Recurrence Times for Parkfield Earthquakes: Actual and Simulated
Paul B. Rundle, Donald L. Turcotte, John B. Rundle, and Gleb Yakovlev
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Abstract In this paper we compare the sequence of recurrence times for
Parkfield earthquakes on the San Andreas fault with the results of
simulations. The coefficient of variation (aperiodicity) of the actual
earthquakes is 0.378. The coefficient of variation of the simulated
earthquakes is 0.354. We conclude that this variability of the recurrence
times of characteristic earthquakes can be attributed to fault segment
interactions. Our simulation results support the applicability of the
Weibull (stretched exponential) distribution to recurrence time statistics
rather than either the lognormal or the Brownian passage time
distributions. The applicability of the stretched exponential distribution for
recurrence times to high-order chaotic systems has been demonstrated by a
variety of simulations.
Introduction
Earthquakes are generally classified as foreshocks, main shocks, or aftershocks,
but this distinction is fuzzy at best. There is also the question of characteristic
earthquakes, which are defined to occur on major faults and to be quasi-periodic. It is
assumed that characteristic earthquakes are the result of near-constant tectonic driving
stress. Standard examples are earthquakes on the northern and southern locked sections
of the San Andreas fault. Recurrence times for great earthquakes on the southern locked
section have been given by Sieh et al. (1989) using paleoearthquake studies. These
authors fit their data to a Weibull distribution with a mean µ = 155 years and a coefficient
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of variation (aperiodicity) Cv = 0.70. The sequence of m ≈ 6 earthquakes on the Parkfield
segment of the San Andreas fault has often been cited as a classic example of
characteristic earthquakes (Bakun and Lindh, 1985).
Slip on the Parkfield section of the San Andreas fault occurred during M ≈ 6
earthquakes that occurred in 1857, 1881, 1901, 1922, 1932, 1966, and 2004 (Roeloffs and
Langbein, 1994). It should be noted, however, that there are uncertainties concerning the
location of early events and the possible occurrence of other Parkfield earthquakes
(Toppozada et al., 2002). For the discussion given in this paper, we will accept the
validity of the sequence of earthquakes given above.
On the basis of the Parkfield earthquakes up to 1966, Bakun and Lindh (1985)
estimated that the recurrence intervals at Parkfield were 22±3 years and that the chances
were 95% that the next earthquake would occur by 1993, but it finally occurred in 2004.
This work provided the basis for the Parkfield Earthquake Prediction Experiment
(Roeloffs and Langbein, 1994). However, in carrying out their study Bakun and Lindh
treated the 1934 event as an early failure and excluded it. Savage (1991, 1993) pointed
out that this exclusion seriously biased their recurrence statistics.
Recurrence statistics at Parkfield
The cumulative probability distribution (cdf) of Parkfield recurrence times (t = 12,
20, 21, 24, 32, 38 years) is given in Figure 1. The mean, standard deviation and
coefficient of variation (aperiodicity) of these recurrence times are µ = 24.5 years,
σ = 9.25 years, and Cv = σ/µ = 0.378 respectively.
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Prior to the 2004 Parkfield earthquake many authors had considered the statistical
variability of the earthquakes. Davis et al. (1989) considered the statistical variability of
the Parkfield sequence in terms of both Poissonian and lognormal distributions. Kagan
(1997) considered the null hypothesis of random (Poissonian) earthquakes at Parkfield.
Because of the small number of earthquakes in the sequence he argued that the null
hypothesis could not be rejected. Ellsworth et al. (1999) have carried out extensive
modeling of the recurrence statistics for Parkfield earthquakes. They considered the
Gaussian lognormal, Weibull, Brownian passage time, and exponential distributions.
They favored the Brownian passage time distribution based on its physical motivation.
Ben-Zion et al. (1993) attributed the variability of recurrence times at Parkfield to
residual effects of the great 1857 earthquake on the southern San Andreas fault. They
argued that the increase in recurrence times is essentially an aftershock decay sequence
and carried out numerical simulations to quantify the effect.
Statistical Distributions
A variety of probability distribution functions (pdf’s) have been used to correlate
earthquake recurrence statistics. Extensive discussions have been given by Sornette and
Knopoff (1997) and by Matthews et al. (2002). Although we will consider several
distributions, we will emphasize the Weibull (stretched exponential) in this paper. The
reason for this emphasis will be given in later sections.
The pdf for a Weibull distribution of interval times is given by (Patel et al., 1976)
β⎛t⎞
p(t ) = ⎜ ⎟
τ ⎝τ ⎠
β −1
⎡ ⎛ t ⎞β ⎤
exp ⎢− ⎜ ⎟ ⎥
⎣⎢ ⎝ τ ⎠ ⎦⎥
(1)
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where β and τ are fitting parameters. The mean µ, standard deviation σ, and the
coefficient of variation Cv of the Weibull distribution are given by
⎛
µ = τ Γ⎜1 +
⎝
1⎞
⎟
β⎠
(2)
1
2 2
⎞
⎛ ⎛
2⎞ ⎡ ⎛
1 ⎞⎤
σ = τ ⎜ Γ⎜ 1 + ⎟ − ⎢ Γ⎜1 + ⎟ ⎥ ⎟
⎜
β ⎠ ⎣ ⎝ β ⎠⎦ ⎟
⎝ ⎝
⎠
(3)
1
⎤2
⎡ ⎛
2⎞
⎥
⎢ Γ⎜1 + ⎟
σ ⎢ ⎝ β⎠
− 1⎥
Cv = =
2
⎥
⎢
µ
⎛
⎞
1
⎛
⎞
⎥
⎢ ⎜ Γ⎜1 + ⎟ ⎟
⎥⎦
⎢⎣ ⎜⎝ ⎝ β ⎠ ⎟⎠
(4)
where Γ(x) is the gamma function of x. The cumulative distribution function (cdf) for a
Weibull distribution is given by
⎡ ⎛ t ⎞β ⎤
P(t ) = 1 − exp ⎢− ⎜ ⎟ ⎥
⎣⎢ ⎝ τ ⎠ ⎦⎥
(5)
where P(t) is the fraction of the recurrence times that are shorter than t. Taking the value
of the mean and coefficient of variation for the Parkfield sequence discussed above, we
find from equations (2) and (4) that the corresponding fitting parameters for the Weibull
distribution are τ = 27.49 yrs and β = 2.88. Using these values the cdf from equation (5) is
also shown in Figure 1. Quite good agreement is found.
Simulation studies
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Savage (1994) has argued convincingly that actual sequences of characteristic
earthquakes on specified faults or fault segments are not sufficiently long to establish
reliable recurrence time statistics. Using numerical simulations, Ward (1992) found that
at least 10 earthquake cycles need to be observed to constrain recurrence intervals to
±10%. In order to overcome this lack of data several numerical simulations of earthquake
statistics on fault systems have been carried out.
In this paper we will utilize results using Virtual California (Rundle, 1988;
Rundle et al. 2004). This model is a topologically realistic numerical simulation of
earthquakes occurring on the San Andreas fault system. It includes the major strike-slip
faults in California and is illustrated in Figure 2. Virtual California is composed of 650
fault segments each with a width of 10 km and a depth of 15 km. The fault segments
interact with each other elastically utilizing elastic dislocation theory.
Virtual California is a backslip model. The accumulation of a slip deficit on each
segment is prescribed using available data. The mean recurrence time of earthquakes on
each segment is also prescribed using available data to give friction law parameters. The
fault interactions lead to complexity and statistical variability. Details of Virtual
California simulations have been given by Rundle et al., 2004.
Our emphasis in this paper will be on the Parkfield segment, but first we will give
results for the northern San Andreas fault (Yakovlev et al., 2005). The pdf for 4606
simulated earthquakes with M > 7.5 on this section is given in Figure 3. The mean
recurrence time is µ = 217 years, the standard deviation is σ = 114.7 years, and the
coefficient of variation is Cv = 0.528. Also included are the corresponding pdf’s for the
Weibull, lognormal, and Brownian passage time distributions. In each case the model
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distributions have the same mean µ and standard deviation σ as the simulation results. It
is clear that the Weibull distribution with τ = 245 yrs and β = 1.976 is in better agreement
with the simulation results than either the lognormal or Brownian passage time
distributions. For the southern San Andreas fault the mean recurrence time was found to
be µ = 196 years and the coefficient of variation was Cv = 0.556. Paleoseismic studies of
paleoearthquakes on the southern San Andreas fault (Sieh et al., 1989) also gives seven
intervals of m = 7 + earthquakes at Pallett Creek. These intervals have a mean
µ = 155 yrs, and a coefficient of variation Cv = 0.70 in reasonably good agreement with
the simulations.
Numerical simulations of earthquake statistics have also been carried out by Goes
and Ward (1994) and Ward (1996, 2000). Goes and Ward (1994) give the results of a
100,000 year simulation of earthquakes on the San Andreas system and found that
Cv = 0.50 – 0.55 for the northern section of the San Andreas fault for earthquakes with
m > 7.5. The two simulations differ in many ways with different faults considered,
different frictional and mean slip velocities, and different numerical approaches. Yet the
variability of interval times as quantified by the coefficients of variation is very similar.
We believe it is reasonable to conclude that this variability is robust and is not sensitive
to details of fault-interaction simulations.
We now turn to the Parkfield section of the San Andreas fault. In Figure 4 we
show a 1000 year simulation of earthquakes on the Parkfield and adjacent sections of the
San Andreas fault. The slip in each earthquake is shown as a function of distance along
the fault measured from the northwest to southeast. The Parkfield section extends from
30 km to 50 km. The southeastern end of the creeping section of the fault extends from 0
7
to 30 km, and the northwestern end of the locked southern section of the San Andreas
fault extends from 50 km to 245 km. An example of a great earthquake on this section is
shown at t = 55 years. The cdf for simulated earthquakes from the Parkfield section of the
San Andreas fault with M > 6 is given in Figure 5. The mean recurrence time is
µ = 22.8 years, the standard deviation σ = 8.09 years, and the coefficient of variation is
Cv = 0.354. The coefficient of variation of the simulated Parkfield earthquakes
(Cv = 0.354) is in remarkably good agreement with the coefficient of variation of the
actual Parkfield earthquakes (Cv = 0.378). It should be emphasized that while the mean
recurrence time is a data input, the variability of recurrence times is a consequence of the
complex interactions between fault segments.
Also included in Figure 5 are the cdf’s for the Weibull, lognormal, and Brownian
passage time distributions. In each case the model distributions have the same mean µ
and standard deviation as the simulation results. Although the simulation results are in
better agreement with the Weibull distribution than the others, there is considerably
larger deviation than for the simulations for the northern San Andreas fault. We attribute
this difference to the small size and few degrees of freedom (2) of the segments
associated with the Parkfield earthquakes relatively to the northern San Andreas fault.
The maximum magnitude earthquake in the simulations is about M ≈ 5.8 when only one
segment slips which occurs less than 1% of the time.
Discussion
A measure of the variability of recurrence interval times on a fault segment is the
coefficient of variation Cv (the ratio of the standard deviation σ to the mean µ). For
8
strictly periodic earthquakes on a fault or fault segment we would have σ = Cv = 0. For
the random (exponential, no memory) distribution of interval times we would have
Cv = 1, (σ = µ). Typically we find Cv’s in the range 0.3 to 0.75. The coefficient of
variation of the six intervals between Parkfield earthquakes is Cv = 0.378, towards the
low or near periodic end of this range. Ellsworth et al. (1999) analyzed 37 series of
characteristic earthquakes and suggested a provision generic value of the coefficient of
variation of Cv = 0.5. However, actual sequences of characteristic earthquakes on a fault
are not sufficiently long to establish reliable statistics (Savage, 1994; Ward, 1992). In
order to overcome this difficulty numerical simulations of earthquakes on the San
Andreas fault system can be used.
In this paper we report on simulations of earthquakes on the San Andreas fault
system utilizing the Virtual California model. For the Parkfield section of the San
Andreas fault our simulations gives a sequence of recurrence times with a coefficient of
variation Cv = 0.354. This is remarkably close to the Cv = 0.378 value found for the actual
sequence. We wish to emphasize that this variability is a consequence of fault segment
interactions. Without these interactions the characteristic earthquakes would be periodic
(Rundle, 1988).
It is now generally accepted that earthquake occurrence on a fault system is an
example of deterministic chaos. The importance of chaotic interactions between the
Parkfield segment and the southern locked segment was demonstrated by Huang and
Turcotte (1990). They used a simple two block, slider-block model. A small block
representing the Parkfield segment and a large block representing the southern locked
section. They showed that the behavior of this model was a classic example of
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deterministic chaos. A direct consequence of this chaotic behavior was a variability in the
recurrence times for slip events on the simulated Parkfield segment.
An important question is whether our simulations provide interval statistics
similar to those on the actual faults. Evidence that this is in fact the case comes from the
similarities between our simulations and the simulation results obtained by Goes and
Ward (1994) and Ward (1996, 2000). The statistical results they obtained using the
SPEM model are quite similar to those reported here using the Virtual California model.
The two simulation models have many differences in details yet the statistical
distributions of recurrence times are quite similar. The principal common feature of both
models is the use of elastic dislocation theory for the stress interactions. This would
indicate that recurrence interevent time scaling may be a universal feature of regional
seismicity in the sense that Gutenberg-Richter frequency-magnitude statistics are
universal.
Another important question is whether there is a scaling of seismicity that requires
the validity of the Weibull distribution. The Weibull distribution is widely used in
engineering to model the statistical distribution of failure times. Its applicability has been
demonstrated by many actual tests (Meekes and Escobar, 1991). In Figure 3 we have
given the pdf of 4606 simulated recurrence times for earthquakes on the northern San
Andreas fault using our Virtual California model. It is seen that the fit of the simulation
data to a Weibull distribution is considerably better than the fit to either the lognormal or
Brownian passage time distributions. We have also shown that the distribution of
recurrence times for both the actual and simulated Parkfield earthquakes are fit quite well
with the Weibull distribution.
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An important difference between the three distributions considered here is their
asymptotic behavior for large recurrence times. The basic question is whether the waiting
time ∆t until the next earthquake increases or decreases as the time since the last
earthquake t0 occurred increases. For the lognormal distribution ∆t increases as t0
becomes larger. For the Brownian passage time distribution, ∆t becomes constant
(Poissonian) for large t0. For the Weibull distribution with β > 1 ∆t decreases as t0
becomes large. For a fault that is continuously being loaded tectonically a decrease in ∆t
would appear to be required (Sornette and Knopoff, 1997). The Weibull is the only one
of the three distributions that satisfies this condition.
Statistical physicists have shown wide applicability of the stretched exponential
(Weibull) distribution for both experimental data and for simulations. This applicability is
found in heterogeneous and homogeneous nucleation, in this context it is known as
Avrami’s law (Avrami, 1940). The similarities between nucleation problems (i.i. droplet
formation in supercooled steam) and the nucleation of earthquakes have been discussed
previously (i.e. Rundle, 1989; Rundle et al. 2003). The sequence of recurrence times on a
fault is a time series. Bunde et al. (2003) and Altmann et al. (2005) have shown that time
series that exhibit long range correlations have interval-time statistics that satisfy
stretched exponential (Weibull) distributions. Specifically Bunde et al. (2003) argue
against distributions in which the extreme event statistics become Poissonian, i.e. like the
Brownian passage time distribution.
Acknowledgments
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This work has been supported by DOE Grant DE-FG02-04ER15568 (GY, JBR,
PBR) and NSF Grant ATM 0327558 (DLT).
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submitted to Bull. Seis. Soc. Am.
Center for Computational Science and Engineering, University of California,
One Shields Ave. Davis, CA 95616 (P.B.R., J.B.R., G.Y.)
Department of Geology, University of California, One Shields Ave. Davis,
CA 95616 (D.L.T.)
15
Figure captions
Figure 1. Cumulative distribution of recurrence times of Parkfield earthquakes. The
discontinuous line is the distribution of the six actual recurrence times. The continuous
line is the best-fit Weibull distribution.
Figure 2. Faults segments making up Virtual California. The model has 650 fault
segments, each approximately 10 km in length along strike and a 15 km depth.
Figure 3. The probability distribution function (pdf) p(t) of recurrence times t for 4606
simulated earthquakes on the northern San Andreas fault with m > 7.5. Also shown are
the corresponding pdf’s for the Weibull, lognormal, and Brownian passage time
distribution.
Figure 4. Illustration of simulated earthquakes on the Parkfield and adjacent sections of
the San Andreas fault. The slip is given as a function of the distance along the fault for
each earthquake over a 1000 year period.
Figure 5. Cumulative distribution of recurrence times of simulated Parkfield earthquakes.
Also shown are the corresponding cdf’s for the Weibull, lognormal, Brownian passage
time distributions.
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Figure 1.
17
Figure 2.
18
Figure 3.
19
Figure 4.
20
Figure 5.
21