Nicholas V. Sarlis

Natural Time Analysis: The New View of Time

Part I Seismic Electric Signals.- 1. Introduction to Seismic Electric Signals-V.- Part II.- 2. Natural Time. Background-F.- 3. Entropy in Natural Time-E.- Part III Natural Time Applications.- 4. Natural Time Analysis of Seismic Electric Signals-AS.- 5. Natural Time Investigation of the Effect of Significant Data Loss on Indentifying Seismic Electric Signals-ASL.- 6. Natural Time Analysis of Seismicity-AEQ.- 7. Indentifying the Occurence Time of an Impending Mainshock-AIM.- 8. Natural Time Analysis of Dynamical Models-AD.- 9. Natural Time Analysis of Electrocardiograms-AEL.- References.

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 of natural time χ, where and pk is the normalized energy released during the kth event of which the natural time is defined as χk = k/N and N stands for the total number of events. Then we show that κ1 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 κ1 = 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.

Fluctuations, under time reversal, of the natural time and the entropy distinguish similar looking electric signals of different dynamics

We show that the scale dependence of the fluctuations of the natural time itself under time reversal provides a useful tool for the discrimination of seismic electric signals (critical dynamics) from noises emitted from man-made sources, as well as for the determination of the scaling exponent. We present recent data of electric signals detected at the Earth’s surface, which confirm that the value of the entropy in natural time as well as its value under time reversal are smaller than that of the entropy of a “uniform” distribution.

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.

Study of the temporal correlations in the magnitude time series before major earthquakes in Japan

A characteristic change of seismicity has been recently uncovered when the precursory Seismic Electric Signals activities initiate before an earthquake occurrence. In particular, the fluctuations of the order parameter of seismicity exhibit a simultaneous distinct minimum upon analyzing the seismic catalog in a new time domain termed natural time and employing a sliding natural time window comprising a number of events that would occur in a few months. Here we focus on the minima preceding all earthquakes of magnitude 8 (and 9) class that occurred in Japanese area from 1 January 1984 to 11 March 2011 (the day of the M9 Tohoku earthquake). By applying Detrended Fluctuation Analysis to the earthquake magnitude time series, we find that each of these minima is preceded as well as followed by characteristic changes of temporal correlations between earthquake magnitudes. In particular, we identify the following three main features. The minima are observed during periods when long-range correlations have been developed, but they are preceded by a stage in which an evident anticorrelated behavior appears. After the minima, the long-range correlations breakdown to an almost random behavior possibly turning to anticorrelation. The minima that precede M ≥ 7.8 earthquakes are distinguished from other minima which are either nonprecursory or followed by smaller earthquakes. Furthermore, it is discussed whether Tsallis statistical mechanics, in the frame of which it has been suggested that kappa distributions arise, can capture the effects of temporal correlations between earthquake magnitudes.

Spatiotemporal variations of seismicity before major earthquakes in the Japanese area and their relation with the epicentral locations.

Significance It was recently found that a few months before major earthquakes, the seismicity in the entire Japanese region exhibits a characteristic change. This change, however, can be identified when seismic data are analyzed in a new time domain termed “natural time.” By dividing the Japanese region into small areas, we find that some small areas show the characteristic change almost simultaneously with the large area and such small areas are clustered within a few hundred kilometers from the actual epicenter of the related major earthquake. This phenomenon may serve for forecasting the epicenter of a future major earthquake. Using the Japan Meteorological Agency earthquake catalog, we investigate the seismicity variations before major earthquakes in the Japanese region. We apply natural time, the new time frame, for calculating the fluctuations, termed β, of a certain parameter of seismicity, termed κ1. In an earlier study, we found that β calculated for the entire Japanese region showed a minimum a few months before the shallow major earthquakes (magnitude larger than 7.6) that occurred in the region during the period from 1 January 1984 to 11 March 2011. In this study, by dividing the Japanese region into small areas, we carry out the β calculation on them. It was found that some small areas show β minimum almost simultaneously with the large area and such small areas clustered within a few hundred kilometers from the actual epicenter of the related main shocks. These results suggest that the present approach may help estimation of the epicentral location of forthcoming major earthquakes.

Estimating the Compressibility of Osmium from Recent Measurements of Ir-Os Alloys under High Pressure.

Several fcc- and hcp-structured Ir-Os alloys have been recently studied up to 30 GPa at room temperature by means of synchrotron-based X-ray powder diffraction in diamond anvil cells. Using their bulk moduli, which increase with increasing osmium content, showing a deviation from linearity, and after employing a thermodynamical model, it was concluded that the bulk modulus for osmium is slightly smaller than that for diamond. Here, a similar conclusion is obtained upon employing an alternative model, thus strengthening the conclusion that osmium is the densest but not the most incompressible element. This is particularly interesting for Earth Sciences because it may be of key importance toward clarifying the anomalous elastic properties of the Earths core.

On the Motivation and Foundation of Natural Time Analysis: Useful Remarks

Since its introduction in 2001, natural time analysis has been applied to diverse fields with remarkable results. Its validity has not been doubted by any publication to date. Here, we indicate that frequently asked questions on the motivation and the foundation of natural time analysis are directly answered if one takes into account the following two key points that we have considered as widely accepted when natural time analysis was proposed: first, the aspects on the energy of a system forwarded by Max Planck in his Treatise on Thermodynamics; second, the theorem on the characteristic functions of probability distributions which Gauss called Ein Schönes Theorem der Wahrscheinlichkeitsrechnung (beautiful theorem of probability calculus). The case of the time series of earthquakes and of the precursory Seismic Electric Signals are discussed as typical examples.

An Application of the Coherent Noise Model for the Prediction of Aftershock Magnitude Time Series

Recently, the study of the coherent noise model has led to a simple (binary) prediction algorithm for the forthcoming earthquake magnitude in aftershock sequences. This algorithm is based on the concept of natural time and exploits the complexity exhibited by the coherent noise model. Here, using the relocated catalogue from Southern California Seismic Network for 1981 to June 2011, we evaluate the application of this algorithm for the aftershocks of strong earthquakes of magnitude Mź6. The study is also extended by using the Global Centroid Moment Tensor Project catalogue to the case of the six strongest earthquakes in the Earth during the last almost forty years. The predictor time series exhibits the ubiquitous 1/f noise behavior.

Entropy in Natural Time

Entropy is a concept equally applicable to deterministic as well as stochastic processes. An entropy S is defined in natural time, which exhibits positivity, concavity and Lesche’s (experimental) stability. The entropy S- deduced from analyzing in natural time the time series obtained upon time reversal, is in general different from S, thus the entropy in natural time does satisfy the condition to be “causal” (while the variance κ1 = x 2 x 2 does not). The physical meaning of the change ΔS ≡ S-S- of the entropy in natural time under time reversal, which is of profound importance for the study of the dynamical evolution of a complex system, is discussed. For a fractional Brownian motion time series with self-similarity exponent H close to unity, as well as for an on– off intermittency model when the critical value is approached from below, both values of S and S- are smaller than the entropy Su~ 0.0966 of a “uniform” distribution. When a (natural) time window of length l is sliding through a time series, the entropy S exhibits fluctuations, a measure of which is the standard deviation δS. Complexity measures are introduced that quantify the δS variability upon changing the length scale l as well as the extent to which δS is affected when shuffling the consecutive events randomly (for l = const.). In a similar fashion, complexity measures can be defined for the fluctuations of the quantity ΔS whose standard deviation is designated σ[ΔS]. For the case that Qk are independent and identically distributed positive random variables, as in the case of data shuffled randomly, their σ/μ value is interrelated with δS and σ[ΔS].