2021/7/3
30 th International Tsunami Symposium
Room1 Hazard and risk assessment e90064

Probabilistic tsunami inundation assessment
using mode decomposition method
ーA case study for Kamakura cityー

Yo Fukutani1 ・Shuji Moriguchi2 ・Kenjiro Terada2 ・Yu Otake2
1 Kanto Gakuin University 2Tohoku University

1

Background and Objective◼ Probabilistic Tsunami Hazard Assessment (PTHA) should be used for a basis ofheight of seawall, nuclear safety, and risk assessment in internationalorganizations, insurance company or real estate sectors.
◼ In PTHA, we need to calculate the possible frequency distributions of tsunami
hazards, which usually requires a significant computational load.

Seawall

Nuclear safety

Global Assessment report

Past studies of Probabilistic Tsunami Hazard Analysis➢ Logic tree approach（Geist & Parsons (2006), Annaka et al. (2007) etc.）➢ Stochastic rupture model (Goda et al. (2014), Miyashita et al.（2020) etc.）➢ KL expansion method (LeVeque et al. (2016), Sepulveda et al. (2017) etc.)➢ Green’s function approach (Løvholt et al. (2015), Lorito et al. (2015) etc.)

Most of them require a large computational load because they
assume hundreds to thousands of earthquake fault cases.
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Background and Objective【Objective】◼ Carry out the probabilistic tsunami inundation assessment
considering the uncertainty of the earthquake fault with
reducing the computational load by evaluating the spatial
correlation mode of tsunami inundation depth using thetsunami
numerical analysis
andthe eigen-orthogonal
decomposition.

3

Tsunami source - Sagami trough earthquake ◼ Three patterns of the Sagami trough megathrust earthquake (Mw 8.7) publishedby the committee of the model for Tokyo Metropolitan Earthquake (2014).
◼ The earthquake has three levels of slip: a very large slip region (37.67 m), alarge slip region (18.83 m), and a background region (9.42 m) to satisfy theMw 8.7 for the entire fault.
(a)

(b)

West
asperity

Centerasperity(c)

The Sagami trough megathrust earthquake（Satake, K. (2015)）

Eastasperity

Asperity location of theSagami troughmegathrust earthquake

Tsunami numerical simulation◼ We calculated the initial tsunami water level from the fault parameters using theequation of Okada (1985), and performed the tsunami numerical simulation for
3 hours after the earthquake occurs under the following conditions.
(a)

Tsunami numerical simulation condition(b２-D nonlinear shallow water equations

Governingequation（TUNAMI-N2）

𝜕𝜂 𝜕𝑀 𝜕𝑁++=0𝜕𝑡 𝜕𝑥 𝜕𝑦Domain 2 (90m)𝜕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+
𝑁 𝑀2 + 𝑁 2 = 0
𝜕𝑡 𝜕𝑥 𝐷
𝜕𝑦 𝐷
𝜕𝑦 𝐷 7ൗ3Integrationmethod

Staggered-leap frog method

Initialcondition

Initial water levelfrom the fault parameters using Okada(1985)

Boundarycondition

Open boundary

Tide
Mesh size
Time interval

Simulationarea(4 nesteddomains)

Domain 1 (270m)(a)

Domain 2 (90m)
T.P. +0.9 m（Kanagawa prefecture）(b)
Domain 4 (10m)Domain 3 (30m)

270 m → 90 m → 30 m → 10 m
0.6 s Domain 1 (270m)

Domain 2 (90m)

5Uncertainty of earthquake fault◼ Consideration of changes in fault slip (Mw8.7±0.1) and fault depth (+2km, ±0km,-1km) → Tsunami numerical simulations of 3 x 3 = 9 cases

West asperity
model（Kamakura）(a)

Mw 8.6
Depth +2.0 km

Mw 8.6
Depth ±0.0 km

Mw 8.6
Depth －1.0 km

Mw 8.7(b)Depth +2.0 km

Mw 8.7
Depth ±0.0 km

Mw 8.7
Depth －1.0 km(c)Calculations forall 27 cases

Inundation depth (m)Westasperity

ControlMw 8.8Depth +2.0 km

Mw 8.8Depth ±0.0 km

Mw 8.8Depth －1.0 km

6Uncertainty of earthquake fault◼ Consideration of changes in fault slip (Mw 8.7±0.1) and fault depth (+2km,±0km, -1km) → Tsunami numerical simulations for 3 x 3 = 9 cases
Mw 8.6
Depth +2.0 km

Mw 8.6
Depth ±0.0 km

Mw 8.6
Depth －1.0 km

Mw 8.7
Depth +2.0 km

Mw 8.7
Depth ±0.0 km

Mw 8.7
Depth －1.0 km

Center asperity
model（Kamakura）(b)

ControlMw 8.8Depth +2.0 km

Calculations forall 27 cases

Inundation depth (m)Centerasperity

Mw 8.8Depth ±0.0 km

Mw 8.8Depth －1.0 km

7Uncertainty of earthquake fault◼ Consideration of changes in fault slip (Mw 8.7±0.1) and fault depth (+2km,±0km, -1km) → Tsunami numerical simulations for 3 x 3 = 9 cases(a)(b)
Mw 8.6
Depth +2.0 km

Mw 8.6
Depth ±0.0 km

Mw 8.6
Depth －1.0 km

Mw 8.7
Depth +2.0 km

Mw 8.7
Depth ±0.0 km

Mw 8.7
Depth －1.0 km

East asperity
model（Kamakura）(c)

ControlMw 8.8Depth +2.0 km

Calculations forall 27 cases

Inundation depth (m)Eastasperity

Mw 8.8Depth ±0.0 km

Mw 8.8Depth －1.0 km8Eigenmode of Inundation Depth Distribution(Singular Value Decomposition)◼ We created a data matrix 𝑿 for the inundation depth value (row direction) andthe analytical case (column direction).

𝒙𝟏𝟏
𝑿=𝒙𝒎𝟏

… 𝒙𝟏𝒏
⋱
… 𝒙𝒎𝒏

𝒎 dimensions（Spatial mesh numbers）301= 301×401 =120701

Mw 8.8Depth +2.0 km

𝒏 dimensions（Analysis case numbers）= 27401◼ We extracted the eigenmodes of the inundation depth by singular value
decomposition of the data matrix 𝑿.

𝑿 = 𝑼𝜮𝑽𝒕
Left singular vector U|
𝑼 = 𝒖𝟏|⋯|𝒖𝒏|

Singular vector𝜆1
𝜮= 000 0⋱ 00 𝜆𝑛Right singular vector Vー𝒗𝟏⋮
𝑽=ー 𝒗𝒏ーー◼ We can understand the information about the spatial correlation between meshesof inundation data by the left singular vector 𝑼9C

Extracted eigenmodes (Left singular vector 𝒖𝑘 ）◼ Positive correlation between meshes of the same sign and negative correlation
between meshes of different signs are expressed.
Mode 1𝒖1Mode 2𝒖2Mode 3𝒖3Mode 4𝒖Mode 5𝒖Mode 6𝒖Mode 7𝒖7Mode 8𝒖

Mode 9𝒖

Spatial mode
only from 1 to 9

※Positive values areshown in red, andnegative values areshown in blue.

Spatial distribution of 𝒖𝑗 ( j = 1, …, 9) constituting the left singular vector 𝑼（Spatial mode）10

Construction of surrogate model◼ The inundation depth data vector 𝒙𝑗 for a given analysis case 𝑗 can be expressed
as follows

𝑿 = 𝑼𝜮𝑽𝒕|
𝑿 = 𝒙1
|

… |
… 𝒙𝑗
… |

𝑛𝒙𝑗 = ෍

𝑘=1|
= 𝒖1
|

… |
… 𝒖𝑗
… |
𝑛

𝑇𝒖𝑘 (𝜆𝑘 𝑣𝑘𝑗
൯=෍𝝀1⋮
0… 0⋱ ⋮… 𝝀𝑗―⋮―𝒗1⋮𝒗𝑗― 𝑇⋮―

𝑛𝑘=1

𝜆𝑘 𝑣𝑗𝑘 𝒖𝑘 = ෍

𝑘=1

𝛼𝑗𝑘 𝒖𝑘𝛼𝑗𝑘 ：Coefficient of 𝑗th case corresponding mode 𝑘n :Number of analysis case （n=27）◼ 𝒙𝑗 can be expressed as a linear sum of the left singular vector 𝒖𝑘 and thecoefficients 𝛼𝑗𝑘 .
◼ Using this relationship, we can obtain the sample vector of 𝒙𝑗 by giving thecoefficients 𝛼𝑗𝑘 through a Monte Carlo simulation that takes into account the11
uncertainty of the fault parameters.

Estimation of coefficients 𝛼𝑗𝑘 corresponding to each mode（in case of 2 variables）
ModeMode(a)(b) 11

𝛼𝑗𝑘Gaussian Kernel（ガウシアンカーネル）(𝒙 𝒙 (𝒙 𝒙𝒙 𝒙 =(2Joint
prior
distributionof and
観測値と予測値の同時分布((𝑁 0𝒎((観測値 と予測値
津波数値計算結果(𝑁 0(の同ModeMode56

Bayesian inference
ベイズ理論
観測値
Posterior与えられたもとでの予測値
predictive distribution | that従う条件付き分布| （事後分布）
follows the prediction
given the training
function|𝑁(𝒎 𝑽
1𝒎 = ((1 (𝑽 =((൯観測値与えられたもと従う条件付き分布 | （事𝛼𝑗𝑘

モード値◼ We estimated the coefficients 𝛼𝑗𝑘 to thefault parameters by using GaussianProcess Regression (GPR).

|𝑁(𝒎 𝑽𝒎 = (𝑽 =((Estimation examples of coefficients 𝛼𝑗𝑘 (a) Mode 1，(b)Mode
5. The curved surface represents the mean value. The red
dots are from the results of tsunami numerical calculation.12

Probability distribution of slip scale, depth and asperity
Variables

Slip scale
factor SFault depth

D [km]

Asperity
locationA

Average

1.00

0.00

Standard
dev.

0.35

2.0

Probability
distribution

Basis forvariability setting

Log normal

Mw is suggested to vary by
about ±0.1, considering
previous earthquakes with thesame fault area. (The Nuclear
Civil Engineering Committee,JSCE, 2011)

NormalThere was a standard
deviation of several kilometers
based on the results ofresearch evaluating the depthof the upper edge of thePhilippine Sea Plate.

Uniform distribution[-1.0~1.0]West asperity：-1
Center asperity：０
East asperity：１◼ Finally, we can reconstruct the inundation depths and distributions by setting
the probability distribution of the fault parameters.
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Tsunami inundation distribution samples from the surrogate model
Monte-Carlo samples of tsunami inundation distributions◼ We estimated the coefficients
𝛼𝑗𝑘using the probabilitydistributionsoftheinputparameters, and generated anumber of samples of tsunami
inundation depth distributions
considering spatial correlationfrom the linear sum of eachmode.

Sample 2

Sample 3

Sample 4

Sample 5

Sample 6

Sample 7

Sample 8

Sample 9

Sample 10

Sample 11

Sample 12

Inundation depth (m)◼ The samples generated by theMonte-Carlo simulation show agood representation of theoriginaltsunami
inundation
depth distribution.

Sample 114

Verification of the surrogate model(a)

Mw: 8.75
Depth: +1.0 km
Asperity: West

Numerical simulation results(a)Mw: 8.7
Depth: ±0.0 km
Asperity: West-Center(b)Surrogate model results(b)(b)
Inundation depth (m)◼ We found good
agreement with theresults of thesurrogate model andtsunami numericalsimulations.

(a)Surrogate
Surrogatemodelmodel
results(b)Inundation depth (m)◼ Comparison of theresults of tsunami
numerical simulationand surrogate modelfor different settings
than the first 27 cases.

Tsunami numerical
Numericalsimulation
simulation results(a)15Probabilistic tsunami inundation assessment◼ We performed a probabilistic tsunami hazard assessment by incorporating atime-dependent probability model (BPT distribution) of the Sagami Troughmegathrust earthquake using 10,000 Monte Carlo simulations.

density
Probability
density
Probability

(a)

(b)

B

Inundation depth (m)Brownian Passage Time
(BPT) Distribution

Numerical simulation results(a)Surrogate model results(b)(b)
50-year 10%
10％ probability(a)50-year 39％
50-year
39%probability
probability
50-yearprobabilityA
Mw: 8.7
Depth: ±0.0
Return periodof km
Asperity:
about 100
yearsWest-Center

Return period of
about 500 years

𝑃 𝑡 =𝜇𝑒𝑥𝑝
2𝜋𝛼 2 𝑡 32𝑡 𝜇2𝜇𝛼 2 𝑡𝑡：time lapse from last earthquake
𝜇：average year of earthquake
occurrence interval
𝛼：variability of earthquake
occurrence interval

50-year frequency of exceedance

Elapsed years from the last event
μ = 180
μ = 200
μ = 250
μ = 300
μ = 350

ATsunami inundation depth (m)μ = 180
μ = 200
μ = 250
μ = 300
μ = 350

BTsunami inundation depth (m)16Summary◼ We generated random distributions of the tsunami inundation depth byapplying SVD to a limited number of tsunami inundation depths andevaluating the spatial correlation of those inundation depths, and weproposed a method to evaluate tsunami hazard curves andprobabilistic tsunami inundation depths within a certain area
considering the imminent occurrence of an earthquake.
◼ Based on the good agreement between the samples generated by the
proposed surrogate model and the nonlinear long wave, we can conclude
that the proposed probabilistic tsunami inundation assessment using theproposed method is sufficiently accurate and practical.
【Future work】◼ We have a plan to use fault
models generated from aphysical stochastic slip modeland fault models with varying
fault parameters, while in thisstudy, the slip distribution offaults was generated simply in
one dimension.

Thank you for your attention.
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