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 defined by Ruangrassamee et al. (2006). To simplify the comparison between the fragility curves, a harmonization of damage scales is proposed (Table 1). In this study, a four-state damage scale ranging from ds0 to ds3 is used.

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.

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.

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 best-estimate 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 ds2 and ds3 and different for ds1. Nonetheless, the curve corresponding to ds1 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 ds2 and ds3. 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.

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 dS1 and dS2 and a negative slope for dS3. 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 (ds1-ds3) to tackle the negative slope for ds3 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.

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.

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 ms-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., ≥ds1, ds2) for both timber and confined masonry buildings is approximately 99 % (Fig. 14a and b). In contrast, the likelihood of complete damage (i.e., ≥ds3) 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 Syamsidik et al. (2020), 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 ds3-curve and the one produced by Syamsidik et al. (2020). 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.

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., ≥ds1), partial damage unrepairable (i.e., ≥ds2), and complete damage (i.e., ≥ds3) 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 ds1 and ds3. The curves based on the flow velocity and the hydrodynamic force are displayed in Fig. 15b and c.

In Fig. 16, we compare (i) the Sunda Strait and Sulawesi–Palu ds3-curves based on the simulated tsunami intensity measures for confined masonry-type buildings, (ii) the 2004 Indian Ocean (Khao Lak–Phuket, Thailand) ds3-curve based on the observed flow depth for reinforced-concrete infilled frames buildings (Foytong and Ruangrassamee, 2007; Rossetto et al., 2007; Ruangrassamee et al., 2006), and (iii) the 2004 Indian Ocean (Banda Aceh, Indonesia) ds3-curves produced by 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, timber-framed, 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 ms-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 6ms-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 25kNm-1, the damage probability is around 99 % in Banda Aceh (Fig. 16c, Table 10).

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 damage. 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 ds3-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 Risi et al. (2017) 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 two-layer 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.

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.