Chapter 5: Calculating Earthquake Probabilities for the SFBR

Mei XueEQW March 16

Outline

• Introduction to Probability Calculations

• Calculating Probabilities• Probability Models Used in the

Calcuations• Final calculation steps

Introduction to Probability Calculations

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

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

Introduction to Probability Calculations

• Express the likelihood of occurrence of one or more M6.7 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

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

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

Calculating Probabilities

Survivor function: gives the probability that at least time T will elapse between successive events

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)

Conditional probability: gives the

hazard function: gives the instantaneous rate offailure at time t conditional upon no event having occurred up to time t

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)

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

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

Summary of rates and 30-year probabilities of EQs (M6.7) in SFBR calculated with various models

Assumptions:

1. Fluctuations in the rate of M 3.0 and M 5.5 EQs reflect fluctuations in the probability of larger events

2. 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)

3. The rate function (t) can be sensibly extrapolated forward in time

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)

1. For smaller α, strongly peaked, remains close to zero longer2. For larger α, “dead time” becomes shorter, increasingly Poisson-

like

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

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

1. 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

2. The effects are strongest when the segment is near failure

Assumptions:

1. The model represents the statistics of recurrence intervals for segment rupture

2. The time of the most recent event is known or constrained

3. The effects of interactions are properly characterized (BPT-step, 1906, 1989 San Andreas only)

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

Five Probability Models:5 Time Predictable Model

Four extensions:1. Model the SFBR as a network of faults2. Strictly gives the probability that a

rupture will start on a segment3. Fault segments can rupture in more

than one way4. Use Monte Carlo sampling of the

parent distributions to propagate uncertainty through the model

Five Probability Models:5 Time Predictable Model

Six-step calculation sequence:1. Slip in the most recent event2. Slip rate of the segment3. Expected time of the next rupture of

the segment4. Probability of a rupture starting on the

segment (4 segments, ignoring interaction, BPT model) – Epicentral probabilities

5. Convert epicentral probabilities into earthquake probabilities

Five Probability Models:5 Time Predictable Model

6. Compute 30-year source probabilities

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

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