Scenario-Based Pyroclastic Density Current Invasion Maps at Poorly Known Volcanoes: A Case Study from Changbaishan (China / North Korea)

Featured Application: Volcanic Abstract: Changbaishan volcano (China / North Korea; last eruption in 1903 AD) was responsible for a Volcanic Explosivity Index (VEI) 7 eruption in 946 AD. Approximately 186,000 people live around Changbaishan and 2,000,000 tourists / year visit the volcano. An unrest occurred between 2002 and 2006. Despite the relevant hazard, the eruptive history is poorly known, a condition common to many volcanoes worldwide. Here, we investigate the extension of the areas potentially a ﬀ ected by pyroclastic density currents (PDCs) in case of future eruptions following a scenario-based approach. We perform energy cone runs referred to four scenarios from columns of height 3, 10, 20 and 30 km at di ﬀ erent vents. By using global datasets on PDCs, we produce spatial probability maps of PDCs invasion. Empirical laws between covered areas, PDC travelled distances, and heights of collapse are provided. In scenarios 3 and 4, PDCs expand at distances up to 42 km and 85 km, respectively. In scenarios 1 and 2, PDCs invade the touristic area and few main roads. Severe e ﬀ ects emerge from scenarios 3 and 4 with the interruption of the China–North Korea land and aerial connections and PDC. Our approach may serve as guide for the rapid evaluation of the PDC-related hazard at poorly known volcanoes.


Introduction
Pyroclastic density currents (PDCs) may cover large distances in short times (few minutes to hours) [1][2][3][4][5]. PDCsform by different mechanisms including the collapse of eruptive columns or domes [6,7], blasts associated to the depressurization of the volcano flanks [8], and syn-eruptive gravitational collapse of hot pyroclasts accumulated over steep slopes [9]. The flow behavior and the emplacement mechanisms of PDCs havebeen reconstructed from field studies [10,11], direct observations [12] as well as numerical and experimental models [13][14][15][16][17]. These studies evidence the complex dynamics of PDCs highlighting the role of different factors (topography, temperature, degree of dilution, humidity) in controlling the path, velocity, impact, and area covered by PDCs.

Geological and Volcanological Setting of the Changbaishan Volcano
Changbaishan volcano (2774 m; ~2 Ma-present) is the largest active volcano of China and lies above the flat portion of the Pacific slab subducting below the Eurasian plate ( Figure 1). The volcano

Geological and Volcanological Setting of the Changbaishan Volcano
Changbaishan volcano (2774 m;~2 Ma-present) is the largest active volcano of China and lies above the flat portion of the Pacific slab subducting below the Eurasian plate ( Figure 1). The volcano is located on the China-North Korea boundary and represents an intraplate, alkaline edifice with a summit caldera filled by a~5 km wide, 300 m deep lake (Heaven Lake) [40,41]. A valley located in the northern flank, where the caldera rim is breached, hosts the only water outlet of the Heaven Lake. Such outlet forms the Erdaobaihe River ( Figure 1). Changbaishan volcano lies on the Gaima basaltic plateau (29 Ma), which has an average altitude of 900-1000 m and covers the Sino-Korean Craton ( Figure 1) [36,42], and it is the northernmost edifice of a group of volcanoes including the 5 to 2 Ma old Wangtian'e and Namphothe strato-volcanoes and smaller scale monogenetic centers [41,43]. The Changbaishan truncated cone has a basal diameter of about 50 km with radial valleys partially modified by glacial processes. Pre-caldera and post-caldera vents are located within and outside the present-day caldera depression (Figure 2).
Appl. Sci. 2020, 10, x FOR PEER REVIEW 4 of 19 is located on the China-North Korea boundary and represents an intraplate, alkaline edifice with a summit caldera filled by a ~5 km wide, 300 m deep lake (Heaven Lake) [40,41]. A valley located in the northern flank, where the caldera rim is breached, hosts the only water outlet of the Heaven Lake. Such outlet forms the Erdaobaihe River ( Figure 1). Changbaishan volcano lies on the Gaima basaltic plateau (29 Ma), which has an average altitude of 900-1000 m and covers the Sino-Korean Craton ( Figure 1) [36,42], and it is the northernmost edifice of a group of volcanoes including the 5 to 2 Ma old Wangtian'e and Namphothe strato-volcanoes and smaller scale monogenetic centers [41,43]. The Changbaishan truncated cone has a basal diameter of about 50 km with radial valleys partially modified by glacial processes. Pre-caldera and post-caldera vents are located within and outside the present-day caldera depression (Figure 2).  The Changbaishan activity is divided in three main stages: a 2-1 Ma old basaltic shield formation stage, a trachytic to rhyolitic composite cone formation stage (1-0.04 Ma), and the latest, caldera-forming explosive stage ( Figure 1) [36,43,44]. The Changbaishan caldera formed by a collapse event related to the VEI ~7, Millennium eruption (946 AD) [33], which ejected tephra of rhyoltic and trachytic composition (DRE ~25 km 3 ) [34,35,37,40]. Other explosive eruptions occurred in Holocene times. These include the 8.1 ka pre-Millennium Plinian Qixiangzhan eruption, and the post-Millennium eruptions recorded in historical documents and recognized in distal tephra (1403, 1668, and the VEI ~5 1702 AD) [42,[45][46][47][48][49]. Wei et al. [36] report that the last eruption at Changbaishan occurred in 1903 AD within the summit caldera lake, however, Pan et al. [37] ascribe the intra-caldera deposits of this event and the proximal deposits of the 1668 and 1702 AD eruptions to the final part of the Millennium eruption. This problem causes a difficulty in evaluating the volcanic hazard correctly [36]. Ash fall deposits of the Millennium and Qixiangzhan eruptions and of the 1215, 1403, 1668 AD eruptions have been found in the lakes around Changbaishan. For the Plinian events, fall deposits have been found in the Japan Sea, Japan mainland and Greenland (Millennium eruption) [49,50]. The PDC deposits of Changbaishan, which frequently cover the fallout layers of the major eruptions, mainly outcrop in river gorges being significantly removed by fluvial and glacial erosion, and rainfall. A map of the spatial distribution of PDC deposits at Changbaishan is, at the present, not available. The few outcrops allow us to estimate that PDCs cover an area of about 2430 km 2 [51]. The PDC deposits of the Millennium eruption can be traced 50-80 km away from the Changbaishan caldera [46,51]. In the valleys around the volcano, most  The Changbaishan activity is divided in three main stages: a 2-1 Ma old basaltic shield formation stage, a trachytic to rhyolitic composite cone formation stage (1-0.04 Ma), and the latest, caldera-forming explosive stage ( Figure 1) [36,43,44]. The Changbaishan caldera formed by a collapse event related to the VEI~7, Millennium eruption (946 AD) [33], which ejected tephra of rhyoltic and trachytic composition (DRE~25 km 3 ) [34,35,37,40]. Other explosive eruptions occurred in Holocene times. These include the 8.1 ka pre-Millennium Plinian Qixiangzhan eruption, and the post-Millennium eruptions recorded in historical documents and recognized in distal tephra (1403, 1668, and the VEI~5 1702 AD) [42,[45][46][47][48][49]. Wei et al. [36] report that the last eruption at Changbaishan occurred in 1903 AD within the summit caldera lake, however, Pan et al. [37] ascribe the intra-caldera deposits of this event and the proximal deposits of the 1668 and 1702 AD eruptions to the final part of the Millennium eruption. This problem causes a difficulty in evaluating the volcanic hazard correctly [36]. Ash fall deposits of the Millennium and Qixiangzhan eruptions and of the 1215, 1403, 1668 AD eruptions have been found in the lakes around Changbaishan. For the Plinian events, fall deposits have been found in the Japan Sea, Japan mainland and Greenland (Millennium eruption) [49,50]. The PDC deposits of Changbaishan, which frequently cover the fallout layers of the major eruptions, mainly outcrop in river gorges being significantly removed by fluvial and glacial erosion, and rainfall. A map of the spatial distribution of PDC deposits at Changbaishan is, at the present, not available. The few outcrops allow us to estimate that PDCs cover an area of about 2430 km 2 [51]. The PDC deposits of the Millennium eruption can be traced 50-80 km away from the Changbaishan caldera [46,51]. In the valleys around the volcano, most PDCs evolve to lahars [51]. These latter have been found at distances up to 83 km north of the Changbaishan caldera. As concerns the next, possible eruption, the emplacement of fall and PDCs deposits with volumes of erupted magma in the range of 0.1-0.5 km 3 , i.e.,VEI~3-4, is expected [36]. However, gravity and tomographic models [52,53] show that two low Vs, low density 50 km (width) × 5 km (height) and 20 km × 2.5 km sills occur at about 18 and 10 km depth. The dimension of these sills suggests the occurrence of magma chambers with a volume comparable to that responsible for the Millennium eruption. Therefore, a future eruption with VEI~5-7 cannot be a priori excluded, also taking into account the poorly known eruptive history of the volcano [37]. Changbaishan was affected by an unrest episode between 2002 and 2006 [38] with hundreds of shallow earthquakes with M L ,max = 4.4. They were accompanied by ground uplift, increase of CO 2 , He, H 2 , and high 3 He/ 4 He possibly related to the refilling of a magma chamber. The 2002-2006 unrest has moved the Chinese and South Korea scientific communities to evaluate the potential hazards. According to Wei et al. [40,54], different hazards characterize the Changbaishan volcano. Avalanches, rock falls, and debris flows, mainly from the inner, steep caldera walls, and explosive interaction between magma and water, as well as CO 2 bursts, and Nyos-like events are well recognized hazards related to the occurrence of the Heaven Lake. Lahars, floods, and rock avalanches associated to the partial water withdrawal by overflows from the lake are also recognized as a primary source of hazard. Other relevant hazards include earthquakes and the emplacement of ash fall and PDCs. The evaluation of the hazard related to ash dispersion and PDC is of primary importance because about 135,000 people of China and 31,000 of North Korea live within 50 km from the Changbaishan caldera, and each year, 2,000,000 tourists visit the Changbaishan volcano National Reserve, a part of the UNESCO Man and Biosphere program [55] (http://www.stats.gov.cn/english/statisticaldata/censusdata/) [56]. In this framework, Lee et al. [57] and Yu et al. [58,59] model the ash dispersion for eruptions with VEI up to 7 and show that, due to the prevailing winds, ash deposition may heavily affect China and North Korea, with also consequences for the aviation routes and agricultural areas in South Korea and north Japan. Yun et al. [60] and Paone and Yun [61], by using the results of TITAN2 simulations for VEI 3 eruptions from 7 vents located along a NE-SW strike inside and outside the Changbaishan caldera, show that PDCs may deposit both inside and outside the caldera to a maximum distance of 7 km from the vent, mostly in the north valley hosting the Erdaobaihe River ( Figure 1). In summary, with all the above discussed uncertainties regarding the eruptive activity of the volcano in historical times and the poorly known distribution of PDC deposits, some constraints are available and may be used as a guide for a scenario-based evaluation of the PDC hazard.

Analytical Methods
The EC model [23] extends the energy line model of Heim [62]. It bases on the conversion of potential energy of pyroclastics emitted from a vent of height H 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 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 (φ) reflects the PDC's resistance to move and it is related to H vent , H 0 , and the distance travelled by PDC L, by: 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 ∆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/). These Ht values are representative of events with VEI~2-7 [39,63,64]. For each Ht, we select different values of ∆H/L as a function of the volume V of the erupted products on the basis of the relation ∆H/L = 0.11 (V −0.14 ) [24], which is based on the FlowDat global database [25]. In particular, we select the best median ∆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 ∆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). These variation ranges are within those estimated by [63] for eruptions with different VEI and column heights. Therefore, the three ∆H/L values, we adopt for each scenario account for the large uncertainties of the relations between Ht and PDCs volumes and for the different mobility of the flows, being smaller volume flows less mobile than the larger ones ( [32] and reference therein). The reference DEM of Changbaishan volcano and surrounding areas used for the EC runs is the 30-m resolution ASTER GDEM V2 elevation model (https://asterweb.jpl.nasa.gov/gdem.asp). In total, we perform 204 runs, 51 for each scenario, and produce the relative invasion maps. We describe the uncertainty about the distribution of sites where PDCs may deposit by estimating the conditional spatial probability that PDCs arrive in a specific site. We assume that all the scenarios have the same probability of occurrence. This simplifying assumption bases on the observation that the available field and chronological data do not allow us, at the present, to determine possible recurrence times or to identify repose time-volume of erupted products relations at Changbaishan. As a result, the spatial probability values, which range from 0 to 1, are computed for each scenario as the proportion of times that a given area is covered by PDCs on the basis of 51 runs. Table 1. Summary of the parameters used in the EC runs for each scenario. The minimum (min), median (med), and maximum (max) ∆H/Lvalues are selected according to the relation∆H/L= 0.11 (V −0.14 ) of Ogburn and Calder [24] and correspond to the lower 95% prediction limit, the best-fitting value, and upper 95% prediction limit.

Results
Examples of the PDC invasion maps for the different scenarios and vent positions are reported in Figures 3 and 4. In scenario 1, these maps show that the PDC invasion area is partly controlled by the presence of the caldera rim ( Figure 3). In particular, PDC are confined within the caldera for vents located in the inner side of the caldera (vents 1 and 6). PDC invasion area of vent 2, which is located within the caldera but close to a zone where the rim is interrupted and the caldera is open toward the north, manly covers the valley located in the northern flank of the volcano and, with a less extension, the caldera floor. The PDC invasion areas related to vents 3 and 7, both located outside caldera, mainly cover the flanks of the volcano, and, in a lesser amount, the inner caldera floor. The distance covered by PDCs in scenario 1 does not exceed about 7 km from the respective vent. This result is consistent with that obtained from TITAND2 simulations by Yun et al. [60] from six vents with a similar position with respect to the caldera rim and column height of 2-5 km. As concerns scenario 2, the PDC invasion area mainly affects the flanks of the volcano and, depending on the vent position, covers different sectors of the volcano (Figure 3).
Examples of the PDC invasion maps for the different scenarios and vent positions are reported in Figures 3 and 4. In scenario 1, these maps show that the PDC invasion area is partly controlled by the presence of the caldera rim ( Figure 3). In particular, PDC are confined within the caldera for vents located in the inner side of the caldera (vents 1 and 6). PDC invasion area of vent 2, which is located within the caldera but close to a zone where the rim is interrupted and the caldera is open toward the north, manly covers the valley located in the northern flank of the volcano and, with a less extension, the caldera floor. The PDC invasion areas related to vents 3 and 7, both located outside caldera, mainly cover the flanks of the volcano, and, in a lesser amount, the inner caldera floor. The distance covered by PDCs in scenario 1 does not exceed about 7 km from the respective vent. This result is consistent with that obtained from TITAND2 simulations by Yun et al. [60] from six vents with a similar position with respect to the caldera rim and column height of 2-5 km. As concerns scenario 2, the PDC invasion area mainly affects the flanks of the volcano and, depending on the vent position, covers different sectors of the volcano (Figure 3).
The caldera floor is always covered by the PDCs. The maximum runout distance from the runs with the smaller ΔH/L value is in the order of 16-20 km, depending on the vent position. The distance from the runs with the larger ΔH/L does not exceed 10 km (Figure 3). The results of scenario 3 show that the distance variesfrom 28 to 37 km for the smallest ΔH/L value, whereas it is in the order of 12-15 km for the largest ΔH/L. In scenario 3 runs, significant effects of the vent position on the PDC dispersion areas are lacking. In scenario 4, the distance of PDCs determined for the smaller ΔH/L is of 80-85 km (Figure 4), a value corresponding to the maximum distance of 83 km measured in the field for the pumice-rich PDCs of the Millennium eruption [51,61].  The caldera floor is always covered by the PDCs. The maximum runout distance from the runs with the smaller ∆H/L value is in the order of 16-20 km, depending on the vent position. The distance from the runs with the larger ∆H/L does not exceed 10 km (Figure 3). The results of scenario 3 show that the distance variesfrom 28 to 37 km for the smallest ∆H/L value, whereas it is in the order of 12-15 km for the largest ∆H/L. In scenario 3 runs, significant effects of the vent position on the PDC dispersion areas are lacking. In scenario 4, the distance of PDCs determined for the smaller ∆H/L is of 80-85 km (Figure 4), a value corresponding to the maximum distance of 83 km measured in the field for the pumice-rich PDCs of the Millennium eruption [51,61].
The maximum distance travelled by PDCs from the runswith the smallest ∆H/L is between 35 and 40 km from the vent. The probability (p) invasion maps are shown in Figures 5-8. As expected, the more relevant feature of resulting probability maps is that the area of invasion and p increases with Ht. Low p-values (≤0.5) and a dependence of invasion probability map from the vent location may be recognized in scenario 1 ( Figure 5). For higher Ht (Scenarios 2-4; Figures 6-8), the probabilities of invasion reach 1 in the proximal and medial zones around the caldera area; p decreases with the distance rather symmetrically around the caldera showing less dependence on vent location with respect to scenario 1. In this latter scenario, the larger p-values may be found within the caldera and in the northern valley, where the caldera rim is breached ( Figure 5).
Appl. Sci. 2020, 10, x FOR PEER REVIEW 9 of 19 The maximum distance travelled by PDCs from the runswith the smallest ΔH/L is between 35 and 40 km from the vent. The probability (p) invasion maps are shown in Figures 5-8. As expected, the more relevant feature of resulting probability maps is that the area of invasion and p increases with Ht. Low p-values (≤0.5) and a dependence of invasion probability map from the vent location may be recognized in scenario 1 ( Figure 5). For higher Ht (Scenarios 2-4; Figures 6-8), the probabilities of invasion reach 1 in the proximal and medial zones around the caldera area; p decreases with the distance rather symmetrically around the caldera showing less dependence on vent location with respect to scenario 1. In this latter scenario, the larger p-values may be found within the caldera and in the northern valley, where the caldera rim is breached ( Figure 5).  part of the plain and surrounding hills, and the northern sectors of the Wangtian'e and Namphothe strato-volcanoes located south of Changbaishan could be affected by PDCs invasion. All the town and infrastructures (roads, highways, railroads, airports, and dams) can be invaded by PDCs. All the main connections between China and North Korea could be interrupted and the headquarters of the Changbaishan Volcano Observatory and Changbaishan Natural Reserve located in the town of Erdaobaihe would be covered by PDCs.  Different types of morphological settings like valleys and walls (caldera rim) influence the deposition and the mobility of PDCs. In scenario 1, only the top of the volcano is potentially affected by PDCs. The area covered by PDC includes the more elevated portion of the Changbaishan Natural Reserve and the road leading to the top of the volcano. In scenario 2, the area with p ≥ 0.5 characterizes the caldera and the upper, outer flanks of the volcano ( Figure 6). The roads connecting China and North Korea located on the southwestern flanks of the volcano are potentially invaded by PDCs. In scenario 3, the area with p ≥ 0.5 covers the caldera and the upper and lower flanks of the volcano (Figure 7). The whole area of the Changbaishan Natural Reserve is potentially covered by PDC as well as the areas close to the Baishan-Changbaishan (China) and Samjiyon (North Korea) airports.Also, the road leading to the top of the volcano from Erdaobaihe, along which the Changbashan Observatory instrument building is located, and the Beigang (China)-Heysan (North Korea) state highway are potentially affected by PDCs. In scenarios 4 (Figure 8), the whole volcano, part of the plain and surrounding hills, and the northern sectors of the Wangtian'e and Namphothe strato-volcanoes located south of Changbaishan could be affected by PDCs invasion. All the town and infrastructures (roads, highways, railroads, airports, and dams) can be invaded by PDCs. All the main connections between China and North Korea could be interrupted and the headquarters of the Changbaishan Volcano Observatory and Changbaishan Natural Reserve located in the town of Erdaobaihe would be covered by PDCs.

Discussion
The results of the EC runs at Changbaishan show that the areas of invasion from PDCs are influenced by the different vent position only in scenarios 1 and 2. This feature reflects the relatively complex morphology of the top of the volcano.Therefore, we conclude that the up-to-400 m high caldera rim surrounding the Heaven Lake and the caldera breach to the north control the distribution of the PDCs invasion areas on the upper flanks of the volcano, at least for PDCs originated by the collapse of 3-10 km high eruptive columns. The water outlet feeding the Erdaobaihe River could play a role in transforming the potential PDCs flowing in the Erdaobaihe valley in lahars. This conclusion is supported by the field data related to the Millennium eruption, where the PDC-lahar transition has been recognized in the field [51]. Therefore, this type of hazard should also be taken into account at Changbaishan. In the runs, we have assumed that the water of the Heaven Lake does not play a role in the dynamics of the eruptive column, e.g., magma-water interaction, or PDC flow dynamics. This is a limit of our analysis which cannot be quantified.

Discussion
The results of the EC runs at Changbaishan show that the areas of invasion from PDCs are influenced by the different vent position only in scenarios 1 and 2. This feature reflects the relatively complex morphology of the top of the volcano.Therefore, we conclude that the up-to-400 m high caldera rim surrounding the Heaven Lake and the caldera breach to the north control the distribution of the PDCs invasion areas on the upper flanks of the volcano, at least for PDCs originated by the collapse of 3-10 km high eruptive columns. The water outlet feeding the Erdaobaihe River could play a role in transforming the potential PDCs flowing in the Erdaobaihe valley in lahars. This conclusion is supported by the field data related to the Millennium eruption, where the PDC-lahar transition has been recognized in the field [51]. Therefore, this type of hazard should also be taken into account at Changbaishan. In the runs, we have assumed that the water of the Heaven Lake does not play a role in the dynamics of the eruptive column, e.g., magma-water interaction, or PDC flow dynamics. This is a limit of our analysis which cannot be quantified. However, with the exception of vent 1, which is located at the center of the lake, the other 16 vents are located outside the lake area ( Figure 3). Our results from runs with the larger ∆H/L values give us a rough estimate of the dispersion areas of potentially denser flows. According to equation 2, the flow mobility is directly expressed by the angle φ, a measure of the energy decay rate. This is because larger values of φ imply faster energy decay rate and less mobile flows [23]. In this regard, we analyze the relationship between the median, larger and smaller ∆H/L values according to the database of Ogburn [25] and Ogburn and Calder [24], and the distance from the center of the Changbaishan caldera taken as common reference point (Figure 9).
Appl. Sci. 2020, 10, x FOR PEER REVIEW 13 of 19 However, with the exception of vent 1, which is located at the center of the lake, the other 16 vents are located outside the lake area ( Figure 3). Our results from runs with the larger ΔH/L values give us a rough estimate of the dispersion areas of potentially denser flows. According to equation 2, the flow mobility is directly expressed by the angle ϕ, a measure of the energy decay rate. This is because larger values of ϕ imply faster energy decay rate and less mobile flows [23]. In this regard, we analyze the relationship between the median, larger and smaller ΔH/L values according to the database of Ogburn [25] and Ogburn and Calder [24], and the distance from the center of the Changbaishan caldera taken as common reference point (Figure 9). In scenario 1, the maximum distance is within 9-12 km, while in scenario 2, it ranges between 13 and 22 km. In scenarios 3 and 4, the distances are in the ranges 22-42 km and 38-85 km. As shown in Figure 9, this increasing difference between the maximum distances theoretically travelled by PDCs with different ΔH/L values in each scenario does reflect not only the different volumes of PDCs, but also the occurrence of a flat, obstacle-free, topography moving away from the caldera. In addition, the results of the scenario 4 runs (Figure 4) show that some PDCs are potentially able to partly fill the valleys located around the Changbaishan plain in the Gaima basaltic plateau and Sino-Korean Craton (Figure 1). The results of the runs allow us to obtain some empirical relations between the calculated area covered by PDCs and Ht as a function of the different vents and selected ΔH/L. These relations are shown in Figure 10, where the values Ht adopted for the different scenarios are reported versus the areas covered by PDCs. In the diagram, the best-fit lines are obtained by interpolating the minimum, median, and maximum areas values in the different scenarios. These lines, even if obtained by interpolating few points, may be used to obtain an order of magnitude estimate of the area potentially affected by PDC invasion if data on the column height are available and the assumption that PDCs originate by column collapse holds. The empirical relations of Figure  10 clearly show that the higher the column height, the smaller the difference between the maximum and minimum area potentially affected by PDC. This feature reflects the effects of the topography on In scenario 1, the maximum distance is within 9-12 km, while in scenario 2, it ranges between 13 and 22 km. In scenarios 3 and 4, the distances are in the ranges 22-42 km and 38-85 km. As shown in Figure 9, this increasing difference between the maximum distances theoretically travelled by PDCs with different ∆H/L values in each scenario does reflect not only the different volumes of PDCs, but also the occurrence of a flat, obstacle-free, topography moving away from the caldera. In addition, the results of the scenario 4 runs (Figure 4) show that some PDCs are potentially able to partly fill the valleys located around the Changbaishan plain in the Gaima basaltic plateau and Sino-Korean Craton (Figure 1). The results of the runs allow us to obtain some empirical relations between the calculated area covered by PDCs and Ht as a function of the different vents and selected ∆H/L. These relations are shown in Figure 10, where the values Ht adopted for the different scenarios are reported versus the areas covered by PDCs. In the diagram, the best-fit lines are obtained by interpolating the minimum, median, and maximum areas values in the different scenarios. These lines, even if obtained by interpolating few points, may be used to obtain an order of magnitude estimate of the area potentially affected by PDC invasion if data on the column height are available and the assumption that PDCs originate by column collapse holds. The empirical relations of Figure 10 clearly show that the higher the column height, the smaller the difference between the maximum and minimum area potentially affected by PDC. 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 ΔH/L regularly superimpose each other (see . This feature is consistent with the observation that smaller PDCs have a more reduced mobility with respect to the larger ones [32]. 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 ∆H/L regularly superimpose each other (see . This feature is consistent with the observation that smaller PDCs have a more reduced mobility with respect to the larger ones [32]. The results from Figures2-9 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)-Heysan (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 ΔH/L is provided. These relations allow us to determine the areas potentially invaded by PDC. Many other PDC invasion maps from column collapses at different height may be provided and the ones presented in our study may be modified.

Conclusions
Changbaishan volcano is characterized by a poorly known eruptive history and by a relevant hazard, a condition common to many active volcanoes and volcanic fields worldwide, and, in particular, those located in South America, Asia (e.g., China, Turkey, Saudi Arabia) and Eastern Europe (Caucasus) [66][67][68]. In this framework, the scenario-based probabilistic approach we adopted for an evaluation of the hazard related to PDCs at Changbaishan, although suffering from afew constraints due to the lack of field data, could be applied to other poorly known and hazardous volcanoes. We produce maps of potential PDCs according to different Ht, PDC volumes and vent location. Probability maps of PDC invasion are also produced for each selected scenario. A contribution of our study is the analysis of the effects of the topography in controlling (or not) the 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)-Heysan (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 ∆H/L is provided. These relations allow us to determine the areas potentially invaded by PDC. Many other PDC invasion maps from column collapses at different height may be provided and the ones presented in our study may be modified.

Conclusions
Changbaishan volcano is characterized by a poorly known eruptive history and by a relevant hazard, a condition common to many active volcanoes and volcanic fields worldwide, and, in particular, those located in South America, Asia (e.g., China, Turkey, Saudi Arabia) and Eastern Europe (Caucasus) [66][67][68]. In this framework, the scenario-based probabilistic approach we adopted for an evaluation of the hazard related to PDCs at Changbaishan, although suffering from afew constraints due to the lack of field data, could be applied to other poorly known and hazardous volcanoes. We produce maps of potential PDCs according to different Ht, PDC volumes and vent location. Probability maps of PDC invasion are also produced for each selected scenario. A contribution of our study is the analysis of the effects of the topography in controlling (or not) the dispersions of PDCs as a function of