- Research Article
- Open Access
Moving Target Indication via RADARSAT-2 Multichannel Synthetic Aperture Radar Processing
© S. Chiu and M. V. Dragošević. 2010
- Received: 29 June 2009
- Accepted: 20 October 2009
- Published: 10 December 2009
With the recent launches of the German TerraSAR-X and the Canadian RADARSAT-2, both equipped with phased array antennas and multiple receiver channels, synthetic aperture radar, ground moving target indication (SAR-GMTI) data are now routinely being acquired from space. Defence R&D Canada has been conducting SAR-GMTI trials to assess the performance and limitations of the RADARSAT-2 GMTI system. Several SAR-GMTI modes developed for RADARSAT-2 are described and preliminary test results of these modes are presented. Detailed equations of motion of a moving target for multiaperture spaceborne SAR geometry are derived and a moving target parameter estimation algorithm developed for RADARSAT-2 (called the Fractrum Estimator) is presented. Limitations of the simple dual-aperture SAR-GMTI mode are analysed as a function of the signal-to-noise ratio and target speed. Recently acquired RADARSAT-2 GMTI data are used to demonstrate the capability of different system modes and to validate the signal model and the algorithm.
- Synthetic Aperture Radar
- Synthetic Aperture Radar Image
- Pulse Repetition Frequency
- Virtual Channel
- Parameter Estimation Algorithm
Due to the significant clutter Doppler spread that is imparted by a fast-moving space-based radar (SBR) platform (typically over 7 km/s) and the large footprints (of the order of kilometers) that result from space observation of the earth, detection of airborne and ground vehicles is a difficult problem. Strong mainbeam clutter can impede even the detection of large targets unless it is suppressed, in which case the detection of small targets might still be hindered by possible sidelobe clutter. Therefore,efficient ground moving target indication (GMTI) and target parameter estimation can be achieved only after sufficient suppression of interfering clutter, particularly for space-based SARs with typically small exoclutter regions (clutter-free Doppler bands in the spectral domain). In its simplest form, this is accomplished using two radar receiver channels, such as the dual-receive antenna mode of RADARSAT-2 (R2) Moving Object Detection EXperiment (MODEX).In this mode of operation, the full antenna is split into two subapertures with two parallel receivers to create two independent phase centers. It is known, however, that two degrees of freedom are suboptimum for simultaneous suppression of the clutter and estimation of targets' properties, such as velocity and position . Parameter estimation is often compromised and limited by clutter contamination of the target signal . This deficiency has led to exploration of means of increasing the spatial diversity for RADARSAT-2. One such method is the so-called "sub-aperture switching" or "toggling" to create virtual channels , a technique originally proposed by Ender . From January to May 2008, the RADARSAT-2 satellite underwent a set of on-orbit commissioning tests, which included the MODEX mode set. Three variants of the originally proposed virtual multichannel concepts  (collectively called MODEX-2) have been successfully evaluated using the RADARSAT-2 satellite, in addition to the standard dual-channel mode (referred to as MODEX-1), and impressive MODEX data sets, to be presented in this paper, have been collected.
This paper first describes the MODEX modes that have been investigated to date in Section 2 with preliminary test results also presented. In Section 3, a set of equations of motion of a ground moving target is derived for a multichannel spaceborne SAR. These equations of motion are shown to be applicable to both airborne and spaceborne stripmap imaging geometries. Assuming that the SAR platform state vectors (position, velocity and acceleration) will be available, these equations serve as a physical basis for the development of a parameter estimation algorithm, called the Fractrum Estimator, in Section 4. The effects of clutter contamination are analysed in Section 5 explaining why MODEX-1 is suboptimum. Fractrum Estimator is then applied to recently acquired RADARSAT-2 MODEX-1 and MODEX-2 data and the results are presented in Section 6, followed by concluding remarks in Section 7.
1.2. Background Work
The history of synthetic aperture radar dates back to 1951 when Carl Wiley of Goodyear postulated the Doppler beam-sharpening concept , but unclassified SAR papers only appeared in the literature a decade later . The effects of moving targets in a SAR image were first discussed and published by Raney  in 1971, twenty years after the conception of SAR. Before the launches of German TerraSAR-X , Canadian RADARSAT-2 , and Italian COSMO-SkyMed  in 2007-2008, spaceborne SARs were only single-aperture systems. Such systems have a very limited GMTI capability due to dominant radar clutter, which prevents slowly moving targets from being detected. The three SAR satellites mentioned above are the first (in the unclassified world) to be equipped with a phased array, programmable antenna, and two physical receiver channels, permitting multiple independent phase centers (or virtual channels) to be synthesized. Although first advertised in  as GMTI capable SAR satellites with a few proposed modifications, COSMO Sky-Med have yet to produce their first GMTI results. TerraSAR-X and RADARSAT-2, on the other hand, have collected numerous GMTI data and the results have been published in several papers, for example, [12–19].
There are two major approaches to detection of ground moving targets with a multichannel SAR: Space-Time Adaptive Processing (STAP) and Along-Track Interferometry (ATI). A comparison of the two techniques has recently been presented in  and an excellent review of these two methods and others is given in . The ATI is a nonadaptive method, which requires proper channel coregistration and balancing for it to work. Many research groups have developed detection algorithms based on these two approaches. The groups adopting mainly the ATI methodology include, for example, [22–25] and those following the STAP stream are, for example, [26–28]. In the following, SAR-GMTI processing algorithms developed by the German Aerospace Center (DLR) and the Institute for High Frequency Physics and Radar Techniques (FGAN-FHR) are discussed in more details, as they have adopted two very different approaches and assumptions for the detection and estimation of ground moving targets.
The DLR researchers have adopted very similar techniques as our group, namely using the ATI and/or the Displaced Phase Center Antenna (DPCA) in combination with a Matched Filter Bank (MFB)  for the detection and estimation of ground movers . The fundamental difference between their approach and ours is the DLR's assumption that vehicles travel on roads of a known road network, which provide a priori information that can be effectively exploited . Although not valid in military scenarios, the assumption is definitely legitimate for civilian applications such as traffic monitoring (except for marine or dense urban traffic). With this a priori knowledge, detections from an ATI (across-track) detector and an MFB (along-track or Doppler rate) detector can be weighted accordingly depending on the road orientation . Also, the target range (across-track) speed can be accurately estimated from the azimuth displacement from the road based on the ATI phase of the target. In addition, the along-track speed can be derived from the range speed using the road orientation as a priori knowledge. Interestingly, once the along-track speed is known the acceleration of the target (if any) can also be inferred based on the estimated Doppler rate (from the MFB) that best focuses or maximizes the target energy. The Fractrum Estimator described in this paper is an alternate way of accomplishing what an MFB does, namely, estimating the true target Doppler (FM) rate by maximizing the target energy.
The FGAN-FHR has adopted primarily the STAP approach to SAR-GMTI for their airborne PAMIR system. A post-Doppler STAP clutter cancellation scheme was implemented, which permits the asymptotic decoupling of the different Doppler frequency contributions given that the time base is sufficiently long for the case of a SAR acquisition [20, 26]. A two-stage detection scheme was applied: the predetection and the postdetection. Since the PAMIR is a multifunction, multifrequency X-band (i.e., five sub-bands) radar, the predetection is performed on each sub-band as part of an elaborate CFAR detector [30, 31]. The target radial speed is estimated from the analysis of the Doppler frequency of the received pulses induced by both the target motion and the known platform velocity. The target localization is accomplished via the estimation of the target azimuth direction in the antenna coordinate system using the maximum-likelihood method . For the existing spaceborne SAR-GMTI systems like TerraSAR-X and RADARSAST-2, equipped with only two physical receiver channels, a similar approach is not very effective, unless a sub-aperture switching (or toggling) scheme is used in order to generate multiple virtual channels. The performance of Direction-Of-Arrival (DOA) approach using such a sub-aperture antenna switching was presented in [32, 33]. We note that similar limitations exist for the ATI method for radial speed estimation and for the DOA-based estimator of the radial speed described in  as the Azimuth Displacement Indicator (ADI).
With a sub-aperture switching or toggling scheme (as presented in the next section for RADARSAT-2), however, there is always a trade-off between more phase centers and a reduced SNR (Signal-to-Noise Ratio) as several transmitter and/or receiver elements are turned off during the switching process. In the case of RADARSAT-2, a duty cycle (or maximum transmit power) constraint forces the pulse length to be reduced by half (from 0.42 to 0.21 s) when a switching mode is employed. This would further reduce the achievable SNR. However, a performance improvement to target parameter estimation using the sub-aperture switching methodology has been established theoretically in [32, 33] and will be here demonstrated using recently acquired RADARSAT-2 MODEX data in Section 6.
The proposed virtual channel modes take advantage of the flexible programming capabilities of the RADARSAT-2 antenna to generate two, three, or four phase centers, as illustrated in Figure 1, using a sub-aperture switching (or toggling) technique originally proposed by Ender . The spatial diversity of the standard dual-receive mode, Figure 1(a), can be increased by either transmitter toggling between pulses, Figures 1(b) and 1(c), or smart receiver excitation schemes, Figure 1(d). These are only a few methods for achieving multichannel capability and are by no means exhaustive. Due to transmitter/receiver toggling between pulses, the pulse repetition frequency (PRF) per virtual channel is effectively cut by one half. This may lead to clutter band aliasing (non-Nyquist sampling), which may be partially compensated for by doubling the original PRF.
The half-aperture, toggled-transmit (toggled-Tx) approach (between fore and aft subapertures), shown in Figure 1(b), has the advantage of maintaining the same phase-center distance (or the along-track baseline) as the standard dual-channel case (Figure 1(a)), which is nominally 3.75 m for RADARSAT-2, and is capable of generating three independent phase centers, shown as down-pointing triangles. The down/up arrows denote the transmitter/receiver physical phase center positions, respectively. However, the two-way beamwidth is significantly increased compared to the standard dual-channel case due to the half-aperture transmit. This could lead to clutter band aliasing (as confirmed by recently acquired MODEX data) even at RADARSAT-2's maximum PRF of 3800 Hz (or 1900 Hz per virtual channel). Also, the half-aperture transmit leads to a decrease in the transmit power and may severely limit the attainable SNR. The proposed solution to mitigate this shortcoming is to increase the transmitter aperture size from half to three-quarter aperture, as depicted in Figure 1(c). This sub-aperture switching configuration generates four independent phase centers (or virtual channels) as represented by down-pointing triangles at four different positions along the antenna.
The last approach is the toggled-receive (toggled-Rx) or sub-aperture switching mode where pulses are transmitted with the full aperture and returns are received using two alternating quarter subapertures as shown in Figure 1(d). Both (c) and (d) modes generate four independent phase centers and produce an effective phase-center distance that is one-half that of the standard dual-receive case. The (d) configuration has a slightly narrower two-way azimuth beam pattern than that of the (c) case.
High resolution Synthetic Aperture Radar (SAR) processing requires that a highly accurate imaging geometry model be first established. For SAR Ground Moving Target Indication (SAR-GMTI), the underlying assumption that the radar scene is stationary must be extended to include nonstationary scenes or moving targets. This can be quite easily accomplished for the case of an airborne platform , which is assumed to be moving along a straight line and transmitting uniformly spaced pulses. This assumption requires good platform motion compensation and good control of the PRF as a function of ground speed. The same cannot be said about a spaceborne platform, where the earth's gravitational force plays a key role in defining the platform trajectory and the velocity of the radar antenna footprint as it sweeps along the surface of the earth. The modeling of a moving target for a single channel spaceborne SAR geometry has already been accomplished to a high degree of accuracy by Eldhuset  and Curlander and McDonough . However, the extension of the model to include a SAR system that is equipped with multiple apertures is evidently absent in the open literature, partly because there were no existing spaceborne SAR systems in the unclassified world equipped with such a capability up until the recent launches of COSMO-SkyMed , TerraSAR-X  and RADARSAT-2  in 2007. In the following, equations of motion of a ground moving target for a multichannel spaceborne SAR are derived. The full derivation is presented here for the first time, although this model has been used in our previous work.
Several assumptions are used to simplify the model. The SAR pointing angles, measured from a reference pointing direction, are assumed to be small. The along-track speed of the target is assumed to be much smaller than the SAR platform speed, which is warranted in the case of spaceborne SAR and typical ground vehicles. It is also assumed that the rate of change is very slow for certain orbital parameters, such as the linear speed, which is true for nearly circular orbits. For the sake of generality, these assumptions are not incorporated in the statement of the problem. They are introduced, where appropriate, only to simplify the final formulae. For a different SAR system, they may be reviewed or removed at the expense of model complexity.
The relative position vector of a moving target with respect to an imaging SAR satellite, in the earth centered earth fixed (ECEF) system, can be written as
where indices "t" and "s" denote "target" and "satellite," respectively. A bold letter indicates a vector and the corresponding regular italic font (of the same symbol) represents the magnitude of the vector, and a bold upper case letter represents a matrix. In the ECEF frame, the earth motion is absorbed into the relative satellite motion.
The Doppler centroid and Doppler rate are proportional to and , respectively, where the dot and double-dot notations indicate first and second derivatives with respect to time. A common approach to the derivation of and is to start from the identity 
and to differentiate it with respect to time, where superscript denotes the vector (or matrix) transpose.
Differentiating both sides of (3) with respect to time, we get
where is the velocity vector of the moving target, is the velocity vector of the satellite, and
and the Doppler shift due to the target's radial speed is
Therefore, the total Doppler shift is given by
Using the following definitions
Equation (10b) can be rewritten as
is the so-called "effective velocity" often used in the spaceborne SAR processing to model the range equation and is the projection of the target velocity onto the direction of platform velocity (also called the along-track direction) and needs not to be parallel to the ground track.
The instantaneous slant range equation (or history) is the key to high precision SAR processing. Accurate estimation of the effective velocity allows complicated mathematical manipulations involving a satellite/earth geometry model to be avoided and a simple hyperbolic approximation to be adopted in most high precision SAR processing algorithms . The hyperbolic model can be further simplified and approximated using a second-order Taylor series expansion or a parabolic model without significantly incurring further loss of accuracy for typical RADARSAT-2 dwell times and resolutions. However, this may not be true in general.
If and in (5a) and (13) are evaluated at some arbitrary time , then the range equation can be approximated by the Taylor series expansion:
where and are now the target's down-range (or across-track) velocity and acceleration components, respectively, and is the broadside range of the moving target. In the vicinity of , therefore, the Taylor series expansion reads
The use of a parabolic model is convenient in the derivation of range equations for multichannel SAR systems. In the following, the range equation for the second aperture of a two-channel SAR is derived.
3.1. Local Frame of Reference
Then the third unit vector, which completes the local reference frame, is given by
We should point out that is not necessarily in the exact same direction as , as illustrated in Figure 4.
3.2. Transformation Matrix
We now derive the transformation matrix from the LF reference frame to the ECEF reference frame. To begin, we express the unit vector in the ECEF frame:
is obviously the horizontal velocity component of the radar and can be easily shown to be
where is the vertical velocity component of the radar platform and is given by
We are now ready to express the forward unit vector in the ECEF frame as
Finally, the transformation matrix from the LF reference frame to the ECEF frame  is simply
3.3. Antenna Look Vector
Let the ideal look direction of the antenna in the LF frame, with an off-nadir angle pointing at a zero Doppler point on the surface of the earth, be
The term, , is considered negligible  and is set to zero in (29b) and (30b). In the ECEF frame, the antenna look vector is then given by
Note that the look vector is not necessarily in the direction of the beam center, rather it points to the direction of the target of interest within the beam footprint.
3.4. Displacement Vector
Let denote the vector pointing from the effective phase center of the aft sub-aperture to the effective phase center of the fore sub-aperture in the LF frame, then can be expressed as
and and are the pitch and yaw angles (or the orientation) of the antenna, representing the attitude of the spacecraft in the LF frame of reference. In the ECEF frame, becomes
3.5. Range Equations for Multiple Phase Centers
A two-aperture SAR-GMTI system is again assumed in the following derivations with the understanding that the derived equations can be generalized to a multiaperture system. Let and denote the position vectors of the antenna's two effective (or two-way) phase centers in the ECEF frame, respectively. The aft antenna phase center is then displaced from the fore antenna phase center by . For the case of RADARSAT-2, the displacement vector is closely aligned with the radar's velocity vector . Perfect alignment would be optimal because it would allow the aft phase center to pass through the same ECEF position as the fore phase center with a time delay of , where is the distance between the two effective phase centers. This perfect alignment would also mean that the whole antenna is ideally steered, generating a zero Doppler centroid in the clutter Doppler spectrum. In the presence of a nonzero Doppler centroid, there exists a nonzero across-track component of , which translates into a small across-track baseline. In the case of a real spaceborne SAR-GMTI system, such as the RADARSAT-2 MODEX, this small cross-track component is always present and, therefore, must be compensated for or taken into account in the system modeling .
The slant-range vector from the aft antenna phase center to the target can, therefore, be expressed as
and are now measured in the slant-range plane. As the antenna footprint sweeps across the target, the pitch angle hardly changes (i.e., remains virtually constant) such that , resulting in . In the case of RADARSAT-2, and are usually small but nonzero such that the beam center is not located exactly at the zero-Doppler point on the surface of the earth (in the ECEF frame). This residual beam squint generates a small constant along-track interferometric phase, which is usually removed by the digital-balance processing of the signal channels and can, therefore, be ignored. For the sake of completeness, however, we shall keep the term in (38c). Then, the zeroth-order coefficient of the Taylor expansion of evaluated at arbitrary time can be expressed as
where it can be shown that , and the term can be neglected.
First, we derive the second term in (41f):
where we have assumed that the spacecraft attitude is not changing in the LF frame such that time derivatives of and (or ) are equal to zero in the imaging time interval. We also assume, for simplicity, that and are small (normally true for RADARSAT-2). Therefore, (42) becomes
Here, we need to find the first time derivative of (i.e., ), which can be shown to be
where terms of the type , , and cancel out in the second column of (44) and are, therefore, dropped. We can further simplify (44) by noting that , , and :
The last term in (46b) is ignored since the look vector is virtually perpendicular to .
where is virtually perpendicular to for ground moving targets, we can rewrite (47b) as
Finally, (49b) can be further simplified by noting that and , yielding
where we make use of (14a) and . The latter is the velocity of the beam footprint that moves along the surface of the earth and the approximation is mainly due to the fact that the satellite orbit is only approximately circular. Therefore, the first-order coefficient of the Taylor expansion of evaluated at time can be written as
Similarly, we derive the second-order coefficient of the Taylor series expansion of by taking the time derivative of (51c), which simply yields . Therefore, the Taylor expansion of (up to the second order) evaluated at arbitrary time can be written as
We are now ready to generalize the moving target range equation (53) for a multichannel SAR system (i.e., with multiple phase centers)
where , ; , , and are defined for time ; (for equidistant phase center 1, 2, 3, etc.); depends on and in a predictable way; , , and vary slowly with time and, therefore, may be evaluated anywhere in the neighborhood of .
If we choose to be the broadside time , then and become zero, resulting in
where subscript "b" denotes the broadside time. In order to generate an interferogram or a SAR-DPCA image, one normally performs coregistration of channels with respect to channel 1 ( ). Therefore, the range history (55) of a coregistered channel becomes
where terms containing or have been neglected in the equation.
where , ( ) is the length of the signal, and, for simplicity, is the idealized two-way antenna pattern.
The accuracy of the equations of motion derived above is validated and demonstrated in Section 6 by applying the parameter estimation algorithm, described in Section 4, to the recently acquired RADARSAT-2 MODEX data.
We describe here a target parameter estimation algorithm based on the fractional Fourier transform (FrFT)  and along-track interferometry (ATI) , also called the Fractrum Estimator, for the RADARSAT-2 imaging geometry. Equations that relate various target parameters are derived.
The FrFT, with fractional frequency variable and rotational angle , of a signal is defined as 
where, for being not equal to zero or a multiple of , the kernel is given by
The FrFT with parameter can be considered as a generalization of the conventional Fourier transform (FT). Thus, the FrFT for and reduces to the conventional and inverse FT, respectively.
where is the wavenumber, is the number of processed pulses, and time scaling has been applied. The scaling is necessary when passing from physical quantities to the normalized unitless variables used in the digital implementation of the FrFT.
The target signal amplitude is maximized in the fractional Fourier domain if in (64c) is set equal to zero, which yields a "sinc" function in (63) and the optimum fractional angle :
Similarly, the optimum fractional Fourier transform of the signal received at channel can be shown to be
Equation (68c) seems to indicate that is sensitive to both the across-track (or slant-range) speed and the along-track speed . This is due to RADARSAT-2's antenna squint, which introduces the along-track velocity dependence of in the direction of radar-to-target line of sight. The result differs from that of the nonsquint case, which depends only on the target's across-track velocity component . However, from (17b) the second term inside the square bracket of (68c) is shown to be
which is negligible compared to the first term in (68c) for typical RADARSAT-2 imaging geometries and common ground vehicles. This important result, which differs from that of an airborne case , essentially decouples the parameter estimation equations, even for large values. Therefore, the formula for estimating the target range speed simplifies to
where in (68c) can be estimated from the phase of the clutter interferogram (for any two coregistered channels) and is removed during the channel-balance processing and, therefore, dropped in (70).
The third and fourth terms in (71b) can be ignored compared to the first two terms, and is smaller than the spatial resolution of "sinc" functions, implying spatial overlap on the fractional Fourier axis or . Therefore, the target broadside azimuth position (= ) can be obtained directly from (71a) by solving for :
where is the measured target position on the unitless fractional Fourier axis and is obtained from the estimated that best focuses the target energy in the fractional Fourier domain using (65):
Finally, the target along-track speed is estimated from (17b):
where a nonaccelerating target is assumed (i.e., ). If the target is accelerating with a constant , then one must use other information to decouple and contained in . This case, however, will not be considered in this paper.
In summary, the equations used for estimating , , and are
satellite effective speed, calculated from satellite state vectors, satellite velocity in ECEF frame, distance from radar to detected target when it is at radar broadside, interferometric phase of coregistered channels (1 and ) focused via FrFT, channel number, wavenumber, distance between two effective phase centers, position of focused target on fractional Fourier axis, azimuth sampling or pulse repetition frequency, target signal length or the number of azimuth pulses, fractional angle or parameter used to best focus the target, target's estimated azimuth speed, target's estimated slant-range speed, and target's estimated broadside time.
The FrFT-ATI parameter estimation algorithm (or the Fractrum Estimator) can be summarized as consisting of the following nine steps
after detection, extract moving target signals from range-compressed, azimuth-uncompressed raw data for channels 1 and ;
apply DPCA processing on the signals;
find fractional parameter that best focuses the DPCA image of the moving target;
measure target position on the fractional Fourier axis;
using the optimum value, calculate ;
form an interferogram from the FrFT focused signals of the two coregistered channels;
measure the interferometric phase ;
solve (75) for , , and using , , , and ;
then , where is the target's broadside azimuth position.
As one of the key steps in the described Fractrum Algorithm, the extraction of the interferometric phase has a strong impact on the estimation accuracy of target's range speed and broadside time. In a dual-aperture configuration, in particular, is inevitably affected by clutter. At positions in the fractional Fourier domain, the focused moving target signature is superimposed on the signature of a stationary background target acquired at a different time, namely, later than the moving target. Since the returns of the two targets partially overlap in the time domain, the samples processed by the FrFT include, at least partially, the interfering stationary clutter. The interferogram is formed between the resulting superimposed samples in the two coregistered channels that can be modelled as
where and are the complex samples from the same location in the two channels, and are the sample amplitudes, and are the corresponding phases, and represent the amplitudes of the clutter and target component, respectively, is the phase of the clutter sample, is the common-mode phase of the target sample, is the channel-to-channel phase difference of the target sample, which is proportional to the target radial speed , and and represent two statistically independent noise processes. In this section, we will discuss some properties of the resulting interferometric phase.
Ignoring the system noise, and , and focusing on the clutter-dominated case, the interferogram can be represented as
where " " denotes complex conjugate, and the interferometric phase is
In this model, both the target and the clutter portion are assumed to be fully coherent because temporal decorrelation is negligible for short ATI baselines and the cross-track baseline has been compensated.
Since is used as an estimate of , the first property that is examined is the bias.
In (78), is the relative phase of the background clutter at the apparent position of the target. It is reasonable to assume that the probability density function (pdf) of the relative clutter phase, , is uniform and independent of the clutter amplitude pdf, , which may be unknown. It can now be shown that the conditional expectation
The case of is obvious from Figure 5(a), which shows a simplified phasor diagram of in the absence of noise, with chosen without loss of generality. Under the assumption of uniform , clutter phase (shown by a solid red line) is equally probable as (indicated by a dotted black line). It is easy to show that these two equally probable clutter realizations cause opposite offsets, , of about (represented by a solid and a dotted line). Clutter also induces an offset in relative to . Similarly, this offset has a zero average because clutter phase and have equal probability. In the case shown in Figure 5(a), for all , that is, is unambiguous and its mean is equal to the sum of the means of and , which is .
The case of can be represented by a phasor of assuming, without loss of generality, that , while is uniformly distributed. As shown in Figure 5(b), now the mover causes (and, similarly, ) to have an offset, , from 0, where for all . Considering that and are equally probable, as are and , and applying the same reasoning as above, both and are found to have a zero mean and so does .
The special case is interesting. It can be shown that for this case
which produces a zero phase on the average.
Unlike all of the above cases, if , neither clutter nor the mover is prevalent. There are some values of for which , thus mapping to a negative value when . In this case, the conditional expectation of the ATI phase depends on the resolution of the ambiguity in the implemented ATI algorithm and in (78). The equality is valid only if the ATI phase is not restricted to the interval and if a suitable ambiguity resolution is applied (e.g., using a priori information or Doppler offsets).
where depends on the ambiguity resolution policy and is bound on both sides by the probability that lies below a certain value. This result is obtained without presuming any particular pdf for . As expected, is a biased estimate of . According to (82b), bias is proportional to , that is, to the probability that exceeds a certain threshold. Thus, bias is more pronounced for long-tailed . Bias is also proportional to ; it is larger for faster moving targets, assuming invariant clutter pdf and signal-to-clutter ratio. The assumption of invariant signal to clutter ratio is true if samples extracted for FrFT processing include the interfering clutter echoes completely. However, as the temporal overlap between the interfering clutter and the moving target decreases in the time domain with increasing of the target, there is, in principle, a possibility to cut out a portion of the interfering clutter returns, thus improving the signal to clutter ratio. This option is of limited benefit in practice (as a consequence of a typically narrow exoclutter zone and the initial uncertainty about the actual target's broadside time ).
By applying the well-known formula , it is easy to show that
which is a monotonically increasing function of both and (on the interval of unambiguous interferometric phases). Therefore, under equal signal-to-noise conditions, we can expect a larger interval of variation of the interferometric phase around the ideal value for faster targets than for slower targets. From (84), it is also clear that suppression of the interfering clutter plays an important role, especially when dealing with faster targets.
The analysis also implies that this dependence on may be mitigated or minimized by canceling the clutter prior to forming the interferogram , using the sub-aperture toggling (or switching) techniques as described in Section 2. An excellent theoretical review of various receiver and transmitter switching strategies is provided in . If clutter is successfully suppressed, the interferometric system becomes noise limited in terms of ATI phase variance.
For the along-track speed estimation, the two trains detected in the Trenton area (Figure 8) have the along-track speed estimates of m/s and m/s for train 1 and train 2, respectively. Based on the geometry of the railway track with respect to the radar, the inferred true range speeds of 3.08 m/s and 12.20 m/s would translate to along-track speeds of 1.09 m/s and 9.15 m/s, respectively, which are consistent with our estimated values. Also, we have observed significantly greater spreads ( ) in measurements than in estimates. Typically, is two to seven times larger than , depending on the range speed of the target. This is expected since the along-track speed is estimated from the slope of the target Doppler history or the Doppler rate, which is directly proportional to , and is in turn dominated by the large satellite effective velocity . Therefore, the target Doppler rate is expected to be not very sensitive to . It is further noted that while the variance of the estimated range speed is a function of both the range speed and the SCR, the variance of the estimated along-track speed does not appear to be sensitive to the target range speed.
This paper first describes the various MODEX modes that were evaluated to date and shows that very high channel correlations ( 0.96) can be achieved by compensating for subbeam pointing errors and by operating the radar at near the DPCA condition. A set of equations of motion that accurately describes a ground moving target in spaceborne multichannel SAR imaging geometry is derived in the ECEF frame of reference using linearization for small angles as a function of the receive phase center. Using these equations of motion, a moving target parameter estimation algorithms is developed. A simple phasor analysis is used to describe the target-clutter interaction. The accuracy of the equations of motion, the effectiveness of the parameter estimation algorithm and the benefits of the advanced antenna configurations have been tested and demonstrated using recently acquired RADARSAT-2 MODEX data.
The authors are grateful to the RADARSAT-2 GMTI team for their support in the experiments. Special thanks go to Dr. Ishuwa Sikaneta who provided us with MATLAB routines to generate antenna patterns and channel correlation plots. They also would like to thank their visiting scientists Dr. Delphine Cerutti-Maori of German FGAN-FHR, who performed theoretical analyses on the MODEX system and led the entire benchmark experimental campaign and Mr. Thomas Jensen of German MOD for supporting all our MODEX trials.
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