## 3 Materials and Methods
### Static, Hydraulic Controlled and Coseismic Cover Stability  
To compute the SF in the soil cover and consider landslide processes featured by shape ratio (\(\mathrm{L/H>10}\)) (i.e., mainly shallow landslides), we used a modified version of the preceding equation which was proposed by [PERSON] and [PERSON] [38] to account for partially saturated soil conditions:  
\[\mathrm{FS}=\frac{\tan\ \varphi^{\prime}}{\tan\ \beta}+\frac{2c^{\prime}}{ \gamma\mathrm{Hsin}2\beta}-\left(\left(\frac{\sigma^{\mathrm{s}}}{\gamma \mathrm{H}}\right)\cdot\left(\tan\ \beta+\cot\beta\right)\cdot\tan\ \varphi^{\prime}\right), \tag{6}\]  
where \(\sigma^{\mathrm{s}}\) is the suction stress; \(\varphi^{\prime}\) is the internal peak or residual friction angle (\({}^{\circ}\)); \(c^{\prime}\) is the cohesion (kPa); \(\gamma\) is the unit weight of soil cover (kN/m\({}^{3}\)); and \(\beta\) is for slope angle (\({}^{\circ}\)). The \(\gamma\) values have been estimated for each geo-lithological unit in the laboratory and referred to literature values reported in [PERSON] et al. [45] and [PERSON] et al. [46]. As the value of H, we considered and fixed different depths within the soil cover to assess the in-depth distribution of SF and weight the contribution of capillarity in the slope stability. Regarding the mechanical parameters, peak strength conditions were respectively assumed for slope portions that did not experience instabilities in the past while residual conditions were adopted within the inventoried landslide perimeters (i.e., already occurred in the past). In the case of shallow landslides, for the calculation of \(\sigma^{\mathrm{s}}\), which accounts for the capillary forces that counterbalance the shear stress in unsaturated soils [47], we assumed the relation formalized in [PERSON] et al. [48]:  
\[\sigma^{\mathrm{s}}=-\frac{\mathrm{Sr}}{\sigma}\left(\mathrm{Sr}\frac{\mathrm{ n}}{1-\mathrm{n}}-1\right)^{\frac{1}{n}}, \tag{7}\]  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Geotechnical Units** & \(\theta_{\mathrm{r}}\) (-) & \(\theta_{\mathrm{s}}\) (-) & **a (m2/kN)** & **n** & **k\({}_{\mathrm{s}}\) (cm/h)** \\ \hline
1 & 0.0967 & 0.4841 & 0.0173 & 1.2641 & 0.16 \\
2 & 0.0982 & 0.4908 & 0.0177 & 1.2535 & 0.08 \\
3 & 0.1030 & 0.5122 & 0.0197 & 1.1963 & 0.  
where Sr is the degree of saturation, defined as in Equation (8) and \(\alpha\) and n represent the fitting parameters of the [PERSON]'s [43] equation reported above (Table 2). The value of Sr was obtained according to the following equation:  
\[\text{Sr}=\frac{\theta_{\text{i}}\ -\ \theta_{\text{r}}}{\theta_{\text{s}}-\ \theta_{\text{r}}}\, \tag{8}\]  
where \(\theta_{\text{i}}\) is the water content (-); \(\theta_{\text{s}}\) is the water content in saturated conditions (-), and \(\theta_{\text{r}}\) is the residual water content (-). To calculate the SF values in seismic conditions, the [PERSON]'s method was applied, based on a yielding acceleration (K\({}_{\text{c}}\)) threshold calculated firstly by a limit-equilibrium pseudo-static analysis through the relations:  
\[\text{K}_{\text{c}}=(\text{SF}-1)\times\sin\beta\text{ or K}_{\text{c}}=(\text{SF}-1)\times\tan\beta, \tag{9}\]  
respectively used if acceleration acts parallel to the slope or horizontally. According to [PERSON] [49], the analysis referred to a simplified solution neglecting the vertical component of the ground motion. Coseismic displacement was computed adopting the semiempirical regression model proposed by [PERSON] and [PERSON] [37] for cohesive soils, which upgrades the original formula by [PERSON] [50; 51] in the following form by considering a worldwide EqTLs dataset:  
\[\log\ \text{D}_{\text{n}}=0.802\text{logI}_{\text{a}}-10.981\text{A}_{\text{c} }+7.377\text{A}_{\text{c}}\text{log I}_{\text{a}}+1.914\pm 0.274, \tag{10}\]  
where I\({}_{\text{a}}\) identifies the Arias intensity (m/s) and A\({}_{\text{c}}\) the critical acceleration (m/s\({}^{2}\)).