1998 Proposal
Model of the Southern California Fault Network
Major Task Addressed: Understanding earthquake rupture dynamics and source physics.
Working Group G
Conclusions:
- The paradigm of self-organized criticality is inconsistent with the observation of two critical scale sizes in earthquake occurrence. Thus SOC is an interesting model that is irrelevant to the earthquake problem.
-Purely elastic loading/brittle fracture models are inadequate to describe clustering on the decadal or shorter time scale.
-Aftershocks are a two-stage process, involving initial damage in the region astride the main fault, and relaxation due to high stress concentrations in a locally weakened damage zone.
-Estimates of stress drop in large earthquakes obtained by inversion of seismic wave traces are probably significantly too low, because they fail to take into account nonlinear deformation in the vicinity of the main fault.
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In earlier work on seismicity on fault networks we have shown that stress redistributions due to fractures on an extended interactive fault network strongly influence local rates of seismicity and slip and cause them to be locally variable quantities[1]. Very long-term lacunae in activity may arise on extended stretches of an interactive fault system. The character of the lacunae is strongly dependent on the distribution of inhomogeneity. Long-term seismicity develops nonstationary statistics, and thus we question whether standard statistical procedures can be applied to analysis of seismicity on the long- time scale[1]. The cause of the variability in the interval times of large earthquakes on the San Andreas Fault as measured at Pallett Creek[2] can be understood in terms of interactions of fractures on adjacent faults of the Southern California Network[1].
Fault-wide lacunarity
As extension of the earlier work, we have applied the quasistatic fuse model with long-range forces (see Annual Report 1996) to the problem of two parallel faults, each with uniform reference fracture threshold B. To model the establishment of new fault topography contacts after slip, the fracture threshold is restored to a value R(x)B on the elements that have most recently fractured; R(x) is a uniformly distributed random variable between 1-r and 1+r. To prevent runaways we use a strength- weakening property so that when the stress exceeds a certain critical threshold, the strength decreases at a predetermined rate of the order of the tectonic loading rate. We find that the localized lacunarity associated with geometrical differences in fracture strength between the two faults[1] (without weakening) disappears, and is replaced by a fault-wide lacunarity. All the activity is concentrated on one fault; after an appropriate interval, associated with the random resetting of the strength, activity flips to the other fault. The distribution of interval times between flips is a power law with exponent that depends on the spacing between the faults; some very long-term intermittency in seismicity has been identified in a few geological observations.
New Magnitude-Frequency Relation for Tectonic Earthquakes
Clustering of seismicity on a single fault or on fault networks is scaled by the only time scale available, which is that of tectonic strain accumulation. Thus most model clustering instabilities take place on time scales that correspond to time scales of tens of thousands of years and even longer, and are not of direct relevance to seismicity on the human time scale. To study clustering on a time scale appropriate to the work of SCEC, we have re-analyzed the familiar log-linear frequency-magnitude distribution for Southern California earthquakes. The power-law distribution implies scale-independence of fracture sizes, which means that there is no overriding imprint of geometry on the distribution, an observation incompatible with the W (vs. the L) model which favors a separate statistical structure for large, characteristic earthquakes from the small ones that fit the power-law distribution. Why is there no signature of the transition between the two populations in the distribution? Part of the difficulty arises from a tradition that plots cumulative distributions rather than differential distributions: cumulative distributions perforce smooth out structure. Nevertheless, the differential distribution for Southern California (upper curve Fig. 1a) does not display any significant deviations from log- linearity. More important, individual aftershock series fit power law distributions of magnitudes with the same b-values as for the total distribution[3,4,5]. We have shown that aftershocks of the Kern County series with M >= 3 fit the Omori rate law to this day(!), show no resumption of earthquake activity at a regular (tectonically driven) rate, and fit the Gutenberg-Richter (G-R) magnitude law with a b-value of 1.03. How much of the distribution for Southern California is due to aftershocks? Detailed manual identification of aftershocks, foreshocks and swarms, which is the statistically preferred method[6], shows that 70% of the catalog with M >= 4 is due to clusters. The residual differential catalog, after clusters are removed (lower curve Fig. 1a), has a clear-cut structural separation in the distribution at about M=6.4, corresponding to a scale size of 15 km. Most of the aftershocks are triggered by the largest earthquakes. There is no scaling of the magnitudes or numbers of aftershocks in relation to the magnitude of the main shock, in agreement with earlier work[3,4].
There is also a second transition in the residual distribution around M=5.0 which corresponds to a fracture dimension of about 3 km. A scale of this dimension can be found in the spatial distribution of aftershocks after the Landers earthquake, in which most of the well-located epicenters are distributed as a cloud of dots straddling the main fault trace, to about this distance[7]. Aki and Li (separate personal communications) indicate that most of the aftershocks occur outside the trapped mode zone. Additionally, if we plot the date of the most recent aftershock of a given magnitude after the (M=7.5) Kern County (1952) earthquake (Fig. 1b), we find that most of the earthquakes with M >= 5 occur within one or two years of the main shock; most of the earthquakes with M <= 4 are still occurring to this day. Thus we suggest that there is a different fracture mechanism for small aftershocks than larger ones, and that the small ones that satisfy the G-R law are identified with a scale size that is less than about 2 or 3 km, which we suggest is a zone astride the main rupture trace and a zone of considerable near-source nonlinearity. In the remainder of this report, we focus on the mechanism of the abundant small aftershocks.
The Mechanism of Small Aftershocks
Aftershocks satisfy not only the G-R and Omori laws, but also the overwhelming number of aftershocks occur in the zone where the stress has been reduced by the main fracture, and hence are the consequence of a relaxation of an average stress that is lower than the stress before the main shock. Some models assume that aftershocks are due to irregular slip on the fault[8], but these models produce aftershocks that obey the G-R law only under special circumstances. In our model, aftershocks arise as a consequence of a two-stage process: first, the region adjoining the main fault is heavily damaged by the stress waves radiated by the main rupture to a distance of the order of 2 to 3 km, an assumption consistent with the observation that small aftershocks from Landers at distances of up to 1000 km are triggered by elastic waves radiated by the main shock[9]. The high damage is characterized by a dense set of unhealed cracks at all scales, much like a fractured windshield (with different symmetry, of course). Second, the aftershocks themselves occur due to delayed fracture of asperities between fractured elements under strength weakening due to high stress concentrations in the asperities between the highly concentrated cracks in the damaged zone. After an aftershock, the asperities are restored to their prior state; the cracks remain open throughout the sequence[10]. The relaxation of stress takes place under a condition of constant stress.
In the quasistatic fuse model, a 2-D damaged elastic system is subjected to a constant load stress along its top and bottom edges; the geometry is periodic in the other direction. The tectonic loading rate is suspended for the duration of the experiment. Damage on a scale larger than the lattice size is modeled by a random distribution of cracks that are never allowed to heal. Damage on a scale smaller than the lattice spacing is modeled by allowing an unbroken lattice element to undergo strength weakening with a decay rate of strength proportional to the stress on that element to some positive non-zero power. The strengths of the unbroken lattice elements have a uniform random distribution. An unbroken lattice site ruptures once the frictional strength decreases to the level of the local stress. Broken intercrack elements heal either instantaneously or on a finite time scale; as above, the initial set of cracks never heal over the time scale of the experiment.
A typical simulation (Figs. 2a,b) exhibits well-defined Omori and G-R relations with exponents of 0.8 and 1.3 respectively, which are in the proper range. These exponents are robust with respect to lattice size, randomization of material strength, and scaling of the stress-dependent decay rate by either the strength before decay, strength at that instant, or by a constant. No a priori distributions of material strengths or fracture lengths were used; the Omori and G-R distributions are generated simply by the self-organization implicit in the process. Figure 2c shows that the larger aftershocks occur early in the series and then die off while the smaller ones persist, a consequence of the relaxation of the regional stress after each event.
The Omori law and the Gutenberg-Richter distribution are the result of two independent, nonlinear, self-organizing processes. The Omori law of aftershock decay rates is the consequence of a dynamical adjustment of strength decay rates to local stress as the aftershock sequence evolves; the decay rates at each site are constantly varying as the stress is changed by neighboring aftershocks. If the stress dependence of the decay rate is removed, and the decay rate at each site is independent of the rates at other sites, we get no Omori law (p=0) while the b-value remains unaffected. On the other hand, healing rates influence the G-R distribution while leaving the p-values in the Omori distribution unaffected. Slower healing rates allow larger aftershocks to occur, with increased slip, thus decreasing the b-value. The Omori Law has a decreasing exponent as the initial damage from the main shock is decreased.
References
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