Ergodicity in Natural Fault Systems

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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

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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?

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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.

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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.

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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.

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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.
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Entire Region, 1932-2001
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Entire Region, 1932-2001
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Variance in Space and Time
Spatial
Temporal
Variance
Cumulative Magnitude
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(No Transcript)
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Varying Region Size, Coalinga
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Varying Region Size, Northridge
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Varying Region Size, Landers
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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.

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Landers Northridge