# Two-Dimensional Linear Inversion of GPR Data with a Shifting Zoom along the Observation Line
## 2 Formulation of the Inverse Scattering Problem  
The problem at hand inserts into the framework of microwave tomography from GPR data, which has been largely investigated in the literature, both from the point of view of the imaging algorithms [30; 31; 32; 33] and from the point of view of the effects of the soil and antennas on the data [34; 35; 36]. Applications to cases in the field have been also presented, e.g., in [37; 38].  
Here, we briefly report a mathematical formulation that describes a well-known model based on the Born approximation. In particular, the problem at hand concerns a two-dimensional (2D) case with a homogenous background scenario. Under the Born inverse model, a linear integral relationship relates the backscattered electric field (\(E_{s}\)) to the unknown contrast function (\(\chi\)). This contrast function is defined as the relative difference between the (complex) permittivity of the target and the reference scenario. Under the here considered multi-monostatic/multi-frequency configuration, such a relationship is given by:  
\[E_{s}(\mathbf{r}_{o},\omega)=k_{b}^{2}\iint\limits_{\Lambda}G(\mathbf{r}_{o}- \mathbf{r}^{\prime},\omega)E_{inc}(\mathbf{r}_{o},\mathbf{r}^{\prime},\omega )\chi(\mathbf{r}^{\prime})d\mathbf{r}^{\prime} \tag{1}\]  
where \(G\) denotes a Green's function, \(E_{inc}\) is the incident field and \(k_{b}\) is the wave number in the relevant medium. In this specific case, the incident field is modeled as the field radiated by a filamentary line source, fed by a unitary electric current located at \(\mathbf{r}_{o}\). \(\Lambda\) is the investigated domain, i.e., the spatial region where we assume that the targets reside, whereas the subscript o denotes the observation point, belonging to the observation domain.  
Here, we make use of a homogeneous reference scenario, and deal with a 2D scalar case. The measurement configuration is multi-monostatic. Consequently, Equation (1) can be specialized as:  
\[E_{s}(\mathbf{r}_{o},\omega)=j\frac{\omega\mu_{0}}{8}k_{b}^{2}\iint\limits_{ \Lambda}\ \left[\mathrm{H}_{0}^{(2)}\left(k_{b}\big{|}\mathbf{r}_{o}-\mathbf{r}^{\prime} \big{|}\right)\right]^{2}\chi(\mathbf{r}^{\prime})d\mathbf{r}^{\prime} \tag{2}\]  
where \(\mathrm{H}_{0}^{(2)}(\cdot)\) denotes a Hankel function of second kind and zero order. Indeed, a theoretically more correct formulation could be done with the Green's function of a half-space, as done in [18]. However, the Green's function of a homogeneous medium (a world \"done of soil\") performs accurately enough (especially if the antennas are shielded, which is the case for the experimental data exploited in this paper), and is computationally more efficient than the Green's function of a half-space. The right-hand side of Equation (2) defines the linear operator (\(\mathbf{L}\)), relating the unknown contrast function to the scattered field data. The linear integral operator (\(\mathbf{L}\)) can be expressed in terms of its Singular Value Decomposition (SVD). Hence, \(\sigma_{n}\) is the \(n\)-th singular value, and \(\mathbf{v}_{n}\) and \(\mathbf{u}_{n}\) are the \(n\)-th left and right singular functions of the operator (\(\mathbf{L}\)). It is worth noting that \(\{\mathbf{v}_{n}\}\) and \(\{\mathbf{u}_{n}\}\) define the orthonormal basis for the object space, and the data space, respectively [13; 22]. A formal solution of the inverse scattering problem is provided by [13; 22]:  
\[\chi(\mathbf{r})=\sum_{n=1}^{\infty}\frac{1}{\sigma_{n}}\langle\mathbf{E}_{s}, \mathbf{u}_{n}\rangle\mathbf{v}_{n}(\mathbf{r})\ \ \ \ \ \mathbf{r}\in\Lambda \tag{3}\]  
The ill-posed nature of the inverse problem imposes the adoption of a regularization scheme. A regularized solution is achieved here by using the Truncated SVD (TSVD):  
\[\hat{\chi}(\mathbf{r})=\sum_{n=1}^{\mathrm{T}}\frac{1}{\sigma_{n}}\langle \mathbf{E}_{s},\mathbf{u}_{n}\rangle\mathbf{v}_{n}(\mathbf{r})\ \ \ \ \ \mathbf{r}\in\Lambda \tag{4}\]  
where the regularization parameter is the truncation index (T) in the finite summation Equation (4).