The seismic cycle

1. The seismic cycle • The elastic rebound theory. • The spring-slider analogy. • Frictional instabilities. • Static-kinetic versus rate-state friction. • Earthquake depth distribution.

2. The elastic rebound theory (according to Raid, 1910)

3. The spring-slider analog

4. Frictional instabilities The common notion is that earthquakes are frictional instabilities. • The condition for instability is simply: • The area between B and C is equal to that between C and D.

5. Frictional instabilities Frictional instabilities are commonly observed in lab experiments and are referred to as stick-slip. Brace and Byerlee, 1966

6. From laboratory scale to crustal scale Figure from http://www.servogrid.org/EarthPredict/

7. Frictional instabilities governed by static-kinetic friction Stress static friction stress kinetic friction Lc slip Slip Time The static-kinetic (or slip-weakening) friction: experiment Constitutive law Ohnaka (2003)

8. Frictional instabilities governed by rate- and state-dependent friction Dieterich-Ruina friction: • were: • V and  are sliding speed and contact state, respectively. • A, B and  are non-dimensional empirical parameters. • Dc is a characteristic sliding distance. • The * stands for a reference value.

9. Frictional instabilities governed by rate- and state-dependent friction State [s] The evolution of sliding the speed and the state throughout the cycles. An earthquake occurs when the sliding speed reaches the seismic speed - say a meter per second. loading point (I.e., plate) velocity

10. According to the spring-slider model earthquake occurrence is periodic, and thus earthquake timing and size are predictable - is that so?

11. The Parkfield example Magnitude Year A sequence of magnitude 6 quakes have occurred in fairly regular intervals. 2004 The next magnitude 6 quake was anticipated to take place within the time frame 1988 to 1993, but ruptured only on 2004.

12. So the occurrence of major quakes is non-periodic - why?

13. The role of stress transfer Stein et al., 1997 • Faults are often segmented, having jogs and steps. • Every earthquake perturb the stress field at the site of future earthquakes. • So it is instructive to examine the implications of stress changes on spring-slider systems. Animation from the USGS site

14. The effect of a stress step The effect of a stress perturbation is to modify the timing of the failure according to: That means that the amount of time advance (or delay) is independent of when in the cycle the stress is applied.

15. The effect of a stress step state [t] The effect of a stress step is to increase the sliding speed, and consequently to advance the failure time.

16. The effect of a stress step The ‘clock advance’ of a fault that is in an early state of the seismic cycle (I.e., far from failure) is greater than the ‘clock advance’ of a fault that is late in the cycle (I.e., close to failure).

17. In summary: • The effect of positive and negative stress steps is to advance and delay the timing of the earthquake, respectively. • While according to the static-kinetic model the time advance depends only on the magnitude of the stress step and the stressing rate, according to the rate-and-state model it depends not only on these parameters, but also on when in the cycle the stress has been perturbed. • Thus, short-term earthquake prediction may be very difficult (if not impossible) if rate-and-state model applies to the earth.

18. What are the conditions for instabilities in the spring-slider system? static friction stress kinetic friction Lc slip The static-kinetic friction: Thus, the condition for instability is:

19. What are the conditions for instabilities in the spring-block system? The rate- and state-dependent friction: The condition for instability is: Thus, a system is inherently unstable if b>a, and conditionally stable if b<a.

20. How b-a changes with depth ? • Note the smallness of b-a. Scholz (1998) and references therein

21. The depth dependence of b-a may explain the seismicity depth distribution Scholz (1998) and references therein

22. But a spring-slider system is too simple… • Fault networks are extremely complex. • More complex models are needed. • In terms of spring-slider system, we need to add many more springs and sliders. Figure from Ward, 1996

23. System of two blocks During static intervals: During dynamic intervals: • To simplify matters we set: We define: Several situations:

24. System of two blocks symmateric ( ) asymmateric ( ) Next we show solutions for: Were: Turcotte, 1997 Breaking the symmetry of the system gives rise to chaotic behavior.

25. Summary • Single spring-slider systems governed by either static-kinetic, or rate- and state-dependent friction give rise to periodic earthquake-like episodes. • The effect of stress change on the system is to modify the timing of the instability. While according to the static-kinetic model the time advance depends only on the magnitude of the stress step and the stressing rate, according to the rate-and-state model it depends not only on these parameters, but also on when in the cycle the stress has been perturbed. • Breaking the symmetry of two spring-slider system results in a chaotic behavior. • If such a simple configuration gives rise to a chaotic behavior - what are the chances that natural fault networks are predictable???

26. Recommended reading • Scholz, C., Earthquakes and friction laws, Nature, 391/1, 1998. • Scholz, C. H., The mechanics of earthquakes and faulting, New-York: Cambridge Univ. Press., 439 p., 1990. • Turcotte, D. L., Fractals and chaos in geology and geophysics, New-York: Cambridge Univ. Press., 398 p., 1997.