Ground-penetrating radar land mine imaging: Two-dimensional seismic migration and three-dimensional inverse scattering in layered media



[1] This paper presents two methods for ground-penetrating radar (GPR) imaging of land mines: a two-dimensional (2-D) seismic migration method and a 3-D nonlinear inverse scattering method. The seismic migration technique has been successfully applied to processing field data sets collected at a test site. The results show that the seismic migration technique is a useful real-time imaging method. To image the 3-D structure of the land mine, we have developed a full 3-D nonlinear inverse scattering algorithm on the basis of the contrast source inversion method. To account for the ground surface and potentially other subsurface layers, the inverse scattering method uses a multilayered medium as a background. Preliminary results demonstrate that the 3-D inverse scattering method can successfully provide high-resolution reconstruction of high-contrast buried objects.

1. Introduction

[2] Ground-penetrating radar (GPR) is an important tool in detecting subsurface buried objects such as land mines and unexploded ordinance. In particular, for demining purpose, one is interested in distinguishing between the signatures of land mines from those of clutters (such as rocks and tree roots). Currently, the most popular schemes for processing GPR land mine data are based on signal processing methods, as successfully demonstrated in the work of Gunatilaka and Baertlein [2000], Gader et al. [2001], and Collins et al. [1999]. However, GPR detection of buried objects is broadly viewed as a class of inverse problem where there exist a number of imaging methods developed to invert data for questing subsurface information. In this work, we consider two complementary methods: (1) a two-dimensional seismic migration technique for GPR data and (2) a 3-D inverse scattering method based on the contrast source inversion and the dyadic Green's function for a layered medium. The migration and inverse scattering methods exploit measured data in a different way to map subsurface in the form of reflectivity and physical parameters of a medium, respectively. The 2-D migration with 1-D array is suitable for real time imaging and can be rapidly used to provide the location and possible shape information of targets. The inverse scattering method is more difficult (as described below) but is capable of determining not only the geometrical information but also the physical properties of the targets. For potentially new GPR systems, one may be able to collect the data using a 2-D antenna array, thus enabling more effective reconstruction of 3-D buried objects than the use of a family of traditional 1-D arrays. It is noted that both methods in principle image the same subsurface structure expressed in different forms and thus are complementary. In this sense, a migrated result may be used as a priori information for inverse scattering problem as in this paper where the dyadic Green's function in layered media is used.

[3] The ground-penetrating radar system under consideration operates in a reflection mode to record the reflected signals from ground surface and buried objects. It is quite similar to the seismic reflection acquisition process used in oil exploration industry. Thus, in principle, the existing seismic reflection imaging methods can be used to processing GPR signals considering that the propagation of electromagnetic and seismic waves obeys similar scalar wave equations, especially under lossless conditions. In reflection seismology, migration methods move the reflections to their true subsurface positions [Robinson, 1983; Claerbout, 1985; Yilmaz and Doherty, 2001] to obtain the subsurface image. Among the most popular migration algorithms are the Kirchhoff, the finite difference, and the frequency-wave number schemes [Robinson, 1983; Claerbout, 1985; Yilmaz and Doherty, 2001].

[4] In recent years, the migration imaging technique has been employed to process GPR data. For example, Lee et al. [1987] proposed a phase-field processing method based on the electromagnetic equivalence of seismic migration. Fisher et al. [1992a, 1992b] applied reverse-time migration to GPR data. Holzrichter and Sleefe [2000] showed the example to locate a metal mine by using phase-shift migration method. Gunatilaka and Baertlein [2000] proposed a signal processing technique to suppress ground-reflection clutters and then employed migration method to image the GPR data. Carin et al. [2002] and Fortuny-Guasch [2002] applied the synthetic aperture radar (SAR) technique, which is similar to seismic migration, to subsurface radar imaging. van der Kruk et al. [2003] presented a 3-D vectorial migration algorithm applied to multicomponent GPR data. Xu et al. [2002] have developed a statistical method to process GPR land mine data. Assuming the scalar case, we utilize the phase-shift migration method, and show some excellent results obtained on field data.

[5] The migration or SAR approach may be performed to form 2-D or 3-D subsurface reflectivity image by using an actual 1-D antenna array or a synthesized 2-D array from 1-D antenna arrays [Fortuny-Guasch, 2002]. Although such migration methods are efficient and can be performed in real time in 2-D, they do not provide quantitative information about the electrical contrast of the scatterer. However, inverse scattering methods do provide such quantitative information about the electrical contrast of the target. Such wave-based inverse scattering techniques have received considerable attention in recent decades, with both 2-D and 3-D nonlinear inversion techniques. It is not the purpose of this work to review the past works in this large area; in the following, we will only outline some works in inverse scattering related to GPR processing.

[6] It is well known that the EM inverse scattering problem for GPR imaging is both nonlinear and highly ill-posed. The presence of the ground surface and possibly subsurface layers makes this problem even more challenging. To make the problem tractable, the Born approximation is often invoked under the assumption of weak scattering. For instance, Molyneux and Witten [1993] and Witten et al. [1994] developed 2-D diffraction tomography (DT) for GPR subsurface imaging based on the Born approximation. Cui and Chew [2000] developed a 2-D diffraction tomography method for a half-space. Hansen and Johansen [2000] and Meincke [2001] proposed a 3-D DT inversion scheme to reconstruct the object located deep in the soil by employing Born approximation and an asymptotic approximation of the dyadic Green function for a half-space. Recently, Galdi et al. [2003] presented a fast 2-D GPR sensing algorithm, which is based on quasi-ray Gaussian beams forward solver under the far-field approximation, to map low contrast objects in the presence of a moderately rough air-soil interface. In these methods, the ill-posed problems are generally tackled by the various regularization techniques embedded in corresponding optimization process. To address the high-contrast problems, a class of nonlinear inverse scattering method called the contrast source inversion (CSI) in 2-D and 3-D have been developed [e.g., van den Berg et al., 1999; Abubakar and van den Berg, 2000; Abubakar et al., 2002a, 2002b; Zhang and Liu, 2001, 2004]. Within the context of the GPR configuration, such nonlinear inverse scattering methods have been developed for 2-D half-space or 3-D assuming no air-soil interface.

[7] In the first part of this paper, we briefly report the application of phase-shift migration [Gazdag, 1978] to processing the field GPR data collected for the detection of land mines by Niitek, Inc., at a U.S. government testing ground. 2-D migration results show excellent focusing effects. The application of migration to processing the real data is presented in section 2.

[8] In the second part of this paper, we report a CSI-based 3-D EM inverse scattering technique in a layered medium for surface GPR survey. The results were obtained for a single frequency and on synthetic data collected either in a family of 1-D arrays or a 2-D array which may be equipped with a newly developed GPR system. A brief formulation and numerical examples for the 3-D CSI method for layered media is presented in section 3. Section 4 summarizes the present and future work.

2. Phase-Shift Migration Technique

2.1. Review of Theory

[9] First, let us review the theory of one commonly used seismic migration method: the phase-shift migration originally proposed by Gazdag [1978]. The method is based on the frequency domain solution of an approximation [Claerbout, 1985] to the one-way wave equation with initial conditions defined by a zero-offset (or called monostatic measurement in GPR) section. The main advantages of this method are its simplicity and robustness.

[10] Consider a 2-D scalar wave field u(x, z, t), where x denotes the horizontal distance, z depth, and t time. The z axis is vertical and oriented downward. Assuming that the permeability μ is constant, the TMy wave field uEy is governed by the wave equation

equation image

where v(z) is the wave velocity in the medium. The Fourier transform of wave field u(x, z, t) over x and t is given by

equation image

where ω is the angular frequency, kx the horizontal wave number, and j = equation image. Then equation (1) is reduced to the Helmholtz equation

equation image

where kz is the vertical wave number given by kz2 = equation imagekx2. Correspondingly, on the ground surface, the Fourier transform of the measurement u(x, z = 0, t) is given by

equation image

It was shown [Robinson, 1983] that the inverse Fourier transform of U(kx, z, ω) with respect to kx and ω is written as

equation image

[11] This integral represents the wave disturbance u(x, z, t) as a superposition of sinusoidal plane waves. Equation (5) holds for a homogeneous half-space but can be approximately used to a model with piecewise constant vertical velocity distribution [Robinson, 1983].

[12] Now we come to the key point in migration according to the exploding reflector model [Robinson, 1983; Claerbout, 1985; Yilmaz and Doherty, 2001]. If the sources underground we are interested in are initiated at time zero, by setting t = 0 in equation (5), we are able to locate those point diffractors, i.e., the reflecting sources appearing as events on the function

equation image

[13] In other words, equation (6) defines the migrated section where the unknown reflecting source distribution can be depicted as a 2-D Fourier transform of the measured data.

[14] From another viewpoint, the migration may be considered as a downward continuation process. Assuming a constant wave velocity, the solution to Helmholtz equation (3) is given by

equation image

With the help of (7), the wave fields at two depth levels, say z and z + Δz where Δz is depth interval, have the relation

equation image

Equation (8) shows that the wave field at the lower level is extrapolated from the wave field at the previous level. This downward continuation is also valid for piecewise constant vertical velocity distribution, i.e., a layered medium. Finally, it is observed that downward continuation of the wave field is achieved by applying a series of phase shifts exp[jΔzkz] to the Fourier transformed data. This is the essence of the so-called phase-shift method [Gazdag, 1978; Claerbout, 1985; Yilmaz and Doherty, 2001].

2.2. Data Acquisition at a Testing Ground

[15] The GPR data processed in this work was collected at a U.S. government testing ground and provided by Niitek, Inc. Collection of the GPR data in the frequency band of 0.2–5 GHz is carried out in this way: a linear array of 1.2 m with 24 sensor pairs is positioned along the cross-track (x) direction. The GPR sensor array is moved at a constant interval (5 cm) in the down-track (y) direction until the whole interested area is scanned. The number of time sampling points is 416 for each trace, with 24 channels in the cross-track direction. The time sampling interval is 0.01 ns. The data set utilized in this work consists of 5112 profiles along down-track direction, and was collected in October 2002. In this experimental site, the soil dielectric constant is believed to be about 4.0, while the conductivity is unknown (neither parameter has been measured in situ); the plastic targets have the dielectric constants of around 3.0. Below, we will illustrate the relevant 2-D imaging results of metallic and plastic land mines. We adapted the Seismic Unix developed by Colorado School of Mines [Cohen and Stockwell, 2002] to process the GPR data using the phase-shift migration technique assuming a half-space. In Figures 15, the left panels show the raw GPR data, while the right panels show the migrated results. On these profiles, the target contours in rectangles are approximately superimposed to show the focusing and localization capabilities of the algorithm. Note that in these profiles the height of the targets are scaled to the two-way travel time with the approximate wave velocity of 15 cm/ns in the soil.

Figure 1.

(left) The raw GPR data and (right) the migrated profile for metallic land mine A. The superimposed rectangle represents the target. Below is the same for Figures 2–5.

Figure 2.

Same as Figure 1 except for metallic land mine B.

Figure 3.

(left) The raw GPR data and (right) the migrated profile for plastic land mine A.

Figure 4.

Same as Figure 3 except for plastic land mine B, cross-track processing.

Figure 5.

The down-track processing of GPR data for plastic land mine B. (left) The raw data and (right) the migrated result.

2.3. Migration Examples for Land Mine Detection

2.3.1. Metallic Land Mine A

[16] Shown in the left panel of Figure 1 is the raw data for the GPR array on top of a antitank metallic land mine. The circular land mine is 21.6 cm high and 23.0 cm in diameter, and is buried 7.62 cm deep. It is observed that there is a strong horizontal reflection event due to the ground bounce, which is “clipped” in order to show the weaker signal from the land mine. More interestingly, a diffraction hyperbola appears in this section. The hyperbola is not symmetric, possibly due to the tilt of the land mine. The right panel of Figure 1 shows the migrated result. Clearly, the diffraction hyperbola in the raw data now collapses into the correct shape approximately. This example, as well as other examples not reported here for metallic land mines, shows that the image of land mine was significantly improved after migration.

2.3.2. Metallic Land Mine B

[17] This example shows the data for another metallic antitank land mine B. It is circular, with a diameter of 32.0 cm and a height of 10.2 cm, and is buried 15.24 cm deep. Shown in Figure 2 are the raw data and migrated result. Again, there is an excellent focusing due to the migration.

2.3.3. Plastic Land Mine A

[18] For plastic land mines, the diffraction from the objects are usually much weaker than metallic land mines. Thus any focusing effects the migration technique can provide will be potentially very useful in land mine detection and discrimination. This example is for a plastic, antitank land mine buried 7.62 cm deep. It is circular, with a height and diameter of 9.02 cm and 22.0 cm, respectively. From the raw data in the left panel of Figure 3, it is observed that there is a weak diffraction hyperbola pattern. The migration result in the right panel of Figure 3 shows the target is focused well by the migration technique even for such a week contrast.

2.3.4. Plastic Land Mine B Cross-Track Processing

[19] This example shows the data for a circular plastic land mine. Its height and diameter are 6.5 cm and 28.0 cm, respectively, and it is buried 10.16 cm deep. In the left panel of Figure 4, it can be seen that the raw section has a diffraction hyperbola pattern. Despite the fact that the land mine signature is close to the edge and is weak, the migrated result in the right panel of Figure 4 displays the focused target fairly well.

2.3.5. Plastic Land Mine B Down-Track Processing

[20] To demonstrate that the migration in the down-track direction can sometimes be used to improve the results when the target is near the edge of the sensor array, we now re-process the data for the above plastic land mine, using the data in the down-track direction. We extract the data at sensor 5, which is directly above the land mine in the cross-track direction in this example, for 24 down-track positions. Now, we can observe a diffraction hyperbola pattern centered in this raw profile of Figure 5. Migrated section in Figure 5 shows that the target is better focused than the previous cross-track processing. This example shows that the profile can be organized along a different direction to improve the overall imaging quality. For instance, targets located at the edge of a normal cross-track profile may not be easily detected. This means that there are detection gaps between two adjacent lanes even if there is some overlap in the survey. To tackle such problem without re-surveying, one may re-organize the data sets along different directions to have better view of the subsurface.

2.4. Discussions on GPR Migration for Land Mine Detection

[21] As shown in the above examples, the metallic mines buried in different depths generally have very clear hyperbolic-type reflection patterns. Thus the migrated images can be well focused. For plastic mines, due to the smaller electrical contrast with respect to the background medium, the relevant reflection patterns of targets may not be very clear. This is the area where the migration technique is likely to provide the most help in improving the ratio of probabilities of detection and false alarm (Pd/Pf ratio). Furthermore, for signals from the edge of the sensor array, it may be advantageous to extract and re-process the data along other directions (such as the down-track direction) to improve the image quality.

[22] In summary, the presented examples show that migration can produce a good improvement in image quality in GPR land mine detection. Such a processing scheme is real time as it only takes 0.06 seconds to process one image on a PC. Four future directions will likely benefit the GPR detection and discrimination of land mines: (1) this migration method can be integrated with signal detection algorithms such as those by Collins et al. [1999] and Gader et al. [2001] to improve the Pd/Pf ratio; (2) with a model of precise information of the wave velocities in the subsurface layers, we can extend this method to a layered earth model to further improve the image quality since some artifacts in the migrated sections may be removed significantly; and (3) seismic migration may be further improved if it is combined with some other signal processing techniques for clutter reduction [e.g., van der Merwe and Gupta, 2000] or statistical analysis methods [e.g., Xu et al., 2002].

3. Three-Dimensional Contrast Source Inversion Method for Targets in Layered Media

3.1. Theory

[23] As a complementary technique to seismic migration, we consider the electromagnetic inverse scattering problem of targets embedded in a layered background medium having M planar layers extending to infinity in x and y directions. Layer 1 is air with permittivity ε0 and permeability μ0, where the GPR transmitting and receiving antennas are located (defined as the measurement domain S). The mth layer is characterized with a dielectric constant εrm, conductivity σm, and permeability μm. We assume that inhomogeneous objects with dielectric constant εr(r), conductivity σ(r) and permeability μi are completely embedded in layer i, where we can define a reconstruction domain D to enclose these unknown objects. Assuming time dependence exp(jωt), the complex permittivity for each background layer and the objects are defined as equation image = ε0εrmjσm/ω and equation image = ε0εrjσ/ω, respectively. Our objective is to reconstruct the complex permittivity equation image given the GPR measurements in air. We apply the contrast source inversion method [van den Berg and Kleinman, 1997] to solve this 3-D inverse scattering problem in a layered medium.

[24] Because of the space limit, we will not detail the formulation. A full description of the method will be reported elsewhere [Song et al., 2004]. For a description of the contrast source inversion method for objects in a homogeneous background medium, the reader is referred to Abubakar et al. [2002a, 2002b] and the references therein. Below we will only summarize the key steps in the inverse method. If the target is embedded in the ith layer, the contrast function χ(r) is defined as χ(r) = equation image − 1. According to Xu and Liu [2002], the scattered electric field measured in the first layer (air) can be written as

equation image

where the contrast source w is defined as w(r) = χ(r)E(r) for rD, ki is the wave number in the ith layer, Gmi is the dyadic Green's function for the magnetic vector potential in the layered medium [Michalski and Mosig, 1997]. Furthermore, from the integral equation, the contrast source can be written as

equation image


equation image

[25] Similar to the contrast source inversion in a homogeneous background [van den Berg and Kleinman, 1997], a cost functional is defined as

equation image

where the first term in the right-hand side of equation (12) measures the misfit in the data equation (9) and the second term measures the misfit in the object equation in (10). The latter term plays a role of regularizing the ill-posed problem. Hence the method has an advantage that there is no need to choose a artificial regularization parameter. For the detailed updating steps to minimize the above functional, the reader is referred to van den Berg et al. [1999] and Abubakar and van den Berg [2000] for a homogeneous background. Note that the major difference between this work and the previous works for 3-D inversion in a homogeneous background [Abubakar and van den Berg, 2000; Zhang and Liu, 2004] are the operations GS,D. They are now involving the dyadic Green's function for the magnetic vector potential in a layered medium, rather than a scalar Green's function in a homogeneous background. The fast method for these operations is described by Xu and Liu [2002] and Millard and Liu [2003, 2004].

[26] The main computational burden of the CSI method for layered media lies on the operations of GS,D and their adjoint operations. GS and its adjoint operator require O(MN) arithmetic operations, where M is the number of data points and N the number of voxels inside D. GD and its adjoint operator require O(MTN log N) arithmetic operations by using the fast Fourier transform (FFT) algorithm, where MT is the number of transmitters. Since the CSI method is performed in a functional formulation instead of matrix formulation, the memory is not a big issue. In the CSI procedure, one only needs to store contrast source vectors w and contrast χ and several auxiliary vectors of the same size; thus the memory requirement is O(MTN).

3.2. Three-Dimensional Inverse Scattering Results

[27] We now examine some numerical results of the 3-D CSI method for reconstructing buried objects in a layered medium. The “measured” scattered fields are obtained by synthetic results using the forward modeling method presented by Millard and Liu [2004]. Since the inverse method does not use the forward solver during inversion, there is no question of the so-called “inverse crime” being committed here.

[28] Figure 6 shows the configuration for a two- or three-layer background characterized by εr1 = 1.0, σ1 = 0 S/m; εrm = 1.5, σm = 0.01 S/m; εr2 = 2.0, σ2 = 0.02 S/m. In this configuration, the positive z direction points upward and the positive x and y directions are defined with the right-hand system. The origin of the coordinates is referred to the center of the D domain. We use this configuration to simulate two scenarios: (1) for a two-layer medium, z1 = z2 = 2.15 m so that the middle layer (index m) is absent and (2) for a three-layer medium, z1 = 2.15 m and z2 = 2.11 m. The center of test domain D is at (1.0, 1.0, 2.04) m and its dimension is 9.2 × 9.2 × 9.2 cm3. The distance from the top side of the D domain to the interface at z1 is d = 6.4 cm. Domain D is divided into a number of cubes, assuming the contrast function to be piecewise constant. In D domain, there are two buried objects to be reconstructed, with the electrical parameters εr = 4.0 and σ = 0.16 S/m. Note that the permeability for background and the objects is all equal to μ0 in free space. Under the operating frequency of 2 GHz, the dielectric wavelength in lower half-space is around 10.6 cm.

Figure 6.

The geometry for reconstructing 3-D objects buried in a two- or three-layer medium characterized by εr1 = 1.0, σ1 = 0; εrm = 1.5, σm = 0.01 S/m; εr2 = 2.0, σ2 = 0.02 S/m. For the two-layer case, z1 = z2 = 2.15; for the three-layer case, z1 = 2.15 m, z2 = 2.11 m.

[29] In the numerical simulation for the synthetic data, the incident electric field is generated by a z-directed electrical dipole J = equation imageδ(rr0), where r0 is the source location. We assume a 2-D planar array with 64 sources and 64 receivers uniformly distributed, with an aperture dimension of 60 × 60 cm2 and 3 cm high above z1. Thus we have complex data of 4096 considering one component Ez of the scattered data in the reconstruction.

[30] In the following simulations, the initial guess is derived from a single frequency migration. Following Claerbout [1992], such migration is performed by applying the adjoint or migration operator GS of the forward modeling operator GS in equation (9) to the data to derive the migrated image of the contrast source w0 = αGSf, where α may be used to scale the migration operation in a least squares sense [van den Berg et al., 1999]. Once w0 is derived, the initial contrast χ0 can be calculated by the constitutive relation w = χE. In the migration, the dyadic kernel and the wave number terms in the integral equations are all changed to their complex conjugates for back-propagating wave fields from receiver positions. Thus such migration operation is also known as back-propagation [van den Berg et al., 1999].

[31] Note that in the first part of this report the 2-D scalar migration is implemented in frequency-wave number domain for a wide-band data under a far field approximation of a free space Green's function. In this part, a migration operation is adapted for easily producing an initial model directly usable in the inverse scattering procedure and performed in space-frequency domain for a single frequency data considering the exact dyadic Green's function in a layered medium.

[32] First, we consider the two-layer case where domain D is divided into 15 × 15 × 15 voxels, thus 3375 complex unknowns to be reconstructed. The problem is the so-called mixed-determined (it is not overdetermined because of the nonuniqueness even though the number of data points is larger than the number of unknowns). Figures 7a and 7b show the ground truth of dielectric constant and conductivity profiles of two buried objects, on three orthogonal cross sections. The simultaneously reconstructed profiles of dielectric constant and conductivity at iteration 50, 100, 300, 500 are displayed in Figures 8 and 9, respectively. By way of comparison, we observe that the dielectric constant profile in Figure 8a at early iteration 50 has revealed the shapes and locations of the objects, although the inverted dielectric constant values are still small compared to the true ones. After 50 more iterations, as illustrated in Figure 8b, the inverted profile is further improved in the level of dielectric constant. At iteration number to 300 and 500, the inverted profiles Figures 8c and 8d become closer and closer to the original profile in the spatial distribution and dielectric constant level. The inverted conductivity profiles of Figure 9 also exhibit a fashion similar to the case of dielectric constant. It is visually demonstrated that the convergence of our algorithm is fairly stable for this problem. To examine the effect of an initial guess, we perform an inversion starting from the migration with multifrequency data (0.50–3 GHz at the interval of 0.50 GHz). After 500 iterations, the corresponding inverted result of dielectric constant and conductivity in Figures 10a and 10b are almost identical to Figures 8d and 9d. In Figure 11, it is observed that the initial guess from the multifrequency data yields the smaller errors at the beginning of the iteration. If the iteration was to be terminated early, say at around 100 iterations, for cases with high noise, this improved initial guess would have accelerated the inversion process. However, under the present lower noise case, the fitting errors for the data and object equations would eventually overlap with the case where the initial guess was provided by a single-frequency back-propagation. In summary, it is seen that the errors are reduced significantly around after 100 iterations and then continue to decrease monotonically with iterations. For this example, the CPU time of 500 iterations is around 5 hours on an IBM 690 and the memory requirement is 80 MB.

Figure 7.

The ground truth of (a) dielectric constant (b) and conductivity for a two-layer case.

Figure 8.

The inverted profiles of dielectric constant at iteration (a) 50, (b) 100, (c) 300, and (d) 500 for the case in Figure 7.

Figure 9.

The inverted profiles of conductivity at iteration (a) 50, (b) 100, (c) 300, and (d) 500 for the case in Figure 7.

Figure 10.

With the initial guess from migration in multifrequency data, the inverted profiles of (a) dielectric constant and (b) conductivity at iteration 500 for the case in Figure 7.

Figure 11.

The fitting errors of (a) the data and (b) the object as the function of iteration number for the two-layer case in Figure 7, corresponding to the single-frequency (SF) and multifrequency (MF) initial guesses.

[33] Now using the same model parameters as the above, we study the performance of the 3-D imaging technique in the form of the actual 1-D array configuration. We simulate the two kinds of 1-D arrays. The first configuration is a multistatic array having 8 sources and 8 receivers uniformly distributed along cross-track (x) direction; this 1-D array is then moved 8 times at constant interval of 8.6 cm along down-track (y) direction to collect data. The total number of data is 8 × 8 × 8 = 512. The second configuration is a monostatic array with 8 sensors, each being a source and receiver to perform a monostatic measurement. In this case, the number of data is only 8 × 8 = 64. After 500 iterations, Figures 12a and 12b illustrate the reconstruction of dielectric constant and conductivity using the multistatic array and Figures 12c and 12d the dielectric constant and the conductivity using the monostatic array. Compared to the results in Figures 8d and 9d using a full 2-D array, it can be observed that the inversion of the locations and shapes of the objects are generally good under such sparse data, although the level of the inverted electric parameters are lower than those of the actual ones, especially in the case of the monostatic array.

Figure 12.

The inverted profiles of (a) dielectric constant and (b) conductivity with 1-D multistatic arrays and (c and d) with 1-D monostatic arrays for the case in Figure 7.

[34] Next we continue the simulation with the 2-D array. In order to examine the effects of the topsoil on the inversion, we perform the imaging for a three-layer case. The configuration geometry is referred to Figure 6. The true and reconstructed profiles (at iteration 500) of dielectric constant and conductivity are shown in Figures 13 and 14, respectively. It appears that the spatial distribution of the two buried objects is generally well reconstructed, although the maximum values of the dielectric and conductivity are somewhat lower than the previous case.

Figure 13.

For a three-layer case, (a) the original and (b) inverted profiles of dielectric constant.

Figure 14.

For a three-layer case, (a) the original and (b) inverted profiles of conductivity.

[35] Finally, we present an example to reconstruct one buried object in the two-layer case using the same configuration geometry as in the first example. The size of the object is 5 × 5 × 5 cm3. In this case, we discretize domain D into 25 × 25 × 25 voxels, thus there are 15625 complex unknowns, but the number of data points remains to be 4096. The purpose of this test is concerned with the performance of the reconstruction algorithm for more unknowns. Figures 15 and 16 depict the true and the reconstructed profiles (at iteration 600) of dielectric constant and conductivity, respectively. It is evident that the buried object is reconstructed but its position appears to shift a little downward in Figure 15b and a little upward in Figure 16b. This may be attributed the fact that the one-side limited view inherently has the potential to introduce various distortions in the inversion, probably leading to the shift in the locations of the reconstructed objects. This phenomenon is somewhat observed in the other examples. Such ambiguity may be reduced by imposing additional a priori information if available or increasing view aperture from the other side. However, we restrain from doing that in this work. In light of the stable convergence behavior in the CSI, it may have possible improvement by continuing more iterations. Overall, it is observed that the reconstruction captures the features of the true object distribution even when the number of unknowns is much larger than the number of independent measurements. For this example, the CPU time of 600 iterations is around 20 hours on an IBM 690 and the memory requirement is 368 MB.

Figure 15.

For a two-layer case with 25 × 25 × 25 voxels, (a) the original and (b) inverted profiles of dielectric constant.

Figure 16.

Same as Figure 15 except for conductivity.

4. Summary and Future Work

[36] We employed two complementary imaging techniques, 2-D seismic migration and 3-D layer-medium inverse scattering, for processing the GPR measurements in land mine detection and discrimination. The seismic migration technique appears to be an effective method to focus the diffraction hyperbola onto the land mine targets. Thus, potentially, this technique can improve the Pd/Pf ratio. The method is efficient and can provide the imaging in real time, although there is no quantitative information obtained for the contrast values of the land mine versus the background medium. It is shown that it may be useful to extract and re-process the data along different directions than the cross-track direction; this can also provide a pseudo-3-D image of subsurface/target. We have tested this technique extensively on field data obtained by Niitek, Inc., on a U.S. government testing ground. Future work in this area will focus on incorporating this technique into signal decision algorithms to improve the Pd/Pf ratio.

[37] However, the 3-D inverse scattering method in a layered medium can greatly improve the image after migration or back-propagation and provides accurate reconstruction of the permittivity and conductivity profiles of buried object, although it is computationally more time consuming. Preliminary numerical results demonstrate the utility of the full 3-D nonlinear GPR imaging algorithm in layered media using a 2-D array and a family of 1-D arrays. In order to apply this technique to field data, we would explore a way to speed up the inverse scattering method incorporating with multifrequency data for further improving subsurface imaging capability.


[38] This work was supported by a DARPA/ARO MURI grant DAAD19-02-1-0252, by Army Night Vision Laboratory, and by NSF through grant CCR-0219528. The authors are grateful to Niitek, Inc., for providing the GPR data; to Sandy Throckmorton, Fred Clodfelter, and Peter Torrione for their help in understanding the data format; to Leslie Collins, Lawrence Carin, and Richard Weaver for the feedback; and to Zhong Qing Zhang, Immo Trinks, and Fenghua Li for their useful discussions on this work. Valuable discussions on the CSI with Aria Abubakar are greatly appreciated. The authors are very grateful for the reviewers' constructive comments, which helped to improve the paper.