1

Tsunami Source Inversion Using Time-derivative Waveform of Offshore Pressure

2

Records to Reduce Effects of Non-Tsunami Components

3
4

Tatsuya Kubota*1, Wataru Suzuki1, Takeshi Nakamura1, Naotaka Y. Chikasada1, Shin5

Aoi1, Narumi Takahashi1,2, Ryota Hino3

61

National Research Institute for Earth Science and Disaster Resilience

7

2

Japan Agency for Marine-Earth Science and Technology

83Graduate School of Science, Tohoku University

910

Abbreviated title: Tsunami source inversion using time-derivative pressure waveform

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12

Corresponding author: Tatsuya Kubota (kubotatsu@bosai.go.jp)

13

National Research Institute for Earth Science and Disaster Resilience, 3-1, Tennodai,

14

Tsukuba, Ibaraki, 305-0006, Japan

15

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Summary

17

Offshore ocean bottom pressure gauges (OBPs) are often used to estimate the18spatial distribution of the initial sea-surface height associated with offshore earthquakes

19(the tsunami source model). However, the sensors sometimes record pressure changes

20

that are neither related to tsunamis nor seafloor coseismic displacements (the non-tsunami

21

components) due to sensor rotation or tilt associated with ground shaking or due to long-22

term mechanical drift. These non-tsunami components can be a source of error when23

accurately estimating the tsunami source model and thus need to be removed to provide

24

reliable coastal tsunami forecasts. This paper proposes a new method that uses time-25

derivative waveforms of the pressure time series from OBP records to robustly estimate

26

the tsunami source model, even when OBP data are perturbed by non-tsunami

27

components. Using OBP data associated with the 2011 Off-Miyagi earthquake (Mw 7.2)

28

and the 2016 Off-Mie earthquake (Mw 5.9), the performance of the method was evaluated29

when reducing artefacts due to non-tsunami components. The tsunami source model was30

found to be largely distorted when a conventional inversion method was used (because31of the non-tsunami components). However, the artefact was dramatically reduced when

32

using time-derivative waveforms, and the predicted coastal tsunami waveforms fitted

33

reasonably with those of observations, thereby suggesting that the new method effectively

34

suppresses artefacts caused by non-tsunami components. As the tsunami source models

35

estimated from pressure and time-derivative waveforms should be similar when OBP data36

are not perturbed by non-tsunami components, we would be able to assess whether OBP

37

data are perturbed by non-tsunami components by evaluating that the tsunami source

38

models estimated from pressure waveforms and from time-derivative waveforms are

39

similar to each other.

40
41

Keywords

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Tsunamis, Waveform inversion, Tsunami warning, Numerical modelling43

1 INTRODUCTION

44

Offshore real-time tsunami observation networks have been established over the45

past few decades (e.g. Kanazawa & Hasegawa 1997; Hino et al. 2001; González et al.

46

2005; Kaneda et al. 2015; Kawaguchi et al. 2015; Kanazawa et al. 2016; Uehira et al.

472016). A cabled tsunami observation network using ocean bottom pressure gauges

48(OBPs), which is known as the Dense Oceanfloor Network System for Earthquakes and49

Tsunamis (DONET), has been constructed off southwestern Japan by the Japan Agency

50

for Marine-Earth Science and Technology (JAMSTEC; Fig. 1a) (Kaneda et al. 2015;51

Kawaguchi et al. 2015). In addition, the National Research Institute for Earth Science and52

Disaster Resilience (NIED) has constructed an observation network, which is known as

53

the Seafloor Observation Network for Earthquakes and Tsunamis along the Japan Trench

54(S-net) (Kanazawa et al. 2016; Uehira et al. 2016) off northeastern Japan. Real-time

55

tsunami records are often used to provide rapid and reliable tsunami forecasts (e.g. Titov56

et al. 2005; Tsushima et al. 2009, 2012; Baba et al. 2014; Gusman et al. 2014; Maeda et57

al. 2015; Yamamoto et al. 2016a; 2016b; Tanioka 2018). For example, Tsushima et al.

58(2009; 2012) developed a tsunami forecasting algorithm (the tsunami Forecasting based

59

on Inversion for initial Sea-surface Height; tFISH) that inverts offshore tsunami data to60

estimate the spatial distribution of initial sea-surface height (hereafter referred to as the61

tsunami source model) and then provide forecasts of coastal tsunamis based on the62

forward calculation.

63

Absolute pressure sensors manufactured by Paroscientific, Inc. (e.g. Watts &64

Kontoyiannis 1990; Eble & Gonzalez 1991) are commonly used for offshore tsunami

65

observations (e.g., Kubota et al. 2015; 2017a; 2017b; Kaneda et al. 2015; Kawaguchi et66

al. 2015; Kanazawa et al. 2016; Uehira et al. 2016). However, it has been reported that67

the pressure outputs of the Paroscientific sensors strongly depend on their orientation

68

relative to the Earth’s gravitational field, and thus their rotation or tilting can become a69source of observational errors (Chadwick et al. 2006). Wallace et al. (2016) investigated70

the Paroscientific OBP data of an Mw 5.9 earthquake that occurred to the southeast off

71

Mie-Prefecture, Japan, on 1 April 2016 (hereafter referred to as the Off-Mie earthquake,72

Fig. 1), and suggested that an OBP observed a pressure offset increase of ~10 hPa nearest73

the epicentre (corresponding to 10 cm of subsidence) (KME18 in Fig. 1a), which was

74

related neither to the tsunami nor to coseismic seafloor displacement and was actually

75

caused by the rotation or tilting of the sensor associated with ground shaking due to76

seismic waves. We here note that a pressure change of 1 hPa is equivalent to a water

77

height change of 1 cm, if assuming a water density of 1.03 g/cm3 and a gravity

78

acceleration of 9.8 m/s2.

79

Pressure sensors manufactured by Hewlett Packard, Inc. (Karrer & Leach 1969)80(hereafter, HP) have also used for offshore tsunami observations (e.g., Takahashi 1981;

81

Kanazawa & Hasegawa 1997; Hino et al. 2001), although they have been reported to have82

long-term mechanical drifts at a maximal rate of approximately 100 hPa/year (Inazu &83

Hino 2011). Kubota et al. (2017a) investigated HP pressure data associated with a Mw

84

7.2 earthquake that occurred off Miyagi-Prefecture, Japan, on 9 March 2011 (hereafter

85

referred to as the Off-Miyagi earthquake, Fig. 2) and found that the HP sensors drifted at

86

a rate of ∼5 hPa/hr (approximately 0.1 hPa/min) within a few hours after the occurrence

87of the earthquake. Long-term trends have also been found in Paroscientific sensors, with88

rates of less than tens of hPa/year (e.g. Watts & Kontoyiannis 1990; Polster et al. 2009;89

Inazu & Hino 2011; Hino et al. 2014). Pressure offset changes and long-term trends

90(hereafter referred to as the non-tsunami components) neither related to tsunamis nor to91seafloor permanent displacement are a large source of error when estimating tsunami

92

source models and providing coastal tsunami forecasts.

93

Some studies have assessed the effects of random observation errors on tsunami

94

forecasts (Takagawa & Tomita 2014; Tatsumi et al. 2014) and dynamic pressure changes

95

associated with seismic waves (Saito & Tsushima 2016). However, it has not yet been96

adequately assessed how the non-tsunami components perturb the tsunami source model,

97

and the impact of the non-tsunami components on coastal tsunami forecasts has not yet

98been investigated. Therefore, to provide accurate coastal tsunami forecasts, it is necessary

99

to develop a method that reduces the perturbation (i.e., the artefacts) of the tsunami source

100

model resulting from non-tsunami components. In the present study, we thus propose

101

such a method that uses time-derivative waveforms of the pressure time series. We also102

use OBP data associated with the 2011 Off-Miyagi earthquake and the 2016 Off-Mie

103

earthquake to assess how the conventional approach used to estimate the tsunami source

104

model is affected when OBP data are perturbed by the non-tsunami components.

105

Furthermore, we assess how the new method proposed in this study effectively reduces

106

the artefacts due to non-tsunami components.

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108

2 METHODS

109

The tsunami waveform inversion used to estimate the tsunami source model110(hereafter referred to as the tsunami source inversion) assumes that observed waveforms

111

can be expressed as a superposition of Green’s function from small unit tsunami source

112

elements (e.g. Baba et al. 2005; Tsushima et al. 2009; 2012; Kubota et al. 2015). Note

113

that this approach does not estimate the slip distribution along the fault plane, which has

114

been employed in many previous tsunami inversion studies (e.g., Satake 1989). An

115

observational equation for the conventional tsunami source inversion using the pressure

116

time series can be expressed as follows,

117𝑑"#$% (𝑡) = ∑.
,/0 𝐺," (𝑡 )𝑚, ,118

where 𝑑"#$% (𝑡) is the observed waveform at the jth station, 𝐺," (𝑡) is Green’s function,

119

which is the response to the ith unit source to the jth station (M is the total number of unit

120

sources) and 𝑚, is the amount of displacement of the ith unit tsunami source element.

121

The inversion approach follows the idea that the least-square objective function, s(m), is122

minimized, which is expressed as,

(1)

8

123

456
.
𝑠(𝐦) = ∑="/03𝑑" (𝑡) − ∑,/0 𝐺," (𝑡 )𝑚, 3 → min,124

where N denotes the total number of the stations. Equation (1) is expressed in a vector

125

form as𝐝#$% = 𝐆𝐦,

126

(2)

(3)

127

where dobs is a vector consisting of observed pressure data, G is a matrix consisting of the

128

Green’s function and m is a vector representing the displacement of the unit source

129

elements. In the tsunami source inversion, a spatial smoothing constraint is often imposed130

as follows,131

𝟎 = 𝐒𝐦,(4)

132

where a matrix, S, denotes the spatial smoothing constraints (e.g. Tsushima et al. 2009;133

Gusman et al. 2013; Kubota et al. 2015). Using equations (1) and (2), a normal equation,

134

which is to be solved, is expressed as follows,

135

B𝐝

𝐆
C = D F 𝐦,
𝛼𝐒
𝟎
#$%

(5)

136

where a constant α indicates the weight of the smoothing constraint. Hereafter, we refer137

to this approach as the conventional inversion.

138

When we assume that the vector m (the amount of the displacement of the unit

139

sources) does not depend on time, we can obtain the following observational equation by140

a temporally differentiating equation (1) as

141

GHIJKLGM(𝑡) = ∑.
,/0

GNOI
GM

( 𝑡 ) 𝑚, ,

(6)

142

where 𝜕/𝜕𝑡 denotes the temporal differentiation. Considering the objective function143(similar to equation (2)) to be minimum, we can obtain the following equation,144

𝐝̇#$% = 𝐆̇𝐦,

(7)

145

where 𝐝̇#$% and 𝐆̇ denote a vector consisting of the time-derivatives of the pressure data

146

(left hand of equation (6)) and Green’s function (right hand of equation (6)), respectively.

147

A normal equation can also be expressed as

148

̇ #$%
𝐆̇
B𝐝 C = B C 𝐦,
𝛽𝐒
𝟎

(8)

149

where β indicates the weight of the smoothing constrain (hereafter this approach is150

referred to as the time-derivative inversion). By solving the normal equations (equations

151(5) and (8)), the tsunami source model (vector m) is obtained.

152

If the pressure data (𝐝#$% in equation (1)) is perturbed by non-tsunami components

153(left panel in Fig. 3), the model vector m will not reflect the true tsunami source model.

154

Approximating the pressure offset changes resulting from ground shaking as a step

155

function (similar to OBP data at station KME18 in Fig. 1b) and the long-term trends as a156

linear function, the associated time-derivative waveforms would be an impulse function

157

and a constant, respectively (right panel in Fig. 3), and the time-derivative waveform of158

the perturbed pressure data (𝐝̇#$% in equation (7)) would be very similar to the original

159

time-derivative waveform. However, the perturbed pressure waveform is quite different

160

from the original pressure data, and it is thus expected that the time-derivative inversion

161

would be less affected by non-tsunami components than conventional inversion.

162
163

3 APPLICATION TO 2011 OFF-MIYAGI EARTHQUAKE

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3.1 Data and methods

165

We applied the time-derivative inversion to OBP data of the Off-Miyagi earthquake166

of 9 March 2011 (Kubota et al. 2017a, 2017b) to assess the performance of the time-167

derivative inversion. Kubota et al. (2017a) estimated the finite fault model of this event

168(red rectangle in Fig. 2a) by inverting OBP data obtained near the focal area (green

169

triangles in Fig. 2a). The spatial distribution of the vertical seafloor displacement

170

calculated from the finite fault model is shown by the black contour lines in Fig. 2a. Using171

the result of Kubota et al. (2017a) as a benchmark, we assessed the performance of the

172time-derivative inversion.

173

We used seven Paroscientific OBPs (GJT3, P02, P03, P06, P07, P08 and P09)

174

installed by Tohoku University and two HP OBPs (TM1 and TM2) installed by the175

Earthquake Research Institute (ERI) of the University of Tokyo (green triangles in Fig.

176

2a; described in detail in Kubota et al. (2017a)) (sampling interval of 1 s). We also used177

data from coastal GPS buoys (Kato et al. 2005; Kawai et al. 2012) of the Port and Airport

178

Research Institute (PARI) of the National Institute of Maritime, Port and Aviation

179

Technology (MPAT) (yellow squares in Fig. 2a) to assess the performance of the coastal

180

tsunami forecasts.

181

We processed tsunami data using the following procedure, which is the same as that

182

presented in Kubota et al. (2017a). We removed ocean-tide components using a

183

theoretical tide model (Matsumoto et al. 2000). To reduce the high-frequency pressure

184

changes attributed to seismic and hydroacoustic waves (e.g., Matsumoto et al. 2012; Saito

185

2013; 2017; Saito & Tsushima 2016; An et al. 2017; Kubota et al. 2017b), we then186

calculated the moving average with a time window of 60 s and applied a causal lowpass

187

filter to the OBP records (a cut-off period of 400 s) and a bandpass filter to GPS buoy

188

records (passband of 400–3600 s) (Saito 1978). Furthermore, we removed hydrostatic

189

pressure due to the water column above the OBPs, using the mean from a 20-min time

190

window recorded prior to the focal time.

191

In the tsunami source inversion, we distributed 12 × 16 small unit source elements

192

with a size of 20 km × 20 km in an area of 130 km E–W × 170 km N–S (rectangular area

193

in Fig. 4a and 4b) with a horizontal spacing of 10 km (overlapping with the adjacent unit

194

sources). Details of the unit source elements are described in Kubota et al. (2015). For

195

simplicity, displacement of initial sea-surface height was assumed to be equal to seafloor

196

displacement. We calculated the tsunami Green’s function using a linear long wave

197

equation with a finite difference method in local Cartesian coordinates (e.g. Satake 1995;

198

Saito et al. 2014), and the equations used in this study were as follows,

199

GV

200

G\

201

GZ

202

GM

GM

GM

GZ

= −𝑔X ℎ G[ ,GZ

= −𝑔X ℎ G] ,
GV

(8)

G\

= − G[ − G] ,where the parameters P and Q are the vertically-averaged horizontal velocity in x- and y-203

directions, respectively; the parameter η is the water height from the static sea surface; h204

is water depth and g0 is the gravitational constant. This equation was discretized on a

205

staggered spatial grid of 2 km by interpolating ETOPO1 1-arcmin bathymetric data

206

(Amante & Eakins 2009). The temporal grid interval was set as 1 s. We assumed that207

deformation of all unit sources started simultaneously (i.e. an infinite rupture propagation

208

velocity) and that the duration of the unit source deformation was 0 s. Static pressure

209

offsets related to seafloor permanent deformation were considered using the method

210

proposed by Tsushima et al. (2012), which subtracts the pressure change components due

211

to seafloor deformation from the pressure change due to sea-surface fluctuation at OBP

212

station points (a schematic illustration of this procedure is shown in Fig. S1). When

213

calculating Green’s function for the time-derivative inversion, we calculated the temporal214

differentiation of the calculated waveforms. Finally, we applied the same filter as those215

applied to observed waveforms.

216

In the inversion, we used a smoothing constraint weight of α = 0.5 for the217

conventional inversion and β = 0.01 for the time-derivative inversion. These values were218

determined so that the maximal displacement of the tsunami source model would be219

equivalent to that of seafloor vertical deformation calculated using the finite fault model220

of Kubota et al. (2017a) (black contours in Fig. 2a), which we considered to be the221benchmark.

222
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3.2 Validation of time-derivative inversion

224

First, to determine whether the time-derivative inversion could provide the same225

performance in resolving the tsunami source model as the conventional inversion, we

226

analysed the OBP data for the Off-Miyagi earthquake. It was considered that if the OBP

227

data had not been perturbed by non-tsunami components, then the tsunami source models

228

estimated by both inversion methods would be similar. We used a time window from 1229

to 20 min after the focal time for the inversion (white background area in Fig. 4c and 4d).

230

Fig. 4a and 4b show the tsunami source models estimated using the conventional

231

inversion and the time-derivative inversion, respectively; the results are seen to be quite

232

similar, and both the calculated pressure and time-derivative waveforms agree well with233

observations (Fig. 4c and 4d). We measured the agreement between the observed and234

calculated waveforms based on variance reduction (VR) as follows,235

VR = `1 −∑O ∑nbHOcde (fgM)hHOijki (fgM)l∑O ∑noHOcde (fgM)pmm

q × 100 (%),

(9)

236

where diobs(kΔt) and dicalc(kΔt) are the observed and calculated OBP data at t = kΔt for ith

237

OBP station, respectively (Δt is the sampling interval). We used a time window of 1 to 20

238

min after the focal time to calculate the VR, and obtained relatively high VRs for both239

pressure and time-derivative waveforms from both the conventional and time-derivative240

inversions (pressure waveform: 99.3% using the conventional inversion and 96.6% using241

the time-derivative inversion; time-derivative waveform: 97.2% using the conventional242

inversion and 97.3% using the time-derivative inversion).

243

Both inversion results effectively reproduce the leading tsunami waves observed by244

coastal GPS buoy waveforms from approximately 0–40 min (Fig. 4e). The discrepancy

245

in the latter part of the GPS buoy waveforms (after ~ 40 min) is probably related to the246

nonlinearity and a lack of fine-scale bathymetry near the coast (e.g. Satake 1995; Saito et247

al. 2014). Fig. 5a and 5b show comparisons of arrival times and maximal tsunami heights

248

of the leading wave between observed and calculated tsunami waveforms (the arrival time249

was defined as the time when the amplitude exceeded 1 cm). The arrival times and250

maximal heights of the conventional inversion (blue bars in Fig. 5a and 5b), the time-

251

derivative inversion (red bars) and the observations (black bars) are all very close to one

252

another. Based on these results, we thus concluded that the time-derivative inversion

253

provided a performance as good as the conventional inversion in estimating the tsunami

254

source model, when the OBP data are not perturbed by the non-tsunami components.

255
256

3.3 Synthetic test using datasets with non-tsunami components

257

We then assessed how the tsunami source model obtained using conventional258

inversion is perturbed by non-tsunami components (when OBP data contain non-tsunami

259

components), and assessed how use of the time-derivative inversion reduces artefacts due

260

to non-tsunami components. We prepared synthetic datasets by adding artificial pressure

261

offset changes to observed OBP data from the 2011 Off-Miyagi earthquake (i.e., pressure

262

data was artificially perturbed) and assuming pressure changes of 20–50 hPa (Fig. 6,

263

Table 1), which correspond to those due to the rotation of Paroscientific pressure sensors

264

with rotation angles of ~30–90° (Chadwick et al. 2006). We assumed the pressure offset

265change was a ramp function with a finite duration of Toffset = 10 s, in consideration of the266

duration of strong ground shaking (a few tens of seconds). The pressure offset change

267

268

poffset(t) is expressed as follows,

M

𝑝#vv%wx (𝑡) = y𝑝# × {0 (𝑡 ≤ 0)(0 < 𝑡 ≤ 𝑇#vv%wx ),(10)

𝑝# (𝑇#vv%wx < 𝑡)269

where po is the given pressure offset value (which is summarised in Table 1). After

270

perturbing the pressure data, they were then processed using the same method as that271

applied to pressure data without the perturbation (hereafter referred to as the original data).

272

After data processing, we considered the perturbed pressure data to be the observed data273

and estimated the tsunami source model. All other settings were the same as those274

employed in the original analysis described in the previous section. Note that the first 1-

275

min of data were not used for the inversion because of the instability of the pressure data.

276

The inversion results are shown in Fig. 6., where it is evident that the estimation of277

the tsunami source model with the conventional inversion (Fig. 6a) is quite different from

278

that estimated using original data (Fig. 4). The pressure waveforms calculated from the279

tsunami source model obtained by the conventional inversion (blue lines in Fig. 6c)

280

explain the artificially-perturbed (i.e., observed) pressure waveforms (grey dashed lines

281

in Fig. 6c) very well (VR = 99%), but the original pressure waveforms (black lines) are282

not explained at all (VR = -1466%). However, the time-derivative calculated pressure

283

waveforms from the tsunami source model (blue lines in Fig. 6d) do not explain the initial

284

part (< approximately 5 min) of the time-derivative waveforms relating to original

285

pressure waveforms, but the latter part is reasonably explained (VR = −142%). Although286

the tsunami source model obtained using the time-derivative inversion (Fig. 6b) is similar287

to that obtained by original data, the results are not exactly the same. The pressure288

waveforms calculated from this tsunami source model explain the original pressure289

waveforms reasonably well (red lines in Fig. 6c, VR = 44.9%), and the discrepancy found290

between the time-derivative waveforms of observational and synthetic data is much291

smaller than that for the conventional inversion (red lines in Fig. 6d, VR = 80.1%). These292

results show that the conventional inversion is unable to remove the artefacts due to293

pressure offset changes, whereas the time-derivative inversion dramatically reduces them.

294

Although the artefact is dramatically reduced by the time-derivative inversion,295

artefacts due to the offset changes are not completely removed; this is considered likely

296

to be related to the temporal smoothing effect due to the moving average and the low-

297

pass filter (grey dashed lines in Fig. 6d). However, we find that the forecasted arrival time298

and maximal amplitude at the coastal GPS buoys (blue bars in Fig. 5c and 5d) tend to be299

early and large (by ~5–10 min and ~5 cm, respectively) compared to observations (black

300bars) when conventional inversion is used, but they are reasonably explained when the301

time-derivative inversion is used (red bars in Fig. 5c and 5d).

302

We also conducted tests assuming a linear pressure trend at a rate of 0.5 hPa/min

303(Fig. S2) and smaller pressure offset values (less than 10 hPa) (Figs S3 and S4). All results

304

show that artefacts due to non-tsunami components are reduced well using the time-305

derivative inversion; however, these results are not achieved when using the conventional306

inversion. The forecast tsunami arrival time and maximal height of the GPS buoys using307

the synthetic dataset containing linear trends (Fig. S2) are shown in Fig. 5e and 5f,

308

respectively. When using the conventional inversion, the forecast arrival time is309

approximately 10 min earlier than the observation, whereas it is nearly similar to the310

observation when the time-derivative inversion is used. These synthetic tests thus

311

demonstrate that the time-derivative inversion effectively reduces the artefacts in the312tsunami source model due to the non-tsunami components and improves the forecast of313

the arrival time and maximal height of the coastal tsunami.

314

315

4 APPLICATION TO 2016 OFF-MIE EARTHQUAKE

316

4.1 Data and analysis

317

In this section, we report results of applying the time-derivative inversion to OBP

318

data from the Mw 5.9 Off-Mie earthquake (Wallace et al. 2016; Asano 2018; Nakano et319

al. 2018; Takemura et al. 2018). The pressure changes due to the tsunami with a maximal

320

amplitude of ~2 hPa (equivalent to a tsunami of ~2 cm) were clearly observed by the321

DONET OBPs, and a few hPa of pressure offset-level changes were also observed at

322

DONET1 stations near the epicentre (for example, KME17, KM19, KME20, and323

KME22) (Fig. 1b). One OBP station nearest the epicentre (KME18) observed a large324pressure offset change of approximately 10 hPa, which could be attributed to the tilting

325

or rotation of the sensors in relation to strong ground shaking, as noted by Wallace et al.

326(2016). Kubo et al. (2018) investigated the site amplification characteristics of DONET1

327

stations and found that station groups KMA and KME (blue and red inverted triangles in

328

Fig. 1a) had large site amplifications due to thick subseafloor sediments. Kubo et al.

329(2018) and Nakamura et al. (2018) reported peak ground accelerations (PGAs) of ~700

330

gal by DONET strong motion seismometers at KME18 during the 2016 Off-Mie

331earthquake, and also found that the site amplification observed at KME18 during this332

event was more than 40 times larger than that expected from the empirical relation. In

333

addition, Kubo et al. (2018) suggested that a nonlinear soil response occurred at DONET1

334

seismometers near the epicentre. These results support the idea that the pressure335

waveform at KME18 station is perturbed by non-tsunami components due to strong

336

ground shaking. Therefore, when estimating the tsunami source model, data from the337

OBP at station KME18 were excluded.

338

We processed DONET OBP data using the same method as used with the 2011 Off-339

Miyagi earthquake (cut-off period of the low-pass filter was 60 s). We estimated the340

tsunami source model (Fig. 7) by manually selecting OBP stations and time windows

341

used for inversion based on a visual inspection of OBP waveforms (drawn by thick black342

lines in Fig. S6). As it was suspected that DONET1 OBP waveforms at stations near the343

epicentre were also perturbed by non-tsunami components (as with station KME18), due344

to the large peak ground acceleration during the earthquake (Kubo et al. 2018; Nakamura

345

et al., 2018), OBP data from stations KMA03, KMD15, KMD16, KME17, KME19,

346

KME20 and KME22 were also excluded (in addition to KME18) from analysis (grey347

inverted triangles in Fig. 7). Furthermore, coastal tsunami data were not used to discuss348

the accuracy of the coastal tsunami forecast, because the observed tsunami height at the349

coast was very small (less than a few cm). We set the analytical area as 100 km × 100 km.

350

To avoid both over-fitting and over-smoothing during analysis, we used smoothing351

constraint weights of α = 0.5 for the conventional inversion and β = 0.01 for the time-352

derivative inversion, which were determined based on the trade-off curve between the353

smoothing weight and the VR values (Fig. S5).

354
355

4.2 Results

356

The estimated tsunami source models obtained from conventional and time-357

derivative inversions were found to be similar to each other (Fig. 7a and 7b, respectively),358

and a pair of uplift and subsidence areas with maximal amplitudes of approximately +3

359

cm and -2 cm, respectively, were estimated. To compare the tsunami source model with360

the seismic analysis, we calculated the seafloor vertical displacement based on the361

centroid moment tensor (CMT) solution of the U.S. Geological Survey (USGS). In this362

calculation, we assumed one planar rectangular fault such that its centre coincided with

363

the USGS centroid. The fault length, width, and slip amount were assumed using the364

scaling law of Wells & Coppersmith (1994), and vertical displacement was calculated

365

using the equations of Okada (1992). We obtained a maximal seafloor vertical366

deformation displacement of approximately 3 cm (green contours in Fig. 7), which is367

consistent with that of the tsunami source model. The vertical displacement of the tsunami

368

source model at OBP station KME18 was approximately +1 cm, which is much smaller369

than that expected from the observed pressure change (~-10 cm). This indicates that the370large offset pressure change at station KME18 was neither due to the tsunami nor to371

seafloor permanent displacement, but due to non-tsunami components. The strike of the372hinge-line between the uplift and subsidence is consistent with the strike angle of the373

finite fault model of Wallace et al. (2016) (215°, yellow rectangles in Fig. 7a and 7b).

374

The vertical displacements expected from our tsunami source model at stations KMA03,

375

KME17 and KME22 are less than 1 cm (Fig. S6a) and are therefore inconsistent with the376

observed pressure changes (displacements of approximately 1 cm). However, the arrival

377

times, amplitudes and durations of the tsunami are reasonably consistent with those of378

the observation. In addition, the expected coastal tsunami heights of the conventional and379

time-derivative inversions are similar (e.g. approximately 1.5 cm at station 301) (Fig.

380

S6e).

381

To investigate the discrepancies of pressure offset changes at stations KMA03,

382

KME17 and KME22 between calculations and observations, we conducted an inversion

383

that included OBP data obtained near the epicentre (KME18 was excluded from analysis)384(Figs S7 and S8) and compared the result with the tsunami source model excluding the385

stations near the epicentre (Fig. 7, hereafter referred to as the reference tsunami source

386

model). We found that the tsunami source model obtained using the time-derivative387

inversion (Fig. S7b) was similar to the reference tsunami source model, whereas the388

tsunami source model using the conventional inversion had a wider subsidence area389

extending around stations KMA03, KME17 and KME22 (Fig. S7a). Since the tsunami

390

source models using the conventional and time-derivative inversions should resemble one391

another when OBP data are not perturbed by non-tsunami components (as shown in the392previous section), the discrepancy of the tsunami source models suggests that OBP data

393

from stations KMA03, KME17 and KME22 are also perturbed by non-tsunami

394

components, although the amplitudes are very small (approximately 1 cm).

395

Tsuji et al. (2017) interpreted the rupture process of this earthquake as being related396

to an ancient splay fault system in an accretionary prism (Tsuji et al. 2014), based on the397

comparison of the strike angles between of the ancient splay fault and of the finite fault

398

model of Wallace et al. (2016) (yellow rectangle in Fig. 7). However, as the tsunami399

source model estimated using the conventional inversion strongly depends on non-400

tsunami components, it is difficult to discuss the rupture process of the Off-Mie

401earthquake, as previously discussed by Tsuji et al. (2014), whereas we can discuss it by402

using the time-derivative inversion, which can reduce the artefacts of the tsunami source

403

model. In addition, conducting a comparison between tsunami source models from both404

inversions is effective for distinguishing whether or not near-source OBPs contain the405

non-tsunami component. The time-derivative inversion is thus useful for discussing the406

detailed rupture processes of tsunami-associated earthquakes, which is not easily

407

achieved using the conventional inversion alone.

408

For a real-time tsunami forecast, we conducted a tsunami source inversion using409

only the early part of DONET OBP data (a time window from 1 to 5 min following the410

focal time) (Fig. 8). As there was no time to inspect OBP data to exclude waveforms

411

containing non-tsunami components from the tsunami source inversion, we also used

412

OBP data from station KME18, which contained a large apparent pressure offset change.

413

When using the conventional inversion, a large amount of subsidence (approximately -414

10 cm) was estimated around station KME18 (Fig. 8a) and results from the tsunami

415

source model were found to be quite different from those of the reference tsunami source

416

model (Fig. 7). However, results from the tsunami source model using the time-derivative

417

inversion (Fig. 8b) were very similar to those of the reference tsunami source model,

418

which suggests that the time-derivative inversion effectively reduces non-tsunami

419

components, even when providing a real-time analysis. The latter part of OBP waveforms

420

calculated from the tsunami source model obtained by conventional inversion (blue lines

421

in Fig. S8) do not match the observations at all, whereas that using the time-derivative422

inversion provide a reasonable fit (red lines in Fig. S9). Although the expected coastal

423

tsunami height is only a few cm, which is less than the noise level (Fig. S9e), the expected424

maximal amplitudes at coastal stations using the conventional inversion (blue lines in Fig.

425S9e) are nearly twice as large as the forecast using the time-derivative inversion (red426lines).

427
428

5 DISCUSSION

429

To provide an accurate and reliable tsunami forecast, it is important to quickly

430

obtain highly accurate information from the tsunami source model. We thus investigated431

the relationship between the tsunami source inversion and the end time of the inversion

432

time window using OBP data associated with the 2011 Off-Miyagi earthquake (Fig. 9).

433

Changing the end time of the inversion time window from 2 min to 20 min after the focal

434

time (the start time of the time window was fixed to 1 min), we conducted a tsunami

435

source inversion. The other settings were the same as those of the original analysis.

436

To evaluate the temporal stability of the inversion, we investigated the temporal

437

evolution of the VR for the observed and calculated waveforms using a time window of438

1–20 min (Fig. 9a). We also calculated the temporal evolution of the total volume of439

displaced seawater (V) (Fig. 9b), which is defined as follows,440

𝑉 = ∑, ∑" 3𝑢," 3 × Δ𝑥 × Δ𝑦,441

where uij is the displacement of the tsunami source at the (i, j)th grid in the x and y442

directions (Δx and Δy are the horizontal grid intervals of 2 km). The temporal evolution

(11)

443

of the VR for pressure waveforms and total volumes was found to be quite similar for the444

two inversion methods; however, the temporal evolution of the total volume was stable

445

after 10 min when the time-derivative inversion was used (Fig. 9b). These results suggest

446

that the time-derivative inversion is not necessarily better than the conventional inversion

447

with respect to convergence time, but it is slightly better in terms of stability.

448

There would be another approach to simultaneously estimate the tsunami source

449

model and the non-tsunami component at each OBP station from pressure waveforms.

450

The advantage of such approach would be that the first few minutes of data following the451

occurrence of the earthquake could be used to estimate the tsunami source model,452

although our approach did not use. However, such an approach should have a trade-off

453

between the estimated seafloor displacement and the estimated non-tsunami components454(an example of this possible trade-off is shown in Fig. S10). If a shorter time window455

were used that did not include the peak amplitude of the tsunami, it would be possible

456

that the gradual pressure change associated with the tsunami is wrongly estimated as a457

linear trend. This would be a disadvantage to provide fast (< ~ 10 min from the458earthquake) and reliable tsunami forecasts. Our approach using time-derivative

459

waveforms is advantageous in avoiding such trade-off, because only the displacement of460

the unit tsunami source elements are the unknown parameters.

461

In a practical tsunami forecast, we also need to consider the artefacts due to high-462

frequency pressure changes associated with seismic and hydroacoustic waves (e.g.,

463

Matsumoto et al. 2012; Saito 2013; 2017; An et al. 2017; Kubota et al. 2017b); although464

these effects are not included in the analysis. As the dominant period of hydroacoustic

465

waves is less than ~ 10 s (e.g., Matsumoto et al. 2012; Saito 2013; 2017), whereas tsunami

466

waves have much longer dominant periods (> ~100 s), hydroacoustic components can be467

removed from OBP waveforms by applying a lowpass filter with an appropriate cut-off

468

period. In addition, previous studies have also reported dynamic pressure changes caused469

by the reaction force from the seawater to the seafloor (in response to the seafloor

470

accelerating the seawater during seafloor displacement) (a dominant period of < ~50 s),

471(e.g., Saito 2017; An et al. 2017; Kubota et al. 2017b). However, although it appears that472

this component may affect the inversion, Saito & Tsushima (2016) found that the effects473

are only minimal, because such short-period pressure components cannot be expressed474

by the superposition of Green’s function of the tsunami (which has much longer dominant475

periods (> ~100 s)). We also note that in the practical tsunami forecast the consideration476

of the additional time to process OBP data is required. But it would not be a major concern477with respect to the proficiency of contemporary hardware, and will be even less of a478

concern when using high-performance computers developed in the future.

479

In the investigation of the temporal evolution of inversion stability, we used OBP

480

data that were not perturbed by non-tsunami components. Our results showed similar

481

temporal evolutions for both the conventional and time-derivative inversions. In the482

synthetic test we assumed that the pressure data were perturbed by non-tsunami

483

components and found that the tsunami source models for conventional and time-484

derivative inversions were very different. Based on these results, it would be very useful485

to compare tsunami source models using conventional and the time-derivative inversions486

to enable a real-time validation of pressure data quality and to distinguish whether or not487

pressure data contain non-tsunami components in real time.

488
489
490

6 CONCLUSIONSWe propose a new method using the time-derivative waveforms of the pressure

491

time series (rather than the raw pressure time series) to estimate the spatial distribution of492

initial sea-surface height (the tsunami source model) using OBP data, with the aim of493

reducing artefacts due to non-tsunami pressure components. Using OBP data associated494

with the Off-Miyagi earthquake that occurred on 9 March 2011 (Mw 7.2), the proposed495

method was found to work as well as the conventional method. We also conducted a496

performance test using a synthetic dataset and artificially perturbing OBP data. The

497

tsunami source model obtained using the conventional inversion approach provided large

498

seafloor displacements around OBPs due to artificial non-tsunami components, and the499

forecast coastal tsunami arrived earlier and had a larger amplitude than the observation.

500

However, when time-derivative waveforms were used for the inversion, artefacts due to501

non-tsunami components were dramatically suppressed, and the forecast coastal tsunami

502

waveforms reasonably matched those of the observation.

503

We also applied the new method to OBP data associated with the 2016 Off-Mie

504

earthquake (Mw 5.9), and the estimated tsunami source model was found to be consistent505

with the USGS CMT solution. The tsunami source model also suggested that OBPs near506

the epicentre contained non-tsunami components (with an amplitude of approximately 1-

507cm) because of sensor tilting or rotation. We then analysed OBP data based on quasi-real-

508

time analysis, and the estimated tsunami source model obtained using the conventional509

method provided very different results from those obtained using post analysis. However,510

the newly developed tsunami source model provided results that were quite similar to511

those obtained by careful post analysis, even when including OBP data perturbed by non-512

tsunami components.

513

We assessed the time window used for inversion to discuss the temporal stability514

of the inversion and found that the tsunami source model obtained using the time-515

derivative inversion was stable after inversion convergence (~ 10 min from the focal time),516

whereas the total volume of displaced seawater was unstable when the conventional517

method was used. For practical tsunami forecasting, it would be useful to compare

518

tsunami source models using both inversion methods to validate real-time OBP data

519

quality, as it is considered that the methods would provide identical results if OBP data520

are not perturbed by non-tsunami components and the results will be different if OBP

521

data are perturbed.

522

523

Acknowledgements

524

The authors thank Editor Frederik Simons and Reviewers Eric Geist and Sebastien

525

Allgeyer for their useful and helpful comments, which improved the manuscript. This

526

study was supported by the Council for Science, Technology and Innovation (CSTI)

527

through the Cross-Ministerial Strategic Innovation Promotion Program (SIP) entitled528“Enhancement of societal resiliency against natural disasters”. This study was also

529

supported by Japan Society for the Promotion of Science (JSPS) KAKENHI Grant

530

Numbers JP20244070 and JP26000002 and the research project “Research concerning

531

Interaction between the Tokai, Tonankai, and Nankai Earthquakes” of the Ministry of532

Education, Culture, Sports, Science and Technology (MEXT), Japan. The authors would

533

like to thank Tatsuhiko Saito, Hiroaki Tsushima, Shunsuke Takemura, and Hisahiko

534

Kubo for the fruitful discussions. The authors would further like to thank the ERI and

535

PARI for providing tsunami data, and Global CMT, USGS and F-net for the CMT

536

solution data. The figures in this paper were prepared using Generic Mapping Tools

537(GMT) (Wessel and Smith, 1998). The authors used the English language editing service

538

Editage (www.editage.jp).

539

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703
704

Wessel, P. & Smith, W.H.F., 1998. New, improved version of generic mapping tools
released, Eos Trans. AGU, 79(47), 579.

705

Yamamoto, N., Aoi, S., Hirata, K., Suzuki, W., Kunugi, T. & Nakamura, H., 2016a.

706

Multi-index method using offshore ocean-bottom pressure data for real-time

707

tsunami forecast, Earth Planets Space, 68, 128.

708

Yamamoto, N., Hirata, K., Aoi, S., Suzuki, W., Nakamura, H., Kunugi, T., 2016b. Rapid

709

estimation of tsunami source centroid location using a dense offshore observation

710

network, Geophys. Res. Lett., 43, 4263–4269.

711

712

713

Table 1. Perturbations to OBP data in synthetic test
Case

Original

Large offset

Trend

Small offset

Offset at P06

Unit

hPa

hPa

hPa/minhPa

hPa

Fig. 4

Fig. 6

Fig. S2

Fig. S3

Fig. S4

GJT3

0

+30

+0.5

+10

0

P02

0

−30

+0.5

+5

0

P03

0

+20

+0.5

+5

0

P06

0

−40

−0.5

+5

−30

P07

0

−20

−0.5

+5

0

P08

0

+50

−0.5

+10

0

P09

0

−50

−0.5

+15

0

TM1

0

0

+0.5

0

0

TM2

0

0

−0.5

0

0

134˚

134.5˚

135˚135.5˚136˚136.5˚

137˚

(a)
34˚

811
KMA01
KMA04
813

33.5˚

MRA01
301
MRA02
MRB05
MRC09
MRB06
MRA03 MRB08
MRB07

KME22

MRC12

817

MRF22
MRF25
MRF23
MRG26
MRG27
MRG29

MRC10 MRD14
MRC11
MRE18

MRE21

KME17
KMB08 KMB05
KME18 KMB07
KMD16 KMB06
KMD15
KMD13
KMD14
KMC12
KMC09
KMC10
KMC11

MRE19

F-net

32.5˚

KMA03

KME20
KME19

815
MRA04

33˚

USGS
KMA02

GCMT

816
812

814
818
0 km

100 km(b) DONET1 (LPF 60s)
KMA01
KMA02
KMA03

48
44

KMA04
40

KME17

36

KME18

1hPa~1cm

Pressure [hPa]322810hPa~10cm

24
KME19
20

KME20
KME22

16

KMD13
KMD14

12

KMD15
KMD16

8

KMB05
KMB06
KMB07
KMB08
KMC09
KMC10
KMC11
KMC12

4
0
0
12

10

20

30

(c) DONET2 (LPF 60s)

1hPa~1cm
MRA01
MRA02
MRA03
MRA04
MRB05
MRB06
MRB07
MRB08
MRC09
MRC10
MRC11
MRC12
MRD14
MRE18
MRE19
MRE21
MRF22
MRF23
MRF25
MRG26
MRG27
MRG29

10

Pressure [hPa]

8

6

4

2

0

714

0

10
Lapse Time from Earthquake [min]

20

30

715

Figure 1. (a) Location map of the 2016 Off-Mie Earthquake (Mw 5.9). Stars denote716

centroids of CMT solutions from Global CMT (Ekström et al. 2012), F-net (Kubo et al.

7172002) and USGS. DONET1 and DONET2 OBP stations are denoted by inverted and718

regular triangles, respectively. OBP symbol colours denote station groups. Each OBP

719

sensor within a group is connected to the same science node, which is a device with the

720

function of a hub that connects the sensors to the main cable system (Kaneda et al. 2015;

721

Kawaguchi et al. 2015). GPS buoys and coastal wave gauges are shown by yellow squares

722

and diamonds, respectively. Pressure time series observed by (b) DONET1 and (c)723

DONET2 OBPs.

141˚
39.5˚

141.5˚

142˚142.5˚143˚143.5˚

144˚

(a)
802

TM2

TM1

39˚

GCMT
803

38.5˚

10

-10

P06

20

P02
P08

P03

38˚

GJT3

P09

P07

804
801
0 km

100 km37.5˚(b) OBP (400s LPF)
180
160

TM2
TM1

140
P06

Pressure [hPa]

120
P02
100
P08
80
P09

20hPa~20cm

60
40
20
0

P03
P07
GJT3
-5

0

5

10

15

20

25

30

35

25

30

35

(c) Time-derivative of OBP (400s LPF)
1.8
1.6

TM2
TM1

1.4
P06
1.2

dP/dt [hPa/s]

P02
1.0
P08
0.8
P09
0.6
0.4
0.2
0.0

724

0.2hPa/s~0.2cm/s

P03
P07
GJT3
-5

0

5

10

15

20

Lapse Time from Earthquake [min]725

Figure 2. (a) Location map of the 2011 Off-Miyagi earthquake (Mw 7.2). Epicentre

726(Suzuki et al. 2012) and main rupture area (Kubota et al. 2017a) are denoted by red star727

and rectangle, respectively. Global CMT solution is also shown. Black contour lines are728

seafloor vertical displacement calculated using fault model of Kubota et al. (2017a); black729

rectangle is analysis area for tsunami source inversion; green triangles and yellow squares

730

denote OBP stations and coastal GPS buoys, respectively. (b) Observed pressure time731

series. (c) Time-derivative waveforms of OBP records.

Pressure changes(Tsunami + Permanent displacement)

Time derivative waveforms
Pressure changes

Pressure change + Offset change

Pressure change + Trend

Pressure change + Offset change

Pressure change + Trend

732
733

Figure 3. Schematic image of seafloor pressure change associated with earthquakes and734

non-tsunami components.

735
736

737

Figure 4. Inversion results for the 2011 Off-Miyagi earthquake using OBP data without

738

adding artificial perturbation (original data). Tsunami source models obtained using (a)739conventional inversion and (b) time-derivative inversion. Uplifted and subsided areas are

740

shown in red and blue, respectively; green contour lines are the seafloor vertical741

displacement expected using the fault model of Kubota et al. (2017a) with 10-cm intervals.

742

Comparisons of (c) pressure and (d) time-derivative waveforms between observed

743

waveforms (black) and calculated waveforms. Waveforms calculated from source models

744

obtained using conventional and time-derivative inversions are shown in blue and red,745

respectively. A time window from 1 to 20 min (white background area) was used in the746

inversion. (e) Comparison of coastal GPS buoy waveforms between observed (black) and747

forecast (blue and red) waveforms.

801(c) Large offset

10

30

804

802

803

801

(e) Trend

20
10
0

804

802

803

801

Maximum Amplitude [cm]

803

20

0

Arrival Time [min]802

Observed
Conv.
Time deriv.

Arrival Time [min]

30

804

20

Maximum Amplitude [cm]10

Observed
Conv.
Time deriv.

20

0

748

(a) Original

Observed
Conv.
Time deriv.

Arrival Time [min]30

20

Maximum Amplitude [cm]Observed
Forecasted (Conventional Method)
Forecasted (Time derivative Method)20(b) Original

10

0

804

802

803

801

802

803

801

802

803

801

(d) Large offset

10

0

804
(f) Trend

10

0

804

749

Figure 5. Comparison of arrival times of first waves and maximal amplitudes between

750

observation (black) and forecasts from conventional inversion (blue) and time-derivative751

inversion (red), using (a, b) original pressure data (Fig. 4), (c, d) synthetic pressure data752

with pressure offset changes (Fig. 6) and (e, f) synthetic pressure data with a long-term

753

trend (Fig. S2).

754
755

756

Figure 6. Inversion results of the 2011 Off-Miyagi earthquake using synthetic OBP data

757

containing artificial pressure offset changes. Tsunami source model from OBP data758

without artificial perturbation using conventional inversion (Fig. 4a) shown by green

759

contours; grey dashed lines in (c) and (d) are synthetic data used in analysis; other760

explanations are same as those in Fig. 4.

UPmax=2.8cm, DOWNmax=-1.4cm(a) Conventional(b) Time derivative UPmax=2.7cm, DOWNmax=-1.4cm

Contour:0.5cm
34˚

Contour:0.5cm
34˚

811

811

KMA01

KMA01
KMA02

KMA04

KMA02
KMA04

USGS

KMA03

KME20

KME20
KME17

KME19

KME18

301
KME22

33˚

KMC12
33˚

MRE19

F-net

0
136˚

1

816

812

2

GCMT

0 km
-2

137˚

KMC12
KMC10
KMC11

MRE18

GCMT

815813817
818 814

KMB06

KMD13
KMC09

MRE21

50 km

KMB07

KMD14

MRD14

MRC10

KMB08 KMB05

KMD16

KMD15

KMD13

KMC10
KMC11

F-net

0 km

761

KME22

KMB06

MRE19

-1

KME18

301

KMC09

MRE21

KME17

KME19

KMB07

KMD14

MRD14
MRE18

-2

KMB08 KMB05

KMD16

KMD15
MRC10

USGS

KMA03

815813817
818 814

50 km
-1

0
136˚

1

816

812

2
137˚

762

Figure 7. Tsunami source model of the 2016 Off-Mie earthquake obtained using (a)763conventional and (b) time-derivative inversions, without OBP data near the epicentre

764(OBPs not used in analysis are shown in grey). Colours of OBPs used in the inversion are765

the same as Fig. 1. The interval of the contour lines is 0.5 cm. Green contours denote766

seafloor vertical displacement expected from the USGS CMT solution; yellow rectangles

767

denote finite fault model of Wallace et al. (2016).

UPmax=3.9cm, DOWNmax=-9.9cm(a) Conventional(b) Time derivative UPmax=3.5cm, DOWNmax=-1.8cm

Contour:0.5cm
34˚

Contour:0.5cm
34˚

811

811
KMA01
KMA02
KMA04

USGS

USGS

KMA03

KME20
KME17

KME19
301

KME18

301
KME22

33˚

33˚

F-net

-2

768

-1

0
136˚

1

KMC10
KMC11

MRE18
MRE19

F-net

815813817
818 814

816

812

2

GCMT

0 km
-2

KMC12
KMC09

GCMT

50 km

KMB06

KMD13

KMD14

MRD14

MRE21

0 km

KMB07
KMD16

KMD15
MRC10

KMB08 KMB05

815813817
818 814

50 km
-1

137˚

0
136˚

1

816

812

2
137˚

769

Figure 8. Tsunami source model of the 2016 Off-Mie earthquake obtained using (a)770conventional and (b) time-derivative inversions with all the OBPs and a time window of771

1 to 5 min. Green contours denote seafloor vertical displacement obtained in post analysis

772(Fig. 7a); other explanations are same as those in Fig. 7.

VR for Pressure waveform (Conventional method)
VR for Pressure waveform (Time derivative method)
VR for Time derivative waveform (Conventional method)
VR for Time derivative waveform (Time derivative method)

100 (a)60

VSubsidence (Conventional method)
VSubsidence (Time derivative method)

(b)

1.0

100

40
20
95

Volume [×109m3]

VR (1-20 min) [%]773

VUplift
VUplift

1.2

80

0.8
0.6

0

0.4

-20

0.2

-40

VTotalVTotal

1.4

ZOOM UP for VR>90%

2

4

6

8

10

12

14

Time Window Length [min]

16

18

20

90

0.0

2

4

6

8

10

12

14

16

18

20

Time Window Length [min]774

Figure 9. (a) Temporal evolution of VR between observed and calculated waveforms

775

using time window from 1 to 20 min. VR of pressure waveform obtained using776

conventional (blue) and time-derivative (red) inversions, and time-derivative waveform

777

obtained using conventional (light blue) and time-derivative (red) inversions. (b)778

Temporal evolution of volumes of displaced seawater. Temporal evolution of total

779

displacement (blue and red), uplifted region (light blue and orange) and subsided region

780(purple and pink) are shown.

781

782

Supplementary materials
EQ(Propagation)
Pressure waveform
Water depth

Sea surface

Sea surface elevation

OBP

Seafloor

Seafloor displacement783

Time-derivative waveform

784

Figure S1. Schematic illustration used to calculate Green’s function for conventional and785

the time-derivative inversions.

TrendUPmax=38.7cm, DOWNmax=-21.7cm
Contour:10cm(a) Conventional

UPmax=37.2cm, DOWNmax=-14.0cm
Contour:10cm(b) Time derivative

TM2 TM1802

TM2 TM1

802

39˚

39˚
803

0.5hPa/min

803

P06
P02

0.5hPa/min
P06
P02

GJT3
P03

GJT3
P03

P09

P07

804

38˚

P08

801

141˚

400

-20

142˚

0

20

143˚

-20

0

801

[cm]
-40

P09

P07

804

38˚

P08

[cm]

40

-40

144˚

141˚

(c) Pressure waveform

142˚

20

143˚

40
144˚

(d) Time derivative waveformm

TM2

360

-0.5hPa/min
0.5hPa/min

1.8

TM2

1.6

TM1

320 TM1
1.4-0.5hPa/min
0.5hPa/min

240
200
160

P02
P08
P09

-0.5hPa/min

120
80

20hPa

-0.5hPa/min
0.5hPa/min-0.5hPa/min
0.5hPa/min

GJT3

-5

0

5

10

15

20

25

1.2
1.0
0.8

P06
P02
P08
P09

0.6

P03
P07

40
0

dP/dt [hPa/s]

Pressure [hPa]

280 P06

30

0.4

P03

0.2

P07

0.0

GJT3

35

-5

0.2hPa/s~0.2cm/s

05Lapse Time from Earthquake [min]
70

10

15

20

25

30

35

Lapse Time from Earthquake [min](e) GPS Buoy waveform
804

60

50
802

Height [cm]

40

30
803
20

10
801
0

-10

786
787

0

10

20

30

40

50

60

Lapse Time from Earthquake [min]Figure S2. Inversion results of the 2011 Off-Miyagi earthquake using synthetic OBP data788

containing artificial long-term trend; other explanations are same as those in Fig. 6.
UPmax=34.7cm, DOWNmax=-21.7cm
Contour:10cm(a) Conventional

UPmax=36.4cm, DOWNmax=-15.7cm
Contour:10cm(b) Time derivative

TM2 TM1802

TM2 TM1

802

39˚

39˚
803

10hPa

803

P06
P02

10hPa
P06
P02

GJT3
P03

GJT3
P03

P09

P07

804

38˚

P08

801

141˚

200
180

-20

142˚

0

20

143˚

-40

144˚

141˚

(c) Pressure waveform

100
80

TM1

TM2

1.6

TM1

1.4

5hPa

1.2

P02
10hPa

P08

1.8

5hPa

P06

P09

60

1.0
0.8

15hPa

0.6

P08
P09

5hPa

0.4

P03

20

P07

5hPa

0.2

P07

GJT3

10hPa

0.0

GJT3

0

5

10

15

20

25

30

35

-5

0.2hPa/s~0.2cm/s

05Lapse Time from Earthquake [min]

10

15

804
60

802

Height [cm]

40

803
20

801
0

0

10

20

20

25

Lapse Time from Earthquake [min](e) GPS Buoy waveform

789

144˚

P02

P03

-5

143˚

40

P06

40

0

142˚

20

(d) Time derivative waveform

TM2

dP/dt [hPa/s]

Pressure [hPa]

120

0

[cm]

40

160
140

-20

801

[cm]
-40

P09

P07

804

38˚

P08

30

Lapse Time from Earthquake [min]

40

50

60

30

35

790

Figure S3. Inversion results for the 2011 Off-Miyagi earthquake using synthetic OBP

791

data containing small (5–10 hPa) artificial pressure offset changes; other explanations are

792

same as those in Fig. 6.

UPmax=35.8cm, DOWNmax=-23.4cm
Contour:10cm(a) Conventional

UPmax=36.2cm, DOWNmax=-17.2cm
Contour:10cm(b) Time derivative

TM2 TM1802

TM2 TM1

802

39˚

39˚
803

30hPa

803

P06
P02

30hPa
P06
P02

GJT3
P03

GJT3
P03

P09

P07

804

38˚

P08

801

141˚

-20

142˚

0

20

143˚

-40

144˚

141˚

(c) Pressure waveform
2.0

TM2

180

TM1

1.8

TM1

1.6

P06

80

P02
P08
P09

1.2
1.0
0.8

P02
P08
P09

40

P03

0.4

P03

20

P07

0.2

P07

0

GJT3

0.0

GJT3

0

5

10

144˚

P06

0.6

60

-5

143˚

40

1.4

-30hPadP/dt [hPa/s]

Pressure [hPa]

140

100

142˚

20

(d) Time derivative waveform

TM2

120

0

[cm]

40

200

160

-20

801

[cm]
-40

P09

P07

804

38˚

P08

15

20

25

30

35

-5

0.2hPa/s~0.2cm/s

05Lapse Time from Earthquake [min]

10

15

20

25

30

35

Lapse Time from Earthquake [min](e) GPS Buoy waveform
804
60

802

Height [cm]

40

803
20

801
0

793
794

0

10

20

30

40

50

60

Lapse Time from Earthquake [min]Figure S4. Inversion results for the 2011 Off-Miyagi earthquake using synthetic OBP

795

data containing large (30 hPa) artificial pressure offset change at station P06; other796

explanations are same as those in Fig. 6.

797

798

(a) Conventional inversion

100

90

90

80

80

70

70

Variance reduction [%]Variance reduction [%]

100

60
50
40

60
50
40

30

30

20

20

10

10

0
0.01

0.02

0.05

0.1

0.2

0.5

Smoothing weight α

1

2

5

10

(b) Time-derivative inversion

00.001 0.002

0.005 0.01

0.02

0.05

0.1

0.2

0.5

1

Smoothing weight β

799

Figure S5. Trade-off curve between smoothing weight and VR, for (a) conventional

800

inversion and (b) time-derivative inversion when analysing the Off-Mie earthquake. Grey801

lines denote weight of smoothing constraint adopted in this study.

(a) Pressure waveform (DONET1)

1.6

KMA01
48 KMA02(b) Time derivative waveform (DONET1)
KMA01
KMA02

KMA03

1.4

44

KMA03

KMA04
KMA04

40
KME17

1.2

KME17

36

1hPa~1cm
1.0

32

0.2hPa/s~0.2cm/s

dP/dt [hPa/s]

Pressure [hPa]

KME18
28

10hPa~10cm

KME18
24
KME19
20 KME20

KME19
KME20
KME22

0.6

KMD13
KMD14
KMD15
KMD16
0.4
KMB05
KMB06
KMB07
KMB08
0.2
KMC09
KMC10
KMC11
KMC12
0.0

KME22
16 KMD13
KMD14
12

0.8

KMD15
KMD16

8 KMB05
KMB06
KMB07
KMB08
4 KMC09
KMC10
KMC11
KMC12
0
0

10

20

0

Lapse Time from Earthquake [min]12(c) Pressure waveform (DONET2)

MRA01

1hPa~1cm
1.0

MRA03

MRA03

MRA04

0.8

MRB06

MRB08
MRC09

MRC09

dP/dt [hPa/s]

Pressure [hPa]

MRB08

MRC10
MRC11
MRC12

0.6

MRC12
MRD14
0.4

MRE18

MRE18
MRE19

MRE19

MRE21

MRE21

MRF22

MRF22
MRF23

0.2

MRF23

MRF25

MRF25

MRG26

MRG26
MRG27

MRG27
0

MRC10
MRC11

MRD14

2

MRB06
MRB07

MRB07

4

0.2hPa/s~0.2cm/s

MRB05

MRB05

6

MRA02

MRA02

MRA04

8

20

(d) Time derivative waveform (DONET2)

MRA01
10

10

Lapse Time from Earthquake [min]

MRG29

0.0
0

10

20

Lapse Time from Earthquake [min]

MRG29

0

10

20

Lapse Time from Earthquake [min](e) GPS Buoy waveform
812
8

816

811

Height [cm]

6

4

2

301

1cm

813
815
817
814

0

802

818
0

10

20

30

Lapse Time from Earthquake [min]

40

50

60

803

Figure S6. Comparisons of (a) pressure waveforms and (b) time-derivative waveforms

804

at DONET1 OBPs, and (c) pressure waveforms and (d) time-derivative waveforms at

805

DONET2 OBPs for the 2016 Off-Mie earthquake between observed tsunami waveforms

806(black) and calculated waveforms calculated from tsunami source model with

807

conventional inversion (blue) and time-derivative inversion (red) with OBP data apart808

from epicentre (Fig. 7). Observed waveforms drawn by thick black lines denote the time809

windows used in the inversion analysis. (e) Comparison of waveforms at coastal GPS

810

buoys.

811
UPmax=2.8cm, DOWNmax=-2.5cm(a) Conventional(b) Time derivative UPmax=2.9cm, DOWNmax=-2.0cm

Contour:0.5cm
34˚

Contour:0.5cm
34˚

811

811

KMA01

KMA01
KMA02

KMA04

KMA02
KMA04

USGS

KMA03

KME20

KME20
KME17

KME19

KME18

301
KME22

33˚

KMC12
33˚

MRE19

F-net

0
136˚

1

816

812

2

GCMT

0 km
-2

137˚

KMC12
KMC10
KMC11

MRE18

GCMT

815813817
818 814

KMB06

KMD13
KMC09

MRE21

50 km

KMB07

KMD14

MRD14

MRC10

KMB08 KMB05

KMD16

KMD15

KMD13

KMC10
KMC11

F-net

0 km

812

KME22

KMB06

MRE19

-1

KME18

301

KMC09

MRE21

KME17

KME19

KMB07

KMD14

MRD14
MRE18

-2

KMB08 KMB05

KMD16

KMD15
MRC10

USGS

KMA03

815813817
818 814

50 km
-1

0
136˚

1

816

812

2
137˚

813

Figure S7. Tsunami source model of the 2016 Off-Mie earthquake obtained using (a)814conventional and (b) time-derivative inversions without OBP data at KME18. Green815

contours denote tsunami source distribution obtained from analysis using OBP data apart816

from epicentre shown in Fig. 7a. Other explanations are the same as those in Fig. 7.

817

(a) Pressure waveform (DONET1)

1.6

KMA01
48 KMA02(b) Time derivative waveform (DONET1)
KMA01
KMA02

KMA03

1.4

44

KMA03

KMA04
KMA04

40
KME17

1.2

KME17

36

1hPa~1cm
1.0

32

0.2hPa/s~0.2cm/s

dP/dt [hPa/s]

Pressure [hPa]

KME18
28

10hPa~10cm

KME18
24
KME19
20 KME20

KME19
KME20
KME22

0.6

KMD13
KMD14
KMD15
KMD16
0.4
KMB05
KMB06
KMB07
KMB08
0.2
KMC09
KMC10
KMC11
KMC12
0.0

KME22
16 KMD13
KMD14
12

0.8

KMD15
KMD16

8 KMB05
KMB06
KMB07
KMB08
4 KMC09
KMC10
KMC11
KMC12
0
0

10

20

0

Lapse Time from Earthquake [min]12(c) Pressure waveform (DONET2)

MRA01

1hPa~1cm
1.0

MRA03

MRA03

MRA04

0.8

MRB06

MRB08
MRC09

MRC09

dP/dt [hPa/s]

Pressure [hPa]

MRB08

MRC10
MRC11
MRC12

0.6

MRC12
MRD14
0.4

MRE18

MRE18
MRE19

MRE19

MRE21

MRE21

MRF22

MRF22
MRF23

0.2

MRF23

MRF25

MRF25

MRG26

MRG26
MRG27

MRG27
0

MRC10
MRC11

MRD14

2

MRB06
MRB07

MRB07

4

0.2hPa/s~0.2cm/s

MRB05

MRB05

6

MRA02

MRA02

MRA04

8

20

(d) Time derivative waveform (DONET2)

MRA01
10

10

Lapse Time from Earthquake [min]

MRG29

0.0
0

10

20

Lapse Time from Earthquake [min]

MRG29

0

10

20

Lapse Time from Earthquake [min](e) GPS Buoy waveform
812
8

816

811

Height [cm]

6

4

2

301

1cm

813
815
817
814

0

818

818

0

10

20

30

Lapse Time from Earthquake [min]

40

50

60

819

Figure S8. Comparisons of waveforms for the 2016 Off-Mie earthquake between

820

observed tsunami waveforms (black) and calculated waveforms calculated from tsunami

821

source model with conventional inversion (blue) and time-derivative inversion (red),822

obtained from OBP data except for KME18 (Fig. S7); other explanations are the same as

823

those in Fig. S6.

(a) Pressure waveform (DONET1)

1.6

KMA01
48 KMA02(b) Time derivative waveform (DONET1)
KMA01
KMA02

KMA03

1.4

44

KMA03

KMA04
KMA04

40
KME17

1.2

KME17

36

1hPa~1cm
1.0

32

0.2hPa/s~0.2cm/s

dP/dt [hPa/s]

Pressure [hPa]

KME18
28

10hPa~10cm

KME18
24
KME19
20 KME20

KME19
KME20
KME22

0.6

KMD13
KMD14
KMD15
KMD16
0.4
KMB05
KMB06
KMB07
KMB08
0.2
KMC09
KMC10
KMC11
KMC12
0.0

KME22
16 KMD13
KMD14
12

0.8

KMD15
KMD16

8 KMB05
KMB06
KMB07
KMB08
4 KMC09
KMC10
KMC11
KMC12
0
0

10

20

0

Lapse Time from Earthquake [min]12(c) Pressure waveform (DONET2)

MRA01

1hPa~1cm
1.0

MRA03

MRA03

MRA04

0.8

MRB06

MRB08
MRC09

MRC09

dP/dt [hPa/s]

Pressure [hPa]

MRB08

MRC10
MRC11
MRC12

0.6

MRC12
MRD14
0.4

MRE18

MRE21

MRE21

MRF22

MRF22
MRF23

0.2

MRF25

MRG26

MRG26
MRG27

MRG29

0.0
0

10

20

Lapse Time from Earthquake [min]14

12

Height [cm]

10

MRG29

0

10

20

Lapse Time from Earthquake [min](e) GPS Buoy waveform

812

816

811

8

6

301

1cm

4

2

0

824

MRF23

MRF25

MRG27

16

MRE18
MRE19

MRE19

0

MRC10
MRC11

MRD14

2

MRB06
MRB07

MRB07

4

0.2hPa/s~0.2cm/s

MRB05

MRB05

6

MRA02

MRA02

MRA04

8

20

(d) Time derivative waveform (DONET2)

MRA01
10

10

Lapse Time from Earthquake [min]

813
815
817
814
818

0

10

20

30

Lapse Time from Earthquake [min]

40

50

60

825

Figure S9. Comparisons waveforms for the 2016 Off-Mie earthquake between observed

826

tsunami waveforms (black) and calculated waveforms calculated from tsunami source

827

model obtained using all OBP data with time window from 1 to 5 min (Fig. 8). White

828

background area denotes time window used for inversion. Note that scale of vertical axis829

is different from that in Figs. S6e and S8e. Other explanations are the same as those in

830

Fig. S6.

831

EQ
Estimated trend

832
833

Time window used for inversion
Figure S10. Schematic illustration of possible trade-off situation in simultaneous

834

estimation of tsunami and linear trend; black line denotes observed tsunami waveform. It

835

is possible to estimate the tsunami as a linear trend (blue dashed line), if tsunami data

836

with a short time window are employed (denoted by red arrow).