Using Wavelet Analysis to Detect Tornadoes from Doppler ...

Using Wavelet Analysis to Detect Tornadoes from Doppler RadarRadial-Velocity Observations

SHUN LIU

Center for Analysis and Prediction of Storms, University of Oklahoma, Norman, Oklahoma, and College of Atmospheric Sciences,Lanzhou University, Lanzhou, China

MING XUE

Center for Analysis and Prediction of Storms and School of Meteorology, University of Oklahoma, Norman, Oklahoma

QIN XU

NOAA/National Severe Storms Laboratory, Norman, Oklahoma

(Manuscript received 25 October 2005, in final form 11 July 2006)

ABSTRACT

A wavelet-based algorithm is developed to detect tornadoes from Doppler weather radar radial-velocityobservations. Within this algorithm, a relative region-to-region velocity difference (RRVD) is defined basedon the scale- and location-dependent wavelet coefficients and this difference represents the relative mag-nitude of the radial velocity shear between two adjacent regions of different scales. The RRVD fields of anidealized tornado and a realistic tornado from a high-resolution numerical simulation are analyzed first. Itis found that the value of RRVD in the tornado region is significantly larger than those at other locationsand large values of RRVD exist at more than one scale. This characteristic forms the basis of the newalgorithm presented in this work for identifying tornadoes. Different from traditional tornadic vortexsignature detection algorithms that typically rely on the velocity difference between adjacent velocity gatepairs at a single spatial scale, the new algorithm examines region-to-region radial wind shears at a numberof different spatial scales. Multiscale regional wind shear examination not only can be used to discard anontornadic vortex signature to reduce the false alert rate of tornado detection but also has the ability ofcapturing tornadic signatures at various scales for improving the detection and warning. The potentialadvantage of the current algorithm is demonstrated by applying it to the radar data collected by OklahomaCity, Oklahoma (KTLX), Weather Surveillance Radar-1988 Doppler (WSR-88D) on 8 May 2003 for acentral Oklahoma tornado case.

1. Introduction

Modern Doppler radars have the ability to scan largevolumes of the atmosphere at high space and time reso-lutions. Such measurements have provided unprec-edented opportunities for detecting small-scale hazard-ous weather phenomena such as tornadoes. A numberof algorithms (e.g., Mitchell et al. 1998; Desrochers andDonaldson 1992; Wieler 1986; Vasiloff 2001) have beendeveloped over the years to automatically identify tor-nado from radar reflectivity and/or radial velocity by

examining hook echoes (Fujita 1958; Browning 1965)and tornadic vortex signatures (TVSs; Burgess et al.1975; Crum and Alberty 1993). Among them, the tor-nado detection algorithms based on hook echoes aloneare usually not reliable enough (e.g., Forbes 1981;Mitchell et al. 1998). Newer algorithms, such as theNational Severe Storms Laboratory Tornado DetectionAlgorithm (NSSL TDA, referred to as NTDA hereaf-ter; Mitchell et al. 1998) and the past operationalWeather Surveillance Radar-1988 Doppler (WSR-88D)TVS algorithm, are based primarily on TVSs identifiedfrom the radial velocity data.

TVS is primarily quantified by gate-to-gate azimuthalshear (Burgess et al. 1975; Brown et al. 1978). With theWSR-88D TVS algorithm, the presence of a mesocy-clone is required. The algorithm is invoked after a me-

Corresponding author address: Shun Liu, Environmental Mod-eling Center, National Centers for Environmental Prediction,Room 207, 5200 Auth Rd., Camp Springs, MD 20746.E-mail: [email protected]

344 J O U R N A L O F A T M O S P H E R I C A N D O C E A N I C T E C H N O L O G Y VOLUME 24

DOI: 10.1175/JTECH1989.1

© 2007 American Meteorological Society

JTECH1989

socyclone is detected by the mesocyclone detection al-gorithm (MDA), and it searches in and around the me-socyclone at each elevation angle for inbound andoutbound velocity extrema. When the shear betweenthe velocity extrema exceeds a certain threshold (de-fault is 0.02 s�1) at two or more elevation angles, a TVSis declared and it is automatically assumed to be tor-nadic (see, e.g., Mitchell et al. 1998). With the newerNTDA, the presence of a mesocyclone is not required,and therefore the detection of nonsupercell tornadoesis not excluded. The algorithm identifies vortices viaexamination of the gate-to-gate azimuthal (adjacent inazimuth and constant in range) velocity differences(GVDs; Mitchell et al. 1998), using low thresholds first;it then attempts to classify them as either tornadic ornontornadic. Three-dimensional (3D) regions withGVD exceeding certain thresholds are constructed andcertain characteristics of such 3D regions are then usedto determine the tornadic nature of the 3D detection.The NTDA was found to have a much higher probabil-ity of detection (POD) than the WSR-88D TVS algo-rithm while still keeping the false alarm rate (FAR)reasonable.

These existing TVS detection algorithms rely heavilyon the velocity differences between two adjacent gatepairs; there exist, however, many uncertainties with theestimation and use of such velocity differences, due toradar data noise, the azimuthal offset of sample vol-umes from the center of the rotation features (Woodand Brown 1997), natural small-scale variations in theflow (Desrochers and Yee 1999), and other data qualityproblems (Zittel et al. 2001). These problems can createspuriously large velocity differences or strong azi-muthal shears that can significantly increase the falsealarm rate. In addition, because tornado size and inten-sity vary, it is difficult to set thresholds based primarilyon GVDs that work for all cases (Desrochers and Yee1999). The discrete sampling of the atmosphere by ra-dar whose sampling resolution changes with the rangeand whose beam positioning relative to the vortex cen-ter affects GVD values further complicates the issue. Itwould be very helpful if additional discriminating pa-rameters could be found that would better define thecharacteristics of tornadoes and their associated vorti-ces. Ideally, such parameters are insensitive to theaforementioned problems. Finding such parameters isthe motive of this study.

Since the azimuthal shear signatures associated withtornadoes as seen in Doppler radial velocity data areoften not confined to two adjacent radar gates in theazimuth, the shear or velocity difference between largerdistances (compared to the adjacent radar gate distancein the azimuth) or between regions can provide valu-

able information that can potentially be utilized to re-duce the false alarm rate of detection. In this paper, weuse the wavelet analysis technique for the quantifica-tion of velocity differences at different spatial scalesand use these measures to help design or improve tor-nado detection algorithms. In particular, a set of wave-let basis functions will be used and each basis functiondescribes the averaged radial-velocity difference be-tween two adjacent regions covered by this basis func-tion. By applying the wavelet analysis to the radial ve-locity field, a region-to-region velocity difference(RVD) can be defined in terms of the wavelet coeffi-cients. When normalized, the relative region-to-regionvelocity difference (RRVD) for each scale can be ob-tained. If the relative difference is larger than a certainthreshold value, then the region at the correspondingscale can be considered as having a significant radialwind shear and a high potential for containing a tor-nado. The normalization allows for a more general de-termination of the threshold without depending toomuch on or being too sensitive to the tornado intensityinformation.

For similar reasons noted above, Smith et al. (2003)and Smith and Elmore (2004) recently showed that alocal, linear, least square approach to calculating theradial velocity derivatives, including the azimuthalshear, leads to a significant improvement over the typi-cally used method of calculating the shear from twodata points. With the approach, a least square estima-tion of the shear in a local region that includes a num-ber of radial velocity data points is performed to im-prove the reliability of the shear estimate. There is acertain similarity between our wavelet-analysis ap-proach and their approach, in that both methods utilizeradial velocity data at more than two neighboring datapoints. Our approach further examines the shear mag-nitudes at more than one scale.

This paper is organized as follows. In section 2, wave-let analysis is introduced and scale-dependent RRVD isdefined. Analyses are performed on radial velocity datasampled from an idealized tornado and from a realistictornado from a high-resolution numerical simulation. Atornado detection algorithm based on RRVD and otherparameters is proposed in section 3 and applied to areal tornado case in section 4. Conclusions are given insection 5.

2. Wavelet analysis and applications toradial-velocity fields

a. Wavelet transform

Similar to the commonly used Fourier transform, thewavelet transform is a method of converting a function

MARCH 2007 L I U E T A L . 345

(or signal) into another form or representation, usuallyin terms of the wavelet coefficients, that would allowcertain features of the original signal more amenable tostudy or enable the original dataset to be describedmore succinctly (Daubechies 1992). In the case of aFourier transform, the coefficients of different Fouriercomponents reveal the strength or intensity of the sig-nal at different wavelengths or frequencies. The mean-ing of the wavelet coefficients dependents very muchon the choice of the wavelet basis function used, whichhas wide variations.

The wavelet transform of a spatial function (or sig-nal) f(x) in two dimensions can be defined in general by

W�s, l� � �f�x��s,l�x� dx, �1�

where x and l are vectors, x � (x, y) and l � (lx, ly), andx and y, lx and ly (all are real) are independent spatialvariables. Here, x � (x, y) defines the spatial coordi-nates of the original function whereas l � (lx, ly) definesthe location at which the local wavelet basis function,�s,l(x), is centered. We use W(s, l) to represent thewavelet transform coefficients at scale s and location l.

The wavelet functions, denoted by �s,l(x), are con-structed by shifting and dilating a wavelet prototypefunction �(x), and �s,l(x) can be expressed as

�s,l�x� � p�s��x � ls �. �2�

The wavelet coefficient W represents the correlationbetween the original signal and the correspondingwavelet basis function or the amount of energy in thesignal at scale s and location l. Here, p(s) is a weightingfunction, which is typically p(s) set to 1/�s for energyconservation.

For discrete signal analyses, we need to use discretewavelet transform algorithms (e.g., Daubechies 1992).The discretization of the wavelet basis function has theform of

�m,n�x� � ��x � n

s0m �, �3�

where integers m and n � (nx, ny) are used instead ofthe continuous s and l in Eq. (2) and they control thewavelet dilation and translation, respectively; s0 is aspecified fixed dilation step parameter (�1). The wave-let transform is then

Wm,n � p�m��f�x��m,n�x� dx, �4�

and Wm,n is the discrete version of the wavelet trans-form coefficient corresponding to scales and locationsrepresented by m and n.

According to the above formulations, the wavelet ba-sis function can be moved to or defined at differentlocations of the data field. In our case, for analyzingradar observations, the basis function can be shifted inthe azimuthal and radial directions. The basis functionitself also can be stretched or compressed, as describedby Eqs. (2) and (3). By using (1) or its discrete versionin (4), the wavelet transform quantifies the local matchof the wavelet function with the original function orfield f(x). When the wavelet matches the shape of thefunction well at a specific scale and location, a largewavelet coefficient W is obtained. Just the oppositetakes place when the wavelet and function do not cor-relate well, as a low value of coefficient W is obtained.Thus, using the wavelet transform, a map of local cor-relations between the wavelet (at various scales andlocations) and the analyzed field can be obtained. Bychecking the wavelet coefficients in the wavelet space,the information of the spatial structures of the field canbe presented in a more useful form.

b. Relative region-to-region velocity difference

Within the Doppler radar radial velocity observa-tions, a tornado usually appears as a couplet of en-hanced incoming and outgoing radial velocities. Be-cause tornadoes usually do not exist in isolation, theshear information associated with a tornado is oftencontained in more than two adjacent radial velocitysampling volumes or pixels of observations (Vasiloff2001; Wood and Brown 1997). Here, a pixel corre-sponds to a radar sampling volume in the image form.Through wavelet analysis, it is possible to quantify azi-muthal radial velocity differences between regions ofdifferent scales, and we can use the shear information atmore than the gate-to-gate azimuthal shear scale aloneto help improve tornado detection.

For the description of the azimuthal shear of radialvelocity in an elevation plane in two dimensions, the 2DHarr wavelet is chosen in this study. Because the mainpurpose of our wavelet analysis is to highlight or extractthe azimuthal shear information at multiple scales, thestep-function-like Haar wavelet appears most suitable.This wavelet function is given by

��x� � ��x, y� � �1 0 � x �

12

, 0 � y � 1

�112

� x � 1, 0 � y � 1

0 elsewhere

.

�5�


Note that in our case, x and y denote the azimuthal andradial directions, respectively. This wavelet basis func-tion can be dilated to construct a set of wavelet func-tions representing different scales in the two dimen-sions. Such 2D wavelet functions can also be expressedas a series of matrices in discrete form using the dyadic-scale discretization. For example, the wavelet functionsfor the first two scales (m � 1 and 2) can be expressedin matrices as follows:

�1 � �1 �11 �1�, �2 ��

1 1 �1 �1

1 1 �1 �1

1 1 �1 �1

1 1 �1 �1� ,

�6�

where the subscripts represent m. As can be seen, a 2Ddiscrete Harr wavelet at scale m � 2 is a dilation in boththe x and y directions from the wavelet at scale m � 1.The matrices in the above equations are mapped to theradial velocity data space, so that each element in thematrices corresponds to one radial velocity measure-ment while the centers of the matrices correspond tothe central location where the wavelets are defined.The columns and rows correspond to the radial andazimuthal directions, respectively. The left half of thematrices is unity while the right half is negative unity.There is a jump between the left and right halves of thematrices, making them suitable for describing azi-muthal shear features.

When a 2D wavelet function at a given scale is used,the wavelet coefficient can be used to measure the dif-ference of the signal f(x) across the two regions corre-sponding to the left and right halves of the discretewavelet function. Applying such a wavelet transform tothe Doppler radial velocity field, a large value of thewavelet coefficient at a given scale indicates the exis-tence of a large velocity difference between the twoadjacent regions. The sizes of the regions are 2m � 2m�1

pixels for scale m, and p(m) in Eq. (4) is given by p(m)� 1/(2m x 2m�1). We can obtain a wavelet coefficient foreach scale m at each particular location that measuresthe region-to-region velocity difference (RRVD) atthat location for the corresponding scale.

However, the intensity of the tornado vortex varieswith cases and the wavelet coefficients vary with thescale so that it is still not easy to determine the generalthresholds for separating a tornadic vortex from non-tornadic shear signatures. To make it easier to deter-mine more general thresholds, we define here the nor-malized or relative wavelet coefficients for each scaleaccording to

RRVDm,n �Wm,n

Wm, max, �7�

where Wm,max is the maximum wavelet coefficient atscale m. RRVDm,n measures the relative region-to-region velocity difference at scale m and location n. Thevalue of RRVD ranges from 0 to 1 at any scale. A largevalue of RRVDm,n indicates the presence of a signifi-cant shear or a jump in the original signal between thetwo regions at scale m.

c. Wavelet analysis for an idealized vortex

To quantify the main RRDV characteristics of tor-nadoes and to determine the suitable RRDV thresholdsfor their detection, we first apply our wavelet analysisto an idealized tornado vortex, given by a modifiedRankine combined vortex model as used by Brown etal. (2002):

V � Vx�R�Rx��, �8�

where V is the rotational velocity at radius R (from thevortex center) and Vx is the maximum rotational veloc-ity at the core radius of Rx, and � � 1 for R � Rx and� � �0.6 for R Rx. For our sampling experiment inthis section, we assume Rx � 200 m and Vx � 50 m s�1,similar to Brown et al. (2002). The WSR-88D radialvelocity simulation method of Wood and Brown (1997)is used here to simulate the radial velocity observations.Specifically, the radial velocity is sampled at 1° azi-muthal intervals and at a 250-m range resolution. Thecenter of the simulated vortex is located 20 km north ofthe radar. For simplicity, a 0° elevation angle (strictlyspeaking, the lowest elevation angle of the WSR-88Dradar is 0.5°) and uniform reflectivity are assumed. Foran effective sampling volume, the mean radial velocityvr(0, r0) at the range, r0, and the azimuth angle, 0, aregiven, following Wood and Brown (1997), by

vr��0, r0� �

�i

I

�j

J

�r��i, rj�W�r�2f 4��

�i

I

�j

J

W�r�2f 4��

, �9�

where I and J are, respectively, the number of samplestaken along the azimuth and range in the sampling vol-ume, �r(i, rj) is the radial velocity in the ith and jthsampling points, and W(r) and f() are the range andantenna pattern weighting function, respectively. Using(8), the u and � components of the wind in Cartesiancoordinates are calculated first on a grid of 25-m reso-lution. The gridpoint values within a radar sample vol-


ume are averaged, according to (9), to obtain the radialvelocity observations, where �r(i, rj) is given by

�r��i, rj� � u��i, rj� cos��i� ��i, rj� sin��i�. �10�

For the simulation of radial velocity data, the angulareffective beamwidth used is 1.39°, that of the WSR-88D, and the (linear) effective beamwidth (EBW) is485 m at the 20-km range, which is larger than the corediameter of 400 m of this idealized vortex. TVS is there-fore expected in the radial velocity field (Brown 1998).The simulated radial velocity field is shown in Fig. 1a.As can be seen, the radial velocity in the left half of theplotting domain is inbound relative to the radar and istherefore negative and that in the right half is positive.A large velocity difference exists across the central axisof the domain, through the vortex center, and the larg-est velocity difference occurs between two pixels sepa-rated by a pixel with nearly zero velocity. This is a TVS

that occurs when one of the beam centers coincidemore or less with the vortex center (Brown 1998). Thewavelet analysis described in section 2a is applied tothis radial velocity field in the radar polar coordinatesand the RRVDs are calculated from Eq. (7) for eachscale and location. The RRVDs for m � 1, 2, and 3 areplotted in Figs. 1b, 1c, and 1d, respectively. As can beseen, the RRVDs near the center of the vortex arelarger than 50% for all three scales and are significantlyhigher than the values away from the vortex. In fact, themaximum values all exceed 90% near the center of thevortex, which should be the case by definition. Thenumber of pixels whose RRVD is larger than 0.5 is 6 forscale m � 1 (Fig. 1b), 12 for scale m � 2 (Fig. 1c), and16 for scale m � 3 (Fig. 1d). Therefore, the signature ofthe vortex increases in size in terms of RRVD as thescale of the RRVD increases. It is clear from this ex-ample that the multiscale characteristics of RRVD in

FIG. 1. (a) Simulated radial velocity field from the modified Rankine combined vortex and the correspondingRRVD (%) at scales m � (b) 1, (c) 2, and (d) 3. The radar is 20 km south of the tornado, located at the coordinateorigin.


Fig 1 live 4/C

the tornado region are different from those in otherregions, suggesting that the RRVDs at multiple scalescan be useful for identifying a tornado vortex in radialvelocity data. In practice, additional criteria will also beneeded to unambiguously identify a tornado.

d. Wavelet analysis for a numerically simulatedtornado

A much more realistic tornado simulated at high spa-tial resolution is available for further quantifying thecharacteristics of RRVD and determining their thresh-olds. This simulation was performed using the Ad-vanced Regional Prediction System (ARPS; Xue et al.2000, 2001), starting from a modified sounding for 20May 1977 during a Del City, Oklahoma, tornadic su-percell storm. A uniform horizontal resolution of 50 mwas used together with a vertical stretched grid with anear-surface vertical resolution of 20 m. Over a half-hour centering on the period of most intense tornadoactivity, a uniform horizontal resolution of 25 m wasused and the model domain was 48 km � 48 km in thehorizontal. A maximum ground-relative wind speed ofover 120 m s�1, located about 30 m above the frictionalground, was obtained in the simulated tornado, with apressure drop of over 80 hPa at the center of the tor-nado vortex. Detailed description and analyses of thesimulations will be reported elsewhere. In this paper,we sample the ground-level flow of this simulated tor-nado in the same way as we did in the earlier idealizedRankine combined vortex case, again at an azimuthalresolution of 1° and at a range gate resolution of 250,with an elevation angle of 0.0°. The same 1.39° effectivebeamwidth is assumed. For the experiment reported inthis section, the radar is located 20 km west of thedomain center. The view from west is chosen becausethe elliptically shaded ground-level tornado vortex isnarrower in the north–south direction (Fig. 2); the tor-nado is smaller when viewed from the east or west andis, therefore harder to detect. As can be seen, the corediameter of the simulated tornado in the north–southdirection, as determined by the distance between theminimum and maximum east–west velocities associatedwith the tornado vortex, of about �20 and 35 m s�1

(Fig. 2a), respectively, is about 200 m. This tornado, as

→

FIG. 2. The (a) east–west and (b) north–south velocity compo-nents of the ground-level winds and (c) wind vectors of a numeri-cally simulated tornado that developed within a numerically simu-lated supercell storm. A 1-km-squared domain centered on thetornado is shown; the horizontal resolution of the numerical simu-lation was 25 m.


seen at the ground level, is weaker than the idealizedtornado of the previous section. The wind field in thiscase is also more complex than the case of the idealizedvortex.

The simulated radial velocity field is shown in Fig. 3a,together with horizontal wind vectors. In this plot, thetornado is centered at (0, 20) km while the radar islocated at the coordinate origin. There exists obviouslystrong convergence in the tornado region and a regionof strong divergence northwest of the tornado that orig-inated from the rear-flank downdraft. Apart from that,the general tornadic vortex features, as revealed by theradial velocity field, are similar to that of the idealizedvortex case. The central part of the tornado is markedby a closed circle in the figure. Similar to the earlieridealized case, we can see a radial velocity couplet witha large velocity difference within the circled tornadicregion. The RRVDs at different scales are calculated

using (7) and are shown in Figs. 3b–d. The RRVDs atthe location of tornado are again larger than 50% andsignificantly higher than those in other regions at allthree scales. The size of the region with large RRVD(0.5) increases with the scale. In contrast to the ide-alized vortex case, relatively large values of RRVD alsoexist away from the tornado due to wind shear andconvergence in other parts of the flow but their valuesare generally significantly below 0.5.

e. Impact of the distance of tornado from radar andtornado size on RRVD

As the distance from the tornado to the radar in-creases, the azimuthal distance at the tornado locationincreases so that the tornado becomes more poorly re-solved in the radar data. The degree of difficulty inunambiguously detecting a tornado also increases. Totest the ability of RRVD in providing distinguishing

FIG. 3. As in Fig. 1, but for simulated radial velocities and the corresponding RRDVs from the numericallysimulated tornado shown in Fig. 2. The radar is located 20 km west of the tornado.


Fig 3 live 4/C

information when the tornado is located at larger dis-tances than examined in the previous subsections, weresampled both the idealized and simulated tornadoes,but with the radar located at distances between 20 and100 km from the tornado, and increased the distance by5 km from experiment to experiment. In addition, forthe case of the idealized tornado, we further decreasedthe core diameter from the original 400 m to 100 m, toimpose more stringent tests on our method. The size ofthe simulated tornado remained the same.

Figure 4a shows the radial velocities sampled alongthe constant range circles that pass through the centerof the idealized vortex when the radar is located 25, 50,and 75 km away from the tornado. As can be seen, thesampled peak velocities are 13.6, 9.3, and 7.3 m s�1 atranges of 25, 50, and 75 km, respectively, even thoughthe true maximum velocity is 50 m s�1. The RRVDs atthe first three scales for the tornado at the three differ-ent distances are shown in Figs. 4b, 4c, and 4d. Clearly,the RRVDs near the center of the vortex are largerthan 50% for all three scales and all three ranges. It isfound that for this idealized tornado, even when therange is 100 km, the RRVDs still show very distinguish-able characteristics in the region of the tornado.

The effectiveness of RRVD with the more realisticnumerically simulated tornado is further examined fordifferent radar ranges. The radial velocity field sampledby a radar located 60 km west of the tornado and thecorresponding RRVDs at the first three scales areshown in Fig. 5. At this range, the sampling volume orpixel appears rectangular, with its azimuthal width be-ing much broader than the range gate length (Fig. 5a),and only a single positive–negative radial velocity cou-

plet shows up near the vortex center. The incoming andoutgoing peak velocities near the tornado center areabout �6 and 10 m s�1, respectively. The TVS is there-fore rather weak. However, the presence of a tornadocan still be revealed by RRVDs at all three scales (Figs.5a–c); the largest RRVDs show up in the tornado re-gion at all three scales. We further found that evenwhen the radar range is larger than 70 km, largeRRVDs still show up in the tornado region, but a strongenough TVS cannot be identified from the radial ve-locity field, making the detection more difficult or lessreliable. The 70-km range corresponds to an EBW tocore diameter ratio of about 8.5. This suggests that forthis realistic simulated tornado, when this ratio issmaller than 8.5, RRVD can be a useful and effectiveparameter.

In summary, the general formulation of the waveletanalysis is first introduced in this section. The wavelettransform based on a 2D Haar wavelet function is thenapplied to the radar radial velocity fields associatedwith two examples of tornado vortices to quantify therelatively region-to-region velocity difference (RRVD)at different scales. For both examples, it is found that 1)in the wavelet domain the tornado vortex is generallyassociated and collocated with large values of RRVD,2) the large values of RRVD (0.5) usually exist atmore than one scale, and 3) the size of the region withlarge RRVD (0.5) typically increases with the waveletscale. These characteristics of RRVD are reliable untilthe ratio of the EBW to the core diameter of the tor-nado exceeds 8.5 for a tornado with a 55 m s�1 peakvelocity difference. This ratio is usually larger for tor-nadoes with a larger peak velocity difference. These

FIG. 4. (a) Sampled radial velocity from modified Rankine combined vortex model and corresponding RRVDs(%) at radar ranges of (b) 25, (c) 50, and (d) 75 km.


Fig 4 live 4/C

characteristics will be used to help identify tornadoes inthe next section.

3. Tornado detection based on wavelet analysis

We present in this section a wavelet-based tornadodetection algorithm, which we call WTDA for short, byusing RRVDs calculated for different scales combinedwith other criteria. If only the RRVD at the smallestscale is used, it becomes similar to algorithms based onthe gate-to-gate velocity difference (GVD) or azi-muthal shear. As demonstrated in the last section, theRRVD is rather effective in identifying regions ofstrong shear, and it can do so at different scales. It istherefore chosen as our primary identification param-eter (ID). However, the RRVD alone is usually insuf-ficient to uniquely and reliably identify tornado vorticeswithout additional information. Two additional identi-fication parameters are also used in our algorithm.They are, respectively, the mean reflectivity (MRF) and

the region-to-region radial velocity difference (RVD)averaged over the same region as that of RRVD at thecorresponding scale. An MRF value larger than 0 dBZ(Mitchell et al. 1998) is required because strong andwell-organized tornadoes rarely occur in the case ofreflectivity below 0 dBZ and radial velocity measure-ments are usually not reliable in the absence of reflec-tivity of sufficient decibel strength. RVD is required tobe larger than a threshold value, which is currently setfor the smallest scale to be the minimum required shearvelocity difference of 11 m s�1 as in NTDA. Typically,RVD at larger scales tends to be smaller, and its thresh-old is therefore set to 9 and 8 m s�1 at scales 2 and 3,respectively. Additional idealized experiments wereperformed to determine the proper choice of RVDthreshold values. It is found that for an idealized tor-nado with a maximum rotational velocity of 50 m s�1,even when the core diameter is as small as 100 m andthe radar is at the far range of 100 km, the RVDs at the

FIG. 5. As in Fig. 3, but for a radar located 60 km west of the tornado.


Fig 5 live 4/C

first three scales are still larger than these thresholds.The function of the RVD parameter is somewhat simi-lar to that of the 2D features used by Mitchell et al.(1998), which were constructed from at least threeshear segments. A shear segment is a velocity pair be-tween adjacent beams whose velocity difference ex-ceeds an adaptable but specified threshold. However,the 2D feature does not really examine the wind shearin two dimensions but the shear between adjacentbeams, in the azimuthal direction only, while the RVDbetter represents 2D features by examining the windshear in square regions of different scales. The MRFand RVD requirements are designed to exclude regionswhere the RRVDs are large but there is no wind rota-tion of significant strength.

A flowchart of this algorithm is given in Fig. 6 toshow how the WTDA is implemented based on, forexample, the WSR-88D level-II data. The first step ofthe algorithm is the quality control (QC) of raw radardata, which includes unfolding aliased radial velocities,removing clutters, range-fold data, and other quality-related problems (Liu et al. 2003; Zhang et al. 2005; Liuet al. 2005). This step is important in avoiding dataquality related false detections (Mitchell et al. 1998).After the QC step, missing values of the radial velocityare filled in with the median of the four adjacent rangegates as is done in Smith et al. (2003) if the number ofcontinuously missing measurements along a beam isless than 3.

The second step is to calculate the three ID param-eters for each of the three smallest scales. Based on ourearlier examinations of idealized and simulated torna-does, large values of RRVD exist at two or more scalesin the presence of a tornado, and therefore a scale con-tinuity check is applied in the third step. For a givenlocation, if the RRVD exceeds a specified threshold,for example, 0.5, in any two adjacent scales, the locationof the corresponding pixel or sampling volume is saved.Note that the threshold of RRVD is an adjustable pa-rameter. In this paper, the threshold is specified to be0.5 based on the cases of idealized and simulated tor-nadoes examined in the previous section. An RRVDvalue larger than 0.5 implies that the radial wind shearin the region is relatively large at the correspondingscale. Thus, such regions deserve special attention.

The pixels or sampling volumes saved in step 3 arefurther checked for their values of MRF and RVD. IfMRF is smaller than 0 dBZ or any of the RVDs at thefirst three scales are smaller than their thresholds, thepixel is discarded. The fifth step builds 2D groups fromthe remaining qualifying pixels. Each pixel is first con-sidered as an individual group. If the distance between

two groups is smaller than two range-gate or azimuthalintervals, they are combined into the same group. Thisprocess is repeated until the distance between any twogroups is larger than two gate intervals. The sixth stepchecks the properties of each group. We require thatthe mean reflectivity, which is averaged over all validobservations in each group, is larger than 0 dBZ, andfurther the peak incoming and outgoing velocities arelarger than 6 m s�1 and that there exists a sign changein the radial velocity in the azimuth direction to ensurethat the vortex is produced by incoming and outgoingradial velocities. The groups that pass all the abovechecks will be considered tornadic. The location of thegroup center is defined as the center of the tornado.The maximum radial velocity and the center locationfor each of the tornadic vortices are then recorded.

As mentioned earlier, in our algorithm, the RRVD atthe smallest scale is similar to the velocity differencebetween adjacent velocity gate pairs, the key identifierin the NTDA. However, as discussed in the introduc-tion, calculating shear from two data points can be un-reliable due to radar data noise arising from data qual-ity problems related to, for example, receiver saturationand moving clutter targets (Zittel et al. 2001). Suchradar data quality problems are difficult to eliminate inlevel II data. Natural small-scale variations in the flow

FIG. 6. Flowchart of the wavelet-based tornado detectionalgorithm.


fields can also create representativeness problems withthe GVD calculations (Desrochers and Yee 1999),thereby causing false alarms. On the other hand, largeGVDs due to data noise tend to be random and local-ized and their spatial scales are small. Such localizedproblems tend to be filtered out effectively in the wave-let components of larger scales. As was shown in sec-tion 3, the information about the tornado usually existsat more than one wavelet analysis scale in the radialvelocity data, and when a tornado is present, it is oftenidentifiable at all three of the smallest scales in thenormalized wavelet amplitudes, or RRVD in our case.When the scale continuity check is applied, featuresdue to small-scale data noise can be filtered out effec-tively, thereby reducing the false alarm rate. As dis-cussed in the introduction, the use of multiple datapoints in the local, linear least square estimation of thevelocity derivative in the work of Smith et al. (2003)and Smith and Elmore (2004) serves a somewhat simi-lar purpose, although the actual approach is different.

It should be pointed out here that Desrochers andYee (1999) also applied wavelet analysis to vortex de-tection and, in their case, to the detection of mesocy-clones. In their method, the wavelet analysis is used toremove small-scale variations and fill data gaps so thatthe mesocyclone-related signals can be retained at thescales of interest. Therefore, the purpose of their ap-plication is very different from ours.

In addition, as pointed out by Desrochers and Yee(1999), traditional tornado detection algorithms do notidentify the 2D features of the velocity couplet directlybut derive them through 1D azimuthal shear segments.This is a factor that potentially increases the false alarmrate (Lee and White 1998). In this paper, the velocitycouplet is described in terms of the region-to-regionvelocity difference and RRVD directly measures the2D radial wind shear at each point and detects the ve-locity couplet instead of 1D shear segments. Also, be-cause the size of tornado vortices varies from tens ofmeters to 1–2 km, the traditional detection algorithmsoften have difficulties in defining suitable criteria forconstructing 2D features from 1D azimuthal shear seg-ments. The mesocyclone detection algorithms (MDAs)often misidentify a tornado vortex as a mesocyclonewhen the tornado is close to a radar and misidentify amesocyclone as a tornado when the mesocyclone is faraway. On the other hand, traditional tornado detectionalgorithms do not attempt to identify mesocyclones.Because our WTDA identifies 2D features in terms ofthe wavelet functions representing different scales, itmay provide a natural way for combining MDA andTDA. This can be an area of future research.

4. Testing of WTDA with the 8 May 2004 centralOklahoma tornado case

a. Statistical characteristics of RRVD

The real test of any detection algorithm should beagainst real tornado cases. Radar data collected by theOklahoma City, Oklahoma, WSR-88D (KTLX) on 8May 2003 are used to examine the performance of ourdetection algorithm. On that day, a weak supercell tor-nado (F0) formed at 2206 UTC about 20 km south ofthe KTLX radar. It reached an intensity of F4 by 2220UTC but dissipated by 2236 UTC after moving to about15 km northeast of the KTLX radar. The tornado trav-eled for about 27 km on the ground (NCDC 2003). Thistornado was observed by the radar in a series of sevenvolume scans. A good description of the case can befound in Burgess (2004). In each volume scan, the tor-nado core region in the lowest elevation of KTLX datais subjectively identified for verification purposes.

To examine the statistical characteristics of radial ve-locity data within the tornadic and nontornadic regions,we separate the radar observations in the lowest eleva-tion into two groups. The observations within the tor-nado region are classified into group I, which contains147 observations from the seven lowest elevations. Thegate-to-gate check of the NTDA is applied to the non-tornadic regions. If the velocity difference is larger than11 m s�1, the corresponding radar observations areclassified into group II so that they will be examined bythe detection algorithm (those with smaller velocity dif-ferences are ignored). Here, the 11 m s�1 threshold isthat used in NTDA (Mitchell et al. 1998). Group IIcontains 9352 observations. As can be seen, the numberof observations in group II is far larger than that ingroup I. All data points in group II should eventually bediscarded by the tornado detection algorithm as weknow they are not tornadic. It will be shown that theWTDA is more effective in eliminating data points ingroup II based on the statistical features of the twogroups of data.

When the wavelet transform is applied to each sweepof radar observations, RRVD can be calculated accord-ing to Eq. (7). The frequency histograms of RRVD areplotted for the first three scales in Fig. 7 for group I dataand in Fig. 8 for group II data. According to Figs. 7aand 8a, the RRVD values for the smallest scale (m � 1)at most of the observation points are smaller than 0.5 inboth groups. This indicates that it is difficult to distin-guish real tornadic from nontornadic wind shear whenusing the smallest scale alone. However, as scale indexm increases, the difference in the histograms betweenthe two groups becomes larger. In group II, most of the


RRVD values remain in the range of 0–0.5 for scaleindices 2 and 3 while a very small percentage (�5%) ofvalues exceed 0.5 for all scales, indicating that at least95% of nontornadic shears can be eliminated when a

FIG. 7. Frequency histograms of RRVDs for data points ingroup I (tornadic shears) at scales m � (a) 1, (b) 2, and (c) 3.

FIG. 8. As in Fig. 7, but for RRVDs for data points in group II(nontornadic shears).


RRVD threshold of 0.5 is used. For group I, however,the distribution is increasingly biased toward large val-ues of RRDV as the scale index increases; in fact, forscale index 3, 90% of RRVD values exceed 0.5. Thesedistinctive characteristics exhibited at the differentscales between the tornadic (group I) and nontornadic(group II) regions agree with the findings obtainedfrom the earlier idealized and simulated tornado vortexexperiments; that is, RRVDs at multiple scales can bevery helpful for separating tornadoes from nontornadicshears.

b. Results of the WTDA

The WTDA algorithm is applied to the seven volumescans of data from the KTLX radar that were collectedon 8 May 2003 to test the effectiveness of the algorithm.To examine the effectiveness of the scale continuitycheck in rejecting nontornadic shears as compared tothe 2D feature construction and elimination procedurein the NTDA, the number of 2D features initially iden-tified by the NTDA procedure and that of 2D groupsidentified after our scale-continuity check are calcu-lated.

A 2D feature is composed of at least three shearsegments. Constructing 2D features is a way of elimi-nating nontornadic shear features by checking the 2Dspatial continuity performed in NTDA. The numbers ofconstructed 2D features in all seven scans examined arelisted in the first row of Table 1. In NTDA, the numberof constructed 2D features in each lowest scan is no lessthan 30. This implies that the tornado-like 2D featureshave to be eliminated by other steps, such as the 3Dfeature construction step performed in NTDA, whichuses data from more elevations. This step tends to besensitive to the number of elevations available, whichmostly depends on the distance of the tornado from theradar. In addition, we found that the 2D feature checkused in NTDA actually missed the true tornadic featureat 2235 UTC among a total of 35 two-dimensional fea-tures it identified.

A scale continuity check is one of the ways of elimi-nating nontornadic shear in WTDA. We show here thatmost of the nontornadic shears can be eliminated byour scale continuity check using the lowest-elevation

scan only. The number of 2D groups remaining afterthe scale continuity check is counted and listed in thesecond row of Table 1. After the check, the remainingnumber of 2D groups in all scans is no more than eightand all true tornadic features are found to remain. Thescale continuity check is therefore very effective ineliminating nontornadic features while keeping the tor-nadic ones.

We note here that NTDA does include additionalsteps, including the construction of 3D features usingmore elevations, which will also eliminate some of the2D features listed in the first row of Table 1. This stepcan also be applied in our algorithm, although we hopethat 3D feature checking can be more effectivelyachieved by performing in the future 3D wavelet analy-sis.

Most of the remaining nontornadic 2D groups can befurther eliminated by the ensuing steps in our algo-rithm; this is illustrated below for one scan time. Thenumbers of remaining 2D groups after passing throughall steps in WTDA are listed in the third row of Table1 for different times. All true tornadic shears in thescans during this test period except that at 2211 UTCare identified uniquely by WTDA, while the two non-tornadic shears remaining at 2211 UTC after all checksteps of WTDA are caused by ground clutter.

An example of applying our WTDA algorithm to thedata from the KTLX radar is shown in Fig. 9. Figure 9ashows the observed radial velocity field roughly be-tween the 10- and 20-km-range circles at 2220 UTC 8May 2003. Nine 2D features in total are identified byNTDA in the plotted region in Fig. 9a and they aremarked by small labeled circles. Among these, only thecircle labeled 1 is the location of a true tornado, whilethe other features are caused by, for example, datanoise and data quality problems.

The corresponding fields of the RRVD at the firstthree scales are plotted in Figs. 9b–d. As shown in Fig.9b, RRVD is larger than 0.5 at circles 1, 6, 7, and 8 butsmall at other circles. This means that the other fivenontornadic 2D features have been effectively elimi-nated even when using RRVDs for the smallest scaleonly. Note that at scale 1, both the maximum value ofRRDVs and the area coverage of the values exceeding

TABLE 1. The number of remaining 2D features (groups) after an NTDA azimuthal gate-to-gate check, after the WTDA scalecontinuity check, and finally after all steps in WTDA.

Time UTC 2206 2211 2215 2220 2225 2230 2235

2D features after NTDA check 32 31 41 46 31 46 352D groups after scale continuity check 4 8 3 2 6 8 72D groups after all WTDA steps 1 3 1 1 1 1 1


0.5 are larger for circle 2 than for circle 1, where thetrue tornado is located. This hints at the unreliability ofRRVD at the smallest scale, which is closest to thegate-to-gate shear. Starting from scale 2, the featureassociated with the true tornado becomes clearly dom-inant in term of the RRVD value. At scale 2, the fea-ture associated with circle 7 is still identifiable (Fig. 9c)but all other features (circles 6 and 8) become veryweak relative to the background values. In particular,all values of RRVD in circle 6 becomes smaller than0.5, and thus this nontornadic feature arising from datanoise is eliminated. At scale 3, only the tornadic fea-ture is clearly identifiable. Thus, there are two features(circles 1 and 7) that pass the scale continuity check.When we further examine the group property of theremaining features in step 6, there is no sign change forthe radial velocity in the group that is indicated bycircle 7. Thus, the nontornadic feature 7 can be dis-carded by the ensuing steps in the WTDA. The above

example demonstrates that the scale continuity check isable to successfully eliminate most of the nontornadicfeatures identified by simple gate-to-gate shear check-ing.

5. Summary and conclusions

In this paper, wavelet analysis is applied to the de-tection of tornadoes from Doppler radial velocity ob-servations. A tornado detection algorithm is developedbased on the wavelet analysis (called WTDA) as aproof of concept. The scale- and location-dependentwavelet coefficients derived from the radial velocityfield are used to define a relative region-to-region ve-locity difference (RRVD). The RRVD is shown to ef-fectively describe the 2D region-to-region wind shear atdifferent spatial scales and is therefore used in WTDAas a key discriminating parameter. This is in contrast tothe gate-to-gate shears corresponding to the (radar)

FIG. 9. (a) Radar radial velocity field observed by the KTLX radar at 2220 UTC 8 May 2003, and thecorresponding RRVD (%) at scales m � (b) 1, (c) 2, and (d) 3.


Fig 9 live 4/C

grid scale used in typical tornado detection algorithms.Multiscale RRVDs make it more flexible to identifytornadic vortices with less sensitivity to the intensity orsize of the tornado and are more effective in eliminat-ing nontornadic shears caused by data noise and/orquality problems.

The wavelet analysis is first applied to radial velocitydata sampled from idealized as well as numericallysimulated tornadoes, for the purpose of understandingthe characteristics of RRVD associated with tornadovortices. It is found that the tornado vortices are collo-cated with areas where the RRVDs are large and theirvalues usually exceed 0.5 at more than one scale. Basedon these characteristics, the WTDA is developed usingRRVD as the primary identification parameter. TheWTDA is further tested with WSR-88D data from an 8May 2003 central Oklahoma tornado case. The scalecontinuity check in the WTDA, in terms of theRRVDs, is found to be effective in eliminating nontor-nadic shears.

The effectiveness of RRVD is also examined bylooking at a smaller idealized tornado, and samplingthe tornadoes from longer ranges. It is found that whenthe ratio of the azimuthal distance between two adja-cent constant-range gates to the tornado core diameteris smaller than 8.5 for a relatively weaker tornado,RRVD can provide very useful information for tornadodetection. The WTDA was further applied to the 10May 2003 Oklahoma tornado case (results not pre-sented in this paper) to examine the capability ofWTDA for a tornado at a larger distance. In this case,an F1 tornado started at 0525 UTC and ended at 0533UTC and traveled for about 2 mi on the ground. Thecore diameter of the tornado was about 280 m (NCDC2003) and the tornado was observed by the KTLX ra-dar at a range of about 90 km at 0528 UTC. The WTDAwas applied to the lowest-elevation scans around thistime and the results show that after a scale continuitycheck, only four 2D features remained and the tornadowas also correctly identified after ensuing steps in theWTDA algorithms.

The thresholds used for the other identification pa-rameters in WTDA are specified based mostly on pre-vious papers (e.g., Mitchell et al. 1998) and the experi-ence of the authors. These thresholds worked the bestfor the cases examined in this paper as well as for thetornado case of 3 May 1999 near Oklahoma City (re-sults not shown in this paper). Optimal values for awide variety of cases would require testing against alarge radar dataset and tornado database. This isplanned for the future but we believe the examplesshown in this paper serve the proof-of-concept purposethat demonstrates the efficacy of the wavelet analysis.

In addition, the current WTDA algorithm does not in-clude a continuity checking across the elevations and itis believed that the vertical check, when implementedin the future, can further improve the reliability of thealgorithm. We may also seek to implement waveletanalysis in the vertical direction and apply the conceptused in the elevation plane to three dimensions. At thesame time, the fact that our multiscale wavelet-basedalgorithm does a reasonably good job without multiel-evation data represents an important advantage, espe-cially for radars not operating in conventional sit-and-spin modes. The latter include mobile radars that oftenfocus on the low levels, and the dynamically adaptiveradars of the Engineering Research Center for Collabo-rative Adaptive Sensing of the Atmosphere (CASA).The rapid detection of tornadoes and other low-levelhazardous weather and the use of the detection infor-mation to adaptively steer the scanning of its networkof radars are important goals of CASA (see, e.g., Xueet al. 2006).

Since no direction comparisons have been performedagainst a large enough tornado database, we do notclaim the superiority of our WTDA over any of theexisting, established tornado detection algorithms. Theneed for tuning and refinement of our algorithms isexpected when we test the algorithm against a largeenough dataset. Eventually, we want to implement ouralgorithm within the NSSL Warning Decision SupportSystem-Integrated Information (WDSS-II) system andperform side-by-side comparisons with the NSSL TDAto further evaluate the performances of WTDA. In ad-dition, the scale-dependent parameter RRVD devel-oped in this paper can also be used as a parameter inprobabilistic tornado detection algorithms, in a mannersimilar to that done in Lakshmanan et al. (2005).

Acknowledgments. This work was primarily sup-ported by NSF Grant EEC-0313747 of the EngineeringResearch Center Program to CASA. S. Liu and M. Xuewere also supported by an FAA grant via DOC-NOAANA17RJ1227 and M. Xue further by NSF GrantsATM0129892, ATM-0331594, and ATM-0331756. Q.Xu acknowledges the support provided by FAA Con-tract IA# DTFA03-01-X-9007, as well as ONR GrantsN000140310822 and N000140410312. Jerry Brotzge isthanked for proofreading and commenting on the origi-nal manuscript.

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