Background

Earthquake triggering and seismicity rate changes following large or great earthquakes have been discussed in terms of static (and/or dynamic) changes in the Coulomb failure function (ΔCFF) (Harris and Simpson 1992; Stein et al. 1992, 1994; Reasenberg and Simpson 1992; Hill et al. 1993; Anderson et al. 1994; Toda et al. 1998; Ogata 2007; Chan and Stein 2009; Chan et al. 2016). The ΔCFF is defined as ΔCFF = Δτ − μ′Δσ, where Δτ is the shear stress change on a given failure plane (assumed to be positive in the fault-slip direction), Δσ is the normal stress change (assumed to be positive in the compressive direction), and μ′ is the apparent coefficient of friction. Positive ΔCFF values enhance failures, whereas negative values suppress failures.

Three megathrust earthquakes, the 2004 Sumatra–Andaman earthquake (Mw 9.1; U. S. Geological Survey), the 2010 Maule earthquake (Mw 8.8), and the 2011 Tohoku-Oki earthquake (Mw 9.0), provided us a great opportunity to investigate the triggering hypothesis that co-seismic stress changes transferred from large earthquakes cause seismicity changes (Fig. 1). For the Tohoku-Oki earthquake, Toda et al. (2011) showed that there is a mean 47% gain in positively stressed aftershock mechanisms over background (1997–2011 March 10) earthquakes. Furthermore, Ishibe et al. (2015) showed that the ΔCFF values following the Tohoku-Oki earthquake have a statistically higher ratio of positively stressed receiver faults compared to those before the Tohoku-Oki earthquake.

Fig. 1
figure 1

a Three megathrust earthquakes investigated in the present study (the 2004 Sumatra–Andaman earthquake (Mw 9.1; orange), the 2010 Maule earthquake (Mw 8.8; dark green), and the 2011 Tohoku-Oki earthquake (Mw 9.0; purple). b–d The fault-slip model for each earthquake as well as the distribution of 1-day aftershocks (open circles) according to the U.S. Geological Survey. The focal mechanism solutions for major earthquakes are from the Global Centroid Moment Tensor catalog. The plate boundary data from Bird (2003) are modified

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Recently, Miao and Zhu (2012) compared the spatial distribution of the ΔCFF imparted by the above three megathrust earthquakes with their aftershock distributions by assuming optimally orientated faults to be receiver faults, and concluded that there is no clear evidence that the Coulomb stress changes enhanced the aftershock activity. The ratios of aftershocks that occurred in the positively stressed regions following the Sumatra–Andaman, Maule, and Tohoku-Oki earthquakes were 47.6, 49.8, and 47.0%, respectively. They reported that the static triggering hypothesis worked well, with more than 85% of aftershocks having positive ΔCFF values, for the two large intraplate earthquakes (the 1999 Chi–Chi and 2008 Wenchuan earthquakes).

Assumption of the receiver fault mechanism sometimes greatly affects estimation of the ΔCFF. The simplest way to assume the specified receiver fault is to fix the strike, dip angle, and rake angle, but this method is valid either for major source faults with well-known fault geometries or a region with a spatially and temporally homogeneous stress field. Another way is to assume optimally oriented receiver faults, which are determined so as to maximize the ΔCFF under the specified local/regional stress field (King et al. 1994). This can also produce large errors in a complex regional stress field in which earthquakes with various focal mechanisms occur, unless the heterogeneity of the local/regional stress field is properly considered.

In the present study, we calculate the ΔCFF for nodal planes of focal mechanism solutions for actual earthquakes. This method has been proven to be effective in reducing the uncertainty of receiver faults in heterogeneous stress fields (Hardebeck et al. 1998; Imanishi et al. 2006; Toda 2008; Ishibe et al. 2011a, b, 2015; Enescu et al. 2012; Heidarzadeh et al. 2016). Following the recent occurrence of megathrust earthquakes, drastic changes in focal mechanism solutions from dominant thrust-type to a variety of types have been observed (e.g. Lay et al. 2005; Asano et al. 2011; Nettles et al. 2011). We investigate the correlation between the Coulomb stress changes transferred from three megathrust earthquakes and seismicity changes before and after the megathrust earthquakes in the neighboring regions.

Data and methodology

We use variable-slip models obtained from tsunami waveforms for the three megathrust earthquakes [the 2004 Sumatra–Andaman earthquake (Fujii and Satake 2007), the 2010 Maule earthquake (Fujii and Satake 2013), and the 2011 Tohoku-Oki earthquake (Satake et al. 2013)] (Fig. 1). In order to examine the sensitivity of ΔCFF computation to slip models, we also use the variable-slip models proposed by Rhie et al. (2007) and Ammon et al. (2005) for the Sumatra–Andaman earthquake, by Delouis et al. (2010) and Luttrell et al. (2011) for the Maule earthquake, and by Yokota et al. (2011) and Gusman et al. (2012) for the Tohoku-Oki earthquake (Additional file 1: Figure S1). For the case of the Sumatra–Andaman earthquake, we include the ΔCFF due to the Mw 8.5 Nias earthquake that occurred in March 2005 southeast of the 2004 source region using the fault-slip model obtained from teleseismic body-wave inversion (Konca et al. 2007).

For receiver faults on which stress changes are calculated, we use the focal mechanism solutions of earthquakes between January 1, 1976 and September 31, 2015 obtained from the Global Centroid Moment Tensor (GCMT) catalog (e.g. Ekström et al. 2012) (Additional file 2: Figure S2). For the case of the Tohoku-Oki earthquake, we also use the F-net focal mechanisms provided by the National Research Institute for Earth Science and Disaster Resilience (Fukuyama et al. 1998). We grouped these mechanisms into pre-seismic and post-seismic periods, before and after the occurrence of the megathrust earthquakes. The number of earthquakes we used to compute the ΔCFF values was 1374 for the Sumatra–Andaman earthquake, 436 for the Maule earthquake, and 1537 for the Tohoku-Oki earthquake. Even if a majority of aftershock nodal planes were positively stressed by the mainshock rupture, this would not necessarily prove the stress triggering hypothesis because the distribution of receiver faults is controlled primarily by predominant stress fields. Thus, we evaluate the triggering hypothesis by comparing the calculated Coulomb stress changes after the megathrusts to those during pre-seismic periods as background.

We calculate the ΔCFF values for two nodal planes of each focal mechanism solution. We assume an elastic half-space with a shear modulus of 40 GPa and a Poisson’s ratio of 0.25. For the apparent friction coefficient (μ′), we adopt an empirically introduced value, μ′ = 0.4, but we also repeat our analyses for two other values (= 0.1 and 0.8). Laboratory rock experiments on frictional slip indicate higher values, e.g., 0.5 ≤ μ′ ≤ 0.8 (e.g., Byerlee and Brace 1968), whereas fluid injection decreases the apparent friction coefficient when the pore–fluid pressure increases (Skempton 1954).

We calculate the median ΔCFF value for receiver faults for 100 consecutive earthquakes by moving the time window for 50 earthquakes. The ΔCFF values for the two nodal planes differ because the unclamping stresses are not the same, but we do not know which of these is the actual receiver fault. In order to consider the arbitrariness of nodal plane selection, we conduct a Monte Carlo simulation in which either the first or the second nodal plane of each focal mechanism solution is randomly selected as a receiver fault. In each time-window, we create 1000 datasets and calculate the average and standard deviation of the median ΔCFF values. We also calculate the ratios of three cases in which (1) both nodal planes are positively stressed; (2) one nodal plane is positively stressed, while the other is negatively stressed; and (3) both nodal planes are negatively stressed. We then evaluate the Coulomb index, which is the ratio of the number of receiver faults with stress increases for at least one nodal plane (e.g., Hardebeck et al. 1998) during both the pre-seismic and post-seismic periods.

Results and discussion

Figure 2 shows the distribution of the ΔCFF values and the focal mechanism solutions during the pre-seismic and post-seismic periods. One of the characteristics of the focal mechanism distribution during the post-seismic period is activation of normal-faulting earthquakes in the shallow crustal region of the overriding plate and the outer-rise region. For the case of the Tohoku-Oki earthquake, shallow normal-faulting earthquakes abruptly began to occur in the Ibaraki/Fukushima prefectural boundary area (coastal region around 37 °N) (e.g., Kato et al. 2011) and the outer-rise region along the Japan Trench. Similar normal-faulting earthquake sequences were also reported for the other two megathrust earthquakes: the Pichilemu region (around 34.5 °S) along the Chilean coast after the Maule earthquake (e.g., Farías et al. 2011) and East offshore of the Nicobar Islands (around 8 °N) after the Sumatra–Andaman earthquake (e.g., Lay et al. 2005; Dewey et al. 2007). Most of these normal-faulting earthquakes were positively stressed due to the occurrence of the megathrust earthquakes.

Fig. 2
figure 2

a Distribution of focal mechanism solutions and their ΔCFF values during the pre-seismic periods before the 2004 Sumatra–Andaman earthquake (left), the 2010 Maule earthquake (center), and the 2011 Tohoku-Oki earthquake (right); and b those during the post-seismic periods

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The temporal changes in the median ΔCFF values for the three megathrust earthquakes are shown in Fig. 3. Here, we focused on the temporal changes since 2000, because the temporal changes during the pre-seismic periods are negligibly small. (See Additional file 3: Figure S3, Additional file 4: Figure S4 for temporal changes in the median Coulomb stress change after January 1976.) The median values of ΔCFF are several bars during post-seismic periods, whereas they are approximately zero during pre-seismic periods for the Sumatra–Andaman and Tohoku-Oki earthquakes (Fig. 3). The rapidly increased median ΔCFF values gradually decayed to background levels for the Sumatra–Andaman earthquake. However, these were still elevated 4 years after the Tohoku-Oki earthquake. The decay rate became smaller approximately 3 years after the mainshock, suggesting that factors other than Coulomb stress changes, which will be discussed later, also contributed to keep the median ΔCFF elevated (e.g., Toda and Stein 2013).

Fig. 3
figure 3

Temporal changes in median Coulomb stress changes after 2000 for a the 2004 Sumatra–Andaman earthquake (orange) b the 2010 Maule earthquake (dark green), and c the 2011 Tohoku-Oki earthquake (purple). Vertical dashed lines indicate the occurrence times of mainshocks. Vertical error bars indicate the standard deviations obtained by Monte Carlo simulation, which randomly selects either the first or the second nodal plane as a receiver fault. See Additional file 3: Figures S3 and Additional file 4: Figure S4 in the electronic supplement for the temporal changes of median Coulomb stress changes after January 1976

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The Coulomb index during the post-seismic period was greater than 50%, with the exception of one of the 27 cases (Table 1). These results are inconsistent with those of a previous study, in which the seismicity was determined to be less than 50% in the positively stressed region by assuming an optimally oriented receiver fault. For 19 cases out of 27 cases, the Coulomb index was higher during the post-seismic period than that during the pre-seismic period. For example, using the fault model of Yokota et al. (2011) with an apparent coefficient friction of 0.4, 79.3% of receiver faults were positively stressed for at least one nodal plane following the Tohoku-Oki earthquake, whereas 64.2% of receiver faults were positive during the pre-seismic period. This is consistent with the findings of previous studies (e.g., Toda et al. 2011; Ishibe et al. 2015).

Table 1 Ratios of receiver faults during the pre-seismic and post-seismic periods with different apparent coefficients of friction and fault-slip models
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For the Maule earthquake, however, temporal changes in the median ΔCFF exhibit various patterns depending on the slip model: an abrupt increase in median ΔCFF value for the slip model of Delouis et al. (2010), an abrupt decrease for that of Fujii and Satake (2013), and almost no change for that of Luttrell et al. (2011). Most of the receiver faults used to compute the ΔCFF for the 2010 Maule earthquake were interplate aftershocks with a thrust mechanism and were concentrated in the mainshock source region (see Additional file 5: Figure S5). The evaluated ΔCFF for these receiver faults may include large uncertainties because slip models using rectangular sub-faults artificially cause anomalous stress concentrations along the edge of each sub-fault (e.g., Woessner et al. 2012), and hence the computed median values are strongly affected by these anomalous values near the source. Another possible reason is the smaller number of available receiver faults compared with the other two megathrust earthquakes.

In order to examine the sensitivity of the ΔCFF calculation to the slip model and the assumed value of the apparent friction coefficient, we repeated our analyses for other slip models (see Additional files 6, 7, 8, 9, 10, 11, 12: Figures S6 through S12) and two other apparent friction coefficients (μ′ = 0.1 and 0.8). The computed ΔCFF value is insensitive to the slip model for the Tohoku-Oki and Sumatra–Andaman earthquakes. The temporal change in the ΔCFF is similar for different values of the friction coefficient, i.e., the abrupt increase in the median value following the occurrence of a megathrust earthquake and the gradual decay toward the background level.

The catalog dependency of our results was also tested for the case of the Tohoku-Oki earthquake by using F-net focal mechanisms as receiver faults. Figure 4 shows the distribution of the receiver faults with the calculated ΔCFF and the temporal changes in the median ΔCFF value using the fault-slip model of Yokota et al. (2011) with μ′ = 0.4. (see Additional file 13: Figure S13, Additional file 14: Figure S14 for different apparent coefficients of friction). The median ΔCFF exhibited similar temporal changes associated with the Tohoku-Oki earthquake: a sudden increase followed by a gradual decrease toward the background level. The median ΔCFF fluctuates more for the F-net catalog than for the GCMT catalog. This may reflect the characteristics of earthquake clustering, because the F-net data include a much larger number of focal mechanism solutions than the GCMT catalog.

Fig. 4
figure 4

a Distribution of focal mechanism solutions and their ΔCFF values calculated for the F-net catalog during the pre-seismic (left) and post-seismic (right) periods of the 2011 Tohoku-Oki earthquake. The slip model of Yokota et al. (2011) was used. b Temporal changes of the median Coulomb stress change. The histograms for receiver faults during the pre-seismic and post-seismic periods are also shown

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Concluding remarks and future developments

We investigated the correlation between static Coulomb stress changes imparted by three megathrust earthquakes and changes in focal mechanism solutions by means of abundant focal mechanism solutions of earthquake catalogs as receiver faults. The median Coulomb stress was approximately zero during the pre-seismic period before the megathrust earthquake, abruptly increased as a result of the earthquake, and then gradually decreased, at least for the Sumatra–Andaman and Tohoku-Oki earthquakes. The ratio of positively to negatively stressed receiver faults increased following these megathrust earthquakes for almost all slip models and apparent friction coefficient values. Changes in the focal mechanism distribution that can be correlated with static Coulomb stress changes due to megathrust earthquakes support the static stress triggering hypothesis. The conclusion of the present study is opposite that of a previous study using optimally orientated receiver faults, and this difference indicates the importance of considering the spatial and temporal heterogeneities of receiver faults.

In the present study, we considered only the static stress changes transferred from co-seismic fault slips of the mainshocks. However, other possible factors for triggering seismicity changes, such as dynamic stress change due to the passage of seismic waves (e.g., Hill et al. 1993; Miyazawa 2011), a decrease in failure strength due to an increase in pore–fluid pressure (e.g., Hubbert and Rubey 1959), post-seismic slip, static stress change from indirectly triggered earthquakes by numerous aftershocks, acceleration of relative plate motion (e.g., Heki and Mitsui 2013; Uchida et al. 2016), and/or viscoelastic relaxation (e.g., Pollitz and Sacks 1995), have been suggested by various studies. These might disturb or mask the correlation between Coulomb stress changes co-seismically transferred from the mainshocks and changes in seismicity following the occurrence of mainshocks.