A Physical Basis for Time Clustering of Large Earthquakes

Abstract

We develop a theory that links stress interaction between earthquakes and the occurrence of temporal clustering. Coseismic static stress change in the vicinity (50 km) of large earthquakes suggests that perturbations of 0.1 to 1 bars may affect the occurrence of other earthquakes. At larger distances, interactions also seem to exist: four M 8 earthquakes have occurred in Mongolia on distant faults (400 km) during the last century. Also, paleoseismic observations documenting much longer time periods display a time clustering of major events. We demonstrate with simple mechanical concepts that postseismic stress relaxation magnifies the coseismic stress change and has a major effect on fault interaction during the seismic cycle. In the simple case where two distant faults are coupled, the probabilistic occurrence of triggered earthquakes may increase dramatically due to long range postseismic coupling.

Full text

1685
Bulletin of the Seismological Society of America, 91, 6, pp. 1685–1693, December 2001
A Physical Basis for Time Clustering of Large Earthquakes
by Jean Che´ry, Se´bastien Merkel, and Ste´phane Bouissou*
Abstract We develop a theory that links stress interaction between earthquakes
and the occurrence of temporal clustering. Coseismic static stress change in the vi-
cinity (50 km) of large earthquakes suggests that perturbations of 0.1 to 1 bars may
affect the occurrence of other earthquakes. At larger distances, interactions also seem
to exist: four M8 earthquakes have occurred in Mongolia on distant faults (400 km)
during the last century. Also, paleoseismic observations documenting much longer
time periods display a time clustering of major events. We demonstrate with simple
mechanical concepts that postseismic stress relaxation magnifies the coseismic stress
change and has a major effect on fault interaction during the seismic cycle. In the
simple case where two distant faults are coupled, the probabilistic occurrence of
triggered earthquakes may increase dramatically due to long range postseismic
coupling.
Introduction
Since large earthquakes have been identified as an elas-
tic rebound of the crust around a fault (Reid, 1910), the
prediction of time and location of large earthquakes hasbeen
a major aim of the earth science community. Unfortunately,
numerous searches for geophysical precursors and extensive
modeling of earthquake mechanics have been unsuccessful
in predicting earthquake occurrence (Geller et al., 1997). In
contrast to volcanic eruptions, which occur at very specific
and identified sites often with clear precursors (i.e., Dvorak
and Dzurizin, 1997), active faults are spread over large
areas and seem to generate earthquakes in an unpredictable
manner. Even in a tectonic zone where detailed fault maps
exist, we always face a triple uncertainty about the next
earthquake: its location, its time, and its size. We study in
the following sections of this article the problem of earth-
quake timing by simulating earthquake occurrence on two
distant faults. We assume in this model that secular stress
accumulation on the faults and earthquake stress transfer
through the crust control the seismic cycle. We specifically
model the postseismic stress change and show that it is the
main factor responsible for earthquake time clustering. The
article is divided into five sections: (1) We first summarize
observations of earthquake-induced seismicity and earth-
quake time clustering. (2) We then present the principle of
earthquake interaction and estimate the magnitude of stress
drop during a large earthquake. (3) We describe a physical
model of the seismic cycle which incorporates local stress
drop on a fault and distant stress change between faults. (4)
We present a numerical device able to simulate earthquake
*Present address: Geosciences Azur, Sophia Antopolis, Valbonne,
France.
series on two coupled faults. (5) Finally, we perform a para-
metric study that shows under which conditions earthquake
time clustering takes place.
Earthquake Stress Change and Earthquake
Time Clustering
Two classes of observations support the idea that a me-
chanical coupling exists between distant earthquakes. First,
local and distant seismicity evolve after a large earthquake.
This change happens immediately after an earthquake and
may last for decades. Second, large earthquakes are some-
times clustered in time, suggesting that fault failure on one
fault may affect earthquake occurrence on another fault.
A large earthquake often induces microseismicity and
other earthquakes during the hours, days, or months follow-
ing the mainshock (Omori law); (see Scholz, 1990). This
effect is generally explained by an elastic stress change in
the crust, which leads to earlier failure at other potential
rupture zones. With the precise knowledge of fault geometry
and coseismic slip distribution, a stress change can be esti-
mated for any part of the crust (e.g., Chinnery, 1963; Okada,
1992). A positive shear stress change Dson a fault plane
should advance the time of the next earthquake on this fault,
and is known as the clock advance. In the case where the
normal stress change is neglected, the clock advance is sim-
ply given by ∆∆t=/
τσ

,where Dsis the shear stress change
and 
σ
the rate of secular stress accumulation onto the fault.
A large earthquake on a fault system may also inhibit
seismicity. A clear example is provided by the 1906 San
Fransisco earthquake, which was followed by the absence
of moderate and large earthquakes in Northern California

1686 J. Che´ry, S. Merkel, and S. Bouissou
until the end of 1970s (Ellsworth, 1990). This behavior is
likely due to sudden elastic stress release in the fault vicinity
(e.g., Jaume´ and Sykes, 1996; Harris and Simpson, 1998).
This stress release is followed by stress reloading due to
either viscous motions in the lower crust (Nur and Mavko,
1974) or creep on deep extension of the rupture surface(Sav-
age and Burford, 1973). The time constant associated with
this relaxation is found on the order of 30 yr (Thatcher,
1983). However, shorter times (80 days) have also been ob-
served by precise geodetic measurements following the1992
Landers earthquake (Savage and Svarc, 1997), suggesting
that the relaxation after an earthquake may be nonlinear.
Clock advance and stress release are observed in the
fault vicinity, typically within 10–50 km of the fault plane
for large (Mⱖ6.5) earthquakes. They are consistent with
positive (unclamping) static stress changes greater than or
equal to 0.1 bar (e.g., King et al., 1994). However, large
earthquakes seem to perturb seismicity at far larger dis-
tances. The best example of this distant triggered seismicity
is probably provided by the 1992 Landers earthquake, which
was followed immediately by a clear increase of microseis-
micity in remote zones (Hill et al., 1993). Such an increase
occurred at the Geysers geothermal area in Northern Cali-
fornia (Gomberg, 1996) at distance exceeding 700 km, 10
times the maximum fault rupture length. The corresponding
static stress change at such distances are less than or equal
to 0.001 bar (Hill et al., 1993). Because such small values
are likely unable to have caused triggering, seismic waves
interacting with crustal fluids may explain such long distance
interactions (Hill et al., 1993; Gomberg, 1996).
While instrumental and short-term historical seismicity
show distant earthquake interactions for a short fraction of
the seismic cycle, archeological and paleoseismic records
span several earthquake cycles. Earthquake occurrence on
large faults is often organized as temporal clusters. For ex-
ample, historical records on rapidly slipping faults, such as
the North Anatolian fault, show periods of great seismic ac-
tivity (967–1105; 1254–1784) followed by long quiescent
times, and a recent earthquake cluster that started in 1939
and has produced 12 M6.7 events until the 1999 Duzce
earthquake (e.g., Stein et al., 1997; Barka, 1999). Near the
Dead Sea fault, a record of lacustrine sediments exhibits
clusters (Marco et al., 1996) with marked peaks of multiple
events separated by a quiescent period of 20,000 yr. Also,
paleoseismic studies of the Central segment of the San An-
dreas Fault reveal temporal earthquake clustering (Grantand
Sieh, 1994).
Another clue of temporal clustering is provided by the
record of earthquake occurrence on groups of faults. In Mon-
golia and surroundings areas, four M8 earthquakes have
occurred on distinct faults during the last century (Baljin-
nyam et al., 1993). Preliminary paleoseismic studies of some
these faults have shown a recurrence time of a few thousands
of years (Schwartz et al., 1996). Also, 100-m offset of
80,000-yr-old alluvial fans along the Bogd fault, along
which the Gobi-Altay M8 earthquake occurred, leads to a
long term slip rate of 1.2 mm/yr (Ritz et al., 1995), consistent
with a time return of a few thousand yr, suggesting that the
four twentieth century earthquakes occurred after a long pe-
riod of quiescence. A similar clustering phenomenon appar-
ently happened in the New Madrid region in the United
States, where at least three M7–8 earthquakes occurred dur-
ing 1811–1812, and previously during approximately A.D.
1530 to A.D. 900, suggesting a recurrence time of about 500
yr (Tuttle et al., 1999). However, precise GPS observations
indicate a low strain rate in this area (Newman et al., 1999),
which rather supports the view that the recurrence time is
much longer, at least 2500 yr (assuming a model appropriate
for a plate boundary, which clearly New Madrid province is
not). The most impressive evidence for large scale earth-
quake clustering exists in an archeological compilation of
Nur and Cline (2000), which suggests that the collapse of
Bronze Age in the Eastern Mediterranean may have been
caused by numerous major earthquakes (so called earth-
quake storm) occurring in a 50-yr period.
We saw in this section that earthquake-induced stress
change is believed to activate or inhibit seismicity. Also,
large earthquakes display time correlation, suggesting a me-
chanical coupling through the crust. The aim of the next
sections is to quantify stress changes during the seismic cy-
cle and to propose a physical model of earthquake interac-
tion. In this article, we refer to three phases of the seismic
cycle: (1) The coseismic phase corresponds to the earth-
quake itself. The crustal motion is generally explained by a
sudden shear stress drop on the fault plane associated with
the onset of slip, which produces an elastic response of the
crust. (2) A postseismic phase follows in the case of large
earthquakes. This corresponds to a slowly decaying crustal
motion in the fault vicinity. (3) A relaxed phase lasts until
the next earthquake. During this phase, the fault only ex-
periences a secular loading due to relative plate motion,
which slowly builds stress until the failure limit.
Stress Interaction Principle and Coseismic
Stress Drop
In order to understand how an earthquake on a fault A
may activate an earthquake on a distant fault B, we define
the following variables at the time of an earthquake on fault
A(illustrated schematically in Fig. 1): (1) the local stress
drop Dsassociated with a large coseismic motion on fault
A; (2) the small remote static stress change ds caused by A
on B; (3) on fault B, the current stress s, the failure stress s
s
,
and the local rate of stress accumulation 
σ
. It is intuitive that
an event on Awill trigger an earthquake on Bif the remote
stress change ds is large enough that it brings the stress level
on Bclose to the static yield stress s
s
. Also, it appears that
the current stress level on Bmust be statistically close to s
s
if the local stress drop Dson Bis small. We first focus on
the expected magnitude of local stress drop after an earth-
quake. Local stress drop during a large earthquake is usually
estimated with a dislocation model that provides a relation

A Physical Basis for Time Clustering of Large Earthquakes 1687
u
∆τ
fault
trace
L
δτ
faultA
stress trigger
fault B
P
P’
W
x
y
z
O
Figure 1. Principle of stress coupling between two
seismogenic faults. Rupture occurs on fault A on a
vertical fault plane Oxz due to a regional loading of
the crust. The local stress drop Dsis controlled by the
rupture dimensions Land Wand the coseismic slip u.
Fault slip causes a small stress change dson the re-
mote fault B. This stress change evolves with time
due to a stress relaxation around the deep part of the
fault A.
between the mean displacement u¯, the mean stress drop ∆
τ
,
and a dislocation length D:
∆
τ
=⋅CG
u
D,(1)
where Cis a geometrical constant of a value close to 1 (Par-
sons et al., 1988), and Gis the shear modulus. Using the
scalar moment magnitude relation MGSu
0=⋅⋅,where Sis
the fault surface rupture area and assuming that G⳱3•
10
10
Pa the combination of these two equations and the hy-
pothesis that the dislocation is circular give an estimate of ∆
τ
in the range 10–100 bars, with a preferred value near 30 bars
(Kanamori and Anderson, 1975; Hanks, 1977). In fact, equa-
tion 1 provides a direct estimation of ∆
τ
if u¯ and Dare
known from local seismological and geodetical observa-
tions. As the Landers surface rupture approximates an elon-
gated rectangle of length Land width W(Fig. 1), this length
scale Dmay correspond to W, L, or a combination of both
(Scholz, 1982a,b). The range of Wand Ldepend on earth-
quake size. For intracontinental earthquakes larger than M
6, instrumental seismicity indicates that Wis between 10 and
15 km and is largely independent of the magnitude (Wells
and Coppersmith, 1994). The value of Lincreases with the
magnitude, with typical values of 10 km to 500 km for M
5.5 to M8 events, respectively. In the case of Landers earth-
quake, u¯ 2.5 m, L75 km, and W15 km. If Wis
chosen for length scale, the stress drop would be then of the
order of 50 bars. This value falls within the range given by
Kanamori and Anderson (1975). The other possible choice
is to take Lfor length scale as suggested by Scholz (1982a)
and Das (1982). In this case where Dis chosen as L, the
mean stress drop is only 10 bars. We will study in the next
section how the length scale used for stress drop estimate
may evolve along the seismic cycle.
Seismic Cycle Model and Postseismic
Stress Change
The previous calculations are only valid for elastic dis-
locations and are restricted to coseismic static motions. As
postseismic relaxation will modify the strain field and there-
fore the crustal stress in the years or decades following an
earthquake (Thatcher, 1983), there is a need to understand
how local and remote stresses evolve during the seismic cy-
cle. Postseismic surface motion is usually explained in two
ways (see Fig. 2). A first model (Savage and Burford, 1973)
assumes a thick elastic lithosphere of thickness Hlarger than
the fault width (HkW). Postseismic slip occurs on the deep
part on the fault. As a result, the long term motion simply
corresponds to a slip of two lithospheric blocks separated by
an infinite fault, without large deformation in the lithosphere.
A second model (Nur and Mavko, 1974) assumes a thin
elastic lithosphere of thickness Hequal to Wfloating on a
viscoelastic asthenosphere. In this case, postseismic motions
are due to a stress relaxation around the fault tip. The long-
term motion of the system corresponds to a slip between the
two thin elastic plates. These models have one main draw-
back: their behavior depends only on Wor Hand do not
incorporate the fault length L. Also, they contain a stress
singularity as the coseismic motion drops abruptly to zero
at the depth W.
We introduce a modified version of the thin lithosphere
model (Fig. 3c) where the fault is a rectangle L⳯W. We
assume that the base of the fault is embedded in the visco-
elastic layer (i.e., His smaller than W) similar to some seis-
mic cycle models of the San Andreas Fault (Li and Rice,
1987). Finally, we assume that the coseismic dislocation
gently decreases from u
0
to 0 between Hand Was suggested
by 2D mechanical models of the seismic cycle (Tse and
Rice, 1986; Huc et al., 1998). Local stress drop during the
cycle can then be controlled by L,H, and W. We assume
that the coseismic dislocation corresponds to a slip patch of
L⳯Wsize with an amplitude u
0
. We choose here L⳱70
km, W⳱15 km and u
0
⳱2.5 m, values that have been
used for Landers earthquake coseismic motion. The choice
of His more difficult, as this quantity cannot be inferred
from the coseismic motion. We choose here H⳱12 km,
but we will demonstrate that this choice is not important.
We compute the shear stress associated with the coseis-
mic slip of a 70 ⳯15 km rectangular patch embedded in
half space as shown in Figure 2c, using the formulation of
Okada (1992). The signal is computed on a profile PP⬘nor-
mal to the fault trace starting from the fault center (Fig. 1,
2c, 3a). Coseismic displacement rapidly decays with dis-
tance from the fault center (Fig. 3a), and horizontal fault-
parallel shear stress sdecreases by 50 bars at a 10 km depth
near the fault center (Fig. 3b). Coseismic stress change at

1688 J. Che´ry, S. Merkel, and S. Bouissou
WH
H = W
b. Thin lithosphere model
Viscous asthenosphere
Elastic lithosphere
infinite fault
H
W
plate
motion
Elastic lithosphere
H >> W
a. Thick lithosphere model
infinite fault
surface
WH
H < W
c. Modified model
elastic lithosphere
viscous asthenosphere
fault length L
PP’
Figure 2. Three seismic cycle models: (a) thick
lithosphere, (b) thin lithosphere, and (c) modified.
These models are loaded laterally by plate motion. (a)
The entire lithosphere is elastic. Coseismic motion u
occurs between depths 0 and W. Interseismic slip oc-
curs below depth W. (b) The elastic lithosphere is thin,
and postseismic relaxation occurs in the viscoelastic
asthenosphere. Models (a) and (b) have a slip singu-
larity at depth W. The only length scale controlling
the local stress drop of these models is W. (c) Modi-
fication of the previous models where the fault has a
finite horizontal length, L. Fault tip penetrates at depth
into the viscous layer as suggested by numerical ex-
periments (e.g., Huc et al., 1998), and uis constant
between 0 and Hand then smoothly decreases be-
tween Hand W. As a consequence, the local stress
drop is mainly controlled by Wduring the coseismic
motion (LkW) but is controlled rather by Lafter
the complete stress relaxation of the lower crust. Re-
mote stress drop is evolving and increases according
to this local stress relaxation.
distance larger than 50 km is close to zero. Display ofremote
stress change between 50 and 1000 km using a log
10
scale
(Fig. 3c) shows a values of 0.045 bar at 150 km. Therefore,
the ratio between absolute values of remote and local stress
change after the coseismic motion is on the order of 10
ⳮ3
at 150 km from the fault. Such a small ratio suggests a low
probability that a large earthquake triggers a distant earth-
quake unless the affected fault was already near failure. If
we assume that the fault strength evolves from 50 bars to 0
during a cycle of 200 yr, the remote stress change corre-
sponds to a clock advance of 10
ⳮ3
⳯200 yr 73 days.
Postseismic motion can be explained by afterslip on the
deep part of the fault plane (e.g., Savage and Prescott, 1978;
Savage and Svarc, 1997; Reilinger et al., 2000). However,
viscoelastic relaxation below the seismogenic layer can also
cause similar surface motion (e.g., Deng et al., 1998; Pollitz
et al., 2000). The equivalence between the two formulations
has been demonstrated by Savage (1990), at least for the
calculation of horizontal displacements (Deng et al., 1998).
For the purpose of reasoning, we assume that no postseismic
slip occurs on the fault rupture plane. Therefore, only stress
release in the viscoelastic part at depth should cause post-
seismic displacements (Nur and Mavko, 1974; Huc et al.,
1998). Although stress diffusion of the lithosphere can delay
the stress transfer from the fault zone to the far field (see
Cohen, 1999), we also assume for simplicity that postseismic
relaxation below the fault zone corresponds to a uniform
exponential change of the stress. Therefore, all stress com-
ponents within the viscoelastic part fall to 0 after more than
two or three relaxation times of the medium. The relaxed
phase that follows the transient relaxation phase should dis-
play any shear traction on the xⳮyplane at the base of the
elastic plate (depth Hon Fig. 2c). Because this plate has
been loaded by a pure strike-slip coseismic motion that is
constant with respect to depth, the relaxed stress state (i.e.,
the xⳮycomponent) in the elastic part is depth invariant.
The earthquake model after the postseismic phase simply
corresponds to a platelike dislocation with scale L(L-model
as defined by Scholz [1982a]). With regard to this symmetry,
we can conclude that the relaxed stress in the elastic plate is
no longer controlled by Wbut rather by L, which is the
characteristic length of the model after the end of the post-
seismic phase. As has been done for other long-term stress
change modeling (e.g., King et al., 1994), we approximate
the stress for the relaxed phase with an elastic model con-
taining a deep dislocation (Savage, 1990) with using the
Okada (1992) formulation by setting WkL.
The local stress drop is only ⳮ10 bars after relaxation
(Fig. 3b), This value is close to the one given using equation
(1). Also, the remote stress at 150 km dramatically rises from
0.045 bar to 0.45 bar due to the disappearance of the length
W(Fig. 3c). The ratio between remote and local stress
change after postseismic relaxation is on the order of 5 •
10
ⳮ2
, 50 times greater than the coseismic value. The cor-
responding clock advance at 150 km is now on the order of
10 yr for a recurrence time of 200 yr.

A Physical Basis for Time Clustering of Large Earthquakes 1689
To summarize, the coseismic stress change is mainly
controlled by the smaller length, W, and the relaxed post-
seismic stress change is controlled by the larger length, L.
Because the postseismic behavior may significantly affect
the clock advance between distant faults, we propose a sim-
ple model that mimics these stress interactions during the
seismic cycle.
A Numerical Model of Earthquake Generation
Earthquake generation by the means of mechanical
models of coupled faults has already been studied (e.g.,
Lomnitz-Adler and Perez Pascual, 1989; Harris and Day,
1993; Senatorsky, 1997), but these studies do not account
for large postseismic stress changes. As we demonstrate that
these changes greatly enhance the efficiency of long-range
interaction, we incorporate coseismic and postseismic stress
evolution in a simple physical model of seismic cycle on
two distant faults.
The simplest earthquake model consists of a frictional
mass Mattached to an elastic spring of stiffness K(Fig. 4a).
Although such a model is a poor representation of the geo-
metrical properties of an earthquake, it provides insight into
the relationship between kinematical variables like the co-
seismic motion uand the plate velocity v, which can be
directly observed, and stress variables, which are only ac-
cessible through a mechanical model. The velocity vrepre-
sents the work of relative plate motion. The frictional slider
accounts for fault friction. If the velocity is constant, and if
the slider abruptly slips by an amount u, the recurrence time
Tis simply T⳱u/v. If u⳱5 m as typical for very large
earthquakes, we find T⳱200 yr for v⳱25 mm/yr. Such
values are typical of the San Andreas Fault. This implies that
if uand vare known, the mechanical properties as Kor the
slider friction lare not needed to compute the recurrence
time. However, urandomly varies for a real spring and fric-
tional mass even for a constant velocity. Because the stiff-
ness Kcan be considered constant, the variability of coseis-
mic motion and recurrence time may be understood in terms
of a friction law (Scholz, 1998). For sake of simplicity, we
assume a very simple friction law in which the yield stress
has a static yield value s
s
and a dynamic value s
d
, giving the
following expression for coseismic motion:
uK
sd
=−2( ) .
ττ
(2)
The stress drop is 2(s
s
ⳮs
d
). Assuming that s
s
is con-
stant and known and that s
d
evolves in a unknown manner
from one earthquake to another leads to the stress evolution
illustrated in Figure 4b. This model is referred as “time pre-
dictable”because the knowledge of the stress at a time t
allows the prediction of the time of the next earthquake (Shi-
mazaki and Nakata, 1980).
We now modify this model to account for postseismic
stress evolution. This is done by adding a viscoelastic body
having stiffness K
W
and relaxation time t
r
(Fig. 4c). The
stress state on the fault at the end of viscoelastic relaxation
is controlled by the stiffness K
L
. We also take into account
a small triggering stress coming from other faults. Four
forces control the motion of the point Pon fault A: the upper
crustal stress f
L
, the viscoelastic stress f
W
, the frictional stress
s, and a small triggering stress d
s
coming from a distant fault
B. The force balance on fault Ais given by
ff Mu
LW
+++=⋅
τδτ
,(3)
where u¨ is the acceleration of the mass M. During the inter-
seismic phase, we have  
uu s
== <0,| |
ττ
(the static yield
stress). The stress evolution of the system is given by
ft ft t
LL
A
() ( ) ,=+⋅
0

σ
(4)
ft fte
WW
Att t
Ar
() ( ) ,
()/
=−−
00(5)
δτ δτ δτ δτ
() ( )( ),
()/
te
co tt t post co
Br
=+− −
−−
10(6)
where ds
co
is the coseismic remote stress change, ds
post
is
the postseismic remote stress change, 
σ
is the secular stress
rate, and t
r
is the relaxation time of the damping zone. tA
0
and tB
0are the time occurrence of last earthquakes on A and
B, respectively. f
L
Ⳮf
W
Ⳮds evolves until s
s
, which cor-
responds to the beginning of coseismic motion at time tA
1.
The force evolution in the springs is then given by
ft ft K u
LL
A
L
() ( ) ,∝−⋅
1(7)
ft ft K u
WW
A
W
() ( ) ,=−⋅
1(8)
(8) and the coseismic motion of Pis given by
uKK
sd
LW
=⋅−
+
2( )
.
ττ
(9)
We now explore with a numerical model how two de-
vices (shown in Fig. 4c) interact when they are coupled with
the remote stress change ds.
Earthquake Clustering and Clock Advance
Mechanical parameters of our model are adjusted to
mimic the behavior of two distant faults capable of gener-
ating M7 earthquakes. The interfault distance is chosen as
150 km, and the coseismic motions vary randomly from1.77
and 3.23 m. We adjust K
L
and K
W
to have a local stress drop
evolving from ⳮ50 to ⳮ10 bars during the postseismic
phase. The remote stress change ds increases from 0.045 to
0.45 bar during this time, as shown in Figure 3. Two values
of 3 and 30 yr are used for t
r
, and 
σ
is set to 0.1 and 0.01
bar/yr in order to produce recurrence times Tof 200 and
2000 yr, respectively. The loading velocity is applied until
10,000 events have been generated. In order to detect earth-
quake clustering, we consider earthquake occurrence on
faults A and B as part of the same time series. Elapsed times

1690 J. Che´ry, S. Merkel, and S. Bouissou
0.0
0.5
1.0
0 50 100 150 200 250
-40
-20
0
0 50 100 150 200 250
10-4
10-3
10-3
10-2
10-1
100
101
102
102103
∆τ postseismic
distance (km)
bars
coseismic
slope -2
postseismic
δτ
P’
P
a. Displacements
meters
coseismic
postseismic
∆τ coseismic
bars
distance (km)
b. Stress changes
c. Log variation of stress changes
δτ
slope -3
distance (km)
150 km
Figure 3. (a) Coseismic and postseismic motion
calculated for a 5-m horizontal motion on a fault plane
having an horizontal length L⳱70 km and a depth
W⳱15 km. Values are computed following Okada
(1992) on a profile normal to the fault plane (PP⬘)in
Figures 1 and 2c. Postseismic motion corresponds to
a stress relaxation at depth, which is simulated by
setting Wto a large value. (b) Stress change at 10 km
depth. Local stress change Dt(distance ⳱0) evolves
from 50 to 10 bars between the coseismic and post-
seismic phase. Coseismic stress change scales is
mainly controlled by W, while postseismic stress
change is controlled by L. ds values (large distance)
are not distinct from 0. (c) Log
10
stress change at 10-
km depth. This reveals the distinct decay of coseismic
and postseismic stress change as a function of dis-
tance from the fault. The arrow reflects the postse-
ismic stress transfer ds at 150 km to the fault, which
corresponds to an increase from 0.045 bar to 0.45 bar.
These values are used for, respectively, dsco and
dspost in our fault interaction model. Slope values
indicate that the postseismic stress change scales with
1/R
2
in the far field, while the coseismic stress change
scales with 1/R
3
. As a consequence, the ratio between
these quantities is 10 at a distance R⳱150 km and
reaches 100 at R⳱1000 km.
t
e
between two successive earthquakes are counted using 10-
yr class intervals and then normalized by the total number
of elapsed times (number of events ⳮ1). Therefore, we ob-
tain histograms of the 10-yr probability of occurrence of an
earthquake in this two fault system (Fig. 5). Such histograms
are used to detect earthquake time clustering: if faults are
not coupled, there is no reason to have an accumulation of
short elapsed times with respect to large elapsed times.
Therefore, the histogram of elapsed times should be flat until
a decay for t
e
T. If faults are coupled due to postseismic
relaxation, the accumulation of short elapsed times should
correspond to a peak in the related histogram.
We first run a case where T⳱200 yr, which typically
corresponds to the recurrence time in rapidly deforming
zones such as the San Andreas fault system. We choose t
r
⳱30 yr such as suggested by postseismic motion associated
with the 1906 earthquake (Thatcher, 1983). The flat aspect
of the recurrence histogram (Fig. 5a) indicates that no mutual
triggering between faults occurs. We now decrease t
r
to 3
yr. This value is compatible with the viscosity used by Deng
et al. (1998) for the postseismic motion of Landers earth-
quake. This a lower bound with respect to studies of Li and
Rice (1987) and Kenner and Segall (1999) that proposed
values of 10–16 yr for the relaxation time of the San Andreas
fault following the 1906 earthquake. This small value results
in a peak for the time [0–10 yr]. We then run a case where
T⳱2000 yr, which is more typical of a slowly deforming
intracontinental fault system. The probability of occurrence
clearly increases by a factor of 4 for t
r
⳱30 yr and by a
factor 10 for t
r
⳱3 yr with respect to the probability for a
normal distribution.
These four examples of stress coupling show that an
increase in postseismic stress change strongly controls the
probability of obtaining two events separated by a short pe-
riod of time on distant faults. This behavior is favored by a
short relaxation time that rapidly leads the fault system to a
stress state close to instability as suggested by Figure 4b and
d. Moreover, a longer individual recurrence time (that cor-
responds to a slower loading rate) still enhances earthquake
time clustering (Fig. 4c, d). This last finding can be heuris-
tically explained using the clock advance concept, which
predicts that the time of a triggered earthquake is advanced
by ∆
τσ
/if the stress change is positive. In the case where
only 
σ
changes between two models, which is the case in
Figure 5b and d, the clock advance is longer when the sec-
ular stress rate is slower. Although this clock advance cal-
culation is not relevant during the postseismic phase because
the stress change is not linear with time, it is meaningful
during the relaxed phase which undergoes only a constant
stress rate. Consequently, this larger clock advance occurs
on a large part of the interseismic phase. This clock advance
effect favors distant triggering of earthquakes and therefore
leads to a clustered seismic cycle.
On the basis of this simple physical model, we suggest
that the interaction between numerous faults coupled by
postseismic stress may lead to generalized earthquake clus-

A Physical Basis for Time Clustering of Large Earthquakes 1691
τs(yield)
τd(slip)
K
v
v
τ
τ
a)
b)
c)
d)
L
δ
{
τ
τ
tr
time
τs
time
τs
post
P
τ
W
f
f
∆τ
relaxed
Figure 4. (a) A simple earthquake model: trep-
resents the continous plate loading, and Krepresents
the elastic stiffness of the portion of the crust. (b)
Variable stress drop, slip, and recurrence time occur
due to a random variation of dynamic stress sd. (c)
Modified model incorporating the relaxation of the
lower crust, which behaves like of a Maxwell body
with a relaxation time tr. This model simulates the
evolution of the local stress drop Ds.(d) Nonlinear
evolution of local stress in the viscoelastic model.
Note that the relaxed stress becomes close to yield ss
after postseismic stress relaxation. The stress rate dur-
ing the relaxed phase is equal to .˙r
tr = 30 yrs
a
0.00
0.05
0.10
Probability
0 100 200 300
tr = 3 yrs
T = 200 yrs
b
0.00
0.05
0.10
Probability
0 100 200 300
c
0.00
0.01
0.02
0.03
Probability
0 100 200 300
time (yrs)
T = 2000 yrs
d
0.00
0.02
0.04
0.06
Probability
0 100 200 300
time (yrs)
Figure 5. 10-yr probabilities of occurrence of M
7 earthquakes for a system made with two faults sepa-
rated by 150 km. Statistics are compiled for 10,000
events for slip values randomly varied (see text). (a)
Histograms corresponding to 200 yr for the individual
mean recurrence time. The normal probability of oc-
currence is 0.05, corresponding to a noncoupled
earthquake model. (b) The probability doubles if a
short relaxation time is used. (c) and (d) Histograms
corresponding to 2000 yr for individual mean recur-
rence times. The probability of occurrence is 4 times
greater than that expected for a normal distribution if
a relaxation time of 30 yr is used and is 10 times
greater for a short relaxation time of 3 yr. Temporal
clustering is very likely in this last case, with a prob-
ability as high as for the short recurrence time model
of Figure 5a (T⳱200 yr; tr ⳱30 yr).
tering as a chain reaction referred to as an earthquake storm
by Nur and Cline (2000). The timescale of this reaction may
be controlled by the relaxation time of the viscoelastic part
of the system as suggested by the decay in the histogram of
Figure 5c. This timescale probably depends on the viscosity
of the lower crust (e.g., Deng et al., 1998) or possibly of the
mantle (Pollitz et al., 2000). A long quiescent period then
follows the cluster. This behavior dramatically contrasts
with the classical recurrence time model (Shimazaki and
Nakata, 1980), which was proposed to explain the variation
of the recurrence time in rapidly deforming subduction do-
mains. Our physical model would be more relevant to ex-
plain the emergence of clustered earthquakes within slower
deforming zones such as the western United States or Mon-
golia. For a specific tectonic province, the emergence of
clustered phenomenon should be controlled by specific val-
ues of L,W,H,u,T, and t
r
. The value of rupture length L
has a specific importance because it imposes a relaxed stress
stress closer to the yield stress value. Therefore, the post-
seismic interaction between large earthquakes such the ones
of Mongolia between 1905 and 1957 should produce signifi-
cant stress change despite the large distances between faults.
Because values of W,H, and useem to have smaller varia-
tions than L, they could not be responsible for dominant
changes of the earthquake system behavior. By contrast, in-
dividual recurrence time Tlargely varies on faults (i.e., 250
yr for the San Andreas fault and thousands of years for Mon-
golian faults). On the basis of Figure 5, it seems that the
clustered aspect of the seismic cycle is dependent of on the
ratio T/t
r
. If we assume that two tectonic zones like the San
Andreas fault system and Mongolia have the same mechan-
ical properties, the slower deforming zone (Mongolia)
should produce more clustered sequences than the rapidly
deforming one. However, the clustering of earthquake series
is dependent from the postseismic relaxation time t
r
of the
lithosphere, which is not known in slowly deforming areas

1692 J. Che´ry, S. Merkel, and S. Bouissou
like the New Madrid seismic zone or in Mongolia. On the
basis on our model, the geodetic assessment of this time
constant appears mandatory for a better understanding of
time clustering of large earthquakes.
Acknowledgments
This article benefited from useful comments from Jean-Franc¸ois Ritz.
Ben Holtzman also provided numerous grammatical and syntax improve-
ments. Reviewers Joan Gomberg, Ruth Harris, and Liz Hearn made fruitful
comments and suggestions that greatly improved the manuscript.
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Laboratoire de Ge´ophysique
Tectonique et Se´dimentologie
Universite´de Montpellier II
34095 Montpellier, France
(J.S., S.B.)
Carnegie Institution
5251 Broad Branch Road
Washington, D.C.
(S.M.)
Manuscript received 1 December 2000.