# GP130 Strong Motion Seismology Laboratory 2

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,

The probability of one or more events (x1) can be shown to be,

In this application u is the product of the annual rate, (number/time), and the interval time, 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 Gutenberg-Richter 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:

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:

Verify that the above relationship is the correct form for P using the P(AB) 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.

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

Log-Normal Distribution

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:

,

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.

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.