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EPS130 – Strong Motion Seismology Laboratory 2
Probability of Occurrence of Mainshocks and Aftershocks
Introduction. In the previous laboratory we learned about the particularly well behaved
statistics of the earthquake magnitude distribution. As we saw it is possible to use the
frequency of event occurrence over a range of magnitudes to extrapolate to the less frequent
large earthquakes of interest. How far this extrapolation may be extended depends upon a
number of factors. It is certainly not unbounded as fault dimension, segmentation, strength
and frictional properties will play a role in the maximum size earthquake that a fault will
produce. Paleoseismic data is used to provide a better understanding of the recurrence of the
large earthquakes of interest. The large earthquakes have greater fault offset, rupture to the
surface of the Earth and leave a telltale geologic record. This record is used to determine the
recurrence of the large characteristic earthquakes and probabilistic earthquake forecasts.
Finally, this type of analysis is perhaps one of the most visible products of earthquake hazard
research in that earthquake forecasts and probabilities of aftershock occurrence are generally
released to the public.
Objective. In this laboratory we will assume a Poisson distribution to determine the
probability of events based on the Gutenberg-Richter recurrence relationship. Given the
statistical aftershock rate model of Reasenberg and Jones (1996) we will forecast the
probability of occurrence of large aftershocks for the 1980 Livermore valley earthquake
sequence. For the Mojave segment of the San Andreas Fault we will compare probability
density models to the recurrence data and use the best fitting model to determine the 30-year
conditionally probability of occurrence of a magnitude 8 earthquake. In order to complete this
laboratory you will need a computer and spread sheet program to analyze the data provided.
Exercise 1. The simplest model to assume is that of random occurrence. In fact when you
examine the earthquake catalog it does in fact appear to be randomly distributed in time with
the exception of aftershocks and a slight tendency of clustering. The Poisson distribution is
often used to examine the probability of occurrence of an event within a given time window
based on the catalog statistics. A Poisson process occurs randomly with no “memory” of time,
size or location of any preceding event. Note that this assumption is inconsistent with the
implications of elastic rebound theory applied to a single fault for large repeating earthquakes,
but is consistent with the gross seismicity catalog. The Poisson distribution is defined as,
pd( x ) 
u x e u
x!
The probability of one or more events (x1) can be shown to be,
p(x  1
) 1.0  e  u
In this application u is the product of the annual rate,  (number/time), and the interval time, t.
p( x  1)  1.0  e   t .
This function describes the probability of 1 or more events in the time interval t relative to the
average annual rate of occurrence =N (where N=number/year).
Using the Poisson model estimate the probability of a magnitude 5 earthquake in a given
week, month, year and 5 year period using the annual rate determined from the GutenbergRichter relationship below.
Log(N)=3.17-0.793M (Greater SF bay area)
Compare the estimated probability of a magnitude 7.0 earthquake for the same time periods.
Compare the recurrence interval for a magnitude 8 (north coast SAF event) from the
Gutenberg-Richter relationship above and that derived assuming that a characteristic
earthquake averages 450 cm of slip and that the loading rate is 1.9 cm/year. Discuss the
importance of the assumed recurrence interval in the forecasting of future large earthquakes.
Exercise 2. The Poisson probability function above may also be used to determine the
probability of one or more aftershocks of given magnitude range and time period following
the mainshock. Typically an estimate of the probability of magnitude 5 and larger earthquakes
is given for the period of 7 days following a large mainshock This aftershock probability
estimate is found to decay rapidly with increasing time. Reasenberg and Jones (1989) studied
the statistics of aftershocks throughout California and arrived at the following equation
describing the rate of occurrence of one or more events as a function of elapsed time for a
generic California earthquake sequence:
rate(t, M )  10 ( 1.670.91*( Mm M )) * (t  0.05) 1.08
This equation describes the daily rate of aftershock production as a function of time after the
mainshock (t) with magnitude Mm, and the aftershock magnitude (M). Elements of both the
Gutenberg-Richter relationship and Omori’s Law are evident in the above equation.
The Poisson probability of 1 or more aftershocks with a magnitude range of M1 < M < M2,
and time range t1 < t < t2 is:
M 2t 2
P( M1, M 2, t1, t 2)  1.0  e
1*
  rate( t , M ) dtdM
M1 t1
Verify that the above relationship is the correct form for P using the P(AB) identities.
The January 24, 1980 Livermore Valley (latitude 37.83o, longitude -121.81o) magnitude 5.8
earthquake has just occurred. The phone is ringing off the hook and the people of Livermore
are demanding to know if this is the end of it or whether there may be other damaging
earthquakes. To provide this information to them use the aftershock production equation and
the Poisson model to estimate the likelihood of one or more magnitude 5 and larger
(potentially damaging) aftershocks in the next 7 days beginning with the elapsed time of 0.1
day. By the end of day two how much has the probability of occurrence of a magnitude 5+
aftershocks decayed?
Compare the estimated probabilities and the observed outcome for the Livermore Valley
sequence. How generic is the Livermore Valley sequence?
The community is also concerned about the chances of an event larger than the mainshock
occurring. The statistics compiled by Reasenberg and Jones take this into account.
Immediately following the Livermore Valley earthquake what is the probability that an event
greater than the mainshock will occur.
Use the results of the exercise to draft a press release to the public that expresses the
likelihood of damaging aftershocks and the occurrence of a larger earthquake in lay terms, but
also conveys the uncertainty in the estimate.
Exercise 3. The following figure and table gives the years of great magnitude 8 earthquakes
on the Mojave segment of the San Andreas fault deduced from the paleoseismic record at
Pallet Creek (e.g. Sieh, K., Stuiver, M. and Brillinger, D., 1989). Determine the mean and
standard deviation of the event interval times.
Pallett Creek Earthquakes
2000
Date (AD)
1500
1000
500
0
0 1 2 3 4 5 6 7 8 9 101112
Event Num ber
Date (AD) Interval Time (yrs)
1857
1812
1480
1346
1100
1048
997
797
734
671
529
Given the event interval times compare them to the Gaussian and Lognormal probability
density models. To do this make a histogram with bins from 1-49, 49-99, etc. The center dates
of the bins will be 26, 76, 126, etc. The probability density models are defined below. They
depend on the mean interval recurrence time (Tave), the standard deviation to the mean (),
and the random variable (u) in this case elapsed time [the memory-less Poisson distribution
excepted].
Gaussian Distribution
 ( u  Tave ) 2
pd ( u) 
e
2 * 2
 2
Log-Normal Distribution
 ln( u / Tave ) 2
e 2 ( / Tave)
pd (u) 
( / Tave)  u  2 
2
Which probability distribution model appears to best fit the data?
Exercise 4. In this problem we will determine the 30-year probability of occurrence of a
magnitude 8 earthquake based on the Pallet Creek recurrence data and the best fitting
probability density model determined in exercise 3. The probability that an event will occur
within a given time window is simply the definite integral over that time window of the
probability density function. Note that the Gaussian and lognormal probability density
functions are normalized to unit area. We are interested in the 30-year probability beginning in
1999 given the time since the previous event (Te). The probability that the event occurs in the
given time window is:
T  T
P (Te  T  Te   T )  T e
pd (u)du ,
e
where T is the length of the forecast window. Determine the variation in probability with
different elapsed time for a T=30 yrs. You will note that probability in any give 30 year
window is small but is greatest near the mean of the distribution.
The next step is to find the probability that the event will occur in the window, T with the
condition that it did not occur before Te. This effectively reduces the sample space and results
in the following normalization for the conditional probability.
Te  T
P(Te  T  Te   T T  Te ) 
Te
pd (u)du
T
10
.  0 e pd (u)du
Estimate the 10-year, 20-year and 30-year probabilities for the Mojave segment event using
your estimates of Tave, , and Te=142 years (time since 1857). Compare these estimates with
those obtained using the Poison model.
Finally, stimate the change in the 30-year probability if the event does not occur next 10 years.