## 2 Materials and Methods
### MTInSAR Time Series Analysis  
As previously mentioned, all data series were processed through the SPINUA MTInSAR processing suite [19]. As with most MTInSAR processing algorithms, the result of the processing consists of a large set of SAR pixels corresponding to coherent objects on the ground, characterized by their coordinates and by a merit figure such as the temporal coherence, and featuring the residual height and the mean line-of-sight (LOS) velocity, in addition to the complete time series of relative LOS displacements at each acquisition date. The temporal coherence is defined as \(\gamma_{t}=\frac{1}{N_{t}}\Big{|}\sum_{j=1}^{N_{t}}\exp{i(\phi_{j}-\hat{\phi} _{j})}\Big{|}\), where \(\phi_{j}\), with \(j=1,\ldots,N_{t}\), are the InSAR phases of the time series, and \(\hat{\phi}_{j}\) are obtained from a model trend. The model has to be postulated a priori in order to define the coherence figure. The most common choice, often adopted to minimize processing times, is to use a linear model, or e.g., a low-order polynomial. Clearly, such models are not suited for the type of nonlinearities involved in ground displacement phenomena such as those investigated here. Therefore, when no ad hoc, computation-intensive processing can be performed on limited datasets involving, e.g., libraries of more complex phase models, selecting targets to investigate time series trends is usually carried out by adopting low threshold values for the provided \(\gamma_{t}\) figure (which results in a large number of selected targets), and relying on the analysis of experts with knowledge of the area, or by integration of ancillary data taken on the ground or by other means. In these cases, automated procedures may help identifying a few targets with peculiar, nonlinear motions that may signal, e.g., previously undetected phenomena, or reactivations/deactivation phases of known landslides. Most approaches adopt some parameterization of the time series trends, relying on statistical tests to assess the most suitable representation of each time series, e.g., through piecewise-linear or polynomial functions (e.g., [33; 34]). In particular, in [34] a procedure is proposed, based on the Fisher distribution, that is able to automatically detect nonlinearities such as piecewise-linear trends, which are typical of warning signals preceding the failure of natural and artificial structures. This nonlinear trend analysis (hereafter referred to as NLTA) processes the displacement time series derived by a generic MTInSAR algorithm and provides as output the coefficients of the polynomial model of degree \(\overline{n}_{p}\) selected for describing the time series, and an index \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Test Site** & **Sensor** & **Geometry** & **N. Images** & **Time Interval** \\ \hline \multirow{3}{*}{Pomarico} & S1 & Ascending & 158 & 15/10/2014-15/05/2018 \\  & S1 & Descending & 141 & 21/10/2014-21/05/2018 \\  & CSK & Ascending & 96 & 28/10/2012–23/06/2018 \\ \hline \multirow{3}{*}{Montescalioso} & S1 & Ascending & 182 & 15/10/2014-29/12/2018 \\  & S1 & Descending & 159 & 21/10/2014-29/12/2018 \\ \cline{1-1}  & CSK & Ascending & 96 & 28/10/2012–23/06/2018 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Dataset characteristics. based on the Fisher statistics, which allows evaluating the quality of the approximation and the temporal coherence \(\gamma_{t,\;OUT}\) computed by using the selected polynomial model. The polynomial degree is selected as:  
\[\overline{n}_{p}=\min\Bigl{(}\left\{n_{p}\middle|F\bigl{(}n_{p}\bigr{)}\leq F_{a} \bigwedge\;F_{A}\bigl{(}n_{p}\bigr{)}\leq F_{Aa}\right\}\Bigr{)} \tag{1}\]  
where \(F\) is the standard Fisher test, and \(F_{a}\) and \(F_{Aa}\) are threshold values that can be set according to both the number of samples (\(N\)) composing the time series and the significance level \(\alpha\) of the statistical test [34]. In this work, the threshold values are set to \(F_{a}=F_{Aa}=4\).