a function of radar-estimated peak velocity and true circulation diameter, showing that the magnitude and sign of the residuals for both equations depend on the strength and size of the circulation.

The residuals for both equations display nonconstant variance, or heteroscedasticity (Wilks 2006). Ideally, the residuals in Fig. 5 would scatter evenly about the zero residual line. Instead, the residuals display fanning for both small and large peak velocities. Unfortunately, this discrepancy in the residual values is an effect of both radar sampling and the LLSD shear method, and would be difficult to remedy. One option could involve creating a different regression equation for every different possible circulation diameter, although this would be computationally expensive and difficult to implement. Another alternative is to use a nonlinear correction technique, such as a neural network, as discussed in the next section.

### c. Artificial neural networks

Nonlinear regression procedures proved to be better suited to the complex nature of the simulated circulation dataset. Using compositions of several functions, ANNs can approximate nonlinear relationships and map virtually any set of input parameters to a set of related output values (Bishop 1995).

To examine the capability of ANNs to estimate the range-corrected shear, all four predictors from the Rankine vortex simulations were entered into the Neural Network Toolbox in MATLAB. A two-layer feed-forward network with a sigmoid transfer function in the inner, hidden layer and a linear transfer function in the output layer was employed. The simulated data were divided into three subsets: 70% of the data were used for training, 15% were used for validation, and the remaining 15% were used for testing. Network weights were initially determined using the training set, then refined using the validation set, and tested on unfamiliar data in the testing set. Five hidden neurons were used in the neural network, and early stopping (e.g., Bishop 1995) was used to prevent overfitting. In the early stopping procedure, the neural network continues to update the network weights until performance on the validation set no longer improves. This prevents the network from overgeneralizing to the training set (Bishop 1995).

The ANN procedure was repeated 10 times with different training, validation, and testing sets. The mean and standard deviation of the *R*^{2} values for the neural network fits were 0.999 69 and 5.5150 × 10^{−5}, respectively, including results from the training, validation, and testing sets. The MSE value for the training sets was about 1 × 10^{−7} s^{−2}, and the MSE value for the validation and testing sets was about 4 × 10^{−8} s^{−2} for all 10 ANNs.

Figure 6 shows the corrected shear values produced by the ANN. The neural network fit is very close to the simulation shear values; the correlation coefficient value exceeded 0.99 for all 10 instances of the ANN. It is not surprising that an ANN provided a better fit for the data, as ANNs can approximate complex, highly nonlinear functions, unlike linear regression techniques.

The residuals for one instance of the ANN fit are shown in Fig. 7. With the exception of a few outliers, the residuals are scattered evenly about the zero residual line, indicating that the residual variance is nearly constant. The ANN took approximately the same amount of computational time for training as the multiple linear regression and calculated the corrected shear faster than the linear regression equations for real radar data. Thus, the ANN appeared to be a more efficient and accurate range-correction method for the simulated circulations.

For comparison with Fig. 2, Fig. 8 shows the ratio of the corrected shear to the true shear for circulations of different diameters located at ranges from 5 to 90 km. Both the linear regression and ANN methods produce corrected shear values that are very close to the original shear values. In particular, the ANN correction produces shear values that degrade very little with range.

## 4. Application of range correction to real data

### a. Range-correction rules

The range-correction methods were also applied to real WSR-88D data. First, a set of rules was defined to locate and range correct the LLSD shear values of significant circulations in real radar data. These rules are described in the following sections and are summarized in Fig. 9.

#### 1) Locating circulations in radar data

Before the range-correction algorithm is applied to a real dataset, initial areas of significant shear are identified by using reflectivity and azimuthal shear criteria. As a first step, the velocity difference, diameter, and range-correction calculations are only applied in regions where the reflectivity is greater than 20 dB*Z* and the LLSD shear exceeds a baseline value (0.005 s^{−1}). A dilation filter is applied to the stamped-out areas before the range-correction parameters are calculated. The dilation filter artificially expands the boundaries of the stamped-out areas to prevent the algorithm from missing peak velocities on the edges of high-shear regions.

The reflectivity and LLSD shear criteria ensure that the range correction is applied to areas of significant shear that are not associated with the noisy data that typically exist outside storm areas. Moreover, the range correction is limited to pixels where the immediate neighboring pixels in range have LLSD shear values higher than 0.007 s^{−1}. This rule is imposed to restrict the application of the range correction to local maxima in the shear field. In a study of 31 tornadoes, discussed later in this section, it was found that nearly all the tornadoes had maximum LLSD shear values that exceeded 0.007 s^{−1}.

#### 2) Maximum velocity and shear diameter estimates

The peak-to-peak velocity difference in the azimuthal direction is calculated for range gates that satisfy the reflectivity and LLSD shear criteria. At each range gate, a 3-km azimuthal search radius is used to find the maximum and minimum velocity values at that range; the difference between these velocities is the peak-to-peak velocity difference. The maximum velocity for range gates in circulation areas is determined by dividing the peak-to-peak velocity difference in half. Since the velocity difference is independent of storm motion, the maximum velocity value computed in this manner is also unaffected by storm motion.

The shear diameter is defined as the azimuthal distance between velocity values corresponding to the peak-to-peak velocity difference. Since the shear correction was primarily developed for radar-resolvable circulations, shear diameters smaller than 1 km, which are often associated with strong, small-scale circulations, are automatically set to 1 km. This diameter represents the smallest circulation that was included in the regression procedure.

The shear that is calculated with the range-correction equations (the “corrected shear”) is evaluated on a pixel-by-pixel basis. The LLSD shear, maximum velocity, and shear diameter for each range gate and azimuth are used to compute a unique corrected shear value for that pixel. Although the same maximum velocity and shear diameter values are typically found at neighboring azimuths, these parameters can vary significantly with range (Fig. 10). Specifically, shear diameter estimates can vary as much as 2 km from one range gate to another along the same azimuth. To smooth the diameter field and potentially provide more accurate estimates of circulation diameters, a median filter is applied to the shear diameter field before the corrected shear is calculated.

One undesirable effect from the diameter median filter is the smoothing of small circulations associated with TVSs, radar signatures that occur when Doppler velocity maxima of opposite sign are located at adjacent azimuths (Brown et al. 1978). Therefore, to allow for small circulations, the range correction has two modes: mesocyclone mode and TVS mode. Usually, the range-correction algorithm operates in mesocyclone mode and uses the median-filtered diameter to calculate the corrected shear. When the gate-to-gate velocity difference at a pixel is approximately equal to the peak-to-peak velocity difference at that pixel (i.e., within 3 m s^{−1}), the range-correction algorithm switches to TVS mode. Instead of incorporating the median-filtered diameter, the algorithm uses the original shear diameter at that pixel to calculate the corrected shear. Consequently, the corrected shear values of small circulations are still calculated.

### b. Examples

Both the linear regression and the ANN range corrections were applied to 31 tornadic circulations from 8 tornado events, and the corrected shear values were compared to the original LLSD shear values along the tornado paths. In addition, both range-correction methods were applied to data from 10 nontornadic mesocyclones to examine the effect of the correction on areas of high shear that were not associated with tornadoes. For both the tornadic and nontornadic cases, maximum LLSD shear, peak velocity, and estimated diameter values from the 0.5° elevation angle were extracted along paths of significant shear clusters and used as input values for the range-correction equations.

#### 1) Tornadic cases

The eight tornado events used for this study are summarized in Table 4; in total, 31 tornadoes were associated with these events. The 31 tornadoes selected for the study occurred in a variety of geographical locations in association with several different convective modes. For each event, data from the nearest WSR-88D were obtained and used to calculate the LLSD shear and the corrected shear. A 3-km search radius was used to find the maximum LLSD shear value and corrected shear value along the approximate tornado path for each 0.5° scan. Paths were determined by manually tracking regions of maximum LLSD shear, using ground truth reports from the Storm Prediction Center or the National Climatic Data Center’s *Storm Data* database for guidance where available. The start and end times of the tornadoes were assumed to correspond to the first and last damage reports, respectively, although it is possible that these damage reports are slightly misplaced in space and/or time (e.g., Witt et al. 1998).

Tornado cases used for the LLSD shear range-correction study.

In general, for the tornadic cases, both range-correction methods increased shear values substantially (Fig. 11) and made it easier, retrospectively, to differentiate between tornadic and nontornadic radar scans. The corrected shear tended to increase substantially when the tornado began to cause damage and decrease significantly after the tornado had dissipated. In contrast, the original LLSD shear showed this temporal trend far less frequently than the corrected shear.

Examples of the tornadic corrected shear trend are shown in Fig. 12 for the 10 February 2009 Oklahoma case. On this day, a cyclic supercell moved through the western side of Oklahoma City and produced tornadoes rated on the enhanced Fujita scale as category 1, 2, and 0 (EF1, EF2, and EF0) at approximately 2034, 2053, and 2123 UTC, respectively. The cyclic supercell was tracked for 1 h while it was located between 35 and 55 km from the Twin Lakes, Oklahoma, WSR-88D. The range correction was applied to maximum low-level LLSD shear values along the supercell’s path.

The aforementioned tornado times correspond well to peaks in the corrected shear time series produced by both range-correction methods; this signal is not as clear in the noisier original LLSD shear field (Fig. 12). In addition, the relative magnitudes of the peak corrected shear values are positively correlated to tornado damage intensity. The peak corrected shear values during the time of the EF2 tornado are much larger than the peak corrected shear values during the time of the EF0 tornado. The largest corrected shear value of the time period was produced at 2101 UTC, when a strong TVS was apparent in the radial velocity field (Fig. 13).

Although both range-correction methods produced similar results, the ANN performed slightly better than the linear regression toward the end of the dataset, when the range of the supercell from the radar had increased. The ANN produced a slightly larger increase in the corrected shear field for the third tornado in comparison to the linear regression (Fig. 12), as it was better able to capture the nonlinear range degradation of the LLSD shear, producing better results at far ranges.

The 22 October 2009 Louisiana case also exemplified the corrected shear trends. The primary convective mode on this day was quasi linear, but a few supercells formed in southeastern Louisiana and produced tornadoes. One of these supercells was tracked for 2 h as it produced three EF0 and two EF1 tornadoes. The supercell was located at ranges between 45 and 92 km from the radar when it was tracked. Tornadoes were produced at approximately 1705, 1722, 1747, 1805, and 1815 UTC.

Based on the shear time series, the original measured shear values between 1730 and 1815 UTC are significantly smaller than those measured at the beginning of the dataset for this case (Fig. 14a). At the end of its lifetime, the supercell had started moving away from the radar; this difference in range could explain the decrease in measured shear magnitude. The range difference appeared to have an impact on the corrected shear values produced by both methods as well (Figs. 14b,c), since more resolution restrictions were applied at farther ranges. However, for this event, the linear regression corrected shear values were not affected as strongly by range compared to the ANN corrected shear values, based on the corrected shear trends toward the end of the time period (Figs. 14b,c). In contrast to the 10 February 2009 case, the linear regression appeared to perform slightly better than the ANN at far ranges for the 22 October 2009 case.

As in the Oklahoma case, the corrected shear values produced by both methods are much larger than the original shear values; at times, the difference is close to an order of magnitude. Furthermore, the peaks in the corrected shear time series correspond well to tornado formation times, appearing in either the volume scan when the tornado began or in the volume scan prior to tornado formation. Unlike the Oklahoma case, tornado damage intensity could not be discerned from the corrected shear magnitudes from either method.

Shear values from all 31 tornadoes were examined to determine how frequently the “tornadic trend” in the corrected shear field occurred. A tornadic trend was defined as a large increase in the 0.5° corrected shear from one scan to the next (approximately 5 min for a WSR-88D operating in precipitation mode). To determine the optimal shear increase for the tornadic trend, the shear increase threshold was varied from 0.0005 to 0.01 s^{−1} for 8 tornadic cases (Table 4) and 5 nontornadic cases, which included 10 nontornadic mesocyclones (Table 6). The tornadic trend shear threshold was defined as the threshold that maximized the Heidke skill score (HSS; Wilks 2006). The HSS was used as the scoring measure because it takes correct forecasts due to chance into account and has been used to score other circulation detection algorithms (e.g., Stumpf et al. 1998). Based on this method, the tornadic trend shear thresholds for the linear regression and ANN were 0.0065 and 0.0075 s^{−1}, respectively.

The tornadic trend results for the corrected and original shear are shown in Table 5. Approximately 52% of the tornadoes in the test dataset were associated with tornadic trends in the linear regression corrected shear field, while only 25.8% of the tornadoes produced tornadic trends in the original shear field when the linear regression threshold of 0.0065 s^{−1} was applied to the uncorrected LLSD shear field. When the ANN was used for the corrected shear field, nearly 42% of the tornadoes showed a tornadic trend in the corrected shear field. Not surprisingly, when the higher ANN tornadic trend threshold of 0.0075 s^{−1} was applied to the uncorrected LLSD shear field, only 12.9% of the tornadoes showed a tornadic trend in the original uncorrected shear field. There were three commonalities associated with the tornadoes that did not produce a corrected shear trend: 1) tornado located at ranges far from the radar (>100 km; Fig. 15), 2) tornado in an area with radar data quality issues, or 3) EF0 or EF1 tornado unresolved by radar (Fig. 15).

Tornadic trend counts for corrected and original shear for 31 tornadoes. Numbers under the corrected trend (LR) and corrected trend (ANN) columns indicate the portion of tornadoes for each event for which a tornadic trend was evident in the corrected shear time series when the linear regression and artificial neural network were used, respectively, with thresholds discussed in the text. Numbers under first and second original trend columns indicate the portion of tornadoes for each event for which a tornadic trend was evident in the original shear time series when the linear regression and artificial neural network thresholds were used, respectively.

Overall, in these cases, trends in the corrected shear field produced by both range-correction methods proved to be a better indicator of tornadogenesis in comparison to the original LLSD shear field. However, the corrected shear trend was not evident for some tornadoes at far ranges, even for the ANN method, which was likely a consequence of the range restrictions imposed on the range-correction algorithm. In addition, the increasing height and width of the radar beam was likely partially responsible for the poor performance of the range correction at far ranges. As the range from the radar increases, the earth’s surface curves beneath the radar beam, causing the beam to sample scatterers that are higher in the atmosphere. The radar beam also expands with increasing range, as discussed in previous sections. Thus, at far ranges, the radar is sampling a large volume of air that is 2–3 km above the ground and is not representative of the small-scale tornadic circulation.

#### 2) Nontornadic cases

Large shear values can be associated with mesocyclones in supercell thunderstorms, but it has been estimated that less than 30% of mesocyclones actually produce tornadoes (Trapp et al. 2005). To investigate the effect of the range correction on areas of nontornadic shear in mesocyclones, the range-correction techniques were tested on 10 mesocyclones from 5 different events.

The nontornadic mesocyclone events are summarized in Table 6. For each event, the LLSD and corrected shear were calculated using data from the closest WSR-88D. Small areas of significant low-level LLSD shear were identified and tracked as the shear increased in response to the strengthening mesocyclone. Maximum original and corrected shear values from the 0.5° elevation angle were extracted in a procedure identical to that used for the tornadic cases.

Summary of nontornadic mesocyclone cases used to test range-correction algorithm.

The results of the range-correction methods on the nontornadic cases are given in Table 7. The linear regression range correction performed well for over two-thirds of the nontornadic cases; for these cases, the corrected shear did not increase or decrease significantly with time and did not produce the tornadic trend seen in the tornadic cases. For many 0.5° scans, the range correction was not applied to areas of high LLSD shear because the shear diameters were large and thus did not require range corrections. However, 3 of the 10 azimuthal shear maxima in the nontornadic cases with mesocyclones displayed tornadic trends with the linear regression method (Table 7). It is likely that further training or more advanced discrimination methods would need to be used in the future to differentiate tornadic mesocyclones from nontornadic mesocyclones with the range-correction method. All of these mesocyclones were also associated with tornadic trends in the original shear field, so the tornadic trend cannot be attributed only to the range correction for these cases. The ANN performed similarly to the linear regression method for the nontornadic cases.

As in Table 5, but for nontornadic cases.

## 5. Summary and conclusions

Motivated by a desire to improve the poor performance of current circulation detection algorithms, the LLSD method was developed as an alternative to the peak-to-peak azimuthal shear calculation. By incorporating a low-order model and several surrounding velocity points, the LLSD method provides more noise-resistant estimates of azimuthal shear and could be the focus of a future circulation detection algorithm.

In this work, an algorithm was developed to correct range-degraded values of LLSD shear. To develop the range correction, thousands of mesocyclone-scale Rankine vortices were simulated and sampled with a synthetic WSR-88D. Regression equations were developed to relate radar-measured parameters to the true azimuthal shear values of the simulated circulations. Two regression techniques were investigated for the range correction: multiple linear regression and an ANN.

Two separate multiple linear regression procedures were used to create two range-correction equations: a small-diameter equation for smaller circulations closer to the radar, and a large-diameter equation for larger circulations. Both equations provided fairly good fits for the simulated circulation data, producing coefficient of determination values greater than 0.9. However, the residuals for both equations were largely heteroscedastic, implying that a linear model was not a good fit for the circulation data.

Owing to these problems, an ANN was then investigated as a potential range nonlinear correction option. The simulated circulation data were used as input parameters for a simple, two-layer feed-forward ANN. The resulting network fit the simulated data extremely well (*R*^{2} > 0.99), and the residuals were scattered evenly around the zero line, indicating that the residual variance was nearly constant and did not vary significantly for different values of the input and output parameters.

To test the range-correction methods on real data, the resulting range corrections were applied to shear values from 31 tornadoes from 8 events. The range-corrected shear from both methods was much higher than the original shear (close to an order of magnitude higher, in some cases) and hence, was more representative of a tornadic circulation. In addition, the range-corrected shear tended to increase substantially during tornadogenesis, displaying a clear tornadic trend for 51.6% of the tornadoes for the linear regression method and 41.9% of the tornadoes for the ANN method. In contrast, this trend only appeared in the original shear time series for 25.8% and 12.9% of the tornadoes in the dataset when the linear regression and ANN thresholds were applied, respectively. Currently, this tornadic corrected shear trend is only evident from one volume scan to the next (about every 5 min). However, as scan update time decreases with future radar systems [e.g., phased-array radar; Zrnić et al. (2007)], it will be easier to detect upward trends in the low-level corrected shear.

Additionally, the range-correction methods were applied to high-shear areas from nontornadic mesocyclones. For the majority of the nontornadic cases, both the linear regression method and the ANN produced corrected shear estimates that were generally on the same order of magnitude as the original shear and did not display large increases with time.

One remaining limitation of the range-correction algorithm is its poor performance at ranges far from the radar; this is likely due to the range restrictions imposed on the algorithm. As tornadoes at far ranges are more likely to occur without official National Weather Service warning (Brotzge and Erickson 2010), it is most important to apply range corrections to these tornadoes. In the future, the restrictions of the range-correction algorithm may need to be adjusted so that the shear signatures of small circulations at far ranges are still corrected. In addition, consideration would need to be given to the radar horizon issue, since at far ranges, the radar beam is too high to sample low-level circulations.

Funding for the authors was provided by NOAA/Office of Oceanic and Atmospheric Research under NOAA–University of Oklahoma Cooperative Agreement NA17RJ1227, U.S. Department of Commerce. The authors thank Bim Wood and Rodger Brown for their assistance with the Rankine vortex simulation and Alan Shapiro and Madison Miller for their valuable input. Comments from two anonymous reviewers helped improve the quality of the manuscript.

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