Radio Science

Comparison of spaced-antenna baseline wind estimators: Theoretical and simulated results

Authors


Abstract

[1] Formulas for the theoretical precision of cross-beam winds measured with spaced-antenna profilers are developed and compared with results obtained from simulations for conditions of high signal-to-noise-ratios. These formulas relate the precision of wind measurement to radar and atmospheric parameters. Formulas for Briggs' Full Correlation Analysis, the Intersection method, and the Slope-at-Zero-Lag method are each presented for two implementations-estimating parameters for assumed Gaussian shaped correlation functions, and a direct finite difference method where this assumption is not necessary. For each wind measurement method and implementation, these formulas are used to evaluate, as an example, the theoretical performance of MAPR, NCAR's 915 MHz, Multiple Antenna Profiling Radar. The theory is also compared with the standard error obtained from simulations presented by Kawano et al. [2002] for the MU radar in Shigaraki, Japan. Comparisons show that the Intersection and Briggs' FCA methods are identical and provide the best performance.

1. Introduction

[2] UHF and VHF spaced-antenna (SA) radars have been used extensively to profile winds in the precipitation-free atmosphere [Larsen and Röttger, 1989]. These wind-profiling radars typically comprise a vertically directed transmitting beam and at least three spaced receiving antennas. Vertical wind along the transmitter beam can be calculated from the argument of the auto- or cross-correlation functions. The horizontal component of the wind, transverse to the radar beam, can be calculated from the magnitude of the auto- and cross-correlation functions of signals from pairs of receiving antennas. The accuracy of the vertical or along-beam wind component is reasonably well established [Doviak and Zrnić, 1993], but formulas to compute uncertainties of the horizontal or cross-beam wind component are not. May [1988] is the first to derive formulas to calculate precision when winds are estimated using the full correlation analysis (FCA) of Briggs [1984]. More recently, Tahara et al. [1997] and Kawano et al. [2002] have used simulations to determine the performance of SA and DBS wind profilers. In this paper we derive theoretical formulas to determine the precision of cross-beam winds calculated from SA data. These theoretical formulas are then compared with results obtained from simulations.

[3] Doviak et al. [1996] developed general formulas that relate the auto- and cross-correlation functions, for signals from pairs of spaced receiving antennas, to the statistical structure of the wind and refractive index field of the scattering medium. These general formulas for stochastic Bragg scatter reduce to relatively simple form if scatter is isotropic as it typically is for relatively short wavelength (i.e., λ < 1 m) radars. These simple forms are used to illustrate several methods for measuring cross-beam wind. Expressions for estimating the precision of wind estimates are developed using propagation of error calculations. For application examples, these expressions are used to estimate the theoretical performance of the UHF wind profiler developed by the National Center for Atmospheric Research (NCAR) [Cohn et al., 1997, 2001], as well as the VHF profiler in Shigaraki, Japan [Kawano et al., 2002]. The theory developed here assumes uniform wind, reflectivity, and isotropic turbulence throughout the radar's resolution volume, as well as infinite signal to noise ratio. Although the formulas are explicitly expressed in terms of profiling radar parameters, they can be applied to any SA radar measuring cross-beam wind.

2. Background Theory

[4] For a receiver pair separation Δx, along the baseline ‘x’ axis, smaller than the horizontal dimension of the transmitting beam at the altitude of measurement, the general expression for the temporal cross-correlation C12 (τ) of the signals echoed from horizontally isotropic scatterers is [Holloway et al., 1997]

equation image

where

equation image

The baseline wind component is vox, and vs is a decorrelating wind component which determines the correlation of signals from the spaced receivers. It is specified by

equation image

where ko = 2π/λ is the radar wavenumber, voy is the cross-baseline wind component, and σt is the root-mean-square velocity associated with turbulence on scales smaller than the dwell time Td. Td is equal to the number of Tse intervals used in processing the signals to derive the correlation functions, and βh is the diffraction pattern wavenumber to be discussed below. In equation (1), Tse is the spacing of the coherently averaged samples (i.e., Tse = McTs, the averaging time, is also an effective sampling time, wherein Ts is the repetition time of the transmitted pulses, and Mc is the number of I, Q echo samples which are averaged), ωd is the Doppler radian frequency, and m is an integer.

[5] If vs = 0, the peak of the cross-correlation magnitude, normalized with equation image, would have a value of one (i.e., the decorrelation parameter η would be zero). Thus, both voy and σt decrease the peak of the cross-correlation function magnitude. Actually ∣C12p)∣ cannot be equal to equation image unless the receiving and transmitting antennas are identical; that is, equations (1) and (2) are approximations, albeit good ones [Doviak et al., 1996].

[6] Although the Gaussian correlation functions have been derived under the assumption that the random field of refractive index perturbations has a spatial spectrum described by a Gaussian function [e.g., Kawano et al., 2002], Doviak et al. [1996] show that the Gaussian form for the signal correlations can also apply when the perturbations are described by the Kolmogorov spectrum. The Kolmogorov spectrum is one commonly thought to hold in the troposphere if refractive index perturbations are generated by turbulence. The Gaussian parameters η, τp, and τc specify, respectively, the decrease of the magnitude of the cross-correlation peak ∣C12p)∣ relative to equation image, the lag to the peak of ∣C12(τ)∣, and the signal coherence time; an example defining the parameters η, τp, and τc, of the normalized magnitudes for the auto- and cross-correlation functions of Gaussian form, is depicted in Figure 1.

Figure 1.

Gaussian shaped and normalized auto- and cross-correlation functions illustrating the various parameters used in this paper.

[7] The parameter βh,

equation image

is related to the diffraction pattern scale ξpat [Doviak et al., 1996, equation (59)], and is a function of the transmitting antenna size, D, and the horizontal correlation scale ρBh of the Bragg scatterers. The parameter α relates the receiving antenna's one-way angular beam width σθR(1), to the one-way beam width, σθT(1), of the transmitter's antenna, through the relation [Doviak et al., 1996],

equation image

Typically, 1.0 < α < equation image. The parameter γ relates D to the transmitter's beam width σθT(1) = γequation imageθT(1) is the square root of the second moment of the one-way angular weighting function of the antenna's power pattern, and is related to the more commonly used one-way half-power width θ1T, as θ1T = 2.35σθT(1) for Gaussian shaped antenna patterns).

2.1. Decorrelating Influence of Cross-Baseline Wind and Turbulence

[8] It is important to note that the influence of turbulence, in decreasing the cross-correlation peak relative to the influence of the cross-baseline wind, is magnified by the factor koh as is evident from equation (3). From equation (4) it is seen that ξpat = equation imageh is about equal to or larger than D. Thus a larger transmitting antenna, measured in units of wavelength, causes a larger effect of turbulence in decreasing the cross-correlation peak and correlation time. For antennas many wavelengths across, σt would have to be more than an order of magnitude less than the mean wind for turbulence to be ignored. Both cross-baseline wind and turbulence cause the signals to lose correlation through the parameters vs and η (equations (2) and (3)), but typically turbulence is the more significant factor, especially if the transmitter beam width is small.

2.2. Full Correlation Methods

[9] In general, βh, vox and vs, need to be calculated from equation (2) using estimates of the Gaussian parameters η, τc, and τp. In this case, all three parameters need to be estimated to determine baseline wind vox. Estimating all three parameters requires a Full Correlation Analysis (FCA) in which both ∣C11(τ)∣ and ∣C12(τ)∣ are used. But if the horizontal correlation scale of the Bragg scatterers is much smaller than the transmitting antenna size D (i.e., ρBh ≪ 0.2D),

equation image

in which we have expressed σθT(1) in terms of the more commonly used one-way 3 dB width θ1T(rad) of the transmitting antenna. Thus, under the stipulated conditions,

equation image

is a constant determined by antenna pattern measurements. If ah is known, there are only two unknowns, vox and vs, in equations (2). Thus, only two of the three equations are needed. If the last two equations are used, the baseline wind can be calculated using only the cross-correlation function. This is the basis for the Slope-at-Zero-Lag (SZL) estimator presented in section 3.4, and for the Cross Correlation Ratio (CCR) method described by Zhang et al. [2003]. Although the focus of this paper is on the accuracy of baseline wind estimates, the results can be used to calculate errors in wind speed and direction by applying the procedures discussed by Doviak et al. [1976].

2.3. Comments on the Gaussian Parameters

[10] Under the condition ρBh ≪ 0.2D, the Fresnel term can be ignored in the scattering equation [Doviak and Zrnić, 1993, section 11.5.2], and thus the solution to the scattering problem is significantly simplified. Observations of tropospheric scatter suggest that scatter is typically isotropic (i.e., independent of the radar beam direction) for wavelengths less than about 1 meter. If scatter is isotropic, the Fresnel term can be ignored, and equation (6) applies. On the other hand, it has been shown [Doviak and Zrnić, 1993, section 11.5.4] that scatter can be anisotropic and yet the Fresnel term can still be ignored. Thus, isotropy of scattering insures that equation (7) applies, but anisotropic scatter does not necessarily imply that Fresnel scatter is occurring.

[11] At wavelengths longer than about 1 m, the Fresnel term usually cannot be ignored and the scatter is surely anisotropic. Under this condition, the horizontal correlation length ρBh of the Bragg scatterers can be calculated from βh using equation (4) as shown by, for example, Chau et al. [2000] and Hassenpflug et al. [2003]. But with radars operating at wavelengths λ < 1 meter, as NCAR's MAPR (i.e., λ = 33 cm) used herein as an example, βh is usually a known constant, ah [e.g., Holloway et al., 1997], and equation (7) applies.

[12] The dependence of the Gaussian correlation functions and their time lag parameters (e.g., τp, τi, etc., Figure 1) on wind and turbulence using, as an example, MAPR parameters (i.e., βh = ah = 1.62 m s−1, Δx = 0.92 m ≈ D/2; see section 5 for discussion of these MAPR parameters) is depicted by Zhang et al. [2003] in a series of figures. From an examination of those figures, it is seen that the correlation functions narrow with increasing σt, and the cross-correlation peak shifts toward zero lag. Also noticed is the invariance, to changes in σt, of the cross-correlation slope at zero lag, as well as the auto- and cross-correlation function intersection point. This invariance forms the basis of the wind estimation techniques described in sections 3.3 and 3.4. Furthermore, it is seen that τi increases as vox decreases; this is important in understanding the dependence of wind precision on wind speed when applying the direct intersection method described in section 4.1.

[13] For large values of vox, (i.e., vsvox, or voh = equation image ≫ (koσtequation image)/ah), equation (2) gives τc ≈ τpvox−1. This is the classical uniform-flow (i.e., laminar flow) limit in which the narrowing of the correlation functions (or, equivalently, the broadening of the corresponding spectra) and the delay to the cross-correlation peak are principally dependent on wind across the radar beam and not on turbulence [Briggs, 1984]. In this limit τc and τp decrease with increasing vox. But as vox approaches small values, τp decreases linearly with vox in contrast to the uniform-flow limit if σt and voy are not both zero. This introduces the interesting result, as pointed out by Holloway et al. [1996], that a single estimate of τp can correspond to two values of vox. This ambiguity of vox given a value of τp does not, however, affect any of the wind estimation methods discussed herein.

[14] If baseline winds are very light (i.e., voxvs), ρ12p) ≈ exp{−(πγα Δx/D)2}. This latter relation illustrates the importance of maintaining the condition Δx < D to ensure measurable cross-correlation levels for light winds. For the uniform-flow limit, however, the peak of the cross-correlation coefficient, ρ12p) = exp{−η}, is equal to 1.

3. Precision of Winds Obtained From Gaussian Parameter Estimates

[15] Equations (1) and (2) are used in illustrating three time domain approaches (i.e., Briggs' FCA; the intersection, INT; the Slope-at-Zero-Lag, SZL) to calculate the wind component vox parallel to the antenna pair baseline aligned in the x direction. For each of these estimators two implementations are used. The first implementation, presented in sections 3.1 to 3.4, is to estimate the Gaussian parameters (τc, τp, and η), and to use these in equation (1) to obtain the slope of ∣C12(τ)∣ at τ = 0 for the SZL estimator (labeled G-SZL), or to obtain τi, the lag to the intersection of ∣C11(τ)∣ with ∣C12(τ)∣ for the INT estimator (labeled G-INT), or to obtain the lag τx at which ∣C11(τ)∣ equals ∣C12(0)∣ for Briggs' FCA method (labeled G-FCA). The derived slopes and time lags are then used to estimate baseline wind. Wind estimate precision is calculated from error variance formulas for the Gaussian parameters using propagation of error calculations.

[16] For the second implementation (section 4), referred to as the Direct method, the pertinent time lags (i.e., τi for the D-INT, and τx, τp for the D-FCA), and slope (for the D-SZL) are directly estimated from the data without assuming a functional form for the correlation functions.

3.1. Variance of the Gaussian Parameters

[17] Expressions for the var[equation imagec], var[equation imagep], and var[equation image] can be developed with the aid of published material. Here var denotes a variance and ^ identifies an estimate (measurement) rather than an exact theoretical value. Although there are alternative methods to estimate Gaussian parameters (e.g., a weighted least squares fitting of correlation data at multiple lags [e.g., Kawano et al., 2002]), the variance formulas presented here apply to two- lag estimators. These estimators, if applied to lags at which ∣C11(τ)∣ and ∣C12(τ)∣ have peak values, have proven to be robust, practical, and are used in many meteorological radars. For example, the spectrum width estimate equation imagev (related to equation imagec by equation imagec = λ/4πequation imagev) is obtained from lag zero and lag one estimates of the auto-correlation function calculated by the USA's WSR-88D weather radar processors [Doviak and Zrnić, 1993]. For typical radar observationts of weather, the performance of these estimators is better than those using more complicated least squares fitting approaches if SNR is high as assumed throughout this paper [May et al., 1989]. The good performance of these simple estimators is due to the fact that the statistical uncertainties at all other lags about the correlation peaks are highly correlated if SNR is high. This will be evident when we compare, in section 3.2, the theoretical results presented here with those standard errors obtained from simulations using fitting approaches. Under these conditions significant substantial variance reduction is not obtained by increasing the number of lags to make an estimate. Nevertheless, as we show in section 3.1.2, there are conditions on the choice of time lags used to calculate τc that give minimum variance of the estimate. In any case, the relative performance of the various estimators, evaluated using data at one and two lags, should be applicable to the least squares fitting approaches.

3.1.1. Variance of the Decorrelation Parameter, η, Estimates

[18] To derive an expression for var[equation image], we consider estimates of the cross-correlation function magnitude at its peak (i.e., ∣C12(τp)∣) and the geometric mean equation image of the signal powers S1 and S2 in receivers 1 and 2. Thus estimates equation image, obtained from equation (1), are

equation image

where equation image1 = ∣equation image11(0)∣, and equation image2 = ∣equation image22(0)∣. The distributions of the estimates equation imagep, ∣equation image12(equation imagep)∣, equation image1, and equation image2 will be concentrated near their expected values if the number of independent signal samples MI (MI = Tdc√π) within the dwell time Td used to make an estimate is sufficiently large. For example, in order to have a sufficient number MI of independent samples so that estimates ∣equation image12p)∣ have deviations about its expected value ∣C12p)∣ that are small compared to its expected value, MI must satisfy the condition,

equation image

where SD[x], obtained from Zhang et al. [2003], is the standard deviation (i.e., the square root of the variance) of the random variable ‘x’. Conditions similar to (9) hold for the other random variables. Condition (9) strongly depends on the peak value of the cross-correlation coefficient ρ12p) = ∣C12p)∣/equation image and is well satisfied for MI larger than 10 if ρ12p) is more than about 0.4. Under these conditions, the deviations of equation imagep about τp will be a fraction of τc, and thus equation imagep in equation (8) can be replaced with τp, and the variance of equation image can be expressed using the first order theory of Papoulis [1965, section 7-3] to obtain,

equation image

Calculating the partial derivatives using equation (1), substituting these into equation (10), and noting that, for identical receiving antennas, 〈equation image1〉 = S1 = 〈equation image2〉 = S2S and var[equation image1] = var[equation image2], the var[equation image2] expression reduces to

equation image

Expressions for the var[∣equation image12p)∣] and cov[equation image,∣C12p)∣] are given by Zhang et al. [2003], and that var[equation image] for by Doviak and Zrnić [1993], but an expression for cov[equation image1,equation image2] needs to be derived.

[19] To obtain an expression for cov[equation image1, equation image2], the procedure developed by Zhang et al. [2003] is used to derive the following expression for the covariance of the auto-correlation functions at lags τ1, τ2,:

equation image

Thus, if τ1 = τ2 = 0, equation (12) reduces to the following simple expression

equation image

But, using equation (1), it can be shown that

equation image

Substituting this into equation (13), the cov[equation image1, equation image2] expression simplifies further to:

equation image

Now from Zhang et al. [2003]:

equation image

and from Doviak and Zrnić [1993, section 6.3.1]:

equation image

Substituting equations (14) into equation (11), and using ρ12p) = C12p)/S, equation (11) reduces to

equation image

Figure 2 compares the standard deviation of equation image (i.e., SD[equation image] = equation image), obtained from the theoretical expression given by equation (15), with the SD[equation image] derived from data obtained through simulations discussed by Zhang et al. [2003] as a function of the normalized cross-correlation peak ρ12p). The standard deviation of the estimates of the peak of the cross-correlation function is obtained from 200 realizations for each set of Gaussian parameters (i.e., η, τp, τc). Figure 2 shows that the theory agrees well with simulation.

Figure 2.

The standard deviation of the decorrelation parameter equation image as a function of the peak correlation coefficient. The Gaussian parameters used in the simulations are τp = 0.008 s, τc = 0.0522 s, and the lag spacing is Tse = 0.002 s.

[20] Alternatively, var[equation image] can be calculated from the variance of the cross correlation coefficient estimates using easily derived formula var[equation image] = ρ12−2p)var[equation image12p)]. Also plotted in Figure 2 is the var[equation image] in which the approximate expression,

equation image

given by May [1988] is used with τ = τp. The number of independent samples is defined by May [1988] as MI(May) = Td0.5 whereas it should be, as defined in this paper and by Awe [1964], MI = Td/equation imageτc. The fading time τ0.5 is defined as the time lag at which the auto- correlation function equals 0.5. In contrast, using the procedure developed in this paper, var[equation image12(τ)] is,

equation image

If this form for var[equation image12p)] is used in the alternate expression for var[equation image], instead of May's approximate expression, we obtain exactly equation (15). Figure 2 shows significant differences between the exact theory derived herein and that obtained using May's [1988] formula and MI as defined herein.

3.1.2. Variance of the Correlation Time Estimates

[21] The correlation time estimate equation imagec, obtained from the auto-correlation function (equation (1)), is

equation image

A like expression for equation imagec can be obtained using equation image22(τ). Using equation (10) applied to the variables ∣C11(0)∣ and ∣C11(τ)∣, the following variance formula for equation imagec is obtained:

equation image

Following the procedure used by Zhang et al. [2003], it can be shown that:

equation image

and

equation image

Thus, substituting τ1 = 0, τ2 = τ in equation (19), and substituting this and equation (18) into equation (17), the following variance of the correlation time is

equation image

[22] Typically (e.g., weather radars) equation imagec is calculated from the correlation functions evaluated at zero and first lag (i.e., at τ = Tse). If Tse ≪ τc, it can be shown, using the relation equation imagec = λ/4πequation imagev, that equation (20a) can also be derived from the variance of the Doppler spectrum width estimate formula given in Doviak and Zrnić [1993, equation (6.30a)]. Under this condition, the standard deviation of equation imagec (i.e., SD[equation imagec]) is, from equation (20a), well approximated by

equation image

[23] On the other hand, simulations show that, if the first lag is used to estimate equation imagec, and Tse ≪ τc, a variance of the estimates larger than calculated from equation (20) will be obtained (Figure 3). The larger variance is due to estimates ∣equation image11(0)∣ being nearly equal to or smaller than the estimates ∣equation image11(Tse)∣. Under this condition a larger variance occurs because the inverse of the logarithmic function, in equation (16), can return exceptionally large values of equation imagec (i.e., outliers) for estimates ∣equation image11(0)∣ ≈ ∣equation image11(Tse)∣, or, if ∣equation image11(Tse)∣ is more than ∣equation image11(0)∣, the estimates equation imagec are imaginary. Melnikov and Doviak [2002] show, for real and simulated data, that imaginary values are also returned for spectrum width estimates; almost 50% of the estimates can be complex if normalized spectrum widths (i.e., 2Tseσv/λ) are less than about 10−2 corresponding to values of Tse less than about 0.06 τc.

Figure 3.

The standard deviation of the correlation time τc estimates versus the number of independent samples. The parameter for each of the four curves is the lag at which auto-correlation data are used in the estimation of τc. Tse = 0.002 s, and τc = 0.0522 s.

[24] If ∣equation image11(0)∣ = ∣equation image11(τ)∣, the singularity in equation (16) does not allow us to use first order theory; this is illustrated in Figure 3 (only real estimates of τc were used in estimating the standard deviation of equation imagec for presentation in this figure). As shown by Figure 3, τc can be better estimated from data at lags equal to or larger than about 0.2 τc. This is in agreement with results shown in Doviak and Zrnić [1993, Figure 6.6] for spectrum widths when they are expressed in terms of τc. However, if Tse > τcequation (20a) shows that the variance will begin to increase exponentially. Thus there is a narrow range of lags (i.e., ≈0.2 τc < τ ≈ τc) which produce a variance of equation imagec given approximately by equation (20b). Typically we do not have prior knowledge of τc from which we could determine the lag to minimize the variance of equation imagec. Nevertheless, we assume a lag can be found whereby the variance of equation imagec is in agreement with equation (20). If this assumption is accepted, we can use our input τc to determine the lag to use so that equation (20) is valid. Alternatively, the two lag Absolute Power Difference (APD) estimator of Melnikov and Doviak [2002] could be used in place of the Pulse Pair estimator assumed here. The APD estimator has significantly better performance if data from the first lag is used to estimate widths of narrow spectra (i.e., those for which σv ≪ λ/4πTse), or correlation times τc for which τcTse.

3.1.3. Variance of the Estimates of the Lag to the Cross-Correlation Peak

[25] Zhang et al. [2003] derive the following theoretical expression for the variance var[equation imagep] of the lag to the cross-correlation peak:

equation image

This theoretical expression has been verified with the variance of τp estimates obtained from simulated data [Zhang et al., 2003]. Note that var[equation imagep] and var[equation image] given by equations (15) and (21) increase as ρ12p)→0 (i.e., as η→∞), but var[equation imagec] is independent of ρ12p) if equation imagec is calculated from the auto-correlation function ∣equation image11(τ)∣ as assumed herein. Note that although equation imagec can also be calculated from the cross-correlation function, its variance would be larger and dependent on ρ12p).

3.2. Briggs' Full Correlation Analysis (G-FCA)

[26] Briggs et al.'s [1950] and Briggs' [1984] FCA and variations of his approach are the most commonly implemented methods for extracting cross-beam wind estimates from SA data. This method is generally a rather complicated procedure, but for horizontally isotropic scatterers the FCA yields the simple expression for baseline wind vox

equation image

for any correlation functional form, and where τx is the lag for which ∣C11(τ)∣ = ∣C12(0)∣. If ∣C11(τ)∣ and ∣C12(τ)∣ are Gaussian in shape, it can be shown from equations (2) that τx2 = 2ητc2 + τp2. Under the stipulated condition, vox(FCA) can be expressed in terms of the Gaussian parameters as

equation image

With this approach, vox is calculated directly from estimates of the three Gaussian parameters τp, τc, η which uniquely and completely specify the correlation functions. Applying equation (10) to the three variables τp, τc, η, and substituting ΔxFp, τc, η)/2 in place of η in that equation, leads to

equation image

in which we have expressed the covariance factors in terms of the correlation coefficients ρ(equation imagep,equation imagec), ρ(equation imagep,equation image), ρ(equation imagec,equation image) and the variances in the estimates of the three Gaussian parameters. Thus the variance in estimating baseline wind vox depends on the variances in all three parameters used to calculate equation imageox and, as well, the covariance between the Gaussian parameter estimates. But, because τp, τc, and η independently and uniquely determine C12(τ), it seems reasonable that the estimates equation imagep, equation imagec, and equation image would also be independent (i.e., the covariance terms would be zero).

[27] To test this hypothesis, data produced from the simulation method described by May [1988] were used to estimate τp, τc, and η using the estimators described herein, while maintaining condition (9). Each parameter estimate was regressed against another, and the correlation coefficients were calculated as a function of ρ12p) for τp = 0.008 s, and τc = 0.0522 & 0.0261s (Figure 4). Other values of τp and τc produce very similar results.

Figure 4.

The correlation among the Gaussian parameter estimates as a function of the peak correlation coefficient with Tse = 0.002 s, τc = 0.0522 (+) & 0.0261s (o), and τp = 0.008 s. (a) ρ(equation imagep, equation imagec), (b) ρ(equation imagep, equation image), and (c) ρ(equation imagec, equation image).

[28] As can be seen from Figure 4, ρ(equation imagep, equation image), and ρ(equation imagep, equation imagec) are nearly zero as hypothesized. Thus the corresponding covariance terms in equation (24) can be set to zero. On the other hand, ρ(equation imagep, equation image) is less than zero by a significant amount, and increases to a maximum negative value of about −0.4 as ρ12p) approaches 1.0. This small, but significant, value of the cross correlation between equation image and equation imagec is caused by the fact that both parameters equation image and equation imagec depend on the estimate ∣equation image11(0)∣ whereas equation imagep does not. Nevertheless, it can be shown that the contributions of the covariance term cov[equation imagec, equation image], to the baseline velocity variance var[equation imageox], is negligible. Thus equation (24) reduces to

equation image

Substituting equations (21), (15), and (20b) into equation (25), a theoretical expression for the baseline wind precision is obtained. Figure 5 shows the normalized standard deviation of the baseline wind estimates, MI1/2SD[equation imageox(G-FCA)]/vox, (i.e., equation (25)) as a function of the peak cross-correlation coefficient ρ12p). It can be shown, for most practical separations of UHF receiving antennas (i.e., ΔxD/2; and βh = ah), that τpc < 1.0. Thus we have plotted in Figure 5 the performance of the G-FCA estimator using τpc ≤ 1.0 as a parameter. Furthermore, for each curve, ρ12p) depends only on the ratio vs/vox. Thus ρ12p) → 1 only if turbulence and the cross-baseline wind component are sufficiently small. In section 5 the dependence of wind estimate precision is presented as a function of the strength of wind and turbulence, but for now, to compare the relative theoretical performance of various estimators, it suffices to show wind precision as a function of ρ12p) and τpc. There is no need to compare results in Figure 5 with simulations because it has already been shown that simulations and theory agree for the variances of the Gaussian parameters (i.e., τp, τc, and η) that comprise equation (25).

Figure 5.

The relative standard deviation of the baseline wind estimated using Briggs' FCA method plotted as a function of ρ12p) for various values of the lag τp.

[29] Recently, Kawano et al. [2002] simulated SA wind measurements and showed the relative performance of FCA and INT (called PCA, Partial-Correlation-Analysis by Kawano et al.) estimators. At this point it would be of interest to also compare the standard errors obtained from their simulations with that obtained from the theory developed herein. Taking the MU radar parameters used by Kawano et al. [2002] to generate the theoretical standard error of baseline wind estimates for vox = 30 m s−1, we computed τc, τp, η, and MI. By substituting these computed values of the Gaussian parameters into equations (15), (21), and (23) to calculate the variances of equation image, equation imagec, and equation imagep, and substituting these variances into equation (25) using Kawano et al.'s 100 s observation time, the standard deviations of the baseline wind is computed to be 2.03, 0.94, and 0.47 m s−1 for T0.5 = 1.0, 2.0, and 4.0 s, whereas Kawano et al.'s simulations produce standard deviations equal to about 2.4, 0.94, and 0.47 m s−1 respectively. T0.5 is a correlation time measured in the coordinate system moving with the velocity 2vox [Kawano et al., 2002].

[30] The near agreement of theory, based on estimates made from data at one and two lags, with simulations, based on multiple lag fitting, confirms that using more data about the peak of the correlation functions provides little or no improvement as is also noted by May et al. [1989] for the condition of large SNR. In fact, the standard deviations derived from the simulations appear to be equal to or a bit worse than those obtained from the theory developed here. The reason why multiple lag data does not improve the estimates is due to the fact that deviations about the peak of the correlation function are highly correlated, and thus significant new information is not obtained by fitting data at other lags near the peaks of the correlation functions.

3.3. Intersection Method (G-INT)

[31] Holloway et al. [1997] show that the lag τi, to the intersection of the auto- and cross-correlation functions, is related to vox as

equation image

A similar relation between the intersection point and the diffraction pattern speed (i.e., vdiff = Δx/2τi) was identified more than 50 years ago by Briggs et al. [1950]. Under conditions of uniform flow, matched transmitting and receiving antennas, etc., Doviak et al. [1996] show that the diffraction pattern only gives the appearance that it is uniformly translating at twice the speed that wind is advecting the scatterers (i.e., vdiff = 2vox). Woodman [1995] proved that equation (26) is valid independent of the functional form of the correlations. Although τi can be directly estimated without assuming a Gaussian model for the correlation functions, both C11 and C12 are required. Thus the Intersection method is also a FCA method. Because vox can be estimated directly from the single parameter τi, there is no need to calculate vs nor βh. Equation (26) is appealing because the value of τi does not depend on σt. On the other hand as turbulence increases, the magnitude of the correlation functions at the intersection decreases to small values; this is troublesome if SNR is small. Expressed in terms of the Gaussian parameters,

equation image

which, if substituted into equation (26) causes it to become identical to equation (23). Thus, the var[equation imageox(G-INT)] is equivalent to the var[equation imageox(G-FCA)] given by equation (25), and thus there is no further need to distinguish these two estimators.

[32] This equivalence in variance is consistent with the simulation results of Kawano et al. [2002]. Kawano et al. show that, for a receiving antenna separation about half the transmitting antenna diameter, the variance of baseline winds estimated using Briggs' FCA estimator is nearly equivalent to the variance of the baseline wind estimated using the Intersection method (i.e., the PCA method). According to the theory presented here, there should be no difference in the two estimators and, as well, the equivalence in performance should be independent of the separation of the two receiving antennas. Differences in the performance of the two estimators, observed by Kawano et al. for other antenna separations, is likely caused by the fact that Briggs' FCA winds are estimated fitting a Gaussian function to simulated data, whereas the time lag τi to the intersection of the auto- and cross-correlation functions is obtained directly by fitting lines connecting the lags either side of the intersection point. That is, Kawano et al. [2002] used the Direct Intersection method (D-INT), described in section 4.1 of this paper, to calculate τi. There can be, as shown in this paper, significant differences in the performances of the G-FCA and the D-INT methods.

3.4. Slope-at-Zero-Lag Method (G-SZL)

[33] Lataitis et al. [1995] present an alternative method for extracting winds from SA signals. They show that the estimate equation imageox is proportional to the slope of the cross-correlation magnitude at zero lag, normalized by the cross-correlation magnitude at zero lag. That is,

equation image

where dτ denotes a derivative with respect to τ. This expression does not make any assumption about the time-lag dependence of the correlation functions, and thus we can directly estimate the slope-at-zero-lag; this is the basis of the Direct-SZL algorithm presented in section 4.2. But to make (28) into an equation, we need to determine the proportionality factor. If the correlation function has a Gaussian form given by (1), the proportionality factor can be determined, and equation (28) can be written as

equation image

Thus the proportionality term needed in equation (28) is a function of the diffraction pattern scale which, in general, is unknown. If βh is evaluated using equations (2), both auto- and cross-correlation functions would be required, and thus equation (29) would become identical to equation (23). In this case, G-SZL would also be an FCA estimator. However, because much of the focus of this paper is on short wavelength (i.e., λ < 1 m) wind profilers, βh = ah (Section 2.3) which is a known or calculable constant. Under this condition, and only under this condition, the SZL method requires only cross-correlation data, and agrees exactly with that derived by Lataitis et al. [1995]. The substitution of ah for βh in the left side of equation (29) forms the basis of the D-SZL method developed in Appendix B. Thus the D-SZL method assumes ρBh ≪ 0.2 D (section 2.2).

[34] Because the estimates of τc and τp are uncorrelated (Figure 4a), the relative error in equation imageox(G-SZL) is

equation image

The error variance computed using equation (30), in which the variances are based on estimation of the parameters τp, τc, is labeled G-SZL. A significant difference between the SZL approach and Briggs' FCA method is that the SZL approach does not require estimation of η, and thus there is no need to use ∣C11(τ)∣ data.

[35] It is of practical interest to show the relative performance of the estimators as a function of wind and turbulence. This will be done in section 5 where we show, for the MAPR profiler, the relative performance as a function of the meteorological variables, vox and σt, as well as the separation Δx of the receiving antennas. But for now, we compare in Figure 6 the relative performances as a function of ρ12p). This latter comparison has broader application.

Figure 6.

Comparison of the normalized relative standard deviations of the baseline wind estimated using different algorithms. (a) τp = τc, and (b) τp = τc/2.

[36] Figure 6 shows the theoretical normalized standard deviation of the baseline wind as a function of ρ12p) for two values of τp, and compares the G-SZL estimator's performance with that for three other estimators (to be presented in section 4), and as well the performance of the G-FCA estimator (i.e., equation (25)). As can be seen from Figure 6, the G-FCA estimator performs, for the most part, better than the G-SZL estimator. Although the G-FCA has the additional variance term, var[equation image], the contribution from var[equation imagep] is usually more important, but its effectiveness is reduced by the factor (1 − τp2/2ητc2)/(1 + τp2/2ητc2). As τpc → 0, this factor approaches 1, and thus is relatively unimportant. On the other hand, var[equation imagep] increases without bound as τpc → 0, and thus the added variance var[equation image] is relatively unimportant and the two estimators (G-FCA and G-SZL) have nearly equal performance as can be seen in Figure 6b.

4. Expressing Precision of SA Wind Estimated Through Direct Methods

[37] Direct methods of estimating τi, τx, τp, and the slope of ∣C12(τ)∣ at zero lag have wider application because they are not restricted by the assumption of a Gaussian shaped correlation function. In this section we develop formulas to compute the variance of SA wind estimates when three direct methods are used to calculate the pertinent time lags or slope.

4.1. Direct Intersection Approach (D-INT)

[38] The intersection algorithm can be applied directly to the measured auto- and cross-correlation data without assuming a functional form for the correlation functions. An expression for the variance of baseline wind, calculated from the directly estimated lag to the intersection of the auto- and cross-correlation functions, is derived in Appendix A. If correlation functions are Gaussian shaped, this formula reduces to

equation image

The normalized S.D.[equation imageox(D-INT)], computed from equation (31), is plotted in Figure 6 as a function of ρ12p), for selected values of τp, τc. Although ρ12p) does not explicitly appear in equation (31), it is a function of η which, in turn, is a function of τp, τc through equation (27). The parameters τi, τp, τc can be expressed in terms of the wind components, vox, voy, turbulence σt, and the spatial scale ξpat of the diffraction pattern. Thus, in section 5, the variances of this and other estimators are plotted as functions of these meteorological variables, as well as the MAPR radar parameters (i.e., receiver antenna spacing). For wider application, but with less meteorological information, it is best to plot the relative performances of the estimators as we have in Figure 6.

[39] From Figure 6 it is obvious that the D-INT estimator performs worse that the G-INT (i.e., equivalent to G-FCA) estimator. The D-INT estimates significantly worsen as τp becomes smaller relative to τc. This is due to the fact that as τp decreases, τi increases (equation (27)) and the slopes of the correlation functions are smaller at larger τi. Smaller slopes create larger displacements or error in equation imagei for a given uncertainty in the estimates of the correlation function magnitudes. On the other hand, if τp ≈ τc the D-INT method has a performance that matches that of the G-FCA method for a wide range of ρ12p), and it outperforms the other direct estimation methods to be discussed in the following sections. But to achieve the condition τp ≈ τc requires a baseline wind component vox that exceeds the decorrelating wind component vs. Such a strong wind results in a “uniform flow” condition.

4.2. Direct Slope-at-Zero-Lag Method (D-SZL)

[40] The SZL technique can also be applied directly to the data without assuming a functional form of the correlation functions. The formula that gives the precision of baseline wind estimates for the direct SZL method is derived in Appendix B. If correlation functions are Gaussian shaped, this formula reduces to

equation image

This result agrees exactly with the CCR method of Zhang et al. [2003] under the condition that the ±lags, at which the ratio of cross correlation functions are computed, are small compared to τc. Although the D-SZL is restricted to the condition Tse ≪ τp (Appendix B), the CCR method is not. The square root of equation (32) is also plotted in Figure 6 to compare its performance with other estimators. It is seen that the precision of the D-SZL estimator has larger variance than the G-SZL estimator, as might be expected, because knowing the functional form of the cross-correlation function is added information that should reduce the variance of the estimates. But in contrast to the D-INT estimator, the D-SZL estimates improve as τp becomes smaller relative to τc. Of course we cannot allow τp to become too small, otherwise we would violate the condition Tse ≪ τp. The D-INT estimator performs better if τp ≈ τc, whereas the D-SZL estimator performs better than the D-INT estimator if τp < τc. In section 5, the performance of the D-SZL estimator will be evaluated as a function of wind, turbulence and radar parameters (i.e., receiver spacing Δx), and results compared with the precision of other estimators.

4.3. Direct Full Correlation Analysis (D-FCA)

[41] In this case the baseline wind vox is estimated from equation (22) in which τp and τx are directly estimated without assumption of the form of the correlation functions. Using equation (10) applied to equation (22), the following formula for the variance of the baseline wind is found:

equation image

The variance terms are evaluated in Appendices C and D using cross-correlation data at three data points about the peak to estimate τp, and at two auto-correlation data points which bracket the value of ∣C11(τ)∣ = ∣C12(0)∣ to estimate τx. If correlation functions are Gaussian shaped, the formula for var[equation imagex], derived in Appendix D, is

equation image

and the formula for var[equation imagep] is, as shown in Appendix C, given by equation (21).

[42] Besides equations (21) and (34) which give the variances of equation imagep and equation imagex needed in equation (33), we also need cov[equation imagep, equation imagex]. This covariance term is proportional to the coefficient of correlation between equation imagep, equation imagex which is evaluated from simulations and plotted in Figure 7 as a function of the peak cross-correlation coefficient ρ12p). We see that it is a small positive value of about 0.1 or less for ρ12p) in the range of interest. Nevertheless, what is of importance is the contribution from the covariance term relative to the two variance terms in equation (33). Because the covariance is proportional to the square root of the product of the variances of the two variables times the coefficient of correlation ρ12(equation imagep, equation imagex), it is seen from an examination of equation (33) that because ρ12(equation imagep, equation imagex) ≤ 0.1 we can safely ignore the covariance term. This conclusion was also verified by simulations, which showed that the covariance term reduces the standard error of baseline wind estimates, but the decrease is negligibly small. Thus, using ρ12(0) = ρ12p)exp{−τp2/2τc2}, equation (33) can be expressed as

equation image

By substituting τx2 = −2τc2 ln{ρ12p)} + τp2, this equation can be expressed entirely in terms of ρ12p), τp, τc. The normalized standard error of baseline winds, estimated with the D-FCA algorithm as a function of ρ12p), is calculated using equation (35), and the results are plotted in Figure 6 along with the performance of the other estimators for comparison. It is seen that the D-FCA estimator performs a bit worse than the G-FCA and G-SZL estimators for τp ≲ τc/2, but it is a poorer performer, for most values of ρ12p), than the D-INT method when τp ≈ τc.

Figure 7.

The coefficient of correlation between estimates of τp and τx, as a function of the peak cross-correlation coefficient.

5. Wind Measurement Precision Dependence on Wind, Turbulence, and Radar Parameters

[43] In this section we examine the dependence of measurement precision as a function of the meteorological parameters (σt, vox) and receiving antenna separation (i.e., Δx).

[44] Our analysis centers on NCAR's Multiple Antenna Profiling Radar (MAPR) [Cohn et al., 1997, 2001], which is a modified version of the commercially available Radian LAP-3000 915 MHZ boundary-layer profiler [Ecklund et al., 1990]. The radar is transportable and is designed as a tool for studying boundary layer dynamics. The transmitting antenna is a four-panel square array, each panel containing a 4 × 4 array of radiating patch elements, 0.7 λ apart. Thus, the transmitting antenna has a size D = 8 × 0.7 λ = 1.84 m on a side. Each of the antenna panels is used to pass back-scattered energy to four receivers. I and Q time series samples in each of the receiving channels are coherently integrated over the time interval Tse and recorded. Thus, the recorded data can be used to extract wind component estimates along any of the six baselines.

[45] The transmitted beam width σθT(1) = 3.76° was computed from a theoretical formula for phased arrays, and by fitting the theoretical pattern with a Gaussian function; the theoretical pattern agreed well with the measured pattern [Cohn et al., 1997]. Thus, γ = 0.37. In terms of the conventional one-way 3 dB beam width θ1, θ1 = 2.35 σθT(1) = 8.83°. The theoretical formulas for the receiver beam width gave σθR(1) = 8.01°, and thus we calculate from equation (5) that α = 1.28. Because this radar operates at a wavelength λ = 0.328 m, we can assume that the scatter is from the equilibrium subrange of the Kolmogorov spectrum of refractive index irregularities, and thus from equation (7) we calculate that ah = 1.62 m−1, in close agreement with the mean value calculated from correlation data [Cohn et al., 2001].

5.1. Precision as a Function of Receiver Antenna Spacing Δx

[46] The wind estimate precision as a function of receiver antenna separation Δx is shown in Figure 8a. We used vox = 10 m s−1, σt = 1 m s−1, Td = 1 s, and voy = 0 in the calculations. To obtain estimate standard error for other dwell times Td, simply divide the ordinate by the square root of Td. If Δx < D/2, it is seen from Figure 8 that the SD[equation imageox] is significantly lower for vox calculated using the G-FCA or G-SZL algorithms than if calculated using the direct estimation algorithms. This result occurs because the Gaussian estimators use the added information of the correlations' functional forms, and use lag data from regions where the correlation coefficients are largest (this gives a small variance of the Gaussian parameter estimates). Although the D-SZL algorithm precision also appears to asymptotically approach a finite value as Δx → 0, it is recalled that in this limit the condition Tse ≪ τp is violated. From equation (2) this latter condition imposes the following condition on Δx:

equation image

Thus, it is likely that the precision of the D-SZL algorithm will worsen as Δx → 0, as do the D-INT and D-FCA estimators. For the parameter values used in constructing Figure 8, the condition on Δx yields Δx ≫ 76 Tse. If Tse = 2 ms, as used in the simulations, then Δx must be much larger than 0.15 m. Limiting the D-SZL data to these large values shows the D-SZL estimate precision is worse, for all values Δx, than any of the estimators which assume a Gaussian functional form for the correlation functions.

Figure 8.

(a) Comparison of standard deviations of baseline wind estimates, obtained with various algorithms, versus Δx, the separation of the receiving antennas. MAPR parameters, D = 1.84 m, ah = 1.62 m−1, λ = 0.328 m, dwell time Td = 1 s, and meteorological parameters, vox = 10.0 m s−1, σt = 1 m s−1, voy = 0. (b) Comparison of theory presented herein for the G-FCA method (continuous curves) applied to the MU radar (λ ≈ 6 m; D ≈ 100 m, Td = 175.1 s), and simulation results from Kawano et al. [2002] (points). Meteorological parameters, vox = 30 m s−1, ξ0.5 = 50 m, and T0.5 is a parameter.

[47] All of the methods produce less precise wind estimates as Δx increases, in particular, as Δx becomes larger than D/2 (for the MAPR parameters this is about 1 m). This increase is a direct consequence of reduced cross-correlation levels for larger receiver separations. The reduced cross-correlation levels yield larger relative uncertainties in the cross-correlation estimates (i.e., equation (9)), which leads to a lower wind estimate precision.

[48] Baseline wind estimate standard deviation versus Δx is presented in Figure 8b for a VHF radar wind profiler. This figure compares G-FCA theory with the simulation results of Kawano et al. [2002]. In constructing Figure 8b, we have taken the lower of the two standard deviations given by Kawano et al.'s Figure 4 for the two estimators (i.e., FCA and PCA). This was done because artifacts in the simulation might cause the standard deviation of wind estimates derived from simulation to be larger than expected. For example, for small antenna spacing, Kawano et al. [2002] shows that the PCA (i.e., the D-INT) method outperforms the G-FCA method, whereas theory (Figure 8a) shows that the D-INT performs worse than the G-FCA estimator. This is the reason why we selected the smaller of the two standard deviation values to be compared with the theoretical results for the G-FCA estimator. The data presented in Figure 8b assumes vox = 30 m s−1, Td = 175.1 s, and ξ0.5 = 50.0 m. ξ0.5 is the spatial lag at which the cross-correlation function at zero lag is reduced to half of its maximum value. This parameter and T0.5 determine the correlation time τc. In the region ΔxD there is good agreement for T0.5 = 2 s, but for T0.5 = 1.0 s theory shows slightly larger errors, whereas for T0.5 = 4 s, theory shows slightly smaller errors.

5.2. Precision as a Function of Turbulence σt and the Decorrelating Wind Component vs

[49] The standard error of the baseline wind estimates equation imageox as a function of σt for the MAPR is presented in Figure 9. As in section 5.1, voy = 0, vox = 10 m s−1, and Td = 1 s, but Δx = 0.92 m is used in the calculations. Although we plot estimate precision as a function of turbulence, it should be noted that the decorrelating wind component vs, a combination of turbulence and cross-baseline wind (equation (3)), is the key parameter. That is, the curves in Figure 9 can be applied equally to vs by multiplying the abscissa values of σt by equation imagekoh = 16.7. Thus, Figure 9 can be applied to combinations of cross-baseline wind and turbulence that give the same vs.

Figure 9.

The standard deviation of baseline wind, estimated with various algorithms, as a function of turbulence σt, or decorrelating wind component vs, for MAPR and meteorological parameters the same as in Figure 8a caption, but Δx = 0.92 m, and voy = 0 for the σt scale.

[50] The most noticeable difference between the performances of the G-FCA and G-SZL algorithms is the trend toward perfect precision (i.e., SD(equation imageox) → 0) of the G-FCA algorithms as σt → 0; this is the uniform-flow limit. Because we have chosen the wind vector to lie along the receiver-pair baseline, vs = 0 if σt = 0. In this case η = 0, and although var[equation image]/η has a zero over zero value, it can be shown that the ratio approaches a limit value of 2/MI. Thus, because the leading multiplication factor in equation (25) vanishes if η = 0, the SD(equation imageox) also vanishes in this limit.

[51] The physical reason why var[equation image] → 0 if uniform-flow is along the baseline, is the signal fluctuations in the receivers are exactly the same but separated in time by τp. Thus the estimates of the cross-correlation are exactly the same as the auto-correlation but displaced by τp (i.e., η = 0 without variance).

[52] The SD(equation imageox) for the D-SZL algorithm is larger than those obtained with the G-SZL algorithm because the D-SZL algorithm uses smaller values of ∣C12(τ)∣ about zero lag, and thus the errors are larger as can be seen from equation (32), whereas the G-SZL algorithm uses larger values of ∣C12(τ)∣ about its peak to estimate τp needed in equation (29). As can be seen from Figure 9, the precision associated with G-SZL and D-SZL wind measurements are relatively independent of σt. Although the expected values of the slope and magnitude of the cross-correlation at zero lag are independent from σt [Zhang et al., 2003, Figure 2c], the estimates of vox obtained from the D-SZL method are affected by turbulence because the estimates of ∣C12(τ)∣ about zero lag are not be perfectly correlated. Thus the corresponding wind estimates, computed from estimates of ∣C12(τ)∣ about zero lag, has a weak dependence on σt.

[53] For the direct application of the intersection approach (i.e., D-INT), we approximate the correlation functions near τi by linearly interpolating between two measured values of the correlation functions, one on either side of τi (see Appendix A). Typically Tse ≪ τc and therefore the uncertainties in the amplitudes for the four required measurements are highly correlated. As a result, the intersection point, although moving up and down (i.e., the estimates of ∣C11i)∣ and ∣C12i)∣ do fluctuate), does not move horizontally (i.e., there is no change in τi). Thus there is corresponding little variance in the wind estimate (i.e., the wind estimate uncertainty for the D-INT method tends to zero) as vs → 0. Perfect precision cannot be obtained in practice in the limit vs = 0 because ∣C11(τ)∣ and ∣C12(τ)∣ are not linear in the interval Tse about τi, and estimates of ∣C11(τ)∣ and ∣C12(τ)∣ about τi are not perfectly correlated.

[54] Next consider the behavior of the curves for larger values of the turbulence intensity, and/or cross-baseline wind. For turbulence intensities around 1 m s−1 and larger (or vs > 16 m s−1), the wind estimate uncertainty associated with four of these techniques (i.e., G-FCA, G-SZL, D-FCA, and D-SZL) increases slowly as σt increases. The exception is the D-INT method for which the associated wind estimate precision decreases dramatically as σt increases. For sufficiently large values of the turbulence intensity (i.e., σt > λvox/D), Equation (2) shows that the normalized cross-correlation amplitude exp(-η) ≈ exp[−(πγαΔx/D)2] = const., τc ∝ σt−1, and τp ∝ σt−2. Therefore, in this limit, the width of the correlation functions, and the delay to the peak of the cross-correlation function, both decrease with increasing σt. Because the time lag τi is proportional to vox−1 = constant (equation (26)), τi lies progressively farther into the skirts of ∣C11(τ)∣ and ∣C12(τ)∣. Thus, the auto- and cross-correlation amplitudes at τi are progressively smaller. Although the uncertainty in the correlation function estimates approaches a constant level according to equations (14b) and (18), the differences Δ1, Δ2 (Appendix A) in the cross- and auto-correlation function at lags τ1, τ2 decrease and, as can be seen from equation (A8), baseline wind variance increases as the inverse square of Δ2 − Δ1, in agreement with what is seen in Figure 9.

5.3. Precision as a Function of Baseline Wind Speed vox

[55] The wind-estimate precision as a function of baseline wind speed vox, for voy = 0, and σt = 1 m s−1, is shown in Figure 10. The precision associated with the G-FCA method using estimated Gaussian parameters varies relatively little with vox, with a precision of about 2 m s−1 over the range vox = 0→50 m s−1. On the other hand, the wind-estimate uncertainty associated with the D-INT method increases dramatically as vox decreases to zero. In this limit, vsvox, the cross-correlation amplitude exp(−η) → const., τpvox → 0, τc = const., and τi → ∞. Because the correlation levels at τi are now very small, the precision of the corresponding wind estimate decreases as explained in section 5.2. With the D-SZL method, SD[equation imageox] increases linearly with vox if voxvs (i.e., the uniform flow condition). Under this condition, η→0 (i.e., ρ12p)→1), and because both τp and τc are inversely proportional to vox, the ratio τpc is a constant. Thus, as can be seen from equation (32), the right side is a constant, and therefore SD[equation imageox(D-SZL)] is proportional to vox as seen in Figure 10.

Figure 10.

The standard deviation of baseline wind, estimated with various algorithms, as a function of baseline wind for MAPR and meteorological parameters the same as in Figure 8a caption, but Δx = 0.92 m.

6. Summary and Conclusions

[56] Three estimators of baseline wind (i.e., the wind component parallel to the baseline of a pair of receiving antennas) are evaluated: (1) the Briggs' FCA, (2) the Intersection, and (3) the slope-at-zero lag methods. For each of these methods, two implementations are examined. In the first one, a Gaussian functional form is assumed for the correlation functions, whereas in the second no such assumption is made. This latter implementation is referred to as a direct one, wherein the required time lags (e.g., τp, τx for the FCA estimator) are directly calculated from the correlation data without assuming a form for the correlation functions.

[57] If auto- and cross-correlation data are required to estimate baseline wind, the estimator is a FCA one. If the diffraction pattern scale is known, then the slope-at-zero lag (SZL) estimator requires only the cross-correlation data and thus is not classified as a FCA estimator. Because both the auto- and cross-correlation data are required for the Intersection method, it is a FCA estimator. If the correlation data can be represented by Gaussian functions, it is found that the Briggs' FCA and the Intersection methods give identical variance for baseline wind estimates, but the SZL method gives estimates with slightly higher variance. Therefore the Intersection and Briggs' FCA methods are labeled G-FCA, whereas the SZL method is labeled G-SZL. The finding that the Intersection and Briggs' FCA estimators produce identical baseline wind estimate variance is supported by the simulations of Kawano et al. [2002].

[58] If the correlation function shapes are known to be Gaussian, Briggs' FCA method (i.e., G-FCA) is the best choice (see Figures 8 to 10). The G-SZL has a performance nearly equal to that of that obtained with the G-FCA estimator, but the diffraction pattern scale must be known; if it has to be calculated from the data, then the SZL estimator also becomes a FCA estimator, and its performance is identical to that of the G-FCA. The theoretical performance of the G-FCA estimator is also compared with simulations made by Kawano et al. [2002], and shows agreement with their results (Section 3.2 and Figure 8b). Although many of our conclusions are based on application of the theory to NCAR's short wavelength (i.e., λ = 33 cm) atmospheric boundary layer wind profiler (i.e., the MAPR), comparison (i.e., Figure 8b) with the simulations of Kawano et al. [2002] show that the theory is also applicable to longer wavelength radars measuring winds in the upper atmosphere.

[59] If the correlation function is not known, a direct method must be used. In this case all three estimators have different baseline wind estimate variance. For the direct Briggs' FCA (i.e., D-FCA), τp and τx are directly calculated, and the variance and covariance formulas for τp, τx are derived in Appendices C and D, respectively. To obtain numerical results that can be compared with those from simulations, a Gaussian functional form is ultimately assumed for the correlation data. Under this condition, it is shown that the general formulas for the variances of τp, τx reduce to that given by Zhang et al. [2003]. These reduced formulas are then evaluated for baseline wind estimate variance to compare with those methods whereby the Gaussian parameters τp, τc, and η, are estimated (i.e., the G-FCA, and G-SZL methods) to calculate needed time lags and the slope at zero lag. For the direct Intersection method (i.e., D-INT), the intersection of the auto- and cross-correlation function is calculated from data at lags either side of τi, the time lag to the intersection. A formula for the variance of τi is derived in Appendix A. Finally, the variance formula for baseline wind estimates derived from the direct-SLZ (D-SZL) method is derived using variance and covariance formulas for cross-correlation data at lags either side of zero.

[60] If a direct method must be used, the theory suggests that the D-FCA or the D-SZL should be chosen. But the choice between these two depends on receiver separation (Figure 8), the strength of the decorrelating wind component vs (a function of cross baseline wind and turbulence; Figure 9), and baseline wind (Figure 10).

[61] Although the theory is based on stochastic Bragg scatter for which the signal-to-noise ratio (SNR) is very large, and for a random field of refractive index perturbations that have horizontally isotropic correlation lengths [Doviak et al., 1996], the theory also can be applied to backscatter from precipitation. Extension of the theory to the more practical case of finite SNR is planned for the future. Furthermore, even though the theory is based on estimating Gaussian parameters from data at few lags around the peak of the correlation functions, the good agreement with the simulation results of Kawano et al. [2002], who fitted data from multiple lags to Gaussian functions, shows that the theory has broad application. The lack of significant improvement from adding data at multiple lags is attributed to the fact that data about the peaks of the correlation functions are strongly correlated; thus little additional information is obtained from including data from multiple lags.

Appendix A:: Wind Estimate Variance for the Direct Intersection Method

[62] A formula for the precision of baseline winds, calculated from a direct estimate of the lag to the intersection of the cross- and auto-correlation functions, is derived in this appendix. Using equation (26), the following expression for the variance of the lag equation imagei is

equation image

To derive an expression for equation imagei, it is noted that there are two estimates of ∣C11(τ)∣ and two of ∣C12(τ)∣ at consecutive lags τ1 and τ2 on either side of τi, (i.e., τ1 < τi < τ2, and τ2 − τ1 = Tse). Under the condition Tse ≪ τc, a straight line connecting the two measurements of each of the correlation functions is a good approximation to these functions about the lag τi. That is,

equation image

and

equation image

[63] An estimate of τi is obtained by equating the above two expressions which gives,

equation image

where,

equation image

is a finite difference estimate of the slope of the cross correlation function ∣C12(τ)∣ about τi. The slope of ∣C11(τ)∣ is estimated in a similar way. Likewise, τi is given by,

equation image

[64] Forming the difference equation imagei − τi, and setting ∣equation image121)∣ = ∣C121)∣ + ε121) and ∣equation image111)∣ = ∣C111)∣ + ε111), etc., the following equation is obtained,

equation image

where the deviations ε are assumed to be small compared to the mean values of the correlation functions, and Δ1 ≡ ∣C121)∣ − ∣C111)∣, and Δ2 ≡ ∣C122)∣ − ∣C112)∣. Awe [1964] shows that the deviations ε12(τ) and ε11(τ) have a correlation time τc. Thus the deviations of, for example, ∣equation image11(τ)∣ from its mean at τ1, τ2 are highly correlated because Tse ≪ τc. It then follows that ε121) ≈ ε122) ≈ ε12i), and ε111) ≈ ε112) ≈ ε11i). Thus, equation (A5) reduces to

equation image

The variance of equation imagei is then,

equation image

Substituting equation (A7) into equation (A1), and noting, from equations (14b) and (18), that at τ = τi var[∣equation image12i)∣] = var[∣equation image11i)∣], the variance of vox(D−INT) is

equation image

The expression for var[∣equation image11i)∣] is given by equation (18), and that for cov[∣equation image12i)∣, ∣equation image11i)∣] is found in Zhang et al. [2003] and is

equation image

where

equation image

To obtain closed form solutions, a functional form for the correlations must be substituted into equations (A9). For example, assuming a Gaussian form (i.e., equation (1)), and performing the integrations, it can be shown that

equation image

Substituting these results into equation (A8), the following solution for var[equation imageox(INT)] is obtained:

equation image

Noting that (Δ2 − Δ1)/Tse is the finite difference approximation of the differences in slopes of ∣C12(τ)∣ and ∣C11(τ)∣ at τ = τi, it can be shown, using equation (1), that

equation image

Therefore, using equations (26) and (A12), equation (A11) can be written as,

equation image

Appendix B:: Wind Estimate Variance for the Direct Slope-at-Zero-Lag Method

[65] Assume we have cross-correlation estimates at plus and minus the first lag (i.e., at ±τ1 = ±Tse). The corresponding finite difference velocity estimate is (section 3.4),

equation image

where K = (ah2Δx)−1. Note that equation (B1) requires that τ1 ≪ τp. Although we could estimate the slope of ∣C12(τ)∣ from its value at zero and another value at +τ1 or −τ1, external noise correlated in each of the receivers must be subtracted from the measured value ∣equation image12(0)∣; using ±τ1 eliminates this noise. The expected value of the velocity estimate is

equation image

[66] Setting ∣equation image12(−τ1)∣ = ∣C12(−τ1)∣ + ε−1 and ∣equation image121)∣ = ∣C121)∣ + ε1, and assuming that we process a sufficiently large number of samples so that the errors are small compared to the expected values (i.e., ε−1 ≪ ∣C12(−τ1)∣, and ε1 ≪ ∣C121)∣), we have, to first order in ε−1 and ε1,

equation image

which yields

equation image

where,

equation image

and,

equation image

The variances of equation image12 (±τ1) are given by equation (14b), and from Zhang et al. [2003], the following formula for the covariance is

equation image

where

equation image

To obtain an analytical solution, functional forms for the correlation functions need to be known. Again, for sake of illustration, assume that the correlation functions are given by equation (1), and recall that Td = MIτc√π. It can be shown from Zhang et al. [2003], that

equation image

Substituting equations (B6), (14b), and (1) into equation (B4a), the variance of the baseline wind calculated with the direct slope of zero lag estimator is

equation image

Because the finite difference approximation for the slope of ∣C12(τ)∣ at τ = 0 requires τ1 ≪ τc, equation (B7) simplifies to

equation image

Appendix C:: Variance of the Directly Estimated τp

[67] Assume that the magnitude of the cross-correlation function is described by a quadratic function in the region close to the lag at which it has its peak. A parabolic function can be written in the form,

equation image

where “a” is the width of the parabola across its focal point, and Cp is the peak value of the cross-correlation function at the lag τp. Both a and Cp can be eliminated by applying equation (C1) to the three contiguous data points having, for example, the largest summed value. Solving the three equations simultaneously, τp is given by

equation image

where, to shorten the notation, we have defined y1 ≡ ∣C121)∣, with similar definitions for y2 and y3, and

equation image

From data τp is determined from the three estimates equation image1, equation image2, equation image3 of the cross-correlation function at lags τ1, τ2, τ3 spaced Tse apart. Assuming that MI is sufficiently large so that equation image1, equation image2, equation image3 are concentrated about their expected values, the first order theory (i.e., equation (10)) can be used to express the variance of equation imagep in terms of the variances and covariances of the cross-correlation estimates equation image1, equation image2, equation image3. Thus we obtain

equation image

[68] We can obtain some simplification by substituting equations (C3) into equation (C4) to obtain

equation image

So that existing formulas can be used to evaluate equation (C5), the variance and covariance terms can be expressed in terms of the variances of each of the variables and the covariances between pairs of variables. Thus equation (C5) can be written as

equation image

Equation (C6) can be simplified by noting that cov[equation image2, equation image3] ≈ cov[equation image2, equation image1], and because Tse ≪ τc, var[equation image1] ≈ var[equation image2] ≈ var[equation image3]. Under these conditions, equation (C6) reduces to

equation image

By using equations given by Zhang et al. [2003], the variance and covariance terms can be evaluated for any functional form of the correlation functions. But to obtain analytical solutions to be compared with simulations, assume that the correlation functions are of a Gaussian form. Thus, using equation (14b) for the variance term, and equation (A14b) in Zhang et al. [2003] for the covariance terms, and noting that a = τc2/2Cp, we obtain equation (21) after noting that, for the stipulated conditions, ∣C121)∣ ≈ ∣C122)∣ ≈ ∣C12p)∣ ≈ Cp. Thus, under the stipulated conditions, equation (C7) reduces exactly to that derived by Zhang et al. [2003, equation (20)], whose derivation initially assumed a Gaussian form for the cross-correlation function and data at two contiguous lags about the estimated peak ∣equation image12p)∣, and is in accord with the simulation results of Zhang et al. [2003, Figure 7].

Appendix D:: Variance of the Directly Estimated τx

[69] The lag τx at which the auto-correlation is equal to the cross-correlation at zero lag is used in Briggs' FCA. By using the values of ∣equation image11(τ)∣ at two consecutive lags τ1, τ2 which bracket the value ∣equation image11(τ)∣ = ∣equation image12(0)∣, an estimate of τx can be obtained. Because Tse ≪ τc, fluctuations of ∣equation image11(τ)∣ spaced Tse apart are highly correlated. Thus the auto-correlation function between τ1, τ2 can be approximated by

equation image

Equating this to ∣equation image12(0)∣, and solving for τ = τx, we obtain

equation image

As before, simplified symbols “y” are used to represent the respective correlation functions. There is a like equation for the expected values of the variables. The variance of equation imagex can be found using equation (10) applied to the three estimates equation image1, equation image2, and equation image0. Thus the variance is given by

equation image

[70] Because Tse ≪ τc, it is seen from equations (14b) and (18) that the variances are nearly equal to var[equation image0]. Thus the variances terms can be combined, and equation (D3) reduces to

equation image

The variance term is directly obtained from equation (14b), and cov[equation image1, equation image0], cov[equation image2, equation image0] are given by Zhang et al. [2003], but the covariance term cov[equation image1, equation image2] needs to be developed for correlation functions of general form. Because we need to assume a functional form to obtain an analytical solution, let us use the known equations for the Gaussian shaped correlation functions. Thus cov[equation image1, equation image2] is given by equation (19), and from Zhang et al. [2003, equation (A15b)] we have

equation image

with a similar expression for cov[equation image2, equation image0]. When these expressions, along with those for yo, y1, and y2, are substituted into equation (D4), the following analytical form

equation image

is obtained after using the approximation τx ≈ τ1, τ2, and ρ12(0) = ρ12p)exp{−τp2/2τc2}. Equation (D6) is exactly that given by Zhang et al. [2003] whose derivation initially assumed a Gaussian form for the cross-correlation function and data at two lags (i.e., zero and τ1 = τx). Thus equation (D6) is supported by simulation results [i.e., Zhang et al., 2003, Figure 7].

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