Maximum Likelihood Estimates of Vortex Parameters from Simulated ...

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Maximum Likelihood Estimates of Vortex Parameters from Simulated Coherent Doppler Lidar Data ROD FREHLICH

AND

ROBERT SHARMAN

Research Application Program, National Center for Atmospheric Research,* Boulder, Colorado (Manuscript received 2 February 2004, in final form 6 August 2004) ABSTRACT The performance of pulsed coherent Doppler lidar in estimating aircraft trailing wake vortices by scanning across the aircraft flight track is evaluated using Monte Carlo lidar simulations of a simple vortex pair in both a nonturbulent and turbulent environment. The performance estimates are based on maximum likelihood estimates of aircraft wake vortex parameters and provide a measure of the ability of the lidar to detect and track wake vortices under the best possible conditions. Two aircraft types are considered: the Boeing 737 and the Boeing 747. Rigorous error analyses are produced by comparing the estimated parameters from numerical simulations of raw lidar data with the known input parameters of the simulation. It is shown that the probability density functions for the estimates are approximately Gaussian and the bias is very small. The main source of the bias was determined to be the movement of the vortex during the lidar scan. The estimation error is increased by the effects of a background turbulent velocity field. The trade-off between lidar pulse energy and pulse repetition frequency for the standard condition of constant laser power is also presented. It is shown that these maximum likelihood estimates provide accurate detection and tracking of the key vortex parameters for a simple vortex model, with and without background turbulence.

1. Introduction The production of aircraft trailing wake vortices is a direct consequence of the lift produced by airfoils. Reviews of the wake generation and evolution processes can be found in Widnall (1975), Spalart (1998), and Gerz et al. (2002). Here it is sufficient to state that the main concern at airports is an aircraft encounter with previously generated wake vortices, especially if produced from a heavy aircraft, during takeoff or landing. Although mandated aircraft separation requirements have been very effective in reducing hazardous wake vortex encounters, that fact also implies that these separations may be overly conservative and, therefore, are needlessly reducing airport capacity. To dynamically optimize separation distances, the ability to track the wake vortex locations and measure the vortex strength in real time is necessary. Because the spatial extent of the vortex is typically small, 5–50 m, high-resolution velocity measurements are essential to accurately re* The National Center for Atmospheric Research is sponsored by the National Science Foundation. Corresponding author address: Dr. Rod Frehlich, Research Application Program, National Center for Atmospheric Research, P.O. Box 3000, Boulder, CO 80307-3000. E-mail: [email protected]

© 2005 American Meteorological Society

JTECH1695

solve the vortex position and strength. From this perspective, an attractive instrument for wake vortex detection is the Doppler lidar. Earlier wake vortex detection strategies used a continuous wave (CW) focused Doppler lidar because of the high spatial resolution possible (Constant et al. 1994; Harris et al. 2002; Köpp et al. 2004b). However, the maximum detection range is limited to a few hundred meters (Köpp et al. 2004a). An alternative approach is to use a pulsed Doppler lidar. Initial results for measuring wake vortex position and circulation strength have been encouraging (e.g., Hannon and Thomson 1994, 1997; Constant et al. 1994, Köpp 1994; Hannon and Henderson 1995; Brockman et al. 1999; Darracq et al. 2000; Keane et al. 2002; Harris et al. 2002; Holzäpfel et al. 2003; Köpp et al. 2004a,b). One Doppler lidar scanning strategy already used in field trials is to point the lidar beam perpendicular to the flight path as shown in Fig. 1. Here profiles of Doppler lidar derived radial velocity (where radial means the velocity component in the direction of the lidar beam axis) measurements perpendicular to the vortex axes at various elevation angles can be used to identify and track wake vortices (Hannon and Thomson 1994; Brockman et al. 1999). Note that in this configuration the radial velocity measured by the lidar in the vicinity of the vortex is primarily due to the tangential component of the vortex pair. Similar analyses have been proposed using CW lidar (Harris et al. 2002;

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FIG. 1. Geometry of wake vortex tracking with a coherent Doppler lidar.

Holzäpfel et al. 2003). Another approach is based on the maximum spread of the signal spectrum in velocity space or “velocity envelopes” (Brockman et al. 1999; Köpp et al. 2004a). All of these techniques emphasize local measurements of the vortex features and must make certain assumptions to connect the information in the spectral estimates to the combined vortex–atmospheric velocity field. A rigorous quantitative analysis of the effect of these assumptions is difficult because the average value of the spectral estimates is a function of the lidar signal-to-noise ratio (SNR), the signal processing window function, and the lidar pulse sensing volume (or lidar focused beam volume for CW lidar) (Frehlich and Cornman 1999). Another approach to vortex detection and tracking is based on global information in the vicinity of the vortex pair. Experiments with the Coherent Technologies Inc. (CTI) 2-␮m coherent Doppler lidar (Henderson et al. 1991, 1993; Hannon and Henderson 1995) demonstrate a clear signature of the vortex can also be produced in multiple coherent Doppler lidar signal spectra when the range gate of the processing interval partially or even completely encompasses the vortex (Hannon and Thomson 1994, 1997; Hannon 2000). Producing a clear signature requires the accumulation of Doppler lidar signal spectra (average signal spectra for N multiple lidar shots) as the lidar scans the vortex region at a beam angle perpendicular to the vortex axis, as shown in Fig. 1. In the CTI technique, the vortex parameters are estimated from the multiple signal spectra using approximations to a maximum likelihood estimator (Hannon and Thomson 1997). Although the lidar detection technique seems to produce very robust results (Hannon and Thomson 1994, 1997; Constant et al. 1994; Köpp 1994; Hannon and Henderson 1995; Brockman et al. 1999; Holzäpfel 2003;

Köpp et al. 2004a,b), verifying the accuracy of the measurements is difficult in an operational setting since there are very few other instruments capable of measuring the vortex position and strength to the accuracies obtainable with lidars. Two independent lidar measurements of the same vortices have been attempted, but assumptions are required to determine the error characteristics of each individual measurement. For example, Köpp et al. (2004b) compared simultaneous CW and pulsed lidar measurements of wake vortices. Assuming the two measurements have equal random error over the observation period, they found a standard deviation of vortex altitude of 6 m, vortex distance of 9 m, and circulation strength of approximately 13 m2 s⫺1 for a spread of circulation strengths from 250 to 450 m2 s⫺1. However, none of the previous works cited above have produced quantitative error analyses of both bias and random error. Holzäpfel et al. (2003) did produce simulations of virtual CW lidar data ignoring the signal processing effects and additive detector noise. Darracq et al. (2000) produced simulated pulsed coherent lidar data for airborne detection of wake vortex pairs but did not extract bias and random error of vortex parameters. Another option for deriving lidar measurement performance statistics, and the strategy used in this paper, is to use many realizations of numerical simulations of pulsed Doppler lidar (Hannon and Thomson 1994; Salamitou et al. 1995; Frehlich 1997, 2000, 2001; Darracq et al. 2000) with a vortex pair of known characteristics (velocity distribution and position). In this Monte Carlo approach, rigorous estimates of lidar measurement error statistics (mean, variance, and bias) of vortex position and strength can be produced since the true vortex parameters are known. Although here only simple atmospheric environments are considered, the Monte Carlo technique is easily extended to more com-

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plex environments by using high-resolution large eddy simulations (LES) to provide more realistic vortex evolution models. In the scanning configuration of Fig. 1, wake vortex parameters can be determined by using a maximum likelihood (ML) algorithm assuming the accumulated signal spectra are statistically uncorrelated and assuming any realistic model for the vortex velocity field. However, the numerical effort of the ML estimator can be prohibitive if one estimates a large number of parameters. An approximation of this ML estimator has been proposed by Hannon and Thomson (1997) to improve the numerical speed, but it is unclear what effect these approximations have on the accuracy of the results. Here we investigate the accuracy of ML algorithms with minimal approximations. By so doing, we are able to derive error statistics of lidar-derived wake vortex parameter estimates for the optimal performance achievable by pulsed Doppler lidar under the best of circumstances. For this we consider a vortex pair with a simple analytic prescription created with aircraftspecific characteristics (such as initial altitude, position, and strength) just after rollup and merging (several seconds) and ignore the effects of atmospheric shear and stability, and proximity to the ground. The ML estimates could be derived for later stages when the vortex pair has been subjected to some amount of atmospheric distortion, provided the vortex model has identifiable free parameters, but these cases are not considered here. Instead, we concentrate on establishing a methodology for deriving error statistics for pulsed Doppler lidar performance and establishing baseline performances. To simplify the analysis of vortex detection, we assume that after wake vortex rollup and merging, the wake vortices are separated by roughly a wingspan (the linear distance between the wingtips), are at the same altitude (the height of the wingtip above the surface), and that each vortex in the pair has the same strength, also a function of the specific aircraft. Also, the signalto-noise ratio of the lidar signal can be assumed known since it can be accurately determined with spectral techniques. Under these assumptions, only three parameters (vortex pair height, horizontal distance to the centroid of the two vortices, and common circulation strength) need to be estimated for vortex pair detection (ML 3 par). However, if individual vortex tracking is desired, the location and strength of each vortex must be estimated and therefore six parameters (vortex height, range, and circulation strength for each individual vortex) must be estimated (ML 6 par) and more computational effort is required. In this case different altitudes and strengths of the two individual vortices are permitted. In this paper the performance of robust ML estimators of wake vortex parameters is determined for both detection mode and tracking mode by Monte Carlo simulations of raw lidar data for a simple vortex pair as

might be generated under typical operating conditions. The accumulated lidar signal spectra are simulated for various times during the vortex pair descent for specified lidar parameters and lidar scan geometry. An analytic model for the vortex is used to provide “truth”; that is, the location and strength of the vortex are known exactly. In addition, the effects of a background turbulent field are investigated by the addition of a homogeneous turbulent velocity field to the analytic vortex model field. Because the simulation domain is not very large, we assume that the aerosol backscatter is uniform. The effects of variation in aerosol backscatter have been shown to be small for typical Doppler lidar velocity measurements (Rye 1990). However, the variations in aerosol backscatter can be easily added to the simulations once a statistical description has been determined. Under these assumptions a rigorous error analysis of the ML estimates can be performed. The layout of the remainder of the paper is as follows. In section 2 the wake vortex analytical model is described and the relevant parameters defined. Section 3 describes the methodology of the lidar simulations. Section 4 describes the maximum likelihood wake vortex estimation algorithms, with results given in section 5. Section 6 provides a summary of the results.

2. Wake vortex model and atmospheric parameters The simulation of lidar data requires a realistic analytical model for the vortex radial distribution and time evolution. A simple postrollup prescription of a vortex that has been successfully used in previous wake vortex studies away from ground effects is [referred to as the Hallock–Burnham model, e.g., by Gerz et al. (2002), Köpp (1994), Proctor (1998), and others or the “dispersion model” by Vaughan and Harris (2001)]

␷␪共r, t兲 ⫽

⌫共t兲 r2 , 2 2␲r r2 ⫹ rC

共2.1兲

where ␷␪(r, t) is the tangential velocity of the vortex, r is the distance from the vortex core, t is time since rollup, ⌫ is the circulation strength, and rC is the vortex core radius, which also indicates the location of the maximum tangential velocity. A vortex pair is established with two counterrotating vortices satisfying Eq. (2.1) and, in the simplest case, are assumed to have the same altitude H(t) for all time. Theoretically, an ideal vortex pair propagates downward with velocity (Lamb 1945, section 155): w共t兲 ⫽

⌫共t兲 dH共t兲 ⫽⫺ , dt 2␲b0

共2.2兲

where b0 is the initial vortex pair horizontal separation. The measurements of Sarpkaya (1983) show this relation holds to within measurement error for early times. The vortex pair description used here includes a time-

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dependent decay of the vortex strength ⌫ for which we assume a simple linear prescription as t ⌫共t兲 ⫽1⫺ , ⌫0 T0

共2.3兲

where ⌫0 is the strength after completion of rollup (defined by t ⫽ 0) and T0 is the decay time of the vortex. For early times in the vortex wake evolution, the linear time decay rate is justified from the approximate solutions of Greene (1986) and Sarpkaya (2000) and the LES results of Holzäpfel (2003), especially for neutral stability conditions as considered here. Note that this is an empirical relation and is intended to roughly model the vortex decay rate from the combined effects of turbulence, crosswind shear, ground effects, mutual diffusion of vorticity, etc. The exact vortex specification used is not important; any specification could be used in the procedure to be described here. However, in the lidar experiments of Holzäpfel et al. (2003) significant errors in circulation estimates were found during the wake rollup stage (for times less than about one time scale, i.e., t* ⫽ 2␲ b20/⌫0 ⬍ 1 or about 10–15 s in their case), so the vortex specification during those times would have to be specified based on observations. In general, the vortex parameters depend on the aircraft type and, to a lesser extent, atmospheric conditions. To reduce the numerical effort, only two aircraft types were considered: a Boeing 737 (B737) and a Boeing 747 (B747). Consistent with the wake vortex simulations of Proctor et al. (1997) the parameters chosen for Boeing 737 aircraft were ⌫0 ⫽ 220 m2 s⫺1, b0 ⫽ 22.3 m, T0 ⫽ 108.0 s, and rC ⫽ 0.669 m and for the Boeing 747 were ⌫0 ⫽ 500 m2 s⫺1, b0 ⫽ 50.5 m, T0 ⫽ 145.0 s, and rC ⫽ 1.5 m. The values used for T0 were suggested by F. H. Proctor (2003, personal communication) based on some of his simulation results. For both aircraft, the center of the vortex pair is set to an operationally realistic distance of 1120 m from the lidar at an altitude of 150.0 m. Another important input is the background wind field. Although any background mean velocity profile could be used, since it can be accurately determined from the lidar data (Frehlich et al. 1994, 1997, 1998), in order to keep the results as simple as possible we include only a constant horizontal ambient velocity ␷ambient ⫽ 3 m s⫺1 away from the lidar to include a realistic vortex horizontal motion. To include more realistic velocity fields, we investigate the effect of turbulence by the addition of a homogeneous isotropic von Kármán radial velocity field with an intensity (as measured by the energy dissipation rate) of ␧ ⫽ 0.001 m2 s⫺3 and an outer length scale L0 ⫽ 50 m. These values are consistent with typical boundary layer measurements of turbulence (Frehlich et al. 1998). Larger values of ␧ could be considered, but they would cause rapid decay of the vortex and would therefore be of less concern to trailing aircraft. The random

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turbulent wind field is generated with algorithms that produce the correct spatial statistics of the von Kármán spectral model (see, e.g., Hinze 1959) at the simulation grid points (Frehlich 1997, 2000; Frehlich et al. 2001). The vortex is then assumed to drift through the frozen turbulent wind field.

3. Lidar simulations The lidar parameters are chosen to match the CTI Windtracer system (Hannon 2000) used during a September 2001 wake vortex measurement campaign at San Francisco International Airport. Thus the lidar wavelength used in the simulations is ␭ ⫽ 2.0225 ␮m, the full width at half maximum (FWHM) Gaussian pulse width is ⌬t ⫽ 0.32 ␮s, the FWHM pulse width in range is ⌬r ⫽ 48 m, the pulse repetition frequency (PRF) is 500 Hz, the sampling rate for real heterodyne lidar data is Fsamp ⫽ 100 MHz, and the data samples per range gate M ⫽ 64, which produces a range gate processing length ⌬p ⫽ M c/(2Fsamp) ⫽ 96 m (where c is the speed of light) and Nbin ⫽ 32 spectral points per range gate. The range gates are offset by 11 data samples (16.5 m). The number of accumulated spectra (lidar shots) per range gate is N ⫽ 20, the number of spectra (range gates) along each line of sight (LOS) is Nspec ⫽ 48, the number of LOSs for each elevation scan is NLOS ⫽ 70, the elevation angle scan rate ␪scan ⫽ 3.3° s⫺1, the first elevation angle ␪1 ⫽ 0.25°, and the time between the start of each elevation scan tscan ⫽ 4.65 s. The lidar is set 3 m above the surface. Note that the 16.5-m offset between adjacent range gates for a given LOS is much less than the range gate length of 96 m and therefore there is some correlation between spectra from adjacent range gates. This small offset is used to increase the sensitivity of the vortex spectral signature to the vortex location. In the simulations, the first lidar LOS starts at t ⫽ 0 and both vortices are assumed to descend at the same rate according to (2.2) with w(t ⫽ 0) ⫽ ⫺1.57 m s⫺1 for both the B737 and B747 cases. Scanning upward from the surface at the chosen elevation scan rate, the lidar beam intersects the centroid of the two vortices at 2.17 s. For the chosen crosswind speed of 3 m s⫺1, the vortex pair drifts 13.95 m away from the lidar between elevation scans. However, during a single scan, for a vortex at 1120-m range and a vertical extent of 10 m, the lidar beam would scan the vortex in 0.155 s and the vortex would only drift 0.465 m. Each realization of raw lidar data is generated assuming constant backscatter over the analysis domain (Salamitou et al. 1995; Frehlich 1997). The signal level is defined by the coherent Doppler lidar SNR (defined as the ratio of the average heterodyne signal power to the average detector noise power in a 50-MHz bandwidth). For the simulations presented here, the radial velocity v(z,␪) as a function of range z from the lidar

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and lidar beam elevation angle ␪ are calculated at a range spacing ⌬z for each LOS based on the wake vortex model and the ambient velocity ␷ambient. An example of the radial velocity profile for three different range gates centered around the vortex at time t ⫽ 0 is shown in Fig. 2. The radial velocity profiles are clearly different for the three LOSs reflecting the lidar beam geometry and the vortex velocity distribution. For example, the two large negative velocity excursions for the lidar beam passing through the centerpoint of the vortex pair (elevation scan number or LOS number 53) results from the two regions of large velocity toward the lidar from the lower section of the nearest vortex (to the lidar) and from the upper section of the farthest vortex (see Fig. 1). Note that the pulse width ⌬r ⫽ 48 m is half the range gate length ⌬p ⫽ 96 m; that is, the vortex size is less than the lidar pulse. Nevertheless, the variations in the radial velocity produce distinctive features in the accumulated Doppler lidar spectra. An example of the accumulated spectra from N ⫽ 20 lidar shots for a single elevation scan is shown in Fig. 3 as a function of the elevation angle or equivalently the LOS index for a range gate centered on the vortex and assuming an SNR ⫽ 10 [see also Hannon and Thomson (1994) Figs. 8–10]. The spectra are normalized by the average spectrum with no signal present (average noise spectrum). Zero velocity is located at bin 16 with positive radial velocity (away from the lidar) for bins less than 16 and negative radial velocity (toward the lidar) for the bins above 16. The large spectral power at bin 13 indicates the aerosol signal from the background ambient velocity. The radial velocity profiles of Fig. 2 are reflected in the spectra at the upper section of the image around LOS index 53. For example, the large nega-

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tive velocity for the lidar beam passing through the centerpoint of the vortex pair is clearly shown in the spectrum of LOS index 53. When the range gate is below the vortex pair (LOS ⫽ 51), the dominant contribution is from the large radial velocity of the far vortex and, in the same manner, the dominant contribution for the range gate above the vortex pair (LOS ⫽ 54) is from the radial velocity of the closest vortex. Note, however, that in either case the vortex spectral signal strength is much weaker than the background aerosol signal produced by the ambient wind. Note in Fig. 3 the presence of random fluctuations in the noise floor from the additive detector noise and the variations in the strong background aerosol signal, which is produced by the random locations of the aerosol particles or “speckle noise.” Because of these random variations, many realizations of the lidar signals are required to produce statistically significant results. For the simulations presented here, each realization consists of independent additive detector noise, aerosol particle location, and turbulent wind field when appropriate. Obviously, the accuracy and numerical effort required for the simulations depends on the chosen grid size ⌬z. As shown in Fig. 3, the most obvious signature of the vortex is contained in the small spectral features just away from the ambient mean velocity. As ⌬z → 0, the statistics of the spectral estimates must approach a constant value. The criterion for selecting an acceptable ⌬z must ensure reliable simulation results. Since the ML estimates are a function of the spectral estimates, we use the average and standard deviation of the spectral estimates S (k, ⌬z) as a performance metric in the limit of ⌬z → 0. This test should be performed for the spectral bins k that are most sensitive to the vortex feature. Figure 4 shows the results of a convergence test using 10 000 simulated spectra for spectral bin k ⫽ 23 and LOS 53 of Fig. 3, which is clearly produced by the vortex. As the grid size ⌬z → 0, both the mean and standard deviation of the spectral estimates approach a constant to within statistical accuracy of the measurements. For this case, within the uncertainty in the measurements indicated by the error bars in Fig. 4, a range grid size of ⌬z ⱕ 0.1875 m will accurately represent the Doppler lidar signal. Note that this accuracy requirement would also be valid for any estimation algorithm that is sensitive to the vortex contribution to the spectral signature (Köpp et al. 2004a,b).

4. Maximum likelihood estimation algorithm

FIG. 2. Radial velocity profiles for three LOS range gates without added turbulence surrounding the vortex pair: range gate with LOS passing through the centerpoint of the vortex pair (solid line), range gate 5 m below vortex (dotted line), and 5 m above vortex (dashed line): V positive indicates motion away from the lidar.

The algorithm chosen for estimating the vortex parameters is an approximation to the optimal maximum likelihood estimator assuming the spectral estimates S(k) are statistically independent ␹2 random variables with mean Sm(k, ⌰) ⫽ 具S(k, ⌰)典 and 2N degrees of freedom (Frehlich 1999), where ⌰ denotes the unknown vortex parameters. The maximum likelihood estimates

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FIG. 3. Accumulated or average lidar signal spectra vs LOS number for a range gate centered on the vortex without added turbulence. Here spectral bin 16 corresponds to zero radial velocity with positive radial velocity (away from the lidar) for bins less than 16 and negative radial velocity (toward the lidar) for the bins above 16.

ˆ of the parameters ⌰ are those values that maximize ⌰ the log likelihood function (Rye and Hardesty 1993): MT

L共⌰兲 ⫽ C ⫺ N

兺 关S共k兲ⲐS

m共k,

⌰兲 ⫹ lnSm共k, ⌰兲兴,

共4.1兲

k⫽1

where C is a constant, k denotes spectral index, MT is the total number of spectral estimates for the range gates in the region around the vortex, and the average spectrum Sm(k, ⌰) is calculated (Frehlich and Cornman 1999) from the radial velocity profiles given by the analytic wake vortex model and the geometry of each lidar shot. For the results presented here, spectra from 12 LOSs and 11 range gates for each LOS were used for a total of MT ⫽ 132 spectra. The log likelihood function L(⌰) was calculated for a fixed grid of the unknown parameters ⌰ and the maximum of L(⌰) was calculated by parabolic interpolation at the grid point with the maximum value. Since the unknown parameters ⌰ depend on the assumptions used for the vortex model, the most efficient ML estimator uses the minimum number of unknown parameters. We will investigate a few ML

estimators. The first ML algorithm is chosen to optimize vortex detection for the early stages of the vortex evolution. Therefore, we assume both vortices have the same altitude H, radius rC, and circulation strength ⌫. We also assume the vortex pair separation b0 and core radius rC are given by the aircraft type, which can be determined by the maximum of the likelihood function from all possible aircraft vortex signatures (Van Trees 1968). For most practical cases of interest, the lidar SNR and background ambient velocity can be accurately estimated from the spectral data (see Fig. 3) and we will therefore assume they are known. Then the only three unknown parameters for the ML estimator are the vortex altitude H, the distance y to the center of the vortex pair, and the vortex circulation strength ⌫. The second ML algorithm investigated is used to optimize vortex tracking of the individual vortices and, therefore, six parameters (vortex position and strength for each vortex) must be estimated for the simple vortex prescriptions used here. More complex vortex descriptions could be accommodated but may require the estimation of more than six parameters.

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FIG. 4. Convergence test for the optimal simulation range grid size ⌬z based on the average spectrum 具S (k, ⌬z)典 and the standard deviation of the spectral estimates SD[S (k, ⌬z)] vs ⌬z with 1 ⫺ ␴ error bars.

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shown in Fig. 5 for a high SNR (⫽2.0) case. The bias of the estimates of all three parameters (altitude H of the vortex pair, horizontal distance y to the vortex pair centerline, and common circulation strength ⌫) is small and the random error is very low. To investigate the source of the bias, the lidar data was simulated for the same parameters but for a nonmoving vortex. The results are shown in Fig. 6. The bias has now been eliminated and a small improvement in the random error is also observed. The motion of the vortex produces a different analytic description of the Doppler lidar spectra Sm (k, ⌰) than assumed in the ML estimator Eq. (4.1). The ML estimates are unbiased when the simulated data matches the functional form of the ML model. The effects of the additive turbulence for the high SNR case are shown in Fig. 7. As expected, the standard deviation of the ML estimates has increased. The performance of the ML estimator in tracking mode (ML 6 par) with six free parameters (vortex height, range, and circulation strength for each vortex) is shown in Fig. 8 for the closest vortex and the simulated data used in Fig. 5, that is, the Boeing 737 with high SNR and without added turbulence. The results with added turbulence are shown in Fig. 9. To permit

5. Results The statistical properties of an estimate x are completely defined by the probability density function (PDF). An important performance statistic is the bias, defined by BIASx ⫽ 具x典 ⫺␮,

共5.1兲

where angle brackets denote ensemble average and ␮ is the correct ensemble average of the random variable x. To produce statistically significant results, all error estimates were produced with 1000 realizations of simulated lidar data. The random error is described by the variance, var关x兴 ⫽ 具关x ⫺ 具x典兴2典

共5.2兲

or the standard deviation SD[x] ⫽ var[x]1/2. The correct values of the estimates of the vortex parameters are difficult to define because the vortex is moving during the time period of the measurement. Therefore, in detection mode, we define the correct vortex parameters as those values of the analytic vortex parameters for the instant in time when the lidar beam passes through the midpoint of the line connecting the centers of the vortex pair. In tracking mode, the correct values are defined as the value of the analytic vortex parameters for the instant in time when the lidar beam passes through the center of each vortex. The PDF of 1000 ML estimates of the vortex parameters for the first lidar scan of the Boeing 737 case is

FIG. 5. PDF of ML estimates of vortex parameters for a B737 assuming three unknown parameters (ML 3 par) and an SNR of 2.0. The correct values are indicated on the right. Also shown is the Gaussian model based on the mean and standard deviation of the estimates.

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FIG. 6. As in Fig. 5 but with no vortex movement.

function of the true SNR. The bias and standard deviation of the ML estimators provides a reference for the best possible performance. The ML estimates with only three parameters estimated (ML 3 par) is ideally suited for the first detection of the vortex. The ML estimator with six parameters estimated (ML 6 par) is required for tracking the vortex since it produces independent estimates of the parameters of each vortex. However, this estimator assumes that the vortex radius rC is known and that it is given by the aircraft type determined from the first detection of the vortex. The results for the Boeing 737 wake vortex simulations are shown in Figs. 12 and 13. As expected, for the ML estimates, the estimation of more independent parameters increases the standard deviation or estimation error of the estimates and the inclusion of turbulence produces an additional increase in the estimation error. Note that the bias decreases slightly with the addition of turbulence because the turbulence smears the peak of the likelihood function, which increases the error in locating the maximum but reduces the average error. Note that the ML 3 parameter estimator has excellent performance even for an SNR ⫽ 0.2. However, the ML 6 parameter estimator has random outliers for SNR ⬍ 1.0, which is observed in the tails of the PDF plots (Figs. 8 and 9). The vortex pair produced by the Boeing 747 is larger and has a larger separation and circulation strength.

comparisons with the ML estimator in detection mode (ML 3 par), the same simulated data were used for the ML estimator in tracking mode (ML 6 par). As expected, the estimation of more independent parameters increases the standard deviation or estimation error of the estimates and the inclusion of turbulence produces an additional increase in the estimation error. However, there is a small impact on the bias of the estimates reflecting the robust nature of the ML estimator. Similar results were produced for the ML estimates of the parameters of the far vortex. To model more realistic situations and to further investigate the robust nature of the ML estimator, simulations were produced with an SNR ⫽ 2.0 and without and with turbulence but with the far vortex initially 2 m higher and at 80% of the circulation strength of the closest vortex. The PDFs obtained for the ML 6 par estimates of the closest vortex are similar to those shown in Figs. 8 and 9, respectively. The PDFs of the ML 6 par estimates of the far vortex are shown in Fig. 10 for the no turbulence case and in Fig. 11 with added turbulence. The PDFs are similar to the estimates of the closest vortex (see Figs. 8 and 9) with little change in statistical accuracy thus indicating the robust nature of the ML estimator. The performance of the ML estimates is conveniently summarized by plotting the bias and the standard deviation of the estimates of the vortex parameters as a

FIG. 7. As in Fig. 5 but with turbulence added.

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FIG. 8. As in Fig. 5 but assuming six unknown parameters for the closest vortex from a B737 and an SNR of 2.0.

FIG. 9. As in Fig. 5 but assuming six unknown parameters for the closest vortex from a B737 and an SNR of 2.0 with added turbulence.

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FIG. 10. As in Fig. 5 but assuming six unknown parameters for the farthest vortex from a B737 and an SNR of 2.0 with the far vortex 2 m higher and 80% of the strength of the closest vortex.

FIG. 11. As in Fig. 10 but for added turbulence.

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FIG. 12. The bias of estimates of vortex parameters as a function of SNR for the B737. The correct values are H ⫽ 146.73 m, y ⫽ 1137.46 m, and ⌫ ⫽ 215.72 m2 s⫺1. The estimates are for the ML 3 par without turbulence (circle) and with turbulence (bullet) and the closest vortex of the ML 6 par without turbulence (square) and with turbulence (filled square). For high SNR, the biases in y are equal for the ML 3 par estimates.

FIG. 13. The standard deviation of estimates of vortex parameters as a function of SNR for the B737. The correct values are H ⫽ 146.65 m, y ⫽ 1115.31 m, and ⌫ ⫽ 215.61 m2 s⫺1. The estimates are for the ML 3 par without turbulence (open circle) and with turbulence (bullet) and the closest vortex of the ML 6 par without turbulence (square) and with turbulence (filled square).

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FIG. 14. As in Fig. 12 but for the B747 estimates.

The PDFs of the ML estimates of the vortex parameters are similar to the Boeing 737 case; that is, they are approximately Gaussian. The performance of the ML estimates of the Boeing 747 vortex parameters is shown in Figs. 14 and 15. As with the Boeing 737 case, the bias

FIG. 15. Standard deviation of estimates of vortex parameters as a function of SNR for the B747. The estimates are for the ML 3 par without turbulence (circle) and with turbulence (bullet) and the closest vortex of the ML 6 par without turbulence (square) and with turbulence (filled square).

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is small and the accuracy (SD) is good. As expected, the accuracy decreases in tracking mode when six parameters are estimated. The random error of the ML estimates is smaller than the results of Köpp et al. (2004b) produced under different conditions; namely, SD[H ] ⬇ 6 m, SD[ y] ⬇ 9 m, and SD[⌫] ⬇ 13 m2 s⫺1 for a spread of circulation strengths from 250 to 450 m2 s⫺1. A critical lidar design issue is the trade-off between lidar pulse energy and PRF. The Monte Carlo approach can be used to investigate this trade-off for the wake vortex application. Typical solid-state lasers produce constant average power as a function of PRF; that is, SNR ⫻ N (where N the number of lidar pulses per LOS) is constant as the PRF or pulse energy varies. The performance of coherent Doppler lidar velocity estimates as a function of PRF for fixed average laser power was determined by Frehlich and Yadlowsky (1994) using computer simulations. For weak signal regimes, higher pulse energy and lower PRF gives better performance, but for high signal regimes, lower pulse energy and higher PRF gives better performance. The same trade-off can be performed for the ML estimates of the vortex parameters, and the results for the Boeing 737 case are shown in Figs. 16 and 17 by plotting the BIAS and SD of the ML estimates as a function of N, for fixed SNR ⫻ N (fixed average laser power). The previous results were all produced with N ⫽ 20. Note that the bias of the estimates has a weak dependence on N. However, the SD of the estimates, especially for the

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FIG. 17. Standard deviation of the ML 3 par estimates of vortex parameters for the B737 case with no turbulence as a function of the number of lidar pulses per LOS for fixed laser power: SNR ⫻ N ⫽ 10 (bullet), SNR ⫻ N ⫽ 20 (open square), SNR ⫻ N ⫽ 40 (open circle), and SNR ⫻ N ⫽ 80 (filled square).

circulation strength ⌫, produce the same performance as for radial velocity estimates (Frehlich and Yadlowsky 1994); that is, for weak signals (low SNR ⫻ N ) the low PRF (small N ) performs better but for higher signal levels (high SNR ⫻ N ) the high PRF (large N ) performs better. The choice of N ⫽ 20 seems to be a good compromise for selecting an operating PRF.

6. Summary and discussion

FIG. 16. Bias of the ML 3 par estimates of vortex parameters for the B737 case with no turbulence as a function of the number of lidar pulses per LOS for fixed laser power: SNR ⫻ N ⫽ 10 (bullet), SNR ⫻ N ⫽ 20 (open square), SNR ⫻ N ⫽ 40 (open circle), and SNR ⫻ N ⫽ 80 (filled square).

We provide here a technique to evaluate lidar performance statistics (bias, average, and standard deviation) based on coupled numerical simulations of pulsed Doppler lidar and a prescription of a wake vortex pair to provide “truth.” In particular, the performance of the ML estimation algorithms of wake vortex parameters was evaluated to establish the optimal performance achievable by a pulsed Doppler lidar scanning perpendicular to the flight path. Using the known characteristics of the analytic vortex representation, rigorous error statistics of the important vortex parameters (position and circulation strength) can be produced with multiple realizations of simulated lidar data that include the additive detector noise, “speckle noise” from the random location of aerosol particles, and an additive turbulent wind field. To produce statistically significant error statistics, 1000 realizations of lidar data were generated for each set of parameters. Two differ-

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ent scenarios were investigated: a detection mode where the vortex radius and separation are assumed known and the location (altitude H and distance y) and circulation strength ⌫ are unknown and a tracking mode where only the vortex radius and SNR are assumed known. For both cases, the ML estimators produced robust measurements of the vortex parameters with small bias and standard deviation and approximately Gaussian probability density functions (Figs. 5–11). The performance of the ML estimator in detection mode is better than for tracking mode because only three unknown parameters need to be estimated. The effects of background turbulence are important as it produces a noticeable decrease in the accuracy of vortex parameter estimates because of the smearing of the spectral signal from the vortex. A similar effect would be produced by a shear in the background velocity. However, the effects of background shear can be removed by adjusting the lidar signal spectra for each line of site based on the velocity from the large aerosol signal (as seen in Fig. 3). This correction will be very accurate under the usual operational conditions where the lidar beam is at small elevation angles so that the radial velocity is nearly equal to the horizontal velocity component and there is a constant background velocity over each range gate. These results are the first quantitative statistical evaluation of the bias and random error of wake vortex parameter estimation obtainable from pulsed Doppler lidar measurements. As demonstrated by the simulations (Figs. 5–15), the random errors are very small in high SNR regimes, and it would be difficult to produce independent verification measurements that surpass this accuracy. In general, in the detection mode, the aircraft type can be determined from the largest likelihood function produced from all possible aircraftrelated vortex models, assuming that the parameters of each vortex pair are determined by the aircraft type. An optimal vortex tracking algorithm is provided by the ML estimator with six unknown parameters assuming the vortex radius rC is given by the aircraft type determined from the first detection. Since the accuracy of the estimates of vortex height H is very good, an alternative estimate of vortex circulation strength ⌫ could also be produced with an estimate of the sink rate w(t) and a knowledge of the vortex pair separation b0 by using Eq. (2.2), and assuming negligible vertical wind velocity. This is the approach used by Hannon and Thomson (1994, 1997). The high accuracy of the lidar estimates can be used to establish accuracy requirements for numerical simulations of aircraft trailing vortices. For example, since the bias in lidar-derived circulation estimates was found to be small, ⬍ 1%, numerical simulations must be able to reproduce this accuracy. Since circulation strength is not a directly computed variable in simulation codes, it must be estimated from other model parameters, and this accuracy requirement may not be easy to satisfy.

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Also, to correctly resolve the scales of motion for accurate lidar simulations that reproduce the critical information in the spectral features produced by the vortices (see Figs. 2 and 3) requires a resolution of at least 0.2 m (Fig. 4). The minimum number of grid points required to resolve important small-scale features in a numerical simulation is discussed, for example, in Margolin et al. (1999) in the context of nonoscillatory forward-in-time schemes (EULAG) for LES of the convective boundary layer. The dissipative effects of such schemes are such that considerable damping of scales smaller than about 6⌬x is observed, although the damping associated with LES models using spectral representations may be less than this (e.g., Moeng and Wyngaard 1988). In practice, careful examination of the small-scale features of a simulation must be made along with convergence tests to identify the required resolution to achieve accurate and stable results. The maximum likelihood estimator is a robust technique for parameter estimation using global information, that is, all the spectral data around the vortex. The main assumption is a realistic prescription of the wake vortex pair as a function of aircraft generating the vortices. The accuracy of the resulting estimates is very good even with added turbulence and low SNR, especially for the detection mode algorithm with only three unknown parameters. The main disadvantage of the ML estimator is the numerical effort required. In contrast, estimation techniques based on radial velocity estimates and the “velocity envelope” extracted from the lidar signal spectra are local estimates (Köpp et al. 2004a,b), and therefore require less numerical effort. However, the effects of the spatial averaging by the lidar pulse and the window function of the signal processing should be included in order to obtain rigorous results. Robust comparisons of the various techniques could be produced with computer simulations that contain all the important physical processes and a priori knowledge of the true vortex. The trade-off between lidar PRF for fixed average laser power (Figs. 16 and 17) indicates that a low PRF system performs better when the SNR is low and a high PRF system performs better with high SNR. However, the PRF dependence is weak and the selection of N ⫽ 20 lidar pulses per LOS is a good compromise. Future work includes more rigorous evaluations of the effects of background velocity and backscatter field including turbulence, all of which can be estimated from the lidar data (Banakh and Smalikho 1997; Frehlich 1997; Frehlich et al. 1998; Hannon et al. 1999; Frehlich and Cornman 2002). Although the current study was limited to vortex decay without significant change in the vortex signature, more realistic models of vortex evolution and distortion produced by the effects of stratification (e.g., Hill 1975; Robins and Delisi 1990; Spalart 1996; Proctor et al. 1997; Holzäpfel and Gerz 2001), wind shear (e.g., Kantha 1996, 1998; Proctor et al. 1997; Hofbauer and Gerz 2000; Mokry 2001), and

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ground effects (e.g., Kantha 1996, 1998; Sarpkaya 2000) could be investigated by using LES of the vortex pair. From the wind fields derived from a suitably high resolution (as defined above) LES, an empirical description of the vortex parameters may be derived that can then be used to perform ML performance estimates of the lidar under these conditions. From a practical point of view, since the coupled vortex–lidar simulations involve very large datasets and intensive numerical effort, both the 3D numerical wind field model of the vortex and the lidar simulation should be performed at the same facility. Acknowledgments. This work was partially supported by the NASA Langley Research Center, contract monitor Dave Rutishauser. The authors wish to acknowledge useful discussions with Steve Hannon, and Fred Proctor for providing the vortex parameter values. Comments by Alex Praskovsky and three anonymous reviewers for their thoughtful comments are gratefully acknowledged and helped to improve the manuscript. REFERENCES Banakh, V. A., and I. N. Smalikho, 1997: Estimation of the turbulent energy dissipation rate from pulsed Doppler lidar data. Atmos. Oceanic Opt., 10, 957–965. Brockman, P., B. C. Barker, G. J. Koch, D. P. C. Nguyen, and C. L. Britt, 1999: Coherent pulsed lidar sensing of wake vortex position and strength, winds and turbulence in the terminal area. Preprints, 10th Biennial Coherent Laser Radar Technology and Applications Conf., Mount Hood, OR, 12–15. Constant, G., R. Foord, P. A. Forrester, and J. M. Vaughan, 1994: Coherent laser radar and the problem of aircraft wake vortices. J. Mod. Opt., 41, 2153–2173. Darracq, D., A. Corjon, F. Ducros, M. Keane, D. Buckton, and M. Redfern, 2000: Simulation of wake vortex detection with airborne Doppler lidar. J. Aircraft, 37, 984–993. Frehlich, R. G., 1997: Effects of wind turbulence on coherent Doppler lidar performance. J. Atmos. Oceanic Technol., 14, 54–75. ——, 1999: Maximum likelihood estimators for Doppler radar and lidar. J. Atmos. Oceanic Technol., 16, 1702–1709. ——, 2000: Simulation of coherent Doppler lidar performance for space-based platforms. J. Appl. Meteor., 39, 245–262. ——, 2001: Estimation of velocity error for Doppler lidar measurements. J. Atmos. Oceanic Technol., 18, 1628–1639. ——, and M. J. Yadlowsky, 1994: Performance of mean frequency estimators for Doppler radar and lidar. J. Atmos. Oceanic Technol., 11, 1217–1230; Corrigendum, 12, 445–446. ——, and L. Cornman, 1999: Coherent Doppler lidar signal spectrum with wind turbulence. Appl. Opt., 38, 7456–7466. ——, and ——, 2002: Estimating spatial velocity statistics with coherent Doppler lidar. J. Atmos. Oceanic Technol., 19, 355– 366. ——, S. Hannon, and S. Henderson, 1994: Performance of a 2-␮m coherent Doppler lidar for wind measurements. J. Atmos. Oceanic Technol., 11, 1517–1528. ——, ——, and ——, 1997: Coherent Doppler lidar measurements of winds in the weak signal regime. Appl. Opt., 36, 3491–3499. ——, ——, and ——, 1998: Coherent Doppler lidar measurements of wind field statistics. Bound.-Layer Meteor., 86, 233–256. ——, L. Cornman, and R. Sharman, 2001: Simulation of threedimensional turbulent velocity fields. J. Appl. Meteor., 40, 246–258.

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