This study presents a framework for rapid tsunami force predictions by the application of mode-decomposition-based surrogate modeling with 2D–3D coupled numerical simulations. A limited number of large-scale numerical analyses are performed for selection scenarios with variations in fault parameters to capture the distribution tendencies of the target risk indicators. Then, the proper orthogonal decomposition (POD) is applied to the analysis results to extract the principal modes that represent the temporal and spatial characteristics of tsunami forces. A surrogate model is then constructed by a linear combination of these modes, whose coefficients are defined as functions of the selected input parameters. A numerical example is presented to demonstrate the applicability of the proposed framework to one of the tsunami-affected areas during the Great East Japan Earthquake of 2011. Combining 2D and 3D versions of the stabilized finite element method, we carry out a series of high-precision numerical analyses with different input parameters to obtain a set of time history data of the tsunami forces acting on buildings and the inundation depths. POD is applied to the data set to construct the surrogate model that is capable of providing the predictions equivalent to the simulation results almost instantaneously. Based on the acceptable accuracy of the obtained results, it was confirmed that the proposed framework is a useful tool for evaluating time-series data of hydrodynamic force acting on buildings.

In order to estimate the potential damage due to a tsunami, predictions need to consider both the global aspect, such as the scale of the inundation areas, and also the local hydrodynamic forces acting on individual houses and each type of infrastructure. In fact, the potential effects of the tsunami force have been implemented in recent design standards

It is also extremely important for adequate disaster responses, such as evacuation actions, that information about the extent of damage can be quickly ascertained. In response to this demand, numerous studies have been made on instantaneous tsunami predictions. For example, NOAA (National Oceanic and Atmospheric Administration) constructed a real-time prediction system for tsunamis based on a long-term observation database

The objective of this study is to evaluate the time series of the spatial distribution of tsunami forces acting on buildings in real time. Because a very high computational cost is required to evaluate the tsunami force over a wide area, it is generally difficult to perform such simulations in real time. Therefore, a surrogate-modeling-based prediction method is employed in this study.
Surrogate modeling has been widely accepted for uncertainty quantification and probabilistic risk assessment, such as studies using response surface

The present study proposes a framework for rapid tsunami force predictions by the application of the POD-based surrogate modeling of numerical simulations. Only a limited number of large-scale numerical analyses are performed for a selection of scenarios with various fault parameters to capture the distribution tendencies of the target risk indicators. After 2D shallow water simulations are performed, the results are used as input data for 3D flow simulations to evaluate the time histories of forces acting on buildings. Then, POD is applied to the evaluation results to extract the principal modes that represent the temporal and spatial characteristics of the forces caused by a target tsunami. A surrogate model can then be constructed by a linear combination of spatial modes whose coefficients are determined by means of the regression analysis and interpolation techniques. To demonstrate the applicability of the proposed framework, one of the tsunami-affected areas during the Great East Japan Earthquake of 2011 is targeted. Using the 2D simulation and the results obtained for different input parameters, we carry out a series of 3D numerical analyses to obtain a set of time history data of the spatial distributions of tsunami force and then construct a surrogate model capable of predicting the time variations of the spatial distributions of tsunami forces almost instantaneously. A surrogate model of the inundation depth is also constructed and compared with that of tsunami force.

The structure of this paper is as follows. Section 2 explains the flow and specific procedures of the proposed framework. In Sect. 3, the proposed framework is applied to a target city to construct the surrogate model that enables rapid predictions of tsunami forces equivalent to the 3D tsunami run-up simulations.

This section describes the flow and methodologies of the proposed framework.
In the framework proposed in this study,
we carry out a series of 2D–3D coupled tsunami analyses with selected sets of fault parameters corresponding to expected scenarios beforehand to obtain a limited number of scenario-specific simulation results.
Then, the principal spatial modes of tsunami forces are extracted from the precomputed simulation data
to construct the surrogate model for the rapid tsunami force prediction. It is to be noted that the coefficients of the modes are interpolated in the parameter space so as to be functions of fault parameters. Thus, because this study stands on the assumption that the fault parameters are given when an actual tsunami event occurs, the tsunami force prediction is made immediately using the constructed surrogate model. Figure

Flowchart of the rapid tsunami force prediction by the mode-decomposition-based surrogate model.

To collect the data necessary for the surrogate modeling explained in the previous section, a set of two sequential numerical analyses are carried out for each case with a selected set of fault parameters. A 2D tsunami analysis is first carried out to obtain the information about the tsunami wave caused by the offshore fault. Then, the time histories of the tsunami height and flow velocity on the specified boundary of the target urban area are extracted for use as input data for a 3D tsunami run-up simulation. A 3D numerical analysis is performed to elaborately evaluate the spatial and temporal distributions of risk indicators within the target area. Since the risk indicators in this study include not only the inundation depth but also the force acting on buildings, the 3D calculation requires a high-fidelity numerical analysis. In this section, only the governing equations for 2D and 3D tsunami simulations are outlined. For the details of their connection,

To numerically analyze tsunami wave propagation over a wide area, 2D shallow water simulations are commonly performed. TUNAMI-N2 (Tohoku University's Numerical Analysis Model for Investigation of Near-field Tsunamis No. 2)

Proper orthogonal decomposition (POD)

Let

Also, based on the theory of singular value decomposition, data matrix

Additionally, from the relational expression of the singular value decomposition, data vector

Once the reduced-order expressions of all the data vectors are obtained, a surrogate model can be obtained straightforwardly. We begin with identifying the relationship between the coefficient for each mode and the set of input parameters

RBF interpolation for

Fujii–Satake model ver. 8.0 (adapted from

In this study, ridge regression

In this section, the proposed method is applied to tsunamis that have occurred. The simulation and the construction of the surrogate model are based on the tsunami induced by the 2011 earthquake that occurred off the Pacific coast of Tohoku.

In this study, as previously mentioned, the calculation was performed in two steps, including a numerical analysis based on the 2011 earthquake that occurred off the Pacific coast of Tohoku using 2D analysis over a wide area and 3D run-up analysis for the target region. To construct a surrogate model that considers uncertainty, it is first necessary to decide what uncertainty should be considered. Next, we considered the dominant factors from the occurrence of the tsunami to its run-up. For example, the epicenter position and magnitude are factors that need to be considered during the earthquake. During tsunami propagation, the fault model that controls the initial waveform of the tsunami and the submarine topography data need to be included. In this study, the 2011 earthquake that occurred off the Pacific coast of Tohoku was used as a basis. That is, the relevant factors in the process of the earthquake itself can be found in the accomplishments of research conducted up to this point. In specific terms, in our study, we used a fault model (Fujii–Satake model ver. 8.0)

Illustration of the fault parameters (adapted from

To obtain the information about the tsunami wave usable for the 3D analysis, a tsunami analysis is carried out utilizing the 2D shallow water flow model. First, a wide-area 2D analysis was performed to obtain time history data for the tsunami height and flow velocity observed within the bay of the target region. Regarding the initial waveform of the tsunami,

Nested analysis regions (adapted from

Comparison of tsunami height between observational data and simulation results (borrowed from

Using the results of the 2D tsunami analysis over a wide area as the input conditions, a 3D tsunami analysis was performed to represent the tsunami run-up in the target region. As mentioned before, the method used in

Boundary between the 2D analysis and 3D analysis areas. Points A to H are used to compare the inundation depths between observational data and simulation results (© Google Maps).

Tsunami run-up was represented by the 3D analysis using SFEM on the

Bird's eye view of FE meshes at two different rates of magnification.

Snapshots of tsunami run-up obtained in 3D analysis. The white-colored area represents the inundation area.

Comparison of inundation heights between observational data and simulation results (observation data are provided by

In order to construct a surrogate model of tsunami force, we can use the hydrodynamic force acting on buildings calculated in the 3D simulation. It is, however, difficult to quantify pointwise tsunami force because the force is strongly affected by the direction of building surfaces. To avoid this problem, we consider a 2D mesh with a grid size of 10 m for evaluating the tsunami force. The tsunami force is evaluated by averaging in each of these sub-domains but not for each building in this study. An image of the mesh is shown in Fig.

An image of mesh for evaluating tsunami force.

In this study, the tsunami force acting on the buildings and the
inundation depth are used as the target risk indicators, and the
values integrated or averaged in the 10 m mesh are used as the mean
values. As the field is

Proper orthogonal decomposition is applied to construct a surrogate model for the results of 40 of the 50 cases shown in Table

Calculation cases.

Next, the data matrix is defined. First, the time-series data for a particular scenario can be defined as follows as matrix

Here, we show the results of applying proper orthogonal decomposition in relation to impact force and inundation data.

First, in relation to the extracted spatial mode, the impact force and water depth for the first mode to the third mode are shown in Fig.

Spatial modes of

The values in Fig.

The contribution rates of the impact force and water depth are shown in Fig.

Contribution rate for each risk index.

The root mean squared error for the results obtained from the numerical analysis and those of the surrogate models for each mode.

Next, by interpolating the proper orthogonal decomposition coefficient as a parameter function, the surrogate models are created. As shown in Eq. (

Here, we compare the results obtained from the numerical analysis and the results from reconstructing the original data using the constructed surrogate models. First, when comparing the time-series data, several evaluation points are set. In this study, the 10 points shown in Fig.

Evaluation points for comparing results obtained from numerical simulation and those of the surrogate models (© Google Maps).

Snapshots of the results obtained from numerical analysis and those reconstructed by using spatial modes (© Google Maps).

Comparison of time-series data obtained from numerical analysis and those reconstructed by using spatial modes.

Comparison of impulses calculated from the results of numerical analysis and the results reconstructed by using spatial modes.

Based on these results, it can be summarized that whereas errors partially occur based on the impact of mode reduction, in general, the original data are being reproduced, and it is possible to express the data as a linear combination of modes. It should also be noted that there are different tendencies between spatial distributions of tsunami force and those of inundation depth. For instance, at time step 65, high values are locally seen in the inundation depth, while that tendency is not seen in the result of tsunami force. This indicates that it is difficult to predict damage to buildings based on only inundation depth, and information on tsunami force is also required for an accurate damage prediction.

By comparing data that were not used for the construction of the surrogate models, it is possible to investigate whether the surrogate models could reproduce the numerical analysis results. A comparison of the results obtained from the numerical analysis and the surrogate models for the two scenarios of S2R3 and S5R7 is shown in Figs.

Snapshots of the results obtained from numerical analysis and those obtained from the surrogate models (S2R3) (© Google Maps).

Snapshots of the results obtained from numerical analysis and those obtained from the surrogate models (S5R7) (© Google Maps).

Comparison of time-series data obtained from the numerical analysis and the surrogate models.

Here, it is confirmed that it is possible to represent the simulation results using arbitrary parameters with the surrogate models.
However, as shown in Figs.

Next, the root mean squared error (RMSE) for the time-series data at all points is calculated for each of the physical quantities by using Eq. (

Error between the results of the numerical analysis and the results obtained from the surrogate models.

With regard to Table

Next, as in the previous section, the impulse is calculated using the time-series data of the impact forces, and a comparison of the impulse calculated from the numerical analysis results and the results obtained from the surrogate model for the 10 evaluation points is shown in Fig.

Figure

Comparison of impulses calculated from the results of
numerical analysis and those obtained from the surrogate models

Furthermore, the surrogate model has distinct advantages in terms of calculation costs. A total of 4 h were required to complete the numerical analysis of one case when using a 16 parallel computing Intel™ Xeon™ CPU E5-2667 v4 (3.20 GHz) for the 2D calculation. In the case of the 3D analysis, using an 8 node 544 core Intel™ Xeon™ Phi KNL (1.4 GHz), the calculation time was approximately 96 h. Once the model was constructed, however, the calculation using the surrogate model took only a few seconds, and because it was possible to calculate the spatiotemporal distribution of the physical quantity using arbitrary parameters, it would be possible to apply the surrogate model using the spatial modes for rapid damage prediction.

This study presents the framework for predicting time-series data of tsunami force based on the results of a high-precision numerical analysis at a low calculation cost. By performing proper orthogonal decomposition in relation to the numerical analysis results while considering uncertainty, we could extract spatial modes and express the spatial distribution of the risk indicators as a linear distribution of the modes. Additionally, by expressing the coefficients as functions of analysis parameters and time in the surrogate models, it was possible to calculate the distribution at extremely low calculation costs for certain cases for which no numerical analysis has been performed. In this study, surrogate models of the impact force and inundation depth of a tsunami running up to a target area were constructed, and it was shown that the results of the numerical analysis could be roughly represented. It is also shown that the impulse calculated from the time-series data of the tsunami force obtained from the surrogate model can sufficiently represent the numerical simulation results. These results indicate that the surrogate models can be efficiently utilized for tsunami risk assessments.

In this study, for tsunami events, only the two fault parameters of slip and rake were considered as uncertainties. However, as many uncertainties are at play in an actual tsunami event, it would be possible to integrate these parameters and perform a numerical analysis and, in the same manner, create a surrogate model that incorporates a larger range of uncertainties in more detail. Nevertheless, when increasing the types of input parameters, as the number of scenarios that need to be considered becomes immense, it is pertinent to use efficient methods for parameter sampling such as Latin hypercubes

While the mechanism shown in this paper was designed for use with tsunamis, it has potential for use in the rapid prediction of various disasters. Since numerical simulations of disasters typically involve high computational costs, it is important to consider uncertainty in advance and perform a numerical analysis before creating a surrogate model. That is, the rapid prediction developed by the surrogate model proposed in this work can be used as a tool to gauge the extent of disasters.

With regard to Eqs. (

In this study, the phase-field approach is employed to capture the complex geometries of free surfaces, such as breaking waves and flow around buildings. In this approach, the locations of free surfaces can be determined by solving the Allen–Cahn advection equation

In this study, we also apply the stabilized finite element method with
the SUPG method to the Allen–Cahn equation above so that the resulting finite element equations are given as

Source code and details of the tsunami simulation were sourced from

The supplement related to this article is available online at:

KTo contributed to analyzing the simulated data and constructing surrogate models. ST, KN and MS provided the simulated data of 2D and 3D analyses. SM, KTe, YO and YF contributed to proposing the framework and supervising the surrogate modeling. KTo, SM and KTe prepared the manuscript. HY led the discussion with practical viewpoints. All authors read and approved the final paper.

The contact author has declared that neither they nor their co-authors have any competing interests.

Publisher's note: Copernicus Publications remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

The authors would like to gratefully acknowledge the support made through Chubu Electric Power (Public Research Concerning Nuclear Power).

This paper was edited by Paolo Tarolli and reviewed by four anonymous referees.