multiple-frequency interferometric velocity sar location and imaging of ...

MULTIPLE-FREQUENCY INTERFEROMETRIC VELOCITY SAR LOCATION AND IMAGING OF ELEVATED MOVING TARGET Xiaowei Li and Xiang-Gen Xia Department of Electrical and Computer Engineering University of Delaware, Newark, DE 19716 USA ABSTRACT

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Locations of moving targets in synthetic aperture radar (SAR) images are determined not only by their geometric locations but also by their velocities that cause their SAR images defocused, smeared, and mis-located. With a linear antenna array, velocity synthetic aperture radar (VSAR) can detect, focus, and locate slowly moving targets well. However, it may mis-locate fast moving targets in the azimuth (cross-range) direction, and sometimes even in the ground range direction if the targets are elevated above the ground. In this paper, we propose an antenna array approach with cross-track interferometry, in which multiple wavelength signals are transmitted. It is shown that our proposed multiple-frequency interferometric velocity SAR (MFIn-VSAR) can locate both slowly and fast moving elevated targets correctly. Index Terms— Moving target imaging, phase unwrapping, radar interferometry, synthetic aperture radar (SAR). 1. INTRODUCTION Synthetic aperture radar (SAR) has attracted much attention in recent decades for moving targets. It is well known that the difficulty of moving target location and imaging is the estimation of moving target position and velocities. The motioninduced phase terms cause images of moving targets mislocated in the azimuth dimension. Moreover, if moving targets are not on the ground and elevated, they are not only shifted in the azimuth, but also migrated in the ground range direction. To deal with the motion-induced migration, many methods were proposed, including single-channel systems [1, 2] based on analysis of the azimuth phase history, and multichannel SAR systems [3] using maximum-likelihood (ML) method. Since the location can only be accurately estimated for slowly moving targets in these methods, some methods based on multi-channel VSAR system have been presented [4,5]. However, they all assume ground moving targets. If the fast moving targets are elevated above the ground, not only the azimuth ambiguity, but also the ground range migration should be rectified. This work was supported in part by the Air Force Office of Scientific Research (AFOSR) under Grant No.FA9550-05-1-0161, a DEPSCoR Grant W911NF-07-1-0422 through ARO, and the World Class University (WCU) Program 2008-000-20014-0, National Research Foundation, Korea. Xia is also with the Institute of Information and Communication, Chonbuk National University, Jeonju 561-756, Korea.

978-1-4244-4296-6/10/$25.00 ©2010 IEEE

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Fig. 1. Imaging geometry In this paper, besides resolving the azimuth location ambiguity, we are also interested in rectifying the ground range migration due to elevation projection. We equip a conventional SAR platform with a linear antenna array, similar to the MFAASAR [4] platform, and we also deploy another crosstrack receiver collaborated with this antenna array in order to retrieve the elevation information, which is used to correct the ground range migration of the targets. The crosstrack interferometry is also a basic concept in interferometric SAR (InSAR) [6, 7] and the use of multi-frequency technique in InSAR can be found in [8]. The newly proposed system is named multi-frequency interferometric velocity SAR (abbreviated as MFIn-VSAR), in which the moving targets may be separated from the stationary clutter with appearance in the corresponding V-images due to VSAR processing. Besides, the elevation information of the targets can be retrieved by using phase interferometry. With the use of multiple frequencies, a robust phase unwrapping (RPUW) algorithm [9] can be used to resolve not only the azimuth ambiguity as in MFAASAR, but also the ground range ambiguity. 2. FORMULATION AND A ROBUST SOLUTION Assuming a general stereo imaging geometry in Fig.1, the radar platform flies parallel along the Y-axis with an altitude 𝐻 and velocity 𝑣. There are totally 𝑀 + 1 antennas onboard, where 𝑀 of them form a uniform linear array (ULA) located along the flight track and all these 𝑀 antennas re-

ICASSP 2010

ceive signals, and the first one is also assumed to be the radar transmitter and located at 𝑥 = 0 in X direction at time 𝑡 = 0; another cross-track receiver is separated from the transmitter vertically by a baseline 𝐵 and is thus normal to the Y-axis. Thus, the instantaneous three-dimensional coordinate (𝑥, 𝑦, 𝑧) of the 𝑚-th receiver of the ULA at time 𝑡 is (𝑥, 𝑦, 𝑧) = (𝑣𝑡 + (𝑚 − 1)𝑑, 0, 𝐻), while the cross-track receiver is at (𝑣𝑡, 0, 𝐻 + 𝐵). At the same time, suppose an elevated point target 𝑃 located at (𝑥 0 , 𝑦0 , ℎ), illuminated by the radar waveforms and is moving with a constant velocity, 𝑣𝑥 in azimuth and 𝑣 𝑦 in ground range.

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2.1. Linear antenna array for azimuth correction Suppose that the radar transmits a linear frequency modulated (LFM) signal with carrier frequency 𝑓 𝑐 . After the range compression and the azimuth focusing [1], the image of 𝑃 formed by the 𝑚-th antenna of the ULA is 4𝜋𝑅0 𝜌𝑥 𝑛 0 𝑑 (𝑚 − 1)] ) ⋅ exp[−𝑗2𝜋 𝑆𝑚 (𝑛,𝑙) = exp(−𝑗 𝜆 𝜆𝑅0 (1) ⋅ 𝛿(𝑛 − 𝑛0 − △𝑠ℎ𝑖𝑓 𝑡,𝑛 , 𝑙 − 𝑙0 − △𝑠ℎ𝑖𝑓 𝑡,𝑙 ) √ where 𝑅0 = 𝑥20 + 𝑦02 + (𝐻 − ℎ)2 is the distance from the transmitter to 𝑃 , 𝜆 = 𝑐/𝑓𝑐 is the carrier wavelength, 𝑛, 𝑙 are the discrete azimuth and ground range position indices, respectively, 𝑙0 and △𝑠ℎ𝑖𝑓 𝑡,𝑙 are quantization results of the true position and shift migration in the ground range domain divided by range resolution 𝜌 𝑦 and will be discussed in the next subsection, 𝑛0 and △𝑠ℎ𝑖𝑓 𝑡,𝑛 are the quantization results of the true position and the shift migration in the azimuth direction divided by azimuth resolution 𝜌 𝑥 , given as 𝑥0 𝑥0 𝑣𝑥 + 𝑦0 𝑣𝑦 (2) , △𝑠ℎ𝑖𝑓 𝑡,𝑛 = − 𝑛0 = 𝜌𝑥 𝜌𝑥 𝑣 After phase compensation as [3], let 𝜁 = 𝜌 𝑥 𝑑 △𝑠ℎ𝑖𝑓 𝑡,𝑛 /(𝜆𝑅0 ), then (1) becomes 4𝜋𝑅0 ) ⋅ exp[𝑗2𝜋𝜁(𝑚 − 1)] 𝑆𝑚 (𝑛,𝑙) = exp(−𝑗 𝜆 (3) ⋅ 𝛿(𝑛 − 𝑛0 − △𝑠ℎ𝑖𝑓 𝑡,𝑛 , 𝑙 − 𝑙0 − △𝑠ℎ𝑖𝑓 𝑡,𝑙 ) for 𝑚 = 1, 2, ..., 𝑀 , which shows that 𝜁 and, thus △ 𝑠ℎ𝑖𝑓 𝑡,𝑛 can be estimated via DFT of 𝑆 𝑚 in terms of 𝑚. In the 𝑀 point DFT results, V-images [3], we have 𝜁 ′ = mod (𝜁, 1), the residue of 𝜁 due to the 2𝜋 folding. If 𝑃 moves slowly such that 0 ≤ 𝜁 ′ = 𝜁 < 1, 𝜁 and thus △𝑠ℎ𝑖𝑓 𝑡,𝑛 can be estimated correctly from the V-images, and the azimuth ambiguity can be removed in this case. Otherwise, if 𝑃 moves fast such that 𝜁 = 𝜁 ′ + 𝐾 for some integer 𝐾, the ambiguity will occur. That is the reason why a fast moving vehicle cannot be correctly positioned in a VSAR system [3]. To overcome this ambiguity, a multi-frequency VSAR has been proposed in [4], where multiple carrier wavelengths 𝜆 are used such that multiple residues of 𝜁 can be obtained, and △ 𝑠ℎ𝑖𝑓 𝑡,𝑛 can then be uniquely determined by the RPUW [9]. Since △𝑠ℎ𝑖𝑓 𝑡,𝑛 is solved, the true azimuth position of 𝑃 , 𝑛0 , can be therefore solved by subtracting the value of △𝑠ℎ𝑖𝑓 𝑡,𝑛 from the detected value 𝑛 0 + △𝑠ℎ𝑖𝑓 𝑡,𝑛 .

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Fig. 2. Geometry for height induced ground range migration 2.2. Cross-track antenna for range rectification In our method, we deploy another cross-track antenna 𝑏 with the ULA, and thus can cooperatively work with the transmitter to use phase interferometry to extract the height information of 𝑃 . Consider the geometry shown in Fig.2, which is basically a Y-Z plane projection of the stereo geometry in Fig.1. Also, consider 𝑃1 , the ground point on the same planar wavefront with 𝑃 , i.e., with the same observed slant range. During the standard imaging process, 𝑃 is incorrectly shifted to position 𝑃1 in ground range due to the projection, imaged at 𝑙0 + △𝑠ℎ𝑖𝑓 𝑡,𝑙 from its true position 𝑙 0 . Similar to (2), we have 𝑦0 ℎ tan 𝜃 , △𝑠ℎ𝑖𝑓 𝑡,𝑙 = − (4) 𝑙0 = 𝜌𝑦 𝜌𝑦 where 𝜃 is the depression angle and can be obtained from 𝐻 Fig.2 as 𝜃 = 12 (arcsin 𝑅𝐻0 + arcsin 𝐻−ℎ 𝑅0 ), and 𝜃 ≈ arcsin 𝑅0 when ℎ is not too large. Thus, in the range cell where a moving target is detected, it is necessary to know △ 𝑠ℎ𝑖𝑓 𝑡,𝑙 , and thus ℎ, to relocate 𝑃 from 𝑃 1 . To retrieve ℎ, we only need to know the phase difference between two independent measurements on a dense grid of sample points, i.e., from the two well-registered complex SAR images. The difference in received phase 𝜙 𝑏1 at the two apertures can also be written as 2𝜋(𝑅𝑏𝑃 (𝑡) − 𝑅1𝑃 (𝑡)) (5) 𝜙𝑏1 (𝑡) = 𝜙𝑏 (𝑡) − 𝜙1 (𝑡) = 𝜆 where 𝜙𝑏 and 𝜙1 are the phases of the received signals at the cross-track receiver 𝑏 and at the first receiver of the ULA (the transmitter), respectively, 𝑅 𝑏𝑝 (𝑡) and 𝑅1𝑃 (𝑡) are distances between the antenna and target 𝑃 at time 𝑡, geometrically. Taking the Taylor expansion of 𝑅 𝑏𝑝 and 𝑅1𝑝 in terms of 𝑡 respectively, then (5) can be approximated as 2𝜋𝐵 𝐵 Δ 𝜙𝑏1 (𝑡) ≈ (𝐻 + − ℎ) = 𝜙𝑟𝑒𝑓 (6) 𝑏1 + 𝜙𝑏1 , 𝜆𝑅0 2 𝑟𝑒𝑓 𝐵 where 𝜙𝑏1 = 2𝜋𝐵 𝜆𝑅0 (𝐻 + 2 ) is the reference term and is removed during the phase processing, and 2𝜋𝐵ℎ 𝜆𝑅0 𝜙𝑏1 . , or ℎ = − (7) 𝜙𝑏1 = − 𝜆𝑅0 2𝜋𝐵

However, the phase 𝜙 𝑏1 might be many radians, such that 2𝜋𝐵ℎ = 𝜙˜𝑏1 − 2𝜋𝑘 (8) 𝜙𝑏1 = − 𝜆𝑅0 for some integer 𝑘, where 𝜙˜𝑏1 is the wrapped value of 𝜙 𝑏1 , and is, the most important, what we can obtain from the two receivers. Thus, unless the phase wrapping is undone, ℎ will 0 be a value modulo 𝜆𝑅 𝐵 . For the azimuth de-ambiguity, 𝐿 different frequencies are already employed to transmit radar probing signals as in [4]. In the range cell where a moving target is detected in Vimages for each wavelength 𝜆 𝑖 , we may possibly obtain its elevation information from its phase difference 𝜙 𝑏1 as in (7). However, we can only have 𝜙˜𝑏1 in (8), the wrapped version 𝜙𝑏1 due to 2𝜋 folding, and thus 𝑓˜𝑖 , an indexed wrapped value of 𝜙𝑏1 𝐵ℎ 𝑓𝑖 = − = 𝑓˜𝑖 + 𝑘𝑖 (9) = 2𝜋 𝜆𝑖 𝑅0 ˜𝑏1 for a series of integers 𝑘 𝑖 . In other words, 𝑓˜𝑖 = − 𝜙2𝜋 is the wrapped value of 𝑓 𝑖 modulo 1. With the use of 𝐿 different frequencies, we can produce significantly different 𝑓 𝑖 for different index 𝑖 such that the RPUW [9] can be used here to resolve the ambiguity due to the wrapping. The most important, note that this algorithm is somewhat robust to the remainder errors of 𝑓˜𝑖 , which is superior to the Chinese remainder theorem (CRT) in [9], and has inspired a robust CRT as in [11]. For convenience, we express 𝑓˜𝑖 in a fraction manner with a denominator 𝑁 , 𝑁𝑖 + 𝜖𝑖 (10) 𝑓˜𝑖 = 𝑁 where 𝑁𝑖 are certain integers with 0 ≤ 𝑁 𝑖 < 𝑁 and can be directly scaled from the values of 𝑓˜𝑖 , 𝜖𝑖 are some fractional 1 errors and upbounded as ∣𝜖 𝑖 ∣ ≤ 2𝑁 , for 𝑖 = 1, 2, ..., 𝐿. Accordingly, we have 𝑁𝑖 Ω𝑖 + 𝑘𝑖 Ω𝑖 + 𝜖𝑖 Ω𝑖 ℎ= (11) 𝑁 where Ω𝑖 = 𝜆𝑖 𝑅0 /𝐵, 𝑖 = 1, 2, ..., 𝐿, are not necessarily integers as in CRT. Let Γ be the smallest positive number such that (12) Γ𝑖 = Γ𝜆𝑖 , 1 ≤ 𝑖 ≤ 𝐿, are all integers and Γ 𝑖 and Γ𝑗 are co-prime 1 for 1 ≤ 𝑖 ∕= 𝑗 ≤ 𝐿. Without loss of generality, we assume Γ 1 < Γ2 < ⋅ ⋅ ⋅ < Γ𝐿 , or equivalently 𝜆 1 < 𝜆2 < ⋅ ⋅ ⋅ < 𝜆𝐿 . For 1 ≤ 𝑖 ≤ 𝐿, let Δ

Δ

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where 𝛾1 = Γ2 ⋅ ⋅ ⋅ Γ𝐿 and 𝛾𝐿 = Γ1 ⋅ ⋅ ⋅ Γ𝐿−1 . When∏the necessary requirements 𝑁 > Γ 1 + Γ𝐿 and 𝐿 𝑅0 ℎ < 𝐵Γ 𝑖=1 Γ𝑖 are satisfied as in [9], the RPUW gives an unbiased estimate as 𝐿 𝐿 ∑ 𝑁𝑖 𝑅0 ∑ 𝑁𝑖 ˆ= 1 ) ⋅ Ω𝑖 = ) ⋅ 𝜆𝑖 (14) (𝑘𝑖 + (𝑘𝑖 + ℎ 𝐿 𝑖=1 𝑁 𝐵𝐿 𝑖=1 𝑁 1 Although this condition may not hold for all possible positive real numbers 𝜆𝑖 , 1 ≤ 𝑖 ≤ 𝐿, there are enough such 𝜆𝑖 in any range that can be easily chosen for radar applications in for example [4, 5].

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where the integer series 𝑘 𝑖 can be uniquely determined as follows. For each { 𝑖 with 2 ≤ 𝑖 ≤ 𝐿, define   𝑁 𝑖 Ω𝑖 Δ (𝑘¯1 , 𝑘¯𝑖 ) = argmin0≤𝑘ˆ1