Dependence of earthquake recurrence times and independence of magnitudes on seismicity history

2006) 177–193www.elsevier.com/locate/tecto

Tectonophysics 424 (

Dependence of earthquake recurrence times and independence ofmagnitudes on seismicity history

Álvaro Corral

Departament de Física, Facultat de Ciències, Universitat Autònoma de Barcelona, E-08193 Bellaterra, Barcelona, Spain

Received 22 July 2005; received in revised form 6 October 2005; accepted 25 March 2006Available online 9 June 2006

Abstract

The fulfillment of a scaling law for earthquake recurrence–time distributions is a clear indication of the importance ofcorrelations in the structure of seismicity. In order to characterize these correlations we measure conditional recurrence–time andmagnitude distributions for worldwide seismicity as well as for Southern California during stationary periods. Disregarding thespatial structure, we conclude that the relevant correlations in seismicity are those of the recurrence time with previous recurrencetimes and magnitudes; in the latter case, the conditional distribution verifies a scaling relation depending on the difference betweenthe magnitudes of the two events defining the recurrence time. In contrast, with our present resolution, magnitude seems to beindependent on the history contained in the seismic catalogs (except perhaps for Southern California for very short time scales, lessthan about 30 min for the magnitude ranges analyzed).© 2006 Elsevier B.V. All rights reserved.

Keywords: Statistical seismology; Recurrence times; Interevent times; Correlations; Scaling and universality; Renormalization group

1. Introduction

Although seismology has intensively analyzed thecharacteristics of single earthquakes with great success,a clear description of the properties of seismicity (i.e.,the structures formed by all earthquakes in space, time,magnitude, etc.) is still lacking. It is true that a numberof empirical laws have been established since the end ofthe XIX century, but these laws are not enough toprovide a coherent phenomenology for the overallpicture of earthquake occurrence (Mulargia and Geller,2003). In general, the approach to this phenomenon hasto be statistical, as the Gutenberg–Richter law for thenumber of occurrences as a function of magnitude

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exemplifies. Moreover, for some studies it is necessaryto assume that an earthquake can be reduced to a pointevent in space and time; despite this drastic simplifica-tion, the resulting point process for seismicity retains ahigh degree of complexity. In this context, the problemturns out to be very similar to those usually studied instatistical physics. Remarkably, in the last years the so-called ETAS model (which combines the Gutenberg–Richter law, the Omori law, and the law of productivityof aftershocks) has been widely studied (Ogata, 1988,1999; Helmstetter and Sornette, 2002); however, thismodel is still a first order approximation to realseismicity, useful mainly as a null-hypothesis model.

In contrast to the issues of the size of earthquakes orthe temporal variations of the rate of occurrence, thetiming of earthquakes (despite its potential interest forforecasting and risk assessment) has been the subject of

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178 Á. Corral / Tectonophysics 424 (2006) 177–193

relatively few empirical statistical studies, with theadditional drawback that no unifying law for the timeoccurrence has emerged, at variance with the generalacceptation of the Gutenberg–Richter law and theOmori law (Gardner and Knopoff, 1974; Udías andRice, 1975; Koyama et al., 1995; Wang and Kuo, 1998;Ellsworth et al., 1999; Helmstetter and Sornette, 2004),see also the citations of Smalley et al. (1987) andSornette and Knopoff (1997). Rather, results turn out tobe in contradiction with each other, from the paradigmof regular cycles and characteristic earthquakes to theview of totally random occurrence (Bakun and Lindh,1985; Sieh et al., 1989; Stein, 1995; Sieh, 1996; Murrayand Segall, 2002; Stein, 2002; Kerr, 2004; Weldon et al.,2004, 2005). In the opposite side of cycle regularity isthe view of earthquake occurrence in the form ofclusters, which has received more support from cataloganalysis (Kagan and Jackson, 1991, 1995; Kagan,1997).

A first step towards the solution of the time-occurrence problem and its insertion in a unifyingdescription of seismicity was taken by Bak et al., whorelated, by means of a unified scaling law, the distributionof earthquake recurrence times with the Gutenberg–Richter law and the fractal distribution of epicenters (Baket al., 2002; Christensen et al., 2002; Corral, 2003, 2004b;Corral and Christensen, 2006; Davidsen and Goltz,2004). One of the goals of this line of research (apart fromthe applications in forecasting) should be to build a theoryof seismicity that clarifies which kind of dynamicalprocess we are dealing with. Simple behaviors such asperiodicity, random occurrence, or even chaos seem toosimplistic to account for the complexity of seismicity; incontrast, self-organized criticality fits better with thispurpose (Bak, 1996; Jensen, 1998; Turcotte, 1999;Sornette, 2000; Hergarten, 2002), although differentalternatives have been proposed more recently, seeRundle et al. (2003) and Sornette and Ouillon (2005).

Nevertheless, the recurrence–time distribution intro-duced by Bak et al. (2002) was defined in a somewhatcomplicated way through the mixing of recurrence timescoming from (more or less) squared equally-sizedregions with disparate seismic rates. A simpler,alternative perspective, in which each (arbitrary) regionhas its own recurrence–time distribution, has beendeveloped (Corral, 2004a, 2005b). In this case, auniversal scaling law involving the Gutenberg–Richterrelation describes these distributions for a wide range ofmagnitudes and sizes of the regions, as long asseismicity shows a stationary behavior.

In this paper we extend this procedure to studyconditional probability distributions. These distributions

will provide important information about correlations inseismicity, which are fundamental for the existence ofthe scaling law for recurrence–time distributions. Itturns out that recurrence times depend on previousrecurrence times and magnitudes, in such a way thatsmall values of the recurrence time and large magni-tudes lead to small recurrence times, and vice versa. Incontrast, the magnitude is fairly independent on history(although obviously we cannot reject the hypothesis of adependence weaker than our uncertainty).

2. Scaling of recurrence-time distributions andrenormalization-group transformations

The main subject of our study is the earthquakerecurrence time (also called waiting time, intereventtime, interoccurrence time, etc.), defined as the timebetween consecutive events, once a window in space,time and magnitude has been selected. In this way, the i-th recurrence time is defined as

siuti−ti−1; i ¼ 1; 2:::

where ti and ti−1 denote, respectively, the time ofoccurrence of the i-th and i−1-th earthquakes in theselected space–time–magnitude window. Note that,following Bak et al., once the window has beenselected, no further elimination of events is performedand then, all events are equally considered, indepen-dently of their hypothetical consideration as main-shocks, aftershocks or foreshocks. This is an importantdifference with respect to the “standard practice,”usually more focused in declustered catalogs.

The recurrence time is a variable quantity and it isconvenient to work with its probability density. Let usrecall that a probability density (in this case ofrecurrence times) is defined as the number of recurrencetimes within a small interval of values, normalized to thetotal number of recurrences and divided by the length ofthe small interval (to remove the dependence on it); to beprecise,

DðsÞu Prob½sVrec: time < sþ ds�ds

; ð1Þ

where Prob denotes probability, τ refers to a precisevalue of the recurrence time, and dτ is the length of theinterval. Strictly, dτ should tend to zero, but in practice itis necessary a compromise to reach some statisticalsignificance for each interval, as the number of data isnot infinite. Moreover, as there are multiple scalesinvolved in the process (from seconds to years) it ismore convenient to consider a variable dτ, with theappropriate size for each scale.

179Á. Corral / Tectonophysics 424 (2006) 177–193

An easy recipe is to consider the different intervals(in seconds with a minimum recurrence time of 1 s, forinstance) growing as [τ, τ+dτ)= [1, c), [c, c2), [c2, c3),etc., with c>1. This procedure is sometimes referred toas logarithmic binning, as in log scale the intervals dτhave the same length. Note that the widely usedcumulative probability distribution is just the integralof D(τ); nevertheless, we find that the characteristics ofthe distributions are more clearly reflected in the density.

The key point of the procedure introduced by Bak etal., which we adopt, is the systematic analysis of thedependence of the probability density on the parametersthat define the magnitude and spatial windows. For aspatial region of arbitrary coordinates, shape and size,independent of tectonic divisions or provinces, onlyevents with magnitude M larger than a threshold valueMc are selected (naturally, the value Mc has to be highenough to guarantee the completeness of the data). Thismeans that the recurrence-time density, D(τ), dependson Mc and on the size, shape, and coordinates of theregion; in order to stress this dependence we add asubscript w (from window) to the density, which turnsout Dw(τ).

An additional point to take into account is theheterogeneity of seismicity in time, with large variationsin the number of events depending on the occurrence oflarge earthquakes (what is usually referred to asaftershock sequences, and also earthquake swarms). Ina first step we only consider time windows with

Fig. 1. Cumulative number of earthquakes N(t) (normalized by the total nuworldwide seismicity withM≥5 and for Southern California seismicity withindication of stationary occurrence, which clearly applies to worldwide seismbehavior, except for some periods of time. Some of these stationary periods

stationary seismicity or, more precisely, seismicityhomogeneous in time. This means that, for a particulartime window, the statistical properties of the process donot change when the window is divided into shorterwindows, except for the fluctuations associated to thefinite size of the windows; in particular, the mean valuedefined for any property of the process will bepractically the same for any of the shorter time windows,provided that these windows are not too short. If, for agiven window, we define its mean seismic rate Rw as thenumber of earthquakes divided by the time periodcovered by the window, i.e.

Rwunumber of events in w

time extent of w;

then, stationarity imposes that the mean rate is roughlyconstant for different windows and a unique seismic rateR can be defined for a long period (however, note thatthere can be different stationary periods with differentrates, although experience tells us that these rates arealso similar). When we plot the accumulated number ofearthquakes versus time, as in Fig. 1, stationarity iseasily identified by a linear behavior in this plot, as therate is just the slope of the curve. In order to quantifystationarity, we may fit straight lines to some part of thebehavior of the accumulated number of earthquakes as afunction of time. For the examples in the figure, whichwill be the data analyzed in this paper as we will explainnext, the correlation coefficient for a minimum-square

mber Ntot in the whole period considered) as a function of time, forM≥2 (Ntot=46,054 and 84,771 respectively). The linear behavior is anicity. In contrast, for Southern California we observe a nonstationaryhave been studied in the text and are marked by a thick line.


procedure is always in between 0.995 and 0.9997,although the quality of the fit can still be improved by abetter selection of the time windows. Nevertheless, thiswill be sufficient for our purposes.

It is illustrative to start our analysis with worldwideearthquakes. In this case, the spatial region of study isfixed to the whole Earth, and only the magnitudethreshold Mc is changed. The main advantage ofworldwide seismicity is that earthquakes occur at auniform rate (at least for the last 30 years). In particular,we use the NEIC-PDE catalog (National EarthquakeInformation Center, Preliminary Determination ofEpicenters, available at http://www.neic.cr.usgs.gov/neis/epic/epic_global.html) for the years 1973 to 2002.

Fig. 2. (top) Probability densities of recurrence times of earthquakes worldwidM≥6.5. (bottom) Same as before but with rescaled axis,D(τ) /R versus Rτ. Thscaling function f. Note that the deficit of occurrences for recurrence timeincompleteness of the catalog at the shortest time scale.

In order to confirm the stationarity of worldwideearthquake occurrence, we plot in Fig. 1 the cumulativenumber of earthquakes with M≥5 since 1/1/1973versus time. A linear trend, characteristic of stationarybehavior, is clearly seen, with a rate of occurrence givenroughly by Rw⋍4 events per day.

Fig. 2 (top) shows the probability densities of therecurrence time in this case for several magnitudethresholds Mc. As Mc increases, the densities becomebroader (due to the decrease in the number of events andthe corresponding increase of the time between them),but all of them share the same qualitative behavior: aslow decrease for most of its temporal range ending witha much sharper decay.

e during 30 years (from 1973 to 2002), withM≥5,M≥5.5,M≥6, ande collapse of all the densities onto a single curve shows the shape of thes smaller than 500 s in the M≥5 case could be an indication of the

http://www.neic.cr.usgs.gov/neis/epic/epic_global.html

http://www.neic.cr.usgs.gov/neis/epic/epic_global.html


Remarkably, when the x-axis is rescaled as Rwτ andthe y-axis as Dw(τ) /Rw, all the different densitiescollapse onto a single curve, see Fig. 2 (bottom). Thisdata collapse allows to write the recurrence-timeprobability density in the form of a scaling law,

DwðsÞ ¼ Rwf ðRwsÞ; ð2Þwhere the scaling function f is the same for all thedifferent Mc's. Then, the above mentioned qualitativesimilarity between the recurrence-time distributionsbecomes clearly quantitative.

Analyzing a great number of windows with station-ary seismicity, for worldwide as well as for localseismicity, for spatial scales as small as 20 km, and withmagnitude threshold ranging from 1.5 to 7.5 (which is afactor 109 in the variation of the minimum energy), itwas found that the scaling law (2) is of general validityfor stationary seismicity, with the same scaling functionf in any case (Corral, 2004a, 2005b). This means thatone single scaling function can describe reasonably wellthe recurrence-time distribution in any stationary region,with small differences in each case. In this way, f can beconsidered a universal scaling function, reflecting theself-similarity in space and magnitude of the stationarytemporal occurrence of earthquakes. Despite the largevariability associated with the earthquake phenomenon(“there are no two identical earthquakes”), it issurprising that the timing of earthquakes is governedby a unique simple law.

The functional form of f turns out to be close to ageneralized gamma distribution,

f ðhÞ ¼ CjdjagCðg=dÞ

1

h1−ge−ðh=aÞ

d

; for h > hminz0; ð3Þ

where the variable θ≡Rwτ constitutes a dimensionlessrecurrence time; the parameter γ turns out to be around0.7, whereas δ is very close to 1 (strictly, δ≡1 for theusual gamma distribution). This means that f decays as apower law ∝ 1 /θ0.3 for θ<1 and the decay isaccelerated by the exponential factor for θ>1. Theconstant a is just a dimensionless scale parameter and itsvalue is around 1.4, Γ is the usual gamma function,which normalizes the distribution, and C is a correctionto normalization due to the fact that this description maynot be valid for very short times, θ<θmin. Let us remarkthat the modeling of the scaling function by means of agamma distribution does not make the scaling law (2)depend on the gamma modeling; the validity of thescaling law is much more general and independent onany model of seismic occurrence, in particular, norenewal model is implicit in our approach.

The slow decay provided by the decreasing powerlaw is a signature of clustering, which means that theevents tend to be closer to each other in the short timescale, in comparison with a Poisson process (given by fexponential, i.e., γ=a=δ=1). In fact, clustering is moreclearly identified by looking at the hazard rate function(containing the same information than the probabilitydensity), which shows a high value for short recurrencetimes and decreases when the recurrence time increases;so, in some sense earthquakes tend to attract each other(Corral, 2005d). This leads to a somewhat paradoxalconsequence for the time one has to wait for the nextearthquake: it turns out that the longer one has beenwaiting for an earthquake, the longer one has still to wait.Note that the clustering we are dealing with takes placefor stationary seismicity, at variance with the clusteringassociated to aftershock sequences due mainly to thelarger seismic rate close to the mainshock. Large ratestend to concentrate events, whereas a subsequentdecrease of rate dilutes the events. This mechanismdoes not operate in stationary seismicity, where the rateof the process stays constant (despite the presence oflocal aftershock sequences originated by small events,which are intertwined with the seismicity of the rest ofthe area to give rise to the observed stationary behavior).

Nevertheless, it is interesting to point out that thescaling law (2) is valid beyond the case of stationaryseismicity. For this purpose we have consideredSouthern-California seismicity for the years 1984–2001, using the SCSN catalog (Southern CaliforniaSeismographic Network) available at the SouthernCalifornia Earthquake Data Center, http://www.scecdc.scec.org/ftp/catalogs/SCSN). It is well known thatseveral important earthquakes occurred in SouthernCalifornia during the selected period, generatingincreased seismic activity and breaking the stationarybehavior of seismicity (see Fig. 1); this leads torecurrence-time distributions dominated by aftershocksequences. Fig. 3 displays these recurrence-time densi-ties for different values of Mc, rescaled by the mean rateRw (which now is not stationary). The nearly perfect datacollapse implies the validity of the scaling law (2), witha scaling function f still given by Eq. (3), although with adecreasing power-law exponent 1−γ closer to 1 (1−γ⋍0.8), clearly different than the one for stationaryseismicity, 1−γ⋍0.3. (Surprisingly, it was shown inCorral (2004a) that when time is nonlinearly trans-formed to obtain a stationary process, the universalscaling function corresponding to stationary seismicityis recovered; this can be done for a single aftershocksequence rescaling not by Rw but by the instantaneousOmori rate, rw(t)∝1 / tp).

http://www.scecdc.scec.org/ftp/catalogs/SCSN

http://www.scecdc.scec.org/ftp/catalogs/SCSN

Fig. 3. Rescaled probability densities of recurrence times of earthquakes in Southern California, (123°W, 113°W)×(30°N, 40°N), for the years 1984–2001, using different magnitude thresholds, from 2 to 4. The good data collapse indicates that a scaling law is valid, although the scaling function isdifferent than in the stationary case. In concrete, the decay of the power law is faster, with an exponent 1−γ⋍0.8. (The distributions only showrecurrence times greater than 10 s.)


In the case in which the spatial window is kept fixedand only Mc is changed, the seismic rate Rw can berelated with the magnitude threshold by means of theGutenberg–Richter law,

Rw ¼ R010−bMc; ð4Þ

where b is the well-known b-value of the Gutenberg–Richter law (usually close to 1) and R0 is anextrapolation of the seismic rate to Mc=0, dependingonly on the spatial part of the window w. Therefore, thescaling law (2) can be written as

DwðsÞ ¼ R010−bMcf ðR010

−bMcsÞ; ð5Þand including here the gamma fit (with δ=1) we get

DwðsÞ ¼ CagCðgÞ

Rg010

−gbMc

s1−gexpð−R010

−bMcs=aÞ; ð6Þ

which is a complicated formula for a simple idea.In addition, it is remarkable the similarity between

our scaling law (2) with Eq. (4) and the scaling ofangular correlation functions in the distribution ofgalaxies (Jones et al., 2004). Also, it has been notedthat the scaling relation (2) corresponds to the Cox–Oakes accelerated-life model, studied in the context ofmedical statistics (Cox and Oakes, 1984; German,2005).

An important consequence of the scaling law (2) isthat it implies that the Gutenberg–Richter law is notonly valid over long periods of time (in the form

Rw=R010−bMc) but it is also fulfilled continuously in

time, provided the times are compared in the appropriateway. Indeed, selecting τw=A /Rw, where A is an arbitraryfixed constant and Rw changes as a function of Mc, weget Dw(τw)=Rwf(A)= f(A)R010

−bMc. In other words,selecting two different thresholds, Mu and Mv, theGutenberg–Richter law is obtained in the form

DvðsvÞDuðsuÞ ¼ 10−bðMv−MuÞ;

when times τu and τv are compared for each threshold,these given bysvsu

¼ 10bðMv−MuÞ:

To make it more concrete, if, for M≥5 we count thenumber of events coming after a recurrence timeτ=104 s (A⋍0.5 in the case of Fig. 2), and comparewith the number of events with M≥6 coming afterτ=105 s, these numbers are given by the Gutenberg–Richter relation. This constitutes somehow an analogousof the well-known law of corresponding states incondensed-matter physics (see for instance Guggen-heim, 1966 or Lee et al., 1973). Two pairs ofconsecutive earthquakes in different magnitude win-dows would be in “corresponding states” if theirrescaled recurrence times are the same (θ=Ruτu=Rvτv).

Interestingly, the scaling relation (2) can be under-stood as arising from a renormalization-group transfor-mation analogous to those studied in the context of


critical phenomena (Corral, 2005c). Let us investigatedeeper the meaning of the scaling analysis we haveperformed and its relation with a renormalization-grouptransformation. Fig. 4 displays the magnitude M versusthe occurrence time t of all worldwide earthquakes withM≥Mc for different periods of time and Mc values. Fig.4 (top) is for earthquakes larger than (or equal to)Mc=5for the year 1990. If we rise the threshold up to Mc=6we get the results shown in Fig. 4 (medium). Obviously,as there are less earthquakes in this case, the distributionof recurrence times (time interval between consecutive“spikes” in the plot) becomes broader with respect to the

Fig. 4. Magnitude of worldwide earthquakes as a function of time of occurrenwith renormalization-group transformations. (top) Earthquakes withM≥5 duin fact a decimation of events). (bottom) Earthquakes withM≥6 expanded toa factor 10). Visual comparison between the top and bottom figures shows t

previous case. The rising of the threshold can be viewedas a mathematical transformation of the seismicity pointprocess, which is referred to as thinning in the context ofstochastic processes (Daley and Vere-Jones, 1988) andis also equivalent to the common decimation performedfor spin systems in renormalization-group transforma-tions (Kadanoff, 2000; McComb, 2004; Christensen andMoloney, 2005). Fig. 4 (bottom) shows the same as Fig.4 (medium) but for ten years, 1990–1999, andrepresents a scale transformation of seismicity (also asin the renormalization group), contracting the time axisby a factor 10 (this factor comes from the Gutenberg–

ce, for different time and magnitude windows to illustrate the analogiesring year 1990. (medium) Same as before but restricted toM≥6 (this is10 years, 1990–1999 (this is equivalent to a contraction of time axis byhat both figures “look the same.”


Richter law with b=1, as it is the ratio of events withM≥5 to M≥6, i.e., RwjMc¼5=RwjMc¼6g10). The simi-larity between Fig. 4 (top) and 4 (bottom) is apparent,and is confirmed when the probability densities of thecorresponding recurrence times are calculated andrescaled following Eq. (2) (see again Fig. 2 (bottom)).

In this way, the existence of the scaling law (2)implies the invariance of the recurrence-time distribu-tion under renormalization-group transformations, i.e.,the rescaled distribution is a fixed point of thetransformation, constituting then a very special function(as in general, a thinning procedure implies a change notonly on the scale of the distribution but also on itsshape). One can realize that there exists a trivial fixed-point solution to this problem, where the recurrencetimes are given by a Poisson process (i.e., independentidentically distributed exponential times) and themagnitudes are generated with total independence aswell; this constitutes what is called a marked Poissonprocess.

Moreover, it is not difficult to show that the markedPoisson process is the only process without correlationswhich is invariant under our renormalization-grouptransformation (Corral, 2005c). But even further, thePoisson process is not only a fixed point of thetransformation, but a stable one, i.e., an attractor:when events are randomly removed from a process(random thinning) the resulting process converges to aPoisson process, under certain conditions (Daley andVere-Jones, 1988).

The fact that for real seismicity the scaling function fis not an exponential tells us that our renormalization-group transformation is not performing a randomthinning; this means that the magnitudes are notassigned independently on the rest of the process andtherefore there exists correlations in seismicity. This ofcourse has been pointed out before, see for instanceKagan and Jackson (1991), but let us stress thatcorrelations are not only fundamental for the existenceof the scaling law (2), but are intimately related to it; theknowledge of correlations determine the precise form ofthe recurrence-time distribution. In any case, it isstriking that seismicity is precisely at the fixed pointof a renormalization-group transformation.

At this point it is worth clarifying that the term“correlations” is generally used with two differentmeanings: First, when the recurrence-time distributionis not an exponential (and the time process is notPoisson), for example in a renewal process. This in factimplies the existence of a memory in the process, asevents do not occur independently at any time, butremember the occurrence of the last event. And second,

when the recurrence time (whatever its distribution)depends on the history of the process, in particular theoccurrence time and magnitude of previous events (ofcourse, this is another form of memory in the process);these are the correlations we are talking about.

There are other arguments that make the existence ofthe scaling law (2) striking. For example, one can arguethat the seismicity of large areas consists on thesuperposition of many more or less independentprocesses, and it is known that the pooled output ofindependent processes converges to a Poisson process(Cox and Lewis, 1966). But Fig. 2 (bottom) demon-strates for the largest possible scale that this is not thecase: worldwide seismicity shows the limited applica-bility of the theorems that guarantee convergence to aPoisson process. (On the other hand, one could arguethat the sum does not have enough samples of unitindependent processes to ensure the convergence.)

In the same line, Molchan (2005) has shown that ifthe scaling law (2) is fulfilled in two regions separately,and also in the “superregion” formed by the two regions,then the scaling function f can only be exponential,provided that seismicity in both regions is independenton each other. As in practice one finds that the scalingfunction clearly departs from a exponential, one mayconclude that seismicity is not independent in theregions considered in our previous analysis. Therefore,the interactions between the regions are importantenough to make the boundary effects not negligible;this could be related with long-range earthquaketriggering (Huc and Main, 2003).

Finally, it is interesting to mention the results ofHainzl et al. (2006). Following theoretical calculationsof Molchan (2005), who assumed that seismicity isformed by Poisson mainshocks generating sequences ofaftershocks, Hainzl et al. use the parameters of thescaling function f to evaluate the ratio between thenumber of mainshocks and the total number of events,given by 1 /a⋍γ, for δ=1. Analyzing a number ofregions these authors find different proportions, butbecause most of the regions analyzed were notstationary their results are not in contradiction with theuniversality of the proposed scaling law (2). Rather, thissuggests that stationary seismicity may keep a constantproportion of mainshocks and aftershocks (in case thatboth entities could be unambiguously defined).

3. Conditional probability densities and correlations

In the previous section we have seen how thefulfillment of a scaling law for the recurrence-timedistributions with a non-exponential scaling function is


an indication of the existence of important correlationsin seismicity. Once a spatial area and a magnitude rangehave been selected for study, we can consider theprocess as taking place in the magnitude–time domain(intentionally disregarding the spatial degrees offreedom, which will be the subject of future research,see also Davidsen and Paczuski (2005)). Given an eventi, its magnitude Mi may depend on the previousmagnitudes Mj with j< i as well as on the previousoccurrence times tj, or equivalently, on the previousrecurrence times, τj= tj− tj−1, with j≤ i now. In its turn,the recurrence time to the next event, τi+1 may dependon Mi, τi, Mi−1, τi−1, etc. (see Fig. 5 for a visualexplanation). This dependence is of course statisticaland the most direct way to measure it is by means of theconditional probability density. In the case of recurrencetimes, the conditional density is defined as

DðsjX Þu Prob½sV rec: time < sþ dsjX �ds

; ð7Þ

where the symbol | states that the probability is onlyconsidered for the cases in which the event or conditionX holds, X being a range of values of previousrecurrence times, or magnitudes, or whatever. In ananalogous way one can define the probability density ofthe magnitude conditioned to X.

It is a fundamental result of probability theory that ifa random variable, for instance the recurrence time, isstatistically independent of an event X, then, thedistribution of the variable conditioned to X is identicalto the unconditional distribution, i.e., D(τ|X)=D(τ) (infact, this is the definition of independence). On the otherside, if both distributions are different, D(τ|X)≠D(τ),there is a dependence between τ and X and somecorrelation coefficient different than zero may be

Fig. 5. Notation used to describe the seismicity point process in timeand magnitude, where only events i−2, i−1, and i are shown.

defined (in general the correlation coefficient must benonlinear; as it is well know dependent variables mayyield a zero value for the linear correlation coefficient).

In this section we will measure the dependence ofseismicity on its history by means of conditionaldistributions of recurrence times and magnitudes,concentrating mainly on the first step backwards, i.e.,we will study the dependence of τi onMi−1 and τi−1, andthe dependence ofMi on τi andMi−1. These dependenceswill be given by the distributions D(τi|Mi−1≥Mc′),D(τi |τa≤ τi−1 < τb), D(Mi |τa≤ τi < τb), and D(Mi |Mi−1≥Mc′), respectively (note that now Mi, τi, Mi−1,etc. do not denote the values of τ and M for a particularevent, they refer to all events i and the correspondingpreceding events i−1). Note also that when the conditionis imposed on the magnitude, we denote its threshold asMc′, in order to distinguish it from the generic condition,M≥Mc (with Mc≥Mc′), which always holds. Theapplication of our method to more steps backwards (i−2, i−3, etc.) is straightforward.

Nevertheless, instead of analyzing D(Mi|τa≤τi<τb)we have found clearer to study D(τi|Mi≥Mc′); althoughcausality is not preserved in this case (as the event ofmagnitude Mi happens after the time τi), from the pointof view of probability theory the equivalence iscomplete, i.e., if we find D(τi|Mi≥Mc′)≠D(τi), then τiand Mi are correlated.

The imposition of a condition X reduces the statisticsand increases the fluctuations of the measured distribu-tions; for this reason we will associate an error oruncertainty to each bin of the densities. Let us considerthe case of the recurrence time τ, with no extracondition. The number of data n in a particular interval[τ, τ+dτ) can be considered as given by the binomialdistribution if the total number of data N is much largerthan the range of the correlations in the data; therefore,its standard deviation will be, as it is well known,rn ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiNpð1−pÞp

. As D(τ)=n / (Ndτ), the error in D(τ)will be

rD ¼ 1ds

ffiffiffiffiffiffiffiffiffiffiffiffiffiffipð1−pÞ

N

r;

where the value of p can be estimated simply as p=n /N=D(τ)dτ. The error in conditional distributions isobtained in the same way, obviously.

The data analyzed in this section comes from the twocatalogs referred to previously. For the worldwideNEIC-PDE catalog we consider the period 1973–2002, which can be considered stationary with a rateof about 4.2 ·10−(Mc− 5) earthquakes per day. As thefulfillment of the Gutenberg–Richter law points out, thiscatalog is reasonably complete for M≥5 (see below,


nevertheless); this yields 46,055 events in the consid-ered magnitude–time window. In contrast, Southern-California seismicity is not stationary in general, butsome stationary periods exist in between large earth-quakes, see Fig. 1. The longest stationary period for thelast years spans from 1994 to 1999, in particular wehave considered the period fromMay 1, 1994 to July 31,1999, for which the number of earthquakes withM≥1.5in the spatial area (123°W, 113°W)×(30°N, 40°N) is46,576; this gives 24.3 earthquakes per day. Otherstationary periods have also been analyzed and will bedetailed below; it is remarkable that all of them sharemore or less the same rate of occurrence.

Fig. 6. Probability densities of earthquake recurrence times τi conditioned to tlead to a systematic variation in the distributions, clearly beyond the uncertainτi are positively correlated. (top) Results for NEIC worldwide catalog usingreported in the text, with M≥2.

3.1. Correlations between recurrence times

Let us start our study of correlations in seismicitywith the temporal sequence of occurrences. Using thetools described above we obtain the conditionaldistributions D(τi|τa≤ τi−1 < τb); in particular wedistinguish two cases; first, the recurrence-time densityconditioned to short preceding recurrence times,D(τi|τi−1<τb), where τb is smaller than the mean ⟨τ⟩of the unconditional distribution, D(τ); and second, therecurrence-time density conditioned to long precedingrecurrences, D(τi|τi−1≥τa), with τa larger than ⟨τ⟩ (or atleast larger than the τb in the first case).

he value of the preceding recurrence time, τi−1. Different ranges for τi−1ty limits given by the error bars, leading to the conclusion that τi−1 andM≥5. (bottom) Southern-California catalog for the stationary period


The results, both for worldwide seismicity and forSouthern-California stationary seismicity, turn out tobe equivalent (see Fig. 6). In any case, a gammadistribution is able to fit the data, with differentparameters though, depending on the condition. Forshort τi−1 (given by short τb), a relative increase in thenumber of short τi and a decrease of long τi isobtained, which leads to a more steep power-lawdecay. In the opposite case, long τi−1 imply a decreasein the number of short τi and an increase in the longerones, in such a way that the power-law part of thegamma distribution becomes flatter, tending to an

Fig. 7. Probability densities of earthquake recurrence times τi conditioned tdistributions are significant (except for the top curves for which the statisticnegative correlation between both variables. (top) Results for the NEIC worlfactors in the density for clarity sake: Mi≥5, factor 1; Mi≥5.5, factor 10;Mi

from Mc′=Mc (unconditioned distribution) to Mc′=7. (bottom) Results for Sofactor 1; Mi≥2, factor 10; Mi≥2.5, factor 100; Mi≥3, factor 1000; Mi≥3.5

exponential distribution for the longest τi−1. Thismeans that short τi−1 imply an average reduction onτi and the opposite for long τi−1 and then bothvariables are positively correlated.

Note that this behavior corresponds to a clustering ofevents, in which short recurrence times tend to be closeto each other, forming clusters of events, while longertimes tend also to be next each other. This clusteringeffect is different from the clustering reported previous-ly, associated to the non-exponential nature of therecurrence-time distribution (Corral, 2004a, 2005d), andalso from the usual clustering of aftershock sequences.

o the value of the preceding magnitude, Mi−1. The differences in thes was poorer), showing that the larger Mi−1, the shorter τi, implying adwide catalog, using different thresholds Mc for Mi and multiplicative≥6, factor 100;Mi≥6.5, factor 1000; the thresholdMc′ for Mi−1 rangesuthern California stationary seismicity, using Mi≥1.5, multiplicative, factor 104; the threshold Mc′ ranges from Mc to 4.


The findings reported here are in agreement withLivina et al. (2005), who studied the same problem forSouthern California, but without separating stationaryperiods from the rest. These authors explain thisbehavior in terms of the persistence of the recurrencetime, which is a concept equivalent to clustering. Theresults could be similar to the long-term persistenceobserved in climate records, but this has not beenstudied up to now. The linear correlation coefficient inthis case was calculated by Helmstetter and Sornette(2004), turning out to be 0.43, clearly supporting thepositive correlation.

We have extended the study to more steps backwardsin time, to be concrete, we have obtained from thecatalogs the distributions D(τi|τa≤τj<τb), using differ-ent values of τj, with j= i−2, i−3, etc., up to j= i−10. Inall cases there are significative differences between theconditional distributions and the unconditional distribu-tion, implying that the temporal correlations extend toup to 10 events, or, in other words, that the recurrencetime is dependent on its own history.

3.2. Correlations between recurrence time andmagnitude

When, in addition to the recurrence times, themagnitudes are taken into account, we can study thedependences between these variables. We consider firsthow the magnitude of one event influences the

Fig. 8. Same as previous figure but rescaling the axis and separating the densit−Mc takes the values 0, 0.5, 1, 1.5, and 2, and the densities in each case havefrom 1.5 to 3.5 for Southern California, and from 5 to 7 for the worldwide datwith a scaling function that depends on ΔMc≡Mc′−Mc. The straight linedistributions with parameter γ⋍0.70, 0.55, 0.48, 0.35, and 0.23, for ΔMc=

recurrence time to the next; in this way we measure D(τi|Mi−1≥Mc′). The results for the case of worldwideseismicity and for Southern California (in the stationarycase) are again similar and show a clear correlationbetween Mi−1 and τi (Corral, 2005a).

Indeed, Fig. 7 shows how larger values of thepreceding magnitudes, given by Mi−1≥Mc′, lead to arelative increase in the number of short τi and a decreasein long τi; this means that Mi−1 and τi are anticorrelated.For the cases for which the statistics is better, thedensities show the behavior typical of the gammadistribution, with a power law that becomes steeper forlargerMc′; the different values of the exponent are givenin the caption of Fig. 8. In a previous work (Corral,2005c) we showed that the conditional distributions weare dealing with could be related with the unconditionalone by means of the scaling relation

DðsijMi−1zMcVÞgKDðKsiÞ ¼ RwKf ðRwKsiÞ;where Rw is the rate of the process forM≥Mc and Λ is afactor depending on the difference Mc′−Mc with Λ=1for Mc′−Mc=0. This is in disagreement with the resultsreported here; indeed, the above relation is onlyapproximately valid for small Mc′−Mc, but for largevalues we have shown clearly that the shape of thedistribution changes, and not only its mean, as wepreviously supposed. Remarkably, this change can bedescribed by a scaling law for which the scaling functiondepends now on the difference between the threshold

ies by the values ofMc′−Mc (rather than byMc). From bottom to top,Mc′been multiplied by 1, 10, 100, 1000, and 104, respectively. Mc′ ranges

a. The figure shows how a scaling law is valid for D(τi|Mi−1≥Mc′ ) buts correspond to the fits of fΔMc

for different ΔMc, yielding gamma0, 0.5, 1, 1.5, and 2, respectively.


magnitude for the i−1 event,Mc′, and the threshold for i,Mc, i.e.,

DðsijMi−1zMcVÞ ¼ RwKfDMcðRwKsiÞ;

with ΔMc=Mc′−Mc, Λ=Rw−1⟨τi(Mc,Mc′)⟩

−1, and ⟨τi(Mc,Mc′)⟩ the mean of the distribution D(τi|Mi−1≥Mc′).Fig. 8 illustrates this new scaling law, setting clearly thatthere is not just a similarity between the distributions forworldwide and Southern-California seismicity, but anidentical behavior described by the same scalingfunctions with the same exponents if ΔMc is the same.

Fig. 9. Probability densities of earthquake recurrence times τi conditioned tprevious figure, now the differences in the distributions are negligible (excconsider τi and Mi statistically independent. (top) Results for the NEIC worldfactors in the density (as in the previous plot):Mi−1≥5, factor 1;Mi−1≥5.5, ffor Mi ranges from Mc′=Mc (unconditioned distribution) to Mc′=7. (bottom) Rfactor 1; Mi−1≥2, factor 10; Mi−1≥2.5, factor 100; Mi−1≥3, factor 1000; M

Further, as in the previous subsection, we have goneseveral more steps backwards in time, measuringconditional distributions up to D(τi|Mi−10≥Mc′) andagain we find significative dependence between τi andthe previous magnitudes, up to Mi−10.

Next, we can consider how the recurrence time to oneevent τi influences its magnitude Mi, or, equivalently,how the recurrence time to one event depends on themagnitude of this event. For this purpose, we measureD(τi|Mi≥Mc′) and the results are shown in Fig. 9. Fromthere it is clear that in most of their range thedistributions are nearly identical, and when some

o the value of the magnitude after the recurrence, Mi. Contrary to theept for very short times in Southern California, see text). So we canwide catalog, using different thresholds Mc for Mi−1 and multiplicativeactor 10;Mi−1≥6, factor 100;Mi−1≥6.5, factor 1000; the thresholdMc′esults for the Southern California stationary period, using Mi−1≥1.5,

i−1≥3.5, factor 104; the threshold Mc′ ranges from Mc to 4.


difference is present this is inside the uncertainty givenby the error bars. However, there is one exception: inCalifornia, very short times, τi⋍200 s, seem to befavored by larger magnitudes, or, in other words, shorttimes lead to larger events. What we are seeing in thiscase may be the foreshocks of small events, which arerestricted to these very short times, but we cannotexclude that this is an artifact due to catalog incom-pleteness at short time scales. With the exception of this,which has a very limited influence, we can consider therecurrence time and the magnitude after it as indepen-dent from a statistical point of view, with our presentresolution (it might be that a very weak dependence ishidden in the error bars of the distributions).

In addition, the extension of the conditional distribu-tions to the future, measuring D(τi|Mj≥Mc′) with j> i,leads to similar results, and from here we can concludethe independence of the magnitude with the sequence ofprevious recurrence times, at least with our presentstatistics and resolution.

3.3. Correlations between magnitudes

Finally, we study the correlations between consecu-tive magnitudes, Mi−1 and Mi, by means of thedistribution D(Mi|Mi−1≥Mc′). As the analysis of the1994–1999 period for Southern California did notprovide enough statistics for the largest events we

Fig. 10. Probability densities of earthquake magnitudesMi conditioned to theseismicity and Southern-California stationary periods (see text). Different thrranges from 5 to 6.5, whereas Mi≥Mc with Mc=5 always. For the Californiacase Mc=Mc′ gives the unconditioned distribution D(M). The top set of cdistributions have been multiplied by 10 for clarity sake. In this case the diffethe statistical independence of consecutive magnitudes.

considered a set of stationary periods, these being: Jan 1,1984–Oct 15, 1984; Oct 15, 1986–Oct 15, 1987; Jan 1,1988–Mar 15, 1992; Mar 15, 1994–Sep 15, 1999; andJul 1, 2000–Jul 1, 2001; all of them displayed in Fig. 1.

Again we find that both worldwide seismicity andstationary Southern-California seismicity share the sameproperties, but with a divergence for short times. Fig. 10shows the distributions corresponding to the tworegions, and whereas for the worldwide case thedifferences in the distributions for different Mc′ arecompatible with their error bars (see bottom-right set ofcurves), for the California case there is a systematicdeviation, implying a possible correlation (bottom-leftset).

In order to find the origin of this discrepancy weinclude an extra condition, which is to restrict the eventsto the case of large enough recurrence times, so weimpose τi≥30 min. In this case, the differences inCalifornian distributions become insignificant (top-leftcurve), which means that the significant correlationsbetween consecutive magnitudes are restricted to shortrecurrence times (Corral, 2005a). Therefore, we con-clude that the Gutenberg–Richter law is valid indepen-dently of the value of the preceding magnitude,provided that short times are not considered. This is inagreement with the usual assumption in the ETASmodel, in which magnitudes are generated from theGutenberg–Richter distribution with total independence

magnitude of the immediately previous event,Mi−1, both for worldwideesholds Mc′ for Mi−1 have been used: for the worldwide case (right) Mc′case (left), Mc′ ranges from 2 to 3.5, with Mc=2 always. Note that theurves corresponds to the extra condition τi≥30 min, for which therences turn out to be not significative and therefore we cannot rule out


of the rest of the process (Ogata, 1999; Helmstetter andSornette, 2002). Of course, this independence isestablished within the errors associated to our finitesample. It could be that the dependence between themagnitudes is weak enough for that the changes indistribution are not larger than the uncertainty. With ouranalysis, only a much larger data set could unmask thishypothetical dependence.

The deviations for short times may be an artifact dueto the incompleteness of earthquake catalogs at shorttime scales, for which small events are not recorded.Helmstetter et al. (2006) propose a formula for themagnitude of completeness in Southern California as afunction of the time since a mainshock and itsmagnitude; applying it to our stationary periods, forwhich the larger earthquakes have magnitudes rangingfrom 5 to 6, we obtain that a time of about 5 h isnecessary in order that the magnitude of completenessreaches a value below 2 after a mainshock of magnitude6. After mainshocks of magnitude 5.5 and 5 this timereduces to about 1 h and 15 min, extrapolating theresults of Helmstetter et al. In any case, it is perfectlypossible that large mainshocks (not necessarily thepreceding event) induce the loss of small events in therecord and are the responsible of the deviations from theGutenberg–Richter law at small magnitudes for shorttimes. If an additional physical mechanism is behind thisbehavior, this is a question that cannot be answered withthis kind of analysis.

As in the previous subsection, we have performedmeasurements of the conditional distributions for world-wide seismicity involving different Mi and Mj, separatedup to 10 events, with no significant variations in thedistributions, as expected. These results confirm theindependence of the magnitude Mi with its own history.

4. Conclusions

We have argued how a scaling law for earthquakerecurrence-time distributions can be understood as aninvariance under a transformation analogous to those ofthe renormalization group (as in the critical phenomenafound in condensed matter physics). Such invariancecan only exist due to an interplay between theprobability distribution and the correlations of theseismicity process. We have seen how the correlationsare reflected on conditional probability densities, whosestudy for stationary seismicity (worldwide or local)leads to the results summarized here:

• The bigger the size of an earthquake, the shortest thetime till next, because of the anticorrelation between

magnitudes and forward recurrence times. Note thatthis result is not trivial, as we are dealing withstationary seismicity.

• The shortest the time to get an earthquake, theshortest the time till the next, as the correlationbetween consecutive recurrence times is positive.

• An earthquake does not “know” how big is going tobe, due to the independence of the magnitude on thehistory of seismicity (at least from the informationrecorded at the catalogs, disregarding spatial struc-ture, and with our present resolution).

These sentences, together with the previouslyestablished: the longer since the last earthquake, thelonger the expected time till the next (which does notdeal with correlations but with a somewhat lessstraightforward concept, see Davis et al., 1989; Sornetteand Knopoff, 1997; Corral, 2005d) form a collection ofaphorisms for seismicity properties. Note however thatthe belief that the longer the time to get an earthquake,the bigger its size, is false, as it would imply an a prioriknowledge of how big the earthquake is going to be, incontradiction with a previous statement, i.e., with theuncorrelated behavior of the magnitude.

This law of independence of the magnitude mayseem very striking; in particular, our results, establishedfor the seismicity recorded at the modern instrumentalcatalogs for extended spatial regions (Southern Califor-nia and worldwide), contrast with common beliefsderived from the earthquake-cycle paradigm. However,historical or paleoearthquake data for single faults, forwhich no correlation is found between the magnitude ofa paleoearthquake and the preceding recurrence time,are surprisingly in concordance with our findings(Weldon et al., 2004; Cisternas et al., 2005).

Other works have found a local variation in theGutenberg–Richter b value for different time periods,depending on the occurrence or not of large earthquakes(Wiemer and Katsumata, 1999). In this case, the maindifference with our approach is that we deal withstationarity seismicity, for which no large earthquakesoccur in Southern California. Nevertheless, one maywonder if the consideration of larger values of Mc′ in themagnitude distributions could lead to a variation in thesedistributions above the statistical uncertainties. In anycase, at this point our method does not allow todistinguish between physical phenomena and catalog-incompleteness artifacts.

Finally, let us mention that very recently themagnitude of an event has been found to be correlatednot with the history of seismicity but with the focalmechanism of that event, in such a way that normal


faulting events yield large b values, whereas thrust eventsyield small b values (Schorlemmer et al., 2005). In a relatedwork it has been shown that a large earthquake is likely tooccur in areas of local low b value (in comparison withsurrounding areas of high b) (see Schorlemmer andWiemer, 2005). Although one could think that this impliesthat the distribution of magnitudes is correlated with themagnitude of a future large event, this is not the case, asdistributions for different areas are involved. So, there is nodisagreement between these recent works and our results.

Acknowledgements

The author is grateful to Y. Y. Kagan for his referral ofa previous work and other interesting comments. M.Kukito, S. Santitoh, and M. Delduero are also acknowl-edged for quick inspiration, as well as the Ramón y Cajalprogram, the National Earthquake Information Center,the Southern California Earthquake Data Center, and theparticipation at the research projects BFM2003-06033(MCyT) and 2001-SGR-00186 (DGRGC).

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Dependence of earthquake recurrence times and independence of magnitudes on seismicity history