## 2 Rivers Water Flow Elevation Retrieval
### The Processing Scheme  
The workflow of the proposed estimation technique is depicted in Figure 2 and comprises 5 main processing blocks. The main rationale of the proposed procedure is based on the pixel tracking. Sub-pixel offset tracking (SPOT) is a relevant technique to measure large-scale ground displacements in both range and azimuth directions. The technique is complementary to differential interferometric SAR and persistent scatterers interferometry when the radar phase information is unstable [23; 24]. In this paper, we apply the SPOT in an alternative way, as explained in the following. Figure 1: Observation geometry for the river water flow elevation estimation: (**a**) Range-azimuth representation (**b**); Range-height representation. The starting point of this algorithm is a long temporal series of InSAR data but, instead of investigating the deformations of the ground, we investigated the movements of the double-bounce scattering effect of man made structures localized on the water surface. Although the variations in the water levels of rivers vary over time much faster with respect to the movements of the ground, we also apply the SPOT technique to trace their hydrometric levels. To this end, it is necessary to design a specific mathematical model that is described in detail below. Indeed, as shown in Figure 2, the starting point (block 1) is the selection of a long temporal series of SAR data, which is based on the desired temporal observation period. This series is, then, processed using images pairs in order to track the hydrometric levels variation over the time. Specifically, the magnitude at the output of the two-dimensional matched filter of the receiver chain can be expressed as the following matrix (for simplicity, we assume that target position in the range-azimuth plane is \((0,0)\)) [25] [Chapter 4], for each interferometric complex pair, indexed by \(i\) and belonging to a time series of length \(G\):  
\[ \begin{split}& r_{c,D}^{i}(n_{c},n_{D})=A\frac{\sin(n_{c}\delta_{R_{ c}}/B_{c})}{(n_{c}\delta_{R_{c}}/B_{c})}\times\frac{\sin(n_{D}\delta_{R_{D}}/B_{ D})}{(n_{D}\delta_{R_{D}}/B_{D})},i=1,\dots,G\\ & n_{c}=-N_{c}/2,\dots,0,\dots,N_{c}/2,\\ & n_{D}=-N_{D}/2,\dots,0,\dots,N_{D}/2,\\ & N_{c},N_{D}\in\mathbb{N}\ (\text{even}),\end{split}  \tag{1}\]  
where:  
* \(A\) is the backscattering coefficient;
* \(N_{c}\) is the number of pixels of the image along the range;
* \(N_{D}\) is the number of pixels of the image along the azimuth;
* \(\delta_{R_{c}}\) is the chirp resolution;
* \(\delta_{R_{D}}\) is the Doppler resolution;
* \(n_{c}\) is the chirp wavenumber;
* \(n_{D}\) is the Doppler wavenumber;
* \(B_{c}\) is the bandwidth of the transmitted chirp signal;
* \(B_{D}\) is the synthesized Doppler band. Figure 2: Flow chart of the main procedure to extract water level information from a temporal series of synthetic aperture radar (SAR) data. Such data will necessarily have to be co-registered (the coregistration process consists of perfectly aligning the pixels of any slave image to the corresponding pixels of the master image. The alignment process is very precise and can also be accurately performed at the sub-pixel level). The co-registration stage is performed as the second stage of the processing chain. After co-registration, the stage 3 exploits the range shifts for an initial coarse estimation of the double-bounce shifts and error correction.