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# Source process with heterogeneous rupture velocity for the 2011 Tohoku-Oki earthquake based on 1-Hz GPS data

- Zhen Wang
^{1}Email authorView ORCID ID profile, - Teruyuki Kato
^{1}, - Xin Zhou
^{1, 2, 3}and - Jun’ichi Fukuda
^{1}

**Received: **2 May 2016

**Accepted: **17 November 2016

**Published: **28 November 2016

### Abstract

^{22}Nm (Mw 9.0), respectively. The large slip, stress drop, and rupture velocity are all concentrated at shallow depths, which indicates that the shallow part of the fault radiated high-frequency as well as low-frequency seismic waves.

## Keywords

## Introduction

Seismological and geodetic observations have been widely used to constrain seismic source parameters in modern seismology. Seismic waveforms including teleseismic waves and strong-motion records at different frequencies have been commonly used to estimate the finite fault rupture process model (Kikuchi and Kanamori 1982; Yoshida et al. 1996; Chi et al. 2001), whereas GPS measurements are used mostly to obtain the coseismic static slip distribution (Yabuki and Matsu’ura 1992). Joint inversion of seismic and geodetic data can integrate the kinematic features of seismic data and the static features of geodetic data (Delouis et al. 2002); however, the weighting between different datasets is always an issue.

Both seismic and geodetic tools for measuring ground motion during earthquakes have inherent advantages and limitations. Far-field seismic waveforms recorded by seismometers provide a smoothed estimate of the moment release rate and constrain the rupture direction but have less information on the near-field terms that reflect the detailed rupture history and static slip on the fault. Near-field strong-motion acceleration waveforms are sensitive to the rupture history of earthquakes and the ground motion during earthquakes; however, they are prone to be affected by saturation for large earthquakes (Bilich et al. 2008), coseismic ground tilts, and errors owing to analog-to-digital conversion (Boore 2003). As a result, displacement waveforms obtained by double integration of strong-motion data often exhibit large drifts, making the accurate recovery of static displacements difficult (Boore et al. 2002; Boore and Bommer 2005). In contrast, high-rate GPS (hr-GPS) waveforms can directly capture both instantaneous and static displacements without saturation and large drift (Larson et al. 2003; Bock et al. 2004).

The rapid development of a high-frequency sampling technique for GPS enables the rupture process inversion for a finite fault by using hr-GPS waveforms. One primary advantage of rupture inversion using hr-GPS data is that it essentially fits both the dynamic and static features to avoid inadequate weighting in different data types as long as the Green’s functions, including both dynamic and static responses, are computed. Thus far, relatively few works exist on rupture process models solely inverted from hr-GPS (Miyazaki et al. 2004; Yokota et al. 2009; Yue and Lay 2011), or on joint inversion combining hr-GPS with other datasets such as seismic waveforms (Yue and Lay 2013) or static GPS displacement (Vigny et al. 2011). The available studies have shown the ability of hr-GPS data to infer the rupture process of medium- and large-sized earthquakes. However, most of the previous works that model the rupture process using hr-GPS data assumed a priori rupture velocity and solved a linear system for slip history during the coseismic period. If an appropriate rupture velocity and a subfault source time function are selected that are long enough to span the actual rupture time, the rupture velocity will have negligible influence on the final cumulative slip (Yue and Lay 2011). However, Lee et al. (2011) showed that heterogeneous rupture velocity could influence the rupture process by the inversion of seismic waveforms. Konca et al. (2007) showed that non-uniform rupture velocities from 1.5 to 2.5 km/s are required to fit the seismic data for the 2005 Mw 8.6 Nias–Simeulue earthquake. In the present study, to better understand the rupture propagation process, we conduct rupture process inversions that allow for heterogeneous rupture velocity.

During the 2011 Mw 9.0 Tohoku-Oki earthquake, abundant continuous GPS stations in the Japan Islands enabled the inversion for slip history and rupture velocity using hr-GPS waveforms. Using hr-GPS data in single inversion (Yue and Lay 2011) or joint inversion with other available data (Yue and Lay 2013; Bletery et al. 2014) for the rupture process has been performed with a constant rupture velocity to obtain a consistent main slip asperity with a simple rupture feature up-dip of the hypocenter to the trench. A rupture model with freely varying rupture velocity for each subfault has been presented by Minson et al. (2014) from joint inversion of geodetic data and tsunami data using a single-time-window method. In the present study, we use hr-GPS data in northeastern Japan and ocean bottom GPS/acoustic (OB-GPS) data to invert for the rupture process of the 2011 Mw 9.0 Tohoku-Oki earthquake. The analysis uses the inherently dynamic and static features of hr-GPS data and considers the inhomogeneous rupture velocity.

In the discussion, the preferred slip model obtained in present study is compared with the corresponding results of previous studies on the 2011 Tohoku earthquake. Our slip model shows that large slip is concentrated in the shallow part of the fault extending from the hypocenter to the trench, which is consistent with several previous studies. Furthermore, the distributions of slip, rupture velocity, and stress drop are compared to discuss the point from which the high-frequency or low-frequency seismic waves are possibly radiated and the corresponding radiation efficiency.

## Observations

The Geospatial Information Authority of Japan (GSI) via Nippon GPS Data Service Company (NGDS) provided 2-h 1-Hz GPS data that span the 2011 Mw 9.0 Tohoku-Oki earthquake from 414 hr-GPS stations of the Global Navigation Satellite System (GNSS) Earth Observation Network System (GEONET) over northeastern Japan. GPS Solutions processed the raw GPS data and verified that the dataset was postprocessed in “real-time mode” with real-time orbits and clock corrections from the Veripos/Apex global service using RTNet software. We used the estimated GPS ground motion time series provided by GPS Solutions. The estimated maximum horizontal displacement exceeded 5 m, and the maximum vertical displacement was larger than 1 m. The average standard deviations of hr-GPS recordings during the 300 s prior to the mainshock were 0.017, 0.019, and 0.038 m for the east, north, and vertical components, respectively. The estimated data error of vertical displacement was larger than twice that for horizontal displacement.

For the 2011 Tohoku-Oki earthquake, all available hr-GPS stations are located west of the rupture area and lack the constraints on the fault rupture near the Japan Trench. Seafloor geodetic measurements using the GPS/acoustic technique above the rupture area reported by Sato et al. (2011) and Kido et al. (2011) with observation errors up to ~0.5 m can fill the gap. The peaks of horizontal and vertical displacements near the hypocenter were ~31 and ~3.9 m, respectively. Because the last preseismic and first postseismic GPS/acoustic measurement were conducted several months before the mainshock and about one month after the mainshock, respectively, the observed displacements include coseismic displacement of the mainshock and postseismic movement during the first month as well as coseismic displacement of foreshocks and aftershocks. However, the total displacement other than that of the coseismic signal by the mainshock has been reported as not larger than 1 m (Sato et al. 2011). Considering the signal-to-noise ratio, the OB-GPS data still provide valuable constraints on the static coseismic behavior near the Japan Trench. We thus used the coseismic static displacements at six stations shown in Fig. 1 with equal weighting in the joint inversion together with hr-GPS data. The relative weight between OB-GPS data and hr-GPS data is analyzed in “Results” section.

## Fault parameters and method

We used a planar fault with a length of 600 km along strike and a width of 240 km along the dip with a uniform strike of 201° and a uniform dip angle of 11° to approximate the subduction plate interface following Zhou et al. (2014) (Fig. 1). Previous studies have demonstrated that the assumption of planar fault geometry does not cause significant modeling effects through inversions of static geodetic displacement (Zhou et al. 2014; Lee et al. 2011). We thus employed a single planar fault in this study. To obtain the seismic slip history, the fault plane was divided into 20 × 8 subfaults with dimensions of 30 km × 30 km. The hypocenter (142.8°E, 38.05°N, 24 km depth) determined by JMA is located at the center of the 11th subfault along the strike and at the center of the 6th subfault along the dip referring to the northwesternmost subfault (Fig. 1).

We used a multi-time-window inversion method (Olson and Apsel 1982; Lee et al. 2006) to estimate the rupture process of the 2011 Tohoku-Oki earthquake. The moment rate function for each subfault was expressed by a linear combination of 20 symmetric triangular basis functions with expansion coefficients. The duration time of each triangle was 10 s, and the shift time of adjacent triangles was 5 s. Suzuki et al. (2011) indicated that the 78-s slip duration for each subfault is sufficient for stable inversion analysis of the 2011 Tohoku-Oki earthquake, whereas the 105-s slip duration in the present study is long enough to span the entire rupture process. The expansion coefficients for the triangular basis functions for all subfaults are unknown parameters in the inversion. To allow for heterogeneous rupture velocity, we assumed the hypocenter to be the initial rupture location. Moreover, we employed a sufficiently large a priori rupture velocity of 3.5 km/s that approaches the shear wave velocity following the “zero rupture delay time” approach of Lee et al. (2006). After the inversion with the a priori rupture velocity, we defined a rupture delay time for each subfault as the time of cumulative slip exceeding a threshold following Lee et al. (2006), which was measured from the initial rupture at the hypocenter. We then calculated the rupture velocity for each subfault as the shortest on-fault distance from the hypocenter to the subfault divided by the rupture delay time. By definition, this estimated rupture velocity cannot exceed the a priori value.

To directly invert for the rupture model from the complete ground motion recordings, it is necessary to use Green’s functions including both dynamic and static displacement responses. We used the frequency–wavenumber (FK) synthetic seismogram package of fk3.1 developed by Zhu to generate the Green’s functions (Zhu and Rivera 2002). We used a layered velocity structure shown in Additional file 2: Table S1, which is based on the local velocity structure of Wu et al. (2008) and the Preliminary Reference Earth Model (PREM; Dziewonski and Anderson 1981) to calculate the Green’s functions. In forward modeling, to compute the synthetic waveforms accurately, we divided each of the 30 km × 30 km subfaults into 25 finer subfaults with dimensions of 6 km × 6 km, and we obtained the Green’s function as the mean of displacements owing to unit slip on the 25 finer subfaults. A 0.05-Hz fourth-order low-pass Butterworth filter, which was applied to the 1-Hz hr-GPS data, was also employed to filter the Green’s functions to ensure common spectral content.

## Results

### Cumulative slip distribution

^{−6}to 10

^{6}was tested. When the smoothing parameter is 10

^{−6}(10

^{6}), \(\lambda^{2}\) is equal to 10

^{−12}(10

^{12}). According to Eq. (1), the result depends almost only on the observed data or the smooth constraint, respectively. A broad range of smoothing parameters was tested to support enough information to plot the trade-off curve and clearly show the location of the bend in the trade-off curve. We subtracted the minimum values from the weighted residual norm and the roughness norm, and we normalized them by their maximum values, respectively. In Fig. 2, the optimal value of 15.0 marked by point B was obtained by measuring the minimum distance from the normalized trade-off curve to the origin of the figure. The smoothing parameter that minimizes the Akaike’s Bayesian Information Criterion (ABIC; Akaike 1980) was also plotted with point A. Point C was selected to show the comparison of points A and B.

^{22}Nm (Mw 9.0). The region with slip of at least 5 m spans 150 km down-dip of the top boundary of the fault, corresponding to a depth of 10–41 km, and extends along the strike for approximately 300 km. The slip model shows that the deepest part of the fault did not slip significantly during the mainshock. Our preferred slip pattern is in good agreement with several previous results, such as the joint inversion of teleseismic body waves and static geodetic displacement (Kubo and Kakehi 2013), the inversion of tsunami waveform data (Satake et al. 2013), the inversion of strong-motion data (Suzuki et al. 2011), and the inversion of high-rate GPS data (Yue and Lay 2011).

### Rupture velocity

As described in “Fault parameters and method” section, the rupture delay time for each subfault can be determined from the time at which the accumulated slip exceeds a threshold, which is regarded as the time required to pass the rupture front. Lee et al. (2006) used a threshold of 0.1 m for the 1999 Chichi, Taiwan, earthquake (Mw 7.6). A higher accumulated slip of 0.5 m was employed as the threshold in this study by considering the poorer temporal resolution of the triangular basis functions with duration of 10 s and relatively large inversion uncertainty for this mega-thrust event. The rupture velocity was estimated as the ratio of the shortest distance between the hypocenter and each subfault to the rupture delay time at the corresponding subfault. Figure 5b shows a heterogeneous rupture velocity distribution ranging from a minimum of 1.3 km/s to a maximum of 3.5 km/s on the fault plane. The rupture velocity within the vicinity of the hypocenter showed a regular pattern, gradually increasing from 1.0 to 2.0 km/s with increasing distance from the hypocenter. The rupture at the shallowest part of the fault significantly accelerated with a velocity larger than 3.0 km/s, which is consistent with that obtained by Lee et al. (2011). The rupture velocity of the southern part of the fault 120 km from the hypocenter was significantly lower than that in other regions, ranging from 1.0 to 2.0 km/s. This indicates that the rupture front first arrived at the shallow part of the fault where large slip occurred and then spread to the surrounding area.

### Temporal rupture process

According to the rupture time periods on each subfault, we marked the rupture regions corresponding to Periods I, II, and III by a red rectangle (Region I), green rectangle (Region II), and blue rectangle (Region III), respectively, in Fig. 8b. The figure shows two significant slip events corresponding to Periods I and II in the proximity of the hypocenter and the shallow part of the fault (Region I) and one slip event in the other regions (Regions II and III). At 40 s from the onset of rupture, the small moment rate in Region I indicates a transition from the first event (Period I) to the second event (Period II). In Period I, ~9% of the entire seismic moment had been released. During Period II, the Region I rupture repeated, and the moment rate rapidly accelerated in Regions I and II, releasing a large amount of moment. During Period III, the slip was concentrated in the southern part of the fault (Region III) and continued until the end of the mainshock with lower rupture velocities of 1.5–2.0 km/s.

### Data fit

Furthermore, we independently inverted the hr-GPS recordings without OB-GPS data to clarify the influence of OB-GPS data to the maximum slip. The inversion result showed a maximum slip of ~50 m, which is significantly less than that of our preferred rupture model. The OB-GPS data significantly increased the maximum slip. In addition, we compared the maximum slip inverted from different relative weights between hr-GPS data and OB-GPS data to determine the degree to which the OB-GPS data affects the maximum slip. The maximum slips were about 60, 64, 65, and 66 m for relative weights of 1:20, 1:50, 1:100, and 1:120 between hr-GPS data and OB-GPS data, respectively. Changing the weight of OB-GPS data from 20 to 120 slightly increased the maximum slip from 60 to 65 m.

## Discussion

Our preferred rupture model had a seismic moment of 4.2 × 10^{22} Nm, whereas other published models have those of 3.4 × 10^{22}–5.8 × 10^{22} Nm (Tajima et al. 2013). The maximum model slip was ~65 m, which is slightly larger than the 25–60 m in most previous results based on seismic waveform, geodetic, and tsunami data (Ozawa et al. 2011; Simons et al. 2011; Koketsu et al. 2011; Yokota et al. 2011; Saito et al. 2011; Suzuki et al. 2011; Lee et al. 2011; Zhou et al. 2014) and approaches that of 64 m obtained by Bletery et al. (2014). The main rupture area spans ~400 km along the strike and ~200 km along the dip. Under the free boundary condition, which allows the slip to reach the free surface, our preferred model showed that the greatest slip occurred up-dip of the hypocenter near the trench. This result is in agreement with that of other works (Lee et al. 2011; Suzuki et al. 2011; Yue and Lay 2011; Bletery et al. 2014), although the amplitudes are inconsistent.

Our moment rate function characterized with three prominent slip events is consistent with that reported in previous research (Lee et al. 2011; Suzuki et al. 2011). The repeating slip behavior in Region I is consistent with the inversion result of solely strong-motion data by Suzuki et al. (2011) and the joint inversion of teleseismic, strong-motion, and geodetic data by Lee et al. (2011). Yue and Lay (2011) inverted for a rupture process assuming a constant rupture velocity from hr-GPS data. Although their result showed only one main slip event on the shallow part of the fault, the cumulative slip distribution was similar to that of our result. This indicates that if a priori homogeneous rupture velocity is selected and the a priori source time function for each subfault is sufficient for spanning the actual rupture time, the cumulative slip distribution is stable, but the rupture process is strongly affected by the rupture velocity. In addition, our results of the moment rate function and the slip images all showed that somewhat small slip at the deepest depths (sixth and seventh rows along dip in Fig. 8b) occurred only at the second rupture stage. These results are in agreement with Suzuki et al. (2011) but differ significantly from Ide et al. (2011). This discrepancy may be attributed to the inversion method. Suzuki et al. (2011) used a method of multiple time windows similar that used in the present study; however, Ide et al. (2011) implemented an empirical Green’s function in the source process.

It is undisputed that a shallow region with large slip radiates very low-frequency seismic waves (Ide et al. 2011; Suzuki et al. 2011; Lee et al. 2011). However, the location in which high-frequency seismic waves are radiated is debatable. The vicinity of the edges of coseismic slip in which the rupture velocity changes abruptly emits high-frequency radiation (Madariaga 1977). Ide et al. (2011) considered that the shallow depths have a quiet rupture and that high-frequency seismic waves are radiated from deep depths. Simons et al. (2011) suggested that relatively high-frequency seismic waves are generated from the edge of coseismic slip at the deepest depths. The results of back-projection imaging show that the short-period energy is radiated down-dip from the hypocenter as well (Koper et al. 2011; Wang and Mori 2011). Dynamic source simulation suggests that the shallowest and deepest edges of coseismic slip all radiate high-frequency seismic waves (Duan 2012). By using kinematic inversion, Lee et al. (2011) demonstrated that deep depths as well as the shallow parts of a fault radiate high-frequency seismic waves.

## Conclusions

Based on the assumption of multiple time windows with a large initial rupture velocity, we inverted hr-GPS data for a rupture model with varying rupture velocity for the 2011 Tohoku earthquake. By comparison of a slip model and rupture velocity distribution, the large slip areas were coupled with areas of large rupture expansion speed. Similar to the result from seismic waveforms, which are not limited by temporal resolution, the temporal rupture process and the total moment rate function showed three energy release stages during the event. The slip propagated from the hypocenter to the shallow part during the first energy release stage. In the second stage, the rupture in Region I that experienced a repeating slip released an extremely large amount of energy. The shallow part ruptured through the overall rupture duration time, forming a large rupture asperity with a maximum slip of about 65 m. The patterns of slip, stress drop, and rupture velocity demonstrate that high-frequency seismic waves radiated from both shallow and deep depths, whereas low-frequency waves radiated from the shallow fault. Therefore, varying the rupture velocity allowed us to obtain a more appropriate rupture model than that when using an assumed constant rupture velocity. Higher temporal resolution is preferred to determine the kinematic rupture process in detail by using hr-GPS data.

## Declarations

### Authors’ contributions

ZW analyzed the data, carried out the source inversions, interpreted the results, and drafted the manuscript. XZ and JF participated in the study design, and drafted the manuscript and critically revised it. TK participated in the study design. All authors read and approved the final manuscript.

### Acknowledgements

The authors thank the GSI of Japan and Nippon GPS Data Service (NGDS) Company for providing the hr-GPS data. We used GMT software for plotting in this study. The authors also thank Dr. Baoping Shi and Dr. Benchun Duan for fruitful discussions. X. Zhou was financially supported by the National Natural Science Foundation of China (No. 41574021) and the Visiting Professor/Post-doctoral Programs of Earthquake Research Institute, University of Tokyo.

### Competing interests

The authors declare that they have no competing interests.

### Availability of data and materials

The hr-GPS GEONET data provided by the Geospatial Information Authority of Japan (GSI) were downloaded from the GPS Solutions Web site at http://www.gps-solutions.com/data_2011_tohoku_eq. The unified hypocenter catalog data reported by JMA were downloaded from the NIED Hi-net Web site at https://hinetwww11.bosai.go.jp/auth/JMA/jmalist.php?LANG=en. Some figures were plotted by using General Mapping Tools (GMT) from the Web site of http://gmt.soest.hawaii.edu. The Green’s functions were calculated by using the program package by Zhu, which was downloaded from http://www.eas.slu.edu/People/LZhu/home.html. All of the aforementioned Web sites were last accessed in October 2015.

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## Authors’ Affiliations

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