1. Introduction
The accurate detection and forecast of severe and hazardous weather such as thunderstorms, downbursts, mesocyclones, and tornadoes are important missions for the National Weather Service (NWS) and have significant impact on society. The Weather Surveillance Radar-1988 Doppler (WSR-88D) has provided operational forecasters and researchers valuable information about these phenomena leading to important nowcasts and short-term forecasts (Serafin and Wilson 2000). It has been shown that the warning lead time has been increased significantly after the installation of the WSR-88D (Polger et al. 1994; Bieringer and Ray 1996; Simmons and Sutter 2005). The current operational tornado detection algorithm relies on the tornadic vortex signature (TVS) in the field of Doppler velocity, which is a strong localized shear between two adjacent gates along azimuth at constant range (i.e., gate-to-gate shear or azimuthal shear; Mitchell et al. 1998). However, the shear signature becomes difficult to identify if the tornado is weak or is located at far range (e.g., Brown and Lemon 1976), even after the statistical error of velocity data is reduced using the 2D linear least squares method (Smith et al. 2004). Recently, it has been shown that the shear signature can be enhanced using half-degree angular sampling (Brown et al. 2002, 2005). In their approach, the effective beamwidth is reduced at the expense of increasing statistical errors because the number of samples used in the estimation is halved. Liu et al. (2007) have proposed a different approach of radar sampling using overlapped beams without comprising statistical accuracy. In their approach, enhanced tornadic vortex structures can be revealed using the variational analysis of oversampled velocity data. Furthermore, distinct polarimetric signatures of debris within a tornado have the potential for the improvement of tornado detection (Ryzhkov et al. 2005). In this work, a higher-order spectrum (HOS) and signal statistics are applied to characterize the tornado signatures in the spectral domain.
A pioneering work in the measurement of tornado spectra was done using a 3-cm continuous wave (CW) radar (Smith and Holmes 1961). Atlas (1963) expected a broad and flat spectrum to be observed by a pulsed radar if a tornado is within the radar resolution volume. Analytical simulations have shown that a tornado spectral signature (TSS) with broad bimodal pattern would result if the tornado is centered close to the radar beam (Zrnić and Doviak 1975). Such broad bimodal signatures were then verified by both pulsed Doppler radar (Zrnić and Istok 1980; Zrnić et al. 1985) and mobile frequency modulated CW (FM-CW) radar (Bluestein et al. 1993, 1997) with extremely high maximum unambiguous velocity of approximately 90 m s−1. Although the history of tornado spectrum measurements is long, the number of observed and studied cases is relatively small. This is largely because neither the technology to process spectra nor the technology to record voluminous amounts of time series data was readily available. Recently, Yu et al. (2003, 2004) reported that tornado spectra observed from systems and settings that are similar to the operational WSR-88D are often flat, similar to white noise spectra. These results have shown that the Doppler spectra from tornadic regions have a distinct character that sets them apart from other spectra. Therefore, it is important to systematically investigate and quantitatively characterize TSS observed by modern radar systems.
HOS, especially the third-order spectrum (termed bispectrum), has been applied to many fields such as pattern recognition and signal classification (e.g., Chandran and Elgar 1993; Shao and Celenk 2001; Zhang et al. 2001). It has been shown that most information about the shape of a signal is often contained in the phase of the Fourier coefficients (Oppenheim and Lim 1981). Unlike the widely used power spectrum (i.e., the second-order spectrum), HOS has the advantage of retaining both the phase and amplitude information. Therefore, HOS has the potential to characterize the distinct features in tornado spectra as a pattern recognition problem. A review of bispectrum estimation and its application can be found in Nikias and Raghuveer (1987). In the present work, a feature parameter derived from the phase of radially integrated bispectrum (PRIB) of the Doppler spectrum in the decibel scale (dB) is introduced to differentiate tornado spectra from typical Gaussian-shaped spectra. PRIB has several interesting invariants including dc-shift invariant, translation (or shift) invariant, amplification invariant, and scale invariant, which are important for pattern recognition (Chandran and Elgar 1993; Shao and Celenk 2001). In addition, the spectrum flatness and the spectrum width are proposed to identify TSS.
The paper is organized as follows. An overview of tornado spectrum is presented in section 2. Numerical simulations of Doppler spectra are developed. The shapes of tornado spectra as a function of several parameters are shown and discussed. The characterization of TSS is developed in section 3 using spectral flatness and HOS. The performance of both parameters and spectrum width for TSS characterization are analyzed statistically using simulations in section 4. In section 5, characterization of TSS is demonstrated and verified using data collected by the research WSR-88D (KOUN) operated by the National Severe Storms Laboratory (NSSL) from a tornado outbreak in central Oklahoma on 10 May 2003. A summary and conclusions are presented in section 6.
2. Tornado Doppler spectrum
Numerical simulation of tornado spectra is developed to include features such as multiple vortices, constant background flow, and horizontal shear. To simplify the problem, it is assumed that the reflectivity and velocity are independent of height. In addition, the velocity field of each vortex is modeled by a combined Rankine vortex (Kundu and Cohen 2002). A virtual WSR-88D with 1° beamwidth (θb) and 250-m range resolution is used. Consequently, the level I time series data are generated with desirable signal-to-noise ratio (SNR) and the three spectrum moments are estimated using the autocovariance method. A detailed description of the simulation is provided in the appendix. A similar approach was used in Bluestein et al. (1993), but the velocity field in that study was modeled by a single vortex, and no level I data were generated. Note that the approach used in this work is flexible enough to simulate other types of radars or a network of radars such as the Collaborative Adaptive Sensing of the Atmosphere (CASA) radars (e.g., Brotzge et al. 2005). Moreover, different vortex model such as Burgers–Rott, which has been shown to better describe some tornado vortices (e.g., Bluestein et al. 2003, 2007), can also be implemented.
Dependence of tornado spectra on the range, background wind, horizontal shear, and reflectivity structure is demonstrated in Fig. 1.
The default parameters in the simulation are presented as follows, with only one parameter changing for each of these cases. The mean background flow is zero (υ0 = u0 = 0 m s−1) and no shear is presented (υ′x = dυ/dx = 0 s−1, u′y = du/dy = 0 s−1). The reflectivity of the tornado is simulated using Eq. (A2) with Wz = 60 m and r0z = 220 m. As a result, a pattern of low reflectivity surrounded by high reflectivity (i.e., doughnut-shaped reflectivity) is produced (as depicted in Fig. A1 of the appendix), whereby the location of maximum reflectivity is slightly outside the tornado core (Dowell et al. 2005). No mesocyclone is included in this case. The maximum unambiguous velocity (υa) is set at 65 m s−1 such that no velocity aliasing occurs. Moreover, the radar resolution volume is collocated with the center of the vortex. Ideal tornado spectra at three different ranges (r′0 = 5, 25, and 75 km) are shown in Fig. 1a. Although a constant noise level of −50 dB is used, the statistical fluctuation in Doppler spectrum (Zrnić 1975) is not included in this case for the purpose of clear demonstration. It can be observed that the tornado spectrum becomes wider and bimodal when the range increases from 5 to 25 km. In other words, larger velocity components can be observed due to the increase of radar resolution volume with range, until the maximum radial velocity from the vortex is reached. Beyond that, the dependence of spectrum pattern on range is relatively small as shown in the figure for r′0 increases from 25 to 75 km. The results are consistent with previous results presented by Zrnić and Doviak (1975) in which a different simulation approach was used. In the upper right panel, Doppler spectra from a tornado within a southerly flow at 25 km with magnitude of 0, 5, and 15 m s−1 are shown. It is clear that the constant background wind shifts the spectrum by its radial component without varying the spectrum pattern. Therefore, it is desirable that the parameter characterizing TSS is not sensitive to the shift of the pattern. In the lower left panel, it is expected that the spectrum can be further broadened by shear (Doviak and Zrnić 1993). On the lower right panel, the results exhibit that different reflectivity structures do not change the maximum radial velocity, but can have a significant impact on the shape of the tornado spectrum. For the case of Wz = ∞, which indicates a uniform reflectivity, a spectrum of approximately rectangular pattern is obtained. Note that other factors such as the location of the tornado within the radar resolution volume and velocity aliasing can further alter the spectrum shape. In practice the statistical fluctuations in the spectrum is inevitable and could degrade the signatures. Moreover, a spectrum is obtained by signals from a 3D radar resolution volumes, and therefore the spectrum shape will also depend on the vertical profile of reflectivity and velocity distribution within the radar resolution. It was shown that reflectivity with weak echo region at low level and higher echoes aloft are observed (e.g., Lemon 1980). Several high-resolution vertical profiles of tornado structure have been reported using mobile Doppler radars (e.g., Wurman and Gill 2000; Alexander and Wurman 2005; Bluestein et al. 2004). The Doppler spectrum from a 3D radar volume can be considered as a superposition of the spectra from the 2D simulations at a number of heights within the radar volume that are weighted by the antenna pattern in the vertical direction.
To further demonstrate the unique features of tornado spectrum, an idealized tornado with a mesocyclone is simulated. The model reflectivity and horizontal velocity fields are shown in the upper left of Fig. 2.
Both the tornado and mesocyclone are modeled by the Rankine vortex with a core radius of 200 m and 2 km, respectively. The centers of the tornado and mesocyclone are separated by 1 km. The maximum tangential velocity for the tornado and mesocyclone is 50 and 25 m s−1, respectively, and a maximum radial velocity of 5 m s−1 is used for both vortices. A constant background wind of (u0 = 5 υ0 = 10 w0 = 0) m s−1 and horizontal shear in the x direction υ′x = 0.01 s−1 are also included. Moreover, the reflectivity structures associated with the mesocyclone and tornado are simulated by a Gaussian and doughnut-shaped function, respectively. Spectra observed by a virtual WSR-88D from the tornadic region, as specified by the black box in Figs. 2a and 2b, are shown in decibels in the lower panels. The bottom panels represent five consecutive radials and ranges. The location of the center of radar resolution volume is denoted by (r0, ϕ0), where r0 is the range and ϕ0 is the azimuth angle of radar beam. The maximum unambiguous velocity is 35 m s−1, and therefore velocity aliasing occurs. Moreover, a constant noise level of −55 dB with the statistical fluctuations is simulated to generate 64 time series data points using the scheme proposed by Zrnić (1975). The field of the mean Doppler velocity observed by the radar is shown in the top right panel and the mean radial velocity is also indicated in each spectrum by the location of triangles. In addition, the inbound and outbound directions are denoted by downward and upward triangles, respectively. Note that no apparent bimodal signature is observed given υa = 35 m s−1. Instead, spectra with signatures similar to white noise spectra, whereby values in (−υa υa) are almost uniform, are observed from the gates close to the tornado center such as those from r0 = 24.75, 25.00, 25.25 km and ϕ0 = −0.2°. The location of tornado center is denoted by (r′0, ϕ′0), where r′0 = 25 km and ϕ′0 = 0°. Note that the mean Doppler velocity at radials of ϕ0 = −1.2°, −0.2°, and 0.8° and range of 25 km are aliased. The mean Doppler velocity at −1.2° and 0.8° can be dealiased by subtracting and adding 2υa, respectively. However, the mean Doppler velocity at ϕ0 = −0.2° and 25 km cannot be dealiased easily due to the aliased version of flat spectrum. Moreover, spectra with narrower widths and Gaussian-like shapes can be observed from the radar resolution volumes that are located away from the tornado center in both azimuth and range direction. Those spectra with broad and flat features may have the potential to facilitate tornado detection and are of interest.
3. Characterization of tornado spectral signatures
The tornado spectrum results from an interplay of the reflectivity, velocity distributions, and radar sampling. It has been shown in the previous simulation that spectra from a tornadic region could significantly deviate from a well-defined Gaussian shape. These spectra are of primary interest and could have the potential to facilitate the detection of tornado vortices if TSS can be characterized and quantified with appropriate parameters. An intuitive parameter to characterize TSS is the spectrum width (συ), which is the square root of the spectral second moment and is derived by considering the normalized Doppler spectrum as a probability density function of the radial velocity υr. The second moment quantifies the spread of the velocity distribution. However, spectrum width has inherent limitations in characterizing TSS. First of all, spectrum width does not provide sufficient information about the shape of the spectrum. In the autocovariance method, a Gaussian-shaped spectrum has been assumed for the estimation of spectrum width. Additionally, the spectrum width estimate is also sensitive to the accuracy of the noise estimation (Doviak and Zrnić 1993). In this work, two additional parameters are developed to characterize TSS based on signal statistics and HOS. To suppress the inevitable random fluctuations in the spectrum and to effectively identify TSS, the spectrum is represented in decibels, x(k) = 10 log10[S(υk)], where S(υk) is the Doppler spectrum at the radial velocity υk. The goal is to identify the distinct signatures possessed by a tornado vortex in the one-dimensional image x(k).
a. Spectral flatness
It has been demonstrated that some tornado spectra are similar to white noise spectra for υa = 35 m s−1 as depicted in Fig. 2. The spectral flatness (σs) is defined as the statistical variance of x(k) to quantify the flatness of the spectrum. The application of σs is exemplified in Fig. 3. A tornado spectrum from the previous simulation for the case of r0 = 25 km and ϕ0 = −0.2° is shown in the left panel. A Gaussian spectrum simulated using the algorithm in Zrnić (1975) with mean Doppler velocity of 5 m s−1 and spectrum width of 3 m s−1 is shown in the right panel.
It is clear that σs is primarily determined by the statistical fluctuations for the case of flattened tornado spectrum. On the contrary, large σs results mostly by contributions from the variation of the Gaussian shape itself. It should be pointed out that if the tornado is weak or the range to the tornado is short such that the tornado spectrum is no longer flat, σs will not help with the identification of a tornado vortex. In general, although a wide spectrum will produce low σs, there are cases where σs can provide information in addition to the spectrum width for the characterization of tornado spectra. Detailed statistical comparisons and the discussions of the limitation of σs as a parameter characterizing TSS are provided in sections 4 and 5.
b. Higher-order spectra
The PRIB of a Gaussian and rectangular pattern as a function of spectrum width is shown in the right panel of Fig. 4 for a = 0.9.
The two patterns have the same spectrum width, which varies from 1 to 30 m s−1. An example of Gaussian and rectangular pulse with a spectrum width of 12 m s−1 is shown in the left panel. The slope of a is selected to provide most distinct P values to identify the two patterns over the entire spectrum widths of interest. This example suggests that the PRIB can be used to differentiate the two patterns for a wide range of spectrum widths, while both patterns have the same spectrum width.
4. Simulation results
The shape of tornado spectra depends on several parameters such as the tornado’s size relative to the resolution volume, its location within the radar volume, and the reflectivity structure. In this work, it is assumed that the maximum wind in tornado exceeds the maximum unambiguous velocity of 35 m s−1. Three parameters—spectrum width, spectral flatness, and PRIB—are proposed to characterize TSS. The three parameters as a function of range will be studied statistically using simulations for various conditions.
A tornado within a mesocyclone as shown in Fig. 2 is located due north from the radar and its position is denoted by (r′0, ϕ′0) where ϕ′0 = 0°. To study radar sampling of different portions of the tornado, 25 radar resolution volumes centered at (r0, ϕ0) are used, where r0 varies from r′0 − 120 m to r′0 + 120 m every 60 m and ϕ0 varies from −0.4° to 0.4° at 0.2° intervals. Spectra from both the uniform reflectivity and doughnut-shaped reflectivity (Wz = 60 m) are simulated and are defined as Tu and Td, respectively. An example of spectra from nine resolution volumes is shown in Fig. 5 for r′0 = 60 km and SNR = 40 dB.
For comparison, Gaussian spectra were reconstructed using the three spectral moments estimated from the tornado spectra of doughnut-shaped and uniform reflectivity, and are denoted by Gd and Gu, respectively. The spectrum for Gu is similar to Tu and therefore is not shown. In the top rightmost panel is the case of the tornado vortex collocated with the center of the radar volume. It is shown that the shapes of tornado spectrum slightly differ if the radar samples different portions of the vortex in both azimuth and range directions. The tornado spectrum from doughnut reflectivity (Td) becomes flatter if the resolution volume is located closer to the vortex center in azimuth and farther from the center in range. For the case of uniform reflectivity, the flatness of the tornado spectrum (Tu) becomes more pronounced if θ0 is closer to zero. Nevertheless, the spectrum of Tu is broader and flatter than the spectrum of Td except for ϕ0 = −0.4° and r0 = r′0 − 120 m. The reconstructed Gaussian spectra have the same spectral moments as those from a tornado and can be considered as extreme cases where the spectra from nontornadic regions have large spectrum widths. The upper leftmost panel exemplifies that the three spectral moments are not sufficient to differentiate these wide Gaussian spectra from the tornado spectra from doughnut reflectivity, even though their patterns are visually different. Thus, it is important to demonstrate that the spectral flatness and/or PRIB can provide additional information for the identification of TSS.
The strength of TVS in the velocity field deteriorates with range due to the increasing size of radar resolution volumes (e.g., Brown et al. 1978, 2002). Thus, it is of interest to investigate the impact of range on the three parameters of spectrum width, spectral flatness, and PRIB. A new series of experiments with the same configuration of tornado and mesocyclone as shown previously was performed at various ranges. The distance between the tornado and the radar is normalized using rn = r′0θb/rt, where rt = 200 m and θb = π/180 (rad). Results of spectrum width, spectral flatness, and PRIB as a function of rn are shown in Figs. 6, 7 and 8, respectively, for the various locations of radar volume defined in Fig. 5.
The spectrum width of tornado spectra is obtained by the autocovariance method and is normalized by 2υa. Note that the reconstructed Gaussian spectra have the same spectrum width as those of tornado spectra. The spectral flatness and PRIB are estimated using the methods presented in section 3. Each data point in the three figures is the mean of 50 realizations, each with a different noise sequence in the generation of time series signals.
Generally speaking, spectrum widths increase rapidly with range for 0 < rn < 1 and decrease relatively smoothly and remain constant for rn > 4. This is also exemplified by the result in the upper left panel of Fig. 1. Figure 6 also indicates that at a given rn the spectrum width from uniform reflectivity decreases if the radar volume is shifted farther from the vortex center in azimuth. However, the shift of radar volume in range does not have significant impact on the spectrum width in this case. These results are also demonstrated in Fig. 5, in which rn = 5.2. In addition, spectrum widths from uniform reflectivity (Tu) are generally larger than or comparable to those from the doughnut-shaped reflectivity (Tu) at all ranges except when a relatively small portion of the tornado is sampled (e.g., ϕ0 = −0.2° and r0 = r′0 − 120 m; ϕ0 = −0.2° and r0 = r′0 − 60 m and r0 = r′0 − 120 m). The smaller spectrum widths can be resulted from velocity aliasing of a relatively flat spectra and suggest that additional parameters are needed to characterize TSS.
Furthermore, spectral flatness decreases rapidly with range for rn < 1 and remains relatively small (<10) beyond rn = 1 for most cases in Fig. 7. Similarly, PRIB in Fig. 8 increases rapidly with range for rn < 1 and remains large. These results are generally consistent with the variation of spectrum width in range, which suggests that TSS can be characterized by large spectrum width, small σs, and high PRIB given rn > 1. Unlike TVS the three parameters for tornado spectra are less sensitive to the increasing size of the radar volume except at close range. Among the three parameters, it has been shown in Fig. 5 that the spectrum width is not sufficient to distinguish the tornado spectrum from the reconstructed Gaussian spectrum even though the patterns are different. On the other hand, Figs. 7 and 8 show that σs and P derived from tornado spectra for the case of doughnut reflectivity (Td) are different from those from the reconstructed Gaussian spectra (Gd) for rn > 1. Although such a difference becomes small when the radar resolution volume is located farther from the tornado’s center in range (i.e., the bottom row in Figs. 7 and 8), high P values and small σs are still observed. It is because the Td at r0 = r′0 − 120 m is the most flattened compared to other cases of r0 as exemplified in Fig. 5. For the case of uniform reflectivity, it is evident that both P and σs can assist with the identification of tornado spectrum especially for ϕ0 = −0.4°. For other cases when the tornado is located closer to the center of radar resolution volume, although the P values of tornado spectra from uniform reflectivity (Tu) are similar to the reconstructed Gaussian spectra (Gu), the P from Tu is higher than or comparable to the one from Td for all locations and ranges. Similar results are observed for σs but with lower values for Tu. These results suggest that if a detection threshold is set based on either of the parameters from Td, tornado spectra from the uniform reflectivity would still be identified. On the other hand, a false detection can occur if only the spectrum width is used and the spectra from nontornadic regions have large spectrum width. Moreover, spectrum width estimate can be significantly biased at large spectrum widths and should be used with caution.
5. Experimental results
The research WSR-88D (KOUN) operated by the NSSL in Norman, Oklahoma, has the unique capability of continuously collecting volumetric level I data. The raw time series data can be postprocessed to generate Doppler spectra. A case of a tornado outbreak in which F2–F3-scale tornadoes were observed in central Oklahoma on 10 May 2003 is used to show the TSS and to demonstrate the application of the three parameters for its characterization. The collection of time series data started at 0318 UTC 10 May and lasted for approximately 2 h.
The reflectivity, mean Doppler velocity, and spectrum width from the lowest elevation angle of 0.5° at 0349 UTC are shown in Fig. 9 from left to right on the top panels, respectively. Results for SNR smaller than 20 dB are not shown.
The hook echo signature centered at approximately 12 km east and 40 km north of the radar is consistent with the tornado damage path and is well correlated with the region of strong azimuthal shear and enhanced spectrum width. Doppler spectra from the region denoted by a black box are shown in Fig. 10.
The spectrum is estimated by the periodogram method of 64 data points, which corresponds to the standard 1° angular sampling. A von Hann window was used to produce a large dynamic range in the Doppler spectrum (Doviak and Zrnić 1993). The maximum unambiguous velocity is approximately 35 m s−1. It is clear that wide and flat spectra are obtained at azimuth of 13.9° and range from 40.625 to 41.375 km. More Gaussian-like and aliased spectra can be observed at radials of 11.9°, 12.9°, 14.9°, and 16.0°. Note that the statistical error of mean Doppler velocity estimates increases exponentially with increasing spectrum width (Doviak and Zrnić 1993). Therefore, the Doppler velocity in the tornadic region is less reliable and the performance of the TVS-based detection algorithm may be degraded. The results of σs and P are shown in the two lower panels of Fig. 9. In the calculation of PRIB, the input sequence was zero padded to 256 points for the estimation of bispectrum to enhance the P values in the tornadic region. The coefficient a is set to 0.9. It is evident that low σs and high P are obtained in the tornadic region. In addition, a few range gates of velocity aliasing can be observed at approximately 5 km north of the box and at azimuth of 12.9° in the field of Doppler velocity, outlined by a circle. In other words, in this region a false detection of a tornado can occur if the velocity is not properly dealiased in the current detection algorithm. Note that the spectrum width and σs in this region are comparable to those in the tornadic region. However, the P parameter in this case still remains relatively low. It suggests that the PRIB is more robust to discriminate a tornado vortex from nontornadic features.
The normalized histograms of reflectivity, spectrum width, spectral flatness, and PRIB are shown in Fig. 11.
The data from 0331 to 0425 UTC, which consist of nine volume scans, are used for the statistical analysis. However, the data at 0407 UTC were not used since no tornado was reported at that time. In addition, only the data from the first two elevation angles and with SNR > 20 dB are considered. The tornadic region is defined at those range gates in which the maximum velocity difference (Δυ = υmax − υmin) is larger than 50 m s−1, where υmax and υmin are the maximum and minimum values in the velocity azimuth profile at a fixed range. This statistical analysis demonstrates that the tornado in this case can be characterized by high reflectivity, larger spectrum width, small σs, and large P. It should be pointed out that prominent feature of low σs and high P can be observed and easily identified in the PPI display for all the dataset (similar to those shown in Fig. 9).
6. Conclusions
Tornadoes have distinct spectral patterns that can potentially be used to facilitate tornado detection while the shear signature is weak. Two parameters, PRIB and spectral flatness, are developed and have shown values in addition to the spectral moments to improve the characterization of the tornado spectral signatures (TSSs). A Doppler spectrum exhibits the power-weighted velocity distribution of scatterers within the radar volume. The three fundamental radar measurements—reflectivity, mean Doppler velocity, and spectrum width—are estimated using the first three moments of a Doppler spectrum. Numerical simulation of Doppler spectrum and level I time series data were developed in which a tornado vortex, mean background wind, horizontal shear, mesocyclone, and various reflectivity structures are included. Spectra with wide and flat signatures, similar to white noise spectra, were obtained in the tornadic region for a typical maximum unambiguous velocity. Moreover, the shape of tornado spectrum is found to be sensitive to the reflectivity structure, the size of the tornado relative to the size of radar resolution volume, and the location of the vortex in the resolution volume. Those results are consistent with early analytical simulation (Zrnić and Doviak 1975).
Previous work has shown that a tornado can be identified by strong azimuthal shear in the Doppler velocity domain and debris signature in the polarimetric variables. In this work, three parameters are proposed to characterize TSS. These are spectrum width (συ), spectral flatness (σ2s), and the phase of radially integrated bispectrum (PRIB, P). Generally speaking, TSS can be recognized by large συ, low σs, and high P. The three parameters as a function of the range between the radar and tornado vortex are also discussed for various conditions. Although only one tornado case is shown using the NSSL research WSR-88D (KOUN), results have shown that both σs and P can assist with the identification of tornado spectra while the information of spectrum width by itself is not sufficient. To fully quantify these spectral features, more observations of tornadoes with various scales are needed and compared with nontornadic thunderstorms. Nevertheless, from the histogram of these parameters it is clear that their distributions for tornado and nontornado cases are overlapped. In other words, the detection result can be ambiguous if a binary decision is made based on only a single parameter such as shear. Therefore, it is desirable to develop an automated expert system such as fuzzy logic and/or neural network that can avoid thresholding and can integrate multiple information to improve the detection of tornadoes.
Acknowledgments
This work was primarily supported by the DOC-NOAA NWS CSTAR program through the grant of NA17RJ1227. In addition, this work was supported in part by the National Science Foundation through ATM-0532107 and the Engineering Research Centers Program of the National Science Foundation under NSF Cooperative Agreement EEC-0313747. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect those of the National Science Foundation. The authors would also like to thanks the NSSL staff for the collection of level I data and the WFO in Norman for providing the ground damage survey.
REFERENCES
Alexander, C. R., and Wurman J. , 2005: The 30 May 1998 Spencer, South Dakota, storm. Part I: The structural evolution and environment of the tornadoes. Mon. Wea. Rev., 133 , 72–96.
Atlas, D., 1963: Radar analysis of severe storms. Severe Local Storms, Meteor. Monogr., Nov. 27, Amer. Meteor. Soc., 177–220.
Bieringer, P., and Ray P. S. , 1996: A comparison of tornado warning lead times with and without NEXRAD Doppler radar. Wea. Forecasting, 11 , 47–52.
Bluestein, H. B., Ladue J. G. , Stein H. , Speheger D. , and Unruh W. P. , 1993: Doppler radar wind spectra of supercell tornadoes. Mon. Wea. Rev., 121 , 2200–2221.
Bluestein, H. B., Unruh W. P. , Dowell D. C. , Hutchinson T. A. , Crawford T. M. , Wood A. C. , and Stein H. , 1997: Doppler radar analysis of the Northfield, Texas, tornado of 25 May 1994. Mon. Wea. Rev., 125 , 212–230.
Bluestein, H. B., Lee W-C. , Bell M. , Weiss C. C. , and Pazmany A. L. , 2003: Mobile Doppler radar observations of a tornado in a supercell near Bassett, Nebraska, on 5 June 1999. Part II: Tornado-vortex structure. Mon. Wea. Rev., 131 , 2968–2984.
Bluestein, H. B., Weiss C. C. , and Pazmany A. L. , 2004: Doppler radar observations of dust devils in Texas. Mon. Wea. Rev., 132 , 209–224.
Bluestein, H. B., Weiss C. C. , French M. M. , Holthaus E. M. , Tananmachi R. L. , Frasier S. , and Pazmany A. L. , 2007: The structure of tornadoes near Attica, Kansas, on 12 May 2004: High-resolution, mobile, Doppler radar observations. Mon. Wea. Rev., 135 , 475–506.
Bringi, V. N., and Chandrasekar V. , 2001: Polarimetric Doppler Weather Radar Principles and Applications. Cambridge University Press, 636 pp.
Brotzge, J. A., Brewster K. , Johnson B. , Philip B. , Preston M. , Westbrook D. , and Zink M. , 2005: CASA’s first test bed: Integrated project #1. Preprints, 32nd Conf. on Radar Meteorology, Albuquerque, NM, Amer. Meteor. Soc., 14R.2.
Brown, R. A., and Lemon L. R. , 1976: Single Doppler radar vortex recognition. Part II: Tornadic vortex signatures. Preprints, 17th Conf. on Radar Meteorology, Seattle, WA, Amer. Meteor. Soc., 104–109.
Brown, R. A., and Wood V. T. , 1991: On the interpretation of single-Doppler velocity patterns within severe thunderstorms. Wea. Forecasting, 6 , 32–48.
Brown, R. A., Lemon L. R. , and Burgess D. W. , 1978: Tornado detection by pulsed Doppler radar. Mon. Wea. Rev., 106 , 29–38.
Brown, R. A., Wood V. T. , and Sirmans D. , 2002: Improved tornado detection using simulated and actual WSR-88D data with enhanced resolution. J. Atmos. Oceanic Technol., 19 , 1759–1771.
Brown, R. A., Flickinger B. A. , Forren E. , Schultz D. M. , Sirmans D. , Spencer P. L. , Wood V. T. , and Ziegler C. L. , 2005: Improved detection of severe storms using experimental fine-resolution WSR-88D measurements. Wea. Forecasting, 20 , 3–14.
Chandran, V., and Elgar S. L. , 1993: Pattern recognition using invariants defined from high order spectra—One dimensional inputs. IEEE Trans. Signal Process., 41 , 205–212.
Doviak, R. J., and Zrnić D. S. , 1993: Doppler Radar and Weather Observations. Academic Press, 562 pp.
Dowell, D. C., Alexander C. R. , Wurman J. M. , and Wicker L. J. , 2005: Centrifuging of hydrometeors and debris in tornadoes: Radar-reflectivity patterns and wind-measurements errors. Mon. Wea. Rev., 133 , 1501–1524.
Janssen, L. H., and Spek G. A. V. D. , 1985: The shape of Doppler spectra from precipitation. IEEE Trans. Aerosp. Electron. Syst., 21 , 208–219.
Kundu, P. K., and Cohen I. M. , 2002: Fluid Mechanics. 2nd ed. Academic Press, 730 pp.
Lemon, L. R., 1980: Severe thunderstorm radar identification techniques and warning criteria. NOAA Tech. Memo. NWS NSSFC-3, 60 pp.
Mitchell, E. D., Vasiloff S. V. , Stumpf G. J. , Witt A. , Eilts M. D. , Johnson J. T. , and Thomas K. W. , 1998: The National Severe Storms Laboratory tornado detection algorithm. Wea. Forecasting, 13 , 352–366.
Nikias, C. L., and Raghuveer M. R. , 1987: Bispectrum estimation: A digital signal processing framework. Proc. IEEE, 75 , 869–891.
Oppenheim, A. V., and Lim J. S. , 1981: The importance of phase in signals. Proc. IEEE, 69 , 529–541.
Papoulis, A., and Pillai S. U. , 2002: Probability, Random Variables, and Stochastic Processes. 4th ed. McGraw-Hill, 852 pp.
Polger, P. D., Goldsmith B. S. , Przywarty R. C. , and Bocchierri J. R. , 1994: National Weather Service warning performance based on the WSR-88D. Bull. Amer. Meteor. Soc., 75 , 203–214.
Ryzhkov, A. V., Schuur T. J. , Burgess D. W. , and Zrnic D. S. , 2005: Polarimetric tornado detection. J. Appl. Meteor., 44 , 557–570.
Serafin, R. J., and Wilson J. W. , 2000: Operational weather radar in the United States: Progress and opportunity. Bull. Amer. Meteor. Soc., 81 , 501–518.
Shao, Y., and Celenk M. , 2001: Higher-order spectra (hos) invariants for shape recognition. Pattern Recognit., 34 , 2097–2113.
Simmons, K. M., and Sutter D. , 2005: WSR-88D radar, tornado warnings, and tornado casualties. Wea. Forecasting, 20 , 301–310.
Smith, R. L., and Holmes D. W. , 1961: Use of Doppler radar in meteorological observations. Mon. Wea. Rev., 89 , 1–7.
Smith, T. M., Elmore K. L. , and Dulin S. A. , 2004: A damaging downburst prediction and detection algorithm for the WSR-88D. Wea. Forecasting, 19 , 240–250.
Wood, V. T., and Brown R. A. , 1997: Effects of radar sampling on single-Doppler velocity signatures of mesocyclones and tornadoes. Wea. Forecasting, 12 , 928–938.
Wurman, J., and Gill S. , 2000: Finescale radar observations of the Dimmitt, Texas (2 June 1995), tornado. Mon. Wea. Rev., 128 , 2135–2164.
Xue, M., Liu S. , and Yu T-Y. , 2007: Variational analysis of oversampled dual-Doppler radial velocity data and application to the analysis of tornado circulations. J. Atmos. Oceanic Technol., 24 , 403–414.
Yu, T-Y., Zrnić D. , Shapiro A. , and Yeary M. B. , 2003: Feasibility of earlier tornado detection using Doppler spectra. Preprints, 31st Conf. on Radar Meteorology, Seattle, WA, Amer. Meteor. Soc., 5A.7.
Yu, T-Y., Shapiro A. , Zrnić D. S. , Foster M. P. , Andra D. L. Jr., Doviak R. J. , and Yeary M. B. , 2004: Tornado spectral signature observed by WSR-88D. Preprints, 22nd Conf. on Severe Local Storms, Hyannis, MA, Amer. Meteor. Soc., 8B.1.
Zhang, X-D., Shi Y. , and Bao Z. , 2001: A new feature vector using selected bispectra for signal classification with application in radar target recognition. IEEE Trans. Signal Process., 49 , 1875–1885.
Zrnić, D. S., 1975: Simulation of weatherlike Doppler spectra and signals. J. Appl. Meteor., 14 , 619–620.
Zrnić, D. S., and Doviak R. J. , 1975: Velocity spectra of vortices scanned with a pulse-Doppler radar. J. Appl. Meteor., 14 , 1531–1539.
Zrnić, D. S., and Istok M. , 1980: Wind speeds in two tornadic storms and a tornado, deduced from Doppler spectra. J. Appl. Meteor., 19 , 1405–1415.
Zrnić, D. S., Burgess D. W. , and Hennington L. , 1985: Doppler spectra and estimated windspeed of a violent tornado. J. Climate Appl. Meteor., 24 , 1068–1081.
APPENDIX
Simulation of Doppler Spectrum
A Doppler spectrum presents the radial velocity distribution of scatterers within the radar volume. The magnitude of Doppler spectrum at a velocity υr is the return power from all scatterers with the same radial velocity υr. The received power from each scatterer is determined by the reflectivity and radar weighting function at the scatterer’s location. A Doppler spectrum can be thought of as a histogram of return power at M discrete velocity bins. For a maximum unambiguous velocity of υa, the velocity bins are defined as υk = −υa + kΔυ, k = 1, 2, . . . , M, where Δυ = 2υa/M. The maximum unambiguous velocity is determined by υa = λ/(4Ts), where λ is the radar wavelength and Ts is the pulse repetition time. For given reflectivity and velocity fields, the Doppler spectrum S(υk) can be simulated by summing the return power from scatterers of radial velocity υk within the radar resolution volume. In this work, it is assumed that the reflectivity and velocity are independent of height. As a result, only 2D horizontal fields are considered. An example of reflectivity and velocity fields are shown in Fig. A1.
It should be noted that this simulation only provides relative return power, which can be scaled according to a desirable signal-to-noise ratio (SNR). The complex time series signals are obtained by taking an inverse Fourier transform of the model spectrum and noise as described in Zrnić (1975). The three spectrum moments (signal power, mean Doppler velocity, and spectrum width) can be estimated using either autocovariance or spectral methods (Doviak and Zrnić 1993). This approach has a number of attractive features: 1) it can include more general environmental conditions such as background flow, shears, and multiple vortices; 2) time series data that can be used for testing various signal processing techniques are generated; 3) it provides desirable statistics of the three moment estimates, which can be used to investigate the statistical performance of a new detection algorithm; 4) a realistic noise corruption is simulated; 5) the velocity aliasing is also simulated in a realistic manner (i.e., in the spectrum domain rather than in the mean Doppler velocity domain).
A demonstration of tornado spectra as a function of range (r0), uniform background wind (υ0), horizontal shear (υ′x), and the width of reflectivity [Wz, defined in Eq. (A2)]. A tornado-like vortex with maximum tangential wind of 50 m s−1 and a radius of 200 m is located at the north of the radar with range of r′0 = 5, 25, and 75 km (the azimuth angle ϕ′0 = 0°).
Citation: Journal of Atmospheric and Oceanic Technology 24, 12; 10.1175/2007JTECHA934.1
(a) Model reflectivity and horizontal wind field. The tornado is located at 25 km north from the radar and a mesocyclone is centered at 1 km northeast from the tornado. (b) The velocity field observed by a virtual WSR-88D. (bottom) Spectra from the region of tornado vortex, denoted by a black box. Here, ϕ0 is the azimuth angle of the radar beam.
Citation: Journal of Atmospheric and Oceanic Technology 24, 12; 10.1175/2007JTECHA934.1
An example to demonstrate the spectral flatness σs for the characterization of TSS. (a) A tornado spectrum with a signature resembling white noise has value of σs, while (b) a Gaussian shape spectrum exhibits high σs. The mean of x(k) is denoted by solid lines, and the range of 2σs is indicated by dashed lines.
Citation: Journal of Atmospheric and Oceanic Technology 24, 12; 10.1175/2007JTECHA934.1
(a) The P value derived from the bispectrum of a Gaussian and rectangular pulse as a function of spectrum width (συ). The Gaussian shape and rectangle have the same second moment. (b) An example of συ = 12 m s−1 is given.
Citation: Journal of Atmospheric and Oceanic Technology 24, 12; 10.1175/2007JTECHA934.1
Simulated spectra for different portions of the tornado sampled by the radar. The tornado vortex is located at (r′0 = 60 km, ϕ′0 = 0°). Radar resolution volumes are centered at azimuths of ϕ = −0.4°, ϕ = −0.2°, and ϕ = 0.0° from the left to the right columns, respectively. Each row represents radar volumes centered at range (r0) as indicated on the right-hand side of each plot. Spectra from doughnut-shaped reflectivity (Td), uniform reflectivity (Tu), and reconstructed Gaussian spectra Gd are denoted by solid, dash–dot, and dashed lines, respectively.
Citation: Journal of Atmospheric and Oceanic Technology 24, 12; 10.1175/2007JTECHA934.1
Range profile of the normalized spectrum widths (συn) from doughnut (Td) and uniform (Tu) reflectivities. Each panel represents the result from different radar volumes as shown in Fig. 5.
Citation: Journal of Atmospheric and Oceanic Technology 24, 12; 10.1175/2007JTECHA934.1
Similar to Fig. 6, but for the spectral flatness (σs). In addition, results from the reconstructed Gaussian spectra (Gd and Gu) are included for comparison.
Citation: Journal of Atmospheric and Oceanic Technology 24, 12; 10.1175/2007JTECHA934.1
Same as in Fig. 7, but for the PRIB P.
Citation: Journal of Atmospheric and Oceanic Technology 24, 12; 10.1175/2007JTECHA934.1
The fields of (a) reflectivity, (b) mean Doppler velocity, and (c) spectrum width from the NSSL’s research WSR-88D (KOUN) at 0349 UTC 10 May 2003. The tornado damage path from the ground survey is superimposed as a light shaded area. The other two parameters for characterizing spectral signature, (d) σs and (e) P. It is evident that the region of hook echo, strong azimuthal shear, large spectrum width, small σs, and large P values are correlated well with the tornado damage path. A region of velocity aliasing is also outlined by circles.
Citation: Journal of Atmospheric and Oceanic Technology 24, 12; 10.1175/2007JTECHA934.1
Spectra from the tornadic region indicated in Fig. 9 by the black box. The mean radial velocity from the pulse-pair method is denoted by triangles. The notation is the same as in Fig. 2.
Citation: Journal of Atmospheric and Oceanic Technology 24, 12; 10.1175/2007JTECHA934.1
Normalized histogram of (a) reflectivity, (b) spectrum width, (c) σs, and (d) P from both tornadic and nontornadic regions.
Citation: Journal of Atmospheric and Oceanic Technology 24, 12; 10.1175/2007JTECHA934.1
Fig. A1. A schematic diagram to demonstrate the simulation of Doppler spectrum. The reflectivity pattern is depicted by the contours and the horizontal wind field is denoted by the arrows.
Citation: Journal of Atmospheric and Oceanic Technology 24, 12; 10.1175/2007JTECHA934.1
