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

€

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