2. Institute of Geophysics, China Earthquake Administration, Beijing 100081, China
The 2008 Wenchuan M_{S}8.0 earthquake was a high angle thrust earthquake in the Chinese mainland, resulting in huge casualties and property losses. Only 5 years after the Wenchuan earthquake, the Lushan M_{S}7.0 earthquake occurred in the southern segment of the Longmenshan fault. Although these two strong earthquakes occurred in the Longmenshan fault, the results of the aftershock relocation showed that the rupture of the Lushan earthquake and the rupture of the Wenchuan earthquake were not connected, and there was a"seismic gap"between them. The rupture space has aroused widespread concern among seismologists, and Chen Yuntai et al. (2013) pointed out that the "seismic gap" has a potential risk of M_{W}6.8 earthquakes. Xu Xiwei et al. (2013) and Gao Yuan et al. (2013) both considered that it was necessary to pay close attention to the risk of rupture of the "seismic gap". Therefore, it is an urgent question to solve the problem of how much the void gap is affected by the stress loading of two earthquakes, and how the probability of the moderate strong earthquakes occurs.
Research shows that the Coulomb failure stress changes produced by strong earthquakes have great influence on the subsequent earthquake location and time (Harris R.A., 1998; Stein R.S., 1999; Das S. et al., 1981; Zhou Longquan et al., 2008; Wan Yongge et al., 2010; Shan Bin et al., 2012, 2013; Song Jin et al., 2011, 2014). Dieterich (Dieterich J.H., 1979, 1981; Dieterich J.H., 1994) puts forward the rate-state friction law and deduces the effect of stress on regional seismicity, which is called the Dieterich model. The cumulative earthquake probability of strong earthquakes in the study area can be estimated by the Coulomb stress change and the regional background seismic activity as input, which provides a reference for studying the regional earthquake risk. After the Wenchuan earthquake in 2008, Toda S. et al. (2008) took into account the co-seismic Coulomb stress change and the regional background seismicity probability of the Wenchuan mainshock and believed that the occurrence rate of M_{W}≥6.0 earthquake in the south section of the Longmenshan fault had obviously increased. Shao Zhigang et al. (2010) used the viscoelastic medium model to calculate the Coulomb stress of the Wenchuan earthquake and the regional background seismicity probability, and believed that the seismic risk of the Dawu-Kangding section of the Xianshuihe fault and Longmenshan fault was higher. Liu Boyan et al. (2013) calculated the variation of the surrounding earthquake occurrence rate caused by the co-seismic Coulomb stress of the Wenchuan earthquake, and the result shows that the occurrence of the Wenchuan earthquake made the occurrence of the Lushan earthquake earlier, and it is necessary to pay close attention to the possibility occurrence of M6.0 earthquakes in the Xianshuihe fault. The Lushan earthquake occurred in 2013. Lei Xinglin et al. (2013) calculated the stress disturbances of the Wenchuan and Lushan earthquakes, and thought that the Wenchuan earthquake did produce the stress triggering effect in the"seismic gap", but the effect of the Dayi earthquake was not completely eliminated till 2013. The results of Xu Jing et al. (2014), Liu Mian et al. (2014) and Jia Ke et al. (2014) showed that the Coulomb stress increased significantly in the"seismic gap"of the two earthquakes of Wenchuan and Lushan, and there was a possibility of an earthquake with M7.0 in the "seismic gap". Most of the previous research focused on the stress loading of the two strong earthquakes in the"seismic gap", but no research has been done in terms of the background seismicity to estimate the cumulative earthquake probability of an M_{W}≥6.0 earthquake in this space under the action of two earthquake stress disturbances. This paper first calculates the cumulative Coulomb stress of the Wenchuan and Lushan earthquakes, based on the layered viscoelastic medium model, and then works out the cumulative occurrence probability of M_{W}≥6.0 earthquakes in the"seismic gap"with the velocity state frictional law of the Dieterich J. (1994) and the regional background seismicity.
1 RESEARCH METHODS 1.1 Coulomb Stress TriggeringCoulomb stress triggering refers to the stress change caused by earlier earthquakes, and plays a role of promotion or inhibition for future earthquake activity in the research area (Harris R.A., 1998). The specific method is to project the stress change tensor of the earlier earthquake to focus on the fault plane and the sliding direction, and calculate the Coulomb fault stress change ΔCFS.
The Coulomb fault stress change ΔCFS is defined as (Harris R.A., 1998;Stein R.S. et al., 1992)
$ \mathit{\Delta }{\rm{CFS = }}\Delta {\tau _{\rm{s}}} + \mu \left({\Delta {\sigma _{\rm{n}}} + \Delta p} \right) $ | (1) |
where μ in the formula is the friction coefficient, Δτ_{s} and Δσ_{n} are respectively the shear stress and the normal stress at the fracture level, Δτ_{s} is positive along the sliding direction, Δσ_{n} is positive, and Δp is the pore fluid pressure change. The effect of the pore pressure reduction coefficient of friction can be expressed by μ′=μ(1-B). Wherein, B is the Skempton coefficient with the range is 0-1 (Rice J.R., 1992). Therefore, formula (1) becomes
$ \Delta {\rm{CFS}} = \Delta {\tau _{\rm{s}}} + \mu '\Delta {\sigma _{\rm{n}}} $ | (2) |
When ΔCFS>0, the stress change is advantageous to subsequent earthquakes, otherwise it is unfavorable to the occurrence of earthquakes.
1.2 Earthquake Probability Based on the Dieterich ModelThere are many frictional constitutive relationships that can be used to explain the laboratory observations of rock friction, which are very consistent with experimental data and are widely accepted as rate-state friction laws (Rate and State variable friction law, or R-S model for short), which can explain a series of laboratory observations such as periodic viscous sliding, self-persistent periodic oscillation, multiple cycle phenomena and chaotic oscillation (Shao Zhigang et al., 2010). One of the important applications of the R-S model in seismic research is the change of regional earthquake probability after simulating stress disturbances. For example, Catalli F. et al. (2008), based on the R-S model, used the Coulomb fault stress changes as stress disturbances to discuss the influence of strong earthquakes on regional seismicity, and the interaction between the events in the 1997 Briatore earthquake group in Italy was analyzed.
The mathematical expression of the R-S model is
$ \tau = \left[ {{\mu _0} + A\ln \left({\frac{V}{{{V_0}}}} \right) + B\ln \left({\frac{{{V_0}\theta }}{{{D_{\rm{c}}}}}} \right)} \right]\bar \sigma $ | (3) |
In formula (3), A and B are empirical constants, representing the short-term direct and long-term effects of the change of sliding rate on the friction coefficient. V and V_{0} are sliding rates and background reference sliding rates respectively. μ is the steady-state coefficient of friction when V=V_{0}. D_{c} is the critical sliding distance, which indicates the distance required to stabilize the particle in the interface after the rate change. The state parameter θ describes the variations of the dynamic friction coefficient with time, and have two forms of the change of the state parameter, namely the"slip"form with the change of the slip distance and the"slow"form of the state parameter and the slip rate.
$ \frac{{{\rm{d}}\theta }}{{{\rm{d}}t}} = - \frac{{V\theta }}{{{D_{\rm{c}}}}}\ln \left({\frac{{V\theta }}{{{D_{\rm{c}}}}}} \right) $ | (4) |
$ \frac{{{\rm{d}}\theta }}{{{\rm{d}}t}} = 1 - \frac{{V\theta }}{{{D_{\rm{c}}}}} $ | (5) |
Some studies have shown that the"slip"form is more suitable for the description of the fault nucleation process (Dieterich J.H., 1979, 1981; Dieterich J.H., 1996).
Based on the R-S model, Dieterich J.H. (1996) further analyzes the influence of stress change on regional seismicity, deduces the earthquake incidence model (i.e. the Dieterich model) and analyzes the influence of the relative parameters on the earthquake incidence. The calculation formula of earthquake probability is
$ R\left(t \right) = \frac{r}{{\left[ {\exp \left({ - \frac{{\Delta {\rm{CFS}}}}{{A\bar \sigma }}} \right) - 1} \right]\exp \left({\frac{t}{{{t_{\rm{n}}}}}} \right) + 1}} $ | (6) |
In formula (6), r is the occurrence rate of the regional earthquake background before the main event, A is the frictional constitutive state parameter, and is the effective positive stress, and t_{a} is the duration of stress disturbance.
Shao Zhigang et al. (2010) points out that the earthquake incidence rate based on the co-seismic and annual Coulomb stress change can be expressed as
$ {R_{{t_k}}} = \sum\limits_{t = 0}^k {R\left({k - i} \right), } \;\;\;\;\;k = 0, 1, \cdots, n\;{t_k} = {t_0} + k $ | (7) |
In formula (7), k plays a role as R in formula (6) and is the first stress loading of the kth year. n is the maximum year length in the calculation. i is the ith year after the main earthquake. t_{0} is the occurrence year of the main earthquake. t_{k} is the earthquake probability calculation year.
The earthquake probability is converted into the cumulative Poisson probability P (Toda S. et al., 2008) for at least one earthquake, and its formula is
$ P = 1 - \exp \left({ - R} \right) $ | (8) |
Based on the R-S model and the Coulomb stress loading results generated by the two strong earthquakes, this paper calculates the M_{W}≥6.0 earthquake occurrence rate of the"seismic gap"of the Longmenshan fault, and finally obtains the change of the cumulative probability of occurrence of the earthquake at any time.
2 CUMULATIVE EARTHQUAKE PROBABILITY 2.1 Static Coulomb StressBased on layered viscoelastic medium model (Xu Jing et al., 2013), we calculate the co-seismic Coulomb stress and the post-earthquake cumulative effect of the Wenchuan earthquake as the Lushan epicenter by using the PSGRN/PSCMP software provided by Wang Rongjiang et al. (2006). The seismic fault parameters are obtained by the inversion of Ji C. et al. (2008) for the limited fault rupture model of the Wenchuan earthquake. The effective friction parameter is set as 0.4, and the depth of calculation is 10km. The result shows that the co-seismic Coulomb stress of the Wenchuan earthquake in the Lushan epicenter is 0.016MPa, greater than the threshold of 0.01MPa which is usually considered to be effective in promoting subsequent earthquakes (Harris R.A., 1998). After 5 years, the viscoelastic effect was accumulating, and the Coulomb stress reached 0.021MPa after the earthquake. Therefore, the Wenchuan earthquake had an obvious stress triggering effect on the occurrence of the Lushan earthquake, and the viscoelastic effect can not be neglected.
In this paper, the effect of the stress disturbance in the"seismic gap"of the middle-south section of the Longmenshan fault is studied by using the inversion result of the finite fault model of Ji C. et al. (2008) and the inversion of the Lushan seismic finite fault rupture model of Zhang Yong et al. (2013). Because of the small area of the"seismic gap", this paper chooses its center point position to estimate the stress disturbance of the"seismic gap". The results (see Fig. 2(b) and Table 2) show that the central point of the"seismic gap"is loaded by the co-seismic Coulomb stress of two strong earthquakes, and the cumulative stress changes are increasing. The co-seismic Coulomb stress produced by the Wenchuan earthquake is 0.053MPa, and that produced by the Lushan earthquake is 0.004MPa. By 2030, the co-seismic cumulative value of the two earthquakes and the post-earthquake stress will reach 0.067MPa.
Toda S. et al. (1998) used the logarithmic coordinate linear fitting time of the number of aftershocks, and the duration of the aftershock is the time when the fitting line returns to the background seismicity level. In this paper, the background seismicity level was computed in the 100km range around the Lushan epicenter. The M_{L}≥3.0 earthquakes occurred 299 times between 1970 and 2007, and the monthly frequency of the M_{L}≥3.0 background seismicity was 0.67. According to the Chinese Seismic Network measurement, as of December 2016 the Lushan earthquake aftershock area has a total of more than 385 M_{L}≥3.0 earthquakes, including 50 M_{L}4.0-4.9 earthquake and 13 M_{L}5.0-5.9 earthquake. With calculation by the unit time (month) and linear fitting in the logarithmic coordinate system, the aftershock frequency attenuation curve is found to return to the background seismicity level, namely the Lushan earthquake aftershock duration (see for Fig. 3). It can be seen from Fig. 3 that the monthly frequency of the sequence is constantly decaying after the main earthquake occurs, and the fitting curve reflects the trend of the aftershock attenuation. The regression of the M_{L}≥3.0 aftershock to the background level needs 10.4 months. The Lushan aftershock duration lasted about 10 months. Using the same method, Shen Wenhao et al. (2013) found that the duration of the Wenchuan aftershock will last for about 80 years. Because of the great difference in the aftershock duration of the two strong earthquakes, the selection of the t_{a} value needs further discussion. The results of the cumulative earthquake probability obtained by different t_{a} values are compared in the following 2.2.3 section, and it is found that the t_{a} value selection has a small influence on the change tendency and magnitude of cumulative earthquake probability in the stress perturbation values produced by the two strong earthquakes. Considering the magnitude of the calculation target is M_{S}6.0, the t_{a} value of the cumulative earthquake probability is assumed to be 10 months in Section 2.22. It should be pointed out that the t_{a} value, in physics, is weakening the possible impact of the Wenchuan earthquake.
From January 1, 1900 to December 31, 2007, the Longmenshan fault only had 3 M_{S}≥6.0 earthquakes, and according to the regional background seismicity formula r_{≥M}=N/Δt (where N is the occurrences of the earthquake with magnitude larger than M, Δt is the time interval), r_{≥6.0}=0.028 t/a can be obtained. The frictional constitutive state parameter A is 0.01 and the effective positive stress σ is 10MPa respectively with the help of the previous practice (Dieterich J.H. et al., 1996).
After the Wenchuan earthquake, the Lushan epicenter was strongly subjected to the co-seismic and post-earthquake stress loading (see Table 1 for the detail), and according to Formulae (6), (7) and (8), we calculate the probability of a greater than 2008-2013 M≥6.0 earthquake occurred in the Lushan epicenter, which is shown in Fig. 6(b). From Fig. 4(b), it can be seen that the stress disturbance by the Wenchuan earthquake causes a significant increase in the cumulative earthquake probability of a M≥6.0 earthquake in the Lushan epicenter, and that the cumulative earthquake probability of a M≥6.0 earthquake in Lushan is up to 18%, which advances the occurrence time of the Lushan earthquake. This is basically consistent with the conclusion of Liu Boyan et al. (2013).
The cumulative earthquake probability of the central point of the"seismic gap"of the Longmenshan fault is shown as follows (see Table 2 and Fig. 5(b)). Due to the co-seismic and continuous loading after the two earthquakes of Wenchuan and Lushan, the cumulative earthquake probability at the center point is increasing, and the probability of a M≥6.0 earthquake will reach 66% at the end of 2030. Based on the above results, we think that there is still a danger of a middle and strong earthquake in the middle-south section of the Longmenshan fault.
In this paper, a logarithmic coordinate linear fitting method (Toda S. et al., 1998) is used and it is estimated that the duration of the Lushan aftershocks lasted for about 10 months. Shen Wenhao et al. (2013) used the same method and found that the aftershock duration of the Wenchuan earthquake will last for about 80 years. Because of the difference in the duration of the aftershocks of the two strong earthquakes, the selection of t_{a} values needs further discussion. In this paper, the center point of the"seismic gap"is taken as the research object, and the cumulative earthquake probability is calculated and compared with different values as t_{a} values in the periods from 10 months to 160 years. The results show that the difference of t_{a} value has a little effect on the cumulative earthquake probability in the stress perturbation values produced by the two strong earthquakes (see Fig. 6(a)). By the end of 2020, when the t_{a} value is set to 10 months, the central point cumulative earthquake probability will be 39.6%, and when the t_{a} value is set to 160 years, the central point cumulative probability of the earthquake will be 40.8%; The result shows that the longer the duration of an aftershock, the greater the cumulative probability.
In this paper, the frictional constitutive state parameter A and the effective positive stress σ are calculated on the basis of the previous practice (Dieterich J.H. et al., 1996), and A=0.01 and σ=10MPa. As for how the A value affects the cumulative earthquake probability, we also use different values for comparison (see Fig. 6(b)). The results show that the Aσ value in the stress disturbances generated by the two strong earthquakes has a smaller impact on the cumulative earthquake probability. After the Lushan earthquake, when Aσ takes 0.05, the cumulative earthquake probability of the center point is 20.3%, and when Aσ takes 10, the result of the cumulative earthquake probability the center point is 20.1%; The result shows that the greater the value of Aσ, the smaller the cumulative probability of the earthquake.
3 DISCUSSION AND CONCLUSIONAfter the 2008 Wenchuan M_{S}8.0 earthquake and the 2013 Lushan M_{S}7.0 earthquake, a seismic gap still exists in the southern section of Longmenshan fault, the scale of which is equivalent to a M_{W}6.0-7.0 earthquake. The seismogenic ability and the current danger have aroused widespread concern. In order to solve the problem, this paper firstly calculates the co-seismic and post stress disturbance of the two earthquakes in Wenchuan and Lushan by the layered viscoelastic medium model. Based on this, the rate-state friction law of Dieterich J. (1994) is combined with the seismic activity level of a M≥6.0 earthquake in the Longmenshan fault, and the cumulative earthquake probability of M6.0 earthquakes in the"gap"is calculated over time. The results show that the cumulative earthquake probability of M6.0 earthquakes increases with time, and will increase to 66% by the end of 2030.
Many uncertainties in the calculation process may result in a certain error in the cumulative earthquake probability. First, the Dayi M_{S}6.2 earthquake occurred in the"seismic gap"in the 1970, which might moderate the stress concentration in the"seismic gap". Then, the Longmenshan thrust tectonic belt is composed of 3 main faults of the Longmenshan fault, the central fault and the Qianshan fault (Zhang Yueqiao et al., 2013), and the"seismic gap"between the Wenchuan and the Lushan earthquakes has certain complexity in geological structure. In this paper, it is simplified as a central point, and the method of receiving the fault parameter is mainly based on the Lushan earthquake focal mechanism. This may have some influence on the final calculation result. Thirdly, the selection of model parameters of the layered viscoelastic medium, especially the setting of viscous coefficients, will introduce some errors to the stress disturbance results of strong earthquakes, thus affecting the final cumulative probability results. At the same time, M6.0 earthquakes have occurred only 3 times in the Longmenshan fault since 1900, and there may be some error in the estimation of background seismicity level.
In addition, the perturbation effect of the Wenchuan earthquake on the Lushan earthquake is calculated by the same method. The results show that the Lushan epicenter is subjected to the continuous co-seismic loading of the Wenchuan earthquake and the stress after the earthquake, the cumulative earthquake probability of M6.0 earthquakes is up to 18%, indicating that the stress disturbance generated by the Wenchuan earthquake may have advanced the Lushan earthquake time, which is basically consistent with the conclusion of Liu Boyan et al. (2013). Therefore, we believe that although there may be some errors in the results, there is still the danger of middle and strong earthquakes in the"seismic gap"in the middle and southern section of the Longmenshan fault, and the risk is increasing with time.
ACKNOWLEDGEMENT
We owe our sincere thanks to Professor Ji Chen and Professor Zhang Yong for their inversion data for the Wenchuan and the Lushan earthquakes, and Professor Wang Rongjiang for the provision of the PSGRN/PSCMP computing software. We especially thank the reviewers for their pertinent amendments and suggestions for this paper.
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