## 3 Analytical Methods  
The EC model [23] extends the energy line model of [PERSON] [62]. It bases on the conversion of potential energy of pyroclastics emitted from a vent of height H\({}_{\text{vent}}\) into kinetic energy as the gravity-driven, cohesionless PDC moves away from the vent. When the energy is fully dissipated by frictions PDC stops. The height of column collapse above the vent is H\({}_{0}\) and H\({}_{\text{t}}\) is the total height of the eruptive column measured from the altitude of the vent [23,28,30,31]. The angle of the energy line (\(\phi\)) reflects the PDC's resistance to move and it is related to H\({}_{\text{vent}}\), H\({}_{0}\), and the distance travelled by PDC L, by:  
\[\Delta\text{H}=\text{H}_{\text{vent}}+\text{H}_{0}\text{ with}\rightarrow\text{H}_{0}=0.1\text{ H}_{\text{t}} \tag{1}\]  
\[\Delta\text{H}/\text{L}=\text{tan}\left(\phi\right) \tag{2}\]  
The energy cone model determines the energy lines on a digital elevation model (DEM) all around the vent area and the results are better constrained for relatively diluted PDC and less constrained for channelized flows. However, Changbaishan volcano shows, like Vesuvius volcano in Italy, a cone-like morphology with PDC deposits radially emplaced around the volcano. According to ref. [31], these features should not affect the results, although a quantification of the uncertainties and a full validation cannot be done on the basis of the available field data. The advantages of the energy cone model are, for various reasons, the ability to perform several runs in relatively short time and to determine the areas covered by large volume flows [24]. In addition, the variability of PDCs may be indirectly taken into account by varying \(\Delta\)H/L [27], which is a function of the volume of the erupted products, by maintaining constant the value of H\({}_{0}\). To evaluate the areas of potential PDC invasion at Changbaishan, we follow a scenario-based approach by varying the vent location, and by selecting columns of different height. We select 17 vents located at the center of the caldera and around the rim inside and outside the caldera depression (see Figure 3). This is because eruptive vents located both inside and outside the caldera have been recognized in the geological record (Figure 2) [36]. We assume that PDCs originate by column collapse and exclude the occurrence of dome collapse or blasts associated to the depressurization of the volcano flanks because domes or weakness zones on the volcano flanks are lacking and not recorded in the past history of the volcano. We also exclude the possible effects of winds on the eruptive column whose collapse may generate pyroclastic PDC. For each of these vents, we select four reference scenarios with column heights Ht of 3 km (scenario 1), 10 km (scenario 2), 20 km (scenario 3) and 30 km (scenario 4). The EC runs are performed by using the G-EVER routine of Takarada [29] ([[http://volcano.g-ever1.org/vhazard/HazardAssessment/](http://volcano.g-ever1.org/vhazard/HazardAssessment/)]([http://volcano.g-ever1.org/vhazard/HazardAssessment/](http://volcano.g-ever1.org/vhazard/HazardAssessment/))). These Ht values are representative of events with VEI \(\sim\) 2-7 [39; 63; 64]. For each Ht, we select different values of \(\Delta\)H/L as a function of the volume V of the erupted products on the basis of the relation \(\Delta\)H/L = 0.11 (V\({}^{-0.14}\)) [24], which is based on the FlowDat global database [25]. In particular, we select the best median \(\Delta\)H/L value and the values of the upper and lower 95% prediction limits of the above reported relation (see Figure 2 of [24]). The selected \(\Delta\)H/L values of each scenario, which are reported in Table 1, take into account the PDCs with the following ranges of volume: \(10^{-4}\) to \(10^{-1.7}\) km\({}^{3}\) (scenario 1), \(10^{-3}\) to \(10^{-0.8}\) km\({}^{3}\)(scenario 2), \(10^{-2}\) to \(10^{0.3}\) km\({}^{3}\)(scenario 3), and \(10^{-0.1}\) to \(10^{1.77}\) km\({}^{3}\)(scenario 4).