Investigation of seismicity after the initiation of a Seismic Electric Signal activity until the main shock.

The behavior of seismicity in the area candidate to suffer a main shock is investigated after the observation of the Seismic Electric Signal activity until the impending main shock. This is based on the view that the occurrence of earthquakes is a critical phenomenon to which statistical dynamics may be applied. In the present work, analysing the time series of small earthquakes, the concept of natural time chi was used and the results revealed that the approach to criticality itself can be manifested by the probability density function (PDF) of kappa(1) calculated over an appropriate statistical ensemble. Here, kappa(1) is the variance kappa(1)(=-(2)) resulting from the power spectrum of a function defined as Phi(omega)= summation operator(k=1)(N) p(k) exp(iomegachi(k)), where p(k) is the normalized energy of the k-th small earthquake and omega the natural frequency. This PDF exhibits a maximum at kappa(1) asymptotically equal to 0.070 a few days before the main shock. Examples are presented, referring to the magnitude 6 approximately 7 class earthquakes that occurred in Greece.

Attempt to distinguish long-range temporal correlations from the statistics of the increments by natural time analysis.

Self-similarity may originate from two origins: i.e., the process memory and the process' increments "infinite" variance. A distinction is attempted by employing the natural time chi . Concerning the first origin, we analyze recent data on seismic electric signals, which support the view that they exhibit infinitely ranged temporal correlations. Concerning the second, slowly driven systems that emit bursts of various energies E obeying the power-law distribution--i.e., P(E) approximately E(-gamma)--are studied. An interrelation between the exponent gamma and the variance kappa1(identical with - ) is obtained for the shuffled (randomized) data. For real earthquake data, the most probable value of kappa1 of the shuffled data is found to be approximately equal to that of the original data, the difference most likely arising from temporal correlation. Finally, it is found that the differential entropy associated with the probability P(kappa1) maximizes for gamma around gamma approximately 1.6-1.7 , which is comparable to the value determined experimentally in diverse phenomena: e.g., solar flares, icequakes, dislocation glide in stressed single crystals of ice, etc. It also agrees with the b value in the Gutenberg-Richter law of earthquakes. In addition, the case of multiplicative cascades is studied in the natural time domain.

Entropy of seismic electric signals: analysis in natural time under time reversal.

Electric signals have been recently recorded at the Earth's surface with amplitudes appreciably larger than those hitherto reported. Their entropy in natural time is smaller than that of a "uniform" distribution. The same holds for their entropy upon time reversal. Such a behavior, which is also found by numerical simulations in fractional Brownian motion time series and in an on-off intermittency model, stems from infinitely ranged long range temporal correlations and hence these signals are probably seismic electric signal activities (critical dynamics). This classification is strikingly confirmed since three strong nearby earthquakes occurred (which is an extremely unusual fact) after the original submission of the present paper. The entropy fluctuations are found to increase upon approaching bursting, which is reminiscent of the behavior identifying sudden cardiac death individuals when analyzing their electrocardiograms.

Natural entropy fluctuations discriminate similar-looking electric signals emitted from systems of different dynamics.

Complexity measures are introduced that quantify the change of the natural entropy fluctuations at different length scales in time series emitted from systems operating far from equilibrium. They identify impending sudden cardiac death (SD) by analyzing 15 min electrocardiograms, and comparing to those of truly healthy humans (H). These measures seem to be complementary to the ones suggested recently [Phys. Rev. E 70, 011106 (2004)]] and altogether enable the classification of individuals into three categories: H, heart disease patients, and SD. All the SD individuals, who exhibit critical dynamics, result in a common behavior.

Entropy in the natural time domain.

A surrogate data analysis is presented, which is based on the fluctuations of the "entropy" S defined in the natural time domain [Phys. Rev. E 68, 031106 (2003)]]. This entropy is not a static one such as, for example, the Shannon entropy. The analysis is applied to three types of time series, i.e., seismic electric signals, "artificial" noises, and electrocardiograms, and it "recognizes" the non-Markovianity in all these signals. Furthermore, it differentiates the electrocardiograms of healthy humans from those of the sudden cardiac death ones. If deltaS and deltaSshuf denote the standard deviation when calculating the entropy by means of a time window sweeping through the original data and the "shuffled" (randomized) data, respectively, it seems that the ratio deltaSshuf /deltaS plays a key role. The physical meaning of deltaSshuf is investigated.