Nonlinear Phenomena in Complex Systems, vol. 11, no. 2 (2008), pp. 274 - 279

Spatiotemporal Chaos into the Hellenic Seismogenesis:Evidence for a Global Seismic Strange Attractor

A. C. Iliopoulos,1 G. P. Pavlos,1 and M. A. Athanasiu11Democritus University of Thrace, Department of Electrical and Computer Engineering, 67100, Xanthi, GREECE∗

(Received 3 January, 2008)

Self-organized criticality (SOC) is one of the most discussed concept for understanding theearthquake process and other scale invariant physical processes but in contrast with chaos. Inthis study, the concept of seismic low dimensional spatiotemporal chaos is supported. For thisreason, we used significant results obtained by the nonlinear analysis of three earthquake data setsa)Latitude, b)Longitude and c)inter-event time intervals of earthquake events obtained in the broadHellenic region. Our results clearly reveal some kind of dynamical synchronization of the local seismicactive regions and the existence of a global seismic strange attractor. Moreover, a unified model ofearthquakes is given by using concepts of the modern physical theory of the far from equilibriumstatistical processes.

PACS numbers: 78.67.Pt+42.65.KyKeywords: earthquakes, spatiotemporal chaos, SOC, nonlinear time series analysis

1. Introduction

According to Richter [1] and his contemporaries,earthquake occurrence seemed to be random andbest modeled by a completely stochastic stationarypoisson process with no memory and no internaldynamics. That is, no regular connection was apparentbetween one large earthquake and another one. Inthis direction, the most extended view is that oftwo separated processes, one for main-shock, whichought to follow a poisson distribution and one foran independent process to generate aftershocks. Incontrast with this reductionistic point of view, thealternative perspective is to look at the seismogenesisas a whole dynamic process and place all the seismicevents on the same dynamical level [2]. Swarms,mainshocks and aftershocks are closely related

FIG. 1. Earthquake Epicenters in Greece during thetwentieth Century for all focal depths (According toBurton et al [45]).

∗E-mail: ailiopou@gmail.com

phenomena in such a way that although seismicityreveals an intrinsic randomness, earthquakes do notoccur randomly in space and time. Seismologistshave discovered that earthquakes can be related bothin near-field and at distances approaching 4000 km[3,4]. The fractal character and the scaling laws ofthe spatiotemporal seismogenesis, including globalregularity and local randomness as well as long-range correlation and space-time clustering, constitutestrong evidence for the existence of an underlyingspatiotemporal nonlinear deterministic dynamicalprocess. This process is unstable, complex andsensitive to initial conditions [4,5,6]. Such a nonlineardeterministic earthquake process, unstable andstochastic, underlying the seismogenesis, produces theobserved space-time behavior, which is mapped intothe self-similar (fractal) properties and power-lawsprofiles, the space-time clustering of seismic eventsand long-range correlations. These characteristicsmake earthquakes a whole [7-11]. This point ofview can be supported a) by the study of waitingtime distributions of seismic events which reveals aunified scaling law of time process in four dimensions(spatial coordinates and magnitude) [2,12-15),b) bythe concept of the seismic cycle supported byFedotov [16], c) the "self-similar"character of thewhole series of the seismic events as an entiremainshock/ aftershock event can be nested inside alarger aftershock sequence. That is, aftershocks canproduce their own aftershock series with the sameproperties. This is a typical self-similar characteristicof the nonlinear dynamical systems [17-20]. Untilnow, several conceptual models have been proposedto describe and explain the seismogenesis and theproperties of the observed seismicity. These modelsare a) deterministic slider-block models [21-23], b)cellular automata models [24-27] and c) statistical-probabilistic models [28-32]. Recently, the concept oflow dimensional chaos in the earthquake process hasalso been supported in many studies both theoreticaland experimental. For a review see Pavlos [33]. Theconcept of low dimensional chaos into the Hellenic

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FIG. 2. (a) Data set of time differences betweensequentially earthquake events with magnitude over 4Richter, occurred in Greece, for the period of 1968-2004.(b)Data set of Latitude of each earthquake event withmagnitude over 4 Richter, occurred in Greece, for theperiod of 1968-2004. (c)Data set of Longitude of eachearthquake event with magnitude over 4 Richter, occurredin Greece,for the period of 1968-2004. (d)AutocorrelationCoefficient estimated for the three seismic Data sets.

seismogenesis has been supported, for the first time,by Pavlos et al [34]. In this study, we extend ourprevious results and we support the concept ofspatiotemporal chaos for the seismic process and theexistence of a global earthquake strange attractor, asthe basic physical process for the seismogenesis ofthe Hellenic broad region. Therefore, we use methodsof nonlinear time series analysis, which are appliedsimultaneously at three seismic data sets, firstlyfor the inter-event time intervals and secondly forthe corresponding Latitude and Longitude sequences,obtained at the Hellenic region. In section 2, weintroduce some theoretical concepts, which concernthe physical system underlying the seismogenesis. Insection 3, we present the results of the nonlinearanalysis applied at the seismic time-series and theirsurrogate data. Finally, in section 4 we introduce theconcept of the chaotic active walker in relation withthe strange attractor process.

2. The seismic physical system

Earthquakes can be viewed as a relaxationprocess of the lithospheric crust loaded by stresses.However, the occurrence of a particular earthquakecannot be completely isolated from the dynamicsof the whole lithosphere. Itv may range fromglobal features, as the motion of tectonic plate,mantle convection, earth rotation, to highly localphenomena as frictional stress and transmissionof stress [35]. There is considerable evidence thatfaults and earthquakes interact on a range ofscales from millimeters or less to thousands ofkilometers [36]. Although most of earthquakes occuralong preexisting faults, when the stress overcomesa threshold of frictional resistance, the nature ofearthquake dynamics is a complex and chaotic

spatiotemporal critical phenomenon. The loadingof faults with stresses is caused by the internalthermal engine of the earth that drives also themantle convection. According to this point ofview, earthquakes is a loading-unloading input-output process, whereas the lithospheric dynamicsis an externally driven and far from equilibriumprocess. When an earthquake (relaxation unloadingevent) occurs, the whole local distribution of thestress threshold is reached [35,37-40]. The intrinsicrandomness of seismicity means that case studies areinsufficient and statistical methods must be used toexplore earthquake occurrence. The scaling properties(scale invariance) of earthquake population, related tomultiple spatiotemporal power laws and long-rangecorrelations, show remarkable similarities to thoseobserved among the critical phenomena of magneticor other complex systems in statistical physics.Furthermore, the extreme randomness and scalingproperties of the earthquakes allow us to compareseismicity with the state of turbulence in distributedand far from equilibrium continuum systems withinfinite numbers of degrees of freedom. Turbulenceis characterized by an energy transport cascade fromlarge-scale to smaller structures with a multiplicityof spatial and temporal scales. This character makesturbulence look equivalent to the critical pointdynamics and to the driven threshold dynamics,which are observed in many self-organizing systems innature. Such a kind of critical process, may includedifferent kind of systems like neuron nets, domainrearrangements in flowing foams, charge density wavesin semiconductors, earthquake fault networks, theworld wide web, as well as many ecological social orpolitical systems. Both phenomena of seismicity andturbulence share inherent randomness, intermittencyin time and space, stochastic scale invariance andhierarchically organized scale invariant structures.For seismicity the scales are extended over manyorders of magnitude, from about a millimeter tothousands of kilometers. Moreover, in turbulenceand critical phenomena generally the fluctuationswhich are associated with correlations in spaceand time are inherent properties characterized bycorrelation lengths and correlation times respectively.The statistical physics approach to the descriptionof multi-scale dynamics of the earthquake faultsystems have led to the conclusion that the observedscaling laws of seismicity can be the result of ageneralized phase transition process, with avalanches(clusters of failed sites), nucleation events and criticalexponents. According to the critical state phasetransition approach, the earthquake fault systemcan be modeled by slider block critical models,SOC models, percolation models or self-organizedspinodal (SOS) models [29,31,41-43]. In this study,we present strong evidence for low dimensionalityand chaoticity of the seismic process. Our resultssupport the theoretical concept of spatiotemporal andsynchronized chaos, which must be included also tothe theoretical description of the earthquake system.In accordance with the above statistical approach thephase transition critical state dynamics ought to beunified with the stochastic description of chaos andgeneral theory of the far from equilibrium stochasticdynamics, presented by Pavlos et al [44].

Nonlinear Phenomena in Complex Systems Vol. 11, no. 2, 2008

276 A. C. Iliopoulos, G. P. Pavlos, and M. A. Athanasiu

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FIG. 3. (a) Slopes of the correlation integrals estimatedfor the Interevent time series for delay time τ=5 andTheiler parameter w=10 and for embedding dimensionsm=3-9, as a function of Ln(r). (b) Slopes of the correlationintegrals estimated for the Interevent time series and its 30surrogates series estimated for embedding dimension m=6and delay time τ=10, as a function of Ln(r). (c) Slopes ofthe correlation integrals estimated for the Latitude timeseries for delay time τ=1 and Theiler parameter w=100and for embedding dimensions m=3-8, as a function ofLn(r). (d) Slopes of the correlation integrals estimatedfor the Latitude time series and its 30 surrogates seriesestimated for embedding dimension m=7 and delay timeτ=10, as a function of Ln(r). (e) Slopes of the correlationintegrals estimated for the Longitude time series fordelay time τ=10 and Theiler parameter w=100 and forembedding dimensions m=3-8, as a function of Ln(r).(f) Slopes of the correlation integrals estimated for theLongitude time series and its 30 surrogates series estimatedfor embedding dimension m=7 and delay time τ=10, as afunction of Ln(r).

3. Data analysis and results

In this section significant results obtained byapplying nonlinear analysis to three seismic data setsare presented. The data series correspond to 5400seismic events that have occurred in the Hellenicregion, using a threshold local magnitude of 4 Richterand were observed during the time period of 1968-2004, within the broad Hellenic area 34.00 − 42.00o

N and 18.00 − 30.00o E. These data series, whichcorrespond to the basic focal parameters (time andspace) of earthquakes, were constructed by thebulletins of the National Observatory of Athens.

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FIG. 4. (a) Show the largest Lyapunov exponent (Lmax )as a function of the seismic events, for the Interevent timeseries for delay time τ=1 and for embedding dimensionm=6. (b) The largest Lyapunov exponent (Lmax ) forthe surrogate data and the Interevent time series as afunction of events estimated for delay time τ=1 andembedding dimension m=6. (c)The largest Lyapunovexponent (Lmax ) as a function of the seismic events, forthe Latitude series for delay time τ=1 and for embeddingdimension m=6. (d) The largest Lyapunov exponent(Lmax ) for the surrogate data and the Latitude series asa function of events estimated for delay time τ=1 andembedding dimension m=6. (e) The largest Lyapunovexponent (Lmax ) as a function of the seismic events,for the Longitude series for delay time τ=1 and forembedding dimension m=6. (f) The largest Lyapunovexponent (Lmax ) for the surrogate data and the Longitudeseries as a function of events estimated for delay time τ=1and embedding dimension m=6.

3.1. The seismic time series

Fig. 1, shows a representative map that containsall the epicenters of earthquakes concerning the Greekcatalogue for the period of 1900-1999, regardless of thefocal depth and magnitude range using a threshold of4 Richter, within the area of 33.00 − 43.00o N and18.00− 30.99o E [45].

In Figure 2(a-d) the three time series of seismicevents are illustrated. In particular, the intereventseries (time intervals between successive earthquakesshown in Fig. 1a), was created following the conceptof Pavlos et al [34] that the earthquake processreveals similarity with the chaotic dynamics of thedripping faucet system according to Shaw [46]. TheLatitude (Fig. 1b) and the Longitude data sets(Fig. 1c) correspond to the spatial distribution ofearthquakes in the Hellenic region. In these figures,it can be observed that the signals maintain a strongstationary profile. The autocorrelation coefficient forthe interevent, latitude and longitude time seriesare shown in Fig. 2d. As we can observe in thisfigure, during ∼ 5-10 lag times the autocorrelationsdecay at a plateau value. In the same figure, theautocorrelation coefficient corresponding to longitude,take higher values than the corresponding values ofthe latitude correlation coefficient. What is more,

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it is observed that interevent time series correlationcoefficient values remain lower than the values ofthe correlation coefficient of the spatial time series.The profiles of the autocorrelation coefficients shownin Fig. 2d, indicate the chaotic and non-periodicalcharacter of the seismic time series, in the three casesthat are studied here and are shown in Fig 2(a-c).

3.2. Correlation dimension of the seismictime series and their surrogate data

Figures 3 illustrates the slopes of the correlationintegral estimated for the seismic timeseries and theirsurrogate data. In particular, Fig. 2a presents theslopes of the correlation integrals estimated for theInterevent time series using delay time τ=5, Theilerparameter w=10 and embedding dimensions m=3-9,as a function of Ln(r). These slopes reveal efficientscaling and low saturation value, D ≈ 3, at lowvalues of Ln(r). According to the embedding theoryof Takens [47], the dynamical degrees of freedom cannot be higher than 2D + 1 ≈ 7 and lower thanDint ≈ 4 where D int is the first integer value ofthe slopes. Moreover, Fig. 2b shows the slopes of thecorrelation integrals estimated for the Interevent timeseries and its 30 surrogates series using parametersembedding dimension m=6 and delay time τ=1, asa function of Ln(r).The low value saturation of theslopes in the reconstructed phase space as well as theclear discrimination, of the Interevent signal from itssurrogate data, at the scaling region ∆ln(r)= 2.75 -4.25, clearly reveal that the underlying dynamics ofthe Interevent earthquake time series correspond to alow dimensional and non linear deterministic process.

Fig. 3c presents the slopes of the correlationintegrals estimated for the Latitude data series usingdelay time τ=10, Theiler parameter w=100 andembedding dimensions m=3-8, as a function of Ln(r).These slopes reveal efficient scaling and low saturationvalue, D ≈ 2.5, at low values of Ln(r). In this case, thedynamical degrees of freedom can not be higher than2D + 1 ≈ 6 and lower than Dint ≈ 3. Fig. 2d showsthe slopes of the correlation integrals estimated for theLatitude time series and its 30 surrogates series usingparameters embedding dimensionm=7 and delay timeτ=10, as a function of Ln(r). The significants of thestatistics was found to be higher than 14 sigmas. Thelow saturation value of the slopes in the reconstructedphase space as well as the clear discrimination ofthe Latitude signal from its surrogate data, at thescaling region ∆ln(r)= 0 - 1.2, clearly reveal thatthe underlying dynamics of the Latitude earthquaketime series correspond to a low dimensional non lineardeterministic process.

Fig. 3e presents the slopes of the correlationintegrals estimated for the Longitude data seriesusing delay time τ=10, Theiler parameter w=100 andembedding dimensions m=3-8, as a function of Ln(r).These slopes reveal efficient scaling and low saturationvalue, D ≈ 2.5, at low values of Ln(r). In this case, thedynamical degrees of freedom can not be higher than2D+1 ≈ 6 and lower than Dint ≈ 3 where D int is thefirst integer value after the saturation value D ≈ 2.

Moreover, Fig. 2f shows the slopes of the correlationintegrals estimated for the Longitude time series andits 30 surrogates series using parameters embeddingdimension m=6 and delay time τ=10, as a function ofLn(r). The significants of the statistics was found tobe higher than 12 sigmas. The low saturation value ofthe slopes in the reconstructed phase space as well asthe clear discrimination, of the Longitude signal fromits surrogate data, at the scaling region ∆ln(r)= 0 -1, clearly reveal that the underlying dynamics of theLatitude earthquake time series correspond to a lowdimensional non linear deterministic process.

3.3. Lyapunov exponents for the seismicsignals and their surrogate data

In Figure 4 we present the estimation of thelargest Lyapunov exponents for the seismic timeseriesand their surrogate data. Fig. 4a presents thelargest Lyapunov exponent (Lmax) for the Intereventtime series and the corresponding surrogate dataas a function of events estimated using delay timeτ=10 and embedding dimension m=6. The largestLyapunov exponent (Lmax ) for the Interevent timeseries was found to be positive, attaining a value ofLmax ≈ 4.2 bit/event clearly discriminated from thevalues of the (Lmax ) estimated for the surrogate data,which take values higher than Lmax ≈ 4.2 bit/event.The significance, S, of the statistical test fluctuatesat value of 3 sigmas and thus, clearly permits therejection of the null hypothesis with confidence greaterthan 95%. These results clearly indicate existence ofsensitivity to initial conditions and so the chaoticcharacter of the temporal earthquake dynamics. Fig.4b presents the largest Lyapunov exponent (Lmax)for the Latitude time series and the correspondingsurrogate data as a function of events estimatedusing delay time τ=10 and embedding dimensionm=6. The largest Lyapunov exponent (Lmax ) forthe Interevent time series was found to be positive,attaining a value of Lmax ≈ 5.3 bit/event clearlydiscriminated from the values of the (Lmax ) estimatedfor the surrogate data, which take values higherthan Lmax ≈ 5.3 bit/event. The significance, S, ofthe statistical test fluctuates at value of 5 sigmasand thus, clearly permits the rejection of the nullhypothesis with confidence greater than 95%. Theseresults clearly indicate existence of sensitivity toinitial conditions and so the chaotic character ofthe earthquake dynamics mapped at the Latitudetime series . Fig. 4c presents the largest Lyapunovexponent (Lmax) for the Longitude time series and thecorresponding surrogate data as a function of eventsestimated using delay time τ=10 and embeddingdimension m=6. The largest Lyapunov exponent(Lmax ) for the Interevent time series was found tobe positive, attaining a value of Lmax ≈ 5.3 bit/eventclearly discriminated from the values of the (Lmax )estimated for the surrogate data, which take valueshigher than Lmax ≈ 5.3 bit/event. The significance,S, of the statistical test fluctuates at value of 5.5sigmas and thus, clearly permits the rejection of the

Nonlinear Phenomena in Complex Systems Vol. 11, no. 2, 2008

278 A. C. Iliopoulos, G. P. Pavlos, and M. A. Athanasiu

null hypothesis with confidence greater than 95%.These results clearly indicate existence of sensitivityto initial conditions and so the chaotic character of theearthquake dynamics mapped at the Longitude timeseries .

4. Summary and discussions

Seismic events are obtained randomly in spaceand time. However, the results presented previously,indicate that the seismic randomness is causedby a low dimensional deterministic and chaoticspatiotemporal process. The spatial and temporaltrace of the seismogenesis, as it has shown by theanalysis of spatial and temporal time series, canbe related with a low dimensional chaotic walker,in contrast with the purely random walker process[48-50]. The hypothesis of seismogenesis, as anactive chaotic walker, corresponds to a point processrealized in the continuously extended lithosphericsystem. Moreover, the results presented in this studyindicate that the lithospheric system is in a stateof weak turbulence, where the existence of thefinite dimensional strange attractor is possible. Weakturbulence is the state of early turbulence in thedistributed system characterized by the existence oflow effective dimensionality and spatial long rangecorrelations. Generally, as the control parametersof the system change, the effective dimensionalitymay increase. As the system arrives at the state of

developed turbulence, then, long-range correlationsare absent and distant elements of the mediumare statistically independent, while the effectivedimensionality becomes practically infinite [51-54].The different possible states of turbulence in adistributed active system (early or developed phases ofturbulence) must also be related with different phasetransition critical states. According to this point ofview, the Hellenic lithospheric system appears to bein a state of early turbulence with a low dimensionaluniversal attractor. Other regions of the earthlithospheric crust can exist in different turbulencestate with different effective dimensionality. Theseismic chaotic walker model supported by the resultsof this study could be related with the chaoticwandering of defects in the turbulence states of thelithospheric medium, according to the theoreticalmodels of distributed and interacted maps [53-55] .Avalanche events or nucleation events in the phasetransition critical state of the lithospheric system,are related also with the dynamics of defects in thelithospheric turbulence state [43,44].

Acknowledgment

We would like to thank undergraduate studentof physics Eugene Pavlos for technical support duringthe preparation of this study.

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