# Normal-faulting stress state associated with low differential stress in an overriding plate in northeast Japan prior to the 2011 Mw 9.0 Tohoku earthquake

**Part of the following topical collections:**

## Abstract

## Keywords

Crustal stress 2011 Tohoku earthquake Deformation Stress tensor inversion Differential stress## Abbreviations

- JMA
Japan Meteorological Agency

- GNSS
global navigation satellite system

- MIM
multiple inverse method

## Introduction

*T*axis (Fig. 1, adapted from Japan Meteorological Agency (JMA) catalog data; see e.g., Lin et al. 2013; Otsubo et al. 2013; Toda and Tsutsumi 2013), despite the fact that crustal normal faulting in response to a trench-type earthquake is very rare (e.g., Farías et al. 2011).

Earthquake focal mechanisms are thought to be the most effective means of identifying crustal stress (e.g., Michael 1984), because they constrain the stress field at depths where earthquakes occur. Imanishi et al. (2012) reported a normal-faulting stress regime around Iwaki City prior to the 2011 Tohoku earthquake by applying the stress tensor inversion method of Michael (1984) to small-magnitude earthquakes. They concluded that the 2011 Tohoku earthquake, in combination with a preexisting normal-faulting stress regime, triggered a normal-faulting earthquake sequence. However, the estimated stress tensor did not account for all focal mechanisms, suggesting the existence of stress heterogeneity.

The aim of this study is to explore spatial and temporal variations in crustal stress around Iwaki City prior to the 2011 Tohoku earthquake. We apply the multiple inverse method (Otsubo et al. 2008) to the earthquake focal mechanisms of Imanishi et al. (2012) and show that two normal-faulting stress states prevailed in different regions. We examine the tectonic background of the spatial and temporal variations in stress and discuss the stress level in the background of the region of the study area.

## Methods

Imanishi et al. (2012) computed the focal mechanisms of 26 small shallow earthquakes (*M*_{j} ≤ 3.2, depth < 20 km) around Iwaki City that occurred between 2003 and 2010. We applied the stress tensor inversion to 12 earthquakes (Imanishi et al. 2012; Additional file 1: Table S1) with normal-faulting and strike-slip faulting mechanisms located within the source region of the normal-faulting earthquake sequence. Following Imanishi et al. (2011), we evaluated focal mechanism uncertainties for each event from the average of their Kagan angles (Kagan 1991), where uncertainties are measured between the best fitting solution and each solution whose residual is less than 1.1 × the minimum residual. Most events show normal faulting with *T* axes that trend WNW–ESE to NNW–SSE (Fig. 1). Imanishi et al. (2012) applied the method of Michael (1984) to these 12 focal mechanisms and obtained a normal-faulting stress regime with a minimum principal stress axis trending NW–SE and a stress ratio Φ = (*σ*_{2 }− *σ*_{3})/(*σ*_{1 }− *σ*_{3}) = 0.6–0.7, where *σ*_{1}, *σ*_{2}, and *σ*_{3} are the maximum, intermediate, and minimum principal stress axes, respectively. However, some focal mechanisms yielded different stress regimes, which may suggest that the assumption of a homogeneous stress state is inappropriate for these data.

In contrast to conventional stress tensor inversion methods (e.g., Gephart and Forsyth 1984; Michael 1984), the multiple inverse method (MIM; Otsubo et al. 2008) is able to separate and isolate stresses from complex focal mechanism data based on a resampling technique without prior information. Significant stresses were categorized into clusters of stress tensors using the resampling technique of Otsubo et al. (2008), which employs *k*-means clustering, similar to Otsubo et al. (2006), to recognize clusters of stress tensors. For the *i*th cluster, the 95% confidence interval (*CI* _{95} ^{(i)} ) is calculated using *CI* _{95} ^{(i)} = *σ*^{(i)} _{mean} ± 1.96 × *SE*^{(i)}, where *σ*^{(i)} _{mean} is the mean of the contributing stresses (i.e., the cluster center) and *SE*^{(i)} is the standard error about the mean. *SE*^{(i)} itself is defined by the equation *SE*^{(i)} = *S*^{(i)}/(*N*^{(i)})^{1/2}, where *S*^{(i)} is the standard deviation of the *i*th cluster and *N*^{(i)} is the number of stress observations contributing to *S*^{(i)}. The standard deviation is calculated from the distance between the cluster center and each stress tensor assigned to the same cluster. We employed the angular stress distance described by Yamaji and Sato (2006) to calculate distances in this study.

When we applied MIM to the 12 focal mechanisms derived by Imanishi et al. (2012), we employed the combination number *k*_{f} = 5 and the enhance factor *e* = 8, following Otsubo et al. (2008, 2013). A value of 4 or 5 is usually assigned to *k*_{f} because the stress tensor inversion is an even-determined problem for four observations and is overdetermined for five (Yamaji 2000; Otsubo et al. 2008). The enhance factor *e* is a parameter used to attenuate the effects of noisy data in the solution space (Yamaji 2000). The default value of *e* is 8; MIM nominally accepts values in the range 0 ≤ *e* ≤ 99.

## Results

*σ*

_{1}and

*σ*

_{3}are orientated at 331°/75° and 140°/15°, respectively, and the stress ratio Φ = 0.54 ± 0.18 (Fig. 2b, c). Thus, regime A corresponds to a NNW–SSE-trending triaxial extensional stress. The second solution, regime B, shows

*σ*

_{1}and

*σ*

_{3}orientations of 147°/87° and 299°/3°, respectively, with a stress ratio Φ = 0.84 ± 0.10 (Fig. 2b, c), which corresponds to NW–SE-trending axial tension (

*σ*

_{1}=

*σ*

_{v}≈

*σ*

_{2}>>

*σ*

_{3}). The angular stress distance between A and B, defined as the dissimilarity between two stresses (Yamaji and Sato 2006), is ~ 39°, suggesting that the stresses are significantly different from each other (Nemcok and Lisle 1995). The

*σ*

_{3}orientations of the stresses do not overlap at the 95% confidence level for the means and standard deviations of the

*k*-means clusters (Fig. 2). The solution of Imanishi et al. (2012), using the method of Michael (1984) and shown by crosses in Fig. 2b, diverges from regime B by ~ 27° and from A by ~ 31°, suggesting a slightly better match to B.

The Wallace–Bott hypothesis (Wallace 1951; Bott 1959) states that an earthquake’s slip vector is parallel to the resolved shear stress on the fault. Based on this hypothesis, we assume that a stress is compatible with a fault slip direction if the misfit angle between the slip direction predicted by the stress and the observed slip direction on the same fault plane is sufficiently small. In this study, we adopt the approach of Otsubo et al. (2008, 2013) for the analysis of stress and associated focal mechanisms in the Iwaki City region. The spatial and temporal changes in stress can be identified by relating observed slip directions from focal mechanisms to a single stress solution (Otsubo et al. 2008, 2013). The threshold of the misfit angle was determined here based on the uncertainties of the strike, dip, and rake (Gephart and Forsyth 1984; Michael 1991). When the misfit angles are smaller than the uncertainties of the stress and focal mechanism solutions, the observed slip directions agree with theoretical values to within the estimated uncertainties.

## Discussion

Using the focal mechanisms of Imanishi et al. (2012), MIM reveals two normal-faulting stress states around Iwaki City before the 2011 Tohoku earthquake (Fig. 2). In this study, we discuss the temporal and spatial stress changes around Iwaki City to explain the stress heterogeneity.

### Temporal and spatial stress changes around Iwaki City

We attribute the variations in the stress field to temporal variations. Stress regime A can be seen to account for all the focal mechanisms derived from earthquakes that occurred before 2005. This implies that only stress regime A should be used as an appropriate solution before 2005 (Fig. 3). In contrast, stress regime B can be seen to account for all the focal mechanisms, except for no. 11, derived from earthquakes occurring after 2008 (Fig. 3). In the intervening period between 2005 and 2008, derived focal mechanisms can be related to both stress regimes A and B. Hence, one possible explanation of the stress heterogeneity is that the stress field around Iwaki City temporally changed from a NNW–SSE-trending triaxial extensional stress (regime A) to a NW–SE-trending axial tension (regime B), with the transition between the two occurring from 2005 to 2008. Hence, we define stress period I from 2003 to 2005 and stress period II from 2008 to 2010 (Fig. 3).

Next, we consider the dynamics for the stress changes around Iwaki City. The *σ*_{3} orientation of stress regime B is subparallel to the orientations of the co- and post-seismic displacements of large earthquakes (Mw ~ 7) that occurred during the period 2005–2010 (Fig. 2b; Suito et al. 2011). The post-seismic deformation determined from continuous GNSS monitoring reveals that the seismic moments released by transient slip following the M7 class earthquakes are much larger than the seismic moment estimates for the earthquakes themselves (Suito et al. 2011). Stress regime B may therefore be the result of accumulated extensional stress associated with co- and post-seismic deformation due to the M7 class earthquakes, which occur more frequently than M9 class earthquakes.

### Estimation of differential stress around Iwaki City

We compared the stress changes caused by the co- and post-seismic deformation with the extensional stresses in stress periods I and II. The direction of extensional stresses induced by the co- and post-seismic deformation is almost parallel to the orientation of the *σ*_{3} axes, while it is perpendicular to the orientations of the *σ*_{1} and *σ*_{2} axes (Fig. 2b). Because *σ*_{1} is the overburden pressure, *σ*_{1} is almost constant during the co- and post-seismic deformation. Hence, the extensional stress induced by the co- and post-seismic deformation should only affect the magnitude of *σ*_{3} rather than those of *σ*_{1} and *σ*_{2}, and in order for the stress ratio to change from 0.54 (± 0.18) to 0.84 (± 0.10), *σ*_{3} must decrease. Here, the reduction in the magnitude of *σ*_{3} corresponds to a build-up of extensional stress.

*σ*

_{1}

^{A}(

*σ*

_{1}

^{B}),

*σ*

_{2}

^{A}(

*σ*

_{2}

^{B}), and

*σ*

_{3}

^{A}(

*σ*

_{3}

^{B}) as the maximum, intermediate, and minimum compressive principal stresses for stress A (and B), respectively. The differential stress for stress A (B) can be expressed by Δ

*σ*

^{A }=

*σ*

_{1}

^{A }−

*σ*

_{3}

^{A}(Δ

*σ*

^{B}=

*σ*

_{1}

^{B }−

*σ*

_{3}

^{B}). From the change in the stress ratio from 0.54 (± 0.18) to 0.84 (± 0.10), we obtain the following relationships:

*σ*

_{3}

^{B}−

*σ*

_{3}

^{A }~ 0.09 − 9.03Δ

*σ*

^{A}and Δ

*σ*

^{B }~ 1.08 – 10.03Δ

*σ*

^{A}(Fig. 6). The former relationship indicates that the extensional stress induced by the co- and post-seismic deformation is approximately 0.1 to 9 times as large as the differential stress for stress A. The latter relationship signifies that the extensional differential stress increases by a factor of almost 1.1 to 10 from stress A to stress B. Based on the amount of displacement from the co- and post-seismic deformation (~ 1–3 cm; Fig. 5), the resulting strain is estimated to be ~ 3 × 10

^{−7}to 2 × 10

^{−6}for the NW–SE component between the two sites near Iwaki City. Assuming that the crust is elastic with a Young’s modulus of 32 GPa, the induced stress,

*σ*

_{3}

^{B}−

*σ*

_{3}

^{A}, is estimated between 0.9 × 10

^{−2}and 6.4 × 10

^{−2}MPa. Inserting this estimated value into the relationships derived above, we obtain differential stresses for stress A, Δ

*σ*

^{A}, and stress B, Δ

*σ*

^{B}, of approximately 1.00 × 10

^{−3}to 7.11 × 10

^{−1}and 1.10 × 10

^{−3}to 7.13 MPa, respectively. Hence, we propose that the differential stress is less than the order of 1 MPa around Iwaki City prior to the 2011 Tohoku earthquake.

### Background of the low differential stress around Iwaki City

We consider the generation of the 2011 Iwaki earthquake in the context of the low differential stress in the study area. We showed the stress heterogeneity around Iwaki City prior to the 2011 Tohoku earthquake as the spatial or temporal stress changes. In this study, we could not identify a possible explanation of the stress heterogeneity. In both cases, we suggest that the stress state around Iwaki City prior to the 2011 Tohoku earthquake might have been extensional with a low differential stress. A low differential stress prior to the 2011 Iwaki earthquake was reported by Yoshida et al. (2015), who estimated differential stress magnitudes ~10^{0}–10^{1} MPa by comparing the stress orientations in the post-Iwaki earthquake period with static stress changes due to the Iwaki earthquake and three nearby M5 class earthquakes. Tomographic studies have imaged low-velocity anomalies beneath the hypocenter of the 2011 Iwaki earthquake (Fig. 1; see also Kato et al. 2013), which may correspond to zones of high pore-fluid pressure. Zhao (2015) showed that a low-velocity zone is visible in the lower crust and mantle wedge, and that it extends beneath the hypocenter of the 2011 Iwaki earthquake, down to the subducting Pacific slab. The discharge of a large amount of thermal water after the 2011 Iwaki earthquake (Sato et al. 2011; Kazahaya et al. 2013) indicates that earthquakes in this region promote the upwelling of deep groundwater. The earthquakes themselves may be triggered by a decrease in effective normal stress and fault strength due to the increased pore-fluid pressure (e.g., Sibson 1990; Micklethwaite and Cox 2006; Terakawa et al. 2013). By examining the fault failure of the 2011 Iwaki earthquake with respect to the change in the state of stress in the Iwaki area produced by the 2011 Tohoku earthquake, Miyakawa and Otsubo (2015) showed that excess fluid pressure is required to explain the 2011 Iwaki earthquake. Therefore, we infer that high pore-fluid pressure could enable the generation of earthquakes under conditions of low differential stress (Sibson 1992).

## Conclusions

An investigation of spatial and temporal changes in extensional stress in the inter-seismic period before the 2011 Mw 9.0 Tohoku earthquake revealed that the pre-Tohoku stress state around Iwaki City was significantly heterogeneous. Our findings support the inference that the stress state around Iwaki City prior to the 2011 Tohoku earthquake was extensional with a low differential stress. The differential stress of the normal-faulting stress regime increased after the 2011 Tohoku earthquake.

The multiple inverse method (MIM) of stress inversion resolves statistically significant stress heterogeneities in the study area. The MIM is capable of detecting stress heterogeneities that cannot be detected using more conventional stress tensor inversion methods. The application of the MIM to other study areas in the future may provide an important monitoring tool for evaluating stress change.

## Notes

### Authors’ contributions

MO led and designed the research and drafted the manuscript. MO carried out the stress tensor inversion analysis, drafted the figures, and contributed to assessing the results. AM and KI contributed to the discussion of the results. All authors read and approved the final manuscript.

## Acknowledgements

We thank two anonymous reviewers for suggestions that led to improvements in the manuscript. We thank Dr. T. Nishimura for discussions on GNSS data, and Dr. J. Hardebeck for fruitful discussions. This study was supported by MEXT KAKENHI (number 26109003). Some of the figures were generated using Generic Mapping Tools (GMT; Wessel and Smith Wessel and Smith 1998).

## Competing interests

The authors declare that they have no competing interests.

## Availability of data and materials

Seismic data are available from the authors upon request.

## Publisher’s Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

## Supplementary material

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