1. Foreshocks, Aftershocks, and Characteristic Earthquakes or Reconciling the Agnew & Jones Model with the Reasenberg and Jones Model Andrew J. Michael

2. Model 1: Reasenberg and Jones, Science, 1989 Probability of earthquakes during an aftershock sequence as a function of time and magnitude. Initial estimates are based on parameters for a “generic” California earthquake sequence. Results start the same for all sequences. Sequence specific parameters are used once they can be determined. Extend aftershocks to foreshocks. Modified-Omori Law Gutenberg-Richter Distribution

3. Agnew and Jones, JGR, 1991: “But it ought to be possible to do better: Should we say the same thing after every event? the probability of a very large earthquake should be higher if the candidate foreshock were to occur near a fault capable of producing that mainshock than if it were located in an area where we believe such a mainshock to be unlikely. Moreover, the chance of a candidate earthquake actually being a foreshock should be higher if the rate of background (nonforeshock) activity were low.”

4. Model 2: Agnew and Jones, JGR, 1991 After discarding aftershocks, earthquakes are divided into three categories for statistical purposes: Mainshocks: which we want to forecast Foreshocks: which are always followed by mainshocks Background Events: which are never followed by mainshocks When a moderate event occurs we can’t tell if it is a foreshock or a background event. We calculate the probability that it is a foreshock by PF = Rate of Foreshocks Rate of Foreshocks + Rate of Background Events Rate of Foreshocks = Rate of Mainshocks * Probability of Foreshocks Before Mainshocks

5. M4.8 Event At Bombay Beach On March 24, 2009 Could It Be A Foreshock To A Larger Earthquake In The Next 3 Days?

6. Mainshock: SAF, Coachella Seg. UCERF2: Length = 69 km M 7 5-yr Prob. = 5% 3-day Prob.= 0.009% M4.8 Event At Bombay Beach On March 24, 2009 Could It Be A Foreshock To A Larger Earthquake In The Next 3 Days?

7. Mainshock: SAF, Coachella Seg. UCERF2: Length = 69 km M 7 5-yr Prob. = 5% 3-day Prob.= 0.009% Reasenberg & Jones, 1989: Probability of M4.8 being followed by an M≥7 event PF = 0.05% M4.8 Event At Bombay Beach On March 24, 2009 Could It Be A Foreshock To A Larger Earthquake In The Next 3 Days?

8. Mainshock: SAF, Coachella Seg. UCERF2: Length = 69 km M 7 5-yr Prob. = 5% 3-day Prob.= 0.009% Reasenberg & Jones, 1989: Probability of M4.8 being followed by an M≥7 event PF = 0.05% M4.8 Event At Bombay Beach On March 24, 2009 Could It Be A Foreshock To A Larger Earthquake In The Next 3 Days? Agnew and Jones, 1991: PF = 4%

10. Reasenberg & Jones with Gutenberg-Richter Rate Overall Productivity Productivity vs. Initiating Event Magnitude Probability of m≥M given an Earthquake P(m≥M|E) (M min =0) modified-Omori Decay Can we modify this to include characteristic behavior?

11. Gutenberg-Richter + Characteristic Earthquake Relationships Rate of Characteristic Earthquake Magnitude of Characteristic Earthquake Heaviside Function

12. Gutenberg-Richter versus Characteristic Clustering Models Rate Overall Productivity Productivity vs. Initiating Event Magnitude Probability of m≥M given an Earthquake P(m≥M|E) (M min =0) modified-Omori Decay

13. Approximate the Probability of an M≥M c event following an M=M i event assuming: rate of M=0 events 10 a >> D the rate of M c events rate of M i events 10 a-bMi >> D the rate of M c events D >> 10 a-bMc the Gutenberg-Richter rate of M≥M c small probabilities so P≈λ

14. Both models are proportional to the rate of characteristic events inversely proportional to the rate of initiating events Characteristic Reasenberg & Jones Approximate Model Agnew & Jones Approximate Model

17. Reasenberg & Jones w/ Characteristic Clustering

18. The behavior of the Agnew and Jones model can be captured by the characteristic clustering version of the Reasenberg and Jones model. The characteristic clustering model covers a wider range of conditions: magnitudes above and below the initiating event times longer than 3 days post-initiating event The characteristic clustering model is therefore more useful. Implications Uncertainty in characteristic earthquake rates is high -> uncertainty in clustering probabilities is high for magnitudes close to the characteristic magnitude. Even if testing guides us to the best clustering model for M < M C the uncertainties for M≥M C will be high For foreshock probabilities of large earthquakes the key question is “do characteristic earthquakes exist and can we determine their long-term probabilities.” Summary