## 4 Combining Ascending and Descending Sentinel-1 Images to Analyze
### Projection to a Common Reference System  
The target zone is located in a stable Paleozoic basement of gentle mountains that characterize a rounded shape landscape. In regard to the potential causes of ground motion, it is worth saying that there are no unconsolidated sand-gravel aquifers, there are no volcanoes, and the tectonic activity is irrelevant in this geological context [41]. Therefore, it is reasonable to assume that the observed displacements in the abandoned elements of the mine, pit, and dumps, are caused by the loss of mechanical integrity. As mentioned earlier, we assume that displacements are exclusively in two directions: (1) Vz (vertical displacement) in zones of a gentle slope, and (2) Vslope (along the slope) in steep zones. The threshold slope value between those two trends is deduced from the representation of slopes vs. Vlos (Figure 6). This graph shows that the largest displacements occur for slopes greater than 13\({}^{\circ}\), as a consequence, we assume that 13\({}^{\circ}\) is the slope threshold between a Vz dominance and a Vslope dominance. The implementation of this strategy can be reduced to solve a simple problem of geometry where the projection of the velocity vector can be found by dividing Vlos by the cosine of the angle \(\sigma\left(\cos\sigma\right)\) formed with the slope in every point. In the flat areas, we  
Figure 6: Cross plot of slopes vs. Vlos. The red line shows the 13\({}^{\circ}\) threshold from which the largest displacements begin. project to the vertical, which is equivalent to assuming a 90\({}^{\circ}\) slope. Therefore, to get \(\cos\sigma\) we define the unit vector uSLOPE in the slope direction, and aLOS (ascending) and dLOS (descending) in the LOS direction. The uSLOPE is defined by the angles S (slope) and A (aspect) (Figure 7a). The aLOS and dLOS are defined by the incidence angle of the beam, \(\theta\), and the azimuth angle of the LOS direction, \(\alpha\) (Figure 7b). In our test-site of Riotinto, with the selected ascending and descending tracks, it turns out to have a very similar incidence angle on ascending and descending geometries, \(\theta\approx 42^{\circ}\) (Figure 1). Besides this, Sentinel-1 LOS azimuth angle, \(\alpha\), is always 80\({}^{\circ}\) for ascending and \(-80^{\circ}\) for descending. The cosine of the angle (\(\sigma\)) between SLOPE and LOS is the result of the scalar product of their unit vectors:  
\[\text{uSLOPE}\cdot\text{uLOS}=\cos\sigma \tag{1}\]  
The geodetic components are:  
\[\text{uSLOPE}\ =(\cos\text{S}\cdot\sin\text{A},\ \cos\text{S}\cdot\cos\text{A},-\sin\text{S}) \tag{2}\]  
where S and A are the slope and aspect angles, respectively. Whereas the uLOS decomposition depends on the acquisition geometry:  
\[\text{uLOS}=\text{aLOS}\ =(\sin\theta_{\text{a}}\cdot\sin\alpha_{\text{a}}, \sin\theta_{\text{a}}\cdot\cos\alpha_{\text{a}},-\cos\theta_{\text{a}}) \tag{3}\]  
\[\text{uLOS}=\text{dLOS}\ =(-\sin\theta_{\text{d}}\cdot\sin\alpha_{\text{d}}, \sin\theta_{\text{d}}\cdot\cos\alpha_{\text{d}},-\cos\theta_{\text{d}}) \tag{4}\]  
Equation (3) is for ascending geometry and Equation (4) is for descending. In the special case of the selected Sentinel-1 tracks in Riotinto (Figure 7b, the incidence angle of the beam and the LOS direction meet:  
\[\theta_{\text{a}}=-\theta_{\text{a}}=42^{\circ}\qquad\text{and}\quad\ \ \ \alpha_{\text{d}}=-\alpha_{\text{d}}=80^{\circ}\]  
Finally, Vslope can be calculated as:  
\[\text{Vslope}\ =\text{Vlos}/\cos\sigma \tag{5}\]  
Figure 7: **(a)** Geodetic components of the slope unit vector in slope, uSLOPE. **(b)** Geometry of the line of sight (LOS) unit vector and geodetic components ascending, aLOS, and descending, dLOS. **(c)** Cone of all possible slopes and two angles (\(\sigma\)) with a LOS vector. Projecting LOS to the slope goes to extremely high values when \(\sigma\) approaches 90\({}^{\circ}\) (\(\sigma\)2).