Migration based SAR imaging for ground penetrating radar systems

Transcription

1 Migration based SAR imaging for ground penetrating radar systems K. Gu, G. Wang and J. Li Abstrat: The authors onsider migration based syntheti aperture radar (SAR imaging of surfae or shallowly buried objets using both down-looking and forward-looking ground penetrating radar (GPR. The well known migration approahes devised to image the interior of the Earth are based on wave equations and have been widely and suessfully used in seismi signal proessing for oil exploration for deades. They have the potential to image underground objets buried in ompliated propagation media. Compared to ray-traing based SAR imaging methods, migration based SAR imaging approahes are more suited to imaging underground objets owing to their simple and diret treatment of oblique inidene at the air ground interfae and propagation veloity variation in the soil. The authors apply the phase-shift migration approah to both onstantoffset and ommon-shot experimental data olleted by PSI GPR systems. They address the spatial aliasing problems related to the appliation of migration to the GPR data and the spatial zeropadding approah to irumvent the problem suessfully. 1 Introdution Migration is a family of imaging tehniques originated from seismi imaging [1]. These imaging tehniques are useful for improving the spatial resolution and inreasing the signal-to-noise ratio of the survey. The goal of migration is to refous the refletion signatures in the reorded data bak to the true positions and, hopefully, physial shape of the target. Conventional syntheti aperture radar (SAR imaging tehniques, suh as the delay-and-sum (DAS approah [2], also ahieve image refousing by oherent summation of the baksattered data after proper phase ompensation. For farfield narrowband data, onventional SAR proessing is idential to migration [3]. However, most migration algorithms differ from onventional SAR proessing beause migration algorithms are wave-equation-based, while onventional SAR algorithms have no diret link to the wave equation [4]. Appliations of migration for spaeborne=airborne radar imaging were pioneered by Roa et al. Detailed disussions on this topi an be found in the literature [3, 5 7]. The use of migration for GPR imaging has beome an ative topi in reent years. Most disussions are based on down-looking GPR systems [8 13], whih plae the GPR antennas vertially above the ground with a ertain elevation. The advantage of migration for down-looking GPR imaging is its high omputational effiieny. It avoids the time-onsuming ray-traing proessing needed by DAS, q IEE, 2004 IEE Proeedings online no doi: /ip-rsn: Paper first reeived 2nd April and in revised form 24th June 2004 K. Gu and J. Li are with the Department of Eletrial and Computer Engineering, P. O. Box , University of Florida, Gainesville, FL 32611, USA G. Wang was with the University of Florida, Gainesville and is now with the Department of Communiation Engineering, Jiangsu University, Zhenjiang , People s Republi of China and reonstruts images iteratively using the fast Fourier transform (FFT. Although migration was suessfully implemented in the past for down-looking GPR systems, the spatial aliasing problem was not fully addressed. Fortunately, the problem is not severe for down-looking systems. Migration algorithms an also be applied to forward-looking GPR systems. Suh a system illuminates a large area in front of the vehile and has the advantage of long standoff distane over its down-looking ounterpart. Spatial aliasing an be a serious problem for forwardlooking systems and must be irumvented. In this paper, a partiular kind of migration, referred to as phase-shift migration [14] is disussed and applied to GPR imaging. To simplify the disussion, propagation loss is ignored. Data used in this paper are olleted by the downlooking and forward-looking systems developed Planning Systems In. (by PSI. 2 Phase-shift migration Phase-shift migration was first proposed by Gazdag in 1978 [14] for wavenumber migration, whih an be used to refous seismi=radar images iteratively. An EM field, u(x, z, t, an be desribed by the following salar wave uðx; z; tþ ¼0 ð1þ where x is the oordinate in the aperture dimension (either syntheti or real, z is the oordinate in the range dimension, t is the time, and is the EM wave veloity in the medium. Note that depending on the onfiguration of the GPR systems, the x-axis an be either along-trak or ross-trak, and the z-axis an be either depth or along-trak. Herein, the y-axis is ignored sine we disuss only 2-D imaging. Taking a three-dimensional Fourier transform over the oordinates yields kx 2 þ kz 2 o2 2 Uðk x ; k z ; oþ ¼0 ð2þ IEE Pro.-Radar Sonar Navig., Vol. 151, No. 5, Otober

2 where k x and k z are the x- and z-diretion omponents of the wavenumber vetor, respetively. Sine Uðk x ; k z ; oþ 6¼ 0; we must have kx 2 þ kz 2 ¼ o2 2 ¼ k2 ð3þ where k is the magnitude of the wavenumber vetor k. Equation (3 is known as the dispersion relation for 2-D wave equations with onstant wave veloity. Now onsider a 2-D Fourier transform of (1 over x and t. Also using the dispersion relation, the resulting frequeny wavenumber domain equation 2 þ k2 z Uðk x ; z; oþ ¼0 ð4þ In radar appliations, we only onsider the wave propagating from the surveyed area to the reeive antennas, i.e. negative diretion of the z-axis. Hene, (4 is Uðk x; z; oþ ¼ jk z Uðk x ; z; oþ The solution to (5 has the form: Uðk x ; z; oþ ¼Uðk x ; z ¼ 0; oþe jk zz ð5þ ð6þ This shows that extrapolation along the z-axis in the frequeny wavenumber domain is a phase-shift operation. By reursively extrapolating the wave field along the z-axis in steps Dz and using the result of eah step as input to the next iteration, the frequeny wavenumber domain distribution of the field an be reonstruted. The proedure for the 2-D phase-shift migration approah is: Step 1. Fourier transform the data over x and t ZZ Uðk x ; z 0 ¼ 0; oþ ¼ uðx; z 0 ¼ 0; tþe jkxx e jot dxdt ð7þ Step 3. Inverse Fourier transform the wavenumber data over k x and o; and set t ¼ t; where t is alled the imaging ondition. For the ommon-offset ases, t ¼ 0: For the ommon-shot ases, t is the propagation delay between the transmit antenna and the foal point. Hene, t is a funtion of the foal point loation ðx; z 1 Þ and an be written as tðx; z 1 Þ: uðx; z ¼ z 1 ; t ¼ tðx; z 1 ÞÞ ¼ 1 ZZ 4p 2 Uðk x ; z ¼ z 1 ; oþe jkxx e jotðx;z1þ dk x do ð11þ Note the integration over o; whih, for GPRs, is the operating frequeny band of the systems. Hene, phase-shift migration is a full-band imaging method. Step 4. Repeat steps 2 and 3 for every depth z mþ1 ¼ z m þ Dz; where m ¼ 0; 1;...; M 1: Note that, for down-looking systems, phase-shift migration an also easily handle veloity variation in the z-diretion by making a funtion of z and assuming onstant ðz m Þ inside eah Dz interval. For forward-looking systems, the imaging area is on the surfae of the ground. Hene, the propagation medium is homogeneous and the propagation veloity is onstant for all z m : 3 Comparison of phase-shift migration and DAS A diagram of DAS is shown in Fig. 1a for a ommon-offset ase. The idea of DAS is to sum all data oherently at one foal point in the ground at a time and repeat for all points of interest. The relevant equation is u DAS ðx; zþ ¼ XP X N Uðx n ; z ¼ 0; o p Þ exp½ jo p t n ðx; zþš p¼1 n¼1 ð12þ where u DAS ðx; zþ is the DAS refoused image, N is the size of the syntheti aperture, Uðx n ; z ¼ 0; o p Þ is the data reorded at position x n and frequeny o p ; and t n ðx; zþ is For stepped frequeny systems, Fourier transform is only needed over x Z Uðk x ; z 0 ¼ 0; oþ ¼ Uðx; z 0 ¼ 0; oþe jkxx dx ð8þ When x is sampled at disrete loations, the disrete-time Fourier transform (DTFT should be used instead. However, in pratie, FFT is used for omputational effiieny. Its impat on the reonstruted image will be disussed in Setion 4. Step 2. Compute the field at a new depth z 1 ¼ z 0 þ Dz by the phase-shift operation. For ommon-offset ases (bistati in radar terminology where transmit and reeive antennas work in pairs and have a onstant offset during data aquisition, ( rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Uðk x ; z 1 ; oþ ¼Uðk x ; z 0 ¼ 0; oþ exp j 2 k2 xdz ð9þ For ommon-shot ases (multistati in radar terminology where one transmit antenna works with an array of reeive antennas, ( rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi o 2 Uðk x ; z 1 ; oþ ¼Uðk x ; z 0 ¼ 0; oþ exp j 2 k2 xdz 318 ð10þ Fig. 1 DAS imaging for ommon-offset ases a Diagram of DAS b Propagation geometry IEE Pro.-Radar Sonar Navig., Vol. 151, No. 5, Otober 2004

3 the travel time from the transmitter to the foal point and bak to the reeiver: t n ðx; zþ ¼ R a;t þ R a;r a þ R g;t þ R g;r g ð13þ where a and g are the propagation veloities in the air and soil, respetively; R a;t and R a;r are the distanes from the two refration points on the ground surfae to the transmitter and reeiver, respetively; R g;t and R g;r are the distanes from the foal point to the two refration points on the surfae, respetively. As shown in (13 and Fig. 1b, to obtain t n ðx; zþ; DAS needs to determine refration points on the ground. The proess requires solving two fourth-order equations, whih is time onsuming (see the Appendix. Sine the loation of the refration point hanges with the loation of foal point and antenna position, the proess must be repeated for every foal point and every antenna pairs. Furthermore, the omputational omplexity will inrease drastially when multilayer ground struture is involved. Phase-shift migration is more effiient in refousing GPR images for ommon-offset ases for the following reasons. First, phase-shift migration does not need to determine any refration points. The non-homogeneity of the medium is refleted in the variation of (z. Hene, only the phase-shift term in step 2 needs updating. Moreover, given (z, the phase-shift term for different media an be omputed offline. Hene, the update only needs a lookup table. Seond, step 4 of phase-shift migration shows that the radar image is reonstruted iteratively. The result of eah step in the z-diretion is used as input to the next step. Third, FFT an be used in steps 1 and 3, whih further inreases the effiieny. For ommon-shot systems, the situation is more ompliated and migration may not be faster than DAS. Now, let us ompare DAS and phase-shift migration theoretially. For the sake of analysis simpliity, our disussion is based on ommon-offset data aquisition in a homogeneous medium. Also, we modify DAS to its ontinuous form: ZZ u DAS ðx; z ¼ DzÞ ¼ Uðx 0 ; z ¼ 0; oþ exp j 2o qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðx x 0 Þ 2 þ Dz 2 dx 0 do ð14þ where u DAS ðx; z ¼ DzÞ is the DAS image at z ¼ Dz: For the sake of notational simpliity, define Dx ¼ D x x 0 and R ¼ D pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi D x 2 þ Dz 2 : Integrating the steps of phase-shift migration, i.e. (8 (9 and (11 yields ZZZ u PSM ðx; z ¼ Dz; t ¼ 0Þ ¼ Uðx 0 ; z ¼ 0; oþ ( rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi exp j 2 k2 xdz þ jk x ðx x 0 Þ dx 0 dk x do ð15þ where u PSM ðx; z ¼ Dz; t ¼ 0Þ is the phase-shift migration image at z ¼ Dz: The integration with respet to k x in (15 is Z ( rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi exp j 2 k2 xdz þ jk x Dx dk x ð16þ This integration has the form, IðaÞ ¼ Z b a f ðvþe jagðvþ dv ð17þ For a!1; the integral an be approximated using the method of stationary phase [15, 16]: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2p n IðaÞ ajg 00 f ðv ðv 0 Þj 0 Þ exp j agðv 0 Þþm p o ð18þ 4 where v 0 is a stationary point of g(v defined by g 0 ðv 0 Þ¼0; and m ¼ signðg 00 ðv 0 ÞÞ: In (16, we have, a ¼ 2o Dz ð19þ f ðk x Þ¼1 ð20þ and sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi gðk x Þ¼ Dx 2oDz k x þ 1 k 2 x ð21þ 2o For =o Dz; we get a 1; and (16 an be asymptotially evaluated by (18. The stationary point is given by 2oDx k x0 ¼ p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð22þ Dx 2 þ Dz 2 ¼ 2oDx ð23þ R Hene, Z ( rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi exp j 2 k2 xdz þ jk x Dx dk x rffiffiffiffiffiffiffi 4p pffiffiffiffiffi 2oR R 3 Dz jo exp j ð24þ Furthermore, when Dz D x; we get Dz R; and Z ( rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi exp j 2 k2 xdz þ jk x D x dk x rffiffiffiffiffi 4ppffiffiffiffiffi 2oR jo exp j ð25þ R Let rffiffiffiffiffi 4ppffiffiffiffiffi WðR; oþ ¼ jo ð26þ R Equation (15 is approximated as ZZ u PSM ðx; z ¼ Dz; t ¼ 0Þ WðR; oþuðx 0 ; z ¼ 0; oþ exp j 2oR dx 0 do ð27þ Note that WðR; oþ is a funtion of both spae and frequeny. Comparing (27 and (14, we see that phase-shift migration an be seen as weighted DAS in spae and frequeny when the imaging area is far away from the antenna array. 4 Spatial aliasing in GPR imaging Due to the data aquisition onfiguration, spatial aliasing ours when phase-shift migration is used on the original data reorded by GPR systems. If not mitigated, it an ause severe performane degradation in the refoused images. Based on the PSI GPR systems, we will disuss the spatial aliasing problem in this setion. After analysing the ause of IEE Pro.-Radar Sonar Navig., Vol. 151, No. 5, Otober

4 spatial aliasing, spatial zero-padding is proposed to mitigate the problem. Although our disussions are based on the PSI GPR systems, they an be extended to other GPR systems using similar data aquisition onfigurations. 4.1 PSI GPR systems To detet buried landmines in lutter, PSI has developed down-looking and forward-looking GPR systems referred to as GPSAR3 and FLGPR, respetively, for the US Army RDECOM CERDEC Night Vision and Eletroni Sensors Diretorate. Both systems are ultra-wideband (UWB stepped frequeny systems. They are urrently onfigured for anti-tank mine detetion and have been tested on pratial mine lanes. A photograph of the GPSAR3 system [17] is shown in Fig. 2. It operates over the frequeny band to GHz. Two independent antenna banks (referred to as bak and front, eah ontaining 15 transmit (Tx and 15 reeive (Rx antennas, are used to aquire data at 58 Tx=Rx pair loations, eah separated by m. The Tx=Rx separation is the same for all 58 hannels, whih forms a ommon-offset data olletion system. Experimental data were olleted every 0.04 m in the along-trak dimension (i.e. the diretion the art moves. Using the syntheti aperture in along-trak and the 58 hannels in ross-trak (the antenna bank diretion, GPSAR3 images buried objets in three dimensions. For GPSAR3, the x- and z-diretions orrespond to ross-trak and depth diretions, respetively. The two-layer model, i.e. air and ground, is used to model the propagation medium. Like other forward-looking systems, the PSI FLGPR system (Fig. 3a has the GPR antennas mounted on the front of a vehile and samples the EM field at equally spae positions (0.1 m as the vehile moves forward in the alongtrak diretion. The beamwidth of the transmitting and reeiving antennas is large enough so that its footprint overs an area of at least 4 m in ross-trak on the ground at 4 m away from the vehile (Fig. 3b. Refousing tehniques, suh as DAS and migration, are used to reonstrut twodimensional images on the ground surfae. The system operates over the frequeny band to GHz. Transmit antennas, elevated about 2.5 m above the ground, and two reeive antenna banks, eah ontaining 15 antennas elevated at 1.9 m and 2.05 m above the ground, are used to aquire stepped frequeny data at 30 ross-trak spatial sampling loations, eah separated by m. Altogether, Fig. 3 PSI FLGPR system a Photograph b Data aquisition geometry the m long antenna array serves as the data aquisition array for the system. Within eah data olletion, all reeive antennas work sequentially with the transmit antenna array to reord data and the vehile is onsidered motionless during the proess, whih forms a ommon-shot data olletion system. For FLGPR, the x- and z-diretion orrespond to ross-trak and along-trak diretions, respetively. The propagation medium in FLGPR imaging is homogeneous, i.e. air only. Fig PSI GPSAR3 system 4.2 Spatial aliasing The idea of phase-shift migration is to refous the image in the wavenumber frequeny domain, and use inverse Fourier transform to obtain the spae time domain image. In Setion 3, the phase-shift migration algorithm is presented based on ontinuously reording data in both spae (the x- diretion and time. In pratie, however, one an only reord disrete data in both spae and time. Hene, the data reorded is uðx n ; z ¼ 0; t i Þ; and for stepped-frequeny IEE Pro.-Radar Sonar Navig., Vol. 151, No. 5, Otober 2004

5 systems, Uðx n ; z ¼ 0; o p Þ; where n¼1;...; N; i ¼ 1;...; I; and p ¼ 1;...; P: Therefore, instead of a ontinuous Fourier transform, the disrete Fourier transform (DFT, or FFT, is used to also avoid the integration in (11. To orretly represent the EM field at z ¼ 0 in the k x o domain, the x t domain sampling rate should be greater than the Nyquist rate to prevent k x o domain aliasing. This is guaranteed by the data aquisition onfiguration of the GPR system. On the other hand, the final step of phaseshift migration requires an inverse Fourier transform from the k x domain to the x domain. Hene, the sampling rate in the k x domain should also be suffiient to prevent spatial aliasing in the x domain. Figure 4 shows the geometry of the antenna array, the illuminated area, and the imaging (survey area for GPSAR3 and FLGPR, where N is the number of antennas in the array, Dx is the separation between antennas, and y is the angle between the antenna array and the boundary of the illuminated area. To simplify the disussion, the illuminated area is approximated as a fan-shaped area. GPSAR3 images a single row of foal points beneath the entre of the antenna array. FLGPR is onfigured to image a survey area wider than the data aquisition array (Fig. 3. The horizontal length of the illuminated area inreases with its distane to the antenna array due to the antenna beamwidth. Only the EM field inside the illuminated area an have a non-zero magnitude, i.e. UðnDx; z;o p Þ 6¼ 0; n ¼ L x ðzþ; L x ðzþþ1;...;n þ L x ðzþ ¼ 0; otherwise ð28þ where ðn þ 2L x ðzþþd x is the length of the illuminated area in the x-diretion, and L x ðzþ inreases with z. Let us onsider the wave field at z ¼ Dz; UðnDx; z ¼ Dz; o p Þ¼0; n 6¼ L;...; N þ L ð29þ where 2LDx is the expanded length of the illuminated area in the x-diretion. Its disrete representation is U½n; z ¼ Dz; o p Š¼0; n 6¼ L;...; N þ L ð30þ It an be shown that, using FFT and the original data in phase-shift migration, the reonstruted wave field ~U½n; z ¼ Dz; o p Š is ~U½n; z ¼ Dz; o p Š¼ X1 ¼ X1 r n ¼ 1 r n ¼ 1 U½n r n N; z ¼ Dz; o p Š Uððn r n NÞDx; z ¼ Dz; o p Þ ð31þ ð32þ Hene, the refoused image at z ¼ Dz using DFT is equivalent to repliating U½n; z ¼ Dz; o p Š with a period of N. Detailed disussion about DFT and its aliasing effet an be found in Chapter 4 of [18]. As mentioned before, U½n; z ¼ Dz; o p Š is a finite length sequene over n with non-zero values at n ¼ L;...; N þ L: Hene, in ~U½n; z ¼ Dz; o p Š; spatial aliasing ours at n ¼ L;...; L and n ¼ N L;...; N þ L: Moreover, as the phase-shift migration iterates down the z diretion, L beomes larger. The spatial aliasing region expands and eventually overlaps with the imaging area of both downlooking and forward-looking systems (Fig. 5. As a result, spatial aliasing ours in the reonstruted image. To avoid spatial aliasing in the x-diretion, the length of the data aquisition array should be enlarged, whih is not possible in this appliation. Hene, zero-padding is performed on the reorded data in the x-diretion: Uðx n ; z ¼ 0; o p Þ ¼ Uðx n; z ¼ 0; o p Þ; n ¼ 1; 2;...; N 0; n ¼ N þ 1;...; N x ð33þ Fig. 4 Geometry of antenna array, illuminated area, and imaging area for GPSAR3 and FLGPR where N x is the predetermined zero-padding length, and U ðx n ; z ¼ 0; o p Þ is the data after zero-padding. Zero-padding is appliable sine given the time=spae sampling rate greater than the Nyquist rate, zero-padding does not hange the spetrum of the signal but only inreases the frequeny=wavenumber domain sampling rate. Intuitively, Fig. 5 Spatial aliasing phenomenon by migration using original reorded data IEE Pro.-Radar Sonar Navig., Vol. 151, No. 5, Otober

6 the proess is equivalent to reording the data by an array with N x antennas. Refletions an only our inside the fan-shaped area shown in Fig. 4. Antennas other than the first N annot illuminate the area, and hene reord zeros in the data. With zero-padding, Uð; ; Þ; U½; ; Š; and N in (31 (32 are substituted by Uð; ; Þ; U½; ; Š; and N x ; respetively. Hene, the phase-shift migration is able to provide an aliasing-free spatial distane of N x Dx in the x-diretion. The spatial aliasing area will beome smaller and smaller as N x inreases. Figure 6 shows the ase where spatial aliasing in the imaging area is just avoided. Avoiding spatial aliasing in the imaging area requires N x Dx>N x;min Dx ð34þ where N x;min is the minimum number of zero-padding. Note that spatial aliasing is aused by the use of FFT in the imaging proessing. Hene, the aliasing effet is not only related to phase-shift migration, but related to all image reonstrution algorithms using FFT. Therefore, for those algorithms, similar zero-padding methods should be used. The only problem left now is to determine the value of N x;min ; or the number of zero-padding in the x-diretion. Theoretially, it is related to the beam pattern of the antennas. If the beam pattern is known, the value of y defined in Fig. 4 an easily be obtained. Hene, the value of N x;min is determined by N x;min ¼ & W x 2 þ z tany þ NDx 2 Dx ð35þ where W x is the width of the imaging area in the x-diretion, z is the position of the imaging area in the z-axis, and de is the eiling operator. However, in GPSAR3 and FLGPR imaging, the beam pattern is unknown. Hene, N x;min an be empirially determined by evaluating the resulting images using different values of N x : Zero-padding avoids the spatial aliasing problem by sarifiing the omputational effiieny of phase-shift migration. Hene, in pratie, the length of zero-padding should be seleted to be as lose to N x;min as possible. Furthermore, N x;min of the down-looking system is smaller than that of the forward-looking system for the following reasons: First, W x is smaller in the down-looking system (Fig. 4; and seondly, the refrations at the ground surfae narrow the beamwidth of the antennas, whih equivalently inreases y: Therefore, phase-shift migration is more effiient for the down-looking system than for the forward-looking system. Figures 7 and 8 show images refoused by phase-shift migration using various zero-padding lengths N x in the x-diretion on data reorded by GPSAR3 and FLGPR, respetively. In Fig. 7, one mine is loated at (2.24, 0.25 m. In Fig. 8, two mines are loated at (3.5, 1 m and ð5:5; 0:5Þ m: Without zero-padding, the target images are distorted due to spatial aliasing, espeially for FLGPR. The imaging areas of Figs. 7a and 8a orrespond to the Fig. 6 Spatial aliasing just avoided in the imaging area Fig. 7 an x ¼ N bn x ¼ 2N N x ¼ 4N Using zero-padding to avoid spatial aliasing in phase-shift migration for GPSAR3: one mine buried at (2.24, 0.05 m 322 IEE Pro.-Radar Sonar Navig., Vol. 151, No. 5, Otober 2004

7 Fig. 8 Using zero-padding to avoid spatial aliasing in phase-shift migration for FLGPR: two mines at (3.5, 1 m and (5.5, m, flushly buried an x ¼ N bn x ¼ 2N N x ¼ 4N dn x ¼ 8N en x ¼ 16N imaging areas shown in Fig. 5, where zero-padding is not onduted and the imaging areas overlap with aliasing area. The more zero-padding is used, the learer the targets in the foused images. When N x ¼ 2N and N x ¼ 16N in Figs. 7 and 8, respetively, the refoused images are free of aliasing. The imaging areas orrespond to those shown in Fig Results on measured data We have used phase-shift migration for SAR imaging for the PSI GPSAR3 and FLGPR systems. The data were olleted in a real landmine test site (overing an area of 462 m 2. Targets (metal and plasti mines were buried at m in depth. 5.1 Results of GPSAR3 data proessing The data onsidered were olleted by GPSAR3 in a test lane with ten landmines (three metal and seven plasti. Owing to the large beamwidth of the GPR antennas, targets are smeared into hyperbolas in radar images. For better visualisation of the hyperbolas the real part of the omplex radar image is shown in Fig. 9. Figure 9a shows the B-san radar image (depth against along-trak of a mine buried at depth 0.05 m. The hyperbolas are learly identifiable, whih shows the neessity for image refousing. Hene, phase-shift migration with a syntheti array in the along-trak diretion is applied to the data. The syntheti array onsists of five onseutive alongtrak sans. The refoused image is shown in Fig. 9b. The hyperbolas disappear after phase-shift migration. Most image refousing methods, inluding phase-shift migration, are apable of inreasing the resolution and SNR of radar images. For GPSAR3, improvement of image resolution is limited beause of the limited size of the syntheti aperture (normally with a length of five. Figure 10 shows the C-san (ross-trak against along-trak of an area ontaining one mine loated at (2, 0.4 m. Note that the SNR inreases signifiantly after phase-shift migration, whih should improve the mine detetion performane (see later on. A simple energy-based detetor is used to examine the benefit of image refousing. The detetor first thresholds the three-dimensional radar image (depth along-trak ross-trak by magnitude, and then projets the image to two dimension (along-trak ross-trak by summing the energy over depth. The two-dimensional energy image is then tested with various thresholds, yielding the reeiver operator harateristi (ROC urve shown in Fig. 11. With phase-shift migration, the false-alarm rate (FAR is redued from more than 0:011=m 2 to less than 0:009=m 2 at 0.9 probability of detetion (Pd. (One small plasti mine annot be deteted at a false alarm rate lower than Using traditional DAS yields the same detetion performane as phase-shift migration. However, with all same parameters (i.e. frequeny steps, size of aperture, number of IEE Pro.-Radar Sonar Navig., Vol. 151, No. 5, Otober

8 Fig. 9 B-san target images before and after phase-shift migration: mine buried at (2, 0.05 m a Before phase-shift migration b After phase-shift migration Fig. 10 C-san target images before and after phase-shift migration: mine buried at (2, 0.4 m a Before phase-shift migration b After phase-shift migration foal points, et., phase-shift migration refouses the radar image more than seven times faster than DAS. 5.2 Results of FLGPR data proessing Data used for refousing ontain twelve metal mines. Image refousing is onduted in both ross-trak and along-trak to form the images shown in Fig. 12. A Taylor window is used for shading in both spae (x-oordinate or ross-trak and frequeny. Details of the Taylor window an be found in Chapter 3 of [2]. Inross-trak,eitherDAS(Fig. 12a or phase-shift migration (Fig. 12b is used, and in along-trak, non-oherent summation, i.e. summation of magnitude only, is used. The targets are loated at (13, 2.5 m, (15, 1 m, ð17; 0:5Þ m; (19,2.5 m, (21,1 m,ð23; 0:5Þ m; (25,2.5 m, (27, 1 m, ð29; 0:5Þ m; (31, 2.5 m, (33, 1 m, and ð35; 0:5Þ m: For FLGPR, phase-shift migration is slower than DAS due to the ommon-shot sheme and large zeropadding needed to avoid spatial aliasing. Note that phase-shift migration yields better lutter suppression than DAS on both the upper and lower portions of the image (vertial oordinate less than 0 or greater than 2, but is worse in the middle (vertial oordinate greater than 0 and less than Fig. 11 ROC of GPSAR3 data before and after phase-shift migration IEE Pro.-Radar Sonar Navig., Vol. 151, No. 5, Otober 2004

9 Fig. 12 Refoused images for FLGPR by DAS and phase-shift migration a DAS b Phase-shift migration 6 Conlusions We have disussed the appliation of phase-shift migration to image refousing in GPR land-mine detetion. Owing to its auray and effiieny, the method an be superior to traditional ray-traing methods, suh as DAS. Theoretial analysis shows that, under ertain onditions, phase-shift migration an be seen as weighted DAS in spae and frequeny. However, the diret implementation of phaseshift migration on GPR data an ause spatial aliasing in the reonstruted image. Zero-padding in spae is used to mitigate the problem at the ost of lower omputational effiieny. Phase-shift migration has been implemented effiiently for the PSI GPSAR3 down-looking system, giving a seven times speed improvement over DAS. Phaseshift migration inreases the SNR of the radar images and improves the detetion performane. For the PSI FLGPR forward-looking system, the omputational effiieny is lower than that of DAS due to the ommon-shot sheme of FLGPR and the large amount of zero-padding required. However, ompared with DAS, phase-shift migration obtains higher quality images in speifi areas of the image. 7 Aknowledgment This work was supported by the U S. Army under Contrat No. DAAB15-00-C array-antenna data. Pro. Eighth Int. Conf. on Ground Penetrating Radar (GPR 2000, Gold Coast, Australia, May Gunatilaka, A., and Baertlein, B.: A subspae deomposition tehnique to improve GPR imaging of anti-personnel mines, Pro. SPIE Int. So. Opt. Eng., 2000, 4038, pp Sheers, B.: Ultra-wideband ground penetrating radar, with appliation to the detetion of anti personnel landmines. PhD thesis, Université Catholique de Louvain, Belgium, Sheers, B., Aheroy, M., and Vander Vorst, A.: Migration tehnique based on the time-domain model of the ground penetrating radar, Pro. SPIE Int. So. Opt. Eng., 2001, 4491, pp Fisher, C., Fortuny, J., and Wiesbek, W.: 3-D imaging for near-range ground-penetrating radar based on v-k migration. Pro. EUSAR 2000, 3rd European Conf. on Syntheti Aperture Radar, Munih, Germany, May Gazdag, J.: Wave equation migration with the phase-shift method, Geophysis, 1978, 43, (7, pp Gunawardena, A., and Longstaff, D.: Ultra-wideband widebeam SAR proessing in the time-spae domain. Pro. Radar 97, Otober 1997, pp Bleistein, N., and Handelsman, R.: Asymptoti expansions of integrals (Holt, Rinehart and Winston, Bradley, M., Witten, T., MCummins, R., and Dunan, M.: Mine detetion with ground penetration syntheti aperture radar, Pro. SPIE Int. So. Opt. Eng., 2002, 4742, pp Oppenheim, A., and Shafer, R.: Disrete-time signal proessing (Prentie Hall, In., Appendix: Determining refration point for two-layer model We determine the refration point for the two-layer model shown in Fig. 13. InFig. 13, oordinates of the radar, the foal point, and the refration point are (0, h, ðx f ; dþ; and (x, 0, respetively; y i and y r denote the inident angle and the refration angle, respetively; 1 and 2 are the dieletri oeffiients of the two layers. Aording to the geometry x sin y i ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð36þ h 2 þ x 2 and x f x sin y t ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð37þ ðx f xþ 2 þ d 2 Moreover, by Snell s law, we have, rffiffiffiffi sin y i ¼ n ¼ 2 ð38þ sin y t 1 where n is the refration oeffiient. Using (36, (37, and (38, we have the following fourth-order equation for the position of refration point x, n 2 x 4 ð2n 2 x f þ 1Þx 3 þ n 2 h 2 þ n 2 x 2 f þ 2x f x 2 2x 2 f n 2 h 2 þ x f þ d 2 x þ n 2 h 2 x 2 f ¼ 0 ð39þ 8 Referenes 1 Claerbout, J.: Imaging the Earth s interior, 1997, stanford.edu/sep/prof/iei/to_html/index.html 2 Van Trees, H.: Optimum array proessing (Wiley-Intersiene, Cafforio, C., Prati, C., and Roa, F.: Syntheti aperture radar: A new appliation for wave equation tehniques, IEEE Trans. Aerosp. Eletron. Syst., 1991, 27, pp Gunawardena, A., and Longstaff, D.: Wave equation formulation of syntheti aperture radar (SAR algorithms in the time-spae domain, IEEE Trans. Geosi. Remote Sens., 1998, 36, pp Prati, C., Roa, F., Guarnieri, A., and Damonti, E.: Seismi migration for SAR fousing: Interferometrial appliations, IEEE Trans. Geosi. Remote Sens., 1990, 28, pp Cafforio, C., Prati, C., and Roa, F.: Syntheti aperture radar: A new appliation for wave equation tehniques, Geophys. Prospet., 1989, 7, pp Ottolini, R.: Syntheti aperture radar data proessing. Stanford Exploration Projet report, 1987, pp Lertniphonphun, W., and MClellan, J.: Migration of underground targets in UWB-SAR systems. Pro. Int. Conf. on Image Proessing, 2000, vol. 1, pp Binningsbø, J., Eide, E., and Hjelmstad, J.: 3D migration of GPR Fig. 13 Refration of two-layer media IEE Pro.-Radar Sonar Navig., Vol. 151, No. 5, Otober