Sendai, Japan - September 13th to 18th 2020
Paper N° C001608, Registration Code: S-A00953

A Study on the Macrospatial Correlation
Characteristics of Tsunami Wave Height andTsunami Inundation Depth

Y. Fukutani(1), T. Nakamura(2), T. Shizuma(3)(1)Associate Professor, Kanto Gakuin University, fukutani@kanto-gakuin.ac.jp(2) Director, Shinozuka Research Institute, Co., Ltd.
(3) Chief Research Engineer, Shinozuka Research Institute, Co., Ltd.

Research background◼ In a probabilistic disaster risk assessment for multiple buildings, weneed to consider hazard correlation among site locations anddamage correlation among multiple buildings.

◼ In the case of probabilistic
seismic hazard assessment
and probabilistic seismic risk
assessment (PSHA and PSRA), Groundwe generally assign various Surfacet y p e s o f u n c e r t a i n t i e s t o Engineering
earthquake ground motions, Bedrock
ground characteristics, building
response characteristics, andAttenuation equationmaterial strengths.
Earthquake source◼ Regarding earthquake groundmotion that is evaluated by an
Ground motion prediction
attenuation formula, we need （CentralResearch Institute of Electric Power Industry
to consider it as having a https://criepi.denken.or.jp/jp/civil/intro/earthquake.html）
spatial correlation according to2
the relative distance.

Objective◼ In the cases of probabilistic tsunami hazard assessment andprobabilistic tsunami risk assessment (PTHA and PTRA), we
also need to consider the spatial correlation
characteristics of tsunami hazard and damage, but there
have been no studies that evaluate this correlation.

The aim of this study is to evaluate the spatial correlation
coefficients (𝜌𝑅 ) of tsunami wave height and tsunami
inundation depth according to the relative distance byusing the results of tsunami numerical simulations withlinear and nonlinear long-wave equations.
We are assumed to be applied to tsunami
damage assessment of building portfolio
considering tsunami hazard correlation.

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An evaluation method for the macrospatial correlation
coefficient of tsunami hazards
Here, we use two buildings as a risk assessment target. Let 𝐶1 and 𝐶2 berandom variables for the strength of the two buildings, and let 𝑅1 and 𝑅2 berandom variables for the hazard value (e.g., peak ground acceleration andtsunami inundation depth) at the two building sites. Assuming that 𝐶1 and 𝐶2
are independent but that 𝑅1 and 𝑅2 have a certain correlation, we obtain
performance functions 𝐹1 and 𝐹2 as follows:
𝐶𝐹𝑖 = 𝑅𝑖 ↔ ln(𝐹𝑖 ) = ln(𝐶𝑖 ) − ln(𝑅𝑖 ), 𝑖 = 1, 2𝑖

The covariance of ln(𝐹𝑖 ) is now the same as the covariance of ln(𝑅𝑖 ); therefore,
we obtain the following relation:cov(𝑙𝑛𝐹1 , 𝑙𝑛𝐹2 ) = cov(𝑙𝑛𝑅1 , 𝑙𝑛𝑅2 )From this relation, the damage correlation coefficient 𝜌𝐹12 is as follows:𝜌𝐹12 =cov(𝑙𝑛𝐹1 , 𝑙𝑛𝐹2 )var ln𝐹1 ∙ var ln𝐹2Then, we obtain the following equation:
𝜌𝑅12 𝜁𝑅1 𝜁𝑅2𝜌𝐹12 =
𝜁𝐶21 + 𝜁𝑅21 ∙ 𝜁𝐶22 + 𝜁𝑅22=cov(𝑙𝑛𝑅1 , 𝑙𝑛𝑅2 )var ln𝐹1 ∙ var ln𝐹2

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An evaluation method for the macrospatial correlation
coefficient of tsunami hazards
Here, let 𝜌𝑅12 (= 𝜌 𝑥 ) be the hazard correlation coefficient with respect to thedistance 𝑥 between two points, let 𝜁𝑛 be the log normal standard deviation of n22and let 𝜁 = 𝜁𝐶𝑖+ 𝜁𝑅𝑖(𝑖 = 1, 2) be the compound deviation. Finally, we obtain thefollowing equation:

𝜌𝐹12 =

𝝆𝑹𝟏𝟐 𝜁𝑅1 𝜁𝑅2
𝜁2◼ For cases of earthquake hazard and damage assessment, we alreadyknow the values of 𝜁 2 , 𝜁𝑅1 and 𝜁𝑅2 from past earthquake records anddamage records; therefore, we can determine 𝜌𝐹12 when we evaluate
𝜌𝑅12 .

◼ However, in the case of tsunami hazards, there are no studies on thecorrelation coefficient of tsunami hazards.
◼ Based on this background, we evaluate the macrospatial
correlation coefficient of a tsunami wave height and tsunami
inundation depth in this study.
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Numerical study on
the macrospatial correlation of tsunami wave height◼ First, we establish a large virtual fault with a length of 250 km and a width of
100 km in the center of a virtual sea area with a water depth of 1000 m. The
fault parameters of the large fault are a slip amount of 10 m, strike of 90°,
dip of 15°, rake of 90° and fault depth of 1 km.
◼ We solved the continuous equation and linear long-wave equation by using
the staggered leapfrog method and plane rectangular coordinates with a 500
m grid.

(a) Initial water displacement (m)due to fault movement(b) Maximum tsunami wave height (m)6Numerical study on
the macrospatial correlation of tsunami wave height◼ We regressed the power function 𝐻 = a𝑥 b + c using the least squares
method to determine the median values.
◼ The regression coefficients were a = -0.0508, b = 0.4384, and c =2.486, and the determination coefficient was 0.598.
◼ The standard deviation of the residual between the numericalsimulation results and the predicted values from the power function
was 𝜎 2 = 0.022.

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Numerical study on
the macrospatial correlation of tsunami inundation depth◼ We first constructed the fault parameters of the earthquake of the Sagami
trough, which has a large slip off the Kanto area in Japan.
◼ Using the initial water displacement calculated from the earthquake
parameters as input data, we solved the continuous equation and nonlinear
long-wave equations by using the staggered leapfrog method and plane
rectangular coordinates. We nested the four grid data with mesh lengths of
270 m, 90 m, 30 m and 10 m.

(a) A slip distribution of the Sagami troughearthquake around the Kanto region(b) Tsunami inundation depth distribution in8
Zushi city, Kanagawa Prefecture

Numerical study on
the macrospatial correlation of tsunami inundation depth◼ For the relationship between the shortest distance from the coastline andthe tsunami inundation depth, we regressed the exponential function 𝐻 =aexp(b𝑥) using the least squares method to determine the median values.
The regression coefficients were a = 4.911 and b = -0.002078◼ For the the correlation coefficients between two points, We regressed thefollowing exponential function using the least squares method:𝜌 𝑥 = aexp b𝑥 + c𝑒𝑥𝑝(d𝑥)a = 0.4555
b = -0.1653
c = 0.5434
d = -0.007345

The correlation length was
53.2 m, which the correlation
coefficient was equal to 1/e.

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Conclusions
We determined the following spatial correlation features of tsunami wave
height and tsunami inundation depth as the results of the tsunami
numerical experiments using linear and nonlinear long-wave equations:◼ The macrospatial correlation coefficients of the tsunami wave height in
the offshore region have a tendency to decrease as the distanceincreases, but the correlation coefficient is as high as approximately
0.74 even at a distance of approximately 100 km.
◼ The macrospatial correlation coefficients of the tsunami inundation
depth in the tsunami run-up region have a tendency to decrease as thedistance increases, but the rate of decrease is greater compared to themacrospatial correlation coefficient results of the tsunami wave height in
the offshore region. The correlation coefficient decreases byapproximately 0.78 at a distance of 10 m, indicating a low correlation of
the tsunami inundation depth. This result occurs because the run-up
tsunamis on the land are affected in various ways by the bottom friction
and attenuate faster compared to those in the offshore region.
◼ The macrospatial correlation coefficient of the tsunami inundation depth
can be evaluated by the exponential function (Eq.(20): 𝜌(𝑥)= 𝑒𝑥𝑝(𝑥)+𝑒𝑥𝑝(𝑥)) regardless of the land gradient.

Thank you for your attention.

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