## 2 Surface Scattering Model  
In this section we briefly recall the relevant scattering equations according to the model presented in [26]. Consider a half-space background medium with the separation interface (i.e., the air/soil interface) at \(z=0\). The scattering targets to be detected are assumed buried in the lower half-space (i.e., for \(z<0\)) and located within the investigation domain \(D=[-x_{in},x_{in}]\times[-y_{in},y_{in}]\times[z_{in},0]\), with \(z_{in}<0\). The electromagnetic features of the soil are unknown. The scattering scene is probed by a single source located at some stand-off distance (from the air/soil interface) from the air side. Its location, timing and radiated waveform are not known. What matters for our detection scheme is only the working frequency band. The field scattered by the buried targets is collected over a set of sensors still located on the air side and all at height \(h\) above the ground. Therefore, all sensors belong to a given measurement plane: say \(\mathbf{r}_{n}\), with \(n=1,2,\cdot\cdot\cdot,N\) their positions. Hence, a single-view/multistatic configuration is considered. The sensors are assumed to be in the far-zone of the scene under investigation. Figure 1 shows a pictorial view of the scattering configuration and illustrates the adopted reference frame. Figure 1: Pictorial view of the scattering configuration and reference system. The air/soil interface is at \(z=0\), the scattering object depth is \(z_{T}<0\) and the equivalent plane at \(z_{p}\). Say the target is buried at the unknown depth \(z_{T}\in[z_{in},0]\). According to the equivalence theorem [30], its scattered field can be considered as being radiated by equivalent surface currents, both electric and magnetic, just supported over a plane at \(z_{p}>z_{T}\). We address this as the equivalent plane (see Figure 1). By filling the half-space \(z<z_{p}\) with a perfect magnetic conductor, only the electric current turns out to be relevant. Moreover, the magnetic conductor can be removed by doubling the electric current. Accordingly, under the far-zone approximation and a proper transformation of the far-field components, the scattered field can be expressed as [26]  
\[ \begin{split}\mathbf{E}_{S}(\phi,\theta)&=\zeta k^ {2}\exp\left(jkz_{p}\cos\theta\right)\\ &\int_{D_{T}}\exp\left[jk\sin\theta(x\cos\phi+y\sin\phi)\right] \mathbf{J}_{eq}(x,y)dxdy\end{split}  \tag{1}\]  
where \(D_{T}=[-x_{in},x_{in}]\times[-y_{in},y_{in}]\) is the investigation domain cross section, the observation point \(\mathbf{r}\) has been expressed in terms of its spherical components, i.e., \(r,\theta\) and \(\phi\), \(\zeta\) and \(k\) are the characteristic impedance and the wavenumber of the free-space. Some explanations are in order. First, we observe that, to be rigorous, surface integration should run over the entire plane \(z=z_{p}\) since equivalent currents do so. However, the target is very close to the air/soil interface, and hence to the plane \(z=z_{p}\), therefore, it is reasonable to assume that the current support is very similar to the target's cross section. Hence, \(D_{T}\) needs to be only large enough to account for all the possible positions of the targets. It must be clarified that the model in Equation (1) assumes a free-space Green function. This is clearly correct when \(z_{p}=0\), that is when the equivalent plane coincides with the air/soil interface. When this is not the case, the model is basically neglecting the part of propagation that takes place in the soil. The main effect of this approximation is that the targets act as if they were more deeply located (because soil is electromagnetically denser than air). However, this is not a serious drawback because the targets are shallowly buried, i.e., the propagation path in the half-space is much shorter than the one in free-space. What is more, choosing \(z_{p}\) not necessarily equal to zero gives a degree of freedom which can be used to adjust focusing in the imaging stage.