Long-range correlations in the electric signals that precede rupture

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

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

Defining the Scale to Build Complex Networks with a 40-Year Norwegian Intraplate Seismicity Dataset

We present a new complex network-based study focused on intraplate earthquakes recorded in southern Norway during the period 1980-2020. One of the most recognized limitations of spatial complex network procedures and analyses concerns the definition of adequate cell size, which is the focus of this approach. In the present study, we analyze the influence of observational errors of hypocentral and epicentral locations of seismic events in the construction of a complex network, looking for the best cell size to build it and to develop a basis for interpreting the results in terms of the structure of the complex network in this seismic region. We focus the analysis on the degree distribution of the complex networks. We observed a strong result of the cell size for the slope of the degree distribution of the nodes, called the critical exponent γ. Based on the Abe-Suzuki method, the slope (γ) showed a negligible variation between the construction of 3- and 2-dimensional complex networks. The results were also very similar for a complex network built with subsets of seismic events. These results suggest a weak influence of observational errors measured for the coordinates latitude, longitude, and depth in the outcomes obtained with this particular methodology and for this high-quality dataset. These results imply stable behavior of the complex network, which shows a structure of hubs for small values of the cell size and a more homogeneous degree distribution when the cell size increases. In all the analyses, the γ parameter showed smaller values of the error bars for greater values of the cell size. To keep the structure of hubs and small error bars, a better range of the side sizes was determined to be between 8 to 16 km. From now on, these values can be used as the most stable cell sizes to perform any kind of study concerning complex network studies in southern Norway.

Tsallis q-Statistics in Seismology

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

Earthquake Nowcasting: Retrospective Testing in Greece

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

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

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

Critical Dynamics in Stratospheric Potential Energy Variations Prior to Significant (M > 6.7) Earthquakes

Lithosphere–atmosphere–ionosphere coupling (LAIC) is studied through various physical or chemical quantities, obtained from different sources, which are observables of the involved complex processes. LAIC has been proposed to be achieved through three major channels: the chemical, the acoustic, and the electromagnetic. Accumulated evidence supporting the acoustic channel hypothesis has been published, while atmospheric gravity waves (AGWs) play a key role in LAIC as the leading mechanism for the transmission of energy from the lower atmosphere to the stratosphere and mesosphere, associated with atmospheric disturbances observed prior to strong earthquakes (EQs). The seismogenic AGW is the result of temperature disturbances, usually studied through stratospheric potential energy (EP). In this work, we examined 11 cases of significant EQs (M > 6.7) that occurred during the last 10 years at different geographic areas by analyzing the temperature profile at the wider location of each one of the examined EQs. The “Sounding of the Atmosphere using Broadband Emission Radiometry” (SABER) instrument, part of the “Thermosphere Ionosphere Mesosphere Energetics Dynamics” (TIMED) satellite, data were employed to compute the potential energy (EP) of the AGW. Using the temperature profile, we first calculated EP and determined the altitudes’ range for which prominent pre-seismic disturbances were observed. Subsequently, the EP time series at specific altitudes, within the determined “disturbed” range, were for the first time analyzed using the criticality analysis method termed the “natural time” (NT) method in order to find any evidence of an approach to a critical state (during a phase transition from a symmetric phase to a low symmetry phase) prior to the EQ occurrence. Our results show criticality indications in the fluctuation of EP a few days (1 to 15 days) prior to the examined EQs, except from one case. In our study, we also examined all of the temperature-related extreme phenomena that have occurred near the examined geographic areas, in order to take into account any possible non-seismic influence on the obtained results.

Natural Time Analysis of Global Seismicity

Natural time analysis enables the introduction of an order parameter for seismicity, which is just the variance of natural time χ, κ1=⟨χ2⟩−⟨χ⟩2. During the last years, there has been significant progress in the natural time analysis of seismicity. Milestones in this progress are the identification of clearly distiguishable minima of the fluctuations of the order parameter κ1 of seismicity both in the regional and global scale, the emergence of an interrelation between the time correlations of the earthquake (EQ) magnitude time series and these minima, and the introduction by Turcotte, Rundle and coworkers of EQ nowcasting. Here, we apply all these recent advances in the global seismicity by employing the Global Centroid Moment Tensor (GCMT) catalog. We show that the combination of the above three milestones may provide useful precursory information for the time of occurrence and epicenter location of strong EQs with M≥8.5 in GCMT. This can be achieved with high statistical significance (p-values of the order of 10−5), while the epicentral areas lie within a region covering only 4% of that investigated.

Order Parameter and Entropy of Seismicity in Natural Time before Major Earthquakes: Recent Results

A lot of work in geosciences has been completed during the last decade on the analysis in the new concept of time, termed natural time, introduced in 2001. The main advances are presented, including, among others, the following: First, the direct experimental verification of the interconnection between a Seismic Electric Signals (SES) activity and seismicity, i.e., the order parameter fluctuations of seismicity exhibit a clearly detectable minimum when an SES activity starts. These two phenomena are also linked closely in space. Second, the identification of the epicentral area and the occurrence time of an impending major earthquake (EQ) by means of the order parameter of seismicity and the entropy change of seismicity under time reversal as well as the extrema of their fluctuations. An indicative example is the M9 Tohoku EQ in Japan on 11 March 2011. Third, to answer the crucial question—when a magnitude 7 class EQ occurs—whether it is a foreshock or a mainshock. This can be answered by means of the key quantities already mentioned, i.e., the order parameter of seismicity and the entropy change of seismicity under time reversal along with their fluctuations. The explanation of the experimental findings identified before major EQs is given in a unified way on the basis of a physical model already proposed in the 1980s.

On the Autocorrelation Function of 1/f Noises

The outputs of many real-world complex dynamical systems are time series characterized by power-law correlations and fractal properties. The first proposed model for such time series comprised fractional Gaussian noise (fGn), defined by an autocorrelation function C(k) with asymptotic power-law behavior, and a complicated power spectrum S(f) with power-law behavior in the small frequency region linked to the power-law behavior of C(k). This connection suggested the use of simpler models for power-law correlated time series: time series with power spectra of the form S(f)∼1/fβ, i.e., with power-law behavior in the entire frequency range and not only near f=0 as fGn. This type of time series, known as 1/fβ noises or simply 1/f noises, can be simulated using the Fourier filtering method and has become a standard model for power-law correlated time series with a wide range of applications. However, despite the simplicity of the power spectrum of 1/fβ noises and of the known relationship between the power-law exponents of S(f) and C(k), to our knowledge, an explicit expression of C(k) for 1/fβ noises has not been previously published. In this work, we provide an analytical derivation of C(k) for 1/fβ noises, and we show the validity of our results by comparing them with the numerical results obtained from synthetically generated 1/fβ time series. We also present two applications of our results: First, we compare the autocorrelation functions of fGn and 1/fβ noises that, despite exhibiting similar power-law behavior, present some clear differences for anticorrelated cases. Secondly, we obtain the exact analytical expression of the Fluctuation Analysis algorithm when applied to 1/fβ noises.

Analysis of Acoustic Emission Activity during Progressive Failure in Heterogeneous Materials: Experimental and Numerical Investigation

This work focuses on an experimental and numerical investigation into monitoring damage in a cube-shaped concrete specimen under compression. Experimental monitoring uses acoustic emission (AE) signals acquired by two independent measurement apparatuses, and the same damage process is numerically simulated with the lattice discrete element method (LDEM). The results from the experiment and simulation are then compared in terms of their failure load, final configurations, and the evolution of global parameters based on AE signals, such as the b-value coefficient and the natural time approach. It is concluded that the results from the AE analysis present a significant sensitivity to the characteristics of the acquisition systems. However, natural time methods are more robust for determining such differences, indicating the same general tendency for all three data sets.

Nonlinear Statistical Features of the Seismicity in the Subduction Zone of Tehuantepec Isthmus, Southern México

After the M8.2 main-shock occurred on 7 September 2017 at the Isthmus of Tehuantepec, Mexico, the spatial distribution of seismicity has showed a clear clusterization of earthquakes along the collision region of the Tehuantepec Transform/Ridge with the Middle America Trench off Chiapas. Furthermore, nowadays, the temporal rate of occurrence in the number of earthquakes has also showed a pronounced increase. On the basis of this behavior, we studied the sequence of magnitudes of the earthquakes which occurred within the Isthmus of Tehuantepec in southern Mexico from 2010 to 2020. Since big earthquakes are considered as a phase transition, after the M8.2 main-shock, one must expect changes in the Tehuantepec ridge dynamics, which can be observed considering that the b-value in the Gutenberg–Richter law, has also showed changes in time. The goal of this paper is to characterize the behavior of the seismic activity by using the Gutenberg–Richter law, multifractal detrended fluctuation analysis, visibility graph and nowcasting method. Those methods have showed important parameters in order to assess risk, the multifractality and connectivity. Our findings indicate, first that b-value shows a dependency on time, which is clearly described by our analyses based on nowcasting method, multifractality and visibility graph.

Long-Range Correlations and Natural Time Series Analyses from Acoustic Emission Signals

This work focuses on analyzing acoustic emission (AE) signals as a means to predict failure in structures. There are two main approaches that are considered: (i) long-range correlation analysis using both the Hurst (H) and the detrended fluctuation analysis (DFA) exponents, and (ii) natural time domain (NT) analysis. These methodologies are applied to the data that were collected from two application examples: a glass fiber-reinforced polymeric plate and a spaghetti bridge model, where both structures were subjected to increasing loads until collapse. A traditional (AE) signal analysis was also performed to reference the study of the other methods. The results indicate that the proposed methods yield reliable indication of failure in the studied structures.

Comparative Assessment of Criticality Indices Extracted from Acoustic and Electrical Signals Detected in Marble Specimens

The quantitative determination of the current load carrying capability of already loaded structural elements and the possibility to detect proper indices that could be considered as signals for timely warning that the load carrying capacity is exhausted is the subject of this study. More specifically, it aims to explore the possibility of detecting signals that can be considered as indices warning about upcoming fracture and then to compare quantitatively such signals provided by different techniques. The novelty of the present study lies exactly in this quantitative comparison of the pre-failure signals provided by various sensing techniques and various methods of analysis of the experimental data. To achieve this target, advantage is taken of data concerning the acoustic and electrical activities produced while marble specimens are subjected to mechanical loading. The respective signals are detected and recorded by means of the acoustic emissions technique and that of the pressure stimulated currents. The signals detected by the acoustic emissions technique are analyzed in terms of three formulations, i.e., the b-value, the F-function and the parameters variance κ1, entropy S and entropy under time reversal S_ according to the natural time analysis. The signals detected by the pressure stimulated currents technique are analyzed by means of the intensity of the electric current recorded. The study indicates that all quantities considered provide promising pre-failure indicators. Furthermore, when the specimen is subjected to near-to-failure load levels, the temporal evolution of three of the quantities studied (b-value, F-function, pressure stimulated currents) is governed by a specific power law. The onset of validity of this law designates some differentiation of the damage mechanisms activated. Quantitative differences are observed between the time instants at which this power law starts dictating the evolution of the above parameters, indicating the imperative need for further investigation, despite the quite encouraging results of the present study.

Detecting Criticality by Exploring the Acoustic Activity in Terms of the “Natural-Time” Concept

The acoustic activity developed in marble specimens under various loading schemes is explored in terms of the recently introduced F-function. The novelty of the study is that instead of describing the temporal evolution of the F-function in terms of conventional time, the Natural Time concept is employed. Although completely different geometries and loading schemes were considered, the evolution of the F-function in the Natural Time domain exhibits a self-consistent motive: its values increase progressively with fluctuations of varying intensity, however, while the fracture is approaching, a power law appears to systematically govern the response of the specimen/structure loaded. The exponent of this law, somehow corresponding to the intensity of the acoustic activity within the loaded complex, varies within broad limits. The onset of validity of the power law designates that the system has entered into its critical stage, namely that of impending fracture, providing a useful pre-failure signal.

Unusual Surface Latent Heat Flux Variations and Their Critical Dynamics Revealed before Strong Earthquakes

We focus on the possible thermal channel of the well-known Lithosphere–Atmosphere–Ionosphere Coupling (LAIC) mechanism to identify the behavior of thermal anomalies during and prior to strong seismic events. For this, we investigate the variation of Surface Latent Heat Flux (SLHF) as resulting from satellite observables. We demonstrate a spatio-temporal variation in the SLHF before and after a set of strong seismic events occurred in Kathmandu, Nepal, and Kumamoto, Japan, having magnitudes of 7.8, 7.3, and 7.0, respectively. Before the studied earthquake cases, significant enhancements in the SLHF were identified near the epicenters. Additionally, in order to check whether critical dynamics, as the signature of a complex phenomenon such as earthquake preparation, are reflected in the SLHF data, we performed a criticality analysis using the natural time analysis method. The approach to criticality was detected within one week before each mainshock.

Estimating the Epicenter of a Future Strong Earthquake in Southern California, Mexico, and Central America by Means of Natural Time Analysis and Earthquake Nowcasting

It has recently been shown in the Eastern Mediterranean that by combining natural time analysis of seismicity with earthquake networks based on similar activity patterns and earthquake nowcasting, an estimate of the epicenter location of a future strong earthquake can be obtained. This is based on the construction of average earthquake potential score maps. Here, we propose a method of obtaining such estimates for a highly seismically active area that includes Southern California, Mexico and part of Central America, i.e., the area N1035W80120. The study includes 28 strong earthquakes of magnitude M ≥7.0 that occurred during the time period from 1989 to 2020. The results indicate that there is a strong correlation between the epicenter of a future strong earthquake and the average earthquake potential score maps. Moreover, the method is also applied to the very recent 7 September 2021 Guerrero, Mexico, M7 earthquake as well as to the 22 September 2021 Jiquilillo, Nicaragua, M6.5 earthquake with successful results. We also show that in 28 out of the 29 strong M ≥7.0 EQs studied, their epicenters lie close to an estimated zone covering only 8.5% of the total area.

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

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

Analysis of Korean Peninsula Earthquake Network Based on Event Shuffling and Network Shuffling

In this work, a Korean peninsula earthquake network, constructed via event-sequential linking known as the Abe–Suzuki method, was investigated in terms of network properties. A significance test for these network properties was performed via comparisons with those of two random networks, constructed from two approaches, that is, EVENT (SEQUENCE) SHUFFLING and NETWORK (MATRIX) SHUFFLING. The Abe–Suzuki earthquake network has a clear difference from the two random networks. However, the two shuffled networks exhibited completely different functions, and even some network properties for one shuffled datum are significantly high and those of the other shuffled data are low compared to actual data. For most cases, the event-shuffled network showed a functional similarity to the real network, but with different exponents/parameters. This result strongly claims that the Korean peninsula earthquake network has a spatiotemporal causal relation. Additionally, the Korean peninsula network properties are mostly similar to those found in previous studies on the US and Japan. Further, the Korean earthquake network showed strong linearity in a specific range of spatial resolution, that is, 0.20°~0.80°, implying that macroscopic properties of the Korean earthquake network are highly regular in this range of resolution.

Seismicity Patterns Prior to the Thessaly (Mw6.3) Strong Earthquake on 3 March 2021 in Terms of Multiresolution Wavelets and Natural Time Analysis

On 3 March 2021, a strong, shallow earthquake of moment magnitude, Mw6.3, occurred in northern Thessaly (Central Greece). To investigate possible complex correlations in the evolution of seismicity in the broader area of Central Greece before the Mw6.3 event, we apply the methods of multiresolution wavelet analysis (MRWA) and natural time (NT) analysis. The description of seismicity evolution by critical parameters defined by NT analysis, integrated with the results of MRWA as the initiation point for the NT analysis, forms a new framework that may possibly lead to new universal principles that describe the generation processes of strong earthquakes. In the present work, we investigate this new framework in the seismicity prior to the Mw6.3 Thessaly earthquake. Initially, we apply MRWA to the interevent time series of the successive regional earthquakes in order to investigate the approach of the regional seismicity at critical stages and to define the starting point of the natural time domain. Then, we apply the NT analysis, showing that the regional seismicity approached criticality a few days before the occurrence of the Mw6.3 earthquake, when the κ1 natural time parameter reached the critical value of κ1 = 0.070.

Discrete Langevin-type equation for p-order persistent time series and procedure of its reconstruction

The stochastic discrete Langevin-type equation, which can describe p-order persistent processes, was introduced. The procedure of reconstruction of the equation from time series was proposed and tested on synthetic data. The approach was applied to hydrological data leading to the stochastic model of the phenomenon. The work is a substantial extension of our paper [Chaos 26, 053109 (2016)], in which the persistence of order 1 was taken into account.

Statistical and Criticality Analysis of the Lower Ionosphere Prior to the 30 October 2020 Samos (Greece) Earthquake (M6.9), Based on VLF Electromagnetic Propagation Data as Recorded by a New VLF/LF Receiver Installed in Athens (Greece)

In this work we present the statistical and criticality analysis of the very low frequency (VLF) sub-ionospheric propagation data recorded by a VLF/LF radio receiver which has recently been established at the University of West Attica in Athens (Greece). We investigate a very recent, strong (M6.9), and shallow earthquake (EQ) that occurred on 30 October 2020, very close to the northern coast of the island of Samos (Greece). We focus on the reception data from two VLF transmitters, located in Turkey and Israel, on the basis that the EQ’s epicenter was located within or very close to the 5th Fresnel zone, respectively, of the corresponding sub-ionospheric propagation path. Firstly, we employed in our study the conventional analyses known as the nighttime fluctuation method (NFM) and the terminator time method (TTM), aiming to reveal any statistical anomalies prior to the EQ’s occurrence. These analyses revealed statistical anomalies in the studied sub-ionospheric propagation paths within ~2 weeks and a few days before the EQ’s occurrence. Secondly, we performed criticality analysis using two well-established complex systems’ time series analysis methods—the natural time (NT) analysis method, and the method of critical fluctuations (MCF). The NT analysis method was applied to the VLF propagation quantities of the NFM, revealing criticality indications over a period of ~2 weeks prior to the Samos EQ, whereas MCF was applied to the raw receiver amplitude data, uncovering the time excerpts of the analyzed time series that present criticality which were closest before the Samos EQ. Interestingly, power-law indications were also found shortly after the EQ’s occurrence. However, it is shown that these do not correspond to criticality related to EQ preparation processes. Finally, it is noted that no other complex space-sourced or geophysical phenomenon that could disturb the lower ionosphere did occur during the studied time period or close after, corroborating the view that our results prior to the Samos EQ are likely related to this mainshock.

Reports on progress in physics the complex dynamics of earthquake fault systems: new approaches to forecasting and nowcasting of earthquakes

Charles Richter's observation that 'only fools and charlatans predict earthquakes,' reflects the fact that despite more than 100 years of effort, seismologists remain unable to do so with reliable and accurate results. Meaningful prediction involves specifying the location, time, and size of an earthquake before it occurs to greater precision than expected purely by chance from the known statistics of earthquakes in an area. In this context, 'forecasting' implies a prediction with a specification of a probability of the time, location, and magnitude. Two general approaches have been used. In one, the rate of motion accumulating across faults and the amount of slip in past earthquakes is used to infer where and when future earthquakes will occur and the shaking that would be expected. Because the intervals between earthquakes are highly variable, these long-term forecasts are accurate to no better than a hundred years. They are thus valuable for earthquake hazard mitigation, given the long lives of structures, but have clear limitations. The second approach is to identify potentially observable changes in the Earth that precede earthquakes. Various precursors have been suggested, and may have been real in certain cases, but none have yet proved to be a general feature preceding all earthquakes or to stand out convincingly from the normal variability of the Earth's behavior. However, new types of data, models, and computational power may provide avenues for progress using machine learning that were not previously available. At present, it is unclear whether deterministic earthquake prediction is possible. The frustrations of this search have led to the observation that (echoing Yogi Berra) 'it is difficult to predict earthquakes, especially before they happen.' However, because success would be of enormous societal benefit, the search for methods of earthquake prediction and forecasting will likely continue. In this review, we note that the focus is on anticipating the earthquake rupture before it occurs, rather than characterizing it rapidly just after it occurs. The latter is the domain of earthquake early warning, which we do not treat in detail here, although we include a short discussion in the machine learning section at the end.

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

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

Transforming Gaussian correlations. Applications to generating long-range power-law correlated time series with arbitrary distribution

The observable outputs of many complex dynamical systems consist of time series exhibiting autocorrelation functions of great diversity of behaviors, including long-range power-law autocorrelation functions, as a signature of interactions operating at many temporal or spatial scales. Often, numerical algorithms able to generate correlated noises reproducing the properties of real time series are used to study and characterize such systems. Typically, many of those algorithms produce a Gaussian time series. However, the real, experimentally observed time series are often non-Gaussian and may follow distributions with a diversity of behaviors concerning the support, the symmetry, or the tail properties. It is always possible to transform a correlated Gaussian time series into a time series with a different marginal distribution, but the question is how this transformation affects the behavior of the autocorrelation function. Here, we study analytically and numerically how the Pearson's correlation of two Gaussian variables changes when the variables are transformed to follow a different destination distribution. Specifically, we consider bounded and unbounded distributions, symmetric and non-symmetric distributions, and distributions with different tail properties from decays faster than exponential to heavy-tail cases including power laws, and we find how these properties affect the correlation of the final variables. We extend these results to a Gaussian time series, which are transformed to have a different marginal distribution, and show how the autocorrelation function of the final non-Gaussian time series depends on the Gaussian correlations and on the final marginal distribution. As an application of our results, we propose how to generalize standard algorithms producing a Gaussian power-law correlated time series in order to create a synthetic time series with an arbitrary distribution and controlled power-law correlations. Finally, we show a practical example of this algorithm by generating time series mimicking the marginal distribution and the power-law tail of the autocorrelation function of real time series: the absolute returns of stock prices.

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

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

Natural Time Analysis of Global Navigation Satellite System Surface Deformation: The Case of the 2016 Kumamoto Earthquakes

In order to have further evidence of the atmospheric oscillation channel of the lithosphere-atmosphere-ionosphere coupling (LAIC), we have studied criticality in global navigation satellite system (GNSS) surface deformation as a possible agent for exciting atmospheric gravity waves (AGWs) in the atmosphere and GNSS fluctuations in the frequency range of AGWs with the use of the natural time (NT) method. The target earthquake (EQ) is the 2016 Kumamoto EQ with its main shock on 15 April 2016 (M = 7.3, universal time). As the result of the application of the NT method to GNSS data, we found that for the one-day sampled GNSS deformation data and its fluctuations in two AGW bands of 20-100 and 100-300 min, we could detect a criticality in the period of 1-14 April, which was one day to two weeks before the EQ. These dates of criticalities are likely to overlap with the time periods of previous results on clear AGW activity in the stratosphere and on the lower ionospheric perturbation. Hence, we suggest that the surface deformation could be a possible candidate for exciting those AGWs in the stratosphere, leading to the lower ionospheric perturbation, which lends further support to the AGW hypothesis of the LAIC process.

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

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

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

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

A Prototype Photoplethysmography Electronic Device that Distinguishes Congestive Heart Failure from Healthy Individuals by Applying Natural Time Analysis

In this paper, a prototype photoplethysmography (PPG) electronic device is presented for the distinction of individuals with congestive heart failure (CHF) from the healthy (H) by applying the concept of Natural Time Analysis (NTA). Data were collected simultaneously with a conventional three-electrode electrocardiography (ECG) system and our prototype PPG electronic device from H and CHF volunteers at the 2nd Department of Cardiology, Medical School of Ioannina, Greece. Statistical analysis of the results show a clear separation of CHF from H subjects by means of NTA for both the conventional ECG system and our PPG prototype system, with a clearly better distinction for the second one which additionally inherits the advantages of a low-cost portable device.

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

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

On Possible Electromagnetic Precursors to a Significant Earthquake (Mw = 6.3) Occurred in Lesvos (Greece) on 12 June 2017

This paper reports an attempt to use ultra-low-frequency (ULF) magnetic field data from a space weather monitoring magnetometer array in the study of earthquake (EQ) precursors in Greece. The data from four magnetometer stations of the HellENIc GeoMagnetic Array (ENIGMA) have been analyzed in the search for possible precursors to a strong EQ that occurred south of Lesvos Island on 12 June 2017, with magnitude Mw = 6.3 and focal depth = 12 km. The analysis includes conventional statistical methods, as well as criticality analysis, using two independent methods, the natural time (NT) method and the method of critical fluctuations (MCF). In terms of conventional statistical methods, it is found that the most convincing ULF precursor was observed in the data of ULF (20–30 mHz) depression (depression of the horizontal component of the magnetic field), which is indicative of lower ionospheric perturbation just 1 day before the EQ. Additionally, there are indications of a precursor in the direct ULF emission from the lithosphere 4 days to 1 day before the EQ. Further study in terms of NT analysis identifies criticality characteristics from 8 to 2 days before the EQ both for lithospheric ULF emission and ULF depression, while MCF reveals indications of criticality in all recorded magnetic field components, extending from 10 to 3 days before the EQ. Beyond the recordings of the geomagnetic stations of ENIGMA, the recordings of the fracto-electromagnetic emission stations of the HELlenic Seismo-ElectroMagnetics Network (ELSEM-Net) in Greece have been analyzed. The MHz recordings at the station that is located on Lesvos Island presented criticality characteristics (by means of both NT analysis and MCF) 11 days before the EQ, while a few days later (7–6 days before the EQ), the kHz recordings of the same station presented tricritical behavior. It is noted that the magnetosphere was quiet for a period of two weeks before the EQ and including its occurrence.

Statistical physics models for aftershocks and induced seismicity

A standard approach to quantifying the seismic hazard is the relative intensity (RI) method. It is assumed that the rate of seismicity is constant in time and the rate of occurrence of small earthquakes is extrapolated to large earthquakes using Gutenberg-Richter scaling. We introduce nowcasting to extend RI forecasting to time-dependent seismicity, for example, during an aftershock sequence. Nowcasting uses 'natural time'; in seismicity natural time is the event count of small earthquakes. The event count for small earthquakes is extrapolated to larger earthquakes using Gutenberg-Richter scaling. We first review the concepts of natural time and nowcasting and then illustrate seismic nowcasting with three examples. We first consider the aftershock sequence of the 2004 Parkfield earthquake on the San Andreas fault in California. Some earthquakes have higher rates of aftershock activity than other earthquakes of the same magnitude. Our approach allows the determination of the rate in real time during the aftershock sequence. We also consider two examples of induced earthquakes. Large injections of waste water from petroleum extraction have generated high rates of induced seismicity in Oklahoma. The extraction of natural gas from the Groningen gas field in The Netherlands has also generated very damaging earthquakes. In order to reduce the seismic activity, rates of injection and withdrawal have been reduced in these two cases. We show how nowcasting can be used to assess the success of these efforts.This article is part of the theme issue 'Statistical physics of fracture and earthquakes'.

Study in Natural Time of Geoelectric Field and Seismicity Changes Preceding the Mw6.8 Earthquake on 25 October 2018 in Greece

A strong earthquake of magnitude M w 6.8 struck Western Greece on 25 October 2018 with an epicenter at 37.515 ∘ N 20.564 ∘ E. It was preceded by an anomalous geolectric signal that was recorded on 2 October 2018 at a measuring station 70 km away from the epicenter. Upon analyzing this signal in natural time, we find that it conforms to the conditions suggested for its identification as precursory Seismic Electric Signal (SES) activity. Notably, the observed lead time of 23 days lies within the range of values that has been very recently identified as being statistically significant for the precursory variations of the electric field of the Earth. Moreover, the analysis in natural time of the seismicity subsequent to the SES activity in the area candidate to suffer this strong earthquake reveals that the criticality conditions were obeyed early in the morning of 18 October 2018, i.e., almost a week before the strong earthquake occurrence, in agreement with earlier findings. Finally, when employing the recent method of nowcasting earthquakes, which is based on natural time, we find an earthquake potential score around 80%.

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

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

Study of Geo-Electric Data Collected by the Joint EMSEV-Bishkek RS-RAS Cooperation: Possible Earthquake Precursors

By employing the cross-correlogram method, in geo-electric data from the area of Kyrgyzstan for the period 30 June 2014–10 June 2015, we identified Anomalous Telluric Currents (ATC). From a total of 32 ATC after taking into consideration the electric current source properties, we found that three of them are possible Seismic Electric Signal (SES) activities. These three SES activities are likely to be linked with three local seismic events. Finally, by studying the corresponding recordings when a DC alternating source injects current into the Earth, we found that the subsurface resistivity seems to be reduced before one of these three earthquakes, but a similar analysis for the other two cannot be done due to their large epicentral distance and the lack of data.

Time series analysis in earthquake complex networks

We introduce a new method of characterizing the seismic complex systems using a procedure of transformation from complex networks into time series. The undirected complex network is constructed from seismic hypocenters data. Network nodes are marked by their connectivity. The walk on the graph following the time of succeeding seismic events generates the connectivity time series which contains, both the space and time, features of seismic processes. This procedure was applied to four seismic data sets registered in Chile. It was shown that multifractality of constructed connectivity time series changes due to the particular geophysics characteristics of the seismic zones-it decreases with the occurrence of large earthquakes-and shows the spatiotemporal organization of these seismic systems.

Statistical Significance of Earth's Electric and Magnetic Field Variations Preceding Earthquakes in Greece and Japan Revisited

By analyzing the seismicity in a new time domain, termed natural time, we recently found that the change of the entropy under time reversal (Physica A2018, 506, 625–634) and the relevant complexity measures (Entropy2018, 20, 477) exhibit pronounced variations before the occurrence of the M8.2 earthquake in Mexico on 7 September 2017. Here, the statistical significance of precursory phenomena associated with other physical properties and in particular the anomalous variations observed in the Earth’s electric and magnetic fields before earthquakes in different regions of the world and in particular in Greece since 1980s and Japan during 2001–2010 are revisited (the latter, i.e., the magnetic field variations are alternatively termed ultra low frequency (ULF) seismo-magnetic phenomena). Along these lines we employ modern statistical tools like the event coincidence analysis and the receiver operating characteristics technique. We find that these precursory variations are far beyond chance and in addition their lead times fully agree with the experimental findings in Greece since the 1980s.

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

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

Criticality Analysis of the Lower Ionosphere Perturbations Prior to the 2016 Kumamoto (Japan) Earthquakes as Based on VLF Electromagnetic Wave Propagation Data Observed at Multiple Stations

The perturbations of the ionosphere which are observed prior to significant earthquakes (EQs) have long been investigated and could be considered promising for short-term EQ prediction. One way to monitor ionospheric perturbations is by studying VLF/LF electromagnetic wave propagation through the lower ionosphere between specific transmitters and receivers. For this purpose, a network of eight receivers has been deployed throughout Japan which receive subionospheric signals from different transmitters located both in the same and other countries. In this study we analyze, in terms of the recently proposed natural time analysis, the data recorded by the above-mentioned network prior to the catastrophic 2016 Kumamoto fault-type EQs, which were as huge as the former 1995 Kobe EQ. These EQs occurred within a two-day period (14 April: MW=6.2 and MW=6.0, 15 April: MW=7.0) at shallow depths (~10 km), while their epicenters were adjacent. Our results show that lower ionospheric perturbations present critical dynamics from two weeks up to two days before the main shock occurrence. The results are compared to those by the conventional nighttime fluctuation method obtained for the same dataset and exhibit consistency. Finally, the temporal evolutions of criticality in ionospheric parameters and those in the lithosphere as seen from the ULF electromagnetic emissions are discussed in the context of the lithosphere-atmosphere-ionosphere coupling.

On the progress of the 2015–2016 El Niño event

Abstract. It has been recently reported that the current 2015–2016 El Niño could become "one of the strongest on record". To further explore this claim, we performed the new analysis described in detail in Varotsos et al. (2015) that allows the detection of precursory signals of the strong El Niño events by using a recently developed non-linear dynamics tool. In this context, the analysis of the Southern Oscillation Index time series for the period 1876–2015 shows that the running 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.

Technical note: On the progress of the 2015–2016 El Niño event.. [DOI: 10.5194/acpd-15-35787-2015]

Abstract. It has been recently reported that the current 2015–2016 El Niño could become "one of the strongest on record". To further explore this claim, we performed the new analysis described in detail in Varotsos et al. (2015) that allows the detection of precursory signals of the strong El Niño events by using a recently developed non-linear dynamics tool. In this context, the analysis of the Southern Oscillation Index time series for the period 1876–2015 shows that the running 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.

Beyond the Markovian assumption: response–excitation probabilistic solution to random nonlinear differential equations in the long time

Uncertainty quantification for dynamical systems under non-white excitation is a difficult problem encountered across many scientific and engineering disciplines. Difficulties originate from the lack of Markovian character of system responses. The response–excitation (RE) theory, recently introduced by Sapsis & Athanassoulis (Sapsis & Athanassoulis 2008 Probabilistic Eng. Mech. 23, 289–306 ( doi:10.1016/j.probengmech.2007.12.028 )) and further studied by Venturi et al. (Venturi et al. 2012 Proc. R. Soc. A 468, 759–783 ( doi:10.1098/rspa.2011.0186 )), is a new approach, based on a simple differential constraint which is exact but non-closed. The evolution equation obtained for the RE probability density function (pdf) has the form of a generalized Liouville equation, with the excitation time frozen in the time-derivative term. In this work, the missing information of the RE differential constraint is identified and a closure scheme is developed for the long-time, stationary, limit-state of scalar nonlinear random differential equations (RDEs) under coloured excitation. The closure scheme does not alter the RE evolution equation, but collects the missing information through the solution of local statistically linearized versions of the nonlinear RDE, and interposes it into the solution scheme. Numerical results are presented for two examples, and compared with Monte Carlo simulations.

Minima of the fluctuations of the order parameter of global seismicity

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

Detrended fluctuation analysis as a regression framework: estimating dependence at different scales

We propose a framework combining detrended fluctuation analysis with standard regression methodology. The method is built on detrended variances and covariances and it is designed to estimate regression parameters at different scales and under potential nonstationarity and power-law correlations. The former feature allows for distinguishing between effects for a pair of variables from different temporal perspectives. The latter ones make the method a significant improvement over the standard least squares estimation. Theoretical claims are supported by Monte Carlo simulations. The method is then applied on selected examples from physics, finance, environmental science, and epidemiology. For most of the studied cases, the relationship between variables of interest varies strongly across scales.

Application of the Huang-Hilbert transform and natural time to the analysis of seismic electric signal activities

The Huang method is applied to Seismic Electric Signal (SES) activities in order to decompose them into their components, named Intrinsic Mode Functions (IMFs). We study which of these components contribute to the basic characteristics of the signal. The Hilbert transform is then applied to the IMFs in order to determine their instantaneous amplitudes. The results are compared with those obtained from the analysis in a new time domain termed natural time, after having subtracted the magnetotelluric background from the original signal. It is shown that these instantaneous amplitudes, when combined with the natural time analysis, can be used for the distinction of SES from artificial noises.

On the increase of the "non-uniform" scaling of the magnetic field variations before the M(w)9.0 earthquake in Japan in 2011

By applying Detrended Fluctuation Analysis (DFA) to the time series of the geomagnetic data recorded at three measuring stations in Japan, Rong et al. in 2012 recently reported that anomalous magnetic field variations were identified well before the occurrence of the disastrous Tohoku Mw9.0 earthquake that occurred on 11 March 2011 in Japan exhibiting increased "non-uniform" scaling behavior. Here, we provide an explanation for the appearance of this increase of "non-uniform" scaling on the following grounds: These magnetic field variations are the ones that accompany the electric field variations termed Seismic Electric Signals (SES) activity which have been repeatedly reported that precede major earthquakes. DFA as well as multifractal DFA reveal that the latter electric field variations exhibit scaling behavior as shown by analyzing SES activities observed before major earthquakes in Greece. Hence, when these variations are superimposed on a background of pseudosinusoidal trend, their long range correlation properties-quantified by DFA-are affected resulting in an increase of the "non-uniform" scaling behavior. The same is expected to hold for the former magnetic field variations. This explanation is strengthened by recent findings showing that the fluctuations of the order parameter of seismicity exhibited an unprecedented minimum almost two months before the Tohoku earthquake occurrence which is characteristic for an almost simultaneous emission of Seismic Electric Signals activity.

Plausible reasons for the inconsistencies between the modeled and observed temperatures in the tropical troposphere

We hereby attempt to detect plausible reasons for the discrepancies between the measured and modeled tropospheric temperature anomalies in the tropics. For this purpose, we calculate the trends of the upper‐minus‐lower tropospheric temperature anomaly differences (TAD) for both the measured and modeled time series during 1979–2010. The modeled TAD trend is significantly higher than that of the measured ones, confirming that the vertical amplification of warming is exaggerated in models. To investigate the cause of this exaggeration, we compare the intrinsic properties of the measured and modeled TAD by employing detrended fluctuation analysis (DFA). The DFA exponent obtained for the measured values reveals white noise behavior, while the exponent for the modeled ones shows that they exhibit long‐range power law correlations. We suggest that the vertical amplification of warming derived from modeled simulations is weighted with a persistent signal, which should be removed in order to achieve better agreement with observations.