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Ergodicity in Natural Fault Systems. K.F. Tiampo, University of Colorado J.B. Rundle, University of Colorado W. Klein, Boston University J. Sá Martins, University of Colorado. Motivation. Earthquakes are a high dimensional complex system having many scales in space and time.
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Ergodicity in Natural Fault Systems K.F. Tiampo, University of Colorado J.B. Rundle, University of Colorado W. Klein, Boston University J. Sá Martins, University of Colorado
Motivation Earthquakes are a high dimensional complex system having many scales in space and time. Recent work with numerical simulations of fault system dynamics suggests that they can be interpreted as mean field threshold systems in metastable equilibrium (Rundle et al., 1995; Klein et al., 1997; Ferguson et al., 1999). In these systems, the time averaged elastic energy of the system fluctuates around a constant value for some period of time, which are punctuated by major events that reorder the system before settling into another metastable energy well. One equilibrium property that can be recovered in simulations of metastable equilibrium systems is ergodicity (Klein, 1996; Egolf, 2000). Can the same be said of the natural fault system?
Ergodicity • A system in metastable equilibrium is locally ergodic, over a large enough spatial and temporal region. • Statistically, a system is said to be effectively ergodic if, for a given time interval, the system has equivalent time-averaged and ensemble-averaged properties. Ergodicity also implies stationarity. • One way to measure the ergodicity of such a system is to check a quantity called the Thirumalai-Mountain (TM) energy metric (Thirumalai & Mountain, 1993; Klein et al., 1996). This energy metric measures the difference between the time average of a quantity, generally related to the energy of the system, and the ensemble average of that same quantity over the entire system.
Thirumalai-Mountain Metric The energy-fluctuation metric , proposed by TM is where is the time average of a particular individual property related to the energy of the system, and is the ensemble average over the entire system. If the system is effectively ergodic at long times, , where D is a constant that measures the rate of ergodic convergence.
Seismicity Data • SCEC earthquake catalog for the period 1932-2001. • Data for analysis: 1932-2001, M ≥ 3.0. • Events are binned into areas 0.1° to a side, over an area ranging from 32° to 40° latitude, -125° to -115° longitude. Lat Long • Total numbers of events per year are calculated for each location, approximately 8000 locations.
TM Metric for Numbers of Events The energy-fluctuation metric, for numbers of earthquakes, is the time average of the seismicity at each site, and is the ensemble average over the entire system, where L is the total number of sites in the region.
Variance in Space and Time Spatial Temporal Variance Cumulative Magnitude
Conclusions • If natural earthquake fault systems are locally ergodic, over a large enough spatial and temporal regime, they can be considered to be in metastable equilibrium over some period of time. • Ergodicity implies stationarity over the same spatial and temporal regions. • This finding validates the use of near mean field models to study earthquake fault networks, and the principles and procedures of statistical mechanics to study both natural and simulated fault systems. • We can use this property to study various aspects of the natural fault network.
