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seismicity on isolated faults that describe self-organization leading to a critical point may be irrelevant to our task. Because our concern is with the problems of large earthquakes, we try to understand why large earthquakes, and possibly others as well, fail to follow the power-law relation. We describe the phenomenological basis for developing an understanding of the physical processes that lead to the occurrence of large earthquakes.

We take as the basis for most of our discussions of phenomenology, observations of earthquakes in Southern California, which are the most extensive local data base we have, and thus the most studied of any history of instrumental seismicity. The magnitude-frequency relations for Southern California are reasonably well-established for the 60 yr of the Southern California catalog. The magnitude distribution of earthquakes in the Southern California catalog with aftershocks removed not only shows that the expected log-linearity for small magnitudes extends, most remarkably, up to the largest earthquakes (Fig. 1), excluding aftershocks. The 60-yr distribution of Fig. 1 does not show any hint of a deviation from linearity of the log-frequency vs. magnitude relation around M=6.4 that has been proposed by a number of authors (13, 1822) to correspond to a transition between two-dimensional and one-dimensional fracture shapes in a seismogenic zone of finite thickness. Despite the reasonableness of the proposal that it is the thickness of the seismogenic zone, which is of the order of 15 km in Southern California, that provides this characteristic dimension, contrary to recent assertions (21), the sharp cutoff to the distribution near M=7.5 and the absence of a rolloff at smaller magnitudes does not support the simplistic proposal. A similar conclusion has been reached from a study of the energy-frequency distribution (23).

Excluding aftershocks, the statistics of the smallest earthquakes, which are by inspection the most numerous, is Poissonian (24); this does not imply that the less frequent stronger earthquakes are also randomly occurring events.

Large Earthquakes

Despite the presence of a cutoff to the distribution in Fig. 1 for the most recent 60 yr, we know that earthquakes with magnitudes greater than M=7.5 occur in Southern California on a longer time scale, as for example the great Fort Tejon earthquake of 1857 on the San Andreas fault (SAF) in this region. Ten prehistoric earthquakes with large slips, and hence presumably with large magnitudes, have been identified by geochronometric methods between 671±13 A.D. and 1857;

FIG. 1. Number of earthquakes in Southern California as a function of magnitude for the interval 1935–1994 in intervals of 0.5 magnitude units (upper curve). Lower curves are the subdivision into 15-yr intervals. Aftershocks have been removed from the distributions (from ref. 42).

the interval times range from roughly 50 to 330 yr, with a mean of about 135 yr (25, 26).

Because the form of the distribution of large earthquakes cannot depend on the power-law statistics of small earthquakes and appears to depend on events that have not happened within the time span of the catalog, we have no seismographic information to bear on this point. We approach the problem of the distribution of large earthquakes from a different point of view. Because the fracture length and the energy released in a large earthquake are roughly related, we assume that the distribution of fracture lengths also has a cutoff or rolloff. If it has a rolloff, then there is a finite probability that a fracture will occur whose length will extend completely across a region as large as Southern California. It has been argued that the fractures in the largest earthquakes are confined by relatively fixed barriers to the extension of growth (27) that may be associated with fault geometry. In this model, these characteristic fault segments have characteristic slips in large earthquakes. But a process of repeated slip between strong barriers must ultimately accumulate stress at the barrier edges, and the barriers in this model must ultimately break as well; under the constraint of a long-term average uniform slip rate at every point in a plate boundary, barriers cannot remain unbroken forever. Thus the stresses at the barriers must ultimately relax, either by fractures in even stronger earthquakes or by some (generalized) viscous relaxation. If the process is viscous, then the relationship between the time constants for stress relaxation and for loading the system becomes important. In most modeling exercises, the restricted view is taken that viscous stress relaxation is extremely slow and, hence, that even long-term stress relaxation can take place by brittle fracture.

If one barrier must ultimately break, does it follow that sooner or later all barriers within a given region must break in the same earthquake, if only we wait long enough? It is sufficient to apply this question to the faults that support the largest of the earthquakes, which are those that occur on the SAF. Hence our question really refers to earthquakes that have not happened in the catalog interval of the most recent 60 yr. If earthquakes on the SAF are stopped according to the same processes as the smaller earthquakes—namely, because of encounters with strong barriers with low stored prestress, then, since all points on an individual plate boundary such as the SAF must sooner or later break, sooner or later two adjacent barriers may reach a nearly threshold state at about the same time—i.e., their prestresses must be close to their strengths at the same time. If this is the case, then, on the model that a stress at the edge of a crack is scaled by the length of the crack, one barrier must surely be triggered into fracture by a rupture on an adjacent barrier and, hence, a superearthquake will be developed.

But this model is valid under the assumption that the earth is homogeneous. The barrier property of the characteristic earthquake model argues for a weaker region of the faults in the reaches between barriers and, hence, that the appropriate scaling distance for redistribution of stress is the size of the nucleation zone of these greatest earthquakes—i.e., the barrier dimension, rather than the length of the crack. Observational support for this point of view is to be found in the self-healing pulses observed in the dynamics of the rupture process described by Heaton (28). We argue in a companion paper (29) that a smaller scaling length is sufficient to prevent an earthquake from tearing through several barriers, if the barriers are widely spaced compared to the scaling distance for the size of the fluctuations in stress outside the edge of the crack and for a sufficiently large ratio between the fracture strengths of the barriers and the relatively smoother segments between them. The scaling distance is the size of the nucleation zone or it is the size of the self-healing pulse, and these two sizes may be the same (30).

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