Application of Capon technique to mitigate bird contamination on a spaced antenna wind profiler

Authors


Abstract

[1] A novel technique is developed for profiling radars to measure atmospheric wind fields when signals are contaminated by migrating birds. It exploits the idea of adaptive beamforming to suppress the interference from birds to provide accurate three-dimensional wind measurements using a spaced antenna system. Numerical simulations based on the configuration of the UHF Multiple Antenna Profiler Radar of the National Center for Atmospheric Research are implemented to investigate the performance and the limitation of the proposed technique. The feasibility of atmospheric wind measurements is further demonstrated by using the experimental data. Wind measurements from the full correlation analysis (FCA) and postset beam steering (PBS) are also provided for comparisons. During the period when a single bird is present in the radar beam, the proposed technique produces wind estimates that are consistent with atmospheric wind field prior to the entry of the bird, while both FCA and PBS wind estimates are biased.

1. Introduction

[2] Wind-profiling radars have been used successfully for almost 3 decades to study the structure and dynamics of the atmosphere since the 1970s [e.g., Woodman and Guillen, 1974; Larsen and Röttger, 1982; Balsley and Gage, 1982]. However, radar signals sometimes are contaminated by interference like aircrafts, birds, and insects [Vaughn, 1985; Pekour and Coulter, 1999] such that the resultant radar measurements are in error. Although interference from migrating birds at VHF band is not as severe as those at UHF band, the echoes reflected from aircrafts are usually much larger than atmospheric backscattered signals and cannot be neglected [Chen et al., 2002]. Birds are often observed by wind profilers, especially at night during the migration season. Birds' signals, which are received from main lobe and/or side lobes and can be much stronger than atmospheric signals, can bias the profiler's wind measurements significantly. Moreover, the interference from birds can smear into adjacent gates through range side lobes introduced in the pulse coding to aggravate the problem. The radar cross section, Doppler spectrum, and the probability density function of birds have been estimated by Vaughn [1985]. The characteristics of UHF radar data contaminated by birds were discussed by Wilczak et al. [1995] in both temporal and spectral domains. Errors of wind measurements up to 15 ms−1 were observed by comparing with simultaneous radiosonde measurements. Similar magnitude of errors have also been reported from a narrow-beam polarimetric weather radar [Zrnić and Ryzhkov, 1998].

[3] Various techniques have been proposed to mitigate bird contamination to estimate the underlying atmospheric wind field. For example, Merritt [1995] proposed a statistical averaging method (SAM) to eliminate the velocity components in Doppler spectra that are contaminated by birds. Pekour and Coulter [1999] removed the effect of migrating birds in profiler data and derived hourly wind profiles by a means of combining spectral data from three adjacent range gates with the assumption of two well-defined signal regions in Doppler spectra. The National Center for Atmospheric Research (NCAR) Improved Moments Algorithm (NIMA) uses fuzzy logic synthesis and global image processing to extract atmospheric information from signals contaminated by ground clutter and/or birds [Morse et al., 2002]. Kretzschmar et al. [2003] proposed a neural network approach to remove bird-contaminated profiler data. Although these techniques have been used to mitigate the interference caused by birds, they are susceptible to a number of factors, including the large dynamic range of the clutter-to-signal ratio (CSR), highly temporal varying and widespread bird spectrum, the overlap of the bird and atmosphere spectra, spread of the bird echo in height, rejection criteria for the contaminated echoes, and the effect of precipitation. Furthermore, wavelet transforms were applied to filter out the contributions of both ground and intermittent clutters like birds from radar returns, on the basis of different temporal characteristics between the clear-air and clutter signals [Jordan et al., 1997]. This technique also has been shown to work well in the presence of rain. However, a set of subjective parameters and predetermined meteorological conditions (dry or wet) are required, which can affect the objectiveness and convenience of the technique.

[4] The Doppler beam-swinging (DBS) technique has been successfully implemented in the operational UHF profilers [Strauch et al., 1984]. On one hand, in DBS, three-dimensional wind vector is derived from the combination of at least three mean Doppler velocities that are observed by the radar beams with different pointing directions. On the other hand, the spaced antenna (SA) technique employs a single vertically pointing transmitting beam. The backscattered signals are received by spatially separated receivers. Consequently, the wind field can be estimated using the full correlation analysis (FCA) [Briggs, 1984] or other approaches on the basis of the autocorrelation and cross-correlation functions [e.g., Doviak et al., 1996; Zhang et al., 2003]. Another approach for SA wind measurements is to form synthesized receiving beams at various directions using beamforming techniques [Röttger and Ierkic, 1985]. Consequently, the wind field is obtained by solving the same equations as those used in DBS. Note that these synthesized beams are produced by constructively combining signals from multiple receivers without physically steering the radar beam. Thus the measurement of mean Doppler velocity at various directions can be obtained simultaneously. As a result, wind measurements with higher temporal resolution than those from DBS can be achieved. For example, the postset beam steering (PBS) [Röttger and Ierkic, 1985] and the poststatistic steering (PSS) [Kudeki and Woodman, 1990] are in this category. A synthesized beam acts as a spatial filter in the angular domain. Thus the goal of this work is to design a synthesized pattern such that the bird contamination is minimized. Beamforming technique has been applied to suppress ground clutter from known directions [Allen and Ghavami, 2005]. However, the angular location of the bird is typically not known a priori. Thus an adaptive beamforming technique is proposed in this article to automatically mitigate bird contamination to retrieve the atmospheric wind fields. A similar technique was proposed by Cheong et al. [2006] to reduce biological clutter using an imaging radar with a special configuration with tens of receivers. The present work focuses on conventional SA systems with a limited number of receivers. The proposed technique provides an alternative wind measurement technique for an SA system, which is less susceptible to intermittent clutter like birds. The organization of this paper is as follows. In section 2, the theory and verification of the proposed technique are established on the basis of an adaptive beamforming technique. The feasibility of the proposed technique to retrieve the three-dimensional wind fields is further demonstrated by experimental data in section 3. A summary and conclusions are drawn in section 4.

2. Theory

[5] In this work an adaptive beamforming technique is proposed to measure the Doppler velocities from an SA system that has a number of beams which are pointed away from the bird. These beams are selected such that the radar returns are dominated by the atmospheric signals. The contamination of bird interference will be further suppressed by the adaptive synthesized pattern. As a result, the three-dimensional atmospheric wind field can be derived from the Doppler velocities with minimum bird contamination.

[6] Let us first model the radar measurements contaminated by the bird echoes. The mean Doppler velocity at a given beam direction (θxp, θyp) is a weighted average of Doppler velocity distribution within the radar volume and can be written in the following form,

display math

where θx and θy are the angular locations in the zonal and meridional directions, respectively, fx, θy) is the transmitting antenna beam pattern centered at (0°, 0°) for SA operation, Zax, θy) and Zbx, θy) represent the angular reflectivity distribution of the atmosphere and the bird, respectively, and Vrax, θy) and Vrbx, θy) are the radial velocity distributions of the atmosphere and the bird. Wpx, θy) is the synthesized receiving beam pattern pointed at (θxp, θyp) and will be determined by the beamforming technique. Note that the range dependence of parameters in (1) is neglected because only the angular distribution is of interest. In addition, it is assumed that only a single bird (or a small flock of birds) is present in the radar volume, which can be considered as a point target since its size is much smaller than the radar resolution volume [Bruderer, 1997a].

[7] In order to simplify (1), let us introduce two variables, Pxp, θyp) and Qxp, θyp), defined as follows:

display math

[8] It is easy to realize that Qxp, θyp) stands for the postbeamforming clutter-to-signal ratio. Note that both P and Q are functions of the pointing direction of the synthesized pattern. If the synthesized pattern is selected such that Pxp, θyp) ≪ 1 and Qxp, θyp) ≪ 1, then (1) can be approximated to the following form,

display math

[9] It is evident in (3) that only the atmospheric terms are remained. If three noncollinear pointing directions for which (3) is valid can be determined, theoretically the wind field derived from these radial velocities using a DBS approach will be free of bird contamination. The validation of Pxp, θyp) ≪ 1 and Qxp, θyp) ≪ 1 will be discussed and verified later.

[10] In DBS the homogeneity of the wind field over the spatial coverage across different beams is assumed [Larsen and Röttger, 1982]. Here the wind field is assumed to be uniform only within the transmitted beam that is a relatively smaller region. In addition, the aspect sensitivity of the backscatter from the atmosphere at UHF band is ignored [Cohn et al., 1997]. Furthermore, we consider uniform reflectivity to facilitate the problem. As a result, the following two equations are obtained,

display math

where u, v, and w represent the zonal, meridional, and vertical components of the atmospheric wind field, respectively, θ is the zenith angle, sin2θ = sin2θx + sin2θy, and c is a constant. Substituting the values of equation (4) into (3), the mean Doppler velocity reduces to the following formula,

display math

where the cross symbol represents the matrix multiplication operator.

[11] Consequently, the three-dimensional wind field can be obtained by solving a set of linear equations as shown in (5) from the measurements of at least three noncollineal beam directions.

[12] Now the problem arises as to how we can design a synthesized pattern to satisfy P ≪ 1 and Q ≪ 1. In order to solve this problem, in this study we propose to use the Capon beamforming [Capon, 1969] because of its adaptive nature and capability of suppressing interference [e.g., Cheong et al., 2006]. The Capon method has been applied to coherent radar imaging (CRI) to estimate the angular power distribution of radar returns with fine resolution, compared to the traditional and nonadaptive Fourier-based techniques [e.g., Palmer et al., 1998; Yu et al., 2000].

[13] The well-known solution to the weighting function of the Capon method, wc, is given by the following equations,

display math

and

display math

where T is the transpose operator, the position vector of receiver i is represented by Di, k represents the wave number vector of the desired pointing direction, and k = (2π/λ)[sinθx sinθy cosθ]. The visibility matrix V(τ) can be defined as follows:

display math

where Vij(τ) is the cross-correlation function of receiving signals for channels i and j. The Capon weighing function is adaptive to the data through the visibility matrix. Consequently, the synthesized pattern can be obtained by performing the spatial Fourier transform of the weighting function.

[14] The Capon method is a constrained optimization in which the output power of the beamformer is minimized to automatically suppress interference from unknown directions. The response of the beamformer is constrained to unity at the direction of interest. As a result, the synthesized pattern of Capon beamforming can adaptively adjust itself to place a null in the angular location of the bird. Therefore the assumptions of P ≪ 1 and Q ≪ 1 could be fulfilled. Consequently, the estimates of mean radial velocities are less contaminated by bird interference, and accurate three-dimensional wind fields can be derived. The assumption of P ≪ 1 and Q ≪ 1 using Capon beamforming for various conditions is now investigated using simulations.

[15] The SA system used in the simulation is the NCAR Multiple Antenna Profiler Radar (MAPR), which is a modified version of Radian LAP-3000 915-MHz boundary layer profiler [Carter et al., 1995]. MAPR has shown to provide reliable wind profiles at high temporal resolution [Cohn et al., 1997, 2001]. MAPR has four receivers located at the four corners of a square. The distance between the centers of the adjacent antenna panels on each side of the square is 0.918 m (corresponding to 2.799 λ) and that of the antenna panels for diagonal pairs is 1.298 m (corresponding to 3.057 λ) [Cohn et al., 1997]. The comparison of the wind field measurements using DBS and SA techniques with MAPR has been shown by Holloway et al. [1997]. In addition, three-dimensional winds with enhanced range resolution can be obtained from the observation using multiple frequencies and multiple receivers [Yu and Brown, 2004].

[16] In simulation the visibility matrix is obtained by performing the spatially inverse Fourier transform of the power distribution in angle, in which a uniform reflectivity, bird signal, and a Gaussian transmitting beam with 9° half-power beam width (HPBW) are contained [Yu et al., 2000]. Consequently, the Capon weighing function can be obtained at a number of directions using (6). The corresponding synthesized receiving beam pattern can be obtained accordingly.

[17] An example of Pxp, θyp) and Qxp, θyp) in decibel (dB) is shown in Figure 1 for the case of a bird located in the first quadrant at (2°, 2°). The bird signal is simulated by a delta function with a flying velocity of [0 5 −0.5] ms−1.

Figure 1.

Magnitude of P and Q in dB at different pointing directions (θxp, θyp) are shown in gray scale contours. A bird is located at 2°, 2° with CSR of 25 dB. The cross signs denote the proposed pointing directions of the synthesized beam. The Doppler velocities estimated from the three directions will be used to derive the three-dimensional wind field.

[18] The atmospheric three-dimensional wind field is assumed to be [8 6 1] ms−1. These vectors represent the velocity components in east, north, and vertical directions, respectively. In addition, the CSR, defined in the following equation, is set to be 25 dB.

display math

[19] In Figure 1, the horizontal and vertical axes represent the zonal and meridional angle (θxp, θyp) of the pointing directions of the synthesized beam, respectively. It is evident that the magnitudes of both P and Q are less than −25 dB in most regions except for a small region where the bird is located. In other words the assumptions of P ≪ 1 and Q ≪ 1 are valid when the Capon synthesized beam is not pointing to the vicinity of the bird. Therefore the beam directions at the other three quadrants can be used for DBS wind measurement with limited bird contamination. For this case the three beam directions of (−4°, 4°), (−4°, −4°), and (4°, −4°) are proposed. Since the HPBW of the transmitted pattern is 9°, further increase of the angular separation of beam directions will degrade the quality of the beamformer's output. Except for the location (2°, 2°), other bird locations are also given to examine the value of P and Q at the three proposed beam directions. The results indicate that the assumptions of P ≪ 1 and Q ≪ 1 are always satisfied for various angular locations as long as the bird is located in the first quadrant. It should be noted that the other beam directions can be used except the vicinity of the location of the bird. It is also possible to improve the estimate of wind field using more than three pointing directions.

[20] The cross sections of the synthesized receiving pattern pointed at the three proposed directions are shown in Figure 2. The cross sections with θy = 2° and θx = 2° are presented on the left and right, respectively, in which the dashed vertical lines indicate the location of the bird. It is evident that the bird signals are almost totally suppressed by a factor of about −90 dB using Capon beamforming. Consequently, the mean Doppler velocities measured by the synthesized beams would be dominated by atmospheric signals without bird contamination. Nevertheless, in practice, the null of the pattern may not be so deep because of radar hardware imprecision.

Figure 2.

Cross sections of the normalized synthesized receiving pattern of Capon beamforming for three pointing directions of (−4°, 4°), (4°, −4°), and (−4°, −4°). The left and right columns show the cross sections of θy = 2° and θx = 2°, respectively. The dashed lines indicate the location of the bird.

[21] To further investigate the effect of the bird's location on wind measurements, 361 bird locations were generated in the first quadrant with uniform grid from 0° to 9° in both θx and θy directions with half-degree increments. The other parameters for the simulation are the same as those used in Figure 1. The histogram of the errors of the wind estimates is shown in Figure 3 for zonal, meridional, and vertical components from top to bottom, respectively. It is apparent that all the errors are extremely small and the maximum error is only about 0.35%. Because the antenna pattern of MAPR is symmetric, results from the cases that a bird is present in other quadrants are the same. Therefore these results suggest that the three-dimensional atmospheric wind field could be recovered by steering the synthesized beam in the proposed three directions, each at a quadrant other than the one with a bird in it. Although the bird's location is not required in Capon beamforming a priori, the quadrant where the bird is present is needed to determine the three pointing directions of the synthesized beam for wind measurement. Because the bird echoes can often be characterized by large radar reflectivity compared to the atmospheric backscatter, the quadrant where the bird is located in the radar volume can be determined from the angle of arrival (AOA) [Chau and Balsley, 1998].

Figure 3.

Histogram of the errors of the wind velocity estimates in (top) zonal, (middle) meridional, and (bottom) vertical directions for the bird's location varies every half degree in the first quadrant within the square region of 9° off zonal and meridional. The east, north, and vertical velocity of the atmosphere u, v, and w are 8 ms−1, 6 ms−1, and 1 ms−1, respectively.

[22] The performance of the proposed algorithm as a function of CSR and signal-to-noise ratio (SNR) is examined using numerical simulation (see Appendix A). The results show that the proposed technique is sensitive to the angular width of the power distribution of the bird echoes, which is a function of range, dwell time, and the speed of the bird. This problem can be overcome by increasing the number of receivers to improve the performance of wind estimation. For example, Cheong et al. [2006] suppressed the effect of birds and obtained the images of echo power and wind velocity by using the Turbulent Eddy Profiler (TEP). The clutter/interference from the main lobe and grating lobes can also be identified by using a special array configuration. It should be mentioned that TEP has up to 60 spatially separated receivers and a flexible array configuration. Most wind profilers with SA configuration have only a limited number of receivers.

3. Experimental Results

[23] The ability of Capon beamforming on a SA system to estimate the three-dimensional atmospheric wind field is further verified using the MAPR data collected on 1 June 2002 during the International H2O Project (IHOP) [Weckwerth et al., 2004]. The data were sampled every 16 ms with 200 m range resolution. For every 256 data points (about 4 s) the quadrant of the bird was obtained from the AOA. The three beam directions were selected from four fixed angles by excluding the angle in the quadrant. Time series signals from the synthesized receiving pattern at the three beams can thus be generated using the adaptive Capon beamforming. Subsequently, the mean Doppler velocity at the three directions can be estimated by using either spectral processing or covariance processing (pulse pair processor) [Doviak and Zrnić, 1993]. The three-dimensional atmospheric wind field can be obtained by solving a set of linear equations as the DBS technique. If no bird is present in the signal, the AOA represents the scattering center of the atmospheric targets and the wind field can still be obtained. If the scattering center is at origin, any three beam directions of the four directions were selected.

[24] Figure 4a shows a time series of echo power from the four receivers at a height of 800 m over a 2-min period. The estimated zonal, meridional, and vertical components of the wind field using Capon beamforming are shown in Figures 4b, 4c, and 4d, respectively. The wind velocities estimated from the FCA method [Briggs, 1984] are denoted by the curves with squares, in which the vertical velocity was obtained from the mean radial velocity of the four receivers. For comparison, the results from PBS, which is the Fourier-based beamforming with the same postbeam directions as Capon beamforming, are shown by the curves with asterisks. As shown in Figure 4a, the magnitude of the echo power was kept at a constant level of approximately −5 dB before 0512:54 UTC. However, the echo power rose rapidly after that time and exhibited fluctuations subsequently. After 0513:35 UTC, the echo power fell to the initial value. During the time period from 0512:54 to 0513:35 UTC, the echo power reached its maximum with peak value of approximately 5 dB. The angular power distributions obtained from Capon CRI are presented in Figure 5 every 4 s during the period in which the location of the peak of each distribution is marked. It is evident that a localized target was moving across the beam from southwest to northeast. Strong signals from the grating lobes are also observed in most of the images because of a spatial aliasing effect caused by long antenna baselines. This aliasing effect is not important here because the synthesized pattern will place the null in the direction of all grating lobes.

Figure 4.

(a) Echo powers detected by four receivers of MAPR from 0511:28 to 0513:43 UTC at a height of 800 m. The (b) zonal, (c) meridional, and (d) vertical wind velocities are obtained from three methods. The results between the two vertical dashed lines represent the data contaminated by the bird.

Figure 5.

(a–k) Sequence (from 0512:54 to 0513:35 UTC) of coherent radar images with Capon's method. The peak of each image is marked with a white circled cross. The horizontal and vertical axes represent zonal and meridional angles in degree, respectively, which are in the range of ±12°. Figure 5l shows the peak locations of the sequence from Figures 5b to 5h, in which the markers are more clear on the images.

[25] In Figure 4, the maximum value of echo power corresponds to the image at 0513:06 UTC presented in Figure 5, when the vertical velocity also shows a pick value of approximately 0.8 ms−1. Moreover, the cross-beam velocity of the target is estimated from the temporal variation of its angular locations shown in Figure 5l and is approximately 4 ms−1. This magnitude of the velocity is consistent with the flying speed of birds [Bruderer, 1997b]. Therefore we believe that the localized target shown in Figure 5 can be attributed to a bird, which is responsible for the increase of the echo power shown in Figure 4 during the period from 0512:54 to 0513:35 UTC. It is very likely that the bird echo came from the antenna side lobes and was aliased into the main imaging region.

[26] Because of the absence of the bird before 0512:54 UTC, the true wind components in the zonal, meridional, and vertical directions can be estimated by using the methods of FCA, PBS, and Capon beamforming. The result shown in Figure 4 suggests that the wind velocities estimated from different methods are in agreement. Because the dwell time for processing each wind velocity is very short, the differences in the wind estimates from these methods are expected [Cohn et al., 2001; Holloway et al., 1997]. Furthermore, the reflectivity distribution of the atmosphere may not be uniform [Pollard et al., 2000]. For nonuniform reflectivity a correction similar to the one used for aspect sensitivity at VHF band needs to be considered [Palmer et al., 1993]. Some amounts of bias may accompany the wind estimation because the reflectivity distribution is unknown.

[27] During the period of the presence of the bird, the FCA and PBS wind estimates exhibit sudden and large deviation from the Capon wind velocities that are consistent in time and good agreement with the background wind field. Although the synthesized beam for PBS method is steered away from the bird location, the interference from the bird can still come in through the side lobes of the Fourier synthesized beam [Palmer et al., 1998; Yu et al., 2000]. Because of the large CSR (about 10 dB or more) in this case, accurate estimation of atmospheric wind field using the conventional techniques, such as PBS and FCA, is very difficult to achieve. However, in light of its excellent performance of interference suppression the Capon beam-forming technique can estimate true wind velocity in susceptible interference from the bird, as shown in Figure 4. Although no radiosonde data are readily available for comparison, these results are supportive of the superiority of Capon beamforming to the conventional methods for retrieving atmospheric wind field contaminated by birds. A preliminary survey of the data set used in this initial study suggests around a fourth of the bird echoes encountered may be mitigated using this technique. The procedure is not yet fully automated, so a more exhaustive survey of the effectiveness of the technique is required.

4. Conclusion

[28] It has long been realized that the data of the wind profilers are frequently contaminated by migrating birds, especially at night, which can lead to large errors in the atmospheric wind estimates. In order to mitigate the problem, a technique based on Capon beamforming is proposed for SA systems. Three synthesized beams are formed at locations that are sufficiently away from the bird but are still within the HPBW of the transmitted beam. The Capon receiving pattern acts as an adaptive spatial filter so the bird echoes can be optimally suppressed. Consequently, the three-dimensional wind field can be estimated using the mean Doppler velocities from the atmosphere at the three angular locations. This approach is similar to the conventional DBS technique except the receiving beams are formed using Capon beamforming rather than physically steered. The procedure of the technique is as follows: (1) Estimate the AOA to determine the quadrant where the bird/scattering center is located. (2) Select three beam directions from four fixed angular directions (each at a quadrant) by excluding the direction of the quadrant where the AOA is located. (3) Compute the Capon weighting function and synthesized pattern for the three beam directions. (4) Generate the synthesized time series data by combining the signals from different receivers with above weights and subsequently estimate the mean Doppler velocities for each beam direction. (5) Estimate the three-dimensional wind field by solving the set of linear equations.

[29] Numerical simulations based on NCAR's MAPR system are first performed to investigate the feasibility and limitation of the proposed technique. It is demonstrated that the atmospheric three-dimensional wind field can be successfully recovered using the proposed technique on a conventional SA system with a limited number of receivers even when radar returns are contaminated by birds. However, the performance is degraded as the bird echoes are smeared into the direction of flight and the CSR is high. From the experimental results it is evident that the proposed technique can produce consistent wind estimates in the presence of the bird, while both FCA and PBS wind estimates exhibit sudden fluctuations that are not continuous in time. However, the issue of optimal section of beam directions, which are free from contamination by multiple dispersed birds, has not been solved. The wind field may not be recovered completely in such cases. To further study this problem, Capon beamforming can be combined with other fuzzy logic approaches to objectively determine the beam directions. Although this technique is demonstrated and verified using a 915 MHz profiler, it is applicable to other SA radar profilers with other frequencies and configurations. However, for VHF profilers the aspect sensitivity of atmospheric echoes needs to be considered.

Appendix A:: Simulation Results of the Performance

[30] The performance of the proposed algorithm as a function of CSR and SNR is examined. The root-mean-square error (RMSE) of the three estimated wind components for various CSR and SNR is shown in Figure A1, and the location of the bird is set as in Figure 1. For each CSR and SNR, 200 realizations were generated, each with an independent set of noise powers. The noise was added to the diagonal of the visibility matrix because the noise on different antenna can be assumed totally uncorrelated [Hysell and Woodman, 1997]. Figure A1 shows that the RMSE of the three wind components are extremely small for all CSR if SNR ≥ 10 dB. The results demonstrate that the atmospheric wind field can be retrieved at high SNR, even CSR as high as 25 dB. Moreover, generally the wind estimates deteriorate faster at low SNR (≤6 dB) than those at high SNR. The rate of degradation is higher at lower CSR. This is because the noise that we added to the visibility matrix is determined by the given SNR, which is dependent on the signals from the atmosphere in the simulation. The visibility matrix includes not only the atmospheric signals and noise but also the bird's echoes. Therefore the noise power becomes more dominant in the visibility matrix as SNR decreases, especially at lower CSR because of weaker bird echoes. As a result, the error is significant for lower CSR at low SNR. Nevertheless, these results suggest that the proposed algorithm is a feasible technique for wind measurement in the condition of high SNR. Even the interference is extremely strong, caused by a point clutter.

Figure A1.

Root-mean-square error (RMSE) of the wind velocity estimates in (top) zonal, (middle) meridional, and (bottom) vertical directions for various SNR and CSR.

[31] Note that previously, the location of the bird was implicitly assumed to be fixed during the dwell time. However, in practice, the movement of the bird will smear the power distribution such that an ideal delta function is not sufficient to characterize it. For example, an angular spread of approximately 0.25° is obtained for a bird flying at a constant range of 1.1 km with a velocity of 5 ms−1 over a 1-s period. This smearing effect is simulated using a boxcar power distribution of the bird in the direction of its motion. The rest parameters are the same as those in Figure A1. The RMSE for the case of the angular width of the bird's power distribution σb = 0.25° is shown in Figure A2. The RMSE increases with the decrease of SNR. However, the pattern of the variation of RMSE with SNR is different from the one shown in Figure A1. For all SNR the errors become larger and they are more severe in the case of high CSR. Furthermore, the RMSE for various widths of the bird's power distribution (σb = 0° − 1°) are shown in Figure A3 for SNR = 15 dB. It is evident that the RMSE increases substantially with increasing CSR and σb. Moreover, the large RMSE values indicate that the bird contamination is not fully removed from the radar returns when σb and CSR are large. It also implies that the performance is degraded as the bird echoes are smeared into the direction of flight and the CSR is high.

Figure A2.

As in Figure A1, except σb = 0.25°, which represents the angular width of the power distribution of the bird echoes.

Figure A3.

RMSE of the wind velocity estimates in (top) zonal, (middle) meridional, and (bottom) vertical directions for various σb and CSR. For all cases, SNR = 15 dB.

Acknowledgments

[32] MYC was supported by the National Space Organization of the Republic of China (Taiwan) under grant 93-NSPO(B)-RS3-FA07-01 during his visit to the School of Electrical and Computer Engineering, University of Oklahoma. TYY was partially supported by the DOD, EPSCOR program under grant N00014-06-1-0590. NCAR is operated by the University Corporation for Atmospheric Research (UCAR) under the sponsorship of the National Science Foundation (NSF). Deployment of MAPR at IHOP was funded by NSF.

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