Chapter 5: Calculating Earthquake Probabilities for the SFBR

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Chapter 5 Calculating Earthquake Probabilities
for the SFBR

• Mei Xue
• EQW March 16

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Outline

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

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Introduction to Probability Calculations
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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

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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

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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

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Calculating Probabilities

• Primary input the rupture source mean occurrence
  rate (Table 4.8)
• The probability rupture source, each fault,
  combined with background -gt the probability for
  the region as a whole

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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

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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)
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Five Probability Models1 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)

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Five Probability Models2 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

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The shape of the magnitude distribution on each
fault remains unchanged the whole distribution
moves up and down in time
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Summary of rates and 30-year probabilities of EQs
(M?6.7) in SFBR calculated with various models
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• 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

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Five Probability Models3 BPT Model
µ the mean recurrence interval, 1/? a the
aperiodicity, the variability of recurrence
times, related to the variance ?2, equals ?/µ
A Poisson process when a1/sqrt(2)
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• For smaller a, strongly peaked, remains close to
  zero longer
• For larger a, dead time becomes shorter,
  increasingly Poisson-like

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Estimates of aperiodicity a obtained by Ellsworth
et al. (1999) for 37 EQ secquences (histogram)
and the WG02 model (solid line)
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Five Probability Models4 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

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• 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

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• 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)

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Five Probability Models5 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

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Five Probability Models5 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

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Five Probability Models5 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

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Five Probability Models5 Time Predictable Model

• 6. Compute 30-year source probabilities

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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

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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