## 5 Discussion
This feature reflects the effects of the topography on the dispersion of PDC, being the areas potentially invaded by PDCs strongly controlled by the relatively complex morphology of the summit portion of the volcano. In this area, PDCs are expected to be confined within the caldera or valleys, or mantle the outer upper flanks of the volcano, depending on the vent position (see scenarios 1 and 2 in Figure 3). In scenarios 3 and 4, these topographic effects are missing because the plain surrounding Changbaishan is free of relevant morphological obstacles (Figure 1), as previously reported. The variation of the probability of PDC invasion with the distance from the center of the caldera for scenarios 1 and 2 shows that the larger probability is attained at a distance of about 5 km from the caldera and not at its center (Figure 11). This feature reflects the previously discussed effect of the topography and different vent position on PDC dispersion for these two scenarios, which may be considered representative of small-intermediate eruptions, i.e., Violent Strombolian, Vulcanian, and Subplinian II [39; 65]. For the scenarios 3 and 4, which are representative of Sub-Plinian and Plinian eruptions, the probability variation with the distance from the vent for the scenarios 3 and 4 decreases moving away from the caldera with a break at distances of 25-30 km and 42-65 km from the vent, respectively (Figure 11). Here, the probability remains constant (\(p=0.33\)), and, at larger distances, decreases. These distances with a \(p=0.33\) constant probability evidence the linear extension of the zones where the simulated PDCs with smaller \(\Delta\)H/L regularly superimpose each other (see Figures 7-9). This feature is consistent with the observation that smaller PDCs have a more reduced mobility with respect to the larger ones [32]. Figure 10: Log-log plots of invasion area (A) versus H\({}_{t}\), for each of the 51 simulations of the scenarios 1–4 and relative linear least square regressions. The regressions are calculated for the median, minimum and maximum values of A in the different scenarios. The results from Figures 2-9 may be used as a guide to evaluate the potential hazard related to PDCs invasion in the case of column collapses from eruptions with VEI 2-7 and Figures 5-8 provide a base to analyze the effects of PDCs invasion on the China and North Korea infrastructural network. In scenarios 1 and 2, only the Changbaishan Natural Reserve, which hosts about 2 milliontourists each year, and the local roads on the upper flanks of the edifice are potentially affected by PDCs. As a result, an interruption of the tourist-related activities with consequent economic loss for the local communities is expected. In scenarios 3 and 4, the Beigang (China)-Heyan (North Korea) state highway, a large part of the residential centers and of the infrastructural network including railroads, airports, and dams could be potentially covered by PDCs. The main limitations of our approach consistofa poor validation of the results due to a lack of knowledge of the timing of volcanic eruptions, their size, and distribution of the products. The latter are available only for the 946 CE 'Millennium' eruption. In that case, our theoretical results of scenario 4 fit the observed distance travelled by PDCs of this eruptionwell. As a result, the main difficulty at Changbaishan is to determinate the frequency of each volcanological event. At present, this lack of data also prevents the use of more sophisticated, probability-based models of hazard map construction [31]. In addition, we do not consider the possible effect of winds on the eruptive column whose collapse may generate PDC. Taking into account these limits, empirical relations between Ht and the PDCs covered area are proposed, and the variation of the expected PDC maximum travelled distance as a function of Ht and \(\Delta\)H/L is provided. These relations allow us to determine the areas potentially invaded by PDC.