Computing Earthquake Probabilities on Global Scales JAMES R. HOLLIDAY, 1 WILLIAM R. GRAVES, 2 JOHN B. RUNDLE, 3,4 and DONALD L. TURCOTTE 5 Abstract—Large devastating events in systems such as earth- quakes, typhoons, market crashes, electricity grid blackouts, floods, droughts, wars and conflicts, and landslides can be unexpected and devastating. Events in many of these systems display frequency- size statistics that are power laws. Previously, we presented a new method for calculating probabilities for large events in systems such as these. This method counts the number of small events since the last large event and then converts this count into a probability by using a Weibull probability law. We applied this method to the calculation of large earthquake probabilities in California-Nevada, USA. In that study, we considered a fixed geographic region and assumed that all earthquakes within that region, large magnitudes as well as small, were perfectly correlated. In the present article, we extend this model to systems in which the events have a finite correlation length. We modify our previous results by employing the correlation function for near mean field systems having long- range interactions, an example of which is earthquakes and elastic interactions. We then construct an application of the method and show examples of computed earthquake probabilities. 1. Introduction Many systems in nature and society, including earthquakes, typhoons, market crashes, electricity grid blackouts, floods, droughts, wars and conflicts, and landslides (SACHS et al. 2012;SCHOLZ 2002; TALEB 2007;TURCOTTE 1997;SORNETTE 2009;MALA- MUD et al. 2005), display power-law statistics for frequency and magnitude. Calculation of event probabilities in these systems is often of considerable importance to reduce the damage (physical, financial and social) that may be associated with the largest events. These power laws appear to be stable, in the sense that including more events in the statistics over larger regions and longer periods of time improves the fit to a power law. In these power-law systems, a fixed number of the frequent small events is associated with one large event. This property suggests a strategy for the computation of the infrequent large event probabili- ties. Let N small events correspond to one large event. Then counting the number of small events occurring since the last large event provides a mea- sure of the ‘‘natural time’’ elapsed between large events (RUNDLE et al. 2012). RUNDLE et al. (2012) used a Weibull probability law to convert the small event count to a probability of occurrence of the next large event during a future time interval Dt. The means of conversion from natural time to calendar time was the long-term average rate of small events. The previous paper applied these ideas to earth- quakes in California and Nevada, USA. An underlying assumption of the analysis presented there was that earthquakes within the analyzed region were perfectly correlated, but that events outside the region had no effect on events within the region. Thus, the region of California-Nevada was considered to be an isolated system having no communication with the external region. In this article, we relax these assumptions and instead consider earthquake systems having a finite correlation length. In this more general model, all events within a finite designated region are affected by events outside the region. Or in other words, the events inside the region interact with external events with a finite external correlation length n. It is assumed that all events within the finite region are highly correlated. There is no longer a privileged 1 Department of Physics, University of California, Davis, CA, USA. 2 Open Hazards Group, Davis, CA, USA. 3 Departments of Physics and Geology, University of Cali- fornia, Davis, CA, USA. E-mail: jbrundle@ucdavis.edu 4 The Santa Fe Institute, Santa Fe, NM, USA. 5 Department of Geology, University of California, Davis, CA, USA. Pure Appl. Geophys. Ó 2014 Springer Basel DOI 10.1007/s00024-014-0951-3 Pure and Applied Geophysics