A New Approximate Method for Quantifying Tsunami Maximum Inundation Height Probability
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Abstract
Regional and global tsunami hazard analysis requires simplified and efficient methods for estimating the tsunami inundation height and its related uncertainty. One such approach is the amplification factor (AF) method. Amplification factors describe the relation between offshore wave height and the maximum inundation height, as predicted by linearized plane wave models employed for incident waves with different wave characteristics. In this study, a new amplification factor method is developed that takes into account the offshore bathymetry proximal to the coastal site. The present AFs cover the NorthEastern Atlantic and Mediterranean (NEAM) region. The model is the first general approximate model that quantifies inundation height uncertainty. Uncertainty quantification is carried out by analyzing the inundation height variability in more than 500 highresolution inundation simulations at six different coastal sites. The inundation simulations are undertaken with different earthquake sources in order to produce different wave period and polarity. We show that the probability density of the maximum inundation height can be modeled with a lognormal distribution, whose median is quite well predicted by the AF. It is further demonstrated that the associated maximum inundation height uncertainties are significant and must be accounted for in tsunami hazard analysis. The application to the recently developed TSUMAPSNEAM probabilistic tsunami hazard analysis (PTHA) is presented as a use case.
Keywords
Tsunamis inundation probabilistic tsunami hazard analysis amplification factors uncertainty quantification1 Introduction
The standard way to estimate tsunami inundation maps is to apply numerical nonlinear shallow water (NLSW) models that include dryingwetting schemes (Titov and Gonzalez 1997; LeVeque and George 2008; Løvholt et al. 2010; Dutykh et al. 2011; de la Asunción et al. 2013; Wronna et al. 2015; Macías et al. 2017). However, if the inundation needs to be quantified over large coastal stretches (e.g. country scale or larger), NLSW inundation simulations are most often not feasible. This is due either to the large computational cost or to the lack of required highresolution topobathymetric models. This issue is particularly relevant for probabilistic tsunami hazard analysis (PTHA, Geist and Parsons 2006; Grezio et al. 2017), in which a large number of scenario simulations must be carried out to take into account the natural potential variability of the tsunami sources (e.g., Selva et al. 2016). Intermediate methods can limit the number of scenario simulations for local PTHA (Gonzalez et al. 2009; Lorito et al. 2015), but they still require highresolution coastal bathymetric and topographical data that are typically not available over large geographical scales. We still need simpler methods for estimating tsunami inundation heights in PTHA, particularly over large regions.
A faster, yet more approximate, method than NLSW is to relate the nearshore surface elevation to the water elevation at the shoreline, which then acts as an approximation for the maximum inundation height (MIH). We refer to this method as the amplification factor (AF) method. Combined with results from offshore tsunami simulations, AF methods can be used to estimate the tsunami maximum inundation height at a coastal location.
The first version of the AF method (Løvholt et al. 2012, 2015) considered a set of only a few amplification factors based on idealized and overly simplified bathymetric profiles. The idea was that, in principle, these could be fitted to a coastline at any given place. A related AF model that also enables a fast calculation of tsunami heights for tsunami early warning was recently developed by Gailler et al. (2018). In contrast to the method of Løvholt et al. (2012), which is the basis for the model in this paper, the method developed by Gailler et al. (2018) needs to be calibrated with detailed bathymetric/topographic information and with highresolution NLSW simulations for each site where it is applied.
For practical purposes, and due to a lack of alternatives, the application of Green’s law amplification to a given reference water depth (often 0.5 m) has previously been employed to estimate tsunami inundation heights from offshore points as an alternative to the AF method (Sørensen et al. 2012; Brizuela et al. 2014; Horspool et al. 2014; Hébert and Schindelé 2015). However, as pointed out by Hébert and Schindelé (2015) and Davies et al. (2018), the choice of the reference water depth in Green’s law makes the method subjective. In this paper, it will be demonstrated that the AF method represents a more accurate method for estimating the coastal amplification than Green’s law, because local factors influencing the runup, such as local bathymetry, wave polarity, and wave length, enable a more accurate approach. In addition, a series of new facilities, which will be described in a moment, makes the new AF method more suitable for use in regional PTHA than other previous approximate methods.
Previous attempts to quantify AFrelated tsunami inundation height uncertainty and bias are limited to a few studies (Davies et al. 2018, and references therein). Davies et al. (2018) estimated the AF uncertainty and bias by comparing offshore tsunami simulations combined with AF analysis against observed runup for four tsunamigenic earthquakes (Chile, 1960; Alaska, 1964; Indian Ocean, 2004; Tohoku, 2011) using the previous AF version based on idealized and much simplified bathymetric transects. The authors found that tsunami simulations combined with the AF gave a relatively small bias compared to observed runup heights, whereas a large lognormal root mean square (\(\sigma \sim 0.9\)) was found. However, their analysis merged uncertainty from the source with the one associated with the AF method. Therefore, their uncertainties combined the variability due to tsunami source complexity (e.g., heterogeneous coseismic slip or other simplifications in seismic source and tsunami generation) and propagation modeling, and possibly even other sources such as landslides, in addition to the inherent variability in the inundation process we try to quantify here. Thus, their variability due to the inundation process was not separated from the variability due to the source and tsunami propagation. In this study, we quantify the uncertainty of the inundation process separately by employing detailed highresolution inundation simulations, tying the observed uncertainties from the inundation simulations directly to the AF method.
This paper is organized as follows: Section 2 describes the computation of the local amplification factors. In Sect. 3, we describe the inundation simulations for six different test sites in southern Europe. In Sect. 4, we estimate and analyze the bias and uncertainty of the method by comparing the AF performance at the test sites with the NLSW inundation models. In Sect. 5, we demonstrate with an example how the results can be used in a PTHA developed for production of tsunami hazard maps in the NorthEastern Atlantic, Mediterranean and connected seas (NEAM) region.
2 Amplification Factors Based on Local Bathymetric Transects
2.1 New Amplification Factors Derived from Transect Simulations
The new method for computing the amplification factors starts with defining the AF points of interest (POIs). The POIs are located at longitudelatitude coordinates inside the computational domain employed for the 2HD tsunami propagation simulations. In this study, amplification factors are computed for almost evenly spaced POIs located every 20 km along the shoreline. For each POI along the 50 m isobath, about 40 depth profiles (exemplified in the upper panels of Figs. 3, 4, 5) are extracted approximately normal to the shoreline, each with a distance of 1 km apart, as described in Appendix 1. However, in the case of complex coastal geometries, deviation from the normal incidence may take place.
The amplification factors are computed along seven subjectively selected profiles out of the 40 profiles. The subjective selection was made to enable computational feasibility. An initial wave of 1 m height in deep water, shaped as a singleperiod sinusoidal wave pulse (Nshaped wave), is fed over the deep water boundary of the model (see Fig. 1). Linear shallow water (LSW) simulations were carried out for all profiles, both for leading trough or leading peak and for a set of wave periods (120, 300, 600, 1000, 1800, and 3600 s).
Maximum surface elevations from the LSW simulation at 50 m depth and at the shoreline (0 m depth) are extracted. For each profile, we compute the amplification factor, which is defined as the ratio of the height between the latter and the former (i.e. \(A_1\) and \(A_2\) in the upper panel of Fig. 1). We use the median value of the seven amplification factors to avoid unrealistic alongshore fluctuations. Examples of amplification factors and related median values as a function of the incident wave periods are shown in the lower panels of Figs. 3, 4, 5. AF values for all combinations of wave polarities and wave periods are stored in lookup tables. As an example, amplification factors for a leading trough polarity and wave period of 600 s for the Mediterranean and Black Sea are depicted in Fig. 6.
As expected, we do observe that the shorter waves and waves with leading trough are amplified more than the longer waves and waves with a leading peak (Figs. 3, 4, 5, lower panels). We also observe that the amplification factors decrease as the period increases; in some cases the behavior appears different and the amplification factors are locally higher with respect to lower periods such as in Fig. 5 (see around period of 1000 s). This stems from interference between incident and reflected waves in the linear shallow water simulations: when the travel time from the POI to the shoreline and back to the POI for the leading trough matches the arrival of the trailing peak here, the measured wave height at the POI is low (trough is canceling the peak) and the amplification factor is high.
3 Local Inundation Simulations for Bias and Uncertainty Estimation
Comparisons between the AF and the NLSW models are undertaken at the six test sites for which a suitable and sufficiently detailed DEM existed. The test sites include one in the Atlantic Ocean, namely Sines, Portugal, with the remainder in the Mediterranean: Colonia Sant Jordi (Mallorca) and SE Iberia in Spain, Siracusa and the Catania plain in Italy, and Heraklion on the island of Crete in Greece (Fig. 8). For each test site, we use 96 earthquake sources with varying magnitude, strike, dip, and rake in order to explore a variety of situations as far as the source mechanism is concerned and, as a consequence, the wave period and wave polarity variability. In particular, we use six different moment magnitudes (7.1, 7.5, 8.1, 8.5, 8.8, 9.0), four strike angles (22.5, 112.5, 202.5, 292.5), and four pairs of dip/rake angles (10/90, 30/90, 50/270, 70/270). Empirical earthquake scaling laws (Strasser et al. 2010) are then employed to define the fault size (length and width) for each earthquake source. The average slip (D) for each scenario is established by considering the classical relationship \(M = \mu AD\), where M is the seismic moment, \(\mu \) is the shear modulus (30 GPa), and A is the rupture area of the seismic source.
We use the NLSW TsunamiHySEA numerical code (see, e.g., de la Asunción et al. 2013; Macías et al. 2017) to simulate the distribution of the tsunami inundation height over the extent of the different sites. All offshore tsunami simulations are conducted on regular grids with a spatial resolution of 30 arcsec. The NLSW models additionally use nested grids to simulate detailed local inundation at test sites. The resolution of the finest grid is about 10 m at all locations. The Manningfriction (n) is set to \(n=0.03\) in all simulations. We note that while \(n=0.03\) is typically a friction value for overland flow (e.g. Kaiser et al. 2011), this value will be sitedependent and vary spatially as well, and hence using a constant value represents a simplification.
For each test site and source scenario we apply the new AF method to quantify MIHs. As prescribed by the method, MIH is quantified by multiplying the maximum offshore surface elevation retrieved at the POI, taking wave characteristics retrieved from time series at the POI, with specific AF retrieved from the relevant lookup table. The method for selecting the maximum offshore surface elevations and related wave periods and polarities at the POI are described in Appendix 2.
4 Maximum Inundation Height Uncertainty and Bias
As explained in Sect. 3, the comparison between the AF method and corresponding NLSW simulations is made from six locations, each site subject to about 96 earthquakeinduced tsunami scenario simulations. The new AFs depend on the coastal slope configuration, the tsunami wave period, and the tsunami wave polarity, as simulated in a single POI located offshore along the 50 m isobath. As the AFs are based on a linear method, they are amplitudeindependent. For each scenario, we extracted the largest MIHs from the NLSW simulations for a given coastline location. MIH maxima were extracted in either the north–south or east–west direction, line by line, for each grid cell along the preferred coordinate axis. That is, if the shoreline is oriented mostly east–west for a given site, we search along lines oriented in the north–south direction, while if the shoreline is predominantly oriented more north–south for another site, we search along lines oriented in the east–west direction. In some special cases, the height of the terrain landside of the shoreline is too high and too steep to be inundated. In these situations, we also include values of maximum surface elevation for a small distance seaside from the shoreline (at least one cell away).
4.1 Inundation Height Statistics for the Full Parameter Set
To gain a deeper and more quantitative insight, we carried out a statistical analysis based on the MIH PDFs, biases, and lognormal uncertainties. In the statistical analysis, both the merged results and statistics isolating the effects of different factors were analyzed.
The bias from various model runs represents just one component of the variability. In addition to the variability of the bias, as shown in Fig. 9, the tsunami inundation height does vary spatially across each site. This variability is related only to the properties of the NLSW simulations at the site, and not to the AF method as such. However, when applying the AF method, we must take this uncertainty into account. For the examples shown in Fig. 9, we obtain lognormal standard deviations \(\sigma \) of 0.26, 0.43, and 0.23, respectively. The distribution of all the \(\sigma \) values is shown in Fig. 12, displaying a mean value of \(\bar{\sigma }\) = 0.25 and standard error of \(\sigma _\sigma \) = 0.41. By visual inspection, the distribution of local standard deviations seems to follow a lognormal distribution relatively closely, despite the data displaying a more peaky behavior than the fitted lognormal PDF. It is worth noting that of the three sources of uncertainty contributing to the AF (bias uncertainty \(\sigma _\epsilon \), mean local uncertainty \(\bar{\sigma }\), and standard error \(\sigma _\sigma \)), the standard error \(\sigma _\sigma \) is the largest contributor. In this study, \(\sigma _\sigma \) reflects the variability owing to different sites, tsunami sources, wave polarity, wave period, etc. In the next subsection, we will demonstrate with examples how bias and variability differ.
Taking the root mean sum of the different \(\sigma \)’s (\(\sigma _\epsilon \) = 0.28, \(\bar{\sigma }\) = 0.25, and \(\sigma _\sigma \) = 0.41) for the full parameter set simulations, we arrive at a root mean square sum of 0.55 due to the inundation process. It is noted that Davies et al. (2018) found an overall uncertainty of 0.92 when comparing PTHA results with observations from past events. If the uncertainty found in this paper is representative for the inundation process, the overall uncertainty of 0.92 in Davies et al. (2018) would be the sum of the inundation uncertainty and a residual uncertainty due to other sources. By removing the inundation uncertainty from the uncertainty obtained by Davies et al. (2018), we obtain a residual uncertainty of 0.73, from which we may speculate that the variability due to the inundation would then contribute to just less than half of the total uncertainty found by Davies et al. (2018). In this context, we note that Geist (2002) reports coefficients of variation in the range of 0.25–0.35 due to variable slip, which is thought to be one of the governing factors in inundation height uncertainty. These aspects deserves to be better investigated. On the other hand, Davies et al. (2018) used idealized AF profiles, which are most likely have a larger overall bias due to a more crude bathymetric representation. In any case, the present analysis shows that the variability in inundation is a strong overall component in the inundation height uncertainty, which needs to be accounted for in a regional PTHA method.
4.2 Inundation Height Statistics for Parameter Subsets
The bias and variability are further examined by carrying out the statistics for certain sets of parameters (e.g. only for positive wave polarity, only for shortperiod waves, for an individual site).
Third, we examine the bias and local variability for three sites (Catania, Heraklion, and SE Iberia) in Fig. 14. Strong negative average biases (AH underestimation) are shown for Catania and Heraklion, whereas a slight positive bias (AH overestimation) is shown for SE Iberia. We further see that the three sites have pronouncedly different spreads with respect to both the bias and local variability, with the Heraklion site being clearly more heterogeneous, and SE Iberia the site with the smallest variability. This is also reflected in the p values from the KS tests using the bias distributions as input, giving lowest values for Heraklion (\(1\%\)) and highest values for SE Iberia (\(61\%\)). It is noted that from a practical point of view, separating the variability and bias from different locations is not feasible, as we cannot perform NLSW simulations at all relevant locations. Consequently, merging uncertainties from different sites would increase the overall MIH uncertainty in a PTHA analysis.
5 Uncertainty Treatment in PTHA
Summarizing from previous sections, the AF depends on both the characteristics of local bathymetry (coastal slope configuration) and the characteristics of the tsunami scenario (dominant period and polarity), as simulated in a single POI located offshore. Multiplying the AF to the tsunami simulated in this POI, we obtain an approximate median MIH for the nearby coast due to the considered individual tsunami scenario.
For the ith tsunami scenario in the jth POI, we indicate the maximum offshore elevation at the POI and the corresponding MIH value with \(\eta _{ij}\) and \(\text {MIH}_{ij}\), respectively. \(\text {MIH}_{ij}\) corresponds, after correction for a generally quite small bias, to the median value of the distribution of MIH values that the ith tsunami scenario causes in stripes of land perpendicular to the coast and located around the projection to the coast and inland of the jth POI.
In fact, as demonstrated by the NLSW simulations, this distribution can be approximated by a lognormal distribution for MIH, with a median that is well approximated by \(MIH_{ij}\) and a reasonably small \(\sigma \) (the standard deviation of the natural logarithm of MIH). These parameters are different for different scenarios and POIs. In particular, for the ith tsunami scenario \(\eta _{ij}\) at the jth POI, we indicated these parameters with \(\mu _{ij}\) and \(\sigma _{ij}\), and, considering the AF \(A_{ij}\) and the normalized bias \(\epsilon _{ij}\) (compare Eq. (2)), we have \(\mu _{ij}=\) ln\((A_{ij}\)\(\eta _{ij}\)/(1+ \(\epsilon _{ij}\))). Among these parameters, the AF \(A_{ij}\) and \(\eta _{ij}\) depend on both the coastal slope around the POI and the tsunami scenario. Instead, the parameters \(\sigma _{ij}\) and \(\epsilon _{ij}\) depend primarily on the local coastal configuration around the jth POI and can be estimated only with an NLSW simulation of the ith tsunami scenario in the jth location.
This curve can be interpreted as the hazard curve for the hazard intensity MIH of one randomly selected point within the stretch of coastline near the jth POI, conditional to the occurrence of the ith tsunami scenario with best guess \(\text {MIH} = \text {MIH}_{ij} = A_{ij} \eta _{ij} /(1+ \epsilon _{ij}\)).

\(A_{ij}\) for a set of different tsunami scenarios has been extracted as shown in previous sections.

The time history of the tsunami and the local \(\eta _{ij}\) may be estimated from linear simulations of the propagation in deep sea of each individual tsunami scenario.

In regional PTHA, these simulations are barely affordable; therefore, both tsunami time history and \(\eta _{ij}\) are here approximated as a linear combination of unit sources (Molinari et al. 2016). Here we introduce \(z_{ij}\) as a further source of epistemic uncertainty, which is introduced by combining unit sources rather than simulating all of them as a single source. The treatment of this uncertainty is elaborated below.

For the estimation of the parameters \(\epsilon _{ij}\) and \(\sigma _{ij}\), we assume that combining all the information from selected test sites reasonably approximates the variability of these parameters for all the POIs.

The correction \(\delta ^*\) from the empirical distribution \(\delta =(z\eta )/\eta \). This is the distribution of the relative error of the approximated \(\eta \) values due to the use of unit sources (as defined in Molinari et al. (2016), see their Figure 4d). This distribution is obtained by aggregation of a very large number of NLSW simulations with variable source and site, over a quite large range of tsunami intensities (up to \(> 10\,m\)), and we may assume that they reasonably approximate this uncertainty source for PTHA purposes.

The parameters \(\epsilon ^*\) and \(\sigma ^*\) from the empirical distributions of b and \(\sigma \) of Fig. 11, upper panel, and Fig. 12.
Repeating this procedure N (e.g. 1000) times, we obtain N alternative conditional hazard curves, representing the sampled epistemic uncertainty in the conditional hazard. To increase the computational efficiency of this estimation, since {\(\delta ^*\),\(\epsilon ^*\),\(\sigma ^*\)} do not depend on the selected POI or tsunami scenario, they can be precomputed for a discrete number of possible MIH values, to be subsequently applied to the different scenarios and POIs.
6 Conclusion
In this paper, a new version of the amplification factor (AF) method for tsunami inundation height estimation has been presented. The new version represents a major upgrade compared to previous AF models, by taking into account the local bathymetry and the uncertainty introduced by using an approximated approach to the inundation process estimation through NLSW models. Because of the new AF capabilities, we can for the first time properly estimate the probability distribution of the maximum inundation height for PTHA. Another step forward is that the resulting MIHs emerging from the model are anchored towards a comprehensive set of more accurate reference models. The new method is implemented for the whole NEAM region, for the purpose of probabilistic tsunami hazard analysis for the TSUMAPSNEAM project.
The AF method is set up to estimate the median value of the maximum inundation height at an area along the coast, when combined with tsunami propagation models. By linking the AF results with local inundation NLSW simulation results, we have also been able to add the inundation uncertainty into the AF model. Statistical analysis of the different sources of uncertainty is performed through 576 local inundation simulations for different earthquake sources and coastal sites. For example, the quantified bias indicates that the AF method may yield a slight underestimation (5%) of the median inundation height at a given site. However, the AF method yields much more accurate results than the Green’s law method with respect to both a smaller overall bias and reduced uncertainty.
By comparing results from the inundation simulations with the AF results, we find mainly three sources of uncertainty related to employing the AF method, namely variability in positive and negative biases, local variability in the maximum inundation height (owing to the fact that AF is nonlocal), and the standard error, i.e. the uncertainty of the variability (because different sites and sources produce different inundation variability). Overall, the results show that the inundation variability is strong and needs to be accounted for in a regional PTHA method.
When the inundation statistics is conducted for smaller subsets of the simulation data, for instance for a single site, statistical distributions tend to be rather heterogeneous. When the results are aggregated, more welldefined normal distributions (for the bias) and lognormal distributions (for the local variability) are obtained. Taking into account that the AF method is designed for regional applications, at present state, merging uncertainties from different sites and sources into a single uncertainty seems to be the most robust option, although this increases the overall uncertainty in a hazard estimate. However, by including many more NLSW simulations in the future, for example covering more sites and sources, we would expect to improve runup estimation capabilities and to reduce related uncertainties, for instance by better taking into account individual bias factors.
Hence, in its present form, the AF method is primarily set up to assist regional hazard quantification, and cannot be used for estimation of maximum runup heights or as a replacement for local detailed inundation maps. For correct use, both the uncertainty and the bias associated with the method and with the specific hazard metric need to be taken into account.
Notes
Acknowledgements
Our warmest thanks go to Roberto Basili, the TSUMAPSNEAM coordinator, who created a pleasant collaborative environment. Test sites used in the present work were part of the European FP7 project ASTARTE (www.astarteproject.eu). For this purpose, we thank the ASTARTE participants, Galderic Lastras, Miquel Canals, and Gerassimos Papadopoulos, for providing input data. Topobathymetric profile extraction and analysis was performed using ArcGIS (www.esri.com). PDF analysis was performed using MATLAB fitting tools (www.mathworks.com/). Figures in the manuscript were produced using GMT (gmt.soest.hawaii.edu), Matplotlib (https://matplotlib.org) and MATLAB. The new amplification factors (i.e. the data emerging from the work leading to this paper) can be found at https://github.com/sylfest/Ampfactors.git. This work received financial support through the EU ASTARTE project within the FP7ENV2013 6.43 Grant 603839, and through the TSUMAPSNEAM project, cofinanced by the European Union Civil Protection Mechanism, Agreement Number ECHO/ SUB/2015/718568/PREV26.
Supplementary material
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