Cross-correlation ratio method to estimate cross-beam wind
and comparison with a full correlation analysis
Guifu Zhang,1 Richard J. Doviak,2J. Vivekanandan,1William O. J. Brown,1 and Stephen A. Cohn1
Received 11 March 2002; revised 18 June 2002; accepted 17 September 2002; published 27 March 2003.
[1] Cross-beam wind is usually estimated using a full correlation analysis (FCA) method applied to signals from spaced antennas. In this paper we present a cross-correlation ratio (CCR) method for wind measurements. The CCR method is illustrated using theory, and data obtained with the National Center for Atmospheric Research’s multiple antenna profiling radar. The standard errors of estimated cross-beam wind using CCR and a FCA are studied based on a rigorous analysis of the variance of the cross-correlation estimates. The results of the analysis are compared with previous works. It is shown that the current method is easy to implement and has smaller error for receiving antenna spacing small compared to the transmitting antenna dimensions. INDEXTERMS:1869 Hydrology: Stochastic processes; 6952 Radio Science: Radar atmospheric physics; 6969 Radio Science: Remote sensing; 6974 Radio Science: Signal processing; 6994 Radio Science: Instruments and techniques;KEYWORDS:spaced antenna technique, cross-beam wind, cross-correlation, cross-correlation ratio (CCR), full correlation analysis (FCA), wind profilers
Citation: Zhang, G., R. J. Doviak, J. Vivekanandan, W. O. J. Brown, and S. A. Cohn, Cross-correlation ratio method to estimate cross-beam wind and comparison with a full correlation analysis,Radio Sci.,38(3), 8052,
doi:10.1029/2002RS002682, 2003.
1. Introduction
[2] Briggs et al.[1950] introduced the spaced antenna (SA) technique in 1950 to determine the velocity of diffraction patterns having random change. If the dif-fraction pattern drifted without change along the baseline of a pair of spaced receiving antennas, the pattern velocity would be given by the antenna spacing divided by the time delay to the peak of the cross-correlation of the two received signals. In order to compensate for random changes in the diffraction pattern, Briggs et al. [1950] used a technique in which both the cross-corre-lation and autocorrecross-corre-lation functions are used. This tech-nique, known as Briggs’ Full Correlation Analysis (FCA), is described in greater detail in a later paper [Briggs, 1984]. Briggs et al. [1950] results are related entirely to the drift and fading of the diffraction pattern, but Liu et al. [1990] first related the properties of the turbulent wind and refractive index perturbations to the complex (i.e., amplitude and phase) electric field of the
diffraction pattern. Doviak et al. [1996] extended this theory to overcome limitations (e.g., receiving antennas symmetrically place about an identical transmitting antenna), and to scattering media having statistical prop-erties described by turbulence theories.
[3] Although Briggs’ FCA method is commonly used by researchers, other methods have been proposed. For example, Holloway et al. [1997] introduced the cross/ autocorrelation ratio (CACR) method, another FCA approach, to determine wind. On the other hand, Law-rence et al. [1972] andLataitis et al. [1995] describe a slope at zero lag (SZL) method in which only the cross-correlation function is required to determine the wind. Shortcomings of these techniques are: Briggs’ FCA and CACR methods require estimation of both auto and cross-correlation functions; the autocorrelation function contains receiver noise at zero lag, whereas, the cross-correlation does not, if external noise is small compared to receiver noise. The SZL method requires sufficiently short lags to ignore the bias in the wind estimates caused by the necessity of approximating the derivative by a finite difference. This is especially a problem if the time lag step is not short compared to the time lag to the correlation peak.
[4] In this paper, we propose and illustrate a method for wind retrieval using only the cross-correlation
func-1
National Center for Atmospheric Research, Boulder, Colorado, USA.
2
National Severe Storms Laboratory, Norman, Oklahoma, USA.
Copyright 2003 by the American Geophysical Union. 0048-6604/03/2002RS002682
tion. This method (CCR) estimates cross-beam wind from the ratio of the cross-correlation functions at positive and negative lags. The turbulence effect is eliminated because the turbulence term in the ratio is canceled. Because the autocorrelation function is not used, renormalization is not needed and the zero-lag autocorrelation noise has no effect on wind estimates. This method is direct and simple to implement.
[5] The paper is organized as follows: In section 2, we present the CCR method for determining wind. We analyze the standard error of estimated wind using the CCR method and compare it with that obtained with Briggs’ FCA method in section 3. The error analysis results for the FCA method are also compared with those presented byMay[1988]. In section 4, the CCR method is applied to process the data collected with NCAR’s multiple antenna profiling radar (MAPR). Finally, we discuss the advantages and disadvantages of the CCR method compared with Briggs’ FCA method.
2. Cross-Correlation Ratio Method
[6] With an SA system, waves from a scattering medium are received in separated antennas. The cross-correlation function of the received signals, based on wave scattering from refractive index perturbations, was first derived by Liu et al. [1990]. Doviak et al. [1996] extended this derivation to receivers not sym-metrically located about the transmitter, and to receiv-ing antennas with a different size from the transmittreceiv-ing one. Furthermore, they considered a scattering media having a Kolmogorov spectrum of refraction index perturbations as well as media filled with point scat-terers such as rain.
[7] A configuration sketch of wind measurement using the SA system is shown in Figure 1. In this paper the ensemble average of the product of two weather signals is defined as the cross-correlation function which is equivalent to the covariance function because weather signals are zero mean Gaussian random processes. For horizontally isotropic refractive index perturbations and a pair of identical receivers with a separation smaller than the horizontal dimension of the transmitting beam at the altitude of measurement, the cross-correlation func-tion of signals from spaced antennas is [Doviak et al., 1996, equation (58)]: C12ð Þ ¼t Sexp b2hðDx=2v0xtÞ2b2h Dy=2v0yt 2 v0zt=2 ffiffiffi 2 p sr 2 2ðkosttÞ22jkov0zt ; ð1Þ
where the time lag between the signal samples ist, the signal power in each of the receivers isS, ko ¼2lp is the
wave number,lis the radar wavelength, (Dx,Dy) is the separation of the two receiving antennas;v0xandv0yare mean cross-beam wind components, and v0zis the mean radial wind, the standard derivation of the radial wind component (associated with turbulence) isst, and range resolution is sr. The parameterbhis given by:
bh2¼ 1:18l paq1T
2
þ2r2Bh; ð2Þ
where pffiffiffi2bh1 is the horizontal (assuming the antenna array is horizontal) scale or correlation length of the diffraction pattern, q1T is the one-way 3 dB width of the transmitting beam, andrBhis the horizontal scale of the Bragg scatterers. This particular form is obtained from Doviak et al. [1996, equation (59)], by expressing the parametergin that equation in terms of the beam width. Because q1T is roughly equal to l/D where D is the transmitting antenna diameter, the first term in equation (2) dominates the second term if DrBh. Under these conditions,bhD1. The parameterais given by:
a¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2q21R q21Rþq21T s ; ð3Þ
where q1R is the receiving antenna’s beam width. Furthermore, for rain scatter, the second term vanishes if the scatter from drops is independent as is typical.
[8] Equation (1) also applies to Doppler weather radars for whichDx=Dy= 0,a= 1, andbhis given by the first term in equation (2). In this case it can be shown that the
Figure 1. Configuration sketch of wind measurement using the spaced antenna technique.
first two terms in the exponent of equation (1) lead to the so-called ‘‘beam broadening’’ contribution to spectrum width. As seen in equation (1), the phase of cross-correlation function is only related to the radial velocity v0z (as in the Doppler method) while the magnitude of the cross-correlation function depends on both the mean wind and turbulence. Three motions contribute to the cross-correlation magnitude: (1) the mean cross-beam wind, (2) the mean radial velocity, and (3) the radial component of turbulence. Turbulence is usually more effective than the mean wind in decreasing the correla-tion between the signals because often k0 bh D1 (if Dl). This makes it difficult to use SA technique with a narrow beam if turbulence is strong.
[9] To illustrate the SA technique, we calculate the correlation coefficients for the MAPR configuration [Cohn et al., 1997]. In this configuration, receivers 1, 2 are aligned along a baseline parallel to x, and thus the baseline wind isvox, and voyis the cross-baseline wind. The results are plotted as functions of time lag and shown in Figure 2. The parameters used for the calcu-lations are: transmitting and receiving beam widthsq1T= 8.89 and q1R = 18.87 [Cohn et al., 1997], antenna separation, Dx= D/2, Dy = 0 where antenna size D = 1.84 m. Figure 2a shows the correlation for various baseline wind components v0x = 5, 10, and 20 m/s if there is no wind across the baseline (i.e.,v0y= 0 m/s) and no turbulence (st= 0 m/s). In this case, the mean wind velocity is simply related to the delay of the cross-correlation peak (tp), and can be easily retrieved as v0,x=Dx/(2tp). When cross-baseline wind or turbulence is present, however, wind retrieval is not as simple. In Figure 2b, we plot correlation coefficient for various cross-baseline wind velocities ofv0y= 5, 10, and 20 m/s while keeping the same mean baseline velocity v0x = 10 m/s. Figure 2c shows the correlation coefficients for various turbulence strengths ofst= 0, 1.0, and 2.0 m/s. As the cross-baseline wind or turbulence increases, the cross-correlation peak delay shifts toward zero lag, and the peak cross-correlation decreases. The autocorrelation is used to account for the turbulence effect in Briggs’ FCA method. Because the location of the intercept (INT) of the autocorrelation and cross-correlation functions, or the slope at zero lag (SZL) of the cross-correlation function, does not depend on turbulence st or
cross-baseline wind, the INT and SZL methods give directly the baseline winds from the correlation functions without the necessity of estimating other variables. Here we propose another method as follows:
Figure 2. (opposite) Correlation coefficients as a function of time lag for various (a) mean baseline velocities v0x= 5, 10, and 20 m/s, with fixed v0y= 0 and st= 0; (b) mean cross-baseline wind v0y= 5, 10, and 20 m/s, with fixed v0x = 10 m/s andst = 0; (c) turbulence intensitiesst= 0, 1, and 2 m/s, with fixed v0x= 10 m/s and v0y= 0 m/s . Other parameters are: voz= 0 m/s,Dx = 0.92 m,bh= 1.62 m1, andl= 0.328 m.
[10] Assuming antennas 1, 2 are aligned along the baselineDx(i.e.,Dy= 0), we take a ratio of the cross-correlation magnitude at positive and negative lags and then its logarithm to obtain:
Lð Þ ¼t ln jC12ð Þtj
C12ðtÞ
j j¼b
2
h2Dxv0xt: ð4Þ Therefore, the logarithm of the CCR is linearly propor-tional to the mean cross-beam windv0xalong the baseline Dx. In this paper we assume that scatter is isotropic so that bhis determined by the antenna parameters and thus is a known constant. If scatter is from anisotropic irregularities of refractive index andbhis not known, then the ratio of the autocorrelation to cross-correlation functions must be used [Holloway et al., 1997]. The wind component in another direction can be obtained by a like derivation for a pair of receivers aligned along that direction.
[11] The results of CCR method are illustrated in Figure 3. It shows the logarithm of CCR for various velocities of 5, 10, and 20 m/s. As expected, a large baseline velocity causes a large slope of the logarithm of CCR. It is noted that the effects due to turbulence and cross-baseline wind are canceled in the cross-correlation ratio. Because of the linear relation, multilag information can be easily combined using least squares fitting to a straight line, or a method of maximum likelihood with a proper weighting function in order to minimize the effects of system noise. The mean cross-beam wind can be obtained by rewriting equation (4) as
v0x¼ Lð Þt
2b2hDxt: ð5Þ
The wind component in any other direction can be obtained by a like analysis for a pair of spaced receiving antennas aligned along that direction.
3. Standard Error of Velocity Estimation
[12] The standard error of wind measurements using the Doppler method has been well studied [Doviak and Zrnic, 1993]. Although there have been approaches to error analysis for Briggs’ FCA method [May, 1988; Tahara et al., 1997], a systematic and rigorous study of the estimation error of retrieved cross-beam wind is not available yet. Here we present a rigorous analysis of the errors in cross-beam winds obtained from the CCR and Briggs’ FCA methods.3.1. Error Analysis for CCR Method
[13] Theoretically, the cross-correlation ratio L(t) at any time lag can be used to determine the baseline velocity v0x. In practice, true L(t) is unknown, and only estimates ˆC12(t) and ˆL(t) are obtained from meas-urements. Following equation (5), and assuming the two
receivers are aligned along the x axis, we write the estimated wind component ˆv0xas
ˆ v0x¼ ˆ Lð Þt 2b2hDxt¼ 1 2b2hDxtln ˆ C12ð Þt ˆ C12ðtÞ ! : ð6Þ
Hence estimation error of the velocity is obtained by a subtraction of equation (5) from (6) yielding
Dv0x¼vˆ0xv0x¼
ˆ
Lð Þ t Lð Þt
2b2hDxt : ð7Þ
The variance of the estimation error for the baseline wind component is var ˆðv0xÞ ¼ jDv0xj2 D E ¼ 1 2b2hDxt 2var ˆLð Þt: ð8Þ
If the distribution of estimates is concentrated near their expected value, the variance of the estimated ˆL(t) can be expressed approximately as [Papoulis, 1965, section 7-3]
var ˆLð Þt @Lð Þt @jC12ð Þtj 2 var Cˆ12ð Þt þ @Lð Þt @jC12ðtÞj 2 var Cˆ12ðtÞ þ2 @Lð Þt @jC12ð Þt j @Lð Þt @jC12ðtÞj cov Cˆ12ð Þt ; Cˆ12ðtÞ
Figure 3. Logarithm of cross-correlation ratio at positive and negative lags as a function of time lag for v0x= 5, 10, and 20 m/s as a parameter. Other parameters are: v0y= v0z= 0 m/s,Dx = 0.92 m,bh= 1.62 m1, andl= 0.328 m.
¼ var ˆ C12ð Þt C12ð Þt j j2 þ var Cˆ12ðtÞ C2 12ðtÞ 2 2cov jC12ð Þt j; ˆ C12ðtÞ C12ð Þt j jjC12ðtÞj ; ð9Þ
where the cross-correlation function is defined in (A1), the variance and covariance of the magnitude of the cross-correlation function estimates are derived in Appendix A. [14] The variance of the cross-correlation function estimates is (equation (A13))
var Cˆ12ð Þt ¼ 1 2MI S 2 þjC12ð Þt j 2 ; ð10Þ
where the number of independent samples within the dwell time isMI. As we expect, the variance is inversely proportional toMI, the same as that for estimated power. BecausejC12(t)j2S2, var[jC12(t)j] lies between 0.5 and 1.0 of that for power estimation. This makes physical sense in the two limiting cases. For example, ifjCˆ12(t)j Cp=S(i.e., the two sample series are perfectly correlated ift=tp),var[jCˆ12(tp)j] =S2/MI. In this case,jC12(tp)jis the powerSand its variance is agreement with that derived by others [e.g.,Doviak and Zrnic, 1993, section 6.3.1.2]. At the other limit, ifjC12(tp)j= 0, var[jC12(tp)j=S
2 /2MI. The factor 2 decrease in variance is caused by the fact that two independent sample series are used to estimate
jC12(tp)j. The factor 2 is in addition to the numberMIof independent samples in each of the time series.
[15] The square root of equation (10) leads to the standard derivation of the cross-correlation function estimates SD Cˆ12ð Þt ¼ ffiffiffiffiffiffiffiffiS 2MI p 1þr212ð Þt1=2; ð11Þ
where the cross-correlation coefficient is
r12ð Þ ¼t jC12ð Þtj
S : ð12Þ
In Appendix A, we also derived the covariance of the
jC12(t)j estimates. This is cov Cˆ12ð Þt1 ; Cˆ12ð Þt2 ¼ 1 2MI " S2exp ðt2t1Þ 2 4t2 c ! þjC12ð Þt1 jjC12ð Þjt2 exp ðt2t1Þ2 4t2 c !# ; ð13Þ
where (tc) is the correlation timetc¼
l
4psv in whichsv
is the spectrum width. Substituting equations (10) and (13)
into (9), the variance of the estimated logarithm of CCR is var ˆLð Þt¼ 1 MI 1exp t2 t2 c þ 1 r2 12ð Þ0 exp t 2 t2 c cosh 2ttp t2 c 1 ; ð14aÞ
where the cross-correlation coefficient at zero lag is r122 (0). If the time lag is much smaller than the coherence time (i.e.,ttc) and iftpis of the order oftcor less, a condition that results ifDxis less than D, equation (14a) reduces to var ˆLð Þt t 2 t2 cMI 1 r2 12ð Þ0 1þ2t 2 p t2 c r212ð Þ0 ! : ð14bÞ
Besides being inversely proportional toMI, the variance of ˆLdecreases as the cross-correlation coefficient r122 (0) increases. When the signals are highly correlated, statistical errors in the estimates of the cross-correlation function are also correlated. Thus the error of estimated cross-correlation function at positive lag and that at negative lag cancel each other, which leads a smaller variance of ˆL.
[16] The standard derivation of ˆL, obtained by taking a square root of equation (14a), is plotted in Figure 4 and compared with results from numerical simulations. To obtain simulated data we followed the approach described byZrnic[1975] and generated two independent time series of complex weather signals, each having the same corre-lation time. Two correlated time series of complex weather signals are then generated from the two uncorrelated series using the technique described byMay[1988]. As seen in the figure, the theoretical and simulation results agree.
[17] Substituting equation (14a) into (8), we obtain standard derivation of the velocity estimate
SDðvˆ0xÞ ¼ 1 2b2hDxt ffiffiffiffiffiffiMI p 1exp t 2 t2 c þ 1 r2 12ð Þ0 exp t 2 t2 c cosh 2ttp t2 c 1 1=2 ; ð15aÞ which reduces to SDðvˆ0xÞ¼ 1 2b2hDxtc ffiffiffiffiffiffi MI p 1 r12ð Þ0 1þ2 t2p t2 c r212ð Þ0 !1=2 ; ð15bÞ
for small time lags such that ttc, and iftp<O(tc). [18] We plotted the standard error of estimating the baseline wind using the CCR method in Figure 5 as a
function of receiver antenna separation for various levels ofst. We see the estimation error increases as the antenna separation and turbulence increase. This is because large Dxandstcause de-correlation of the signals and also of the error of the cross-correlation function estimates at different lags, leading to less cancellation of estimation error at ±t, and hence larger error in the velocity estimates. On the other hand, small Dx and st lead to high correlation of the estimate errors at ±tand hence they cancel each other. This suggests that an optimum radar configuration could be antenna separation smaller than one half the transmitting antenna size. This also allows receiver antennas to be overlapped increasing the receiver antenna size which also increases the SNR.
3.2. Error Analysis for a FCA Method
[19] Briggs’ FCA method [Briggs et al., 1950;Briggs, 1984], and its variations, are the most commonly imple-mented methods for estimating the cross-beam wind using the SA technique. The idea of the FCA is to use both autocorrelation and cross-correlation coefficients to estimate wind and turbulence. It is generally a rather complicated procedure, but for horizontally isotropic scatterers Briggs’ FCA yields the simple equation
ˆ v0x¼ Dxtˆp 2 ˆt2 x ; ð16Þ
where ˆtxis an estimate oftxwhich is the lag such that jC11(tx)j = jC12(0)j, and ˆtp is the estimated lag to the cross-correlation peak. Both lag estimates ˆtxand ˆtpdo not depend on the normalization of the cross-correlation
functions, and thus these lags would be the same whether the correlation function estimates or the correlation coefficient estimates are used. May [1988] used the cross-correlation coefficients to determine the errors in estimating tˆx and tˆp, whereas here we use the correlation function.
[20] We ran numerical simulations and found that the contribution from the covariance term of ˆtp and ˆtx is negligible as compared with the variance terms. Hence the standard deviation of the baseline velocity, estimated using the Briggs’ FCA method, is
SDðˆv0xÞ ¼v0x var ˆtp t2 p þ4var ˆð Þtx t2 x !1=2 ; ð17Þ
where the variances of tp and tx are derived in Appendix B and are
var ˆtp ¼ t2 c 4MI 1r2 12 tp r2 12 tp ð18Þ var ˆð Þ ¼tx t4 c t2 xMIr 2 12ð Þ0 1þr212ð Þ 0 2r212ð Þ0 exp t 2 xþt2p 4t2 c ! cosh txtp 2t2 c ; ð19Þ
where the normalized peak cross-correlation coefficient is r12(tp). Equations (17) to (19) constitute the error analysis for velocity retrieval using Briggs’ FCA.
Figure 4. Comparison between simulation and theory: standard derivation of the estimated cross-correlation ratio as a function of the cross-correlation coefficient at zero lag.
Figure 5. Standard error of the CCR estimated baseline wind as a function of receiver antenna separation for v0x = 10.0 m/s, and various levels of turbulence. Other parameters are: v0y= v0z= 0 m/s, D = 1.84 m,bh= 1.62 m1,l= 0.328 m, and dwell time Td= 1 s.
[21] We calculate the standard error of Briggs’ FCA velocity estimates using equation (17), and compare it with that for CCR. The results are shown in Figure 6. The same parameters as that in Figure 5 are used except for a fixedst= 1.0 m/s. We see that the error of velocity estimation for the FCA is smaller than CCR for large antenna separation (i.e.,DxD/2). The CCR, however, performs better than Briggs’ FCA for a small antenna separation (Dx< D/2).
[22] To compare our results for Briggs’ FCA with previous works, we obtain expressions of standard der-ivation of tpˆ and txˆ by taking the square root of equations (18) and (19) SD tˆp ¼0:5 tc ffiffiffiffiffiffi MI p 1r 2 12 tp 1=2 r12 tp ð20Þ SDð Þ ¼tˆx t2 c tx ffiffiffiffiffiffi MI p 1 r12ð Þ0 1þr 2 12ð Þ 0 2r 2 12ð Þ0 exp t 2 xþt2p 4t2 c ! cosh txtp 2t2 c 1=2 : ð21Þ
[23] Previously, the error analysis of baseline wind estimates using Briggs’ FCA method has mostly depended on formulas derived by May [1988]. These formulas for the standard error of ˆtp and ˆtxare
SD tˆp ¼0:501 tc ffiffiffiffiffiffi MI p 1r 2 12 tp r12 tp ð22Þ SDð Þ ¼tˆx 0:568 tc2 tx ffiffiffiffiffiffiMI p 1r 2 12ð Þ0 r12ð Þ0 : ð23Þ
Comparing equations (20) and (21) with (22) and (23), respectively, we see some similarities, but substantial differences. The standard errors are calculated and compared with numerical simulations in Figure 7. The results are plotted as a function of the correlation coefficients (r12(tp),r12(0)), and shown in Figure 7a for
ˆ
tpand Figure 7b for ˆtx. Present results agree better with the simulations. This is because we derive the standard error in a rigorous way based on an analysis of the
Figure 6. Comparison of standard error of baseline wind estimates obtained with the CCR and FCA methods. v0x = 10.0 m/s, st = 1 m/s. Other parameters are: v0y= v0z= 0 m/s, D = 1.84 m,bh= 1.62 m1,l= 0.328 m, and dwell time Td= 1 s.
Figure 7. Standard errors of the FCA parameters (a) ˆtp and (b) ˆtx versus peak cross-correlation coefficient r12(tp), and the cross-correlation coefficient at zero lag r12(0). Results are from simulations (*), the theory in this paper, andMay’s [1988] theory.
variance of the cross-correlation function magnitude without making significant approximations as needed if the cross-correlation coefficient is used.
[24] In May’s derivation, he assumes the variances of ˆ
tp and ˆtx are proportional to the variance of the esti-mated cross-correlation coefficient (i.e., var ˆ½r12ð Þt ) while neglecting the covariance terms. May then resorts to simulations to determine the constant of proportion-ality. But strictly, neither the variance of ˆtp nor that of ˆ
txis proportional to the variance of the cross-correlation coefficient because the errors in the estimates are corre-lated. Thus it is necessary to include and calculate the covariance terms.
[25] Furthermore, May uses Awe’s [1964] expression for the standard derivation of the cross-correlation
coefficient SD½rˆ12ð Þt ¼1r 2 12ð Þt ffiffiffiffiffiffi MI p ; ð24Þ
which is not valid when the cross-correlation coefficient is near one, as noted by May [1988]. This is because the sample population distribution is not normally distributed for a finite number, and especially for a small number, of independent samples as the cross-correlation coefficient approaches one. But the sample population of the cross-correlation function is normally distributed, and our results in equations (11) and (12), are valid if the cross-correlation coefficient is equal to 1. As shown in Figure 7, our theoretical expressions agree much better with the simulated results for SD tˆp in the region where the cross-correlation coefficient is near 1 and for SD½ txˆ in almost all the range ofr12(0). In Figure 7a, we also plotted the May’s simulation results taken from Figure 3 of the paper by May [1988]. It shows that May’s simulation results (4 squares) are close to current simulation results and agree with current theory better than May’s theory.
4. Application in MAPR Data Processing
[26] Data used to illustrate and test the CCR method were collected by NCAR’s MAPR on 16 November 1998 [Cohn et al., 1997, 2001; W. Brown et al., NCAR’s multiple antenna profiler radar, Preprints of the 29th International Conference On Radar Meteorology, 12 – 16 July 1999, hereinafter referred to as Brown et al., preprint, 1999]. MAPR is a modified 915 MHz profiler (Radian LAP-3000) designed to measure wind and turbulence at a rate considerably faster than the tradi-tional Doppler Beam Swing (DBS) wind profiler. The radar has four closely spaced antennas. They are com-bined to form one transmitting antenna and then act separately as four receiving antennas so that spaced antenna techniques can be applied. The MAPR trans-mitted 1.4 ms width pulses with an inter pulse period (IPP) of 25ms, and backscattered waves were received by the four receiving antennas. 254 signal samples are uniformly boxed averaged to give an effective lag spac-ing of 6.35 ms between samples. The time series of averaged samples is then used to calculate correlation functions for velocity retrieval. The radar was deployed in Erie, Colorado, approximately 600 m south of the instrumented tower of the NOAA Boulder Atmospheric Observatory (BAO). The wind was also measured by sonic and propeller anemometers at the 300 m level.[27] Figure 8 shows an example of velocity retrieval using the CCR method. The time series averaged signal samples E1 and E2 received at one of the 6 pairs of receivers are processed. Both autocorrelations (C11) and cross-correlation (C12) are estimated and shown in
Figure 8. An example of: (a) the correlation estimates normalized by signal plus noise, and (b) the logarithm ˆL of the ratio of cross correlations at positive and negative lag using MAPR measurements.
Figure 8a. The autocorrelations have a spike at zero lag because of noise, and its height can be used to estimate the signal-noise ratio. The cross-correlation just has a minor noise spike (due to external interference that is correlated between the receivers) as well as other fluc-tuation errors due to noise, as does the autocorrelation function. (The noise effect is planned to be discussed in a future paper). An estimated ratio of cross-correlation magnitude at positive and negative lags can be obtained as described in section 2. Figure 8b shows a logarithm of the cross-correlation ratio.
[28] Figure 9 shows results of cross-beam wind retriev-als using the CCR and Briggs’ FCA methods, along with in situ anemometer measurements. Two pairs of received in- and quadrature-phase signals, obtained from receiver pairs whose baselines are perpendicular to each other, are used in the cross-correlation processing as described in this paper to obtain the corresponding CCR baseline winds. The CCR baseline wind components are then used to compute the north and east winds to compare with anemometer measurements and the FCA winds presented by Brown et al. ( preprint, 1999). Figure 9 is not intended to show a rigorous comparison of the two techniques, but is only presented to show that the CCR method gives results that are in relatively close agree-ment to the more commonly used Briggs’ FCA method, as well as with anemometer measurements. Because wind estimates errors are strongly affected by the
sig-nal-to-noise ratio, a theoretical error analysis of the wind estimates using CCR technique with signals in the presence of noise will be studied, and rigorously com-pared with observations in a subsequent paper.
5. Summary and Discussions
[29] We presented a cross-correlation ratio method for cross-beam wind measurements. The standard error of the cross-beam wind estimates has been studied for the CCR method and the commonly used Briggs’ FCA method. It is shown that the CCR method has a smaller standard error for velocity estimation than the FCA method for highly correlated signals, i.e., for receiver antennas with small separation. This allows signals be coherently combined to increase SNR [Zhang et al., 2001]. The CCR method is illustrated with data collected by NCAR’s MAPR. It shows that the CCR gives reliable estimates of wind velocity. The estimated velocity agrees with that obtained with Briggs’ FCA method and as well with in situ ane-mometer measurements.
[30] The variance and covariance of the magnitude of estimated correlation functions given in Appendix A are general and can be applied to the error analysis of other methods using the SA technique. The results of our error analysis for Briggs’ FCA method are different from that provided by May [1988], and agree better with simula-tions. This is because we included the covariance term and used an accurate error analysis for the cross-corre-lation function magnitude in our derivation, whereas May neglected the covariance terms and used an approx-imate expression for the variance of cross-correlation coefficient, an expression not valid when the normalized cross-correlation coefficient is near one.
Appendix A. Variance and Covariance of
the Magnitude of the Correlation Estimates
[31] To study the variance and the covariance of the estimated correlation function, we examine two corre-lated Gaussian random processes, E1(t) and E2(t). They are assumed to be stationary with the expected cross-correlation function C12ð Þ ¼t E1ð Þt E2*ðtþtÞ ¼CpF ttp;tc ejwdt; ðA1Þ
whereCpF(ttp;tc) is any real function oft,tcis the correlation time,tpis the shift of cross-correlation peak, wdis the mean Doppler shift and Cpis the peak value of the magnitude of the cross-correlation function. Equation (A1) becomes the autocorrelation function iftp= 0; then Cp= S, where S is the signal power.
Figure 9. Results of cross-beam winds using the CCR and FCA methods, and comparisons with anemometer measurements, at a 300 m height, for the period starting from 00:00 on 16 November 1998. (a) Eastward wind. (b) Northward wind.
[32] Using SA radar measurements, the time series signal samples from two receivers are recorded and the correlation functions are estimated by processing a finite number of samples for each estimate. Suppose the signals are sampled at time steps of Ts, the sample interval, and let t= nTs, we have the cross-correlation estimator written as ˆ C12ð Þ ¼t 1 M XM m¼1 E1ð ÞmE2*ðmþnÞ; ðA2Þ where the hat symbol signifies an estimated value. We write ˆC12(t) as a sum ofC12(t) and the fluctuationDC12as
ˆ
C12ð Þ ¼t C12ð Þ þt DC12ð Þt: ðA3Þ [33] In the Doppler radar technique for wind measure-ments, the statistics of the phase of the autocorrelation estimator have been studied [Doviak and Zrnic, 1993]. In the SA technique, the magnitude of the estimated cross-correlation function is used (along with the autocorrela-tion funcautocorrela-tion for some estimators) to retrieve the cross-beam wind. Hence the variance and covariance of the magnitude of the estimated cross-correlation function is now examined. The autocorrelation function is a special case ifDx = 0.
[34] We first assume a small fluctuation approximation for the correlation function estimator, i.e., DC12(t) C12(t), and use a polynomial expansion up to the second order. The subscript ‘‘12’’ and the cross-correlation argument are implicit in the following equations.
ˆ C ¼Cˆ ˆ C* ˆ C 0 @ 1 A 1=2 ¼ðCþDCÞ C *þDC* CþDC !1=2 ¼j jC 1þDC C 1=2 1þDC * C* !1=2 j jC 1þDC 2Cþ DC* 2C*þ DC j j2 4j jC2 DC2 8C2 DC*2 8C*2 ! : ðA4Þ
Because DC and DC* have zero expected values, the expectation of the magnitude of the correlation function estimate is ˆ C j jC 1þ DC j j2 D E 4j jC2 0 @ DC 2 8C2 DC*2 D E 8C*2 1 A: ðA5Þ
[35] From (A2) and (A3), we have DC12ð Þ ¼n 1 M XM m¼1 E1ð Þm E2*ðmþnÞ C12ð Þn : ðA6Þ The second moments of the fluctuation are
DC12ð Þn j j2 D E ¼ 1 M XM m¼1 E1ð ÞmE*2ðmþnÞ C12ð Þn 2 * + ¼ 1 M2 XM m¼1 XM m0¼1 E1ð Þm E2*ðmþnÞE*1ð Þm0 h E2ðm0þnÞi jC12ð Þn j2
Because the In-phase and Quadrature-phase components of E(n) are zero mean Gaussian distributed random variables, the fourth moment of the complex signal can be expressed in terms of the products of the second moments [Doviak and Zrnic, 1993, section 5.1.6] which yields, ¼ 1 M2 XM m¼1 XM m0¼1 E1ð Þm E*2ðmþnÞ D E E1*ð Þm0 E2ðm0þnÞ D E h þ E1ð ÞmE*1 m 0 ð Þ D E E2*ðmþnÞE2ðm0þnÞ D Ei jC12ð Þn j 2 : It can be shown that this equation reduces to the compact form, ¼ 1 M2 XM m¼1 XM m0¼1 C11ðm0mÞC22*ðm0mÞ: Thus hjDC12(n)j2i is independent of n. This condition holds provided tTdso that approximately the same number of samples are used to estimate C12(t) for all lags of interest (i.e., t= 0, andt= tp).
[36] Assume receivers are identical [i.e.,C22(m0m) = C11(m0m)]. Because there areM jm0mjidentical terms along the diagonals, jm0 mj = constant, of the
jC11(m0m)jmatrix, the double sum can be reduced to the following single sum,
¼ 1 M XM m¼1 1j jm M C11ð Þm j j2: ðA7Þ
[37] We assume the time stepTsis much shorter than the correlation time tc (i.e., Ts tc), and that the averaging or dwell time Td is much longer than tc (Td tc). Under these conditions, the summation in (A7) can be evaluated with an integral as
DC12ð Þt j j2 D E 1 Td Z C11ð Þt j j2dt¼S 2 Td Z r211ð Þt dt ¼ ffiffiffi p p tc Td S2;
wherer11(t) is the autocorrelation coefficient, and tc 1 ffiffiffi p p Z r211ð Þt dt:
Because S2 is the variance of a single sample of signal power [Doviak and Zrnic, 1993, section 6.3], the factor Td= ffiffiffip
p tc
ð Þ MI is the equivalent number of indepen-dent samples in the dwell timeTd. Thus
DC12ð Þt j j2 D E ¼ 1 MI S2: ðA8Þ
[38] In the similar way as above, we obtain DC122 ð Þt ffiffiffi p p tc Td C 2 12ð Þ ¼t 1 MIC 2 12ð Þt ðA9Þ and DC12* 2 ðtÞ D E ffiffiffi p p tc Td C *2 12ð Þ ¼t 1 MIC12 *2ð Þt : ðA10Þ
Note that although hjDC12(t)j2i is independent of t,
hDC122(t)idoes depend ont.
[39] Substitution of (A8) – (A10) into (A5) yields
ˆ C12ð Þt jC12ð Þt j 1þ 1 4MI S2 C12ð Þt j j21 !! ; ðA11Þ so that ˆ C12ð Þt 2 jC12ð Þtj2 1þ 1 2MI S2 C12ð Þt j j21 !! ; ðA12Þ
under the condition MI >> 1 4
S2 C12ð Þt
j j2. That is, we
apply our results to those regions ofjC12(t)jwhere it is not too small compared toS. Typically these are at lags close to tp. Thus the variance of the magnitude of the estimated correlation function is obtained by subtracting (A12) from the second moment,hjDC12(t)j2i. That is,
var Cˆ12ð Þt Cˆ12ð Þt 2 D E Cˆ12ð Þt 2 : [40] It can be shown that hjC12(t)j2i = jC12(t)j2 + hjDC12(t)j2i, and using (A8) and (A12), the variance can then be written as var Cˆ12ð Þt ¼jC12ð Þt j 2 þS 2 MI jC12ð Þt j 2 1þ 1 2MI S2 C12ð Þt j j21 !! ¼ 1 2MI S 2þ C 12ð Þt j j2 : ðA13Þ
The variance of the magnitude of the correlation function estimates reduces to that S2
MI for estimated power when signals are perfectly correlated (i.e.,jC12ð Þtj
S !1) in
agree-ment with known results. It is half that value, as expected, when signals have no correlation (i.e.,
C12ð Þt
j j
S !0).
[41] Following the same procedure as above, the covariance of the magnitude of the correlation functions can be obtained. We have the covariance of the cross-correlation function at different lags given by
cov Cˆ12ð Þt1 ; Cˆ12ð Þt2 ¼ Cˆ12ð Þt1 Cˆ12ð Þt2 Cˆ12ð Þt1 ˆ C12ð Þt2 ¼jC12ð Þt1 jjC12ð Þt2 j 2 ðRef g þA Ref gB Þ; ðA14aÞ where A¼ R C12ðtþt1ÞC12ðtþt2Þdt TdC12ð Þt1 C12ð Þt2 ; and B¼ R C11ð Þt C11*ðtþt2t1Þdt TdC12ð Þt1 C12*ð Þt2 :
IfC11(t) andC12(t) have the Gaussian functional form given by C11ð Þ ¼t Sexp t2 2t2 c jwt ; and C12ð Þ ¼t Cpexp ttp 2 2t2 c jwt ! ;
then the expression for the covariance reduces to
cov Cˆ12ð Þt1 ; Cˆ12ð Þt2 ¼ 1 2MI S 2exp ðt2t1Þ 2 4t2 c ! þjC12ð Þt1 jjC12ð Þt2 jexp t2t1 ð Þ2 4t2 c !! : ðA14bÞ
[42] The covariance for the auto/cross-correlation func-tions is cov Cˆ11ð Þt1 Cˆ12ð Þt2 ¼ Cˆ11ð Þt1 Cˆ12ð Þt2 Cˆ11ð Þt1 ˆ C12ð Þt2 ¼jC11ð Þjjt1 C12ð Þt2 j 2 ðRef g þC Ref gD Þ; ðA15aÞ
where C¼ R C11ðtþt1ÞC12ðtþt2Þdt TdC11ð Þt1 C12ð Þt2 ; and D¼ R C11ð Þt C12*ðtþt2t1Þdt TdC11ð Þt1 C*12ð Þt2 :
Substituting equation (1) into the above leads to
cov Cˆ11ð Þt1 Cˆ12ð Þt2 ¼ 1 2MI C11ð Þt1 j jjC12ð Þt2 j exp t2t1tp 2 4t2 c ! þexp t2þt1tp 2 4t2 c !! : ðA15bÞ
[43] Hence equations (A13) to (A15) constitute the variance and covariance for the magnitude of autocorre-lation/cross-correlation estimators. The standard error of the wind estimates can be derived from them.
Appendix B. Variances of
T
ˆ
pand
T
ˆ
xin
Briggs’ FCA Method
[44] In Briggs’ FCA method, baseline wind is estimated from the lag location of the cross-correlation peak tpand the time lag ˆtxwhere Cˆ11ð Þtˆx ¼ Cˆ12ð Þ0 . The varian-ces of tpˆ and txˆ are derived in this appendix. The approach is somewhat similar to the one described by May [1988], but with a more rigorous derivation that includes the covariance term and an accurate calculation for the variance of the cross-correlation function estimates.
B1. Variance of Tˆp
[45] We first write the magnitude of the cross-correla-tion estimate ˆC12(t) as ˆ C12ð Þt ¼Cpˆ exp ttˆp 2 2 ˆt2c ! : ðB1Þ
A ratio of the cross-correlations,jC12(t1)jandjC12(t2)j at two lags, gives
ˆ C12ð Þt1 ˆ C12ð Þt2 ¼exp t 2 2t 2 1 2ðt2t1Þtˆp 2 ˆt2 c ¼exp t2t1 2 ˆt2c t2þt12 ˆtp : ðB2Þ
The estimated tp, obtained from solving (B2), is
ˆ tp¼ t2þt1 2 þ ˆ t2c Ts ln ˆ C12ð Þt2 ˆ C12ð Þt1 ! : ðB3Þ
[46] If estimate variances are small compared to expected values, and t2 t1 = Ts tc (as expected in practice), where Ts is the sample spacing, it can be shown that var ˆ½ tc is a second order contribution to var ˆtp . Thus assuming perturbations are normally dis-tributed about their expected values, the first order perturbation analysis yields
var ˆtp ¼ t2 c Ts 2 var jCˆ12ð Þjt1 jC12ð Þjt1 2 þvar j ˆ C12ð Þjt2 jC12ð Þjt2 2 2cov j ˆ C12ð Þjjt1 Cˆ12ð Þjt2 jC12ð Þjjt1 C12ð Þjt2 ! : ðB4Þ
Substituting (A13) and (A14) into (B4) and finding t1 andt2that lie next totp(e.g., by taking two contiguous estimates ˆC12(t) that have the largest summed value) the variance of tp reduces to var ˆtp ¼ t2 c 4MI 1r2 12 tp r2 12 tp ; ðB5Þ
where the normalized peak cross-correlation coefficient is r12 tp C12 tp
S , and MI is the number of independent samples. equation (B5) differs from that presented byMay [1988]; May’s expression squares the term (1 r122 (tp)), otherwise they agree.
B2. Variance of Tˆx
[47] To find the estimation error of ˆtx, we follow the procedure ofMay[1988] and write the magnitude of the estimated autocorrelation function as
ˆ C11ð Þt ¼Sexp t2 2 ˆt2 c þe11ð Þt; ðB6Þ
and that for cross-correlation function as,
ˆ C12ð Þt ¼ C12 tp exp ttˆp 2 2 ˆt2c ! þe12ð Þt: ðB7Þ As with var ˆtp
, it is also expected that the contribu-tion of var ˆ½tcto var ˆ½txis second order and can be ignored. Thus ˆtx is obtained by equating (B6) to (B7)
ˆ
Substitution of (B6) and (B7) into (B8) yields Sexp tˆ 2 x 2t2 c þe11ð Þ ¼tx C12 tp exp t2p 2t2 c ! þe12ð Þ ¼0 jC12ð Þ0 j þe12ð Þ0 ; and thus exp tˆ 2 x 2t2 c ¼jC12ð Þ0 j S þ e12ð Þ 0 e11ð Þtx S : ðB9Þ Taking the logarithms of the both sides and using a first order expansion, we have
tˆ 2 x 2t2 c ln jC12ð Þ0 j S þe12ð Þ 0 e11ð Þtx C12ð Þ0 j j : ðB10Þ
An error analysis yields the following expression for the variance of tx: var ˆð Þ ¼tx t4 c t2 xjC12ð Þ0 j 2varðe11ð Þ tx e12ð Þ0 Þ ¼ t 4 c t2 xjC12ð Þ0 j2 var Cˆ11ð Þtx Cˆ12ð Þ0 ¼ t 4 c t2 xjC12ð Þ0 j2 var Cˆ11ð Þtx þvar Cˆ12ð Þ0 2cov Cˆ11ð Þtx ; Cˆ12ð Þ0 : ðB11Þ
Letting t1 = tx and t2 = 0 in (A15) and substituting (A13) and (A15) into (B11), we obtain
var ˆðtx¼ t4 c t2 xjC12ð Þ0 j2 1 2MI 2S 2 þ Cˆ11ð Þtx 2 þCˆ12ð Þ0 2 2jC11ð Þtx jjC12ð Þ0 j exp txþtp 2 4t2 c ! þexp txtp 2 4t2 c !!! ¼ t 4 c t2 xMIr212ð Þ0 1þr212ð Þ0 1exp txtp 2 4t2 c ! exp txtp 2 4t2 c !!! ¼ t 4 c t2 xMIr212ð Þ0 1þr2 12ð Þ 0 2r 2 12ð Þ0 exp t2 xþt 2 p 4t2 c ! cosh txtp 2t2 c ! : ðB12Þ
Hence (B5) and (B12) are expressions for the variances of ˆtpand ˆtx, and constitute a basis for error analysis for Briggs’ FCA method.
[48] Acknowledgments. The authors appreciate helpful discussions with Akira Ishimaru, Dick Strauch, Dusan S. Zrnic, and Larry Cornman. P. May’s review of the manuscript has considerably improved the paper. We sincerely appreciate the support provided by NCAR’s Research Application Program. MAPR is partially supported by the Department of Energy VTMX program.
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W. O. J. Brown, S. A. Cohn, J. Vivekanandan, and G. Zhang, NCAR, P. O. Box 3000, Boulder, CO 80307, USA. (wbrown@ ucar.edu; [email protected]; [email protected]; [email protected]) R. J. Doviak, NSSL, 1313 Halley Circle, Norman, OK 73069, USA. ([email protected])
