池田菜穂

IG04-D4-PM2-P-021 Tsunami Research in the 21st Century

A Stochastic Analysis and an Uncertainty Assessment of Tsunami Wave Height
Using a Random Source Parameter ModelYo Fukutani, Anawat Suppasri, Fumihiko Imamura
Earthquake induced Tsunami Risk Evaluation, International Research Institute of Disaster Science, Tohoku University

Research BackgroundAs for stochastic analysis methods for tsunami height, a method considering epistemic uncertainty as logic trees and aleatory uncertainty as
probabilistic distributions is general in Japan. In this study, we focused on variability of tsunami height in case that we deal with heterogeneous slip
on a fault as epistemic uncertainty by using the logic trees. In addition, we quantitatively evaluated the uncertainty of tsunami height.
①Correlated Random Source Parameter Model②Quantitative assessment for uncertainty of tsunami wave heightFig.3 (a) shows examples of slip distributions for a type of the 3.11 Tohoku earthquake fault
We used CRSP (Correlated Random Source Parameter) model
model using the CRSP model, and Fig.3 (b) shows a logic tree including these slip distributions
developed by Liu et al. (2006)) in order to generate
as epistemic uncertainty.
heterogeneous slip on a fault. If we use the model, we canFig.4 shows tsunami hazard curves when the number of included slip distributions change to (a)generate hypothetical two dimensional slip distributions on the5 cases, (b) 3 cases and (c) 1 case. In case of 1 case, we can see that range of tsunami wave
fault based on a probabilistic distribution of slip rate estimatedheight for one return period is less than that of 3 cases and 5 cases.
from past seismic recording data.
An overview of the model is as follows. First, a twodimensional random Gaussian distribution is converted to wave
number space by using 2 dimensional Fourier transform after
multiplying below filter (Mai and Beroza (2002))(a)

𝐹 𝑘𝑥 , 𝑘𝑦 = 1 + 𝑘𝑥 𝐶𝐿

22+ (𝑘𝑦 𝐶𝑊 )𝑙𝑜𝑔10 𝐶𝐿 = −2.5 + 𝑀𝑤/2
𝑙𝑜𝑔10 𝐶𝑊 = −1.5 + 𝑀𝑤/3

−𝜈/2⑤(1)(2)(3)

Where, kx: wave number along fault strike, ky: wave number along fault dip,
ν: fractal dimension, CL: correlated length along fault strike, Cw: correlated
length along fault dip, Mw: moment magnitude①(b)Source type

Next, after we apply Fourier inverse transform to theconverted distribution, we can get random Gaussian
distributions, which theirs amplitude in the wave number spaceattenuate following k-ν/2. Finally, we covert the distributions
to a slip rate distributions using NORTA method (Cairo andNeilson (1997)) because the slip rate distribution follows
Truncated-Cauchy Distribution (Fig.1). In this study, wedivided the fault into shallow area and deep area, and apply the
Truncated-Cauchy distribution to each area. We adopt threetimes by Ishii et al. (2013) as magnification factor of
dislocation of shallow area and deep area. In each area, we use
below value as average slip Dave and maximum slip Dmax;②
Range of
moment magnitude

Position ofasperity

Average return
period①②③④⑤Log normal standard
Truncated range ofdeviation
uncertainty(Aleatory uncertainty)※The appended numbers in thebranches of the logic tree denoteweights of each branch

Figure.3 (a) Examples of slip distributions for the 3.11 Tohoku earthquake and (b) a logic tree

(b)

(a)

(c)

90％ confidence interval

Slip distribution of
only ③was considered
Slip distributions of①・②・③・④・⑤
were considered

𝐷𝑚𝑎𝑥 = 2.8 ∗ 𝐷𝑎𝑣𝑒④③

Slip distributions of①・③・⑤
were considered(4)

We referenced slip rate value 2.8 of maximum slip and averageslip in shallow area of the 3.11 Tohoku earthquake (Yoshida etal.(2013)).

Figure.4 Hazard curves in case that (a) five, (b) three and (c) one slip distribution include the logic tree

Fig.5 shows (a) 90% confidence interval, (b)(a)(b) Coefficient of variation of stochastic wave heightfor 1,000 years return period along the Tohoku
coastal area at the water depth of 50 m. The below
1Cauchy distribution 𝑝 𝐷 = 𝐶
figure of Fig.5 shows frequency distribution of2
1 + 𝐷 − 𝐷𝑜/𝜅stochastic wave height for 1,000 return period
𝐶：regularization of probabilistic density function
where the coefficient of variation is (c) maximum
𝐷𝑜 ：location parameter of Cauchy distributionand(d)minimum.
(d)κ：scale parameter of Cauchy distribution
Basically, the 90% confidence interval is highFigure.1 Truncated-Cauchy Distribution
where the wave height of each fractile point is high.
(c)
Thevalues
ofcoefficientofvariation
clearly
Trench Axis 3.7 5 5.2 5.1 5.5 6.4 8.5 12 16 19 18 15 13 13 13 13 11 6.3 4.4 5.5 8.5 13 14 11 5.7 4 4.8 5 3.8 2.9 2.5 2.7
1812 5.8 2.7 2.4 area
3.4 5.9 11 16 18 18 16 13 12 14 15 13 11 10 13 16 19 17 12 8 7.2 9.4 13 17 20 21 20
Shallow
10 9.3 7 5.4 5.6 7.2 9.1 11 14 17 19 19 19 20 21 20 16 10 6 6 9.2 13 11 7.9 5.9 6.639 times
12 14 14 11 10
indicate regional difference of wave height. Off the(d)(c)
8.2 10 12 12 14 13 12 12 14 15 16 18 19 21 21 18 11 5 2 1 1.2 2.2 3.6 5.4 6.5 7 8.8 12 15 13 9.5 7.2
Off
2.6 3.5 3.8 3.6 4.8 7.3 11 15 17 17 17 15 11 9 8.8 9.7 9.4 6 2.4 1.1 1.4 2.1 3.6 7.9 14 19 22 22 22 18 9.1 3.7
Boso peninsula in Chiba prefecture where the
9 6 3.9 3.4 4.1 6.3 11 16 19 22 22 22 21 17 11 8.3 7.6 7.1 6.8 7.7 9.3 10 13 16 19 20 20 20 19 18 16 13Fukushima
12 10 8.3 7.9 8.9 8.8 8 9.9 14 16 17 15 13 9.9 9.3 12 15 16 14 11 11 14 18 20 19 16 12 11 11 11 12 12
coefficient of variation is maximum, the variance
Boundary line 0.8 1.2 1.8 2.5 3.3 3.7 3.9 4.5 4.6 3.4 1.8 1 0.9 1.4 2.6 4.3 5 4.7 4.1 3.6 4 5.6 7 7.4 7.3 6.6 5 3.5 2.2 1.5 1 0.7
Off the Boso
peninsula inofmaximum
wave
heightdue
tothedifferenceof(10km-20km) Deep
0 0.1 0.3 1.4
3.3 5 6 6.3 5.8 4.1 1.9 1.1 1.2 2 3.1 4 4.5 4.8 5.2 5.6 6.1 6.7 7 7.1 6.9 6.2 4.9 3.4 2.2 1 0.2 0
area
Chiba
0 0 0.1 0.7 1.9 2.9 3.4 3.4 2.9 2.2 1.8 2.2 3.1 3.8 4 4.1 4.2 4.4 5.2 5.8 5.7 5.2 5.2 5.7 6.2 6.41
6.1 time
4.6 2.3 0.8 0.2 0slip distribution on the fault greatly influence on
0.1 0.1 0.3 0.6 1.4 2.3 3.3 4.3 5 6 6.8 6.9 6.8 6.4 5 3.7 2.5 1.6 1.4 1.5 1.9 3 3.8 3.4 2.4 1.8 1.5 1.3 0.8 0.5 0.3 0.1
Wave height (m)Wave height (m)2.1 1.9 1.4 1 1.4 2.2 3.4 3.9 3.4 3.3 3.7 4 3.8 2.9 1.8 1 0.7 0.5 0.3 0.2 0.5 1.5 2.1 1.2 0.3 0.1 0.1 0.5 1.4 2.2 2.2 2.1
the coefficient of variation. The spot where the
3.4 2 1.4 1.8 3.1 4.1 3.7 2.9 2.2 1.9 1.8 1.9 2.3 2.3 1.7 1.2 0.8 0.5 0.2 0.1 0.2 0.3 0.3 0.3 0.3 0.5 1.2 2.5 4.5 5.7 5.8 4.9
Figure.5 (a) 90% confidence interval of wave height, (b) Coefficient of
1.3 0.8 0.7 1.4 2.6 3.7 4.4 4 3 2.1 1.8 2.4 3.1 2.8 2.1 1.9 1.9 1.5 1.1 1 0.9 0.8 0.8 0.5 0.5 0.6 1.3 2.5 3.7 4 3.2 2.2
coefficient of variation is minimum is off
variation, (c)・(d) Frequency distribution of wave height0.3 0.2 0.4 0.9 1.8 2.5 2.1 1.2 0.8 1.3 2.9 5.3 6.8 6.8 5.9 4.3 2.9 1.8 1.5 1.6 2 2.1 1.4 0.6 0.5 0.6 1.5 3.1 4 3 1.5 0.7
0 0.1 0.1 0.2 0.5 0.9 1.1 1.3 2.2 3.8 4.8 4.8 5 5.2 4.6 3 1.5 0.6 0.3 0.4 0.6 0.9 1.2 1.2 1.1 1.2 1.4 1.6 1 0.3 0.1 0
Fukushima prefecture. Fig.6 shows the results forFigure.2 An example of slip distribution in case of dividing a fault into deep(a)(b) rias coastal area in Iwate prefecture. In the rias area,
are and shallow area
the coefficient of variation is high off the edge(d)References
point of the peninsula while the coefficient of(c)Cairo, M. C., and B. L. Neilson (1997), Modeling and generation random vectors withvariation is low inside the bay area.
arbitrary marginal distributions and correlation matrix, Technical Report, Department ofIndustrial Engineering and Management Sciences, Northwestern University, Evanston, Illinois.
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assessment of tsunami wave height using a random slip source parameter model, Stochastic
Environmental Research and Risk Assessment. (submitted)
Ishii, Y., K. Dan, J. Miyakoshi, H. Takahashi, M. Mori, and N. Fukuwa (2013), Source modelingof hypothetical Nankai Trough, Japan, earthquake and strong ground motion prediction using
the empirical Green’s functions, Shimizu Corporation Research Report, 90, 17-26.
Liu, P. C., R. Archuleta, and S. Hartzell (2006), Prediction of broadband ground-motion time
Peninsula
Inside bay
histories: hybrid low/high frequency method with correlated random source parameters,edge area
area
Bulletin of the Seismological Society of America, 96, 2118-2130.
Mai, P., and G. Beroza (2002), A spatial random-field Model to characterize complexity inWave height (m)Wave height (m)earthquake slip, J. Geophys. Res., 107(B11), 2308, doi:10.1029/2001JB000588.
Yoshida, K., K. Miyakoshi, and K. Irikura (2011), Source process of the 2011 off the PacificFigure.6 (a) 90% confidence interval of tsunami wave height, (b)coast of Tohoku Earthquake inferred from waveform inversion with long-period strongCoefficient of variation, (c)・(d) Frequency distribution of wave heightmotion records, Earth Planet Space, 63, 577-582.

(c)

(d)

③Conclusion
・ We conducted the uncertainty assessment oftsunami wave height using CRSP model and thelogic tree, then, we were able to obviously confirm
the regional difference of wave height uncertainty.
・It is important for us to quantitatively assess theuncertainty of tsunami wave height and to visualize
it on the topographic maps, because the uncertainty
of stochastic tsunami wave height calculated fromstochastic assessment might be higher than people
expected.