1 INTRODUCTION

Coulomb stress triggering model has been found to play an important role in the production of aftershocks and subsequent mainshocks on surrounding faults (Harris, 1998; Stein, 1999; King and Cocco, 2001; Freed, 2005). A number of workers have investigated earthquake triggering and achieved marvelous progresses (i.e., King et al., 1994; Harris, 1998; Stein, 1999; King and Cocco, 2001; Freed, 2005; Toda et al., 2008; Parsons et al., 2008; Wan et al., 2000; Shi, 2001; Zhang et al., 2010; Miao and Zhu, 2012, 2013). In particular, Stein et al.(1997)successfully predicted the 1999 Izmit earthquake by calculating the Coulomb stress changes for ten earthquakes with magnitude over 6.7 in North Anatolian fault from 1939 to 1992, in which 90 per cent of earthquakes were triggered by previous events. Similarly, Parsons et al. presented the distribution of the Coulomb stress changes caused by the 2008 Wenchuan earthquake, and found that the Ya0an region is located in the area of Coulomb stress increase, which had been regarded as a risk area. Subsequently, the 2013 Lushan Ms7.0 event occurred in the Ya0an region, proving that Coulomb model is of predictive power(Miao and Zhu, 2013).

According to previous studies (e.g., King et al., 1994; Harris et al., 1998), the Coulomb failure stress changes (ΔCFS) can be calculated by the following simplified formula:

whereΔγ is the shear stress change on a given fault plane (positive in the sense of fault slip), Δσ_n is the fault normal stress change (positive for fault unclamping), and $\mu '$ is an apparent coefficient of friction. In addition, most investigators compared Coulomb stress models with different apparent friction coefficients to examine the model parameters. In general, a number of apparent friction coefficients $\mu '$ are taken into account from 0.0 to 0.8 in order to better explain the correspondence between the main shock and its triggered aftershocks. For example, Parsons et al.(1999)calculated stress changes for four faults in the Santa Clara Valley thrust belt, and demonstrated that the strongest correlation exists between seismicity and normal stress changes or Coulomb stress changes for large coefficients of friction. Bilek & Bertelloni (2005) computed ΔCFS at 15 km depth in the Costa Rica subduction zone caused by the 1999 M_w=6.9 Quepos earthquake due to variations in frictional coefficients between 0.1 and 0.9, and showed that the ΔCFS increased with the rise of friction coefficients. Toda et al.(2011)also calculated ΔCFS imparted to the nodal planes of aftershocks of the 2011 Tohoku earthquake with frictional coefficients of 0.4 and 0.8, respectively. The computed results displayed that ΔCFS with large friction coefficient are larger than those with small friction when the normal stress change is positive (Δσ_n > 0). In the same way, Gahalaut & Gahalaut (2008) found that change in static stress on normal faults is positive for all values of fault friction, and is maximum for high fault friction.

As a matter of fact, Coulomb stress changes will increase with frictional coefficients as long as the variation in normal stress is positive based on Eq.(1). In this case, the effect of earthquake triggering becomes high with the increase of frictional coefficients. That is to say, the larger the frictional coefficient on the fault, the more easily the fault slips to produce earthquakes. Obviously, this result violated our common sense in which friction always resists fault slips and inhibits earthquakes no matter what the situation is. In this study, we will present an interpretation for this contradiction and give an alternative way to calculate the Coulomb stress changes with different frictional coefficients.

2 COULOMB FAILURE STRESS CHANGES DUE TO VARIATIONS OF FRICTIONAL COEFFICIENT

In general, Coulomb failure stress on the fault can be formulated as the following expression (e.g., Harris and Simpson, 1992; Reasenberg and Simpson, 1992; Stein et al., 1992, 1994; Simpson and Reasenberg, 1994; King et al., 1994; Harris et al., 1995; Nostro et al., 1997):

where γ is the shear stress on a target fault plane (positive in the sense of fault slip), σ_n is the normal stress on the fault (positive for extension), $\mu '$ is the friction coefficient, p is the pore fluid pressure (p = -Bσ_n, B Skempton coefficient, which varies from 0 to 1 generally), and ${\mu '_0}$ is the apparent coefficient of friction that is defined by B is used to take into account the modifications of the effective normal stress caused by pore fluid pressure.

However, when the apparent friction coefficient on the fault varies from ${\mu '_0}$ to ${mu '_1}$, the Coulomb failure stress will become

Obviously, the Coulomb failure stress changes due to the variation of frictional coefficient can be described by the following formula:

Suppose the frictional coefficient varies from ${\mu '_0}$ = 0.3 to ${mu '_1}$ = 0.4, the ΔCFS₀ will be ~ -400 MPa (=0.1×400) in the case when the fault depth is ~15 km and the corresponding σ_n is ~ -400 MPa. It can be seen that, as the frictional coefficient varies a little, the Coulomb failure stress changes a great deal, and ΔCFS₀ decreases with the increase of the apparent frictional coefficient in the Earth’s crust. The additional ΔCFS₀ imparted by the variations of frictional coefficient is often ignored by almost all authors in calculation of Coulomb failure stress.

3 COULOMB STRESS CHANGES DUE TO COSEISMIC DISLOCATIONS

When an earthquake occurs, it modifies the state of stress on nearby faults due to coseismic dislocations. Then, based upon Eq.(2), the Coulomb failure stress on the fault will become

whereΔγ is the shear stress change on a given fault plane (positive in the sense of fault slip) due to the coseismic slips, and Δσ_n is the fault normal stress change (positive for fault unclamping) imparted by the main shock.

Under these circumstances, if the apparent friction coefficient on the fault varies from ${\mu '_0}$ to ${\mu '_1}$, the Coulomb failure stress will be

Thus, the combined Coulomb failure stress changes due to both the occurrence of the earthquake and the variation of apparent frictional coefficient are

However, almost all previous workers neglected the first term (ΔCFS₀) in formula (8) when they compared Coulomb stress change models with different frictional coefficients (e.g., Deng and Sykes, 1997; Parsons et al., 1999; Toda and Stein, 2000; Bilek and Bertelloni, 2005; Parsons et al., 2008), they only took account of the stress changes of the last term (${\mu '_1}$ Δσ_n), which is no more than several MPa in general.

Suppose changes of normal stresses on the fault (Δσ_n) is 2 MPa, when the frictional coefficient varies from ${\mu '_0}$ = 0.3 to ${mu '_1}$ = 0.4, traditional Coulomb stress changes, i.e., the term (${mu '_1}$Δσ_n) in Eq.(8), only increase 0.8 MPa, but Coulomb stresses caused by tectonic force will decrease by 40 MPa. Thus, the combined ΔCFS will be as large as –39.2 MPa, rather than 0.8 MPa (in light of traditional Coulobm stress). Clearly, the increase of apparent frictional coefficient gives rise to the large decrease of Coulomb stress changes, not an increase of stress changes, which is in good agreement with our common knowledge. If we incorporated the additional ΔCFS in calculation, the above contradiction will disappear completely. Therefore, it is suggested that we should consider changes of combined ΔCFS due to the variation of friction coefficient, especially when we compare different Coulomb stress models with different apparent frictional coefficients.

4 DISCUSSION AND CONCLUSIONS

Studying earthquake triggering has attracted more and more attention both at home and abroad. The triggering model has also been improved recently, from the initial Okada analytical model (Okada, 1985, 1992) to complex one in which material is not homogeneous. However, combined ΔCFS is usually not taken into account in studying earthquake triggering presently by most workers, which is one of the disadvantages. With the introduction of numerical methods, Coulomb stress model will be more perfect, and more helpful to earthquake prediction and seismic hazard assessment.

By the way, almost all models in investigating earthquake triggering do not consider initial stresses. In fact, if the stress regime is far away from a rupture state, a high Coulomb stress change will not trigger any event at all. In contrast, when the stress on the fault is very high, approaching critical rupture level, a very small stress increase may trigger a large event.

By means of analyses above, preliminary conclusions are reached in the following:

(1) We can see that a little variation of apparent frictional coefficient on the fault will give rise to great changes in Coulomb failure stress, which is ignored by many previous investigators in comparing different Coulomb stress models. The additional ΔCFS, produced by changes of friction coefficients, will be as large as 40 MPa if the variation of apparent friction coefficient is 0.1 given the depth of receiver fault of 15 km.

(2) When the apparent frictional coefficient changes, the combined CFS is composed of 2 parts, one from tectonic forces, and the other from coseismic dislocations. We should note that the stress changes caused by tectonics are much larger than those by coseismic slips. The combined ΔCFS will decrease as large as 39.2 MPa, whereas traditional ΔCFS increase only 0.8 MPa, if the variation of apparent friction coefficient is 0.1 given the depth of receiver fault of 15 km. This presents a solution to the paradox in which Coulomb failure stress changes increase with frictional coefficients.

Therefore, we should be careful in stress triggering study, in particular when comparing with different models by changing frictional coefficients on faults. Evidently, initial stresses may be considered in Coulomb stress change calculations in order to improve Coulomb stress triggering model.

This work was financially supported by the National Natural Science Foundation of China (41574041), and by Natural Science Foundation of Beijing (8152034).