WHAT CONTROLS THE SPATIAL DISTRIBUTION OF REMOTE AFTERSHOCKS? · WHAT CONTROLS THE SPATIAL DISTRIBUTION OF REMOTE AFTERSHOCKS? A. Ziv ... Frictional resistance on cell ... ij being

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WHAT CONTROLS THE SPATIAL DISTRIBUTION OF

REMOTE AFTERSHOCKS?

A. Ziv

Dept. of Geological and Environmental Sciences, Ben-Gurion University of the Negev,

Beer-Sheva, Israel

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]

Short title: AFTERSHOCKS OF AFTERSHOCKS

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Abstract.

Motivated by recent studies showing that stress modifications due to small quakes

are important in earthquake triggering, we use a quasi-dynamic spatially discrete model

of a rate- and state-dependent fault to examine the effect of multiple elastic stress

transfers on the spatial extent of aftershock activity. We show that multiple stress

transfers may significantly alter the spatial distribution of aftershock sequences, and

give rise to stress-seismicity relations that are inconsistent with Dieterich’s aftershock

model. Specifically, we present an example in which, owing to the effect of multiple

elastic stress transfers, the area experiencing seismicity rate change is much larger

than that subjected to a stress change. We define a parameter that quantifies the

proximity to the failure in the model, and show that differences in the magnitude of

aftershock rates are due to differences in the proximity to failure that prevailed prior

to the mainshock. Furthermore, we show that accelerating and decelerating cumulative

Benioff strains are indicative of a region approaching to or moving away from failure,

respectively. Finally we compare the cumulative Benioff strains in remote sites triggered

either by the Landers or the Hector Mine earthquake, and infer that differences in the

state of stress that prevailed prior to these mainshocks may explain the different spatial

distribution of their remote aftershocks.

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1. Introduction

There is a growing body of evidence indicating that stress modifications due to

small-to-moderate quakes are important in earthquake triggering. For example, Felzer

et al. [2002] examined the possibility that the timing of the Mw 7.1 Hector Mine rupture

has been advanced by the Mw 7.3 Landers earthquake, and concluded that the triggering

effect of the Mw 5.4 Pisga earthquake, which ruptured 7 days after the Landers, was

more important than the triggering effect of Landers at the future location of Hector

Mine. In addition, Ziv [2006] introduced a new method for quantifying the degree to

which the triggering effect of an aftershock is locally more important than that of the

mainshock. Application of this method to the Landers and the Hector Mine remote

aftershock sequences showed that many remote aftershocks were not directly triggered

by the Landers and Hector Mine, but were instead triggered by previous aftershocks.

Despite the numerous studies showing that stress modifications due to small quakes

are important in the redistribution of stresses in the brittle crust [Hanks, 1992; Felzer

et al., 2002; Felzer et al., 2003; Helmstetter et al., 2003; Ziv, 2003; Marsan, 2005; Ziv,

2006], probabilistic earthquake models that neglect the effect of multiple stress transfers

are widely used [e.g., Dieterich and Kilgore, 1996; Stein et al., 1997; Toda et al., 1998;

Parsons et al., 2000; Toda and Stein, 2002; Hardebeck, 2004; Parsons, 2004; Gomberg

et al., 2005]. It is therefore instructive to asses the degree to which probabilistic models

neglecting multiple interactions may differ from those accounting for that effect. In

this study we take a step towards resolving this question, and examine the effect of

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multiple stress transfers on the spatial extent of aftershock activity. This question is

addressed numerically using a quasi-dynamic spatially discrete model of a rate- and

state-dependent fault.

This paper is organized as follows. In Section 2 we describe the quasi-dynamic rate-

and state-dependent seismicity model. In Section 3 we compare properties of aftershock

sequences simulated with the quasi-dynamic model to those predicted by Dieterich’s

aftershock model [Dieterich, 1994]. We show that multiple interactions could give rise

to stress-seismicity relations that are inconsistent with Dieterich’s model, if the stress

perturbation is heterogeneous. In Section 4 we show that the magnitude of aftershock

rate depends on the degree to which the region in question was close to failure prior

to the application of stress. In Section 5 we demonstrate that the rate at which the

cumulative Benioff strain (and possibly other cumulative seismicity functions) increases

is indicative of proximity to the failure threshold. Specifically we show that accelerating

and decelerating cumulative Benioff strains in a given region of the model are indicative

of that region approaching to or moving away from failure, respectively. Finally in

Section 6, we compare the cumulative Benioff strains in remote sites triggered by the

Landers and the Hector Mine earthquakes. Based of this analysis we suggest that

differences in the state of stress that prevailed prior to these mainshocks may explain

the different spatial distribution of their remote aftershocks.

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2. Overview of the Quasi-Dynamic Rate- and State-Dependent

Seismicity Model

Here we detail the model’s governing equations and boundary conditions. We

wish to emphasize that the model used in this study incorporates elements that exist

in several previous models. Fault configuration and boundary conditions are as in the

quasi-static model of Dieterich [1995]. The motion is quasi-dynamic due to the inclusion

of a radiation dumping term as in Rice [1993]. Finally, the time stepping procedure is

as in Perfettini et al. [2003].

We model an infinite 2D fault that is embedded in an infinite elastic medium. The

fault is subjected to a 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. The fault

plane is discretized by a periodic grid of 64 by 64 square cells. Slip 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 ln

Vplateθi(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, A and B are unitless constitutive parameters,

and Dc is a characteristic distance for the evolution of the state from one steady state

to another. The state evolves with slip and time according to [Ruina, 1980]:

dθi

dt= 1 −

θiVi

Dc

. (2)

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Stress is related to slip and sliding speed according to:

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. The second term represents the driving stresses

imparted on the fault surface due to differences between the total displacement on the

point in question, δi, and the cumulative tectonic slip, Vplatet, with G being the shear

modulus. The third term accounts for elastic interaction, with L being the length of

the computational cell and Kij being a scalar non-dimensional elasto-static kernel (as

in Ziv and Cochard [submitted]). 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’. Stress balance and derivation with

respect to time yields:

dVi

dt=

[

GW

(Vplate − Vi(t)) + GL

∑

j Kij(Vj(t) − Vplate) −σiBi

θi(t)

(

1 −Vi(t)θi(t)

Dc

)]

(

G2β

+ σiAi

Vi(t)

) . (4)

The evolution of V and θ is fully described by (2) and (4), which we solve simultaneously

at successive time steps using a fifth-order adaptive time step Runge-Kutta algorithm

[Press et al., 1992].

We consider that an earthquake starts each time the sliding speed on a cell exceeds

a centimeter per second, and stops after the state variable passes through a steady

state (i.e., when the sign of dθ/dt changes from negative to positive). In principle more

than one cell may undergo seismic slip at the same time. While in reality simultaneous

ruptures are recorded as a single event, with the current algorithm they are recorded

separately. For the choice of parameters used in this study (detailed below), however,

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simultaneous ruptures are very rare. Because such events are rare, treating them as

separate events is not a problem.

The constitutive parameters are distributed uniformly. Results presented here

were obtained with model parameters as follows: A = 0.01, B = 0.06, Dc = 10−2m,

W = 2.5Km, L = 500m, β = 103m/s, Vplate = 10cm/year, G = 10GPa, and

σ = 100MPa. The simulation begins with random initial distribution of Vi and θi, and

is run long enough until a macroscopic steady-state is reached. When analyzing the

results, the output of the transient regime is disregarded. As we shall see later, even

in the absence of an external stress perturbation, earthquake production rate is not

always constant. Like natural seismicity, simulated seismicity too exhibits intervals of

accelerating and decelerating activity.

3. Aftershocks of Aftershocks

We compare properties of aftershock sequences simulated with the quasi-dynamic

model to those predicted by Dieterich’s aftershock model [Dieterich, 1994]. While the

first accounts for the effect of elasto-static stress transfers from early aftershocks, the

latter does not. Next, in order to provide context for the analysis that follows, we

summarize Dieterich’s main results.

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3.1. The reference model

According to Dieterich [1994], aftershock production rate, R, depends on time and

stress history as:

R =R0

γτand dγ =

1

Aσ(∂t − γ∂τ), (5)

where R0 is the background earthquake rate and τ is the stressing rate, which in our

model equals GVplate/W . Solution of (5) for a stress step of ∆τ applied at t = 0 that

is preceded and followed by a constant stressing rate of τ yields (i.e., equation 12 of

Dieterich [1994]):

R = R0

[

[

exp(−∆τ

Aσ

)

− 1]

exp(−tτ

Aσ

)

+ 1

]

−1

. (6)

At t = 0, i.e. immediately following the stress application, R = R0 exp ∆τ/Aσ.

Additionally, R → R0 for t > Aσ/τ . It is therefore a common practice to normalize

the stress change by Aσ, and to non-dimensionalize the time with ta = Aσ/τ . In the

quasi-dynamic model, τ = VplateG/W , and ta = 2.5 years.

3.2. Consequences of heterogeneous stress change

Ziv and Rubin [2003] investigated the effect on aftershock production rate of a static

stress step applied uniformly on a fault. They found a close match between simulated

and Dieterich’s predicted rates. This result could be misinterpreted as indicating that

multiple interactions are unimportant. We next show that multiple interactions could

significantly alter the spatial extent of the aftershock activity if the stress perturbation

is heterogeneous.

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In order to examine the effect of a heterogeneous stress step on aftershock

distribution, we have imposed a uniform stress step of 4Aσ on a sub-region with

dimensions of 32 by 32 cells, referred to as zone-1 in the inset of Figure 1a. Outside this

region, in zone-2 and zone-3, the stress has been left unchanged. While zone-1 occupies

25% of the fault total area, zone-2 and zone-3 occupy 31% and 44%, respectively. Thus,

in this example, the stress perturbation is heterogeneous in that it is only applied on a

part of the model. In Figure 1 we compare simulated time-dependent aftershock rate

(solid) induced by a permanent stress step to the corresponding solutions of 6 (dashed)

for the three regions. Note that seismicity curve in the perturbed area (in zone-1) is in

good agreement with the predicted response. On the other hand, the seismicity curve in

zone-2 shows a significant increase in earthquake rate with respect to the background

rate. There seems to be also a small increase in the seismicity rate of zone-3, but

because this change is so subtle we do not investigate it further. Aftershocks outside

zone-1 are aftershocks of aftershocks, caused by stress perturbations induced by early

aftershocks. Thus in this example, owing to the effect of multiple elastic stress transfers,

the area experiencing seismicity rate change is much larger than that subjected to a

stress change.

During aftershock activity, the average slip rate is greater than Vplate. This access

slip is compensated for by intervals of quiescence, during which earthquake production

rate falls below the long-term average rate. Interestingly, intervals of quiescence

following aftershock activities are seen in all three regions.

Given an earthquake catalog and an estimate of τ , Equation (5) may be employed

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to infer changes in normalized stress [Dieterich et al., 2000]. Application of this concept

to the aftershock data in zone-2 indicates that the stress in this area increased gradually

and reached a maximum after about 0.02Aσ/τ (20 days).

Clearly, the results presented above are sensitive to the model parameters.

Inspection of (3) reveals that while the tectonic stressing is modulated by G/W , the

contribution due to stress transfer induced by slip on the fault is modulated by G/L.

Thus, W/L measures the relative importance of multiple stress transfers relative to

the tectonic stressing. Consequently, larger W/L gives rise to a greater seismicity rate

change in Zone-2. Another quantity that affects the result is (B − A). This is because

increasing (B −A) increases the co-seismic stress drop proportionally [e.g., Beeler et al.,

2001], and therefore increases the effect of multiple interactions. Finally, owing to the

discreteness of the computational grid, the near-tip stress concentration in our model is

under-represented, and the stress perturbation that early shocks induce at the site of

later shocks is lower than what it should be. Thus, the result presented above provides

only a minimum estimate of the effect of multiple stress transfers.

In reality, the stress perturbation induced by an earthquake decays with distance

from the rupture plane. Here, in order to facilitate the analysis of the result and simplify

the comparison between simulated and predicted seismicity response to a stress change,

we examined an unrealistic case. Because the stress perturbation is unrealistic, it is

of course impossible to scale it to a certain earthquake magnitude. Nevertheless, Ziv

[2003] have already demonstrated that modeled earthquakes, which perturb the static

stress field in a realistic manner, can trigger aftershocks in areas that are located several

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mainshock radii away from the mainshock hypocenter. What is still not understood is

what controls the spatial distribution of these aftershocks, and why some earthquakes

trigger remote aftershocks while others do not.

4. What Controls the Magnitude of Aftershock Rate in Areas

of Zero Stress Application?

Each simulated curve in Figure 1 was obtained by averaging the result of fifty

independent simulations, having different distribution of initial slip rate and state.

The averaging over many simulations was necessary due to the great variability of the

seismicity response to a stress step from one simulation to another. What gives rise

to this variability? Here we show that two main factors affect the the magnitude of

aftershock rate in zone-2; one is the magnitude of earthquake rate increase in zone-1,

and the other is the degree to which zone-2 was close to the failure threshold prior to

the application of stress.

Since the aftershocks in zone-2 are aftershocks of the aftershocks in zone-1, the

greater the number of aftershocks in zone-1, the greater should be the number of

aftershocks in zone-2. Guided by this reasoning, we compare, for each simulation, the

total number of quakes that occurred in zone-2 during the first ta after the application of

stress with the corresponding number of quakes in zone-1 (Figure 2a). Indeed, a general

tendency is apparent for the number of aftershocks in zone-2 to increase proportionally

to the number of aftershocks in zone-1. Yet, the large scattering of the data indicates

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that the number of aftershocks in zone-1 is not the only factor affecting the magnitude

of aftershock triggering in zone-2. Furthermore, several inconsistencies may be seen.

For example, while the number of aftershocks in zone-1 produced by simulations 27 and

31 were about the same, the number of aftershocks in zone-2 produced by simulation 31

was an order of magnitude larger than that produced by simulation 27. Additionally,

simulations 30 and 36 produced similar number of aftershocks in zone-2, but the latter

produced an order of magnitude more aftershocks in zone-1 than the former.

The other factor affecting the magnitude of aftershock triggering in zone-2 is

the degree to which this area was close to failure before the application of stress.

What would be a sensible measure of proximity to failure in this model? Because

the characteristic time scale for aftershock relaxation in the model, ta, is equal to 2.5

years, it can be determined that a given segment (i.e., a computational cell) would

rupture during the aftershock sequence if it was 2.5 years away from failure prior to

the application of stress. In Figure 3 we show the evolution of V/Vplate, θVplate/Dc

and V θ/Dc as a function of time on an arbitrary cell. Note that these time series are

non-periodic. Abrupt jumps in the rate-versus-time curve are due to stress steps induced

by co-seismic slip on nearby cells. These time series are different from those calculated

for a single block-slider models, where the evolution of V (and therefore V θ/Dc) between

subsequent ruptures is continuous, and enters the self-accelerating stage when V θ/Dc

exceeds unity. By inspection of time series that are much longer than the one shown in

Figure 3 we determine that, on average, a segment is less than 2.5 years from failure if

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V θ/Dc > 1.E − 10. We thus define a proximity to failure parameter, Ω, to be:

Ω = Ω + 1 if (Viθi/Dc > 1.E − 10) for i ∈ zone-2. (7)

In other words, Ω is the number of cells in zone-2 satisfying V θ/Dc > 1.E − 10 prior to

the application of stress in zone-1. In Figure 2b we show the total number of quakes

that occurred in zone-2 during the first ta after the application of stress against Ω in

zone-2. Note that simulations with large Ω (e.g., simulations 22 and 32) produced

more aftershocks in zone-2 than simulations with small Ω (e.g., simulations 39 and

17). It is now possible to resolve the apparent discrepancies pointed out above. The

reason simulation 31 produced more aftershocks in zone-2 than simulation 27, despite

having the same number of aftershocks in zone-1, is because zone-2 in simulation 31

was closer to failure than it was in simulation 27. Similarly, the reason simulations 30

and 36 produced equal number of aftershocks in zone-2, despite the larger number of

aftershocks in zone-1 produced by simulation 36, is because the distribution of rate and

state parameters prior to the application of stress was more favorable for failure at the

beginning of simulation 30 than at the onset of simulation 36.

Note that neither of the effects discussed above can explain the unexpectedly large

number of aftershocks produced by simulation 49. More detailed examination of that

result revealed that the vast majority of the aftershocks in zone-2 were in fact triggered

by a spontaneous burst of seismicity in zone-3.

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5. Correlation Between the Rate of the Cumulative Benioff

Strain and the Proximity to the Failure Threshold

For practical purposes, since neither the slip rate nor the fault state can be

measured directly, it is necessary to identify a measurable geophysical quantity that

is indicative of the proximity to failure. Changes in the proximity to failure may be

reflected by changes in the seismicity. A function that is commonly used to quantify

changes in the state of seismicity prior to mainshock origin times is the cumulative

Benioff strain, CBS, defined as [Bufe and Varnes, 1993]:

CBS(t) =∑

t

M1/20 (t), (8)

where M0 is the seismic moment. The seismicity is said to be accelerating if this

quantity grows at an increasing rate, and decelerating if it grows at a decreasing rate.

In order to see if the rate at which the CBS grows is correlated with changes in Ω, we

fit the synthetic CBS prior to the the stress application with a power-law function of

the form:

ε(t) = a + b∆tP , (9)

where ∆t is time before the mainshock, and a, b, and P are fitting coefficients, with b

being negative. The exponent, P , is less than unity if CBS rises at an accelerating rate,

and more than unity if it rises at a decelerating rate. To further asses the degree to

which CBS is curved, it is useful to define a curvature parameter, Γ, such that:

Γ =χ2-test of linear fit

χ2-test of power-law fit. (10)

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The larger the Γ, the more curved CBS is.

For each simulation we calculated CBS in zone-2 prior to the stress application

over zone-1, and employed (9) to fit the data. Representative examples of synthetic

CBS and their corresponding power-law (dashed) and linear (dotted) fits are shown

in Figure 4, with P , Γ and Ω indicated in each panel. The interval for which CBS is

calculated is 30 years. The rationale for choosing this time window is that it is less than

the recurrence interval of the model, and yet it is long enough to provide statistically

meaningful fit. We experimented with various intervals and obtained essentially the

same result for time windows between 25 and 35 years.

In Figure 5a we show the logarithm of P as a function of Ω. We use bold indexes to

highlight simulations for which Γ is greater than 2, i.e. with CBS that is either highly

concave or highly convex. Also shown is the least-squares linear fit. Note the negative

correlation between the logarithm of P and Ω. The correlation coefficient (caclulated

according to 11) is equal to −0.55, and in the Appendix we present the result of

randomization tests verifying that this correlation is very unlikely to arise by chance

(less than 1%). We conclude that accelerating and decelerating cumulative Benioff

strains in a given region in the model are indicative of that region approaching to or

moving away from failure, respectively. In Figure 5b we show the logarithm of P as a

function of aftershock count in zone-2. Despite the considerable scatter of the data, we

find significant negative correlation between the logarithm of P and zone-2 aftershock

count. Finally, the results presented above prove the viability of a commonly invoked

(but thus far speculative) idea, according to which triggering following negligible stress

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changes are indicative of a fault system that is close to the failure threshold [e.g., Hill et

al., 1995].

6. Possible Cause for the Different Spatial Distribution of the

Landers and the Hector Mine Remote Aftershocks

We now test the applicability of the previous results to natural seismicity. Both

the M7.3, 1992, Landers and the M7.1, 1998, Hector Mine ruptured northwest-trending

right-lateral faults in the eastern California shear zone. Their temporal and spatial

separations were seven years and fifty kilometers, respectively. Despite these similarities,

the spatial distributions of their remote aftershocks were very different. While remote

sites triggered by the Landers earthquake were located primarily to the north of the

mainshock hypocenter over a large area that extended up to 50 km north of Lake Tahoe,

remote sites triggered by the Hector Mine earthquake were located to the south and

occupied a much smaller area in the Salton Sea and the Imperial Valley. What gave rise

to the dramatic differences in the spatial distribution of remote aftershock sequences of

two fairly similar mainshocks? Gomberg et al. [2001] pointed out that the asymmetry in

the location of remotely triggered sites is similar to the asymmetry in the distribution

of the peak dynamic stress, due to the effect of rupture directivity. This similarity has

been interpreted as a fingerprint of the dynamic role in remote aftershock triggering.

An alternative explanation is put forward here.

Figure 6 shows remote sites triggered either by the Hector Mine (South) or the

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Landers (North2, North1 and LVC) earthquake. Since seismicity in the Long Valley

Caldera (i.e., LVC) is coupled with volcanic activity, it is widely believed that the

physical processes governing earthquake triggering in this site are different than those

in other sites [e.g., Hill et al., 1995]. Thus, CBS were analyzed in regions South,

North2 and North1, but not in LVC (Figure 7). In order to ensure that the analysis is

unaffected by the catalog incompleteness, CBS were computed for magnitudes ≥ 3.

We find that seismicity in South decelerated during the 4 years before the Landers

earthquake, but accelerated during the 4 years prior to the Hector Mine earthquake

(Figure 7a). Furthermore, seismicity in North2 accelerated during the 4 years before

the Landers earthquake, but decelerated during the 4 years that preceded the Hector

Mine earthquake (Figure 7b). These results are consistent with the hypothesis that

remotely triggered sites are areas that were close to failure prior to the mainshock. In

contrast, the evolution of the CBS in area North1 neither accelerated prior to Landers

nor decelerated prior to Hector Mine (Figure 7c), and is therefore inconsistent with the

hypothesis. We suggest that differences in the spatial distribution of remote aftershocks

may be partly due to differences in the state of stress that prevailed prior to each

mainshock, and that remote aftershocks occur in areas that are close to failure.

It is interesting to asses the possibility that the acceleration/deceleration reported

above may arise by chance. To that end we have designed a randomization test. The

probabilities that the acceleration and deceleration of the CBS in North2 before the

Landers and the Hector Mine earthquakes, respectively, may arise by chance were

calculated as follows:

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(a) Randomly choose spatial patches with dimensions of 0.5 × 0.5 degrees within the

studied area (i.e., regions North1, North2 and South combined), which together

makeup the total area of North2.

(b) Combine the seismicity contained in these patches, and compute the CBS for the

combined data set.

(c) Randomly choose an interval of 4 years, apply a least-square algorithm to fit the

data with Equation (9) and record the power exponent, P , and the curvature

parameter, Γ.

(d) Repeat steps (a) through (c) a large number of times.

In Figure 8a we compare the observed log(P ) before Landers (dashed line) and Hector

Mine (dotted line) with that of 100 randomization tests. The numbers indicate the

curvature parameter of each test. Note that amongst the 100 tests, only five gave a

power exponent that are smaller than or equal to the observed exponent before the

Landers earthquake. Of these, only three tests gave a curvature parameter, Γ, that is

greater than or equal to the observed Γ (circles). We thus infer that the probability

that the acceleration of seismicity in North2 before the Landers earthquake may arise

by chance is about 3%. Similarly we find that the probability that the deceleration of

seismicity in North2 before the Hector Mine earthquake may arise by chance is about

5%. In Figure 8b we present the result of similar tests for region South. Based on this

analysis we conclude that the probabilities that the observed CBS deceleration prior to

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the Landers earthquakes and the CBS acceleration prior to the Hector Mine earthquake

may arise in a random data is less than 1%.

Admittedly, while the conclusions of the preceding sections were drawn form

numerous simulations and were confirmed by thorough statistical tests, the results

presented here, since they rely on only two earthquakes, are less conclusive. Clearly,

further investigation is needed in order to ascertain a tendency for distant aftershocks

to occur in sites that experienced accelerating seismicity prior to the mainshock, and be

absent from areas that experienced decelerating seismicity.

7. Summary and Conclusions

We used a quasi-dynamic spatially discrete model of a rate- and state-dependent

fault to examine the effect of multiple elastic stress transfers on the spatial extent of

aftershock activity. We presented an example in which, owing to the effect of multiple

elastic stress transfers, the area that experienced seismicity rate change was much

larger than that subjected to a stress change. On the basis of this example we conclude

that multiple stress transfers from early aftershocks may significantly affect aftershock

distribution, and give rise to complex time-dependent aftershock rates.

We showed that differences in the magnitude of aftershock rates are partly due to

differences in the proximity to failure that prevailed prior to the mainshock, and that

accelerating and decelerating cumulative Benioff strains in a given area are indicative of

that region approaching to or moving away from failure, respectively.

Often, natural aftershock sequences occupy areas were the stress perturbation due

20

to the mainshock alone is negligible [Miller et al., 2004]. The result presented here

indicates that triggering of aftershocks in sites that are located far from the mainshock

could be the result of multiple stress transfers [Ziv, 2003]. Motivated by this view

we compared the cumulative Benioff strains in remote sites triggered by the Landers

and the Hector Mine earthquakes, and infer that differences in the state of stress that

prevailed prior to these mainshocks may explain the different spatial distribution of

their remote aftershocks.

Appendix: Randomization Tests

Inspection of Figure 5a reveals a negative correlation between the logarithm of P

and Ω, with a linear correlation coefficient that is equal to −0.55. Here we perform

randomization tests to test the null hypothesis that such a correlation in the data could

arise by pure chance. The test follows these steps:

(a) Randomly shuffle the data.

(b) Use a least-squares algorithm to fit a straight line to the randomized data set, and

calculate a linear correlation coefficient according to:

r =

∑

i(Ωi − Ω)(log Pi − log P )√

∑

i(Ωi − Ω)2√

∑

i(log Pi − log P )2, (11)

where the bar indicates an average and the subscript i indicates a test index.

(c) Repeat steps a and b a large number of times.

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In Figure 9a we compare the least-squares linear fit of the actual data (solid

line) with that of 100 randomized data sets. In Figure 9b we compare the correlation

coefficient of the actual data set with the correlation coefficients of 100 tests. Since

both the slope of the linear fit and the correlation factor of the actual data set exceed

those of the 100 randomized data sets we can determine that the null hypothesis can be

rejected with a confidence level that is greater than 99%.

22

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Ben-Gurion University of the Negev, Beer-Sheva, Dept. of Geological and

Environmaental Sciences, 84105, Israel. (e-mail: [email protected])

Received

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Figure 1. Earthquake rate as a function of time, calculated for three non-overlapping

regions: (a) Zone-1, (b) Zone-2, and (c) Zone-3. The three regions are shown in the inset

of (a). While zone-1 experienced an instantaneous stress step of 4Aσ at t = 0, zone-2 and

zone-3 did not. Rates and times are normalized by the background rate and ta (see text),

respectively. Dashed lines are Dieterich’s predicted aftershock rates [Dieterich, 1994].

Figure 2. The total number of quakes that occurred in zone-2 during the first ta after the

application of stress as a function of: (a) The corresponding number of quakes in zone-1,

and (b) Ω in zone-2, defined as the number of cells in zone-2 satisfying V θ/Dc > 1.−10

prior to the application of stress in zone-1. The numbers correspond to simulation’s

index. Simulations with bold indexes are discussed in the text. Dashed lines are least-

squares linear fits with linear correlation coefficients in (a) and (b) that are equal to 0.78

and 0.7, respectively.

Figure 3. The evolution as a function of time of: (a) V/Vplate, (b) θVplate/Dc and (c)

V θ/Dc on an arbitrary cell.

Figure 4. Representative examples of synthetic CBS and their corresponding power-law

(dashed) and linear (dotted) fits, with P , Γ and Ω indicated in each panel.

Figure 5. The logarithm of P as a function of (a) Ω and (b) aftershock count in zone-2.

Bold indexes highlight simulations for which Γ is greater than 2.

Figure 6. A location map showing remote sites triggered either by the Hector Mine

(South) or the Landers (North2, North1 and LVC) earthquake.

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Figure 7. Cumulative Benioff strain as a function of time in three regions: (a) South,

(b) North2, and (c) North1. Vertical dashed and dotted lines indicate the times of the

Landers and the Hector Mine earthquakes, respectively. The solid curves are best fit to

Equation (9). The exponent of the best fits, P , and the curvature parameter, Γ, are

indicated next to each curve.

Figure 8. Comparison between observed log(P ) before Landers (dashed line) and Hector

Mine (dotted line) earthquakes and that of 100 randomization tests (numbers), for areas

with dimensions that are equal to: (a) North2 and (b) South. The numbers indicate Γ of

each test (with values that are rounded to the nearest integer).

Figure 9. (a) Comparison between the least-squares linear fit in Figure 5a (solid line)

with that of 100 randomized data sets (dashed lines). (b) Comparison between the

absolute linear correlation coefficient of the actual data set (dashed line) with that of 100

tests (circles). Linear correlation coefficients were calculated according to (11).