Natural Time Analysis: The New View of Time, Part II: Advances in Disaster Prediction Using Complex Systems 3031260058, 9783031260056

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Panayiotis Varotsos · Nicholas Sarlis · Efthimios Skordas

Natural Time Analysis: The New View of Time, PART II ARISTOTELES

EINSTEIN

SCHRÖDINGER

Advances in Disaster Prediction Using Complex Systems

Natural Time Analysis: The New View of Time, Part II

Panayiotis Varotsos · Nicholas Sarlis · Efthimios Skordas

Natural Time Analysis: The New View of Time, Part II Advances in Disaster Prediction Using Complex Systems

Panayiotis Varotsos Department of Physics National and Kapodistrian University of Athens Athens, Greece

Nicholas Sarlis Department of Physics National and Kapodistrian University of Athens Athens, Greece

Efthimios Skordas Department of Physics National and Kapodistrian University of Athens Athens, Greece

ISBN 978-3-031-26005-6 ISBN 978-3-031-26006-3 (eBook) https://doi.org/10.1007/978-3-031-26006-3 © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors, and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

To the memory of my parents (Antonios and Efrosyni) and my wife (Mary) Panayiotis Varotsos

Preface

Since 2001, it has been shown that unique dynamic features hidden behind can be revealed from the time series of complex systems, if we analyze them in terms of a new concept of time termed natural time χ (from the Greek word “χ ρóνoς ” which means “time”), which was introduced by the authors in 2001 [1]. In this new view, time is not continuous, thus being in sharp contrast with the hitherto used conventional time t which is modeled as the one-dimensional continuum R of the real numbers. Examples of data analysis in this new time domain have appeared in diverse fields, including biology, cardiology, condensed matter physics, environmental sciences, geophysics, physics of complex systems, statistical physics, seismology, and volcanology. Several of these applications have been compiled in 2011 in a monograph [2], where the foundations of natural time analysis have been also explained in detail by providing the necessary mathematical background in each step. It was also shown [1] that the analysis in natural time enables the study of the dynamical evolution of a complex system and identifies when the system enters a critical stage. This reflects that natural time plays a key role in predicting impending catastrophic events in general, examples of which have been published in Physical Review and Physical Review Letters. In the latter journal was demonstrated that natural time is optimal for enhancing the signals in time–frequency space when employing the Wigner function and measuring its localization property (see Sect. 2.6 of Ref. [2] as well as in the Appendix of the present monograph). In other words, natural time analysis conforms to the desire to reduce uncertainty and extract signal information as much as possible. Since its introduction, the validity of natural time analysis has not been doubted by any publication to date. 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 have been considered by the authors as widely accepted when natural time analysis was proposed in 2001 [1]: first, the aspects on the energy of a system forwarded by Max Planck in his Treatise on Thermodynamics in 1945, and second, the theorem on the characteristic functions of probability distributions which Gauss called Ein Schönes Theorem der Wahrscheinlichkeitsrechnung (beautiful theorem of probability calculus), as will become clear below: vii

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For a time series comprising N events, we define as natural time χk for the occurrence of the k-th event the quantity χk = k/N . In doing so, we ignore the time intervals between consecutive events, but preserve their order and energy Q k . The analysis in natural time is carried by studying the evolution of the pair / out ∑N (χk , pk ), where the quantity pk = Q k n=1 Q n is the normalized energy for the 2 k-th event, and using the normalized power spectrum ∏(ω) ∑ N≡ |ϕ(ω)| (cf. ω stands for the angular natural frequency) defined by ϕ(ω) = k=1 pk exp(i ω χk ). ϕ(ω) is the characteristic function of pk for all ω ∈ R, since pk can be regarded as a probability for the occurrence of the k-th event at χk . In natural time analysis, of the distrithe behavior of ϕ(ω) is studied at ω → 0, because all the moments / bution of pk can be estimated from the derivatives d m ϕ(ω) dωm (for m positive integer) of the characteristic function ϕ(ω) at ω → 0. For this purpose, a quantity κ1 was defined from the Taylor expansion ∏(ω) = 1 − κ1 ω2 + κ2 ω4 + . . . where 2 ∑ ∑N N κ1 = ⟨χ 2 ⟩ − ⟨χ ⟩2 = k=1 pk (χk )2 − p χ . k k k=1 It has been shown that κ1 becomes equal to 0.070 at the critical state for a variety of dynamical systems (e.g., [2]). In general, this quantity is useful in identifying the approach to a critical point. In the natural time analysis of the seismicity, a careful inspection reveals (see Chap. 6 of [2]) that the quantity κ1 may be considered as an order parameter of seismicity. Furthermore, in [2] (Chap. 3), the entropy S in natural time was defined and additionally was shown that the entropy S− deduced from the natural time analysis of the time series obtained upon time reversal (i.e., reversing the time arrow) is in general different from S; thus, the entropy in natural time does satisfy the condition to be “causal”. In addition, the physical meaning of the change ΔS ≡ S − S− of the entropy in natural time under time reversal was discussed, which is of profound importance when studying the evolution of complex systems and the approach to a dynamic phase transition. Complexity measures were introduced that quantify the fluctuations of the entropy S and of the quantity ΔS upon changing the length scale as well as the extent to which they are affected when shuffling randomly the consecutive events. Since the publication of Ref. [2], a lot of work on the natural time analysis of seismicity has been published during the last decade 2011–2022, which shed light on the crucial importance of the order parameter of seismicity (κ1 ) and on the entropy change ΔS under time reversal, along with their fluctuations, in the identification of important changes before major earthquakes (EQs). These changes are described in the thirteen chapters of the present monograph and constitute key precursory phenomena of broad interest in the geosciences. The contents of these chapters could be summarized as follows: In Chap. 1, upon analyzing the seismic catalog of the Japanese Meteorological Agency (JMA) in natural time and considering all EQs of magnitude M JMA ≥ 3.5 from 1 January 1984 to 11 March 2011 (the time of the M9 Tohoku EQ) within 148 the entire Japanese area N46 25 E125 , we find that all shallow EQs of magnitude 7.6 or larger during this 27-year period were preceded by minima βW,min (or simply βmin ) of the fluctuations of the order parameter of seismicity 1 to around 3 months before

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these large EQs. Remarkably, among the minima, which are shown to be statistically significant by various techniques, the deepest minimum was observed around 5 January 2011, i.e., around two months before the M9 Tohoku EQ. In addition, a minimum of the order parameter fluctuations of seismicity was observed almost three and a half months before an ultra-deep EQ on 30 May 2015 in Japan, which is the strongest EQ (M W 7.9) after the M9 Tohoku EQ on 11 March 2011 and, strikingly, the deepest EQ ever detected (https://www.nationalgeographic.com/science/article/ deepest-earthquake-ever-detected-struck-467-miles-beneath-japan). In Chap. 2, it was found experimentally that a seismic electric signal (SES) activity (cf. the properties of SES activities have been recapitulated in Ref. [2]) started on 26 April 2000, i.e., around 2 months before the volcanic-seismic swarm activity in 2000 in the Izu Island region, Japan. Approximately at the time of the initiation of this pronounced SES activity, the fluctuations β of the order parameter of seismicity exhibited a clearly detectable minimum. These two phenomena are also linked in space as shown by analyzing in natural time the EQs reported during this period in the JMA seismic catalog. This is the first time that, well before the occurrence of major EQs, anomalous changes were found to appear approximately simultaneously in two independent datasets of different geophysical observables (geoelectrical measurements, seismicity). This simultaneous appearance is shown to be far beyond chance. 148 In Chap. 3, by dividing the Japanese region N46 25 E125 into small areas, we find that some small areas show minimum of the fluctuations β of the order parameter of seismicity almost simultaneously with the entire Japanese region. Such small areas clustered within a few hundred kilometers from the actual epicenter of the related mainshocks a few months before all the shallow major EQs of magnitude larger than 7.6 that took place in the region from 1 January 1984 to 11 March 2011. By means of this procedure, the epicenter of an impending major EQ can be estimated well in advance. In Chap. 4, the following three main features were identified upon studying the minima βW,min of the order parameter fluctuations of seismicity preceding all EQs of magnitude 8 (and 9) class that occurred in the Japanese area from 1 January 1984 to the M9 Tohoku EQ occurrence on 11 March 2011: Applying detrended fluctuation analysis (DFA) to the EQ magnitude time series we find that, when the minima βW,min are observed, long-range correlations are developed, and in addition, they are preceded by a stage in which an evident anti-correlated behavior appears. After the minima βW,min , the long-range correlations break down to an almost random behavior possibly turning to anti-correlation. In Chap. 5, we first recall that in general, two procedures had been suggested in order to determine the occurrence time of an impending mainshock: A procedure, which solely for the sake of convenience was called preliminary procedure, has been followed for several cases by starting the natural time analysis of seismicity in the candidate area A immediately after the SES activity initiation, because the latter signals that the system enters the critical stage and the time variation of parameters should be traced only on the single candidate area A. To identify, however, such a single area, one should know the properties of the preceding SES activity in general,

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or, in particular, the so-called selectivity map of the measuring station that recorded the SES activity (see Ref. [3]). An alternative procedure, called “updated” procedure, was later developed that considers the natural time analysis of the seismicity in all the possible subareas of A, instead of a single area identified by the SES selectivity map. When geoelectrical measurements are not available, however, we cannot identify directly the initiation time of a SES activity, but since this initiation occurs almost at the time of the minimum of the fluctuations of the order parameter of seismicity in a large area (as found in Chaps. 1 and 2 above), we take advantage of the occurrence time of this minimum. Furthermore, a spatiotemporal study of this minimum described above in Chap. 3 enables an estimation of the epicentral area of the impending major EQ. This has been applied for example, to the case of the identification of the occurrence time of the impending M9 Tohoku EQ that occurred on 11 March 2011, in Japan as follows: Starting the natural time analysis of seismicity at the date of 5 January 2011, at which the fluctuations of the order parameter of seismicity exhibited the deepest minimum ever observed during the period 1984– 2011 in Japan (as mentioned in Chap. 1) and computing the κ1 values of seismicity in the candidate epicentral area identified by means of the procedure described in Chap. 3, we found that the κ1 values converge to 0.070 almost one day before the Tohoku EQ occurrence, i.e., the system approaches the critical point almost one day before the mainshock. In Chap. 6, investigating what happens upon the occurrence of each of the major (M ≥ 7.6) shallow EQs in Japan from 1 January 1984 until the M9 Tohoku EQ occurrence on 11 March 2011, we find that the increase of the fluctuation β of the order parameter of seismicity on 22 December 2010 upon the occurrence on the same day of the M7.8 near Chichi-jima EQ is distinctly larger compared to the β fluctuations increase upon the occurrences of all other shallow EQs in Japan of magnitude 7.6 or larger from 1 January 1984 to the time of the M9 Tohoku EQ. In addition, studying the interrelation between this β fluctuation increase on 22 December 2010 versus the scale (number of events above a magnitude threshold) we find a functional form strikingly reminiscent of the one discussed by Penrose et al. in computer simulations of phase separation kinetics using the ideas of Lifshitz and Slyozov and independently of Wagner for the time growth of the characteristic size of the minority phase droplets in phase transitions. This also holds for the 2019 M6.4 Ridgecrest EQ in California. In Chap. 7, we focus on the entropy change ΔS, i.e., the difference S − S− upon considering the time reversal, i.e., T pk = p N −k+1 . ΔS may also have a subscript (ΔSi ) meaning that the calculation is made (for each S and S_) with a sliding natural time window of length i (=number of successive events), i.e., at scale i. It was found that ΔSi is a key measure which may identify when the system approaches the critical point (dynamic phase transition). Upon analyzing in natural time, all EQs of magnitude M3.5 or larger in Japan from 1 January 1984 until the occurrence of the M9 Tohoku EQ on 11 March 2011, we find that at scales like 4 × 103 , 5 × 103 and 7 × 103 events, a pronounced minimum of ΔS is observed on 22 December 2010 (on this date, the M7.8 Near Chichi-jima EQ occurred as mentioned in the preceding

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Chapter), which is statistically significant, upon changing either the EQ depth (from 0 to 300 km) or the magnitude threshold as well as the size of the area. In Chap. 8, using a sliding window of length i (number of consecutive events) through the time series of consecutive M ≥ 3.5 EQs and considering the standard deviation σ (ΔSi ) of the time series of ΔSi , we study the complexity measure ∆i , (ΔSi ) ∆i = σσ(ΔS , which quantifies how the results are affected upon changing the scale 100 ) 2 from 10 events to a longer scale, e.g., i = 103 events. Plotting the ∆i values, for example for the scales i = 2000, 3000, and 4000 events versus the conventional time t from 1 January 1984 until the occurrence of the M9 Tohoku EQ on 11 March 2011, we observe for all the three scales studied an abrupt increase Δ∆i of the ∆i values on 22 December 2010. This happens upon the aforementioned M7.8 EQ occurrence exhibiting a scaling behavior of the form Δ∆i = A(t − t0 )c where the exponent c is approximately equal to 1/3 independently of the scale i, while the pre-factors A are proportional to i and t 0 is almost 0.2 days after this M7.8 EQ occurrence. Such a behavior confirms the seminal work by Lifshitz and Slyozov and independently by Wagner on phase transitions, which predicts that the time growth of the characteristic size of the minority phase droplets grows with time as t 1/3 . In Chap. 9, we suggest a new procedure to identify the approach of the critical point (mainshock) without making use of the criticality condition κ1 = 0.070 mentioned in Chap. 5 for the evolution of seismicity after the initiation of the SES activity in the candidate epicentral area. In particular, we show that in natural time analysis, the complexity measure ∆i that quantifies the fluctuations of the entropy change ΔS of the seismicity in the Japanese area under time reversal plays a key role in identifying the occurrence time of the M9 Tohoku EQ that occurred on 11 March 2011 in Japan. The physical basis for the procedure followed is that, when analyzing in natural time the most reliable EQ model the quantity ΔS exhibits an evident minimum [2] (or maximum upon defining ΔS ≡ S− − S instead of ΔS ≡ S − S− ) before a large avalanche, i.e., a large EQ. Concerning the evolution of the ∆i values obtained from 148 the natural time analysis of the M ≥ 3.5 EQs inside the entire Japanese region N46 25 E125 , the following facts have been secured for (each) of the longer scales i: The values of ∆i start to increase after 22 December 2010, when ΔS reached a minimum, and attain a maximum around the first days of January 2011, very close to the beginning of the SES activity (that gave rise—according to Maxwell equations—to the observed anomalous variations of the magnetic field of the Earth recorded predominantly on the vertical component). Subsequently, ∆i decrease gradually up to the 11 March 2011 M9 Tohoku EQ. On the other hand, ∆i calculated by considering the seismicity inside the candidate epicentral area behaves differently, increasing abruptly after the M7.3 EQ on 9 March 2011 until the Tohoku M9 EQ. This important difference unveiled by natural time analysis leads, well in advance, to the characterization that the M7.3 EQ was a foreshock. More or less, similar results are obtained before the M8.2 EQ in Mexico on 7 September 2017 as well as before the M7.1 Ridgecrest EQ in California on 6 July 2019. In Chap. 10, it is investigated whether a physical model based on thermodynamics of point defects in solids, that was introduced in the 1980s, is compatible with

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the precursory phenomena associated with the M9 Tohoku mega-EQ occurrence in 2011. This physical model termed “pressure stimulated polarization currents (PSPC) model” suggests the following: In the future, focal region of an EQ, where the electric dipoles, due to lattice imperfections (point and linear defects) in the ionic constituents of rocks, has initially random orientations, the stress, σ, starts to gradually increase due to an excess stress disturbance (stage A). When this gradually increasing stress reaches a critical value (σ cr ), the electric dipoles exhibit a cooperative orientation resulting in the emission of a transient electric signal which constitutes an SES (stage B). (We emphasize that “cooperativity” is a hallmark of criticality and that several SES recorded within a short time are termed SES activity.) Actually, we find that precursory phenomena were mainly accumulated around two dates, i.e., 22 December 2010 and 5 January 2011, which strikingly concur with the two stages A and B of the SES generation PSPC physical model, respectively. These phenomena include the following: (A) Around 22 December 2010, the entropy change of seismicity under time reversal was minimized and the DFA exponent diminished in the value αmin, be f = 0.35, which is the lowest value observed during the period 1984–2011 pointing to an evident anti-correlated behavior in the EQ magnitude time series, as well as the horizontal GPS azimuths started to become gradually oriented toward the southern direction (while during the preceding period 12–22 December 2010, they had random orientations). (B) Around 5 January 2011, unprecedented minimum of the fluctuations of the order parameter of seismicity and Earth’s magnetic field started exhibiting anomalous variations primarily on the vertical component (which, according to Maxwell equations, should be accompanied by a strong SES activity) as well as full alignment of the orientations of the GPS azimuths southward that was accompanied by the most intense crust uplift; in addition, long-range temporal correlations appear in the EQ magnitude time series. In Chap. 11, the following matter is treated: There are areas in the globe which are not covered with a dense and accurate seismological network such as Japan. In these cases, the method suggested in Chap. 3, based on the spatiotemporal variations of the order parameter of seismicity for the determination of the epicenter of an impending major EQ, cannot be applied. Fortunately, an answer to this question can arise if we combine the fluctuation analysis of the order parameter of seismicity in natural time together with the theory of EQ networks based on similar activity patterns and with the newly introduced method of EQ nowcasting, which is also based on the natural time concept. The latter introduced the so-called EQ potential score for the estimation of the seismic risk in a region. In this chapter, we discuss the construction of self-consistent average EQ potential score maps and their ability to provide an estimate of the epicenter of a future strong EQ. Encouraging results related to the regional studies of the Eastern Mediterranean, the Southern California, Mexico and a part of Central America are presented. In Chap. 12, we present the natural time analysis of global seismicity by means of the global centroid moment tensor (GCMT) and the Centennial Earthquake Catalog (CEC). This analysis reveals the presence of nontrivial magnitude correlations for strong EQs with magnitudes M ≥ 6.5 for GCMT and M ≥ 7.0 for CEC. Characteristic precursory minima of the order parameter fluctuations of seismicity appear almost

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five and a half months on average before all major EQs of magnitude M ≥ 8.5. The empirical mode decomposition and Hurst analysis, when applied in natural time in the GCMT catalog, reveals the existence of micro-, mid- and macro-scale time series of global seismicity with different multifractal behaviors. Upon using the mid-scale time series only, we obtain EQ prediction results similar to those obtained by the full GCMT catalog. In addition, by combining the results—presented in Chap. 4— concerning the temporal correlations between EQ magnitudes with the minima of the fluctuations of the seismicity order parameter and EQ nowcasting, we find useful precursory information for the occurrence time and epicenter location of all EQs with M ≥ 8.5 in GCMT. This can be achieved with high statistical significance, while the epicentral areas lie within a region covering only 4% of that investigated. In Chap. 13, we focus on additional applications of natural time analysis which have not been addressed in the previous monograph [2] or mentioned so far in the present one. These cover a wide range of phenomena that precede EQs (like the fracture-induced electromagnetic emissions in the MHz range or the sub-ionospheric very low-frequency propagation anomalies or the precursory ultra-low-frequency magnetic field variations), acoustic emission precursors before fracture of brittle materials and other related precursors. Moreover, we present a method to estimate the occurrence time of strong aftershocks based on natural time. In addition, nowcasting volcano eruptions as well as atmospheric physics and climate applications are discussed. Finally, the recent advances in the analysis of photoplethysmography signals for the study of heart rate variability in natural time are also considered. An appendix is also included in this monograph where we briefly answer fundamental questions about natural time as well as summarize the key quantities of natural time analysis so that the reader may obtain clarifications without resorting to our previous monograph [2]. Athens, Greece December 2022

Panayiotis Varotsos Nicholas Sarlis Efthimios Skordas

References 1. Varotsos, P.A., Sarlis,N.V., Skordas, E.S.: Spatio-temporal complexity aspects on the interrelation between seismic electric signals and seismicity. Pract. Athens Acad. 76, 294–321 (2001) 2. Varotsos, P.A., Sarlis, N.V., Skordas, E.S.: Natural Time Analysis: The New View of Time. Precursory Seismic Electric Signals, Earthquakes and other Complex Time-Series. Springer, Berlin (2011). https://doi.org/10.1007/978-3-642-16449-1 3. Varotsos, P.: The Physics of Seismic Electric Signals. TerraPub, Tokyo (2005)

Contents

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2

The Order Parameter Fluctuations of Seismicity Are Minimized Before Major Earthquakes . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Computing the Order Parameter Fluctuations of Seismicity . . . . 1.3 Data and Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4 Minimum of the Variability β Before the M9 Tohoku EQ . . . . . 1.5 Minima of the Variability β Before Other Major EQs in Japan . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5.1 Minimum of the Order Parameter of Seismicity Preceding the Ultra-deep EQ (MW 7.9) in Japan on 30 May 2015 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5.2 Statistical Significance of the Minima of the Order Parameter Fluctuations from 1 January 1984 to 11 March 2011 (The Time of the M9 Tohoku EQ) . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5.3 More Recent and Useful Remarks on the Receiver Operating Characteristics Technique . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.6 Summary and Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A Straightforward Experimental Fact Demonstrating the Physical Interconnection of a SES Activity with Seismicity . . . . . 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Data and Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Studying the Robustness of the Simultaneous Appearance of the Two Phenomena with Respect to the Choice of the Area Selection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5 The Two Phenomena Are Linked Also in Space . . . . . . . . . . . . .

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2.6

3

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Statistical Significance of the Simultaneity of the Minimum βW,min with the Initiation of SES Activities . . . 2.7 Main Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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The Spatiotemporal Variations of the Minimum of the Seismicity Order Parameter Fluctuations May Estimate the Epicenter of an Impending Major Earthquake . . . . . . . . . . . . . . . 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Data and Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

45 45 48 52 52 53

Temporal Correlations in the Magnitude Time Series Before and After the Minimum of the Order Parameter Fluctuations of Seismicity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 The Procedure Followed in the Analysis and the Data Analyzed . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.1 The M9 Tohoku EQ on 11 March 2011 . . . . . . . . . . . . 4.3.2 The M8 Off Tokachi EQ on 26 September 2003 . . . . . 4.3.3 The Case of the Volcanic-Seismic Swarm in 2000 in the Izu Island Region . . . . . . . . . . . . . . . . . . 4.3.4 The Other MJMA ≥ 7.8 EQs in Japan . . . . . . . . . . . . . . 4.3.5 Distinguishing Precursory β Minima Associated with M ≥ 7.8 EQs from Other Minima Followed by M < 7.8 EQs . . . . . . . . . . . . . . . 4.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5 Main Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Identifying the Occurrence Time of a Mainshock by Means of the Minimum of the Seismicity Order Parameter Fluctuations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Earlier Procedures to Identify the Occurrence Time of an Impending Mainshock . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 The Criteria to Assure a True Coincidence of the EQ Time Series with that of Critical State . . . . . . . . . . . . . . . . . . . . . . 5.4 An Example for the Identification of the Occurrence Time of a Mainshock . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4.1 On the Deadly Mexico M8.2 EQ on 7 September 2017. Background . . . . . . . . . . . . . . . . . . . .

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5.4.2

Identifying the Occurrence Time of the M8.2 EQ on 7 September 2017 by Applying the Criteria to Assure a True Coincidence with the Critical State . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 5.5 A Second Example: The M9 Tohoku EQ on 11 March 2011 in Japan . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 5.6 Additional Comments on the Identification of the Occurrence Time of the M9 Tohoku EQ on 11 March 2011 in Japan . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 5.6.1 On the Difference in the Usefulness of the Seismicity Occurring Just Before and Just After the Initiation of a SES Activity . . . . . . . . . . . . . . 97 5.6.2 On the Accuracy in the Estimation of the Epicentral Area of an Impending Mainshock . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 5.7 Recapitulating the Explanation of the Criticality Condition κ1 = 0.070 to Identify the Occurrence Time of an Impending Mainshock . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98 5.8 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 6

7

On a Unique Increase of the Seismicity Order Parameter Fluctuations Before the M9 Tohoku Earthquake in Japan in 2011 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Data and Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4 Summarizing the Lifshitz-Slyozov-Wagner Theory. Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4.1 Introduction of the Becker-Döring Equations and the LSW Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4.2 Additional Comments Discussed by Penrose . . . . . . . . 6.5 Aggregation According to Classical Kinetics: From Nucleation to Coarsening . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.6 A Global Asymptotic Stability Result for Solutions to the LSW System of Equations . . . . . . . . . . . . . . . . . . . . . . . . . . 6.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Minimum of the Seismicity Entropy Change Under Time Reversal Before Major Earthquakes in Natural Time Analysis . . . . . 7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 Data and Methodology to Analyse the Data . . . . . . . . . . . . . . . . . 7.3 Results for the M9 Tohoku EQ in Japan in 2011 . . . . . . . . . . . . . 7.4 Results for The Mexico M8.2 EQ on 7 September 2017 . . . . . . 7.5 Main Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

105 105 106 106 111 111 112 112 113 115 115 119 119 120 123 140 147 149

xviii

8

9

Contents

Fluctuations of the Entropy Change of Seismicity Under Time Reversal Before Major Earthquakes in Natural Time Analysis . . . . . 8.1 Introduction. Motivation of This Research . . . . . . . . . . . . . . . . . . 8.2 The Case of the M9 Tohoku EQ in Japan in 2011 . . . . . . . . . . . . 8.3 Main Conclusions Related to the M9 Tohoku EQ in Japan in 2011 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.4 The Case of the Deadly Mexico M8.2 EQ on 7 September 2017 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.5 Main Conclusion for the Deadly Mexico M8.2 EQ on 7 September 2017 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Identifying the Occurrence Time of a Mainshock by Means of the Fluctuations of the Seismicity Entropy Change Under Time Reversal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2 Results from the Case of the M9 Tohoku EQ in Japan in 2011 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3 Main Conclusions Related to the M9 Tohoku EQ in Japan in 2011 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.4 Results from the Deadly Mexico M8.2 EQ on 7 September 2017 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.5 Results from the M7.1 Ridgecrest EQ on 6 July 2019 in California . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.6 Main Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

10 Compatibility of the SES Generation Model with the Precursory Phenomena Before the Tohoku M9 Earthquake in Japan in 2011 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.2 Precursory Phenomena Before the M9 Tohoku EQ in 2011 in Brief . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.2.1 Earth’s Magnetic Field Anomalous Variations . . . . . . 10.2.2 Earth’s Surface Displacements . . . . . . . . . . . . . . . . . . . . 10.2.3 Results from Natural Time Analysis of Seismicity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.3 Compatibility of the Physical Model with the Observed Precursory Phenomena Before the M9 Tohoku EQ in 2011 . . . . 10.4 Complementary Comments on the Compatibility of the Physical Model with the Observed Precursory Phenomena Before the M9 Tohoku EQ in 2011 . . . . . . . . . . . . . . 10.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

153 153 154 163 163 170 171

173 173 174 179 180 181 187 187

189 189 192 193 193 194 197

199 203 204

Contents

11 Recent Advances on the Estimation of a Future Earthquake Epicenter Based on Natural Time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Fluctuations of the Order Parameter of Seismicity . . . . . . . . . . . . 11.3 Earthquake Nowcasting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.4 Average EQ Potential Score Maps . . . . . . . . . . . . . . . . . . . . . . . . . 11.5 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.5.1 Results for Strong EQs In Eastern Mediterranean . . . . 120 11.5.2 Results for Strong EQs in the Area N35 10 W80 . . . . . . . . 11.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 Study of the Global Seismicity Using Natural Time Analysis . . . . . . . 12.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.2 Fluctuations of the Order Parameter of Seismicity . . . . . . . . . . . . 12.3 Identification of Long Range Correlations in the Presence of Heavy Tails . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.4 Earthquake Nowcasting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.5 Average EQ Potential Score Maps . . . . . . . . . . . . . . . . . . . . . . . . . 12.6 Empirical Mode Decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.6.1 Ensemble Empirical Mode Decomposition . . . . . . . . . 12.7 Hurst or R/S Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.8 Detrended Fluctuation Analysis of EQ Magnitude Time Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.9 Receiver Operating Characteristics . . . . . . . . . . . . . . . . . . . . . . . . . 12.10 Event Coincidence Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.11 Magnitude Correlations in Global Seismicity . . . . . . . . . . . . . . . . 12.12 Micro-scale, Mid-scale, and Macroscale in Global Seismicity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.12.1 Identification of the Micro-, Midand Macro-scale Components in Global Seismicity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.12.2 Spectral Study of the Micro-, Midand Macro-scale Time Series . . . . . . . . . . . . . . . . . . . . . 12.12.3 The Importance of the Mid-scale . . . . . . . . . . . . . . . . . . 12.13 Minima βmin of the Fluctuations of the Order Parameter of Global Seismicity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.13.1 Estimation of the Statistical Significance by Means of Receiver Operating Characteristics . . . . . 12.13.2 Estimation of the Statistical Significance by Means of Event Coincidence Analysis . . . . . . . . . . 12.14 Results from the Combination of βmin , Detrended Fluctuation Analysis and Earthquake Nowcasting . . . . . . . . . . . . 12.15 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

xix

209 209 211 212 215 219 219 226 231 234 239 239 241 242 244 245 246 247 247 247 248 249 251 255

255 258 259 265 271 273 277 284 284

xx

Contents

13 Applications of Natural Time Analysis to Disaster Prediction in Other Disciplines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.2 Fracture-induced Electromagnetic Emissions Before EQs . . . . . 13.2.1 Laboratory Experiments on LiF . . . . . . . . . . . . . . . . . . . 13.3 Subionospheric Very Low Frequency Propagation Anomalies Before EQs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.4 Ultra-low Frequency Magnetic Field Variations Before EQs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.5 Global Navigation Satellite System Surface Deformation Before the 2016 Kumamoto Earthquakes . . . . . . . . . . . . . . . . . . . 13.6 Predicting the Occurrence Time of Strong Aftershocks . . . . . . . 13.7 Combination of Multiresolution Wavelets and Natural Time Analysis Before Earthquakes . . . . . . . . . . . . . . . . . . . . . . . . 13.8 Nowcasting: Volcanic Eruptions, Arctic Low Temperatures, and Extreme Cosmic Ray Events . . . . . . . . . . . . . 13.9 Acoustic Emissions Before Fracture . . . . . . . . . . . . . . . . . . . . . . . 13.10 Acoustic Emissions in an Experiment in Plunged Granular Bed . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.11 Electrical Resistance Fluctuations Before Fracture . . . . . . . . . . . 13.12 Ozone Hole Dynamics Over Antartica . . . . . . . . . . . . . . . . . . . . . . 13.13 Precursory Signals of Major El Niño Southern Oscillation Events . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.14 Heart Dynamics Monitored Through Photoplethysmography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.15 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

293 293 298 299 299 301 303 305 311 313 315 319 321 322 323 326 332 332

Appendix A: Optimality of Natural Time Representation and a Brief Introduction to Natural Time Analysis . . . . . . . . . . . . . . . . . . . . 341 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 355

Acronyms

AE AGW AUC BC BD BK BTW CDF CEC CHF CH CMT DFA ECA ECG EEMD EMD EME EN ENBOSAP ENSO EPS EQ ETAS FA foF2 FP G GCMT GEONET GNSS

Acoustic emission Atmospheric gravity wave Area under the (ROC) curve Baja California Becker-Döring Burridge–Knopoff Bak-Tang-Wiesenfeld Cumulative distribution function Centennial earthquake catalog Congestive heart failure Chiapas Centroid moment tensor Detrended fluctuation analysis Event coincidence analysis Electrocardiogram Ensemble empirical mode decomposition Empirical mode decomposition Electromagnetic emissions Earthquake nowcasting Earthquake network based on similar activity patterns El Niño–Southern oscillation Earthquake potential score Earthquake Epidemic type aftershock sequence False alarm Critical frequency of ionospheric F2 layer False positive Guerrero Global centroid moment tensor catalog GNSS Earth observation network system Global navigation satellite system xxi

xxii

GPS GSI H IMF J JMA LAIC LF LS LSW LT M M MD-OHA MEM MRWA NEIC O OFC PDE PDF PP PPG PPP PSPC QBO rms ROC RTK SC SCD SCEC SD SES SOC SOI SSN SVM TP ULF USGS

Acronyms

Global positioning system Geospatial information authority of Japan Healthy Intrinsic mode function Jalisco-Colima Japanese meteorological agency Lithosphere-atmosphere–ionosphere coupling Low frequency Lifshitz-Slyozov Lifshitz-Slyozov-Wagner Local time Earthquake magnitude Michoacán Maximum daily ozone hole area Maximum entropy method Multiresolution wavelet analysis National earthquake information center of the United States of America Oaxaca Olami-Feder-Christensen Preliminary determination of epicenters (bulletin) Probability density function Peak to peak Photoplethesmography Precise point positioning Pressure stimulated polarization currents Quasi-biennial oscillation Root mean square Receiver operating characteristics Real-time kinematic Southern California Sudden cardiac death Southern California earthquake catalog or center Sudden cardiac death Seismic electric signal Self-organized criticality Southern oscillation index National seismic service of the Universidad Nacional Autónoma de México Support vector machines True positive Ultra-low frequency United States Geological Survey

Acronyms

UTC VF VLF

xxiii

Coordinated universal time Ventricular fibrillation Very low frequency

Chapter 1

The Order Parameter Fluctuations of Seismicity Are Minimized Before Major Earthquakes

1.1 Introduction As mentioned in the Preface, for a time series comprising N earthquakes (EQs), we define the natural time for the occurrence of the k-th event by χk = k/N which means that we ignore the time intervals between consecutive events, but preserve their order. We also preserve∑ their energy Q k . We then study the evolution of the N Q n is the normalized energy. The approach of a pair χk , pk , where pk = Q k / n=1 dynamical system to criticality can be identified by the variance κ1 of natural time χ weighted for pk , namely κ1 =

N ∑ k=1

( pk (χk ) − 2

N ∑

)2 p k χk

≡ − 2 .

(1.1)

k=1

Earthquakes exhibit complex correlations in time, space, and magnitude, and the opinion prevails (e.g., Ref. [8] and references therein) that the EQs are critical phenomena. In natural time analysis of seismicity, the quantity κ1 calculated from seismic catalogs serves as an order parameter [36, 43]. Experiences have shown that the mainshock occurs in a few days to around 1 week after the κ1 value in the candidate epicentral area approaches 0.070 [27]. This was found useful in narrowing the lead time of EQ prediction. However, to trace the time evolution of κ1 value, one needs to start the analysis of the seismic catalog at some time before the yet-to-occur mainshock. We chose, for the starting time for analysis, the initiation time of Seismic Electric Signal (SES) activity. SESs are low-frequency (≤ 1 Hz) electric signals that precede EQs [33]. The reason for this choice was based on the consideration that SESs are emitted when the focal zone enters the critical stage [34]. In the case of the lack of SES data we cannot adopt this approach, but we instead examine the fluctuations of κ1 near criticality, i.e., near the EQ occurrence. To compute the fluctuations, we apply the procedure that follows. © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 P. Varotsos et al., Natural Time Analysis: The New View of Time, Part II, https://doi.org/10.1007/978-3-031-26006-3_1

1

2

1 The Order Parameter Fluctuations of Seismicity Are Minimized …

1.2 Computing the Order Parameter Fluctuations of Seismicity To obtain the fluctuation β of κ1 , we need many values of κ1 for each target EQ. For this purpose, we first take an excerpt comprising W successive EQs just before a target EQ from the seismic catalog. The number W is chosen to cover a period of a few months. Thus, W is the number of EQs that on average occur during the average lead time of an SES activity which is around a few months (since the lead time of SES activity ranges from a few weeks to 5 21 months, see Chap. 7 of Ref. [36]). } { For this excerpt, we form its sub-excerpts S j = Q j+k−1 k=1,2,...,N of consecutive N = 6 EQs (since at least six EQs are needed [43] for obtaining reliable ∑ N κ1 ) of energy Q j+k−1 , and Q j+k−1 and natural time χk = k/N each. Further, pk = Q j+k−1 / k=1 by sliding S j over the excerpt of W EQs, j = 1, 2, . . . , W − N + 1(= W − 5), we calculate κ1 using Eq. (1.1) for each j. We repeat this calculation for N = 7, 8, . . . , W , thus obtaining an ensemble of [(W − 4)(W − 5)] /2(= 1 + 2 + · · · + W − 5) κ1 values. Then, we compute the average μ(κ1 ) and the standard deviation σ (κ1 ) of thus obtained ensemble of [(W − 4)(W − 5)] /2 κ1 values. The variability β of κ1 for this excerpt W is defined to be: β ≡ σ (κ1 )/μ(κ1 )

(1.2)

and is assigned to the (W + 1)-th EQ in the catalog, i.e., the target EQ. The time evolution of the β value can be pursued by sliding the excerpt through the EQ catalog. Namely, through the same process as above, β values assigned to (W + 2)th, (W + 3)th, . . . EQs in the catalog can be obtained. Note that β may bear a subscript βW to clarify the window length W used for its calculation. Since 2013 (see Refs. [28, 29]) following the procedure described above, the calculations have been made by taking all natural time windows comprising 6 to W events. Note, however, that before 2013 (as for example is the case in [41] discussed in the next Chapter), the calculations were made as follows: First, for an event considering natural time windows from 6 to 40 events (i.e., selecting the precise value 40 as an upper limit). Second, this process was performed for all the events sliding through the whole EQ catalog excerpt.

1.3 Data and Analysis In Ref. [28], analysis was made by using the Japan Meteorological Agency (JMA) seismic catalog and considering all of the EQs in the period from 1984 to the time of the M9 Tohoku EQ, within the area 25◦ –46◦ N, 125◦ –148◦ E, which covers the whole Japanese region (Fig. 1.1). The energy of EQs was obtained from MJMA after converting [30] to the moment magnitude MW [10]. Setting a threshold MJMA = 3.5 to assure the data completeness, we are left with 47,204 EQs in the concerned period

1.4 Minimum of the Variability β Before the M9 Tohoku EQ

3

48˚N

46˚N

44˚N

1993−07−12 1994−10−04

42˚N

2003−09−26 40˚N

1994−12−28

38˚N

2011−03−11 36˚N 34˚N 32˚N 30˚N

1984−03−06 28˚N 26˚N

2010−12−22

24˚N 126˚E

129˚E

132˚E

135˚E

138˚E

141˚E

144˚E

147˚E

150˚E

46 E148 from Fig. 1.1 Earthquakes of magnitude 7.6 or greater whose epicentres (stars) lie within N25 125 1 January 1984 until the M9 Tohoku EQ. The triangle is a deep EQ that occurred on 6 March 1984. Taken from Ref. [28]

of about 326 months. Thus, we have on the average ∼ 102 EQs per month. We chose the values W = 200, 300, and 400, which would cover a period of a few months before each target EQ. This choice of a few months is based on the experience that the lead time of SES activities is of this order both in Japan [32] and Greece [34–36].

1.4 Minimum of the Variability β Before the M9 Tohoku EQ Figure 1.2a depicts about 47,200 β-values calculated for W = 300 versus the target EQ number from 1984 to the day of the Tohoku EQ, 11 March 2011. EQs with MJMA ≥ 6.9 (MJMA in the right scale) are shown by blue asterisks. One can see that the β-values show a deep minimum value just before the Tohoku EQ (rightmost side of Fig. 1.2a). This observation prompted us to investigate more about this β minimum. Figure 1.2b is an expanded version, in the conventional time, of the concerned part of

4

1 The Order Parameter Fluctuations of Seismicity Are Minimized …

1.4 1.2 1 0.8 0.6 0.4 0.2 0

9.5 9 8.5 8

MJMA

β for W=300

(a) 1.6

7.5 0

10,000

20,000

30,000

40,000

7 50,000

No of earthquakes since 1984

β

(c)

1.6 1.4 1.2 1 0.8 0.6 0.4 0.2 0

β 2000 β 3000

9.5 9 8.5 8

MJMA

β 200 1.4 0.160 β 300 1.2 β 400 1 0.150 0.8 0.6 0.4 0.2 0.157 0 May01 Jun01 Jul01 Aug01 Sep01 Oct01 Nov01 Dec01 Jan01 Feb01 Mar01 2010 2010 2010 2010 2010 2010 2010 2010 2011 2011 2011

7.5 7

9.5 9 8.5 8

MJMA

β

(b) 1.6

7.5 0

10,000

20,000

30,000

40,000

7 50,000

No of earthquakes since 1984

Fig. 1.2 Plot of the variability β of κ1 (left scale) together with all MJMA ≥ 6.9 EQs (blue asterisks, MJMA in the right scale). a Versus EQ number when a natural time window comprising W = 300 events is moving through the JMA catalog from 1984 until the M9 Tohoku EQ occurrence. b Versus the conventional time during the period of the last 10 months (depicted by yellow in (a)). Red for W = 200, blue for W = 300 and green for W = 400. Every tick corresponds to 10 days in the horizontal scale. c Variability β for natural time window W = 2000 (green) and W = 3000 (blue). Taken from Ref. [28]

Fig. 1.2a (the last 10-months period shown in yellow). The red, blue, and green curves show what happened to β for W = 200, 300, and 400. For brevity, we use hereafter the symbols βW and βW,min as needed. A careful inspection of Fig. 1.2b reveals that after around 1 September 2010 a decrease of βW became evident and βW went down to a minimum (β200,min ∼ 0.157, β300,min ∼ 0.160, and β400,min ∼ 0.150) in early January 2011, about 2 months before the mainshock. (The abrupt increase of β around 22 December 2010 was due to the M7.8 near Chichi-jima EQ on this date and constitutes an important increase that will be discussed in detail in Chap. 6.) Results of the computation on this minimum of β are summarized as follows (Fig. 1.2a, b, and Table 1.1). First, such a β minimum almost two months before the Tohoku EQ was not observed again during the 27-year period studied. Second, the ratio β300,min /β200,min is near unity. Third, the dates of βW,min for W = 200, 300 and 400 are 5 January,

2011-03-11

f

Tohoku

Near Chichi-jima 38.10

27.05

41.78

40.43

43.38

42.78

Lat., ◦ N

142.86

143.94

144.08

143.75

147.67

139.18

Long, ◦ E

9.0

7.8

8.0

7.6

8.2

7.8

M

0.157 (2011-01-05)

0.232 (2010-11-30)

0.289 (2003-07-03)

0.196 (1994-10-15)

0.295 (1994-06-30)

0.293 (1993-05-23)

β200,min

0.160 (2011-01-05)

0.248 (2010-11-30)

0.306 (2003-07-14)

0.197 (1994-10-19)

0.319 (1994-07-22)

0.278 (1993-06-07)

β300,min

1.02

1.07

1.06

1.01

1.08

0.95

β300,min β200,min

2

1

3

2-3

3

2

△t200

2

1

2

2

2–3

1

△t300

The symbols βW,min are the minima of the κ1 variability that preceded these EQs along with their dates. △t200 and △t300 are the differences in months between the dates of β200,min , β300,min and the EQ; Lat., latitude; Long., longitude Compiled from Ref. [28]

2010-12-22

e

Off Tokachi

Far-Off Sanriku

1994-12-28

2003-09-26

c

d

East-Off Hokkaido

1994-10-04

b

Southwest-off Hokkaido

1993-07-12

EQ name

a

EQ date

46 E 148 together with their corresponding Table 1.1 All shallow EQs with magnitude 7.6 or larger since 1 January 1984 until M9 Tohoku EQ within the area N25 125 variability minima

1.4 Minimum of the Variability β Before the M9 Tohoku EQ 5

6

1 The Order Parameter Fluctuations of Seismicity Are Minimized …

5 January and 10 January 2011, respectively, i.e., almost the same. Fourth, this minimum is less clear for greater W corresponding to time intervals longer than a few months. It is almost invisible for W = 2000 and 3000 (Fig. 1.2c). The same applies to all other βW,min , as seen in Fig. 1.2c. In what follows, we restrict ourselves to W = 200 and W = 300.

1.5 Minima of the Variability β Before Other Major EQs in Japan During the 27-year study period investigated six shallow EQs with MJMA 7.6 or larger occurred (Fig. 1.1 and Table 1.1). They are EQ a 1993-07-12: 1993 Southwest-Off Hokkaido EQ (MJMA = 7.8); EQ b 1994-10-04: 1994 East-Off Hokkaido EQ (MJMA = 8.2); EQ c 1994-12-28: 1994 Far-Off Sanriku EQ (MJMA = 7.6); EQ d 2003-09-26: 2003 Off Tokachi EQ (MJMA = 8.0); EQ e 2010-12-22: 2010 Near Chichi-jima EQ (MJMA = 7.8); EQ f 2011-03-11: 2011 Tohoku EQ (MW = 9.0). In the following, we examine if minimum of β exists before these EQs also. Figure 1.3a–c are the expanded versions of Fig. 1.2a in the conventional time in three 10-year periods. The EQs are marked by a–f. Because these figures are still too small, we expanded the time axis for each EQ as shown in Fig. 1.4a–e, just as we did in Fig. 1.2b for the Tohoku EQ. We can now see β minima within 1–3 months before every one of the six mainshocks as shown in Table 1 of Ref. [28], which is reproduced here in Table 1.1. In this Table, the β300,min /β200,min ratio and the △t200 as well as △t300 of β minima before each of these EQs are very similar to those before the Tohoku EQ, e.g., the ratios lie in the range 0.95–1.08. It was then concluded that these minima are precursory to the time-correlated EQs. Beyond these β minima before the six MJMA ≥ 7.6 EQs, there were many more minima during the 27-year period, as seen in Fig. 1.3. We examined whether they were also followed by EQs. Along these lines, the minima were chosen deeper than the shallowest one of the six β200,min in Table 1.1, which happened before EQ b, the 1994 East-Off Hokkaido EQ (MJMA = 8.2), yielding 0.295 for β200,min . We found that, out of the thus chosen 31 minima, 9 (labeled 1–9 in Fig. 1.3) also displayed β300,min /β200,min ratio (Fig. 1.5 and Table 1.2) in the range 0.95–1.08, thus almost equal to those observed before the six MJMA ≥ 7.6 EQs (Table 1.1). These nine minima have been followed by MJMA ≥ 6.4 EQs within 3 months (Fig. 1.5 and △t200 in Table 1.2). Such correspondences are naturally less certain, because of a larger number of EQs. In particular, there were 139 MJMA ≥ 6.4 EQs during the 27-year period studied. The cases mentioned here, however, may have some reason to be reliable as explained below (Table 1.2). For example, the EQ that apparently followed βW,min No. 6 in April 2000 was the largest EQ of the volcano-seismic activity in the Izu Island area [32],

1.5 Minima of the Variability β Before Other Major EQs in Japan

β

(c)

1.6 1.4 β200 d 9.5 β300 1.2 9 b 1 8.5 0.8 0.6 8 0.4 7.5 0.2 67 c 8 5 7 0 Jan01 Jan01 Jan01 Jan01 Jan01 Jan01 Jan01 Jan01 Jan01 Jan01 Jan01 1994 1995 1996 1997 1998 1999 2000 2001 2002 2003 2004 1.6 1.4 β200 9.5 β300 1.2 e 9 f 1 8.5 0.8 0.6 8 0.4 7.5 0.2 9 7 0 Jan01 Jan01 Jan01 Jan01 Jan01 Jan01 Jan01 Jan01 Jan01 Jan01 Jan01 2004 2005 2006 2007 2008 2009 2010 2011 2012 2013 2014

MJMA MJMA

β

(b)

1.6 a 1.4 β200 9.5 β300 1.2 9 1 8.5 0.8 0.6 8 0.4 7.5 0.2 4 2 1 3 7 0 Jan01 Jan01 Jan01 Jan01 Jan01 Jan01 Jan01 Jan01 Jan01 Jan01 Jan01 1984 1985 1986 1987 1988 1989 1990 1991 1992 1993 1994

MJMA

β

(a)

7

Fig. 1.3 Variability β versus the conventional time given in three consecutive 10-year periods in (a–c), respectively. Every tick corresponds to 3 months in the horizontal scale. No data are plotted in (c) after M9 Tohoku EQ. The EQ magnitudes MJMA are shown by black asterisks read in the right scale. Taken from Ref. [28]

the largest EQ swarm ever recorded in Japan. This EQ is of specific importance since it has been preceded by a very clear SES activity detected by Uyeda et al. [31, 32] the study of which, as it will be explained in Chap. 6, shed light on the fact that there exists a physical interconnection of a SES activity and seismicity. For a second example, βW,min No. 7 in July 2000 was followed by MJMA 7.3 Western Tottori EQ. The period after this EQ in the year 2000 was completely free from shallow EQ greater than MJMA 6. Other examples are the minima Nos. 9, 2, 8, 3, and 4 which were followed by EQs of MJMA 7.2, 7.1, 7.0, 6.9, and 6.9. For brevity, each case has not been described in Ref. [28], but it was inferred that these βW,min might have also been precursory to sizable EQs. In fact, there were only 43 MJMA ≥ 6.9 EQs during the 27-year period. Likewise, the βW,min Nos. 1 and 5 seemed to be followed by EQs of MJMA 6.6 and MJMA 6.4, respectively, although their correlations are even less certain. After handling these, we are still left with 22 minima unnumbered or unmarked in Fig. 1.3. Each of them has been checked in Ref. [28] for their β300,min /β200,min ratio. For example, βW,min (β200,min = 0.213 and β300,min = 0.259) observed on December 4, 2008 (Fig. 1.3c) exhibited a ratio β300,min /β200,min (= 1.22), which lies outside

1 The Order Parameter Fluctuations of Seismicity Are Minimized …

β

(c)

β

(d)

β

(e)

1.6 1.4 β200 β300 1.2 1 0.8 0.6 0.4 0.2 0 Aug01 1994 1.6 1.4 β200 β300 1.2 1 0.8 0.6 0.4 0.2 0 Apr01 2003 1.6 1.4 β200 β300 1.2 1 0.8 0.6 0.4 0.2 0 Sep01 2010

8.5 8

a

MJMA

9

7.5 Mar01 1993

Apr01 1993

May01 1993

Jun01 1993

7 Aug01 1993

Jul01 1993

9.5 9 8.5

b

8

MJMA

1.6 1.4 β200 β300 1.2 1 0.8 0.6 0.4 0.2 0 May01 1994

9.5

7.5 7 Jun01 1994

Oct01 1994

Sep01 1994

Aug01 1994

Jul01 1994

9.5 9 8.5

b

8

c Sep01 1994

Oct01 1994

Nov01 1994

Dec01 1994

MJMA

β

(b)

1.6 1.4 β200 β300 1.2 1 0.8 0.6 0.4 0.2 0 Feb01 1993

7.5

7 Jan01 1995

9.5 9 8.5

d

8

MJMA

β

(a)

7.5 7 May01 2003

Jun01 2003

Aug01 2003

Jul01 2003

Sep01 2003

9.5

f

9 8.5 8

e

MJMA

8

7.5 7

Oct01 2010

Nov01 2010

Dec01 2010

Jan01 2011

Feb01 2011

Mar01 2011

Fig. 1.4 Excerpts of Fig. 1.3 plotted in an expanded time scale showing what happened for the variability β before the occurrence of a EQ a, b EQ b, c EQ b and EQ c, d EQ d, and e EQ e and EQ f (the M9 Tohoku EQ). Every tick corresponds to 10 days in the horizontal scale. The red arrow heads show β200,min along with EQs (in black a–f, MJMA in the right scale). Taken from Ref. [28]

1.5 Minima of the Variability β Before Other Major EQs in Japan β200 β300

9 8.5 8

1

β β

(d)

Jan01 1987

Mar01 1987

May01 1987

β200 β300

9.5 9 8.5 8

2

7.5 7 Sep01 1989

Jul01 1989

(c)

Nov01 1986

MJMA

β

1.6 1.4 1.2 1 0.8 0.6 0.4 0.2 0

7.5

EQ1

7 Sep01 1986

Jul01 1986

(b)

MJMA

9.5

Nov01 1989

Jan01 1990

Mar01 1990

May01 1990

1.6 4 1.4 β200 β300 1.2 1 0.8 0.6 0.4 3 0.2 0 Jan01 Mar01 May01 Jul01 Sep01 Nov01 Jan01 Mar01 May01 Jul01 Sep01 1992 1992 1992 1992 1992 1992 1993 1993 1993 1993 1993

1.6 1.4 β200 β300 1.2 1 0.8 0.6 0.4 0.2 0 Jan01 1998

9.5 9 8.5 8

MJMA

1.6 1.4 1.2 1 0.8 0.6 0.4 0.2 0

7.5 7

9.5 9 8.5

5

8 7.5

EQ5 Mar01 1998

May01 1998

MJMA

β

(a)

9

Jul01 1998

Sep01 1998

Nov01 1998

7 Jan01 1999

Fig. 1.5 Excerpts of Fig. 1.3 but corresponding to each of the nine cases of β200,min given in Table 1.2: a 1, b 2, c 3 and 4, d 5, e 6 and 7, f 8, and g 9. Every tick is 10 days in the horizontal scale. Numbers 1–9 correspond to β200,min in Fig. 1.3 and Table 1.2. EQs time-correlated to minima 1, 5, and 6 are shown with the vertical black arrow heads pointing downwards. Taken from Ref. [28]

10

β

(g)

1.6 1.4 β200 β300 1.2 1 0.8 0.6 0.4 0.2 0 Jan01 2002

1.6 1.4 β200 β300 1.2 1 0.8 0.6 0.4 0.2 0 Jan01 2005

9 8.5 8

MJMA

9.5

EQ6

7.5

6

7 May01 2000

Mar01 2000

Jul01 2000

Sep01 2000

Nov01 2000

7 Jan01 2001

9.5 9 8.5 8

MJMA

β

(f)

1.6 1.4 β200 β300 1.2 1 0.8 0.6 0.4 0.2 0 Jan01 2000

7.5

8 Mar01 2002

May01 2002

Jul01 2002

Sep01 2002

Nov01 2002

7 Jan01 2003

9.5 9 8.5 8

MJMA

β

(e)

1 The Order Parameter Fluctuations of Seismicity Are Minimized …

7.5

9 Mar01 2005

May01 2005

Jul01 2005

Sep01 2005

Nov01 2005

7 Jan01 2006

Fig. 1.5 (continued)

the range 0.95–1.08. Figure 1.6 is the histogram of the β300,min /β200,min ratio for all of the 37 minima examined so far, consisting of the 6 in Table 1.1 marked a–f, 9 in Table 1.2 marked 1–9, and the 22 additionally chosen minima. From this figure, interestingly, none of these additional 22 minima exhibits the ratio within the range 0.95–1.08.

1.5.1 Minimum of the Order Parameter of Seismicity Preceding the Ultra-deep EQ (MW 7.9) in Japan on 30 May 2015 On 30 May 2015, a powerful EQ (MW 7.9, JMA reported magnitude M8.1) struck west of Japan’s remote Ogasawara (Bonin) island chain, which lies more than 800 km

1.5 Minima of the Variability β Before Other Major EQs in Japan

11

Table 1.2 Nine βW,min of the κ1 variability not included in Table 1.1 but chosen by the procedure described in the text No. β200,min

β300,min β200,min

β300,min

EQ date

Lat., ◦N

Long, ◦E

M

△t200

△t300

1

0.254 (1986-10-13)

0.257 (1986-11-15)

1.01

1987-01-14

42.45

142.93 6.6

3

2

2

0.278 (1989-08-08)

0.292 (1989-09-15)

1.05

1989-11-02

39.86

143.05 7.1

3

2

3

0.250 (1992-04-05)

0.253 (1992-05-10)

1.01

1992-07-18

39.37

143.67 6.9

3

2

4

0.188 (1993-07-13)

0.182 (1993-07-15)

0.97

1993-10-12

32.03

138.24 6.9

3

3

5

0.237 (1998-02-17)

0.233 (1998-03-12)

0.98

1998-05-31

39.03

143.85 6.4

3.5

2.5

6

0.229 (2000-04-12)

0.219 (2000-05-06)

0.96

2000-07-01

34.19

139.19 6.5

3

2

7

0.243 (2000-07-09)

0.258 (2000-07-09)

1.06

2000-10-06

35.27

133.35 7.3

3

3

8

0.244 (2002-05-12)

0.252 (2002-06-03)

1.03

2002-06-29

43.50

131.39 7.0

2

1

9

0.286 (2005-06-11)

0.309 (2005-07-01)

1.08

2005-08-16

38.15

142.28 7.2

2

1.5

The βW,min exhibit β200,min deeper than 0.295, which corresponds to the shallowest β200,min in Table 1.1. EQs are time-correlated to the β200,min Taken from Ref. [28] 7 6

Number of minima

Fig. 1.6 Histogram of the β300,min /β200,min ratio for the 37 minima in Fig. 1.3 which are deeper than the shallowest β200,min of Table 1.1. The minima marked a–f or numbered 1–9 in Fig. 1.3 are placed vertically in the corresponding column according to their β200,min values. Taken from Ref. [28]

b

5

1 d

4

a 3 9

3

5 8 2 2

6 c 7 1

4 f e 0 0.9

1

1.1

1.2

1.3

1.4

1.5

β300,min/β200,min

south of Tokyo. It occurred at 680 km depth in an area without any known historical seismicity and caused significant shaking over a broad area of Japan at epicentral distances in the range 1000–2000 km. Specifically, the maximum shaking intensity was reached at both Hahajima Island (above the hypocenter) and Kanagawa (near Tokyo) located over 800 km from the hypocenter. It was the first EQ felt in every Japanese prefecture since intensity observations began in 1884. By applying natural time analysis, Varotsos et al. [42] found that almost three and a half months before this powerful EQ, which is the strongest one after the M9 Tohoku EQ on 11 March 2011, the fluctuations of the order parameter of seismicity were minimized around 17 February 2015. No SES has been reported for this undersea ultra-deep EQ. We emphasize that this is the deepest EQ ever detected (https://www.nationalgeographic.com/science/article/ deepest-earthquake-ever-detected-struck-467-miles-beneath-japan). In addition, in Ref. [45] it was noted that globally, this is the deepest (680 km centroid depth) event

12

1 The Order Parameter Fluctuations of Seismicity Are Minimized …

with MW > 7.8 in the seismological records and that, in general, deep focus EQs located in high pressure conditions 300 to 700 km below Earth’s surface within sinking slabs of relatively cold oceanic lithosphere, are mysterious phenomena. Hence, the identification of the minimum of the order parameter of seismicity makes it more challenging especially because its lead time (∼ 3 21 months) does not differ significantly than that (1–3 months) of the shallow EQs in the Japanese area. We now turn to a few details of this minimum. 148 In Fig. 1.7 we plot, for the area N46 25 E125 , the variability βW for W = 200 (red) and 300 (blue) events versus the conventional time. In Fig. 1.7a, we present the results for the whole eight year period from 1 January 2011 until the end of 2018, while in Fig. 1.7b and in order to better visualize what happened close to the Ogasawara EQ, an excerpt of fourteen month duration, i.e., from 1 May 2014 until 1 July 2015, is depicted. A careful inspection of the latter figure reveals that a broad βW minimum is observed, for both natural time window lengths W = 200 and 300 events, around 17 February 2015 shown by a green arrow head preceding the Ogasawara EQ almost by 3 21 months. (This date, quite interestingly, is compatible with what was mentioned by Gardonio et al. [6] that an accelerating preseismic phase started almost 3 months prior to the mainshock.) The ratio β300,min /β200,min actually lies in the range 0.95 to 1.08 and in addition β200,min ≤ 0.295 as found in Ref. [28] for the precursory βW minima 1–3 months before all shallow EQs of magnitude 7.6 or larger in Japan during the period from 1 January 1984 until the M9 Tohoku EQ in 2011. Practically the same results—as far as the existence of the aforementioned precursory βW broad minimum is concerned—are obtained, see Fig. 1.8, if we repeat the above βW computation for 146 the smaller area N46 25 E125 . The Ogasawara EQ has not been followed yet by an appreciably stronger EQ in contrast to the M7.8 Chichi-jima shallow EQ which occurred also at Bonin islands at 27.05◦ N 143.94◦ E almost three months before the super-giant M9 Tohoku EQ on 11 March 2011. This could be understood in the following context [42]: Upon the occurrence of the Chichi-jima EQ the following facts have been observed that will be discussed in detail in later Chapters: First, according to Ref. [38] the complexity measures △2000 , △3000 and △4000 , i.e., the △i values at the natural time window lengths (scales) i = 2000, 3000 and 4000 events, respectively, show in Fig. 7 of Ref. [38] a strong abrupt increase △△i on 22 December 2010 and after the EQ occurrence exhibit a scaling behavior of the form △△i = A(t − t0 )c where the exponent c is independent of i with a value very close to 1/3, while the pre-factors A are proportional to i (see Fig. 3 of Ref. [38]) and t0 is approximately 0.2 days after the M7.8 EQ occurrence. This conforms to the seminal work by Lifshitz and Slyozov [13] and independently by Wagner [44] (LSW) on phase transitions which shows that the characteristic size of the minority phase droplets exhibits a scaling behavior in which time growth varies with time as t 1/3 (a behavior reminiscent of Eqs. (100.14) and (100.23) in the chapter entitled “Kinetics of Phase Transitions” of Vol. 10 of Landau and Lifshitz Course of Theoretical Physics [22]). Second, the order parameter fluctuations exhibited a unique change [39], i.e., an increase △βW which exhibits a functional form consistent with the LSW theory and the subsequent work of Penrose et al. [21] (as also discussed in detail in Chap. 6). In particular, △βW obeys

1.5 Minima of the Variability β Before Other Major EQs in Japan

5

9

104

8

[λ(t)-μ]/K0

10

7

103

6

102

MJMA

9 1.6 W=200 8.8 1.4 W=300 8.6 1.2 1 8.4 0.8 8.2 0.6 8 0.4 7.8 0.2 7.6 0 01 Jan 01 Jan 01 Jan 01 Jan 01 Jan 01 Jan 01 Jan 01 Jan 01 Jan 2011 2012 2013 2014 2015 2016 2017 2018 2019

MJMA

βW

(a)

13

5

1

1.6 W=200 1.4 W=300 1.2 1 0.8 0.6 0.4 0.2 0 01 May 01 Jul 2014 2014

01 Sep 2014

01 Nov 2014

01 Jan 01 Mar 01 May 2015 2015 2015

9 8.8 8.6 8.4 8.2 8 7.8 7.6 01 Jul 2015

105

9

10

4

8

10

3

10

2

7 6 5

1

10 01 May 2014

MJMA

[λ(t)-μ]/K0

βW

(b)

MJMA

4 10 01 Jan 01 Jan 01 Jan 01 Jan 01 Jan 01 Jan 01 Jan 01 Jan 01 Jan 2013 2012 2011 2017 2015 2018 2016 2014 2019

01 Jul 2014

01 Sep 2014

01 Nov 2014

01 Jan 2015

01 Mar 2015

01 May 2015

4 01 Jul 2015

Fig. 1.7 The variability βW for W = 200 (red) and 300 (blue) versus the conventional time for the 148 area N46 25 E125 since 1 January 1984 until 1 January 2019 (a), or since 1 May 2014 until 1 July 2015 (b). In the lower panel of (a) and (b), we plot the time dependent seismicity rate (λ(t) − μ)/K 0 = ∑ p ti 90%) discrimination [25, 43]. In Ref. [10] the GCMT catalog for the period 1 January 1976 until 31 January 2022 was analysed. The authors employed natural time analysis of the order parameter of seismicity in order to identify the fluctuation minima that are precursory to EQs of M ≥ 8.5, detrended fluctuation analysis for the identification of long range correlations in the magnitude time series at the time of the minimum, and plot, at that time, the average EQ potential score maps for providing information about the epicenter location. The results show [10] that with statistical significance of the order of 10−5 , the time of occurrence of the strongest M ≥ 8.5 EQs can be determined with a maximum lead time of nine months with outstanding discrimination, while their epicenters lie in a region covering 4% of the total studied area.

284

12 Study of the Global Seismicity Using Natural Time Analysis

12.15 Conclusions We have shown how natural time analysis may provide evidence that EQs of M ≥ 7.0 are globally correlated (see §12.11). We have also shown in the §12.12 that global seismicity can be decomposed—when combining natural time, EMD (see §12.6), and Hurst analysis (see §12.7)—into three component time series, the micro-, mid-, and macro-scale time series which have unique spectral characteristics (see §12.12.2). Out of the three, the mid-scale time series is of profound importance as discussed in §12.12.3.

Finally, the minima of the fluctuations (see §12.2) of the order parameter of seismicity in global scale have been analyzed in §12.13 and §12.14, in conjunction with the conditions (12.21), (12.22), and (12.23) complemented by an estimation of the future EQ epicenter location. They lead to an EQ prediction method that may identify globally all M ≥ 8.5 EQs with a maximum lead time of 9 months and a false alarm rate of 3.64% within a region covering only 4% of the total area studied.

References 1. Ahn, S., Fessler, A.: Standard errors of mean, variance, and standard deviation estimators. Technical Report. Comm. and Sign. Proc. Lab., Department of EECS, University of Michigan, Ann Arbor, MI, USA, July 2003. http://web.eecs.umich.edu/~fessler/papers/lists/files/ tr/stderr.pdf 2. Amante, C., Eakins, B.W.: ETOPO1 1 arc-minute global relief model: procedures, data sources and analysis. NOAA Technical Memorandum NESDIS NGDC-24. National Geophysical Data Center, Marine Geology and Geophysics Division, Boulder, Colorado (2009). https://doi.org/ 10.7289/V5C8276M 3. Båth, M.: Lateral inhomogeneities of the upper mantle. Tectonophysics 2, 483–514 (1965). https://doi.org/10.1016/0040-1951(65)90003-X 4. Bernardi, A., Fraser-Smith, A.C., McGill, P.R., Villard, O.G.: ULF magnetic field measurements near the epicenter of the Ms 7.1 Loma Prieta earthquake. Phys. Earth Planet. Int. 68, 45–63 (1991). https://doi.org/10.1016/0031-9201(91)90006-4 5. Bowman, D.C., Lees, J.M.: The Hilbert-Huang transform: a high resolution spectral method for nonlinear and nonstationary time series. Seismol. Res. Lett. 84(6), 1074–1080 (2013). https://doi.org/10.1785/0220130025

12.4 Results from the Combination of βmin , Detrended Fluctuation …

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99. Varotsos, P., Alexopoulos, K.: Physical properties of the variations of the electric field of the earth preceding earthquakes. I. Tectonophysics 110, 73–98 (1984). https://doi.org/10.1016/ 0040-1951(84)90059-3 100. Varotsos, P., Alexopoulos, K., Lazaridou, M.: Latest aspects of earthquake prediction in Greece based on seismic electric signals. II. Tectonophysics 224, 1–37 (1993). https://doi.org/10. 1016/0040-1951(93)90055-O 101. Varotsos, P., Alexopoulos, K., Nomicos, K., Lazaridou, M.: Earthquake prediction and electric signals. Nature (London) 322, 120 (1986). https://doi.org/10.1038/322120a0 102. Varotsos, P., Eftaxias, K., Lazaridou, M., Nomicos, K., Sarlis, N., Bogris, N., Makris, J., Antonopoulos, G., Kopanas, J.: Recent earthquake prediction results in Greece based on the observation of seismic electric signals. Acta Geophys. Pol. 44, 301–327 (1996) 103. Varotsos, P., Lazaridou, M.: Latest aspects of earthquake prediction in Greece based on seismic electric signals. Tectonophysics 188, 321–347 (1991). https://doi.org/10.1016/00401951(91)90462-2 104. Varotsos, P., Sarlis, N., Skordas, E.: Scale-specific order parameter fluctuations of seismicity before mainshocks: natural time and detrended fluctuation analysis. EPL 99, 59001 (2012). https://doi.org/10.1209/0295-5075/99/59001 105. Varotsos, P.A., Sarlis, N.V., Skordas, E.S.: Spatio-temporal complexity aspects on the interrelation between seismic electric signals and seismicity. Pract. Athens Acad. 76, 294–321 (2001) 106. Varotsos, P.A., Sarlis, N.V., Skordas, E.S.: Electric fields that “arrive” before the time derivative of the magnetic field prior to major earthquakes. Phys. Rev. Lett. 91, 148501 (2003). https://doi.org/10.1103/PhysRevLett.91.148501 107. Varotsos, P.A., Sarlis, N.V., Skordas, E.S.: Detrended fluctuation analysis of the magnetic and electric field variations that precede rupture. Chaos 19, 023114 (2009). https://doi.org/ 10.1063/1.3130931 108. Varotsos, P.A., Sarlis, N.V., Skordas, E.S.: Natural time analysis: the new view of time. Precursory Seismic Electric Signals, Earthquakes and other Complex Time-Series. Springer, Berlin (2011). https://doi.org/10.1007/978-3-642-16449-1 109. Varotsos, P.A., Sarlis, N.V., Skordas, E.S.: Scale-specific order parameter fluctuations of seismicity in natural time before mainshocks. EPL 96, 59002 (2011). https://doi.org/10.1209/ 0295-5075/96/59002 110. Varotsos, P.A., Sarlis, N.V., Skordas, E.S.: Order parameter fluctuations in natural time and bvalue variation before large earthquakes. Nat. Hazards Earth Syst. Sci. 12, 3473–3481 (2012). https://doi.org/10.5194/nhess-12-3473-2012 111. Varotsos, P.A., Sarlis, N.V., Skordas, E.S.: Study of the temporal correlations in the magnitude time series before major earthquakes in Japan. J. Geophys. Res.: Space Phys. 119, 9192–9206 (2014). https://doi.org/10.1002/2014JA020580 112. Varotsos, P.A., Sarlis, N.V., Skordas, E.S.: Phenomena preceding major earthquakes interconnected through a physical model. Ann. Geophys. 37(3), 315–324 (2019). https://doi.org/ 10.5194/angeo-37-315-2019 113. Varotsos, P.A., Sarlis, N.V., Skordas, E.S.: Order parameter and entropy of seismicity in natural time before major earthquakes: recent results. Geosciences 12(6), 225 (2022). https:// doi.org/10.3390/geosciences12060225 114. Varotsos, P.A., Sarlis, N.V., Skordas, E.S., Lazaridou, M.S.: Fluctuations, under time reversal, of the natural time and the entropy distinguish similar looking electric signals of different dynamics. J. Appl. Phys. 103, 014906 (2008). https://doi.org/10.1063/1.2827363 115. Varotsos, P.A., Sarlis, N.V., Skordas, E.S., Lazaridou, M.S.: Seismic electric signals: an additional fact showing their physical interconnection with seismicity. Tectonophysics 589, 116– 125 (2013). https://doi.org/10.1016/j.tecto.2012.12.020 116. Varotsos, P.A., Sarlis, N.V., Skordas, E.S., Tanaka, H.K., Lazaridou, M.S.: Attempt to distinguish long-range temporal correlations from the statistics of the increments by natural time analysis. Phys. Rev. E 74, 021123 (2006). https://doi.org/10.1103/physreve.74.021123

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117. Varotsos, P.A., Sarlis, N.V., Skordas, E.S., Uyeda, S., Kamogawa, M.: Natural time analysis of critical phenomena. the case of seismicity. EPL 92, 29002 (2010). https://doi.org/10.1209/ 0295-5075/92/29002 118. Varotsos, P.A., Sarlis, N.V., Skordas, E.S., Uyeda, S., Kamogawa, M.: Natural time analysis of critical phenomena. Proc. Natl. Acad. Sci. U.S.A. 108, 11361–11364 (2011). https://doi. org/10.1073/pnas.1108138108 119. Varotsos, P.A., Sarlis, N.V., Tanaka, H.K., Skordas, E.S.: Similarity of fluctuations in correlated systems: the case of seismicity. Phys. Rev. E 72, 041103 (2005). https://doi.org/10. 1103/physreve.72.041103 120. Wang, Y.H., Yeh, C.H., Young, H.W.V., Hu, K., Lo, M.T.: On the computational complexity of the empirical mode decomposition algorithm. Phys. A 400, 159–167 (2014). https://doi. org/10.1016/j.physa.2014.01.020 121. Wessel, P., Luis, J.F., Uieda, L., Scharroo, R., Wobbe, F., Smith, W.H.F., Tian, D.: The generic mapping tools version 6. Geochem. Geophys. Geosyst. 20(11), 5556–5564 (2019). https:// doi.org/10.1029/2019GC008515 122. Wu, H., Huangs, N.E.: Ensemble empirical mode decomposition: a noise-assisted data analysis method. Adv. Adapt. Data Anal. 01, 1–41 (2009). https://doi.org/10.1142/ S1793536909000047 123. Xie, H., Wang, Z.: Mean frequency derived via Hilbert-Huang transform with application to fatigue EMG signal analysis. Comput. Methods Programs Biomed. 82, 114–120 (2006). https://doi.org/10.1016/j.cmpb.2006.02.009 124. Xu, G., Han, P., Huang, Q., Hattori, K., Febriani, F., Yamaguchi, H.: Anomalous behaviors of geomagnetic diurnal variations prior to the 2011 off the Pacific coast of Tohoku earthquake (Mw9.0). J. Asian Earth Sci. 77, 59–65 (2013). https://doi.org/10.1016/j.jseaes.2013.08.011 125. Yang, J.N., Lei, Y., Lin, S., Huang, N.: Hilbert-Huang based approach for structural damage detection. J. Eng. Mech. 130(1), 85–95 (2004). https://doi.org/10.1061/(ASCE)07339399(2004)130:1(85) 126. Zaliapin, I., Gabrielov, A., Keilis-Borok, V., Wong, H.: Clustering analysis of seismicity and aftershock identification. Phys. Rev. Lett. 101, 018501 (2008). https://doi.org/10.1103/ PhysRevLett.101.018501

Chapter 13

Applications of Natural Time Analysis to Disaster Prediction in Other Disciplines

13.1 Introduction Here, we will review various recent applications of natural time analysis in diverse fields which are mostly directed to disaster prediction. More specifically, in §13.2 the natural time analysis of the preseismic fractureinduced electromagnetic emissions in the MHz range will be discussed, while §13.3 and §13.4 present the natural time analysis of the preseismic subionospheric Very Low Frequency (VLF) propagation anomalies and the preseismic Ultra Low Frequency (ULF) magnetic field variations, respectively. The natural time analysis of the surface deformations before the 2016 Kumamoto earthquakes (EQs) is summarized in §13.5. Section §13.6 demonstrates an algorithm for the prediction of the occurrence time of strong aftershocks on the basis of natural time analysis. The combination of multiresolution wavelets and natural time analysis for the estimation of the occurrence time of an impending strong EQ is the subject of §13.7. The very recent application of nowcasting volcanic eruptions is presented in §13.8. Results from laboratory studies of acoustic emission are exposed in §13.9 and §13.10. The natural time analysis of the fluctuations of the electrical resistance before fracture are summarized in §13.11. Sections §13.12 and §13.13 demonstrate natural time analysis applications in atmospheric physics, namely in the ozone hole dynamics over Antartica and the prediction of major El Niño Southern Oscillation events, respectively. Finally, the recent introduction of natural time in the analysis of photoplethysmography signals for the study of heart rate variability is also discussed in §13.14. Since many of the applications presented in the present Chapter focus on the identification of the approach to criticality in various time series of physical observables, the following background information is recapitulated for readers’ convenience: For a time series comprising of N events of type A (each one of which is identified define the natural time for the occurrence [139, 145] by means of a threshold Ath ), we ∑ N Q n is the normalized “energy” of of the k-th event χk = k/N and pk = Q k / n=1 the k-th event. The quantity Q k is proportional to the energy emitted during the k-th © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 P. Varotsos et al., Natural Time Analysis: The New View of Time, Part II, https://doi.org/10.1007/978-3-031-26006-3_13

293

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event. The behavior of the normalized power spectrum [140–142] ∏(ω) ≡ |Φ(ω)|2

(13.1)

defined by Φ(ω) =

N ∑

pk exp(i ωχk )

(13.2)

k=1

where ω stands for the angular frequency, is studied at ω close to zero for capturing the dynamic evolution. This so because all the moments of the distribution of the pk can be estimated from Φ(ω) at ω → 0 (see p. 499 of Ref. [25]). For this purpose, a quantity κ1 is defined from the Taylor expansion ∏(ω) = 1 − κ1 ω2 + κ2 ω4 + . . .. The relation for the critical state that has been shown for Seismic Electric Signal (SES) activities [140, 145]: ∏(ω) = for ω → 0, simplifies to

6 cos ω 12 sin ω 18 − − . 5ω2 5ω3 5ω2

∏(ω) ≈ 1 − 0.07ω2

(13.3)

(13.4)

which shows that the second-order Taylor expansion coefficient of ∏(ω), i.e., κ1 , is equal to 0.070, see Fig. 13.1. Equation (13.4) has been shown to hold also for EQ models like the Burridge-Knopoff model [109, 154] and the Olami Feder Christensen model [69, 145] when they approach the critical point (cf. for other critical systems where the condition κ1 = 0.070 is valid see Ref. [155]; see also p. 343 Ref. [145]). The entropy S in natural time is given by [140, 143, 150] S = ⟨χ ln χ⟩ − ⟨χ⟩ ln⟨χ⟩,

(13.5)

 ∑  N . . . pk denote averages with respect to the diswhere the brackets ⟨. . .⟩ ≡ k=1 ∑N tribution pk (= Q k / n=1 Q n ). The entropy S is a dynamic entropy that exhibits [157] positivity, concavity and Lesche [49, 50] experimental stability. When Q k are independent and identically distributed random variables, S reaches [150] the value Su ≡ ln22 − 41 ≈ 0.0966 that corresponds to the “uniform” distribution. Moreover, upon reversing the time arrow and hence applying time reversal T , i.e., T pk = p N −k+1 , the value of S changes to a value S− , see Fig. 13.2. Hence, the entropy S in natural time does satisfy the condition to be “causal”. Since the concept of entropy is equally applicable to deterministic as well as stochastic processes, the natural time entropies S and S− may provide a useful tool [5–7, 103, 105, 145, 150–152] for the analysis of physiological time series (cf. the latter in most cases are due to processes involving both stochastic and deterministic components).

13.1 Introduction

295 1

critical κ1 < 0.070 κ1 > 0.070

0.9 0.8

Π (ω)

0.7 0.6 0.5 0.4 0.3

0

π/2

π/4

ω

3π/4

π

Fig. 13.1 Schematic diagram showing the normalized power spectrum ∏(ω) in natural time for ω ∈ [0, π ]. The red curve is ∏(ω) of Eq. (13.3) which holds for the critical stage (κ1 = 0.070), whereas the two other curves are for κ1 > 0.070 (blue) and κ1 < 0.070 (green). The red arrow indicates how ∏(ω) approaches the critical one from below which is the second criterion that should be fulfilled for a true coincidence of the evolving seismicity in the candidate epicentral area with the critical state. Taken from Ref. [28]. This figure is a slightly different version of Fig. 5.1

It is worthwhile to mention that natural time entropy S has a particularly simple physical meaning since it may capture small trends existing in Q k . For example, when considering a small increasing trend ∈(> 0) for pk versus k by studying the parametric family p(χ ; ∈) ≡ 1 + ∈ (χ − 1/2) of continuous distributions for pk , one can show [152] -using the definition of Eq. (13.5)—that 1 S(∈) ≡

⎡ 1 ⎤ ⎡ 1 ⎤   p(χ ; ∈)χ ln χ dχ − ⎣ p(χ ; ∈)χ dχ ⎦ ln ⎣ p(χ ; ∈)χ dχ ⎦

0

1 ∈ =− + − 4 72



0

 1 ∈ ∈ 1 + + ln . 2 12 2 12

0

(13.6)

Expanding the last term of Eq. (13.6) around ∈ = 0, we obtain that  S(∈) = Su +

6 ln 2 − 5 ∈ + O(∈ 2 ). 72

(13.7)

Since for this family of continuous distributions S− (∈) simply equals S(−∈), we observe that an increasing trend in p(χ ; ∈), i.e., ∈ > 0, corresponds to S− (∈) values higher than S(∈). This result indicates that if we study the entropy change in natural time under time reversal:

296

13 Applications of Natural Time Analysis to Disaster Prediction in Other Disciplines 0.16

100

S SQk

0.14

80

0.12

60

0.1

40

0.08

20

0

0.06 0

10

20

30

40

50

60

70

80

N 0.16

100

S SQk

0.14

80

0.12

60

0.1

40

0.08

20

0

0.06 0

10

20

30

40

50

60

70

80

N Fig. 13.2 Important properties of the entropy in natural time S and the entropy in natural time under time reversal S− : In both panels, a signal consisting of 84 pulses is analyzed in natural time, the green and blue lines indicate the values (left scale) of S (green) and S− (blue) obtained for N = 10, 11, 12, . . . , 84. The signal is composed from 80 pulses of equal energy and 4 pulses which are ten times stronger (right scale, arbitrary units). In the upper panel, the stronger pulses are emitted periodically, while in the lower panel consecutively in the middle of the process. Although, the Shannon entropies for both panels are equal, the entropies in natural time differ essentially. Moreover, when using the entropy in natural time under time reversal S− , we obtain values which are in general different from those of S. Taken from Ref. [100]

13.1 Introduction

297

△S ≡ S − S−

(13.8)

increasing or decreasing trends transform to negative or positive values of △S, respectively. Finally, since S(∈) in Eq. (13.6) is a nonlinear function of ∈ we observe that the entropy change △S under time reversal is a nonlinear tool capturing alternations in the dynamics of the complex system, see also Chap. 3 of Ref. [145]. Using S and △S within a specified natural time window of length l, one can study time series resulting from complex systems by constructing the corresponding time series of S and △S obtained upon estimating these two quantities every l events as the natural time window of length l is sliding through the time series. This inspired the introduction of complexity measures that quantify the fluctuations of the entropy S and of the quantity △S upon changing the length scale as well as the extent to which these quantities are affected when randomly shuffling the consecutive events [145]. In most of the applications presented in this Chapter, the criticality has been identified by securing a true coincidence of the analyzed time series with that of critical state in natural time (recall that in such an analysis a threshold Ath is used for the identification of the individual events [139]). The criteria for such a coincidence are given explicitly in §5.3, but in view of their frequent use in this Chapter for readers’ convenience are also recapitulated below: 1. The “average” distance ⟨D⟩ between the curves of ∏(ω) of the evolving seismicity and Eq. (13.3) for ω ≤ π should be ⟨D⟩ < 10−2 , for details, see Refs. [140, 145]. 2. The final approach of the evolving ∏(ω) to that of Eq. (13.3) must be from below, see Fig. 13.1. This reflects that κ1 gradually decreases with time before strong EQs finally approaching from above that of the critical state, i.e., (13.9) κ1 = 0.070. 3. At the coincidence, both entropies S and S− in natural time must be smaller than Su , i.e., (13.10) S, S− < Su .

4. Since this process (critical dynamics) is considered to be self-similar, the occurrence time of the true coincidence should not markedly vary upon changing the threshold Ath used in the natural time analysis of the original time series.

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In addition, we mention that the second criterion for a true coincidence starts to be fulfilled when the entropy change △S exhibits a (local) minimum (this is the fifth criterion mentioned in §5.3 when the seismicity in the candidate epicentral area approaches the critical point, i.e., the mainshock occurrence).

13.2 Fracture-induced Electromagnetic Emissions Before EQs Fracture-induced (or fracto-) electromagnetic emissions (EME) have been repeatedly recorded [83] before strong EQs in Greece. For example, the cases before the strong EQs of Grevena-Kozani (MW = 6.6) on 13 May 1995, Athens (MW = 6.0) on 7 September 1999, and of the two western Greece (close to Methoni) (MW = 6.9 and MW = 6.5) EQs on 14 February 2008 were presented in Ref. [85]. Another recent example, is the fracto-EME before the 12 June 2017 Lesvos (MW = 6.3) EQ [87]. The fracto-EME lie in the range of MHz and are collected by customdesigned receivers and λ/2 electric dipole antennas [83]. In short, a network of ground-based observatories called ELSEM-Net (hELlenic Seismo-ElectroMagnetics Network), recording electromagnetic emissions has been operating in Greece since 1994, see the Supplementary Material of Ref. [83]: In the exemplary station located at Zakynthos (Zante) island in Greece, three vertical λ/2 electric dipole antennas operate for the detection of the electric field variations in the bands of 41, 54 and 135 MHz, while at the other observatories the electric field variations at 41 and 46 MHz are monitored. All the time series are sampled with a sampling frequency Fs = 1Hz. In order to apply natural time analysis in such time series, the notion of an EME event has to be defined. For this purpose, Potirakis, Karadimitrakis and Eftaxias [85] introduced the following procedure: Let us consider an EME time series of amplitude A(ti ). One first selects a threshold amplitude Ath , which is indicative of the maximum amplitude value of the background noise, and examines the difference A(ti ) − Ath . If m consecutive values (samples) of A(ti ) exceed the threshold Ath , say from i = l to i = l + m − 1, then they are considered to belong to the same EME event. The energy∑ of this EME event, which is assumed that it occurred k-th in order, is given by |A(ti ) − Ath |2 . Natural time analysis can be now applied [81–83, 85, Q k = l+m−1 i=l 87] and criticality is identified by means of the validity of Eq. (13.9) and conditions (13.10) for a wide range of values of the threshold Ath , as in the case of long-duration SES activities [144] (see also Sect. 4.11 of Ref. [145]). Finally, we note that the criticality of fracto-EME before the MW 6.4 EQ on 12 October, 2013, in the South West segment of the Hellenic Arc has been shown in Ref. [81] while that of the fracto-EME before the Cephalonia MW 6.0 EQ on 26 January 2014 in Ref. [82].

13.3 Subionospheric Very Low Frequency Propagation Anomalies Before EQs

299

13.2.1 Laboratory Experiments on LiF As mentioned in Ref. [114], lithium fluoride (LiF) is a unique crystal possessing the largest reported bandgap of any material and is predicted to remain transparent to visible light under stresses in excess of 1000 GPa [29]. In Ref. [79], EME were recorded (simultaneously with acoustic emission) during fracture experiments in non-irradiated and irradiated by gamma rays LiF. Natural time analysis has revealed criticality, which was secured by the criteria mentioned in §13.1, in the EME before fracture for various thresholds. The same holds [79] for the acoustic emissions indicating that critical behavior naturally arises during the fracture process. The experiments showed that the EME criticality is observed later than that of the acoustic emissions and just before the global fracture for both the non-irradiated and irradiated samples.

13.3 Subionospheric Very Low Frequency Propagation Anomalies Before EQs For the detection [83] of the subionospheric VLF propagation anomalies observed before EQs, custom designed VLF/LF receivers (usually employing simple electrical rod -monopole- antennas) are installed at stations distributed [80] all around Japan (see Fig. 13.3). These stations monitor the signal transmitted (at discrete frequencies) from specific transmitters located both in Japan and abroad and the receiver amplitude A(t) is continuously recorded at a sampling frequency Fs = 1Hz. From these recordings, only the (local) nighttime data have been used, which, depending on the time periods around the year, correspond to [80]: 10:00–20:00 UTC for 22/11–21/02, 11:00–19:00 UTC for 22/02–21/05, 11:30–17:30 UTC for 22/05–21/09, and 10:30–19:00 UTC for 22/09–21/11. From the time series A(ti ) of the raw measurements of the amplitude, one first calculates [80] the residue d A(ti ) between the received signal amplitude A(ti ) and an average signal amplitude ⟨A(t)⟩ calculated by means of a running average over ±15 days as d A(ti ) = A(ti ) − ⟨A(t)⟩. Then, only the (local) nighttime data are used to calculate daily values (1 value/day) for the following three quantities TR (‘trend’), DP (‘dispersion’), and NF (‘nighttime fluctuation’): ∑ Ne i=Ns d A(ti ) , (13.11) TR = Ne − Ns /∑ DP =

Ne i=Ns

[d A(ti ) − TR]2 Ne − Ns

,

(13.12)

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13 Applications of Natural Time Analysis to Disaster Prediction in Other Disciplines

Fig. 13.3 Map of the wider area of Japan, showing the network of VLF/LF receivers (triangles), the VLF transmitters (rectangles), the 5-th Fresnel zones for the JJI-NSB, JJI-STU and JJI-KTU paths, as well as all MW > 5.5 EQs, which happened during the time period from 1 January 2016 to 30 April 2016. The circle size is proportional to EQ magnitude and its color refers to the hypocenter depth. The occurrence date appears only for the EQs with hypocenter at depth 0.3, see Fig. 13.7a. The 14 operating points in this case lead to an average hit rate 61% and an average false alarm rate 16%. Finally, we comment on the recent application [113] of this method [16] in the case of the 2019 Ridgecrest EQs. In this case, Fig. 13.8 depicts the “aftershock” area that corresponds to the M6.4 Ridgecrest EQ on 4 July 2019 together with its

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13 Applications of Natural Time Analysis to Disaster Prediction in Other Disciplines

Fig. 13.8 The small EQs with M ≥ 2.0 that occurred after the 4 July 2019 M6.4 EQ (blue star), and before the M7.1 Ridgecrest EQ (magenta star) within a 0.4◦ × 0.4◦ rectangle (red dashed lines) centered at the first event (following the rules of selecting the size of the aftershock set in Ref. [16] and discussed in the text). Taken from Ref. [113]

−118˚00'

−117˚30'

−117˚00'

36˚00'

36˚00'

35˚30'

35˚30'

−118˚00'

−117˚30'

−117˚00'

“aftershocks” that occurred there until the M7.1 Ridgecrest EQ on 6 July 2019 (cf. it has been finally clarified that the M6.4 EQ on 4 July 2019 was a foreshock of the M7.1 mainshock on 6 July 2019, see §9.5). The construction of the ek time series is very simple because only two EQs that preceded the M7.1 EQ have exhibited “extreme” magnitudes as can be seen in Fig. 3c of Ref. [113]: the M5.36 EQ at 07:53 UTC on 5 July 2019, and the M4.97 EQ at 03:16 UTC on 6 July 2019. Hence ek = 2 before the M7.1 Ridgecrest EQ that occurred at 03:20 UTC. This result combined with the fact that seismicity has reached [113] criticality at 22:41 UTC on 2 July 2019 lead us to the conclusion that the occurrence time of the M7.1 Ridgecrest EQ might have been anticipated well in advance, while its epicenter could have been determined on the basis of the M6.4 Ridgecrest EQ epicenter on 4 July 2019 (see Fig. 13.8), see also §9.5.

The anticipation well in advance is further strengthened by the following comments based on the content of Chap. 6 as well as on Ref. [149]. Upon increasing the scale i (number of events), it is observed (see Figs. 2B and 4E of Ref. [108]) that the increase △βi of the βi fluctuation, upon the occurrence of the M7.8 EQ on 22 December 2010, becomes distinctly larger which does not happen (see Figs. 4A–D of Ref. [108]) for the increases of the β fluctuations upon the occurrences of all other shallow EQs in Japan of magnitude 7.6 or larger during the period from 1 January 1984 to the time of the M9 Tohoku EQ. Such a behavior that obeys the interrelation

13.7 Combination of Multiresolution Wavelets and Natural …

311

△βi = 0.5 ln(i /114.3), see Fig. 2g, h of Varotsos et al. [148], has a functional form strikingly reminiscent of the one discussed by Penrose et al. [75] in computer simulations of phase separation kinetics using the ideas of Lifshitz and Slyozov [52], see their Eq. 33 which is also due to Lifshitz and Slyozov. Remarkably, after the publication of Ref. [148], an increase of similar form, i.e. △βi ≈ 0.5 ln(i /68), was also observed (see Fig. 6.4) upon the occurrence of the M6.4 EQ almost 34 h before the M7.1 Ridgecrest EQ in California that struck on 6 July 2019 at 03:20 UTC. The β fluctuation on 22 December 2010 accompanying the minimum △Smin is unique as it was also shown by employing the event coincidence analysis [21], which considers a time lag τ and a window △T (> 0) between the precursor and the event to be predicted, which demonstrated the profound statistical significance of this unique result that has been further assured in Ref. [148].

13.7 Combination of Multiresolution Wavelets and Natural Time Analysis Before Earthquakes In Refs. [125–127], a procedure based on natural time analysis has been suggested for estimating the occurrence time of a strong EQ, which, however, is applied in retrospect, i.e., the epicenter is already known. Specifically, at the first stage the evolution of possible preseismic patterns in the area of a future strong EQ is investigated by applying the method of multiresolution wavelet analysis (MRWA) [127]. This consists of the analysis of time intervals between successive seismic events in various seismic catalogs that employ different magnitude thresholds Mthr es in various areas around the future strong EQ epicenter; for example see Table 1 of Ref. [127]. This analysis is applied retrospect hence the epicenter is already known, otherwise the future epicenter has to be estimated by other means, see, e.g., Chap. 3 or on the basis of a recorded SES activity by using the selectivity map if it is available from the previous geolectric data. The time series of the interevent times δti is then analysed by the discrete wavelet transform [125–127] wav Wm,n =

L 1 ∑

2m/2

i=1

 δti ψ

i − n , 2m

(13.18)

where m is the scale variable, n the translation variable, L the total number of interevent times under study, and ψ the analyzing wavelet (e.g., Haar, db2, db3, sym2, sym3, coif2). The standard deviation of wavelet coefficients σwav,m as a function of scale for the smaller scales (m ≤ 4) are computed for moving windows comprising of some number of events along the time series and are plotted versus conventional time as we approach to the strong EQ, see, e.g., Fig. 13.9 (see also Figs. 5 and 6 of Ref. [125] or Fig. 5 of Ref. [126]). A significant temporal variability in the strength of the multiscale properties of the interevent times can be seen in such figures. However, there always [125–127]

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13 Applications of Natural Time Analysis to Disaster Prediction in Other Disciplines

Fig. 13.9 Time variation of σwav,m with scale m ranging from 1 up to 4, for moving windows with length of 16 events and a shift of 3 events for the interevent times of EQs with magnitudes larger than or equal to the threshold Mthres = 3.0 within a circular area of radius R = 25 km (SC5) or R = 50 km (SC6) around the epicenter of the central Crete (MW 6.0) EQ on 27 September 2021. The red vertical lines indicate the day of minimum in variance, observed at each scale. Taken from Ref. [127]

13.8 Nowcasting: Volcanic Eruptions, Arctic Low…

313

appear a significant decrease in the temporal evolution of σwav,m at lower scales. For example, for the results presented in Fig. 13.9 (as well as in Figs. 6 and 7 of Ref. [127]) a dominating candidate for the time marker of this decrease is 24 July 2021. Translating the results from lower scales in an alternative way, it was proposed [127] the use of the observed time marker of 24 July 2021, which appears several weeks before the strong EQ that took place on 27 September 2021, as the initiation point for the natural time analysis of the subsequent seismicity. This lead time is consistent with the fact that, the natural time analysis of EQ magnitude time series, has revealed[106, 138, 147] clear changes in the temporal correlations a few months before major EQs in California and Japan, as it was also discussed in Chap. 4. Using as the initiation point of the natural time analysis the time marker of the minimization of σwav,m at lower scales (by the same token as we did with βmin in Chap. 5), the seismicity for various magnitude thresholds and for various areas around the epicenter is studied and the criticality is determined by the criteria listed in §13.1. Such an analysis has revealed that the regional seismicity approached criticality for a prolonged period of approximately 40 days before the occurrence of the 27 September 2021 central Crete (MW 6.0) EQ [127], while this criticality is reached a few days before the occurrence of the 3 March 2021 Thessaly (MW 6.3) strong EQ [126] as well as a few days before the January-February 2014 Cephalonia (MW 6.1 and 6.0) EQs [125]. Finally, we note that by employing natural time analysis of seismicity and the criteria mentioned in §13.1 for various areas around the future epicenter, in Ref. [128] criticality was identified a few days before the 12 October 2013 (MW = 6.4) EQ in the south west segment of the Hellenic arc.

13.8 Nowcasting: Volcanic Eruptions, Arctic Low Temperatures, and Extreme Cosmic Ray Events As already mentioned in §11.3 and §12.4, EQ nowcasting (EN) was introduced by Rundle et al. [98] for estimating the seismic risk in fault systems by monitoring the progress of the EQ cycle on the basis of natural time. EN was applied to tectonic [15, 17, 59, 71–74, 76, 92–98, 104, 158] and induced [58, 60, 61] seismicity so far. In 2022, Fildes et al. [27] tested the nowcasting method for the first time in an active volcanic eruption and caldera collapse sequence. The subject of this study was the 2018 K¯ılauea volcano eruptive sequence (cf. K¯ılauea is the southeastern most volcano of the five volcanic systems comprising the island of Hawai’i, USA). From mid-May to early-August 62 collapse events occurred about every 1–2 days each one releasing seismic energy larger than or equal to an EQ of magnitude M ≥

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4.7. This was also the first application of nowcasting for such a short time interval (90 days compared to years in the tectonic and induced seismicity applications) during which the dynamic volcanic system changes the mechanisms and the potential drivers behind the EQ system. For these reasons, Fildes et al. [27] employed nowcasting over three different time-windows: (a)17 May–2 August 2018, (b)29 May–2 August 2018, and (c)14 June–2 August 2018, which correspond to times when other observations indicate a change in the eruption sequence, which could correspond to changes in the stress state. Two different areas: (A) within 5km of the summit caldera and (B) the whole island of Hawai’i have been also studied, see Ref. [26] for the corresponding EQ catalogs and how they were obtained from the United States Geological Survey EQ catalog. Instead of using the EQ potential score (EPS) approach of Eqs. (11.4) and (12.10), Fildes et al. [27] followed the method suggested by Luginbuhl et al. [58]: 1. A small Mσ and a large Mλ threshold was selected. Mσ = 2.5 was set due to magnitude completeness reasons, while Mλ = 4.7 for caldera collapse events or Mλ = 3.5. The latter threshold is the upper limit of the validity of the Gutenberg-Richter law for the area within 5 km from the center of the summit caldera, see, e.g., Fig. 3 of Ref. [27]. 2. The natural time relationship between Mσ and Mλ sized events was determined. This means an estimate has been provided on how the number Ncλ of events of magnitude larger than or equal to Mλ depends on the number Ncσ of events with magnitudes M ∈ [Mσ , Mλ ). For example, see the sixth column of Table 1 of Ref. [27]. 3. The conventional time t evolution of Ncσ (t) was found from the seismic catalog. 4. The nowcasted number Ncλ (t) of large events was plotted versus the conventional time t in view of the previous steps 2 and 3.

Fildes et al. [27] show that the nowcasting results obtained for the last timewindow c (14 June–2 August 2018) exhibit the strongest agreement between the actual and nowcasted collapse events. A result suggesting that once the eruption reaches this more stable stress state, then a successful nowcast can be made even if the Gutenberg-Richter assumption is not fully met.

There are, however, poor nowcasting results for the time-window that includes the earlier phases of the eruption (17 May–2 August 2018) leading to the conclusion that the technique produces limited “success” nowcasting 37 collapse events that agreed with the catalog of actual events.

13.9 Acoustic Emissions Before Fracture

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However, this should be seen as a promising result since, as stated by Fildes et al. [27], further investigations may help determine the necessity and strictness of the method assumptions and if there are adaptions that could be made for applying nowcasting to future EQ sequences associated with volcanic eruptions. We also note that the nowcasting method [58] has been also recently applied for nowcasting extreme low daily minimum air temperatures in the Arctic [132] as well as for nowcasting extreme cosmic ray events [133]. In particular, in Ref. [132] using the natural time concept, a new nowcasting tool of the Arctic stratospheric regime was developed, which showed a striking agreement between the real values of minimum stratospheric temperature and high stratospheric ozon depletion values with those derived from the nowcasting model.

Furthermore, Varotsos et al. [133] using natural time developed a new nowcasting method for extreme cosmic rays’ events that can cause several problems (e.g., in telecommunications, transportation, water supply). This tool calculates the duration of high intensity cosmic ray events as well as the average time interval between two successive extremes, depending on their intensity.

13.9 Acoustic Emissions Before Fracture In Ref. [124], acoustic emission (AE) before fracture was studied by analyzing in natural time an AE catalog collected in laboratory experiments on Etna basalt. The method presented in §12.11 was employed, which revealed the presence of magnitude correlations in the evolution of AE. Moreover, the scaled distribution of the order parameter κ1 of the AE exhibited [124] a characteristic feature similar to that of seismicity and other equilibrium or non-equilibrium critical systems including selforganized critical systems discussed in detail in Sect. 6.2.2 of Ref. [145] and Ref. [107], see also Refs. [99, 101, 146, 156] (cf. we will return to this subject later in this Section, see, e.g., Fig. 13.10). These results obtained by natural time analysis support the presence of a universal behavior of the fracturing processes from the laboratory to the global scale [124].

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Fig. 13.10 The scaled PDFs σ (κ1 ) P (y) versus y = (μ (κ1 ) − κ1 )/σ (κ1 ) obtained in natural time when analyzing either the avalanche sizes of the OFC and BK models (magenta, cyan, blue and brown lines) or the AE events of the marble (GDM1, GDM2, DENT, CSR) and cement (CEM1 and CEM2) specimens studied in Ref. [56] (points, see the legend). The thick red and green lines have been drawn as guide to the eye and correspond to the scaled PDFs of the order parameter of the 2D Ising model below the critical temperature at an inverse temperature β ≡ 1/(k B T ) = 0.4707 (cf. βc ≈ 0.4407, e.g., see Eq. (15.146) of Ref. [43]) for linear dimensions L = 128 and L = 256, respectively. Taken from Ref. [56]

In general, analysing in natural time AE signals recorded when various materials are driven to fracture, provides the means to examine how the specimens under intense mechanical load as dynamical systems, initially at equilibrium, gradually degenerate into a non-equilibrium one, as the applied mechanical load increases. In this view, impending fracture can be considered as a phase change. For this reason, natural time of AE has been previously studied in a variety of fracture experiments for various materials like marble [42, 48, 55, 57], LiF [79], steel [67], cement mortar [53, 54, 56], rods of Luserna stone [68] as well as structures of technological interest [30]. Consider a time series comprising N acoustic events [53]. For this case, the natural , see Fig. 13.11a. time of the k-th event of AE energy Ak is defined by χk = ∑k/N N An represents the Attention is focused on the pair (χk , pk ) where pk = Ak / n=1 normalized AE energy. The variance κ1 of the natural time χ weighted for pk is then given by Eq. (13.9). In general, the scaled Probability Density Function (PDF) of κ1 displays [54, 56, 107, 156] a behavior similar to the order parameter of many equilibrium and non-equilibrium systems [8–12, 18, 159, 162, 163]. In order to calculate the PDF of κ1 , an ensemble of κ1 values is needed. In the case of AE [54, 56], the procedure described in Section IV of Ref. [101] (see also Sect. 2.5.4 of Ref. [145]) was followed and a natural time window sliding, event by event, in the AE energy time series was applied: Starting from the first AE event, the

13.9 Acoustic Emissions Before Fracture

317

Fig. 13.11 Results obtained from the natural time analysis of the AE recorded during the threepoint bending loading protocol described in Ref. [55]. a The temporal evolution of the AE energies recorded (orange squares, left scale) in combination with the normalized applied load L/L f (%) (green line, right scale). b The temporal evolution of the parameters κ1 (red pluses), S (blue squares), S− (cyan squares) in combination with the normalized applied load L/L f (%), as in (a). See the legend for the values of the horizontal lines plotted. c The same as in (b) but versus natural time. The colored areas in (c) indicate when the natural time analysis results resemble those of the onedimensional or two-dimensional BTW model and are shown by yellow or blue shade, respectively. Taken from Ref. [55]

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κ1 values were calculated using a window of 6 consecutive events (including the first one). Next, the window increases by one AE event, and a new value of κ1 is calculated by taking into consideration, also the 7th event. The window keeps increasing until its length becomes 40. Then, the calculation is repeated starting from the second AE event, etc. After sliding, event by event, through the AE energy time series, the calculated κ1 values enable the construction of the PDF P (κ1 ). The scaled PDF is then defined as S(y) = σ (κ1 ) P (y) with y = [μ (κ1 ) − κ1 ]/σ (κ1 ).

(13.19)

It is the latter PDF that when studying long-term seismicity [89, 101, 145, 156] exhibits a left exponential tail similar to that obtained upon studying the order parameter fluctuations for several equilibrium systems, like the 2D Ising model [18, 162, 163], and non-equilibrium systems, such as 3D rice piles, magnetic vortices penetration into type II superconductors and other self-organised critical systems [107]. The identification of this left exponential tail is of prominent importance for impending strong EQs as discussed in §5.4.1.

Within this context and by using Eq. (13.9) and condition (13.10), Loukidis et al. [53, 54, 56] examined the possibility of determining the criticality in cement mortar and Dionysos marble (of the kind used in the restoration project of the Athenian Acropolis [46, 47]) specimens under severe mechanical load until their fracture. The results showed [53, 54, 56] that the variance κ1 of natural time may be of practical importance for identifying the entrance of mechanically loaded specimens in the stage of impending fracture. During different stages (regimes) of loading, the behavior of κ1 , S, and S− could be understood either by the Burridge-Knopoff (BK) train model [13] or the Olami-Feder-Christensen (OFC) EQ model [69] when they are analysed in natural time (see Sects. 8.2 and 8.3 of Ref. [145], respectively). Figure 13.10 shows that the AE results in the corresponding regimes are also compatible with those obtained by the BK and OFC model in terms of the scaled distribution of κ1 that exhibits the universal left exponential tail which is a common feature in equilibrium and non-equilibrium critical systems, as already mentioned. Recently, AE data recorded when notched fiber-reinforced concrete specimens were subjected to three-point bending until fracture, were analysed in natural time [55], see Fig. 13.11b, c. This analysis leads to κ1 , S, and S− values that are compatible with those obtained by a centrally fed Bak-Tang-Wiesenfeld (BTW) sandpile model [4], which was theoretically studied in natural time almost a decade ago (see Sect. 8.4.2 of Refs. [145] and [155]).

13.10 Acoustic Emissions in an Experiment in Plunged Granular Bed

319

Additionally, the identification of criticality in natural time analysis by means of Eq. (13.9) and the condition (13.10) of AE before fracture might be of significant technological interest in structural integrity and engineering: Kourkoulis et al. [48] showed that in notched marble specimens criticality was delayed when analyzing data for an AE sensor close to the notch from which the onset of macroscopic fracture was delayed. Niccolini et al. [67] found that the AE monitoring of an in-service double girder bridge reveals the existence of an asymmetric damage pattern. The damage zone is consistently characterized by natural time analysis entrance to a critical state. On the other hand, despite the presence of a comparatively larger number of localized AE events, the other monitored portions of the girder crane are characterized by less intense AE activity with natural time analysis indicating non-critical stages.

13.10 Acoustic Emissions in an Experiment in Plunged Granular Bed The experimental methodology used by Tsuji and Katsuragi [120] is the following: Glass beads (of grain diameter d = 0.4, 0.8, or 2.0 mm) are poured into a cylindrical Plexiglas container. A steel sphere (of radius r = 5, 10, or 20 mm) is then penetrated into the granular bed. The penetration speed is fixed as v = 0.5, 1.0, or 5.0 mm/s. The top surface of the granular bed is open to the atmosphere and any confining pressure is not applied to the bed. An AE sensor (NF AE-900s-WB) is buried and fixed in the granular bed to capture AE events created by the penetration. Because the AE signals are very weak, they are amplified by an amplifier (NF AE-9913) and a discriminator (NF AE-9922). The sampling rate of the AE data is 1 MHz. Three experimental realizations for each set of experimental conditions were performed to check the reproducibility. Since granular behaviors have a strong memory effect and history dependence, a fresh granular bed was deposited before every experimental run to erase the memory of penetration. Tsuji and Katsuragi [120] analyzed the experimental data in natural time by using the squared maximum amplitude of each AE event for its released energy value Q k . They examined the criticality condition κ1 = 0.070 of Eq. (13.9) for each experimental setup. As expected, κ1 appears to fluctuate around 0.07 in some data, which indicates the criticality of the system, see, e.g., Fig. 13.12a, b. However, this tendency is not universal. For instance, the asymptotic value of κ1 in Fig. 13.12c appears to be different from 0.07. In order to study the κ1 behavior in more detail, the PDF of κ1 was computed by following the procedure suggested in Refs. [153, 156], i.e., the κ1 value was computed for moving natural time windows of 6–40 consecutive events sliding through the whole data set. Figure 13.13 shows the PDF computed

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Fig. 13.12 Evolution of κ1 as a function of the number of AE events k for various experimental conditions: a d = 2.0 mm, b d = 0.8 mm, and c d = 0.4 mm. Other parameters are fixed at r = 10 mm and v = 5.0 mm/s. Dashed horizontal black lines represent κ1 = 0.07, indicating the criticality of the system. For the cases (a, b) the data fluctuate around 0.07, while for (c) data are clearly offset. Reprinted figure with permission from Ref. [120]. Copyright (2015) by the American Physical Society Fig. 13.13 PDF of the κ1 value under the same experimental conditions as in Fig. 13.12. The color code is the same as in Fig. 13.12. The dashed vertical line indicates κ1 = 0.07. Peak values of the blue (d = 2.0 mm) and red (d = 0.8 mm) curves are around 0.07. Reprinted figure with permission from Ref. [120]. Copyright (2015) by the American Physical Society

13.11 Electrical Resistance Fluctuations Before Fracture

321

from the data depicted in Fig. 13.12. The most probable value κ1, p estimated by the peak location of the PDF depends on the experimental conditions. Particularly, the grain diameter d seems to be an important parameter. This d-dependent tendency is similar to the behavior of the event-size distribution exponent γ [62]. Tsuji and Katsuragi [120] in their concluding remarks state that natural time analyses revealed that the κ1 value is distributed around 0.07–0.083 and κ1 ≈ 0.07 can be established in the brittle-like regime (d/r > 0.04). They also stress that the statistical properties of seismic activity can be mimicked by granular AE events in the range d/r > 0.04 and 0.25 < v/r < 1.0s−1 . Although their experimental system is different from the microfracturing of rocks and geological-scale phenomena, the AE data obtained from plunged granular matter exhibit some similarities to geological-scale phenomena like EQs in terms of conventional time and natural time analyses.

13.11 Electrical Resistance Fluctuations Before Fracture Natural time has been applied [68] to the analysis of the electric resistance changes before fracture of cement mortar and Luserna stone specimens. In these laboratory experiments, the acoustic emissions have been also analyzed in natural time as mentioned in §13.9. Assuming an undamaged specimean before the loading protocol starts, the electrical resistance R0 between the electrodes is given by l R0 = ρ , s

(13.20)

where ρ is the resistivity, l is the distance between the electrodes, and s is the crosssectional area defined by the electrodes surfaces, through which most of the current flows within the samples, see, e.g., Fig. 1 of Ref. [68]. Making the reasonable assumption that the changes in l and ρ (the latter due to changes in porosity) are expected to be minimal before the fracture of the sample, Niccolini et al. [68] suggested that the electrical resistance of the damaged sample during the test is R' = ρ

l l =ρ , s(1 − D) s'

(13.21)

where s ' is the active cross-sectional area which is reduced compared to s due to the freshly formed microcracks. This reduction can be quantified by D which is a continuous damage variable lying in the range [0,1], with D = 1 representing the maximum damage at the moment of failure. It represents the surface density

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of microcracks and cavities in any plane of a representative volume element V . Equations (13.20) and (13.21) imply that D =1−

△R R0 = ' , R R'

(13.22)

where △R = R ' − R0 is the change in the electrical resistance between the undamaged state and a generic damaged state. By considering the time series of resistance measurements ri = R ' (ti )/R0 , Niccolini et al. [68] estimate the time series of energy events to be considered in natural time by assuming that the amount of energy dissipated over a cracking step is calculated by means of fracture mechanics: Wi = G C △si

(13.23)

where G C is the toughness of the material and △si = si − si−1 is the surface change. The latter quantity can be estimated by means of Eqs. (13.20), (13.21) as  △si =

ρl R0 ri−1



ri−1 1− ri

,

(13.24)

and hence Wi ∝ (1/ri−1 − 1/ri ). For the natural time analysis of electrical resistance time series [68], one considers Q k = |Wk |, provided that this is higher than a certain threshold Wth , and the criteria discussed in §13.1 are applied for the determination of criticality. The natural time analysis reveals that criticality in terms of electrical resistance changes systematically precedes acoustic emission criticality for all investigated specimens [68].

13.12 Ozone Hole Dynamics Over Antartica In Ref. [135], natural time analysis has been employed for the study of the ozone hole dynamics over Antarctica through the maximum daily ozone hole area (MD-OHA). Especially, the notion of entropy S in natural time (see, e.g., Chaps. 7, 8, and 9 for applications in seismicity) has been used. Each event was considered to be a year, and the yearly maximum MD-OHA value was considered as Q k . The entropy S in natural time has been studied [135] for various windows of lengths l = 3–15 years that slide along the yearly MD-OHA values since 1977 to 2009, for example see their Fig. 2. Each value of S is attributed to the last year involved in

13.13 Precursory Signals of Major El Niño Southern Oscillation events

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its calculation. The same holds for S− and △S, which were also calculated in Ref. [135]. Three precursory changes have been identified [135] after the analysis of MDOHA time series, notably: • For scales larger than 8 years, the entropy S in natural time exhibits a gradual increase after around 1999. • From 2000 to 2001, the entropy S− in natural time under time reversal shows an increase for all scales (3–15 years) except for the scale of almost 13 years. • The values of the entropy change △S in natural time under time reversal almost coincide at 2000 for the short scales 3–7 years and then decrease. The coexistence of these three facts is consistent with the approach of the system to a critical point (dynamic phase transition) almost two years before the September 2002 ozone hole split [131] over Antarctica.

13.13 Precursory Signals of Major El Niño Southern Oscillation Events Varotsos et al. [136] analyzed in natural time the temporal evolution of the El Niño Southern Oscillation (ENSO) from January 1976 to November 2011. For this purpose they used the Southern Oscillation Index (SOI), which is computed from the difference in the monthly surface air pressure between Tahiti and Darwin and measures the strength of the ENSO phenomenon [88]. The data used are publicly available by the Queensland Goverment at https://longpaddock.qld.gov.au/. The monthly SOI values (calculated [136] by employing Troup’s formula [119] which are dimensionless) were considered as Q k in natural time analysis, assuming a new event each month, after subtracting the minimum monthly SOI value observed since 1876 which was SOImin = −38.8 and was observed during May 1896. This way all monthly values SOIk −SOImin (= Q k ) become positive and can be analyzed in natural time. For this purpose, the entropy change △Sl in natural time under time reversal has been calculated for moving window lengths of l months starting from 1876. Each value of △Sl was assigned to the last month used in its calculation and has been employed as a binary predictor for the SOI value of the next month. This means that if △Sl is greater than or equal to a threshold △Sth , i.e., △Sl ≥ △Sth , an alarm is set ON that the next month SOI value will be smaller than or equal to a threshold value T . Two SOI thresholds T = −5.0 and T = −15.0 have been used in Ref. [136] as corresponding to weak and strong El Niño maximum SOI values. Various window lenghts l have been studied and the quality of the predictions has been evaluated in terms of ROC [24], see, e.g. §12.9.

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The results drawn from the ROC analysis indicate that the highest skill is obtained [136] for △S20 to △S24 . In other words, △S20 to △S24 (i.e. the change △S of the entropy values under time reversal estimated by the SOI data almost 2 years before) have been found to be the optimal binary predictors for weak and strong El Niño events.

Varotsos et al. [136] noted that a plausible mechanism which dictates the time window of around 2 years could be the quasi-biennial oscillation (QBO) in the zonal wind of the tropical stratosphere, which drives the mean meridional circulation inducing a warm or cold anomaly during its descending zonal mean westerly or easterly shear, respectively. A further investigation of this link employing natural time analysis has been made by Varotsos et al. [134] in the study of the 2016 episode of anomalous QBO and the strong 2015–2016 El Niño event. The above presented method has been also used in Ref. [137] as a method for predicting the strength of an ongoing El Niño event.

More specifically, it has been reported that the aforementioned 2015–2016 El Niño event could become “one of the strongest on record”. Varotsos et al. [137] employed △S20 and showed that the 2015–2016 El Niño would be rather a “moderate to strong” or even a “strong” event and not “one of the strongest on record”, as that of 1997– 1998. Specifically, Fig. 13.14 shows the time series of △S20 for the case of 2015 in comparison with that of the very strong El Niño events of 1982–1983 and 1997–1998. As can be clearly seen in Fig. 13.14, the SOI values during the last three months of the 2015 El Niño event remain in the green band and in the limits of the yellow one, indicating that this El Niño should be rather characterized as a “moderate to strong” or even “strong” and not “one of the strongest on record”, as also shown by comparing with the El Niño events of 1982–1983 and 1997–1998. Furthermore, the variation of △S20 during the 2015 El Niño in comparison with 1982–1983 and 1997–1998 El Niño events is not as sharp, also confirming that the 2015 one is not “one of the strongest on record”. In order to estimate the extent of this variation, Varotsos et al. [137] plotted the PDF of △S20 (see the black curve in Fig. 13.15) to show that △S20 exceeded the value 0.0205 only in the three strong El Niño events of 1905–1906, 1982–1983 and 1997–1998. Figure 13.14 shows that this was not fulfilled in 2015 El Niño, since the observed values were [137] close to 0.01, i.e., markedly smaller than the value of 0.0205.

13.13 Precursory Signals of Major El Niño Southern Oscillation events

325

Fig. 13.14 The entropy change △S20 in natural time for the window length l = 20 months (red line, left scale) along with SOI monthly values (blue line, right scale) for the 1982–1983, 1997–1998 (the two strongest in the last century) and the 2015 El Niño event for the period January 2014–October 2015. The alarm for a strong or very strong event is set ON (black line different from 0), when △S20 exceeds the threshold value △Sthres = 0.0035. The colored areas represent the mean minimum negative values of SOI along with the one standard deviation (1σ ) bands for the two cases of “weak, weak to moderate, moderate, moderate to strong” (green band) and “strong, very strong” (yellow band) El Niño events. Taken from Ref. [137]

Indeed, the minimum SOI value reported during the whole 2015–2016 El Niño event was −21.75, i.e., very close to the last SOI value depicted in the lower panel of Fig. 13.14, and the whole event was “moderate to strong” or even “strong” as suggested well in advance in Ref. [137].

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Fig. 13.15 The PDF of △S20 (black curve, left scale) together with the corresponding histogram (red bars, left scale) obtained from the time series of △S20 , which is also plotted versus time (read in the right scale) from the bottom to the top with the blue crosses. The arrows indicate when △S20 exceeds 0.0205 and are labeled by the corresponding ongoing strong El Niño events. Taken from Ref. [137]

13.14 Heart Dynamics Monitored Through Photoplethysmography The application of natural time in heart dynamics and especially in the analysis of electrocardiograms (ECGs) has been extensively discussed in Chap. 9 of Ref. [145]. In a single sinus (normal) cycle of an ECG, the turning points are traditionally labelled with the letters Q, R, S, T, see Fig. 13.16a. How an ECG is read in natural time can be seen in Fig. 13.16b. In 2015, the ability of the complexity measure Ʌl ≡

σ (△Sl ) , σ (△S3 )

(13.25)

where σ (△Sl ) is the standard deviation of the time series △Sl of the change of the entropy under time reversal (see Eq. (13.8)) obtained when a moving window of length of l heartbeats is sliding through the natural time domain representation of the ECG, see Fig. 13.16b, for the separation of healthy individuals (H) from those suffering from congestive heart failure (CHF) and sudden cardiac death (SCD), was investigated [103]. For that analysis, the NN intervals of the ECG have been used. These intervals are obtained from the ECG annotation files of Physiobank [34] by using the option [64] “-c -PN -pN”, which yields only intervals between consecutive normal beats, while intervals between pairs of normal beats surrounding an ectopic beat are discarded. The data analyzed came from 134 long-lasting (from several hours

13.14 Heart Dynamics Monitored Through Photoplethysmography

327

Fig. 13.16 How an ECG in conventional time (a) is read in natural time (b) when Q k = RR k . Panels (c) and (d) correspond to (a) and (b), respectively, under time reversal. Taken from Ref. [103]

Fig. 13.17 The complexity measure Ʌ49 versus Ʌ7 for all the individuals studied in Ref. [103]. The (red) horizontal line corresponds to [Ʌ49 ]c = 2.07, while the vertical (blue) one to [Ʌ7 ]c = 1.97. The labels of SCD that mix with H are also shown (these labels are the ones used in Sudden Cardiac Death Holter Database, sddb, available from the Physiobank[34]). Taken from Ref. [103]

to around 24 h) ECG available from Physiobank databases [34]. The results showed that two optimal scales l = 7 and l = 49 arise when focusing on the distinction of H from both CHF and SCD as shown in Fig. 13.17. Beyond the standard ECG, in recent years, a technique termed photoelectric plethysmography, also known as photoplethysmography (PPG), has simplified the recording of heart rate in an easy and reliable way, see, e.g., Refs. [1, 110, 116]. From

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13 Applications of Natural Time Analysis to Disaster Prediction in Other Disciplines

Fig. 13.18 A captured PPG signal (healthy male, 34 years of age) where the PP intervals to be analyzed in natural time are also shown. Taken from Ref. [6]

1980 the PPG method is used as pulse oximeter for monitoring a person’s oxygen saturation into clinical care [6]. Nowadays the PPG technology is included in many modern affordable devices such as smart phones, smart watches, tablets, bracelets, rings etc. It is a simple technique with low cost and give us health-related information such as heart rate variability, blood oxygen saturation (SpO2 ), blood pressure and the respiratory rate [23, 70]. In 2018, Baldoumas et al. [7] presented a comparison of ECG and oximeter signals to be used in the entropy based complexity measures of natural time analysis. They presented a comparison of the RR intervals in ECG and the peak to peak (PP) intervals of the oximeter signals, see Fig. 13.18, and found that the difference in the intervals between the two setups is of the order of a few milliseconds in root mean square (rms) value. They also presented simulations for the complexity measures Ʌ7 and Ʌ49 which showed that the PPG technique may be as sufficient as that of ECG for their calculation. In the natural time analysis of PPG signals, the PP intervals (of Fig. 13.18) are considered as Q k and the standard analysis is applied to obtain Ʌ7 and Ʌ49 according to Eq. (13.25). In 2019, a prototype PPG electronic device was presented [6] for the distinction of individuals with CHF from the healthy by applying the complexity measures Ʌ7 and Ʌ49 . For this reason, data were collected simultaneously with a conventional three-electrode electrocardiography system and the prototype PPG electronic device from H and CHF voluntaries at the 2nd Department of Cardiology, Medical School of Ioannina, Greece [6]. The measurements took place on the bed of the patients who

13.14 Heart Dynamics Monitored Through Photoplethysmography

329

PPG

5

Λ49

1.5

[Λ7 ]c = 1.55

6

1

4

0.5

3

0

2

-0.5 30

1

57

29 43

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36

-1

19

-1.5

0 1

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Λ7 Fig. 13.19 The complexity measure Ʌ49 versus Ʌ7 obtained from the prototype PPG device concerning the 32 H (blue circles) individuals and 67 CHF (red crosses) patients. The color contours denote the value of the function in the signum of Eq. (13.26) as it resulted from the SVM. The black dotted curves surround the H region identified by SVM in Ref. [6]. The yellow dots stand for the CHF patients who died during the follow up study [5]. Taken from Ref. [5]

were lying quiet for 20 min in the supine position. The ECG leads were obtained using 3 transdermal patches (the same patches were applied to all patients during the study) placed over a bone formation: two on the upper right and left thorax and one over the lower ribs. At the same time the PPG signals were recorded using the prototype PPG device applied usually to the right index figure. Statistical analysis of the results has shown a clear separation of CHF from H subjects by means of natural time analysis for both the conventional ECG system and the PPG prototype system. The PPG method, however, additionally inherits the advantages of a low-cost portable device. To optimize this separation, Baldoumas et al. [6] employed Support Vector Machines (SVM) [2, 19, 20, 44, 129, 130] by using the computer code SMVlight [44, 45] with a Gaussian radial basis function K (a, b) = exp(−γ |a − b|2 ), e.g., see Eq. (5.35) on page 145 of Ref. [129], for the construction of the decision function  f (x) = sign

m ∑

 [yk αk K (xk , x)] − b .

(13.26)

k=1

SVM optimally provide [19, 129, 130] the support vectors {xk }m k=1 , the quantities yk , αk , and the bias b. For applying the SVM method, Baldoumas et al. [6] assigned the value +1 to the points with coordinates x = (Ʌ7 , Ʌ49 ) corresponding to CHF, while a value −1 was assigned to H. In Fig. 13.19, the complexity measures Ʌ7 and

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13 Applications of Natural Time Analysis to Disaster Prediction in Other Disciplines

Ʌ49 estimated by the PP time series are shown together with the color contours that correspond to the values of the function in the signum of Eq. (13.26). In the same figure, two threshold lines have been also inserted [6]: a vertical [Ʌ7 ]c straight line with value Ʌ7 = 1.55 and a horizontal [Ʌ49 ]c straight line with value Ʌ49 = 1.48. An inspection of Fig. 13.19 reveals that the vast majority of H (29 out of 32) lies in the right of the [Ʌ7 ]c line and above the [Ʌ49 ]c line. Especially, only 3 of the 32 healthy individuals are mixed with CHF and hence the sensitivity s H (≡ TP/P in terms of the ROC jargon of §12.9) in this region (the H region) when the Ʌ7 > [Ʌ7 ]c and Ʌ49 > [Ʌ49 ]c , is s H = 90.6%. In the remaining region, the vast majority of CHF is located, , i.e., 54 out of 67, thus the sensitivity for CHF is sC H F = 80.6% (and only three H are mixed with CHF). This is the CHF region. The introduction of SVM greatly improves the sensitivity for the distinction between CHF and H in Fig. 13.19 since SVM lead [6] to an appreciably higher sensitivity, i.e., sC H F = 97.7%, upon using the two complexity measures Ʌ7 and Ʌ49 . PPG devices, based on the prototype of Ref. [6], may transfer via Bluetooth the heartbeat data to a user portable device (smatphone, tablet etc.) which can do the calculation of the complexity measures and provide the result. The results can also be sent via wifi to a central recording system for continuous monitoring if needed. Such portable PPG devices in view of their high connectivity have been also suggested [5] for remote heartbeat monitoring. Moreover, in Ref. [5] a follow up study of the original database comprising 67 CHF and 32 H was presented. Challenging results were obtained since during the subsequent period six individuals died. The complexity measure N3 ≡ σ [△S3,shuff ]/σ [△S3 ] together with σ [△S7 ], both used in Ref. [152] for the distinction of SCD from H and CHF (see also Chap. 9 of Ref. [145]), have been applied to the PP time series obtained.

All six patients who died satisfied both conditions (i.e., their N3 and σ [△S7 ] values lie outside the limits defined by the H individuals and have σ [△S7 ] larger than the minimum value Hmin of the H, see Fig. 13.20) that may distinguish the majority of SCD from CHF, as found in Ref. [152]. Note that an inspection of Fig. 3 of Ref. [152] reveals that 17 out of the 18 SCD individuals satisfied both these conditions.

13.14 Heart Dynamics Monitored Through Photoplethysmography

331

ECG

(a)

2.6

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0.8 7

6

5

4

3

2

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(σ [ΔS7]) x 10

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(b)

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30

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19

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3

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6

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(σ [ΔS7]) x 103

Fig. 13.20 The complexity measure N3 versus the complexity measure σ [△S7 ] for the RR time series of a ECG: concerning the 28 H (blue circles) and 67 CHF (red crosses) individuals, b PPG: concerning the 26 H (blue circles) and 67 CHF (red crosses) individuals. The six (out of 67) CHF who died during the follow up study, are also plotted being marked with solid blue dots. In the rectangle enclosed by black lines the lower horizontal line corresponds to the minimum N3 value computed in H. Concerning the vertical lines in this rectangle they correspond to the minimum (left) and the maximum (right) σ [△S7 ] values computed in H. Taken from Ref. [5]

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13.15 Conclusions As already mentioned in the Preface, examples of data analysis in natural time have appeared in diverse fields, including Biology, Cardiology, Condensed Matter Physics, Environmental Sciences, Geophysics, Physics of Complex Systems, Statistical Physics, Seismology and Volcanology. Natural time analysis has been also applied in several other disciplines. For example, Mintzelas and Kiriakopoulos [63] introduced natural time analysis in financial markets by applying it to price prediction and algorithmic trading. They tested it through a trading strategy and concluded that they found encouraging results. We hope that in future additional applications of natural time analysis will appear.

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Appendix A

Optimality of Natural Time Representation and a Brief Introduction to Natural Time Analysis

Aiming at a better readers’ understanding of the advances in natural time analysis during the last decade, discussed in the present monograph (Part II), we recapitulate here the knowledge emerged from our previous monograph [1] along the following three directions: First, time and not space poses the greatest challenge in Science (see §A.1). Second, the optimality of natural time representation (which is, of course, not continuous) has been demonstrated, while the concept of continuous observation (based on the conventional time t, which is currently modelled as the one dimensional continuum R) meets enormous difficulties when studying according to Schrödinger very modern mathematics (see pp. 62–63 of Ref. [2]), e.g. Cantor set theory (see §A.2). Third, a brief introduction to natural time analysis is given in §A.3 for readers’ convenience.

A.1 Time and Not Space Poses the Greatest Challenge to Science Here, we follow Sect. 2.1.1 of Ref. [1]. Time, according to Weyl (see p.5 of Ref. [3]) for example, is “the primitive form of the stream of consciousness. It is a fact, however, obscure and perplexing to our minds, that …one does not say this is but this is now, yet no more” or according to Gödel “that mysterious and seemingly self-contradictory being which, on the other hand, seems to form the basis of the world’s and our own existence.” (p.111 of Ref. [4]). The challenge seems to stem from the fact that special relativity and quantum mechanics, which are the two great (and successful) theories of twentiethcentury physics, are based on entirely different ideas, which are not easy to reconcile (in general, the former theory, according to Einstein [5], is an example of “principled theory” in the sense that you start with the principles that

© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 P. Varotsos et al., Natural Time Analysis: The New View of Time, Part II, https://doi.org/10.1007/978-3-031-26006-3

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underlie the theory and then work down to deduce the facts, while the latter is a “constructive theory” meaning that it describes phenomena based on some known facts but an underlying principle to explain the strangeness of the quantum world has not yet been found). In particular, special relativity puts space and time on the same footing, but quantum mechanics treats them very differently, e.g., see p. 858 of Ref. [6]. (In quantum gravity, space is fluctuating and time is hard to define, e.g., Ref. [7]). More precisely, as far as the theory of special relativity is concerned, let us recall the following wording of Einstein [8]: “Later, H. Minkowski found a particularly elegant and suggestive expression…, which reveals a formal relationship between Euclidean geometry of three dimensions and the space time continuum of physics…. From this it follows that, in respect to its role in the equations of physics, though not with regard to its physical significance, time is equivalent to the space co-ordinates (apart from the relations of reality). From this point of view, physics is, as it were, Euclidean geometry of four dimensions, or, more correctly, a static in a four-dimensional Euclidean continuum.” whereas in quantum mechanics, Von Neumann complains [9]: “First of all we must admit that this objection points at an essential weakness which is, in fact, the chief weakness of quantum mechanics: its non-relativistic character, which distinguishes the time t from the three space coordinates x, y, z, and presupposes an objective simultaneity concept. In fact, while all other quantities (especially those x, y, z, closely connected with t by the Lorentz transformation) are represented by operators, there corresponds to the time an ordinary number-parameter t, just as in classical mechanics”. Note also that Pauli [10] has earlier shown that there is no operator canonically conjugate to the Hamiltonian, if the latter is bounded from below. This means that for many systems a time operator does not exist. In other words, the introduction of an operator t is basically forbidden and the time must necessarily be considered as an ordinary number (but recall the long standing question that Schrödinger’s equation, as well as Einstein’s general theory of relativity, is symmetric under time reversal in contrast to the fact that our world is not, e.g., Ref. [11]). These observations have led to a quite extensive literature mainly focused on time-energy (as well as on “phase-action”) uncertainty relation, proposing a variety of attempts to overcome these obstacles.

A.2 Optimality of the Natural Time Representation. Is Time Continuous? As mentioned in the Preface, it has been demonstrated in Physical Review Letters [12] that natural time is optimal for enhancing the signals in time–frequency space when employing the Wigner function and measuring its localization property. Specifically,

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in order to address the problem[12] of optimality of the natural time representation of time series resulting from complex systems, we first study the structures of the time–frequency representations[13] of the signals (cf. comprising four SES activities and six “artificial” noises depicted in Fig. 3 of Ref. [12]) by employing the Wigner function [14] to compare the natural time representation with the ones, either in conventional time or in other possible reparametrizations. We find [12] that significant enhancement of the signals is observed in the time-frequency space if natural time is used, in marked contrast to a multitude of other time domains, see, e.g., Figs. 2.7 and 2.8 of Ref. [1]. We recall that in time series analysis, it is desired to reduce uncertainty and extract signal information as much as possible. In other words, the most useful time domain should maximize the information measure. Natural time χ , from its definition, is not continuous and takes values which are rational numbers in the range (0,1]. (In these numbers, as the complex system evolves, the numerators are just the natural numbers (except 0), which denote the order of appearance of the consecutive events.) Hence, one of the fundamental differences between (conventional) time and natural time refers to the fact that the former is based on the idea of continuum, while the latter is not. Recall that the conventional time t is currently modelled as the one-dimensional continuum R of the real numbers, e.g., p.10 of Ref. [7] (or p.12 of Ref. [3] in which it is stated that “…the straight line …is homogeneous and a linear continuum just like time”). In the following, we aim at raising some consequences of this difference, and in particular those that stem from the set theory developed by Cantor, inspired from the following crucial remark made by Schrödinger (see pp. 62–63 of Ref. [2]): “We are familiar with the idea of continuum, or we believe ourselves to be. We are not familiar with the enormous difficulty this concept presents to the mind, unless we have studied very modern mathematics (Dirichlet, Dedekind, Cantor).”

A.2.1 Natural Time and the Cantor Set Theory Here, we follow Ref. [15] as well as Sect. 2.7 of Ref. [1] in order to recapitulate some points of the Cantor set theory that are relevant to our present discussion. A transfinite number or transfinite cardinal is the cardinality of some infinite set, where the term cardinality of a set stands for the number of members it contains, e.g., pp. 2–3 of Ref. [16]. The set of natural numbers is labelled by N , while the number of natural numbers is designated by ℵ0 , i.e., ℵ0 = |N | (cf. the cardinality of a set S is labelled |S|). In

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this transfinite number, the zero subscript is justified by the fact that, as proved by Cantor, no infinite set has a smaller cardinality than the set of natural numbers. It can be shown that the set of rational numbers designated by Q has the same cardinality as the set of natural numbers, or |N | = |Q| (e.g., Theorem 2 in Ref. [16]). In other words, the rationals are exactly as numerous as the naturals. Note that a set is countable i f f its cardinality is either finite or equal to ℵ0 and in particular is termed denumerable i f f its cardinality is exactly ℵ0 (cf. as usually, for “if and only if” we write simply “i f f ”). A set is uncountable i f f its cardinality is greater than ℵ0 , see also below. Hence, natural time takes values (which, as mentioned, are rational numbers) that form in general a countable set; this becomes a denumerable set in the limit of infinitely large number of events. Further, since in natural time analysis we consider the pairs (χk , Q k ), the values of the quantity Q k should form a set with cardinality smaller than (or equal to) ℵ0 . In other words, the values of the energy also form a countable set, which reflects of course that the energy is not continuous, thus the quantization of energy seems to emerge. The fact that |N | = |Q| is an astounding result in view of the following: The rational numbers are dense in the real numbers, which means that between any two rational numbers on the real number line we can find infinitely more rational numbers. In other words, although the set of rational numbers seems to contain infinities within infinities, there are just as many natural numbers as there are rational numbers. This reflects the following point. Let us assume that we follow the evolution of a system with some (experimental) accuracy, in which, as mentioned, in the limit of infinitely large number of events the cardinality of the set of the values of natural time is ℵ0 . Let us assume that we now repeat the measurement with more sensitive instrumentation, i.e., counting events above an appreciably smaller energy threshold (which should be constrained by the uncertainty principle, but a further discussion on this point lies beyond the scope of the present monograph); hence between two consecutive events of the former measurement a considerable number of appreciably smaller events may be monitored. The corresponding cardinality, in contrast to our intuition, is again ℵ0 . (The inverse, i.e., when the instrumentation becomes less sensitive, may correspond to a “coarse-graining” procedure.)

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In other words, when considering the limit of infinitely large number of consecutive events, the natural time takes values that form a denumerable set and this remains so even upon increasing the accuracy (and hence lowering the uncertainty) of our measurement.

We now turn to the aspects of Cantor set theory related to the real numbers, which as mentioned are associated with the conventional time. It is shown that the number of points on a finite line segment is the same as the number of points on an infinite line (e.g., Theorem 13 in Ref. [16]). Considering the definition: The number of real numbers is the same as the number of points on an infinite line (or in the jargon, the numerical continuum has the same cardinality as the linear continuum), let “c” designate the cardinality of the continuum -or equivalently the cardinality of the set of real numbers. (Hence c = |R| by definition). It is proven (e.g., Theorem 16 in Ref. [16]) that the set of real numbers is uncountable, or |R| > ℵ0 . (Equivalently, this theorem asserts that c > ℵ0 ). Hence, the values of conventional time form an uncountable set, in contrast to that of natural time which in general as mentioned is countable.

In order to further inspect this fundamental difference, we resort to the continuum hypothesis which was formulated (but not proved) by Cantor. Continuum hypothesis, after Euclid’s parallel postulate, was the first major conjecture to be proved undecidable by standard mathematics. We first clarify that the power set ∗ S of a set S, which is the set of all subsets of S, has a cardinality |∗ S| = 2|S| when S is finite. According to Cantor’s Theorem the cardinality of the power set of an arbitrary set has a greater cardinality than the original arbitrary set, i.e., |∗ S| > |S| (e.g., Theorem 4 in Ref. [16]). This theorem is trivial for finite sets, but fundamental for infinite sets. Hence, for any infinite cardinality, there is a larger infinite cardinality, namely, the cardinality of its power set. The continuum hypothesis asserts that there is no cardinal number α such that ℵ0 < α < c. Then it follows that the next largest transfinite cardinal after ℵ0 (labelled ℵ1 ) is c, thus c = ℵ1 . Since Cantor proved (e.g., Theorem 17 in [16]) that ℵ1 = 2ℵ0 ,

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continuum hypothesis leads to: c = 2ℵ0 (thus, this is the number of points on an infinite line). Hence, if we assume the continuum hypothesis, the cardinality of the set of the values of natural time -in the limit of infinitely large number of eventscorresponds to ℵ0 , while that of the conventional time is 2ℵ0 . The values of the former, as mentioned, are rational numbers, while almost all the values of the latter are irrational, because, since 2ℵ0 ≫ ℵ0 , almost all reals are irrational numbers. (On the other hand, without assuming the continuum hypothesis we have essentially no idea which transfinite number corresponds to c, and we would know the cardinality of the naturals, integers, and rationals, but not the cardinality of the reals, e.g., p. 15 of Ref. [16].) As for the values of Q k , they are not necessarily rational, because in general when taking ℵ0 (at the most) out of 2ℵ0 values they may all be irrational. Hence, in the limit of infinitely large number of events, even upon gradually improving the accuracy of our measurements, both sets {χk } and {Q k } remain denumerable, the former consisting of rational numbers only.

A proof of the cardinality of the set of the values of natural time in the limit of infinitely large number of events can be found in Sect. 2.7.1 of Ref. [1], where we conclude that the cardinality of the set of the values of natural time equals to ℵ0 .

A.2.2 Is Natural Time Compatible with Schrödinger’s Point of View? We now focus on the question whether natural time is compatible with Schrödinger’s point of view. Schrödinger, in order to point out “The intricacy of the continuum”, used the following example (see pp. 138–143 of Ref. [17]): Let us consider the interval [0,1], you first take away the whole middle third including its left border point, thus the points from 1/3 to 2/3 (but you leave 2/3). Of the remaining two thirds you again take away “the middle thirds”, including their left border points, but leaving their right border points. With the remaining “four ninths” you proceed in the same way and so on. The cardinality of the set that remains ad infinitum is no less than that of [0,1] because it can be shown [17] that there is an one to one correspondence between their elements. Moreover, since it is a subset of [0,1], its cardinality is also no greater, so it must in fact be equal. In particular, Schrödinger concluded[17] as follows:

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“The remarkable fact about our remaining set is that, although it covers no measurable interval, yet it has still the vast extension of any continuous range. This astonishing combination of properties is, in mathematical language, expressed by saying that our set has still the ‘potency’ of the continuum, although it is ‘of measure zero’.” In other words, the cardinality of the aforementioned remaining set considered by Schrödinger exceeds drastically that of the set of the values of natural time.

Let us now comment on the common view that (conventional) time is continuous, keeping in the frame that, as pointed out by Schrödinger (p. 145 of Ref. [18]) “our sense perceptions constitute our sole knowledge about things”. In short, it seems that the continuity of time does not stem from any fundamental principle, but probably originates from the following demand on continuity discussed by Schrödinger (see p. 130 of [17]):

“From our experiences on a large scale…physicists had distilled the one clear-cut demand that a truly clear and complete description of any physical happening has to fulfill: it ought to inform you precisely of what happens at any point in space at any moment of time …We may call this demand the postulate of continuity of the description.” Schrödinger, however, subsequently commented on this demand as follows (see p. 131 of Ref. [17]): “It is this postulate of continuity that appears to be unfulfillable!…’ and furthermore added: “We must not admit the possibility of continuous observation.” Considering these important remarks, we may say that the concept of natural time is not inconsistent with Schrödinger’s point of view. Let us summarize: Conventional time is currently assumed continuous, but this does not necessarily result from any fundamental principle. Its values form an uncountable set, almost all of which may be irrational numbers. On the other hand, natural time is not continuous, and its values form a countable set consisting of rational numbers only; further, the values of the energy also form a countable set but they are not necessarily rational.

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In the limit of infinitely large number of events, the cardinality of the set of the values of natural time is ℵ0 (irrespective if we increase the accuracy of the measurement), thus being drastically smaller than that of conventional time, which equals to 2ℵ0 if we accept the validity of the continuum hypothesis.

A.3 Brief Introduction to Natural Time Analysis For a time series comprising of N events, we define [1, 19, 20] the natural ∑ Ntime for Q n is the occurrence of the k-th event by χk = k/N . The quantity pk = Q k / n=1 the normalized “energy” of the k-th event while Q k is proportional to the energy emitted during the k-th event. Figure A.1 depicts how a time series from a variety of complex systems can be read in natural time. We clarify that we cannot apply natural time analysis to the case (which is inapplicable to our universe [30]) raised by Gödel in 1949 who discovered [31] unexpected solutions to the equations of general relativity corresponding to universes in which no universal temporal ordering is possible (see also Refs. [4, 32] and references therein). This is so, because the concept of natural time is applicable only in cases in which we have to follow the order of the occurrence of the events. Since the positive pk for k = 1, . . . N sum up to unity, they can be considered as probabilities (for more details, see Ref. [33]). Hence, the behavior of the normalized power spectrum [19, 20, 34] ∏(ω) ≡ |ϕ(ω)|2

(A.1)

defined by ϕ(ω) =

N ∑

pk exp(i ωχk )

(A.2)

k=1

where ω stands for the angular frequency, is studied at ω close to zero for capturing the dynamic evolution of the complex system. This is so because all the moments of the distribution of pk can be estimated from ϕ(ω) at ω → 0 (see p. 499 of Ref. [35]). For this purpose, a quantity κ1 is defined from the Taylor expansion ∏(ω) = 1 − κ1 ω2 + κ2 ω4 + . . . which equals: κ1 =

) N ( ∑ k 2 k=1

N

(

N ∑ k pk − pk N k=1

)2 .

(A.3)

As can be seen from Eq.(A.3), the quantity κ1 is just the variance (κ1 = ⟨χ 2 ⟩ − ⟨χ⟩2 ) of natural time χ with respect to the distribution pk .

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Fig. A.1 How a time series of a dichotomous (e.g., zero or one) electric signals [20], b earthquakes [21], c avalanches in 3D rice piles [22–24], d an ECG (for the so-called QT intervals) [25], e a not obviously dichotomous electric (or magnetic) signal [26] and f monthly Southern Oscillation Index [27, 28] values can be visualized in natural time. The meaning of the symbols, as well as the mathematical details for each case are discussed in Sect. 5 of Ref. [29]. Taken from Ref. [29]

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Varotsos et al. [36] proposed that κ1 given by Eq. (A.3) can be considered as an order parameter for seismicity since its value changes abruptly when a strong earthquake (EQ) occurs and its fluctuations have statistical properties similar to other non-equilibrium and equilibrium critical systems [36, 37]. When considering an excerpt of an EQ catalog comprising W consecutive events, we can estimate various κ1 values that correspond to subexcerpts of consecutive EQs of length 6 to W EQs, e.g., by using the first 6 EQs, the first 7 EQs, …, the first W − 1 EQs, all the W EQs, the second 6 EQs, the second 7 EQs, etc. (cf. in some early works as in Ref. [38] the maximum length of the subsexcerpts considered was 40 EQs). This multitude of κ1 values enable the calculation of their average value μ(κ1 ) and their standard deviation σ (κ1 ). We then determine the variability β of κ1 [38]: βW =

σ (κ1 ) . μ(κ1 )

(A.4)

that corresponds to this natural time window of length W . Note that βW of Eq. (A.4) could be identified [39] as effectively the square root of the Ginzburg criterion, e.g., see Eq.(6.25) on p. 175 of Ref. [40], the importance of which in EQ processes has been discussed, for example, in Ref. [41]. The time evolution of βW is followed by sliding the window of W consecutive EQs, event by event, through the EQ catalog and assigning to its value the occurrence time of the EQ which follows the last EQ of the window studied. See Chap. 2 of Ref. [1] for more details on the properties of κ1 . Apart from κ1 , another useful quantity in natural time is the entropy S given by [19, 42, 43]: S = ⟨χ ln χ⟩ − ⟨χ ⟩ ln⟨χ⟩,

(A.5)

( ∑ ) N where the brackets ⟨. . .⟩ ≡ k=1 . . . pk denote averages with respect to the distribution pk . The entropy S is a dynamic entropy that exhibits [44] positivity, concavity and Lesche [45, 46] experimental stability. When Q k are independent and identicallydistributed random variables, S reaches [43] the value Su ≡ ln22 − 41 ≈ 0.0966 that corresponds to the “uniform” distribution. Moreover, upon reversing the time arrow and hence applying time reversal T , i.e., T pk = p N −k+1 , the value of S changes to a value S− . Hence, the entropy S in natural time does satisfy the condition to be “causal”.

It is worthwhile to mention that natural time entropy S has a particularly simple physical meaning since it may capture small trends existing in Q k . For example, when considering a small increasing trend ∈(> 0) for pk versus k by studying the parametric family p(χ ; ∈) ≡ 1 + ∈ (χ − 1/2) of continuous distributions for pk , one can show[47] -using the definition of Eq. (A.5)—that

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{1 S(∈) ≡

351

⎡ 1 ⎤ ⎡ 1 ⎤ { { p(χ ; ∈)χ ln χ dχ − ⎣ p(χ ; ∈)χ dχ ⎦ ln ⎣ p(χ ; ∈)χ dχ ⎦

0

∈ 1 =− + − 4 72

(

0

) ( ) 1 ∈ ∈ 1 + + ln . 2 12 2 12

0

(A.6)

Expanding the last term of Eq.(A.6) around ∈ = 0, we obtain that ( S(∈) = Su +

) 6 ln 2 − 5 ∈ + O(∈ 2 ). 72

(A.7)

Since for this family of continuous distributions S− (∈) simply equals S(−∈), we observe that an increasing trend in p(χ ; ∈), i.e., ∈ > 0, corresponds to S− (∈) values higher than S(∈) (cf. 6 ln 2 < 5). This result indicates that if we study the change of the entropy in natural time under time reversal ΔS ≡ S − S−

(A.8)

increasing or decreasing trends transform to negative or positive values of ΔS, respectively. Finally, since S(∈) in Eq.(A.6) is a nonlinear function of ∈ we observe that the change of the entropy under time reversal ΔS is a nonlinear tool capturing alternations in the dynamics of the complex system playing a key role when a complex system approaches criticality or, in general, a dynamic phase transition [1]. Using S and ΔS within a specified natural time window of length i (number of events), one can study time series resulting from complex systems by constructing the corresponding time series of Si and ΔSi obtained upon estimating these two quantities as the natural time window of length i is sliding event by event along the time series [1]. This inspired (see, e.g., Ref. [1]) the introduction of complexity measures that quantify the fluctuations of the entropy S and of the quantity ΔS upon changing the length scale as well as the extent to which these quantities are affected when randomly shuffling the consecutive events. By considering the standard deviation σ (ΔSi ) of the time series of ΔSi ≡ Si − (S− )i , we define [1, 48, 49] the complexity measure Ʌi =

σ (ΔSi ) σ (ΔS100 )

(A.9)

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when a moving window of i consecutive events is sliding through the time series and the denominator has been selected [48] to correspond to the standard deviation σ (ΔS100 ) of the time series of ΔSi of i = 100 events. The complexity measure Ʌi quantifies how the statistics of ΔSi time series changes upon increasing the scale from 100 events to a longer scale, e.g., i = 103 events. Additionally, we define the measure shu f

Ni = shu f

where σ [ΔSi

σ [ΔSi ] , σ [ΔSi ]

(A.10)

] is the standard deviation upon randomly shuffling (shuf) the events.

The measure Ni quantifies the extend to which the ordering of the events contributes to the ΔSi values being equal to unity for a random process. See Chap. 3 of Ref. [1] for more details on the properties of S and ΔS.

References 1. Varotsos, P.A., Sarlis, N.V., Skordas, E.S.: Natural time analysis: the new view of time. Precursory Seismic Electric Signals, Earthquakes and other Complex Time-Series. Springer, Heidelberg (2011). https://doi.org/10.1007/978-3-642-16449-1 2. Schrödinger, E.: Nature and the Greeks. Cambridge University Press, Cambridge 1954; Canto Edition with Science and Humanism (1996) 3. Weyl, H.: Space-Time-Matter. Dover, New York (1952) 4. Yourgrau, P.: A World Without Time: The Forgotten Legacy of Gödel and Einstein. Basic Books, Cambridge MA (2005) 5. Einstein, A.: Ideas and Opinions (Crown Publishers, 1954; new edition by Souvenir Press, 2005); see also p. 54 in A. Gefter in New Scientist, 10 Dec 2005 6. Wilczek, F.: Nobel lecture: asymptotic freedom: from paradox to paradigm. Rev. Mod. Phys. 77, 857–870 (2005). https://doi.org/10.1103/RevModPhys.77.857 7. Wilczek, F.: Whence the force F=ma? Phys. Today 58, 10–11 (2005). https://doi.org/10.1063/ 1.1825251 8. Einstein, A.: A brief outline of the developement of the theory of relativity. Nature (London) 106, 782–784 (1921) 9. von Neumann, J.: Mathematical Foundations of Quantum Mechanics. Princeton University Press, Princeton NJ (1955) 10. Pauli, W.: Die allgemeinen Prinzipien der Wellenmechanik. In: Geiger, K., Scheel, H. (eds.), Handbuck der Physik, 2nd edn, vol. 245, p. 140. Springer, Berlin (1933)

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Index

Symbols 1/ f δ behavior, 240, 248, 259, 306 S. see Entropy in natural time S− . see Entropy in natural time, under time reversal Su . see Entropy in natural time, “uniform” distribution ΔS. see Entropy in natural time, change of the entropy under time reversal ΔS ΔSl . see Entropy in natural time, evaluation of the change ΔSl at various length scales ΔSi n. see Entropy in natural time, change of the entropy under time reversal ΔS minimum of α value, 58, 60, 63–68, 73, 131, 132, 142, 147, 182–185, 194, 195, 197, 198, 202, 203, 248, 259, 279–281 beta, beta. see Variability betam in. see Variability, minimum κu . see Variance κ1 in natural time, “uniform” distribution κ1, p . see Variance κ1 in natural time, the most probable value of d f see Fractal dimension k B Boltzmann constant, 316

A Acoustic emission, 293, 299, 315–322 Acronyms, list of, xvii, xviii Albert Einstein, 341, 342 Antipersistent time series, 247, 262 Area under the (ROC) curve, AUC, see Receiver operating characteristics, ROC, area under the curve Avalanches, xi, 120, 147, 305–307, 316, 349

B Bak-Tang-Wiesenfeld sandpile model, 99, 317, 318 Bimodal feature, 82, 85, 86, 140, 226 Burridge-Knopoff model, 79, 119, 294, 316, 318

C Cantor set theory, 341, 343, 345 Theorem, 345 Cardinality, 307, 308, 343–348 Central America, xii, 209, 216, 218, 219, 226, 232 Characteristic function, vii, viii Climatology, 247 Coarse-grain, 215, 216, 218–220, 222, 245, 280, 281, 344 Coherent noise model, 305–307 Congestive heart failure, CHF, xxi, 326–331 Constructive theory, 342 Continuum hypothesis, 345, 346, 348 Cooperative orientation of electric dipoles, xii, 190, 197 Cooperativity, xii, 190 Correlated signals, see Long range correlations Correlation length, 98, 99 Coulomb failure stress, 192 Countable set, 344, 345, 347 Critical condition, 98 dynamics, x–xii, 1, 29, 30, 46, 55, 56, 78, 80, 98, 99, 122, 154, 190, 199, 202, 260,

© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 P. Varotsos et al., Natural Time Analysis: The New View of Time, Part II, https://doi.org/10.1007/978-3-031-26006-3

355

356 267, 293, 297, 298, 315, 318, 319, 322, 350 exponent, 99 frequency, xxi point, viii, x, 56, 79, 91, 98, 120, 167, 187, 294, 298, 323 process, 70, 71, 272, 277 stage, vii, ix, 1, 46, 77–80, 85, 94, 96, 98, 105, 160, 240, 266, 294, 295, 297, 299– 305, 310, 313, 318, 319, 351 temperature, 316 value, xii, 190–192 Criticality condition, xi, xvii, 98, 173, 268, 319 Cross-correlation, 30, 122, 220 Crust uplift, xii, 193, 197, 198, 203 Cumulative distribution function, CDF, xxi, 213, 214, 222, 244, 245, 250, 254

D Defects, xi, 41, 190, 266 Denumerable set, 344–346 Detrended fluctuation analysis, DFA, ix, xii, xxi, 56, 88, 130, 182, 194, 196, 198, 202, 203, 240, 247, 248, 258, 259, 271, 277, 278, 280, 282, 283 DFA exponent, see α value Dynamic exponent, 98, 99 phase transition, viii, 99, 120, 187, 323, 351 scaling, 99 Dynamic scaling, see Scaling, dynamic

E Earth’s magnetic field anomalous variations, 41, 86, 179, 190, 193, 196, 197, 201––203 Earthquake network based on similar activity patterns, ENBOSAP, xii, xvii, 30, 31, 34, 40, 56, 58, 122, 162, 209–211, 220–222, 226, 232 Earthquake, EQ Afyon, 225 Aqaba, 210, 216 Athens, 223–225, 298 Chiapas, xvii, 78, 81, 82, 84–86, 88–91, 100, 140–146, 148, 154, 164–168, 170, 173, 180, 181, 187, 279, 281 Crete, 312, 313 Düzce, 210, 216, 221, 223, 225

Index Eastern Mediterranean, 211, 213, 218, 219 Guerrero, 217 Halabja, 210, 222, 225 Hector mine, 181, 182, 308, 309 120 in the study area N35 10 E80 , 217, 218, 226, 229 Izmit, 210, 216, 224 Kefalonia, 225 Kobe, 302 Kumamoto, 293, 300–303, 305 Landers, 182, 307–309 Lefkada, 225 Lemnos, 225 Lesvos, 210, 216, 298, 302 Methoni, 298 Mexican flat slab, 142, 147–149 Michoacan, 228 nowcasting, xii, xiii, xvii, 122, 189, 212– 214, 221, 241, 244, 255, 277, 313 Ogasawara, 10, 12––15 potential score, xvii, 213–219, 221–229, 231–234, 244–246, 255, 277, 279–281, 314 maps, see self-consistent, average earthquake potential score maps Racha, 225 Ridgecrest, x, xi, 79, 109, 110, 115, 173, 181–187, 199, 309–311 Skyros, 225 Tohoku, viii–xi, 2–8, 11–14, 16, 17, 19, 21, 26, 29, 39–41, 46–48, 50, 52, 55– 58, 63, 64, 69, 70, 72, 73, 78, 88, 91–97, 100, 105–107, 109, 115, 119– 121, 123–125, 127–131, 140, 147, 154, 155, 159–164, 167, 168, 170, 173–176, 178–181, 185, 187, 189, 192–199, 202, 203, 266, 268, 270, 272, 279, 280, 302, 303, 309, 310 Van, 210, 216 Vrancea, 210, 216, 225 Zakynthos, 225 Eastern Mediterranean, 209, 211, 213, 216, 218–221, 232 Economics, 332 Einstein, see Albert Einstein Electrocardiogram, ECG, xxi, 199, 326– 329, 331, 349 Electrokinetic effect, 192 Entropy Shannon, 296 Entropy in natural time, 88–90, 317, 318, 322

Index “uniform” distribution, 80, 86, 294, 295, 297, 350, 351 change of the entropy under time reversal ΔS, viii, x, xi, 82, 85, 109, 120––124, 128––143, 148, 149, 153, 154, 164, 165, 167, 174, 175, 323, 351, 352 minimum of, 108, 109, 174, 180, 181, 185, 195, 197, 311 complexity measures of the fluctuations of the change ΔS of the entropy under time reversal, xi, 12, 109, 153–156, 158, 163, 166–169, 352 complexity measures of the fluctuations of the entropy S, 163–167, 170 concavity, 153, 294, 350 definition of, 79, 119, 153 evaluation of the change ΔSl at various length scales, x, xi, 82, 85, 120–124, 128–143, 148, 149, 153, 154, 164, 165, 167, 174, 175, 322, 323, 351, 352 in Environmental Sciences, 322 interconnection with a small linear trend, 295, 297, 350, 351 Lesche (experimental) stability, 153, 294, 350 of seismicity after SES and before mainshock, 80, 86, 297 physical meaning, viii, 153, 295, 297, 350, 351 positivity, 153, 294, 350 under time reversal, viii, xi, 80, 86, 88–90, 109, 120, 153, 297, 317, 318, 322, 323, 352 EPS, see Earthquake, EQ, potential score Erwin Schrödinger, 341–343, 346, 347 Esashi geomagnetic station, 92, 97, 193, 268 Event coincidence analysis, ECA, xxi, 16, 109, 200–202, 249, 273–276 Experimental robustness, see Lesche F Faults, 159, 216, 244 Fractal, 215, 234, 257 Fractal dimension, 215, 234 G Gauss, vii Gaussian distribution, 84, 244, 251–253 Geoelectrical measurements, ix, x, 39, 63, 192 Ginzburg

357 criterion, 212, 350 Landau theory of phase transitions, 212 Global seismicity, xii, xiii, 11, 213, 239–241, 244, 251–253, 255, 258–261, 263–266, 268, 270, 273, 275, 277–282, 284 Global positioning system, GPS, xxii, 189, 193, 196 horizontal azimuths, xii, 193, 197, 198, 203 Greece, 3, 19, 25, 39, 41, 45, 47, 187, 192, 200–202, 213, 216, 225, 264, 266, 273, 298, 301, 328 Groundwater changes, 189, 196, 197, 203, 268, 270 Gutenberg-Richter law, 314 H Haar, 311 Heart rate, 327 variability, xiii, 73, 198, 202, 293, 328 Heaviside function, 250, 306 Herman Weyl, 341 Hurst analysis, xiii, 240, 247, 257, 262, 284 exponent, 247, 257, 262 index, 57 I Izu Island area, 6 peninsula, 189 region, ix, 25, 37, 39, 40, 56, 67, 72, 78 volcanic-seismic swarm activity, 46, 64, 67, 72, 194 J Japan, viii–xii, xviii, 2, 3, 6, 7, 10––13, 16, 19––21, 25, 30, 31, 39, 40, 45–48, 52, 55, 63, 64, 69, 72, 73, 78, 88, 91, 92, 97, 100, 105–109, 119, 120, 122, 123, 130, 140, 147, 153, 154, 159, 161, 163, 164, 173–175, 178–180, 182, 185, 187, 189, 192–195, 197, 199, 201–203, 209, 213, 259, 264–266, 268, 270–273, 279, 280, 299–301, 303, 309, 310, 313 K Kolmogorov-Smirnov-Lilliefors test, 252, 253

358 L Laboratory experiments, 293, 299, 315, 321 Lesche, 153, 294, 350 Lifshitz-Slyozov, xxii, 112, 115 Lifshitz-Slyozov-Wagner, 12, 105, 111–114, 156 Long range correlations, ix, xii, 56, 58, 65– 68, 73, 88, 98, 122, 131, 132, 142, 154, 175, 179, 184, 185, 194, 195, 197–199, 202, 203, 222, 240, 242, 247, 254, 257– 259, 270, 280, 283 Lorentz transformation, 342

M Magnetic field, xi, 47, 48, 199, 301 anomaly, 184 component, 268, 302 data, 201 variations, xii, xiii, 29, 41, 56, 65, 86, 91, 92, 97, 174, 179, 190, 192, 196, 197, 202, 203, 266, 293, 301, 302 Magnetic storms, 193, 300 Magnitude of earthquake, viii–xiii, xxii, 1–3, 5, 7, 10, 12–14, 16, 17, 26–28, 39–41, 45, 47, 48, 50–52, 55–58, 67, 68, 70– 74, 77, 79–82, 85, 86, 88–92, 94, 105– 110, 119–123, 125, 127, 128, 130, 131, 140–146, 154, 155, 159, 160, 164–166, 168, 170, 173, 174, 179–184, 187, 189, 191, 194, 195, 197, 198, 200, 202, 203, 209, 211–214, 217, 220, 222, 226–229, 232, 239–241, 243–245, 247, 250–260, 262, 268, 270, 272, 273, 277–280, 283, 300, 307, 310–315 Magnitude threshold invariance, 86, 164 Maxwell’s equations, xi, xii, 41, 65, 203 Mean field theory, 212 Meteorology, 247 Mexico, xi, xii, 78, 81, 83, 85–87, 91, 100, 140–146, 148, 153, 154, 163, 164, 170, 173, 180, 181, 187, 192, 209, 216–219, 226, 232, 266 Minkowski, 342 Mizusawa geomagnetic station, 92, 97, 193 Monte Carlo calculations, 16, 38, 39 steps, 99 Multifractal, xiii, 240, 258

N

Index National earthquake information center of the United States of America, NEIC, xxii, 213, 219, 226 National Observatory of Athens, 200 Natural time, vii–x, xii, xiii, 40, 46, 47, 56, 62, 77, 79–81, 98, 100, 105, 119, 153, 163, 173, 194, 196, 211–213, 220, 239–242, 244, 284, 293–297, 300, 302, 303, 305, 306, 313–319, 321–323, 332, 341–351 analysis, vii, viii, x, xi, xiii, 85, 121, 173, 181, 187 identification of long range correlations, 242 in financial markets, 332 of acoustic emissions, 315–319, 321 of aftershock time series, 293, 305 of atmospheric gravity waves, 303 of Bak-Tang-Wiesenfeld model, 317, 318 of Burridge-Knopoff model, 316, 318 of El Niño Southern oscillation, 323–325 of electrical resistance fluctuations, 293, 321, 322 of electrocardiograms, 198, 199, 326– 329 of fracture-induced electromagnetic emissions, 298 of heart rate variability by photoplethysmography, 293, 326, 328, 329 of Olami-Feder-Christensen model, 109, 119, 195, 316, 318 of ozone hole dynamics, 322, 323 of seismicity, xi–xiii, 1, 11, 14, 21, 26, 29, 46, 47, 56, 65, 73, 77, 78, 82, 85– 87, 90–92, 94, 97, 100, 105, 106, 122, 123, 142, 148, 149, 160, 163, 174, 179, 180, 189, 194, 203, 209, 220–222, 232, 239, 240, 251–253, 255, 260, 262, 264, 268, 281, 282, 284, 293, 311, 313 of subionospheric VLF propagation anomalies, 293, 300, 301 of surface deformations, 293, 304, 305 of ULF magnetic field variations, 293, 302 definition, 1, 2, 119, 211, 241 dynamic exponent, 99 entropy, see Entropy in natural time optimality of, 341–343 variance, see Variance κ1 in natural time window, 2, 4, 10, 12, 15, 21, 26–31, 40, 49, 55, 58–61, 105, 106, 120, 161, 211,

Index 212, 239, 241, 242, 297, 316, 319, 326 Noise, 40, 193, 247, 248, 257, 298, 301, 302, 305–307, 343 Non-equilibrium, 29, 30, 55, 98, 315, 316, 318, 350 Nonextensive statistical mechanics, 159 Normalized power spectrum, 79, 98, 294, 295, 348 Nowcasting extreme cosmic ray events, 315 extreme low daily minimum air temperature in the Artctic, 315 volcano eruptions, 293, 313–315 O Olami-Feder-Christensen (OFC) model, 79, 109, 119, 195, 294, 316, 318 Omori-Utsu law, 306 Optimality of natural time representation, see Natural time, optimality of Order parameter, 30, 46, 105, 106, 316, 318 of acoustic emission, 315 of seismicity, 1, 2, 10, 11, 12, 14, 18, 20, 21, 25, 27, 29, 30, 39–41, 45–47, 55– 57, 72, 73, 78, 85, 86, 88, 91, 94, 97, 100, 105–107, 110, 115, 121, 142, 154, 160–162, 173, 185, 194, 197, 203, 209, 211, 216, 219, 221, 222, 226, 232, 239– 241, 259, 260, 264–266, 273, 278–280, 282–284, 350 of the 2D Ising model, 316 P Pauli, see Wolfgang Pauli Persistent time series, 247, 264 Physiobank, 326, 327 Piezoelectric effect, 192 Polarization currents, see Pressure stimulated polarization currents, PSPC ratio, 321 Porosity, 321 Postulate of continuity, 347 of Euclid, 345 Power law, 216, 219, 257, 306 distribution, 99 exponent, 246 Power spectrum, 240, 258, 259 exponent, 240, 259 PP interval, 328, 330

359 Prediction, 19, 77, 112, 200, 248, 249, 271, 273, 274, 283, 293 based on Seismic Electric Signals, 200 of disasters, 293 of earthquake, 1, 57, 77, 239, 240, 248, 260–265, 272, 284 of major El Niño Southern Oscillation events, 293, 323 of price, 332 of the occurrence time of strong aftershocks, 293, 308 of the ozone hole dynamics over Antartica, 293 results, xiii scheme, 16, 239, 272 threshold, 272 window, 279 Pressure stimulated polarization currents, PSPC, xii, xxii, 41, 190, 197, 199, 203 Principled theory, 342 Probability density function, PDF, xxii, 29, 82, 83, 114, 316, 319–321, 324, 326 bimodal, 86 scaled, 316, 318 Public warning, 199 Q QT interval, 349 Quasi-biennial oscillation, QBO, xxii, 324 R Radon increase, 189, 196, 198, 203 Random, 58, 114, 193, 220, 249, 273, 274, 283, 306 behavior, ix, 65, 66, 68, 69, 71, 73, 142, 184, 185, 194, 195, 198, 202 number, 306 orientations, xii, 190, 193, 197, 203 predictor, 20, 272 process, 352 shuffling, 123, 144, 145, 243, 244, 250– 254, 297, 307, 351, 352 stress, 306, 307 variable, 114, 251, 294, 350 Randomness uncorrelated, 73, 198, 202 Receiver operating characteristics, 273 Receiver operating characteristics, ROC, xxii, 16, 19, 20, 38, 39, 200, 201, 248, 249, 271, 272, 274, 282, 283, 307, 309, 324, 330

360 area under the curve, xxii, 16, 19, 20, 38, 249, 272–274 Regional seismicity, xii, 225, 240, 246, 259, 268, 270, 272, 280, 313 Rescaled range (R/S) see Hurst, analysis Respiratory rate, 328 RR interval, 327, 328, 331 S Sampling frequency, 298, 299, 319 Scaling, 307 behavior, xi, 12, 105, 109, 155, 156, 159, 163, 168, 195 dynamic, 98 exponent, 248 function, 98 hypothesis, 98, 99 law, 55, 167, 234, 306 theory, 98 Schrödinger, see Erwin Schrödinger Seismic catalog, viii, 1, 2, 55, 57, 251, 311, 314 Japan Meteorological Agency, ix, 2, 14, 19, 25–27, 40, 48, 58, 105, 106, 120, 154, 194 National Seismic Service (SSN) of the Universidad National Autónoma de México, 163 United States National Earthquake Information Center (NEIC) PDE, 213 risk, xii Seismic Electric Signals, ix, xii, xxii, 11, 25, 27, 37, 45–47, 53, 77, 86, 92, 97, 100, 160, 190–192, 199–202, 228, 264–266, 270, 273 activity, ix, xxii, 1–3, 7, 25–27, 29, 39–41, 46, 47, 52, 55, 56, 63–65, 67, 68, 73, 77–79, 86, 91, 92, 97, 100, 105, 121, 122, 140, 154, 173–175, 179, 187, 190– 192, 194, 197–200, 202, 203, 211, 225, 240, 241, 262, 266, 268, 270, 280, 294, 298, 311, 343 generation model, xii, 190–192, 197, 280 selectivity map, x, 45, 78, 92, 97, 100 Seismoelectric effect, 192 Seiya Uyeda, 7, 25–27, 29, 40, 68, 194 Self-consistent, 70, 74 average earthquake potential score maps, xii, 215–219, 221–223, 225–228, 231– 234, 245, 246, 277, 279–281, 283 Self-organized criticality, xviii, 100, 119, 195, 315

Index SES see Seismic Electric Signals Short term forecasting, 45, 47 Shuffling, see Random, shuffling Similarity, 258 self-similarity, 247 Southern California, xii, xxii, 29, 110, 182, 209, 216, 218, 219, 226, 232, 251, 258, 305, 309 earthquake center, 110, 182, 183, 186, 251, 252 Spatial clustering, 305 correlations, 270 window, 37, 40, 307 Spatiotemporal variation of seismic quiescence, 45 variation of the order parameter of seismicity, x, xii, 40, 45, 72, 78, 97, 100, 194, 209 Statistical analysis, 329 Physics, vii, 332 properties, 321, 350 Seismology, 45 significance, xiii, 14, 16, 19–21, 29, 38, 39, 108, 109, 123, 166, 167, 195, 200, 201, 241, 248, 249, 271–273, 282, 283, 311 tests, 250 threshold, 193 tools, 200 Successful prediction, 39, 309 Sudden cardiac death, xxii, 198, 199, 202, 326, 327, 330 Surface air pressure, 323 deformation, 293, 303, 305 density of microcrcks, 322 displacements, 189, 193 of a fault, 192 wave, 254 Surrogate test, 250 time series, 250

T Temporal clustering, 244 correlations, xii, 55–58, 68, 73, 122, 130, 142, 154, 175, 184, 194, 197–199, 202, 203, 251, 259, 313

Index epidemic-type aftershock sequence (ETAS) model, 13, 17 evolution, 212, 317, 323 ordering, 348 variability, 311 Thermodynamics, vii, 69 of point defects, xi, 41, 190 Time coincidence, see Event coincidence analysis, ECA Time reversal, viii, x–xii, 78, 80, 81, 108, 109, 115, 119, 142, 147–149, 153, 163, 165, 167, 170, 173, 174, 181, 187, 195, 197, 203, 294–297, 323, 324, 326, 327, 342, 350, 351 Transfinite cardinal, 343, 345 number, 343, 344, 346 True coincidence, 79–81, 85, 86, 91, 94, 295, 297, 298 positive, 39, 40, 279 SES, 40 Tsallis, 159 entropic index q, 158–160, 163, 168, 170, 195, 197, 203

U Uncertainty, vii, 27, 29, 35, 89, 91, 343, 345 relation, 342, 344 Uncountable set, 344, 345, 347 “Uniform” distribution, 294, 350 Updated procedure, x, 78 Uyeda, see Seiya Uyeda

V Variability, ix, x, 2–4, 7, 12, 13, 15, 19, 27, 29, 31, 40, 48, 57, 58, 63, 65, 71, 72, 106–110, 120, 142, 154, 161, 162, 175, 182, 183, 194, 195, 211, 212, 221, 222, 227, 232, 239, 242, 260, 262, 265–269, 274, 277, 278, 310, 350

361 increase scaling of, 12, 13, 15, 107–110, 195, 310, 311 minimum, viii, ix, 3–14, 16, 17, 19–21, 25, 27–30, 32, 34–36, 38–41, 45, 47, 48, 52, 53, 57–62, 64–68, 70–74, 86, 88, 89, 91, 100, 105, 109, 121–123, 140, 142, 154, 162, 173, 183, 185, 194, 195, 197, 198, 203, 211, 212, 216, 217, 221– 229, 242, 251, 260–274, 277, 279–282, 313 Variance κ1 in natural time, viii, x, xi, 1, 2, 21, 26, 27, 29, 31, 37, 39, 40, 48, 56, 57, 63, 71, 77–86, 88–94, 96, 98–100, 106, 140, 154, 160, 161, 163, 173, 194, 195, 211, 212, 222, 232, 239–244, 253, 254, 265, 266, 268, 294, 295, 297, 315, 316, 318–320 fluctuations of, see Variability the most probable value of, 321 Ventricular fibrillation, 198 Volcano eruptions, xiii, 244, 293, 300, 313 K¯ılauea, 313 seismic activity, 6

W Wavelet analyzing function, 311 Multiresolution analysis, MRWA, xviii, 293, 311 transform, 257, 303, 311 coefficients, 311 Weibull distribution, 213–215, 222, 245, 255 Weyl, see Herman Weyl Wigner function, vii, 343 Wolfgang Pauli, 342 Worldwide seismicity, see Global, seismicity

Z Z-score, 244, 251, 252, 254