## 3 Methodology
### Definition of Reference Period-Dependent M-R Scenarios  
Which _M-R_ pair should be considered as the reference scenario for earthquake-induced landslides: the mean, the mode, or one of the modes (in the case of multi-modal distributions)? While the mode (\(M^{*}\), \(R^{*}\)) undoubtedly represents the scenario with the highest contribution to the hazard, it is sensitive to the size of magnitude and distance bins. In contrast, the mean (\(\overline{M}\), \(\overline{R}\)) is independent of the bin size, but it might represent an unlikely scenario (i.e., it could correspond to a scenario with a small contribution to the overall hazard), particularly in the case of multi-modal distributions [28; 34]. In the present work, the selection of the reference scenarios was conducted regardless of the advantages and disadvantages of the mean and mode but was guided by the conservatism criterion whereby if at least one _M-R_ pair among those selected from a PMF (i.e., mean and modal pairs) lies below the upper-bound curve of Keefer [1] for a given landslide type, then the triggering of earthquake-induced landslides cannot be neglected for that site. In order to avoid the selection of unlikely scenarios in the case of multi-modal distributions, following [PERSON] et al. [39], the minor modes (i.e., 2 nd, 3 rd, \(\ldots\), mode) were considered in subsequent computations only if their contribution to the hazard is greater than the contribution associated with the mean scenario (in this case, the mean scenario was disregarded). If the contributions associated with the minor modes equal that of the mean, then all selected scenarios were considered. Then, for each PMF, the selected _M-R_ scenarios were compared to the upper-bound curves proposed by [PERSON] [1], separately for the three types of earthquake-induced landslides. Specifically, for each computation node considered by [PERSON] et al. [29], the reference _M-R_ pair was selected as follows:  
* if all _M-R_ pairs selected from the PMF stand above the reference upper-bound curve, then the triggering of earthquake-induced landslides can be neglected; in these cases, the modal _M-R_ pair (\(M^{*},R^{*}\)) is stored as the reference scenario, because of its larger contribution to the hazard. Figure 2: Sensitivity of the maximum filtering operation to the size of the structuring element: (**a**) a 3-by-3 window and (**b**) a 5-by-5 window. The mean and modal scenarios are shown in yellow and red, respectively. if only one of the \(M\)-\(R\) pairs is below the reference upper-bound curve, then the triggering of earthquake-induced landslides cannot be discounted and that \(M\)-\(R\) pair is taken as the reference. * if more than one \(M\)-\(R\) pair lies below the reference upper-bound curve, then the triggering of earthquake-induced landslides cannot be excluded and the \(M\)-\(R\) pair that contributes the most to hazard (i.e., the \(M\)-\(R\) scenario with the greatest PMF value) is selected as the reference. Figure 3 compares the mean and modal scenarios associated with an example PMF to the upper-bound curves of Keefer [1] for the three different types of landslides. In Figure 3a, as both the (primary) mode and the mean lie below the curve, the reference \(M\)-\(R\) pair is chosen according to the highest PMF value. The criteria above were applied to each computation node for the four considered spectral periods (\(T\) = 0.01 s, \(T\) = 0.2 s, \(T\) = 0.5 s, and \(T\) = 1.0 s). As a result, for each landslide type and for each return period, geographic distributions of the reference magnitude (\(\hat{M}(T)\)) and distance (\(\hat{R}(T)\)) were obtained, as well as maps showing the contribution of the selected scenarios to the hazard (i.e., PMF values). For the case of coherent slides and a return period of 475 years, Figure 4 shows the resulting maps of (\(\hat{M}(T)\)) and (\(\hat{R}(T)\)) for spectral periods of \(T\) = 0.01 s and \(T\) = 1.0 s, together with those of the related contribution to the hazard.