Ground moving target parameter estimation for two-channel SAR C.H. Gierull Abstract: The author introduces and analyses the performance of different techniques to estimate the parameters of ground moving targets in multi-channel synthetic aperture radar (SAR) data. Candidates are matched filter banks, the along-track interferometric phase and direction-ofarrival estimation methods, which can work in either the raw data or in the compressed SAR image domain. Of particular interest are systems with only two channels because many existing or near-future SAR systems, such as RADARSAT-2 and TerraSAR-X, are restricted to a maximum of two sub-apertures. Desired parameters are the two velocity components (alongand across-track), acceleration if present and the true azimuth location. Theoretical results are evaluated and illustrated with experimental airborne SAR data.
1
Introduction
In many civilian and military applications of airborne and spaceborne synthetic aperture radar (SAR) imaging, it is highly desirable to simultaneously monitor traffic on Earth’s surface. The measurement of object motion using SAR requires two consecutive operations. First, the detection in the SAR data and, secondly, target parameter estimation such as location, speed and course. Target detection and estimation can either be performed incoherently with a single SAR sensor, or coherently, with much higher fidelity, with two or more apertures. Although many authors have investigated the detection part in the past, the estimation problem has been rarely dealt with [1]. Great importance is attached to the practical applicability of the presented techniques, particularly with regard to the experimental ground moving target indication (GMTI)mode of RADARSAT-2 [2]. The techniques presented in this paper are analysed with real data acquired during an experiment conducted at Canadian Forces Base (CFB) Petawawa on 14 July 1999. The SAR data were acquired by the Environment Canada’s CV 580 C-band SAR configured in its along-track interferometric (ATI) mode [3]. The radar, operating at C-band (5.30 GHz), was configured to use a transmitting antenna situated on the belly of the aircraft and a two-aperture (0.27 m effective phase center separation) microstrip receiving antenna situated on the starboard side of the aircraft above the transmit antenna. Horizontally polarised radiation was transmitted and received by the system running in the nadir mode, with most targets of interest situated at incidence angles of approximately 508. To minimise azimuth ambiguity effects, the radar system was run at twice the usual pulse repetition frequency (PRF), that is, at PRF ¼ 678 Hz. At a # Canadian Crown copyright 2006 IEE Proceedings online no. 20045094 doi:10.1049/ip-rsn:20045094 Paper first received 1st October 2004 and in revised form 11th October 2005 The author is with the Defence Research and Development Canada – Ottawa (DRDC Ottawa), Radar Systems Section, 3701 Carling Avenue, Ottawa, ON, Canada K1A 0Z4 E-mail:
[email protected]
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508 incidence angle, and an altitude of approximately 7 km, targets of interest were offset from the ground projection of the radar flight track by approximately 9 km. The system parameters such as incidence angle and PRF (related to the clutter bandwidth) are chosen to resemble RADARSAT-2’s geometry. As RADARSAT-2’s spatial resolution of about 3 m is better than the Convair’s (4 m), a similar signal-to-cutter ratio (SCR) is anticipated. The only major difference is the lower SNR because of the significantly increased ranges to the targets with the corresponding signal power loss.
2
Moving target detection
The first step in any MTI mode is the actual test for the presence of movers in the measured data. In air-to-ground applications, such as SAR, this task is challenging because of the simultaneously overlayed clutter power, which can, in the mainbeam regions of the antenna pattern, be a multiple of the signal power. As the clutter must be suppressed prior to target testing over the entire clutter Doppler spectrum, one-channel based techniques are typically very limited in performance. Although, real multi-channel SAR – GMTI is known to enhance the detection and estimation performance significantly [4], many present and future radar systems are restricted to only two parallel receiver channels because of limited resources. This is particularly true for space-based sensors, such as upcoming RADARSAT-2 and TerraSAR-X. Ground moving target detection (GMTD) can be either achieved in the raw data domain, that is without actually focusing/compressing the data in along-track (azimuth direction) or it can be done on the processed data, that is in the image domain [5, 6]. Either domain has its advantages and disadvantages. While raw data techniques usually suffer from low SNR they usually allow detection of targets in a much wider velocity range than image-based techniques. This is due to the smearing effects of the point spread function for fast moving targets when they are filtered with the optimum filter for the stationary world during SAR processing [3]. Processing two SAR images with ATI for slow moving vehicles, that are still focused but displaced, has IEE Proc.-Radar Sonar Navig., Vol. 153, No. 3, June 2006
been proven to work successfully in that a slowly moving control vehicle was detected [7]. For the subsequent moving target parameter estimation, it is assumed in this paper that the detection has been done in the raw data domain, as described in [5]. However, if the SNR is too low in the raw data, the detection may be done after compression/focusing and the entire corresponding line (at range in which the target was detected) can be analysed as described below. However, the clutter contribution within this azimuth line is larger, which leads to a lower SCR. In the approach used here, the entire azimuth interval is segmented into small, typically highly overlapping, pieces, where the data of each segment are successively analysed for movers. The data segment is transformed into the Fourier or Doppler domain; the length of the segment is chosen so that a moving target is confined to one Doppler cell (see Fig. 2 in [5]). Adequate CFAR thresholds are then applied to each range-Doppler data segment separately and the results are stored in lists. Recently, several new metrics for two channel SAR – GMTD have been proposed and compared with wellknown techniques [8], such as ATI and displaced phase centre antenna (DPCA) [9, 10]. The most promising group of methods, are two-step detectors sometimes called hyperbolic detectors. These are based on the combination of two independent metrics in such a way that thresholding with the first metric detects targets with a high probability of detection while allowing a rather large number of false alarms. Computing and testing a second metric on the
data which passed the first detection step ought to decrease the number of false alarms while preserving the probability of detection. Using DPCA as the primary detector typically results in many false alarms particularly in very heterogeneous terrain such as urban areas. This is due to the presence of many very strong point-like stationary scatterers which are, after clutter suppression (i.e. subtraction of the two channels), still strong enough to exceed the detection threshold. Knowing that stationary targets are typically suppressed to a greater degree than moving targets, recommends application of a second test which only retains those detections with a relatively small amplitude change before and after suppression. An example of the impact of such a second step is illustrated in the range-Doppler plot in Fig. 1 for one segment of the highway sub-scene image in Fig. 2a. The left side shows the DPCA magnitude with its still relatively high clutter power with respect to some slow and low radar cross section (RCS) targets in the front part of the plot. Applying the second step now reveals the small or lowspeed targets. The peaks in the back part of the plot are several fast targets traveling on the highway that are not visible on or nearby the highway in the SAR image, even though approximately ten different cars and trucks were traveling on that street during the observation period. 3
Fig. 2a shows the detected target histories superimposed onto the stationary SAR image of the scene. One particular problem faced by any MTI technique is the clustering of individual detections to target tracks. The conceived algorithm starts with detections at the far left side in Fig. 2a and checks whether adjacent range cells for the next time sample contain a detection as well. If positive, both are assigned a common target track ID and the next pixels to the right are checked until no direct neighbour includes a detection anymore. At this point, this specific ID expires and the track ID counter is incremented by one. After reaching the right end of the scene only those tracks are kept whose number of detections exceeds a pre-defined value. This value is based on physical parameters such as antenna beamwidth and a range for reasonable target speeds. The corresponding Doppler histories, Fig. 2b, can be used to further eliminate more false alarms, for instance,
Fig. 1 Demonstration of the reduction of false alarms through an additional detector step
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by rejecting all those tracks which do not follow an allowed Doppler-time trajectory, for example, the pixel clusters at roughly 2225 Hz on the left side of the plot. 4
4.2
Target parameter estimation
In order to estimate the target parameter, the individual (for each target) one-dimensional raw data chips are extracted corresponding to the detected target trajectories in Fig. 2a. This is done separately for both channels, where the resulting time signals are stored in the two-dimensional vectors s(t). 4.1
Signal model for constant velocities
The parameters describing the moving target, such as velocity in along-track direction vx , velocity in across-track or range direction vy and its location x0 at time t ¼ 0 are combined in the vector j ¼ [vx , vy , x0]T. Alternatively, the position xb , when the platform is broadside to the target can be used as the location parameter. This position depends on the time tb , which depends on the aircraft velocity va tb ¼
x0 vx va
or xb ¼ va tb ¼
va x0 vx va
ð1Þ
It is evident that both positions are identical for stationary scatterer with vx ¼ 0. Note that the range-position y0 is assumed to be known because of the range-resolution of the radar. The far-field model for the received target signals measured at two sensors aligned in flight direction and separated by distance d is D1 ðuðt; j ÞÞ ð2Þ sðt; j Þ ¼ aej2bRðt;jÞ D2 ðuðt; j ÞÞe jbuðt;jÞd where a is the complex amplitude describing the reflectivity of the scatterer, R(t, j ) denotes the slant range distance to a reference channel and u(t, j ) is the directional cosine from the reference antenna to the moving target on the ground. Di(u) describes the two-way antenna pattern of the ith channel and b ¼ 2p/l is the wavenumber. The distance can be expressed as qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð3Þ Rðt; j Þ ¼ ðx0 þ ðvx va ÞtÞ2 þ ð y0 þ vy tÞ2 þ H 2 where H denotes the constant altitude of the aircraft. For small antenna azimuth beam-widths, that is, a relatively small period of time in which a particular scatterer is ‘seen’ by the radar, it is possible to approximate (3) by a second-order Taylor series around broadside-time tb RðtÞ ffi Rðtb ; jÞ þ R0 ðtb ; jÞðt tb Þ þ ¼ Rb þ vs ðt tb Þ þ
v2rel 2Rb
ð4Þ
where vs ¼ avy , v2rel ¼ (vx 2 va)2 þ (1 2 a2)v2y, a ¼ yb/Rb , and p the broadside distance p can be expressed as Rb ¼ (( y0 2 vytb)2 þ H 2) ¼ (y2b þ H 2). The point of closest approach tc between the radar and the target, where the Doppler frequency vanishes, can be determined via R 0 (tc , j ) ¼ 0 giving
226
v s Rb v2rel
ð5Þ
Doppler rate (along-track velocity)
4.2.1 Parameter fit: A simple but crude estimation scheme is to utilise the range walk of the individual tracks (if any are present), that is, to apply a direct parameter fit of (4) to the detected range-time trajectories, Fig. 2a. As this technique is solely based on range information (resolution) it is not particularly accurate. These initial estimates can, however, help to resolve phase ambiguities, such as the correct sign for vy , that is, approaching or receding target. 4.2.2 Matched filter banks: Estimation of the alongtrack velocity component vx can be achieved by applying a matched filter bank r(t, j 0 ) ¼ exp(22jbR(t, j 0 )) with vari0 , v s0 ¼ 0, t b0 ¼ 0]T to the indiable parameter vector j 0 ¼ [v rel vidual target tracks (maximum-likelihood estimator (MLE)) [1]. Fig. 3a shows the matched filter responses for the dark blue highway track (at roughly 8500 m range in the center 0 was varied between 80 and 182 m/s of Fig. 2), where v rel (relates to vx between 250 and 50 m/s). Although the maximum appears at a reasonable velocity of v^ x ffi v^ rel 2 va ¼ 22.7 m/s, the clutter contamination within the data can be clearly seen. The time tc ¼ 1.95 s at which the maximum occurs (16), represents the closest approach of platform and target. Around the the optimum vrel the matched filter responses shift away from tc and broaden; the ridge is tilted and fans out at the edges. These effects are caused by the mismatch of the linear phase terms in the matched filter v s0 and the data vs . After co-registration, that is, the time shift of the aft channel by d/(2va) Appendix and [11], the DPCA difference of the two channels in (12) and (13) can be written as pðt; jÞ ¼ ejbw1 ðt;jÞ ejbw2 ðtþðd=2va Þ;jÞ ¼ 2ja sinðbgðt; jÞÞe2jbRðt;jÞ ejbgðt;jÞ
ð6Þ
where g(t, j) ¼ vs d/(2va) þ e(j)(t 2 tb)/2 with e(j) given in (19). Therefore, p(t, j) is of identical structure as either channel signal but with the stationary clutter components significantly suppressed. Applying the matched filter bank onto the DPCA output results in Fig. 3b, where a clutter amplitude suppression of about 10–15 dB can be recognised. It is important to note that this sensitivity of the quadratic phase match (i.e. to the parameter vrel and hence mainly vx) is sustained even for space-based systems, because
b
R00 ðtb ; jÞ ðt tb Þ2 2
ðt tb Þ2
tc ¼ tb
Note that the position xc ¼ tcva is the displaced location in azimuth where the moving target appears in the conventionally processed SAR image.
v2 2vx va þ v2a þ ð1 a2 Þvy v2rel ¼b x Rb Rb va ffi 2b vx þ const: Rb
ð7Þ
for Rb vx , vy , where the ratio va/Rb is of the same order for airborne and space-borne systems. 4.3
Doppler centroid (across-track velocity)
4.3.1 Matched filter banks: Analogously, a matched filter bank for the across-track velocity component vs can now be used with varying parameter vector j 0 ¼ [^vrel , vs0 , 0]T. In order to avoid ambiguous impulse responses, the length of the reference function should not exceed the length of the target trajectory. Fig. 4a shows the filter bank map where vs0 is varied from 210 to 10 m/s IEE Proc.-Radar Sonar Navig., Vol. 153, No. 3, June 2006
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Fig. 3 Normalised amplitude (dB) of along-track MFM a Fore channel b Channel difference (DPCA)
with the maximum of v^ s ¼ 3.85 m/s at time tb ¼ 20.825 s, which is in fact identical to broadside time (provided the track is perfectly centered around tb , see Appendix). However, since the magnitude of the point spread function depends only slightly on vs , this technique tends to be less accurate and more sensitive to errors than methods based on the phase information, such as ATI phase (Section 4.3.2). Inserting this MLE, that is v^ s , into the reference function and repeating the vrel matched filter bank results in Fig. 4b. The ridge is now correctly shifted to the broadside time tb and also straightened out. One can show that the center of the impulse response is independent of a varying mismatch in vrel as long as v s0 2 vs ¼ 0, that is the correct vs has been included in the reference function. This property can be utilised in an adaptive estimation scheme which varies v s0 until the fan in the vrel matched filter map (MFM) is straightened. This so-called ‘straightening filter’ is described in more detail and applied to experimental data in Section 4.5. 4.3.2 ATI phase: Alongside matched filter banks there also exists the possibility to estimate the velocity vs more robustly from the interferometric phase f after compression of the target trajectory with optimum estimate v^ rel and, for instance, v s0 ¼ 0. Using the Taylor series expansion of the
x0 þ ðvx va Þt vx va ’ ðt tb Þ Rðt; jÞ Rb
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the ATI phase at t ¼ tc is calculated in Appendix to be bd 1 ðvx va Þ vs fðjÞ ¼ vs þ eðjÞ ffi b d ð9Þ 2 va va v2rel |fflfflfflfflffl{zfflfflfflfflffl} ’v1 a
Since the error term e (j) ¼ ((be(j))/2)dt, caused by an unsymmetrical target trajectory around broadside time, is usually small, it can be neglected. Such non-symmetries can be caused by RCS variation due to fading of the vehicles’ signature. It is interesting to note that according to (18), the ATI phase only equals the well-published term b dvs/va exactly when the filter is also perfectly matched to the across-track velocity, that is, v s0 ¼ vs . This, however, is impossible since vs itself is the desired parameter. Taking, for instance, the phase value at maximum magnitude in Fig. 5, vs can be estimated by solving (9) using the obtained estimate v^ rel . It has been shown that the ATI phase is compromised by strong clutter [9], that is a relatively strong bias of the estimate may be introduced for low SCR. However, because of a commonly large compression gain of the
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targets location but will also enhance the clutter suppression performance. However, even with fully adaptive STAP the DoA estimation performance is significantly compromised by the fact that the optimum Wiener filter (inverse of covariance matrix) is not known and has to be estimated as well.
matched filter, the SCR is expected to be sufficient for most practical cases. 4.4
Target broadside location
4.4.1 Direction-of-arrival estimation: As it can be seen from (5), the estimated velocity v^ s is directly coupled to the broadside time estimate ^tb; small errors in v^ s usually translate into relatively large position errors x^ b , particularly for space-borne systems (since this time must be multiplied with the platform speed va). A multi-channel SAR can overcome this shortfall by adding additional degrees of freedom to the estimation process. Here, the angle u(t, j) can be independently estimated by beamforming techniques to reveal the target position [4]. These direction-of-arrival (DoA) estimators cannot directly be applied to two-channel SAR systems, since only one degree-of-freedom is left after clutter suppression, for example, via DPCA. This general dilemma can be (at least in principle) overcome by fully adaptive STAP, that is, using multiple time sample along with the two spatial samples to form a larger space-time vector [12]. The increased dimensionality offers the possibility to suppress the clutter and at the same time preserve (to a certain degree) the two spatial DoF necessary to estimate the angle to the target. One practical way in the Doppler domain to achieve this is the so-called PRI-staggering scheme. Herein, several segments of measured channel output data are Fourier transformed, whereby each segment is exactly shifted by one pulse (staggered by one PRI). Although these signals are mutually shifted to each other in the time domain they do coincide exactly in the frequency domain. By concatenating the values of all segments for one particular frequency bin to a large vector, we get the increased dimensionality. Choosing the segment length sufficiently long, the individual frequency samples will be (approximately) mutually stochastically independent, which offers the opportunity to suppress the clutter at each frequency separately. As the staggered segments are highly overlapped (correlated) it has been shown that the rank of the underlying spacetime covariance matrix does not grow at the same rate as the dimensionality of the data vector [13]. This technique does not only allow (in principle) the estimation of the 228
4.5
Cubic phase term (along-track acceleration)
As recently shown, [15], the along-track acceleration ax is another important parameter which has to be taken into account both for detection and parameter estimation. Since this parameter shows up as an additional cubic phase term it does not only significantly effect the geometric resolution (e.g. the width of the point spread function or the sidelobe distribution), but it compromises the estimation accuracy of the aforementioned parameters. In contrast, the across-track acceleration will be an additional component of the quadratic coefficient and hence is ambiguous with the along-track velocity vx . In the presence of along-track acceleration, the measured signal has to be extended to sðt; jÞ ¼ expðj2bRðt; jÞÞ v2rel 2 3 ¼ exp j 2bvs ðt tb Þ þ b ðt tb Þ þ a3 ðt tb Þ Rb ð10Þ where j ¼ [vrel , vs , tb , a3]0 . In general, a3 comprises all parameters leading to the appearance of a cubic phase term; an analytical expression is given in [11, 15]. 0 90 –5
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Fig. 5 Polar plot of ATI phase against normalised interfero0 metric magnitude of each pixel after compression with v rel ¼ v^ rel and v s0 ¼ 0
4.4.2 Fractional Fourier transform: A different class of techniques tries to suppress the clutter for each channel individually without compromising the two degrees of freedom required for DoA estimation. Since classical spectral filtering of the clutter is impossible because of the extended target bandwidth (that usually covers most of the unambiguous bandwidth), it has been proposed to suppress the clutter in the fractional Fourier (FrF) domain, or ‘fractrum’ [14]. If the optimum rotation angle for the FrF transform is used, most of the target’s energy will be compressed in a few fractrum bins and a relatively narrow band-pass filter around the target location will remove most clutter energy. After inverse FrFT, the DoA time history can be retrieved. Fig. 6 illustrates the ability to reduce the clutter contamination by ‘fractral’ suppression on the signal of the fore channel; an almost 30 dB suppression gain in the MFM compared to Fig. 3a is observed.
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Fig. 7 Delta control target from Petawawa 1999 experiment
The impact of acceleration will generally be less severe for space-based sensors than for airborne SARs because of the different synthetic aperture times and the much higher platform velocity of a satellite [11]. A major problem one faces is ignorance as to whether a significant amount of acceleration occurs during the observation time, that is, a non-trivial detection of acceleration is required. One possibility is mentioned in [11] or [16], where a time-frequency analysis is applied to estimate the instantaneous frequency history. A subsequent polynomial fit may yield an estimate about the cubic coefficient. However, this technique may suffer from limited space – time resolution. Acceleration can also be detected by analysing the matched filter output or MFM introduced in Section 4.2.2. More accurately, one can detect typical deterioration patterns and deviations from the undisturbed ‘fan’ structure, for instance, plotted in Fig. 6. As a case study an exemplary control target is investigated with respect to acceleration. This controlled moving target (denoted as Delta) consisted of a corner reflector and a GPS system mounted on a cart that moved on a 650 m long rail track. This transporter was a remotely controlled, enginepowered car guided by the rail system, which were programmed to move at predetermined speeds. Fig. 7 shows the used Delta system. Some accelerations are expected due to the relatively short track length, which required significant acceleration to achieve the desired test speeds followed by significant deceleration to slow them down before the end of the track. Further, it was very difficult
to coordinate and synchronise the trials exactly such that the airborne system was imaging these targets only when they moved at constant velocities. Fig. 8 (left) shows the MFM of the Delta target for a 0 , 0, 0, 0]0 . matched filter parameter vector j 0 ¼ [v rel Typical deterioration of the ‘fan’ because of the cubic term mismatch is recognisable, namely a one-sided bent and smeared energy distribution of the main peak. An estimation of vs via across-track matched filter yields 0 , 26.5, 0, 0]0 v^ s ¼ 26.5 m/s, which inserted into j 0 ¼ [v rel causes the unsymmetrical ‘fan’ to straighten out as good as possible (Fig. 8 (right)). Fig. 9 (left) shows the point spread function at the optimal value v^ rel ¼ 126.36 m/s (cut through MFM at this value); again, the typical raised one-sided peak sidelobes are clearly visible. The polar plot on the right identifies a corresponding ATI phase of fˆ ¼ 233.528, which relates to v^ s ¼ 24.86 m/s via (9). The MLE for the parameter vector demands a global maximisation of the point spread function with respect to all parameters [1]. Obviously, such maximisation is not only numerically difficult but the computational load is intolerable. The complexity of the problem can be significantly 0 simultaneously for reduced by only considering a30 and vrel the maximisation, because vs0 merely causes a shift, and hence virtually has no impact on the magnitude. Fig. 10a shows the peak magnitudes of the point spread function 0 [ f90, 175g m/s and a30 [ f25, 15g Hz/s2, partialong vrel tioned into 300 and 200 values, respectively. The maximum value occurs at a^ 3 ¼ 8.65 Hz/s2, indicating a significant acceleration of Delta. As the computation time was about 10 min on a Pentium IV in MATLAB, faster techniques must be developed for real-time applications. Inserting the optimum value a^ 3 into the matched filter yields a slightly different ATI phase estimate of fˆ ¼ 236.18, which relates to v^ s ¼ 24.76 m/s (see Fig. 12 (right)). Alternatively, the output function of the ‘straightening filter’ is plotted after insertion of a 30 ¼ a^ 3 ¼ 8.65 Hz/s2 in Fig. 10b. This function represents the standard deviation 0 0 [ f^vrel 2 between M peak positions tm for varying v rel 0.5 m/s, v^ rel þ 0.5 m/sg in close neighbourhood around position ^tb corresponding to the optimum estimate v^ rel:
s2 ¼
ð11Þ
The standard deviation s is now calculated for different values of v s0 in the matched filter. As mentioned earlier,
b3 = 0, vs = 0
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Fig. 9 Point spread function of Delta target (left) and polar plot of ATI phase against interferometric magnitude (right) at optimum relative velocity and v s0 ¼ 0
the closer the matched filter parameter v s0 gets to vs , the straighter the ‘fan’ in the MFM and, hence, the smaller the deviations from the optimum peak position ^tb . This ‘fan’ behaviour is illustrated in Fig. 11 by means of two examples, v s0 ¼ 0 and v s0 ¼ 25.106 m/s, where the latter relates to the optimum value read from Fig. 10b, which appears to be a bit smaller than the ATI phase-based estimations. Two observations are apparent. First, the smearing and bending of the energy distribution along vrel-direction is virtually eliminated compared to Fig. 8. Second, after inserting the optimum across-track velocity estimate v^ s into the matched filter the ‘straightest line’ occurs at a different point in time ^tb ¼ 24.3752 s. A different look of the improvement is given in Fig. 12 (left), where the cut through the MFM at the optimum vrel estimate (Fig. 11 (right)) is plotted two-dimensionally. The peak magnitude of the point spread function is about 12% larger than the corresponding one in Fig. 9 (left) and the one-sided sidelobes have almost entirely disappeared. The 3 dB resolution was measured as 0.38 m. This, in fact, is close to the theoretically expected value of va dx ffi 0:866lRb 2 T 2vrel This theoretical value for the 3 dB resolution of the compressed target can be calculated from (14) in Appendix.
After putting v s0 ¼ 0 and e(j) ¼ 0 and approximate b(t) 2 a(t) ffi T (coherent processing time), the width of the impulse response is given as 0.866 times the time tx of the first zero-crossing of the sinc-function, that is, when its argument equals p. Hence, tx ¼ 0.866lRb/ (2Tv2rel) or for the spatial width dx ¼ txva . For the Convair SAR and the maximum coherent processing time T ¼ 5.66 s for Delta (detected trajectory duration) the resolution should be about 0.33 m. The ATI phase for values within the mainbeam are now almost identical, leading to an almost perfect straight line in the polar plot (Fig. 12 (right)). Fig. 13 shows a close-up of the region around Delta rail track. Superimposed onto the stationary SAR image are the estimated locations listed in Table 1 along with the other corresponding parameter estimates. It is evident that neglecting significant acceleration can lead to replacement errors of about 120 m for this example. In contrast, the broadside time in the last row (‘straightening filter’) agrees perfectly with the track position, which is located immediatly to the left of the white line in the image. The ATI phase results in an estimate which is about 20 m off the track. It is interesting to note that in contrast to the constant velocity case (first two rows), the optimum estimate for the relative motion depends on the choice of the across-track velocity vs in the presence of a cubic phase term.
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140
b3 = 8.65, vs = –5.1
125
142 144 130
148
vrel [m/s]
vrel [m/s]
146
150 152
135
154 156 158 160 4.5
140 5 Slow time [s]
-4.55 -4.5 -4.45 -4.4 -4.35 -4.3 -4.25 -4.2 Slow time [s]
5.5
Fig. 11 MFM of Delta target for a 30 ¼ 8.65: v s0 ¼ 0 (left) and v s0 ¼ 25.1 m/s (right)
Fig. 12 Point spread function of Delta target (left) and polar plot of ATI phase against interferometric magnitude (right) with a 30 ¼ 8.65 Hz/s2 and v s0 ¼ 25.1 m/s
0.98
¥104
Slant Range [m]
0.99 1 1.01 1.02 1.03 1.04 1.05 1.06 –800
–600
–400
–200
0 Along Track [m]
200
400
600
800
Fig. 13 SAR image of Delta track including the different tb estimates
5
Table 1: Parameter estimates a3 , s23
vrel , m/s
0
126.36
0
0
126.36
26.5
8.65
152.26
0
8.65
131.92
24.86
24.52
8.65
130.81
25.1
24.37
vs , m/s
IEE Proc.-Radar Sonar Navig., Vol. 153, No. 3, June 2006
tb , s 4.47 23.49 5.13
Conclusions
This paper studied different techniques to estimate moving target parameters in two-channel SAR – GMTI. All techniques are evaluated on the basis of experimental airborne SAR data. Several conclusions can be drawn for the different parameter estimators. The relative velocity vrel can be reliably and relatively quickly estimated via a matched filter bank, which is valid both for airborne and spacebased systems. Estimation of the across-track velocity vs 231
can be achieved through the ATI phase which is more accurate than a matched filter. A novel strategy for estimation of vs called the ‘straightening filter’ has been shown to yield an even more accurate estimate, for the particular airborne data set used, at the cost of higher computational load. However, more testing and comparing of the two methods is required for conclusive answers, particularly for targets of opportunity (non-ground truth targets). In presence of significant acceleration of the target, a simple but time-consuming maximum-likelihood estimation scheme was proposed to accurately estimate the cubic phase term. Reducing the computational complexity of such an estimator will be left for future work. The ATI phase appeared to work robustly even in cases of accelerating targets. The most challenging parameter to estimate is arguably the correct broadside location, since its strongly correlated to the vs estimate. Small errors in the latter will incorporate into rather large displacements. Techniques, which try to de-couple the two parameters, such as the mentioned multi-aperturebased DoA estimation procedures, are currently under investigation. 6
References
1 Ender, J.H.G.: ‘Azimutpositionierung bewegter Ziele mit MehrkanalSAR’. U.R.S.I. Jahrestagung (Kleinheubacher Berichte), Kleinheubach, Oktober 1995, vol. 38, pp. 749 –758 2 Gierull, C.H., and Livingstone, C.: ‘SAR-GMTI concept for RADARSAT-2’, in Klemm, R.: ‘The applications of space-time processing’ (IEE Press, Stevenage, UK, 2004) 3 Livingstone, C., Sikaneta, I., Gierull, C.H., Chiu, S., Beaudoin, A., Campbell, J., Beaudoin, J., Gong, S., and Knight, T.: ‘An airborne SAR experiment to support RADARSAT-2 GMTI’, Can. J. Remote Sens., 2002, 28, (6), pp. 1–20 4 Ender, J.H.G.: ‘Space-time processing for multichannel synthetic aperture radar’, Electron. Commun. Eng. J., 1999, 11, (1), pp. 29–38 5 Gierull, C.H., and Sikaneta, I.: ‘Raw data based two-aperture SAR ground moving target indication’. Proc. IGARSS’03, Toulouse, France, 2003 6 Meyer-Hilberg, J., Bickert, B., and Schmid, J.: ‘Flight test results of a multi-channel SAR/MTI real-time system’. Proc. EUSAR’02, Cologne, Germany, 2002, pp. 209–212 7 Gierull, C.H.: ‘Statistical analysis of multilook SAR interferograms for CFAR detection of ground moving targets’, IEEE Trans. Geosci. Remote Sens., 2004, 42, (4), pp. 691– 701 8 Sikaneta, I.C., and Chouinard, J.-Y.: ‘Eigendecomposition of the multi-channel covariance matrix with applications to SAR-GMTI’, Signal Process., 2004, 84, (9), (special issue), pp. 1501–1535 9 Gierull, C.H.: ‘Moving target detection with along-track SAR interferometry – a theoretical analysis’. Tech. Rep. DRDC TR 2002-084, Defence Research & Development Canada – Ottawa, Canada, 2002 10 Stockburger, E.F., and Held, D.N.: ‘Interferometric moving ground target imaging’. Proc. IEEE Int. Radar Conf., 1995, pp. 438 –443 11 Sharma, J.: ‘The influence of target acceleration on parameter estimation and focusing in SAR imagery’. MS thesis, University of Calgary, 2004 12 Klemm, R.: ‘Space-time adaptive processing’ (IEE Press, Stevenage, UK, 1998) 13 Ward, J.: ‘Space-time adaptive processing for airborne radar’. Tech Rep. 1015, Lincoln Laboratory MIT, USA. 1994 14 Gierull, C.H.: ‘Azimuth positioning of moving targets in two-channel sar by direction-of-arrival estimation’, Electron. Lett., 2004, 40, (21), pp. 1380–1381 15 Sharma, J., and Collins, M.J.: ‘Simulations of SAR signals from moving vehicles (focussing accelerating ground moving targets)’. Proc. EUSAR’04, Ulm, Germany, 2004, pp. 841 –844 16 Rieck, W.: ‘Zeit-Frequenz-Signal-Analyse fu¨r Radaranwendungen mit synthetischer Apertur’. PhD thesis, Rheinisch-Westfa¨alische Technische Hochschule Aachen, 1998
7
w2 ðt; jÞ ¼ 2bRðt; jÞ þ buðt; jÞd
ð12Þ
where the distance R(t, j) is given in (4) and the directional cosine u(t, j) in (8). Please note, the phase signal for the first channel is determined by simply setting d ¼ 0 in (12). In order to co-register the two channels the aft channel must be time-shifted by d/(2va) d d w2 t þ ; j ¼ 2 bRb þ 2 bv s t t b þ 2va 2va 2 2 v d þ 2b rel t tb þ 2va 2Rb vx va d þb t tb þ d 2va Rb vs ð13Þ ¼ w1 ðt; jÞ þ b d þ beðjÞðt tb Þ va where e(j) ¼ ((v2reld/Rbva) þ ((vx 2 va)/Rb)d). The signal of channel two is then s2(t) ¼ exp( jw2(t þ (d/2va), j)) rectf(t 2 tb 2 dt)/Tg, where dt ¼ (Tu þ To)/2 2 tb describes the deviation of the target trajectory center from broadside time tb; Tu and To are the start and end time of the target track. Such a deviation has been found to be common in practical data because of the aspect angle dependency of the RCS and imbalanced clutter contamination along the target track. Azimuth compression is done with the conventional filter h(t, j 0 ) ¼ exp( jw1(t, j 0 )) rectft/Tg with j 0 ¼ [v2rel, v s0 , 0]. Note that the Doppler rate parameter is assumed to be known, that is, matched to the true one. Under this assumption, the matched filter output p(t) ¼ s2(t) h (2t) is derived to ð1 0 e jw2 ðttþðd=2va Þ;jÞ ejw1 ðt;j Þ pðtÞ ¼ 1
nto t t tb dt rect rect dt T T ¼ e jw1 ðt;jÞ ebððvs =va ÞdþeðjÞðttb ÞÞ ð bðtÞ 2 0 ej2bðvrel =Rb Þðttb Þt ej2bðvs vs Þt ejbeðjÞt dt aðtÞ
¼e
jw1 ðt;jÞ bððvs =va ÞdþeðjÞðttb ÞÞ
e
2
0
ej2bððvrel =Rb Þðttb Þþðvs vs ÞþðeðjÞ=2ÞÞðbðtÞþaðtÞ=2Þ ðbðtÞ aðtÞÞ 2 vrel eðjÞ bðtÞ aðtÞ 0 ðt tb Þ þ ðvs vs Þ þ sinc 2b 2 2 Rb t t b dt rect ð14Þ 2T As illustrated in Fig. 14, the sum and difference of the integration limits can be analytically expressed as t tb dt bðtÞ þ aðtÞ ¼ ðt tb dtÞ rect 2T
t t b dt bðtÞ aðtÞ ¼ T jt tb dtj rect 2T
ð15Þ
Appendix: ATI phase
This appendix calculates the exact expression for the ATIphase for the far-field model (2). Re-calling the phase of 232
the aft channel to be
Inserting (15) into (14) yields for the peak location of the aft channel tc as the zero of the argument of the sinc-function ((v2rel/Rb)(tc 2 tb) þ (vs 2 v s0 ) þ (e(j)/2)) IEE Proc.-Radar Sonar Navig., Vol. 153, No. 3, June 2006
Fig. 14 Illustration of integration limits for dt ¼ 0
(T 2 jtc 2 tb 2 dtj) ¼ 0,
and by inserting (16) leads to
ðv0 vs ÞRb eðjÞRb tc ¼ tb þ s 2 vrel 2v2rel ¼ tb þ
ðv0s
vs ÞRb d ðvx va Þd 2 2va 2v2rel vrel |fflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflffl ffl}
ð16Þ
’0
Equation (16) confirms that the peaks of both channel signals occur virtually at the same location and the ATI phase can be computed as the difference between the phases at this location. Hence, using (14) and (15), the ATI phase f(j) ¼ w2(tc , j) 2 w1(tc , j) at peak location tc can be expressed as vs fðjÞ ¼ b d þ eðjÞðtc tb Þ va 2 v eðjÞ b rel ðtc tb Þ þ ðvs v0s Þ þ ðtc tb dtÞ ð17Þ 2 Rb
IEE Proc.-Radar Sonar Navig., Vol. 153, No. 3, June 2006
vs eðjÞ ðv0s vs ÞRb eðjÞ fðjÞ ¼ b d þ b þb dt 2 2 va v2rel vs d 1 vx va eðjÞ þ 2 dt ¼b d þb ðv0s vs Þ þ b 2 va 2 va vrel ð18Þ where eðjÞ ¼
v2rel d d þ ðvx va Þ ffi vx Rb Rb va
ð19Þ
for va vx .
233
