Sampling Strategies for Tornado and Mesocyclone Detection Using Dynamically Adaptive Doppler Radars: A Simulation Study

Transcription

1 492 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 26 Samplng Strateges for Tornado and Mesocyclone Detecton Usng Dynamcally Adaptve Doppler Radars: A Smulaton Study JESSICA L. PROUD* AND KELVIN K. DROEGEMEIER Center for Analyss and Predcton of Storms, and School of Meteorology, Unversty of Oklahoma, Norman, Oklahoma VINCENT T. WOOD AND RODGER A. BROWN NOAA/Natonal Severe Storms Laboratory, Norman, Oklahoma (Manuscrpt receved 29 October 2007, n fnal form 25 July 2008) ABSTRACT Increasng tornado and severe storm warnng lead tme (lead tme s defned here as the elapsed tme between the ssuance of a watch or warnng and the tme at whch the antcpated weather event frst mpacts the specfed regon) through the use of radar observatons has long been a challenge for researchers and operatonal forecasters. To mprove lead tme and the probablty of detectng tornadoes whle decreasng the false alarm rato, a greater understandng, obtaned n part by more complete observatons, s needed about the regon of storms wthn whch tornadoes form and persst. Drven n large part by ths need, but also by the goal of usng numercal models to explctly predct ntense local weather such as thunderstorms, the Natonal Scence Foundaton establshed, n fall 2003, the Engneerng Research Center for Collaboratve Adaptve Sensng of the Atmosphere (CASA). CASA s developng a revolutonary new paradgm of usng a network of small, closely spaced, nexpensve, low-power dual-polarzaton Doppler weather radars to overcome the nablty of wdely spaced, hgh-power radars to sample large regons of the lower atmosphere owng to the curvature of earth gven that zero or negatve beam elevaton angles are not allowed. Also, current radar technology operates mostly ndependently of the weather and end-user needs, thus producng valuable nformaton on storms as a whole but not focused on any specfc phenomenon or need. Conversely, CASA utlzes a dynamcally adaptve sensng paradgm to dentfy, and optmally sample, multple targets based upon ther observed characterstcs n order to meet a varety of often competng end-user needs. The goal of ths study s to evaluate a varety of adaptve samplng strateges for CASA radars to assess ther effectveness n dentfyng ntense low-alttude vortces. Such dentfcaton, for the purposes of ths study, s defned as achevng a best ft of smulated observatons to an analytc model of a tornado or mesocyclone. Several parameters are vared n ths study ncludng the sze of the vortex, azmuthal samplng nterval, dstance of the vortex from the radar, and radar beamwdth. Results show that, n the case of small vortces, adaptvely decreasng the azmuthal samplng nterval (.e., overlappng beams) s benefcal n comparson to conventonal azmuthal samplng that s approxmately equal to the beamwdth. However, the beneft s lmted to factors of 2 n overlappng. When smulatng the performance of a CASA radar n comparson to that of a Weather Survellance Radar-1988 Doppler (WSR- 88D) at close range, wth both operatng n the conventonal nonoverlappng mode, the WSR-88D (wth a beamwdth about half that of a CASA radar) performs better. However, when overlappng s appled to the CASA radar, for whch lttle addtonal processng tme s requred, the results are comparable. In effect, the samplng resoluton of a radar can be ncreased smply by decreasng the azmuthal samplng nterval as opposed to nstallng a larger antenna. * Current afflaton: Renassance Computng Insttute, Chapel Hll, North Carolna. Correspondng author address: Jessca L. Proud, Renassance Computng Insttute, 100 Europa Drve, Sute 540, Chapel Hll, NC E-mal: proud@renc.org 1. Introducton The current Natonal Weather Servce (NWS) Weather Survellance Radar-1988 Doppler (WSR-88D; Crum and Alberty 1993) radar network s the prncpal tool used for detectng severe storms and tornadoes and for ssung warnngs. Ths network has been crtcal to the forecastng of severe weather and has saved DOI: /2008JTECHA Ó 2009 Amercan Meteorologcal Socety

2 MARCH 2009 P R O U D E T A L. 493 FIG. 1. Shown here s the probablty of detecton, false alarm rato, and lead tme of tornado warnngs from 1986 to (B. MacAloney Jr., Natonal Weather Servce, 2007, personal communcaton). Installaton of WSR-88Ds took place durng countless lves snce ts formal commssonng n 1994 (Smmons and Sutter 2005). To mprove lead tme and the probablty of detectng tornadoes whle decreasng false alarms (Fg. 1), a greater understandng, obtaned n part by more complete observatons, s needed about the regon of storms wthn whch tornadoes form and persst; that s, the regon wthn a few klometers of the ground. Drven n large part by ths need, but also by the goal of usng numercal models to explctly predct ntense local weather such as thunderstorms, the Natonal Scence Foundaton (NSF) establshed n fall 2003 the Engneerng Research Center for Collaboratve Adaptve Sensng of the Atmosphere (CASA). CASA s led by the Unversty of Massachusetts at Amherst wth several academc partners, ncludng the Unversty of Oklahoma, Colorado State Unversty, and the Unversty of Puerto Rco at Mayaguez, along wth ndustral, educatonal, and end-user partners. It s developng fve radar test beds, the frst of whch s located n southwestern Oklahoma and conssts of four radars and emphaszes the detecton of storms and tornadoes. Ths test bed collected numerous datasets durng the sprng 2007 and 2008 severe weather seasons, and the data now are beng analyzed (e.g., Kong et al. 2007; Wess et al. 2007; Kan et al. 2009). The second test bed, whch s beng developed n downtown Houston, emphaszes quanttatve precptaton estmaton for hydrology. Located n Puerto Rco, the thrd test bed s beng developed entrely by students and utlzes solar power and wreless communcaton technologes to study quanttatve precptaton estmaton and forecasts n mountanous terran. The fourth and ffth test beds focus, respectvely, on clear-ar sensng and phased array technology, and are now beng planned. In contrast to most current weather radars, whch operate n a st and spn mode ndependent of evolvng weather, 1 CASA radars are desgned to operate n a dstrbuted, collaboratve, adaptve framework (DCAS; McLaughln et al. 2005). Dstrbuted refers to placng radars n clusters wth a spacng much closer (;25 km) than that of the WSR-88D network so as to overcome samplng problems (e.g., azmuthal resoluton degradaton or beam locaton above ground) that occur at a long range owng to beam spreadng and the earth s curvature. Collaboratve connotes the coordnated targetng of multple radar beams based on atmospherc and hydrologc analyss tools, such as detecton, predctng, and trackng algorthms (McLaughln et al. 2005). By utlzng ths collaboraton, the system allocates resources such as radated power, beam poston, and polarzaton dversty for optmally samplng regons of the atmosphere where a partcular threat exsts. The term adaptve refers to the ablty of the CASA radars and the assocated nfrastructure to quckly reconfgure (e.g., begn usng a smaller azmuthal samplng nterval) n response to changng weather and to meet varous end-user needs. Another unque adaptve characterstc of CASA radars s the ablty to change from samplng a large area to much smaller sector scans, say of a tornadc regon, ndependently or collaboratvely wth other CASA radars. In contrast, WSR-88D radars scan ndependently of one other and contnuously survel the entre volume largely ndependent of weather occurrng wthn t. Gven these features, a fundamental research challenge for CASA concerns understandng whch modes of radar operaton wll yeld the maxmum amount of useful nformaton about a partcular weather phenomenon whle mnmzng the use of avalable resources (so they are avalable for other phenomena or use by other neghborng radars). As a frst step toward addressng ths and related challenges, the goal of the present study s to use pseudo-observatons of an dealzed vertcal vortex to evaluate a varety of samplng strateges for CASA radars n order to determne whch mght be most effectve for real tornadoes and mesocyclones. Here, effectveness s defned as the best ft of the pseudo-observatons to an analytc model of tornadoes and mesocyclones. 1 The Termnal Doppler Weather Radar (TDWR; e.g., Weler and Shrader 1991), operates n a sector scan mode to detect mcrobursts and other avaton hazards. The WSR-88D utlzes a varety of volume coverage patterns (e.g., Klazura and Imy 1993; Brown et al. 2005) that are chosen based upon weather characterstcs; however, none nvolve sector scannng or other modes of adaptaton to specfc weather features.

3 494 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 26 Several parameters are tested, ncludng the radus of the vortex, azmuthal samplng nterval, and the dstance of the vortex center from the radar. It may seem logcal to sample a vortex or other atmospherc phenomena wth as much spatal and temporal resoluton as possble, but dong so may actually waste avalable radar resources 2 whle provdng no useful added nformaton. The allocaton of resources s especally mportant n DCAS, where multple, frequently competng end-user goals must be met n an optmal manner. In the current radar test bed located n Oklahoma, ths allocaton s performed va a polcy medaton framework n whch detecton algorthms run n real tme on radar moment data categorze observed sgnatures and store ther attrbutes (e.g., feature type, tme of development, locaton) n a so-called feature repostory. Ths nformaton s combned wth user-specfed prortes (Phlps et al. 2007) to produce scannng pattern commands that are communcated to the radars by the Meteorologcal Command and Control (MC&C) system. The frequency of ths communcaton s determned by the radar heartbeat, whch presently s 60 s. That s, every 60 s, the radars can be retasked to scan dfferent regons at dfferent rates. For more nformaton on CASA scannng strateges, see Brotzge et al. (2005, 2008), and Gagne et al. (2008). The results of the present study are expected to help optmze ths process for dentfyng potentally hazardous atmospherc vortces early n ther lfetme. Secton 2 descrbes the methodology used, ncludng a detaled explanaton of the vortex model, radar emulator, and retreval technque, along wth parameters vared to study varous samplng strateges. The results are dscussed n secton 3, and a summary and suggestons for future work are presented n secton Methodology a. Vortex model To create a representatve tornadc or mesocyclonc flow feld approprate for samplng by a vrtual CASA radar, an analytc vortex model s used to generate an dealzed one-dmensonal (horzontally through the center of the vortex) azmuthal velocty profle. An analytc vortex model, for whch an exact soluton s avalable, s approprate for use n assessng the potental value of varous adaptve strateges pror to ther applcaton wth real data. One famlar model s the Rankne (1901, ) combned vortex (RCV), whch s gven by v t 5 V x f ðr, R x Þ, (1) where v t s the tangental velocty, V x s the maxmum tangental velocty, and f(r,r x ) s the dmensonless velocty profle gven by 8 r ><, f ðr, R x Þ 5 R x >: R x r, 0 # r, R x. R x # r (2) Here, r s the radal dstance from the vortex center and R x s the core radus at whch V x occurs. The core of the vortex (r # R x ) s characterzed by sold body rotaton, whle for r $ R x, potental flow s assumed. The RCV profle was not used n the present work owng to the dscontnuty n velocty at r 5 R x and ts effects on the mnmzaton algorthm used here. The observed Doppler core radus s usually larger than R x (whch needs to be known to make an ntal guess) because of smearng effects wthn the radar beam. When applyng the mnmzaton algorthm, an updated R x must be known before the velocty profle s scanned agan, and t s dffcult for the algorthm to determne whch part of the velocty profle (core or potental flow) the radar s samplng. Ths can cause nstablty n the algorthm and a lack of convergence to a soluton. Also, accordng to hgh-resoluton tornado observatons (e.g., Wurman and Gll 2000), ths dscontnuty does not exst n real tornadoes, and the decay of tangental wnd speed wth dstance outsde the core of sold body rotaton occurs less rapdly than wth the RCV. For ths reason, we use a more flexble analytc model (developed by L. Whte, Unversty of Oklahoma, personal communcaton 2005) that does not contan a dscontnuty and that decays more slowly than the RCV. Referred to as the three-parameter vortex model (TPVM), t s characterzed by a functon that has no dscontnutes yet retans the prncpal features of and provdes greater flexblty than the RCV: 3 V n ðr, R x Þ 5 V x F n, (3) where F n 5 2nR 2n 1 x r/½ð2n 1ÞR 2n x 1 r2n Š s the velocty profle of the vortex, V x s tangental velocty, r the radus from the vortex center, and n s a nonzero nteger value that controls the velocty profle beyond the core radus. After testng varous combnatons of (3) wth 2 Note that maxmzng resources, n a framework such as CASA n real tme, s nherently a property of dynamcally adaptve systems. 3 Unlke the RCV, the TPVM does not conserve angular momentum. However, that property s unmportant for the purposes of ths study.

4 MARCH 2009 P R O U D E T A L. 495 modfed the emulator to accommodate CASA radar characterstcs usng velocty felds based upon the TPVM. The three-dmensonal Doppler velocty (V d ) can be expressed n terms of the radal (v r ), tangental (v t ), and vertcal (w 1 V T ) components of hydrometeor moton and s gven by V d 5 v r sn g cos u 9 d 1 v t cos g cos u 9 d 1 ðw 1 V TÞ sn u 9 d, (5) FIG. 2. Shown here are three dfferent tangental velocty profles usng Eq. (4). A velocty profle of the RCV s shown for comparson. dfferent values of n, three dfferent tangental velocty profles were selected for ths study: V 1 ðr, R x Þ 5 V x V 2 ðr, R x Þ 5 V x! 2rR x R 2 x 1 r2, (4a)! 4rR 3 x 3R 4 x 1, and (4b) r4 V 12 ðr, R x Þ 5 1 ½ 2 V 1ðr, R x Þ 1 V 2 ðr, R x Þ 5 V x R x r R 2 x 1 r2 1 Š! 2R3 x r 3R 4 x 1. (4c) r4 Profle V 12 was selected as the control tangental velocty profle. Fgure 2 shows a comparson of V 12 and the tradtonal RCV, as well as two pseudo-observaton profles V 1 and V 2 (descrbed further below) that were used to evaluate varous scannng strateges used n ths study. b. Doppler radar smulaton Havng defned the analytc vortex, we focus now on generatng from t Doppler radar pseudo-observatons and usng the varatonal retreval technque to ft them to a dfferent vortex model. The radar emulator used here orgnally was desgned to mmc a WSR-88D radar (e.g., Wood 1997; Wood et al. 2001; Brown et al. 2002) and was appled to an RCV based upon the work of Dovak and Zrnc (1993, pp ). We have where g s the angle between the radar vewng drecton f d at a pont (R d, f d, ud 9 ) and the tangental velocty component, V T s the termnal fall speed of hydrometeors (a negatve quantty), and ud 9 s the beam elevaton angle plus the angle between the vertcal axs at the radar and the vertcal axs at the locaton of the vortex s gven by Dovak and Zrnc (1993, p. 307): ud 9 5 u d 1 tan 1 R d cos u d, (6) a e 1 R d sn u d where a e s the effectve earth radus (1.33 tmes the mean earth radus) that s used to account for beam curvature due to beam refracton. The azmuthal wdth of a radar beam ncreases lnearly wth ncreasng dstance from the radar; however, n addton to ths broadenng wth range, an effectve azmuthal broadenng also occurs for a horzontally rotatng beam, defnng the effectve half-power beamwdth. That s, when a radar antenna scans azmuthally through a fnte samplng nterval whle transmttng and recevng pulses, the feature beng sampled s smeared n the azmuthal drecton, as f the antenna were wder. Three radar parameters determne the degree broadenng (Dovak and Zrnc 1993): 1) antenna rotaton rate, 2) number of pulses transmtted and receved, and 3) tme nterval between pulses. Effectve half-power beamwdth can be reduced by decreasng one or more of these three parameters. The mean Doppler velocty component V d ðf o, R o, u o Þ at the center azmuthal angle f o, center range R o, and the center elevaton angle u o of the effectve resoluton volume of the radar beam s gven by V d ðf o, R o, u 9 o Þ 5 å K k å K k V d Gðf, u 9 d ÞWðR dþ 2 Z Gðf, u 9 d ÞWðR dþ 2 Z, (7)

5 496 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 26 where,, k are the azmuth, range, and elevaton drectons, respectvely; f s the azmuthal angle; and u s the elevaton angle. Reflectvty (Z) s assumed unform across the vortex; G(f, u 9 d) s the two-way Gaussan pattern weghtng functon used to weght Doppler velocty at the (f, u k ) data pont and s gven by " # Gðf, ud 9 Þ 5 exp ðf f o Þ 2 ðu9 k u9 o Þ2 2su 2. (8) 2s 2 f In (8), s f 2 and s u 2 are the standard devatons of the Gaussan densty n the f and u drectons, respectvely, and are specfed by s 2 f 5 f2 e 16 ln 2, (9) s 2 u 5 u ln 2, (10) where f e s the effectve half-power beamwdth n the azmuthal drecton and u 1 s the vertcal half-power beamwdth n the elevaton drecton. Another term n (7) s the Gaussan-shaped range weghtng functon, WðR d Þ 2, used to weght Doppler velocty n range and s gven by where " # WðR d Þ 2 5 exp ðr R o Þ 2, (11) 2s 2 R s 2 R 5 0:35ct, (12) 2 and where R s a gven range, R o s the center range, c s the speed of lght, and t s the pulse wdth (Dovak and Zrnc 1993). The expresson n (7) s a general equaton for mean Doppler velocty. In ths work, a smplfed twodmensonal geometry n the x y plane s employed usng assumptons shown n Table 1. In ths case, (7) reduces to a two-dmensonal problem wth the ad of (1), (3), and (5) and s gven by V d ðf o, R o Þ 5 V n ðr max, rþ cos ggðfþwðr d Þ 2 GðfÞWðR d Þ 2, (13) where V n s the tangental velocty profle beng used where n 5 1, 2, or 12 and cos g s defned as TABLE 1. Assumptons used n ths study. Assumpton No. 1 Tangental velocty feld s unform wth heght 2 Vortex s steady state and does not translate 3 Radal and vertcal velocty components of the vortex n (5) are zero 4 Radar beam pattern s Gaussan shaped and there s no attenuaton 5 Effectve half-power beamwdth vares for each azmuthal samplng nterval 6 Unform reflectvty across the vortex 7 Beam axs s horzontal so ud 9 s approxmately zero 8 CASA radars have a constant rotaton rate 9 No Nyqust velocty lmt cos g 5 R c snðf d f c Þ r 5 sn c, (14) and where f d s the radar vewng drecton, f c s the azmuth of the model vortex center from the radar, and c s the angle between f d and the radal velocty component (va the laws of snes). c. Retreval technque The retreval technque used n ths study s based on a varatonal method smlar to that employed by Wood (1997) and estmates vortex radus (R x ), and maxmum wnd speed (V x ), from pseudo-observatons of the specfed axsymmetrc vortex. Other retreval methods, such as the prncpal component analyss (PCA) method (Harast and Lst 2005), could be employed as well. The technque nvolves frst developng ntal guesses for V x and R x that are used to solve a set of nonlnear equatons. Ther soluton yelds a set of lnear equatons from whch the retreved values of V x and R x are obtaned. Detals about the retreval technque are dscussed n the appendx. d. Input parameters Several nput parameters are held constant n our experments, ncludng antenna elevaton angle and the angle between the vortex center and the closest data pont to the vortex center (both were 08). Holdng these two parameters constant allows us to sample the vortex close to the ground and at ts center. The parameters vared (Table 2) nclude the radus of maxmum azmuthal wnd (R x ) to encompass an array of vortces rangng from a small tornado to an average-szed mesocyclone. The analytc profle used to generate the pseudoobservatons s dfferent from that used for the control profle. Analytc azmuthal velocty profle V 1 typcally

6 MARCH 2009 P R O U D E T A L. 497 TABLE 2. Parameters used throughout study to test varous samplng strateges for the CASA radars. The bottom row gves values for a typcal WSR-88D. Parameter name Values tested Maxmum tangental velocty (V x ) 40 m s 21 for maorty of tests. Also tested 60 and 80 m s 21. Radus of maxmum wnds (R x ) 0.1, 0.5, 1, 1.5, 2, 2.5 km. A few tests used 0.05 km. Pseudo-observatons generated V 1 was used for maorty of tests; V 2 and V 12 were tested also. by: V 1, V 2,orV 12 Range km (ncrements of 2.5 km) for maorty of tests km was tested also. Beamwdth (bw) 28 for maorty of tests; 18 was tested also. Azmuthal samplng nterval (daz) 0.58, 18, and 28 (based on beamwdth used) Effectve half-power beamwdth (ebw) For bw 5 18, daz , ebw bw 5 18, daz , ebw bw 5 28, daz , ebw bw 5 28, daz , ebw bw 5 28, daz , ebw Range samplng nterval (drng) 0.10 km for maorty of tests; 0.25 km was tested also. Nose Specfed standard devaton of whte nose for most tests as 1 m s 21 ; also tred 0, 2, 4, 6, 8, and 10 m s 21. WSR-88D bw , daz , ebw , elevaton angle drng km. s used to generate pseudo-observatons (Fg. 2), and s stronger everywhere than the control profle V 12 beyond the core radus, whereas the opposte s true for V 2.Profle V 2 has such a steep gradent beyond R x that convergence s rarely achevable n the retreval or a very poor estmate s attaned. The true tangental velocty profle V 12 s used n only a few experments to test the algorthm because t yelds overly optmstc results for obvous reasons. The azmuthal samplng nterval s defned as the angular dstance from the center of one beam n the scan pattern to the center of the next adacent beam. Because the half-power beamwdth of the CASA radar antenna s 28, creatng a scan wth no mssng or overlappng radals requres an azmuthal samplng nterval of 28 (Fg. 3, top); n realty, the effectve beamwdth of the scannng antenna s greater than 28 (owng to the factors dscussed above), but to smplfy the current dscusson we assume that t s 28. However, because the CASA radars can scan adaptvely, for example, by changng the azmuthal samplng nterval based upon the phenomena beng sampled, examnng mpacts assocated wth varatons n ths nterval s an mportant part of the present study. For ths reason, smaller azmuthal samplng ntervals of 18 (Fg. 3, mddle) and 0.58 (Fg. 3, bottom) were tested; that s, for an azmuthal samplng nterval of 18 (0.58), 18 (0.58) separates the centers of two adacent 28 beamwdths. Usng a 18 or 0.58 azmuthal samplng nterval s referred to here as overlappng 4 because the beam moves less than one beamwdth from one azmuthal samplng nterval to the next. A lmtaton of overlappng s that data felds may appear to be noser than the more heavly smoothed felds, as the overlapped beams provde less smoothng and better resoluton of smaller-scale features. If overlappng were acheved by usng fewer samples (pulses) nstead of slowng down the antenna, the data felds would appear slghtly noser. 4 Some studes use the term oversamplng, but ths also relates to the number of pulses transmtted wthn a gven tmeframe, thus confusng the ssue. FIG. 3. Illustraton of three azmuthal samplng ntervals for a 28 half-power beamwdth: (top) daz 5 28, (mddle) daz 5 18, (bottom) daz The 23-dB ponts on each curve represent the halfpower beamwdth.

7 498 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 26 FIG. 4. Plot of percent error n V x vs range for R x km, where bw s beamwdth and daz s azmuthal samplng nterval. 3. Results In ths secton we present results of varous smulated samplng strateges for CASA radars, ncludng azmuthal overlappng wth varyng beamwdths and vortex szes. Only results for V x are shown because the results for R x are qualtatvely smlar. We also compare the performance of a sngle CASA radar wth a WSR- 88D radar. a. Azmuthal overlappng wth varyng beamwdths The results presented n ths subsecton utlze four dfferent samplng strateges appled to three dfferentszed vortces. Two samplng strateges use the CASA conventonal beamwdth of 28 whle the other two use a 18 beamwdth (note that the beamwdth of CASA radars currently s 28). For each par of samplng strateges, one uses an azmuthal samplng nterval equal to the half-power beamwdth (referred to as conventonal samplng), whle the other uses an nterval half as large (referred to azmuthal overlappng). The goal s to evaluate any mprovement acheved by azmuthal overlappng. Fgure 4 shows percent error n V x for a moderateszed tornado-scale vortex havng a core radus (R x )of 0.1 km. Percent error s defned as the dfference between the retreved tangental wnd V x and the true value (obtaned from pseudo-observatons based upon the V 12 model profle) dvded by the true value. Note that the error s postve near the radar, ndcatng that the retreved profle has slghtly stronger veloctes than the true profle. Conversely, because the vortex s sampled poorly wth ncreasng range, the assocated error becomes ncreasngly negatve. The conventonal samplng strategy, n whch both the beamwdth and azmuthal samplng nterval are 28, has the largest overall error whle a 18 beamwdth and overlapped samplng nterval of 0.58 has the smallest (Fg. 4). One mght notce that the two mddle curves are essentally dentcal. The azmuthal samplng nterval s the same (18), but the beamwdths dffer by a factor of 2. It s curous that two dfferent beamwdths would produce nearly the same results. So, to determne whether or not ths stuaton s typcal, we reran the computatons for the same vortex wthout nose and wth varous combnatons of beamwdth (0.58, 18, 28) and azmuthal samplng nterval (0.58, 18, 28). We found that the tendency was for curves wth the same azmuthal samplng nterval to cluster together, regardless of beamwdth (not shown). Ths fndng ndcates that, for a gven-szed vortex, the azmuthal spacng of data ponts s more mportant than the beamwdth n resolvng the sgnature of a vortex whose core dameter s less than the beamwdth. An adaptve strategy of overlappng provdes notable mprovement, thereby llustratng that adaptve samplng would be useful for the CASA radars when samplng tornado-scale vortces. However, to mantan the same number of samples for computng Doppler moments and thus mantan data qualty, the antenna rotaton rate must be reduced wth overlappng, thus ncreasng data collecton tme. Shftng a CASA radar nto an overlappng mode s an mportant capablty of the radar, partcularly when a conventonal survellance mode s only margnally able to detect the sgnature of a small vortex. The retreval error s smaller when usng overlappng because the densty of azmuthal data ponts s greater, as llustrated n Fg. 5. Also, the effectve half-power beamwdth s smaller wth decreased azmuthal samplng and, thus, less smoothng/smearng of the true velocty profle occurs. In both the top and bottom mages, none of the data ponts used to determne the Doppler velocty peak occurred wthn the vortex core (shaded band). Ths n part explans why the overall retreval for ths small vortex exhbts sgnfcant error regardless of the azmuthal samplng nterval. However, for the smaller azmuthal (overlapped) nterval, the data densty s greater and thus the profle s closer to the model Doppler velocty peaks (dashed black lne), thereby allowng for a better retreval of V x and R x. Fgure 6 shows percent error n V x for a large tornadoscale vortex havng a core radus of 0.25 km. As n Fg. 4, conventonal samplng produces the greatest error of the four samplng strateges tested wth overlappng

8 MARCH 2009 P R O U D E T A L. 499 As R x ncreases, the error approaches a mnmum value of 13% to 15% for velocty profle V 1 (Fg. 8) for a mesocyclone-szed vortex havng a core radus of 2 km. The small and progressvely smaller errors wth ncreasng vortex radus result from an ncreased number of data ponts wthn and beyond the core of the vortex (Fgs. 5, 7, and 9). The data ponts n Fg. 9 have the same azmuthal spacng (DAZ) as n Fg. 5, but the greater number of ponts across the larger vortex results n a better overall retreval. Not only do the errors become progressvely smaller wth ncreasng vortex radus, they also become ncreasngly postve. Had the pseudo-observatons been generated usng control profle V 12 nstead of profle V 1, the errors would have approached zero. However, wth the V 1 profle havng stronger veloctes than the V 12 profle beyond R x, the retreved profle V RET has stronger veloctes than the control profle and therefore the errors are postve. b. Azmuthal overlappng wth a constant beamwdth FIG. 5. Relatonshps of data ponts relatve to the azmuthal profles of a Doppler velocty sgnature at a range of 15 km for azmuthal samplng ntervals (DAZ) of (top) 2.08 and (bottom) 1.08 for a beamwdth of 28 and a vortex havng a core radus of 0.1 km. The shaded regon ndcates the core dameter of the vortex. The dashed lne represents the model profle V 12 ; the sold black lne represents the pseudo-observaton profle (V 1 ), the thck lne on whch the data ponts fall s the Doppler velocty (V d ) azmuthal profle of the sgnature. The data ponts (black large dots) ndcate the locatons of successve Doppler velocty measurements (V d ) collected at azmuthal samplng ncrements as the radar beam scans across the vortex when one data pont s concdent wth the vortex center. The thck curve ndcates the profle of the retreved tangental velocty (V RET ) data. The RMS represents a root-meansquare dfference between pseudo-observed Doppler velocty data ponts and the retreved data ponts usng (A9) n azmuthal and range drectons. The results descrbed prevously show that overlappng s ndeed a benefcal adaptve samplng strategy, 5 especally for small vortces that mght be only margnally detected usng conventonal scannng. Therefore, the goal of experments n ths subsecton s to determne the degree of overlappng needed to show sgnfcant mprovement n the retreved results. A beamwdth of 28 s used for all cases and overlappng factors of 2 and 4 are tested. Only one plot, for R x km, s shown because overlappng has the bggest mpact on ths sze vortex. For larger vortces, all of the samplng strateges produce nearly equal results. Fgure 10 shows the percent error n V x for R x km usng three samplng strateges. Any factor of overlappng yelds an mprovement over conventonal samplng. A much larger mprovement n error s evdent when the azmuthal samplng nterval s reduced from 28 to 18 than from 18 to Resoluton mprovements owng to overlappng can be calculated by takng the rato of the effectve half-power beamwdths (e.g., Brown et al. 2002). Therefore, when reducng the azmuthal samplng nterval from 28 to 18, the relatve ncrease n azmuthal resoluton s gven by showng a great deal of mprovement. As expected, the 18 beamwdth, coupled wth a 0.58 azmuthal samplng nterval, produces the smallest error; however, all strateges except for conventonal 28 samplng begn to exhbt smlar errors wth ths larger vortex. Compared to the vortex shown n Fg. 5, Fg. 7 llustrates that a larger vortex results n an mproved retreval. resoluton mprovement 5 ebw for daz 5 28 ebw for daz :908 2: : %. ð15þ 5 Although we focus here on vortces, CASA radars can survel other features wthn a storm (e.g., gust fronts, heavy precptaton cores) n a temporally nterleaved manner.

9 500 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 26 FIG. 6. Plot of percent error n V x vs range for R x km. Smlarly, when reducng from 18 to 0.58, the ncrease s resoluton mprovement 5 ebw for daz 5 18 ebw for daz 5 0: :238 2: :08 5 8%. ð16þ Consstent wth our results, ths suggests that a factor of more than 2 n overlappng would most lkely be a waste of resources. c. Azmuthal overlappng for a small vortex In addton to the vortex szes evaluated prevously, a small vortex of radus R x km, whch s closer to the average sze of a tornado, was tested. Table 3 (Table 4) shows the results for a beamwdth of 28 (18). As expected, a 28 beamwdth, even wth overlappng, does not provde suffcent resoluton to retreve ths vortex at most ranges. That s, the tolerance values n the mnmzaton algorthm were not met and thus convergence s not acheved, most lkely caused by a lack nformaton caused by the relatvely small number of data ponts avalable across the vortex. For a beamwdth of 18 wth no overlappng, values are retreved but exhbt large errors. When factors of 2 and 4 overlappng are appled, the retreval fals n most cases, lkely because of the small number of samples gven the azmuthal samplng ntervals. From ths we conclude that n the context of our experment desgn, the retreval technque works best for vortces havng core rad of at least 0.1 km (.e., equvalent to large tornadoes). Another FIG. 7. Same as n Fg. 5, except for R x km. mnmzaton algorthm, such as one that uses the conugate gradent method, may allow for convergence n more cases. d. WSR-88D and CASA radar comparsons A secondary goal of these experments s to determne how the WSR-88D, wth a half-power beamwdth of and conventonal 18 azmuthal samplng, compares to the CASA radar wth a 28 beamwdth and both conventonal 28 azmuthal samplng as well as overlappng. Fgure 11 shows the percent error for a vortex of radus R x km. Conventonal samplng by a CASA radar, as expected, exhbts the hghest percent error. Both the CASA radar n overlappng mode and the WSR-88D have a lower error. For such a small vortex, the WSR-88D curve and CASA curve wth overlappng are nearly equal at most ranges because the overlapped CASA azmuthal samplng nterval s the same as the WSR-88D samplng nterval (see dscusson n secton 3a). Fgure 12 shows results for a WSR-88D

10 MARCH 2009 P R O U D E T A L. 501 FIG. 8. Plot of percent error n V x vs range for R x 5 2 km. radar and a CASA radar, both samplng a vortex where R x km. For such large vortces, both radars exhbt smlar error at all ranges wthn the 30-km lmt of the CASA system. e. Comparson of ntal guess to retreved value Gven the possble senstvty of the retreval to the ntal guess documented n the prevous secton, we recomputed the results wth a dfferent ntal guess of V x (Fg. 13) for three dfferent vortex rad. For vortces wth R x km and R x km, the retreval has a lower percent error than the ntal guess, thus showng that the retreval represents an mprovement over raw radar observatons. However, for the largest vortex sze of R x km, the retreved value s nearly equal to the ntal guess at all ranges wthn 30 km. Because ths vortex s large compared to the sze of the radar beam, the ntal guess wth the correcton factor appled to t [see Eq. (A6)] s very close to the true value. 4. Dscusson We evaluated strateges for retrevng the core radus and maxmum tangental velocty of smulated vertcal vortces, as proxes for tornadoes and mesocyclones, to determne whch strateges mght be most effectve for a real CASA radar, whch s a dynamcally adaptve system. The measure of effectveness was defned as the best ft of pseudo-observatons to an analytc vortex model. The model used here to create the true and FIG. 9. Same as n Fg. 5, except for R x km. pseudo-observatons, known as the three-parameter vortex model (TPVM), does not contan a sngularty at the core radus (as does the Rankne combned vortex). The TPVM was appled to a Doppler radar emulator that sampled analytc tangental velocty profles usng data generated by the TPVM. A varatonal retreval model was employed to optmally estmate two control varables of the vortex: the maxmum tangental velocty V x and ts radus R x. Only results for V x were shown because the R x results show comparable behavor. Several parameters were vared to evaluate the effectveness of CASA samplng strateges on varous parameters ncludng vortex sze, range from the radar, azmuthal samplng nterval, and radar beamwdth. Comparsons of CASA and WSR-88D radars samplng at close ranges also were shown. For all ranges tested ( km from the radar) for a sngle CASA radar, vortces of radus 0.1 km and larger are detectable usng ts conventonal 28 beamwdth, and a 28 azmuthal samplng nterval. Overlappng of radar

11 502 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 26 TABLE 4. Percent error n V x for a vortex of radus R x km, beamwdth (bw) of 18, and an azmuthal samplng nterval (daz) of 18, 0.58, and for varous ranges; 2999 ndcates that no value of V x could be retreved for that range. Range (km) PE V x bw 5 18, daz 5 18 PE V x bw 5 18, daz PE V x bw 5 18, daz FIG. 10. Plot of percent error n V x vs range for R x km. Three samplng strateges are tested usng the same beamwdth (bw) of 28 wth varous factors of overlappng (see Fg. 3). Increased overlappng results n decreased effectve beamwdths (ebw). beams was shown to be an mportant adaptve samplng strategy for a CASA radar, especally n capturng the behavor of small vortces of radus km (medum to large tornadoes). Fnally, overlappng does not yeld any notceable mprovement for vortces of radus 1 km and larger at any ranges tested for a CASA radar. For the vortex szes tested, there appears to exst a lmt beyond whch addtonal overlappng (a factor greater than 2) yelds no consderable mprovement n the retreval of small or large vortces because the ncrease n resoluton dmnshes quckly wth decreasng azmuthal samplng nterval. The results usng a 18 beamwdth, especally wth overlappng, do yeld better retrevals for smaller vortex szes ncludng R x km. However, usng a 18 beamwdth for the CASA radars would be nconsstent wth the goal of developng small, nexpensve radars because a larger antenna would be requred. When comparng results of retrevals usng conventonal CASA parameters to those of the WSR-88D at close ranges, t s not surprsng that the latter, wth ts narrower beamwdth, produces smaller errors. However, when a CASA radar uses overlappng, the results are nearly equal to those for a WSR-88D. Ths agan confrms the beneft of overlappng to CASA. TABLE 3. Percent error n V x for a vortex of radus R x km, beamwdth (bw) of 28, and an azmuthal samplng nterval (daz) of 28 (center column) and 18 (rght column) for varous ranges; 2999 ndcates that no value of V x could be retreved for that range. Range (km) PE V x bw 5 28, daz 5 28 PE V x bw 5 28, daz FIG. 11. Comparson of percent error of a WSR-88D radar to a CASA radar for ranges less than 30 km for R x km.

12 MARCH 2009 P R O U D E T A L. 503 FIG. 12. Comparson of percent error of a WSR-88D radar to a CASA radar for ranges less than 30 km for R x km. Several lmtatons exst for the present study and could be examned n future work. One s the TPVM vortex model, whch s smple and may not capture the ntrcaces of true tornadoes and mesocyclones, ncludng asymmetry and nonvertcal orentaton. The use of more complcated analytcal models would overcome ths lmtaton. Another lmtaton of the present study s the assumpton that the CASA radars have a constant rotaton rate regardless of ther samplng strategy. Ths s not true durng operaton as the radars decrease or ncrease ther rotaton rate based on the phenomenon beng scanned and the samplng strategy employed. A thrd lmtaton s that the scannng radar always has one azmuthal samplng volume that concdes wth the center of the vortex, whch does not occur very often durng actual data collecton. Also, ths study uses a sngle radar to sample a sngle vortex. In realty, CASA s a network of radars that work together to sample multple atmospherc phenomena. Another lmtaton s the use of Newton s method n the retreval, whch can be extremely senstve to the ntal guess feld. Fnally, we dd not take nto account the effects of attenuaton, whch s an ssue of great sgnfcance to CASA gven ts operaton at X band. Many extensons of ths work could be undertaken to better understand how the CASA radars mght sample tornadoes and mesocyclones n an optmal manner. For example, the dealzed vortces used to create pseudoobservatons could be replaced wth hgh-resoluton numercal model smulatons of vortces, thus provdng more realstc multdmensonal wnd profles. To do so, a two-dmensonal horzontal cross secton of wnd (u FIG. 13. Plot of percent error of V x for the CASA radar usng three dfferent vortex szes: R x 5 0.1, 0.5, and 2.5 km. The percent error for the retreved value (Vx Retr) for each sze s denoted by the heavy lne, whereas the percent error for the ntal guess (Vx Int G) s denoted by the lght lne. and y) centered on the vortex at low elevatons would be needed and could be used n the present code qute easly. Also, t would be nterestng to use smulated data at varous tmes so that dfferent stages of the tornado s lfe cycle (.e., whle ts sze s changng) could be studed. Another obvous extenson s the use of real CASA data n the retreval program. Agan, ths would be more realstc than the dealzed vortex model used and could nvolve dealng wth multple phenomena smultaneously. The code would most lkely have to be altered n order to accommodate the latter, though of course no control soluton would be avalable for comparson. Fnally, ths study could be extended by conductng experments wth more than one CASA radar. Ths would be a relatvely straghtforward extenson by smply addng more terms to the cost functon. Understandng how multple radars work together collaboratvely and adaptvely n order to extract maxmum nformaton whle mnmzng the use of radar resources s a fundamental challenge for CASA radars and must be studed further. Acknowledgments. Ths research was supported n part by the Engneerng Research Centers Program of the Natonal Scence Foundaton under NSF Cooperatve Agreement EEC The authors thank Professor Luther Whte of the Unversty of Oklahoma for hs nput n the mathematcal development of the threeparameter vortex model.

13 504 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 26 FIG. A1. Geometry for computng the apparent dameter D A [Eq. (18)]; N and P are the extreme negatve and postve Doppler velocty values, respectvely, at the core radus r c and occur on the dotted black crcle of maxmum wnds; (R N, f N ) and (R P, f P ) are the range and azmuth locatons of the Doppler velocty peaks. The sold crcle passes through the radar locaton, ponts N and P, and the true crculaton center C T. The azmuthal dfference between N and P s gven by Df (after Wood and Brown 1992). APPENDIX Retreval Technque The retreval technque used n ths study s based on a varatonal method smlar to that used by Wood (1997) and estmates the vortex core radus (R x ) and maxmum tangental velocty (V x ) from the rangedegraded Doppler velocty sgnature of the axsymmetrc vortex. Brefly, the technque nvolves frst developng ntal guesses of V x and R x that wll be used to solve a set of nonlnear equatons. Ther soluton yelds a set of lnear equatons (also dscussed below) from whch the retreved values of V x and R x are obtaned. Several steps are used to determne the ntal guesses of V x and R x ; these guesses must be suffcently close to the true value n order to acheve convergence n the retreval, wth close dependng upon a number of factors as descrbed later. The ntal value for V x (V x 9) s calculated from the Doppler rotatonal velocty (V rot ), gven by V 9 x 5 V rot ðv P V N Þ. (A1) In (A1), V P s the extreme postve Doppler velocty value away from the radar and V N s the extreme negatve Doppler velocty value toward the radar. The ntal guess for R x (R x 9) s gven by R 9 x 5 D E 2, (A2) FIG. A2. Varaton of sgnature wth varable aspect ratos (a) for fxed velocty rato of zero (pure rotaton). The sold crcle centered on the true crculaton center C T represents the crcle of maxmum wnds. The heavy arrows ndcate the drectons that the apparent crculaton center C A moves toward the radar locaton as the radar becomes closer to the crculaton. The apparent dameter decreases as the aspect rato ncreases (Wood and Brown 1992). where D E s the estmated core dameter and s expressed as (Wood and Brown 1992) D E 5 FD A. (A3) In (A3), D A s the apparent dameter shown n Fg. A1 between the locaton of V P and V N (Wood and Brown 1992) and s gven by D A 5 ðr 2 N 1 R2 P 2R NR P cos DfÞ 1/2, (A4) where R N and R P are the ranges of the extreme negatve and postve Doppler velocty values, respectvely; and Df s the dfference between the azmuths of the extreme postve and negatve Doppler velocty values gven by Df 5 f P f N. (A5) In (A3), F s the correcton factor (Wood and Brown 1992) expressed as

14 MARCH 2009 P R O U D E T A L. 505 F 5 sec Df 2. (A6) Ths factor s needed because of what s called the aspect rato a (Wood and Brown 1992), gven by a 5 D T R T, (A7) where D T s the true dameter of the vortex and R T s the true range. When ths rato s relatvely large (close ranges, large dameters as shown n Fg. A2), the apparent center of the vortex s not concdent wth the true center and the ntal guess for R x s not close to the true soluton. When ths rato s small (far ranges, small dameters), the apparent center of the vortex s near the true center and a good ntal guess close to the true soluton s possble. Fgure A2 llustrates the effect of the aspect rato on the apparent core dameter. When the correcton factor s appled to the apparent dameter [as shown n (A3)], the apparent dameter becomes closer to the true dameter, thereby ncreasng the qualty of the frst guess. After the correcton factor s appled, we seek the values of V x and R x that yeld the mnmum value of acost functon JðmÞ. Mathematcally, the functon s wrtten as JðmÞ 5å½V d ðmþ V ~ d Š 2, (A8) where V d ðmþ s the model mean Doppler radal velocty, ~V d s the pseudo-observed mean Doppler velocty, and summaton s taken over the number of Doppler velocty data ponts. We take m 5 ½m 1,m 2 Š T 5 ½V x,r x Š T to be a vector of model parameters that defne the vortex strength (V x ) and sze (R x ), respectvely. In (A8), Vd ~ s gven by ~V d 5 ~V d 5 V x F 1 ðr x, rþ cos ggðfþwðr d Þ 2 GðfÞWðR d Þ 2, (A9) V x F 2 ðr x, rþ cos ggðfþwðr d Þ 2 GðfÞWðR d Þ 2, (A10) ~V d 5 1 V x ½ 2 ðf 1ðR x, rþ 1F 2 ðr x, rþšcos ggðfþwðr d Þ 2 GðfÞWðR d Þ 2, (A11) whch represent V 1, V 2 and V 12, respectvely. In (A9) (A11), s the azmuthal data subpont, s the range data subpont for a pseudo-observaton, F 1 5 V 1 /V x, and F 2 5 V 2 /V x [see Eq. (3), p. 7]. The parameters n (A9) (A11) have been defned n subsecton 2b. The azmuth and range must be computed before gong on to the next subpont value wthn a beamwdth area. To determne the optmal estmate, the cost functon J s mnmzed va a frst dervatve test, yeldng the followng necessary condtons: J m å ½ V d ðmþ V ~ d Š V d ðmþ m. (A12) We generate a sequence of models m 0 and m 1, wth the hope that Jðm Þ!mn m JðmÞ as the number of teratons approaches nfnty. In (A12), we expand n scalar form [readng JðmÞ as Jðm 1, m 2 Þ]as where Jðm 1 Þ V x å ½ V d ðm 1 Þ ~ V d Š V d ðm 1 Þ V x, and (A13) Jðm 2 Þ R x å ½ V d ðm 2 Þ ~ V d Š V d ðm 2 Þ R x, (A14) V d ðm 1 Þ 5 V x 1 ½ 2 F 1ðR x, rþ 1F 2 ðr x, rþšcos ggðfþwðr d Þ 2 GðfÞWðR d Þ 2, (A15)

15 506 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 26 V d ðm 2 Þ 5 R x 1 F 1 ðr x, rþ V x 1 F 2ðR x, rþ cos ggðfþwðr d Þ 2 2 R x R x GðfÞWðR d Þ 2, (A16) where I and J are the upper lmts of the number of azmuthal and range data subponts wthn the beamwdth area, respectvely. Therefore, usng n 5 1 and n 5 2 n (3) and dfferentatng (3) wth respect to R x yelds F 1 ðr x, rþ R x 5 2r3 2R 2 x r ðr 2 x 1 r2 Þ 2 and (A17) F 2 ðr x, rþ R x 5 ð12r2 x r5 12R 6 x rþ ð3r 4 x 1 r4 Þ 2. (A18) To solve (A12), Newton s method s appled. We begn by wrtng these equatons n another form (Gerald and Wheatley 1984, ) Fðm Þ 5 JðmÞ 5 0, (A19) m where m 5 ½V x, R x Š T are the local mnmzers that satsfy (A19). We expand the equatons as a Taylor seres about the pont m 5 ½V x, R x Š T n terms of (m m), where m s a pont near m. The Taylor seres expanson of (A19) s gven by F 1 ðv x, R x Þ F 1ðV x, R x Þ 1 F 1ðV x, R x Þ ðv x V V xþ x 1 F 1ðV x, R x Þ ðr x R R xþ 1 hgher-order terms, x and (A20) F 2 ðv x, R x Þ F 2ðV x, R x Þ 1 F 2ðV x, R x Þ ðv x V V xþ x 1 F 2ðV x, R x Þ ðr x R R xþ 1 hgher-order terms. x (A21) In (A20) and (A21), each functon s evaluated at the approxmate root ðv x, R x Þ. By usng the Taylor seres expanson, we have reduced the problem from a set of two nonlnear equatons to a set of two lnear equatons. The unknown values are the mprovements n each estmated varable ðv x V xþ and ðr x R xþ. To mplement (A20) and (A21), the partal dervatves may be approxmated by recalculatng the functons wth a small perturbaton d to each of the varables n turn F 1 ðv x, R x Þ ff F 1ðV x 1 d, R x Þ F 1 ðv x, R x Þ, (A22) V x d F 1 ðv x, R x Þ ff F 1ðV x, R x 1 dþ F 1 ðv x, R x Þ, (A23) R x d F 2 ðv x, R x Þ ff F 2ðV x 1 d, R x Þ F 2 ðv x, R x Þ, (A24) V x d F 2 ðv x, R x Þ ff F 2ðV x, R x 1 dþ F 2 ðv x, R x Þ. (A25) R x d If m s suffcently smlar to m, we can truncate after the frst dervatve terms n (A20) and (A21) and solve for the unknowns (m m). Also note that we take the dervatves of F 1 and F 2, whch s the second dervatve of J; therefore, we also are performng a second dervatve test of J to determne whether the extrema are a mnmum, whch s the desred condton. Thus, by Cramer s rule F F 1 1 R x F F 2 V 11 x 5 V x 1 2 R x, (A26) det F 1 F 1 V x F 2 F l R 11 x 5 R x 1 2 V x, (A27) det where the superscrpt s the teraton number and the determnant s gven by F 1 F 1 det 5 V x R x F 2 F 6¼ 0. (A28) 2 V x R x To acheve convergence, the teratons are termnated when the functon values Fðm Þ n (A19) are suffcently small or the changes n the m values are suffcently small. The maxmum number of teratons s set to 50 n ths study.