## 2 Rivers Water Flow Elevation Retrieval
### The Processing Scheme
To give a sketch of these steps, let consider the offset components of the sub-pixel normalized cross-correlation, that according to References [15; 26] are described by the complex parameter \(\boldsymbol{D^{i}}_{\text{tot}_{(c,D)}}\) referred to as total displacement, which is given by:  
\[\boldsymbol{D^{i}}_{\text{tot}_{(c,D)}} =\boldsymbol{D^{i}}_{\text{disp}_{(c,D)}}+\boldsymbol{D^{i}}_{ \text{topo}_{(c,D)}}\] \[+\boldsymbol{D^{i}}_{\text{orbit}_{(c,D)}}+\boldsymbol{D^{i}}_{ \text{control}_{(c,D)}}+\boldsymbol{D^{i}}_{\text{atmosphere}_{(c,D)}}+ \boldsymbol{D^{i}}_{\text{noise}_{(c,D)}},i=1,\ldots,G, \tag{2}\]  
where:  
* \(\boldsymbol{D^{i}}_{\text{disp}_{(c,D)}}\) is the offset component of the signal position presented in (1), generated by the variation of the river water level and detected as a sub-pixel misalignment existing between the first SAR image (master) and the \(i\)-th slave SAR image;
* \(\boldsymbol{D^{i}}_{\text{topo}_{(c,D)}}\) is the offset component generated by the earth displacement when located on highly sloped terrain;
* \(\boldsymbol{D^{i}}_{\text{orbit}_{(c,D)}}\) is the offset caused by residual errors of the satellite orbits;
* \(\boldsymbol{D^{i}}_{\text{control}_{(c,D)}}\) is the offset component generated by general attitude and control errors of the flying satellite trajectory;
* \(\boldsymbol{D^{i}}_{\text{atmosphere}_{(c,D)}}\) and \(\boldsymbol{D^{i}}_{\text{noise}_{(c,D)}}\) are the contributions accounting for change in the atmospheric and ionospheric dielectric constant and for decorrelation phenomena (spatial, temporal, thermal, etc. ), respectively. Note that the above equation accounts for the general case where displacement exists in both range and azimuth dimensions. In the operating scenario considered here, the displacement generated by the double-bounce scattering component of the bridge, when perturbed by the temporal variations of the river water levels, is significantly greater with respect to any other sporadic displacements due to physical phenomena. As depicted in Figure 1, this phenomenon is maximum when the longitudinal axis of the bridge is observed perpendicular to the range direction. With the above remarks in mind, we can assume that the contribution in the displacement \(\boldsymbol{D^{i}}_{\text{tot}_{(c,D)}}\) is greater along the range dimension. Thus, neglecting the azimuth component, we can consider \(\boldsymbol{D^{i}}_{\text{tot}_{(c,D)}}=\boldsymbol{D^{i}}_{\text{tot}_{c}}\). Figure 3 is a schematic representation of the parameters estimated by the coregistration procedure. The square number one is a focused pixel of the master image and the square number two is the same pixel but located on the slave image. The parameters \(\boldsymbol{D^{i}}_{\text{tot}_{c}}\) and \(\theta^{i}\) are the distance between the master and slave pixel centers and the angle respect to the horizontal axis respectively. In the present case, since the shift of the double-bounce scattering occurs in range, the parameter \(\theta^{i}=0\) so \(\boldsymbol{D^{i}}_{\text{tot}_{D}}=0\). Moreover, the atmospheric time-variation during the very short acquisition time interval has little influence on the temporal component of the last displacement parameters because of its low accuracy. All errors are compensated for, choosing only high energy and stable points and subtracting the initial offsets in order to retrieve the shifts contributions only generated by the target displacement. Once all the image pairs are co-registered and \(\mathbf{D}^{i}_{\text{tot}_{c}}\) is computed, we proceed with the estimation of the water level on a specific region of interest (ROI). The ROI extraction is performed by the block number 4 and the last computational block (number 5) performs a precise range shifts estimation and water level estimation as detailed in the sub-blocks (5.1-5.6). In particular, the data-input ROI are fed to the block 5.1 that extracts the sub-ROI, containing only the bridge.