## 5 Numerical Examples  
In order to check the proposed approach, in this section, we report some numerical examples. We start by considering first the targets embedded in free-space. This, of course is the simplest scenario against which to check the detection and the localization ability of the method and represents a benchmark for the more interesting half-space background medium. The incident field is provided by a \(y\) polarized plane-wave impinging along \(\theta=\pi\) direction, the equivalent (reconstruction) plane is considered at \(z_{p}\leq 0\) that can be varied in order to explore the depth as well. The investigation domain is a square of side 4 m (that is \(x_{in}=y_{in}=2\) m ) whereas rectangular metallic plates are considered as targets. Finally, the scattered field is observed over the observation domain \(\theta\leq\theta_{max}=\pi/4\). For this simple case, the scattered field is computed by adopting a standard method of moment based on the Physical Optics approximation. However, all the data are corrupted by additive white noise with \(10 dB\) of signal-to-noise ratio (SNR). In order to perform the single-frequency reconstruction, first, the linear integral operator Equation (1) is discretized by a standard method of moment [26]. To this end, here, Fourier harmonics are used to expand the unknown source function and a point matching procedure is set in the data space. Thus, the reconstruction is obtained by a truncated singular values decomposition procedure. Note that, even though both \(J_{seq}(x,y,f)\) and \(J_{seq}(x,y,f)\) are reconstructed (to get the reconstruction \(\|J_{eq}(x,y,f)\|\)), according to the presented model, the singular value decomposition needs to be computed only once. Moreover, this holds true even when reconstructions at different depths are desired. Indeed, in this case it is sufficient to change the \(z_{p}\) parameter in Equation (4), correspondingly normalize the scattered field, and rerun the reconstruction procedure. We start by considering two metallic plates of \(0.30\times 0.15\) m\({}^{2}\). The first one is centered at \((-0.5,-0.1,-0.2)\) m (center point) whereas the second one at \((0.6,0.1,-0.2)\) m. Hence, the two targets are at the same depth. For the configuration described above, the results in Section 3 say that, at \(f=1 GHz\), more than 200 observation points would be needed. In order to have a benchmark, Figure 6 shows the normalized single frequency reconstruction, \(\|J_{eq}(x,y,f)\|\) (with \(f=1\) GHz), of the two plates obtained by using full data, i.e., by sampling the \(u-v\) domain as dictated by Section 3. In particular, we fixed \(N=15\), so that the overall observations are \(N^{2}\) and set \(z_{p}=0\). As can be seen, both the targets are clearly detected and localized. Figure 6: Normalized linear scale) single-frequency reconstruction of two metallic rectangular plates both located at the same depth of \(-0.2\) m. Full data (i.e., \(15\times 15\) observations evenly taken in the \(u-v\) plane) at \(f=1\) GHz have been used and \(z_{p}=0\). Black solid line denotes actual plates’ shape and position. Now, we switch to considering under-sampled data. To this end, the observations are reduced to \(4\times 5=20\) (still evenly taken in the \(u-v\) plane), that is, less than \(1/10\) of the required ones. The corresponding reconstruction is shown in Figure 7. As expected, the reconstruction is corrupted by many aliasing artifacts that make it impossible to discern where the targets actually are. To overcome this drawback, we use the multi-frequency approach. In particular, here we choose \(\alpha=0.4\) in Equation (21) so that the corresponding frequencies turn out to be \((0.6694,0.8182,1.0000,1.2222,1.4938)\) GHz. Note that we kept \(f=1\) GHz as the central frequency. The corresponding multi-frequency reconstruction \(I(x,y)\) is shown in Figure 8. Now, the image is much less cluttered with artifacts than Figure 7 and the two targets are clearly detected and accurately located.