Natural Time Analysis of Seismicity within the Mexican Flat Slab before the M7.1 Earthquake on 19 September 2017.

One of the most important subduction zones in the world is located in the Mexican Pacific Coast, where the Cocos plate inserts beneath the North American plate. One part of it is located in the Mexican Pacific Coast, where the Cocos plate inserts beneath the North American plate with different dip angles, showing important seismicity. Under the central Mexican area, such a dip angle becomes practically horizontal and such an area is known as flat slab. An earthquake of magnitude M7.1 occurred on 19 September 2017, the epicenter of which was located in this flat slab. It caused important human and material losses of urban communities including a large area of Mexico City. The seismicity recorded in the flat slab region is analyzed here in natural time from 1995 until the occurrence of this M7.1 earthquake in 2017 by studying the entropy change under time reversal and the variability ß of the order parameter of seismicity as well as characterize the risk of an impending earthquake by applying the nowcasting method. The entropy change ΔS under time reversal minimizes on 21 June 2017 that is almost one week after the observation of such a minimum in the Chiapas region where a magnitude M8.2 earthquake took place on 7 September 2017 being Mexico's largest quake in more than a century. A minimum of ß was also observed during the period February-March 2017. Moreover, we show that, after the minimum of ΔS, the order parameter of seismicity starts diminishing, thus approaching gradually the critical value 0.070 around the end of August and the beginning of September 2017, which signals that a strong earthquake is anticipated shortly in the flat slab.

Effect of significant data loss on identifying electric signals that precede rupture estimated by detrended fluctuation analysis in natural time.

Electric field variations that appear before rupture have been recently studied by employing the detrended fluctuation analysis (DFA) to quantify their long-range temporal correlations. These studies revealed that seismic electric signal (SES) activities exhibit a scale invariant feature with an exponent αDFA≈1 over all scales investigated (around five orders of magnitude). Here, we study what happens upon significant data loss, which is a question of primary practical importance, and show that the DFA applied to the natural time representation of the remaining data still reveals for SES activities an exponent close to 1.0, which markedly exceeds the exponent found in artificial (man-made) noises. This enables the identification of a SES activity with probability of 75% even after a significant (70%) data loss. The probability increases to 90% or larger for 50% data loss.

Nonextensivity and natural time: The case of seismicity.

Nonextensive statistical mechanics, pioneered by Tsallis, has recently achieved a generalization of the Gutenberg-Richter law for earthquakes. This remarkable generalization is combined here with natural time analysis, which enables the distinction of two origins of self-similarity, i.e., the process' memory and the process' increments infinite variance. By using also detrended fluctuation analysis for the detection of long-range temporal correlations, we demonstrate the existence of both temporal and magnitude correlations in real seismic data of California and Japan. Natural time analysis reveals that the nonextensivity parameter q , in contrast to some published claims, cannot be considered as a measure of temporal organization, but the Tsallis formulation does achieve a satisfactory description of real seismic data for Japan for q=1.66 when supplemented by long-range temporal correlations.

Multiplicative cascades and seismicity in natural time.

Natural time chi enables the distinction of two origins of self-similarity, i.e., the process memory and the process increments infinite variance. Employing multiplicative cascades in natural time, the most probable value of the variance kappa(1)(is identical to chi(2)-chi(2))is explicitly related with the parameter b of the Gutenberg-Richter law of randomly shuffled earthquake data. Moreover, the existence of temporal and magnitude correlations is studied in the original earthquake data. Magnitude correlations are larger for closer in time earthquakes, when the maximum interoccurrence time varies from half a day to 1 min.

Detrended fluctuation analysis of the magnetic and electric field variations that precede rupture.

Magnetic field variations are detected before rupture in the form of "spikes" of alternating sign. The distinction of these spikes from random noise is of major practical importance since it is easier to conduct magnetic field measurements than electric field ones. Applying detrended fluctuation analysis (DFA), these spikes look to be random at short time lags. On the other hand, long-range correlations prevail at time lags larger than the average time interval between consecutive spikes with a scaling exponent alpha around 0.9. In addition, DFA is applied to recent preseismic electric field variations in long duration (several hours to a couple of days) and reveals a scale invariant feature with an exponent alpha approximately 1 over all scales available (around five orders of magnitude).

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.

Similarity of fluctuations in correlated systems: the case of seismicity.

We report a similarity of fluctuations in equilibrium critical phenomena and nonequilibrium systems, which is based on the concept of natural time. The worldwide seismicity as well as that of the San Andreas fault system and Japan are analyzed. An order parameter is chosen and its fluctuations relative to the standard deviation of the distribution are studied. We find that the scaled distributions fall on the same curve, which interestingly exhibits, over four orders of magnitude, features similar to those in several equilibrium critical phenomena (e.g., two-dimensional Ising model) as well as in nonequilibrium systems (e.g., three-dimensional turbulent flow).

Some properties of the entropy in the natural time.

We show that the entropy S , defined as S identical with chi ln chi - chi ln chi [Phys. Rev. E 68, 031106 (2003)] where chi stands for the natural time [Phys. Rev. E 66, 011902 (2002)], exhibits positivity and concavity as well as stability or experimental robustness. Furthermore, the distinction between the seismic electric signal activities and "artificial" noises, based on the classification of their S values, is lost when studying the time-reversed signals. This reveals the profound importance of considering the (true) time arrow.

Origin of the usefulness of the natural-time representation of complex time series.

The concept of natural time turned out to be useful in revealing dynamical features behind complex time series including electrocardiograms, ionic current fluctuations of membrane channels, seismic electric signals, and seismic event correlation. However, the origin of this empirical usefulness is yet to be clarified. Here, it is shown that this time domain is in fact optimal for enhancing the signals in time-frequency space by employing the Wigner function and measuring its localization property.

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.

Electric fields that "arrive" before the time derivative of the magnetic field prior to major earthquakes.

The low frequency electric signals (emitted from the focal area when the stress reaches a critical value) that precede major earthquakes, are recorded at distances approximately 100 km being accompanied by magnetic field variations. The electric field "arrives" 1 to 2 s before the time derivative of the horizontal magnetic field. An explanation, which is still awaiting, should consider, beyond criticality, the large spatial scale as well as that the transmission of the electromagnetic fields (through an inhomogeneous weakly conductive medium like the Earth) obeys diffusion type equations.

Attempt to distinguish electric signals of a dichotomous nature.

Three types of electric signals were analyzed: Ion current fluctuations in membrane channels (ICFMC), Seismic electric signals activities (SES), and "artificial" noises (AN). The wavelet transform, when applied to the conventional time domain, does not allow a classification of these signals, but does so in the "natural" time domain. A classification also becomes possible, if we study -(q) versus q, where chi stands for the "natural" time. For q values approximately between 1 and 2 the signals are classified and ICFMC lies between the other two types. For q=1, the "entropy" S identical with -ln of ICFMC almost equals that of a "uniform" distribution, while the AN and SES have larger and smaller S values, respectively. The recent [P. Varotsos, N. Sarlis, and E. Skordas, Phys. Rev. E 67, 021109 (2003)] finding that, in short time scales, both SES and AN (which are shown to be non-Markovian) result in comparable detrended fluctuation analysis exponents alpha in (1.0,1.5) is revisited. Even a Markovian dichotomous time series, in short time scales, leads to similar alpha exponents.

Long-range correlations in the electric signals that precede rupture: further investigations.

The correlations within the time series of the seismic electric signal (SES) activities have been studied in a previous paper [P. Varotsos, N. Sarlis, and E. Skordas, Phys. Rev. E 66, 011902 (2002)]. Here, we analyze the time series of successive high- and low-level states' durations. The existence of correlation between the states is investigated by means of Hurst and detrended fluctuation analysis (DFA). The multifractal DFA (MF-DFA) is also employed. The results point to a stronger correlation, and hence longer memory, in the series of the high-level states. Furthermore, an analysis in the "natural" time domain reveals that certain power spectrum characteristics seem to distinguish SES activities from "artificial" (man-made) electric noises. More precisely, for natural frequencies 0

Long-range correlations in the electric signals that precede rupture.

The Smoluchowski-Chapman-Kolmogorov functional equation is applied to the electric signals that precede rupture. The results suggest a non-Markovian character of the analyzed data. The rescaled range Hurst and detrended fluctuation analyses, as well as that related with the "mean distance a walker spanned," lead to power-law exponents, which are consistent with the existence of long-range correlations. A "universality" in the power spectrum characteristics of these signals emerges, if an analysis is made (not in the conventional time frame, but) in the "natural" time domain. Within this frame, it seems that certain power spectrum characteristics of ion current fluctuations in membrane channels distinguish them from the electric signals preceding rupture. The latter exhibit a behavior compatible with that expected from a model based on the random field Ising Hamiltonian at the critical point.