## 3 SAR Datasets and Methodology
### Vertical and Horizontal Component Analysis  
SAR satellites follow specific semi-polar (North-South) orbits and acquire information in the West-East direction (Figure 4) [38]. This means that, for ascending data, positive values move toward the satellite in the East-West direction, while negative data moves away from the satellite in the West-East direction. On the other hand, the positive descending data moves toward the satellite in a West-East direction, and the negative data in an East-West direction. It is possible to obtain the displacement components in the West-East and vertical directions by knowing the direct cosines of image acquisition, as shown in Figure 5. In this work, a MATLAB script [43] was implemented to decompose the movement. The first step requires the production of a grid (the MATLAB script allows one to choose the grid dimension).  
After the first step, the script calculates the number of ascending and descending RTs located within each cell and their mean rate value. Once this grid has been produced, three different formulas are applied, depending on the amount of ascending and descending RTs in the cell.  
In case there are both ascending and descending RTs inside of a cell, the formula [30] used is:  
\[v_{z}\;(Vertical\;Component)=\frac{(Rate_{desc}\times sx_{asc})-(Rate_{asc} \times sx_{desc})}{(sx_{asc}\times sz_{desc})-(sx_{desc}\times sz_{asc})} \tag{1}\]  
\[v_{x}\;(Horizontal\;Component)=\frac{(Rate_{asc}-v_{z}\times sz_{asc})}{sx_{asc}} \tag{2}\]  
where:  
* \(Rate_{asc}\) and \(Rate_{desc}\) are the velocities obtained from the ascending and descending data;
* \(Sx_{asc}\) is equal to \(-\)sin(\(\theta_{asc}\)) \(\times\)\(cosin\)(\(\tau_{\text{asc}}\));
* \(Sz_{asc}\) is \(cosin\)(\(\theta_{asc}\));
* \(Sx_{desc}\) is equal to \(-\)sin(\(\theta_{desc}\)) \(\times\)\(cosin\)(\(\tau_{desc}\));
* \(Sz_{desc}\) is \(cosin\)(\(\theta_{desc}\));
* \(\tau\) is the angle between the orbit of the satellite and the North (heading angle);
* \(\vartheta\) is the incident angle. If inside of the cell there are ascending RTs and no descending RTs, the formula used is: \[v_{zasc}=\frac{Rate_{asc}}{cosin(\vartheta_{asc})}\] (3)  
\[v_{zasc}=-\frac{Rate_{asc}}{\sin(\vartheta_{asc})} \tag{4}\]  
Figure 4: The geometry of acquisition: (**a**) ascending, (**b**) descending. The warm color palette indicates the direction of movement away from the satellite (–) while the cold color palette indicates the direction of movement towards the satellite (+).  
Figure 5: Geometric representation of the decomposing of the data.  
On the other hand, when there are only descending RTs, the formula does not change:  
\[v_{\textit{zdesc}}=\frac{Rate_{\textit{desc}}}{\textit{cosin}(\vartheta_{ \textit{desc}})} \tag{5}\]  
\[v_{\textit{xdesc}}=\frac{Rate_{\textit{desc}}}{\text{sin}(\vartheta_{ \textit{desc}})} \tag{6}\]  
From these formulas, we obtained the values related to the satellite \"reference\" system (positive when it moves toward the satellite, negative when it moves away from it). As shown in Figure 5, to obtain a value suitable for our cartesian coordinate system, we have to multiply the value obtained in Vx\({}_{\text{asc}}\) with \(-1\). That is because it is customary to have the positive side of the \(x\)-axis going toward the first and fourth quadrants. Instead, the values obtained from the ascending data are negative when they move toward the East. As for the vertical components, there is no need to intervene on the formulas because, in both cases, the movement towards the satellites corresponds to moving along the \(z\)-axis towards its positive values.