Numerical tsunami simulation method
Based on the Tohoku-type earthquake fault model generated in Sect. 2, we calculated tsunami wave height at each point as follows. First, the points of a 10-km mesh encompassing a Tohoku-type earthquake fault zone (500 km × 200 km) were generated; then, using the fault parameters in Table 2 and the fault slip artificially generated by the CRSP model as input values, the initial displaced water height was calculated at those points using the formula of Okada (1985). Here, the slip angle was assumed to be a uniform 90°. Using the calculated initial displaced water height as input values, time integration was performed with a time interval of 0.9 s and a grid spacing of 15 s (approximately 450 m) at the longitude/latitude origin by a nonlinear long-wave equation using the TUNAMI program (Tohoku University’s Numerical Analysis Model for Investigation of Far-field Tsunamis, IUGG/IOC TIME Project 1997).
Logic tree construction
Next, to stochastically analyze the tsunami hazard, we constructed a logic tree with five branch levels (range of moment magnitude, position of asperity, earthquake return period, log normal standard deviation expressed as aleatory uncertainty, and truncated uncertainty) based on the technique of Annaka et al. (2007) (Fig. 3).
Here, we would like to clarify the definitions and handling methods regarding aleatory uncertainty and epistemic uncertainty in this study. Annaka et al. (2007) explain that aleatory uncertainty is due to the random nature of earthquake occurrence and its effects. Its nature can be determined from the variation in the ratios of observed to numerically calculated tsunami heights for historical tsunami sources (Aida 1978) in their model, which is named “Aida’s kappa (κ)”. They also stated that epistemic uncertainty is due to incomplete knowledge and data about the earthquake process. Uncertainties in various model parameters and various alternatives are treated as epistemic uncertainty using the logic-tree approach.
Aida’s kappa, corresponding to aleatory uncertainty in this study, is determined from a numerical simulation for the optimal fault models of 11 tsunamigenic earthquakes with many historical run-up data (Table 4). However, the uncertainty that can be captured by this calculation only represents the difference between tsunami observation data due to an earthquake that occurred in the past and the tsunami value estimated from the earthquake model using numerical simulation. For the reason stated above, the Japan Society of Civil Engineers (2011) mentioned that parameters, which could vary even though the earthquakes occurred in the same place, should also be considered as “variability due to changing of fault models” (aleatory variability) or “uncertainty of fault models” (epistemic uncertainty). Therefore, in this study, we considered earthquake source models having different slip distributions that have not occurred in the past, as described in the second chapter, the moment magnitude of the earthquakes, and the recurrence interval as epistemic uncertainty. In addition, we treated variability of Aida’s kappa and its truncation as epistemic uncertainty.
Table 4 Values of Aida’s kappa (κ) and logarithm of kappa (σ) for 11 historical tsunamigenic earthquakes listed by Annaka et al. (2007)
For the moment magnitude (M
w), three cases of M
w 8.9, 9.0, and 9.1 were considered. Variations in moment magnitude are obtained by multiplying the slip across the fault by a coefficient calculated from seismic moment M
o (N*m) = μDS, where μ is the fault rigidity (N/m2), S is the fault area (m2), and D is the average displacement (m) across the fault.
For the position of the asperity, we initially generated many two-dimensional distributions of slip on the fault using the CRSP model. From these, we subjectively selected just five patterns of slip distribution such that the position of asperity was dispersed throughout the entire fault. Variations in the tsunami water level accompanying the variations in slip distribution in the strike direction of the fault are greater than the variations in the tsunami water level accompanying the variations in slip distribution in the dip direction (Geist and Dmowska 1999). Therefore, we considered fault patterns in which the variations in slip distribution in the strike direction were expressed more distinctly than were the variations in the slip distribution in the dip direction of the fault. In this study, we dealt with five, three, and one pattern of slip distributions, but we will confirm the large influence on the hazard curve, as will be discussed later.
For the return period of the Tohoku-type earthquake, we used the value of 600 years employed by Fujiwara et al. (2013). In their analysis, a model of the average return period for the Tohoku earthquake was treated as the BPT distribution. In this study, we decided on a deviation of the BPT distribution as 0.3 and set an upper limit and lower limit confidence interval based on its error.
For the log-normal standard deviation (aleatory uncertainty) and truncated uncertainty, we used the values of Annaka et al. (2007), which were determined from a numerical simulation for the optimal fault models of 11 tsunamigenic earthquakes with many historical run-up data (Table 4).
The number appended to each branch of the logic tree in Fig. 3 indicates the weight of that branch. For the branches related to the range of moment magnitude and position of the asperity, the values were equally divided weights. For the weights for each branch of the recurrence interval, we set the basis value as 0.5 and both end values as 0.25 because we set the branch by considering the confidence interval. Finally, regarding the weights for the branches of the standard deviation of the log-normal distribution and truncation of the log-normal distribution, we adapted the weighting values used in the study of Annaka et al. (2007); these values were used when they used a variable fault model in their study.
Hazard curves
Based on the constructed logic tree and the simulated tsunami wave heights, we created hazard curves (Fig. 4a). Each curve corresponds to a branch of epistemic uncertainty (logic tree) with the aleatory uncertainty for each outcome shown in Fig. 4a. The method by which each hazard curve is generated is as follows. First, we can derive a probability density function of a log-normal distribution, expressed as Eq. (7), when we set a maximum tsunami height value, calculated from the tsunami numerical simulation, using the condition of each branch of the logic tree as a median value (μ) and set a logarithm of Aida’s kappa as the log-normal standard deviation (σ):
$$ f\left( x \right) = \frac{1}{{\sqrt {2\pi } \sigma x}}{ \exp }\left\{ {\frac{{ - \left( {logx - \mu } \right)^{2} }}{{2\sigma^{2} }}} \right\},\quad 0 \le {\text{x}} $$
(7)
Second, we can obtain a tsunami hazard curve by converting the probability density function into an exceedance probability distribution on the assumption of an ergodic hypothesis. The ergodic hypothesis is a statistical assumption that spatial variability is equal to temporal variability. Therefore, if we use the log-normal distribution that indicated spatial variability, we can obtain a tsunami hazard curve that shows the relationship between tsunami height and the annual probability of exceedance.
The hazard curves are drawn to correspond to one specific location. Here, for example, the hazard curves at the installation site (39.259°N, 142.097°E) of the south Iwate offshore GPS wave gauge set up by the Ministry of Land, Infrastructure, Transport and Tourism (MLIT) in April 2007 are shown. The same number of hazard curves as the number of branches of the logic tree (3 × 5 × 3 × 4 × 2 = 360) was created.
From these 360 hazard curves, statistical processing was performed in consideration of branch weight. Annual probability of exceedance at a certain tsunami wave height and cumulative branch weight are shown in Fig. 4b. These were calculated in the same manner as the tsunami wave height from 0.0 to 0.1 m, and annual probability of exceedance was determined at the points of 95, 50, and 5 % cumulative weight. From this, the relationship between tsunami wave height and annual probability of exceedance was re-drawn, resulting in the 95, 50, and 5 % fractile curves shown in Fig. 4c. The fractile curves simultaneously include epistemic uncertainty and aleatory uncertainty in earthquake occurrence and tsunami propagation. The analysis of this hazard curve, which includes the various types of uncertainty, enables a stochastic interpretation of a tsunami hazard. Additionally, a curve produced by simply averaging the annual probability of exceedance without considering branch weight was also generated as a simple average curve. We note that the tsunami hazard curves presented in this study are actually the conditional hazard exceedance curves because only 3.11 Tohoku-type earthquake events are included in the logic tree. However, we can say that we conducted stochastic tsunami hazard analysis in terms of considering the variability of the moment magnitude, position of asperity, and recurrence interval of the target earthquake. In addition, there is a possibility that other parameters such as the log-normal standard deviation or weights based on expert judgment strongly influence the shape of the hazard curves (e.g., Annaka et al. 2007; Iwabuchi et al. 2014). However, we would like to note that we focused on how the hazard curves change in response to the different positions of asperity in the logic tree, on the premise that parameters such as the log-normal standard deviation and weights based on expert judgment are fixed.
Comparison with Tohoku earthquake observation results and a past study
In this section, we compare the results of the stochastic tsunami hazard analysis performed in Sect. 3.1 with the observation results of the Tohoku earthquake that occurred in 2011 and a past study performed by Sakai et al. (2006). The measurement data of the south Iwate offshore GPS wave gauge and the Fukushima Prefecture offshore GPS wave gauge set up by MLIT and analyzed by Port and Airport Research Institute (PARI) were used as the Tohoku earthquake observation data (Kawai et al. 2013).
Comparison with the south Iwate offshore GPS wave gauge observation results
Figure 5a shows the waveform observed during the Tohoku earthquake, measured by the south Iwate offshore GPS wave gauge. The position of the south Iwate offshore wave gauge is indicated by the yellow star in Fig. 2a. According to information published by MLIT, the maximum tsunami wave height at the south Iwate offshore GPS wave gauge was 6.7 m. The results of stochastic tsunami hazard analysis at the same location are shown in Fig. 4c. In addition, the relationship between the return period and fractile point at the maximum wave height of 6.7 m observed during the Tohoku earthquake is indicated by the red line in Fig. 6. The results show that the range of the return period is approximately 1,780 years at the 0.50 fractile wave height and 450 years at the 0.95 fractile wave height. Furthermore, the return period at the 0.50 fractile wave height is 1,709 years. In a report by Fujiwara et al. (2013), the return period of the 2011 Tohoku earthquake was estimated at approximately 600 years, but in the results of our analysis, approximately 600 years is equivalent to a fractile from 0.81 to 0.91, which is a relatively high fractile point.
Comparison with the Fukushima Prefecture offshore GPS wave gauge observation results
Figure 5b shows the waveform observed during the Tohoku earthquake, measured by the Fukushima Prefecture offshore GPS wave gauge. The position of the Fukushima Prefecture offshore wave gauge is indicated by the red star in Fig. 2a. According to information published by MLIT, the maximum tsunami wave height at the Fukushima Prefecture offshore GPS wave gauge was 2.6 m. The results of stochastic tsunami hazard analysis at the same location are shown in Fig. 7. In addition, the relationship between the return period and fractile point at the maximum wave height of 2.6 m observed during the Tohoku earthquake is indicated by the blue line in Fig. 6. The results show that the range of the return period is 600 years at the 0.50 fractile wave height and 450 years at the 0.95 fractile wave height. The return period at the 0.50 fractile wave height is 600 years. According to our analysis, the return period of the Tohoku earthquake of approximately 600 years is equivalent to a fractile between 0.35 and 0.75; this is a relatively wide fractile range.
Comparison with a past study
Sakai et al. (2006) also conducted probabilistic tsunami hazard analysis using the logic-tree approach, targeting the coast of Fukushima Prefecture. Before comparing our research with their work, it is important to note that one of the fundamental differences between the two studies is that our research considers only the Tohoku earthquake type fault (M
w 8.9–9.1), while their research considered nine earthquake faults (M
w 7.7–8.6) along the Japan trench for the near-field tsunami. In addition, the contents and weights of the branches in the logic tree differ between the two studies. Thus, although there are some differences in the conditions of the studies, we discuss the results of the analyses in the following.
In the long-term averaged hazard curve of Sakai et al. (2006), wave heights with a return period of 1,000 years had a range of about 3.5 m (0.05 fractile wave) to about 5.6 m (0.95 fractile wave). However, in our study, wave heights with a similar return period had a range of about 4.0 m (0.05 fractile wave) to about 17.3 m (0.95 fractile wave). It is indicated that both the calculated stochastic tsunami height and its range in our analysis are larger than in their analysis, which is mainly because the moment magnitude of the assumed earthquake fault in our analysis is larger. Furthermore, the difference of the 0.05 fractile wave between the analyses is about 0.5 m, while the difference of the 0.95 fractile wave is about 13.3 m. We can also understand from the results that the difference in tsunami height between the analyses has a larger value as the fractile point is large.
Influence of the number of random slip distribution patterns on the results of hazard analysis
In this section, we examine the influence of the number of random slip distribution patterns in a fault zone on the results of stochastic tsunami hazard analysis. In the logic tree, variations in slip distribution in the fault zone are handled by a separate branch as epistemic uncertainty. Thus, we examined the effect of the number of slip distribution patterns in the fault zone on the stochastic tsunami hazard results by comparing the analysis results when using five slip distribution patterns (standard case), three patterns, and one pattern. For the case of three patterns, we considered the cases where the slip distribution was at the northernmost part, in the center, and at the southernmost part of the hypothetical fault. For the one pattern case, we only considered the case where the slip distribution was in the center. Since the number of branches of the logic tree changes if the number of slip distribution patterns is changed, the number of hazard curves also changes. When there are three slip distribution patterns, the number of branches is 216 (3 × 3 × 3 × 4 × 2), and when there is one, the number of branches is 72 (3 × 1 × 3 × 4 × 2).
Figure 8 shows the hazard curves at the site of the south Iwate offshore GPS wave gauge for the cases of three slip distribution patterns and one slip distribution pattern. Figure 9 shows the relationship between the return period and fractile at wave heights of 10 and 2.0 m in addition to the maximum wave height of 6.7 m measured during the Tohoku earthquake. These figures show that although there is almost no influence of the number of slip distribution patterns on the results of the hazard analysis in the case of a relatively small wave height (2.0 m), when the wave height is relatively large (6.7, 10 m), the number of slip distribution patterns greatly influences the results of the hazard analysis. In the case of the 10-m wave height, when there is only one slip distribution pattern, the return period at each fractile point provides the largest estimate (low annual probability of exceedance), whereas in the cases of three and five slip distribution patterns, the return period at each fractile point tends to provide an estimate smaller than that in the case of only one pattern (high annual probability of exceedance). In the case of the 6.7 m wave height, no particular trend is seen, but the variability in the estimated return period becomes relatively large compared to when the wave height is 2.0 m. Based on the approximate wave heights, we speculate that there is a great difference in how the number of slip distribution patterns influences the results of the hazard analysis depending on the hypothesized wave source.
Figure 10 shows the tsunami wave height frequency distribution taking into consideration the uncertainty of a 1,000-year return period at the same location. In the cases of the five and three slip distributions, the standard deviation and coefficient of variation are about the same, but when there is only one slip distribution, the frequency of wave heights 10 m and above becomes extremely small. Accordingly, the standard deviation becomes smaller and the coefficient of variation decreases. This is because, in the cases of the five and three slip distributions, a high tsunami wave height was calculated at the south Iwate offshore GPS wave gauge because a large amount of slip was present in the northern portion of the fault, but in the case of only one slip distribution, a high tsunami wave height was not observed at the south Iwate offshore GPS wave gauge because a large amount of slip was present only in the center of the fault.
Quantitative assessment of uncertainty of tsunami wave height
In this section, we discuss the uncertainty of tsunami wave height in each region of the Tohoku earthquake using the results of stochastic tsunami hazard analysis and the logic tree of Fig. 3.
Uncertainty of tsunami wave height in the Tohoku offshore region
Considering the results of stochastic tsunami hazard analysis at the south Iwate offshore GPS wave gauge (Fig. 4), when we consider the tsunami wave height of a certain annual probability of exceedance, we see that various values can be obtained depending on the fractile point. Figure 11a shows the tsunami wave heights at the 5 % fractile point of a 1,000-year return period at a depth of 50 m underwater, offshore from the Tohoku coast, and Fig. 11b shows the tsunami wave heights at the 95 % fractile point at the same locations. For the 5 % fractile wave height, a relatively high wave height from 5.0 m to less than 10 m was calculated primarily offshore from Miyagi Prefecture and Fukushima Prefecture. On the other hand, for the 95 % fractile wave height, a high wave height from 20.0 m to less than 40.0 m was calculated in the coastal regions of the Iwate and Miyagi Prefectures. If we limit the wave height to 5.0 m and above, the offshore area from southern Aomori Prefecture in the north to southern Ibaraki Prefecture in the south was analyzed. In this study, as an index of uncertainty, we defined the wave height equivalent in the range from the 5 % fractile point to the 95 % fractile point (a value obtained by subtracting the 5 % fractile wave height from the 95 % fractile wave height) as the 90 % confidence interval of the tsunami wave height. Figure 11c shows the 90 % confidence interval of tsunami wave height of a 1,000-year return period at the same locations. The figure indicates that, in general, at locations where wave height at each fractile point is high, the 90 % confidence interval is simultaneously high. In particular, a large range in wave height, from 30.0 m to less than 40.0 m, is seen offshore from Iwate Prefecture and at offshore parts in the vicinity of the Oshika Peninsula of Miyagi Prefecture. In the offshore area of Tohoku as a whole, it ranges from a minimum of 0.6 m to a maximum of 36.5 m.
As another index of uncertainty, we defined the coefficient of variation obtained from the standard deviation and average of various tsunami wave heights obtained while focusing on a certain annual probability of exceedance. Figure 11d shows the coefficient of variation of tsunami wave height of a 1,000-year return period. Some very interesting observations can be made from this figure. First, there were multiple locations offshore of the Boso Peninsula of Chiba Prefecture that exhibited high coefficient of variation values, from 0.91 to less than 1.10. When this is considered in combination with Fig. 11a, b, and when we compare the region offshore of the Boso Peninsula of Chiba Prefecture with other regions off the Tohoku coast, the uncertainty of the hypothesized wave height is large, although small wave heights were predicted on average, and there is a high probability of incursion of a wave height divergent from the average wave height. Table 5a shows stochastically processed wave heights at the location at which the maximum coefficient of variation was recorded offshore from the Boso Peninsula of Chiba Prefecture and the maximum wave height of each tsunami calculation. Figure 12a shows the tsunami wave height frequency distribution when considering the uncertainty at the same location. The maximum wave height data calculated directly from the tsunami simulation of Table 5a show a large difference between the maximum wave height when the position of the asperity is placed at the southernmost edge of the fault zone (4.3 m for the case of M
w 9.0) and when the position of asperity is placed at the northernmost edge of the fault zone (1.0 m for the case of M
w 9.0); we believe this is the cause for the increase in the coefficient of variation. Figure 12a shows a frequency distribution with a relatively wide range at higher wave heights, up to a maximum of 6.8 m with an average of 1.3 m. The second interesting fact is that the point where the coefficient of variation reached a minimum was a point offshore of the Fukushima Prefecture coast. Similarly, Table 5b shows the data at the same location, and Fig. 12b shows the tsunami wave height frequency distribution when considering the uncertainty at the same location. The wave height data for the case of Mw 9.0 calculated directly from the tsunami simulation of Table 5b shows that the wave height was highest when the position of the asperity was placed at the southernmost edge of the fault zone; however, relatively high tsunami wave heights were recorded in all cases. The wave height frequency distribution in Fig. 12b is distributed relatively widely, from a minimum of 4.5 m to a maximum of 18.9 m, but the average wave height was at 10.5 m and the coefficient of variation had the minimum value.
Table 5 Stochastically processed wave heights and maximum wave heights for each case of tsunami simulation at (a) the location of the maximum coefficient of variation and (b) the location of the minimum coefficient of variation
Uncertainty of tsunami wave height at the ria coastline of the Tohoku region
Finally, we look at the results for the ria coastline of Iwate Prefecture. Figure 13a shows 0.05 fractile wave height, Fig. 13b shows 0.95 fractile wave height, Fig. 13c shows 90 % confidence interval wave height, and Fig. 13d shows the coefficient of variation of a 1,000-year return period at a depth of 50 m underwater offshore of the ria coastline of Iwate Prefecture. No noticeable trend is seen for the 0.05 fractile wave height, but the values of the 0.95 fractile wave height, 90 % confidence interval, and coefficient of variation show that the values tend to be higher offshore from the peninsular tips of the ria coastline and lower in the closed-off sections of bays. Table 6a shows the data at the location of the maximum coefficient of variation offshore from the tip of a peninsula of the ria coastline, and Table 6b shows the location of the minimum coefficient of variation in a closed-off section of the bay shown in the yellow boxes in Fig. 13. Figure 14 shows the tsunami wave height frequency distribution when considering the uncertainty at the same locations. These data show that, in the vicinity of the peninsular tip of the ria coastline, the standard deviation is larger and the average is smaller than that in the vicinity of the closed-off section of the bay; however, in the vicinity of the closed-off section of the bay, the standard deviation is smaller and the average is larger, and this determines the magnitude of the coefficient of variation. In the vicinity of the peninsular tip of the ria coastline, there is a trend of tsunami waves with high to low wave heights due to the difference in slip distribution within the fault observed widely, and the uncertainty of the tsunami wave height is large. However, in the closed-off section of the bay, the average tsunami wave height is larger than that near the peninsular tip because of effects of the submarine topography in the vicinity, but the range of the obtained wave height is small, i.e., the uncertainty of the tsunami wave height is small. These results clearly demonstrate the influence of topographical effects specific to ria coastlines.
Table 6 Stochastically processed wave heights and maximum wave heights for each case of calculation at (a) the location of the maximum coefficient of variation offshore from the peninsular tip of a coastline and (b) the location of the minimum coefficient of variation in the closed-off section of a bay shown in the yellow boxes in Fig. 13
The above results show that when the results of stochastic tsunami wave height analysis limited to a Tohoku-type earthquake fault are analyzed, regional differences in uncertainty at the 90 % confidence interval and coefficient of variation of tsunami wave height clearly exist. In this study, we analyzed a Tohoku-type earthquake fault, but we believe that if the subject fault is changed, regional variations in the uncertainty of tsunami wave heights will also change.