Once an estimate of the coarse-scale air–soil profile is available, the subsurface imaging problem can be addressed in a quasi-deterministic fashion. In this connection, we restrict consideration to the problem of subsurface sensing in the presence of a known roughness profile. Statistical models can subsequently be used to account for both noise and residual unmodeled effects.
4.1. Forward Scattering Model
 As customary in GPR problems, we split the y-directed total backscattered field E observed in free space at r = (x, z) into the background field Eb (the field in the absence of the target) and the scattered field Es (due to the target)
where, via the Lippman–Schwinger equation [cf. Chew, 1996],
In (7), E represents the total field in the target region, Gb denotes the FD Green's function of the rough-interface dielectric half-space, and
is the so-called object function, with εr(r′) and σ(r′) denoting the local relative dielectric permittivity and electrical conductivity, respectively, and Δεr, Δσ defined via the last equality in (8). The integration in (7) is extended over the target region () (Figure 1) wherein the object function in (8) is nonzero.
 The first step in our procedure consists of using the estimated surface profile to generate predictions of the background field Eb via the forward GB model of Galdi et al. [2002b], thereby isolating the scattered contribution Es, which contains the information needed for imaging the target. Next, in view of the typically low contrast in the target scenarios of interest here (plastic antipersonnel land mines), we use the linearizing Born approximation which replaces the total field E inside the target by the transmitted field Et in in the absence of the target (see the work of Keller  for the inherent limitations). One obtains
where the (weak) conductivity contrast contribution Δσ in (8) has also been neglected. The inversion of the linearized model in (9) will be addressed in section 4.2. The unperturbed transmitted field Et and the Green's function Gb in (9), which account for further distortion of the useful signal due to the twice-traversed rough air–ground interface, are computed efficiently via the PO GB syntheses detailed by Galdi et al. [2001b, 2002b], which are conceptually analogous to that in (5).
4.2. Object-Based Target Reconstruction: Curve Evolution
 The inversion of the forward scattering model in (9) is an inherently ill-posed problem, thereby implying limits of retrievable information through inverse scattering. Moreover, only limited-viewing noisy observations and approximate forward modelings are available in our problem conditions. Therefore, it is essential to incorporate stabilization via regularization methods [cf. Bertero, 1989; Karl, 2000] which, through exploiting possible a priori information, restore well-posedness by suitably restricting the solution space. In the work of Galdi et al. [2001b], with reference to pulsed GPRs, various pixel-based and object-based reconstruction and regularization approaches have been explored for reliable inversion of the TD counterpart of the Born-linearized forward scattering model in (9). Here, we focus on the object-based approaches. Unlike pixel-based approaches, where one tries to retrieve the unknown dielectric contrast Δεr at a number of suitably small pixels in a mosaiced test domain, object-based approaches rely on the use of parametric or semiparametric deformable shape models for the object function, which implicitly incorporate possible a priori information about the target geometry. For our specific problem, the target being homogeneous, it is suggestive to estimate the key features of the target (shape and dielectric contrast) directly rather than reconstruct them pointwise (with all the inherent problems of a posteriori edge detection). Several applications of such approaches to EM inverse scattering have been proposed in the recent past. Chiu and Kiang [1991a, 1991b, 1992a, 1992b] have used a shape-based formalism in conjunction with the Newton–Kantorovich technique for inverse scattering from (possibly imperfect) conducting cylinders buried in a homogeneous (possibly lossy) half-space. Budko and van den Berg [1998, 1999] have used an effective circular cylinder scattering model, with radius, permittivity and center position to be retrieved, to simulate inverse scattering due to subsurface dielectric targets. Miller et al.  have used low-order polynomial expansions for background and target permittivity and quadratic B-spline parameterizations for the target boundary to deal with inhomogeneous background and targets.
 A particularly interesting class of object-based inversion techniques is represented by curve evolution (CE) [Yezzi et al., 1997; Shah, 2000], where a gradient flow is designed which attracts initial closed curves to the target boundary. Such techniques, widely used in image processing, have recently been explored in EM inverse scattering problems, and have revealed attractive features from both the computational and reconstruction quality viewpoints. For instance, in the works of Santosa , Litman et al. , Dorn et al. , and Ramananjaona et al. [2001a, 2001b], CE techniques have been applied to FD nonlinear inverse scattering problems involving penetrable targets with known electric properties in a homogeneous background. Feng et al. [2000a] and Galdi et al. [2001b] have explored applications to subsurface imaging of low-contrast targets in the presence of flat and moderately rough air–soil interfaces, using TD data, and we retain their approach for the latter geometry here to perform FD imaging of subsurface targets with unknown dielectric properties in the presence of a moderately rough air–soil interface. Although, in principle, the method can handle multiple targets, here we consider the simplest scenario with a single homogeneous target occupying the region () (Figure 1) bounded by a continuous curve . To proceed, the object function in (8) is rewritten as
with , and with Π denoting the characteristic function of the target region ,
Here, prior-information-based implicit regularization is implemented through the homogeneity condition in (10), with the unknowns of the inverse problem now becoming the target boundary and the single value of permittivity contrast .
 Given a set of Nr × Nω backscattered field observations at angular frequencies ω1, …, ωNω at receiver locations r1r, …, rNrr (Figure 1), the problem of estimating target boundary and dielectric contrast is posed as an optimization problem involving minimization of the cost functional
where ℰpqs ≡ ℰs(rpr, ωq) represents the observed target-scattered contribution (i.e., after background field removal) at angular frequency ωq at receiver location rpr, and
with k0q2 = ωq2ε0μ0. The first term in the cost functional in (12) enforces fidelity to the data, whereas the second term provides additional regularization by penalizing the arc length of the estimated curve, with the choice of the regularization parameter β affecting its smoothness. The proper choice of β is an important issue, and several strategies have been proposed [cf. Hansen, 1992]. In our implementation, β is selected empirically by trial and error, taking into account prior expectations about target geometry (e.g., convexity). Given a family of smooth curves (τ) parameterized by τ, the minimization of the cost functional in (12) is achieved by evolving the curve (τ) along the negative gradient of JCE with respect to (τ) (steepest descent). Using arguments similar to those of Feng et al. [2000a] one finds
where , r′c denotes points on the curve (τ), and and κc indicate the outward normal and the signed curvature of at r′c, respectively. For a given τ, the optimal value of the contrast is estimated by enforcing the stationarity of (12),
For numerical convenience, the evolution in (14) needs to be discretized in τ and stepped forward. The procedure is initialized through a rough initial guess of the target boundary, which is subsequently used to obtain, via (15), the initial contrast estimate. The procedure is thus evolved, with alternative updating of the curve (τ) (via (14)) and of the contrast estimate (via (15)) until convergence is achieved. In this connection, the target boundary initial guess was found to be not particularly critical for the overall accuracy, affecting primarily the convergence rate. Our numerical implementation is based on the level set method [Osher and Sethian, 1988; Santosa, 1996], which was found to provide numerically efficient and stable evolution. Implementation details are similar to those of Feng et al. [2000a] and are not discussed here.