## 4 Data Processing and Experimental Results: Sounder Mode
### Data Processing
This approximation is acceptable when the goal is to locate shallow targets;
* the measurement configuration is multimonostatic/multifrequency so that, at each measurement point \(\underline{r}_{m}\), the radar illuminates the scene at nadir and collects the field scattered by the targets at the same measurement point \(\underline{r}_{m}\), within the frequency band of the transmitted signal;
* a linearized model of the electromagnetic scattering equations is exploited [67]. According to the above assumptions, the electromagnetic scattering phenomenon is governed by the integral equation [67]:  
\[E_{s}\Big{(}\underline{r}_{m},\omega\Big{)}=k_{0}^{2}\iint_{D}g\Big{(} \underline{r}_{m},\underline{r}\Big{)}E_{inc}(\underline{r})\chi(\underline{r })d\underline{r}=\mathcal{A}\chi \tag{1}\]  
where \(E_{s}\) is the co-polar component of the field scattered by the target (data) and received by the antenna in \(\underline{r}_{m}\) at the frequency \(\omega\), \(k_{0}\) is the propagation constant in free-space, \(E_{inc}\) is the incident electric field in \(D\), \(\chi\) is the so called contrast function, which is different from zero in the target regions and zero elsewhere. Moreover, \(\mathcal{A}\) is a linear operator that maps the space of the unknowns in the data space and  
\[g\Big{(}\underline{r}_{m},\underline{r}\Big{)}\approx\frac{e^{-|\underline{r }_{m}-\underline{r}_{m}|}}{\left|\underline{r}-\underline{r}_{m}\right|} \tag{2}\]  
accounts for the scalar Green's function in free-space [67]. The inverse problem defined by the Equation (1) is ill posed; adoption of a regularization scheme is thus necessary to obtain a stable and robust solution with respect to the data noise [68]. A simple inversion scheme that provides a regularized solution to the inverse problem is that based on the adjoint operator \(\mathcal{A}^{+}\)[68]:  
\[\chi(\underline{r})=\mathcal{A}^{+}E_{s}=\iint\Bigl{(}g\bigl{(}\underline{r}_{ mr},\underline{r}\bigr{)}E_{inc}(\underline{r})\Bigr{)}^{*}E_{s}\bigl{(} \underline{r}_{m},\omega\bigr{)}\underline{d}_{m}\underline{d}\omega \tag{3}\]  
where the symbol \({}^{\star\prime}\) denotes the conjugation operator. The module of the contrast function \(\bigl{|}\chi(\underline{r})\bigr{|}\), obtained through the Equation (3), represents the focused tomographic image \(I\) to be used for the subsequent analysis and interpretation. It is worth stressing that, despite the adjoint inversion scheme defined by Equation (3) is computationally efficient, the computational resources required are in any case high for the imaging problem at hand since large-scale (in terms of probing wavelength) data have to be processed. For this reason, we decided to adopt a shifting zoom approach [69], whose main steps are shown in Figure 5. More specifically, the measurement domain \(\Gamma\) and the survey area \(D\) are first partitioned into \(N\) partially overlapping subdomains, say \(\Gamma_{i}\) and \(D_{i}\), \(i=1\),...,\(N\). Then, for each subdomain \(D_{i}\) the reconstruction \(I_{i}\) is obtained by the adjoint inversion. The tomographic image relevant to the whole survey area \(D\) is finally obtained by combining the reconstructions \(I_{i}\) relating to the subdomains \(D_{i}\). Note that splitting the data processing domain in partially overlapping subintervals reduces the computational burden and in particular the memory occupation. In our case, the dimension of the azimuth partitioning has been chosen in the interval [150; 200] m.  
Note also that the inversion procedure described above makes it possible to process the data acquired in the presence of arbitrary flight paths, provided that the measurement positions \(\underline{r}_{m}\) are known. In this regard, it is remarked that the accurate position of the measurement points is provided  
Figure 4: Sounder acquisition geometry. Figure 5: Shifting zoom imaging approach. by the embedded INS-GPS device mounted onboard the platform.