Conclusions:
Precursors to large earthquakes involve the disappearance of small-scale asperities between existing cracks in a nucleation zone.
Cracks, once formed in a nucleation zone, do not heal over the time that it takes to break through the nucleation zone.
The unreleased seismic moment obeys a power law as a function of remaining time before breakthrough of the nucleation zone, and is thus a predictor of breakthrough.
There is no evidence of log-periodic fluctuations prior to breakthrough of a nucleation zone.
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Studies of self-organizing systems that make use of a simple model of elastic loading and brittle fracture have focused on issues of clustering on times that are scaled by the intervals between the largest events in the system, which can be of the order of thousands of years[1]. Other physical properties of fracture of earth materials must be incorporated into our models if we are to understand clustering on shorter time scales, such as that of precursory slip weakening before breakthrough of asperities and breakout in a strong earthquake.
Most studies of deformation prior to rupture involve a parametric description of weakening. Can we derive the properties of slip-weakening from the mechanics of elastic/brittle fracture on rough fault interfaces? We show below that we can. A simple argument is the following: the deformation due to earthquakes having magnitudes smaller than the detection threshold of a network will be interpreted as creep, since no earthquakes will have been recorded; thus both creep and small earthquakes may be manifestations of the same process. Can we account for deformation in accelerated creep, and in particular strength weakening, by small fractures? The answer is yes, but we must take into account the physics of the healing of fractures, not ordinarily considered in models of seismicity.
Healing:
Recent analyses of near-source velocity seismograms have shown that many, if not all, earthquakes exhibit a distinct seismic nucleation phase prior to ultimate breakout[2,3]. A preferred explanation of nucleation is that failure initiates aseismically in a well-defined zone with stable, gradually accelerated sliding which becomes unstable once the slipping patch reaches a critical size. The crossover between the stable aseismic phase and the unstable breakout phase is marked by a seismic nucleation phase in which the patch growth has become unstable but remains confined to the nucleation zone. We present a model that may shed light on the nucleation process.
The fracture of the topographic contacts between two surfaces, which we call microasperities, are the mechanistic (microscopic) ingredients of the larger-scale processes of precursory creep, described parametrically by rate-state, slip-weakening, or other models of strength weakening. In either the microfracture or the parametric weakening cases, there is an accelerated slip or seismic moment release with time, prior to throughgoing rupture of all contacts in the nucleation zone. The zone of nucleation of earthquakes is a ``macroasperity" on a scale of many meters or even kilometers; it has significant intrinsic strength and hence can store significant amounts of prestress[4].
We model the macroasperity as an intermeshing of two irregular fault surfaces, described as a series of microasperities; we imagine the interface to consist of a series of alternating contacts and gaps. The strength of a nucleation zone is identified with the set of asperities (we omit the prefix `micro') and the gaps between them. We consider a purely elastic model of the loading and progressive rupture of the asperities in the nucleation zone; when an asperity disappears through brittle fracture, its strength is not (usually) restored over the lifetime of the nucleation zone, i.e. healing does not take place, and the net area of contacts decreases[5]. The load stress is therefore transferred to the remaining unbroken asperities, which then break at an accelerated rate because of the increased load, and so on until all asperities are broken and the nucleation zone breaks completely through, and breakout into the weaker fault zone surrounding the nucleation zone takes place.
On the usual model of self-organization of fractures, friction on a crack is re-established on a time scale that is short compared to the interval time between successive events. On our precursor model the friction is not re-established upon the disappearance of an individual asperity in a time scale comparable to the fracture of all asperities in the nucleation zone. This model of the absence of healing has been used [6,7,8] to account for a variety of short-term seismic processes including precursory foreshocks, intermediate-time quiescence and aftershocks. The fiber bundle model [9] of fracture of an individual element, and consequent transfer of load stress to neighboring members of the bundle, is a much simplified version of the above. An intrinsic ingredient in the fiber bundle model is that the strength of an already broken fiber is not re-established over the lifetime of the fracture of the ensemble in either model.
An attempt has been made[10] to use the fiber bundle model to establish a connection with a presumed log-periodic occurrence of precursory seismicity observed prior to the Loma Prieta earthquake[11] (but not observed before other large earthquakes.) The version of the fiber bundle model in [10] is a quasistatic, ultrametric, hierarchical, scale-independent structure, in which stress from a broken element is redistributed asymmetrically to a surviving twin of its own scale size. In our model, we introduce time delays to fracture of individual bonds as in [10] through a stress-dependent decay rate of strength that varies as sigma^p. We have used an inhomogeneous, quasistatic crack model, in which stresses are redistributed symmetrically over distances beyond the crack on scales gauged by the fracture size; once a crack is formed, it is not allowed to heal. We randomize the initial distribution of threshold strengths. The failure sequence of a typical system is shown in Fig. 1a for a case p=4. Failure is mainly uncorrelated for the first 90% of time since start of loading; the last 10% is strongly correlated. Fig. 1b shows the cumulative number of broken bonds as a function of elapsed time, and shows an accelerated moment release up to breakthrough, and is consistent with predictions from the rate-state model[12]. In Fig. 1c we show that the number of unbroken bonds as a function of remaining time has a power law behavior and is therefore a predictor of the breakthrough instability. The power law behavior is found only for cases p>1; the exponent in Fig. 1c is 0.22, in good agreement with a theory which gives 1/p in general. We find no log-periodic behavior in these simulations.
Microfracture/Strength Weakening Equivalence:
In a second set of experiments to simulate precursory creep, we have applied the dynamic vane model discussed in Annual Report 1996 and [13]. In the vane model we consider the elastic interaction of two blocks, each having an irregular surface. Under an increasing load stress, the blocks deform according to static elasticity at the points of contact, modeled by the deformation of overlapping elastic vanes that protrude from opposing blocks. The frictional resistance is gauged by the topographic overlap between opposing vanes; nonoverlapping topographies offer no frictional resistance and are accommodated trivially. When vane contacts break they transfer stress to their neighbors dynamically. Unlike the Burridge-Knopoff model[14], the static and dynamic frictions are not predefined separately, but both are determined by the overlap of the vanes on the two opposing blocks. The dynamic friction is a consequence of the decrease of momentum of the (massive) vanes upon collision with opposing vanes during sliding. In this model we do not invoke any intrinsic (creep) strength weakening, with its implicit time scale, for any asperity contact.
The temporal history of microfractures in one example of a model of topographic overlap is shown in Fig. 2a; the distribution of vane heights is a uniform random number chosen from a power law distribution with exponent 1.5. (We have used other distributions.) At the beginning of the sequence, the sites with the weakest bond strengths (overlaps) break, but these are geometrically isolated from one another; thus in the early stages these microfractures are independent events with small moments. The moment release accelerates as stronger sites break and the fractures now occupy a significant fraction of the asperity; the cracks become more strongly interactive. When these stronger sites break, they have a greater chance to break into the neighboring open regions that have been generated by earlier microevents, and thus they generate larger moment release than the preceding ones.
We stack the results from a number of numerical simulations (Fig. 2b). The moment release at the start of the tearing process, when the fractures are far apart and are non-interactive, has a power law behavior, but this has no predictive capability because there is no early signature of future breakthrough. For times close to breakthrough, the cracks are strongly interactive, with accelerated moment release; at the time of breakthrough, the as yet unreleased moment curve is also a power law, with time measured backwards from the breakthrough time, which does have predictive capability (Fig. 2c). As above, moment release is consistent with predictions from rate-state models[12]. Thus a microcracking model under steady tectonic loading, with absence of healing, not only provides an understanding of the precursory process, but also provides a prediction criterion: It is the unreleased seismic moment in the nucleation zone that has predictive power, since the moment release will have infinite slope at the moment of breakthrough, with power law behavior up to the time of breakthrough of the nucleation zone.
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