The first step in defining vulnerability is to develop fragility functions, which require spatial hazard metrics (HIMs), measured or descriptive spatial asset data (both damaged and undamaged), and asset attribute information. The HIM and asset attribute information are the two key variables when considering vulnerability, and both are considered, independently and in tandem, to define vulnerability of assets. The most common HIM is inundation depth, and the first step was to use this data to calculate fragility functions for mixed-construction assets. With the Coquimbo dataset, roads were separated into approximately 50 m sections and assigned the corresponding damage level (DL0-DL3) and inundation depth, through the watermark interpolation, at the centre of each feature (Fig. 2) using inundation depth bins of 0.25 m (0.0–0.25, 0.25–0.5 m, etc.). The total length of road (in km) for each depth bin and for each damage level was tabulated for each HIM by count and proportion (Fig. 3). Once inundation depth had been considered, other HIMs were used to define vulnerability more holistically. Fragility functions were then developed, as described in more detail in Sect. 2.3.

The Tōhoku dataset lacked spatial undamaged asset data (DL0), which is crucial in defining proportional damage probabilities. Therefore, all roads within the inundation area were extracted from OpenStreetMap (OSM; OpenStreetMap contributors, 2015) or were digitised from aerial imagery, and those which were not recorded in the MLIT data were assumed undamaged (DL0). This resulted in a complete dataset of roads and bridges exposed to the tsunami, each with an observed damage level (DL0–DL3). Figure 4 shows an example of observed damage levels for roads in the town of Ishinomaki within the study area. Tsunami inundation depths MLIT (2012) were then assigned to each road section using 1 m inundation depth bins (i.e. 0.0–1.0, 1.0–2.0 m, etc.). Larger inundation depth bins were used compared with the Illapel dataset (i.e. 1 m vs. 0.25 m), as there were greater hazard intensity values (>10 m vs. <4 m). Road length totals in each hazard intensity bin were tabulated for each damage level) by count and proportion (Fig. 5). The results of this analysis are presented in Sect. 3 as fragility functions.

Asset attribute information should include road construction type, allowing for the development of construction specific fragility functions. As this was not included in the MLIT (2012) dataset, the closest equivalent was a road-use-type category based on jurisdiction (0 – unclassified; 1 – state road; 2 – main local road; 3 – general prefectural road; 4 – municipality road; 5 – lowest-class road). These classifications were then converted to road-use-type equivalent categories (0 – unclassified; 1 – motorway, trunk, primary; 2 – secondary; 3 – residential, road; 4 – tertiary; 5 – construction, service, unsurfaced) to ensure compatibility with OSM (OpenStreetMap contributors, 2015) data (DL0). Roads digitised from satellite imagery were assumed to be in class 3. However, those that could not be classified were “unclassified” (0), which did not contribute towards the resulting fragility functions. These road-use classes link to different traffic loading levels, which inform road design; therefore these data broadly encompass differences in construction type, but some degree of overlap is assumed.

The analysis used for Tōhoku bridge vulnerability was similar to that of roads; however inundation depth was normalised to the height above the base of a bridge deck. OSM data already included bridges as a separate road attribute and so were easily integrated, and satellite imagery was used only to validate that all bridges were included. Neither bridge construction materials nor bridge deck base height above ground was available, both of which would be necessary for a higher-resolution fragility function (Horspool and Fraser, 2016; Shoji and Moriyama, 2007).

The MLIT dataset had an assigned bridge damage level between DL1 and DL3 (Table 1). All non-surveyed bridges in the inundation area were assumed to be undamaged and consequently assigned DL0. Figure 6 shows an example of observed damage levels for bridges in the town of Ishinomaki within the study area. Modelled tsunami inundation depth was assigned at the centre point of each bridge to avoid a bridge falling within multiple inundation depth bins. Since deck base height was not included in the dataset and in many cases the hazard layer did not include depths within river channels, to estimate the inundation depth above deck base height the inundation depth at the bridge abutment was used, and the assumption was made that in most cases the deck would be relatively level with the abutment, although the deck height (thickness of beams and roadway) is still not considered. Bridges in each hazard intensity bin were tabulated for each damage level by count and proportion (Fig. 5), and resulting fragility functions are presented in Sect. 3.

The asset damage probabilities for each damage level were calculated and shown against a median value within increasing HIM bins to account for lower amounts of data at a higher HIM. Following the methods of Koshimura et al. (2009) linear regression analysis was performed to develop the log-normal cumulative-distribution-function vulnerability curves. A probability P of reaching or exceeding a damage level for a given hazard intensity value is given by either Eqs. (1) or (2): 1Px=Φx-μσ,2Px=Φln⁡x-μ′σ′, where Φ is the standardised normal (log-normal) distribution function, x is the HIM (i.e.. inundation depth), and μ and σ (μ′ and σ′) are the mean and standard deviation of x (ln⁡x), respectively. Two statistical parameters of fragility function, i.e. μ and σ (μ′ andσ′), are obtained by plotting x (ln⁡x) and the inverse of Φ-1 on normal or log-normal plots and performing the least-squares fitting of this plot. Two parameters are obtained by taking the intercept (=μ or μ′) and the slope (=σ or σ′) in either Eqs. (3) or (4), depending on the result of the least-squares fitting: 3x=σΦ-1+μ,4ln⁡x=σ′Φ-1+μ′.

The resulting fragility functions from each dataset are presented in the following section, although not all of the data analysed in this study are applicable to the development of fragility functions.

The exposed roads assessed in this study perform well in general, even under the highest inundation depths. There is less than 0.2 and less than 0.3 probability of complete washout (DL3) at 15 m inundation depth for roads and bridges, respectively, in the Tōhoku dataset (Figs. 13a and 14). Roads in Coquimbo have less than 0.25 probability of complete damage (DL3) at 15 m (Fig. 13b). By comparison, a reinforced concrete building has a 0.4 probability of reaching or exceeding complete damage at the same inundation depth (Suppasri et al., 2013). All Tōhoku road damage levels are at a lower probability than that of the equivalent Tōhoku bridges. This is to be expected, as each asset has a different tsunami loading regime, with road impacts associated with scour, while bridge impacts are related to horizontal loading across piers and the superstructure as well as vertical loading across the bridge superstructure. Bridges are typically exposed to higher levels of hydrodynamic forces (both horizontal and vertical), as tsunami flows are concentrated in the channels these bridges span. Although not considered in this study, flexible bridge connections will reduce tension from tsunami loadings when compared to rigid (i.e. steel) connections. A higher flexibility in the substructure will also reduce horizontal tsunami loadings (Istrati et al., 2017; Istrati and Buckle, 2014). The Coquimbo roads are at a higher probability of damage compared with the Tōhoku roads and bridges. This is to be expected given the differing levels of construction standards in Japan in comparison to Chile. The Illapel study area did not contain any roads that could be considered equivalent in capacity to the likes of Japanese state roads, which would be given the highest design standards in terms of maximum flexibility and loading design. This will have resulted in a lower overall vulnerability for mixed-construction Tōhoku roads (Fig. 13a) when compared with the mixed-construction road vulnerability for Illapel (Fig. 13b). None of the functions have a probability of 1.0 within the parameters of the presented results (i.e. up to 15m inundation). This is a reasonable interpretation, as roads and bridges, although particularly vulnerable under certain conditions, are far more resistant to tsunami impacts than many other assets (Williams et al., 2019). As a comparison, mixed-construction buildings in the 2011 Tōhoku earthquake and tsunami had a probability of 1.0 at all damage levels when inundation exceeds 10 m (Suppasri et al., 2013). Bridge piers and abutments are designed with scour, horizontal loading and vertical loading from moving water in mind, although not specifically for tsunami forces, whereas the foundations and structures of buildings are typically not designed for this purpose, making them more vulnerable to tsunamis.

The results of the analysis on culvert locations and the associated road damage levels in Coquimbo indicate a correlation with the presence of a culvert and an increased damage level (Fig. 15). This indicates that all instances of a culvert in this event have resulted in road damage to some extent and in most cases moderate or severe damage (DL2 and DL3).

The analysis for distance from the coast, as a proxy for inundation energy, did not warrant the development of vulnerability curves (Sect. 2.2.2). However, the results (Fig. 16) show a clear trend between higher probabilities of damage occurring closer to the coastline. This may be an indicator for deteriorating wave energy (due to surface friction and gravity) but could simply be an indicator of increased inundation depths at the coast, since there is no empirical evidence of hydrodynamic forces in the Illapel event. The same was noted in a study of building vulnerability in the Illapel event (Aránguiz et al., 2018), particularly with less damage occurring beyond a wetland area and behind a raised railway ballast, when compared to those nearer the coast.

Tsunami debris transport is a function of inundation depth, inundation velocity and debris size, resulting in horizontal sorting of objects toward the inland inundation extent as larger materials fall out of suspension (Charvet et al., 2014; Evans and McGhie, 2011; Naito et al., 2014; Prasetya et al., 2012). In Coquimbo the debris-based level of service analysis is indicative of this statement, as a higher proportion of roads have debris deposited on them between 0 and 150 m from the landward extent of tsunami inundation (0 %–22 % of maximum inland inundation extent). This indicates that debris carried inland falls out of horizontal suspension prior to reaching the maximum inland extent. The results from the debris density analysis show that there is higher debris density between approximately 75.0 and 150.0 m (11 % and 22 %) from the inland inundation extent (Fig. 17). Debris density probability is consistently lower, for all levels of service, between 0.0 and 75.0 m (0 % and 11 %) from the inland inundation extent and 200.0 and 672 m (30 % and 100 %) from the inland inundation extent. SL1 has a much higher probability of occurrence than SL2 and SL3 at distances >200 m from the inland inundation extent (Fig. 17). This is consistent with previous studies and field observations where debris is consistently distributed across an inundation area during landward and seaward inundations.

Using the mixed-construction road data for exposed areas of the Tōhoku event, the different structural types are split into broader use categories, as the closest approximation of construction material, type and method, for the development of fragility functions (Fig. 18). The most resistant use category, with respect to tsunami impacts, is state roads (Fig. 18a), with all damage levels being of a lower probability than the other use categories. This is followed by main local roads and general prefectural roads (Fig. 18b and c), each with very similar probabilities of DL1 and DL3, although main local roads have a higher probability of reaching or exceeding DL3 at less than 7 m inundation depth. Main local roads (Fig. 18b) also have a higher probability of reaching or exceeding DL2 in comparison to general prefectural roads (Fig. 18c). This can be interpreted as main local roads having a certain characteristic that pushes them from DL1 to DL2 much faster than with general prefectural roads. It is likely these two classes share similar construction standards and materials. Municipality roads (Fig. 18d) have considerably higher probabilities of reaching or exceeding each DL, with a much steeper gradient when compared to road class 1, 2 and 3 (Fig. 18a, b and c). The most vulnerable roads are the lowest-class roads (Fig. 18e), which we cautiously assume here to be unsealed based on pre-event satellite imagery. Given that these roads are scarce in the mostly urban environment of the study area and the unknown nature of their construction, the data for this road class were not sufficient to classify DL1 and DL2. At 3 m of inundation depth, this road class already exceeds a probability of 0.5 of complete damage (DL3).

The previous fragility functions for the Tōhoku event, presented above, represent an average of the data for the whole of the tsunami-exposed area. This section presents the results of the analysis looking at the effects of two different topographic settings on tsunami damage to roads. An example of the difference in these coastal topographic settings is that at 2 m of inundation depth there is ∼0.09 probability of DL3 on plains, whereas this is only ∼0.05 in valleys (Fig. 19a and b). The damage probability in plains increases to ∼0.11 at 10 m inundation depth, while the damage probability is ∼0.16 for valleys. It is noted that the damage probability for the plains abruptly increases from 0 to 0.08 (at around 0.03 m), while for valleys 0.08 is not reached until 3 m.

Since it was already established in Sects. 1 and 2, road-use type (as an estimation of construction type and material) is an important factor in defining tsunami vulnerability. Therefore, the Tōhoku data are again split into different road classes to compare, in more detail, the effects of coastal topography on tsunami vulnerability. Two examples are presented below, for state roads and main local roads (Fig. 20). Road classes 3–5 are not presented here, as these did not have sufficient data to warrant fragility functions. In general, the damage probabilities for roads in the valleys are higher than those on plains for each inundation depth. However, the r2 values for coastal valleys are particularly low, so the comparisons between each coastal setting may not be entirely representative of true vulnerability.

The Illapel data are from a relatively small tsunami event in a localised area of one single coastline. This represents a small sample size but a high level of detail applied though a census-style survey. The Tōhoku data are from a considerably larger sample size, but asset characteristics provided to the authors of this study are in some cases recorded at a low resolution (e.g. the lack of recorded construction material). The quantity of data is important when developing vulnerability curves for a dataset of the Tōhoku scale. For example, fragility functions for road class 3–5 on the two coastal topographies would not have been applicably comparable to that of class 1 and 2 roads due to the different quantities of data and particularly at higher inundation depths. However the more localised data of Illapel have more consistent quantities of data across the range of inundation depths but are also limited by the overall data size. For example, the dataset did not warrant the comparison of different coastal settings, since only flat topography was represented in the study area, which is also noted by Aránguiz et al. (2018) in the context of building vulnerability. The Tōhoku survey also did not include undamaged (DL0) roads, which, as outlined in Sect. 2, were remotely sensed through this study on the assumption that any roads not included in the data were DL0. This assumption depends on a range of factors, including scope of survey, access, classification criteria and the accuracy of shapefiles used to classify DL0. DL0 is more likely over-represented in this study than under-represented.

The Tōhoku data represent not only the effects of tsunami hazards but also those from seismically induced shaking, including soil instability. Although in most cases the observed damage will be characteristic of tsunami impacts, some assets may have been initially weakened by seismically induced hazards, including shaking, liquefaction, landslides, lateral spreading and differential settlement. This can be interpreted as the fragility functions potentially over-estimating the actual vulnerability of Japanese transportation assets to tsunamis.

As with any field survey data, there is inherent bias in each individual surveyor's assignment of damage levels. With respect to the Illapel data collection this was controlled to some extent by using one consistent survey team member for each asset recorded. It is more difficult to evaluate the Tōhoku dataset in this regard, but it is reasonable to assume that the MLIT survey team grappled with and attempted to mitigate similar issues. However, it is reasonable to note that a subjective bias is possible with both datasets, particularly when comparing with equivalent damage classifications in other tsunami events. The methods used to spatially define the Tōhoku dataset (i.e. 50 m sections of road) are not as applicable with smaller datasets, such as with the Illapel event. Since some of the DL3 road washouts were smaller than 50 m in Coquimbo, there is potentially an over-representation of this damage level and potentially even DL1 and DL2. This means that the Illapel curves may be an over-estimation in terms of damage. In addition, some of the worst-damaged roads had already begun repair works and may have been over-represented in the survey.

The results of this study are also limited, to a certain extent, by the methodology used to fit the impact data to fragility curves. Some curves overlap at the lower hazard intensities (e.g. Fig. 14), since the data are treated nominally when fitting curves. This could be addressed by adopting a cumulative link model which fits raw asset impact and hazard intensity data to fragility curves simultaneously (Lallemant et al., 2015).

Information on Japanese culvert locations was not available to the authors of this study. It would have been useful to validate the positive relationship between culverts and road damage in a tsunami against that identified in the Coquimbo case study. We note this may be a fruitful future study. The results for increased road vulnerability associated with the presence of a culvert may be under-represented. The field survey was thorough in its collection of data; however, if culverts and outfall pipes were covered with debris or sediment, they would not have been recorded. Similarly, regarding the classifications for levels of service for debris-affected roads in Coquimbo, the drone footage used covered approximately 90 % of the inundation zone. Therefore, some areas may be under-represented for debris deposition as a result.

Another limitation of using another team's survey data is the assumptions made around asset classifications. In the road-use-type fragility functions, road class 1 shows very low vulnerability to inundation depth. This would include Japan's most highly engineered road assets, implying higher construction standards compared with other road-use classes. Road classes 2, 3 and 4 trend very similarly and likely share a similar spread of road construction standards. These also show considerably lower vulnerability to inundation depth than that of class 5 roads. Class 5 roads show high vulnerability at even low inundation depths. This class likely includes roads highly susceptible to erosion. This suggests that at the resolution of this road classification method, there is potentially only a need for three broader classes – highly engineered, standard and low-grade. However ideally the various range of road types considered in this range of data would be separated, which is outside the scope of the present research. The subtleties of use-type classification vulnerability may be a function of geophysical setting of each road class, which could not be tested due to a lack of available high-resolution soil data.

This study presents a full analysis of empirical post-event tsunami impact data from field survey to refined fragility functions. It can therefore be used as a framework for similar analysis of transportation impact data from future tsunami events. Data from future tsunamis may also provide some degree of refinement to the results presented in this study. Future work should also consider the data collection and analysis of a range of other critical infrastructure assets, such as electricity, telecommunications, water and fuel. There is a considerable knowledge gap on tsunami impacts on infrastructure, which should be better addressed to inform risk reduction strategies.

The Tōhoku dataset of tsunami-damaged roads remains the most extensive in the world, and the vulnerability curves developed from them in this study could be even further refined through more complete data. One particular limitation addressed above is the high concentration of co-seismic hazards the roads were exposed to prior to tsunami inundation. It would be possible to eliminate some assets which were likely earthquake-damaged by using high-resolution geomorphic, soil and liquefaction hazard data from 2011, which were not available to the authors of this study. Given the results of the culvert analysis, future post-event survey data can be used to corroborate the increased vulnerability of roads associated with culverts. This can also be used to inform potential mitigation strategies to increase the resilience of roads and culverts alike. This could also be done for the Tōhoku dataset if pre-event surface drainage channel data (i.e. an indicator of culverts location) were used, which were not available to the authors of this study. Similarly, the Tōhoku dataset could be further refined by eliminating potentially undamaged roads (those covered in debris during the survey but not physically damaged). Aerial or satellite imagery could be used for this or using the observations from the Illapel debris analysis (Fig. 17) in this study to apply a proportional alteration to the dataset (e.g. removing an equivalent proportion of roads within 11 %–22 % of the inundation extent).

Future post-event tsunami surveys should include data on debris dispersal and deposition if possible. It is acknowledged that a lack of time and resources often plays a defining role in the type and quantity of data collected by survey teams, but if technology such as high-definition (HD) drone footage or rapid HD aerial photography were conducted after tsunami events, this could be combined with ground-level observations on debris. This is often not possible, as communities begin cleaning debris almost immediately after an event. In the case of Coquimbo, all roads were cleared of debris by the field team's arrival, 6 d after the event.

During the analysis, several interesting damage characteristics were potentially identified, although there were not enough data to develop robust fragility functions, particularly as the small sample size at Coquimbo reduces the ability to derive a robust statistical sample. Therefore, the observation remains qualitative and the parameters require further investigation from future events. One such observation is the potentially increased chance of road damage in Coquimbo given the presence of a culvert (91 % roads with culverts ≥ DL2). Future post-event tsunami surveys should consider the collection of data for these types of observations to validate against those presented in this study.