Probabilistic Tsunami Hazard Model of Building Inundation Following a Subaqueous Volcanic Explosion Based on the 1716 Tsunami Scenario in Taal Lake, Philippines


 A probabilistic hazard analysis of a tsunami generated by a subaqueous volcanic explosion is performed for Taal Lake in the Philippines. The Taal volcano in Taal Lake is an active volcano on Luzon Island in the Philippines, and its eruption would have a strong impact on humans around the coastal area of the lake. This study aims to develop a probabilistic tsunami hazard model of inundated buildings for tsunami mitigation in future scenarios. To develop the probabilistic tsunami hazard model, different explosion diameters were used to generate tsunamis of different magnitudes in the TUNAMI-N2 model. The initial water level in the tsunami model was estimated based on the explosion energy as a function of the explosion diameter. The tsunami-induced inundation from the TUNAMI-N2 model was overlaid on the distribution of buildings. The statistical distribution of inundated buildings can be modeled with the lognormal distribution, which exhibits the best fit among nine candidate statistical distributions. The tsunami hazard analysis is explained by using the conditional hazard curve and map. These products were used to calculate the probability of building inundation given the occurrence of a subaqueous explosion. The results from this study can be used for future tsunami mitigation in the case of a tsunami generated by a subaqueous volcanic explosion.


Introduction
Tsunamis generated by subaqueous volcanic eruptions represent only a few of all recorded tsunamis, which are more commonly generated by earthquakes or landslides. However, a tsunami can be generated by an eruption, and the surface water and tsunami hazards related to subaqueous eruptions are principally interesting to evaluate and forecast because there is a lack of geographical, observational, and contributory data. For example, in Taal Lake in the Philippines, as shown in Figure  However, PTHA methods for a volcanic source of tsunamis are more limited than those for an earthquake source (Grezio et al. 2017). Here, tsunamis are produced by a volcanic source, and this work focuses on a conditional PTHA in which the volcanic mechanism has the potential to generate a tsunami from a subaqueous explosion.
In this study, tsunamis generated by a subaqueous explosion were considered under different scenarios in Taal Lake (Luzon Island, Philippines). The scenarios vary in the size of the eruption, with the eruption location confined to that of the 1716 event, which represents one of the largest submarine explosions in the lake. Previous studies have focused only on tsunami hazard assessment and mapping, while tsunami hazard assessment involving building exposure has not been studied. Thus, to address this gap, we first prepare a probabilistic tsunami inundation model of building exposure based on different scenarios of subaqueous explosion-induced tsunamis affecting the coastal area around the lake.
The paper is presented as follows: the methodology is explained in section 2. The beginning of section 2 presents the hypothesis that a subaqueous explosion generates an initial water level for tsunami propagation (section 2.1). Then, the tsunami modeling is described in section 2.2. The probability of hazard analysis is explained in section 2.3. Finally, the modeling results are presented in section 3, the results are discussed in section 4, and conclusions are presented in section 5.

Methodology
The methodology presented in this paper is based on using the Monte Carlo technique (Li et al. 2016) to quantify the tsunami hazard posed to buildings in a subaqueous explosion scenario. This method is based on 4 steps. First, the Monte Carlo method involved identifying the potential vent area (explosion size). The vent diameter, according to Paris and Ulvrova (2019), was approximately 600 m, and the submerged volcanic cone (vent size) was assumed to be the same size. For this study, we varied the vent diameter from 100 -1000 m into 99 scenarios as conditional scenarios with an eruption location corresponding to the position of the 1716 event. Second, the 99 scenarios were used to generate the initial water level by using the method explained in section 2.1. Third, the initial water levels of the 99 scenarios were used to simulate the tsunami waves reaching the shores of Taal Lake. The tsunami modeling also incorporates the effect of building (i.e., the friction of surface resistance) into the model (see detail in section 2.2). The modeled buildings in this study were extracted from OpenStreetMap (OSM) though QGIS software. Fourth, the conditional probability of each demonstrative scenario was calculated and combined with the building inundation results for each simulated tsunami scenario to present the PTHA for exposed buildings. In this step, we used 10 candidate distributions to verify the best fit of the probability modeling of the number of inundated buildings, as presented in detail in section 2.3.

Model of subaqueous explosion and tsunami generation
We selected 99 different sizes of simulated explosions in Taal Lake with the eruption location corresponding to the location of the 1716 subaqueous eruption. The energy of an explosion can be estimated with the empirical formula provided by Sato and Taniguchi (1997).
if > , = 0 where is the distance from the explosion center in meters.

Numerical simulation of a tsunami
To obtain the tsunami inundations for different explosion sizes, a numerical tsunami simulation was driven with the TUNAMI-N2 model (Imamura 1995;Goto et al 1997). The TUNAMI-N2 model was first developed at Tohoku University to model tsunami propagation and inundation on land and operates using the nonlinear theory of the shallow water equation, which is solved using a leap-frog scheme. The nonlinear shallow water equation is presented as Equations 6 -8, wherein the finite difference method is applied to run the nonlinear equation with a bottom fiction represented by Manning's roughness coefficient.
where is the water level, and are the fluxes of water in the and directions, is the total depth, is gravitational acceleration, and is Manning's roughness coefficient. The preparation process was performed for the bathymetry grid for the tsunami propagation and inundation simulations.
In the TUNAMI-N2 model, buildings are represented by the roughness as a function of the drag coefficient in the last term in Equations 7 and 8. The coefficient is evaluated by the setting the roughness coefficient to that of surface types other than buildings. The composite equivalent roughness coefficient from other surface types and buildings is estimated by using Equation 9 and the shore areas had no specific boundary conditions for wet/dry fronts (Imamura, 1995). Note that the simulated grid without buildings was assigned a constant Manning's coefficient of 0.025.
Building shapes were collected from OpenStreetMap (OSM) though QGIS open-source software as shown in Figure 1. In this study, we analyzed buildings in an area close to the shore of Taal Lake within a distance of approximately 2 km. The shapes of the collected buildings were overlaid on the computation grid to estimate the building area percentage in each grid. The building percentage is used to estimate the roughness in Equation 9, and the building location is used to determine the maximum flow depth for probability hazard analysis in section 2.3.

Probabilistic hazard analysis
For each scenario, the maximum inundation depth at all inland computation grids was estimated by subtracting the topography (from SRTM DEM) data from the simulated maximum water level. The inundation depth map from the 99 scenarios of the explosion area was overlaid on the building locations from the OSM data to determine the inundated buildings. The inundated buildings were counted in each inundation depth range.
The probability of explosion size in this study is based on the vent diameter in the 99 scenarios. The distribution function reveals a decreasing probability of increasing explosion size. For example, the 1 st scenario features the smallest explosion size with a probability value of 1. The 99 th scenario features the largest explosion size and a probability value of approximately 0.01.
Hazard curves were established based on the total number of inundated buildings and each depth range.
The curve for each scenario is described by the best-fit distribution among 9 candidate statistical distributions. The 9 candidate distributions were beta, exponential, gamma, lognormal, normal, where is the referenced frequency for bin and is the expected frequency for bin .

Tsunami wave generation and inundation extent
Numerical Tsunami generation was based on the maximum initial water level and size of the water cavity, and the diameter of the cavity was calculated with Equation 2. The distribution of the initial water level was identified as a parabolic shape that was generated with Equations 4 and 5. Figure 3 presents the tsunami generation in the 99 th scenario with a vent diameter of 1000 m and a maximum initial water level of 135 m; the propagation is shown at the different times, namely, 0 s, 10 s, 30 s, and 60 s. The tsunami initially features a parabolic shape, after which the second wave forms and moves from the center of the explosion point. Figure 4 presents the maximum water level after 60 minutes in the tsunami model for three different scenarios, namely, the 1 st , 50 th , and 90 th scenarios, as well as the maximum amplitude distribution in the lake. The difference between the maximum water level and the topography reflects the inundation extent and flow depth based on the runup process. The right column in Figure 4 shows the inundation extent in the three different scenarios and its overlap with the buildings. The 99 th scenario is the greatest hazard scenario and produces the largest inundation area in this study.
The inundated area and buildings are presented in Figure 5. The relationship between the maximum initial water level and the inundated area can be seen in Figure 5a: the increase in the inundation area was dependent on the increase in the maximum initial water level. Additionally, the number of inundated buildings increases with the increase in the inundation area, as can be observed in Figure 5b.
The maximum initial water level of 25 m in the 1 st scenario, the lowest of the water levels, generates an inundation area of approximately 1.5 km 2 and inundates 3500 buildings. On the other hand, the greatest hazard (the 99th scenario) generates an inundation area of approximately 69 km 2 and inundates approximately 11500 buildings.

Probabilistic condition hazard assessment of inundated buildings
We have produced tsunami hazard maps and curves of building exposure in which the scenarios are conditional on the magnitude of the subaqueous explosion based solely on the vent size. The number of inundated buildings was first counted based on the inundation depth, and the results are represented with a histogram. Then, the histogram was compared to the nine candidate statistical distributions mentioned in the methodology. The best-fit distribution was evaluated with the chi-square test and was selected based on the minimum value of the test. Finally, the parameter of the best-fit distribution was used to model the hazard curve to determine the relationship between probability and the cumulative number of inundated buildings in each inundation depth range. Additionally, the hazard map was estimated by overlaying all 99 scenarios and counting how many buildings were inundated to generate the hazard curve.
A building hazard map for some example scenarios is shown in Figure 6. This reveals that the inundation depth of the buildings increases depending on the level of hazard. For example, considering the same building, the 1st scenario (the smallest hazard in Figure 6a) has a lower water depth than the 99th scenario (the greatest hazard in Figure 6e). The number of inundated buildings in each scenario was used to develop the probability hazard assessment, which is the goal of this study. In all scenarios, the highest number of the inundated building corresponds to low inundation depths between 1 and 10 m. For example, the peak of the 1st scenario is at a water depth of approximately 1 m, while the peak of the 99th scenario is at a water depth of approximately 5 m.
According to the model, application of the nine statistical distributions to the inundated buildings in each inundation depth range can be observed in Figure 7, which shows a comparison between the observed and predicted data. As shown in Figure 7c, the lognormal and Pearson-III distributions can capture the peaks of the reported scenarios.
A chi-square test was used to identify the best-fit statistical distribution among the nine candidate distributions to predict the probability hazard curve. Figure 8 shows the comparison of the nine candidate statistical distributions. The lognormal distribution is the best-fit distribution among the nine distributions for all scenarios, followed by Pearson-III and Weibull as second and third, respectively.
The lognormal distribution has also been used to model collapsed buildings due to tsunamis in several These final results can be useful for stakeholders on the local side and policymakers on the government side. Stakeholders can use the probability map to be more aware of the hazard level of their own residential areas, and the map might also be a useful tool in an evacuation. For example, if a house is located in a zone with a probability of 1, the people in the house need to evacuate immediately to a safe zone when a tsunami occurs. In contrast, if the probability of a house is approximately 0.01, the people in the house may need to evacuate only in some limited situations.
Policymakers can apply this map for planning the residential tsunami safety zone for future tsunami prevention, and they can also use the map for estimating the direct disaster losses immediately when a tsunami occurs.

Discussion
Tsunami hazards related to subaqueous volcanic explosions in Taal

Conclusion
Taal is one of the active volcanoes located in Taal Lake on Luzon Island in the Philippines, and this volcano presents risk mitigation challenges to humans. In its history, eruptions have occurred in the central crater, such as the 1749, 1754, 1911, and 1965 eruptions, while the 1716 eruption was a subaqueous volcanic explosion. We simulated different scenarios of subaqueous explosions based on the location of the 1716 event and the tsunami disaster related to this explosion. In this study, the conditional probability hazard assessment of tsunamis generated by subaqueous volcanic explosions is performed for Taal Lake. The scenarios all show that the tsunamis generated by subaqueous