Foreshocks, Aftershocks, and Characteristic Earthquakes or Reconciling the Agnew & Jones Model with the Reasenberg and Jones Model Andrew J. Michael
Jan 07, 2016
Foreshocks, Aftershocks, and Characteristic Earthquakes
or
Reconciling the Agnew & Jones Model with the Reasenberg and Jones Model
Andrew J. Michael
Model 1: Reasenberg and Jones, Science, 1989
Probability of earthquakesduring an aftershock sequenceas a function of time andmagnitude.
Initial estimates are based onparameters for a “generic”California earthquake sequence.
Results start the same for allsequences.
Sequence specific parametersare used once they can bedetermined.
Extend aftershocks to foreshocks.
Modified-Omori Law
Gutenberg-RichterDistribution
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.”
Model 2: Agnew and Jones, JGR, 1991After discarding aftershocks,earthquakes are divided into three categories for statistical purposes:
Mainshocks: which we want to forecastForeshocks: which are always followed by mainshocksBackground Events: which are never followed by mainshocks
When a moderate event occurs we can’t tell if it isa 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
M4.8 Event At Bombay Beach On March 24, 2009Could It Be A Foreshock To A Larger Earthquake In The Next 3 Days?
Mainshock:SAF, Coachella Seg.UCERF2:Length = 69 kmM 75-yr Prob. = 5%3-day Prob.= 0.009%
M4.8 Event At Bombay Beach On March 24, 2009Could It Be A Foreshock To A Larger Earthquake In The Next 3 Days?
Mainshock:SAF, Coachella Seg.UCERF2:Length = 69 kmM 75-yr Prob. = 5%3-day Prob.= 0.009%
Reasenberg &Jones, 1989:Probabilityof M4.8 beingfollowed byan M≥7 eventPF = 0.05%
M4.8 Event At Bombay Beach On March 24, 2009Could It Be A Foreshock To A Larger Earthquake In The Next 3 Days?
Mainshock:SAF, Coachella Seg.UCERF2:Length = 69 kmM 75-yr Prob. = 5%3-day Prob.= 0.009%
Reasenberg &Jones, 1989:Probabilityof M4.8 beingfollowed byan M≥7 eventPF = 0.05%
M4.8 Event At Bombay Beach On March 24, 2009Could It Be A Foreshock To A Larger Earthquake In The Next 3 Days?
Agnew andJones, 1991:PF = 4%
Reasenberg & Jones with Gutenberg-Richter
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λ t,M( ) = k10bM i10−bM (t + c)− p
RateOverall
Productivity
Productivity vs.Initiating Event
Magnitude
Probability of m≥M given an Earthquake
P(m≥M|E)(Mmin=0)
modified-OmoriDecay
Can we modify this to include characteristic behavior?
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N(m ≥ M ) =10a−bM + DH (M c −M )
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P(m ≥M | E) =10a−bM +DH(Mc −M)
10a−bM min +D
Gutenberg-Richter + Characteristic Earthquake Relationships
Rate ofCharacteristic
Earthquake
Magnitude ofCharacteristic
Earthquake
HeavisideFunction
Gutenberg-Richter versus Characteristic Clustering Models
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λ t,M( ) = k10bM i10−bM (t + c)− p
RateOverall
Productivity
Productivity vs.Initiating Event
Magnitude
Probability of m≥M given an Earthquake
P(m≥M|E)(Mmin=0)
modified-OmoriDecay
€
λ t,M( ) = k10bM i10a−bM +DH(Mc −M)
10a +D(t + c)−p
Approximate the Probability of an M≥Mc eventfollowing an M=Mi event
assuming:rate of M=0 events 10a >> D the rate of Mc events
rate of Mi events 10a-bMi >> D the rate of Mc events
D >> 10a-bMc the Gutenberg-Richter rate of M≥Mc
small probabilities so P≈λ
Both models are proportional to the rate of characteristic eventsinversely proportional to the rate of initiating events
Characteristic Reasenberg & Jones Approximate Model
Agnew & Jones Approximate Model
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P(C |F ∪B) ≈2Nm
(10bμ −10−bμ )
D
10a−bM i€
P(M ≥Mc ) ≈ kItD
10a−bM i
Reasenberg & Jones w/Characteristic Clustering
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 eventtimes 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 < MC the uncertainties for M≥MC 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