We investigate the rupture process of the M9.0 Tohoku-Oki mega-thrust earthquake using the relatively low-frequency strong-motion records (0.01–0.125 Hz) observed at 36 K-NET and KiK-net stations, the epicentral distances of which range from 120 km to 400 km. The fault model is a rectangular plane, the length and width of which are 510 km along the Japan Trench and 210 km along subducting direction of the Pacific Plate, respectively. We perform the multi-time-window inversion analysis with a 30 × 30 km2 subfault. The derived slip model has one large slip area. This area extends from the region around the hypocenter to the shallow part of the fault plane and further to the north and south along the trench axis, located far off southern Iwate, Miyagi, and northern Fukushima prefectures. The seismic moment is 4.42 × 1022 Nm (Mw 9.0) and the maximum slip is 48 m. The slips near the coast are relatively small, except off Miyagi prefecture, which experienced a slip greater than 5 m. The shallow large slip area, which continuously ruptured from 60 s to 100 s after the initial break, radiated seismic waves rich in very-low-frequency content (<0.02 Hz). The rupture after 100 s propagating to the southern fault area, contributes to the distinct phases observed for Fukushima and Ibaraki prefectures. The relationship between the proposed rupture model and the feature of the acceleration waveforms is not straightforward and suggests the frequency dependency of the seismic wave radiation.
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 The M9.0 Tohoku-Oki earthquake of March 11th, 2011, which is officially referred to as the off the Pacific coast of Tohoku earthquake by the Japan Meteorological Agency (JMA), is the largest earthquake in Japan since the start of instrumental observation in the late 19th century. The focal mechanism and aftershock distribution indicate that the Tohoku-Oki earthquake is a mega-thrust earthquake rupturing a 500-km-long plate boundary between the subducting Pacific Plate and the Eurasian Plate (Figure 1a). This huge earthquake shook almost all of Japan, and severe ground shaking of JMA seismic intensity of 6- or greater was observed from the Tohoku district to the Kanto district of eastern Japan. A huge tsunami generated by the event caused catastrophic damage to the Pacific coast of eastern Japan, particularly to Iwate, Miyagi, and Fukushima prefectures.
 As a first step in elucidating the generation mechanisms of this devastating earthquake, we should understand the features of the rupture process. Whereas the teleseismic data are widely used to quickly obtain the overall features of the seismic source [e.g., Hayes, 2011; Shao et al., 2011], the strong-motion data observed at closer distances contain spatially and temporally more detailed information about the rupture process. The K-NET and KiK-net nation-wide strong-motion seismograph networks [Aoi et al., 2004, 2011] maintained by the National Research Institute for Earth Science and Disaster Prevention (NIED) succeeded in recording the acceleration waveforms at more than one thousand stations across Japan.
Figure 1b shows the record section of the strong-motion waveforms ordered by station latitude. For the acceleration waveforms, there are two distinct wave groups (groups 1 and 2) that arrive first at stations in Miyagi prefecture and then at northern and southern stations, separated by approximately 40 s, suggesting that two rupture events occurred just off Miyagi prefecture. Another distinct wave group, which appears after group 2, propagates from Fukushima prefecture (group 3), implying the possibility of the southern rupture event. Thus, the record section of the observed acceleration waveforms appears to provide an image of the rupture process. However, these acceleration wave groups do not exactly correspond to the distinctive phases observed in the low-frequency (0.01–0.05 Hz) velocity waveforms, which means that the acceleration record section shows only one aspect of the entire rupture. In the present study, we invert the rupture process of the Tohoku-Oki earthquake from the relatively low-frequency strong-motion waveforms and examine the characteristics of the wave radiation process.
2. Source Inversion Method
 We employed the multi-time-window linear waveform inversion method to derive the rupture process [Olson and Apsel, 1982; Hartzell and Heaton, 1983]. We assumed a rectangular fault model (Figure 1a), constructed to follow the geometry of the Pacific Plate [Hasegawa et al., 1994]. The strike and dip angles of the fault plane were set to 195° and 13°, respectively. The fault plane had a length of 510 km and a width of 210 km and was divided into 30 × 30 km2 subfaults. The projection of the assumed fault plane covers the epicenters of the earthquakes that occurred in 24 h following the mainshock. The depths of the top and bottom edges of the fault plane are 7.1 km and 54.4 km, respectively. The slip history of each subfault was represented by 25 6-s time windows, each of which was separated by 3 s, allowing slip for 78 s. The rupture starting point was set to 38.10°N, 142.85°E, 24 km deep, referring to the hypocenter determined by the NIED and JMA.
 Strong-motion data used for the inversion analysis was the S-wave portion of the 0.01- to 0.125-Hz velocity waveforms of 10 K-NET stations observed on the ground surface and of 26 KiK-net stations recorded at boreholes. The S-wave arrival times were manually picked by carefully comparing the waveforms of nearby stations. The epicentral distances of these stations range from 120 km to 400 km. The Green's functions were calculated using the discrete wavenumber method [Bouchon, 1981] and the reflection/transmission matrix method [Kennett and Kerry, 1979]. The rupture propagation effect inside subfault was considered by convolving the moving dislocation effect [Sekiguchi et al., 2002] calculated from the first time window triggering velocity. A one-dimensional underground structure model for the calculation of the Green's function was constructed for each station considering the three-dimensional crustal structure model [Fujiwara et al., 2009]. The rupture process was inverted using the least squares method with an inequality constraint [Lawson and Hanson, 1974], to limit the variation of the rake angle to within 90° centered at 90°, i.e., the pure dip slip angle. The smoothing constraint on slips was applied following the procedure proposed by Sekiguchi et al. .
3. Inversion Result
Figures 2a and 2b show the slip distribution estimated from the inversion analysis. A large slip area, in which the slip is larger than 20 m, extends from the area around the hypocenter to the shallower part of the fault plane. A slip of 20 m is approximately twice as large as the average slip over the entire fault plane. A maximum slip of 48 m is estimated to the east of the hypocenter near the trench axis, far off Miyagi prefecture. The slips near the coastline are relatively small, except off Miyagi prefecture, where the slip is greater than 5 m from the coastline to the trench. The total seismic moment of the derived model is 4.42 × 1022 Nm (Mw9.0). The first time window triggering velocity of 3.2 km/s was selected because this velocity gives the smallest misfit (Figure S1 in the auxiliary material). The observed and synthetic waveforms are compared in Figure 2c. The derived rupture model is able to reproduce the observed data very well.
 In order to observe the temporal characteristics, in Figure 3, we show the slip distribution every 10 s, the total moment rate function, and the slip velocity of each subfault. The total moment rate indicates that first remarkable moment release started 20 s after the initial break, when the rupture occurred around the hypocenter. Then, at approximately 40 s, the rupture proceeded northward along the trench axis and towards the down-dip direction. Somewhat later, the rupture also extends southward along the trench axis. The largest slip event occurred from 60 s to 100 s, with the rupture expanding toward the down-dip direction from the area along the trench axis. In this stage, large slip occurred continuously far offshore of southern Iwate, Miyagi, and northern Fukushima prefectures. The last stage starts at around 100 s, where the rupture propagated southward in the area off Fukushima and Ibaraki prefectures. The entire rupture almost ceased within 150 s.
 The slip-velocity time function (Figure 3c) shows that the subfaults around the hypocenter and the shallow part of the fault experienced two slip events. In order to represent this complex rupture, we allowed a 78-s slip for each subfault, which is much longer than that assumed in previous studies [e.g., Suzuki et al., 2010]. Comparison with the results inverted allowing a 93-s slip indicates that the slip duration of 78 s allowed in the present study is sufficient for the shallow slip area and enables stable inversion analysis, although the preferred duration appears to be rather short for the area around the hypocenter (Figure 3a).
4. Discussion and Conclusions
 One prominent large slip area (delimited by the rectangle as area 1 in Figure 2b) extends into the shallow part of the fault plane along the trench axis, far off the coast of southern Iwate, Miyagi, and northern Fukushima prefectures, where the tsunami damage was extensive. Analysis using the tsunami records [Fujii et al., 2011] indicated a large slip at a location similar to the results of the present study, which implies that the large slip within area 1 caused the devastating tsunami. The large slip near the trench is also consistent with the slip model estimated from the seafloor displacement [Ito et al., 2011], although the resolution for the shallow part of the fault plane may be lower than that for the deeper area in the present inversion analysis. We performed the inversion analysis using the strong-motion data bandpass-filtered at 0.02–0.125 Hz (Inv002) in order to compare the results obtained in the present study using the 0.01–0.125 Hz data (Inv001). The comparison clarifies the characteristics of the wave radiation of area 1. Inv001 and Inv002 estimated a large slip area at the same location, although the amount of slip during the rupture from 60 to 100 s and the final slip for area 1 derived from Inv001 are significantly larger than those from Inv002. This indicates that the rupture of area 1 for 60-100 s with a large slip and a long duration radiated seismic waves rich in the lower-frequency components (<0.02 Hz). Figure 4 shows that this rupture event contributes to the phase with the largest amplitude observed at Iwate and Miyagi prefectures arriving at approximately 50 s. For the stations in Miyagi prefecture, the slip of the area between the hypocenter and the coastline (area 2 in Figure 2b) after 60 s also contributes to this phase. The waveforms generated from area 1 fluctuate more gradually or exhibit lower-frequency behavior than the waveforms generated from area 2.
 Although the slip amount in area 2 is less significant than that in area 1, the slip amount in area 2 is larger than 5 m and makes a large contribution to the synthetic waveforms. The rupture propagating toward the down-dip in area 2 occurred from 20 to 50 s and from 60 to 100 s (Figure 3a). Figure 4 shows that the first 20- to 50-s rupture event reproduces the velocity phases that appear at almost the same time as the first acceleration wave group, indicated as group 1 in Figure 1b. The S-wave radiated at 20 s from point 2 within area 2 (Figure 2b) arrives at the time consistent with the onset of group 1. Based on these results, we can infer that the rupture of area 2 after 20 s generated group 1. On the other hand, it is difficult to speculate on the source of group 2 in Figure 1b based on the waveform synthesis because the correspondence between the velocity phase and the acceleration wave group is not clear. If we consider the possible initiation points of two distinct rupture events after 60 s (areas 1 and 2), the S-wave arrival times from the rupture of point 2 within area 2 at 60 s correspond to the onset of group 2 more closely than those from the rupture of point 1 within area 1 at 60 s (Figure 4). This is consistent with studies that simulate the higher-frequency strong motions using the empirical Green's functions, which located the sources of the two acceleration wave groups within the area between the hypocenter and the coast of Miyagi prefecture [e.g., Kurahashi and Irikura, 2011].
 The slips in the southern fault area (area 3 in Figure 2b), which ruptured later than 100 s, are also less significant than those of area 1, but greatly contribute to the velocity phases appearing 80 s or later at the stations of Fukushima, Ibaraki, and Chiba prefectures near the southern part of the fault (Figure 4). Group 3 in Figure 1b arrives at roughly the same time as the above-mentioned velocity phases. Based on this result and the S-wave arrival time from the rupture of point 3 at 100 s, we can consider that group 3 may be generated by the rupture after 100 s propagating to the southern fault area. Thus far, the relation between the rupture model of the present study and the observed acceleration record section is not straightforward and indicates the frequency-dependent characteristics of the seismic wave radiation of the Tohoku-Oki earthquake. Since the detailed knowledge of the frequency-dependent wave radiation processes provides a clue to understanding the hierarchy of the rupture heterogeneities, it is necessary to examine the wave radiations process for the different frequency bands of the strong motions as consistently as possible [e.g., Suzuki and Iwata, 2009].
 We would like to thank the two anonymous reviewers for providing helpful comments. We used generic mapping tools (GMT) to draw figures [Wessel and Smith, 1991].
 The Editor thanks the two anonymous reviewers for their assistance in evaluating this paper.