Earthquake Statistics Research Articles

A three‐dimensional fluid‐controlled earthquake model: Behavior and implications

We describe the behavior of a three‐dimensional, fluid‐controlled fault model that couples the dominant mechanical effects of fluid within a cellular fault zone with shear stress accumulation from constant plate motion applied at the downward continuation of the fault. Improvements from a previous model include long‐term plate motion loading and porosity creation through dilatant slip, which allow the model to evolve to its steady state dynamic equilibrium. The examined results include slip and slip‐deficit accumulation, pore pressure buildup and release, stress states, the emergence of seismic scaling relationships, and frequency‐size statistics of model earthquakes. We find that asperities develop naturally within the model, reflecting the disorganization of the evolving stress state in Mohr space. The dynamical interaction of shear stress and effective normal stress perturbs the initial uniform stress state to a complex state that produces transient asperity development along the fault. These “Mohr‐space” asperities spontaneously evolve, disintegrate, reemerge, and migrate along the fault plane. The general model behavior is independent of the state of the fluid pressure. In four examined cases, which span the range of possible fault zone overpressures, the equilibrium condition is that which occupies all of the available Mohr space. Maximum slip deficits along the fault depend on the degree of fault weakness, ranging from about 3 m for a weak fault to over 30 m for a strong fault after 4000 years of model evolution. For events that breach the surface the seismic moment scales with the cube of the source dimension Mo ∼L3, which reflects the slipped area times the depth extent of the rupture. This scaling crosses isolines of stress drop. For confined events, Mo ∼L2 along isolines of stress drop, but no general scaling emerges. Clusters emerge between stress drop versus seismic moment and stress drop versus source dimension, with large events converging to average stress drops of about 8 MPa for a weak fault and about 20 MPa for a strong fault.

Regional Thresholds of the ISC Bulletin

INTRODUCTION In any analysis of a seismicity catalog to learn about the Earth's behavior, unintended effects from the process of compiling the catalog are pernicious. It is often difficult, sometimes impossible, to separate compilation effects from features of the catalog that reflect the Earth's behavior. A comparison of earthquake parameters from different regions or time periods risks mistaking differences in detection capability for real differences in seismicity. Such mistakes can result in misleading results, for example, in earthquake prediction (Wyss et al., 1992) and statistics of aftershocks and foreshocks (Jones, 1994). A frequently invoked way of minimizing this influence is to exclude small earthquakes. A larger earthquake, recorded by more stations, is less likely to be badly mislocated, assigned a grossly inaccurate magnitude, or missed entirely. Exclusion of small earthquakes only partly addresses the problems, of course, since systematic changes in magnitude measurement or location bias can also give rise...

Testing earthquake prediction algorithms: statistically significant advance prediction of the largest earthquakes in the Circum-Pacific, 1992–1997

Algorithms M8 and MSc (i.e., the Mendocino Scenario) were used in a real-time intermediate-term research prediction of the strongest earthquakes in the Circum-Pacific seismic belt. Predictions are made by M8 first. Then, the areas of alarm are reduced by MSc at the cost that some earthquakes are missed in the second approximation of prediction. In 1992–1997, five earthquakes of magnitude 8 and above occurred in the test area: all of them were predicted by M8 and MSc identified correctly the locations of four of them. The space–time volume of the alarms is 36% and 18%, correspondingly, when estimated with a normalized product measure of empirical distribution of epicenters and uniform time. The statistical significance of the achieved results is beyond 99% both for M8 and MSc. For magnitude 7.5+, 10 out of 19 earthquakes were predicted by M8 in 40% and five were predicted by M8–MSc in 13% of the total volume considered. This implies a significance level of 81% for M8 and 92% for M8–MSc. The lower significance levels might result from a global change in seismic regime in 1993–1996, when the rate of the largest events has doubled and all of them become exclusively normal or reversed faults. The predictions are fully reproducible; the algorithms M8 and MSc in complete formal definitions were published before we started our experiment [Keilis-Borok, V.I., Kossobokov, V.G., 1990. Premonitory activation of seismic flow: Algorithm M8, Phys. Earth and Planet. Inter. 61, 73–83; Kossobokov, V.G., Keilis-Borok, V.I., Smith, S.W., 1990. Localization of intermediate-term earthquake prediction, J. Geophys. Res., 95, 19763–19772; Healy, J.H., Kossobokov, V.G., Dewey, J.W., 1992. A test to evaluate the earthquake prediction algorithm, M8. U.S. Geol. Surv. OFR 92-401]. M8 is available from the IASPEI Software Library [Healy, J.H., Keilis-Borok, V.I., Lee, W.H.K. (Eds.), 1997. Algorithms for Earthquake Statistics and Prediction, Vol. 6. IASPEI Software Library].

Changes in the Frequency-Energy Distribution of Seismic Events During Mining in the Ostrava-Karviná Coal Field

The relation between the number of seismic events N and their magnitude and/or energetic magnitude is one of the basic relations in the statistics of earthquakes and mining-induced events. A detailed analysis of the b-values in the relation log N =a−b logE for the induced seismic events occurring in the Ostrava-Karvina Coal Basin is given in this paper. The least-squares method, as well as the maximum likelihood method were applied in solving this problem mathematically. Based on the results of frequency-energy distribution analysis, it was proved that the variations of b-values in space and time do exist, and that these values are inversely proportional to the level of induced seismicity influenced by the various mining activities being carried out in-situ. This statement could, in general, be supported by the fact that lower b-values correspond to higher seismic activity, while higher b-values correspond to low and moderate seismic activity.

Gutenberg-Richter and characteristic earthquake behavior in simple mean-field models of heterogeneous faults

The statistics of earthquakes in a heterogeneous fault zone is studied analytically and numerically in the mean field version of a model for a segmented fault system in a three-dimensional elastic solid. The studies focus on the interplay between the roles of disorder, dynamical effects, and driving mechanisms. A two-parameter phase diagram is found, spanned by the amplitude of dynamical weakening (or ``overshoot'') effects (epsilon) and the normal distance (L) of the driving forces from the fault. In general, small epsilon and small L are found to produce Gutenberg-Richter type power law statistics with an exponential cutoff, while large epsilon and large L lead to a distribution of small events combined with characteristic system-size events. In a certain parameter regime the behavior is bistable, with transitions back and forth from one phase to the other on time scales determined by the fault size and other model parameters. The implications for realistic earthquake statistics are discussed.

Earthquake statistics and fractal faults

We introduce a Self-affine Asperity Model (SAM) for the seismicity that mimics the fault friction by means of two fractional Brownian profiles (fBm) that slide one over the other. An earthquake occurs when there is an overlap of the two profiles representing the two fault faces and its energy is assumed proportional to the overlap surface. The SAM exhibits the Gutenberg-Richter law with an exponent $\beta$ related to the roughness index of the profiles. Apart from being analytically treatable, the model exhibits a non-trivial clustering in the spatio-temporal distribution of epicenters that strongly resembles the experimentally observed one. A generalized and more realistic version of the model exhibits the Omori scaling for the distribution of the aftershocks. The SAM lies in a different perspective with respect to usual models for seismicity. In this case, in fact, the critical behaviour is not Self-Organized but stems from the fractal geometry of the faults, which, on its turn, is supposed to arise as a consequence of geological processes on very long time scales with respect to the seismic dynamics. The explicit introduction of the fault geometry, as an active element of this complex phenomenology, represents the real novelty of our approach.

Depth dependence of earthquake frequency‐magnitude distributions in California: Implications for rupture initiation

Statistics of earthquakes in California show linear frequency‐magnitude relationships in the range of M2.0 to M5.5 for various data sets. Assuming Gutenberg‐Richter distributions, there is a systematic decrease in b value with increasing depth of earthquakes. We find consistent results for various data sets from northern and southern California that both include and exclude the larger aftershock sequences. We suggest that at shallow depth (∼0 to 6 km) conditions with more heterogeneous material properties and lower lithospheric stress prevail. Rupture initiations are more likely to stop before growing into large earthquakes, producing relatively more smaller earthquakes and consequently higher b values. These ideas help to explain the depth‐dependent observations of foreshocks in the western United States. The higher occurrence rate of foreshocks preceding shallow earthquakes can be interpreted in terms of rupture initiations that are stopped before growing into the mainshock. At greater depth (9–15 km), any rupture initiation is more likely to continue growing into a larger event, so there are fewer foreshocks. If one assumes that frequency‐magnitude statistics can be used to estimate probabilities of a small rupture initiation growing into a larger earthquake, then a small (M2) rupture initiation at 9 to 12 km depth is 18 times more likely to grow into a M5.5 or larger event, compared to the same small rupture initiation at 0 to 3 km.

Statistics of Earthquakes in Simple Models of Heterogeneous Faults

Simple models for ruptures along a heterogeneous earthquake fault zone are studied, focusing on the interplay between the roles of disorder and dynamical effects. A class of models are found to operate naturally at a critical point whose properties yield power-law scaling of earthquake statistics. Various dynamical effects can change the behavior to a distribution of small events combined with characteristic system size events. The studies employ various analytic methods as well as simulations.

Earthquake size‐frequency relations from an analytical stochastic rupture model

An analytical stochastic model of earthquake rupture is presented. Although the general formulation makes no a priori self‐similarity assumption, scale independent earthquake size‐frequency (ESF) statistics emerge from the interaction between fault strength heterogeneity and rupture mechanics. The model also gives theoretical derivations for scale dependent ESF statistics, including gradual frequency saturation for small earthquakes, and a continuum of possible transitions in the statistics of large earthquakes. A model fault is characterized as a surface composed of a large number of asperities, defined as small, discrete contact surfaces of finite strength. An earthquake rupture grows via successive asperity‐breaking subevents. The probability of rupture to a certain size earthquake or greater is the total product of incremental rupture growth probabilities. These are governed by a new asperity failure probability distribution, derived as a mixture of Weibull distributions. By including fracture mechanics, the resulting mixture distribution, characterized by the scale and shape parameters Λ and Γ, is written in terms of the rupture size. The important difference between the mixture distribution and the Weibull distribution is that the mixture results in power law ESF statistics. Self‐similarity occurs when material heterogeneity and rupture mechanics are mutually “tuned” to give Λ = Γ = 1. Frequency saturation for small earthquakes is shown to be a simple consequence of a general aseismic nucleation process. The nucleation magnitude mν defines the roll‐off point in ESF statistics that occurs when the area of seismic rupture equals the area of precursory, aseismic rupture growth. By fitting the model to the Parkfield earthquake data set, mν = 0.9 is obtained. The change in ESF scaling for the transition from moderate to large earthquakes depends on the changes in stress at the growing rupture front. The Northern California Earthquake Data Center seismicity data set is fitted for a transition at magnitude m ≃ 5 with Λ′ = 0.4 and Γ′ = 0.75. This result indicates that the change in the ESF relation associated with the transition from moderate to large earthquakes in California and Nevada is caused by surface rupture accompanied by partial width and slip saturation.

Stress, slip, and earthquakes in models of complex single‐fault systems incorporating brittle and creep deformations

Numerical simulations of slip evolution along a cellular vertical strike‐slip fault in an elastic half‐space are performed for several models representing discrete fault systems embedded in three‐dimensional elastic continua. The geometry and imposed boundary conditions correspond approximately to the central San Andreas fault. The simulations incorporate brittle and creep deformations in series; the net fault zone deformation rate is the sum of creep rate and frictional slip rate. Brittle fault properties are given by various distributions of earthquake stress drops on failing segments (numerical cells). The assumed distributions represent two idealized situations corresponding to different extreme states along an evolutionary path of a fault: (1) a strongly disordered state characterized by a wide range of size scales, representing immature fault zones and extended spatial domains, and (2) a relatively regular state having a narrow range of size scales, representing mature highly‐slipped faults. The assumed creep properties are identical in all cases. These are prescribed in terms of coefficients characterizing a power law dependency of creep‐slip rate on stress. The combined brittle‐creep process and employed parameters lead to an overall “pine‐tree” stress‐depth profile with a “brittle‐ductile” transition depth of about 12.5 km, and variable stress‐along‐strike profiles with “brittle‐creep” transition around 65 km NW of the 1857 rupture. The spatial patterns of simulated hypocenters are statistically similar to observed data. The results indicate that the range of size scales characterizing strong fault zone heterogeneities has important manifestations on the seismic response of a fault system. A narrow range of size scales leads to frequency‐size statistics of earthquakes resembling the characteristic earthquake distribution, and quasi‐periodic temporal distribution of large events as in the seismic gap hypothesis. On the other hand, a wide range of size scales leads to Gutenberg‐Richter power law frequency‐size statistics, and random or clustered temporal distribution of large events. The simulations demonstrate that treatment of the various observed forms of frequency‐size and temporal statistics of earthquakes can be unified through the concept of range of size scales characterizing fault zone heterogeneities. This has a clear physical interpretation in terms of structural properties of a given fault zone or broad lithospheric domain, and is supported by observed earthquake and fault data. In some simulated cases the frequency‐size statistics of small earthquakes fall sharply below the self‐similar Gutenberg‐Richter line. The results indicate that small earthquakes prepare the fault for the occurrence of a large event by smoothing, during gradual tectonic loading, the long‐wavelength components of stress on the fault. This is done through short‐wavelength stress roughening associated with the numerous ruptures of the small events. The above pattern of smoothing/roughening of long/short wavelengths of stress on the fault is reversed during large‐scale ruptures of the big events.

Slip patterns and earthquake populations along different classes of faults in elastic solids

Numerical simulations of slip instabilities on a vertical strike‐slip fault in an elastic half‐space are performed for various models belonging to two different categories. The first category consists of inherently discrete cellular fault models. Such are used to represent fault systems made of segments (modeled by numerical cells) that can fail independently of one another. Their quasi‐independence is assumed to provide an approximate representation of strong fault heterogeneity, due to geometric or material property disorder, that can arrest ruptures at segment boundaries. The second category consists of models having a well‐defined continuum limit. These involve a fault governed by rate‐ and state‐dependent friction and are used to evaluate what types of property heterogeneity could lead to the quasi‐independent behavior of neighboring fault segments assumed in the first category. The cases examined include models of a cellular fault subjected to various complex spatial distributions of static to kinetic strength drops, and models incorporating rate‐ and state‐dependent friction subjected to various spatial distributions of effective stress (normal stress minus pore pressure). The results indicate that gradual effective stress variations do not provide a sufficient mechanism for the generation of observed seismic response. Strong and abrupt fault heterogeneity, as envisioned in the inherently discrete category, is required for the generation of complex slip patterns and a wide spectrum of event sizes. Strong fault heterogeneity also facilitates the generation of rough rupture fronts capable of radiating high‐frequency seismic waves. The large earthquakes in both categories of models occur on a quasi‐periodic basis; the degree of periodicity increases with event size and decreases with model complexity. However, in all discrete segmented cases the models generate nonrepeating sequences of earthquakes, and the nature of the large (quasi‐periodic) events is highly variable. The results indicate that expectations for regular sequences of earthquakes and/or simple repetitive precursory slip patterns are unrealistic. The frequency‐size (FS) statistics of the small failure episodes simulated by the cellular fault models are approximately self‐similar with b ≈ 1.2 and bA ≈ 1, where b and bA are b values based on magnitude and rupture area, respectively. For failure episodes larger than a critical size, however, the simulated statistics are strongly enhanced with respect to self‐similar distributions defined by the small events. This is due to the fact that the stress concentrated at the edge of a rupture expanding in an elastic solid grows with the rupture size. When the fault properties (e.g., geometric irregularities) are characterized by a narrow range of size scales, the scaling of stress concentrations with the size of the failure zone creates a critical rupture area terminating the self‐similar earthquake statistics. In such systems, events reaching the critical size become (on the average) unstoppable, and they continue to grow to a size limited by a characteristic model dimension. When, however, the system is characterized by a broad spectrum of size scales, the above phenomena are suppressed and the range of (apparent) self‐similar FS statistics is broad and characterized by average b and bA values of about 1. The simulations indicate that power law extrapolations of low‐magnitude seismicity will often underestimate the rate of occurrence of moderate and large earthquakes. The models establish connections between features of FS statistics of earthquakes (range of self‐similar regimes, local maxima) and structural properties of faults (dominant size scales of heterogeneities, dimensions of coherent brittle zones). The results suggest that observed FS statistics can be used to obtain information on crustal thickness and fault zone structure.

Statistics of the catastrophic slip events in the Portevin-Le Châtelier effect.

The statistics of catastrophic slip associated with discontinuous yielding (the Portevin--Le Ch\^atelier effect) is investigated on an Al-Mg alloy. It is shown that a peaked distribution of stress drops changes towards a monotonic one, characterized by a power law with an exponent of $\ensuremath{-}$1.1, with increasing temperature and rate of straining. A model describing this behavior is studied, and it is suggested that there exists a formal similarity of the governing equations with those describing instabilities in frictional sliding. The use of discontinuous yielding in a tensile Al-Mg specimen is proposed as a candidate for modeling of earthquake statistics on a crustal fault.

Positive and negative feedback in the earthquake cycIe: the role of pore fluids on states of criticality in the crust

Fluids exert a strong physical and chemical control on local processes of rock fracture and friction. For example they may accelerate fracture by stress corrosion reactions or the development of overpressure (a form of positive feedback), or retard fracture by time-dependent stress relaxation or dilatant hardening (negative feed-back), thereby introducing a variable degree of local force conservation into the process. In particular the valve action of dynamic faulting may be important in tuning the Earth to a metastable state of incipient failure on all scales over several cycles, similar to current models of Self-Organised Criticality (SOC) as a paradigm for eartiquakes However laboratory results suggest that ordered fluctuations about this state may occur in a single cycle due to non conservative processes involving fluids which have the potential to be recognised, at least in the short term, in the scaling properties of earthquake statistics. Here we describe a 2-D cellular automaton which uses local rules of positive and negative feedback to model the effect of fluids on failure in a heterogeneous medium in a single earthquake cycle. The model successfully predicts the observed fractal distribution of fractures, with a negative correlation between the predicted seismic b-value and the local crack extension force G. Such a negative correlation is found in laboratory tests involving (a) fluid-assisted crack growth in tension (b) water-saturated compressional deformation, and (c) in field results on an intermediate scale from hydraulic mining-induced seismicity all cases where G can be determined independently, and where the physical and chemical action of pore fluids is to varying degrees a controlled variable. For a finite local hardening mechanism (negative feedback), the model exhibits a systematic increase followed by a decrease in the seismic b-value as macroscopic failure is approached, similar to that found in water-saturated laboratory tests under controlled «undrained» conditions, and where dilatancy hardening is independently known to be a local mechanism of negative feedback. A similar pattern is suggested from selected field observations from natural seismicity, albeit with a lesser degree of statistical significance.

A multidisciplinary approach to seismic hazard in southern California

Abstract A serious obstacle facing seismic hazard assessment in southern California has been the characterization of earthquake potential in areas far from known major faults where historical seismicity and paleoseismic data are sparse. This article attempts to fill the voids in earthquake statistics by generating “master model” maps of seismic hazard that blend information from geology, paleoseismology, space geodesy, observational seismology, and synthetic seismicity. The current model suggests that about 40% of the seismic moment release in southern California could occur in widely scattered areas away from the principal faults. As a result, over a 30-yr period, nearly all of the region from the Pacific Ocean to 50 km east of the San Andreas Fault has a greater than 50/50 chance of experiencing moderate shaking of 0.1 g or greater, and about a 1 in 20 chance of suffering levels exceeding 0.3 g. For most of the residents of southern California, thelion's share of hazard from moderate earthquake shaking over a 30-yr period derives from smaller, closer, more frequent earthquakes in the magnitude range (5 ≦ M ≦ 7) rather than from large San Andreas ruptures, whatever their likelihood.

An exact renormalization model for earthquakes and material failure statics and dynamics

Earthquake events are well-known to posses a variety of empirical scaling laws. Accordingly, renormalization methods offer some hope for understanding why earthquake statistics behave in a similar way over orders of magnitude of energy. We review the progress made in the use of renormalization methods in approaching the earthquake problem. In particular, earthquake events have been modeled by previous investigators as hierarchically organized bundles of fibers with equal load sharing. We consider by computational and analytic means the failure properties of such bundles of fibers, a problem that may be treated exactly by renormalization methods. We show, independent of the specific properties of an individual fiber, that the stress and time thresholds for failure of fiber bundles obey universal, albeit different, scaling laws with respect to the size of the bundles. The application of these results to fracture processes in earthquake events and in engineering materials helps to provide insight into some of the observed patterns and scaling - in particular, the apparent weakening of earthquake faults and composite materials with respect to size, and the apparent emergence of relatively well-defined stresses and times when failure is seemingly assured.

A four-parameter, two degree-of-freedom block-spring model: Effect of the driver velocity

We analyze the effect of tectonic plate velocities in the earthquake pattern using a simple mass-spring model of the Burridge and Knopoff type with two blocks and a velocity-weakening friction law. Previous versions of the two-block model assume a steady driver during slip events (limit of zero driver velocity), which, in some cases makes necessary the introduction of artificial parameters to start the numerical integration of the equations of motion at impending slip of any block. Still maintaining the condition of zero driver velocity during slip, we shall introduce a procedure to start the numerical integration without introducing artificial parameters and this will be done by using a linearized version of the equations of motion valid for small velocities and considering nonzero driver velocity. We also introduce a four parameter model in which the driver velocity enters the equations during the whole simulation, and analyze the effect of the new parameter, the driver velocity, in the displacement and time patterns of blocks motion, directly related to earthquake statistics such as coseismic slips and average repeat times.

FRACTAL TECTONICS AND EROSION

Tectonic processes build landforms that are subsequently destroyed by erosional processes. Landforms exhibit fractal statistics in a variety of ways; examples include (1) lengths of coast lines; (2) number-size statistics of lakes and islands; (3) spectral behavior of topography and bathymetry both globally and locally; and (4) branching statistics of drainage networks. Erosional processes are dominant in the development of many landforms on this planet, but similar fractal statistics are also applicable to the surface of Venus where minimal erosion has occurred. A number of dynamical systems models for landforms have been proposed, including (1) cellular automata; (2) diffusion limited aggregation; (3) self-avoiding percolation; and (4) advective-diffusion equations. The fractal statistics and validity of these models will be discussed. Earthquakes also exhibit fractal statistics. The frequency-magnitude statistics of earthquakes satisfy the fractal Gutenberg-Richter relation both globally and locally. Earthquakes are believed to be a classic example of self-organized criticality. One model for earthquakes utilizes interacting slider-blocks. These slider block models have been shown to behave chaotically and to exhibit self-organized criticality. The applicability of these models will be discussed and alternative approaches will be presented. Fragmentation has been demonstrated to produce fractal statistics in many cases. Comminution is one model for fragmentation that yields fractal statistics. It has been proposed that comminution is also responsible for much of the deformation in the earth’s crust. The brittle disruption of the crust and the resulting earthquakes present an integrated problem with many fractal aspects.

Statistics of characteristic earthquakes

Statistical methods are used to test the characteristic earthquake hypothesis. Several distributions of earthquake size (seismic moment-frequency relations) are described. Based on the results of other researchers as well as my own tests, evidence of the characteristic earthquake hypothesis can be explained either by statistical bias or statistical artifact. Since other distributions of earthquake size provide a simpler explanation for available information, the hypothesis cannot be regarded as proven.

Relation between the earthquake statistics and fault patterns, and fractals and percolation

Using computer simulations and analyzing experimental data, we investigate the structure of fault patterns in heterogeneous rock, and the distribution of earthquake hypocentral locations. At large length scales, the fracture networks of rock are fractal with a fractal dimension D⋍1.9 and 2.5, in 2D and 3D, respectively. We present a computer simulation model that explains these results in terms of percolation fractals. Three-dimensional fractal analysis of regional hypocentral locations yields a fractal dimension of about 1.8. This result can be explained if earthquakes are distributed on the backbone of the fractal network of fault structure with a fractal dimension equal to that of the percolation backbone. Thus, percolation provides a unified theory for the geometry of fault patterns and the spatial distribution of earthquakes.

A cellular-automata, slider-block model for earthquakes II. Demonstration of self-organized criticality for a 2-D system

SUMMARY It has been suggested that distributed seismicity is an example of self-organized criticality. If this is the case, the Earth’s crust in an active tectonic zone is in a near-critical state and faults can interact over large distances. Observed seismicity and earthquake statistics are consequences of the dynamical interaction of seismic faulting over a wide range of scales. We address this problem by considering a two-dimensional array of slider blocks with static/dynamic friction. The twodimensional system is treated as a cellular automaton such that only one slider block is allowed to slip at a given time and interacts only with its nearest neighbours through connecting springs. Because of this treatment, the amount of slip for each failed block can be obtained analytically, and the system is deterministic with no stochastic inputs or spatial heterogeneities. In many cases, the slip of one block induces the slip of adjacent blocks. The size of an event is specified by the number of blocks that participate in the event. The number of small events with a specified size has a power-law dependence on the size with a power close to -1.36. The distributions of normalized recurrence times for most events are close to a Poisson process, and gradually deviate towards periodicity for large events. The recurrence time statistics are generally insensitive to parameter variations. Large events may occur at stress levels considerably lower than the failure strength of an individual block, and the stress drops associated with large events are generally small. This may provide an explanation for observed low stress levels in tectonically active