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 :2 earthquakes gives a reasonably good prediction of m :5 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 time-independent 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 <= 5.5) 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.
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