1. Introduction

1.1. Motivation

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 [1]. Parameter estimation is often compromised and limited by clutter contamination of the target signal [2]. 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 [3], a technique originally proposed by Ender [4]. 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 [5] (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 [6], but unclassified SAR papers only appeared in the literature a decade later [7]. The effects of moving targets in a SAR image were first discussed and published by Raney [8] in 1971, twenty years after the conception of SAR. Before the launches of German TerraSAR-X [9], Canadian RADARSAT-2 [10], and Italian COSMO-SkyMed [11] 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 [11] 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 [20] and an excellent review of these two methods and others is given in [21]. 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) [29] for the detection and estimation of ground movers [24]. 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 [9]. 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 [24]. 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 [31]. 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 [13] 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.

2. RADARSAT-2 MODEX Modes

The 512 transmit-receive modules (TRMs) in the RADARSAT-2 two-dimensional active phased array are organized as 16 columns, as depicted by little rectangles in Figure 1, with 32 TRMs per column. All TRMs have independent control of transmitter phase and receiver phase and amplitude for both vertical and horizontal polarizations [34]. The phase and amplitude controls in the elevation dimension allow for the formation and steering of all beams. Transmitter phase control in the azimuth dimension allows the formation of the wider beams required for the Ultrafine resolution mode. This is accomplished by the deliberate defocussing of the beam [35].

Figure 1: RADARSAT-2 MODEX modes: (a) standard two-channel receive mode, (b) three-channel half-aperture toggle-transmit mode, (c) four-channel 3/4-aperture toggle-transmit mode, and (d) four-channel quarter-aperture toggle-receive mode. Shaded rectangles constitute active antenna panels with different shades representing different channels; down/up arrows represent transmitter/receiver physical center positions, respectively; black down-pointing triangles denote two-way effective phase centers.

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 [4]. 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.

The actual antenna patterns of the first three MODEX modes of Figure 1 have been estimated from recently acquired RADARSAT-2 MODEX data and are shown in Figure 2. The corresponding correlation plots between coregistered channels are also shown. The antenna patterns for the standard dual-receive mode (Figure 2(a)) and the -aperture toggled-Tx mode (Figure 2(e)) show that the clutter bands are adequately sampled using a PRF of 1900 Hz (per channel). The -aperture toggled-Tx mode, on the other hand, shows a 3 dB beamwidth of about 1800 Hz, which is just below the maximum sampling frequency of 1900 Hz (per channel). Often, the maximum PRF is not achievable due to the duty cycle limitation of RADARSAT-2. More realistic maximum PRF values often fall in the range of 3600–3700 Hz (or 1800–1850 Hz per channel). Therefore, clutter ambiguities can become quite severe for the -aperture toggled-Tx mode. It is important to note that ambiguities must be avoided or minimized, because they cause decorrelation between coregistered channels due to interpolation errors [36, 37] as seen in Figure 2(d) and often generate false moving targets as a result of the erroneous phase imparted on the ambiguous clutter. These interpolation errors are illustrated in Figure 3. To coregister channels, the spatially displaced channel signals are time-shifted, via interpolation, to align them in space. This time shift is accomplished by applying a phase ramp on the signals in the frequency domain as illustrated by a solid peach line in the figure. As the ambiguities fold back into the Doppler band, the phase ramp is incorrectly applied and imparts a positive or negative constant phase error (or bias) on these ambiguous clutter signals, depending on the sign of their original frequencies. Therefore, ambiguous clutter shows up as false moving targets in an interferometric SAR image. The constant phase errors imparted on the ambiguities can be derived from the Fourier transformation pair: , where is the slope of the phase ramp applied to signal in the frequency domain to effect the desired time shift on in the time domain. Therefore, the constant phase error can be shown to be where corresponds to sign in the original frequency of ambiguous signals, assuming a positive ramp slope, and and are the pulse repetition frequency (PRF) and pulse repetition interval (PRI), respectively. Signs are reversed in case of a negative ramp. Moreover, one observes from (1) that the interpolation does not lead to channel decorrelation if the PRF is chosen to be , such that the so-called Displaced Phase Center Antenna (DPCA) condition is met. Under this condition, there is no sub-sample interpolation (only integer sample shifting) and the phase errors imparted on the ambiguities are exactly multiples of , which have no effect on the signal (including both main and side beams). The channel-to-channel decorrelation can also be caused by beam pointing errors, as the beam footprints for different channels do not coincide perfectly, generating slightly different clutter Doppler centroid for each channel. This effect is clearly seen in Figure 2(c), where the Doppler centroids of different channels differ by up to a few hundreds of Hertz. The decorrelation caused by the beam pointing errors is most noticeable for the toggled modes, as seen in Figure 2(d), where a drop in the correlation is observed for the interpulse channels. Fortunately, the beam pointing errors can be easily compensated for by applying a corrective phase ramp across the elements of a phased array, as was done for the last case shown in Figure 2(e), where the errors are reduced down to less than a few tens of Hertz. With this corrective measure, there is now virtually no drop in the correlation between the interpulse channels and all the channels have now correlations over 0.96, as shown in Figure 2(f).

Figure 2: Estimated antenna patterns and channel correlations for three MODEX modes. (a) and (b) Standard dual-receive mode, (c) and (d) half-aperture toggle-transmit mode, and (e) and (f) 3/4-aperture toggle-transmit mode.

Figure 3: Illustrating the interpolation (or time shift) of an ambiguous clutter signal via the application of a frequency phase ramp.

3. Equations of Motion of a Moving Target

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 [38], 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 [39] and Curlander and McDonough [40]. 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 [11], TerraSAR-X [41] and RADARSAT-2 [13] 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 [40]

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

Equation (4b) can be rewritten as where is the velocity vector of the moving target, is the velocity vector of the satellite, and

are the projections of the target and satellite velocity vectors, respectively, onto the line of sight (LOS) or the radial direction. Also, the radial speed of a stationary target as “seen” by the radar due to the platform motion is equal to . Therefore, the Doppler shift at the beam center induced by the motion of the platform (or the stationary clutter Doppler centroid) is given by and the Doppler shift due to the target's radial speed is

Therefore, the total Doppler shift is given by

Again, differentiating both sides of (4a) with respect to time yields

Using the following definitions

Equation (10b) can be rewritten as

Therefore,

where 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 [42]. 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 considered negligible with respect to and, therefore, are dropped in (16d). If is chosen to be the broadside time , then by definition and the radial direction (subscripted by “”) becomes exactly perpendicular to the flight direction or the along-track direction (subscripted by “”). Under this condition, (16b) and (16d) become 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

In order to continue with our derivations, we first define a local flight (LF) frame of reference for the radar as shown in Figure 4, where is defined as the unit vector pointing down from the radar's center of gravity to the center of the earth. To define the second axis, we cross (vector) multiply with the radar's velocity vector to form the right pointing unit vector :

Figure 4: Local reference frame of radar.

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:

Then becomes

where

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 [43] 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

Then the actual antenna look vector (or pointing vector) in the local reference frame of the radar is given by where is the yaw-pitch rotation matrix, and and are the yaw and pitch angles about the axes and , respectively. We are assuming that and correspond to a LOS within the beam, but not necessarily at its center. For RADARSAT-2, and are typically small (1) in the ECEF frame due to the mechanical zero-Doppler beam steering. The rotation matrix can, therefore, be shown to be where

The term, , is considered negligible [43] 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

where

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 [19].

The slant-range vector from the aft antenna phase center to the target can, therefore, be expressed as

where and . Then the projections of these slant-range vectors, and , along the look vector direction are given by From (29b) and (33), (37d) becomes where

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

Next, we derive the first-order coefficient of the Taylor series expansion of . From (36b), we obtain 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 :

Therefore, (43) becomes The last term in (46b) is ignored since the look vector is virtually perpendicular to .

We now derive the last term of (41f). From (27b) and (35), where , and they are neglected in (47b). Also by noting that 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

Putting everything together, (41f) becomes 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