Chapter 5: Calculating Earthquake Probabilities for the SFBR

1. Chapter 5: Calculating Earthquake Probabilities for the SFBR Mei Xue EQW March 16

2. Outline • Introduction to Probability Calculations • Calculating Probabilities • Probability Models Used in the Calcuations • Final calculation steps

3. Introduction to Probability Calculations

4. Introduction to Probability Calculations • Earthquake probability is calculated over the time periods of 1-, 5-, 10-, 20-, 30- and 100-year-long intervals beginning in 2002 • The input is a regional model of the long-term production rate of earthquakes in the SFBR (Chapter 4) • The second part of the calculation sequence is where the time-dependent effects enter into the WG02 model

5. Introduction to Probability Calculations • Review what time-dependent factors are believed to be important and introduce several models for quantifying their effects • The models involve two inter-related areas: recurrence and interaction

6. Introduction to Probability Calculations • Express the likelihood of occurrence of one or more M6.7 EQs in the SFBR in the time periods in five ways: • The probability for each characterized large EQ rupture source (35) • The probability that a particular fault segment will be ruptured by a large EQ (18) • The probability that a large EQ will occur on any of the 7 characterized fault systems • The probability of a background EQ (on faults in the SFBR, but not on one of the 7) • The probability that a large EQ will occur somewhere in the region

7. Calculating Probabilities • Primary input: the rupture source mean occurrence rate (Table 4.8) • The probability rupture source, each fault, combined with background -> the probability for the region as a whole

8. Calculating Probabilities • They model EQs that rupture a fault segment as a renewal process: independent • Probability Models • Poisson: the probability is constant in time and thus fully determined by the long-term rate of occurrence of the rupture source • Empirical model: a variant of the Poisson, the recent regional rate of EQs • Time-varying probability models: BPT, BPT-step (1906, 1989), and Time-predictable (1906), take into account information about the last EQ

9. Calculating Probabilities Survivor function: gives the probability that at least time T will elapse between successive events hazard function: gives the instantaneous rate of failure at time t conditional upon no event having occurred up to time t Conditional probability: gives the Probability that one or more EQs will occur on a rupture source of interest during an interval of interest, conditional upon it not having occurred by T (year 2002)

10. Five Probability Models:1 Poisoon Model  - the mean rupture rate of each rupture source • The hazard function is constant • Fails to incorporate the most basic physics of the earthquake process: reloading • Fails to account for stress shadow • Reflects only the long-term rates • Conservative estimate for faults that time-dependent models are either too poorly constrained or missing some critical physics (interaction)

12. Five Probability Models:2 Empirical Model • (t) – not stationary, estimated from the historical EQ record (M  3.0 since 1942 and M  5.5 since 1906) • - the long term mean rate • Complements the other models as M  5.5 is not used by other models • Take into account the effect of the 1906 EQ stress shadow • Specifies only time-dependence, preserving the magnitude distribution of the rupture sources

13. The shape of the magnitude distribution on each fault remains unchanged; the whole distribution moves up and down in time

15. Summary of rates and 30-year probabilities of EQs (M6.7) in SFBR calculated with various models

16. Assumptions: • Fluctuations in the rate of M  3.0 and M  5.5 EQs reflect fluctuations in the probability of larger events • Fluctuations in rate on individual faults are correlated (though stress shadow is not homogeneous in space, affected seismicity on all magjor faults in the region) • The rate function (t) can be sensibly extrapolated forward in time

17. Five Probability Models:3 BPT Model μ – the mean recurrence interval, 1/ α – the aperiodicity, the variability of recurrence times, related to the variance 2, equals /μ A Poisson process when α=1/sqrt(2)

18. For smaller α, strongly peaked, remains close to zero longer • For larger α, “dead time” becomes shorter, increasingly Poisson-like

19. Estimates of aperiodicity α obtained by Ellsworth et al. (1999) for 37 EQ secquences (histogram) and the WG02 model (solid line)

20. Five Probability Models:4 BPT-step Model • A variation of the BPT model • Account for the effects of stress changes caused by other earthquakes on the segment under consideration (1906, 1989) • Interaction in the BPT-step model occurs through the state variable

21. A decrease in the average stress on a segment lowers the probability of failure, while an increase in average stress causes an increase in probability • The effects are strongest when the segment is near failure

22. Assumptions: • The model represents the statistics of recurrence intervals for segment rupture • The time of the most recent event is known or constrained • The effects of interactions are properly characterized (BPT-step, 1906, 1989 San Andreas only)

23. Five Probability Models:5 Time Predictable Model • The next EQ will occur when tectonic loading restores the stress released in the most recent EQ • Dividing the slip in the most recent EQ by the fault slip rate approximates the expected time to the next earthquake • Only time of next EQ not size • Only for the SAF fault

24. Five Probability Models:5 Time Predictable Model Four extensions: • Model the SFBR as a network of faults • Strictly gives the probability that a rupture will start on a segment • Fault segments can rupture in more than one way • Use Monte Carlo sampling of the parent distributions to propagate uncertainty through the model

25. Five Probability Models:5 Time Predictable Model Six-step calculation sequence: • Slip in the most recent event • Slip rate of the segment • Expected time of the next rupture of the segment • Probability of a rupture starting on the segment (4 segments, ignoring interaction, BPT model) – Epicentral probabilities • Convert epicentral probabilities into earthquake probabilities

26. Five Probability Models:5 Time Predictable Model 6. Compute 30-year source probabilities

27. Final calculation steps • Probabilities for fault segments and fault systems • Probabilities for earthquakes in the background • Weighting alternative probability models (VOTE) • Probabilities for the SFBR model

31. Paper recommendataions • Reasenberg, P.A., Hanks, T.C., and Bakun, W.H., 2003, An empirical model for earthquake probabilities in the San Francisco Bay region, California, 2002-2031, BSSA 93 (1): 1-13 FEB 2003 • Shimazaki, K., and Nakata, T., Time-predictable recurrence model for large earthquakes: Geophysical Research Letters, v. 7, p. 279-282, 1980 • Ellsworth, W.L., Matthews, M.V., Nadeau, R.M., Nishenko, S.P., Reasenberg, P.A., and Simpson, R.W., A physically-based earthquake recurrence model for estimation of long-term earthquake probabilities: USGS, OFR 99-522, 23 p., 1999 • Cornell, C.A., and Winterstein, S.R., Temporal and magnitude dependence in earthquake recurrence models: BSSA, v. 78, no. 4, p. 1522-1537, 1988 • Harris, R.A., and Simpson, R.W., Changes in static stress on Southern California faults after the 1992 Landers earthquake: Nature, v. 360, no. 6401, p. 251-254, 1992