Angular and range interferometry to measure wind

Transcription

1 RADIO SCIENCE, VOL. 38, NO. 6, 116, doi:1.129/23rs2927, 23 Angula and ange intefeomety to measue wind Guifu Zhang, 1 Richad J. Doviak, 2 J. Vivekanandan, 1 and Tian-You Yu 3 Received 27 June 23; evised 8 Octobe 23; accepted 2 Octobe 23; published 18 Decembe 23. [1] Radial wind is outinely measued by Dopple method, wheeas winds tansvese to the ada beam ae measued using an intefeometic techniue in which thee o moe spaced antennas ae used (i.e., the Spaced Antenna (SA) techniue). In this pape, an intefeometic techniue is examined wheeby a single antenna is used to measue both adial and tansvese winds. Angula Intefeomety (AI) detemines tansvese wind, and Range Intefeomety (RI) detemines adial wind. The coss-coelation of signals, eceived fom diffeent angles and fom diffeent anges by a single antenna, is deived based on wave scatteing fom andom fluctuations of efactive index. The adial and tansvese wind components ae estimated fom the coss-coelation of signals eceived fom diffeent anges and diffeent diections. The theoetical standad deviation of the estimated wind is deived, and its dependence on spatial esolution, obsevation time, and tubulence is pesented. The theoy shows that AI euies small beam size to measue tansvese wind accuately, contay to the SA techniue, wheeas RI euies fine ange esolution to pefom well. INDEX TERMS: 6969 Radio Science: Remote Sensing; 699 Electomagnetics: Geneal o miscellaneous; 6952 Radio Science: Rada atmospheic physics; 6974 Radio Science: Signal pocessing; KEYWORDS: single antenna intefeomety, wind measuement, coss-coelation atio, weathe ada, MST ada, wind pofile Citation: Zhang, G., R. J. Doviak, J. Vivekanandan, and T.-Y. Yu, Angula and ange intefeomety to measue wind, Radio Sci., 38(6), 116, doi:1.129/23rs2927, Intoduction [2] Components of the wind can be measued by the Dopple method, intefeometic techniues, o by Tacking Reflectivity Echoes by Coelation (TREC) [Doviak and Znic, 1993; Biggs et al., 195; Rinehat, 1979]. The Dopple method measues adial velocity, and intefeometic techniues measue the wind component tansvese to the beam axis. Both the Dopple and the intefeometic techniues diectly measue wind components (i.e., the velocity of andomly distibuted scattees advected by wind). The vecto wind can be detemined using the Dopple method (e.g., though a VAD type analysis) if the wind is unifom ove a lage aea, but the intefeometic techniue can measue the vecto wind within the esolution volume of the ada, thus poviding fine esolution. Both of these techniues can detemine the wind even if the eflectivity field is 1 Reseach Application Pogam, National Cente fo Atmospheic Reseach, Boulde, Coloado, USA. 2 National Sevee Stoms Laboatoy, Noman, Oklahoma, USA. 3 School of Electical and Compute Engineeing, Univesity of Oklahoma, Noman, Oklahoma, USA. Copyight 23 by the Ameican Geophysical Union /3/23RS statistically homogenous. On the othe hand, the TREC method tacks featues in the eflectivity field to detemine featue motions [Cane, 1979; Rinehat, 1979; Tuttle and Foote, 199]. Thus, if eflectivity is not conseved, eflectivity motion is not necessaily the same as the wind velocity; futhemoe the TREC method does not wok if the eflectivity field is homogenous. [3] The Dopple method uses the phase of the autocoelation function of eceived signals to estimate adial wind. Because the phase can only be measued within an inteval of 2p, the Dopple method is pone to aliasing which causes the estimated wind velocity to alias with a peiod of l/2t s, whee T s is the pulse epetition time. Cuent dealiasing techniues ae based on spatial/time continuity of the wind field. We develop an altenative method, Range Intefeomety (RI), that is immune to aliasing, and which could supplement Dopple wind measuements. Intefeometic techniues use the magnitude of the coss-coelation function, and thus the poblem of phase aliasing does not exist. [4] Intefeomety had been successfully developed and applied to measue tansvese wind [Biggs et al., 195; Biggs, 1984; Vincent and Rottge, 198; May et al., 1989; Doviak et al., 1996]. Intefeomety fo wind measuement is based on the spaced antenna (SA) techniue, in which scatteed signals eceived at sepaate

2 14-2 ZHANG ET AL.: SAMPLE VOLUME TRACKING TO MEASURE WIND antennas ae coss-coelated to estimate the tansvese wind. A single antenna intefeomete was poposed to measue ocean suface featues using ai-bone synthetic apetue adas (SAR) [Fitch, 1991]. To ou knowledge, howeve, single antenna intefeomety has not been used in the field of atmospheic emote sensing with gound-based adas. Rada intefeomety with a single antenna/eceive fo wind measuement has not been investigated o demonstated. [5] In this pape, we intoduce single-antenna intefeomety to measue the wind vecto, and study its feasibility fo pactical applications. We popose Angula Intefeomety (AI) to detemine the tansvese wind components fom the angula coss-coelation function, and Range Intefeomety (RI) to measue the adial wind component fom the ange coss-coelation function. It is noted that neithe AI no RI ae TREC methods because both AI and RI apply even if the eflectivity field is unifom. [6] This pape is oganized as follows: In section 2 a conceptual desciption of single antenna intefeomety fo wind measuement is pesented; In section 3 the poblem is fomulated, and the angula and ange cosscoelation functions ae deived fo signals eceived by a single-antenna/eceive; In section 4 the coss-coelation atio (CCR) method [Zhang et al., 23] is used to detemine the tansvese and adial wind components, and to analyze the standad eos of the wind estimates. Finally, the feasibility of the method fo pactical applications is discussed. 2. A Conceptual Desciption of Single- Antenna Intefeomety fo Wind Measuements [7] The accepted explanation fo ada intefeometic measuement of cossbeam wind is that the echo diffaction patten advects acoss an aay of eceiving antennas at twice the speed that scattees ae advected acoss the beam [Biggs, 1984]. Doviak et al. [1996], howeve, conside pais of scattees and eceives to pove that diffaction pattens do not necessaily advect at twice the speed of wind. Consideing a pai of eceives, symmetically placed about a tansmitting antenna to fom a pai of side-by-side bistatic scatteing volumes, and wind along the baseline of the eceives, they showed by applying the ecipocity theoem that echoes in the two eceives ae pefectly coelated when signals fom one eceive is lagged a time diffeence eual to the time it takes the scattees to advect fom one scatteing volume to the next. [8] This late intepetation (i.e., spaced esolution volumes, athe than spaced antennas), leads us to an altenative explanation of intefeomety applied to the measuement of wind advecting eithe discete scattees (e.g., ain dops) o Bagg scattees [Doviak and Znic, 1993]. Ignoing the effects of tubulence, wind simply advects scattees without changing thei elative displacements. If the ada s esolution volume V 6 [Doviak and Znic, 1993, section 4.4.4] can be stictly tanslated the same vecto distance that the scattees have moved, the phase path (i.e., fom the tansmitte to each of the scattees and back to the eceives) diffeences would be the same fo all the scattees in each of the V 6 s. Fo example, if scattees ae hoizontally advected, V 6 must also be stictly tanslated hoizontally (i.e., without otation as would occu if the cente of V 6 is tanslated hoizontally and the ada is at a fixed location). If V 6 is to be stictly tanslated hoizontally, both the tansmitte and eceive must be tanslated hoizontally the same vecto distance as the scattees o, if the tansmitte is fixed, the eceive must be symmetically and hoizontally tanslated acoss the tansmitting antenna twice the distance that the scattees have moved (i.e., a bistatic ada configuation is euied). This explanation also applies if the scattees and sample volumes advect adially. If the esolution volume stictly follows the motion of the scattees such that the phases to all scattees ae fixed (i.e., ignoing tubulence), the magnitude of coss-coelation function is maximal. Othewise, the coss-coelation of the eceived signals deceases. [9] Based on the above intepetation of SA intefeomety, we popose a single-antenna/eceive intefeometic techniue to measue wind. As shown in Figue 1a, AI locates the angula displacement of the scattees to measue tansvese wind. An angula coss-coelation function of the signals fom diffeent diections is constucted to estimate tansvese wind. Similaly, RI locates the adial displacement of the scattees along ange (Figue 1b), and estimates the adial wind fom the coss-coelation of signals fom diffeent anges. [1] The ada that uses AI can have a phased aay antenna o single dish eflecto povided the signals fom two adacent diections ae acuied apidly to maintain signal coelation, and epeatedly to have many independent samples. The phased aay weathe ada being assembled by National Sevee Stoms Laboatoy (NSSL) has beam agility and is a good candidate to test AI fo wind measuement. The angula coelation functions at positive and negative angles about the tansmitting beam can also be obtained with a single eflecto of a monopulse ada. If the beam shifts in the diection that follows the displacement of the scattees, the coelation of the signals is high. Beam shifts in the opposite diection leads to a lowe coelation. Hence, the atio of the coelation functions of signals fom V 6 s displaced in the diection of the wind and against it should give tansvese wind infomation. [11] RI can be ealized with a pulsed Dopple ada with a wide band eceive that samples signals fom two

3 ZHANG ET AL.: SAMPLE VOLUME TRACKING TO MEASURE WIND 14-3 Figue 1. Configuation of single antenna intefeomety. (a) Angula Intefeomety (AI) of signals fom diffeent diections, and (b) Range Intefeomety (RI) of signals fom diffeent anges 1 and 2, espectively. Range sepaation is d. Range esolution is s. adacent anges with esolution volumes highly ovelapped, and hence, the ange coss-coelation function is estimated. The adial wind is estimated fom the atio of the coss-coelation functions at positive and negative lags. 3. Fomulation fo Single Antenna Rada Intefeomety [12] Conside a ada located at the oigin (,, ) of a spheical coodinate system, and a esolution volume V 6 located at ange ~ and at an angula displacement ~s fom a efeence axis (Figue 2; two V 6 s ae shown fo a late efeence). The eceived complex signal V(~, t), backscatteed fom a esolution volume V 6 centeed at ~, can be obtained by integating the spatial distibution of efactive index fluctuations Dn(~, t) weighted with angula and ange weighting functions [Doviak and Znic, 1993, section 11.5]. That is, V ð~; tþ ¼A Z Dn ð ~ ; tþ 2 exp ~s ~s 2 2kð Þ! d~ ; 4s 2 ð Þ 2 4s 2 ð1þ between two spaced esolution volumes, is the ange to the cente of V 6, and a pime defines the location of Dn(~, t). The fist tem in the exponent is the angula weighting function, the second tem is the ange weighting function, and 2k( ) is the phase path diffeence between the location of Dn(~, t) and the cente of V 6. The ange esolution, s (Figue 2), is defined as the suae oot of the second moment of the ange weighting function [Doviak and Znic, 1993, section 11.5]. The suae oot of the second moment of the two-way angula weighting function s is elated to that p ffiffi of one-way angula weighting function s 1 as s = s 1 / 2, and the standad deviation of the one-way adiation patten, s 1 = gl/d, is elated to the commonly used one-way 3dB beam width as 1 = 2.36s 1. Paametes, l, D, and g ae wavelength, antenna size and antenna efficiency facto, espectively. [13] The coss-coelation function is the ensemble aveage of the poduct, V(~ 1, t 1 )V*(~ 2, t 2 ), in which V(~ 1, t) and V(~ 2, t) ae the eceived signals fom the esolution volumes centeed at ~ 1 and ~ 2. The cosscoelation function of the eceived signals is Z Z C ð~ 1 ; t 1 ;~ 2 ; t 2 Þ ¼ A 2 Dn ~ 1 ; t 1 Dn ~ 2 ; t exp ~s 1 ~s 2 1 þ ~s 2 ~s 2 4s 2 1 2þ s 2 2k þ! 2 d~ 1 d~ 2 : ð2þ To simplify the integations in (2): 2 whee A is a constant dependent on ada paametes,~s is the angula distance vecto fom the efeence axis located Figue 2. wind. Configuation of esolution volumes fo

4 14-4 ZHANG ET AL.: SAMPLE VOLUME TRACKING TO MEASURE WIND [14] 1. Assume that the field of Dn(~, t) is statistically homogeneous and stationay so that its coelation function is only a function of the distance ~ d = ~ 2 ~ 1 between the two locations of Dn(~, t), and the time lag diffeence t = t 2 t 1. Thus this coelation function can be witten as, B n ~ d ; t ¼ Dn ~ 1 ; t 1 Dn ~ 2 ; t 2 : ð3þ Because we only have samples of the signal spaced T s = PRT (i.e., the Pulse Repetition Time), t changes incementally in T s steps along sample-time t = mt s [Doviak and Znic, 1993, section 4.1]. [15] 2. Assume a Taylo seies expansion fo the phase tem up to the second ode [Doviak and Znic, 1993, section 11.5], i.e., ¼ þ 2 2 : ð4þ 2 [16] 3. Assume a naow ada beam and a small angula diffeence (i.e., s 1 and ~s 1 ~s 2 1), so that the angula displacement vecto, ~s, can be epesented as, ~s ~s ~ ~ ; ð5þ whee ~ is the ectilinea displacement tansvese to the efeence axis. [17] 4. Assume a fine ange esolution, so that the appoximations, 1 2, and = can be used in euation (2), whee is the longitudinal displacement paallel to the efeence axis. [18] Using the assumptions 1 4 in (2) and noting that and ae independent vaiables, we sepaate the integals and have ZZZZ C ð~ 1 ; t 1 ;~ 2 ; t 2 Þ A2 4 B n ~ d ; t exp ~ 1 ~ 2 1 þ ~ 2 ~ s 2 k þ! 2 2 exp 1 2þ s 2! 2k d~ 1 d 1 d~ 2 d 2 : ð6þ Euation (6) constitutes the geneal fomulation fo the coss-coelation function of the signals eceived by a single antenna eceive fom two spaced esolution volumes. [19] Making the following coodinate tansfomation to cente and diffeence coodinates, c ¼ 1 ð 2 1 þ 2 Þ; and d ¼ 1 2 ; ð7þ fo the vaiables ~, ~,, and, and pefoming the integations ove the cente vaiables c and d in (6), we obtain, C ð~ d ; tþ A2 ffiffiffiffiffi ZZ 3p 2 2p s 2 s B n ~ d ; t exp exp ~ d ~ 2 d 8 2 s 2 d 2 d 8s 2 2k 2 s 2 2 d! 2k d! d d~ d d d ; ð8þ whee the fist exponential contains both the angula esolution volume weighting tem and the Fesnel tem. It can be shown that (8) educes to euation (11.122) in Doviak and Znic [1993] when ~ d = d = t =.Inthis case the coss-coelation function becomes the autocoelation function at zeo lag, and is simply the backscatteed powe. [2] We use the following Gaussian coelation function fo the efactive index fluctuation field, Dn [Doviak et al., 1996], B n ~ d ; t 2 ¼ c? c b 2 T b exp whee b 2 2 T = c? ~ d ~v Tt 2 d v 2! t ; 2b 2 T 2b 2 ð9þ + s 2 t t 2, and b 2 = 2 c + s 2 t t 2, s t is the standad deviation of the adial component of wind (i.e., tubulence), and c?, c ae the coelation lengths of Dn in diections tansvese and longitudinal to the efeence axis (Figue 2). ~v T = v x^x + v y^y and v ae wind components tansvese and along the efeence axis, espectively. Substituting (9) into (8), and integating ove ~ d and d, yields the following coss-coelation function, C ð~ d ; tþ ¼ 8p 3 A 2 s 2 n s2 s 2 c? c þ b 2 T = 42 s 2 þ 4k2 s 2 b2 T exp ~ d ~v T t 2 = 8 2 s 2 þ k 2 b 2 T 2 d =! ð 22 Þþ2k 2 s 2 v2 T t2 1 þ b 2 T = 42 s 2 þ 4k2 s 2 b2 T 1 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 þ b 2 = 4s2 ð exp d v tþ 2 = 8s 2 þ 2k 2 b 2 2k! ð d v tþ 1 þ b 2 = ;ð1þ 4s2

5 ZHANG ET AL.: SAMPLE VOLUME TRACKING TO MEASURE WIND 14-5 tem, the Fesnel tem, accounts fo the decoelation due to the scattees motion in a tansvese ectilinea diection athe than along acs at constant ange, known as ange defocusing. The last tem shows the de-coelation due to tubulence. Tubulence and Fesnel tems usually cause faste de-coelation of eceived signals than the displacement of esolution volumes. [22] By letting d = in (11) we obtain the magnitude of the angula coss-coelation function of signals fom sample volumes at the same ange. This is given by C A ð d ; tþ S exp ~s d ~v T t s 2 v2 t2 8s 2 2k 2 s 2 v2 T t2 2k 2 s 2 t t2!: ð12þ Figue 3. Angula coelation coefficient as a function of time lag and angula sepaation. Paametes ae s t =.1 m/s, s 1 =.42 degee, v x = 5 m/s, v y = m/s, l =.1 m, and ange = 1 km. See colo vesion of this figue in the HTML. whee s 2 n is the mean suae of Dn(~, t). Assume that the tansvese coelation length of Dn is much smalle than the antenna diamete (D) (i.e., c? D), and that b satisfies the condition, b s. Because only the magnitude of the coss-coelation function is used to estimate the wind using intefeometic techniues, we can simplify (1) by neglecting the small contibution tems, taking the magnitude, and witing the esult in spheical coodinates as, C ð~ d ; tþ S exp ~s d ~v T t 2 ð 8 2 s 2 d v tþ 2 8s 2 2k 2 s 2 v2 T t2 2k 2 s 2 t t2!; ð11þ whee the angula sepaation of the two esolution volumes is~s d =~s 1 ~s 2 d^s d (^s d is the unit vecto in the diection of angula displacement d ), the ange sepaation is d d, and adial wind is v v l. [21] Euation (11) shows that the coss-coelation of signals sepaated in time fom sepaated esolution volumes depends on mean wind and tubulence as well as ada chaacteistics. The fist exponential tem shows the effect of the angula sepaation of the esolution volumes and it depends on the tansvese wind component. The second tem shows the effect of ange sepaation and it depends on the adial component of wind. The de-coelation caused by the esolution volume sepaation is compensated by the scattees motion matching the displacement of the esolution volumes. The thid It can be seen that if the scattees displacement in sample-time t matches the sepaation of the esolution volume cente, the coelation eaches a maximum. Figue 3 shows the nomalized coss-coelation coefficient as a function of the angula sepaation d and sample-time lag t. The othe paametes used fo the calculation ae v x = 5 m/s, v y = m/s, v = m/s, s 1 =.42 degee, = 1 km, s t =.1 m/s, and l =.1 m. The coelation ellipse width pependicula to the mao axis (i.e., the line along d = v x t/) is oughly 2s, and its width along the mao axis depends pincipally on the tubulence and Fesnel tems in euation (12); the smalle ae the coefficients of t 2, the longe is the mao axis. Fo lamina flow (i.e., s t = ), the length is limited to 1/(2ks v T ) by the Fesnel tem. Thus, the smalle is v T,the longe is the mao axis, and thus a longe lag time can be used to estimate the tilt o slope v T / of the mao axis. A longe mao axis also makes easie the estimation of the tansvese wind fom the tilt v T /. Likewise, the smalle is s the naowe and longe is the coelation ellipse; thus naow beams ae favoed fo accuate tansvese wind measuement using the AI appoach. [23] Similaly, letting ~s d = in (11), we obtain the coss-coelation magnitude as a function of ange sepaation and the adial wind component, ð C R ð d ; tþ S exp d v tþ 2 8s 2 v2 T t2 8 2 s 2 2k 2 s 2 v2 T t2 2k 2 s 2 t t2!: ð13þ The nomalized coss-coelation coefficient is calculated and shown in Figue 4 as a function of ange spacing and time lag. The paametes used fo the calculation ae v T = m/s, v = 5 m/s, s 1 =.42 degee, s =1m,s t =.1 m/s and l =.1 m. Again we have an ellipse with an

6 14-6 ZHANG ET AL.: SAMPLE VOLUME TRACKING TO MEASURE WIND Figue 4. Range coss-coelation function as a function of time lag and ange sepaation. Paametes ae s t =.1 m/s, s =1m,v =5m/s,v T = m/s, l =.1m. See colo vesion of this figue in the HTML. axis tilt that depends on the ange spacing and the adial wind component. 4. Wind Estimate Eos 4.1. Wind Estimation Fom Coelation Ratios [24] To measue the wind vecto, we calculate the wind components fom estimates of the angula and ange coss-coelation functions. We use the coss-coelation atio (CCR) method developed to estimate tansvese wind fom SA measuements [Zhang et al., 23]. The logaithm of the atio of angula coss-coelation functions(12) at positive and negative lags leads to L A ð d ; tþ ¼ ln C Að d ; tþ C A ð d ; tþ ¼ 1 2 d ~v T ^s d s 2 t; ð14þ whee ~v T ^s d = v x is the poection of the tansvese wind on the diection of the angulaly displaced esolution volumes. Hencefoth, we shall dop the inne poduct notation, and assume it to be undestood that the angula displacement d of the esolution volumes is in the diection of the component v x of the tansvese wind. The angula coss-coelation function can be estimated fom two time seies that ae measued at the same ange but two adacent diections with thei esolution volumes highly ovelapped. Fom the above euation, the tansvese wind component v x is given by v x ¼ 2s2 L Að d ; tþ : ð15þ d t With the antenna pointing to a fixed diection, d =, time seies data ae collected at two adacent anges with thei esolution volumes highly ovelapped, and hence, the ange coss-coelation function is estimated. Similaly, the logaithm of the atio of ange cosscoelation functions (i.e., euation (13)) at positive and negative lags gives, L R ð d ; tþ ¼ ln C Rð d ; tþ C R ð d ; tþ ¼ dv t ; ð16þ 2s 2 fom which the adial velocity can be expessed as: v ¼ 2s2 L Rð d ; tþ : ð17þ d t In both (15) and (17), the wind components ae etieved though the magnitudes of coss-coelation functions of signals fom a single antenna without using the phase infomation as in the Dopple techniue Standad Deviation of Wind Estimates [25] In theoy, the wind components can always be detemined using (15) and (17). In pactice, howeve, the angula and ange coss-coelation functions ae not pecisely known; only estimates can be obtained fom measuements. [26] Because the wind components ae linealy popotional to the logaithm of the coss-coelation atios, the standad deviation of the wind estimates ^v T and ^v ae simply elated to the SD of ^L A ( d, t) and ^L R ( d, t) estimates, espectively. That is, and SDð^v x Þ ¼ 2s2 d t SD ^L A ð d ; tþ ð18þ SDð^v Þ ¼ 2s2 d t SD ^L R ð d ; tþ : ð19þ The coss-coelation function magnitude can be witten in the fom, CðtÞ ¼ S exp t t 2! p h ; ð2þ 2t 2 c whee the Gaussian paametes t c, t p and h ae the coheence time, the time lag to the peak of the cosscoelation function, and the de-coelation paamete. The SD of the estimated coss-coelation atio ^L(t) has been deived in Zhang et al. [23] and is SD ^L ¼ t c t 1 pffiffiffiffiffiffi 1 þ 2t2 p t 2 c 2! 1=2 ; ð21þ

7 ZHANG ET AL.: SAMPLE VOLUME TRACKING TO MEASURE WIND 14-7 pffiffiffi whee = MT s / p tc is the numbe of independent samples, M is the total numbe of coelated samples, and is the coss-coelation coefficient at zeo lag. Because the angula and ange coss-coelation functions can be witten in the fom (2), the standad deviations of the estimates fo the tansvese and adial wind components can be deived as shown in the next section Standad Deviation of Tansvese Wind Estimates [27] Rewiting the angula coelation function (11) in the fom of (2) fo d = leads to the Gaussian paametes t c, t p and h having the following expessions: t ð c AÞ v 2 1=2 T ¼ 4 2 s 2 þ v2 4s 2 þ 4k 2 s 2 v2 T þ 4k2 s 2 t ð22þ and t ð p AÞ ¼ dv T t 2 c 4s 2 h ðaþ ¼ 2 d 8s 2 2 d v2 T tð c AÞ s 4 ð23þ : ð24þ Unde fa field conditions (i.e., 2D2 l ) assumed in the deivations, the Fesnel tem (i.e., the thid tem in (22) is much lage than the fist tem. Thus the expession fo the coelation time can be educed to, t ð c AÞ ¼ v2 4s 2 þ 4k 2 s 2 v2 T þ 1=2 s2 t : ð25þ Because s is much smalle than 1, it is easily seen that unless tubulence is extemely small compaed to the tansvese wind, the coelation time is pincipally contolled by tubulence. If s t s v T, then the coelation time is contolled by wind; othewise it becomes (2ks t ) 1. [28] The tansvese coss-coelation coefficient at zeo lag, (A), can be obtained fom (12) by setting t =. Assuming esolution volume sepaation is small ( d s ), the appoximate expession ðaþ ¼ exp 2 d 8s d 8s 2 ð26þ can be used in (21). Substituting (26) into (21), along with the expessions fo t c (A) and t p (A), and then using these in (18), SDð^v x Þ ¼ t ð c AÞ A s ð Þ pffiffiffiffiffiffi 1 þ v2 T tðaþ2 1=2 c 2 2 s 2 : ð27þ If t c (A) is contolled by the tansvese wind, then t c (A) (2ks v T ) 1. Fo this elation to be valid, the adial wind component and tubulence must satisfy the conditions, v 4pg s D vt, and s t gp l ffiffi 2D v T. In this case, it can be shown that, because the solutions ae valid fo the fa field (i.e., > 2D 2 /l = f ), the second tem in the adical of (27) is small compaed to the fist, and the standad eo of the tansvese wind estimates SD(^v x ) educes to SDð^v x Þ 4pg2 pffiffiffiffiffiffi ðaþ f v T ; > f : ð28þ Thus, unde the specified conditions, the tansvese wind is estimated moe accuately at close anges. But satisfying the condition on tubulence is unlikely unless the flow is nealy lamina (i.e., tubulence is much weake than the mean flow). [29] If the coelation time is mainly detemined by tubulence, then t (A) c =(2ks t ) 1 and the standad eo in estimating the tansvese wind component becomes p 2 pg SDð^v x Þ ¼ 2 p ffiffiffi ðaþ ffiffiffiffiffiffi D s t : ð29þ Because /D 1, satisfactoy measuements of tansvese wind euies obsevations nea the ada, and a lage. The standad deviation of the tansvese wind estimates as a function of tubulence at vaious anges calculated fom (27) is plotted in Figue 5. This figue shows that the standad deviation inceases as the tubulence becomes stonge, and as ange inceases. [3] Fo compaison, the standad deviation of the wind estimates obtained with Spaced Antenna (SA) techniues is also shown in Figue 5. The ada chaacteistics ae the same as that used in the calculation fo the AI appoach (i.e., the tansmitte antenna size D T 6. m) except the two eceiving antennas have a smalle size (i.e., D R = 3. m) and ae sepaated by 3 m (i.e., the same apetue is used fo tansmitting and eceiving). The accuacy of the SA techniue is independent of ange and, as shown, povides moe accuate wind measuement than the AI appoach. This is due to the diffeence in the tansvese coelation lengths as explained in the following paagaph. [31] The SA techniue fo wind measuement can be undestood as tacking intefeence pattens on the antenna plane, wheeas the AI techniue tacks the movement of the scattees acoss the esolution volume at ange. The coelation length of the intefeence pattens on the antenna plane is on the ode of the antenna diamete [Doviak et al., 1996]. Howeve, the tansvese coelation length fo AI is at the ode of the angula width of the beam at ange. Both SA and AI use the coelation change due to spatial sepaation as elated to the intefeence patten s motion (o the scattees motion) fo wind measuements. A small coelation

8 14-8 ZHANG ET AL.: SAMPLE VOLUME TRACKING TO MEASURE WIND (R) The ange coelation coefficient can be obtained fom (13) by setting t =. Thus ðrþ ¼ exp 2 d 8s d 8s 2 : ð32þ Substituting (3), (31), and (32) into (21), and the esult into (19) yields: Figue 5. Standad deviation of tansvese wind estimates as a function of tubulence at anges.72, 2, 1, and 5 km. Othe paametes ae s 1 =.42 degee, v x = 5 m/s, v y = m/s, v = m/s, T d = 6 s. See colo vesion of this figue in the HTML. length allows a shot time fo scattees to advect to maintain signal coheence, and thus an easie measuement of wind. The beam size fo a collimated beam in the fa field is much lage than the antenna size. Theefoe, the wind measuement with the AI techniue is less accuate than with the SA techniue in the fa field. In nea field, howeve, SA and AI techniues should povide compaable accuacy of wind measuements when the eceiving SA antennas ae highly ovelapped. Less (o no) ovelapping in SA antennas degades its pefomance when the CCR method is used fo wind estimation. AI can outpefom the SA techniue fo a focused beam when the beam size (i.e., at the focal point) is smalle than antenna size Standad Deviation of Radial Wind Estimates [32] Thee ae two appoaches in using euation (13) to obtain the adial wind fom RI: (1) the ange-time seies fom a pai, o multiple pais, of tansmitted pulses could be lagged with espect to one anothe, o (2) a sample-time seies at a pai of gates, o multiple gates, sepaated by d could be lagged with espect to one anothe. In the second appoach, if the ange coelation function (13) is witten in the Gaussian coelation fom of (2), it can be shown that t c (R) = t c (A) given by (22), but the time-lag to coelation peak t p (R) and peak magnitude h (R) ae given by expessions, t ð p RÞ h ðrþ ¼ 2 d 8s 2 ¼ dv t ð c AÞ2 4s 2 2 d v2 tð c AÞ2 32s 4 ð3þ : ð31þ SDð^v Þ ¼ s ð Þ t c R p ffiffiffiffiffiffi 1=2 1 þ v2 t2 c 2s 2 : ð33þ If the flow is lamina (i.e., s t = ), it can be shown that SDð^v Þ ¼ 1 ffiffiffiffiffiffi 3 v 2 ðrþ ; if v 4pg s v T ð34þ D and SDð^v Þ ¼ 4pg s ð Þ pffiffiffiffiffiffi D R v T ; if v 4pg s D If tubulence contols the coelation time, then SDð^v Þ ¼ 4p pffiffiffiffiffiffi ðrþ s l s t : v T : ð35þ ð36þ If a pai of ange-time seies ae lagged with espect to one anothe (i.e., appoach 1), the above euations pffiffiffi also hold, but must be eplaced with R = Mdt / p tc, whee dt 2s /c is the gate spacing, and t c =4s /c is the coelation time along ange-time. The standad deviation calculated fom (33) is shown in Figue 6 as a function of tubulence fo vaious ange esolutions of 2, 4, 8 and 16 m. The othe paametes ae v T = m/s, v = 5 m/s, d = 1 m, and l =.1 m. As expected, eo in wind estimate inceases as s and tubulence s t incease. [33] Fo typical Dopple ada measuements of wind, RI gives adial wind estimates with accuacies much wose than that obtained fom Dopple measuements. But RI is not subect to aliasing eos that often plague Dopple measuements. Thus RI could be an altenative appoach to esolving velocity ambiguities. In this case, accuacy euiements ae elaxed and we only need to estimate v sufficiently well to esolve the ambiguity. A valid adial wind measuement would euie the estimation eo to be smalle than the unambiguous velocity inteval 2v a = l/2t s (i.e., SD(^v )<l/2t s ). If flow is lamina (i.e., s t = ), (35) shows, fo typical weathe and ada paametes (i.e., s = 1 m; v T 1 m s 1 ; T s 1 3 s; D 1 m; l =.1m;g.5), that at most a few hunded independent samples would have to be aveaged. Howeve, if coelation time is dominated by

9 ZHANG ET AL.: SAMPLE VOLUME TRACKING TO MEASURE WIND 14-9 Figue 6. Standad deviation of adial wind estimates as a function of tubulence fo vaious ange esolutions of 2, 4, 8, and 16 m. Othe paametes ae v T = m/s, v =5m/s, d =1 m, and T d = 6 s. See colo vesion of this figue in the HTML. tubulence, t c = l/(4ps t ), and then the following condition on is obtained: p ffiffiffiffiffiffi s t T s s > 8p l l : ð37þ Fo the assumed ada paametes, and s t = 1 m/s, the numbe of independent samples euied is almost 1,! [34] Although this might be achievable fo some eseach adas, it is not pactical fo opeational weathe adas. On the othe hand, if one is allowed to decease the pulse width by a facto of 1 (and incease the bandwidth by ten to about 1 MHz, a easonable value) the numbe of independent samples euied educes to about 63. Consideing that we can obtain about 6 independent samples duing a typical dwell time to estimate moments in a esolution volume, we need only 1 moe independent samples. Howeve, we can aveage about 1 estimates in ange, which would educe ange esolution to about 1 km. That is, assuming the conditions hold ove this ange inteval, we should be able to estimate the high-speed adial wind within the typical dwell time of weathe adas. Wosening the ange esolution to 1 km should be acceptable because the angula esolution of opeational weathe adas (e.g., the WSR-88D) is moe than 1 km fo anges lage than 6 km, and at pesent, the ange esolution of eflectivity estimates is also 1 km. 5. Summay and Conclusions [35] We develop and discuss single antenna intefeomety to measue thee-dimensional wind. We povide an altenative explanation of intefeomety fo wind measuement fom that commonly used (i.e., detecting the diffaction patten displacement acoss pais of eceiving antennas) to explain Spaced Antenna (SA) measuements. We extend the SA intefeometic techniue that uses multiple antennas to single antenna adas. We popose Angula Intefeomety (AI) to detemine tansvese wind and Range Intefeomety (RI) to detemine adial wind using time seies data collected fom diffeent beam positions and fom diffeent ange gates. Angula and ange coss-coelation functions of signals eceived by a single antenna ada ae deived based on wave scatteing fom andom fluctuations of efaction index. The wind vecto is estimated fom the atio of the coelation function at positive and negative lags to estimate the tansvese wind fom the angula coss-coelation function, and the adial wind fom the ange coss-coelation function. The feasibility of the method was studied though eo analysis. The standad deviations of the estimated wind velocities deived and thei sensitivity to esolution, dwell time, and tubulence ae analyzed. [36] It has been shown that AI woks best with naow beams at close ange, wheeas RI woks best with fine ange esolution; both wok bette if tubulence is weak. The advantage of the single antenna intefeomety techniue fo wind measuement is that it does not euie the multi-eceives that a SA system needs, it has highe signal to noise atio and thee is no aliasing poblem that the Dopple method encountes. The data collection and pocessing ae simple and staightfowad. RI and AI can be applied to wind pofiles, weathe adas, and middle and uppe atmospheic adas to measue the wind vecto field at high esolution without significant hadwae advancement. A limitation of RI and AI is the euiement of small esolution volumes and weak tubulence. [37] The eo analysis is done fo sampling eo only and using a single lag estimates. Eos due to noise, inhomogeneity, and clutte can futhe degade the pefomance, wheeas educing esolution volume dimensions and using advanced signal pocessing can impove the techniue. Futhe studies ae needed to uantify the othe eo effects and fully undestand the limitation of the poposed techniue. The study fo othe applications of angula and ange intefeomety such as esolution impovement is undetaken. [38] Acknowledgments. Authos geatly appeciate helpful discussions with Ds. Dusan S. Znic, Akia Ishimau, Bant Foote, and Robet J. Seafin and the suppot povided by NCAR s Reseach Application Pogam. The caeful eading of the manuscipt and detailed eview comments by D. Mamou Yamamoto ae geatly appeciated. Refeences Biggs, B. H., The analysis of spaced senso data by coelation techniues, in MAP Handbook, vol. 13, edited by R. A.

10 14-1 ZHANG ET AL.: SAMPLE VOLUME TRACKING TO MEASURE WIND Vincent, pp , Sci. Comm. on Sol.-Te. Phys. Sec., Univ. of Ill., Ubana, Biggs, B. H., G. J. Phillips, and D. H. Shinn, The analysis of obsevation on spaced eceive of the fading adio signals, Poc. Phys. Soc. London, 63, , 195. Cane, R. K., Automatic cell detection and tacking, IEEE Tans. Geosci. Electon., 17, , Doviak, R. J., and D. S. Znic, Dopple Rada and Weathe Obsevations, Academic, San Diego, Calif., Doviak, R. J., R. J. Lataitis, and C. L. Holloway, Coss coelation and coss specta fo spaced antenna wind pofiles: 1. Theoetical analysis, Radio Sci., 31, , Fitch, J. P., The single antenna intefeomete, Poc. IGARSS 91, 4, , May, P. T., R. G. Stauch, R. J. Lataitis, K. P. Moan, and D. A. Meitt, Single station ocean cuent vecto measuement: Application of the spaced antenna (SA) techniue, Geophys. Res. Lett., 16, , Rinehat, R. E., Intenal stom motion fom single non-dopple weathe ada, Ph.D. dissetation, Colo. State Univ., Fot Collins, Tuttle, J. D., and G. B. Foote, Detemination of the bounday laye aiflow fom a single Dopple ada, J. Atmos. Oceanic Technol., 7, , 199. Vincent, R. A., and J. Rottge, Spaced antenna VHF ada obsevations of topospheic velocities and iegulaities, Radio Sci., 15, , 198. Zhang, G., R. J. Doviak, J. Vivekanandan, W. O. J. Bown, and S. Cohn, Coss-coelation atio method to estimate cossbeam wind and compaison with a full coelation analysis, Radio Sci., 38(3), 852, doi:1.129/22rs2682, 23. R. J. Doviak, National Sevee Stoms Laboatoy, 1313 Halley Cicle, Noman, OK 7369, USA. J. Vivekanandan and G. Zhang, Reseach Application Pogam, National Cente fo Atmospheic Reseach, P.O. Box 3, Boulde, CO 8516, USA. (guzhang@uca.edu) T.-Y. Yu, School of Electical and Compute Engineeing, Univesity of Oklahoma, Noman, OK 7319, USA.