## 4 Discussion  
The application of the proposed methodology based on the Heim method [27] for the evaluation of the velocity of fast-moving landslides on a regional scale provided positive results in terms of the velocity value accuracy and the calculation times obtained. Moreover, it is a simple application method that overcomes the traditional difficulties of regional scale studies. Figure 12: Boxplots of the velocity values of the selected sample of rockfalls. The full-colour boxplots refer to the proposed method velocity values, and the dashed ones to the GPP velocity values. The colours correspond to those used in Figure 10. The boxplot consists of a box, delimited by the first and third quartiles, with the median value reported inside, and two whiskers that connect the box to the minimum and maximum value. Figure 13: Boxplots of the velocity values of the selected sample of shallow landslides. The full-colour boxplots refer to the proposed method velocity values, and the dashed ones to the GPP velocity values. The colours correspond to those used in Figure 11. The boxplot consists of a box, delimited by the first and third quartiles, with the median value reported inside, and two whiskers that connect the box to the minimum and maximum value. In particular, as mentioned by [PERSON] [50], the dynamics of rapid landslides depend on the mass movement type. Rockfalls are characterised by an initial acceleration caused by a loss of cohesion on very steep slopes, followed by free fall and rebound of rock material. These phases are characterised by extremely rapid velocity, which then decreases as a result of various impacts and rolling on low slope surfaces. On the other hand, landslides characterised by rapid flow have a high front velocity, and they tend to stop in the presence of a change in slope. The dynamics of these fast-moving landslides were confirmed by the obtained results. In fact, as shown in Figures 5-9, the distributions of the velocity values within the landslide polygons are consistent with the topographic characteristics of the area. In particular, the highest velocities were observed in correspondence with significant altitude differences, decreasing to zero in the proximity of the deposit areas. In order to verify the velocity values obtained, a GPP application was performed; it is a simple model, but it allows for obtaining good results that are useful for validating the velocity values obtained through the proposed method. The comparison between the maximum velocity values obtained with these two methods showed good results, as demonstrated by the statistical correlation coefficients (Pearson, Kendall, and Spearman). In addition, a comparison between the distributions of all the velocity values of the selected landslide polygons was performed; this was carried out by the implementation of boxplots, which showed a good correspondence between the proposed approach and the GPP model, especially for rockfalls. In the case of shallow landslides, the particular shape of the landslide polygons, which are often curved and very small in width, could explain the not optimal result. Uncertainties can come from the spatial resolution of the DEM, which could influence the results, potentially reducing the accuracy of the results or masking some defects of both the used models. Other uncertainties are derived from the use of a remote-sensed inventory that, as stated by [PERSON] et al. [51], is characterised by several limitations. In particular, this type of inventory does not distinguish landslides from their runout, it is not updated, and it does not include accurate geomorphological data. Given these limitations, the inventory used was suitable for the purposes of this paper, and working at a regional scale, it provided good results. The application of the Heim method [27] is characterised by some disadvantages. Indeed, it must be considered that it is a simplified morphometric method and so it is based exclusively on altitude differences.