In this section, we aim to evaluate the probabilistic coastal tsunami height in the Tohoku region adjacent to the Pacific Ocean. First, we need to select an appropriate earthquake-generating fault to produce a tsunami. We select ten regions and one region of interlocking earthquakes from among the occurrence areas of trench-type earthquakes along the Japan Trench used in the probabilistic earthquake prediction map released by the NIED [

14], all of which are shown in

Figure 4 and

Table 2 as earthquake faults that could generate tsunamis. These selections exclude both earthquakes with moment magnitudes (

Mw) reaching 7.4 or less in consideration of a variation of ±0.1

Mw and earthquakes for which the source fault is unlikely to be predicted beforehand. To evaluate the epistemic uncertainties for these eleven fault regions, we use the logic tree method proposed in Annaka’s study [

10].

Figure 5 shows the logic tree constructed for these regions. We establish five branches within the logic tree: The

Mw range of the earthquake, the asperity position of the earthquake fault, the average occurrence interval (return period) of the earthquake, the standard deviation of the lognormal distribution followed by the error of the tsunami wave height and the truncation range of the lognormal distribution. Except for the asperity position of the earthquake fault, the other four branches follow the branches shown in Annaka’s study [

10]. The total number of branches in the logic tree constructed using this approach is 3384 branches. The outline of setting for each branch is as follows.

The

Mw range of each earthquake is varied by ±0.1. This

Mw variation is accomplished by changing the average slip amount along the entire fault. For the asperity position of the fault, three types of branches with asperities located at the center of the fault and near both ends of the fault are established when the fault length is 150 km or more. Since only Tohoku-type earthquakes have long lengths (approximately 500 km), we set five branches by adding two cases where an additional asperity is located between the three asperities. The method utilized to establish the asperity positions along the fault is detailed in Fukutani et al. [

15]. Regarding the occurrence probability of the earthquake, we construct three types of branches that take into consideration the confidence interval of the occurrence probability determined by the probabilistic seismic motion prediction evaluation published by the NIED [

14].

Table 3 shows the model name for the generation interval of the earthquake, α value of the BPT distribution, average return period, sample period, earthquake generation time within the period used to determine the average occurrence interval and lower and upper limits of the confidence interval for each earthquake fault, which are shown in the Headquarters for Earthquake Research Promotion [

16]. See

Appendix A for the method used to establish the confidence intervals. Although the probability of the occurrence of an earthquake with a relatively small magnitude can be evaluated with high accuracy using the general Gutenberg-Richter rule, it is known that the probability of occurrence of a relatively high-magnitude earthquake that causes a large tsunami cannot be perfectly evaluated using the Gutenberg-Richter rule. Based upon this background, in this study, we note that data of the return period collected as the result of a detailed examination of the historical earthquake record are used. Regarding the standard deviation of the lognormal distribution (i.e. aleatory uncertainty) followed by the error of the tsunami wave height, we use the modeling error in the tsunami numerical simulation results and observation records, that is, the geometrical standard deviation

κ of Aida [

17], for the past eleven historical earthquakes represented by the following expression based on Annaka’s study [

10]:

where

n is the number of observation points,

i is the observation point,

βi = (

Ri/

Hi),

Ri is the observed tsunami height at the

i-th point and

Hi is the simulated value at the

i-th point. The values of

κ are evaluated from the past eleven historical earthquakes. The minimum value is the result for the 1707 Hoei earthquake, where

κ = 1.35 (

σ = log

κ = 0.300) and the maximum value is the result for the 1946 Nankai earthquake, where

κ = 1.60 (

σ = log

κ = 0.470). Finally, two types of branches with ±2.3

σ and ±10

σ are established as truncation values at both ends of the lognormal distribution.

The numbers attached to the branches of the logic trees are the weights of each branch and the sum of all of the weights is set to 1.0. The weights of the branches containing the

Mw range and asperity positions along the fault are set by equally dividing their weights. Regarding the branch consisting of the occurrence probability of the earthquakes, we use a weight of 0.50 for the central branch and 0.25 for the branches at both ends to consider the confidence interval. The weight value adopted in Annaka’s study [

10] is also adopted for the weights of the branches for the standard deviation of the lognormal distribution and the truncation range.