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

2. 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

3. 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

4. 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 Cumulative moment Accelerating Moment Release Bufe & Varnes, 1993 Year 1920 1940 1960 1980 EQ magnitude, M Number of Eqs, 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.

5. Archetypical Earthquake Model: Burridge-Knopoff Block-Slider (Figure thanks to J.Rundle, ICCS 2003 presentation)

6. 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?

7. 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

8. 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 -> 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

9. 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

10. 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

11. ● 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

12. 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

13. 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