One commonly accepted model for the behavior of fault ruptures is the cascading rupture model [Bak and Tang, 1989; Steacy and McCloskey, 1998; Ide and Aochi, 2005; Otsuki and Dilov, 2005; Sato and Mori, 2006], which describes a cascading failure of successive fault patches as a result of loading by the rupture of preceding patches. Because the rupture of a given patch is unaffected by those patches ahead of the rupture front, this model predicts that earthquakes should be non-deterministic for the duration of their rupture. That is, there should be no discernible information about the extent of the final rupture until it has completely finished propagating.
Recent observations of the spectral character of P-waves [Olson and Allen, 2005; Lockman and Allen, 2005, 2007; Wurman et al., 2007; Lewis and Ben-Zion, 2007] suggest that there is some information in the early seismic arrivals that may be correlated to the final magnitude of the event, and that in many cases this information is available before the rupture has completed. These data suggest that there may be an additional effect which is not accounted for by the cascading rupture model, which lends the rupture at least a degree of determinism.
Olson and Allen  hypothesized that this determinism is provided by the intensity of the early rupture history of the earthquake. A higher stress concentration at the focus of the earthquake generates a stronger early rupture phase, which imparts sufficient energy to the rupture to overcome barriers on the fault surface and produce a large earthquake. Conversely, an area of lower stress around the focus generates a comparatively weak early rupture, and the rupture may not gain enough energy to overcome the same barriers, and stops before becoming a large earthquake. Henceforth we use the term ``nucleation'' to describe the early dynamic (radiative) rupture history of the event, rather than a long-term (quasistatic) aseismic nucleation.
We test this concept by modeling the dynamics of a planar fault with stochastic initial stress conditions, with systematically varying stress in the nucleation region. We use the Support Operators Rupture Dynamics (SORD) dynamic fault code [Ely et al., 2008]. The initial shear stress field is generated randomly using the method of Ripperger et al. . One realization of an initial stress field is shown in Figure 2.70a.
We select an appropriate nucleation point by examining the asperities in our stress realization and selecting the highest-stress asperity on the fault that is broad enough to accommodate the 3 km diameter of the nucleation zone. To simulate varying nucleation stresses, we take a contour around the nucleation point at 64 MPa, approximately halfway between static and dynamic yield stress. Within that contour, we scale the stresses up or down such that the maximum stress is between 64 MPa and 80 MPa as desired. This process provides strong differences in the strength of nucleation, while largely preserving the spatial characteristics of the stress field. Outside the 64 MPa contour, the stress field remains identical between model runs. Figure 2.70b shows the stress field for the same realization as in Figure 2.70a, but with a weak nucleation.
We vary the peak nucleation stress between 64 and 80 MPa in increments of 1.6 MPa and allow the rupture to propagate to completion. We consider the moment release of the nucleation phase to be all moment released within the first 2 seconds of the rupture. This allows time for the effects of artificial nucleation to end, as well as for the rupture to propagate across the region of scaled nucleation stress. Figure 2.71 shows the total moment release in each run as a function of the nucleation moment release. Seismic moment is calculated at each element as , where D is the slip in cm at the element, S is the area of each element, , and . The total moment release is the sum of the moment at all nodes on the fault plane.
Figure 2.72 shows the final stress distribution and final slip on the fault for a high-stress nucleation case (80 MPa peak stress, 2.72a and 2.72c) and a low-stress case (65.6 MPa peak stress, 2.72b and 2.72d). A comparison of these figures to Figures 2.70a and b and the plot in Figure 2.71 shows that there is a point at which the initial nucleation is strong enough to overcome the low-stress barrier in the middle of the fault, and the rupture propagates through to the asperity on the other side. The behavior exhibited in this example is highly sensitive to the parameterization of the initial stress distribution. A distribution with slightly higher minimum shear stress will rupture through the entire fault for all initial conditions, while a distribution with lower minimum shear stress will never proceed past the nucleation asperity. This entire range is observed over a variation of less than 0.5 MPa in minimum initial shear stress.
Because the moment release after 2 seconds of rupture can be correlated to the moment release at the end of the rupture, this model supports the concept of determinism in the rupture resulting from the strength of the nucleation phase. However, the rupture area in this realization has only two asperities and one barrier, which does not demonstrate that this behavior can extend to more heterogeneous faults. Because barriers require additional crack energy to propagate through, they reduce the available energy for the continuation of the rupture on the other side of the barrier.
The behavior presented here is only stable for a narrow range of stress parameters. It is particularly sensitive to the degree to which stresses on the fault plane are allowed to be less than the dynamic yield stress. Thus, only finely-tuned stress parameters create faults whose modeled behavior explains the observations of determinism in real earthquakes. If we regard the observations of earthquake scaling as reliable, and the mechanism shown in this study is responsible for the observed scaling behavior, our results may serve to constrain the possible state of in situ stress in real faults. However, this result is dependent on the methodology we use with regard to how we vary the intensity of nucleation. A different choice in terms of frictional parameterization, such as rate-and-state weakening, or spatially varying friction coefficients, may also have an effect on the range of possible initial shear stresses. Even within the bounds of the method used in this study, other realizations of initial stress conditions will also produce slightly varying results. Indeed, given the dearth of observations of stress on real faults, it is difficult to know what a reasonable state of initial shear stress is to begin with. A suite of models and stress patterns will provide a greater understanding of the state of stress on real faults.
The authors thank Luis Dalguer and Geoff Ely for assistance with dynamic rupture codes. This work was partially supported by USGS NEHRP Grant 06HQAG0147.
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