Interacting Earthquake Fault Systems:Cellular Automata and beyond...

D. WeatherleyQUAKES & AccESS

3rd ACES Working Group MeetingBrisbane, Aust. 5th June, 2003.

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

Scope of the Problem

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?)

X

Multi-Fractal fault heirarchy

Complicated interactions between faults due to stress transfer during Eqs

Nonlinear Rheology

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

Cu

mu

lati

ve m

om

ent

Accelerating MomentRelease

Bufe & Varnes, 1993

Year1920 1940 1960 1980

EQ magnitude, M

Nu

mb

er o

f E

qs,

N(M

)

N(M) ~ M-b

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.

Archetypical Earthquake Model: Burridge-Knopoff Block-Slider

(Figure thanks to J.Rundle, ICCS 2003 presentation)

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.

What has the BK model taught us?

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 -> avalanches may be any size between one site and the entire grid

Thermodynamic Criticality & Self-Organised CriticalityTHERMODYNAMIC 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 -> 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

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

Statistical Fractal Earthquake Automata

● Statistical fractal distribution of site strengths,

f = {0.1,1.0}

● Stress is incremented uniformly until a site has

i >=

fi

● 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

● 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

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

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