## 1. Introduction

[2] In many ground-penetrating radar (GPR) applications, such as antipersonnel land mine remediation, one is typically interested in detecting, localizing and characterizing shallowly buried small targets with constitutive properties very close to those of the background soil [*Dubey et al.*, 2001]. In such applications, a major source of corruption and distortion in the interrogating signals is related to reflection from, and (double) transmission through, the irregular unknown air–soil interface. Traditional approaches to cope with this problem tend to model the roughness effect through a purely statistical Monte Carlo-based approach, e.g., via additive colored Gaussian noise [*Dogaru and Carin*, 1998; *Yang and Rappaport*, 2001]. Such techniques perform reasonably well in detection problems in the presence of small roughness [*Dogaru et al.*, 2001; *Zhan et al.*, 2001], but have been found to yield limited accuracy and reliability concerning localization and classification for moderate roughness [*Feng et al.*, 2000b].

[3] In an ongoing series of recent investigations, so far restricted to 2-D geometries, we have been working toward a more robust *adaptive* approach for imaging low-contrast mine-like targets shallowly buried under a moderately rough air–soil interface, using both frequency-stepped and pulsed GPRs. Our approach is based on prior estimation of the coarse-scale roughness profile. In this connection, a low-dimensional parameterization of the unknown interface is used in conjunction with physics-based approximate forward scattering models based on *narrow-waisted* quasi-ray Gaussian beams (GB) [*Galdi et al.*, 2001a, 2002b], and the interface estimation problem is posed as a nonlinear optimization problem [*Galdi et al.*, 2002a, 2003]. The estimated roughness profile is then used to compensate for the effect of the air–soil interface in the received data. The corrected data are used subsequently to image the subsurface region and localize possible anomalies. At this stage, statistical models may be useful to account for noise, measurement uncertainty, and residual unmodeled effects.

[4] In our investigation, we have proceeded along two parallel routes, in the frequency domain (FD) and time domain (TD), in order to explore both frequency-stepped and direct TD pulsed operation for eventual time-resolved GPR scenarios. In particular, the present paper (together with the work of *Galdi et al.* [2002a]) deals with the frequency-stepped FD formulation, in contrast with the direct TD formulation of *Galdi et al.* [2001b, 2003]. Both the FD and TD formulations have most of the technical background related to the GB forward solvers in common. The data processing, however, is significantly different. Although the FD approach in the present paper and in the work of *Galdi et al.* [2002a] could be applied, via Fourier inversion, also to *wideband* TD data, *Galdi et al.* [2001b, 2003] have pursued the above alternative *direct* TD processing route, which they believe to be better matched to the wideband *physics* of the problem, with better insight as a consequence.

[5] Referring to the works of *Galdi et al.* [2001a, 2001b, 2002b] for technical background on the GB forward scattering algorithms, our focus in this paper is on the description and evaluation of the *FD inversion* scheme, in particular on its performance in connection with *object-based* reconstruction. While the inverse scattering algorithms utilized here are *formally* independent of the chosen forward solver, the *overall performance* depends *strongly* on this choice, which in our case is based on GBs [for details, see *Galdi et al.*, 2001a, 2001b, 2002b].

[6] The rest of this paper is organized as follows. In section 2, the problem is formulated and the underlying geometry is described. In section 3, the multifrequency rough surface estimation algorithm of *Galdi et al.* [2002a] is briefly reviewed. Section 4 deals with the subsurface imaging problem in the presence of a known air–soil moderately rough interface. Numerical results, limitations of the approach, and computational issues are discussed in section 5. Brief conclusions are given in section 6.