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
Modified-Omori Law
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.
Gutenberg-Richter
Distribution
Results start the same for all
sequences.
Sequence specific parameters
are used once they can be
determined.
Extend aftershocks to foreshocks.
Should we say the same thing after every event?
Agnew and Jones, JGR, 1991:
“But it ought to be possible to do better:
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, 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
M4.8 Event At Bombay Beach On March 24, 2009
Could It Be A Foreshock To A Larger Earthquake In The Next 3 Days?
M4.8 Event At Bombay Beach On March 24, 2009
Could It Be A Foreshock To A Larger Earthquake In The Next 3 Days?
Mainshock:
SAF, Coachella Seg.
UCERF2:
Length = 69 km
M7
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?
Mainshock:
SAF, Coachella Seg.
UCERF2:
Length = 69 km
M7
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?
Mainshock:
SAF, Coachella Seg.
UCERF2:
Length = 69 km
M7
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%
Agnew and
Jones, 1991:
PF = 4%
Reasenberg & Jones with Gutenberg-Richter
t, M  k10 10
bM i
Rate
Overall
Productivity
Productivity vs.
Initiating Event
Magnitude
bM
(t  c)
p
modified-Omori
Decay
Probability of m≥M
given an Earthquake
P(m≥M|E)
(Mmin=0)
Can we modify this to include characteristic behavior?
Gutenberg-Richter + Characteristic Earthquake Relationships
N(m  M )  10
abM
Rate of
Characteristic
Earthquake
P(m  M | E) 
10
 DH (M c  M )
Heaviside
Function
a bM
Magnitude of
Characteristic
Earthquake
 DH(M c  M)
a bM min
10
D
Gutenberg-Richter versus Characteristic Clustering Models
t, M  k10 10
bM i
Rate
Overall
Productivity
Productivity vs.
Initiating Event
Magnitude
t, M   k10
bM i
10
a bM
bM
(t  c)
p
modified-Omori
Decay
Probability of m≥M
given an Earthquake
P(m≥M|E)
(Mmin=0)
 DH(M c  M)
p
(t  c)
a
10  D
Approximate the Probability of an M≥Mc event
following 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
Characteristic Reasenberg & Jones Approximate Model
P(M  M c )  kIt
D
10
a bM i
Agnew & Jones Approximate Model
2N m
D
 P(C | F  B)  (10 b 10 b ) 10 a bM i
Both models are
proportional to the rate of characteristic events
inversely proportional to the rate of initiating events
Reasenberg & Jones w/
Characteristic Clustering
Summary
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 < 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.”