Dynamical complexity detection in pre-seismic emissions using nonadditive Tsallis entropy

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Abstract

We investigate, within the framework of nonextensive Tsallis entropy, particular aspects of the relation between signal analysis and signals originating in a very special nonlinear system: the focal area of an impending earthquake. The Tsallis-like time-dependent entropy is shown to be a rather powerful tool for providing a novel quantitative strategy for monitoring the focal area states just before the earthquake occurrence.

Introduction

Earthquakes (EQs) are large-scale fracture phenomena. Despite the large amount of experimental data and the considerable effort that has been undertaken by material scientists, many questions about fracture processes remain standing. Especially, many aspects of EQ generation still escape our full understanding.

Fracture-induced physical fields allow a real-time monitoring of damage evolution in materials during mechanical loading. Electromagnetic (EM) emissions in a wide frequency spectrum ranging from kHz to MHz are produced by opening cracks, which can be considered as the so-called precursors of general fracture. These precursors are detectable both at a laboratory and a geophysical scale [1], [2], [3], [4], [5], [6], [7], [8], [9], [10]. Being non-destructive, monitoring techniques based on these fracture-induced fields are a basis for a fundamental understanding of fracture mechanism and for developing consecutive models of rock/focal area behavior [1].

Our main observational tool is the monitoring of the fractures which occur in the focal area before the final break-up by recording their kHz–MHz EM emissions. Recent studies imply that these pre-seismic EM signals not only contain information characteristic of an ensuing EQ but also yield clues regarding the underlying fracture dynamics [1], [2], [3], [4], [5], [6], [7], [8], [9], [10]. It is crucial to point out that the pre-fracture MHz EM radiation appears earlier than the kHz one both in the laboratory and geophysical scale [6].

An important challenge in this field of research is to distinguish characteristic epochs in the evolution of precursory EM activity and identify them with the equivalent last stages in the EQ preparation process. Recently, we attempted to approach this challenge bringing together experimental pre-seismic EM data and aspects having their roots in statistical physics [3]. Our model of the focal area consists of: (i) a backbone of strong and large asperities distributed along the fault that sustains the system; (ii) a strongly heterogeneous material that surrounds the family of asperities. Analyses by means of critical phenomena have been applied to pre-seismic MHz–kHz EM fluctuations indicating a possible two stage transition from the normal state to the catastrophic seismic event [2], [3]. The first epoch, which includes the initially emerged MHz part, originates during cracking in the heterogeneous component of the focal area, while the underlying fracto-electromagnetic mechanism can be described as a generalized continuous phase transition [3]. The abrupt emergence of strong impulsive kHz EM activity in the tail of the precursory activity is thought to be due to the fracture of the family of main asperities that sustain the system. The kHz EM radiation evolves as a phase transition far from equilibrium [3].

Herein we focus on the finally emerged kHz EM emission: we study whether novel signatures further indicate the transition to the last stage of the EQ preparation process, namely the fracture of the main asperities. A time-dependent Tsallis entropy is employed to characterize the level of precursory “crust injury”. The analysis reveals that this entropy detects the pattern of alterations in pre-seismic kHz EM signals and is able to discriminate between the “injury levels” of the focal area. We compare the results of the aforementioned analysis with ones resulting from a fractal spectral analysis performed in terms of wavelets. The results suggest that a significant increase of the degree of organization coupled with appearance of persistency can be confirmed at the tail of the detected pre-seismic kHz EM emission. We claim that this feature might be used as diagnostic tools for the fracture of the backbone of strong and large asperities that sustain the system.

Section snippets

Candidate precursory EM anomalies

In this work, we concentrate on the candidate kHz EM precursors associated with the Athens and Kozani–Grevena EQs. We mainly focus on the precursor of the Athens EQ, for the following reasons. (i) It has a rather long duration, thus it provides sufficient data for a valuable statistical analysis; the data have been recorded with a sampling rate of 1 sample/s while the duration of the candidate EM precursor is more than six days. (ii) A multidisciplinary analysis in terms of fault modelling,

Foundation of Tsallis entropy

The aim of statistical mechanics is to establish a direct link between the mechanical laws and classical thermodynamics. The most famous classical theory in this field has been developed by Boltzmann and Gibbs (B–G). One of the crucial properties of the entropy in the context of classical thermodynamics is extensivity, namely proportionality with the number of elements of the system. The B–G entropy satisfies this prescription if the subsystems are statistically (quasi-) independent, or

The Tsallis entropy in terms of symbolic dynamics

Herein, we estimate the Tsallis entropy based on the concept of symbolic dynamics: from the initial measurements we generate a sequence of symbols, where the dynamics of the original (under analysis) system has been projected [4], [7], [14]. More precisely, the original EM time series of length N, (X1,X2,,XN), is projected to a symbolic time series (A1,A2,,AN) with An from a finite alphabet of λ letters (0,,λ1) with An=0for Xmin<Xn<x(0)An=1for x(0)<Xn<x(1)An=λ1for x(λ2)<Xn<Xmax, where xk

A comparison of complexity and anti-persistency/persistency in their time-dependent fashion

It would be highly desirable to confirm the above mentioned emergence of two different patterns from the EM background in the pre-seismic time series under study based on an independent analysis. For this purpose, we compare the time-dependent fashion of the Tsallis entropy with that of anti-persistency/persistency extracted by a fractal spectral analysis by means of wavelets.

A time series behaves as a temporal fractal when the power spectrum S(f) has a scaling form: S(f)fβ where f is the

On the appropriate choice of the q-index

As is mentioned, although the appropriate choice of the entropic index q is significant, it still remains to be studied [15]. Here, we found that the q-values restricted in the range 1<q<2 magnify differences of the Tsallis entropy as the catastrophic event is approaching. However, a question arises: what is the precise q value that describes the system under study?

A model for EQ dynamics coming from a nonadditive Tsallis formulation, starting from first principles, has been recently introduced 

Conclusions and discussion

We have shown that a combination of nonadditive Tsallis statistics [11], [12], [13], [23], [24], [25] with a fractal spectral analysis in terms of wavelets allows one to extract rich information hidden in pre-seismic EM time series.

The main thesis underlying the present paper is that the temporal evolution of Tsallis entropy and Hurst exponent imply the occurrence of two distinct precursory epochs in the emerged kHz EM fluctuations possibly associated with the fracture of asperities that

Acknowledgement

The project is co-funded by the European Social Fund and National Resources—(EPEAEK II) PYTHAGORAS (70/3/7357).

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