Shinmoto and Baba ﻿Progress in Earth and Planetary Science(2024) 11:50
https://doi.org/10.1186/s40645-024-00647-6

RESEARCH ARTICLE

Progress in Earth andPlanetary Science

Open AccessA methodology for appropriate withdrawalof tsunami warnings based on numericalsimulations
Naoko Shinmoto1 and Toshitaka Baba2*   

Abstract
This study performed large-scale numerical simulations for predicting the attenuation of tsunamis caused by the 2011
Tohoku, Japan, and 2010 Maule, Chile earthquakes, recorded at Japanese tide gauges. Tsunami amplitude waveforms
were generated by computing the moving root-mean-square of the data for quantitative analyses. Sensitivity analysis
showed that tsunami nonlinearity and computational grid intervals near the tide gauges significantly impact the prediction of tsunami attenuation. The predicted withdrawal times of tsunami warnings agreed with the observations;however, time discrepancies were observed for advisory withdrawals at several stations. Using the proposed method,
we predicted the warning period of a great interplate earthquake in the Nankai Trough to be approximately one day.
These findings can provide critical information for disaster prevention because the withdrawal of warnings is directly
related to permission to enter coastal areas affected by the tsunami, whereas unnecessarily long warnings hinder
rescue operations.
1 Introduction
Tsunami disaster mitigation requires an understanding of
the generation, propagation, and attenuation processes oftsunamis. Understanding tsunami generation and propagation enables the prediction of tsunami arrival times
and their heights. The withdrawal of tsunami warnings
is directly related to the attenuation process. According
to the Japan Meteorological Agency (JMA) (2006) andKoshimura et al. (2007), the maximum tsunami due tothe 2006 Kuril earthquake occurred after the withdrawalof tsunami warnings in many locations along the Japanese coast. Inappropriate withdrawal may cause severe
damage because the movement of people resumes in*Correspondence:
Toshitaka Babababa.toshi@tokushima-u.ac.jp
1
Graduate School for Sciences and Technology for Innovation, Tokushima
University, Tokushima, Japan
2
Graduate School for Social and Industrial Science and Technology,
Tokushima University, 2‑1 Minami‑jyosanjima‑cho, Tokushima 770‑8506,
Japan

coastal areas after the withdrawal of tsunami warnings.
However, unnecessary long tsunami warnings should be
avoided because their withdrawal signals the end of thetsunami threat and the beginning of full-scale rescue
operations, sheltering, and other recovery activities. Prolonged evacuation under a tsunami warning can causesignificant stress to the residents. Timely withdrawal will
reduce this psychological burden and bring mental stability. Additionally, economic activities in coastal areas
are suspended during a tsunami warning, and prolonged
warnings exacerbate the economic impact. Proper withdrawal of warnings helps maintain residents’ confidencein public disaster information, resulting in prompt andappropriate evacuation actions in future disasters. Therefore, accurately predicting the tsunami attenuation process is crucial.
Hayashi et al. (2010) defined the moving root-meansquare (MRMS) amplitude of tsunami waveforms for performing a quantitative analysis of tsunami attenuation.
Using MRMS amplitude waveforms, Hayashi et al. (2010)© The Author(s) 2024. Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which
permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to theoriginal author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images orother third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line
to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory
regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of thislicence, visit http://​creat​iveco​mmons.​org/​licen​ses/​by/4.​0/.

Shinmoto and Baba ﻿Progress in Earth and Planetary Science(2024) 11:50

analyzed the tsunamis of the 2006 and 2007 Kuril earthquakes, which stuck on the Japanese coast. The MRMS
amplitude waveforms in the decay process recorded bytide gauges were approximated by an exponential function, with the time constant of decay determined using
regression analysis. JMA (2021) attempted to predicttsunami attenuation by fitting an approximate curve to
the MRMS amplitudes of the observed waveforms. Saito
et al. (2013) used numerical tsunami simulations andadjusted coefficients of bottom friction terms to reproduce the tsunami attenuation recorded at ocean bottompressure gauges for the 2011 Tohoku earthquake. Davies
et al. (2020) also demonstrated the importance of bottom
friction terms using distant tsunamis that were observedin Australia. Imai et al. (2014) found that the attenuationin coastal areas was slower than that in offshore areas.
Tanioka et al. (2019) highlighted the effect of seiche on
long-lasting tsunamis in a bay.
Despite the above-mentioned efforts, the tsunami
attenuation processes remain unclear. Currently, tsunami warnings are withdrawn based on empirical
judgments and real-time tsunami monitoring. The
empirical method by JMA (2021) can be used to predict tsunami attenuation quantitatively; however, thismethod can only be applied after tsunami decay starts.
Therefore, in this study, we developed an alternative
method based on large-scale tsunami simulations toPage 2 of 9predict the attenuation of the tsunami MRMS amplitudes. Using this method, we reproduced the attenuation of the tsunami MRMS amplitudes recorded at
Japanese tide gauges for the 2011 Tohoku, Japan, and2010 Maule, Chile earthquakes (Fig. 1). The March
11, 2011, Tohoku earthquake, which was an interplate
earthquake, occurred in the Japan Trench at a depthof 24 km with a moment magnitude (Mw) of 9.0. The
earthquake generated tsunamis with a height of morethan 9.3 m as recorded at a tide gauge in Soma City
(JMA 2011) and with a maximum run-up of approximately 40 m as measured by post-earthquake surveys (Mori et al. 2012). The February 27, 2010, Maule
earthquake, which was also an interplate earthquake,occurred in the Chile Trench, at a depth of 20 km with
Mw 8.8 (JMA 2010a). The earthquake generated a tsunami that reached the coast of Japan approximately
23 h later. A tsunami of 1.2 m height was recorded at a
port in Japan (JMA 2010b).
We performed sensitivity tests by varying the calculation conditions to understand the key factors that affectthe attenuation process. Simulation-based withdrawals of tsunami warnings and advisories were comparedwith those by the observed waveforms. Finally, we predicted the withdrawal times of tsunami warnings andadvisories for a potential future earthquake that canoccur in the Nankai Trough.

Fig. 1 The nested grid system used for tsunami simulation. a The whole computational area. Yellow dots indicate the epicenters of the earthquakes.
Moment tensor solutions were obtained from the U.S. Geological Survey catalog. b The area around Japan showing the locations of tide stations(red dots) used in this study

Shinmoto and Baba ﻿Progress in Earth and Planetary Science(2024) 11:50

2 Data and methods
We acquired tsunami waveforms of the 2011 Tohoku
and 2010 Chile earthquakes recorded by JMA tidegauges with the data sampling interval of 15 s (Fig. 1).
The tidal components were removed from the original
data to retrieve tsunami waveforms using the tide prediction software NAOTIDEJ (Matsumoto et al. 2000)
and applying a 3-h high-pass filter using the seismic
analysis code (George et al. 2013). We generated tsunami MRMS waveforms using the method described by
Hayashi et al. (2010), which considered a moving average with a time width of 64 min on the squared rootof the tsunami waveforms. The background amplitudeof the MRMS waveform calculated from data for the
3 h before the tsunami arrival was subtracted fromthe entire tsunami MRMS waveform to remove background noise.
For tsunami simulations, we solved the linear (Eqs. 1–3) and nonlinear (Eqs. 4–6) long-wave equations asfollows:
gH ∂η
∂Hu
=−
− fHv
∂t
R sin θ ∂ϕ

(1)

gH ∂η
∂Hv
=−
+ fHu
∂t
R ∂θ

(2)

1
∂η
=−
∂t
R sin θ

∂Hu ∂Hv sin θ
+
∂ϕ
∂θ



+

1 ∂
R ∂θ



MN
H +η



1 ∂
+
R ∂θ





1
∂
∂N
+
∂t
R sin θ ∂ϕ





(3)

M2
H +η

1
∂
∂M
+
∂t
R sin θ ∂ϕ

1
∂η
=−
∂t
R sin θ∂M
∂N sin θ
+
∂ϕ
∂θ



Page 3 of 9

was used to simulate tsunami inundation on land when
the water level rose above the coastal elevation.
We conducted tsunami simulations using the opensource tsunami software JAGURS (Baba et al. 2015,2017), which is a finite-difference code of the leapfrog,
staggered grid with nested domains. We used bathymetric datasets consisting of nested domains at six levels (Fig. 1b) with the grid intervals of 180, 60, 20, 20/3,
20/9, and 20/27 arc sec (approximately 23 m along thelatitude, 17–20 m along longitude directions), respectively. Gridded bathymetry data were constructedusing a combination of ETOPO1 (Amante and Eakins
2009), the Global Tsunami Terrain Model (GtTM, Chikasada 2020), and digital elevation models gridded at
5-m and 10-m intervals by the Geospatial Information
Authority of Japan (see Availability of data and materials). We simulated the coarsest domain gridded at 180
arc sec using the linear long-wave equations to stabilize the computation. We applied an open boundary
condition that assumed that tsunami propagated with√phase velocity of gH at the outer edges of the coarsest domain. For the remaining domains, we switchedbetween the linear and nonlinear equations, as shownin Table 1, to investigate the effects of tsunami nonlinearity and grid intervals on tsunami attenuation predictions. We used constant bottom friction coefficient( n = 0.025) in the nonlinear computation domains.
For the tsunami sources, we used the fault models
derived from tsunami inversion analyses (Satake et al.
2013; Yoshimoto et al. 2016) and an analytic solutiongn2g(H + η) ∂η− fN −
M M2 + N 2
7/3
R sin θ ∂ϕ
(H + η)(4)g(H + η) ∂η
gn2+ fM −
N
M2 + N 2R∂θ(H + η)7/3

(5)

MN
H +η



=−

N2
H +η



=−

(6)

where u and v are the corresponding horizontal flow
velocities along the longitude (ϕ) and co-latitude (θ )directions, η is the water level, H is the depth of the ocean
at rest, t is the time, R is the Earth’s radius, g is the gravitational acceleration, f is the Coriolis parameter andn is Manning’s roughness coefficient. M and N are thedepth-integrated flow quantities equal to (H + η)v and(H + η)u, respectively. The linear tsunami model did notinclude tsunami inundation on land, resulting in perfectreflection at the boundary between land and sea. In the
nonlinear tsunami model, a moving boundary algorithm

Table 1 Descriptions for tsunami propagation models
Model 1Model 2Model 3Model 4Model 5180 arc-sec
gridLinearLinearLinearLinearLinear60 arc-sec
gridLinearNonlinear

Nonlinear

Nonlinear Nonlinear

20 arc-sec
grid

Nonlinear NonLinear NonLinear Nonlinear Nonlinear

20/3 arc-sec
grid

Not usedNot used

NonLinear Nonlinear Nonlinear

20/9 arc-sec
grid

Not usedNot usedNot used

Nonlinear Nonlinear

20/27 arcsec gridNot usedNot usedNot usedNot used

NonlinearShinmoto and Baba ﻿Progress in Earth and Planetary Science(2024) 11:50

to calculate crustal deformation due to fault motion(Okada 1985). We also included the additional tsunami
generation by the horizontal movement of the seafloor
slope (Tanioka and Satake 1996). A hydraulic filter proposed by Kajiura (1963) was applied to the calculated
vertical deformation to account for the non-hydrostatic
effect of fluids in tsunami generation. The time step
width was at 0.1 s to satisfy the computational stability condition in the tsunami simulation. The model
was integrated for 60 and 80 h for the 2011 Tohoku
and 2010 Maule earthquakes, respectively. Simulations were performed on the FUJITSU Supercomputer
PRIMEHPC FX1000 at the Information Technology
Center, at the University of Tokyo.

3 Simulation results
The nonlinearity of tsunamis is significant in shallow
coastal areas; therefore, we inevitably used the nonlinear
long-wave equations on the domains with grid intervals
smaller than 20 arc sec. Previous studies suggested thatbottom friction in the Pacific Ocean could affect tsunami
attenuation (Saito et al. 2013; Davies et al. 2020). Therefore, we calculated tsunami propagation in the domainof 60 arc-sec grid intervals by solving the linear (Model1) and nonlinear long-wave equations (Model 2, see
Table 1). In Model 1, numerical instabilities occurred at
the coastline in the domain of the 60 arc-sec grid interval
after 48 and 72 h for the 2011 Tohoku and 2010 Maule
earthquakes, respectively. Unfortunately, the reason for
the numerical instability is unclear. Since wave energy
does not decay in the linear model, numerical errors
might have accumulated during the long calculations.
In the 2011 Tohoku tsunami simulations, the presence(red in Fig. 2a) or absence (blue) of nonlinearity did nothave a significant effect on MRMS amplitude immediately after the earthquake. However, the differences in
the MRMS amplitudes became significant approximately
24 h after the earthquake. In the linear long-wave calculations, the attenuation of the MRMS amplitude waveforms was weak. At 24 h, the tsunami waves reflectedfrom the North American continent reached the coast ofJapan. Because bottom friction is significant in shallow
waters, we speculated on energy dissipation in shallow
waters off the North American coast. Although nonlinear
effects were included in Model 2 in all the domains, theMRMS amplitudes of the simulated waveforms remainedlarger than those of the observed at Awayuki and Chichijima tide gauge stations. Conversely, the MRMS amplitude of the simulated waveforms was smaller than that of
the observed during the first 20 h at Amami.
Figure 2b shows the MRMS amplitude waveforms forthe 2010 Maule earthquake. In this case, in contrast to
the 2011 Tohoku earthquake, the MRMS amplitudes ofPage 4 of 9

Models 1 and 2 differed immediately after the arrival of
the tsunami at the gauge stations around Japan. As the2010 Maule earthquake was a far-field one, the first wave
was followed by tsunamis that propagated via various
paths, including reflections on the Chilean coast where
the nonlinear effect was significant. At Hanasaki, Mera,and Awayuki, the MRMS amplitude waveforms of Model
2 were not in agreement with those of the observed waveforms, similar to the case of the 2011 Tohoku earthquake.
Next, we used fine-grid intervals with a nested algorithm (Models 3–5 in Table 1, Fig. 3a) for the 2011 Tohoku
earthquake, which improved the agreement between
simulations and observations at Chichijima and Awayuki.
For the 2010 Maule earthquake (Fig. 3b), the effects of thefine-grid intervals were less apparent than those of the2011 Tohoku earthquake. However, the normalized rootmean-square errors improved slightly for Awayuki. Basedon these results, we conclude that the nonlinear longwave equations and fine-grid intervals near tide stations
are necessary to predict MRMS amplitude waveforms.
However, it is challenging to determine the appropriate
grid intervals because the characteristic wavelengths ofthe topography near the tide stations play a key role.

4 Discussion
4.1 Predicted and observed withdrawal times

Hayashi et al. (2010) compared the original waveformsof the tsunamis with its MRMS amplitude waveforms
and found that the peak values of the original waveforms
were always less than three times those of the MRMS
amplitudes. Based on this relationship, we set the withdrawal time of tsunami warnings to be when the MRMS
amplitude was consistently below 25.0 cm. The withdrawal time of tsunami advisories was set to be when the
MRMS amplitude was below 8.3 cm. We compared theobserved withdrawal times of warnings and advisories
with those estimated using the MRMS amplitude waveforms (Fig. 4). For tsunami warnings, the standard deviation of the difference between the calculated using Model
5 (Table 1) and observed withdrawal times was estimatedto be 4.5 h. The error was approximately 12 h when using
a prediction method in previous studies (e.g., JMA 2021).
Thus, our method can predict more accurately than previous studies.
For the 2010 Maule earthquake, the calculated withdrawals of advisories were inconsistent with the observations at Hanasaki and Amami. The standard deviation
was estimated to be 14.1 h, which was much higher thanthat of the warnings. For the 2011 Tohoku earthquake,the MRMS amplitudes did not fall below 8.3 cm even 60 h
after the earthquake at Hanasaki, Mera, and Chichijima;
therefore, we could not predict the withdrawal times.
Although we can adjust the decay curves empirically byShinmoto and Baba ﻿Progress in Earth and Planetary Science(2024) 11:50

Page 5 of 9

Fig. 2 Comparisons between moving root-mean-square (MRMS) amplitude waveforms computed using the observation (gray), Model 1 (blue),and Model 2 (red). a MRMS amplitude waveforms for the 2011 Tohoku earthquake and b MRMS amplitude waveforms for the 2010 Maule
earthquake. The values in figures are the normalized root-mean-square errors between the observed and calculated MRMS amplitude waveforms.
The MRMS amplitude waveforms for Model 1 are truncated at 48 (a) and 72 (b) hours due to numerical instabilities

Shinmoto and Baba ﻿Progress in Earth and Planetary Science(2024) 11:50

Page 6 of 9

Fig. 3 Comparisons between moving root-mean-square (MRMS) amplitude waveforms obtained using the observation (gray) and Model 2–5.
a MRMS amplitude waveforms for the 2011 Tohoku earthquake and b MRMS amplitude waveforms for the 2010 Maule earthquake. The values
in figures are the normalized root-mean-square errors between the observed and calculated MRMS

Shinmoto and Baba ﻿Progress in Earth and Planetary Science(2024) 11:50

Page 7 of 9changing the coefficients in friction terms, the physicalmechanism of changing the coefficients should be scientifically understood. In addition, because the accuracy of
the tsunami warning withdrawal was good, adjusting thecoefficient would be challenging.
4.2 Withdrawal times of tsunami warnings caused
by a great earthquake in the Nankai TroughFig. 4 Observed and calculated withdrawal times of tsunami
warnings and advisories. Red and yellow circles represent the tsunami
warnings for the 2011 Tohoku and 2010 Maule earthquakes,
respectively. Red and yellow squares represent the tsunami advisories
for the 2011 Tohoku and 2010 Maule earthquakes, respectively

Interplate earthquakes occur repeatedly in the Nankai Trough, where the Philippine Sea Plate is subducting beneath the Eurasian Plate. The recurrence intervalsof the earthquakes were estimated to be approximately
100–150 years, and almost 80 years have passed since themost recent earthquake in 1946. Therefore, we anticipate
an earthquake accompanied by strong seismic shaking anda giant tsunami. The Cabinet Office (2012) has predictedtsunamis of up to 34 m in height on the southwestern
coast of Japan. Consequently, tsunami warnings and advisories will be issued for an extended period. Therefore,
we predicted the withdrawal timing of tsunami warnings
and advisories in the event of an earthquake in the Nankai Trough. For the tsunami simulation, we used the fault

Fig. 5 Calculated moving root-mean-square (MRMS) amplitude waveforms using a fault model of the 1707 Hoei earthquake proposedby Furumura et al. (2011). The waveforms of Model 2–5 (Table 1) are shown. Dashed line indicates a threshold to withdraw the tsunami warnings.
The tsunami warnings periods were estimated to be 18.3, 15.5, 20.2, and 10.3 h at Mera, Awayuki, Chichijima and Amami, respectively

Shinmoto and Baba ﻿Progress in Earth and Planetary Science(2024) 11:50

model proposed by Furumura et al. (2011), which was used
to simulate the tsunami caused by the 1707 Hoei earthquake, the most devastating earthquake in the Nankai
Trough. We chose the MRMS amplitudes estimated using
Model 5, based on the sensitivity analysis results. Hanasaki
was excluded from the analysis because its energy attenuation was too low to calculate the advisory time.
Based on the MRMS amplitude waveforms, the tsunami warnings periods were estimated to be 18.3, 15.5,
10.3, and 20.2 h at Mera, Awayuki, Amami, and Chichijima, respectively (Fig. 5). The tsunami advisory periods
ranged from 36.3 to 48.9 h. Considering the prediction
errors, it was predicted that tsunami warnings would last
for approximately one day and advisories would last from
2 to 2.5 days on the Japanese coast for a great earthquake
in the Nankai Trough.

5 Conclusion
This study conducted large-scale tsunami simulations to
identify tsunami calculation models that can simulate
tsunami attenuation in coastal areas. We found that thesimulation-based methods could predict the withdrawal
times of tsunami warnings and advisories. Further, weinvestigated the effects of nonlinearity and computational grid intervals on tsunami simulations. The conclusions of this study are as follows.
1. Numerical simulations using the nonlinear longwave equations with nested grids were successfully
performed for 60 and 80 h for the tsunamis causedby the 2011 Tohoku and 2010 Maule earthquakes,
respectively.
2. The effect of nonlinearity was significant in the prediction of tsunami attenuation.
3. Our simulation-based method predicted the tsunami
warning withdrawal time with a standard deviation
error of 4.5 h, but the accuracy of the advisory withdrawal was not as good as that of the warning.
For tsunami advisories, the prediction errors were largeat Amami and Hanasaki. In this study, the effects of otherparameters, such as horizontal eddy viscosity and dispersion, which may play essential roles in the tsunami attenuation process, were not investigated. In future research,
we will incorporate these parameters and continue developing a long-term tsunami simulation model.
Acknowledgements
We appreciate considerate comments of anonymous reviewers and thehandling editor, Dr. Takuto Maeda. These comments were very helpful inimproving the manuscript. For our tsunami calculations, we used the FUJITSUSupercomputer PRIMEHPC FX1000 and the FUJITSU Server PRIMERGY GX2570(Wisteria/BDEC-01) at the Information Technology Center, the University ofTokyo. The Japan Meteorological Agency provided tide-level observation
records from tide stations.

Page 8 of 9

Author contributions
N.S. performed the tsunami simulations and wrote the draft of the manuscript.
T.B. managed the entire study and wrote the manuscript.
Funding
This work was supported by KAKENHI grants-in-aid from the Japan Society
for the Promotion of Science [grant numbers JP22H01308, JP22H01742 andJP22H01750] and by the Earthquake Research Institute of the University ofTokyo [ERI JURP 2023-S-B101].
Availability of data and materials
We used the tsunami software JAGURS (Baba et al. 2015, 2017) provided in
an online repository at https://​doi.​org/​10.​5281/​zenodo.​37378​16. Tidal level
observation data at 15-s sampling (tsunami) were obtained from the Japan
Meteorological Business Support Center at http://​www.​jmbsc.​or.​jp/​en/​indexe.​html. GtTM (Chikasada 2020) is available at https://​kiyuu.​bosai.​go.​jp/​GtTM/.
Digital elevation models gridded at 5 m and 10 m intervals by the Geospatial
Information Authority of Japan are available at https://​www.​gsi.​go.​jp/​kiban.
The tide prediction model NAOTIDEJ (Matsumoto et al. 2000) is available athttps://​www.​miz.​nao.​ac.​jp/​staffs/​nao99/​index_​En.​html.

Declarations
Competing interests
The authors declare that they have no competing interests.
Received: 11 March 2024 Accepted: 25 July 2024

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