## 4 Data Processing and Experimental Results: Sounder Mode
### Data Processing  
Like any penetrating radar, the Sounder data collected onboard, usually named raw-data, are arranged in a two-dimensional matrix representing the radar echoes as a function of the fast time and the slow time, which are related to the two-way sensor-to-target distance (range) and the position of the sensor along the flight trajectory (azimuth), respectively. The main rationale of the processing chain applied to focus these raw data is shown in Figure 3 and consists of the cascade of four steps. The first step is an azimuth pre-summing aimed at improving the signal-to-noise ratio. In this regard, it is noted that the Sounder PRF (500 Hz, see Table 1) leads to a spectral window significantly larger than the Doppler bandwidth (about 40 Hz) achievable with the available azimuth beam (50\({}^{\circ}\), see again Table 1) and the platform velocity (on the order of 40 m/s). Accordingly, the pre-summing factor can be safely set on the order of 10 or even more. In the cases at hand, a factor of 20 has been applied along with a proper anti-aliasing filtering procedure. Figure 3: Sounder data processing. Figure 2: Erfoud area, southeastern Morocco. The red, green and blue lines represent the projections on the ground of the flight tracks relevant to the acquisitions carried out with the Sounder, SAR-Low and SAR-High modes, respectively. The second processing step (see again Figure 3) is the chirp pulse compression [65], which includes also the application of a spectral windowing procedure (in the case at hand, we use Hamming window) aimed at reducing the level of the secondary lobes of the impulse response relevant to the range-focused image. Following this step, the data are still not focused along the azimuth direction: they are thus generally characterized by the presence of diffraction hyperbolas that can make very difficult the interpretation of the radar image. The data interpretation is further complicated by the fact that the Sounder acquisition is typically corrupted by the so called motion errors, that is, the deviations of the flown flight track with respect to an ideal rectilinear path. Indeed, the motion errors, if not properly accounted for, produce distortion effects on the final radar image. For instance, in the presence of altitude variations, a flat surface (for instance a lake) is imaged in the Sounder data as a curve specular to the flight path instead of being seen as flat [66]. Application of a proper data processing step aimed at reducing all these effects is thus needed at this stage. In our case, the azimuth focusing procedure, including the compensation of the motion errors, is carried out in the frequency domain. In particular, following the application of a Fast Fourier Transform (FFT) algorithm along the range direction (see again the processing chain of Figure 3), the microwave tomography algorithm described below is applied. To this aim, let us consider the two-dimensional Sounder acquisition geometry shown in Figure 4. \(\Gamma\) is the flight path that, as observed above, is typically not a straight line, that is, the flight altitude with respect to the ground is variable. \(\underline{r}_{m}\) represents the generic measurement position, that is, the antenna phase center position at a generic azimuth time, say \(t_{m}\). Let \(D\) be the 2D domain of investigation within which we assume the targets of interest are present and let \(\underline{r}\) be the position of a generic point in \(D\). The applied tomographic reconstruction approach is based on the following assumptions:  
* the scene under investigation is invariant along the direction perpendicular to the plane defined by the normal to the ground and the (nominal) flight direction (2D geometry);
* the radar antenna is located in the far zone with respect to the investigated area. Moreover, the transmitting antenna is modeled as an electrical line source perpendicular to the survey plane;
* the propagation in the soil is negligible because the flight altitude is much greater than the depth of the target of interest.