[1] We present an analytical solution and numerical tests of the epidemic-type aftershock (ETAS) model for aftershocks, which describes foreshocks, aftershocks, and main shocks on the same footing. In this model, each earthquake of magnitude m triggers aftershocks with a rate proportional to 10 am . The occurrence rate of direct aftershocks triggered by a single main shock decreases with the time from the main shock according to the ''local'' modified Omori law K/(t + c) p with p = 1 + q. Contrary to the usual definition, the ETAS model does not impose an aftershock to have a magnitude smaller than the main shock. Starting with a main shock at time t = 0 that triggers aftershocks according to the local Omori law, which in turn trigger their own aftershocks and so on, we study the seismicity rate of the global aftershock sequence composed of all the secondary and subsequent aftershock sequences. The effective branching parameter n, defined as the mean aftershock number triggered per event, controls the transition between a subcritical regime n < 1 and a supercritical regime n > 1. A characteristic time t*, function of all the ETAS parameters, marks the transition from the early time behavior to the large time behavior. In the subcritical regime, we recover and document the crossover from an Omori exponent 1 À q for t < t* to 1 + q for t > t* found previously in the work of Sornette and Sornette for a special case of the ETAS model. In the supercritical regime n > 1 and q > 0, we find a novel transition from an Omori decay law with exponent 1 À q for t < t* to an explosive exponential increase of the seismicity rate for t > t*. The case q < 0 yields an infinite nvalue. In this case, we find another characteristic time t controlling the crossover from an Omori law with exponent 1 À |q| for t < t, similar to the local law, to an exponential increase at large times. These results can rationalize many of the stylized facts reported for aftershock and foreshock sequences, such as (1) the suggestion that a small p-value may be a precursor of a large earthquake, (2) the relative seismic quiescence sometimes observed before large aftershocks, (3) the positive correlation between b and p values, (4) the observation that great earthquakes are sometimes preceded by a decrease of b-value, and (5) the acceleration of the seismicity preceding great earthquakes.

We have initially developed a time-independent forecast for southern California by smoothing the locations of magnitude 2 and larger earthquakes. We show that using small m Ն2 earthquakes gives a reasonably good prediction of m Ն5 earthquakes. Our forecast outperforms other time-independent models (Kagan and Jackson, 1994;Frankel et al., 1997), mostly because it has higher spatial resolution. We have then developed a method to estimate daily earthquake probabilities in southern California by using the Epidemic Type Earthquake Sequence model (Kagan and Knopoff, 1987;Ogata, 1988;Kagan and Jackson, 2000). The forecasted seismicity rate is the sum of a constant background seismicity, proportional to our timeindependent model, and of the aftershocks of all past earthquakes. Each earthquake triggers aftershocks with a rate that increases exponentially with its magnitude and decreases with time following Omori's law. We use an isotropic kernel to model the spatial distribution of aftershocks for small (m Յ5.5) mainshocks. For larger events, we smooth the density of early aftershocks to model the density of future aftershocks. The model also assumes that all earthquake magnitudes follow the Gutenberg-Richter law with a uniform b-value. We use a maximum likelihood method to estimate the model parameters and test the short-term and time-independent forecasts. A retrospective test using a daily update of the forecasts between 1 January 1985 and 10 March 2004 shows that the short-term model increases the average probability of an earthquake occurrence by a factor 11.5 compared with the time-independent forecast.

[1] We estimate the relative importance of small and large earthquakes for static stress changes and for earthquake triggering, assuming that earthquakes are triggered by static stress changes and that earthquakes are located on a fractal network of dimension D. This model predicts that both the number of events triggered by an earthquake of magnitude m and the stress change induced by this earthquake at the location of other earthquakes increase with m as $10 Dm/2 . The stronger the spatial clustering, the larger the influence of small earthquakes on stress changes at the location of a future event as well as earthquake triggering. If earthquake magnitudes follow the Gutenberg-Richter law with b > D/2, small earthquakes collectively dominate stress transfer and earthquake triggering because their greater frequency overcomes their smaller individual triggering potential. Using a southern California catalog, we observe that the rate of seismicity triggered by an earthquake of magnitude m increases with m as 10 am , where a = 1.05 ± 0.05. We also find that the magnitude distribution of triggered earthquakes is independent of the triggering earthquake's magnitude m. When a % b, small earthquakes are roughly as important to earthquake triggering as larger ones. We evaluate the fractal correlation dimension D of hypocenters using two relocated catalogs for southern California. The value of D measured for distances 0.1 < r < 5 km is D = 1.54 for the Shearer et al. catalog and D = 1.73 for the Hauksson et al. catalog. The value of D reflects both the structure of the fault network and the nature of earthquake interactions. By considering only those earthquake pairs with interevent times larger than 1000 days, we can largely remove the effects of short-term clustering. Then D % 2, close to the value D = 2a = 2.1 predicted by assuming that earthquake triggering is due to static stress. The value D % 2b implies that small earthquakes are as important as larger ones for stress transfers between earthquakes and that considering stress changes induced by small earthquakes should improve models of earthquake interactions.