Space-time clustering and correlations of major earthquakes

Complexity measure in natural time analysis identifying the accumulation of stresses before major earthquakes

Here, we suggest a procedure through which one can identify when the accumulation of stresses before major earthquakes (EQs) (of magnitude M 8.2 or larger) occurs. Analyzing the seismicity in natural time, which is a new concept of time, we study the evolution of the fluctuations of the entropy change of seismicity under time reversal for various scales of different length i (number of events). Although the stress might be accumulating throughout the entire process of EQ preparation due to tectonic loading, here we find that the proposed complexity measure reveals different stress accumulation characteristics from those in the long-term background when the system approaches the critical stage. Specifically, we find that anomalous intersections between scales of different i are observed upon approaching a major EQ occurrence. The investigation is presented for the seismicity in Japan since 1984 including the M9 Tohoku EQ on 11 March 2011, which is the largest EQ ever recorded there, as well as for the seismicity before 2017 Chiapas M8.2 EQ, which is Mexico's largest EQ in more than a century. Based on this new complexity measure, a preprint submitted on 5 December 2023 anticipated the 1 January 2024 M7.6 EQ in Japan.

Visibility Graph Analysis of the Seismic Activity of Three Areas of the Cocos Plate Mexican Subduction Where the Last Three Large Earthquakes (M > 7) Occurred in 2017 and 2022

The understanding of the dynamical behavior of seismic phenomena is currently an open problem, mainly because seismic series can be considered to be produced by phenomena exhibiting dynamic phase transitions; that is, with some complexity. For this purpose, the Middle America Trench in central Mexico is considered a natural laboratory for examining subduction because of its heterogenous natural structure. In this study, the Visibility Graph method was applied to study the seismic activity of three regions within the Cocos plate: the Tehuantepec Isthmus, the Flat slab and Michoacan, each one with a different level of seismicity. The method maps time series into graphs, and it is possible to connect the topological properties of the graph with the dynamical features underlying the time series. The seismicity analyzed was monitored in the three areas studied between 2010 and 2022. At the Flat Slab and Tehuantepec Isthmus, two intense earthquakes occurred on 7 and 19 September 2017, respectively, and, on 19 September 2022, another one occurred at Michoacan. The aim of this study was to determine the dynamical features and the possible differences between the three areas by applying the following method. First, the time evolution of the a- and b-values in the Gutenberg-Richter law was analyzed, followed by the relationship between the seismic properties and topological features using the VG method, the k-M slope and the characterization of the temporal correlations from the γ-exponent of the power law distribution, P(k) ∼ k-γ, and its relationship with the Hurst parameter, which allowed us to identify the correlation and persistence of each zone.

Tsallis q-Statistics in Seismology

Non-extensive statistical mechanics (or q-statistics) is based on the so-called non-additive Tsallis entropy. Since its introduction by Tsallis, in 1988, as a generalization of the Boltzmann-Gibbs equilibrium statistical mechanics, it has steadily gained ground as a suitable theory for the description of the statistical properties of non-equilibrium complex systems. Therefore, it has been applied to numerous phenomena, including real seismicity. In particular, Tsallis entropy is expected to provide a guiding principle to reveal novel aspects of complex dynamical systems with catastrophes, such as seismic events. The exploration of the existing connections between Tsallis formalism and real seismicity has been the focus of extensive research activity in the last two decades. In particular, Tsallis q-statistics has provided a unified framework for the description of the collective properties of earthquakes and faults. Despite this progress, our present knowledge of the physical processes leading to the initiation of a rupture, and its subsequent growth through a fault system, remains quite limited. The aim of this paper was to provide an overview of the non-extensive interpretation of seismicity, along with the contributions of the Tsallis formalism to the statistical description of seismic events.

Earthquake Nowcasting: Retrospective Testing in Greece

Earthquake nowcasting (EN) is a modern method of estimating seismic risk by evaluating the progress of the earthquake (EQ) cycle in fault systems. EN evaluation is based on a new concept of time, termed 'natural time'. EN employs natural time, and uniquely estimates seismic risk by means of the earthquake potential score (EPS), which has been found to have useful applications both regionally and globally. Amongst these applications, here we focused on Greece since 2019, for the estimation of the EPS for the largest-magnitude events, MW(USGS) ≥ 6, that occurred during our study period: for example, the MW= 6.0 WNW-of-Kissamos EQ on 27 November 2019, the MW= 6.5 off-shore Southern Crete EQ on 2 May 2020, the MW= 7.0 Samos EQ on 30 October 2020, the MW= 6.3 Tyrnavos EQ on 3 March 2021, the MW= 6.0 Arkalohorion Crete EQ on 27 September 2021, and the MW= 6.4 Sitia Crete EQ on 12 October 2021. The results are promising, and reveal that the EPS provides useful information on impending seismicity.

Optimizing Earthquake Nowcasting With Machine Learning: The Role of Strain Hardening in the Earthquake Cycle

Nowcasting is a term originating from economics, finance, and meteorology. It refers to the process of determining the uncertain state of the economy, markets or the weather at the current time by indirect means. In this paper, we describe a simple two-parameter data analysis that reveals hidden order in otherwise seemingly chaotic earthquake seismicity. One of these parameters relates to a mechanism of seismic quiescence arising from the physics of strain-hardening of the crust prior to major events. We observe an earthquake cycle associated with major earthquakes in California, similar to what has long been postulated. An estimate of the earthquake hazard revealed by this state variable time series can be optimized by the use of machine learning in the form of the Receiver Operating Characteristic skill score. The ROC skill is used here as a loss function in a supervised learning mode. Our analysis is conducted in the region of 5° × 5° in latitude-longitude centered on Los Angeles, a region which we used in previous papers to build similar time series using more involved methods (Rundle & Donnellan, 2020, https://doi.org/10.1029/2020EA001097; Rundle, Donnellan et al., 2021, https://doi.org/10.1029/2021EA001757; Rundle, Stein et al., 2021, https://doi.org/10.1088/1361-6633/abf893). Here we show that not only does the state variable time series have forecast skill, the associated spatial probability densities have skill as well. In addition, use of the standard ROC and Precision (PPV) metrics allow probabilities of current earthquake hazard to be defined in a simple, straightforward, and rigorous way.

Kaniadakis Functions beyond Statistical Mechanics: Weakest-Link Scaling, Power-Law Tails, and Modified Lognormal Distribution

Probabilistic models with flexible tail behavior have important applications in engineering and earth science. We introduce a nonlinear normalizing transformation and its inverse based on the deformed lognormal and exponential functions proposed by Kaniadakis. The deformed exponential transform can be used to generate skewed data from normal variates. We apply this transform to a censored autoregressive model for the generation of precipitation time series. We also highlight the connection between the heavy-tailed κ-Weibull distribution and weakest-link scaling theory, which makes the κ-Weibull suitable for modeling the mechanical strength distribution of materials. Finally, we introduce the κ-lognormal probability distribution and calculate the generalized (power) mean of κ-lognormal variables. The κ-lognormal distribution is a suitable candidate for the permeability of random porous media. In summary, the κ-deformations allow for the modification of tails of classical distribution models (e.g., Weibull, lognormal), thus enabling new directions of research in the analysis of spatiotemporal data with skewed distributions.

Estimating the Epicenter of an Impending Strong Earthquake by Combining the Seismicity Order Parameter Variability Analysis with Earthquake Networks and Nowcasting: Application in the Eastern Mediterranean

The variance κ1 of the natural time analysis of earthquake catalogs was proposed in 2005 as an order parameter for seismicity, whose fluctuations proved, in 2011, to be minimized a few months before the strongest mainshock when studying the earthquakes in a given area. After the introduction of earthquake networks based on similar activity patterns, in 2012, the study of their higher order cores revealed, in 2019, the selection of appropriate areas in which the precursory minima βmin of the fluctuations β of the seismicity order parameter κ1 could be observed up to six months before all strong earthquakes above a certain threshold. The eastern Mediterranean region was studied in 2019, where all earthquakes of magnitude M≥7.1 were found to be preceded by βmin without any false alarm. Combining these results with the method of nowcasting earthquakes, introduced in 2016, for seismic risk estimation, here, we show that the epicenter of an impending strong earthquake can be estimated. This is achieved by employing—at the time of observing the βmin—nowcasting earthquakes in a square lattice grid in the study area and by averaging, self-consistently, the results obtained for the earthquake potential score. This is understood in the following context: The minimum βmin is ascertained to almost coincide with the onset of Seismic Electric Signals activity, which is accompanied by the development of long range correlations between earthquake magnitudes in the area that is a candidate for a mainshock.

Nowcasting Avalanches as Earthquakes and the Predictability of Strong Avalanches in the Olami-Feder-Christensen Model

Nowcasting earthquakes, suggested recently as a method to estimate the state of a fault and hence the seismic risk, is based on the concept of natural time. Here, we generalize nowcasting to a prediction method the merits of which are evaluated by means of the receiver operating characteristics. This new prediction method is applied to a simple (toy) model for the waiting (natural) time of the stronger earthquakes, real seismicity, and the Olami-Feder-Christensen earthquake model with interesting results revealing acceptable to excellent or even outstanding performance.

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 β 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 β 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.

Receiver operating characteristic curves for a simple stochastic process that carries a static signal

The detection of a weak signal in the presence of noise is an important problem that is often studied in terms of the receiver operating characteristic (ROC) curve, in which the probability of correct detection is plotted against the probability for a false positive. This kind of analysis is typically applied to the situation in which signal and noise are stochastic variables; the detection problem emerges, however, also often in a context in which both signal and noise are stochastic processes and the (correct or false) detection is said to take place when the process crosses a threshold in a given time window. Here we consider the problem for a combination of a static signal which has to be detected against a dynamic noise process, the well-known Ornstein-Uhlenbeck process. We give exact (but difficult to evaluate) quadrature expressions for the detection rates for false positives and correct detections, investigate systematically a simple sampling approximation suggested earlier, compare to an approximation by Stratonovich for the limit of high threshold, and briefly explore the case of multiplicative signal; all theoretical results are compared to extensive numerical simulations of the corresponding Langevin equation. Our results demonstrate that the sampling approximation provides a reasonable description of the ROC curve for this system, and it clarifies limit cases for the ROC curve.

Natural Time Analysis: The Area under the Receiver Operating Characteristic Curve of the Order Parameter Fluctuations Minima Preceding Major Earthquakes

It has been reported that major earthquakes are preceded by Seismic Electric Signals (SES). Observations show that in the natural time analysis of an earthquake (EQ) catalog, an SES activity starts when the fluctuations of the order parameter of seismicity exhibit a minimum. Fifteen distinct minima—observed simultaneously at two different natural time scales and deeper than a certain threshold—are found on analyzing the seismicity of Japan from 1 January 1984 to 11 March 2011 (the time of the M9 Tohoku EQ occurrence) 1 to 3 months before large EQs. Six (out of 15) of these minima preceded all shallow EQs of magnitude 7.6 or larger, while nine are followed by smaller EQs. The latter false positives can be excluded by a proper procedure (J. Geophys. Res. Space Physics 2014, 119, 9192–9206) that considers aspects of EQ networks based on similar activity patterns. These results are studied here by means of the receiver operating characteristics (ROC) technique by focusing on the area under the ROC curve (AUC). If this area, which is currently considered an effective way to summarize the overall diagnostic accuracy of a test, has the value 1, it corresponds to a perfectly accurate test. Here, we find that the AUC is around 0.95 which is evaluated as outstanding.

On the Statistical Significance of the Variability Minima of the Order Parameter of Seismicity by Means of Event Coincidence Analysis

Natural time analysis has led to the introduction of an order parameter for seismicity when considering earthquakes as critical phenomena. The study of the fluctuations of this order parameter has shown that its variability exhibits minima before strong earthquakes. In this paper, we evaluate the statistical significance of such minima by using the recent method of event coincidence analysis. Our study includes the variability minima identified before major earthquakes in Japan and Eastern Mediterranean as well as in global seismicity.

Identifying the Occurrence Time of the Deadly Mexico M8.2 Earthquake on 7 September 2017

It has been shown that some dynamic features hidden in the time series of complex systems can be unveiled if we analyze them in a time domain termed natural time. In this analysis, we can identify when a system approaches a critical point (dynamic phase transition). Here, based on natural time analysis, which enables the introduction of an order parameter for seismicity, we discuss a procedure through which we could achieve the identification of the occurrence time of the M8.2 earthquake that occurred on 7 September 2017 in Mexico in Chiapas region, which is the largest magnitude event recorded in Mexico in more than a century. In particular, we first investigated the order parameter fluctuations of seismicity in the entire Mexico and found that, during an almost 30-year period, i.e., from 1 January 1988 until the M8.2 earthquake occurrence, they were minimized around 27 July 2017. From this date, we started computing the variance of seismicity in Chiapas region and found that it approached the critical value 0.070 on 6 September 2017, almost one day before this M8.2 earthquake occurrence.

Global Seismic Nowcasting With Shannon Information Entropy

Seismic nowcasting uses counts of small earthquakes as proxy data to estimate the current dynamical state of an earthquake fault system. The result is an earthquake potential score that characterizes the current state of progress of a defined geographic region through its nominal earthquake "cycle." The count of small earthquakes since the last large earthquake is the natural time that has elapsed since the last large earthquake (Varotsos et al., 2006, https://doi.org/10.1103/PhysRevE.74.021123). In addition to natural time, earthquake sequences can also be analyzed using Shannon information entropy ("information"), an idea that was pioneered by Shannon (1948, https://doi.org/10.1002/j.1538-7305.1948.tb01338.x). As a first step to add seismic information entropy into the nowcasting method, we incorporate magnitude information into the natural time counts by using event self-information. We find in this first application of seismic information entropy that the earthquake potential score values are similar to the values using only natural time. However, other characteristics of earthquake sequences, including the interevent time intervals, or the departure of higher magnitude events from the magnitude-frequency scaling line, may contain additional information.

Tsallis Entropy Index q and the Complexity Measure of Seismicity in Natural Time under Time Reversal before the M9 Tohoku Earthquake in 2011

The observed earthquake scaling laws indicate the existence of phenomena closely associated with the proximity of the system to a critical point. Taking this view that earthquakes are critical phenomena (dynamic phase transitions), here we investigate whether in this case the Lifshitz–Slyozov–Wagner (LSW) theory for phase transitions showing that the characteristic size of the minority phase droplets grows with time as t1/3 is applicable. To achieve this goal, we analyzed the Japanese seismic data in a new time domain termed natural time and find that an LSW behavior is actually obeyed by a precursory change of seismicity and in particular by the fluctuations of the entropy change of seismicity under time reversal before the Tohoku earthquake of magnitude 9.0 that occurred on 11 March 2011 in Japan. Furthermore, the Tsallis entropic index q is found to exhibit a precursory increase.

Micro-scale, mid-scale, and macro-scale in global seismicity identified by empirical mode decomposition and their multifractal characteristics

The magnitude time-series of the global seismicity is analyzed by the empirical mode decomposition giving rise to 14 intrinsic mode functions (IMF) and a trend. Using Hurst analysis one can identify three different sums of these IMFs and the trend which exhibit distinct multifractal behaviour and correspond to micro-, mid- and macro-scales. Their multifractal detrended fluctuation analysis reveals that the micro-scale time-series exhibits anticorrelated behaviour in contrast to the mid-scale one which is long-range correlated. Concerning the mid-scale one, in the range of 30 to 300 consecutive events the maximum entropy method power spectra indicates that it exhibits an 1/f^α behaviour with α close to 1/3 which is compatible with the long-range correlations identified by detrended fluctuation analysis during periods of stationary seismicity. The results have been also verified to hold regionally for the earthquakes in Japan and shed light on the significance of the mid-scale of 30 to 300 events in the natural time analysis of global (and regional) seismicity. It is shown that when using the mid-scale time-series only, we can obtain results similar to those obtained by the natural time analysis of global seismicity when focusing on the prediction of earthquakes with M ≥ 8.4.

Hierarchical block model for earthquakes

The presented model for earthquakes is based on two fundamental principles: the hierarchical structure of seismic areas and the concept of self-organized criticality. The model reproduces the basic empirical properties of seismic processes: the frequency-energy scaling relation (the Gutenberg-Richter law), the generalized Omori law for temporal decay of aftershocks, the aftershock productivity law, the fractal distributions of hypocenters (epicenters) with power-law dependencies of the number of events on distances between hypocenters (epicenters), and, finally, the γ distribution for waiting times. In the model, the threshold energies depend on the block sizes and are distributed according to the Gauss law. After strong earthquakes they are redistributed at the decreasing average values. The change of threshold energies leads to the triggering of aftershock series.

Minima of the fluctuations of the order parameter of global seismicity

It has been recently shown [N. V. Sarlis, Phys. Rev. E 84, 022101 (2011) and N. V. Sarlis and S.-R. G. Christopoulos, Chaos 22, 023123 (2012)] that earthquakes of magnitude M greater or equal to 7 are globally correlated. Such correlations were identified by studying the variance κ1 of natural time which has been proposed as an order parameter for seismicity. Here, we study the fluctuations of this order parameter using the Global Centroid Moment Tensor catalog for a magnitude threshold Mthres = 5.0 and focus on its behavior before major earthquakes. Natural time analysis reveals that distinct minima of the fluctuations of the order parameter of seismicity appear within almost five and a half months on average before all major earthquakes of magnitude larger than 8.4. This phenomenon corroborates the recent finding [N. V. Sarlis et al., Proc. Natl. Acad. Sci. U.S.A. 110, 13734 (2013)] that similar minima of the seismicity order parameter fluctuations had preceded all major shallow earthquakes in Japan. Moreover, on the basis of these minima a statistically significant binary prediction method for earthquakes of magnitude larger than 8.4 with hit rate 100% and false alarm rate 6.67% is suggested.

Finite-size effects on return interval distributions for weakest-link-scaling systems

The Weibull distribution is a commonly used model for the strength of brittle materials and earthquake return intervals. Deviations from Weibull scaling, however, have been observed in earthquake return intervals and the fracture strength of quasibrittle materials. We investigate weakest-link scaling in finite-size systems and deviations of empirical return interval distributions from the Weibull distribution function. Our analysis employs the ansatz that the survival probability function of a system with complex interactions among its units can be expressed as the product of the survival probability functions for an ensemble of representative volume elements (RVEs). We show that if the system comprises a finite number of RVEs, it obeys the κ-Weibull distribution. The upper tail of the κ-Weibull distribution declines as a power law in contrast with Weibull scaling. The hazard rate function of the κ-Weibull distribution decreases linearly after a waiting time τ(c) ∝ n(1/m), where m is the Weibull modulus and n is the system size in terms of representative volume elements. We conduct statistical analysis of experimental data and simulations which show that the κ Weibull provides competitive fits to the return interval distributions of seismic data and of avalanches in a fiber bundle model. In conclusion, using theoretical and statistical analysis of real and simulated data, we demonstrate that the κ-Weibull distribution is a useful model for extreme-event return intervals in finite-size systems.

Minimum of the order parameter fluctuations of seismicity before major earthquakes in Japan

It has been shown that some dynamic features hidden in the time series of complex systems can be uncovered if we analyze them in a time domain called natural time χ. The order parameter of seismicity introduced in this time domain is the variance of χ weighted for normalized energy of each earthquake. Here, we analyze the Japan seismic catalog in natural time from January 1, 1984 to March 11, 2011, the day of the M9 Tohoku earthquake, by considering a sliding natural time window of fixed length comprised of the number of events that would occur in a few months. We find that the fluctuations of the order parameter of seismicity exhibit distinct minima a few months before all of the shallow earthquakes of magnitude 7.6 or larger that occurred during this 27-y period in the Japanese area. Among the minima, the minimum before the M9 Tohoku earthquake was the deepest. It appears that there are two kinds of minima, namely precursory and nonprecursory, to large earthquakes.

On earthquake prediction in Japan

Japan's National Project for Earthquake Prediction has been conducted since 1965 without success. An earthquake prediction should be a short-term prediction based on observable physical phenomena or precursors. The main reason of no success is the failure to capture precursors. Most of the financial resources and manpower of the National Project have been devoted to strengthening the seismographs networks, which are not generally effective for detecting precursors since many of precursors are non-seismic. The precursor research has never been supported appropriately because the project has always been run by a group of seismologists who, in the present author's view, are mainly interested in securing funds for seismology - on pretense of prediction. After the 1995 Kobe disaster, the project decided to give up short-term prediction and this decision has been further fortified by the 2011 M9 Tohoku Mega-quake. On top of the National Project, there are other government projects, not formally but vaguely related to earthquake prediction, that consume many orders of magnitude more funds. They are also un-interested in short-term prediction. Financially, they are giants and the National Project is a dwarf. Thus, in Japan now, there is practically no support for short-term prediction research. Recently, however, substantial progress has been made in real short-term prediction by scientists of diverse disciplines. Some promising signs are also arising even from cooperation with private sectors.

Probabilities for large events in driven threshold systems

Many driven threshold systems display a spectrum of avalanche event sizes, often characterized by power-law scaling. An important problem is to compute probabilities of the largest events ("Black Swans"). We develop a data-driven approach to the problem by transforming to the event index frame, and relating this to Shannon information. For earthquakes, we find the 12-month probability for magnitude m>6 earthquakes in California increases from about 30% after the last event, to 40%-50% prior to the next one.

Natural time analysis of the Centennial Earthquake Catalog

By using the most recent version (1900-2007) of the Centennial Earthquake Catalog, we examine the properties of the global seismicity. Natural time analysis reveals that the fluctuations of the order parameter κ(1) of seismicity exhibit for at least three orders of magnitude a characteristic feature similar to that of the order parameter for other equilibrium or non-equilibrium critical systems-including self-organized critical systems. Moreover, we find non-trivial magnitude correlations for earthquakes of magnitude greater than or equal to 7.

Predictability of the coherent-noise model and its applications

We study the threshold distribution function of the coherent-noise model for the case of infinite number of agents. This function is piecewise constant with a finite number of steps n. The latter exhibits a 1/f behavior as a function of the order of occurrence of an avalanche and hence versus natural time. An analytic expression of the expectation value E(S) for the size S of the next avalanche is obtained and used for the prediction of the next avalanche. Apart from E(S), the number of steps n can also serve for this purpose. This enables the construction of a similar prediction scheme which can be applied to real earthquake aftershock data.

Magnitude correlations in global seismicity

By employing natural time analysis, we analyze the worldwide seismicity and study the existence of correlations between earthquake magnitudes. We find that global seismicity exhibits nontrivial magnitude correlations for earthquake magnitudes greater than M(w) 6.5.

Natural time analysis of critical phenomena

A quantity exists by which one can identify the approach of a dynamical system to the state of criticality, which is hard to identify otherwise. This quantity is the variance κ(1)(≡<χ(2)> - <χ>(2)) of natural time χ, where <f(χ)> = Σp(k)f(χ(k)) and p(k) is the normalized energy released during the kth event of which the natural time is defined as χ(k) = k/N and N stands for the total number of events. Then we show that κ(1) becomes equal to 0.070 at the critical state for a variety of dynamical systems. This holds for criticality models such as 2D Ising and the Bak-Tang-Wiesenfeld sandpile, which is the standard example of self-organized criticality. This condition of κ(1) = 0.070 holds for experimental results of critical phenomena such as growth of rice piles, seismic electric signals, and the subsequent seismicity before the associated main shock.

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.

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)(=<chi(2)>-<chi>(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.

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.