## 6 Discussion  
Despite the congruence of our triggering maps and past observations, some questions can be raised, especially in view of the lack of historical events in many areas where landslide triggering is expected to occur (this could also be due to the incompleteness of historical accounts or to geological conditions that do not favor the occurrence of slope failures) but also considering the occurrence of earthquake-induced landslides (although rare) in areas where landslide triggering could be excluded based on our results. Thus, the following questions need to be answered. First, what is the (conditional) probability that an earthquake of magnitude \(\hat{M}=m\) triggers a landslide at a source-to-site distance \(\hat{R}=r\)? The knowledge of this probability in conjunction with the triggering maps presented above can guide priorities for action within the framework of risk mitigation programs and the scheduling of focused geological and geotechnical investigations aimed at assessing landslide susceptibility (or, conversely, rule out further studies if the seismic triggering of landslides is very unlikely). Second, can we actually exclude the occurrence of landslide triggering at sites where the \(\hat{M}\) and \(\hat{R}\) pairs are above the upper bounds of Keefer [1] (we recall that, in these cases, the modal \(M\)-\(R\) pair \((M^{*},R^{*})\) is assumed as the reference scenario)? Although rare, [PERSON] [8] documented cases of earthquakes that produced landslides at distances significantly greater than indicated by the upper bounds. To answer the above questions, for each computation node and for each failure mechanism, we have computed the conditional probability of exceeding specific Arias intensity [49] thresholds for landslide triggering given an earthquake scenario of magnitude \(\hat{M}=m\) and distance \(\hat{R}=r\): 0.1 m/s for disrupted slides and falls, 0.32 m/s for coherent slides, and 0.54 m/s for lateral spreads and flows [35; 36]. We recall that the Arias intensity (\(I_{a}\)) was found to correlate well with landslide displacement [18; 50], especially in the case of stiff and weak slopes [51]. Compared to peak acceleration as a descriptor of ground motion, \(I_{a}\) indeed has the advantage of reflecting the energy released by an earthquake, being calculated as the integral of an acceleration time history \(a(t)\) over its duration (Arias, 1970):  
\[I_{a}=\frac{\pi}{2g}{\int_{0}^{\infty}{[a(t)]^{2}dt}} \tag{3}\]  
The conditional probability of exceeding an Arias intensity value \(i_{a}\) can be computed as:  
\[P(I_{a}>i_{a}|\hat{M}=m,\ \hat{R}=r)=1-\Phi\left(\frac{\log{i_{a}-\mu_{\log{I_{a} }}}}{\sigma_{\log{I_{a}}}}\right) \tag{4}\]  
where \(\Phi\) indicates the standard normal cumulative distribution function, \(\mu_{\log{I_{a}}}\) is the mean logarithmic Arias intensity predicted by the ground motion prediction equation (GMPE) at hand, and \(\sigma_{\log{I_{a}}}\) is the associated standard deviation. In this application, the GMPE of [PERSON] et al. [52] derived for Italy was used. This GMPE uses a linear site-response term to account for ground motion amplification as a function of \(V_{5,30}\) (time-averaged shear wave velocity above 30 m depth). The values of \(V_{5,30}\) employed here were those provided by [PERSON] et al. [48]. Basically, Equation (4) can be interpreted as the conditional probability of triggering a seismic landslide at a site. Once the conditional probability for each type of landslide was assessed (an example is shown in Figure 9 for a return period of 475 years), the probability of triggering a landslide classified as \"disrupted slides or falls\", \"coherent slides\", or \"lateral spreads and flows\" was computed by using the addition rule for probability (this is equivalent to calculating the probability of at least one triggering, whatever the failure mechanism). To this end, we assumed that events are independent.