Which is heterogeneous, stress or strength? An estimation from high-density seismic observations
© The Author(s) 2017
Received: 22 April 2017
Accepted: 11 October 2017
Published: 20 October 2017
It has long been believed that stress in the Earth’s crust is highly heterogeneous. Borehole measurements show orientations of maximum horizontal compressive stress varying by tens of degrees in the upper several kilometers (e.g., Barton and Zoback 1994; Wilde and Stock 1997). Heterogeneous slip distributions along faults are attributed to heterogeneity in the stress acting on those faults, as well as in fault strength (e.g., Andrews 1980; Ben-Zion and Sammis 2003). Furthermore, heterogeneous focal mechanisms have been observed in aftershock areas close to faults associated with large earthquakes, which suggests local stress heterogeneity (e.g., Michael et al. 1990; Hauksson 1994).
However, these observations do not necessarily indicate that stress in the crust is typically heterogeneous, particularly in seismogenic regions. Borehole measurements reflect the stress state only in the shallow crust, where earthquakes rarely occur. Heterogeneous slip distributions and focal mechanisms are often observed in limited portions on or near earthquake faults. Furthermore, because heterogeneous slip distributions are generally estimated assuming smooth, planar faults, heterogeneous stress and strength are necessary to explain observations, whereas complex fault geometries, such as offset or bend of faults, generally are not considered. In addition, variations of focal mechanisms may be explained not only by heterogeneous stress but also by heterogeneous strength of faults (e.g., Rivera and Kanamori 2002).
However, there is a fundamental question under debate about what the results of stress inversion reflect. Smith and Heaton (2011) proposed that they reflect not the orientations of the three principal stresses but those of incremental stresses, as shown in Fig. 1b. Because it is assumed in stress inversion that stress is uniform throughout an analysis area, fault strength is inferred to vary depending on the orientations of faults (Rivera and Kanamori 2002). However, Smith and Heaton (2011) proposed that each fault is subjected to individual stress to which that fault is most favorably oriented. In other words, if an earthquake occurs on a fault, a fault plane with a different orientation is subjected to a different stress that produces maximum shear along the slip direction. In this case, it is inferred that the orientations of P-axes of focal mechanisms are in accordance with those of the maximum compressional stress, particularly if the fault strength is small. Furthermore, each fault is assumed to exist under a critical stress state in which the magnitude of shear stress on the fault is comparable to its strength. In this case, if incremental stress is added, only a fault with stress that is favorable to the incremental stress can break.
Recently, changes in stress before and after large earthquakes have been observed for several aftershock areas (Hardebeck and Hauksson 2001; Yoshida et al. 2014, 2015). The findings suggest that the magnitudes of differential stresses are much smaller than expected from results of frictional experiments in laboratories (Byerlee 1978), which Smith and Heaton (2011) regard as artifacts, as they reflect stress changes (incremental stress) generated by earthquake slips. However, Hardebeck (2010) reported results that contradict the view of Smith and Heaton (2011). They estimated stress fields before and after large earthquakes based on stress inversion of focal mechanisms and showed that the focal mechanisms of aftershocks are mainly controlled by the stress before large earthquakes. Furthermore, Hardebeck (2015) argued that the assumption of errors in focal mechanisms in the simulation by Smith and Heaton (2011) was inadequate.
If the hypothesis of Smith and Heaton (2011) holds and focal mechanisms of earthquakes that occur close to each other are significantly different, stress must change over small distances. However, such change is not easy to occur because stress should satisfy the condition of continuity. Actually, Smith and Heaton (2011) assumed a spatial correlation in stress distribution at the kilometer scale. Therefore, to resolve this problem, it is important to clarify how stress changes in space. In this study, we precisely determined focal mechanisms and estimated spatial changes in stress using high-quality data from a dense seismic observation network installed in the western Nagano prefecture region, Japan.
Geophysical setting of western Nagano prefecture region
Dense seismic observations
The second network, the Manten seismic observation network, has operated from August 2008 to the present with 29 stations equipped with the Manten system, which is a newly developed seismic observation instrument composed of a small, light, and portable velocity-type sensor (2 Hz) and an extremely low-power data logger (Iio 2011). Waveforms are recorded continuously at a 250 Hz sampling rate with 18-bit resolution. The clocks are corrected by GPS every 1 h such that uncertainties in the absolute time are less than 1 ms. These stations are installed mainly in the central part of the area with swarm activity, as indicated by green inverted triangles in Fig. 2. At stations marked by both a red cross and a green inverted triangle, the Manten system was installed after removal of the high-frequency observation system. Permanent stations (cyan diamonds) operated by the National Institute for Earth Science and Disaster Prevention (NIED), the Japan Meteorological Agency (JMA) and Nagoya University were also used. These continuous recordings enabled us to determine focal mechanisms of very small earthquakes down to magnitudes of zero.
We used the data collected from June 1995 to June 2010. For before August 2008, only data from the high-sampling observation network were used, whereas data from both networks were used for August 2008 and later. We determined focal mechanisms as follows. P- and S-wave arrival times were picked visually, as shown in Fig. 3. Transverse components were used for S-wave picking to reduce the effects of converted waves such as Sp phases. The picking errors of P- and S-wave arrival times for the high-sampling data were estimated as 2 and 30 ms, respectively (Doi et al. 2013), which indicates very high accuracy, whereas the picking errors of P-wave arrival times for the Manten observation were slightly larger because of the lower sampling rate of 250 Hz. Hypocenters were determined using the modified Hypomh program (Hirata and Matsu’ura 1987) in which S-wave velocity structure can be set independently of P-wave structure (Kawanishi et al. 2009). We employed a one-dimensional velocity structure with 10 layers of widths of 0.1–2 km estimated using the joint hypocenter determination (JHD) method (Asaka et al. 2005). First, we calculated station corrections as averages of O–C for P- and S-wave arrival times at each station, using only large events for which the number of P-wave arrivals was greater than 20. Standard deviations of the station corrections were 25 and 46 ms for P- and S-wave arrival times, respectively. After these corrections, the RMSs of O-C of P- and S-arrival times for each event were less than 10 ms and a few tens of ms, respectively. Thus, we could determine relative hypocenters very accurately, and many relative errors were estimated as several tens of m and about 100 m in the horizontal and vertical directions, respectively.
Focal mechanisms were determined using the program of Maeda (1992), in which all the fault plane solutions with the least inconsistent polarities are extracted by a grid search with a spacing of about 8° in azimuth and dip directions of B-axis on the focal sphere. The best fault plane solution is then selected from these numeral solutions as follows. In the case of no inconsistent polarity, the fault plane solution is drawn apart from locations of consistent stations projected on the focal sphere, whereas in other cases, it is drawn as close to the locations of inconsistent stations as possible.
We estimated the stress field in the area shown in Fig. 2 using a standard stress inversion method in which the difference between the observed and calculated slip directions (misfit) is minimized, assuming uniform stress in an individual analysis area (e.g., Gephart and Forsyth 1984; Michael 1987). In this method, the root mean square of misfits is minimized. The fault plane is distinguished from the auxiliary plane based on the difference in misfits calculated for both the planes. When multiple focal mechanisms are determined, the focal mechanism with the least misfit is selected.
In the same study region, Yukutake et al. (2010) conducted a stress inversion analysis using high-sampling data from 1999 to 2011. They estimated stress at points on a three-dimensional grid with 1 km spacing using the focal mechanisms for which hypocenters were located in a cubic subregion with dimensions of 2 km. In this study, we estimated stress precisely in a subregion with dimensions of 1 km at grid points with 500-m spacing. We first performed a grid search for directions of the principal stress axes with 10° grid intervals and the stress ratio, R (= (σ 1 − σ 2)/(σ 1 − σ 3)), with 0.1 grid intervals. We repeated the grid search to seek better estimates of the directions of the principal stress axes with 5° grid intervals and the stress ratio with 0.05 grid intervals in the vicinity of the values estimated using the first grid search. Ninety-five percent confidence limits are calculated by a boot strap resampling method, repeating the above procedure 2000 times, following Michael (1987).
Figure 10a shows that the plots are scattered; however, half of the data with large P p (i.e., larger than zero) have large differences of misfits between the fault plane and auxiliary plane (which we define as the ‘misfitdiff’), which suggests that the selection of the fault plane is reliable for these data. Furthermore, Fig. 10b shows that a large proportion of the misfits are smaller than the error in focal mechanisms. Figure 10c shows that orientations of fault planes are distributed very widely, and any special focal mechanisms do not have large misfits. These plots suggest that stress in the subregion was well determined and can be regarded as uniform there. As shown in Fig. 10d, misfits did not show clear dependence on magnitude. Figure 10e shows that P p values reach a peak around − 0.4, but more than half of them are widely distributed. Thus, the results likely do not reflect a special stress state in a small portion of this subregion.
The results shown in Figs. 9 and 10 were estimated for a single grid point; however, similar results were estimated for many grid points away from the earthquake fault. As shown in Fig. 8b, RMS misfits were small, and the orientations of fault planes are varied widely in these regions.
It is estimated from these analyses that the stress field can be regarded as uniform at a length scale of 1 km away from the mainshock fault, whereas it appears to be disordered in a smaller portion of the area, even in the subregions with dimensions of 1 km near the mainshock fault.
Variety of focal mechanisms
It was found that focal mechanisms vary widely and that the observed data are well explained by uniform stress in a cubic subregion with dimensions of 1 km, except for a portion of the data for grid points near the earthquake faults. These results indicate that stress can be regarded as uniform over a small region at the kilometer scale, which suggests that the strength of faults varies greatly. Such uniformity of stress supports the validity of stress inversion analysis; however, it is thought to conflict with the hypothesis of Smith and Heaton (2011), which assumes heterogeneous stress. If their hypothesis holds, each fault is basically subjected to the stress to which it is most favorably oriented, and the orientations of the P-axes of focal mechanisms are thought to be in accordance with those of the maximum compressional stress axes. In particular, fault strength is small.
The orientations of P-axes varied widely even over very short distances. This finding strongly indicates that stress was not heterogeneous, but that strength was, because stress should hold for the condition of continuity and show some correlation with focal distance.
In this study, it was found that a major proportion of the data had misfits smaller than 10°. These results indicate that the observed focal mechanisms were well explained by uniform stress estimated for each grid point. Thus, it is thought that the variety of focal mechanisms shown in Figs. 9 and 11 are attributed to the strength of the faults. The variation was quantitatively measured based on the parameter P p, and even P p values larger than zero, namely larger than σ 3, were estimated, as shown in Figs. 10 and 12. Recently, high pore pressures have been estimated from analyses of focal mechanisms (e.g., Terakawa et al. 2010; Teradata et al. 2012). A portion of P p values estimated in this study were too large to be regarded as pore pressure, because they exceeded the magnitude of the minimum compressional stress. These results suggest the possibility that there exists a mechanism by which fault strength is reduced without high pore pressure.
Using data from the high-density seismic observation networks installed in the western Nagano prefecture region, we precisely determined focal mechanisms and estimated the high-resolution stress field at a scale of 1 km. We found that nearly vertical σ 2 axes are concentrated around the mainshock fault, which is attributed to aseismic slip in the downward extension of the mainshock fault, as pointed out by Yukutake et al. (2010). The root mean squares (RMSs) of differences between the observed and calculated slip directions (misfit) are smaller than the errors for focal mechanisms at grid points away from the mainshock fault. These findings clearly indicate that the estimated uniform stress well explains focal mechanisms in each subregion away from the mainshock fault. Although it appears at grid points near the mainshock fault that more than half of RMS misfits are larger than the errors in focal mechanisms attributed to the mainshock slip, close inspections of misfits for individual subregions revealed that many of the misfits are smaller than the error in focal mechanisms, and that stress can be regarded as uniform for a larger portion within each subregion. However, we found that focal mechanisms and P-axes vary widely and differ from each other over a short focal distance of 100 m. These results clearly show that stress can generally be regarded as uniform, but that strength is heterogeneous.
YI carried out data acquisition, processing and analysis. IY, TM, and YT carried out data acquisition. MS and SG contributed to data processing. KO and HK supervised the research. All authors read and approved the final manuscript.
We are grateful to the landowners and people in Ohtaki Village and Kiso Town in Nagano Prefecture for their great help in conducting seismic observations. We are also grateful to the staffs of the local and prefectural governments and for their assistance with our study. We used seismic data from the National Research Institute for Earth Science and Disaster Prevention (NIED), the Japan Meteorological Agency (JMA) and Nagoya University.
YI, IY, MS, TM, and HK are members of the Manten seismic observation project. YT worked at the Kamitakara observatory near the Western Nagano prefecture region and joined in maintenances of the dense network. KO is the responsible person of the NIED joint project. HK is a gifted programmer and developed automated and manual phase picking programs.
The authors declare that they have no competing interests.
Availability of data and materials
The data are basically utilized through cooperative studies.
This study was partly supported by the Ministry of Education, Culture, Sports, Science and Technology (MEXT) of Japan, under the Earthquake and Volcano Hazards Observation and Research Program, and KAKENHI Grant Number 26109006.
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