The seismic cycle and the difference between foreshocks

and aftershocks in a mechanical fault model

A. ZivBen-Gurion University of the Negev, Beer-Sheva, Israel

J. SchmittbuhlEcole Normale Superieure, Paris, France

Received 19 September 2003; revised 5 November 2003; accepted 11 November 2003; published 16 December 2003.

[1] We examine the evolution of and the exchangebetween two forms of elastic energies stored in the quasi-static fault model of Ziv and Rubin [2003]. The first, Etect, isdue to the integrated slip deficit accumulated between theplate boundaries and the fault surface, and the second, Efault,is the result of differential slip along the fault surface. Theresults of our analysis reveal cyclic exchange between thetwo energies. On a Efault versus Etect plot, the seismic cyclehas a triangular shape with the large earthquakes occurringat the top corner of the triangle (where Efault is maximum),and the foreshocks and the aftershocks occupying the rightside and left side, respectively. While both foreshocksand aftershocks dissipate tectonic energies, the cumulativeeffect of the foreshocks is to increase the potentialelastic energy along the fault plane and the cumulativeeffect of the aftershocks is to reduce it. INDEX TERMS:

3210 Mathematical Geophysics: Modeling; 7209 Seismology:

Earthquake dynamics and mechanics; 7230 Seismology:

Seismicity and seismotectonics; 8168 Tectonophysics: Stresses—

general. Citation: Ziv, A., and J. Schmittbuhl, The seismic cycle

and the difference between foreshocks and aftershocks in a

mechanical fault model, Geophys. Res. Lett., 30(24), 2237,

doi:10.1029/2003GL018665, 2003.

1. Introduction

[2] Since Reid’s [1910] elastic rebound theory was firstproposed, earthquakes are viewed as sudden relaxations ofelastic strains that accumulate steadily over time due torelative motion of the adjacent plates. This concept impliesa cyclic behavior, in which large earthquakes are separatedby interseismic periods during which stress builds uplinearly with time. Indeed, such stress histories are observedin spring-slider systems [e.g., Schmittbuhl et al., 1996] andin 2-D fault models [e.g., Ben-Zion et al., 2003], whereoccasional system-size earthquakes break periodically orquasi-periodically. If Reid’s model applies to the earth, largeearthquakes are time predictable. Yet, neither paleo-seismic[e.g., Marco et al., 1996] nor modern data sets (e.g., seeScholz, 1990 and references therein on the Parkfield exper-iment) confirm cyclic repetition of large earthquakes. Infact, the picture that emerges from many studies is of verycomplex and non-periodic earthquake occurrence.[3] In this study we examine the evolution of elastic

energies in the quasi-static fault model of Ziv and Rubin

[2003]. This model produces non-periodic behavior withforeshocks, aftershocks and close to power law earthquakesize distribution. The results of our analysis reveals cyclic(but not periodic) exchange between two types of elasticenergies: the first arising from slip deficit accumulatedbetween the plate boundaries and the fault surface; thesecond being the result of slip disorder along the faultsurface. We find that while both foreshocks and aftershocksdischarge tectonic energies, the total effect of the foreshocksis to increase the fault disorder and the effect of the after-shocks is to reduce it.

2. Overview of the Quasi-Static Fault Model

[4] To provide a context for the analysis that follows, wefirst summarize the main model ingredients [for moredetails the reader is referred to Ziv and Rubin, 2003] andprevious results relevant for this study. The fault is modeledas a shear crack that is embedded in a homogeneous infiniteelastic medium. The fault surface is represented by aperiodic grid of 50 � 50 square computational cells. Motionon the fault is driven by steady slip imposed on rigidboundaries located at distance W on either side of the fault.Slip on the fault is resisted by friction that is a function ofslip rate _d and fault state q as follow [Dieterich, 1979;Ruina, 1983]:

m _d; q� �

¼ m* þ A ln _d=_d*� �

þ B ln q=q*� �

; ð1Þ

where m is the coefficient of friction, A and B aredimensionless constitutive parameters, m*, _d* and q* arereference values of friction coefficient, velocity and state,respectively. The state, q, evolves with time and slipaccording to [Ruina, 1980]:

dqdt

¼ 1� q_dDc

; ð2Þ

where t is time and Dc is a characteristic sliding distance forthe evolution of q from one steady state to another. Theevolution of state and slip rate is approximated according toa computational procedure developed by Dieterich [1995].The characteristic length-scale Dc gives rise to a criticallength-scale Lc, which defines the minimum dimension of acrack below which instability cannot develop [Dieterich,1992]. Here the size of the computational cells, L, is muchlarger than Lc. Constitutive properties and normal stress areuniform.

GEOPHYSICAL RESEARCH LETTERS, VOL. 30, NO. 24, 2237, doi:10.1029/2003GL018665, 2003

Copyright 2003 by the American Geophysical Union.0094-8276/03/2003GL018665$05.00

SDE 2 -- 1

[5] On a given cell, steady stress increase arising fromtectonic displacement is interrupted by stress steps imposedby coseismic slips. Static stress transfer due to coseismicslip is computed at the center of the cell using the stressfield of a uniform square dislocation in 3-D elastic space. Arupture may grow beyond the size of a single cell if thestress change that is induced by that cell is large enough toinstantaneously bring one or more cells from their currentsliding speed to the seismic sliding speed. The subset ofcells that comprises the rupture set is determined through aniterative procedure.[6] The model is controlled by three nondimensional

numbers [Ziv and Rubin, 2003]. The controlling parame-ters related to the constitutive law are A/B and ta/tc.The latter is a ratio between two time scales: (1) ta isDieterich’s [1994] characteristic time for the return ofseismicity rate to the background rate following a stressstep; and (2) tc = Dc=_dseis, where _dseis is the seismic slipspeed, may be interpreted as the average contact life-timeduring seismic slip. An additional controlling parameterarising from the model discreteness is W/L, whichmeasures the coarseness of the computational grid withrespect to loading distance W. The synthetic cataloginvestigated here was generated with B/A = 8, ta/tc =1010, and W/L = 10. With this choice of W/L, thecontribution to the long-term stressing rate from elasticinteraction with other fault elements is about 10 timeslarger than the contribution from tectonic slip [Equation28 in Ziv and Rubin, 2003].[7] Despite the simplicity of the model, the synthetic

seismicity that we record is extremely complex, and exhibitsmany of the characteristics of natural seismicity [Ziv et al.,2003; Ziv, 2003]. These include the increase of foreshocksrate and the decay of aftershocks rate according to themodified Omori law [Omori, 1894; Utsu, 1961], a close topower law distribution of earthquake sizes, and remoteaftershock triggering. While the Omori law is a conse-quence of the constitutive law [Dieterich, 1994], theincrease in seismicity rate far from the mainshock (wherethe static stress changes imposed by the mainshock weresmall) is a consequence of multiple stress transfers, i.e., thevery distant aftershocks are not directly triggered by themainshock, but instead they are aftershocks of previousaftershocks.

3. Elastic Energies in a Quasi-Static Seismic Fault

[8] Following Schmittbuhl et al. [1996], we introducetwo elastic energy densities per unit area in the systemimmediately after an earthquake k, when no slip takes place:Etectk and Efault

k . The first is due to the slip deficit accumu-lated between the plate boundaries and the fault surface, andis defined as:

Ektect ¼

1

N

G

W

X

i

Utect tkð Þ � Ui tkð Þ½ �2 ð3Þ

where G is the shear modulus, N is the number ofcomputational cells, Utect is the cumulative tectonicdisplacement increasing linearly with time, Ui is thecumulative slip on cell i increasing coseismically, and tk istime immediately after event k. The second energy is the

result of stress transfer due to slip on the fault, and isdefined as:

Ekfault ¼ � 1

N

X

i

X

j

gijUj tkð ÞUi tkð Þ; ð4Þ

where gij is the elastic influence coefficient, relating slip oncell j with stress on cell i in an infinite elastic medium[Equation (4) in Ziv and Rubin, 2003]. Here the influencecoefficient is a scalar function that depends only on distancebetween the two cells, thus gij may be expressed as aproduct of (G/L) and �gij, where the bar indicates anondimensional number. Additionally, gij is symmetricand the sum over all influence coefficients adds up to zero.Using these properties, (4) may be rewritten as:

Ekfault ¼

1

2N

G

L

X

i

X

j

�gij Ui tkð Þ � Uj tkð Þ� �2

: ð40Þ

Written this way, it is easy to see that Efault is a consequenceof the differential displacements along the fault surface.Thus, while Etect is a measure of the mismatch between totalslip of the plate boundary and cumulative slip on the fault,Efault is a measure of the heterogeneity of the slipdistribution. In the followings, results are presented innondimensional form, with both Etect and Efault normalizedby [s2(B � A)2L]/G, and times normalized by ta.[9] The evolution as a function of time of Etect and Efault

is shown in Figure 1 for a sample of 2 � 104 events.Although each cell in this example broke at least 8 times(recall that the grid consist of 50 � 50 cells), no obviousperiodicity may be noticed. The time of occurrence of the9 largest events is depicted by starts. Notice that largeearthquakes occur during the descending portions of theEtect-curve, and close to the maxima of the Efault-curve.

4. Earthquake Classification and the SeismicCycle

[10] Additional insight is gained through inspection ofenergy changes,�Ek, defined as: Ek� Ek�1. Strain energy ischarged if �Etect

k is positive, and dissipated if it is negative.Positive and negative values of �Efault

k correspond to dis-ordering (i.e., storing elastic energy at small scale) andordering of slip distribution, respectively. A plot of �Efault

k -versus-�Etect

k is shown in Figure 2. In this plot, events withrupture area greater than 20 cells are labeled by stars,whereas events with rupture area less or equal to 20 cellsare labeled by dots if the lag-time to previous event isshorter than 0.1ta, and by crosses if the lag-time is longerthan 0.1ta. Three distinct classes of earthquake may berecognized. The first group is characterized by positive�Etect

k and small positive or negative �Efaultk values

(crosses). These earthquakes, since they are small and occurat very low rate during which the fault is being charged, arereferred to as interseismic earthquakes. Earthquakes withlarge negative �Etect

k and large positive �Efaultk form a

second group (stars). Earthquakes belonging to this groupare relatively large, and are therefore referred to as main-shocks. Finally, a third group (dots), which consists ofearthquakes with small negative values of �Etect

k and with

SDE 2 - 2 ZIV AND SCHMITTBUHL: THE SEISMIC CYCLE

small negative or positive �Efaultk values, includes the

foreshocks and the aftershocks. Although in Figure 2 bothforeshocks and aftershocks seem to occupy the same regionand may thus appear to be indistinguishable, the resultsshown in the next section indicate that they play verydifferent roles in the seismic cycle.[11] In Figure 3 we plot phase diagrams of Efault

k -versus-Etectk for the entire data set and for four time windows. The

four time windows, indicated by dashed lines in Figure 1,were chosen such that they contain at least one of the largestearthquakes in the sample. While a phase diagram of theentire data set is rather difficult to interpret, similar dia-grams for shorter time windows provide a clearer view. Thepicture that emerges is of cyclic exchange between the twoenergies, which has a roughly triangular shape (this isillustrated schematically in Figure 3a). Each cycle startswith an interseismic period, during which tectonic energyincreases while fault disorder remains constant. During thattime, small earthquakes occur at very low rate. Later duringthe cycle, tectonic energy discharges and fault disorderincreases. This trend, of converting tectonic energy intofault disorder, ends with a large earthquake. Following thelarge earthquake, tectonic energy decrease is accompaniedby smoothing and homogenization of the slip distributionby aftershocks.[12] It is evident from this representation that foreshocks

and aftershocks play different roles in the seismic cycle.Foreshocks occupy the right side of the triangle, and theircumulative effect is to dissipate tectonic energy and enhanceslip heterogeneity. Aftershocks, on the other hand, aresituated on the left side, they continue to dissipate tectonicenergy, but at the same time they smoothen the slipdistribution.[13] Sometimes smaller cycles are superimposed on large

cycles. This is most apparent in cycle-3 in Figure 3e, whereeach of the two sub-cycles that are superimposed on a larger

Figure 1. Plots of Etect (top) and Efault (bottom) as afunction of time for a sample of 2 � 104 events. The time ofoccurrence of the 9 largest events is depicted by stars. Thevertical dashed lines and the numbers indicate the 4 timewindows for which phase diagrams are shown in Figure 3.

Figure 2. A plot of �Efaultk -versus-�Etect

k . Events withrupture area greater than 20 cells are labeled by stars,whereas events with rupture area less or equal to 20 cells arelabeled by dots if the lag-time to previous event is shorterthan 0.1ta, and by crosses if the lag-time is longer than 0.1ta.Vertical and horizontal lines that pass through the originwere added for reference.

Figure 3. Phase diagrams of Efaultk -versus-Etect

k for theentire data set (a), and for four time windows (b–e). Thefour time windows, indicated by dashed lines in Figure 1,were chosen such that they contain at least one of the largestearthquakes in the sample. The largest earthquakes aredepicted by stars, and the starting point of each path isindicated.

ZIV AND SCHMITTBUHL: THE SEISMIC CYCLE SDE 2 - 3

cycle are culminating in a large event. In some cases, as incycle-3 and cycle-4, the largest earthquake is followedimmediately by one or two smaller earthquakes that adddisorder to the system. These events may be viewed as sub-events, that in a more realistic simulation (quasi-dynamic orfully-dynamic) would become part of the mainshockrupture.

5. Conclusions

[14] We examine the evolution of and the exchangebetween two forms of elastic energies stored in the quasi-static fault model of Ziv and Rubin [2003]. The first, Etect,arises from slip deficit accumulated between the plateboundaries and the fault surface, and the second, Efault, isthe result of differential displacements along the faultsurface. Time series of these energies are very complex,and yet show very clearly that large earthquakes occurduring the descending portions of the Etect-curve, and closeto the maxima of the Efault-curve. This result is not inagreement with the classical view of the elastic reboundtheory, which predicts that large earthquakes occur at themaxima of the Etect-curve, and does not account for thestrain that is stored on the fault due to heterogeneous slipdistribution. On a Efault-versus-Etect plot, the seismic cyclehas a roughly triangular shape with large earthquakesoccurring at the top corner of the triangle, and the fore-shocks and the aftershocks occupying the right side and leftside, respectively. While both foreshocks and aftershocksdissipate tectonic energies, the cumulative effect of theforeshocks is to increase the fault disorder and the cumu-lative effect of the aftershocks is to reduce it.[15] Clearly these results are model dependent, and it is

desirable to examine the evolution of Etect and Efault in amodel that incorporates a creeping substrate. The inclusionof such substrate may be important since the stress pertur-bations that large earthquakes induce on the creepingregions initiate post-seismic slips that may relax long afterthe mainshock. Such a feedback process between creepingand stick-slip regions may play an important role in theseismic cycle, and may modify the evolution of both Etect

and Efault during the post-mainshock and the inter-seismicperiods.[16] Since neither Etect nor Efault is a directly measurable

or easily computable geophysical quantity, testing thevalidity of these results for geological faults is not a simpletask. Both geodetic (primarily GPS) and seismological dataare necessary in order to compute these parameters. Becausethe result of this study indicate that small earthquakes canplay an important role in ordering and disordering the slip

distribution, one should seek a catalog that is both completedown to very small magnitudes and for which relativeearthquake location is very precise.

[17] Acknowledgments. This study benefited from discussions withR. Madariaga. Comments from Associate Editor Aldo Zollo and twoanonymous reviewers helped to improve the manuscript. A. Z. acknowl-edges support from the Marie Curie postdoctoral fellowship and J. S.acknowledges support from the ACI ‘Risques Natwels’ of the FrenchMinistry of Education.

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Ziv, A., Foreshocks, aftershocks and remote triggering in quasi-static faultmodels, J. Geopys. Res., 108(B10), 2498, doi:10.1029/2002JB002318,2003.

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�����������������������A. Ziv, Ben-Gurion University of the Negev, Beer-Sheva, 84105, Israel.

(zival@bgumail.bgu.ac.il)J. Schmittbuhl, Ecole Normale Superieure, Paris, 75231, France.

(schmittbuhl@geologie.ens.fr)

SDE 2 - 4 ZIV AND SCHMITTBUHL: THE SEISMIC CYCLE