## 3 Methodology
### Definition of Reference Period-Independent M-R Scenarios  
As is largely known, sites resonate at different fundamental periods. Hence, whereas PGA hazard disaggregation (\(T\approx 0.01\) s) may be appropriate for rock sites (which are known to resonate at high frequencies), it can provide ineffective (i.e., non-conservative) results in the case of soil sites, which generally have resonance periods in the 0.1-1.0 s range (this range may extend up to longer periods in the case of deep alluvial valleys). As observed above, the contribution from larger magnitudes and distant sources indeed increases as \(T\) increases. Therefore, to improve the evaluation of the triggering potential related to earthquake-induced landslides, it is crucial to relate the preliminary period-dependent triggering maps to ground response, which can be roughly considered through the use of simple site classification systems, such as those proposed by national building codes. In other words, for each site, the reference \(M\)-\(R\) pair should be selected so that the spectral period considered in the disaggregation analysis is compatible with ground response. This allows for the reference \(M\)-\(R\) pair to be selected for each site in relation to geological conditions.  
Figure 5: Preliminary period-dependent maps of the seismic triggering of coherent landslides for a 475-year return period: (**a**) \(T=0.01\) s, (**b**) \(T=0.2\) s, (**c**) \(T=0.5\) s, and (**d**) \(T=1.0\) s. Sites where landslide triggering is expected to occur are displayed in red.  
Recently, [PERSON] et al. [48] developed a soil classification map for all of Italy (see Figure 9 in [48]) in compliance with the ground types (also referred to as \"soil types\" or \"subsoil classes\") defined by the Italian building code [29] as a function of the \(V_{\text{S,eq}}\) parameter (i.e., time-averaged shear wave velocity above the seismic bedrock, the latter being defined as a rock formation or rigid soil with \(V_{\text{S}}\geq 800\) m/s). This map is used in conjunction with the maps described in the previous section to define the final maps of \(\hat{M}\) and \(\hat{R}\) (from here on, we drop the dependence of \(\hat{M}\) and \(\hat{R}\) on \(T\) to indicate the reference period-independent \(\hat{M}\)-\(\hat{R}\) scenarios). To this end, the following criteria were applied:  
* \(\left(\hat{M},\hat{R}\right)=\left(\hat{M}(T=0.01\text{ s}),\hat{R}(T=0.01\text{ s})\right)\) for sites classified as ground type A (i.e., rock sites or stiff soils with \(V_{\text{S,eq}}\geq 800\) m/s).
* \(\left(\hat{M},\hat{R}\right)=\left(\hat{M}(T=0.2\text{ s}),\hat{R}(T=0.2\text{ s})\right)\) for sites classified as ground type B (i.e., soft rock or deposits of dense soil characterized by a gradual increase in the mechanical properties with depth, where \(360\leq V_{\text{S,eq}}<800\) m/s).
* \(\left(\hat{M},\hat{R}\right)=\left(\hat{M}(T=1.0\text{ s}),\hat{R}(T=1.0\text{ s})\right)\) for sites classified as ground type C or D (i.e., sites characterized by deposits of loose-to-medium cohesionless soil with thickness greater than 30 m, where \(100\leq V_{\text{S,eq}}<360\) m/s).
* \(\left(\hat{M},\hat{R}\right)=\left(\hat{M}(T=0.5\text{ s}),\hat{R}(T=0.5\text{ s})\right)\) for sites classified as ground type E (i.e., sites of type C or D but where the thickness of soil deposits is smaller than 30 m).  
The maps resulting from the application of the criteria above are described in the next section along with the corresponding maps of landslide triggering.