## 6 Conclusions  
A method for detecting and localizing shallowly buried targets from under-sampled multi-frequency far-zone scattered field data has been presented. Basically, the method relies on two main ingredients: a suitable scattering model and a simple procedure to process multi-frequency under-sampled data. The scattering model is derived according to the equivalence theorem which allows to cast the detection as the reconstruction of equivalent sources supported over a reference plane (above the targets). Inverting this model offers a number of advantages. First, the detection becomes a 2D problem with obvious reduction of the computational burden. Also, since the incident field is embodied into the unknown equivalent sources, coherence (i.e., synchronization) between the TXs and Rxs is not necessary when single-frequency data are employed. This in principle makes the overall system more cost effective. Figure 16: The same as Figure 15 but with \(z_{p}=-0.4\) m.  
Figure 15: Normalized (linear scale) multi-frequency reconstruction of two metallic cylinders with axis parallel to the \(x\)-axis, buried within a half-space. Black solid line denotes actual cylinders’ position. The parameter of configuration are the same as Figure 8, the considered depth is \(z_{p}=-0.1\) m.  
Even under this model and single-frequency data the study of the NDF of the scattered field has revealed that the number of far-zone spatial measurements required to correctly image the equivalent currents may become impractically large, especially for electrically large investigation domains. In order to reduce the need for such a high number of spatial measurements (and, hence, the system cost) we have proposed to use multi-frequency data. In particular, here frequencies are used to mitigate artifacts that arise when data are reduced with respect to the NDF. Since artifacts are frequency dependent, images obtained at different frequencies are fused via a simple multiplicative strategy. A criterion to choose the frequencies in order to have a spatial region free from artifacts has been provided as well. A drawback of the proposed method is that the target's depth is not _directly_ retrieved. However, this appears a minor drawback when the targets are shallowly buried beneath the air/soil interface (this is just the case we focused on herein). Moreover, by changing the reference depth \(z_{p}\) the proposed method exhibits a certain ability to retrieve the targets' depths with a minor impact on the computational cost. Numerical results for scattering objects embedded in free-space and within a half-space medium confirmed the feasibility of the approach. The presented results appear promising but represent only a preliminary step towards the experimental validation of the method, since on-field conditions will present harder challenges to be faced. However, the proposed strategy is linear and intrinsically very robust and stable against uncertainties on data; hence, we are confident that our method will work also for realistic scenarios. Also, we have to mention that we already made some experiments for the case of a homogeneous medium (targets were not buried) as reported in [26]. Actually, to test the proposed strategy against experimental data for a subsurface scattering is an our commitment that will be addressed in future developments. Finally, we observe that it would be interesting to compare the proposed method with others present in the literature. Most of those methods employ multi-frequency coherent processing, hence the comparison with them would not be fair. On the other hand, comparison with a single-frequency (full-data) method is already contained in this paper. To our knowledge, a fair comparison could be achieved only against compressed sensing methods [37], which can work with a reduced number of measurements and by considering one frequency at time. Each author contributed extensively to the preparation of this manuscript. In particular, conceptualization, A.B., G.L. and R.S. ; methodology, A.B., G.L. and R.S.