JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 108, NO. B10, 2498, doi:10.1029/2002JB002318, 2003
Foreshocks, aftershocks, and remote triggering
in quasi-static fault models
A. Ziv
Laboratoire de Géologie, École Normale Supérieure, Paris, France
Received 22 November 2002; revised 23 May 2003; accepted 23 June 2003; published 22 October 2003.
[1] We present the results of time-space analysis of synthetic catalogs, generated using a
quasi-static discrete fault model that is governed by rate- and state-dependent friction.
These analyses reveal that increase in the postmain shock earthquake production rate
occurs over an area surrounding the main shock rupture with dimensions that are several
times larger than the main shock dimensions. We show that the increase in seismicity
rate far from the main shock (where the static stress changes imposed by the main shock
were small) is a consequence of multiple stress transfers and that the very distant
aftershocks are not directly triggered by the main shock but that instead they are
aftershocks of previous aftershocks. Snapshots of the fault state at progressive times prior
to each main shock show that the regions which accelerated toward failure covered areas
that were much larger than the final size of the main shock ruptures. As a result, as
time approached the main shock time, the seismicity rate in these regions increased
while the B value of the earthquake size distribution decreased. It has been previously
suggested that remote seismicity rate increase could be triggered by the passage of seismic
waves. Remote triggering in a quasi-static model indicates that it is not necessary to
INDEX TERMS:
invoke such a ‘‘dynamic effect’’ in order to explain distant aftershocks.
1734 History of Geophysics: Seismology; 3220 Mathematical Geophysics: Nonlinear dynamics; 7209
Seismology: Earthquake dynamics and mechanics; 7230 Seismology: Seismicity and seismotectonics;
KEYWORDS: earthquakes, remote triggering, earthquake interaction, Omori law
Citation: Ziv, A., Foreshocks, aftershocks, and remote triggering in quasi-static fault models, J. Geophys. Res., 108(B10), 2498,
doi:10.1029/2002JB002318, 2003.
1. Introduction
[2] That earthquakes are clustered in time and space has
been recognized long ago. Nevertheless, many fundamental
questions related to earthquake clustering have remained
unresolved. Do foreshock sequences follow the same
empirical laws of aftershock sequences, what controls the
timescale over which they occur, and what determines their
spatial distribution are just some of the questions that we
would like to understand. Here we present the results of a
numerical study that aims to address some of these questions. Special attention is given to the issue of remote
aftershock triggering.
[3] Seismicity rate increase following large earthquakes
has been observed in sites that are located more than a
source length from the main shock centroids. For example,
the Mw = 7.3 Landers, California, earthquake, triggered
aftershock activity over an area that extends up to 1000 km
away from the main shock rupture [Hill et al., 1993, 1995].
Significant seismicity rate increase following the Mw = 7
Hector Mine, California, earthquake, has been observed as
far as 250 km to the south (in Cerro Prieto) and 400 km to
the north (in Long Valley) [Gomberg et al., 2001]. FollowCopyright 2003 by the American Geophysical Union.
0148-0227/03/2002JB002318$09.00
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ing the Mw = 7.4 Izmit, Turkey, earthquake, seismicity rate
increased in a region on mainland Greece extending from
400 km to nearly 1000 km away from the main shock
hypocenter [Brodsky et al., 2000]. While some of the remote
seismicity in California occurred in or in the vicinity of
volcanic regions, that on mainland Greece did not. Thus
the occurrence of remote aftershocks on mainland Greece
is neither attributable to the effect of shaking-induced
de-gassing in magma chambers [e.g., Linde et al., 1994],
nor to the effect of a stress corrosion acting on cracks that are
embedded in hot and wet rock [e.g., Brodsky et al., 2000].
[4] According to Dieterich’s [1994] aftershock model, the
seismicity rate change should scale with the amplitude of
the stress change. More than one source length away from
an earthquake centroid, the static stress change induced by
that event is not only a small fraction of the stress drop, but
it is also smaller than typical values of tidal stress changes.
The small size of static stress changes, the absence of clear
temporal correlation between tidal stresses and seismicity
[Emter, 1997; Vidale et al., 1998a, 1998b] and the belief
that 0.01 MPa is a lower threshold for earthquake triggering
[Reasenberg and Simpson, 1992], prompted a search for an
alternative triggering mechanism. This led to the idea that
dynamic stresses, since they decay slower with distance
than static stresses, may be the cause for remote triggering
[e.g., Anderson et al., 1994; Gomberg and Bodin, 1994]. An
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ZIV: REMOTE TRIGGERING
additional evidence for the role of ‘‘dynamic effect’’ in
earthquake triggering comes from the Landers and the
Hector Mine earthquakes, for which the asymmetries of
the peak dynamic stress pattern, due to the effect of rupture
directivity, and that of the aftershock distribution are similar
[Kilb et al., 2000, 2002; Gomberg et al., 2001].
[5] Gomberg [2001] demonstrated that various failure
models, including the rate- and state-dependent friction,
cannot explain the persistence of aftershock activity after
the passage of the seismic waves. Thus the occurrence of
remote aftershock activity is not yet understood. The main
objective of this study is to revisit the notion that remote
aftershock activity is triggered by dynamic stresses. This
problem is addressed numerically, using the quasi-static
discrete fault model of Ziv and Rubin [2003]. The strategy
is simple; if the aftershock activity in a quasi-static model
extends to the far field, the occurrence of remote triggering
is explainable without dynamic stresses.
2. Background
[6] Several aftershock models that employ constitutive
failure criteria have been proposed in recent years [e.g.,
Dieterich, 1994; Gomberg et al., 1998]. These models rely
on several simplifying assumptions. For example, in these
models only the effect of the main shock is accounted for,
while the effect of each aftershock on successive aftershocks is disregarded. Thus, although these models provide
important insights, their applicability to natural aftershock
sequences is not straight forward. In order to relax the
assumptions underlying these models we turn to numerical
calculations. We study simulated earthquake clustering in a
quasi-static fault model that is governed by a constitutive
friction law [Ziv and Rubin, 2003]. We shell see that the
spatial dimensions of aftershock sequences produced by this
model are much larger than that predicted by seismicity
models, which neglect elastic interaction among earthquakes. Below we provide some background that is necessary for the understanding of the model and for the
interpretation of the results.
2.1. Constitutive Friction Law
[7] It is believed that motion on geological faults is
mainly controlled by frictional resistance. Laboratory
experiments show that the friction coefficient m is a function
of slip rate d_ and fault state q as follow [Dieterich, 1979;
Ruina, 1983]:
_ q ¼ m þ A ln d=
_ d_ þ B ln q=q ;
m d;
0
*
*
Inspection of equation (2) reveals the existence of two
extreme conditions that a fault may experience during a
seismic cycle. Early in the seismic cycle, since the fault is
almost locked, the first term on the right-hand side of
equation (2) dominates. During this phase, the state
increases approximately as elapsed time and the fault is
undergoing strengthening. This phase of the cycle is
referred to as the ‘‘locked phase.’’ Later in the cycle, sliding
velocity increases to a point where the second term on the
right-hand side of equation (2) dominates. At that time, the
state may be approximated as decreasing exponentially with
slip [Dieterich, 1992] and the fault is undergoing weakening. This property of the evolution law gives rise to selfaccelerating slip prior to an instability that is concentrated
on a crack with dimension Lc that scales like G Dc/sB,
where s is the effective normal stress and G is the shear
modulus [Dieterich, 1992]. Thus this phase of the seismic
cycle is referred to as the ‘‘self-accelerating’’ phase. Since
the duration of the locked phase is much longer than that of
the self-accelerating phase, at any given time a given point
on the fault is much more likely to be in the locked phase
than in the self-accelerating phase.
2.2. Implications of Dieterich’s Aftershock Model
[8] The effect of a stress change is to modify the expected
time of the failure. Owing to the nonlinearity of the
evolution law, the change in the expected failure time
depends upon when in the seismic cycle the stress is
applied. Consequently, a stress step induced on a population
of faults, that is assigned an initially uniform distribution of
times to failure, gives rise to a modification of the earthquake production rate that relaxes with time. Dieterich
[1994] modeled exactly this, starting with a population of
faults that is already in the self-accelerating phase (i.e., the
evolution of the state may be approximated as decreasing
exponentially with slip). With this simplification, Dieterich
was able to obtain a simple expression relating timedependent earthquake production rate to a stressing history.
The ratio between the perturbed seismicity rate N_ and the
reference seismicity rate N_ 0 is given by
N_
1
¼
;
_
g
t_ 0
N0
where t_ 0 is a reference stressing rate for which N_ = N_ 0, and
g is a state variable for the fault population that evolves with
time and shear stress, t, according to
ð1Þ
dg ¼
where A and B are dimensionless constitutive parameters,
m0 is the nominal coefficient of friction, and d_ * and q* are
reference velocity and state, respectively. Various physical
properties of the fault surface evolve with slip and time.
The evolution of these properties is embodied in the state
parameter q. The following evolution law is adopted in this
study [Ruina, 1980]:
dq
qd_
¼1 ;
dt
Dc
ð2Þ
where t is time and Dc is a characteristic sliding distance for
the evolution of q from one steady state to another.
ð3Þ
1
½dt gdt:
As
ð4Þ
Equation (4) may be applied recursively in order to approximate the effect of an arbitrary complex stressing history.
Ziv et al. [2003] found that the main features of natural
aftershock sequences are explainable in terms of Dieterich’s
model. These include the decay of aftershock rate according
to the modified Omori’s law, and the independence of
aftershock duration on distance from the main shock.
[9] Can this model explain a significant seismicity rate
increase at distances from the main shock that are greater
than the main shock length? To calculate the instantaneous
seismicity rate change induced by a stress step we first
ZIV: REMOTE TRIGGERING
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14 - 3
employ equation (4) to evolve g through a stress step of t
to get
t
;
g ¼ g0 exp
As
ð5Þ
where g0 is the value of the state variable before the
application of the stress step, and in steady state it is equal
to 1/t_ 0. Replacing equation (5) into equation (3) and setting
g0 to be equal to its steady state value leads to
N_
t
:
¼ exp
As
N_0
ð6Þ
Notice the trade-off between t and As, and that it is t/As
which determines the magnitude of the seismicity rate
change. For a stress change of 0.01 MPa (which is of the
order of the static stress change at a distance of 5 rupture radii
from a penny-shaped crack with a stress drop of 1 MPa), a
midcrustal normal stress of 100 MPa and a laboratory value
for A of 0.01, the expected seismicity rate change is 1%.
Such a seismicity rate change is too small to be detected, and
therefore this model cannot explain remote triggering. This
exercise, however, does explain why it is so difficult to
resolve a statistically meaningful correlation between stress
changes smaller than 0.01 MPa and seismicity [Reasenberg
and Simpson, 1992].
[10] What is the effect of dynamic stress changes on
earthquake production rate? As in the work by Gomberg
et al. [1998], the stress perturbation that is caused by the
passage of the seismic waves is idealized as a positive stress
pulse with a shape of a boxcar. Using equation (4), the
effect of a pulse with a magnitude of t and a duration of
t that is superposed on a long-term stressing rate of t_ 0
may be calculated in four steps. For 0 < t < t, one finds
g¼
1
t_ 0
t
t
1 exp
þ1 ;
exp
As
ta
ð7Þ
and for t > t,
g¼
t
t
þ1
1 exp
exp
As
ta
t
t t
exp
þ1 ;
1 exp
As
ta
1
t_ 0
ð8Þ
where ta = As/t_ 0 is a characteristic relaxation time
[Dieterich, 1994]. The seismicity rate change is then
obtained by replacing g in equation (3). Equation (7)
applied for all times gives the response of g to a stress step
superimposed on a constant stressing rate, and leads to a
result that is identical to equation (12) of Dieterich [1994].
Examples for seismicity rate changes caused by stress
pulses of different amplitudes and durations are shown in
Figure 1. These show that the effect of a positive stress
pulse is to increase the seismicity rate during the pulse, but
depending on the pulse amplitude and duration, the seismicity
rate after the pulse may either immediately return to
background level, or it may first drop below background
level and return to it at later times. A similar result has been
obtained by Gomberg [2001], who concluded that since the
Figure 1. Log-log diagram of normalized seismicity rate as
a function of normalized time, showing analytical (lines) and
numerical (symbols) solutions for seismicity rate change due
to a stress pulse of a shape of a boxcar with amplitudes and
durations as indicated. Analytical rates are obtained by
evolving g according to equations (7) and (8) and
substitution into equation (3). In calculating the numerical
result, elastic interaction has been disabled. The open
symbols at the bottom indicate, for each stress pulse, the
lag time to the last aftershock that nucleated on a cell that was
in the self-accelerating phase at the beginning of the pulse.
rate- and state-dependent friction (and other constitutive
failure laws) cannot explain the persistence of aftershock
activity beyond the pulse duration, it cannot explain remote
triggering either.
[11] Since the model’s starting point is a population of
faults that is already in the self-accelerating phase, an
important assumption inherited in the model is that aftershocks nucleate over areas on the fault that were already
close to failure before the application of the stress. Additionally, the self-accelerating fault population is responding
only to an externally imposed load, while the stress changes
induced by every aftershock at the site of all later aftershocks
are not accounted for. Finally, while large earthquakes are
often preceded by foreshocks and other precursory phenomena, that occupy a volume with dimensions that may be
much larger than the final size of the main shock [Knopoff et
al., 1996; Bowman et al., 1998], Dieterich [1994] assumed
that aftershock sequences are preceded by constant seismicity. The numerical fault model described below is more
realistic, since the assumption underlying Dieterich’s [1994]
model are completely relaxed, and since it gives rise to
foreshocks and to a close to power law earthquake size
distribution.
3. Quasi-Static Discrete Fault Model
[12] Here we provide only a short description of the
model ingredients, for more details the reader is referred
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ZIV: REMOTE TRIGGERING
to Ziv and Rubin [2003]. The fault is modeled as a shear
crack that is embedded in a homogeneous infinite elastic
medium. The fault surface is represented by an array of
100 100 square computational cells that is periodic in
both Cartesian directions. Constitutive properties and
normal stress are uniform. Motion on the fault is driven
by slip imposed at constant rate at distance W on either
side of the fault. The model is inherently discrete, in the
sense that the size of the computational cell L is much
larger than the size of the critical crack Lc. It is this
discreteness which gives rise to slip complexity in the
model [Rice, 1993].
[13] A time stepping algorithm of Dieterich [1995] is
adopted, according to which the evolution of a seismic
element that is governed by equations (1) and (2) is
separated into three distinct phases. The three phases are:
the locked phase, the self-accelerating phase and the seismic
phase. The latter is assumed to be instantaneous and at
constant sliding speed. For each phase, approximations are
made that simplify the governing equations. Using these
equations, the transition time from one phase to another
may be calculated. At every time step, the transition time of
each cell to switch from its current phase to the next phase is
calculated. The simulation is then advanced by a time step
that is equal to the minimum of all transition times in the
model.
[14] Stress transfer due to coseismic slip is computed at
the center of the cell using the stress field of a uniform
square dislocation [Erickson, 1986]. In calculating the
interaction coefficients, we set Poisson’s ratio in the elastic
Green function to be equal to zero. With this choice of
Poisson’s ratio, the stress changes at equal distances ahead
of the mode 2 and the mode 3 fronts are equal. A
consequence of the time-advancing scheme is that a seismic
event always begins with a failure of a single cell. A rupture
may grow beyond the size of a single cell if the stress
change that is induced by that cell is large enough to
instantaneously bring one or more cells from their current
sliding speed to the seismic sliding speed. The subset of
cells that comprises the rupture set is determined through an
iterative procedure. Owing to the azimuthal independency
of the interaction coefficients, large ruptures have a general
tendency to become circular.
[15] The controlling frictional parameters are A/B and ta/tc.
The latter is a ratio between two timescales: ta is Dieterich’s
[1994] characteristic time for the return of seismicity rate to
the background rate following a stress step, and tc = Dc/d_ seis,
where d_ seis is the seismic slip speed, may be interpreted as the
average contact lifetime during seismic slip. Additional
controlling parameters are W/L, which measures the coarseness of the computational grid with respect to W, and a
prespecified range from which for each cell stress overshoot
is picked randomly.
[16] Initial conditions are assigned such that in the
absence of any earthquake interaction, earthquakes occur
at constant rate and at random location. Since one of the
effects of earthquake interaction is to increase the effective
stressing rate, earthquake production rate with interaction is
much higher than without. It is therefore necessary to let the
simulation run long enough, so that a steady state is
reached, i.e., seismicity rate that is more or less constant,
and an average cumulative slip that matches with the total
slip imposed. After this condition is met, event times,
locations, and sizes are recorded.
[17] Before proceeding, we perform a simple exercise that
helps to clarify the important differences between Dieterich’s
[1994] model and our numerical simulation. If elastic interaction is turned off, the only assumption of Dieterich’s model
that is being relaxed is that aftershocks nucleate on cells that
were already in the self-accelerating phase prior to perturbing
the stress. To examine the effect of relaxing this assumption,
we disabled stress transfer and imposed three boxcar stress
pulses with amplitudes and durations as in section 2. The
time-dependent seismicity rate change that is caused by these
pulses are shown in Figure 1. We find close match between
the analytical solution that employs equations (7) and (8) and
the simulation result. Later, we will show that the numerical
model, with stress transfers turned on, gives rise to a
stressing-seismicity relationship that is not predicted by
Dieterich’s [1994] model, and on the basis of this exercise
we will argue that these differences are due to the effect of
internal stress interactions.
[18] A consequence of not allowing for internal stress
interaction is that cells which at the start of the pulse were
in the self-accelerating phase fail before cells that were in
the locked phase at that time. The empty symbols at the
bottom of the diagram indicate, for each stress pulse, the
lag time since the beginning of the pulse and the rupture
of the last cell that was in the self-accelerating phase at the
start of the pulse. For example, for a pulse with an
amplitude of t/As = 7 and a duration of 0.1 ta all cells
that were in the self-accelerating phase at the beginning of
the pulse have ruptured within the first 0.01 ta. Even in
this case, for times greater than 0.01 ta, we find the
simulated and predicted seismicity rate changes to be in
good agreement. This shows that Dieterich’s self-accelerating approximation still holds if one starts with a population of faults with times to failure distributed evenly
throughout the seismic cycle.
4. Data Set
4.1. Synthetic Catalog
[19] The synthetic catalog contains 2.5 106 events and
was generated using B/A = 10, ta/tc = 1010, random stress
overshoot between 5% and 30%, and W/L = 10. A histogram of rupture areas (measured in number of cells) is
shown in Figure 2 (left). Only a couple of dozens of events
exceeded the size of 100 cells, with the largest rupture
occupying about 220 cells. It is possible to control the range
of the size distribution by changing B/A and W/L. From our
experience, however, increasing B/A beyond 10 and W/L
beyond 10 quickly leads to a rupture that is unstoppable. A
histogram of the log10 of the seismic moment is shown in
Figure 2 (right). Apart from a small number of events with
an abnormally small seismic moment release, the distribution of the logarithm of the seismic moment exhibits a close
to power law distribution that extends over more than 2
logarithmic units. Events with very small seismic moment
are the result of cells that failed twice within a very short
time interval, due to experiencing a very large stress
step during that interval. As a result, these cells did not
strengthen, and released abnormally small stress drop when
failed. The breakdown of the power law distribution for
small earthquakes in natural catalogs is often interpreted as
ZIV: REMOTE TRIGGERING
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Figure 2. Histograms of (left) rupture dimensions and (right) the logarithm of the nondimensional
moment. The subset of earthquakes that satisfies the main shock criteria is highlighted in black.
resulting entirely from completeness threshold of the
recording system. The result of our model suggest that
some of this effect may be due to the time-dependent
strengthening of the frictional resistance. Testing this possibility, however, may be very difficult and is beyond the
scope of this study.
[20] Time-space diagrams for events with rupture dimensions greater than 20 cells and greater than 50 cells are
shown in Figure 3. Each point on these plots corresponds to
a cell on a 100 100 grid whose position is projected onto
a single axis. The catalog contains more than 2500 events
that satisfy this size criterion, but here we show only a small
fraction of the catalog. This plot reveals great irregularity in
both the spatial distribution of these events, and the time
intervals separating them. A similar degree of irregularity
was found in catalogs produced with a fixed stress overshoot of 17.5% (which is exactly equal to the average
overshoot used here). In order to increase the irregularity,
it is necessary to increase both the average and the range of
the stress overshoot.
[21] In order to illustrate the spatiotemporal earthquake
clustering in the model, we plot in Figure 4 the distribution
of ruptures at various time intervals before and after one of
the largest model earthquakes. In this plot, a cell that did not
rupture during that time interval is assigned a white color, it
is assigned a grey shading if it ruptured once and dark grey
if it ruptured more than once. The edge of the main shock
rupture is indicated by the solid line. Before the main shock
time, the main shock rupture area was seismically inactive,
while at the same time regions adjacent to the main shock
rupture experienced high seismic activity (Figure 4a).
Shortly before the main shock time, seismic activity started
inside the main shock rupture (Figure 4c). Shortly after the
main shock, seismic activity increased both inside and
outside the rupture zone (Figure 4d). The spatial distribution
of aftershock activity outside the main shock rupture is very
irregular. Note that aftershocks did not occur in regions that
experienced intense seismic activity before the main shock
(indicated by arrows in Figure 4d). The aftershock activity
inside the main shock area was triggered by the aftershock
sequence outside that area (results shown in section 6.2
confirm this conclusion). This region, since it has ruptured
shortly before, did not recover its strength and released a
relatively small seismic moment during the aftershock
Figure 3. Time-space diagram for events with rupture area
(left) greater than 20 cells and (right) greater than 50 cells.
Each point on these plots corresponds to a cell on a 100 100 grid whose position is projected onto a single axis.
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Figure 4. Spatial distributions of ruptures at various time intervals before and after one of the largest
model earthquakes. White is assigned to cells that did not rupture during that time interval, grey is
assigned to cells that ruptured once, and dark grey is assigned to cells that ruptured more than once. The
solid line marks the edge of the main shock rupture.
sequence. After about 1 ta, the spatial distribution of the
earthquakes was uncorrelated with the main shock, and
seismic activity started to be localized in regions that are far
away from the main shock (Figure 4f ).
[22] The properties of earthquake clustering may vary
significantly from one sequence to another. In order to
obtain a more general and statistically meaningful view of
earthquake distribution before and after large earthquakes, it
is necessary to average the properties of earthquake clustering with respect to many main shocks. This approach is
explained below.
4.2. Stacking Procedure
[23 ] The simulated catalog was transformed into a
stacked catalog. In producing the stacked catalog we follow
these steps:
[24] 1. We define as a main shock an event that ruptured
at least 20 cells, and that released a seismic moment that
was by at least 0.5 logarithmic unit larger than that of all
other events which occurred during the time 1 ta that
preceded and followed it. The requirement that this event
will be the largest during the 1 ta before and after its rupture
time, intends to eliminate the possibility of treating as a
main shock an event that is either a foreshock or an
aftershock of a larger main shock. This is because the
change in seismicity rate induced by a stress step relaxes
almost entirely after 1 ta. Only 500 of 2.5 106 events in
the catalog satisfy these criteria. The subset of events that
was treated as main shocks is highlighted in black in the
histograms of Figure 2.
[25] 2. We treat as potential foreshocks or aftershocks
events that occurred within a certain time window that is
centered about the main shock time. Since we wish to
resolve the departure of the foreshock rate from the background rate as well as the return of the aftershock rate
toward the background rate, it is necessary to choose a time
window that is long compared to the foreshock and aftershock durations. The time window chosen for this study is
1000 ta. In order to ensure that the analysis is not affected
by the finiteness of the catalog, events that satisfied the
main shock criteria but occurred within less than half of the
time window from the start or the end of the catalog were
removed from the main shock list.
[26] 3. For each main shock, we calculate interevent
times and normalized distances R/C to all premain shock/
postmain shock events, where R is the distance between
centroids and C is the main shock radius (approximated as
the square root of its rupture area).
[27] 4. Finally, we bin the list of premain shocks and
postmain shocks according to normalized distances, and
stack those lists to get a foreshock-aftershock catalog.
[28] Interevent distances are normalized by the main
shock radius because the seismicity rate change at a given
point is expected to scale with the size of the stress change
that is imposed by the main shock at that point, and since
the decay of that stress change is a function of R/C. Because
ZIV: REMOTE TRIGGERING
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we would like to compare the spatial distribution of foreshocks with that of the aftershocks, and since we would like
to know if and how the distribution of foreshocks scales
with the main shock rupture dimensions, distances to
premain shocks are also normalized by the main shock
radius.
5. Results
[29] It is a common practice to quantify properties of
earthquake catalogs in terms of fitting coefficients of
empirical laws. Earthquake frequency-size distributions
are fit with
N ðM0 Þ ¼ aM0B ;
ð9Þ
where N(M0) is the number of events with seismic moment
M0, a is a measure of the earthquake productivity, and B
value (not to be confused with the B of the rate-and-state
friction) is a measure of the ratio between large and small
events. Usually, the decay of seismicity rate with time
following large earthquakes may be fitted with the modified
Omori law [Omori, 1894; Utsu, 1961]:
N_
1
¼
p;
N_0 ðt0 þ tÞ
ð10Þ
where N_ and N_ 0 are the aftershock and background rates,
respectively, t is time since the main shock, p is a positive
number, and t0 is a constant that intends to remove the
singularity at t = 0, and to correct for the worsening of the
recording completeness shortly after a large earthquake.
More recently, it was found that seismicity rate change
during foreshock activity may also be fitted with a function
of that form, but with t being the time before the main shock
[Papazachos, 1975; Kagan and Knopoff, 1978; Jones and
Molnar, 1979; Shaw, 1993]. We describe spatiotemporal
variations in the a value and the p value in section 5.1 and
report variations in the B value in section 5.2.
5.1. Foreshocks and Aftershocks
[30] Plots of cumulative number of events as a function of
time are shown in Figure 5 for four circular regions. The
smallest region is inside the main shock rupture, whereas
the largest region, for which R/C is between 3 and 4, is in
the far field where stress changes induced by the main shock
are negligible. The horizontal dashed line in each panel
separates premain shock events from postmain shock
events, and the vertical dashed line indicates the main shock
time (i.e., t = 0). At all distance ranges, there is an
immediate positive jump in the earthquake production rate
following the main shock that relaxes with time toward the
long-term rate. In addition, except for the most distant
region, there is clear evidence for an acceleration of the
seismicity rate prior to the main shock. The departure of
seismicity rate from the long-term rate is largest in the
region adjacent to the main shock rupture, and decreases
with distance away from it.
[31] Cumulative earthquake count may be transformed
into earthquake rate. Log-log diagrams of earthquake rate
changes as a function of time are shown in Figure 6a for
circular regions outside the main shock rupture that extend
Figure 5. Cumulative number of events as a function of
lag time, with respect to the main shock time, in four
circular regions. The horizontal dashed line in each panel
separates premain shock events from postmain shock
events, and the vertical dashed line indicates the main
shock time.
up to 6 main shock radii, and in Figure 6b for the region
inside the main shock rupture. These reveal seismicity rates
just before and right after the main shock that are several
orders of magnitude larger than those of the background
level.
[32] For the postmain shock activity, aftershock duration
is independent of distance from the main shock and is equal
to ta. In addition, the aftershock decay rate inside the main
shock rupture and in the nearest region surrounding the main
shock, where the stress changes are largest, is inversely
proportional to time. These properties of the aftershock
sequence are predicted by Dieterich [1994] solution for the
seismicity response to a stress step (i.e., application of
equation (7)). Since, as we explained before, the derivation
of this result relies on several simplifying assumptions that
are completely relaxed in the numerical model, Ziv and Rubin
[2003] concluded that these properties of the aftershock
sequence remain valid even when the model assumptions
are relaxed.
[33] The secondary peaks interrupting the long-term seismicity rates, at about ±100 ta and ±200 ta, are caused by
other large events that occurred near the site of the main
shock in question. The tendency of large quakes to occur at
the sites of previous large events is due to the smoothing
effect of these ruptures on the spatial distribution of the
physical conditions left behind them. Large events smoothen
ESE
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ZIV: REMOTE TRIGGERING
Figure 6. Plots of change in earthquake production rate as a function of time (a) for four circular
regions outside the main shock rupture and (b) inside the main shock rupture. For reference, we added
thin lines that indicate slopes of 1/time, and triangles that mark Dieterich’s aftershock relaxation time.
the stress and state fields over large areas, thus setting the
conditions for another large event in the future. This effect
introduces some degree of periodicity into the catalog. We
note that similar quasiperiodicity has also been observed in
some natural catalogs [e.g., Nishenko and Buland, 1987].
[34] During the premain shock seismicity, earthquake
rates at all distance ranges increase with time approaching
to the main shock time. The duration of foreshock activity is
longer than ta in the nearest region and is much shorter than
ta in the most distant region. Thus ta is not a characteristic
timescale for foreshocks activity, and foreshock duration is
distance-dependent. As with the postmain shock rates, the
increase in the seismicity rate in the nearest region is
inversely proportional to time before main shock.
[35] Inside the main shock rupture (Figure 6b), the main
shock is preceded by a long interval of quiescence, during
which earthquake production rate falls about 80% below its
long-term average rate. This quiescence is then followed by
a period during which seismicity rate increases linearly with
time. Following the main shock, seismicity rate in this
region decays according to 1/time, and drops to a value
that is well below the long-term average before returning to
it at later times. Similar pattern of premain shock and
postmain shock quiescence was documented by Mogi
[1981], who gave it the name ‘‘doughnut pattern’’. Mogi
interpreted the foreshock activity in the periphery of the
quiet zone as resulting from a stress concentration around a
locked patch.
[36] What emerges from this analysis is the existence of
an upper bound on Omori’s p value of the foreshock
sequence that is identical to that of the aftershock sequence.
That aftershock rate decay follows a modified Omori’s law,
with p value less or equal to 1, is a well known consequence
of the rate- and state-dependent friction [Dieterich, 1994].
On the other hand, the recognition that the same is true for
the foreshock sequence is a new result of this study.
[37] The two ingredients of Ziv and Rubin [2003] fault
model that act jointly to produce the type of time-space
clustering that is described here are the rate- and statedependent friction and the elastic interaction. The first gives
rise to a temporal clustering that follows the Omori’s law,
whereas the latter is responsible for the emergence of spatial
clustering over large distance range.
5.2. Time-Space Variations in the B Value
[38] The B values of 1000 consecutive events contained
in a sliding window that was moved by one event each time
were estimated based on the maximum-likelihood approach
[Aki, 1965; Utsu, 1965]. To each population, a lag time was
assigned that is equal to the median of all lag times of that
population. The B values are plotted in Figure 7 as a
function of lag time for premain shock (Figure 7, left) and
ZIV: REMOTE TRIGGERING
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Figure 7. Plots of B value as a function of time for (left) premain shock and (right) postmain shock
events that occurred within 1 < R/C < 2. Each point represents a B value estimate of a population of 1000
consecutive events that are contained in a sliding window that was moved by one event each time.
postmain shock (Figure 7, right) events that occurred within
1 < R/C < 2. We find that the B value before the main shock
decreases with time approaching the main shock time,
whereas B value after the main shock increases with elapsed
time. This result can explain the decrease in b value before the
Mw = 7.6 Izmit, Turkey, earthquake, and the decrease in the
mean magnitude following the main shock [Karakaisis,
2003]. Similarly, it can explain smaller b value for foreshocks
than for aftershocks in central Japan [Suyehiro et al., 1964],
and the faster decay rate for large aftershock than for small
ones [Hosono and Yoshida, 2002]. The origin of these timedependent variations is discussed in section 6.
[39] Although B value may vary from one location to
another, its value is often close to 2/3 [Kagan, 1997]. The B
values of the synthetic catalog are, thus, much larger than
what is observed in natural catalogs. It is possible to lower B
value by increasing the ratio B/A. However, as we have
explained before, with larger ratios of B/A the simulation
quickly evolves to an unstoppable rupture. A possible way
out of this problem is to add creeping regions (e.g., cells for
which B/A is less than 1) that will act as barriers. Indeed,
relocated catalogs from California reveal a tendency for
stick-slip patches to be surrounded or bounded by aseismic
regions [Rubin et al., 1999; Waldhauser and Ellsworth,
2002; Schaff et al., 2002]. With the current algorithm,
however, creeping elements cannot be modeled.
6. Discussion
6.1. Practical Implications
[40] A common approach in studies of earthquake clustering is to examine spatiotemporal variations of seismic
activity with respect to a single large earthquake. For the
sake of comparison between this approach and the one used
here, we show in Figure 8 earthquake rates as a function of
time with respect to the event that is examined in Figure 4.
Comparison between Figures 8 and 6 shows that the
stacking approach improves the statistics and allows to
resolve features that are otherwise unresolvable. These
include the increase of foreshock rate according to the
modified Omori law far away from the main shock, and
the independence of aftershock duration on distance from
the main shock.
6.2. Aftershocks of Aftershocks
[41] According to Dieterich’s [1994] aftershock model,
the immediate seismicity rate change caused by a stress step
Figure 8. Plots of earthquake production rate as a function of time with respect to the main shock that is
shown in Figure 4. Line styles indicate R/C intervals as in Figure 5. For reference, we added thin lines
that indicate slopes of 1/time.
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ZIV: REMOTE TRIGGERING
of t imposed on a population of faults with a uniform
distribution of times to failure is equal to exp(t/As) (i.e.,
equation (6)). Although the results presented in section 6.1.
show that the distribution of failure times prior to main
shocks is not uniform, it is interesting to quantify the extent
to which the empirical rates differ from the predicted rates.
In order to address this question, we calculated the average
of all t/As in circular regions around the main shock
ruptures. Empirical and predicted seismicity rates are shown
in Figure 9, for regions of R/C between 2 and 3 (dotted
line), between 3 and 4 (dashed-dotted line), and between 4
and 6 (gray line). This comparison shows that actual
seismicity rate changes calculated for these regions are by
several orders of magnitude higher than what is predicted by
Dieterich’s model.
[42] What gives rise to this discrepancy? Here we show
that to a large extent the discrepancy between predicted and
simulated seismicity rate changes is due to the effect of
stress transfers, which is accounted for in the simulation but
not in Dieterich’s model. In order to illustrate the role of
stress transfers we perform a simple exercise. First, we run
the simulation until the main shock in Figure 4 is reached.
Then, after accounting for the effect of the stress change
induced by that event, we turn off elastic interaction and
begin catalog recording. A comparison between postmain
shock earthquake production rates with (Figure 4) and
without stress transfers (Figure 10) shows that the area
which experienced seismicity rate increase is much smaller
when elastic interaction is turned off. This result indicates
that the increase in seismicity rate far from the main shock
(where the static stress changes imposed by the main shock
were small) is a consequence of multiple stress transfers,
and that the very distant aftershocks are not directly
triggered by the main shock, but instead they are aftershocks of previous aftershocks. In addition, as in Dieterich’s
model, there is no aftershock activity inside the main shock
rupture if stress transfer is turned off. This shows that
aftershock activity inside the main shock area was triggered
by the aftershock sequence outside that area. The absence of
Figure 9. A comparison between the time-dependent
simulated seismicity rates and the instantaneous predicted
rates (horizontal lines). Line styles indicate ranges of
normalized distance as in Figure 1.
Figure 10. A plot of aftershock rate as a function of time
with respect to the main shock that is shown in Figure 4, but
with postmain shock elastic interaction turned off. Line
styles indicate R/C intervals as in Figure 5.
aftershocks inside the rupture is due to the stress decrease in
that region.
6.3. Premain Shock Strain Localization
[43] Additional insight into the process that precedes the
occurrence of large earthquakes may be gained through
examination of the fault properties at various times prior to
the main shock. In Figure 11 (top) we plot snapshots of the
fraction of the self-accelerating elements (i.e., elements for
which dq/dt < 0) as a function of distance from the main
shock and for progressive times before the main shock time.
Each data point represents the average value of cells that are
within a circular region of a thickness of one R/C unit, and
is plotted at the midpoint of that range. At 10 ta before the
main shock, the fraction of self-accelerating elements in the
area that will become the main shock rupture (i.e., R/C < 1)
is abnormally low compared to that of the surroundings and
the long-term average. From that time on the fraction of
self-accelerating elements in that region increases, and by
the time of 1 ta before the main shock the fraction of these
elements is significantly higher than that of the background.
Just prior to the main shock rupture, an area surrounding the
main shock with a radial dimension that is about 4 times
larger than that of the main shock radius is accelerating
toward failure.
[44] In Figure 11 (bottom) we plot snapshots at various
times before the main shock of the average time since the last
rupture as a function of normalized distance from the main
shock. As before, each data point represents the average
value of cells that are contained within circular regions with
a thickness of one R/C unit. For large values of R/C, all
curves converge to a single value of time since failure that is
equal to 30 ta, and that is indicative of the half recurrence
interval of the entire grid. Between 10 ta and 1 ta, the average
time since previous failure is decreasing outside the main
ZIV: REMOTE TRIGGERING
ESE
14 - 11
shock rupture, due to the increase in the seismicity rate in
that region, but increasing inside the rupture, due to the long
period of quiescence that this region is experiencing. At
shorter lag times before the main shock, e.g., t = 0.001 ta,
average time since last failure decreases everywhere. Thus
the picture which emerges is that of strain localization around
a highly resistant fault patch (i.e., a patch that did not break
for long time), that ends abruptly with the breaking of that
patch.
Figure 11. Plots of (top) the fraction of self-accelerating
elements and (bottom) average time since last failure as a
function of normalized distance from the main shock and
for different times before the main shock time. Each data
point represents the average value of cells that are contained
within circular regions with a thickness of one R/C unit and
is plotted at the midpoint of that range.
6.4. Origin of the B Value Changes
[45] A rupture, starting with the failure of a single cell, is
more likely to expand if it is surrounded by cells that are near
failure. For that reason, an increase in the fraction of the selfaccelerating cells is expected to lower the B value, and vice
versa. In Figure 12 we plot the fraction of the self-accelerating elements as a function of time before and after the main
shock, in a region confined to the main shock rupture area for
which R/C < 1 (Figure 12, top) and in a region surrounding
the main shock rupture for which 1 < R/C < 2 (Figure 12,
bottom). Before the main shock, the fraction of self-accelerating elements increases with time approaching the main
shock time. The rate of that increase is faster in the inner
region than in the outer region. After the main shock, the
fraction of self-accelerating elements decrease with time.
Inside the main shock rupture, since most of the cells in that
region failed together with the main shock, the fraction of
self-accelerating elements drops almost immediately to a
value that is near zero. Comparison between Figure 12 and
Figure 7 reveals that changes in the fraction of the selfaccelerating elements and changes in the B values are of
opposite trend. Thus the observed time-dependent modifications of the B value are indicative of the fault physical
state. More specifically, a decrease of the B value indicates
that the region is approaching failure and is undergoing
weakening that may lead to a large earthquake, whereas an
Figure 12. Plots of the fraction of self-accelerating elements as a function of time before and after the
main shock time and for normalized distance of (top) R/C < 1 and (bottom) 1 < R/C < 2.
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ZIV: REMOTE TRIGGERING
increase of the B value indicates that the region is undergoing strengthening.
6.5. A Comment on the Acceleration of the Benioff
Strain Before Large Earthquakes
[46] Large earthquakes are often preceded by intervals of
accelerating seismic release, that occur over areas with
dimensions that are much larger than their rupture dimensions [Knopoff et al., 1996; Bowman et al., 1998]. A
measure for the total seismic release is the cumulative
Benioff strain, defined as
eðt Þ ¼
X
1=2
M0 ðt Þ:
ð11Þ
In principle, two processes may contribute to the acceleration
of the cumulative Benioff strain. The acceleration of this
function may either reflect an increase with time of
earthquake occurrence rate with size distribution remained
fixed, or it may be the result of an increase with time of the
average seismic moment released of a single event with
earthquake production rate remained fixed, or some combination of the two. Several observational studies showed that
the earthquake production rate is increasing with time before
a main shock [Bufe and Varnes, 1993; Knopoff et al., 1996]. If
our fault model applied to the earth, the result that the
increase in earthquake production rate is accompanied by a
decrease in the B value suggests that a systematic change in
the earthquake size distribution is also adding to the
acceleration of the Benioff strain. Some evidence for this
effect comes from a study of Karakaisis [2003], who
identified a region of accelerating seismicity prior to the
Izmit, Turkey, earthquake, and found that the b value of the
magnitudes greater than 4 within that region has dropped
from 1.1 in the early 1980s to 0.6 prior to the main shock.
7. Summary
[47] We present the results of time-space analyses of
synthetic catalogs, generated using a quasi-static discrete
fault model of Ziv and Rubin [2003], where slip is governed
by the Dieterich-Ruina rate-and-state friction law [Dieterich,
1979; Ruina, 1980]. We find that while aftershock duration is
independent of distance from the main shock and is equal to
ta, foreshock duration is distance-dependent. The upper
bound on Omori’s p value of foreshock and aftershock
sequences is identical, and is equal to 1. The increase in the
postmain shock earthquake production rate occurs over an
area surrounding the main shock rupture with dimensions
that are several times larger than the main shock dimensions.
We show that the increase in seismicity rate far from the main
shock (where the static stress changes imposed by the main
shock were small) is a consequence of multiple stress transfers, and that the very distant aftershocks are not directly
triggered by the main shock, but instead they are aftershocks
of previous aftershocks. Before the main shock, the seismicity rate in distant regions increases and the B value decreases.
It has been previously suggested that remote seismicity rate
increase may be triggered by stress changes associated with
the passage of seismic waves. The existence of remote
triggering in a quasi-static model indicates that it is not
necessary to invoke such a ‘‘dynamic effect’’ in order to
explain distant aftershocks.
[48] Acknowledgments. This study benefited from discussions at
various stages with J.-P. Ampuero, A. Cochard, J. Dieterich, R. Madariaga,
A. M. Rubin, and J. Schmittbuhl. I thank R. Madariaga for a constructive
review of an early version of the manuscript. Comments from J. Gomberg
and two anonymous reviewers helped to improve the manuscript.
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A. Ziv, Laboratoire de Géologie, École Normale Supérieure, 24 rue
Lhomond, F-75231 Paris Cedex 05, France. ([email protected])