## 3 Materials and Methods
### Overview of the Proposed Methodology  
As previously discussed, the main aim of this work was to develop an approach for the risk assessment of slope instability driven by glacier melting. The best-known and most suitable definition of \"risk\" for Disaster Risk Management is the one adopted in 1972 by UNESCO in the \"Convention Concerning the Protection of the World Cultural and Natural Heritage\" [78], defining \"risk\" (R) as a combination of hazard (H), vulnerability (V), and worth exposed (W) using the following formula:  
\[\text{R (E, I)}=\text{H (I)}\times\text{V (I, E)}\times\text{W (E)}\]  
where E represents the type of element at risk, and I represents the intensity of the event. In this work, we are adopting the same definition, and we are presenting two different levels of analysis: the first one at the regional level and the second one at the local level (Lys Valley). For the regional analysis, the specification of the different elements in the risk definition is summarized in Figure 2. The first step in the generation of the landslide risk map was the selection of hazard indicators (H) and their representation to find indices that could summarize the main factors, as highlighted in the literature review (see Section 2), that are involved in the occurrence of a landslide due to ice melting. The hazard component, in fact, only represents landslide susceptibility, as it does not include the time factor required for estimating probability. In this study, the hazard map reported was obtained by combining two intermediate maps corresponding to two main groups of indicators: environmental conditions and triggering factors [79]. Environmental conditions take into account activity, slope aspect, paragalcal areas, glacial geomorphological features, vegetation and protection forests, and existing lithology and geological setting, while triggering factors are identified as permafrost degradation, gravitational processes, and the distribution of precipitation (more in Section 4.1). The second element used to determine the landslide risk is the worth exposed (W) element, which identifies the assets potentially affected by landslides, such as built-up areas and population (social assets); transportation networks (physical assets); tourism and services (economic assets); and sites of community importance, important bird areas, nature reserves, and land cover (environmental assets); more details are in Section 4.2. The third element involved in risk assessment is vulnerability (V). Vulnerability assessment involves, in many cases, the evaluation of several different parameters and factors such as the building materials and techniques, state of conservation, presence of protection structures, presence of warning systems, and so on [80; 81]. For the lack of information on expected intensities of landslides, the lack of accurate information on the state of conservation of buildings and presence of protection structures at the regional scale, the maximum value (equal to 1) was considered in this study. The proposed approach is certainly a preliminary phase that requires in-depth analysis at the local scale,as presented for the Lys Valley, but can give a general view that also allows comparisons at the regional scale to identify planning priorities and determine future evolutionary scenarios of these environments. In conclusion, the methodological steps, as illustrated in Figure 2, are as follows:  
* Step 01. Development of hazard indicators based on environmental conditions and triggering factors, as illustrated in Section 4.1;
* Step 02. Aggregation of the nine indicators in a glacier melting landslide susceptibility map;
* Step 03. Development of worth exposed layers, divided into social, physical, environmental, and economic aspects (explained in Section 4.2);
* Step 04. Aggregation of the worth exposed value in the Valle d'Aosta region;
* Step 05. Development of a methodology for risk assessment at the regional level (discussed in Section 4.3);
* Step 06.