1. Introduction

A giant earthquake off the Pacific coast of Tohoku, Japan (38.1035°N, 142.861°E, M 9.0 at 14:46:18 JST according to Japan Meteorological Agency) on March 11, 2011, generated a huge tsunami and caused 15,729 fatalities and 4,539 missing in the Hokkaido, Tohoku and Kanto regions (The National Police Agency, as of 24 Aug. 2011). The USGS W-phase moment tensor solution shows a shallow dipping thrust mechanism with a strike parallel to the Japan trench, indicating an interplate earthquake associated with the subduction of the Pacific plate (Fig. 1). Continuous GPS data revealed coastal subsidence as great as one meter along the coast of the Tohoku area (Geospatial Information Authority of Japan (GSI), 2011).

Fig. 1.
figure 1

Epicenter of the 2011 Tohoku Earthquake (red star), W-phase MT solution by USGS, and stations that recorded the tsunami. Triangles, diamonds and squares indicate the locations of coastal (tide or wave) gauges, offshore GPS wave gauges and ocean-bottom tsunami sensors or DART (Deep-ocean Assessment and Reporting of Tsunamis) systems, respectively. Colors indicate operating agencies (yellow: Japan Meteorological Agency (JMA), blue: Japan Coast Guard (JCG), green: Japan Agency for Marine-Earth Science and Technology (JAMSTEC), orange: National Oceanic & Atmospheric Administration (NOAA), light blue: Japan’s Nationwide Ocean Wave information network for Ports and HArbourS (NOWPHAS), and purple: University of Tokyo). The thin rectangle indicates the region shown in Fig. 3.

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The Pacific coasts of Tohoku have suffered from large tsunamis in the past. The 1896 Sanriku earthquake caused large (up to 38 m of run-up height) tsunamis on the Sanriku coast with 22,000 casualties, although the seismic shaking was not very strong. The source of the tsunami of this “tsunami earthquake” (Kanamori, 1972) was located near the trench axis (Tanioka and Satake, 1996b). The 1933 Sanriku earthquake, a large normal-fault earthquake (Kanamori, 1971) also caused a large (up to 29 m) tsunami with about 3,000 casualties. The tsunami heights from these Sanriku earthquakes were less than a few meters in the Sendai plain. However, the 869 Jogan earthquake produced a large tsunami inundation up to a few kilometers (Minoura and Nakaya, 1991), which was modeled by an interplate earthquake (Satake et al., 2008).

On the basis of large historical earthquakes, The Earthquake Research Committee (ERC) (2009) made long-term forecasts in northern, central, and southern Sanriku-oki, Miyagi-oki, Fukushima-oki and Ibaraki-oki regions, as well as tsunami earthquakes near the trench axis. The estimated earthquake size was M ~ 7.7 for southern Sanriku-oki, M ~ 7.5 for Miyagi-oki, M ~ 7.4 for Fukushima-oki, M ~ 6.7–7.2 for Ibaraki-oki, and M ~ 8.2 for offshore tsunami earthquakes. The ERC also considered a multiple segment rupture of the Miyagi-oki and southern Sanriku-oki regions with an estimated size of M ~ 8.0.

In this paper, we estimate the tsunami source of the 2011 Tohoku Earthquake by inverting tsunami waveforms recorded at tide and wave gauges, GPS wave gauges and ocean-bottom tsunami sensors.

2. Tsunami Data

Because of the severity and wide extent of the tsunami damage, tsunami field surveys are still ongoing. Preliminary surveys reported tsunami run-up heights exceeding 30 m (The 2011 Tohoku Earthquake Tsunami Joint Survey Group, 2011).

The 2011 tsunami was also recorded instrumentally at various gauges. Many coastal tide gauges on the Pacific coast stopped recording after the first tsunami with >9 m amplitude, because of power failure or because the stations were washed away by the tsunami. Three offshore gauges, one GPS wave gauge (Iwate S at ~200 m water depth) and two cabled pressure-gauges (TM-2 at ~1,000 m and TM-1 at 1,600 m depth), recorded the two-stage tsunamis (Fig. 2). The water level gradually rose up to 2 m during the first 10 minutes, then an impulsive tsunami wave with 3–5 m amplitude having a shorter period (~8 min) was recorded. At southern GPS (Fukushima) and coastal (Onahama) gauges, two similar pulses were recorded, although their periods were more similar to each other.

Fig. 2.
figure 2

The observed tsunami waveforms (red curves) at offshore bottom-pressure gauges (TM-1, TM-2), GPS wave gauges, and a coastal tide gauge. The original data were resampled at 1 min intervals. Blue and green curves are computed tsunami waveforms from the large slip near the trench (blue) and deeper interplate slip (green) as shown in Fig. 3(b).

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We use tsunami waveforms recorded at coastal tide and wave gauges, and offshore GPS wave gauges, deep-ocean bottom-pressure gauges (Fig. 1). While most of the coastal tide gauge stations went off-scale and did not record the tsunami peak, the arrival time and the initial slope recorded provide information on the tsunami source, hence we include them in the inversion. In order to retrieve a tsunami signal, we first approximate a tidal component by fitting a polynomial function, and then remove it from the original record.

3. Waveform Inversion

We divide the tsunami source into 40 subfaults (50 km × 50 km) to cover the aftershock area (Fig. 3 and Table 1). Strike 193°, dip 14°, slip angle 81° are from the USGS’s W-phase moment tensor solution. Static deformation of the seafloor is calculated for a rectangular fault model (Okada, 1985). We also consider the effects of coseismic horizontal displacement in regions of steep bathymetric slopes (Tanioka and Satake, 1996a). Tsunami waveforms are calculated assuming a constant rise time (or slip duration) of 30 s on each subfault, considering that the duration of the first pulse of source time function was ~ 1 min (e.g., USGS). We assume an instantaneous rupture. The slip distribution estimated from tsunami waveforms, however, is not sensitive to the choice of rise time or rupture velocity (Fujii and Satake, 2007).

Fig. 3.
figure 3

(a) Slip distributions estimated by tsunami waveform inversion. The color bars are shown below. The subfault numbers are shown in the northernmost and southernmost subfaults. The star shows the mainshock epicenter. Circles indicate aftershocks within one day after the mainshock (JMA data). Dashed lines indicate regions where the probabilities and size of future subduction-zone earthquakes were estimated by the Earthquake Research Committee (2009). Coastal and offshore stations (the same symbol as Fig. 1) are also shown. (b) Seafloor deformation computed from the estimated slip distribution. The red solid contours indicate uplift with a contour interval of 1.0 m, whereas the blue dashed contours indicate subsidence, with a contour interval of 0.5 m. The light blue and dark green frames show the subfaults with >2 m slips. The synthetic tsunami waveforms are computed from these two regions (near the trench axis and interplate slip) and shown in Fig. 2.

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Table 1. Slip distributions estimated by tsunami waveform inversion.
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To calculate tsunami propagation from each subfault to gauge stations, the linear shallow-water, or longwave, equations are numerically solved by using a finite-difference method (Fujii and Satake, 2007). Whilst the non-linearity becomes important around coastal tide gauge stations, we have confirmed, by comparing the nonlinear and linear computations, that they produce similar arrival times and initial slopes, while the peak amplitudes (not observed) are different. We use two sets of bathymetric data for calculating tsunami waveforms or Green’s functions. For the DART stations in the Pacific ocean, a 2′-interval grid for 125°–175°E and 15°–55°N is resampled from GEBCO_08 30″-grid data. For stations around Japan, a 30″-interval grid of JTOPO30, provided by the Marine Information Research Center, is used for 128°–150°E and 25°–45°N. Time steps of 3 s and 1 s are used to satisfy the stability condition for the finite-difference method in the far-field and near-field computations, respectively.

For the inversion, we use a non-negative least-square method (Lawson and Hanson, 1974) and a delete-half jack-knife method (Tichelaar and Ruff, 1989) to estimate slips and errors, respectively. The observed tsunami waveforms were resampled at 1 min intervals, hence synthetic waveforms are also computed at 1 min intervals. The total number of data points used for the inversion is 2818. Considering the different amplitude and signal durations of tsunami waveforms recorded on DART gauges, GPS gauges, and coastal tide gauge data, we use variable weights so that the relative powers become similar. We weight nearby GPS gauge and bottom-pressure data 10, 20, or 30 times, and near-source tide gauge data 2 times or 10 times, because the duration of these data are shorter than the far-field stations. We also weight the DART data ten times, because the amplitudes are much smaller.

4. The Slip Distribution

The inversion results are shown in Fig. 3(a) and Table 1. The result shows a tsunami source length (with >2 m slip) of about 350 km, extending from over southern Sanriku-oki, Miyagi-oki, Fukushima-oki as well as near the trench axis. The largest slips of more than 40 m are estimated along the Japan trench axis off southern Sanriku-oki (subfaults 4 and 5). Around the epicenter, in the southern Sanriku region (subfaults 14 and 15), the estimated slip is 28–34 m. On the deeper subfault in the Miyagi-oki region (subfaults 24, 25, 34 and 35), the slip is 9–23 m. To the north of the epicenter, 5–11 m slip is estimated in a part of central Sanriku region (subfault 13 and 23). To the south, the slip is ~ 10 m in the Fukushima-oki region (subfaults 27 and 28), and <3 m in the Ibaraki-oki region (subfaults 29 and 30). The total seismic moment is calculated from these slip distributions as 3.8 × 1021 N m (Mw = 9.0) by assuming a rigidity of 5.0 × 1010 N/m2 for all the subfaults. The slip distribution and the size clearly indicate that multiple segments of the ERC’s long-term forecast ruptured simultaneously.

The synthetic waveforms generally agree with the observed ones at most stations (Fig. 4). The two-stage tsunami observed at offshore gauges (TM-1, TM-2, Iwate M, and Iwate S) is well reproduced. The initial part of the tsunami, a small negative wave followed by a gradual increase at coastal tide gauge stations (Miyako, Kamaishi, Ofunato and Soma) is also reproduced.

Fig. 4.
figure 4

Comparisons of the observed (red curves) and synthetic (blue curves) tsunami waveforms computed from the estimated slip distribution. Time ranges shown by solid curves are used for the inversions; the dashed parts are not used for the inversions, but shown for comparison. Note the same vertical scales for bottom-pressure gauges (the upper left two columns), GPS wave gauges (upper central two columns) and coastal tide and wave gauges (upper right one column and bottom columns). See Figs. 1 and 3(a) for the station locations.

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Seafloor deformation calculated from the estimated slip distribution (Fig. 3(b)) shows a very large uplift (>10 m) near the trench axis and about 5-m uplift near the epicenter. Near the coast, subsidence up to 2 m, as observed by the GPS data (GSI, 2011) is predicted.

5. Discussions and Conclusions

The large slip estimated near the trench axis is similar to the 1896 Sanriku earthquake (Tanioka and Satake, 1996b), although the 2011 slip was much larger. Large offshore slip is responsible for the very large and destructive tsunami on the Sanriku coast. To confirm this, we computed tsunami waveforms from the large slip near the trench axis, and compared it with the tsunami waveforms at selected offshore stations (Fig. 2). The computed tsunami waveforms show later arrivals than the observed initial tsunami arrivals, but reproduce the large impulsive tsunamis.

Unlike the 1896 earthquake, large slip was also estimated on the plate interface at a deeper depth in the southern Sanriku-oki and Miyagi-oki regions. The initial gradual rise of sea level observed at some stations is due to interplate slip (Fig. 2). Large tsunami inundation (>5 km) in the Sendai plain may be due to such a nearshore, deeper slip on the plate interface. For the 869 Jogan tsunami, a similar large inundation (a few km) estimated from the tsunami deposits was reproduced only by an interplate model. A narrow fault near the trench axis (“tsunami earthquake” model) or an outer-rise normal fault model could not produce a large inundation, because of the shorter wavelength of seafloor deformation (Satake et al., 2008).

The inversion of available tsunami waveforms showed that the large tsunami from the 2011 off the Pacific coast of Tohoku Earthquake was produced by both a very large displacement near the trench axis and a deeper interplate slip in the southern Sanriku-oki, Miyagi-oki, and Fukushima-oki regions. The former explains the largest and impulsive tsunami waveforms, while the latter reproduces the initial part of the tsunami waveforms, as well as the large inundation on the Sendai plain. The survey results of coastal tsunami run-up, for example >30 m near the central Sanriku-oki region, may require additional tsunami source(s), not revealed by available tsunami waveform data.