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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-

[email protected])

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-



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



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



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



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.



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



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



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.



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.



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



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.



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.



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



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



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



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



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.,



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



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



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)



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).



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



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.



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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.



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.



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.



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.



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.



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.



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.



-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.



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).


quasi-dynamic modeling of seismicity on a ... - bgu.ac.il · e-mail: [email protected] a. ziv,...

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