The time dynamics of the aftershock sequence of the Al-Hoceima (Morocco) earthquake of 24 February 2004 has been investigated. The sequence of the occurrence times of the events with threshold magnitude &lt;i&gt;M&lt;sub&gt;th&lt;/sub&gt;&lt;/i&gt;&amp;ge;3.2 is characterized by a time-clustering behavior, identified using different fractal methods (Fano Factor, Allan Factor, Count-based Periodogram), well suited to reveal scaling features in point processes. The obtained results not only show the presence of memory phenomena and correlation structures in the Al-Hoceima aftershocks, but also furnish quantitatively the estimate of the magnitude of such correlation by means of the estimate of the scaling exponent &amp;alpha;

L. Telesca

M. Rouai

T. E. Cherkaoui

Natural Hazards and Earth System Science

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Nat. Hazards Earth Syst. Sci., 9, 2063–2066, 2009www.nat-hazards-earth-syst-sci.net/9/2063/2009/© Author(s) 2009. This work is distributed underthe Creative Commons Attribution 3.0 License.Natural Hazardsand EarthSystem SciencesTime-clustering behavior in the sequence of the aftershocks of theAl-Hoceima (Morocco) 24 February 2004 earthquakeL. Telesca1, M. Rouai2, and T. E. Cherkaoui31Istituto di Metodologie per l’Analisi Ambientale, CNR, Tito Scalo (PZ), Italy2Laboratory of Enginnering Geology, Faculty of Sciences, Meknes Universiy, P.O. Box 11201, Zitoune, Meknes, Morocco3Institut Scientifique, Mohamed V – Agdal University, Rabat, MoroccoReceived: 3 August 2009 – Accepted: 30 September 2009 – Published: 8 December 2009Abstract. The time dynamics of the aftershock sequence ofthe Al-Hoceima (Morocco) earthquake of 24 February 2004has been investigated. The sequence of the occurrence timesof the events with threshold magnitudeMth≥3.2 is charac-terized by a time-clustering behavior, identified using differ-ent fractal methods (Fano Factor, Allan Factor, Count-basedPeriodogram), well suited to reveal scaling features in pointprocesses. The obtained results not only show the presenceof memory phenomena and correlation structures in the Al-Hoceima aftershocks, but also furnish quantitatively the es-timate of the magnitude of such correlation by means of theestimate of the scaling exponentα.1 IntroductionAnalyzing the temporal distribution of seismic sequencesplays an important role in the general framework of the stud-ies devoted to seismicity. The time-correlation structuresgoverning observational time series can give useful informa-tion on the dynamical features of seismic processes and onthe involved geodynamical mechanisms. In particular ana-lyzing the temporal distribution of earthquakes could havegreat implications to improve the probabilistic techniques forseismic hazard analysis (Main, 1995, 1996).Seismic processes display a scale-invariant behavior inseveral statistical features. Gutenberg and Richter (1944)found that earthquake size follows a power-law distribution.Scale-invariant features were determined in Kagan (1992)and Kagan and Jackson (1991); Kagan (1994) reviewed ex-perimental evidences for earthquake scale-invariance. A the-ory to explain the presence of scale-invariance was proposedby Bak et al. (1988) in a pioneer work; they introducedthe idea of self-organized criticality (SOC) starting from aCorrespondence to:L. Telesca(luciano.telesca@imaa.cnr.it)simple cellular automaton model, namely sandpile. In thismodel, sand is dropped slowly into random locations on alattice, one grain at a time. For a critical slope, when a newgrain is added many avalanches are triggered. Numericallythey obtained that the distribution of avalanche sizes presentsa power-law spectrum.The analysis of scaling laws concerning earthquakes hasled to the development of a wide variety of physical modelsof seismogenesis used to better characterize the seismicitypatterns (Sornette et al., 1990; Scholz, 1991).Several distributions have been used to model seismic ac-tivity. The Poisson distribution implies the independence ofeach event from the time elapsed since the previous event.However, earthquake occurrence is characterized by cluster-ing properties with both short and long timescales (Kaganand Jackson, 1991) that imply a temporal correlation amongthe seismic events. A complex phenomenon, like earth-quakes, can be deeply known only if the different timescalesgoverning its dynamics are well understood. Statistical ap-proaches like the Fano Factor, Allan Factor, Count-based Pe-riodogram, can furnish an exhaustive description of the tem-poral properties of point processes, and, in particular, earth-quakes. Using those methodologies, the aftershock sequenceof the Al-Hoceima (Morocco) 24 February 2004 earthquakeis investigated.Aftershock sequences, apparently related to the faultplane that slipped during the main event (Lay and Wal-lace, 1995), are the relevant characteristic of shallow earth-quakes. An aftershock series displays temporal fractal prop-erties, by means of the Omori’s law (Utsu et al., 1995). TheOmori’s law gives only preliminary information of the time-correlation structures that characterises the aftershocks of amain event. A deep investigation of these structures is pro-vided by the analysis of second-order statistics. For this pur-pose, in this paper the aftershock sequence is modeled bya point process. A point process describes events that oc-cur at some random locations in space and time, and it isPublished by Copernicus Publications on behalf of the European Geosciences Union.2064 L. Telesca et al.: Time-clustering behavior of the aftershocks of the Al-Hoceima earthquakecompletely defined by the set of event times and hypocenterlocations. Dividing the time axis into equally spaced contigu-ous counting windows of durationτ , we produce a sequenceof counts{Nk}, whereNk represents the number of eventsfalling into the kth window. This sequence is a discrete-random process of non-negative integers. An important fea-ture of this representation is that it preserves the correspon-dence between the discrete time axis of the counting process{Nk} and the “real” time axis of the underlying point pro-cess, and the correlation in the process{Nk} refers to corre-lation in the underlying point process (Thurner et al., 1997).Such a process may be called fractal when a number of rele-vant statistics exhibit scaling with related scaling exponents,that indicate the represented phenomenon contain clusters ofpoints over a relatively large set of timescales (Thurner et al.,1997).The temporal fluctuations of a process are investigated bythe analysis of its power spectral density. The power spectraldensity provides a measure of how the power in a processis concentrated in various frequency bands, and for a pro-cess with scaling properties it follows a power-law behaviorS(f )∝f −α, with the exponentα measuring the strength ofthe cauterization. Of course, for a finite size fractal real pro-cess the power spectral density behaves as an inverse power-law function in a limited range of frequency, approaching anasymptotic value at high frequencies, at which the behaviorof the process can be considered Poissonian. The cluster-ing behavior of a point process leads to a power-law (fractal)shape of some of the statistics, used to describe its properties;we refer, in particular, to the Allan Factor (AF), that allowsto estimate the spectral exponentα; the numerical value ofαis an indicator of the presence of clusterisation in the process(Thurner et al., 1997). If the point process is Poissonian,the occurrence times are uncorrelated; for this memorylessprocessα≈0. On the other side,α 6=0 is typical of point pro-cesses with self-similar behavior; self-similar meaning thatparts of the whole can be made to fit to the whole in someway by scaling (Mandelbrot, 1983).2 The methodsThe sequence of the recorded earthquakes can be expressedby a finite sum of Dirac’s delta functions centered on the oc-currence timesti :y(t) =n∑i=1δ(t − ti) . (1)Dividing the time axis into equally spaced contiguous count-ing windows of durationτ , we can produce a sequence ofcounts{Nk (τ )}, with Nk (τ ) denoting the number of earth-quakes in thek-th window:Nk(τ ) =tk∫tk−1n∑j=1δ(t − tj)dt . (2)A measure, related to the variability of successive counts,useful to detect the event clustering in a point process, is theAllan Factor (AF) (Allan, 1966), defined as the variance ofsuccessive counts for a specified counting timeτ divided bytwice the mean number of events in that counting timeAF(τ ) =< (Nk+1(τ )−Nk(τ ))2 >2< Nk(τ ) >. (3)This measure reduces the effect of possible nonstationarityof the point process, because it is defined in terms of thedifference of successive counts. The AF of a fractal pointprocess varies with the counting timeτ with a power-lawform:AF(τ ) = 1+(ττ1)α(4)with 0<α<3 over a large range of counting timesτ (Lowenand Teich, 1995 ). The monotonic power-law increaseis representative of the presence of fluctuations on manytimescales. The scaling exponentα is the so-called fractalexponent. Ifα>0 then the represented phenomenon containsclusters of points over a relatively large set of timescales.If α≈0, the earthquake occurrence process is Poissonianand the occurrence times are uncorrelated. The crossovertimescaleτ1 is the fractal onset time, and marks the lowerlimit for significant scaling behavior in the AF (Teich et al.,1996), so that forττ1 the clustering property becomesnegligible at these timescales. Thus,τ1 is estimated as thetimescale over which the AF increases as a power-law func-tion of the timescaleτ . AF assumes values near unity forPoisson processes.The count-based Periodogram (PG) (Teich et al., 1996),that is the periodogram of the sequence of the counts, is an-other statistical measure that allows estimating the fractal ex-ponentα. This measure performs an estimate of the powerspectral density, which gives information on how the powerof the process is concentrated at various frequency bands.The calculation of the PG by means of a count-based ap-proach implies the division of the total observation periodinto a numberN of no overlapping windows of lengthw.The sequence of the counts is then Fourier transformed andthe PG is obtained by the squares of the coefficients of theFourier representation of the series of counts. Generally, inorder to use the Fast Fourier Transform (FFT) the numberN of the windows of durationw is a power of two. Forpoint processes with scaling properties the PG decreases asa power-law function of the frequencyf over a significantrange of frequencies,S(f )∝f −α. Of course, for a finite sizefractal real process the power spectral density behaves as aninverse power-law function in a limited range of frequen-cies, approaching an asymptotic value at high frequencies,at which the behavior of the process can be considered Pois-sonian. The numerical value ofα is an indicator of the pres-ence of clusterization in the process (Thurner et al., 1997).If the point process is Poissonian, the occurrence times areNat. Hazards Earth Syst. Sci., 9, 2063–2066, 2009 www.nat-hazards-earth-syst-sci.net/9/2063/2009/L. Telesca et al.: Time-clustering behavior of the aftershocks of the Al-Hoceima earthquake 2065 Fig. 1    1 2 3 4 5 6 7100101102103a=6.2+0.2b=1.04+0.05Mc=3.2N(M>Mth)Mth Fig. 2  Fig. 1. Hypocenter of 938 aftershocks recorded with a minimum offive stations.uncorrelated; for this memoryless process, the PG is approx-imately flat for any frequency bands andα≈0. On the otherside,α 6=0 is typical of point processes with scaling behavior.This method introduces a bias at higher frequencies, sincethe fine time resolution information is lost as a result of theminimum count-window size. But, the estimation of the ex-ponentα principally involves lower frequencies where thisbias is negligible.3 Analysis of Al-Hoceima aftershock seriesAl-Hoceima region (NE Morocco, Fig. 1) is the most seis-mically active zone of the country (Cherkaoui, 1991). It islocated in the eastern Mediterranean collision boundary be-tween African and Eurasian plates. The average displace-ment is about 5 mm/y (Argus et al., 1989). This convergentmovement of Africa towards Eurasia induces high and dif-fuse seismicity, with a strike-slip dominant mechanism. On24 February 2004 at 2 h 27 mn 47 s, Northern Morocco wasagain struck by a shallow earthquake of magnitudeMw=6.3(Stich et al., 2005) and intensity IX (MSK scale).The earth-quake occurred in Ait Qamra, a village located 15km south-west of Al-Hoceima. The earthquake shook the region, caus-ing the considerable damages to the Ait Qamra, Imzourenand its surrounding regions. The toll of earthquake is catas-trophic: 629 dead, 926 injured, 15 000 of people lost theirhomes and more than 2500 houses destroyed. The mainshock was located at 35.129◦ N–4.011◦ W and depth=5.5 km.Investigations carried out rapidly in the region just after theearthquake resulted in serious damages to the constructions:hoses, schools, mosques (Cherkaoui, 2005). The earthquakemain event was followed by a long aftershock sequence dur- Fig. 1    1 2 3 4 5 6 7100101102103a=6.2+0.2b=1.04+0.05Mc=3.2N(M>Mth)Mth Fig. 2  Fig. 2. Gutenberg-Richter analysis of the aftershock sequence. Theminimum magnitude of completeness is 3.2. The b-value, estimatedas the slope of the line fitting the cumulative number of events ver-sus threshold magnitude, forMth≥3.2 is 1.04.ing the time from 24 February to 31 May. During this pe-riod six strong aftershocks occurred withmb≥5.0. A totalof 838 shocks have been identified as having occurred in theneighbourhood of the main shock and recorded with of aminimum of five stations. These events were located usingHypo71PC program (Lee and Valdés., 1985) with theP andS arrival times collected at all the available stations (Fig. 1).In order to study the temporal distribution of a com-plete catalogue, we performed the Gutenberg-Richter anal-ysis (Fig. 2), obtaining a b-value of 1.04 for the cumulativedistribution of events with magnitudeM≥3.2, that can beconsidered the minimum magnitude of completeness. Wecomputed the AF(τ ) for counting times of duration of 50 sto T/10, whereT represents the total observation period;the upper cut-off is due to the finite data length (Thurner etal., 1997) . The AF plot (Fig. 3) gives clear indication ofthe fractal behaviour; the AF is seen to increase with lin-ear behaviour for counting times greater than approximatelyτ1∼104 s in bilogarithmic scales, which can give a rough es-timate of the fractal onset time. The slope of the straightline that fits the AF in the linear range gives the estimateof the exponentα. In particular we estimatedα∼0.9. Fig-ure 4 shows the results obtained performing the PG analysis;the estimated scaling exponent (α∼1.09) and the crossovertimescale are consistent with those calculated by the AFmethod.4 ConclusionsThe occurrence time sequence of aftershocks of the Al-Hoceima (Morocco) 24 February 2004 earthquake has beentudied by means of quantitatively fractal tools (AF and PG).These methods, extensively used in other scientific fields, arevery useful to reveal the presence of clustering behavior inwww.nat-hazards-earth-syst-sci.net/9/2063/2009/ Nat. Hazards Earth Syst. Sci., 9, 2063–2066, 20092066 L. Telesca et al.: Time-clustering behavior of the aftershocks of the Al-Hoceima earthquake102 103 104 105 106100101102τ1=104sα=0.90+0.01AF(τ)τ (s) Fig. 3  10-6 10-5 10-410-1210-1110-1010-910-810-710-610-510-410-3α=1.09+0.04τ=104.3sf (Hz)Power Fig. 4  Fig. 3. Doubly logarithmic plot of the Allan Factor curve versus thecounting timeτ for the aftershock sequence, with threshold magni-tudeMth=3.2.102 103 104 105 106100101102τ1=104sα=0.90+0.01AF(τ)τ (s) Fig. 3  10-6 10-5 10-410-1210-1110-1010-910-810-710-610-510-410-3α=1.09+0.04τ=104.3sf (Hz)Power Fig. 4  Fig. 4. Doubly logarithmic plot of the Count-based Periodogramversus the frequency f for the aftershock sequence, with thresholdmagnitudeMth=3.2.time series represented as realizations of point processes, andto determine, for the point process under investigation, theexponent of its power spectral density, which behaves likea power-law function if governed by time-clustering proper-ties.Time-occurrence sequences of Al-Hoceima 24 February2004 aftershocks characterized by magnitude greater orequal to 3.2 show the presence of power-law behavior in theperformed statistics, indicating the existence of fluctuationson many timescales and therefore of fractal clustering.Acknowledgements.This study was supported by the bilateralproject Italy-Morocco in the frame of the CNR/CNRST bilateralagreement for scientific cooperation.Edited by: M. E. ContadakisReviewed by: J. Sebbani and another anonymous refereeReferencesAllan, D. W.: Statistics of atomic frequency standards, Proc. 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Time-clustering behavior in the sequence of the aftershocks of the Al-Hoceima (Morocco) 24 February 2004 earthquake

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Time-clustering behavior in the sequence of the aftershocks of the Al-Hoceima (Morocco) 24 February 2004 earthquake

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