## 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.