Upload
others
View
0
Download
0
Embed Size (px)
Citation preview
JOURNAL OF GEOPHYSICAL RESEARCH, VOL. ???, NO. , PAGES 1–32,
QUASI-DYNAMIC MODELING OF
SEISMICITY ON A FAULT WITH
DEPTH-VARIABLE RATE- AND
STATE-DEPENDENT FRICTION
A. Ziv, and A. Cochard
Corresponding author:
Alon Ziv
Dept. of Geological and Environmental Sciences
Ben-Gurion University of the Negev
P.O.Box 653, Beer-Sheva 84105, ISRAEL
E-mail: [email protected]
A. Ziv, Ben-Gurion University of the Negev, Beer-Sheva, 84105, Israel. (e-mail: zi-
A. Cochard, Ludwig-Maximilians University, Munich, Germany. (e-mail: [email protected]
muenchen.de)
D R A F T April 30, 2006, 3:33pm D R A F T
2 ZIV AND COCHARD: QUASI-DYNAMIC MODELING OF SEISMICITY
Abstract.
Neither the Omori type of clustering prior to and following large earth-
quakes, nor the Gutenberg-Richter distribution are reproducible by present
continuous models. Discrete models, on the other hand, give rise to more com-
plex, and closer to realistic earthquake clustering. The objective of this study
is two-fold, to explore the consequences of spatial discreteness on the distri-
bution in time and space of earthquake activity on a fault governed by rate-
and-state friction, and to examine the effect of interaction between seismic
slip and aseismic creep on aftershock sequences. To that end we model a long,
vertical, 2D strike-slip fault that is governed by rate- and state-dependent
friction, and is embedded in a 3D elastic half-space. Quasi-dynamic motion
on the fault is driven by steady displacement applied below the fault, and
at distance W/2 on either side of the fault plane. The model is said to be
spatially discrete in that the computational cells are oversized with respect
to the critical length-scale that is implied by the friction law. The model re-
produces some of the characteristics of natural seismicity, including the non-
periodic recurrence times and the Omori type of clustering prior to and fol-
lowing large earthquakes. It also produces a wide range of earthquake mag-
nitudes, but with a ratio of small to large earthquakes that is in excess with
respect to what is inferred form natural catalogs. We examined the effect of
a stress step applied on the creeping portions of the model, and confirmed
that aftershock sequences resulting from stress relaxation on creeping seg-
ments decay asymptotically to 1/time. Finally we discuss fundamental dif-
ferences between seismicity models employing rate-and-state friction and those
employing static/kinetic friction.D R A F T April 30, 2006, 3:33pm D R A F T
ZIV AND COCHARD: QUASI-DYNAMIC MODELING OF SEISMICITY 3
1. Introduction
Earthquake depth distributions on well-developed faults show upper and lower seismic-
ity cutoffs [Marone and Scholz, 1988]. For example, 80% of the earthquakes along the
Parkfield section of the San Andreas fault (California) occur in a depth interval between 4
and 11 kilometers, and almost none are observed deeper than 15 km. Yet, exhumed faults
indicate that faults do not terminate at these depths, but continue downward, forming
ductile fabric. What gives rise to the transition from aseismic to seismic slip at shallow
depth, and to the transition from seismic to aseismic slip at greater depth? A common
view is that earthquakes are frictional instabilities [Brace and Byerlee, 1966], and that
transitions from stable to unstable sliding, and vice versa, are consequences of the fric-
tional constitutive parameters being sensitive to changes in pressure [Shimamoto, 1986],
temperature [Blanpied et al., 1998] and lithology [Marone and Scholz, 1988]. For exam-
ple, friction experiments on granite and quartz gouge under hydrothermal conditions show
transition from velocity weakening to velocity strengthening (i.e., a transition from unsta-
ble to stable sliding) at 325C, corresponding to a depth of about 11 km [Blanpied et al.,
1991, 1998]. Thus, the effect of temperature alone can explain the transition to aseismic
slip at depth (this and other mechanisms are discussed by Scholz [2002]). Motivated by
this view, several researchers modeled slip on faults as being governed by friction, and
distributed frictional properties in a manner that favors aseismic creep near the surface
and at great depth, but stick-slip behavior in between [e.g., Tse and Rice, 1986; Rice,
1993; Lapusta et al., 2000; Shibazaki and Iio, 2003; Perfettini et al., 2003].
For a fault to fail repeatedly, there must exist both weakening and restrengthening
mechanisms. Such is the case with laboratory inferred friction laws, in which frictional re-
D R A F T April 30, 2006, 3:33pm D R A F T
4 ZIV AND COCHARD: QUASI-DYNAMIC MODELING OF SEISMICITY
sistance depends upon the logarithm of sliding speed and sliding history [Dieterich, 1979].
Dieterich [1994] modeled the effect of a stress perturbation on a population of faults that
are governed by rate-and-state friction, and showed that observable quantities, such as the
duration and the magnitude of aftershock activity, can be related to constitutive param-
eters. Time-space analyzes of microseismicity along several faults in Northern California
confirm that the main features of aftershock sequences are explainable in terms of Di-
eterich’s model [Ziv et al., 2003]. These include the decay of aftershock rate according to
the modified Omori law [Omori, 1894; Utsu, 1961], and the independence of aftershock
sequence duration on distance from the mainshock. In addition, Schaff et al. [1998]
showed that the decay with time of recurrence intervals of repeating earthquakes along
the creeping section of the San Andreas fault following the Loma Prieta earthquake is con-
sistent with the friction being a logarithmic function of the creep rate. Finally, the most
straightforward prediction of the rate-and-state friction is a scaling between the stress
drop of repeating earthquakes and the logarithm of their recurrence time [e.g., Beeler et
al., 2001]. Indeed, several seismological studies inferred that stress drop increases with
recurrence interval in a manner that is consistent with this prediction [Kanamori and
Allen, 1986; Scholz et al., 1986; Vidale et al., 1994; Marone et al., 1995; Marone, 1988].
Thus, despite some skepticism that arises mainly from the extrapolation of laboratory
results to geological scale, rate-and-state friction laws are gaining considerable popularity
among earthquake scientists.
Spatio-temporal analyzes of natural earthquake catalogs reveal strong tendency for
earthquakes to cluster in time and space. This property is hereafter referred to as ‘earth-
quake complexity’ or ‘slip complexity’. Despite considerable efforts to model earthquake
D R A F T April 30, 2006, 3:33pm D R A F T
ZIV AND COCHARD: QUASI-DYNAMIC MODELING OF SEISMICITY 5
complexity, the origin of earthquake complexity has remained uncertain. Neither the
Omori type of clustering prior to and following large earthquakes, nor the Gutenberg-
Richter distribution over wide range of sizes are reproducible by present continuous mod-
els. Laboratory derived friction laws contain a characteristic sliding distance [Ohnaka,
1975; Dieterich, 1979]. This characteristic length-scale gives rise to a critical length-scale
Lc, which defines the minimum dimension of a crack below which instability cannot de-
velop [Dieterich, 1992]. When modeling a seismic fault, the set of governing equations is
discretized and solved on a computational grid of cells. Rice [1993] warned that a proper
solution of these equations requires a proper refinement of the computational grid, such
that the size of an individual computational cell, L, is much smaller than Lc. At the
present time, simulations with L Lc result in a little, far from realistic, slip complexity
[3D quasi-static: Tse and Rice, 1986, 2D quasi-static: Horowitz and Ruina, 1989, 3D
quasi-dynamic: Rice, 1993, and 2D elastodynamic: Cochard and Madariaga, 1996; Shaw
and Rice, 2000; Lapusta et al., 2000; Lapusta and Rice, 2003]. Faults are not perfectly
planar, and Rice [1993] speculated that setting L to be larger than Lc may be equivalent
to mapping 3D geometrical irregularities onto a 2D plane. Interestingly, simulations with
L > Lc give rise to considerable slip complexity [3D quasi-static: Dieterich, 1995; Ziv and
Rubin, 2003, and 3D quasi-dynamic: Rice, 1993]. The problem with L > Lc is that the
details of the nucleation process at the rupture tip are not resolved. There is, however,
a host of interesting problems for which the details of the nucleation phase are unimpor-
tant; for example, how stress and strain evolve during the interseismic periods, what is
the effect of stress transfer due to aseismic creep, and many more. Other properties of
D R A F T April 30, 2006, 3:33pm D R A F T
6 ZIV AND COCHARD: QUASI-DYNAMIC MODELING OF SEISMICITY
the rate-and-state friction, namely, the time-dependent re-strengthening following rupture
and the viscous response are preserved in such models.
The objective of this study is two-fold. The first is to explore the consequences of
spatial discreteness on the distribution in time and space of earthquake activity on a fault
governed by rate-and-state friction. The second is to examine the effect of interaction
between seismic slip and aseismic creep on aftershock sequences. To that end we model a
long, vertical, 2D strike-slip fault that is governed by rate- and state-dependent friction,
and is embedded in a 3D elastic half-space. Quasi-dynamic motion on the fault is driven
by steady displacement applied below the fault, and at distance W/2 on either side of the
fault plane. The present study represent a major improvement over previous studies of
seismicity (as opposed to individual ruptures) on a rate-and-state fault [Dieterich, 1995;
Ziv and Rubin, 2003]. The improvements with respect to Dieterich, [1995] and Ziv and
Rubin, [2003] are: (a) In going from quasi-static to quasi-dynamic motion; (b) While in
those studies the set of governing equations is solved approximately, here they are solved
exactly; and (c) The previous point enables the incorporation of aseismic creep. From a
practical stand point, switching from quasi-static to quasi-dynamic significantly prolongs
the computation time per event. Nevertheless, thanks to the improvement of computation
speed and numerical algorithms, it now possible to simulate a few tens of thousands of
earthquakes per day.
We describe the model and the computational approach in Section 2. In Section 3 we
examine various properties of a ‘case-study’ catalog, including earthquake size distribution
and temporal clustering. In Section 4 we explore the effect of three model parameters on
the distribution of earthquake sizes. In Section 5 we simulate an aftershock sequence
D R A F T April 30, 2006, 3:33pm D R A F T
ZIV AND COCHARD: QUASI-DYNAMIC MODELING OF SEISMICITY 7
resulting from stress relaxation of the model’s creeping portions. Finally, in Section 6
we discuss fundamental differences between this model and seismicity models employing
static/kinetic friction.
2. The Model
We model a long, vertical, 2D strike-slip fault that is embedded in a 3D scalar elastic
half-space (Figure 1). Similar to Rice [1993] we calculate quasi-dynamic slip in a region
that extends down to 32km. Below that depth the fault slips steadily at the plate velocity,
Vplate. In addition to being loaded by steady slip from below, the fault is subjected to an
additional constant stressing rate due to displacement applied at Vplate rate on parallel
planes located at distance W/2 on either side of the fault plane. Motion on the fault is
resisted by rate- and state-dependent friction. Frictional proprieties are depth-variable,
and are distributed in a manner that favors aseismic creep near the surface and at great
depth, but stick-slip behavior in between. The fault is discretized, and is covered by
a computational grid of square cells that is periodic along strike. The horizontal and
vertical dimensions of the computational grid are 128 cells and 64 cells, respectively. The
computational cells are oversized with respect to the minimum dimensions of a crack
below which instability cannot develop. Such a grid is believed to be a way of mapping
3D structural irregularities (kinks, jogs, etc.) onto a 2D plane [Rice, 1993]. Often, the
tectonic setting is more complex than that, with several faults at different orientations
operating simultaneously. In such cases, a large earthquake on one fault may either trigger
aftershocks or shut off activity on other faults. Admittedly, the present model does not
account for the effect of stress transfers among faults of different orientations.
D R A F T April 30, 2006, 3:33pm D R A F T
8 ZIV AND COCHARD: QUASI-DYNAMIC MODELING OF SEISMICITY
2.1. The Friction
Slip on the fault is resisted by rate- and state-dependent friction [Dieterich, 1979; Ruina,
1983]. Frictional resistance on cell i, is given by:
τi(t) = σi
(
µssi + Ai ln
Vi(t)
Vplate
+ Bi lnVplateθi(t)
Dc
)
, (1)
where t is time, σ is the effective normal stress, µss is the friction coefficient when the fault
slides steadily at the plate velocity Vplate, A and B are unitless constitutive parameters, Dc
is a characteristic distance for the evolution of the state from one steady state to another,
V and θ being the slip rate and fault state, respectively. The state evolves with slip and
time according to [Ruina, 1980]:
dθi
dt= 1 −
θiVi
Dc
. (2)
2.2. Elastic Interaction
The shear stress on cell i is a function of time, slip and slip speed as follows:
τi(t) = τ 0i +
G
W(Vplatet − δi(t)) +
G
L
∑
j
Kij(δj(t) − Vplatet) −G
2β(Vi(t) − Vplate). (3)
The first term, τ 0i , is a constant. It has no effect on the dynamics, once the model has
evolved through a self-organization period beyond which the result is statistically insensi-
tive to the initial conditions. The second term represents the driving stresses imparted on
the fault surface due to mismatch between the total displacement on the point in question,
δi, and the cumulative tectonic slip imposed at rate Vplate at distance W/2 on either side
of the fault plane, with G being the shear modulus. This term is also present in other
fault models [e.g., Horowitz and Ruina, 1989; Dieterich, 1995; Ziv and Rubin, 2003]. The
third term accounts for elastic interaction, with L being the length of the computational
cell, Kij being a scalar non-dimensional elastic kernel (see Appendix A) and δj being slip
D R A F T April 30, 2006, 3:33pm D R A F T
ZIV AND COCHARD: QUASI-DYNAMIC MODELING OF SEISMICITY 9
on j. Finally, the fourth term embodies the quasi-dynamic approximation of Rice [1993].
The factor G/2β, with β being the shear wave speed, is often referred to as the ‘radiation
damping term’. The inclusion of this term is necessary in order for the slip speed to
remain bounded. Note that if the second term is omitted, the above expression is equiva-
lent to the stress-slip relation of Rice [1993]. The inclusion of this term introduces a new
length-scale, which does not exist in Rice’s model. Thus, Rice’s model may be viewed
as a special case of our model for which W is infinitely large. Reducing W increases the
stiffness of the system, and enhances the complexity [Ziv and Rubin, 2003].
2.3. Computational Approach
Stress balance and derivation with respect to time yields:
dVi
dt=
[ G
W(Vplate − Vi(t)) +
G
L
∑
j Kij(Vj(t) − Vplate) −σiBi
θi(t)
(
1 −Vi(t)θi(t)
Dc
)]
( G
2β+
σiAi
Vi(t)
)
. (4)
Note that the evolution of V and θ is fully described by a set of two first order ordinary
differential equations (2) and (4). The governing equations are solved simultaneously at
successive time steps using a fifth-order adaptive time step Runge-Kutta algorithm [Press
et al., 1992]]. This approach results in fewer and longer time steps during periods of
quasi-static loading, and numerous shorter time steps during the nucleation and instability
phases.
By far, the most time consuming step in such models is the convolution between slip
speed and the elastic kernel, i.e., the second term in the numerator of (4). Since here the
elastic kernel is translational invariant, the convolution theorem may be implemented,
and this term may be computed using fast Fourier transforms. Additionally, the free
surface may be modeled by an insertion of a mirror image. We use zero-padding along the
D R A F T April 30, 2006, 3:33pm D R A F T
10 ZIV AND COCHARD: QUASI-DYNAMIC MODELING OF SEISMICITY
z-direction (depth) to properly compute the convolution without replication; we do not
use zero-padding along the x-direction (strike) to effectively model periodic replications
along that direction.
In order to identify the controlling parameters we have carried out a dimensional analy-
sis. It turned out that the governing equations are a function of only four non-dimensional
parameters: Bi/Ai, W/L, LBiσi/GDc, and VplateL/Dcβ. Their physical significance is dis-
cussed in Appendix B.
2.4. Recording Procedure and the Building of the Synthetic Catalog
We consider that the recording of a single-cell seismic episode starts once the sliding
speed on that cell exceeds a centimeter per second, and ends once the state variable passes
through a steady state (i.e., when the sign of dθ/dt changes from negative to positive).
The information that we output at the end of each such episode includes its starting time,
its ending time, total slip between these times, and coordinates. After this computation
is finished, the simulation output is analyzed and a synthetic catalog is constructed. In
constructing the synthetic catalog, we follow these steps:
1. Two seismic events are merged to give a single event if the later of the two started
before the earlier ended. This process is repeated so long as the criterion for event merging
is satisfied.
2. The subset of cells that comprise the final event are assigned an origin time that is
equal to the earliest origin time of the subset, and a seismic moment that is equal to the
product of the shear modulus and the integral of co-seismic slip over the rupture area.
This information is written to the synthetic catalog.
3. Move to the next single-cell event on the list, and return to step-1.
D R A F T April 30, 2006, 3:33pm D R A F T
ZIV AND COCHARD: QUASI-DYNAMIC MODELING OF SEISMICITY 11
Each simulation starts with arbitrary non-uniform initial conditions. Following an initial
self-organization stage, the results appear to be statistically independent of the specific
choice of the starting conditions. The results from the self-organization were excluded
from the synthetic catalog.
3. A Case-Study Catalog
In this section we examine various properties of a ‘case-study’ catalog. We shall see
that the model produces a wide range of event sizes, and clustered activity before and
after large events.
3.1. Model Parameters
While it is reasonable to assume that the physical properties on geological faults are
heterogeneous and fluctuate in both the down-dip and the along-strike directions, we
think that in this early stage of the investigation it is more constructive to consider
simple models. The distribution of the constitutive parameters is uniform along strike
and varies either linearly or step-wise with depth. Thus, the system investigated here is
far less complex than the highly disordered systems studied by some investigators [e.g.,
Ben-Zion, 1996; Zoller et al., 2005a and 2005b].
Model parameters are as follows: W = 5000 m, L = 500 m, G = 10 GPa, β = 1000m/s,
Vplate = 0.03m/year, and Dc = 0.02m. Effective normal compression, σ, increases linearly
with depth according to: 7.e106[Pa]+z[m]×2.e104[Pa/m], where z is depth. For 2.5km <
z < 15km, A and B equal 0.005 and 0.04, respectively. Above and below that depth, A and
B equal 0.01 and 0.005, respectively. Thus, steady-state friction is velocity-strengthening
near the surface and at great depth, and velocity-weakening in between.
D R A F T April 30, 2006, 3:33pm D R A F T
12 ZIV AND COCHARD: QUASI-DYNAMIC MODELING OF SEISMICITY
The size of the critical crack, Lc, is approximately equal to GDc/Bσi [Dieterich, 1992].
With the above parameters, the computational cells are everywhere oversized with respect
to Lc.
3.2. Size Distribution
Earthquake magnitudes were calculated according to the magnitude-moment relation
of Purcaru and Berckhemer [1978]: log M0 = 1.5M + 9.1, where M0 is seismic moment in
Nm and M is the magnitude. Magnitude versus cumulative frequency is shown in Figure
2. Note that the distribution of magnitudes exhibits a wide range of sizes that extends
over 2.5 magnitude units, with the largest rupture occupying about 500 computational
cells.
The flattening of the curve for magnitude smaller than 5.3 (see arrow) is the result of
cells that failed twice within a very short time interval (but numerically perfectly resolved),
due to experiencing a large stress step during that interval. As a result, these cells did
not strengthen, and released abnormally small seismic moment during failure.
A consequence of the spatial discreteness is under representation of the stress concen-
tration ahead of the rupture front. This under representation is more severe for small
ruptures than for large ones, thus making the growth (by coalescence with adjacent cells)
of a single-cell rupture much more difficult than larger ruptures. This effect causes a
notable break in the curve at M = 5.7 (see arrow).
It is evident that the distribution cannot be fitted with a single Gutenberg-Richter curve,
and that the ratio of smallest to largest magnitudes is slightly in excess with respect to
what is inferred form natural catalogs. Lower ratios of small to large magnitudes may be
obtained with larger ratios of B to A (see Section 4). The problem with larger B/A is
D R A F T April 30, 2006, 3:33pm D R A F T
ZIV AND COCHARD: QUASI-DYNAMIC MODELING OF SEISMICITY 13
that the simulation quickly evolves to an unstoppable rupture. A way out of this problem
is to embed patches of velocity strengthening that would act as barriers and help to arrest
large ruptures from breaking the entire fault.
3.3. Distribution of Inter-Event Times
In order to illustrate the complexity in the model, we plot in Figure 3 the distribution
of time intervals between successive events with magnitude equal to or greater than 6,
6.5 and 7. Temporal distributions may be classified on the basis of the ratio between the
distribution average, α, and its standard deviation, Ω [Ben-Zion, 1996]. With α/Ω < 1
the distribution is clustered in time. The distribution is random Poissonian if α/Ω ∼ 1.
Finally, the distribution of time intervals is periodic if α/Ω 1. The result of this analysis
shows that small-moderate earthquakes are clustered in time (Figure 3a-b). In contrast,
the occurrence of the largest model earthquakes is approximately random (Figure 3c).
3.4. Temporal Clustering Before and After Large Events
In order to obtain a statistically meaningful view of earthquake clustering before and
after large earthquakes, we stacked segments of the catalog around the 5 largest earth-
quakes. A cumulative event count as a function of time of the stacked catalog is shown
in Figure 4. Note that the stacked curve departs from a constant slope; it jumps to a
higher seismicity rate following the mainshock, and decays toward a constant rate. Note
that earthquake production rates at long times before and after the mainshock are nearly
constant. These rates are representative of the seismicity level during interseismic periods.
D R A F T April 30, 2006, 3:33pm D R A F T
14 ZIV AND COCHARD: QUASI-DYNAMIC MODELING OF SEISMICITY
We transformed the cumulative stacked curve of Figure 4 into rate diagrams. Log-log
diagrams of earthquake rate changes as a function of time are shown in Figure 5. For
reference we added dashed lines that indicate slopes of 1/time.
Note that the seismicity rate following the mainshock decays asymptotically according
to the modified Omori law [Omori, 1894; Utsu, 1961], with a decay exponent that is
slightly less than 1. That aftershock production rate decays asymptotically according to
the modified Omori law is a well known consequence of the rate- and state-dependent
friction [Dieterich, 1994]. Dietrich [1994] showed that the exponent of the decay rate
equals 1 if the fault is subjected to a uniform stress change, but it equals 0.8 if the stress
change imposed by the mainshock rupture decays as 1/r0.5 near the rupture tip and as
1/r3 at larger distance (see figure 5a of Dieterich [1994]). Also in our model the stress
change induced by the mainshock rupture is heterogeneous. As a result, the exponent
of the decay rate is less than 1. Note the flattening of the seismicity curve early in the
aftershock sequence. This too is predicted by Dietrich’s aftershock model.
The ongoing improvement in the recording completeness of seismicity led to an improved
view of foreshock activity, and to the recognition that foreshocks rate may also be fitted
with a power law [e.g., Papazachos, 1975; Kagan and Knopoff, 1978; Jones and Molnar,
1979; Shaw, 1993, Helmstetter and Sornette, 2003]. Here too (see also Ziv 2003) we find
that the increase in the seismicity rate before the mainshock is inversely proportional to
time before mainshock (Figure 5). Finally, note that what appears to be a subtle increase
in earthquake production rate prior to the mainshock in Figure 4, turned out to be a few
orders of magnitude increase in foreshock rate due to transforming the cumulative count
into earthquake rate, and plotting the time axis in logarithmic scale.
D R A F T April 30, 2006, 3:33pm D R A F T
ZIV AND COCHARD: QUASI-DYNAMIC MODELING OF SEISMICITY 15
4. The Controls on Earthquake Size Distribution
4.1. The Effect of Varying B Throughout the Seismogenic Depth
In the previous section we examined the results of a simulation for which the adopted B
throughout the velocity-weakening depth interval is equal to 0.04; here we examine how
varying this parameter affects the distribution of earthquake sizes. Because both static
and dynamic stress drops scale linearly with B−A, increasing B increases the stress drop
and prolongs the recurrence interval.
In Figure 6a we compare magnitude statistics for various B. Note that larger values of
B result in flatter slopes, i.e., an increase in the ratio of large-to-small events. A plot of
average rupture dimensions as a function of B is shown in Figure 7a, which shows that the
effect of increasing B is not only to increase the average magnitude, but also to increase
the average dimension of ruptures.
What gives rise to the flattening of the magnitude versus frequency curves and the in-
crease of rupture dimensions with increasing B? An instability, starting from a single cell,
may grow larger and occupy additional cells if it triggers unstable slip over adjacent cells.
The ability of one rupture to trigger additional ruptures is related to the instantaneous
change in sliding speed that this rupture induces on nearby cells. Indeed, larger B pro-
duces larger stress drop, and therefore also greater increase in sliding speed on adjacent
segments.
4.2. The Effect of Varying A Throughout the Seismogenic Depth
Changing A has two effects. Since the stress drop is proportional to B −A, reducing A
results in larger stress drops. Additionally, since the sliding speed increase due to a stress
step of ∆τ is proportional to exp(∆τ/Aσ), reducing A results in larger velocity increase
D R A F T April 30, 2006, 3:33pm D R A F T
16 ZIV AND COCHARD: QUASI-DYNAMIC MODELING OF SEISMICITY
due to a positive stress step. Consequently, reducing A increases the range of magnitudes
(Figure 6b), and the average dimension of ruptures (Figure 7b).
4.3. The Effect of W
Let us now examine how W/2, the distance from the fault to the rigid loading blocks
moving at Vplate, affects the distribution of earthquake sizes. From Equation (3), quasi-
static slip on a given cell reduces the stress on that cell by an amount that is proportional
to Kii−G/W , where Kii ≈ −G/L. Thus, increasing W reduces the stiffness. With smaller
stiffness, the co-seismic slip is greater, and stress transfer during an earthquake is stronger.
Since larger W results in stronger stress transfer during seismic slip, ruptures are more
difficult to stop, and rupture dimensions are on average larger (Figure 7c). Additionally,
increasing W causes the magnitude statistics to switch from a close to power-law to a
characteristic distribution (Figure 6c).
5. Interaction Between Seismic Slip and Aseismic Creep
Following large earthquakes, the stress field is being perturbed over areas in which
steady-state friction is partly velocity strengthening and partly velocity weakening. Such
a perturbation relaxes seismically on segments where steady-state friction is velocity weak-
ening, and aseismically where it is velocity strengthening. There are two different physical
mechanisms by which this situation may trigger aftershock activity that decays according
to the modified Omori law, one that has been proposed by Dieterich [1994] and another
one by Schaff et al. [1998].
According to Dieterich [1994] aftershock model, a stress step applied on a collection
of patches of velocity weakening advances the timing of their failure by an amount that
D R A F T April 30, 2006, 3:33pm D R A F T
ZIV AND COCHARD: QUASI-DYNAMIC MODELING OF SEISMICITY 17
depends on their time to failure prior to the application of the stress. Specifically, the
failure of fault patches that were initially far from failure is advanced more than the failure
of patches that were closer to failure. Owing to this non-linear effect, an instantaneous
stress step causes an immediate increase in the earthquake production rate. Dieterich
showed that this perturbation relaxes asymptotically according to the modified Omori law,
with a relaxation time that depends on the stressing rate and the constitutive parameters,
but independent of the magnitude of the stress step. Ziv et al. [2003] showed that
aftershocks of small earthquakes decay in a manner that is consistent with the these
predictions. Several researchers have interpreted aftershock sequences of large earthquakes
in terms of this model [Gross and Kisslinger, 1997; Toda et al., 1998; Stein, 1999].
An alternative model for aftershock triggering has been proposed by Schaff et al. [1998].
This model applies for cases where patches that fail in stick-slip are surrounded by areas of
creep, so that the stressing rate acting on the stick-slip patches is directly proportional to
the creep rate of the surroundings (e.g., Loma Prieta repeating aftershocks along the San
Andreas fault: Schaff et al., 1998; Morgan Hills repeating aftershocks along the Calaveras
fault: Peng et al., 2005). If the friction on the creeping regions is proportional to the
logarithm of the sliding speed, as in the rate-and-state, a positive stress step applied
on the creeping regions causes a velocity jump that decays as 1/time. This results in
repeating earthquakes with recurrence intervals inversely proportional to the time since
the stress step. An extension of this model is discussed by Perfettini and Avouac [2004],
who suggest that the Chi-Chi aftershocks (Taiwan) resulted from post-seismic relaxation
of a velocity strengthening substrate. Furthermore, they show that the mathematical
D R A F T April 30, 2006, 3:33pm D R A F T
18 ZIV AND COCHARD: QUASI-DYNAMIC MODELING OF SEISMICITY
expression describing the decay of aftershock rate predicted for this model is identical to
that predicted by Dieterich’s model.
Aftershock sequences resulting from Dieterich’s model were simulated previously by
Dieterich [1995] and Ziv and Rubin [2003]. Here we simulate aftershock sequences resulting
from the relaxation of creeping segments. From Equations 1 and 2, the effect of an
instantaneous stress step of ∆τ is to increase the sliding speed by a factor of exp(∆τ/Aσ),
and to leave the state variable unchanged. Thus, in order to initiate a post-seismic stress
relaxation on the creeping portions of the model, we multiplied the sliding speed on these
areas by exp(∆τ/Aσ). In Figure 8 we show earthquake rate as a function of time following
a stress step of 1 MPa applied on the creeping portions of the model (solid). This result
was obtained by averaging the result of twenty independent simulations, having different
distribution of initial slip rate and state. The averaging over many simulations is necessary
due to the great variability of the seismicity response to a stress step from one simulation
to another. We find that aftershock sequences resulting from stress relaxation on creeping
segments decay in a manner that is consistent with that predicted by Perfettini and
Avouac [2004]. For comparison we also show aftershock rate versus time since a stress step
of 1 MPa applied on the entire model (dashed). This result suggests that in reality both
aftershock mechanisms may operate simultaneously and contribute to the total number
of aftershocks.
6. Comparison With Models Employing Static/Kinetic Friction
In Figure 9 we show the evolution of friction as a function of time on a cell that is located
in the middle of the seismogenic depth. Abrupt drops from peak to residual strength
indicate episodes of seismic slip. Note that while the residual strength is constant, the
D R A F T April 30, 2006, 3:33pm D R A F T
ZIV AND COCHARD: QUASI-DYNAMIC MODELING OF SEISMICITY 19
peak strength is not. Episodes of seismic slip are followed by interseismic periods, during
which strength is recovered gradually. Positive jumps in friction during the interseismic
intervals are the result of positive stress perturbations induced by co-seismic slip on other
cells. Some workers that study intrinsically discrete 2-D faults adopt a static/kinetic plus
viscous response laws [e.g., Ben-Zion, 1996]. This approach too results in friction-versus-
time curves that are similar to those produced by rate-and-state models. Nevertheless,
seismicity models employing rate-and-state friction differ fundamentally from seismicity
models employing static/kinetic friction.
The first difference is related to the dependence of fault strength on recurrence interval.
In Figure 10 we plot peak strength as a function of the logarithm of recurrence times,
for the data shown in Figure 9. Note that peak strength increases proportionally to the
logarithm of recurrence interval. This proportionality is a direct consequence of the friction
law that we use. During interseismic times the fault is nearly locked, and substituting
δi = 0 into (2) yields: θi = θ0i +t, where θ0 is the initial state. Because θ → 0 during seismic
slip, from (1) the contribution of the state to the peak strength, µpeak, is proportional
to B ln(trDc/Vplate), where tr is the recurrence time, i.e. the time since the previous
seismic slip. For comparison with the data in Figure 10, we added a line of µpeak − µss =
B ln(trDc/Vplate). The modest scattering of the data about this line is the result of the
complex sliding histories. Laboratory experiments with transparent materials revealed
that the increase of strength with hold time is due to the increase of the true contact area
with time [Dieterich and Kilgore, 1994 and 1996]. Several seismological studies inferred
that stress drop increases with recurrence interval in a manner that is consistent with the
rate-and-state prediction [Kanamori and Allen, 1986; Scholz et al., 1986; Vidale et al.,
D R A F T April 30, 2006, 3:33pm D R A F T
20 ZIV AND COCHARD: QUASI-DYNAMIC MODELING OF SEISMICITY
1994; Marone et al., 1995; Marone, 1988]. Models employing static/kinetic friction do not
reproduce this result, and are therefore incompatible with the aforementioned laboratory
experiments and seismological studies.
Another difference between models employing rate-and-state friction and those employ-
ing static/kinetic friction is related to the mechanism of aftershock production. In Section
5 we showed that aftershocks in the model may arise from two different sources; the first
is due to the effect of a stress step on the failure time of seismogenic cells (where B > A),
the second is due to the relaxation of stresses that are stored in the creeping cells (where
B < A). Models adopting static/kinetic friction are not capable of producing aftershock
sequences that decay according to the modified Omori law. Aftershocks in models that in-
corporate static/kinetic friction and creep laws arise solely from the relaxation of stresses
in the creeping regions [e.g., Zoller et al., 2005a and 2005b]. In such models aftershocks
are concentrated near the edges of creeping segments.
7. Summary and Conclusions
The objective of this study is two-fold. The first is to explore the consequences of
L > Lc on the distribution in time and space of earthquake activity on a fault governed
by rate-and-state friction. The second is to examine the effect of interaction between
seismic slip and aseismic creep on aftershock sequences. To that end we modeled a long,
vertical, 2D strike-slip fault that is embedded in a 3D scalar elastic half-space. Quasi-
dynamic motion on the fault is driven by steady displacement applied below the fault, and
at distance W/2 on either side of the fault plane. On the fault, slip is resisted by rate- and
state-dependent friction. Frictional proprieties are depth-variable, and are distributed in
D R A F T April 30, 2006, 3:33pm D R A F T
ZIV AND COCHARD: QUASI-DYNAMIC MODELING OF SEISMICITY 21
a manner that favors aseismic creep near the surface and at great depth, but stick-slip
behavior in between.
We show that the evolution of sliding speed and fault state is fully described by a set
of only two first order ordinary differential equations that are a function of only four
non-dimensional parameters: B/A, W/L, LBσ/GDc, and VplateL/Dcβ.
Magnitude statistics and time-space analyzes of a case-study catalog reveal that the
model reproduces the major characteristics of natural seismicity, including the close to
power-law distribution of event sizes, non-periodic recurrence times, and an Omori type
of temporal clustering prior to and following large earthquakes. Since the model lacks any
stochastic forcing, the emergence of complexity is attributed entirely to the discreteness
of the computational grid, and the non-linearity of the underlying physics.
We examined the effect of changing W and varying A and B throughout the seismogenic
depth on the distribution of earthquake magnitudes and rupture dimensions. We find that
the effect of increasing W , B, and reducing A is to increase the average dimension of the
rupture, and to reduce the ratio of small to large events.
We examined the effect of a stress step applied on the creeping portions of the model. We
confirmed that aftershock sequences resulting from stress relaxation on creeping segments
decay asymptotically to 1/time.
Finally, we pointed out two fundamental differences between seismicity models em-
ploying rate-and-state friction differ fundamentally from seismicity models employing
static/kinetic friction. The first is that the peak strength in models adopting rate-and-
state friction is proportional to the logarithm of the recurrence interval. This is not the
case for models adopting static/kinetic friction. The second difference is that aftershocks
D R A F T April 30, 2006, 3:33pm D R A F T
22 ZIV AND COCHARD: QUASI-DYNAMIC MODELING OF SEISMICITY
in models employing rate-and-state friction model may arise from two different sources;
the first is due to the effect of a stress step on the failure time of seismogenic segments, the
second is due to the relaxation of stresses that are stored in creeping segments. Models
adopting static/kinetic friction are not capable of producing aftershock sequences that
decay according to the modified Omori law. Aftershocks in models that incorporate
static/kinetic friction and creep laws arise solely from the relaxation of stresses in the
creeping segments.
Appendix A: The Scalar Elastostatic Kernel
The expression for the static stress interaction is extracted from the full elastodynamic
stress transfer function, f(x, z, t), that for the 3D scalar model being used here is given
by (see eq. (49) and (57) of Cochard and Rice [1997], and explanation below (57) and in
their Appendix C.1.):
f(x, z, t) =Gβ
2
(
∂2
∂x2+
∂2
∂z2
)∫∫∫
Γ
1
2πβ2(t − t′)2δ(x′, z′, t′) dx′ dz′ dt′ , (5)
where δ(x′, z′, t′) is prior slip, and the integration is carried out over a causality cone, Γ,
satisfying t − t′ >√
(x − x′)2 + (z − z′)2/β.
The elastostatic contribution, f stat(x, z, t), is then extracted by retaining the term with-
out time convolution in the integration by parts of (5) with respect to time (see, e.g.,
Perrin et al., 1995). Transferring the differentiation operators inside the integrals, we get:
f stat(x, z, t) =G
4π
∫∫
x-z plane
1√
(x − x′)2 + (z − z′)2
(
∂2δ(x′, z′, t)
∂x′2+
∂2δ(x′, z′, t)
∂z′2
)
dx′ dz′ .
(6)
D R A F T April 30, 2006, 3:33pm D R A F T
ZIV AND COCHARD: QUASI-DYNAMIC MODELING OF SEISMICITY 23
We introduce the following spatial discretization in which displacement is uniform over
each cell with horizontal and vertical dimensions ∆x and ∆z, respectively:
δ(x′, z′, t) =∑
m
∑
n
δmn(t)(H[x′ − (m −1
2)∆x, z′ − (n −
1
2)∆z]
+H[x′ − (m +1
2)∆x, z′ − (n +
1
2)∆z]
−H[x′ − (m −1
2)∆x, z′ − (n +
1
2)∆z]
−H[x′ − (m +1
2)∆x, z′ − (n −
1
2)∆z]) ,
(7)
with H being the Heaviside function, satisfying H[x, y] = H[x]H[y]. Inserting this ex-
pression in (6) gives:
f statkl (t) =
∑
m
∑
n
Kklmnδmn(t), (8)
where:
Kklmn =g(k − m +1
2, l − n +
1
2)
+g(k − m −1
2, l − n −
1
2)
−g(k − m +1
2, l − n −
1
2)
−g(k − m −1
2, l − n +
1
2) ,
(9)
and for ∆x = ∆z = L:
g(a, b) =−G
4πL
(1 +b
√a2 + b2
a+
1 +a
√a2 + b2
b
)
. (10)
Note that Kklmn is a product of G/L and a non-dimensional number that depends on
k−m and l−n , hence can be formally written with 2 indices only: Kij. Thus, the scalar
kernel is translational invariant along both x and z directions, and the spatial convolution
may be carried out using Fourier methods. In actual calculations, this convolution is done
with δ − Vplatet, rather than δ (i.e., Equation 4).
D R A F T April 30, 2006, 3:33pm D R A F T
24 ZIV AND COCHARD: QUASI-DYNAMIC MODELING OF SEISMICITY
Appendix B: Scaling
The scaling of the dependent variables t, θ and V are chosen as follows: t = t(Dc/Vplate)
, θ = θ(Dc/Vplate), and V = V Vplate, where non-dimensional variables are indicated by a
bar. Equation (2) in non-dimensional form reads:
dθi
dt= 1 − Viθi, (11)
and (4) becomes:
dVi
dt=
[ L
W(1 − Vi) +
∑
j Kij(Vj − 1) −LBiσi
DcG
1
θi
(
1 − Viθi
)]
(1
2
VplateL
Dcβ+
LBiσi
DcG
Ai
Bi
1
Vi
)
. (12)
Note that the governing equations are a function of four dimensionless parameters: Bi/Ai,
W/L, LBiσi/GDc, and VplateL/Dcβ. Additional factors that affect the result of such mod-
els are the model dimensions and the spatial distribution of the constitutive parameters.
We now discuss the physical significance of these parameters.
VplateL/Dcβ: The seismic cycle consists of two major phases, the interseismic phase
during which the fault is stressed slowly, and the seismic phase during which stress is
unloaded abruptly. While the duration of the interseismic phase is governed by Dc/Vplate,
that of the seismic phase is controlled by L/β. Thus, VplateL/Dcβ is a ratio between
two very different time scales. From a computational standpoint, since VplateL/Dcβ 1,
simulating many seismic cycles requires a time stepping algorithm with an adaptive time
step.
LBiσi/GDc: Dieterich [1992] examined the evolution of slip on a crack that is subjected
to slow stressing rate, and showed (both analytically and numerically) that unstable slip
is preceded by self-accelerating creep that is concentrated on a patch with dimensions that
D R A F T April 30, 2006, 3:33pm D R A F T
ZIV AND COCHARD: QUASI-DYNAMIC MODELING OF SEISMICITY 25
are equal to ηGDc/Bσ, where η is a dimensionless constant with value near one. Thus,
GDc/Bσ may be identified as Lc, and LBiσi/GDc as the ratio between L and Lc.
W/L: During interseismic periods, the stress on a given cell varies due to tectonic
displacement imposed on parallel planes, situated at distance W/2 on either side of the
fault plane, and due to slip on the fault plane. While the contribution from tectonic slip
applied on parallel planes is modulated by G/W , that from interaction with co-planar
slip is modulated by G/L. Thus, W/L may be viewed as a ratio between two elastic
coefficients, and as a measure of the importance of interaction with co-planar slip (seismic
and aseismic) relative to tectonic stressing.
B/A: If B/A > 1, friction is velocity-strengthening and it favors stick-slip behavior. If
on the other hand B/A < 1, friction is velocity-weakening and it favors aseismic creep.
D R A F T April 30, 2006, 3:33pm D R A F T
26 ZIV AND COCHARD: QUASI-DYNAMIC MODELING OF SEISMICITY
References
Beeler, N. M., S. H. Hickman, and T. -F. Wong, Earthquake stress drop and laboratory-
inferred interseismic strength recovery, J. Geophys. Res., 106, 30,701–30,713, 2001.
Ben-Zion, Y., Stress, slip, and earthquakes in models of complex single-fault systems
incorporating brittle and creep deformations, J. Geophys. Res., 101, 5677–5706, 1996.
Blanpied, M. L., D. A. Lockner, and J. D. Byerlee, Fault stability inferred from granite
sliding experiments at hydrothermal conditions, Geophys. Res. Lett., 18, 609-612, 1991.
Blanpied, M. L., C. J. Marone, D. A. Lockner, J. D. Byerlee, and D. P. King, Quantitative
measure of the variation in fault rheology due to fluid-rock interactions, J. Geophys.
Res., 103, 9691–9712, 1998.
Brace, W. F., and J. D. Byerlee, Stick slip as a mechanism for earthquakes, Science, 153,
990–992, 1966.
Cochard, A., and R. Madariaga, Dynamic faulting under rate-dependent friction, PA-
GEOPH, 142, 419–445, 1994.
Cochard, A., and R. Madariaga, Complexity of seismicity due to highly rate-dependent
friction, J. Geophys. Res., 101, 25321–25336, 1996.
Cochard, A., and J. R. Rice, A spectral method for numerical elastodynamic fracture
analysis without spatial replication of the rupture event, J. Mech. Phys. Solids, 45,
1393–1418, 1997.
Dieterich, J. H., Modeling of rock friction, 1. Experimental results and constitutive equa-
tions, J. Geophys. Res., 84, 2161–2168, 1979.
Dieterich, J. H., Earthquake nucleation on faults with rate- and state-dependent strength,
Tectonophysics, 211, 115–134, 1992.
D R A F T April 30, 2006, 3:33pm D R A F T
ZIV AND COCHARD: QUASI-DYNAMIC MODELING OF SEISMICITY 27
Dieterich, J., A constitutive law for rate of earthquake production and its application to
earthquake clustering, J. Geophys. Res., 99, 2601–2618, 1994.
Dieterich, J. H., Earthquake simulations with time-dependent nucleation and long-range
interaction, Nonlinear Proc. Geophys., 2, 109–120, 1995.
Dieterich, J. H., and B. Kilgore, Direct observation of frictional contacts: New insights
for state-dependent properties, Pure Apply. Geophys., 143, 283–302, 1994.
Dieterich, J. H., and B. Kilgore, Implications of fault constitutive properties for earthquake
prediction, Proc. Natl. Acad. Sci., 93, 3787–3794, 1996.
Gross, S., and C. Kisslinger, Estimating tectonic stress rate and state with Landers after-
shocks, J. Geophys. Res., 102, 7603–7612, 1997.
Hanks, T. C., and H. Kanamori, A moment magnitude scale, J. Geophys. Res., 84, 2348–
2350, 1979.
Helmstetter, A., and D. Sornette, Foreshocks explained by cascades of triggered seismicity,
J. Geophys. Res., 108, 2457, doi:10.1029/2003JB002409, 2003.
Horowitz, F. G., and A. Ruina, Slip patterns in a spatially homogeneous fault model, J.
Geophys. Res., 94, 10279-10298, 1989.
Jones, L. M., and P. Molnar, Some characteristics of foreshocks and their possible rela-
tionship to earthquake prediction and premonitory slip on fault, J. Geophys. Res., 84,
3,596–3,608, 1979.
Kagan, Y. Y., and L. Knopoff, Statistical study of the occurrence of shallow earthquakes,
Geophys. J. R. Astr. Soc., 55, 67–86, 1978.
Kanamori, H., and C. Allen, Earthquake repeat time and average stress drop, In Earth-
quake Source Mechanics, AGU Geophys. Mono. 37, eds. S. Das, J. Boatwright, and C.
D R A F T April 30, 2006, 3:33pm D R A F T
28 ZIV AND COCHARD: QUASI-DYNAMIC MODELING OF SEISMICITY
Scholz. Washington, DC: American Geophysical Union, pp. 227–236, 1986.
Lapusta, N., J. R. Rice, Y. Ben-Zion, and G. Zheng, Elastodynamic analysis for slow
tectonic loading with spontaneous rupture episodes on faults with rate- and state-
dependent friction, J. Geophys. Res., 105, 23765–23789, 2000.
Lapusta, N., and J. R. Rice, Nucleation and early seismic propagation of small
and large events in a crustal earthquake model, J. Geophys. Res., 108, 2205,
doi:10.1029/2001JB000793, 2003.
Marone, C., Laboratory-derived friction laws and their applications to seismic faulting,
Annu. Rev. Earth Planet. Sci., 26, 643–696, 1998.
Marone, C., and C. Scholz, The depth of seismic faulting and the upper transition from
stable to unstable slip regimes, Geophys. Res. Lett., 15, 621–624, 1988.
Marone, C., W. Ellsworth, and J. Vidale, Fault healing inferred from time-dependent
variations of source properties of repeating earthquakes, Geophys. Res. Lett., 22, 3095–
3099, 1995.
Ohnaka, M., Frictional characteristics of typical rocks, J. Phys. Earth, 23, 87–102, 1975.
Omori, F., On the aftershocks of earthquakes, J. Coll. Sci. Imp. Univ. Tokyo, 7, 111–120,
1894.
Papazachos, B., Foreshocks and earthquake prediction, Tectonophysics, 28, 213-226, 1975.
Peng, Z., J.E. Vidale, C. Marone and A. Rubin, Systemic variations in recurrence
interval and moment of repeating aftershocks, Geophys. Res. Lett., 32, L15301,
doi:10.1029/2005GL022626, 2005.
Perrin, G. and J. R. Rice, Disordering of a dynamic planar crack front in a model elastic
medium of randomly variable toughness, J. Mech. Phys. Solids, 42, 1047–1064, 1994.
D R A F T April 30, 2006, 3:33pm D R A F T
ZIV AND COCHARD: QUASI-DYNAMIC MODELING OF SEISMICITY 29
Perrin, G., J. R. Rice, and G. Zheng, Self-healing slip pulse on a frictional surface, J.
Mech. Phys. Solids, 43, 1461–1495, 1995.
Perfettini, H., J. Schmittbuhl, and A. Cochard, Shear and normal load perturbations on
a two-dimensional continuous fault: 1. Static triggering, J. Geophys. Res., 108, 2408,
doi:10.1029/2002JB001804, 2003.
Perfettini H, Avouac JP, Postseismic relaxation driven by brittle creep: A possible
mechanism to reconcile geodetic measurements and the decay rate of aftershocks,
application to the Chi-Chi earthquake, Taiwan, J. Geophys. Res., 109, B02304,
doi:10.1029/2003JB002488, 2004.
Press, W. H., B. P. Flannery, S. A. Teukolsky, and W. T. Vetterling, Numerical Recipes
in Fortran 77, 933 pp., Cambridge University Press, 1992.
Purcaru, G., and H. Berckhemer, A magnitude scale for very large earthquakes, Tectono-
physics, 49, 189-198, 1978.
Rice, J. R., Spatio-temporal complexity of slip on a fault, J. Geophys. Res., 98, 9885-9907,
1993.
Rice, J. R., and Y. Ben-Zion, Slip complexity in earthquake fault models, Proc. Natl.
Acad. Sci. USA, 93, 3811–3818, 1996.
Rubin, A. M., Aftershocks of microearthquakes as probes of the mechanics of rupture, J.
Geophys. Res., 107, doi:10.1029/2001JB000496, 2002.
Ruina, A. L., Friction laws and instabilities: A quasistatic analysis of some dry frictional
behavior, Ph.D. thesis, Brown Univ., Providence, R. I., 1980.
Ruina, A., Slip instability and state variable friction laws, J. Geophys. Res., 88, 10359–
10370, 1983.
D R A F T April 30, 2006, 3:33pm D R A F T
30 ZIV AND COCHARD: QUASI-DYNAMIC MODELING OF SEISMICITY
Schaff, D. P., G. C. Beroza, and B. E. Shaw, Postseismic response of repeating aftershocks,
Geophys. Res. Lett., 25, 4549–4552, 1998.
Scholz, C. H., Size distributions for large and small earthquakes, Bull. Seism. Soc. Am.,
87, 1074–1077, 1997.
Scholz, C. H., The Mechanics of Earthquakes and Faulting, 2nd ed., Cambridge University
Press, New York, 2002.
Scholz, C. H., C. A. Aviles, and S. G. Wesnousky, Scaling differences between large
intraplate earthquakes, Bull. Seism. Soc. Am., 76, 65–70, 1986.
Shaw, B. E., Generalized Omori law for foreshocks and aftershocks from a simple dynam-
ics, Geophys. Res. Lett., 20, 907–910, 1993.
Shaw, B. E., and J. R. Rice, Existence of continuum complexity in the elastodynamics of
repeated fault ruptures, J. Geophys. Res., 105, 23791–23810, 2000.
Shibazaki, B., and Y. Iio, On the physical mechanism of silent slip events along the deeper
part of the seismogenic zone, Geophys. Res. Lett., 30, 1489, doi:10.1029/2003GL017047,
2003.
Shimamoto, T., A transition between frictional slip and ductile flow undergoing large
shearing deformation at room temperature, Science, 231, 711–714, 1986.
Stein, R. S., The role of stress transfer in earthquake occurrence, Nature, 402, 605–609,
1999.
Thatcher, H., and T. C. Hanks, Source parameters of southern California earthquakes, J.
Geophys. Res., 78, 8547–8576, 1973.
Toda, S., R. S. Stein, P. A. Reasenberg, J. H. Dieterich, and A. Yoshida, Stress transferred
by the 1995 Mw = 6.9 Kobe, Japan, shock: Effect on aftershocks and future earthquake
D R A F T April 30, 2006, 3:33pm D R A F T
ZIV AND COCHARD: QUASI-DYNAMIC MODELING OF SEISMICITY 31
probabilities, J. Geophys. Res., 103, 24543–24565, 1998.
Tse, S. T., and J. R. Rice, Crustal earthquake instability in relation to depth variation of
frictional slip properties, J. Geophys. Res., 91, 9452–9472, 1986.
Utsu, T., A statistical study on the occurrence of aftershocks, Geophys. Mag, 30, 521–605,
1961.
Vidale, J. E., W. L. Ellsworth, A. Cole, and C. Marone, Variations in rupture processes
with recurrence interval in a repeated small earthquake, Nature, 368, 624–626, 1994.
Ziv, A., Foreshocks, aftershocks and remote triggering in quasi-static fault models, J.
Geophys. Res., 108, 2498, doi:10.1029/2002JB002318, 2003.
Ziv, A., and A. M. Rubin, Implications of rate-and-state friction for properties of af-
tershock sequence: quasi-static inherently discrete simulations, J. Geophys. Res., 108,
2051, doi:10.1029/2001JB001219, 2003.
Ziv, A., A. M. Rubin, and D. Kilb, Spatio-temporal analyzes of earthquake productivity
and size distribution: Observations and simulations, Bull. Seism. Soc. Am., 93(5),
2069-2081, 2003.
Zoller G., S. Hainzl, M. Holschneider, Y. Ben-Zion 2005, Aftershocks resulting
from creeping sections in a heterogeneous fault, Geophys. Res. Lett., 32, L03308,
doi:10.1029/2004GL021871, 2005a.
Zoller, G., M. Holschneider, and Y. Ben-Zion, The role of heterogeneities as a tun-
ing parameter of earthquake dynamics, Pure Appl. Geophys., 162, 1077-1111, DOI
10.1007/s00024-004-2662-7, 2005b.
D R A F T April 30, 2006, 3:33pm D R A F T
32 ZIV AND COCHARD: QUASI-DYNAMIC MODELING OF SEISMICITY
128 cells, 64 km
64 cells, 32 km
32 km
PSfrag replacements
:rate-and state-dependent frictionapplies; depth-variable properties
:slip is imposed at
slip is imposed at
uniform rate that is
uniform rate that is
equal to plate
equal to plate
:
:equal to plate
periodic in -direction
depth
Figure 1. Schematic diagram showing the fault. The region over which motion is
calculated is covered by 128 × 64 computational cells. Displacement at rate Vplate is
imposed on a co-planar substrate below the computational grid, and on fault-parallel
planes located at distance W/2 on either side of the fault plane.
D R A F T April 30, 2006, 3:33pm D R A F T
ZIV AND COCHARD: QUASI-DYNAMIC MODELING OF SEISMICITY 33
0
1
2
3
4
log 1
0 cu
mul
ativ
e nu
mbe
r
5 6 7 8M
slope = -1
slope = -2
Figure 2. Cumulative diagram of earthquakes with magnitude greater than M . For
reference we added lines indicating slopes of −1 and −2.
D R A F T April 30, 2006, 3:33pm D R A F T
34 ZIV AND COCHARD: QUASI-DYNAMIC MODELING OF SEISMICITY
10-1
100100
101
102
103
# co
unt
(a) M > 6.0, α/Ω = 0.40
10-1
100100
101
102
# co
unt
(b) M > 6.5, α/Ω = 0.55
10-1
100100
101
# co
unt
0 100 200recurrence time
(c) M > 7.0, α/Ω = 1.25
Figure 3. Histograms showing the distribution of time intervals between successive
events with magnitude equal to or greater than: (a) 6, (b) 6.5, and (c) 7. The ratio of α
to Ω is indicated in each frame.
D R A F T April 30, 2006, 3:33pm D R A F T
ZIV AND COCHARD: QUASI-DYNAMIC MODELING OF SEISMICITY 35
0
500
1000
coun
t
-8 -4 0 4 8time from mainshock [year]
Figure 4. Cumulative number of events as a function of lag-time, with respect to the
time of the stacked mainshock. The stacked mainshock is obtained by stacking the five
largest earthquakes in the catalog.
101
102102
103
104
105
106
107
eart
hqua
ke r
ate
[1/y
ear]
10-610-510-510-410-310-210-1100101102
time before [year]
101
102102
103
104
105
106
107
eart
hqua
ke r
ate
[1/y
ear]
10-6 10-510-5 10-4 10-3 10-2 10-1 100 101 102
time after [year]
Figure 5. A Plot of earthquake production rate as a function of time, calculated for
a composite catalog. For reference, we added dashed lines that indicate a decay rate of
1/time.
D R A F T April 30, 2006, 3:33pm D R A F T
36 ZIV AND COCHARD: QUASI-DYNAMIC MODELING OF SEISMICITY
0
1
2
3
4lo
g 10
cum
ulat
ive
num
ber
5 6 7 8M
a) B = 0.0400.0350.030
5 6 7 8M
b) A = 0.0050.0070.009
5 6 7 8M
c) W = 5 Km7 Km9 Km
Figure 6. Cumulative diagram of earthquakes with magnitude greater than M for
various values of: (a) B, (b) A, and (c) W . The total number of events for each curve is
equal to 104.
D R A F T April 30, 2006, 3:33pm D R A F T
ZIV AND COCHARD: QUASI-DYNAMIC MODELING OF SEISMICITY 37
1.5
2.0
Ave
rage
are
a/#
of c
ells
0.030 0.035 0.040B
a)
1
2A
vera
ge a
rea/
# of
cel
ls
0.006 0.008A
b)
2
3
4
Ave
rage
are
a/#
of c
ells
2500 3000 3500 4000 4500W [m]
c)
Figure 7. Diagrams showing average rupture dimensions normalized by the cell di-
mensions as a function of: (a) B, (b) A, and (c) W . The star denotes the case-study
catalog.
D R A F T April 30, 2006, 3:33pm D R A F T
38 ZIV AND COCHARD: QUASI-DYNAMIC MODELING OF SEISMICITY
100
101101
102
103
104
105
106
eart
hqua
ke r
ate
[1/y
ear]
10-6 10-510-5 10-4 10-3 10-2 10-1 100 101 102
time since the stress application [year]
Figure 8. Aftershock rates as a function of time since a stress step of 1 MPa applied
on the creeping substrate (solid) and the entire model (dashed). For reference, we added
a dotted line that indicates a decay rate of 1/time.
D R A F T April 30, 2006, 3:33pm D R A F T
ZIV AND COCHARD: QUASI-DYNAMIC MODELING OF SEISMICITY 39
-1.0
-0.8
-0.6
-0.4
-0.2
0.0
0.2
0.4
µ-µss
6000 7000 8000 9000Years
Figure 9. The evolution as a function of time of µ− µss on a single cell that is located
in the middle of the seismogenic layer.
D R A F T April 30, 2006, 3:33pm D R A F T
40 ZIV AND COCHARD: QUASI-DYNAMIC MODELING OF SEISMICITY
0.0
0.1
0.2
0.3
0.4
µpeak
2 3 4 5 6 7 8ln(tr Vplate/Dc)
Figure 10. A plot showing the peak strength as a function of the logarithm of recurrence
times (normalized by Vplate/Dc), for the data shown in Figure 9. The solid line is a solution
of: µpeak − µss = B ln(trDc/Vplate).
D R A F T April 30, 2006, 3:33pm D R A F T