Physica A 340 (2004) 590–597www.elsevier.com/locate/physa

Universal local versus uni!ed global scaling lawsin the statistics of seismicity

&Alvaro CorralDepartament de F��sica, Facultat de Ci�encies, Universitat Aut�onoma de Barcelona,

E-08193 Bellaterra, Spain

Received 15 February 2004; received in revised form 27 February 2004

Abstract

The uni!ed scaling law for earthquakes, proposed by Bak, Christensen, Danon and Scanlon, isshown to hold worldwide, as well as for areas as diverse as Japan, New Zealand, Spain or NewMadrid. The scaling functions that account for the rescaled recurrence-time probability densitiesshow a power-law behavior for long times, with a universal exponent about (minus) 2.2. Anotherdecreasing power law governs short times, but with an exponent that may change from one areato another. This is in contrast with a local, time-homogenized version of Bak et al.’s procedure,which seems to present a universal scaling behavior.c© 2004 Elsevier B.V. All rights reserved.

PACS: 91.30.Dk; 05.65.+b; 64.60.Ht; 89.75.Da

Keywords: Statistical seismology; Marked point processes; Complex systems

1. Introduction

Earthquakes are complex phenomenon, where one single event may generate anavalanche of scienti!c papers, each telling a part of the story of the quake. Essentially,these papers argue that a speci!c set of tectonic forces and other diverse factors can beused to arrive at the characteristics of the earthquake discussed. This research providesimportant information for speci!c mechanisms triggering earthquakes, “one explanationfor each earthquake”.

E-mail address: Alvaro.Corral@uab.es ( &A. Corral).

0378-4371/$ - see front matter c© 2004 Elsevier B.V. All rights reserved.doi:10.1016/j.physa.2004.05.010

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However, another point of view is possible. As an alternative to the described reduc-tionism, Bak was claiming the necessity of a general theory encompassing all earth-quakes [1], from the very small (imperceptible by humans) to the largest, catastrophicones (“killing hundreds of thousands of people”), irrespective of their location at theboundaries or interior of the plates, depth, and other tectonic details. A key clue sig-naling the unity of the phenomenon is the Gutenberg–Richter law [2,3], which statesthat, for any region, the number of earthquakes decreases exponentially as a func-tion of their magnitude. If the seismic processes did not form a whole, how couldit be that all that variety of events conspire together to align onto such a simplecurve?The !rst step towards a theory of earthquakes should consist on identifying what

kind of dynamical process we are dealing with [4]: “Is it periodic? Is it chaotic?Is it random in space and time?” From our present knowledge, the best candidateis self-organized criticality (SOC) [1,5–7]. The analogies between earthquakes andSOC systems are clear [8–10]: the Earth crust accumulates energy (supplied by slowconvective motion in the mantle) in the form of elastic deformation at a very slowrate. At some point the stress cannot be sustained and a rupture initiates, propagatingvery fast through a fault by means of a domino eMect, giving rise to an earthquake.So, we have the basic ingredients for SOC, i.e., a long-term balance between slowdriving and fast avalanches in a spatially extended system consisting of many in-teracting parts. The Gutenberg–Richter law is again prominent in this picture, as itimplies that there is not a characteristic scale for the energy dissipated during an earth-quake; this is because the energy increases exponentially with the magnitude (abouta factor 30 in the energy for each unit in the magnitude), and then, the exponen-tial frequency-magnitude relation transforms into a power-law distribution of dissipatedenergies [3], which is the indication of scale invariance, and may be of criticality.Consequently, one may talk about the crust as being at a critical state, but in contrastto equilibrium critical points, this state has to arise spontaneously, as an attractor ofthe dynamics.In addition to the Gutenberg–Richter law, there are other indicators of scale invari-

ance in seismicity, such as the fractal distribution of hypocenters or epicenters [2,3,11],and the Omori law, which tells us that the decay of the rate of seismic activity af-ter a large event does not present any characteristic time [12]. Further, the structureof faults and tectonic plates is also fractal [3,13,14]. For these reasons, although theSOC paradigm is represented by a sandpile [15], earthquakes may be considered asthe clearest illustration of how a real SOC system would look [1].Despite the initial opposition of the “very conservative” geophysics community to

the idea of earthquakes as a SOC phenomenon [1], nowadays SOC is very seriouslyconsidered by many professional seismologists. It seems that scienti!c evolution inthe solid-Earth sciences takes place mostly after great painful controversies [16]; andearthquakes are a !eld where the debate is open [17].A great hallmark in the earthquake-as-a-SOC-phenomenon development was the re-

lease of the Olami, Feder, and Christensen (OFC) model [18] (for a summary of therich dynamics of its variations see Ref. [19]). However, it is a subsequent surprisingproposal by Ito which calls our interest here [20]: the well-studied Bak–Sneppen (BS)

592 �A. Corral / Physica A 340 (2004) 590–597

model [21], introduced to account for biological evolution, reproduced some otherproperties of earthquakes. Ito measured for a California catalog the same quantitiesused to characterize the BS model [22], in particular, the !rst-return-time distribution:this is the probability that the activity returns (for the !rst time) at a given spatiallocation after a certain time. As the locations of earthquake occurrence are continu-ous (in contrast to the BS model), Ito divided the area covered by the catalog intosmall regions of 1◦ latitude × 1◦ longitude and measured the return times to theseregions. The results seemed compatible with a power-law distribution (as in the BSmodel), but clearly, a more in-depth investigation was needed. Note also another im-portant diMerence between the BS model and real earthquake occurrence: the formeris spatially homogeneous, whereas earthquake epicenters draw a fractal over the Earthsurface.It was Bak, together with Christensen, Danon and Scanlon, who re-opened the prob-

lem [23,24]. In essence, they used Ito’s procedure with the addition of a lower boundMc for the magnitude, in such a way that events with magnitude M below Mc weredisregarded. This is crucial to avoid spatial and time variations in the completeness ofthe catalogs and to ensure that no events (or not many) with M¿Mc are missing. Apower-law !rst-return-time distribution, followed by a faster decay, was indeed found,but the exponent was diMerent from Ito’s exponent; the faster decay was later identi-!ed as another power law [25]. However, Bak et al. also introduced a crucial element,which was the study of the distribution under the variation of its two parameters: thelower bound Mc and the size L of the small regions. Remarkably, a scaling analysisshowed that diMerent distributions corresponding to diMerent values of L and Mc col-lapsed onto a single curve under rescaling of the axes by a factor 10bMc =Ldf , where thenumerator comes from the Gutenberg–Richter relation and the denominator accountsfor the fractal distribution of epicenters. In this way, it is appropriate to talk about auni1ed scaling law for earthquakes.This approach reNects Bak’s philosophy applied to earthquakes, which can be sum-

marized as: (1) Do not bother about the tectonic environment: the small regions inwhich California is divided are independent of it, in contrast with traditional studies.(2) Do not bother about aftershocks and foreshocks, none of these events are removed,all are equally treated, again at variance with usual approaches. (3) Do not botherabout temporal heterogeneity.In this paper, I explore in detail Bak et al.’s uni!ed scaling law; !rst, introduc-

ing some variations to their procedure [26,27], and next, turning to their method andextending it to several regions in the world. We will see how scaling is an intrinsiccharacteristic of seismicity, which supports the view of a critical crust.

2. Universal scaling law for local distributions of recurrence times

Let us consider spatial regions of arbitrary shape, which can be of small size (as inIto’s paper [20]) but also large. In contrast to Ito and Bak et al. [23,24], we concentrateonly on one of these regions, where (in the same way as Bak et al.) earthquakes withmagnitude M above a lower bound Mc are selected. Then, if ti denotes the time of

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occurrence of the ith earthquake in the spatial and magnitude windows considered, wecalculate the recurrence time between events i and i − 1 as i = ti − ti−1. This timeis the same !rst-return time de!ned by Ito, or the waiting time in the language ofBak et al.The probability density of the recurrence times can easily be obtained; however, in

order to pay attention to all the time scales involved in the process we will look attime logarithmically, estimating the densities using an exponentially increasing binning.The !rst data set to start these measurements is a global earthquake catalog, as itsrate of seismic occurrence, de!ned as number of earthquakes per unit time, is fairlyconstant, and therefore stationary. Using the NEIC worldwide catalog [26] (1973–2002) for several very large regions, and rescaling the axes by the seismic rate, weget the results displayed in Fig. 1 (top curve). All the distributions lie on one uniquecurve, so,

Dxy() = Rxyf(Rxy) ; (1)

where, for a region of spatial coordinates xy, Dxy() is the probability density that therecurrence time is around a value and Rxy is the mean rate of seismic occurrence (oractivity). It is implicit that both Dxy() and Rxy depend as well on Mc and the size ofthe region; to be concrete, if the region is kept !xed the rate depends exponentiallyon the magnitude, following the Gutenberg–Richter relation: Rxy = Nxy10−bMc , withNxy the (hypothetical) number of events per unit time in the region with magnitudeabove 0.The scaling function f can be represented by

f(�) =C|�|a�(�=�)

(�a

)�−1

e−(�=a)� ; (2)

which has a very general shape. If � and � are positive, the former controls the shapefor small � and � the shape at large �; the situation is reversed if both parameters arenegative; a is a scale parameter and C a normalization correction. Note that � ≡ Rxyis a dimensionless time.If regions of smaller size are considered, the rate turns nonstationary, giving rise

to heterogeneities in time. This is due to large earthquakes, which provoke a kind of“avalanches of earthquakes”, i.e., the aftershock sequences. In order to compare withthe worldwide case, we consider space-time windows in which the rate stays station-ary; in other words, we stay away from time periods which include very prominentaftershock sequences, in opposition to Bak et al. (later we will eliminate this restric-tion). The simplest way to recognize stationarity is by a linear dependence between theaccumulated number of earthquakes in a region and time. Using the NEIC data andseveral regional and local catalogs (Southern California, Japan, Spain, Great Britain[26], New Zealand, New Madrid [27], and also Northern California [28]), we !nd theresults displayed in Fig. 1, taking regions of L degrees in longitude and L degrees inlatitude. The behavior of the distributions is identical to the previous case, collaps-ing under rescaling onto the same universal curve f. The deviations found at shorttimes are due to a nonstationary rate at this time scale, provoked by small aftershocksequences.

594 �A. Corral / Physica A 340 (2004) 590–597

Fig. 1. Single-region recurrence-time probability densities, Dxy , after rescaling with the rate Rxy . The !vedata sets correspond, from top to bottom, to: 1, the NEIC worldwide catalog for regions with L¿ 180◦,1973–2002; 2, NEIC with L6 90◦ (same period of time); 3, Southern California, 1984–2001, 1988–1991,and 1995–1998; 4, Northern California, 1998–2002, 5, Japan, 1995–1998, and New Zealand, 1996–2001; 6(bottom), Spain, 1993–1997, New Madrid, 1975–2002, and Great Britain, 1991–2001. The distribution setshave been multiplied by 100; 10−2; 10−4; 10−6, 10−8, and 10−10, for clarity sake. A total of 82 diMerentdistributions are shown, with the size of the regions from 0:16◦ to the whole world, and Mc from 1.5 to7.5. Recurrence times go from 2 min to about 1.5 years; values of ¡ 2 min are not shown. The linescorrespond to f(�).

A least-square !t using many regions from the NEIC catalog and several values ofMc and L yields [26] �= 0:67 ± 0:05, �= 1:05 ± 0:05 and a= 1:64 ± 0:15; so, � canbe considered to be one and we have a decreasing power law with exponent 1 − �about 0.3 accelerated by an exponential term at large times. This type of distributionindicates that earthquakes cluster in time, not only for sequences of aftershocks (asit is well known) but even for “background seismicity”. The counterintuitive conse-quences of this phenomenon for the time evolution of seismic hazard are analyzedin Ref. [27].Finally, the scaling law we propose is valid beyond the stationary limit, replacing the

mean rate Rxy by its instantaneous value, rxy(t). In this way, the probability density� of � ≡ rxy(t) satis!es � = f(rxy(t)), for aftershock sequences where the ratedecays following the modi!ed Omori law, rxy(t) = Axy=tp for long enough time, withthe origin of time at the mainshock [26].As all data analyzed are well !t by the same function f, using the same parameter

values, this implies a universal scaling law. Nevertheless, this law cannot be designated

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Fig. 2. Recurrence-time probability densities, calculated following Bak et al.’s procedure, after rescalingby R; 84 distributions are shown, with L ranging from 0:039◦ to 45◦, and 1:56Mc6 6. The curves areshifted by factors 100; 10−2; 10−4 and 10−6, and correspond to: 1 (top), Southern California, 1984–2001;2, Northern California, 1985–2003; 3, Southern California, 1988–1991 (stationary rate), NEIC, 1973–2002,Japan, 1995–1998, and Spain, 1993–1997; 4 (bottom), New Zealand, 1996–2001, and New Madrid, 1975–2002. Short or intermediate times are !t by: 1 (top), 0:12=� 0:95; 2, 0:15=� 0:9, 3 and 4 (bottom), 0:05=� 0:95

and 0:5=� 0:5, with � ≡ R (dimensionless). In all cases the long-time tail is !t by 0:25=� 2:2. The times inthe horizontal axis span from 2 min to about 20 years. Recurrence times smaller than 4, 10, and 2 min arenot shown, for Japan, NEIC, and the rest of catalogs, respectively.

as uni!ed, since the scaling only includes the Gutenberg–Richter law, but not thefractal dimension of the epicenters, in contrast to Bak et al.’s law. And, in terms ofits de!nition in space, Dxy may be referred to as a local distribution, although oneshould note that the size of the xy-region can grow to reach the total area covered bythe catalog.

3. Uni�ed scaling law beyond Southern California

We now return to the original approach of Bak et al. We only have to considerall the (non-overlapping) regions of size L necessary to cover completely a muchlarger area (Southern California, Japan, the whole world, or New Madrid) and expandthe time windows without bothering about the nonstationarity of the seismic rate. Wemeasure for each xy-region the recurrence times (xy)i in the same way as before,

596 �A. Corral / Physica A 340 (2004) 590–597

with the diMerence that all these series of recurrence times are counted into one singleprobability density, D(), which we may call global (in contrast with our local version).It is found that D() scales, under the change of L and the lower bound Mc, as

D() =RF(R) (3)

with R the mean value of the local mean rate Rxy, calculated over all the xy-regionsof size L with seismic activity, i.e., R =

∑xy Rxy=n, where n is the number of such

regions. From the equation for Rxy and the scaling of n with L, n = (‘=L)df , we get,R = N (L=‘)df10−bMc , with N =

∑xy Nxy and ‘ a rough measure of the linear size of

the total area under study (in degrees).In addition to Southern California [23,24], the scaling relation for D is valid for

the catalogs studied in the previous section, see Fig. 2. However, the scaling functionF does not seem to be universal; the clear decreasing two-power-law behavior ofSouthern California [25], with exponent for small about 0.95 and 2.2 for large ,changes slightly for Northern California (exponents 0.9 and 2.2), but becomes morecomplicated for the other catalogs, with the appearance of an intermediate bump. Fig. 2shows how this bump can be approximated by a third decreasing power law, withexponent roughly 0.5. Further, for New Madrid there is no trace of the 0.95 exponent,and for New Zealand the behavior is not clear. The only exponent that seems to beuniversal is the one for large times, 2.2 in all cases. The uni!ed scaling law couldalso hold for Great Britain, but the few data considered there, only about 500 events,makes the statistics too poor for small and intermediate times.

4. Conclusions

The probability densities of Bak et al. and ours, in addition to quite diMerent shapes,have diMerent meanings. In our case, we provide the probability of return of an earth-quake for a given xy-region of size L, with the only information required being themean seismic rate there, Rxy. On the other hand, Bak et al.’s distribution gives thereturn probability if one does not know in which region of size L of a much largerarea (like Southern California) one is (or you know the region but you do not haveknowledge about the rate Rxy), and it is necessary to know the average rate R forregions of size L. The relation between these two approaches is studied in Ref. [25].Summarizing, both the uni!ed scaling law and our local approach show the scaling

of the recurrence-time distributions with some average value of the rate of seismicactivity. This is a clear consequence of the scale-invariant structure of seismicity intime, space and magnitude. Whatever triggers earthquakes operates in the same way atall spatial and temporal scales.

Acknowledgements

The author would have been unable to undertake this research without the seminalcontribution of Per Bak, whose personal warmth was also very much appreciated.

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M. Bogun&a, K. Christensen, and the Ram&on y Cajal program have been important atdiMerent stages of this process.

References

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