## 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

*C*is a parameter that is a function of radar wavelength, peak transmitted power, range, and antenna gain;

*W*(

_{r}*r*) is the range weighing function;

*f*

^{4}

_{b}(

*θ*) is the two-way antenna pattern;

*Z*(

**r**) and

*υ*(

**r**) are the 3D reflectivity and radial velocity fields, respectively; and

*ds*

_{1}and

*ds*

_{2}are two orthogonal differential lengths on the surface of a constant

*υ*(

**r**) (isodop). The radar is located at the origin and the radar resolution volume is centered at

**r**

_{0}. The term of velocity gradient is used to adjust the density of scatterers between the two isodop surfaces. The magnitude of Doppler spectrum represents the return power from all the scatterers within the radar resolution volume that have the same radial velocity. Moreover, the mean Doppler velocity and spectrum width are defined as the first and second moments of a Doppler spectrum (Doviak and Zrnić 1993). In practice a Doppler spectrum can be estimated using the periodogram method, which is the square of the magnitude of the Fourier transform of the level I time series data (e.g., Bringi and Chandrasekar 2001). The three spectral moments (defined as level II data) can be estimated from the resultant spectrum using the moment method and are comparable to those obtained by the autocovariance method (Doviak and Zrnić 1993).

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 (*θ _{b}*) 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 (*υ*_{0} = *u*_{0} = 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 *W _{z}* = 60 m and

*r*

_{0}

*= 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 (*

_{z}*υ*) is set at 65 m s

_{a}^{−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

*W*= ∞, 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.

_{z}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 (*u*_{0} = 5 *υ*_{0} = 10 *w*_{0} = 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 (*r*_{0}, *ϕ*_{0}), where *r*_{0} is the range and *ϕ*_{0} 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 *υ _{a}* = 35 m s

^{−1}. Instead, spectra with signatures similar to white noise spectra, whereby values in (−

*υ*) are almost uniform, are observed from the gates close to the tornado center such as those from

_{a}υ_{a}*r*

_{0}= 24.75, 25.00, 25.25 km and

*ϕ*

_{0}= −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

*ϕ*

_{0}= −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 2

*υ*, respectively. However, the mean Doppler velocity at

_{a}*ϕ*

_{0}= −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 (*σ _{υ}*), 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

*υ*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,

_{r}.*x*(

*k*) = 10 log

_{10}[

*S*(

*υ*)], where

_{k}*S*(

*υ*) is the Doppler spectrum at the radial velocity

_{k}*υ*The goal is to identify the distinct signatures possessed by a tornado vortex in the one-dimensional image

_{k}.*x*(

*k*).

### a. Spectral flatness

It has been demonstrated that some tornado spectra are similar to white noise spectra for *υ _{a}* = 35 m s

^{−1}as depicted in Fig. 2. The spectral flatness (

*σ*) is defined as the statistical variance of

_{s}*x*(

*k*) to quantify the flatness of the spectrum. The application of

*σ*is exemplified in Fig. 3. A tornado spectrum from the previous simulation for the case of

_{s}*r*

_{0}= 25 km and

*ϕ*

_{0}= −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 *σ _{s}* is primarily determined by the statistical fluctuations for the case of flattened tornado spectrum. On the contrary, large

*σ*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,

_{s}*σ*will not help with the identification of a tornado vortex. In general, although a wide spectrum will produce low

_{s}*σ*, there are cases where

_{s}*σ*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

_{s}*σ*as a parameter characterizing TSS are provided in sections 4 and 5.

_{s}### b. Higher-order spectra

*x*(

*k*), as the 1D image. The power spectrum of

*x*(

*k*) is given by

*S*(

_{x}*f*) =

*X*(

*f*)

*X**(

*f*), and the bispectrum is obtained by the following form (Nikias and Raghuveer 1987):

*X*(

*f*) is the discrete Fourier transform of

*x*(

*k*). The bispectrum can be determined unambiguously within the region of 0 <

*f*

_{2}<

*f*

_{1}<

*f*

_{1}+

*f*

_{2}< 1, where

*f*

_{1}and

*f*

_{2}are the normalized frequencies. Oppenheim and Lim (1981) have shown that the shape information is often contained in the phases of the Fourier transform of the pattern or image. The application of bispectrum to identify TSS is motivated by the fact that bispectrum can retain both the phase and amplitude of the Fourier coefficients, while the commonly used power spectrum suppresses the phases due to the multiplication of Fourier coefficients and their complex conjugates. Moreover, bispectrum is invariant to the shift of the pattern because the resultant phase shifts in

*X*(

*f*) are cancelled in the triple product. The bispectrum is also insensitive to additive Gaussian noise because theoretically the third-moment sequence of a stationary Gaussian process is zero (Papoulis and Pillai 2002) and the bispectrum is defined as its Fourier transform. It also has been shown that bispectrum is useful in the identification of nonlinearity and non-Gaussianarity of the process (e.g., Nikias and Raghuveer 1987).

*P*, is proposed to effectively extract the shape information in a bispectrum (e.g., Chandran and Elgar 1993; Shao and Celenk 2001; Zhang et al. 2001). PRIB is defined in the following forms:

*I*is the integration of bispectrum along the path of

*f*

_{1}−

*f*

_{2}= 0 and

*f*

_{2}=

*af*

_{1}in the bispectrum domain, 0 <

*a*≤ 1 is the slope of the path, and PRIB is the phase of the complex variable

*I.*Parameters obtained by different integration paths are discussed in Zhang et al. (2001) and the references therein. It is shown in Chandran and Elgar (1993) that the phase of the Fourier transform of a asymmetric pattern is a nonlinear function of frequency, and PRIB can isolate this nonlinearity to provide information about the shape. In addition, it has been proven that PRIB is invariant to translation, amplification, scaling, and dc shifting of the input signals. The translation or shift invariance has a great advantage in our application because spectra with different mean Doppler velocities will produce the same value of PRIB. It should be noted that

*P*does not provide any shape information if the input sequence has even or odd symmetry, because the resultant bispectrum is either real or purely imaginary, respectively. An alternative approach was developed by Chandran and Elgar (1993) to alleviate this problem. A new sequence is first generated that is the amplitude of the Fourier transform of the original signal

*x*(

*k*). Consequently, the bispectrum is calculated using only the first half of the new sequence such that an asymmetry pattern is obtained. This approach can still maintain those invariance properties (Chandran and Elgar 1993; Shao and Celenk 2001).

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 (*r*_{0}, *ϕ*_{0}) are used, where *r*_{0} varies from *r*′_{0} − 120 m to *r*′_{0} + 120 m every 60 m and *ϕ*_{0} varies from −0.4° to 0.4° at 0.2° intervals. Spectra from both the uniform reflectivity and doughnut-shaped reflectivity (*W _{z}* = 60 m) are simulated and are defined as

*T*and

_{u}*T*, respectively. An example of spectra from nine resolution volumes is shown in Fig. 5 for

_{d}*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 *G _{d}* and

*G*, respectively. The spectrum for

_{u}*G*is similar to

_{u}*T*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 (

_{u}*T*) 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 (

_{d}*T*) becomes more pronounced if

_{u}*θ*

_{0}is closer to zero. Nevertheless, the spectrum of

*T*is broader and flatter than the spectrum of

_{u}*T*except for

_{d}*ϕ*

_{0}= −0.4° and

*r*

_{0}=

*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 *r _{n}* =

*r*′

_{0}

*θ*/

_{b}*r*, where

_{t}*r*= 200 m and

_{t}*θ*=

_{b}*π*/180 (rad). Results of spectrum width, spectral flatness, and PRIB as a function of

*r*are shown in Figs. 6, 7 and 8, respectively, for the various locations of radar volume defined in Fig. 5.

_{n}The spectrum width of tornado spectra is obtained by the autocovariance method and is normalized by 2*υ _{a}.* 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 < *r _{n}* < 1 and decrease relatively smoothly and remain constant for

*r*> 4. This is also exemplified by the result in the upper left panel of Fig. 1. Figure 6 also indicates that at a given

_{n}*r*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

_{n}*r*= 5.2. In addition, spectrum widths from uniform reflectivity (

_{n}*T*) are generally larger than or comparable to those from the doughnut-shaped reflectivity (

_{u}*T*) at all ranges except when a relatively small portion of the tornado is sampled (e.g.,

_{u}*ϕ*

_{0}= −0.2° and

*r*

_{0}=

*r*′

_{0}− 120 m;

*ϕ*

_{0}= −0.2° and

*r*

_{0}=

*r*′

_{0}− 60 m and

*r*

_{0}=

*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 *r _{n}* < 1 and remains relatively small (<10) beyond

*r*= 1 for most cases in Fig. 7. Similarly, PRIB in Fig. 8 increases rapidly with range for

_{n}*r*< 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

_{n}*σ*, and high PRIB given

_{s}*r*> 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

_{n}*σ*and

_{s}*P*derived from tornado spectra for the case of doughnut reflectivity (

*T*) are different from those from the reconstructed Gaussian spectra (

_{d}*G*) for

_{d}*r*> 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

_{n}*P*values and small

*σ*are still observed. It is because the

_{s}*T*at

_{d}*r*

_{0}=

*r*′

_{0}− 120 m is the most flattened compared to other cases of

*r*

_{0}as exemplified in Fig. 5. For the case of uniform reflectivity, it is evident that both

*P*and

*σ*can assist with the identification of tornado spectrum especially for

_{s}*ϕ*

_{0}= −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 (

*T*) are similar to the reconstructed Gaussian spectra (

_{u}*G*), the

_{u}*P*from

*T*is higher than or comparable to the one from

_{u}*T*for all locations and ranges. Similar results are observed for

_{d}*σ*but with lower values for

_{s}*T*These results suggest that if a detection threshold is set based on either of the parameters from

_{u}.*T*, 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.

_{d}## 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 *σ _{s}* 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

*σ*and high

_{s}*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

*σ*in this region are comparable to those in the tornadic region. However, the

_{s}*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 (Δ*υ* = *υ*_{max} − *υ*_{min}) is larger than 50 m s^{−1}, where *υ*_{max} and *υ*_{min} 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 *σ _{s}*, and large

*P.*It should be pointed out that prominent feature of low

*σ*and high

_{s}*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 (*σ _{υ}*), spectral flatness (

*σ*

^{2}

_{s}), and the phase of radially integrated bispectrum (PRIB,

*P*). Generally speaking, TSS can be recognized by large

*σ*, low

_{υ}*σ*, and high

_{s}*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

*σ*and

_{s}*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.

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## 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 *υ _{r}* is the return power from all scatterers with the same radial velocity

*υ*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

_{r}.*M*discrete velocity bins. For a maximum unambiguous velocity of

*υ*, the velocity bins are defined as

_{a}*υ*= −

_{k}*υ*+

_{a}*k*Δ

*υ*,

*k*= 1, 2, . . . ,

*M*, where Δ

*υ*=

*2υ*/

_{a}*M.*The maximum unambiguous velocity is determined by

*υ*=

_{a}*λ*/(4

*T*), where

_{s}*λ*is the radar wavelength and

*T*is the pulse repetition time. For given reflectivity and velocity fields, the Doppler spectrum

_{s}*S*(

*υ*) can be simulated by summing the return power from scatterers of radial velocity

_{k}*υ*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.

_{k}**r**

_{0}is defined by the range weighting function

*W*(

*r*) and antenna pattern

*f*

^{2}

_{b}(

*θ*) in range and azimuth, respectively, as depicted by the white box. Each radar volume consists of 1500 × 500 grid points in the

*x*and

*y*directions, respectively. A Doppler spectrum is simulated using the following equation:

**r**

_{0}is the center of radar volume;

*i*and

*j*are indices for the grid points in the

*x*and

*y*directions; and

*Z*(

*i*,

*j*),

*W*(

_{r}*i*,

*j*), and

*f*

^{4}

_{b}(

*i*,

*j*) are the reflectivity, range weighing function, and two-way antenna pattern at a Cartesian grid. In this work, the reflectivity pattern associated with a tornado vortex is modeled by the following equation (Zrnić and Doviak 1975; Bluestein et al. 1993):

*z*is the maximum reflectivity,

_{m}*r*

_{0z}is the radius of maximum reflectivity,

*W*is the width of reflectivity, and

_{z}*r*is the distance between the grid point (

_{ij}*i*,

*j*) and the center of tornado. As a result, a doughnut-shaped reflectivity can be produced as shown in Fig. 12. Moreover, a uniform reflectivity field can be simulated by letting

*W*≈ ∞ and a Gaussian-shaped reflectivity can be obtained by setting

_{z}*r*

_{0z}= 0. A reflectivity pattern for multiple vortices can be obtained by the superposition of each individual reflectivity pattern.

**V**

_{0}= [

*u*

_{0}

*υ*

_{0}

*w*

_{0}] is the uniform background flow,

**V**

*is the velocity from one or multiple vortices, and*

_{υ}**V**

*is the contribution by the 2D horizontal shear. In this work, both the tangential (*

_{s}*V*) and radial flow (

_{T}*V*) of the tornado are simulated using the following combined Rankine vortex model:

_{R}*V*and

_{Tm}*V*are the maximum tangential and radial velocity,

_{Rm}*r*is the radius of maximum wind, and

_{t}*γ*= 1 for

*r*≤

_{ij}*r*and

_{t}*γ*= −1 for

*r*>

_{ij}*r*The radial velocity

_{t}.*υ*(

_{r}*i*,

*j*) is obtained by

**V**·

**a**

*, where*

_{r}**a**

*is the unit vector pointing from the grid point to the radar. If*

_{r}*υ*(

_{r}*i*,

*j*) is outside the range of (−

*υ*) (velocity aliasing), then it will be adjusted by

_{a}υ_{a}*υ*(

_{r}*i*,

*j*) ±2

*υ*until

_{a}*υ*(

_{r}*i*,

*j*) is between (−

*υ*). Consequently, the Doppler spectrum at radial velocity

_{a}υ_{a}*υ*is the sum of

_{k}*W*

^{2}

_{r}(

*i*,

*j*)

*f*

^{4}

_{b}(

*i*,

*j*)

*Z*(

*i*,

*j*) at those grids with radial velocity between

*υ*− (Δ

_{k}*υ*/2) and

*υ*+ (Δ

_{k}*υ*/2). The resultant spectrum is defined as the model spectrum. A similar scheme was used by Bluestein et al. (1993) to simulate Doppler spectrum and by Brown and Wood (1991) and Wood and Brown (1997) to simulate the mean Doppler velocity, which is the first spectral moment of

*S*(

*υ*; Doviak and Zrnić 1993).

_{k}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).