Characteristics of building fragility curves for seismic and non-seismic tsunamis: case studies of the 2018 Sunda Strait, 2018 Sulawesi–Palu, and 2004 Indian Ocean tsunamis

. Indonesia has experienced several tsunamis triggered by seismic and non-seismic (i.e., landslides) sources. These events damaged or destroyed coastal buildings and infrastructure and caused considerable loss of life. Based on the Global Earthquake Model (GEM) guidelines, this study assesses the empirical tsunami fragility to the buildings in-ventory of the 2018 Sunda Strait, 2018 Sulawesi–Palu, and 2004 Indian Ocean (Khao Lak–Phuket, Thailand) tsunamis. Fragility curves represent the impact of tsunami characteristics on structural components and express the likelihood of a structure reaching or exceeding a damage state in response to a tsunami intensity measure. The Sunda


Introduction
Indonesia regularly faces natural disasters such as earthquakes, volcanic eruptions, and tsunamis because of its geographic location in a subduction zone of three tectonic plates (Eurasian, Indo-Australian, and Pacific plates) (Marfai et al., 2008;Sutikno, 2016). The Sunda Arc extends for 6000 km from the north of Sumatra to Sumbawa Island (Lauterjung et al., 2010) (Fig. 1a). Megathrust earthquakes regularly occur in this region, causing horizontal and vertical movement of the ocean floor which tends to be tsunamigenic (Mc-Closkey et al., 2008;Nalbant et al., 2005;Rastogi, 2007). These tsunamis are likely to cause greater destruction as they can follow prior damaging earthquake ground shaking and/or liquefaction (Sumer et al., 2007;Sutikno, 2016). Earthquake- generated tsunamis also tend to have longer wave periods affecting the coast than non-seismic ones (Day, 2015;Grezio et al., 2017). On 26 December 2004, the Sumatra-Andaman earthquake (M w = 9.0-9.3) hit the north of Sumatra, Indonesia (Fig. 1b). The rupture of the seafloor is estimated at 1200 km length and around 200 km width (Ammon et al., 2005;Krüger and Ohrnberger, 2005;Lay et al., 2005). In the city of Banda Aceh, a strong ground shaking was recorded (Lavigne et al., 2009). This megathrust earthquake was the second largest ever recorded (Løvholt et al., 2006) and caused the deadliest tsunami in the world. Overall, a dozen Asian and African countries were devastated, with around 280 000 casualties (Asian Disaster Preparedness Center, 2007;Suppasri et al., 2011). Although earthquakes represent the main cause of tsunamis, non-seismic events such as landslides can also initiate tsunami waves (Grezio et al., 2017;Ward, 2001). After a few months of volcanic activity in the Sunda Strait, Indonesia, the Anak Krakatau volcano erupted on 22 December 2018, leading to its southwestern flank failure (Fig. 1c). It triggered a relatively short wave period tsunami (∼ 7 min) , which devastated the western coast of Banten and the southern coast of Lampung with a death toll of 437 (Heidarzadeh et al., 2020;Muhari et al., 2019; National Agency for Disaster Management (BNPB), 2018; . The tsunami generation process is unclear. The subaerial and submarine landslide volume is still being investigated and ranges between 0.10 and 0.30 km 3 according to recent studies (Dogan et al., 2021;Grilli et al., 2019;Omira and Ramalho, 2020;Paris et al., 2020;Williams et al., 2019). Almost 2 months before this event, an unexpected tsunami struck Palu Bay, on Sulawesi Island, claiming 2000 lives and considerable loss to property (Association of Southeast Asian Nations (ASEAN)-Coordinating Centre for Humanitarian Assistance on disaster, 2018; Omira et al., 2019). The Sulawesi earthquake (M w = 7.5) occurred near the Palu-Koro strike-slip fault, 50 km northwest of Palu Bay (Fig. 1d) (Socquet et al., 2019). Ground shaking led to significant liquefaction along the coast (Paulik et al., 2019;Sassa and Takagawa, 2019). The fault mechanism did not suggest that the tsunami would be so destructive. The wave rapidly reached Palu (∼ 8 min), implying that its source was inside or near the bay Omira et al., 2019). Its short wave period (∼ 3.5 min) also indicates a non-seismic source (i.e., landslide). Some studies suggested that submarine landslides are responsible for the main tsunami. Moreover, a dozen coastal landslides were reported during field surveys and likely contributed to amplify tsunami waves Muhari et al., 2018;Omira et al., 2019;Pakoksung et al., 2019). However, according to Ulrich et al. (2019), those subaerial and submarine landslides may not be the only tsunami source as the Sulawesi earthquake rupture may have also induced a large portion of the tsunami waves.
The term "tsunami fragility" is a measure recently proposed to estimate structural damage and casualties caused by a tsunami, as mentioned by Koshimura et al. (2009b). Tsunami fragility curves are functions expressing the damage probability of structures (or death ratio) based on the hydrodynamic characteristics of the tsunami inundation flow (Koshimura et al., 2009a, b). These functions have been widely developed after tsunami events such as the 2004 Indian Ocean tsunami (IOT; Koshimura et al., 2009a, b;Murao and Nakazato, 2010;Suppasri et al., 2011), the 2006 Java tsunami (Reese et al., 2007), the 2010 Chilean tsunami , or the 2011 great eastern Japan tsunami (Suppasri et al., , 2013. Several methods aim to develop building fragility curves based on (i) a statistical analysis of on-site observations during field surveys of damage and flow depth data (empirical methods) (Suppasri et al., 2015, (ii) the interpretation of damage data from remote sensing coupled with tsunami inundation modelling (hybrid methods) (Koshimura et al., 2009a;Mas et al., 2020;Suppasri et al., 2011), or (iii) structural modelling and response simulations (analytical methods) (Attary et al., 2017;Macabuag et al., 2014).
Here, we empirically developed building fragility curves for the 2018 Sunda Strait, 2018 Sulawesi- Palu, and 2004 Indian Ocean (Khao Lak-Phuket, Thailand) tsunamis based on the Global Earthquake Model (GEM) guidelines . From the field surveys conducted after the 2018 Sunda Strait (Syamsidik et al., 2019), 2018 Sulawesi-Palu (Paulik et al., 2019), and 2004 Indian Ocean (Khao Lak-Phuket, Thailand) (Foytong and Ruangrassamee, 2007;Ruangrassamee et al., 2006) events, we utilize three databases called DB_Sunda2018, DB_Palu2018, and DB_Thailand2004, respectively. In the literature, tsunami inundation modelling has been performed many times to better understand the tsunami hydrodynamics, especially for earthquake-generated tsunamis (Charvet et al., 2014;Gokon et al., 2011;Koshimura et al., 2009a;Macabuag et al., 2016;De Risi et al., 2017;Suppasri et al., 2011). Compared to the 2004 IOT, the 2018 Indonesian tsunamis are uncommon events remaining less understood. Therefore, to improve our understanding of the structural damage caused by the Sunda Strait and Sulawesi-Palu tsunamis and to discuss the impact of wave period, ground shaking, and liquefaction events, we reproduce their tsunami intensity measures (i.e., flow depth, flow velocity, and hydrodynamic force) based on twolayer modelling (TUNAMI two-layer). We then compared the fragility curves of the Sunda Strait, Sulawesi-Palu, and Indian Ocean (Khao Lak-Phuket) tsunamis to those derived for the 2004 IOT in Banda Aceh (Indonesia), produced by Koshimura et al. (2009a). In this study, we explore the characteristics of building fragility curves for the 2018 Sunda Strait event and 2004 IOT in Khao Lak-Phuket, as well as for complex events, such as the 2018 Sulawesi-Palu tsunami in Palu City and the 2004 IOT in Banda Aceh, where the tsunamis may not be the only cause of structural destruction. Studying the impact of the wave period, ground shaking, and liquefaction events on the structural performance of buildings aims to improve our knowledge on the relationship between local vulnerability and tsunami hazard in Indonesia.

Post-tsunami databases
A post-tsunami database has been established for the Sunda Strait area by Syamsidik et al. (2019), Palu Bay by Paulik et al. (2019), and Khao Lak-Phuket by Ruangrassamee et al. (2006) and Foytong and Ruangrassamee (2007) in urban areas strongly affected by these events. These databases include 98, 371, and 120 observed flow depth traces at buildings, respectively. Here, the tsunami fragility analysis stands on subsets of the original databases of the 2018 Sunda Strait, 2018 Sulawesi-Palu, and 2004 Indian Ocean (Khao Lak-Phuket) tsunamis, as explained in Sects. 3.2.2, 3.2.3, and 2.2, respectively. We define these subsets as "new" databases, and we call them DB_Sunda2018, DB_Palu2008, and DB_Thailand2004, respectively. We note that the use of smaller databases for the fragility assessment is expected to increase the uncertainty in the exact shape of the fragility curves. Each database gathers exclusive information regarding the degree of damage, the building characteristics, and the flow depth traces (Tables 1 and 2). A brief analysis of the key variables (i.e., damage scale, building class, and tsunami intensity) are presented below.

Damage state
Each field survey adopted a different scale to record the degree of structural damage. In DB_Sunda2018, the five-state damage scale proposed by Macabuag et al. (2016) and Suppasri et al. (2020) is adopted, ranging from no damage to complete damage or washed away. In DB_Palu2018, the observed damage was classified into four states: no damage, partial damage repairable, partial damage unrepairable, and complete damage, as proposed by Paulik et al. (2019). Finally, in DB_Thailand2004, a four-state damage scale is  (2006). To simplify the comparison between the fragility curves, a harmonization of damage scales is proposed (Table 1). In this study, a fourstate damage scale ranging from ds 0 to ds 3 is used.

Building characteristics
Each survey also recorded the building construction type, which influences the damage probability (Suppasri et al., 2013). In Table 2, among the 94 buildings included in DB_Sunda2018, 67 are confined masonry, 26 are timber, and 1 is a steel frame building. In DB_Palu2018, most of the buildings are confined masonry with unreinforced clay bricks (∼ 95 %). The database also includes reinforced concrete and timber buildings. Finally, DB_Thailand2004 contains only reinforced concrete buildings. We note that after the 2004 IOT, 120 flow depth traces were recorded at reinforced concrete structures (e.g., residence, hotel, school, shop, bridge) in the Khao Lak-Phuket area. As we are not considering the data regarding the surveyed bridges, DB_Thailand2004 includes only 117 reinforced concrete buildings.

Tsunami intensity
The tsunami intensity has been measured in terms of flow depth level. Table 2 also presents the number of flow depth traces at surveyed buildings and the range of flow depth levels for each database.

Tsunami intensity simulations
3.1 Tsunami numerical modelling with a landslide source

Tsunami inundation model
The TUNAMI two-layer tsunami model used in the Sunda Strait and Palu areas relies on a two-layer numerical model solving non-linear shallow water equations. It considers two interfacing layers and appropriate kinematic and dynamic boundary conditions at the seafloor, interface, and water surface (Imamura and Imteaz, 1995;Pakoksung et al., 2019). To reproduce the landslide-generated tsunami, we model the interactions between tsunami generation and submarine landslides as upper and lower layers. The mathematical model performed in the landslide-tsunami code is obtained from a stratified medium with two layers. The first layer, composed of a homogeneous inviscid fluid with constant density, ρ 1 , represents the seawater, and the second layer is composed of a fluidized granular material with a density, ρ s , and porosity, ϕ. As assumed by Macías et al. (2015), the mean density of the fluidized sliding mass is constant and equals ρ 2 = (1 − ϕ)ρ s + ϕρ 1 . We consider the two layers immiscible. The governing equations are written as follows.
The continuity equation of the seawater (first layer) is The momentum equations of the seawater in the x and y directions are The continuity equation of the landslide (second layer) is The momentum equations of the landslide in the x and y directions are Index 1 and 2 refer to the first and the second layers, respectively, and ρ 1 and ρ 2 are the densities of the seawater and the landslide. Z i (x, y, t), Q i (x, y, t), and τ i (x, y, t) represent the level of the layer based on the mean water level, the vertically integrated discharge, and the bottom stress in each layer at each point (x, y) over the time t, respectively ( Fig. A1). D i denotes the thickness of each layer. The fifth term of the momentum equations (Eqs. 2, 3, 5, and 6) represents the interaction between the two layers. The tsunami model provides the maximum water flow depth and flow velocity along the coast during the tsunami inundation. The hydrodynamic force acting on buildings and infrastructure is defined as the drag force per unit width of the structure, as shown in Eq. (7) (Koshimura et al., 2009b).
C D represents the drag coefficient (C D = 1.0 for simplicity), ρ is the seawater density (ρ = 1000 kg m −3 ), u stands for the current velocity (m s −1 ), and D is the inundation depth (m).

Flow resistance within a tsunami inundation area
BATNAS and DEMNAS, Indonesia, provided the bathymetric and topographic data at 180 and 8 m resolutions, respectively. The data were established from synthetic aperture radar (SAR) images (http://tides.big.go.id/DEMNAS/index. html, last access: 2 February 2020). Both datasets were resampled to three computational domains with a grid size of 20 m resolution ( Fig. 2a and b). In Palu City, the bathymetric and topographic data at 1 m resolution were obtained through lidar images and supplied by the Geospatial Information Agency (BIG), Indonesia ( Fig. 2c and d).
For tsunami inundation modelling in a densely populated area, we apply a resistance law with the composite equivalent roughness coefficient depending on the land use and building conditions, as shown in Eq. (8) (Aburaya and Imamura, 2002;Koshimura et al., 2009a).
where n o corresponds to the Manning's roughness coefficient (n o = 0.025 s m −1/3 ), C D represents the drag coefficient (C D = 1.5; Federal Emergency Management Agency (FEMA), 2003), and the constant d signifies the horizontal scale of buildings (∼ 15 m). θ is the building occupation ratio in percent (0 %-100 %) for each computational cell of 20 m × 20 m and 1 m × 1 m resolutions in Sunda Strait and Palu areas, respectively. θ is obtained by computing the building area over each pixel using geographic information system (GIS) data. The computational cell corresponding to buildings can be inundated by the n Manning coefficient through the term D, which represents the simulated flow depth (m). In the urban areas of Sunda Strait and Palu, the average occupation ratios are 24 % and 84 %, respectively ( Fig. 2b and d). In non-residential areas, we set the Manning's roughness coefficients inland and on the seafloor to 0.03 and 0.025, respectively, which are typical values for vegetated and shallow water areas (Kotani, 1998).

Performance parameters
The tsunami inundation model is calibrated using two performances parameters: K and κ proposed by AIDA (1978), as defined below: where x i and y i are the recorded and simulated tsunami flow depths at location i. K is defined as the geometrical mean of K i , and κ is defined as deviation or variance from K. The Japan Society of Civil Engineers (JSCE) (2002) recommends 0.95 < K < 1.05 and κ < 1.45 for the model results to achieve "good agreement" in the tsunami source model and propagation and inundation model evaluation (Otake et al., 2020;.

The 2018 Sunda Strait tsunami inundation model
To correct the digital surface model (DSM), we removed the vegetation, building, and infrastructure elevations based on the linear smoothing method and used the resulting digital elevation model (1st DEM) as topography in the tsunami inundation model (Fig. 3). The vertical accuracy of the DSM and DEM is about 4 m. The 2018 Sunda Strait tsunami model depends on the density of the landslide (ρ 2 ), its stable slope (α), its volume (V L ), and its sliding time (t S ). As proposed by Paris et al. (2020), the low sensitivity parameters are set as follows: ρ 2 = 1500 kg m −3 , α = 5 • , and V S = 0.15 km 3 . We reach the best fit between the simulated and observed flow depths at buildings for 10 min sliding time. Nevertheless, most of the simulated flow depths are underestimated compared to the observed ones, with a mean difference of 0.28 ± 1 m. Using quantum GIS (QGIS) software, we smoothed the 1st DEM to remove these mean differences in elevation at buildings where the flow depth is underestimated. The resulting DEM (2nd DEM) provides a topography more reliable at buildings (Fig. 3). We completed three cross sections along the Sunda Strait coasts to show the different corrections applied to the DSM ( Fig. 4a-g). K and κ values for damaged buildings are 0.99 and 1.11, respectively, which means that we achieve "good agreement" for the Sunda Strait tsunami model, displayed in Fig. 5a-f. We note that the simulated inundation zone overlays 94 buildings out of 98. In Sect. 4.1, the Sunda Strait tsunami fragility assessment is based on these 94 buildings (DB_Sunda2018). Simulation snapshots of the Sunda Strait tsunami propagation are shown in Fig. B1 10, 20, 60, and 120 s after the tsunami generation. In Fig. B2, the simulated tsunami height based on the best-fitting parameters is also displayed.

The 2018 Sulawesi-Palu tsunami inundation model
We increased the mean sea level (MSL) by 2.3 m to reproduce the high tide during the 2018 Sulawesi-Palu tsunami. As shown by Pakoksung et al. (2019), the observed waveform at Pantoloan tidal gauge does not fit the simulated one with the finite fault model of TUNAMI-N2. Although recent studies show that seismic seafloor deformation may be the primary cause of the tsunami Ulrich et al., 2019), in this study, the main assumption is that the 2018 Sulawesi-Palu event was triggered by subaerial and submarine landslides. According to Heidarzadeh et al. (2019), a large landslide to the north or the south of Pantoloan tidal gauge is responsible for the significant height wave recorded. Arikawa et al. (2018) also identified several sites of potential subsidence in the northern part of Palu Bay. Based on these previous studies, we assume two large landslides: L1 and L2. Small landslides (S1-S12) also occurred in the bay; their location is known from observations from satellite imagery, field surveys, and video footage Carvajal et al., 2019) (Fig. 6). From the trial and error method and the topographic and bathymetric data provided by the Geospatial Information Agency (BIG), we determined the soil property and achieved the volume of the landslides (Table 3). In Fig. 7, the submarine landslides model reproduces well the tsunami observations at Pantoloan.
The calibration of the model depends on the landslide S8 because (i) as a small landslide, its volume is too small to distort the simulated wave height at the Pantoloan tidal gauge, (ii) it has the largest volume among the other small landslides, and (iii) it is close and ideally oriented to Palu City; the slide direction, captured by an aircraft pilot, is perpendicular to the bay (Carvajal et al., 2019). The density of the landslides (ρ 2 ), their stable slope (α), and their sliding time (t s ) are set as follows: ρ 2 = 2000 kg m −3 (Palu Bay receives a large amount of fine continental deposits such as clay-sized sediments; Frederik et al., 2019), α = 14 • (Chakrabarti, 2005), and t S = 10 min. For a landslide ratio of 1.2 (i.e., S8 volume is multiplied by 1.2), the tsunami model shows a great similarity between observed and simulated flow depths (a = 1.027). The simulated tsunami inundation zone overlays 175 traces out of 371 because (i) 151 buildings with flow depth traces are not included in our computational area (Fig. 2c) and (ii) 45 buildings are outside the simulated tsunami envelope, which is shorter than the surveyed one (Fig. 8). The geometric mean is near the recommended values (K = 0.93), while the standard deviation and the root mean square error (RMSE) are high (κ = 2.18, RMSE = 0.92 m). Therefore, to develop accurate and reliable curves, we set a 1 m confidence interval including 124 flow depth traces at buildings out of 175 ( Fig. 9). In Sect. 4.2, the Sulawesi-Palu tsunami fragility assessment is based on these 124 buildings (DB_Palu2018). K and κ values for damaged buildings are 0.93 and 2.14, respectively, with a root mean square error of 0.26 m. The validity of the model is mainly based on the geometric mean K, close to 0.95, so we consider the tsunami inundation model accurate enough (Fig. 8). In Fig. C1, the simulation snapshots of the Sulawesi-Palu tsunami propagation are shown 2, 10, 30, and 60 s after the tsunami generation. The simulated tsunami height based on the best-fitting parameters is also displayed in Fig. C2. Figure C3 illustrates the maximum simulated flow velocity of the 2018 Sulawesi-Palu tsunami inundation model.

Tsunami fragility assessment
The proposed fragility assessment framework has two main steps. In the first step, an exploratory analysis aims to (i) assess the trends that the available data follow and (ii) determine the main explanatory variables that need to be included in the statistical model and their influence on the slope and intercept of the fragility curves. Then, we select a statistical model and examine its goodness-of-fit to the data based on the observations of the exploratory analysis. We note that the development of the computed fragility curves for the 2018 Sunda Strait and 2018 Sulawesi-Palu tsunamis is directly based on DB_Sunda2018 and DB_Palu2018, in which each building has both observed and simulated flow depth values (Table 4).
To explore the relationship between the tsunami intensity and the probability of damage, we fit a generalized linear model (GLM) to the data of each database, as proposed by the GEM guidelines . A GLM assumes that the response variable y ij is assigned 1 if the building   j sustained damage DS ≥ ds i and 0 otherwise. The variable follows a Bernoulli distribution: where π i (x j ) is the probability that a building j will reach or exceed the "true" damage state ds i given estimated tsunami intensity levelx j . The Bernoulli distribution is characterized by its mean, which is expressed here in terms of a probit model, commonly used to express the mean in the empirical fragility assessment field (Rossetto et al., 2013), defined in terms of [.], the cumulative distribution function of a standard normal distribution: where η ij is the linear predictor, which can be written in the form where θ 1i and θ 0i are the two regression coefficients representing the slope and the intercept, respectively, of the fragility curve corresponding to damage state ds i . For the exploratory analysis, the tsunami intensity is measured in terms of observed flow depth levels. We also fit the GLM models to subsets of data of each database to explore the importance of the construction type to the shape of the fragility curves. The confidence in the exact shape of the mean curves is estimated and presented in terms of the 90 % confidence intervals around the best-estimate curves. Based on the aforementioned observations, we construct parametric statistical models for the three databases to (i) identify the simulated tsunami measure type that fits the data best and (ii) construct fragility curves for the tsunami intensity type that fits the data best.
Ideally, the response variable y ij of an appropriate statistical model is the damage state i = {0, 1, 2, 3} sustained by a building j . The damage state follows a categorical distribution (i.e. also called a generalized Bernoulli distribution)  which describes the possible levels of damage i = {0, 1, 2, 3} sustained by a given building (Table 1). The random component of this model can be written as where P (DS = ds i x j ) is the probability that a building j will reach the "true" damage state ds i given estimated tsunami intensity levelx j .
Multiple expressions of the systematic component are constructed to test their goodness of fit. With regard to the link function, apart from the commonly used probit function, two alternative expressions found in the GEM guidelines for empirical vulnerability assessment (Rossetto et al., 2013), namely the logit and complementary loglog (termed here "cloglog"), are considered in the form The linear predictor is also expressed in various forms of increasing complexity, as depicted in Eq. (19).
Class is a categorical unordered variable which expresses here the construction type. θ 0−3 are the unknown regression coefficients of the model. Equations (19a) and (19b) assume that the fragility curves are only influenced by the tsunami intensity. Equation (19a) assumes that the slope of the fragility curves is the same for all damage states. In contrast, Eq. (19b) allows the slope of each curve to vary for each damage state; the slope varies for each fragility curve. The following three equations account for the influence of the building class (i.e. the construction type) in the shape of the fragility curves. All three equations assume that the construction type affects the intercept of the fragility curves, and only Eq. (19e) assumes that the construction type affects both the intercept and the slope of the curves. Finally, Eqs. (19c) and (19e) assume identical slopes for all fragility curves irrespective of the damage state. In contrast, Eq. (19d) relaxes this assumption and considers that the slope changes for each damage state. The combinations of random and systematic components result in five distinct models (Table 5).
In what follows, we fit multiple models to each database based on the observations of the exploratory analysis. We examine the goodness of fit of these models for a given tsunami intensity measure and link function with two formal tests, as proposed in the GEM guidelines . Eq. (19e) Figure 10. Probit functions fitted for each individual damage state to DB_Sunda2018 (a) to assess whether the observed flow depth is an efficient descriptor of damage and (b) to assess whether the construction type affected the shape of fragility curves for ds 2 and ds 3 . In both cases, the 90 % confidence interval is plotted.
Firstly, we compare the Akaike information criterion (AIC) values, which estimates the prediction error of the examined models (Akaike, 1974). The model with the lowest value fits the data best. The alternative models used in this study are nested, which means that the more complex model includes all the terms of the simpler ones plus an additional term. For this reason, we also perform a series of likelihood ratio tests to examine whether the fit provided by the model with the lowest AIC value is statistically significant over its alternative nested models, which relaxes its assumptions . We also use the AIC value to determine which of these simulated intensity measures fits the data best. Furthermore, the 90 % confidence intervals of the best-estimate fragility curves are constructed using bootstrap analysis. According to the latter analysis, 1000 samples of the database are obtained with a replacement, and the selected model is refitted to each sample.

DB_Sunda2018
We fit the GLM models to the data in DB_Sunda2018 (irrespective of their structural characteristics), and we plot the obtained probit functions against the natural logarithm of the observed flow depth to explore how the slope and the intercept of the models change for each damage state (Fig. 10a). The 90 % confidence intervals around the bestestimate curves are also included. All three curves have positive slopes, which indicates that the flow depth is an adequate descriptor of the damage caused by a tsunami as the probability of a given damage state being reached or exceeded increases with the increase in the flow depth. The slope of each function is similar for ds 2 and ds 3 and different for ds 1 . Nonetheless, the curve corresponding to ds 1 is also associated with substantial uncertainty. In Fig. 10b, we fit probit models to subsets of the available data for the two main construction types. One of the drawbacks of the small database is that not all damage states were observed for each building class. Therefore, the comparison of probit models is limited for damage states ds 2 and ds 3. The curves for the two construction types appear to be substantially different. As expected, the timber buildings are more vulnerable than the confined masonry buildings. Their intercept is responsible for the difference as the two curves are parallel. It indicates the need to develop a statistical model which allows only the intercept to change with the construction type, and the slope should be identical. Following the main observations of the exploratory analysis, we consider that M3 is an acceptable model with two explanatory variables: the tsunami intensity and the construction type. To assess its goodness of fit, we consider each link function with three alternatives for the linear predictor (i.e., M4, M5, and M1) which relax some of its assumptions. In Table 6, we compare the AIC values of the three models to assess the fit of the different models for the observed flow depth levels assuming the probit link function. M3 has the smallest AIC value compared to its alternatives, which indicates that it fits the data better than the remaining three models. Nonetheless, some of these differences are rather small, and it raises the question of whether the improvement in the fit provided by M3 is statistically significant over its alternatives. To address this, we perform likelihood ratio tests, and the results are reported in Table 7. We note that the p values vary for the three comparisons. The p value is significantly above the 0.05 threshold when the identical slope for each fragility curve assumption (i.e. comparison of M3 and M4) is tested. This means that M4 (which assumes varying slopes for each damage state) does not provide a statistically significant improvement than its alternative. Therefore, the fit of M3 is the best. Similarly, the p value is well above the threshold for M3 vs. M5, highlighting that the construction type does not affect the slope of the fragility curves. In contrast, the p value is well below the threshold for the comparison of M3 and M1, indicating that the construction type is an important variable and affects only the intercept. Having concluded that M3 based on the observed flow depth data fits the data better than its alternatives (i.e. M4, M5, and M1), we repeat the procedure to identify which simulated intensity type fits the data best. Table 6 also shows the comparison of the AIC values for the three simulated tsunami intensity types. For all simulated intensity types, M3 is identified as the model which fits the data better than its alternatives, and this conclusion is further reinforced by the likelihood ratio tests presented in Table 7. By comparing the AIC values for M3 for all three simulated intensity types, we note that the simulated flow depth is the tsunami intensity that fits the data best. The aforementioned observations can also be made if instead of the probit link function, the two alternative functions (i.e. logit and cloglog) are considered, as depicted in Table D1. The comparison of the AIC values of M3 for the three link functions identifies the probit link function as the one that fits the data best. The regression coefficients of the 2018 Sunda Strait fragility curves based on the best-fitted M3 model with a probit link function are listed in Table E1. An advantage of constructing a complex model that accounts for the ordinal nature of the damage and for the two main construction types in the systematic component is that fragility curves for timber buildings can be obtained even for the states for which there are available data. A timber building is found to sustain more damage than confined masonry buildings for the more intense damage states. Nonetheless, there is substantially more uncertainty in the prediction of the likelihood of damage, and this can be attributed to the rather small sample size.

DB_Palu2018
We also fit GLM models to the data in DB_Palu2018 using the observed tsunami flow depth to express the tsunami intensity and then to construct fragility curves and their 90 % confidence intervals for the three individual damage states (Fig. 11). The data seem to produce fragility curves with positive slopes for d S1 and d S2 and a negative slope for d S3 . This latter observation is counter-intuitive as it is expected that the likelihood of collapse will grow with the increase in the tsunami depth. This outcome could be attributed to the collected sample, which includes very few collapsed buildings observed at low flow depth levels.
Based on the observations of the exploratory analysis, we use identical slopes for the fragility curves for all three damage states (ds 1 − ds 3 ) to tackle the negative slope for ds 3 and three link functions. Therefore, model M1 is fitted to DB_Palu2018 assuming that the tsunami intensity is expressed in terms of simulated flow depth, flow velocity, and hydrodynamic force. Table 8 depicts the AIC values for each model. We note that for all cases the flow depth fits the data the best. Table 8 also shows that the logit function fits the data best. The regression coefficients of the 2018 Sulawesi-Palu fragility curves for the logit function are depicted in Table E2.

DB_Thailand2004
The exploratory analysis aims to identify trends in the shape of the fragility curves for each damage state. Thus, we fit GLM models to DB_Thailand2004 to construct fragility curves for the three individual damage states, and we plot them with their 90 % confidence interval in Fig. 12. The data seem to produce fragility curves with positive slopes for all three damage states and also are parallel to each other, which suggests that the slope should be identical for all three curves.
Based on the observations of the exploratory analysis, we consider model M1 as the most suitable. To test its goodness of fit, model M2, which relaxes the assumption that the slope of all three curves is identical, is also fitted to the data. In Table 9, the comparison of the AIC values for the two models also shows that M1 is the model which fits the data best for all three link functions considered in this study (i.e., probit, logit, and cloglog). We also perform a likelihood ratio test to confirm that the improvement in the fit provided by the more complex M2 model over M1 is not statistically significant. The p value is found to be equal to 0.76, 0.95, and 0.33 for the probit, logit, and cloglog functions, respectively, which is significantly above the 0.05 threshold. This suggests that M2 does not provide a statistically better fit to the data; therefore, the less complex M1 model fits the data best. The regression coefficients of the 2004 Indian Ocean (Khao Lak-Phuket) fragility curves for the best-fitted model M1 with logit link function can be found in Table E3.

Building fragility curves of the 2018 Sunda Strait tsunami
The fragility curves determine conditional damage probabilities according to the tsunami intensity measures of the 2018 Sunda Strait event for both confined masonry concrete (Fig. 13a-c) and timber (Fig. 14a-c) buildings of DB_Sunda2018. In Fig. 14a and b, there are no data to predict the shape of the curves between 0-1 m flow depth and 0-1 m s −1 flow velocity. The curves as a function of the observed flow depth reveal a great similarity with the ones based on the simulated flow depth from the TUNAMI two-layer model (Figs. 13a and 14a). For instance, when the observed and simulated flow depths reach 3 m, the likelihood of minor to major damage (i.e., ≥ ds 1 , ds 2 ) for both timber and confined masonry buildings is approximately 99 % ( Fig. 14a  and b). In contrast, the likelihood of complete damage (i.e., ≥ ds 3 ) is 70 % for timber buildings and only 10 % for confined masonry buildings. Consequently, the tsunami functions based on observation and simulation are highly similar, which illustrates the accuracy and the reliability of the tsunami inundation model. The curves show that confined masonry-type buildings have higher performance than timber structures. When the flow depth is greater than 5 m and 2.5 m, the probability of complete damage is around 99 % for confined masonry and timber buildings, respectively. We also compare the completely damaged or washed away fragility curve for confined-masonry buildings to , who developed the curve as a function of observed flow depth for these buildings, as depicted in Fig. 13a. Fragility curves representing complete damage or washed away are similar up to 4.5 m flow depth. Each curve estimates a 15 % building damage probability at 3.5 m flow depth. However, a few data points are available beyond 5 m in the Sunda Strait area. Therefore, the damage probability uncertainty is greater for this value, hence the difference between our ds 3 -curve and the one produced by . The curves as functions of the maximum simulated flow velocity and the hydrodynamic force are displayed in Figs. 13b and 14b and Figs. 13c and 14c for confined masonry concrete and timber buildings, respectively.

Building fragility curves of the 2018 Sulawesi-Palu tsunami
The 2018 Sulawesi-Palu tsunami curves are developed for confined masonry buildings with unreinforced clay brick of DB_Palu2018. The computed and surveyed curves show a similar damage trend. When the observed and simulated flow depths reach 1.5 m, the building damage probabilities for partial damage repairable (i.e., ≥ ds 1 ), partial damage unrepairable (i.e., ≥ ds 2 ), and complete damage (i.e., ≥ ds 3 ) are around 90 %, 40 %, and 15 %, respectively (Fig. 15a). The fragility curves based on the observed and simulated flow depths are relatively similar, especially for ds 1 and ds 3 . The curves based on the flow velocity and the hydrodynamic force are displayed in Fig. 15b and c.

Comparison between the 2018 and 2004 building fragility curves
In Fig. 16, we compare (i) the Sunda Strait and Sulawesi-Palu ds 3 -curves based on the simulated tsunami intensity measures for confined masonry-type buildings, (ii) the 2004 Indian Ocean (Khao Lak-Phuket, Thailand) ds 3 -curve based on the observed flow depth for reinforced-concrete infilled frames buildings (Foytong and Ruangrassamee, 2007;Ros-Figure 11. Probit functions fitted for each individual damage state to DB_Palu2018 to assess whether the observed flow depth is an efficient descriptor of damage. The 90 % confidence interval is plotted.  Koshimura et al. (2009a). The curves are based on a visual damage interpretation of remaining roofs using the pre-and post-tsunami satellite data (IKONOS) and are thus developed for mixed buildings (low-rise wooden, timberframed, and non-engineered reinforced concrete constructions; Koshimura et al., 2009a;Saatcioglu et al., 2006). For 1 m flow depth, the likelihood of complete damage is greater in Palu (10 %) than in Banda Aceh, Khao Lak-Phuket, and Sunda Strait (Fig. 16a, Table 10). However, when the flow depth reaches 3 m, the damage probability is about 50 % in Banda Aceh, 25 % in Palu City, and less than 20 % in Khao Lak-Phuket. We also note that the likelihood of completely damaged or washed away buildings is higher in Sunda Strait than in Khao Lak-Phuket above 4 m flow depth. However, the data points in Thailand are mostly ranging from 0 to 5 m, and the 90 % confidence interval upon this value is constantly increasing with the flow depth. Below 1 m s −1 , the flow velocity has a low impact on the damage probability in Banda Aceh (< 1 %). However, beyond this value, the probability of damage becomes very sensitive to the current velocity (Fig. 16b, Table 10). As an example, when the flow velocity attains 6 m s −1 , the curve estimates 99 % building damage probability in Banda Aceh. The hydrodynamic force also contributes to increase the probability of complete damage in Banda Aceh. For example, when the force reaches 25 kN m −1 , the damage probability is around 99 % in Banda Aceh (Fig. 16c, Table 10).

Reliability of the building fragility curves
The reliability of the curves depends mainly on (i) the quality and the quantity of post-tsunami data and (ii) whether the tsunami intensity measures are efficient predictors of damage. With regard to the first factor, DB_Sunda2018, DB_Palu2018, and DB_Thailand2004 include relatively little data (Table 2). For each database, the relatively broad confidence intervals around the best-estimate fragility curves reflect the small sample size. Moreover, the complexity of each studied event also plays a role in how well the selected tsunami intensity measure can represent the tsunami damage. In particular, in DB_Sunda2018 and DB_Thailand2004, only the tsunami load is responsible for the building dam- age. In contrast, in DB_Palu2018, buildings may have suffered prior damage due to ground shaking and liquefaction (Kijewski-Correa and Robertson, 2018;Sassa and Takagawa, 2019). Nonetheless, we are not able to establish precisely which of the surveyed buildings had suffered prior damage in the database and to what extent. The complexity of the 2018 Sulawesi-Palu event could introduce a bias in the tsunami fragility assessment, and this has also been mentioned for other events such as the 2011 great eastern Japan tsunami (Charvet et al., 2014). This bias could explain why we observed a negative slope for our ds 3 -curves based on the observed flow depth combined with very few collapsed buildings, especially for very low intensity levels (Fig. 11). Despite the aforementioned reservations, the adopted statistical tests identified that the flow depth is consistently the best descriptor of the tsunami damage for both the DB_Sunda2018 and DB_Palu2018 data, while the flow velocity is the worst. This finding is in line with similar observations made by Macabuag et al. (2016). De  illustrated well the influence of the DEM resolution and the model sources on the efficiency of the flow velocity as a tsunami intensity measure. In Sunda Strait, the DEM resolution is relatively high (20 m), and it could explain why the flow velocity is not a good descriptor of damage. In Palu City, we perform twolayer numerical modelling using the finest grid size of 1 m. However, the 2018 Palu tsunami is a complex event. The subaerial and submarine landslides may not be the main cause of the tsunami, as shown by Ulrich et al. (2019), and this could have affected the flow velocity data. As the flow velocity of the Sunda Strait and Sulawesi-Palu tsunamis does not provide a good description of the damage, we cannot evaluate the impact of floating debris on Indonesian structures (Song et al., 2017). The hydrodynamic force of these events, computed from the flow velocity and the flow depth, does not provide a good description of the tsunami damage either.
6.2 Impact of the wave period, ground shaking, and liquefaction events on the building damage probability The curve comparison illustrates well the relationship between the 2004 Indian Ocean, the 2018 Sunda Strait, and the 2018 Sulawesi-Palu tsunamis characteristics, summarized in Table 11, and the structural performance of buildings.

Impact of the wave period
The 2018 Sunda Strait tsunami and the 2004 IOT (Khao Lak-Phuket, Thailand) are characterized by dominant wave periods of about 7 min  and 40 min (Karlsson et al., 2009;Puspito and Gunawan, 2005;Tsuji et al., 2006), respectively (Table 11). Damage from ground shaking or liquefaction episodes was not reported, so the tsunami is the main cause of building damage. We compare the Sunda Strait and the Indian Ocean (Khao Lak-Phuket) curves based on the flow depth to investigate the impact of the tsunami wave period on buildings. In Fig. 16a, the curves showed that the short wave period tsunami in the Sunda Strait is less damaging than the 2004 IOT below 5 m flow depth. For instance, for 3 m flow depth, the likelihood of complete damage is around 20 % in Khao Lak-Phuket against only 10 % in the Sunda Strait area (Table 10). On the other hand, above 5 m flow depth, the structures in Khao Lak-Phuket reveal a better performance than the ones in the Sunda Strait area. As few data points are available beyond this value for completely damaged buildings, the Sunda Strait and the Indian Ocean (Khao Lak-Phuket) curve reliability is insufficient. Even though the long wave periods of the IOT seem to increase the likelihood of building damage, the sample size of collapsed buildings beyond 5 m flow depth is too small to validate this assumption.

Impact of ground shaking and liquefaction events
The city of Banda Aceh and the Khao Lak-Phuket area were damaged by the 2004 IOT. Along Banda Aceh shores, the simulated tsunami wave period ranges from 40 to 45 min (Prasetya et al., 2011;Puspito and Gunawan, 2005), and the one simulated in Khao Lak-Phuket is estimated at approximatively 40 min (Karlsson et al., 2009;Puspito and Gunawan, 2005;Tsuji et al., 2006). Although the tsunami wave periods are similar at both locations, the 2004 Indian Ocean earthquake was strongly felt in the city of Banda Aceh, where it lasted about 10 min (Table 11). The earthquake intensity is estimated at VII to VIII on the Modified Mercalli Scale Saatcioglu et al., 2006). Despite the ground acceleration not being recorded in the damage zones, seismic failure was distinguished from tsunami damage. For example, buildings with three to five stories were heavily damaged by the ground motion, which was amplified by the soft soil characteristics, compared to low-rise structures. In Fig. 16a, the curves estimate about 50 % and 20 % of building damage probabilities for complete damage in Banda Aceh and Khao Lak-Phuket, respectively, for 3 m flow depth (Table 10). Therefore, the building resilience is higher in Khao Lak-Phuket than in Banda Aceh. It comes from the fact that the Khao Lak-Phuket curve is developed for reinforced concrete buildings, while the ones in Banda Aceh are produced for mixed buildings (Koshimura et al., 2009a). Another reason is that the 2004 Indian Ocean earthquake was not recorded in Khao Lak-Phuket, so the ground motion did not damage the buildings before the tsunami's arrival. Furthermore, the likelihood of complete damage is very high for low inundation depth levels in Banda Aceh. This feature is usually observed for buildings suffering prior damage such as ground shaking and/or liquefactions episodes, as mentioned by Charvet et al. (2014) for the 2011 great eastern Japan event.
The 2018 Sulawesi-Palu event is characterized by short wave periods of about 3.5 min according to Syamsidik et al. (2019), like the 2018 Sunda Strait tsunami (Table 11). However, the curves based on the flow depth are remarkably different (Fig. 16a). For instance, for 3 m flow depth, the likelihood of complete damage is 25 % in Palu against 10 % in Sunda Strait, which means that buildings affected by the Sulawesi-Palu tsunami were more susceptible to complete damage. Most importantly, up to 2 m flow depth, the building damage probability is higher in Palu than in Banda Aceh, affected by ground shaking and then being hit by a long wave period tsunami. As an example, for 1 m flow depth, the building damage probability of complete damage is about 10 % in Palu against less than 5 % in Banda Aceh (Table 10). The main cause of structural damage caused by the Sulawesi-Palu tsunami is still being investigated. Mas et al. (2020) sug-gested that the tsunami hydrodynamic or debris impact might be the main cause of structural destruction in the waterfront area of Palu Bay. Here, the flow velocity and the hydrodynamic force are not good descriptors of damage, so we cannot support this assumption (Song et al., 2017). On the other hand, Palu City sits on alluvial soil layers from Palu River and is thereby vulnerable to liquefaction disaster (Darma and Sulistyantara, 2020;Goda et al., 2019;Kijewski-Correa and Robertson, 2018). Even though the largest liquefaction areas were recorded outside the inundation zone (Watkinson and Hall, 2019), Sassa and Takagawa (2019) and Kijewski-Correa and Robertson (2018) observed land retreats along the coastal area of Palu City ( Fig. 17a and b). Most of the masonry-type buildings completely damaged are very close to these coastal retreats. Some of them were washed away by the tsunami. Therefore, these buildings do not have flow depth values and could not be used for the tsunami fragility assessment (Fig. 17b). Furthermore, in Palu, the earthquake intensity is estimated at VII to VIII on the Modified Mercalli Scale, but ground shaking was not the main cause of structural destruction (Kijewski-Correa and Robertson, 2018;Supendi et al., 2019). The likelihood of complete damage is also relatively high for low flow depth levels, so ground motion could have triggered liquefaction events and enhanced the building susceptibility to tsunami damage in the waterfront of Palu City. This assumption cannot be verified through satellite images; it needs direct and close observations, which might be erased by the tsunami.

Conclusions
According to the GEM guidelines, building fragility curves of the 2018 Sunda Strait, 2018 Sulawesi- Palu, and 2004 Indian Ocean (Khao Lak-Phuket, Thailand) tsunamis are empirically developed from post-tsunami databases respectively called DB_Sunda2018, DB_Palu2018, and DB_Thailand2004. To improve our understanding of the   structural damage caused by the Sunda Strait and Sulawesi-Palu tsunamis, we reproduce their tsunami intensity measures (i.e., flow depth, flow velocity, and hydrodynamic force) with the TUNAMI two-layer model for the first time.
The flow depth is constantly the best descriptor of tsunami damage for each event. The building fragility curves for complete damage reveal the following. (i) The buildings affected by the Sunda Strait tsunami sustained less damage than the ones in Khao Lak-Phuket (IOT). For example, for 3 m flow depth, the building damage probability is around 20 % in Khao Lak-Phuket against 10 % in the Sunda Strait area hit by a short wave period tsunami (landslide source). Considering that the tsunami was the main cause of structural damage (i.e., damage related to ground shaking and/or liquefaction was not recorded), the longer wave period of the 2004 IOT may have increased the likelihood of complete damage, and (ii) the building resilience is weaker in Banda Aceh than in Khao Lak-Phuket. For 3 m flow depth, the likelihood of complete damage is about 50 % in Banda Aceh and 20 % in Khao Lak-Phuket. Although both locations were hit by the 2004 IOT, Banda Aceh was strongly affected by ground shaking before the tsunami's arrival, and (iii) the buildings affected by the Sulawesi-Palu tsunami were more susceptible to be completely damaged than the ones affected by the IOT in Banda Aceh (i.e., ≤ 2 m). As an example, for 1 m flow depth, the building damage probability of complete damage is about 10 % in Palu and 5 % in Banda Aceh. The Sulawesi-Palu tsunami is a complex event as it may not be the only cause of structural destruction. The 2018 Sulawesi earthquake caused minor damage to buildings and most importantly could have triggered liquefaction events in the waterfront of Palu City where coastal retreats have been observed, increasing the susceptibility of buildings to tsunami damage.      Code and data availability. Post-tsunami field survey data are available from references cited in the text. The bathymetric and topographic data for the Sunda Strait area were provided by BAT-NAS and DEMNAS, Indonesia, respectively (http://tides.big.go.id/ DEMNAS/index.html, last access: 1 February 2020) (DEMNAS, 2020). The Geospatial Information Agency (BIG), Indonesia, provided the bathymetric and topographic data for Palu Bay. The tidal gauge records were supplied by the Coastal Disaster Mitigation Division, Ministry of Marine Affairs and Fisheries, Jakarta, Indonesia. Spatial data in this study are depicted through QGIS software.
Author contributions. FI, AS, KP, EL, II, and FB designed and coordinated this research. AS, KP, and EL performed the tsunami simulations and participated in the calibration of the inundation models in Palu Bay and in Sunda Strait. SS and RP contributed to the tsunami data collection in the Sunda Strait and Palu areas, respectively. II developed the fragility functions through advanced statistical analysis. All authors contributed to the drafting of the manuscript.