Accelerating Precursory Activity

in Statistical Fractal Automata

Dion Weatherley and Peter MoraQUAKES, Univ. of Qld., Australia.

Outline

• Model Description• Results from a parameter space study

• Event size scaling• Evolution of mean stress• Stress field evolution• Accelerating precursory activity

• Summary of results• Effect of varying the fractal dimension• Conclusions and questions

Model Description

• Uniform rectangular grid of cells

• Each cell has two properties• Constant, scalar Strength• Variable, scale Stress

• Uniform loading of all cells• Cell failure when a cell’s

stress exceeds its strength• Nearest Neighbour stress

redistribution• Periodic boundary conditions

Event size scaling

Mean Stress Evolution• models with a char. event distribution (A)

• relatively low time-averaged mean stress

• large, saw-tooth fluctuations

• models with a roll-over distribution (D)

• high time-averaged mean stress

• small, irregular fluctuations

• models with high dissipation (C)

• near-constant mean stress

• models with GR-scaling (D)

• correspond with drop in time-averaged mean stress

• mark a transition from large to small fluctuations

Snapshots of Stress Evolution

Accelerating Precursory Activity

1. Select the 100 largest events in each simulation for analysis.

2. Define Tf as the time of the large event.

3. Vary To and m within a range.

4. Compute A,B via least squares fit.

5. Compare power-law with a linear fit.

6. Select best power-law fit for each event.

7. Count number of good power-law fits for each simulation = fit probability.

Summary of Results

• Accelerating Precursory Activity in models with:• overabundance of larger events• relatively low time-averaged mean stress• large, saw-tooth fluctuations in mean stress• relatively smooth stress field over broad regions

• Constant rate of activity in models with:• an under-abundance of large events• high, near-constant mean stress• stress field dominated by small-scale heterogeneity

• Evidence for a distinct transition between two modes of activity• GR-scaling only for simulations along a line through parameter space• change in character of mean stress fluctuations across this line• change in stress field heterogeneity across this line• probability of accelerating precursory activity drops to near-zero at this line

Effect of varying the Fractal Dimension of cell Strengths

D=1.875

D=2.250

D=2.625

Conclusions and some Questions• Accelerating Precursory Activity occurs in models in which:

• there is an overabundance of larger events• these large events perturb the system from a state of high stress• stress redistribution smoothes the stress field within failed region• long-range clustering of cell strengths (smaller fractal dimension)

• Stress field heterogeneity apparently a factor determining the mode of activity

• relatively smooth stress field corresponds with accelerating activity• a stress field dominated by small-scale heterogeneity corresponds with a

constant rate of activity

• Some questions to motivate discussions• Is the transition line a SPINODAL i.e. a line of Critical Points?• How might one formulate a mathematical description for these models?• Are these models any more or less appropriate as earthquake analogues

than pseudo-spinodal or SOC models?