## 3 Methods and Data
### Validation of the PS-MTI Results by Comparison to GPS Data  
In order to be able to directly compare the PS-MTI derived displacements to the GPS reference ones, the GPS 3D displacement vectors (\(U_{\mathrm{G}}\)) were projected along the SAR line of sight (LOS) (\(D_{\mathrm{G}}\), LOS) through the standard dot product:  
\[D_{\mathrm{G}}=U_{\mathrm{G}}\times n_{\mathrm{LOS}} \tag{1}\]  
where \(n_{\mathrm{LOS}}\) is the unit vector along the LOS direction, positively orientated between the satellite and the ground direction. The \(n_{\mathrm{LOS}}\) unit vector could then be described in the east, north, up reference system by:  
\[n_{\mathrm{LOS}}=(\mathrm{sin}\alpha_{\mathrm{s}}\;\mathrm{sin}\theta;\; \mathrm{cos}\alpha_{\mathrm{s}}\;\mathrm{sin}\theta;\;\mathrm{cos}\theta) \tag{2}\]  
where \(\alpha_{\mathrm{s}}\) is the clock-wise angle between the north and the scan line direction (i.e., 96.75\({}^{\circ}\)) and \(\theta\) is the mean incidence angle (i.e., 45.77\({}^{\circ}\)). Considering the geometry of these acquisitions, the unit vector was defined with \(n_{\mathrm{LOS}}=(0.712;\,-0.084;\,0.696)\).  
\begin{table}
\begin{tabular}{c l l} \hline \hline
**Year** & \multicolumn{2}{c}{**Month and Day of Acquisition**} & **Total** \\ \hline
2013 & 12 September; 30 September; 14 October; 26 October; 13 November; 27 November; & \\  & 10 December; 29 December & \\ \hline
2014 & 9 January; 22 January; 18 February; 3 March; 18 March; 1 April; 23 April; 14 May; & \\  & 27 May; 26 June; 11 July; 16 July; 8 August; 26 August; 11 September; 23 September; & \\  & 10 October; 21 October; 6 November; 25 November; 12 December; 28 December & \\ \hline
2015 & 12 January; 27 January; 11 February; 3 March; 18 March; 1 April; 16 April; 29 April; & \\  & 12 May; 27 May; 12 June; 25 June; 6 July; 22 July; 4 August.; 15 September & \\ \hline \hline \end{tabular}
\end{table}
Table 2: List of periodic GPS data.  
For a better 3D visual comparison of \(U_{\text{G}}\) in relation to this unit vector, we decided to compare it to a fixed \(n^{\prime}_{\text{LOS}}\) vector negatively oriented along the x-axis (i.e., western direction) such as:  
\[n^{\prime}_{\text{LOS}}=\text{sqrt}(x^{2}+y^{2}+z^{2})=100\text{ mm} \tag{3}\]  
A first assessment to quantitatively define the differences between GPS-LOS and PS-MTI results is based on the percentage of mean velocity retrieved by PS-MTI in comparison to GPS-LOS and GPS-3D, by using the following formulations:  
\[\%\text{ VEL}_{\text{LOS}}=(\text{PS-MTI}_{\text{mean velocity}}/\text{GPS-LOS}_{\text{mean velocity}})\times 100\% \tag{4}\]  
\[\%\text{ VEL}_{\text{3D}}=(\text{PS-MTI}_{\text{mean velocity}}/\text{GPS-3D}_{\text{mean velocity}})\times 100\% \tag{5}\]  
Moreover, the cumulative and differential displacement time-series have been analyzed in terms of absolute offset and relative offset by using the following formulations:  
\[\text{OFFSET}_{\text{abs}}=\text{PS-MTI}_{\text{displacement}}-\text{GPS-LOS}_{ \text{displacement}} \tag{6}\]  
\[\text{OFFSET}_{\text{rel}}=\frac{\text{PS-MTI}_{\text{displacement}}-\text{GPS-LOS}_{ \text{displacement}}}{\text{GPS-LOS}_{\text{displacement}}}100\% \tag{7}\]