It is necessary to evaluate aggregate damage probability to multiple buildings when performing probabilistic risk assessment for the buildings. The purpose of this study is to demonstrate a method of tsunami hazard and risk assessment for two buildings far away from each other, using copulas of tsunami hazards that consider the nonlinear spatial correlation of tsunami wave heights. First, we simulated the wave heights considering uncertainty by varying the slip amount and fault depths. The frequency distributions of the wave heights were evaluated via the response surface method. Based on the distributions and numerically simulated wave heights, we estimated the optimal copula via maximum likelihood estimation. Subsequently, we evaluated the joint distributions of the wave heights and the aggregate damage probabilities via the marginal distributions and the estimated copulas. As a result, the aggregate damage probability of the 99th percentile value was approximately 1.0 % higher and the maximum value was approximately 3.0 % higher while considering the wave height correlation. We clearly showed the usefulness of copula modeling considering the wave height correlation in evaluating the probabilistic risk of multiple buildings. We only demonstrated the risk evaluation method for two buildings, but the effect of the wave height correlation on the results is expected to increase if more points are targeted.

Probabilistic hazard and risk assessment methods of disasters are developed mainly in the field of nuclear safety focused on countermeasures relative to severe accidents at nuclear power plants. Among them, a variety of probabilistic tsunami hazard assessment (PTHA) and probabilistic tsunami risk assessment (PTRA) methods for tsunami disasters have been rapidly developed since the 2000s (e.g., Geist and Parsons, 2006; Annaka et al., 2007; González et al., 2009; Thio et al., 2010; Løvholt et al., 2012, 2015; Goda et al., 2014; Fukutani et al., 2015; Park and Cox, 2016; De Risi and Goda, 2017; Grezio et al., 2017; Davies et al., 2018). The main purpose of a PTHA is to assess the likelihood of a given measure of tsunami hazard metrics (e.g., maximum tsunami wave height) being exceeded at a particular location within a given time period. The most basic outcome of such an analysis is typically expressed as a hazard curve, which shows the exceedance level of the hazard metric with the probability. This is often expressed as a rate of exceedance per year. A PTHA can be expanded to a PTRA by combining hazard assessment with loss evaluation of a target. Several studies have proposed a method of PTRA for an individual site in a local area. Detailed risk assessment is undoubtedly important in terms of grasping the risk of exposing assets located in a local area.

However, probabilistic risk evaluation methods are also utilized in cases to evaluate risks for multiple buildings (e.g., Kleindorfer and Kunreuther, 1999; Chang et al., 2000; Grossi and Kunreuther, 2005; Goda and Hong, 2008; Salgado-Gálvez et al., 2014; Scheingraber and Käser, 2019). With respect to businesses that own a building portfolio, including factories and offices over a wide area, it is extremely important in risk-based management decisions to evaluate the detailed risks posed by the building portfolio. A portfolio means a collection of assets held by an institution or a private individual. By quantitatively assessing the risks posed by the building portfolio, for example, it is possible to identify assets held that have a large impact on the overall risk and to compare the amount of risk held over time, which leads to support for decision-makers.

When evaluating physical risks for multiple buildings over a wide area, it is necessary to evaluate the aggregate risk for the buildings that are located at a distance. In these types of cases, it is necessary to evaluate the risk by considering the spatial correlation of hazards. For example, let us consider assessing the risk of two buildings located at two sites. When the positive correlation of hazards between two sites is strong, the hazard at one site tends to be large if the hazard at another site is large. In this case, the hazards at the two target sites both increase, and as a result, the aggregate risk for the two buildings considering the hazard correlation increases. Conversely, when the positive correlation of hazards is small, the hazard at one site is not necessarily large, even if the hazard at another site is large. In this case, compared to the former case, the hazards at the two target sites are smaller, and as a result, the aggregate risk for the two buildings is smaller if we assume that the vulnerability of the two buildings is equal. Therefore, analyses that do not consider the spatial correlation of hazards involve the risk of underestimating the risk over a wide area. It is clear that the difference of aggregate risk between two cases becomes more prominent as the number of target sites increases. Analyses that consider the spatial correlation of hazards are relatively advanced in the field of earthquake hazard and risk assessment (e.g., Boore et al., 2003; Wang and Takada, 2005; Park et al., 2007) albeit insufficient in the field of tsunami hazard and risk assessment. Analyses that consider the hazard correlation using copulas are used in hydrological/earthquake modeling (e.g., Goda and Ren, 2010; Goda and Tesfamariam, 2015; Salvadori et al., 2016) although there is a paucity of the same in tsunami modeling.

In this study, we assume the occurrence of a large earthquake in the Sagami Trough in Japan that significantly affects the metropolitan area and evaluate the tsunami risk of two buildings located at distant locations by considering the spatial correlation of the tsunami wave height between the two sites. The objective of this study involves evaluating the frequency distribution of the tsunami height via the response surface method and evaluating the spatial correlation of the tsunami heights and damages by using various copulas. Specifically, we analyze the frequency distribution (marginal distribution) of tsunami height via the response surface method and target two steel buildings located at Oiso and Miura along the Sagami Bay, Kanagawa Prefecture, in Japan. Subsequently, we derive the joint distribution of tsunami wave heights between two sites by using various copulas and the marginal distributions, convert it to the joint distribution of damage by applying a damage function, and evaluate the expected value of the aggregate damage probability for the target buildings. Finally, we confirm the extent to which the expected value of the aggregate damage probability fluctuates in a case where the spatial correlation of tsunami wave height is considered and a case where it is not considered.

Section 2 provides an outline of the response surface method and tsunami hazard and risk assessment method for multiple buildings using copulas. Section 3 describes a case where the proposed method is applied to the Sagami Trough area. The final conclusions are discussed in Sect. 4.

Figure 1 shows a flowchart of tsunami hazard and risk assessment considering the correlation of tsunami wave heights in this study. Herein, the risk assessment target points only correspond to two points: Oiso and Miura, Kanagawa Prefecture, in Japan. Figure 2 shows the location of these points. First, we simulate the tsunami wave heights considering the uncertainty at the target sites by numerical tsunami simulations via nonlinear long-wave equations. Based on this, we construct a response surface and apply probability distributions to obtain a frequency distribution of tsunami wave heights. This distribution becomes a marginal distribution for a joint distribution of tsunami wave heights of two target points. Separately, we estimate appropriate copula via maximum likelihood estimation from the simulation results of the tsunami wave height considering uncertainty. Subsequently, we obtain a joint distribution of tsunami wave heights from the estimated copula and the marginal distributions of tsunami wave height. Furthermore, we obtain a joint distribution of damage probabilities by applying the tsunami damage function.

Flowchart of probabilistic tsunami hazard and risk assessment considering the spatial correlation of tsunami wave height. Numbers in the parentheses indicate the section numbers escribed.

The outline of the response surface method and copula modeling used in this
study is explained below. The response surface method is a statistical
combination method to determine an optimum solution using the lowest number
of measurement data possible. The basic idea is based on a reliability-based
design scheme developed in the research field of geomechanics (e.g., Honjo,
2011). Generally, the response surface model is given by Eq. (1) as follows:

This response surface method has an advantage that the probability distribution of the objective variable can be easily evaluated by applying an appropriate probability distribution to the explanatory variable and performing a Monte Carlo simulation. Although the tsunami numerical simulation considering uncertainty usually has a high calculation cost to conduct vast numbers of simulation cases, it is possible to significantly reduce the simulation cost by using the response surface method.

A simple synthetic example of a copula in a bivariate case.

The foundation of the copula theory corresponds to the Sklar theorem (Sklar,
1959). A copula is a multivariate distribution whose marginals are all
uniform over [0, 1]. Given this in combination with the fact that any
continuous random variable can be transformed to be uniform over [0, 1] by
its probability integral transformation, copulas are used to separately
provide multivariate dependence structure from the marginal distributions.
Let

In this study, we use the bivariate case as the tsunami wave height at two target
points and model the correlation using a copula. The linear correlation
coefficient (Pearson's correlation coefficient) is an index that captures
the linear relation between variables and essentially cannot express the
dependency between variables that are not in linear relation. Conversely,
the copula is a function that expresses the correlation based on the order
of the data of each variable rather than the data themselves. The order of the
data is expressed by Kendall's

Bivariate copula, parameter vectors, and Kendall's

In this chapter, we demonstrate a case study where the hazard and risk assessment method described in the previous chapter is applied for two buildings located on the coast of Sagami Bay, Kanagawa Prefecture, in Japan. Section 3.1 shows the assessment target points, Sect. 3.2 shows the tsunami numerical simulation considering uncertainties, Sect. 3.3 constructs the response surface, Sect. 3.4 shows the modeling of tsunami wave height correlation using copulas, and Sect. 3.5 shows the results of the evaluation and discussion.

Figure 2a shows major subduction-zone earthquakes around the Japanese islands, namely the Sagami Trough earthquake, the Nankai Trough earthquake, and the Tohoku-type earthquake announced by NIED (2017). Figure 2b shows the located points of tsunami hazard and risk assessment targets, namely Oiso and Miura, Kanagawa Prefecture, in Japan. The Sagami Trough earthquake covers most of the Kanto region, including the target points. Oiso is located at the approximate center of Sagami Bay coast, and Miura is located at the tip of the Miura Peninsula, which is located between Tokyo Bay and Sagami Bay. We assume a steel-framed building located at these two points and evaluate the tsunami damage probability for the two buildings.

In this section, we evaluate the tsunami wave heights by considering the uncertainty at the target points.

We selected 10 earthquake occurrence sources of the moment magnitude (

Tsunami numerical simulation results (

Moment magnitude, average slip, number of faults, and area in each earthquake source of the Sagami Trough earthquake.

Figure 4b shows the calculation results of the initial water level
distribution of the tsunami using the Okada (1985) equation. The initial
water level of up to approximately

There are a total of 10 earthquake sources; thus, we implemented a total of
250 cases of tsunami numerical simulation nested in four stages of 270, 90, 30, and 10 m in the Japanese plane rectangular coordinate system IX for
each simulation and executed the simulation for 3 h from the earthquake
occurrence. As an example, Fig. 5 shows the numerical simulation results of
nine cases around Oiso and Miura in which the

Maximum tsunami wave heights simulated from the tsunami numerical simulation at Miura and Oiso and Pearson's correlation coefficients in each earthquake source.

In this section, we construct response surfaces, which indicate maximum wave height at target sites.

With respect to the results of the maximum wave height of the tsunami
numerical simulation, we regressed the response surface (Eq. 2) using the
least-squares method. The explanatory variables correspond to the fault slip
and the fault depth, and the objective variable denotes the maximum wave
height at the target sites. We performed the regression analysis based on
all combinations of four explanatory variables (2

Response surfaces at

Akaike information criterion (AIC) results of the regression analyses. The regression analyses were performed based on all combinations of four explanatory variables.

Regression coefficients of each selected response surface for each earthquake source.

Histograms of tsunami wave height simulated from the response surface at

As reported by Japan Society of Civil Engineers (2002), the estimated variation of

To ascertain the normality of the frequency distributions, we performed the
Kolmogorov–Smirnov test. Table 5 shows the results of

Kolmogorov–Smirnov test results.

In this section, we estimate appropriate copulas from the results of the tsunami numerical simulation considering uncertainties and evaluate the spatial correlation structure of tsunami wave height between two sites.

As confirmed in the previous section, despite the high linear correlation of
the frequency distribution of the tsunami wave height in Miura and Oiso, it
is observed that the normality of tsunami wave height for several sources
was not secured by the normality test. The Pearson correlation coefficient
did not accurately grasp the spatial correlation structure of tsunami wave
height, and thus we attempt modeling using a copula. Hereafter, we only
illustrate the analysis results of the earthquake source 8 (

Table 6 shows the results of estimating copulas by maximum likelihood estimation for the distribution obtained by converting the numerical simulation results over [0, 1]. We considered a copula associated with the minimum AIC and Bayesian information criterion (BIC) (Schwarz, 1978) as the best-fit copula. The BIC is more useful in selecting a correct model, while the AIC is more appropriate in finding the best model for predicting future observations. In source 8, the copula with the minimum AIC and BIC corresponded to the Frank copula. We derived the joint distribution of the tsunami wave heights considering the wave height correlation using the Frank copula and the empirical cumulative distributions obtained from the histogram of the tsunami wave height evaluated in the previous section. Figure 9 shows the Frank copula over [0, 1] with 10 000 trials, Fig. 10a and b show the empirical cumulative distributions of tsunami wave height for Oiso and Miura, and Fig. 11a shows the results considering the wave height correlation. The black points denote the results of the Monte Carlo simulation. The number of simulations is 10 000. The red points denote the results of the tsunami numerical simulation using the nonlinear long-wave equation. To compare with this result, Fig. 11b shows the results without considering the wave height correlation. We independently generated the tsunami wave height by using a uniform random number and the cumulative frequency distribution of the tsunami wave height at each site without using a copula. By considering the spatial correlation of the tsunami wave heights using copula, we performed a Monte Carlo simulation that appropriately captures the nonlinear spatial correlation of the tsunami wave height. We clearly showed the usefulness of copula modeling considering the wave height correlation.

Selected Frank copula for source 8.

Empirical cumulative distributions of tsunami wave height
(

Monte Carlo simulation results for source 8. The black points denote the results with 10 000 trials

Maximum likelihood estimation results of each copula for source 8.

Table 7 shows the result of estimating copulas under the same procedure for
other earthquake sources. In the earthquake sources targeted in this study,
four types of copula were estimated, namely the rotated Gumbel copula,
asymmetric Gumbel copula, Frank copula, and Gumbel copula. The rotated Gumbel
copula corresponds to a copula that rotates the ordinary Gumbel copula by
180

Estimated optimal copulas distributed on [0, 1]

Estimated optimal copulas, copula parameters, and Kendall's

In this section, we evaluate the joint distribution of tsunami wave heights and damage probability of target buildings for the entire area of the Sagami Trough earthquake using the occurrence probability weights of each earthquake source.

Table 8 shows the occurrence probability weights of each source of the Sagami Trough earthquake published by NIED (2017). We first determine the earthquake occurrence source via uniform random numbers using the weights and then evaluate the joint distribution of the tsunami wave heights due to the determined earthquake using the estimated copula. Figure 13 shows the results of evaluation by Monte Carlo simulation with 10 000 trials. Figure 13a shows the joint distribution of the tsunami wave heights considering the spatial correlation of the wave height, and Fig. 13b shows the results without considering the spatial correlation of the tsunami wave height. Furthermore, Fig. 13c shows the joint damage probability of two buildings that transform both axes of tsunami wave heights in Fig. 13b into the damage probability by using the damage function of the steel frame (Suppasri et al., 2013) based on the assumption that a steel building exists at the evaluation target point. Table 9 shows the average value of the aggregate damage probability of two buildings, 95th percentile value, 99th percentile value, and maximum value assuming that the two buildings exhibit the same asset value. Although the expected value of the aggregate damage probability barely changed when compared with that of the no-correlation case, the aggregate damage probability of the 99th percentile value was approximately 1.0 % higher and the maximum value was approximately 3.0 % higher when considering the hazard correlation utilizing the copulas. We clearly showed the significance of considering the spatial correlation structure of tsunami wave height in evaluating tsunami risks for a building portfolio. In this study we only demonstrated the evaluation method for two points, but the effect of the wave height correlation on the evaluation result is expected to increase if more points are targeted.

Occurrence probability weights of each source of the Sagami Trough earthquake (NIED, 2017).

Tsunami risk assessment results.

In this study, we evaluated the aggregate tsunami damage probability of two buildings located at two relatively remote locations based on the frequency distribution of the tsunami height via the response surface method and the spatial correlation of the tsunami height by using various copulas, assuming the occurrence of the Sagami Trough earthquake that significantly affects the metropolitan area in Japan. The 99th percentile value of the aggregate damage probability was approximately 1.0 % higher, and the maximum value was approximately 3.0 % higher in the evaluation considering the spatial correlation of the tsunami wave height when compared with the evaluation without considering the spatial correlation. The results clearly show the significance of considering the spatial correlation of the tsunami hazard in evaluating tsunami risks for a building portfolio and suggest that spatial correlation modeling by copulas is effective in the case wherein nonlinear correlation of the tsunami hazard exists. In addition, the response surface method used in this study significantly reduces the numerical simulation costs for probabilistic tsunami hazard assessment considering uncertainty. In this study, we only focused on the slip amount and fault depth among many tsunami hazard uncertainties, and we evaluated them using the response surface method. It has been reported that the heterogeneity of the slip distribution of the fault has a great influence on tsunami intensity. It is a future issue to evaluate these effects with a response surface method.

The evaluation result was shown for only two buildings, but when an entity evaluates the risk of assets it owns it is assumed that there will be more target sites. It is clear that as the number of target assets increases, the percentile value and maximum value of the aggregate damage of assets become more prominent. Risk assessment that does not consider the spatial correlation of wave heights will lead to the underestimation of the risks held. The basic method shown in this study can be applied even when the number of target assets increases. It is also important to avoid underestimating the assessed risk by considering the wave height correlation using a copula. It is expected that the tsunami risk assessment method for a building portfolio over a wide area as proposed in this study can be used for probabilistic tsunami risk assessment of real-estate portfolios or business continuity plans by parties such as large companies, insurance companies, and real-estate agencies.

The earthquake source parameters of the Sagami
Trough model used in this study are freely available at

YF conceived and designed the experiments, analyzed the data, and wrote the paper with assistance and input from SM, KT, TK, YO, and TK.

The authors declare that they have no conflict of interest.

We thank two reviewers who provided us valuable comments and helped improve the manuscript. This research was partially supported by funding from the International Research Institute of Disaster Science (IRIDeS) at Tohoku University.

This research has been supported by the International Research Institute of Disaster Science (IRIDeS) at Tohoku University (Tsunami mitigation research 2).

This paper was edited by Ira Didenkulova and reviewed by Elena Suleimani and one anonymous referee.