## 4 Multi-Frequency Method
The reconstructions now consist of sinc pulses of different width, say \(2l_{i}\) with \(i\in(1,2)\), both have a peak at the source position \(x=0\) but the replicas appear at different rates. This simple framework suggests the adoption of a multiplicative fusion scheme to mitigate artifacts. Indeed, it is apparent that multiplying \(|J_{eq}(x,f_{1})|\) and \(|J_{eq}(x,f_{2})|\) still retains the peak at \(x=0\) (i.e., the actual source position) where the two contributions overlap, while other peaks may be reduced with a proper choice of the frequency step. In particular, in order to completely avoid the first replicas of \(|I_{eq}(x,f_{1})|\) and \(|I_{eq}(x,f_{2})|\) overlap, it must hold that  
\[\Delta x_{1}-\Delta x_{2}\geq l_{1}+l_{2} \tag{14}\]  
i.e.,  
\[\lambda_{1}\frac{N}{2u_{max}}-\lambda_{2}\frac{N}{2u_{max}}\geq\frac{\lambda_{ 1}}{2u_{max}}+\frac{\lambda_{2}}{2u_{max}} \tag{15}\]  
which leads to the condition  
\[f_{2}\geq f_{1}\frac{N+1}{N-1} \tag{16}\]  
Figure 5: Sketch of main lobe, first and second replicas at different frequencies \(f_{1}\) (blue) and \(f_{2}>f_{1}\) (red). which in turn dictates how to choose the subsequent higher frequency. In the same way, if we want that the second replica at \(f_{2}\) does not overlap with the first one at \(f_{1}\), the condition is  
Figure 4:0  
that leads to the condition  
Figure 4:1  
which can be satisfied when \(N>5\). How must conditions in Equations (16) and (18) be interpreted and what do they guarantee about artifacts in the reconstruction? Equation (16) basically represents the frequency step, so that the frequencies can be chosen as  
Figure 4:2  
whereas Equation (18) sets somehow the frequency band, because once \(f_{1}\) has been fixed,  
Figure 4:3  
If Equations (19) and (20) are both fulfilled, then an artifact-free region of about \([-\Delta x_{1},\Delta x_{1}]\) is guaranteed around the target. What is more, previous criteria depend on the number of data \(N\) and, hence, points out how spatial under-sampling can be somehow and partially counteracted by collecting data at different frequencies. In particular, a lower \(N\) requires choosing subsequent frequencies at higher multiplicative steps, which roughly can be thought of as a trade-off between data reduction and frequency bandwidth requirements. In order to more easily get the required bandwidth, condition (Equation (14)) can be relaxed by accepting a partial superposition of the aliasing replicas. More in detail, by setting \(\Delta x_{1}-\Delta x_{2}\geq\alpha(l_{1}+l_{2})\) with \(\alpha<1\), the subsequent frequencies can be chosen according to the criterion  
Figure 4:4  
Note that if a larger free of artifacts region should be required, the same reasoning as above can be followed to derive different, and generally more stringent, conditions. Finally, we stress that similar arguments can be applied for the other spatial dimension. Eventually, one adopts the more severe requirements, among the two, so that for both dimensions, the non-overlapping criteria are satisfied. According to previous results, the detection and localization of the buried target can be achieved by the following fusion strategy  
Figure 4:5  
where \(J_{eq}(x,y,f_{n})\) is obtained by a standard linear inversion scheme based on the truncated singular values decomposition procedure and \(\|I_{eq}(x,y,f_{n})\|=\sqrt{|J_{eq}(x,y,f_{n})|^{2}+|J_{eq}(x,y,f_{n})|^{2}}\) accounts for both the reconstructed surface current components. It is worth noting that the choice of a multiplication based fusion strategy, of course is not the only way to merge the different reconstructions. Indeed, the optimization of this point is beyond the scope of the presented paper as the focus is on the possibility of exploiting non-coherent reconstructions to mitigate spatial under-sampling.