Interacting Earthquake Fault Systems:

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Interacting Earthquake Fault Systems Cellular
Automata and beyond...
D. Weatherley QUAKES AccESS 3rd ACES Working
Group Meeting Brisbane, Aust. 5th June, 2003.
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Overview

• Introduction and Scope
• A Brief History of Earthquake Physics
• Burridge-Knopoff block-slider
• Sandpile automaton and SOC
• Two statistical fractal automata
• Current developments in the statistical physics
  of Eqs
• Conclusions

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Scope of the Problem
Complicated interactions between faults due to
stress transfer during Eqs
X
Nonlinear Rheology

• Earthquake Fault systems are COMPLEX
• Many degrees of freedom
• Strongly coupled spatial and temporal scales
• Nonlinear dynamical equations constitutive laws
• Multi-physics mechanical, chemical, thermal,
  fluids, (EM?)

Multi-Fractal fault heirarchy
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Accelerating Moment Release

• To make matters worse, opportunities for direct
  observations are limited
• Seismometers only record the aftermath
• GPS/InSARs/Geodetic not continuous in
  space/time/both
• Paleoseismology imprecise and only identifies
  the big guys
• Geological near-surface only

Bufe Varnes, 1993
Cumulative moment
1920 1940 1960 1980
N(M) M-b
Year
Number of Eqs, N(M)
Not all doom and gloom though! These limited
observations may be sufficient if we understand
the underlying dynamical processes, at least for
reliable probabilistic forecasting.
EQ magnitude, M
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Archetypical Earthquake Model Burridge-Knopoff
Block-Slider
(Figure thanks to J.Rundle, ICCS 2003
presentation)
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What has the BK model taught us?

• The Block-Slider model can reproduce the
  power-law earthquake size-distribution without
  prescribing any power-law correlations/structure.
  
• Power-law distributions are a natural consequence
  of the dynamics of systems with
• Large numbers of elements (DOFs)
• Nonlinear interactions between elements
• External loading of elements
• Energy dissipation during interaction cascades
• This conclusion was drawn by Per Bak et al by
  studying an analogous model, the so-called
  Sandpile Automaton. Per Bak proposed the concept
  of self-organised criticality as a description of
  the dynamics of such systems.

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Per Bak's Sand-pile Automaton

• Rectangular grid of sites
• Each site may support a maximum of 4 grains of
  sand
• Sand is added to sites at random
• Sites with 4 grains avalanche i.e the sand
  cascades to the nearest neighbouring sites
• Redistribution of sand can trigger neighbouring
  sites to fail which in turn may trigger failure
  of their neighbours -gt avalanches may be any size
  between one site and the entire grid

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Thermodynamic Criticality Self-Organised
Criticality

• THERMODYNAMIC CRITICALITY
• Occurs when thermodynamic systems are driven
  through a phase transition by varying properties
  such as temperature, pressure etc.
• Characterised by a sudden change is macroscopic
  properties of the system
• As a critical point is approached, long-range
  spatial and temporal correlations emerge -gt
  power-laws
• Thanks to mean-field theory etc. thermodynamic
  criticality is relatively well understood and the
  values of various measurable quantities (e.g
  power-law exponents) can be predicted
• SELF-ORGANISED CRITICALITY
• Certain classes of systems do not require
  tuning to go critical
• Criticality represents an attractor for the
  dynamics of said systems
• SOC is elegent because it can explain
  observations of power-law correlations in natural
  systems without needing to hypothesize the
  existence of a god-like system-tuner who turns
  the knobs to cause criticality

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Where to go next?

• The Block-Slider and Sandpile automaton are
  hardly rigourous models for interacting fault
  systems, however their simplicity is
  advantageous...we can study the long-term system
  behaviour of such models relatively easily
• The simplicity of the models allows one to
  experiment with various different approaches for
  failing sites, redistribution of energy,
  dissipation, healing of failed sites etc.
• Doing so reveals that SOC is not as universal
  as first thought...models in different regimes of
  parameter space may have significantly different
  long-term dynamics

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Statistical Fractal Earthquake Automata

• Statistical fractal distribution of site
  strengths, sf 0.1,1.0
• Stress is incremented uniformly until a site has
  si gt sfi
• The stress of the failed site is redistributed to
  surrounding sites according to a particular
  stress redistribution mechanism
• Stress redistribution may trigger failure of
  additional sites
• Redistribution continues until no more sites fail

• CASE ONE Nearest Neighbour Automaton
• Dissipation factor
• Fraction of stress redistributed is dissipated
• Stress transfer ratio
• Previously failed sites receive less stress than
  unfailed sites
• Healing of sites subsequent to failure cascade

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• CASE TWO Longer Range Interactions
• Stress is redistributed to all cells within a
  square transfer region, with an R-p weighting
• Failed sites do not support stress until they
  heal
• Healing occurs after a specified number of
  cascade iterations, the healing time
• Thermal noise is added by choosing a random
  residual stress for failed sites

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Statistical Physics of EQ automata

• Mean-field theory for an instantaneous-healing BK
  automaton (Klein et al. 1995) revealed that such
  automata are Spinodal rather than SOC
• The theory provides an accurate description for
  this model BUT,
• The theory requires modification to include
  memory effects and healing to obtain the
  intermittent criticality observed in slow-healing
  automata

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CONCLUSIONS

• Cellular automata have provided some insight into
  the statistical physics of interacting fault
  systems
• How much can we draw from studies of these
  simplified models though?
• Presuming equivalent dynamical modes occur in the
  Earth's crust, the prospects for earthquake
  forecasting are relatively bright...at least for
  some fault systems (some of the time)
• Need more realistic models to verify whether
  these modes are reasonable