Simulated aftershock sequences for a M 7.8 earthquake on the

Southern San Andreas Fault

Karen Felzer

U.S. Geological Survey

525 S. Wilson

Pasadena, CA 91106

**1 Introduction**

Aftershock activity constitutes one of the largest risks in the
aftermath of an earthquake. Aftershocks shake already weakened
structures, and if an aftershock is closer to a population center than
the original rupture it may cause even more severe local shaking. The
1992 M 6.4 Big Bear aftershock, for example, which occurred several
hours after and 40 km to the west of the 1992 M 7.3 Landers mainshock,
caused substantially more damage to the city of Big Bear than the
Landers earthquake. Even more damaging was the August 22, 1952 M 5.8
Bakersfield aftershock of the M 7.5 Kern county earthquake, which
occurred about a month after the mainshock. Due to the proximity of the
aftershock to Bakersfield and the weakened condition of the buildings,
this aftershock killed 2, injured 35, and caused \$10 million in
property damage. More recently the 2008 May 12 M 7.9 Sichuan Province,
China earthquake has been followed, as of August 1, 2008, by 5
aftershocks that caused significant additional injuries, fatalities
and/or major damage.

Given the danger posed by aftershocks it is important to model what type
of aftershock sequence might follow the next large earthquake in
southern California. The largest earthquake to threaten southern
California or the "Big One", is often presumed to be a M \~8 on the
southern San Andreas fault, and a statewide simulation exercise, called
ShakeOut, is being held in California in November 2008 to practice
response to such a quake. Here I present ten different random
simulations of the first week of aftershocks that could accompany the
earthquake being modeled for the preparation exercises. One of these
simulations will be used for the actual exercises.

No physics is used in the modeling here since aftershock physics are
both very complex and controversial. Instead the aftershocks are
generated stochastically using established empirical relationships for
the distribution of aftershock magnitudes, times, and locations. In
addition, each aftershock generates its own aftershocks (secondary
aftershocks), an important process which occurs in real sequences
(Felzer et al. 2003). This type of statistical simulation is known as
ETAS (Epidemic Type Aftershock Sequences) modeling (Ogata 1998). Only
one week of aftershock activity is simulated because we do not expect
the simulation exercises to last longer than this, and because this will
be the most intense period of seismicity. It is important to note,
however, that a mainshock this size is expected to produce some
aftershocks for years, even decades, and that an aftershock produced at
any time may be large (Lomnitz 1966). Thus the long-term aftershock risk
should be kept in mind as the simulation exercises are brought to a
close.

**2 Method**

We simulate the aftershocks for the ShakeOut Scenario earthquake with
the version of the ETAS model developed by Felzer et al. (2002), with
the addition of an aftershock distribution in space. First we simulate a
set of primary, or direct, aftershocks produced by the mainshock over a
duration of one week. For this exercise the mainshock is modeled as 22
rectangular subfaults, each with their own strikes and dips. The
aftershocks of the direct aftershocks are then generated, and
aftershocks of these aftershocks, etc. until no new earthquakes are
produced within the seven-day time period. In order to keep the number
of calculations reasonable we only include aftershocks that are at least
M 2.5. Earthquakes smaller than this certainly exist and produce
aftershocks, but we found that M 2.5 is small enough to make the
simulations realistic while keeping the calculations tractable. The
parameter values used for the simulation are adjusted for this minimum
magnitude.

The aftershock rate as a function of time is given by the modified Omori
law (Utsu 1961) expressed in the following form (Reasenberg and Jones
1989; Felzer et al. 2004),

$n(t) = k\text{10}^{(M_{\text{main}} - M_{\text{min}})}(c + t)^{- p}$
(1)

where *n* is the rate of aftershocks larger than or equal to *M~min~*,
*M~main~* is mainshock magnitude, *t* is time, and *k*, *c*, and *p* are
constants. Note that the *k*, *c*, and *p* parameters used for the ETAS
model must be for direct aftershock sequences, not for the complete
sequences made up of direct plus secondary aftershocks. Our best-fit
direct modified Omori Law parameters for California are *p*=1.34, *c* =
0.95 days and *k*=0.008 where *k* is in units of the number of
aftershocks ≥ M~main~ produced per day (for details of the parameter
solution see Hardebeck *et al.* (2008)).

The magnitude of each simulated aftershock is chosen randomly from the
Gutenberg-Richter magnitude frequency distribution, which gives that
*N*, the number of earthquakes larger than or equal to magnitude *M*, is
equal to

$\text{log}(N) = a - \text{bM}$ (2)

(Gutenberg and Richter 1944). Here *b* is a constant that we set equal
to 1.0, and *a* is a constant that varies with the total number of
aftershocks. The law is also truncated for the purposes of the
simulation such that no aftershocks larger than M 8 are allowed. Very
large aftershocks increase the simulation run time substantially. It
should be noted, however, that there is an approximately 4% probability
that an M 7.8 mainshock could trigger an M \> 8 aftershock.
Theoretically the largest possible aftershock that could be triggered is
equal to the magnitude of the largest possible earthquake that could
occur in California.

The distribution of aftershocks in space is modeled using the equation
of Felzer and Brodsky (2006), which states that the aftershock density
*(r)* decays with distance from the nearest point on the mainshock fault
plane, *r*, as

$\rho(r) = \text{cr}^{- n}$ (3)

where *n* = 1.37 ± 0.1 at 98% confidence for southern California when
*(r)* is given in 1D (linear aftershock density -- see Felzer and
Brodsky (2006)). All M ≥ 5.5 earthquakes in the simulation are modeled
as extended fault planes. The strike of these planes is assigned to be
parallel to the strike of the nearest portion of the nearest major
California fault. Fault dip is set randomly between 60 deg and 90 deg
(reflecting that many southern California faults are strike-slip or
thrust), and fault dimensions are taken from the magnitude-area
relationships of Wells and Coppersmith (1995). Mainshocks M \< 5.5 are
modeled as point sources. Aftershocks are allowed to be up to 1000 km
away from their mainshock, but are rarely generated at such large
distances due to the inverse power law decay of aftershock density. To
keep the model simple, extra remotely triggered earthquakes are not
assigned to volcanic or geothermal areas, although it has been observed
that distant triggering may be more energetic in such regions (Hill et
al. 1993).

Finally we note that the SAF scenario mainshock has a unilateral rupture
from south to north, and that several studies have shown that triggered
earthquakes tend to be more prevalent in the direction of rupture
(Gomberg et al. 2003). Exactly how much more prevalent, though, is
difficult to quantify since there are few well-constrained mainshocks
with known unilateral rupture. We roughly infer from inspecting the
aftershocks of the fairly unilateral 1992 M 7.3 Landers and 1999 M 7.1
Hector Mine earthquakes that it

may be appropriate to place 30% of the aftershocks within 15 degrees of
the rupture direction of each fault segment, with the remainder randomly
assigned to other azimuths. We do this in our simulations, with the
result that a slightly higher portion of aftershocks end up in the SAF
Big Bend area and offshore of Central and Northern California than to
the south. Note that this one part of the simulation is rather ad hoc as
it is based on little quantitative data, however the affect on the
earthquake distribution is minimal, simply adding to the aftershock
sequence some of the directionality which has been observed in other
sequences.

The SAF scenario mainshock also specifies different amounts of slip on
the different fault segments, which may be important for aftershock
locations. A robust empirical relationship between the amount of slip
and the numbers and locations of aftershocks generated has not yet been
established, however. So we do not apply any mainshock slip-dependent
aftershock density variations here, pending future research.

**3 Results**

Not surprisingly, the majority of the aftershocks in all of the
sequences occur near the main fault trace. Communities commonly affected
by local M ≥ 5.5 aftershocks include Palm Springs, San Bernardino,
Coachella, Wrightwood, Cathedral City, Lancaster, Palmdale, Desert Hot
Springs, Mentone, Mecca, and Indio. Looking at some of the larger
communities near the fault, 6 out of the 10 simulations produced one or
more M ≥ 5.5 aftershocks within 20 km of the center of San Bernardino
(population 198,000), and 5 out of the 10 produced one or more such
aftershocks within 20 km of the center of Redlands (population 70,000).
In addition, all of the scenarios include a minimum of 376 M ≥ 4
earthquakes in the first week, and a minimum of 241 M ≥ 4 on the first
day. The main feature of the results, however, is a strong degree of
variability between the different sequences. The tamest of our
sequences, for example, has only one M≥6 aftershock in the first week
whereas the most active has 13. The average magnitude of the largest
aftershock in each sequence is M 6.9, but the largest aftershock in
individual sequences varies from M 6.4 to M 7.7. The total number of M ≥
5 aftershocks ranges from 30 to 92, and the total number of M ≥ 3.0
ranges from 3812 to 8380. This variability results from the
non-stationary Poissonian process model used to choose the exact timing
and magnitude of each aftershock from the empirical statistical
distributions (e.g. see Felzer et al. (2002)), and from the positive
feedback of the secondary aftershock triggering process. If a sequence
starts out somewhat less active than average, for example, few secondary
aftershocks will be triggered, resulting in an even lower activity
level, whereas a somewhat more active initial sequence or larger than
usual aftershock will lead to more and more secondary aftershock
generation. Variability is a well known feature of real aftershock
sequences in California; compare the 1990 M~W~ 5.7 Upland Earthquake,
for example, with a maximum magnitude aftershock of M 4.72 and 25 M ≥ 3
aftershocks in the first week with the anemic aftershock sequence of the
1988 M~L~ 5.02 Pasadena earthquake which had a maximum magnitude
aftershock of M 2.57 and only 6 M ≥ 2 earthquakes over the first 7 days.
Local geology, stress, and variation in mainshock characteristics may
account for some of the variation between sequences, but these
simulations demonstrate that identical mainshocks and initial parameters
can also result in a wide range of aftershock outcomes.

Another important result is that in addition to the clearly significant
aftershock risk to communities immediately adjacent to the San Andreas
fault, significant aftershocks occasionally happen at greater distances.
In one of the simulated scenarios, for example, an M 6.95 occurs east of
Sacramento, near the Sierra Nevada, and in another an M 7.2 rips along a
parallel trend to the Sierra Madre fault strongly affecting the San
Gabriel Valley, a densely populated region containing over 40
municipalities and about 2 million people. There is clear precedent for
such triggering of distant aftershocks by large San Andreas earthquakes;
within two days of the 1906 San Francisco San Andreas earthquake distant
aftershocks occurred in or near the Imperial Valley, Pomona Valley,
Santa Monica Bay, western Nevada, and western Arizona (Meltzner and Wald
2003) and shortly after the 1857 Ft. Tejon earthquake additional
earthquakes were felt in the Northern California cities of Martinez,
Benecia, Santa Cruz, San Juan Batista, San Benito, and Mariposa (Townley
and Allen 1939). Overall 4 out of our 10 simulations had one or more M ≥
5 aftershocks triggered somewhere north of the central California city
of Parkfield. Closer to the mainshock but still removed from the
immediate fault trace, half of the simulations produced a M ≥6
earthquake within 50 km of the city of Pasadena, where efforts to
collect and catalog earthquake data are centered. None of the
simulations produced a M ≥ 6 within 50 km of the center of Los Angeles,
but by extrapolating from the rate of M ≥ 2.5 occurring in this area we
estimate a 1% probability of such an event. A short tabulation of the
largest earthquakes in each sequence and the communities most affected
is given in Table 1. Average aftershock densities (tabulated in 25 by 25
km bins) are mapped in Figure 1.

It is also important to compare our simulated sequences with known
aftershock sequences of similar mainshocks. The last major San Andreas
Fault earthquake to occur in southern California was the January 9, 1857
M 7.9 Ft. Tejon earthquake, which ruptured southwards from Parkfield to
San Bernardino. A contemporary report, written by Mr. Barrows for
publication in the San Francisco Bulletin and marked "Los Angeles,
January 28, 1857", reported, "We had at Los Angeles five or six shocks
during the same day and night and within about eight days time we had
twenty shocks -- some violent, some light." (Wood 1955). Meltzner and
Wald (1999), estimating the magnitudes of the aftershocks from
historical reports, found M 6.25 and M 6.7 aftershocks in the first 8
days, occurring near the southern end of the rupture, and a later M 6
near San Bernardino and M 6.3 near Parkfield. The simulated scenario 6
came out quite similar to this historic sequence, with with one M 6.2
and one M 6.7, both close to the mainshock fault trace, occurring in the
first week. The other sequences ranged around this activity rate.

We can also compare our simulation results with the aftershock
production from another modern continental strike slip earthquake, the
2002 M 7.9 Denali earthquake in Alaska, which is perhaps one of the best
instrumentally recorded analogs of the simulated ShakeOut Southern San
Andreas event. The time series and a map view of the simulated
aftershocks are compared against the Denali aftershocks in Figure 2. The
comparison is hampered, however, by the fact that the local activity of
the Denali aftershock sequence was very low -- the largest aftershock
produced was M 5.8, whereas on average we would normally expect a
largest aftershock of *M~main~* -- 1.2 = M 6.7 (Båth 1965). By chance
none of the ten simulations run here came out with a productivity rate
this low.

**4 Conclusions**

Our stochastic ETAS simulations indicate that a wide variety of
aftershock sequences could accompany the next M \~7.8 earthquake on the
southern San Andreas fault. Some of our ten simulated sequences are
similar to the aftershock sequence of the 1857 Ft. Tejon earthquake in
terms of a similar number and magnitude of M≥6 shocks; some are less
active, and a few are much more active. One simulation contains an M 7.7
aftershock -- nearly as large as the original mainshock. Most simulated
large aftershocks

affect near-fault communities, with San Bernardino being the largest
municipality with a high probability of damaging aftershock activity.
One simulation, however, has a distant M 6.95 aftershock located to the
east of Sacramento, and another has an M 7.22 along the trend of the
Sierra Madre fault, which could be very damaging to a number of San
Gabriel Valley communities, including Pomona and Pasadena.

The diversity of the aftershock sequences produced, even though the same
mainshock rupture and aftershock production parameters were used for
each simulation, results from the randomness of the process and the
positive feedback that occurs during secondary aftershock production.
Given the inherent variability of the aftershock production process, it
is critical to be prepared for the possibility of an active and damaging
aftershock sequence after the next Big One -- and for the possibility of
large aftershocks in unexpected places.

**Acknowledgements** I am grateful to M. Page, E. Pounders, and an
anonymous reviewer for thoughtful and thorough reviews and to A. and L.
Felzer for editing and assistance. I thank L. Jones and K. Hudnut for
inviting me to do the aftershock simulations for the ShakeOut scenario
and K. Hudnut for providing the data for the mainshock rupture planes.
Thank you also to everyone who has been doing such hard and quality

work for the ShakeOut scenario exercises.

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**Table and Figure Captions**

**Table 1**

Summary of the results from the ten aftershock simulations. The first
column gives simulation number, the second the largest aftershock
produced, the third the day of this largest aftershock (Day 1 is the day
of the mainshock), fourth is the total number of M ≥ 5.5 aftershocks
over the 7-day simulation, and the final column gives some of the
communities expected to be most affected by the largest aftershock.

**Figure 1**

The logarithm of the average (mean) number of M ≥ 2.5 aftershocks
produced at different locations by the ten simulations. Aftershocks are
counted in 25 by 25 km bins. The logarithm is used so that regions with
both high and low aftershock rates can be seen without saturation of the
color map. Bright spots of activity at distant locations from the fault
occur where a large aftershock has occurred in a single simulation and
produced many secondary aftershocks.

**Figure 2**

This figure provides the time series (rate of aftershocks with time
after the mainshock) for the ten simulations and a map of the
aftershocks produced in Simulation 4. Both the time series and the maps
are compared to the first week of aftershocks of the 2002 M 7.9 Denali
earthquake, a contemporary earthquake comparable in magnitude and
mechanism to the simulated ShakeOut southern San Andreas event. Because
the aftershock sequence of the Denali earthquake is complete to M 4 (as
estimated from comparison with the Gutenberg-Richter magnitude-frequency
relationship (Gutenberg and Richter 1944)), only M ≥ 4 earthquakes are
shown in both plots. (a) Time series of the ten simulated southern SAF
aftershock sequences (in gray) and the time series of the first week of
the Denali aftershock sequence (in black). Aftershock rate is given as
the number of M ≥ 4 aftershocks measured each quarter day (6 hours).
Note that the Denali sequence had a very low activity rate in comparison
to the world-wide average (see text). For the simulated aftershock
sequences, Simulation 4 had the fewest aftershocks (376 M ≥ 4 compared
to the 126 detected at Denali), so this simulation is overlain in map
view with the first week of M ≥ 4 Denali aftershocks in (b).

Table 1

  -------- ------------ ------------ ----------- ---------------------------
  **Sim.   **Lgst       **Day of     **No.       **Most affected cities**
  \#**     aftshck**    lgst.**      M\>=5**     

  1        6.95         4            20          Sacramento, Modesto,
                                                 Mariposa

  2        6.87         1            9           San Bernardino, Crestline

  3        7.09         3            14          Palmdale, Lancaster

  4        6.39         1            9           Lancaster

  5        6.75         1            21          Wrightwood

  6        6.73         1            10          San Bernardino, Yucaipa

  7        7.71         4            30          Palm Springs, El Centro

  8        6.48         1            13          Little Rock, Lancaster

  9        7.28         2            24          Little Rock, Palmdale,
                                                 Lancaster

  10       7.22         1            23          San Gabriel Valley cities
  -------- ------------ ------------ ----------- ---------------------------