1. What can we learn about dynamic triggering in the the lab? Lockner and Beeler, 1999

2. Questions… Is the relationship between periodic stress loading and timing of failure threshold or time dependent? (i.e. Coulomb or rate & state) What amplitudes and frequencies cause correlated failure? How can these results be extended to dynamic triggering in the real world? (seismic waves, earth tides, seasonal stress variations)

3. Experimental design 30  highly polished cut in a 190 x 76 mm core of Westerly granite Teflon “shim” minimizes resistance to lateral movement Polyurethane jacket undergoes elastic recovery when pressure removed, sample “jumps back”

4. Experimental procedure 3 background rates (V r ): 0.001 0.01 and 0.1  m/s Cosine periodic forcing: 2.8 hrs to 10 sec Amplitude ranges: 1.1 – 11  m (0.07- 0.7 MPa Shear or 0.04- 0.4 MPa Normal)  is constant at 0.68 Confining pressure constant at 50 MPa ???

5. Experimental procedure 1.5 to 2 mm of pre-shortening to reach steady state 14 mm of axial shortening, yields ~ 50 events (peak stress constant… stress drop increasing??) Fault gouge continually removed, repolished

6. Long period forcing… V r =0.01  m/s, period ~ 38 min  m ~ 0.12 MPa Short period forcing… V r =0.001  m/s, period ~ 50 sec  m ~ 0.07 MPa Lockner and Beeler, 1999

7. Determine the phase of each failure (  ), and determine the offset from the peak stress (phase shift,  ) Create a PDF use “cosine weighting functions” and normalizing by average seismicity rate (area = 1) Analyzing results … u m = U m Cos(  t) Lockner and Beeler, 1999

8. PDF’s are fit to: P(  )=P 0 + P m Cos(  -  ) Where…. P 0 = probability w/out forcing (1/2  ) P m = amplitude  phase shift (why no frequency term??) Define a metric: When P m /P 0 > 1, the correlated EQ signal is above the random noise Analyzing results … u m = U m Cos(  t) Lockner and Beeler, 1999

9. Assessing “correlation” between phase of failure and forcing period “random walk” statistics

10. Coulomb failure model predicts: R =R 0 (Ŝ/S 0 ) (Seismicity rate is proportional to stressing rate) (R max -R min )/R 0 = (Ŝ max -Ŝ min )/Ŝ 0 = (2  m  Ŝ 0 where… Ŝ is seismicity rate,  m is max shear stress amplitude,  is frequency Thus, if (R max -R min )/R 0 = 1 is threshold for correlation of seismicity, then…  m = (1/  Ŝ 0 /2  ) is a line predicting correlation…

11.  m = (1/  Ŝ 0 /2  We can plot the line representing predicted Coulomb failure Notice that required  m for correlation decreases at high frequency (  ) mm Beeler and Lockner, 2003

12. Periodic displacement: u m = U m Cos(  t) (differentiate to get periodic stressing rate) Where… U m is amplitude of periodic displacement (in  m) Full stressing rate (V t ): V t = V r -  U m Sin(  t) Where… V r is the backround rate, and  is frequency  U m /V r < 1 -stressing rate is always positive (compressive)  U m /V r > 1 - stressing rate becomes negative for part of the cycle When in the cycle does Coulomb model predict failure? Lockner and Beeler, 1999

13. When ….  U m /V r > 1 Peak seismicity migrates towards the peak stress because the stress “shadow” Is becoming larger and larger When…  U m /V r < 1 Peak seismicity rate corresponds to peak stressing rate at  = -90  When in the cycle does Coulomb model predict failure? Beeler and Lockner, 2003

14.  m =0.69  m =0.35  m =0.17 Observed in Experiments Coulomb Model Coulomb model doesn’t explain experimental observations at short period …. Both examples ~~ V r = 0.01  m/s, period = 25 sec,  =.68,  n =86 MPa Lockner and Beeler, 1999

15. Dieterich (1987) did numerical simulations showing that if faults obey the rate and state model, then seismicity rate is… What does the rate and state model predict? R a = (R max -R min )/R 0 = 2  m /A 1  n = P m /P 0 1 yr cycle Dieterich (1987)

16. Thus, R a is Dependent on: Shear stress amplitude (  m ), and the constituitive parameter A 1 such that… R a is Independent of: frequency (  ), characteristic length (Dc), stiffness (K), and parameter B When R a = 1…  m = A 1  n / 2 What does the rate and state model predict? R a = 2  m /A 1  n = P m /P 0

17. We expect different responses at different points in frequency-amplitude space 100% probability of correlation using Coulomb model R&S model (independent of period) Coulomb : (dependent on period) Model predictions for V l = 0.1  m/s mm Beeler and Lockner, 2003

18. Dependence on both frequency (  ) and amplitude (  m ) This study observes… Lockner and Beeler, 1999

19. The response mode is determined by relationship of nucleation time (t n ) to period (t w ) Coulomb (if t n < t w ) Rate & State (if t n > t w ) Nucleation Time (t n ) mm Beeler and Lockner, 2003

20. Positive slope at high frequency is not predicted by rate & state model… Positive Slope!! Perhaps a lower value of A 1 is necessary to explain this?

21. Dieterich (1987) R&S theory predicts: R a /2 =  m /A 1  n = P m /P 0 R a is seismicity rate,  m is shear stress, A 1 is the R&S constituitive parameter Plugging in experimental observations to back-calculate A 1 gives: A 1 = 0.001 -0.004 (measured) A 1 = 0.003 – 0.009 (known) But… R&S model can correctly back Calculate the parameter A 1 ….

22.  m =0.69  m =0.35  m =0.17 Observed in Experiments Coulomb ModelRate & State Model And…R&S model correctly predicts onset of Correlation as function of  and  m All three examples ~~ V r = 0.01  m/s, period = 25 sec,  =.68,  n =86 MPa (note different phase predictions for Coulomb vs. R&S)

23. Lab results suggest we should see dynamic triggering of EQ’s by seismic waves Three studies have observed a threshold stress amplitude for dynamic triggering: Anderson et al., 1994 - 0.6 - 0.9 MPa ( 350 >  n < 50) Hill et al., 1993 - 0.1 - 0.2 MPa ( 350 >  n < 50) Gomberg and Davis, 1996 - 0.003 - 0.012 MPa ( 50 >  n < 0) This Study (experimental) - 0.4 – 1 MPa (  n ~ 50 MPa) But… do experimental results extrapolate to higher confining pressures (higher  n at depth) Do results under estimate the stress amplitude necessary to induce EQ correlation… assuming  is constant?

24. Implications for dynamic triggering by earth tides Because earth tides have a longer period than nucleation time, they fall in the Coulomb regime, and stresses are not high enough This study: Lowest amplitude is ~ 0.05 MPa Longest period is ~ 2.8 hours Earth tides: Amplitudes ~ 0.001 – 0.004 MPa Period ~ 12.5 hours Assuming linear relationship between seismicity correlation and stress amplitude, > 20,000 EQ’s are required to see a correlation between earth tides and seismicity What assumption are required to extrapolate to frequencies and amplitudes of earth tides? (i.e. constant A 1 constant  )

25. Key results… Premonitory slip is observed in lab, suggesting failure is a time dependent (R&S) process Two distinct linear relationships between  m and  required for correlation, dependent on nucleation time (t n ) Coulomb or R&S models work for long period forcing, only R&S works for short period Because period of earth tides is much shorter than nucleation times, R&S model is appropriate, tides don’t exert enough shear stress amplitude to cause failure in R&S model Seismic waves may impart enough shear stress, earth tides do not, except in rare cases (~1% caused by earth tides…)

26. Some questions… Is the confining pressure (50 MPa) in this experiment reasonable… and does it allow extrapolation of this data set to real EQ’s ? Do the high frequencies used in this study allow extrapolation to triggering by earth tides? Does the lack of fault gouge accumulation cause these results to underestimate the necessary stress amplitudes for EQ intitiation?