Aftershock sequence simulations using synthetic earthquakes and rate-state seismicity formulation
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We use an efficient earthquake simulator that incorporates rate-state constitutive properties and uses boundary element method to discretize the fault surfaces, to generate the synthetic earthquakes in the fault system. Rate-and-state seismicity equation is subsequently employed to calculate the seismicity rate in a region of interest using the Coulomb stress transfer from the main shocks in the fault system. The Coulomb stress transfer is obtained by resolving the induced stresses due to the fault patch slips onto the optimal-oriented fault planes. The example results show that immediately after a main shock the aftershocks are concentrated in the vicinity of the rupture area due to positive stress transfers and then disperse away into the surrounding region toward the background rate distribution. The number of aftershocks near the rupture region is found to decay with time as Omori aftershock decay law predicts. The example results demonstrate that the rate-and-state fault system earthquake simulator and the seismicity equations based on the rate-state friction nucleation of earthquake are well posited to characterize the aftershock distribution in regional assessments of earthquake probabilities.
KeywordsEarthquake simulator Rate-and-state seismicity Aftershock
Earthquakes occur in response to continuous tectonic forcing and from interactions among elements of fault networks that slip and transfer stresses. Following a large earthquake occurrence of aftershocks in time and space is related to that main shock. Those aftershocks could last for days or years and pose as seismic hazard after a main shock. Aftershock sequences or earthquake interactions, such as earthquake triggering and earthquake suppressing, have been widely investigated in the context of earthquake-induced static or dynamic stress changes. It has been shown for decades that large earthquakes can inhibit or promote failure on nearby faults in form of such stress transfer. The abundance of phenomenological investigations has shown that spatial patterns of static or dynamic stress changes seem to correlate well with spatial patterns of aftershocks for natural earthquakes (e.g. Harris 1998; Kilb et al. 2002; Steacy et al. 2005a; Durand et al. 2013). It becomes a commonplace understanding that positive stress changes tend to activate faults, triggering more failures, resulting in more aftershocks, whereas negative stress changes tend to relax faults, suppressing further failures, diminishing the possible aftershocks.
Generally, the Coulomb stress transfer due to an earthquake is calculated by summing the stress changes due to all slipping fault patches using elastic dislocation theory (Okada 1992) and resolving the resulting stresses onto the plane of interest such as pre-existing fault planes or optimally oriented planes (OOP) determined by stress states. The OOP are strongly dependent on the orientation of the regional stress field and can vary widely in an active seismic region. The predictive stress transfer map might look quite different depending on assumption about the orientation of the active structures. The optimal planes could be determined from a combination of regional stress field and main shock-induced stress field using the maximum possible Coulomb failure plane orientation. This method shows that distribution of aftershocks of natural earthquakes can be explained by the Coulomb criterion (e.g. King et al. 1994; Toda et al. 2005). The optimal planes might be computed in a 3D stress field with some constraints placed on the fault orientation (McCloskey et al. 2003; Steacy et al. 2005b; Xu et al. 2010). Such a method uses the available geological structure information in a region of interest and fixes one of the fault orientations. The predictions by such a method would fit the spatial distribution of the events better than using unconstrained fault orientation in case studies (e.g. McCloskey et al. 2003; Steacy et al. 2005a).
The Coulomb failure hypothesis is conceptually simple and convenient to implement. It has been used for decades to explain the characteristics of aftershocks, however, it does have limitations. In the investigation of suppression of large earthquakes, Harris and Simpson (1998) found that a new failure model, rate-state formulation, offers a consistent explanation for the aftershocks after 1906 San Francisco earthquake whereas the Coulomb failure model does not. Unlike the Coulomb failure model, the new failure model is based on rate- and state-dependent friction constitutive representation of laboratory observations, and appears more complex by describing the evolution of quantities as they approach failure limits though it was shown that rate-state models might asymptotically become equivalent to Coulomb models under a variety of conditions (Gomberg et al. 2000). It has also been used to examine the earthquake probabilities by transient rate-and-state triggering effects and used to forecast the evolution of seismicity in southern California (Toda et al. 1998, 2005) and to model the aftershock sequences with 3D stress heterogeneities and planar/rough fault models (Smith and Dieterich 2010). The rate-and-state model has been compared with the Coulomb stress model on predictions of clock advance (Gomberg et al. 1998, 2000). The rate-and-state parameters can be found by fitting aftershock sequences (Gross and Burgmann 1998; Harris and Simpson 1998). Favorably, the rate-and-state formulation explains temporal features of aftershocks, such as the Omori law decay in aftershock seismicity rate as consequence of Coulomb stress transfers and predicts that aftershock duration is proportional to mainshock recurrence time (Dieterich 1994, 2007). Based on the rate-and-state constitutive model, the seismicity rate equation describing the evolution of seismicity rate with stress transfer could be used to determine earthquake probabilities in a region of interest and that the seismicity rate is a function of time and the stress transfer (Harris and Simpson 1998; Toda et al. 2005; Hainzl et al. 2009). Such a synthetic aftershock catalog might be obtained using the rate-state seismicity rate as a non-stationary statistical process and the catalog might serve as a regional seismic hazard estimator which depicts some characteristics of the aftershock occurrence in time and space after an event (Dieterich 1994; Smith and Dieterich 2010).
The rate-state off-fault seismicity rate is determined in term of the geophysical variables such as the stressing rate that determines the seismicity decay characteristics with time and Coulomb stress transfer that changes the seismicity rate instantaneously at earthquake occurrence. In addition to the rate-state representation model, earthquake occurrence can be described as a stochastic process. One is the stress release model (Vere-Jones 1978) based on elastic rebound theory. This model assumes the occurrence probability depends on some quantity which may be interpreted as the mean-stress level in a region and increases with an increasing stress level and drops when an earthquake happens (e.g. Keuhn et al. 2008). Stress transfer and stress-triggering between distant faults can be included to simulate realistic scenarios. Note that the stress release is connected with the earthquake magnitude empirically. The other is the epidemic-type aftershock sequence (ETAS) model (Ogata 1988). The ETAS model describes earthquake activity as a point process, and the earthquake occurrence rates are history dependent and depend on four constants that are aftershock productivity factor, power-law exponent of the event rate decay in the Omori law, magnitude sensitivity parameter, and characteristic time shift. The four model constants and the background seismic rate can be estimated from the regional aftershock seismicity observations. This model and its extensions have been shown useful for quantifying the seismic activation and quiescence in earthquake active regions, and some results suggest the background seismicity might increase with time in some regions (e.g. Ogata et al. 2003; Bansal and Ogata 2013; Kumazawa and Ogata 2013).
We investigate off-fault aftershock sequences from the simulated earthquakes in the fault system calculated by an efficient earthquake simulator (Dieterich and Richards-Dinger 2010) in this article. The simulator is based on rate-state friction nucleation of earthquake and can resolve the discrete fault-slip events across the scale range needed to track the state evolution for the brittle regions of the solid Earth using appropriate constitutive friction law so that we could better understand the short-term dynamics of how fault-system state evolves with long-term tectonic behavior. This type of multi-scale simulation spans a continuous range of time scales from <10−3 s for slip evolution during fault rupture to ~103 s for the inertial dynamics of large-scale ruptures, to ≥1011 s for the characterization of long-term deformation and large-event statistics. Corresponding spatial scales extend from ≤100 μm to describe the evolution of fault properties during slip to >100 km to represent plate boundary fault systems. Such an earthquake simulator generates an earthquake sequence catalog over a long-period window and has been used in study of characteristic earthquake recurrence in a nonplanar fault system (e.g. Dieterich and Richards-Dinger 2010). The produced earthquakes (slips on the fault surface) in the fault system are then used to calculate the static stress changes off-fault in a region in a physical material model (Okada 1992). The stress transfers depend on the source/receiver locations, source mechanism and receiver fault’s orientation that is determined by the stress field (e.g. King et al. 1994). The seismicity rate is calculated using the rate-state formulation and the aftershock sequence is obtained by drawing from the seismicity rate at each grid point (cell center). The background seismicity rate in this study is set to be constant and uniform. Finally, the synthetic regional aftershock catalog is obtained by merging and sorting the aftershock sequences at each grid point for further regional aftershock analysis.
In this study, we first summarize the method to calculate the synthetic earthquake catalog in a fault system using the earthquake simulator code RSQSim (Dieterich and Richards-Dinger 2010; Richards-Dinger and Dieterich 2012). The earthquake catalog consists of earthquake times and faulting information. Then the induced seismicity rate off-fault in a region is calculated using the Coulomb stress change in the rate-state seismicity equations. The seismicity rate is both time-dependent and space-dependent.
2.1 Synthetic earthquake catalog calculation
Summary of RSQSim computational algorithm and key features
1. Model setup: define the system geometry and initial conditions, precompute interaction matrices, calculate loading conditions
Interactions calculated via an elastostatic boundary element method (Okada 1992). Stress loading conditions via backslip
Read in input file and set up the initial conditions in the simulator for a fault system such as San Andreas fault system
2. Determine time of next sliding state transition for every fault element
Very fast due to use of analytic approximations. Scales very well to many cores
3. Find next system-wide transition time: the minimum of all the times from step 2
This step consumes much time, dependent on the platform performance
4. Evolve each fault element to the transition time from step 3
Same analytic expressions used in step 2
As with step 2, very fast and scales well to many cores
5. Transition the fault element whose time was found in step 3 to its new sliding state. Loop to step 2
If sliding state change involves a change in slip speed, use interaction matrices to update stressing rates on all other fault elements
For a state 0 to state 1 transition, no system-wide update is needed. Scales well to many cores
The simulator code has been benchmarked against a fully dynamic finite element calculation that takes a completely different calculation approach, and the agreement between the two codes is found to be quite satisfactory (Dieterich and Richards-Dinger 2010; Richards-Dinger and Dieterich 2012).
2.2 Rate-and-state seismicity rate calculation
In creating the synthetic aftershock sequences, we follow the steps given by Smith and Dieterich (2010), assuming that the aftershocks at each grid point follow a random non-stationary Poisson process. By using the seismicity rate R (Eq. 7) as the earthquake occurrence rate at each grid point in a region, we draw the earthquake occurrence times from this rate distribution to generate an aftershock sequence at a grid point. Eventually, we merge and sort all the synthetic aftershock sequences from each grid point into a regional catalog for analysis of the aftershock distribution characteristics in this region.
3.1 Synthetic earthquakes in the ALLCAL2 fault system
3.2 Calculation of seismicity
Using the synthetic earthquake catalog in the ALLCAL2 fault system obtained from above, we calculate the seismicity rate on surface in a region of 200 km × 200 km in southern California (box in Fig. 1). The region is divided into 100 × 100 cells so each cell is of size 2 km × 2 km. At each grid point (cell center), the optimally oriented fault planes are determined using the sum of two stress rate tensors of (1) the stress rate field generated by the plate tectonic rate 49 mm/a (Argus and Gordon 2001) in the direction of N38.6°W and (2) the stress rate field generated by the long-term loading rate on the fault surface from the ALLCAL2 model. This approach is different from using the sum of the regional stress fiseld and coseismic stress field (e.g. King et al. 1994), and the optimal-orientated planes stay fixed since we aim to understand the long-term seismic hazard characteristics in this region of interest. In this region, the focal mechanism is of essentially strike-slip type, and the right-lateral fault orientation is selected in this study to be consistent with the dominant fault orientations in the fault system. The stressing rate is calculated by resolving the plate tectonic rate tensor onto the OOP at each cell center. The static stress transfer due to each earthquake in the catalog is conveniently calculated using the dislocation theory in the uniform half-space (Okada 1992) and resolved onto the determined optimal fault plane. All the cells are presumably independent thus no stress coupling between cells is involved in this study.
4 Discussion and conclusions
We employ the efficient rate-and-state earthquake simulator RSQSim to generate synthetic earthquakes in the fault system (ALLCAL2) and calculate the earthquake probabilities in a region from the synthetic earthquakes. The effect of the earthquakes in the fault model on the off-fault seismicity is obtained by calculating the Coulomb stress transfer on the optimal-oriented planes. The rate-state seismicity equation relates the Coulomb stress transfer to the seismicity rate change. The factors controlling the seismicity rate include the stressing rate, Coulomb stress transfer as well as the background rate that is constant in this study. The possible earthquake occurrence in a region is estimated using the seismicity rate as non-stationary Poisson rate. The results show that the aftershocks immediately after a main shock are concentrated near the rupture area, and then disperse into the surrounding area to reach uniformity eventually within a variable time frame. The total number of aftershocks near the rupture region decay with time, also resembling the Omori decay law, with a power exponent of 0.9 similar to the decay obtained for the uniform slip on finite faults (Smith and Dieterich 2010). The decay rate is close to the decay rates found for some big earthquakes in the region (Toda et al. 2005; Hainzl et al. 2009). The example results strongly demonstrate that the rate-and-state fault system earthquake simulator and the seismicity equations that are based on the rate-state friction nucleation of earthquake are well posited to characterize the aftershock distribution in regional assessments of earthquake probabilities.
The method described above to calculate the off-fault seismicity in a region is somewhat similar to that used in the stress release model of statistical earthquake occurrence (Keuhn et al. 2008) and to that used in the ETAS model (e.g. Bansal and Ogata 2013) but the main difference is that the stress in the stress release model is only a symbolic scalar related to the magnitude and not the true stress tensor as used in the rate-state seismicity rate model that can be connected with a physics-based earthquake simulator like RSQSim to better model the entire fault system and off-fault regions, and the ESTAS model uses the spatially distributed four model constants for seismicity rates. The background seismicity is simply set constant in this study but might be time-dependent as suggested by the ETAS model results in optimal estimates from hypocenter data in some regions (e. g. Bansal and Ogata 2013; Kumazawa and Ogata 2013). The same calculation procedure described above also applies to the scenarios with non-stationary background seismicity to investigate the phenomena such as seismic activation and quiescence but how to determine the time dependence will be a future effort.
It need to be pointed out that in generating the aftershocks in a region, their magnitudes or rupture characteristics cannot be directly derived from the stress transfer model and rate-state seismicity equations. The simulation of the magnitude distribution may be carried out using the Gutenberg-Richter distribution (e.g. Keuhn et al. 2008; Gu et al. 2013) to complete the earthquake probability estimation. Other additions, such as realistic background seismicity rate distribution, stress-interactions between adjacent regions instead of assuming the independence among regions, and dynamic stress transfer, would help improve evaluations of seismic hazard in a region (e.g. Kilb et al. 2002; Keuhn et al. 2008; Durand et al. 2013).
This research is supported by the NSF Frontiers in Earth-System Dynamics (EAR-1135455). The computational work used the Extreme Science and Engineering Discovery Environment (XSEDE), which is supported by National Natural Science Foundation grant No. OCI-1053575. This research is part of the Blue Waters sustained-petascale computing project, which is supported by the National Natural Science Foundation (award No. OCI 07-25070) and the state of Illinois. Blue Waters is a joint effort of the University of Illinois at Urbana-Champaign and its National Center for Supercomputing Applications. We are grateful to the anonymous reviewers for critical suggestions and comments which have greatly improved the manuscript.
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