Ergodicity and earthquake catalogs: Forecast testing and resulting implications

Recently the equilibrium property of ergodicity was identified in an earthquake fault system (Tiampoet al., Phys. Rev. Lett. 91, 238501, 2003; Phys. Rev. E 75, 066107, 2007). Ergodicity in this context not only requires that the system is stationary for these networks at the applicable spatial and temporal scales, but also implies that they are in a state of metastable equilibrium, one in which the ensemble averages can be substituted for temporal averages when studying their behavior in space and time. In this work we show that this property can be used to identify those regions of parameter space which are stationary when applied to the seismicity of two naturally-occurring earthquake fault networks. We apply this measure to one particular seismicity-based forecasting tool, the Pattern Informatics index (Tiampoet al., Europhys. Lett. 60, 481-487, 2002; Rundleet al., Proc. National Acad. Sci., U. S. A., Suppl. 1, 99, 2463, 2002), in order to test the hypothesis that the identification of ergodic regions can be used to improve and optimize forecasts that rely on historic seismicity catalogs. We also apply the same measure to synthetic catalogs in order to better understand the physical process that affects this accuracy. We show that, in particular, ergodic regions defined by magnitude and time period provide more reliable forecasts of future events in both natural and synthetic catalogs, and that these improvements can be directly related to specific features or properties of the catalogs that impact the behavior of their spatial and temporal statistics.