May 2020
The EGU General Assembly 2020 online activities
Sharing Geoscience Online
EGU2020-4033

Numerical Study on Spatial
Correlation Characteristics ofTsunami Inundation Depth
Yo Fukutani1
1Kanto Gakuin University

College of Science and Engineering1

Background◼ In a probabilistic disaster risk assessment for multiple buildings, weneed to consider hazard correlation among site locations and damage
correlation among multiple buildings because the probability
distribution of aggregate damage for multiple buildings is affected.
◼ Analyses that consider the spatial correlation of hazards have beenrelatively advanced in the fields of probabilistic seismic hazard
assessment and probabilistic seismic risk assessment (PSHA and PSRA,respectively) (e.g. [1]-[3]).
Strong motion simulation
Observation

Bedrock(Depth 2-3 km)

Seismic wave propagation characteristics
Earthquake sources

Prediction of Strong Motion（Reference:CRIEPI HP/https://criepi.denken.or.jp/jp/civil/intro/earthquake.html

2

Purpose of this study◼ On the other hand, in the cases of probabilistic tsunami hazard
assessment and probabilistic tsunami risk assessment (PTHA andPTRA, respectively), we also need to consider the spatial
correlation characteristics of tsunami hazard and damage, but there
have been no studies that evaluate these correlation.
The purpose of this study is to evaluate the spatial correlation
coefficients of tsunami inundation depth according to the relative
distance by using the results of tsunami numerical simulations withnonlinear long-wave equations.

The results of this study, the correlation coefficients of tsunami
inundation depth is assumed to be applied to the tsunami damage
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assessment of the building portfolio.

An evaluation method forthe macrospatial correlation coefficient of tsunami hazards◼ Based on past studies (e.g. [4]-[7]) that evaluated the spatial correlation
coefficients of seismic ground motions, we evaluated the spatial correlation
coefficients of tsunami phenomena.
◼ Now we define 𝐻𝑠𝑖𝑚 (𝑟𝑖 ) as the maximum tsunami wave heights obtained bynumerical simulations on a grid 𝑖 and 𝐻𝑝𝑟𝑒 (𝑟𝑖 ) as the predicted wave heights
obtained by an evaluation formula on a grid 𝑖. Here, 𝑟𝑖 is the shortest
distance between each grid and the fault end point, or the coastline. Theresidual 𝜀 𝑟𝑖 between the numerical simulation result and the predictedvalue is evaluated as follows:
𝜀 𝑟𝑖 = log

𝐻𝑠𝑖𝑚 (𝑟𝑖 )𝐻𝑝𝑟𝑒 (𝑟𝑖 )(1)◼ Using this residual 𝜀 𝑟𝑖 , we evaluate the variance in the residual (𝜎 2 ) as
follows:1(2)
𝜎 2 = σ𝑛𝑖=1(𝜀 𝑟𝑖 )2𝑛Here, 𝑛 is the number of target grids.
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An evaluation method forthe macrospatial correlation coefficient of tsunami hazards◼ Next, we evaluate the macrospatial correlation characteristics.
◼ The covariance of the 𝑖-th observation point 1 and observation point 2 can bewritten as follows from the definition of covariance:
cov 𝑟𝑖1 , 𝑟𝑖2 = 𝜌 𝑟𝑖1 , 𝑟𝑖2 𝜎 𝑟𝑖1 𝜎 𝑟𝑖2(3)Here, 𝜌 𝑟𝑖1 , 𝑟𝑖2 is a correlation coefficient between two points, and σ is astandard deviation in each grid. Assuming that the distance 𝑥 between thetwo observation points 1 and 2 and the standard deviation 𝜎 are constantregardless of the point, we can reconstruct Eq. (3) to the Eq. (4):cov 𝑟𝑖1 , 𝑟𝑖2 = 𝜌 𝑥

𝜎 𝑟𝑖

2(4)

◼ cov 𝑟𝑖1 , 𝑟𝑖2 can also be written as follows by using the residual 𝜀 𝑟𝑖
between the numerical simulation result and the predicted value:
cov 𝑟𝑖1 , 𝑟𝑖2 =

1
σ𝑁(𝑥)
𝜀 𝑟𝑖1
𝑁(𝑥) 𝑖=1

𝜀 𝑟𝑖2(5)Here, 𝑁(𝑥) is the number of pairs in which the distance between the two5
points satisfies the following formula:An evaluation method forthe macrospatial correlation coefficient of tsunami hazards
𝛥𝑥𝛥𝑥(6)𝑥−≤ 𝑟𝑖1 − 𝑟𝑖2 ≤ 𝑥 +22Here, 𝛥𝑥 was set to 500 m for a numerical simulation of tsunami wave
height and 10 m for a numerical simulation of tsunami inundation depth.
◼ We can derive the following formula from Eq. (4) and Eq. (5):𝑁(𝑥)

𝜌 𝑥

𝜎 𝑟𝑖

2

1
=
෍ 𝜀 𝑟𝑖1 𝜀 𝑟𝑖2)
𝑁(𝑥(7)

𝑖=1◼ Finally, the correlation coefficient 𝜌 𝑥 with respect to the distance 𝑥
between points 1 and 2 can be evaluated as follows:𝜌 𝑥 =1
𝜎 𝑟𝑖

2𝑁(𝑥)

σ𝑁(𝑥)
𝑖=1 𝜀 𝑟𝑖1 𝜀 𝑟𝑖2(8)We can evaluate the spatial correlation coefficient 𝜌 𝑥 of tsunami inundation depth bycalculating each variable on the right side of the formula. If we know the spatial
correlation coefficient of the tsunami hazard, we can evaluate the spatial correlation
coefficient of damage among multiple buildings using the relationship in the next
6paragraph and perform a stochastic damage evaluation for the building portfolio.

Numerical study ofthe 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, with reference to the earthquake
parameter published by the Japan Seismic Hazard Information.
◼ The moment magnitude (Mw) of the earthquake is 8.7, and there are 6,149 small
faults. The slip distribution of the earthquake was set to three levels for the superlarge slip region (23.5 m), large slip region (11.7 m), and background slip region(1.94 m) to satisfy the Mw 8.7 of the earthquake [8].

Slip distribution of the Sagami trough（Mw 8.7）

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Numerical study ofthe macrospatial correlation of tsunami inundation depth◼ Using the initial water displacement calculated from the earthquake parameters as
input data [9], 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. We
used the topography and roughness data published by the Cabinet Office in Japanand conducted the tsunami numerical simulation for 3 hours after the earthquake
occurrence.
Zushi cityRegion 1Region 2Region 3

10m
90m

Slip distribution of the Sagami trough（Mw 8.7）30m

Target region of numerical simulation

270m8Numerical study ofthe macrospatial correlation of tsunami inundation depth◼ We used the two-dimensional continuous equation and nonlinear long-wave
equations are as follows:
Calculation condition of the numerical simulation
The continuous equation and nonlinear long-wave equations

Governing equation𝜕𝜂 𝜕𝑀 𝜕𝑁++=0𝜕𝑡 𝜕𝑥 𝜕𝑦𝜕𝑀 𝜕 𝑀2
𝜕 𝑀𝑁
𝜕𝜂 𝑔𝑛2+++ 𝑔𝐷+𝑀 𝑀2 + 𝑁 2 = 0
𝜕𝑡 𝜕𝑥 𝐷𝜕𝑦 𝐷
𝜕𝑥 𝐷 7ൗ3𝜕𝑁 𝜕 𝑀𝑁
𝜕 𝑁2𝜕𝜂 𝑔𝑛2+++ 𝑔𝐷+ 7 𝑁 𝑀2 + 𝑁 2 = 0𝜕𝑡 𝜕𝑥 𝐷
𝜕𝑦 𝐷
𝜕𝑦 𝐷 ൗ3
where 𝜂 is the water level, D is the total water depth, g is the gravitational acceleration, n is
Manning’s roughness coefficient and M and N are the flow fluxes in the x and y directions.

numerical
Staggered-leap frog method
integration method
Initial water level as evaluated from fault parameters by Okada's (1985) [9] formula
Initial condition
Boundary condition
Open boundry
Coordinate system
Tide level
Mesh size (ΔxΔy)
Calculation timePlane rectangler coordinate system
T.P. +0.9 m
270 m → 90 m → 30 m → 10 m
3 hours

9Numerical study ofthe macrospatial correlation of tsunami inundation depth◼ Below figure shows the results of the tsunami inundation simulation in Zushi city,
Kanagawa Prefecture. The maximum tsunami inundation depth was 8.71 m. The
results are almost consistent with the inundation area and depth of the tsunami
hazard map published by Zushi city.

(b)

Tsunamiinundation
depth (m)Slipmount (m)500 mTsunami inundation depth distribution in Zushi city, Kanagawa Prefecture 10Numerical study ofthe macrospatial correlation of tsunami inundation depth◼ Below figure shows the relationship between the shortest distance from the coastline
and the 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, and the determination coefficient was 0.538.

(a)

(b)
𝜌=𝐻 = 𝑥 ( 𝑥)=11
= −0 0020
Determination coefficient====ee2 =0Tsunami inundation depth attenuation

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Numerical study ofthe macrospatial correlation of tsunami inundation depth◼ The standard deviation of the residual 𝜀 𝑟𝑖 between the numerical simulation results
and the predicted values from the exponential function is σ2 = 0.794.
◼ Below figure shows the spatial distribution of the residual 𝜀 𝑟𝑖 between thenumerical simulation results and the predicted values. There is an area where thenumerical calculation is overestimated along the river; in addition, there is an area
where the estimation by the evaluation formula is overestimated at the tip of the runup.

Residual𝜀(𝑟𝑖 )200 m

Spatial distribution of the residual 𝜀 𝑟𝑖
between the numerical simulation results and the predicted values

120

Numerical study ofthe macrospatial correlation of tsunami inundation depth◼ Below figure shows the result of the correlation coefficients between two points.
The correlation coefficient decreased by approximately 0.78 at a distance of 1 mesh
(10 m), indicating a low correlation of tsunami inundation depth.
◼ This result occurs because the run-up tsunamis were greatly affected by the bottom
friction on the land and attenuated the inundation depth faster compared to theoffshore area.
◼ We regressed the following exponential function using the least squares method:(9)𝜌 𝑥 = aexp b𝑥 + c 𝑥 (d𝑥 )(b)
𝜌=𝑥 𝑥 + 𝑥 ( 𝑥)=0
= −0 1=0= −0 00
2 =0eea o coe c e2

◼ As a result, we obtained an evaluation
formula with relatively high accuracy.
The regression coefficients were a =0.4555, b = -0.1653, c = 0.5434, and d= -0.007345, and the determinationcoefficient was 0.992.
◼ The correlation length was 53.2 m, andthe correlation coefficient was equal to1/e.

Correlation coefficient of the tsunami inundation depth

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Numerical study ofthe macrospatial correlation of tsunami inundation depth◼ However, the values of the spatial correlation coefficient could exhibit various
fluctuations due to changes in the assumed faults, seabed and land topography.
◼ In the following, we set the slope of the land to 1/2 or 1/4, we re-evaluated thespatial correlation coefficient without changing the other conditions, and wedetermined how the values change.

◼ Below Figs. (a) and (b) show the simulation results of the tsunami inundation depth.
When the slope of the land is set to 1/2 or 1/4, the tsunami penetrates and floods into
the inland area where the terrain is more complicated. For this reason, the maximum
tsunami wave height along the coastline is reduced by approximately 1-2 m.

(a)

(b)
Tsunamiinundation
depth (m)500 m

Tsunamiinundation
depth (m)500 m(a) – Tsunami inundation depth in the case of a 1/2 gradient (b) – 1/4 gradient14

Numerical study ofthe macrospatial correlation of tsunami inundation depth◼ Below Figs. 8 (a) and (b) show the results of the relationship between the shortest
distance from the coastline and the tsunami inundation depth.
◼ We regressed the exponential function 𝐻 = aexp(b𝑥) using the least squares method
to determine the median values.
◼ The standard deviation of the residual 𝜀 𝑟𝑖 between the numerical simulation results
and the predicted values from the exponential function is σ2 = 1.309 in the case witha 1/2 gradient and is σ2 = 0.825 in the case with a 1/4 gradient.

(b)

(a)
𝐻 = 𝑥 ( 𝑥)= 221
= −0 001
Determination coefficient

2=01

𝐻 = 𝑥 ( 𝑥)= 1= −0 00110
Determination coefficient

2=015(a) – Tsunami inundation depth attenuation in the case of changes with a 1/2 gradient, (b) – 1/4 gradientNumerical study ofthe macrospatial correlation of tsunami inundation depth◼ Below Fig. (a) shows the result of the correlation coefficients between two points,and Fig. (b) shows the difference in the correlation coefficient from the results of thecontrol simulation without changing the gradient.
◼ From these figures, the fluctuation of ± 0.05 can be confirmed up to the distance ofapproximately 400 m between the two points; however, the spatial correlation
coefficient hardly changes even if the topographic gradient is changed. This result
indicates that we can evaluate the macrospatial correlation coefficient of tsunami
inundation depth regardless of the land gradient by using Eq. (9).

(a)

1/2 Gradient
1/4 Gradient(b)

1/2 Gradient
1/4 GradientFig. (a) – Correlation coefficients of the tsunami inundation depth in the case of the control results,
1/2
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gradient and 1/4 gradient, (b) – Differences in correlation coefficients from the control results

Conclusions
We determined the following spatial correlation features of tsunami inundation
depth as the results of the tsunami numerical experiments using nonlinear longwave equations:◼ The macrospatial correlation coefficients of the tsunami inundation depth in thetsunami run-up region have a tendency to decrease as the distance increases.
◼ The correlation coefficient decreases by approximately 0.78 at a distance of 10m, indicating a low correlation of the tsunami inundation depth. This resultoccurs 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 beevaluated by the exponential function (Eq. (9): 𝜌 𝑥 = 𝑥
𝑥 +𝑥 ( 𝑥)) regardless of the land gradient.
The results of this study can be used for a probabilistic tsunami risk assessment ofreal estate building portfolios by parties such as large companies, insurancecompanies, reinsurance companies, and real estate agencies.
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