Overview of Statistical Stress/Strain Release Model (Vere-Jones, 1978; Jaume ´ &Bebbington, 2000)

1. Does the Scaling of Strain Energy Release with Event Size Control the Temporal Evolution of Seismicity? Steven C. Jaumé Department of Geology And Environmental Geosciences College of Charleston Charleston, South Carolina, USA

2. Overview of Statistical Stress/Strain Release Model(Vere-Jones, 1978; Jaume´&Bebbington, 2000) • Stress/strain history: X(t) = X(0) + t - (t) t = time;  = stressing rate; (t) = stress release of earthquake at time t • Earthquake magnitude distribution: Prob ( > y) = (1 + y)- e-y/U  = linear slope at small magnitudes; U = magnitude threshold for rolloff • Rate of earthquakes: (t) = exp[ + X(t)]  and  are constants

3. Overview of Statistical Stress/Strain Release Model(continued) • Location of magnitude rolloff: U = A + Bexp[X(t)] A and B are constants • Stress/strain release as a function of magnitude: a. i = 10 0.75Mi + 2.4 “Benioff strain” stress drops b. i = 10 1.5Mi + 9.0 Seismic moment stress drops • Simulations: A series of simulations were run where either the event rate (), magnitude distribution (U) or both were a function of regional stress (X). All resulting catalogs were then tested to see if they contained accelerating moment release sequences.

4. Example of an accelerating moment release sequence from a stress release model simulation. Solid circles are simulated earthquakes and the line is a fit to the equation e(t) = A – B(tf – t)m where e is the cumulative Benioff strain, t is time, and tf is the time of the mainshock (Bufe & Varnes, 1993).

5. Magnitude-frequency distributions resulting from stress release model simulations where both the rate and size of earthquakes are a function of regional stress. Note that the relative scaling of event size vs. number is different even though starting distribution is the same.

6. Summary of Stress Release Models and Accelerating Moment Release (AMR) Model Type% with AMR • “Benioff strain” stress drops 0-30% • Seismic moment stress drops 78-100% • Randomized catalogs 38-69% Tentative Conclusion: Scaling of stress/strain release with magnitude influences the dynamics of earthquake systems.

7. Comparison of stress history (red line) to earthquake history (vertical black lines) for a model where both event rate and size are controlled by X(t) and stress drop scaling follows 5a. Compare with cellular automaton model which shows SOC behavior (next slide).

8. Comparison of stress history (red line) to earthquake history (vertical black lines) in a cellular automaton model that shows SOC behavior.

9. Comparison of stress history (red line) to earthquake history (vertical black lines) for a model where both event rate and size are controlled by X(t) and stress drop scaling follows 5b. Compare with cellular automaton model which shows Intermittent Critical behavior (next slide).

10. Comparison of stress history (red line) to earthquake history (vertical black lines)in a cellular automaton model which shows intermittent critical behavior.

11. How Does Strain Energy Release Scale With Earthquake Magnitude? • From Benioff (1951): Strain energy release (Est) scales as the square root of radiated seismic energy(Es) - Est Es0.5 • Common Present Assumption:Strain energy release (Est) scales with seismic moment release (M0) - Est M0 Strain energy (potential energy stored in elastically deformed crust), W = ½V ij ij dV ij = stress tensor ij = strain tensor

12. Estimating the Scaling of Strain Energy Release Scale With Earthquake Magnitude Calculated W for a pre-strained (i.e., locked fault model) volume (Wint) and then for the same volume after an earthquake event (Wfin). The strain energy release by the event (Est) should then be: Est = Wint – Wfin In practice the strain energy release by a particular event depends upon the assumed loading model.

13. Sketch of Strike-Slip Fault Model Model strike-slip fault buried in an elastic halfspace. Strain energy is calculated (Okada, 1992) at the center of 1 km3 cubes to a depth of 20 km; horizontal grid is scaled to be 5 times rupture length.

14. Both the “virtual slip” loadingmodel and the earthquake slip models consist of 10 over-lapping dislocations that taper to zero at the edges. For Mw < 6.5 earthquakes the slip model tapers in all directions; for Mw 6.5 earthquakes they mimic the loading in depth and taper in the horizontal dimension.

15. Strain energy released as a function of Mw in the strike-slip faulting model described above. The scaling of seismic moment with magnitude is shown for comparison.

16. Release of strain energy stored in shear component parallel to assumed fault motion in model. Note change in scale from previous figure. Again, seismic moment scaling shown for comparison.

17. Energy dissipation as a function of magnitude from a pair of stress-release models. In those models where energy release is scaled as seismic moment large events dominate the energy dissipation, whereas small events dominate when energy release scales as Benoiff strain.

18. Energy dissipation versus event size (Energy “Release”) in two cellular automaton models. Although seismicity evolves very differently in the two models, in both cases most of the energy dissipation occurs in the largest events.

19. Conclusions? • Results from modified stress-release models suggest that scaling of strain energy release with magnitude helps control the temporal evolution of seismicity. • An elastic dislocation model of a strike-slip fault suggests that strain energy release may scale as Log10 Est 2.1-2.2 Mw. • However…a check back with cellular automaton models suggest that strain energy scaling may not be the only or even most important factor.