Scaling behaviour in nonstationary time series can be successfully detected using the detrended fluctuation analysis (DFA). Observational time series often do not show a stable and uniform scaling behaviour, given by the presence of a unique clear scaling region. The deviations from uniform power-law scaling, which suggest the presence of changing dynamics in the system under study, can be identified and quantified using an appropriate instability index. In this framework, the scaling behaviour of the 1981–2007 seismicity in Umbria-Marche (central Italy), which is one of the most seismically active areas in Italy, was investigated. Significant deviations from uniform power-law scaling in the seismic temporal fluctuations were revealed mostly linked with the occurrence of rather large earthquakes or seismic clusters

L. Telesca

M. Lovallo

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Nat. Hazards Earth Syst. Sci., 8, 973–976, 2008www.nat-hazards-earth-syst-sci.net/8/973/2008/© Author(s) 2008. This work is distributed underthe Creative Commons Attribution 3.0 License.Natural Hazardsand EarthSystem SciencesInvestigating non-uniform scaling behaviour in temporalﬂuctuations of seismicityL. Telesca and M. LovalloDipartimento Infrastrutture e Mobilit` a, Ufﬁcio Difesa del Suolo, Regione Basilicata, Potenza, ItalyReceived: 30 June 2008 – Revised: 28 July 2008 – Accepted: 28 July 2008 – Published: 1 September 2008Abstract. Scaling behaviour in nonstationary time seriescan be successfully detected using the detrended ﬂuctuationanalysis (DFA). Observational time series often do not showa stable and uniform scaling behaviour, given by the pres-ence of a unique clear scaling region. The deviations fromuniform power-law scaling, which suggest the presence ofchanging dynamics in the system under study, can be iden-tiﬁed and quantiﬁed using an appropriate instability index.In this framework, the scaling behaviour of the 1981–2007seismicity in Umbria-Marche (central Italy), which is one ofthe most seismically active areas in Italy, was investigated.Signiﬁcant deviations from uniform power-law scaling in theseismic temporal ﬂuctuations were revealed mostly linkedwith the occurrence of rather large earthquakes or seismicclusters.1 IntroductionThe complexity of the dynamics of geophysical processesis often characterized by self-similarity, which suggests thattheir dynamics may be interpreted as due to many compo-nents interacting over a wide range of time or space scales(Ashkenazy et al., 2003). A typical example of complex sys-tems are earthquakes, characterized by power-law distribu-tion of spatial, temporal and energy parameters. The exis-tence of power-laws also qualiﬁes earthquakes as a good ex-ample of fractal systems.The interevent intervals (times between two successiveseismic events) follow an exponential decreasing distribution(Poissonian) for completely random sequences, while theyare generally power-law distributed for time-clusterized se-quences (Telesca et al., 2001). But the probability densityCorrespondence to: L. Telesca(luciano.telesca@imaa.cnr.it)function of the interevent intervals is only one feature of theprocess and it reveals none about the correlation properties(Turcott et al., 1994). Therefore, time-fractal second-ordermethods are necessary to investigate the temporal ﬂuctua-tions of seismic sequences more deeply. The use of statisticslike the Allan Factor (Allan, 1966), the Fano Factor (Lowenand Teich, 1995), the Detrended Fluctuation Analysis (DFA)(Peng et al., 1995), has allowed getting more insight into thetime dynamics of seismicity. All these measures lead to thedetermination of a scaling exponent, estimated by a linearﬁtting procedure performed on the power-law statistics plot-ted in log-log scales. Such scaling exponent informs aboutthe dynamical properties of an earthquake sequence, in termsof its correlation structures and its memory phenomena andseveral studies have been performed aiming at the: (i) dis-crimination between Poissonian and clusterized sequences,(ii) spatial variability of time-clustering behaviour, and (iii)magnitude-variability of the property of time-clusterization(Telesca et al., 2003). Furthermore, the numerical value ofthe scaling exponent allows to determine quantitatively thestrength of the temporal ﬂuctuations of seismicity.In the present paper, the scaling behaviour of the seriesof earthquakes occurred in Umbria-Marche, central Italy(Fig. 1) was performed. This area was struck by a strongearthquake occurred on 26 September 1997, with a dura-tion magnitude of 5.8 (this event will be indicated as EQ0hereafter). After that, the Umbria-Marche region, centralItaly, was deeply studied and many investigations have beenperformed regarding its geodynamical features (Boncio andLavecchia, 2000; Calamita et al., 2000), active tectonics(Galadini et al., 1999; Cinti et al., 2000), spatio-temporalseismic distribution (Ripepe et al., 2000; Di Giovambat-tista and Tyupkin, 2000), induced geophysical effects (Es-posito et al., 2000). This area is characterized by a well-documented historical and instrumental seismicity, mainlyconﬁned within the upper part of the crust (<16km) (Lavec-chia et al., 1994).Published by Copernicus Publications on behalf of the European Geosciences Union.974 L. Telesca and M. Lovallo: Non-uniform scaling behaviour in seismicity  9  7.00 8.00 9.00 10.00 11.00 12.00 13.00 14.00 15.00 16.00 17.00 18.0036.0037.0038.0039.0040.0041.0042.0043.0044.0045.0046.0047.00EQ0  Fig. 1 Fig. 1. Spatial distribution of seismicity in Umbria-Marche areafrom 1981 to 2007.2 The methodA series of earthquakes can be represented by a temporalpoint process, that describes events that occur at some ran-dom locations in time (Cox and Isham, 1980), and it can beexpressed by a ﬁnite sum of Dirac’s delta functions centeredon the occurrence times ti, with amplitude Ai proportional tothe magnitude of the earthquake:y(t) =N+1 Xi=1Aiδ (t − ti). (1)(N+1) represents the number of events recorded. This pro-cess can be described by the set of the interevent times, that isthe series of the time intervals between two successive earth-quakes.The DFA was proposed by Peng et al. (1995), and it avoidsspurious detection of correlations that are artifacts of nonsta-tionarity, that often affects experimental data. Such trendshave to be well distinguished from the intrinsic ﬂuctuationsof the system in order to ﬁnd the correct scaling behaviour ofthe ﬂuctuations. Very often we do not know the reasons forunderlying trends in collected data and we do not know thescales of underlying trends. DFA is a method for determin-ing the scaling behaviour of data in the presence of possibletrends without knowing their origin and shape.The methodology operates on the series of the intereventtimes τi, where i=1,2,...,N and N is the length of the series.With ¯ τ we indicate the mean interevent time.The series is ﬁrst integratedy(k) =k Xi=1[τi − ¯ τ], (2)with k=1..N.Next, the integrated time series is divided into boxes ofequal length, n. In each box a least-squares polynomial yn(k)of degree p is ﬁt to the data, representing the trend or order pin that box. Next we detrend the integrated time series y(k)by subtracting the local trend yn(k) in each box. The root-mean-squareﬂuctuationofthisintegratedanddetrendedtimeseries is calculated byF(n) =v uu t 1NN Xk=1[y(k) − yn(k)]2. (3)Repeating this calculation over all box sizes, we obtain a re-lationship between F(n), that represents the average ﬂuctu-ation as a function of box size, and the box size n. If F(n)behaves as a power-law function of n, data present scaling:F(n) ∝ nd. (4)Under these conditions the ﬂuctuations can be described bythe scaling exponent d, representing the slope of the lineﬁtting logF(n) to logn. The estimate of the d-value, per-formedbyaleastsquaremethod, furnishesinformationaboutthe type of correlations in the interevent series; furthermore,it quantiﬁes the growth of the root mean square ﬂuctuationsF(n). For an uncorrelated seismic sequence, d=0.5. If thereare only short-range correlations, the initial slope may be dif-ferent from 0.5 but will approach 0.5 for large window sizesn. d>0.5 indicates the presence of persistent long-range cor-relations, meaning that a large (compared to the average) in-terevent interval is more likely to be followed by large oneandviceversa. d<0.5indicatesthepresenceofantipersistentlong-range correlations, meaning that a large (compared tothe average) interevent interval is more likely to be followedby small one and vice versa. d=1 indicates ﬂicker-noise dy-namics, typical of systems in a self-organized critical state.d=1.5 characterizes processes with Brownian-like dynamics.3 Results and discussionWe investigated the interevent time series of earthquakes oc-curred in Umbria-Marche area, central Italy, from 1981 to2007. We considered the earthquakes (M≥2.1) whose epi-centers were located within a 80km-radius circular area cen-tered on the epicenter of the strongest event of the series(EQ0) (Fig. 1). The seismicity data were extracted from theINGV catalogue, publicly available on http://www.ingv.it.The choice of such selection was given by the great seis-mological importance given by this event, and by the highactivity characterizing such seismic area, featured by severalrather large events during the investigation period.Nat. Hazards Earth Syst. Sci., 8, 973–976, 2008 www.nat-hazards-earth-syst-sci.net/8/973/2008/L. Telesca and M. Lovallo: Non-uniform scaling behaviour in seismicity 975To examine transient phenomena, the seismic intereventtime series was analysed in overlapping windows of ﬁxednumber of events. Therefore the DFA method was per-formed in a sliding window of 300 events, in order to havea sufﬁcient amount of data. The shift between two suc-cessive windows was set to one event, in order to have agood smoothing among the calculated values. In each win-dow, we applied the DFA, removing the polynomial trendsup to the 4th order. Furthermore, after calculating the ﬂuc-tuation curve log[F(n)]∼log(n), using the DFA method,in each window we calculated the scaling instability indexβ, as follows: the local scaling exponent χ(n) can be de-ﬁned as the local derivative (slope) of the ﬂuctuation functionlog[F(n)]∼log(n),χ(n) ≡ d[log(F(n)]/d[log(n)]. (5)If χ(n) is constant for different timescales n, the scaling isstable. Substantial variation of χ(n) with n indicates that thescaling behaviour is unstable and deviates from the uniformpower-law(Viswanathanetal., 1997). Toquantitativelymea-sure the deviation from stable power-law scaling, the scalinginstability index β can be deﬁned as the standard deviationof the local scaling exponents χ(n), in the timescales of in-terest. High values of the index β indicate large deviationsfrom uniform power-law. In some cases the high value ofthe instability index is also given by the presence of differ-ent stable power-law regimes, characterized each of them bywell deﬁned scaling exponent. Using this deﬁnition of scal-ing instability index, each calculated β value was associatedwith the time of occurrence of the last event in the window.Figure 2 shows the time variation of β. It is also shown thedotted line µ+2σ, which can be considered like the thresh-old for a β value to be taken as anomalous. Several featuresare revealed by this analysis: i) the variation of the index βis characterized by a spiky behaviour, where the highest val-ues denote increase of instability; ii) three clusters of eventswith magnitude M>4 (which can be considered as a rela-tively high magnitude) were identiﬁed in the seismicity ofthe area, and are indicated in Fig. 2 by vertical red arrows;iii) in correspondence with these clusters the instability in-dex β shows an increasing spiky behaviour indicated by let-ters a to d; the anomalous β values indicated by letters a,c and d are clearly linked with the earthquakes; the spike bcould be connected with the aftershock activity following thestrong earthquake EQ0.These results evidence that rather large events are corre-lated with an anomalous increase of the instability index,which indicates that the time-scaling behaviour of the seis-micity, given by the power-law shape of the ﬂuctuation func-tion F(n), becomes unstable. The occurrence of an intenseseismic event contributes to break the stability of the scalingbehaviour of the background seismicity, mainly given by thesmaller events. The rupture of the scaling stability indicatesthat the earthquake distribution is like to be “disturbed” bythe occurrence of a large event. Such result seems to conﬁrm  102x108 3x108 4x108 5x108 6x108 7x108 8x108 9x1080.00.51.01.52.02.53.03.5246βt (S)EQ0d cba M Fig. 2  Fig. 2. Time variation of β for the seismicity data plotted in Fig. 1.The values of β were calculated in a window of 300 events shiftingthrough the entire catalogue by one event. Each value was asso-ciated to the time of occurrence of the last event in the window.The red vertical arrows indicate the seismic events with magnitudeM>4.what has been obtained in a rock fracture experiment withdrive control (Kuksenko et al., 2007). In fact, it was foundthatattheinitialstage, theeventsoccurpredominantlyincer-tain “weak points” whose distribution is mainly determinedby the structural heterogeneity and mechanical prehistory ofsamples. The system is dynamically connected through elas-tic interactions and oscillation processes and its scaling be-haviour shows its critical state that could be maintained aslong as the structural rearrangements of the given scale levelwould enable to compensate the energy release. The appear-ance of larger-scale events (primary damages of upper hier-archic level) reduces the contribution of smaller-scale eventsto the formation of the fault geometry; and in this period ofhierarchic transition the scaling properties of the system be-come disturbed due to low occurrence of large events thatdetermine the further evolution of the system4 ConclusionsThe detrended ﬂuctuation analysis of seismicity data ob-served in Umbria-Marche area, central Italy from 1981 to2007, reveals scaling behaviour in the temporal dynamics ofthe seismic interevent time series. The instability parameterβ identiﬁes and quantiﬁes the deviations from uniformpower-law scaling. Its time variation reveals a signiﬁcantincrease connected with the occurrence of the larger earth-quakes.Edited by: P. F. BiagiReviewed by: M. Rouai and another anonymous refereewww.nat-hazards-earth-syst-sci.net/8/973/2008/ Nat. Hazards Earth Syst. Sci., 8, 973–976, 2008976 L. Telesca and M. 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Investigating non-uniform scaling behaviour in temporal fluctuations of seismicity

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