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Lévy and Gauss statistics in the preparation of an earthquake
Physica A 528 (2019) 121360
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Lévy and Gauss statistics in the preparation of an earthquake ∗
Stelios M. Potirakis a , , Yiannis Contoyiannis a , Konstantinos Eftaxias b a
Department of Electrical and Electronics Engineering, University of West Attica, Campus 2, 250 Thivon and P. Ralli, Aigaleo, Athens GR-12244, Greece b Department of Physics, University of Athens, Panepistimiopolis, Zografos, Athens GR-15784, Greece
highlights • • • •
Fracture-induced electromagnetic emissions in the MHz band are analyzed. Intermittency-induced criticality has repeatedly been identified. The results are re-evaluated in view of Lévy flight and Gaussian processes. Power law exponent p2 ∈ (1.5,2) indicates that no strong EQ is expected.
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Article history: Received 27 February 2019 Received in revised form 20 April 2019 Available online 8 May 2019 Keywords: Critical dynamics Lévy flight Gaussian processes Fracture-induced electromagnetic emissions Time-series analysis
a b s t r a c t In course of the preparation of a strong earthquake (EQ), the fracture of the highly heterogeneous system surrounding the fault has been suggested to follow the theory of critical phenomena and continuous phase transitions. Among the variety of electromagnetic phenomena which have been reported to be observed prior to a strong EQ, the ground-observed fracture-induced electromagnetic emissions (EME) in the MHz band have repeatedly been reported to present intermittency-induced criticality, while it has been suggested that reflect the specific stage of EQ preparation. Moreover, based on the analysis of pre-EQ MHz EME time series, it has been suggested that the underlying cracking procedure is organized to criticality by means of the Lévy flight mechanism. In an attempt to gain a deeper understanding to the involved EQ preparation processes and the organization to critical state, we study here the pre-EQ MHz EME presenting critical characteristics in terms of the mechanism of Lévy flight and its possible engagement with Gaussian stochastic processes. We suggest that if the analysis of the MHz EME time series according to the method of critical fluctuations indicates critical characteristics with a power law exponent, p2 , value in the range (1.5, 2), then a strong EQ should not be expected to follow. This is because such a result indicates a slow transition from the Lévy to the Gaussian process, cutting-off the long scales due to the Gaussian mechanism. This prevents the organization of the critical state to be completed according to the Lévy flight scenario and the spreading of the correlations all over the focal area. The above hypothesis is corroborated by a number of MHz EME time series analyses. © 2019 Elsevier B.V. All rights reserved.
1. Introduction Along with many other phenomena observed prior to earthquakes (EQs), a wide variety of electromagnetic (EM) phenomena possibly related with EQ preparation processes have been reported in the literature during the last few ∗ Corresponding author. E-mail addresses: [email protected] (S.M. Potirakis), [email protected] (Y. Contoyiannis), [email protected] (K. Eftaxias). https://doi.org/10.1016/j.physa.2019.121360 0378-4371/© 2019 Elsevier B.V. All rights reserved.
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decades, e.g., [1–7]. Such a group of phenomena concern the referred to as the MHz–kHz fracture-induced electromagnetic emissions, or fracto-electromagnetic emissions, (EME). Summarizing twenty years research results on this field, based on a multidisciplinary analysis of the recorded time series, a four-stage model of EQ generation in terms of fracture-induced electromagnetic emissions has recently been proposed, e.g., [8–12]: First stage: The initially observed MHz EM anomaly is due to the fracture of the highly heterogeneous system that surrounds the formation of strong brittle and high-strength entities (asperities) distributed along the rough surfaces of the main fault sustaining the system. The MHz EME can be described by means of a second-order phase transition in equilibrium. The appearance of the symmetry breaking phenomenon signifies the departure from critical state. Second stage: The appearance of tri-critical behavior in the final stage of MHz EME, or in the initial stage of kHz EME, or in both, signalizes a next, distinct, state of the EQ preparation process. Third stage: The finally abruptly emerging strong sequence of kHz EM avalanches originates in the stage of stick–slip-like plastic flow, namely, the fracture of asperities themselves. The burst-like kHz EME does not present any footprint of a second-order transition in equilibrium. Fourth stage: Finally, the systematically observed EM silence in all frequency bands before the time of the EQ occurrence is sourced in the process of preparation of the dynamical slip which results to the fast, even super-shear, mode that surpasses the shear wave speed and corresponds to the observed EQ tremor. We focus here on the first stage of the above model. The associated MHz EM precursors include a ‘‘critical epoch’’ when the ‘‘short-range’’ correlations evolve into ‘‘long-range’’ ones, while an epoch of localization of the damage (departure from critical state) signalizes their cease [8,11,12]. In the framework of the strong view that the fracture of the highly heterogeneous system is a result which follows the theory of critical phenomena and continuous phase transitions (e.g., [13,14]), we have argued that this organization, as reflected on the MHz EME observable, is governed by intermittent dynamics. More precisely, we have found, by applying the method critical fluctuations (MCF) [9,15–17] to the pre-EQ MHz EME time series, that the distribution of appropriately defined intervals, known as laminar lengths (or waiting times), follow a power-law [8,18–25]. At this point, we note that, as we already have clarified, the identification of a valid (presenting critical characteristics) MHz EME anomaly is a necessary but not sufficient condition for the occurrence of a strong EQ [8,11,12]. This happens if and only if the EQ preparation process evolves up to the third stage of the abovementioned four-stage model during which valid kHz EME anomaly is emitted. In this work we attempt to establish a criterion, in terms of the organization of the critical state, for discriminating valid pre-EQ MHz EME into two categories: (i) those corresponding to geophysical processes that may evolve to strong EQs and (ii) those corresponding to processes that are not expected to evolve to strong EQs. The establishment of such a criterion is done through the MCF analysis of MHz EME time series, the mechanism of Lévy flight and its engagement with Gaussian stochastic processes according to Koponen [26], and correlation analysis. 2. Method of critical fluctuations The description of phase transition in equilibrium is a statistical description in the configuration space. In [15] it has been shown that a dynamical in time description in terms of the order parameter fluctuations can be obtained. This dynamics is expressed by an one-dimensional intermittent map. The invariant density ρ (φ ) of such a map is characterized by a plateau which decays in a super-exponential way. Importantly, the exact dynamics at the critical point can be determined analytically for a large class of critical systems introducing the so-called critical map. This critical map can be approximated as:
φn+1 = φn + uφnz + ϵn
(1)
The shift parameter ϵn introduces a non-universal stochastic noise which is necessary for the creation of ergodicity [16]. Each physical system has its characteristic ‘‘noise’’, which is expressed through the shift parameter ϵn . Notice, for thermal systems the exponent z is connected with the isothermal critical exponent δ as z = δ + 1. We emphasize that the plateau region of the invariant density ρ (φ ) corresponds to the laminar region of the critical map where fully correlated dynamics takes place. The laminar region ends when the second nonlinear term in Eq. (1) becomes dominant. However, due to the fact that this dynamical law changes continuously with φ , the end of the laminar region φl cannot be easily defined based on a strictly quantitative criterion. Thus, the end of the laminar region should be generally treated as a varying parameter. Based on the aforementioned description of the critical fluctuations, the Method of Critical Fluctuations (MCF) develops an algorithm permitting the extraction of the critical fluctuations, if any, in a measured time series. The crucial observation in this approach is the fact that the distribution P(l) of the suitable defined laminar lengths l (waiting times in the laminar region) of the above mentioned intermittent map of Eq. (1) in the limit ϵn → 0 is given by the power law [27]: P(l) ∼ l−pl
(2)
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where the exponent pl is connected with the exponent z by pl = z −z 1 . Therefore the exponent pl is connected with the isothermal exponent δ by pl = 1 + 1δ . Inversely, the existence of a power law such as in Eq. (2), accompanied by a plateau form of the corresponding density ρ (φ ), is a signature of underlying correlated dynamics similar to the critical behavior [28]. We emphasize the fact that it is possible, in the frame of universality characteristic for critical phenomena, to give meaning to the exponent pl beyond the thermal phase transitions [17]. Up to now, the MCF has been applied on a wide variety of complex systems such as numerical experiments of thermal systems [17], pre-EQ electromagnetic signals [8,18–25,29–31], electrocardiac signals from biological tissues [20,28], resistor-inductor-diode circuits [32], and economical systems [33]. We present in the following the general characteristics of MCF. The reader is referred to the above mentioned works for more details, while a step-by-step procedure for the application of MCF comprising six simple steps can be found in [30,31]. The main aim of the MCF is to estimate the exponent pl . In summary, we produce the laminar lengths l as the lengths that result from successive φ -values obeying the condition φo ≤ φ ≤ φl , where φo is the marginally stable fixed point and φl is the end of laminar lengths. In others words, the laminar lengths are the waiting times within the laminar region. The distribution of the laminar lengths of fluctuations is then fitted by the function: g(l) ∼ l−p2 e−p3 l
(3)
We focus on the exponents p2 and p3 . If the exponent p3 is zero, then, the exponent p2 is equal to the exponent pl . The relation pl = z −z 1 suggests that the exponent pl (or p2 ) should be higher than 1. On the other hand, from the theory of critical phenomena [34] it results that the isothermal exponent δ is higher than 1. Consequently, as results from pl = 1 + 1δ , pl (= p2 ) ∈ (1, 2). In conclusion, the critical profile of the temporal fluctuations is restored for the conditions: p2 > 1 and p3 ≈ 0. A time series excerpt satisfying the abovementioned criticality conditions is often referred to as ‘‘critical window’’. As the system departs from the critical state, then, the exponent p2 decreases while simultaneously the exponent p3 increases reinforcing, in this way, the exponential character of the laminar lengths distribution. It should be noted that Eq. (3) seems to be an appropriate choice for the fitting of the laminar lengths distribution of the order parameter fluctuations of possibly EQ-related phenomena. Its main feature is the combination of a power law with an exponential factor as happens with generalized gamma distributions that have been found to successfully describe the statistical properties of waiting-times arising from fracture systems, both at the laboratory scale (acoustic emission, AE, waiting-times distributions) and the geophysical scale (earthquake waiting-times distributions) [35–38]. 3. Lévy and Gauss statistics in fracture organization The Lévy statistics is applied to physical systems which develop stable processes with main characteristic the infinite variance and infinite moments of all orders. The characteristic function of this statistics has the form Pˆ LF (k) = e−c |k|a , where 0 < a < 2 [39]. This characteristic function is the Fourier transform of the jump probability PLF (x). The later function behaves for large |x| as [39]: PLF (x) ∼ |x|−(1+a) .
(4)
The random walk process, which has the above jump probability, is known as Lévy ‘‘flight’’. In [22] the strong relation between the Lévy statistics and critical phenomena has been shown. It has been shown that the Lévy random walk is a way to organize the fractal geometry of the critical state. In real world processes the Lévy walk obeys to the requirement of finite variance of the process. The topic has been faced by Mantegna and Stanley [40] introducing the truncated Lévy flight (TLF) where an upper cutoff is introduced to the values of random variables. The TLF introduces an exponential cutoff factor to the distribution of Eq. (4) and so the corresponding expression for TLF is written as: PTLF (x) ∼ e−λ|x| |x|−(1+a)
(5)
This form is similar to the function g (l) (Eq. (3)) proposed for the fitting of the distribution of the laminar lengths of fluctuations, where p2 corresponds to 1 + a and p3 corresponds to λ. However, there is a difference in the limits of the corresponding power law exponents. Specifically, in TLF the exponent 1 + a lies between 1 and 3 while the exponent p2 of critical phenomena lies between 1 and 2. This means that the critical phenomena is a subspace of the application of TLF in nature. As it is known [26] for values α ≥ 2 one obtains a Gaussian process. In [26] the TLF model has been modified in such way as to allow an analytical calculation of the characteristic function of the distribution PTLF (x). By using solely the characteristic function Pˆ TLF (k), the existence of a smooth but slow transition of a Lévy stable process towards a Gaussian process has been demonstrated. The transition of the density of the distribution from a Lévy form to a Gaussian form can be achieved by increasing the exponent a while keeping the exponent λ close to zero. Specifically, in shorter time-scales the Lévy distribution dominates while in longer timescales the convergence to a Gaussian distribution is accomplished. In [26] this transition is obtained numerically from the probability distribution (Eq. (5)) by increasing the exponent a starting from the value 0.5 for λ = 0.05. The value of the critical exponent p2 which corresponds to a = 0.5 is p2 = 1.5, while the condition λ ≈ 0 indicates proximity to the critical state. In critical phenomena, the specific value of p2 exponent (p2 = 1.5) has a particular
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importance. Generally, the exponent z of the non-linear term of the intermittent map of Eq. (1) determines the dynamics of intermittency. As we have already mentioned in Section 2, exponent z is connected with the critical exponent δ as z = δ + 1. So, since in critical phenomena δ > 1 it follows that z > 2. On the other hand, in terms of the power spectrum Sf ∼ f −β there is a relation of exponent β with the exponent z which is different for different ranges of the value of z [27]. The z value which separates two different power spectrum behaviors in critical phenomena is z = 3 [27], discriminating therefore two z value intervals: z ∈ (2, 3) and z ∈ (3, +∞). As we already mentioned in Section 2, pl = z −z 1 = 1 + z −1 1 , while at the critical state it is p2 (= pl ) = 1 + z −1 1 , which means that the corresponding intervals for p2 value are p2 ∈ (1.5, 2) and p2 ∈ (1, 1.5), respectively. Thereupon, the value p2 = 1.5 is dividing the range of p2 exponent values, (1, 2), into two intervals corresponding to different behaviors. In this way, it is confirmed that the value a = 0.5 discriminates two statistical behaviors corresponding to different dynamics, specifically Lévy dynamics from TLF (Lévy-Gaussian) dynamics. Thus, in the following we divide the p2 exponent range into two regions, namely p2 ∈ (1, 1.5) and p2 ∈ (1.5, 2). In the first region, the critical organization is achieved by the domination of the Lévy process while in the second range a slow transition from the Lévy to the Gaussian process appears. According to the four-stage model that has been proposed (see Section 1 and [12]), the preparation of a strong EQ begins when the heterogeneous focal area is at the critical state. At this state a micro-cracking procedure is performed during which both the times and lengths of the micro-cracking are organized according to the Lévy mechanism as has been shown in [22]. The micro-fracture organization is spread by Lévy steps across all space–time scales of the focal area. This mechanism, according to Lévy walk, begins to occur at a region where the resistance to rupture growth has the minimum value. Failure nucleation develops in this area with micro-cracking and then, after a great deal of such short steps, a local fracture occurs and the walk jumps to another point to repeat the process. This picture is compatible with the view that a major fault is activated after the activation of a population of local fractures (corresponding to small EQs) in the heterogeneous region that surrounds the major fault. Note that EQ triggering has been suggested to be driven by the smallest EQs at all scales, even for the largest EQs [12,41], while the small EQs are considered the agents by which longer stress correlations are established [12,42]. Note that EQ precursors in general are expected to be sourced from a wide area around the epicenter referred to as ‘‘EQ preparation zone’’ [43,44], while the MHz EME, that are in the focus of the present study and refer to the fractures in the heterogeneous region that surrounds the major fault during the critical phase, are expected to be sourced from the so called ‘‘critical region’’ [42]. In an attempt to describe, in terms of the MHz EME observable and using intermittency terminology, the organization to the critical state through Lévy walk mechanism, one gets the following picture: In the weaker section of the focal area local micro-cracking begins. This can be observed through MHz EME amplitude fluctuations, the waiting times of which within the laminar region are called laminar lengths. The short departures of the MHz EME amplitude fluctuations from the laminar region could be attributed to the creation of new fresh surfaces through ruptures occurring within the specific section. The waiting times constitute the observable Lévy flight (temporal) steps. At some point, when the stresses exceed the fracture threshold of the boundaries of the specific section, a local fracture occurs corresponding to a small EQ (which might be under the magnitude of completeness of the recorded EQ catalog), transmitting the stress to another section of the focal area. Such a local fracture is manifested by a relatively high amplitude MHz EME burst outside the laminar region. During the development of the cracking network within a specific section of the focal area, the rupture of the weaker parts of the heterogeneous material happens first and progressively, as the stress increases, the heterogeneity is reduced as the weaker parts are fractured and the stronger parts remain. In course of this process, the longer waiting times indicate higher fracture threshold differences. Under the hypothesis that the fracture threshold difference increases as the heterogeneity reduces, it is expected that the longer waiting time related to a specific section appears just before the local fracture, signifying a Lévy flight (temporal) jump. As long as the stress has been transferred to a new section the whole process is repeated until it spans over the whole focal area. Through this mechanism, the entire focal area is engaged in this cracking organization at the critical state, while, according to what we have previously outlined about the dynamics of the critical state, the distribution of the waiting times (laminar lengths) of the emitted MHz EME follows power-law. Note that the foreshock seismicity before a strong EQ has been found to reach criticality within the time frame during which MHz EME present critical characteristics [24,25,45]. Moreover, since at the critical state it is expected the appearance of scale-free structures both in time and space, one could imply that the observed temporal Lévy flight steps are expected to correspond to spatial steps the lengths of which also follow Lévy statistics. In the framework of this scenario, one understands that the critical organization we have described for the development of the cracking structure is a necessary condition for the final occurrence of a strong EQ. At the observation level, such a process is manifested by MHz EME recordings the laminar lengths distribution of which follow a power law according to what we have set out previously. According to the above mentioned about the discrimination of two regions for the observed p2 exponent range in the laminar phase, i.e., p2 ∈ (1, 1.5) and p2 ∈ (1.5, 2), we can consider that if the exponent p2 lies within the interval (1.5, 2), a slow transition from the Lévy to the Gaussian process appears where the long scales are cut-off due to the Gaussian mechanism, so the organization of the critical state is not completed according to the Lévy flight scenario since the correlations do not spread all over the focal area. On the contrary, if the exponent p2 lies within the interval (1, 1.5) then the organization of the critical state is completed according to the Lévy flight scenario. The results obtained after the analysis of the MHz EME observed before 15 strong EQs which occurred in Greece (see Section 4) are compatible with the above suggestion.
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Table 1 15 strong EQs, which have occurred on-land or near coastline in Greece during the time period 1995–2015 for which there are available MHz EME recordings, and the associated MCF analysis results. The reader is referred to [21,24] for more details on our EME telemetric stations network deployed throughout Greece. EQ details
Critical window details
Place of occurrence (near or at)
ML
Epicenter
Kozani-Grevena * Athens * Zante * Cephalonia * Methoni * Methoni Andravida * Lamia * Agios Nikolaos * Strofades * Karpathos Limnos Chania** Cephalonia*** Lefkada
6.1 5.4 5.4 5.5 6.2 6.0 6.5 5.2 5.2 5.5 6.2 5.8 6.2 5.8 6.0
(40.18 (38.15 (37.68 (38.34 (36.50 (36.18 (37.98 (38.72 (35.66 (37.13 (35.64 (39.67 (35.50 (38.22 (38.67
◦
N, N, ◦ N, ◦ N, ◦ N, ◦ N, ◦ N, ◦ N, ◦ N, ◦ N, ◦ N, ◦ N, ◦ N, ◦ N, ◦ N, ◦
21.71 23.62 20.91 20.42 21.78 21.72 21.51 22.57 26.39 20.78 26.56 25.56 23.28 20.53 20.60
◦
E) E) ◦ E) ◦ E) ◦ E) ◦ E) ◦ E) ◦ E) ◦ E) ◦ E) ◦ E) ◦ E) ◦ E) ◦ E) ◦ E) ◦
Depth (km)
Date
Window width (s)
# days before EQ
p2max
p3min
Recorded at the station
39 30 18 15 41 25 25 24 42 15 63 31 65 16 11
13-05-1995 07-09-1999 11-4-2006 25-03-2007 14-02-2008 20-02-2008 08-06-2008 13-12-2008 13-1-2009 16-02-2009 01-04-2011 08-01-2013 12-10-2013 26-1-2014 17-11-2015
11000 20000 17000 12000 19000 20000 18000 24000 20000 10000 16000 23000 27000 10000 12000
12 9 1 7 2 5 5 1 5 1 5 2 5.5 2 3
1.49±0.06 1.31±0.046 1.37+-0.044 1.43±0.065 1.37±0.044 1.35±0.07 1.41±0.047 1.48±0.04 1.48±0.01 1.37±0.049 1.48±0.05 1.38±0.04 1.51±0.026 1.35±0.047 1.28±0.05
0.009 0.005 0.001 0.009 0 0.002 0 0 0.04 0 0.048 0.001 0.035 0 0
Zakynthos Zakynthos Zakynthos Zakynthos Zakynthos Zakynthos Zakynthos Zakynthos Neapoli Zakynthos Vamos Lesvos Vamos Cephalonia Zakynthos
4. MHz fracture-induced electromagnetic emissions analysis results In the following, we present results concerning the 15 strong, relatively shallow, seismic events which have occurred on-land or near coastline in Greece during the time period 1995–2015 and for which we have available MHz EME recordings (see Table 1). Note that the MHz EME have been analyzed by MCF for 9 of them (see events with asterisk in Table 1) in [19], while the MHz EME for the 2013 Chania and 2014 Cephalonia EQ (see events with two and three asterisks in Table 1, respectively) have been analyzed in [24,25], respectively. All details concerning EQ have been retrieved from the EQ catalogs of the Institute of Geodynamics of the National Observatory of Athens (http://www.gein.noa.gr/en/ seismicity/earthquake-catalogs). As a representative example, we present in Figs. 1 and 2 the MCF analysis results for the 2013 Limnos and the 2015 Lefkada EQs (see also Table 1), respectively. Note that we have proved [8] that the electric field amplitude of MHz EME plays the role of an order parameter. It is noted that the critical windows (time series excerpts satisfying the MCF criticality conditions presented in Section 2) presented in Table 1 were not the only ones identified prior to the specific strong EQs. The criterion for the choice of the presented results was the lowest exponent p3 in the laminar phase which corresponds to the best effective laminar region, i.e., closer to critical state. As can be seen from Table 1, for all the events the estimated p2 max exponent values, within the margin of the estimation error, belong to the interval (1.1.5). According to the theoretical analysis presented in Section 3, this result signifies that the organization of these strong events to critical state was accomplished according to the Lévy scenario in which correlations are spread throughout the whole focal area, at all scales in agreement with the theory of critical phenomena. In a seismogenic country such as Greece many small EQs occur every day. So, a question effortlessly arises on whether the emergence of such a critical window could refer to the appearance of high seismic activity during which energy is released not through a strong EQ but through many weaker EQs within a specific area. The answer is positive. One such example of high seismic activity happened between the days from 22-5-2010 to 27-5-2010 around the neighboring islands of Zakynthos (Zante) and Cephalonia. Specifically, during the specific time period and within the area roughly defined by the triangle Cephalonia–Zakynthos–Western Greece coastline, a swarm of 55 seismic events with magnitudes from 2.5 to 3.5 were recorded within the specific area (35 of them occurred on a single day!). A few days before, on 20-5-2010, the Zakynthos station recorded a critical window of 12000 s duration, for which the MCF analysis resulted to p2 max value in the range (1, 1.5), specifically the analysis yielded p2 max = 1.17 and p3 min = 0.02. We suggest that it is possible, by means of the autocorrelation function, to discriminate the criticality detected in MHz EME which is related with an upcoming strong EQ from the criticality related with a high seismic activity of weaker events within a specific geographic area. According to our suggestion presented in Section 3, a swarm of weak EQs could correspond to local fractures. However, the appearance of local fractures per se is not enough to create a strong EQ. Local fractures must be correlated to lead to a strong EQ. Since we are interested in the correlations at the laminar phase where the critical dynamics have been developed, we herein estimate the ‘‘laminar’’ autocorrelation function, CL (m), where m denotes the time-lag (time-length), by using only the parts of the critical window of the MHz EME time series that belong to the laminar phase, i.e., by using the values φo ≤ φ ≤ φl . Fig. 3 depicts the plot of CL (m) vs. m on a log–log plot for the critical windows of the 7 stronger EQ events of Table 1 (with ML ≥ 6.0) and the abovementioned swarm of weaker EQs.
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Fig. 1. (a) The time series excerpt of the MHz EME recordings of 06-01-2013, at Lesvos station associated with the 2013 Limnos EQ; the time scale refers to the time in sec starting from 00:00:00 UT of the specific day. (b) The values’ distribution for the time-series excerpt of panel (a); the value φo = 450 was chosen as the start of laminar regions. (c) A representative example of laminar lengths distribution for the end point φl = 473, i.e., for the laminar region (450, 473). The continuous line corresponds to the fitted function of the form of Eq. (3). A clear power-law is identified. (d) The exponents p2 , p3 vs. the end point φl . The validity of criticality condition p2 > 1, p3 ≈ 0 for a wide range of end point values, and consequently for a wide range of laminar regions, is evident.
From Fig. 3, it is evident that there is a qualitative differentiation between the results obtained for the weak EQs swarm and the results obtained for the strong EQs in terms of the autocorrelation function. A ‘‘collapse’’ of autocorrelation is observed on the longer scales in the case of the weak EQs swarm, thus declaring the lack of long correlations among the local fractures (attributed to the specific seismic activity) which would be necessary for the preparation of a strong event. On the contrary, the behavior observed for the strong EQ cases, namely the fact that autocorrelation function indicates long-range correlations extending up to long scales (up to a different scale for each one of them), is compatible with a full deployment of critical dynamics throughout the whole focal area including not only short correlations between micro-fractures but also long correlations among the local fractures. Moving further with our investigation, we formulate the following questions: Have there been identified any MHz EME time series excerpts for which, by application of the MCF, power law can be found in the distribution of laminar lengths with exponent p2 in the range (1.5, 2)? Did a strong EQ followed such cases? In a country of high seismic activity such as Greece, it is difficult to find relatively long time periods without any strong EQ in order to check the answer to the above posed questions. However, we were able to find such a time period of 8 months duration (April to December 2009) during which no EQs with ML > 4.5 have been recorded in Greece. Nevertheless, during the specific time period 4 critical windows were identified among the MHz EME recordings of the telemetric stations of our network as shown in Table 2. According to the MCF analysis results presented in Table 2, for all cases p2 max was estimated in the range (1.5, 2), while no strong EQ or weaker EQ swarm did happen following any of the specific critical windows. According to the theoretical analysis presented in Section 3, p2 in the range (1.5, 2), signifies a slow transition from Lévy to Gaussian process, in which the later cuts-off the long spatial and temporal steps which are necessary to be developed in the micro-cracking network at the critical state, according to what we have explained in Section 3. 5. Conclusions In this work we focused on the analysis of MHz EME using the MCF time series analysis method which has been known to identify the approach of a system to critical state as well as its departure from it. Specifically, we further investigated the hypothesis that the MHz EME recorded prior to the occurrence of a strong EQ reflect the organization of the fracture of the highly heterogeneous system surrounding the fault to criticality according to the Lévy flight mechanism.
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Fig. 2. (a) The time series excerpt of the MHz EME recordings of 14-11-2015, at Zakynthos station associated with the 2015 Lefkada EQ; the time scale refers to the time in sec starting from 06:23:20 UT of the specific day. (b) The values’ distribution for the time-series excerpt of panel (a); the value φo = 590 was chosen as the start of laminar regions. (c) A representative example of laminar lengths distribution for the end point φl = 610, i.e., for the laminar region (590, 610). The continuous line corresponds to the fitted function of the form of Eq. (3). A clear power-law is identified. (d) The exponents p2 , p3 vs. the end point φl . The validity of criticality condition p2 > 1, p3 ≈ 0 for a wide range of end point values, and consequently for a wide range of laminar regions, is evident.
Fig. 3. The log–log plots of ‘‘laminar’’ autocorrelation function CL (m) vs. m of the MHz EME critical windows detected prior to the 7 stronger EQs events of Table 1 (ML ≥ 6.0) and a swarm of weaker EQs which occurred between 22-5-2010 and 27-5-2010 around the neighboring islands of Zakynthos (Zante) and Cephalonia (see text). Note that m denotes the time-lag in time units equal to the sampling period of the time series, here 1 s.
We theoretically proved that the MCF power law exponent p2 range can be divided into two regions, namely p2 ∈ (1, 1.5) and p2 ∈ (1.5, 2). In the first region, the critical organization is achieved by the domination of the Lévy process, while in the second region a slow transition from the Lévy to the Gaussian process appears. Based on this theoretic result, we checked the hypothesis that valid pre-EQ MHz EME, i.e., a MHz EME presenting critical characteristics, could be discriminating into two categories in terms of the organization of the critical state: (i) those corresponding to geophysical processes that may evolve to strong EQs and (ii) those corresponding to processes that are not expected to evolve to strong EQs. The MCF analysis of several MHz EME which have been recorded by our ground-based observatories indicates that such a discrimination is possible.
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S.M. Potirakis, Y. Contoyiannis and K. Eftaxias / Physica A 528 (2019) 121360 Table 2 MCF analysis results for the critical windows identified in the MHz EME recordings of our telemetric stations network from April to December 2009. Date
Window duration (s)
p2max
p3min
Recorded at the station
25-04-2009 08-07-2009 15-12-2009 25-12-2009
11000 18000 18000 19000
1.71 1.80 1.80 1.80
0 0 0 0.005
Zakynthos Zakynthos Zakynthos Komotini
Specifically, we presented the MCF analysis results for the pre-EQ MHz EME recorded a few days prior to each one of 15 strong EQ events which took place in Greece. The results show that in all these cases the values of p2max exponent (the p2 value corresponding to the best effective laminar region, i.e., closer to the critical state) lies within the interval (1, 1.5). In such cases, the organization of the critical state is considered to be completed according to the Lévy flight scenario spreading correlations all over the focal area during the fracture of the highly heterogeneous system surrounding the fault. Attention is also drawn to the fact that, as we demonstrated, the MCF analysis of a MHz EME time series yielding a p2max value in the interval (1, 1.5) may also be observed prior to a swarm of smaller events. A qualitative criterion which discerns a swarm of smaller events from a strong EQ is that in the first case a ‘‘collapse’’ of autocorrelation is observed on the longer scales of the critical MHz EME recorded prior to the EQ swarm appearance. On the other hand, we demonstrated that in one relatively long period during which no strong EQs (all recorded EQs had ML ≤ 4.5) or EQ swarms occurred in Greece, MHz EME time series excerpts with critical characteristics were identified by MCF, but in all cases p2 max was estimated in the range (1.5, 2). In such cases, it is considered that the organization of the critical state is not completed according to the Lévy flight scenario since the long scales are cut-off due to the Gaussian mechanism, and accordingly the correlations do not spread all over the focal area. Thereupon, in such a case a strong EQ should not be expected to follow. It has to be mentioned that long time periods without any strong EQ are rare in Greece, rendering very difficult to check more similar cases within the frame of the present work. We consider that the main outcome of the present work is that MHz EME time series excerpts presenting, by means of the MCF analysis, critical characteristics but with p2max values in the interval (1.5, 2) are not expected to be followed by a strong EQ. However, more MHz EME results in terms of the presented analysis would be necessary to reinforce the findings of this work. It should finally be underlined that a MHz anomaly for which critical characteristics with p2,max ∈ (1, 1.5) have been identified by MCF analysis, is considered a ‘‘necessary’’ condition for the occurrence of a strong EQ event, it is not ‘‘a necessary and sufficient’’ one. Thereupon, after its appearance, which corresponds to the first stage of the proposed fourstage model, one should investigate MHz and kHz EME for the possible identification of the features corresponding to the next stages of the specific model. It is well known that EME as well as AE fracture precursors from rocks during laboratory loading tests are widely investigated (e.g., [46–52]). We consider that in the future it would be very interesting to study EME and AE time series recorded during laboratory loading experiments within the frame of the hypothesis that was investigated in the present work. References [1] P.A. Varotsos, The Physics of Seismic Electric, Terrapub, Tokyo, 2005. [2] O.A. Molchanov, M. Hayakawa, Seismo Electromagnetics and Related Phenomena: History and Latest Results, Terrapub, Tokyo, 2008. [3] S. Uyeda, T. Nagao, M. 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Lévy and Gauss statistics in the preparation of an earthquake ∗
Stelios M. Potirakis a , , Yiannis Contoyiannis a , Konstantinos Eftaxias b a
Department of Electrical and Electronics Engineering, University of West Attica, Campus 2, 250 Thivon and P. Ralli, Aigaleo, Athens GR-12244, Greece b Department of Physics, University of Athens, Panepistimiopolis, Zografos, Athens GR-15784, Greece
highlights • • • •
Fracture-induced electromagnetic emissions in the MHz band are analyzed. Intermittency-induced criticality has repeatedly been identified. The results are re-evaluated in view of Lévy flight and Gaussian processes. Power law exponent p2 ∈ (1.5,2) indicates that no strong EQ is expected.
article
info
Article history: Received 27 February 2019 Received in revised form 20 April 2019 Available online 8 May 2019 Keywords: Critical dynamics Lévy flight Gaussian processes Fracture-induced electromagnetic emissions Time-series analysis
a b s t r a c t In course of the preparation of a strong earthquake (EQ), the fracture of the highly heterogeneous system surrounding the fault has been suggested to follow the theory of critical phenomena and continuous phase transitions. Among the variety of electromagnetic phenomena which have been reported to be observed prior to a strong EQ, the ground-observed fracture-induced electromagnetic emissions (EME) in the MHz band have repeatedly been reported to present intermittency-induced criticality, while it has been suggested that reflect the specific stage of EQ preparation. Moreover, based on the analysis of pre-EQ MHz EME time series, it has been suggested that the underlying cracking procedure is organized to criticality by means of the Lévy flight mechanism. In an attempt to gain a deeper understanding to the involved EQ preparation processes and the organization to critical state, we study here the pre-EQ MHz EME presenting critical characteristics in terms of the mechanism of Lévy flight and its possible engagement with Gaussian stochastic processes. We suggest that if the analysis of the MHz EME time series according to the method of critical fluctuations indicates critical characteristics with a power law exponent, p2 , value in the range (1.5, 2), then a strong EQ should not be expected to follow. This is because such a result indicates a slow transition from the Lévy to the Gaussian process, cutting-off the long scales due to the Gaussian mechanism. This prevents the organization of the critical state to be completed according to the Lévy flight scenario and the spreading of the correlations all over the focal area. The above hypothesis is corroborated by a number of MHz EME time series analyses. © 2019 Elsevier B.V. All rights reserved.
1. Introduction Along with many other phenomena observed prior to earthquakes (EQs), a wide variety of electromagnetic (EM) phenomena possibly related with EQ preparation processes have been reported in the literature during the last few ∗ Corresponding author. E-mail addresses: [email protected] (S.M. Potirakis), [email protected] (Y. Contoyiannis), [email protected] (K. Eftaxias). https://doi.org/10.1016/j.physa.2019.121360 0378-4371/© 2019 Elsevier B.V. All rights reserved.
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decades, e.g., [1–7]. Such a group of phenomena concern the referred to as the MHz–kHz fracture-induced electromagnetic emissions, or fracto-electromagnetic emissions, (EME). Summarizing twenty years research results on this field, based on a multidisciplinary analysis of the recorded time series, a four-stage model of EQ generation in terms of fracture-induced electromagnetic emissions has recently been proposed, e.g., [8–12]: First stage: The initially observed MHz EM anomaly is due to the fracture of the highly heterogeneous system that surrounds the formation of strong brittle and high-strength entities (asperities) distributed along the rough surfaces of the main fault sustaining the system. The MHz EME can be described by means of a second-order phase transition in equilibrium. The appearance of the symmetry breaking phenomenon signifies the departure from critical state. Second stage: The appearance of tri-critical behavior in the final stage of MHz EME, or in the initial stage of kHz EME, or in both, signalizes a next, distinct, state of the EQ preparation process. Third stage: The finally abruptly emerging strong sequence of kHz EM avalanches originates in the stage of stick–slip-like plastic flow, namely, the fracture of asperities themselves. The burst-like kHz EME does not present any footprint of a second-order transition in equilibrium. Fourth stage: Finally, the systematically observed EM silence in all frequency bands before the time of the EQ occurrence is sourced in the process of preparation of the dynamical slip which results to the fast, even super-shear, mode that surpasses the shear wave speed and corresponds to the observed EQ tremor. We focus here on the first stage of the above model. The associated MHz EM precursors include a ‘‘critical epoch’’ when the ‘‘short-range’’ correlations evolve into ‘‘long-range’’ ones, while an epoch of localization of the damage (departure from critical state) signalizes their cease [8,11,12]. In the framework of the strong view that the fracture of the highly heterogeneous system is a result which follows the theory of critical phenomena and continuous phase transitions (e.g., [13,14]), we have argued that this organization, as reflected on the MHz EME observable, is governed by intermittent dynamics. More precisely, we have found, by applying the method critical fluctuations (MCF) [9,15–17] to the pre-EQ MHz EME time series, that the distribution of appropriately defined intervals, known as laminar lengths (or waiting times), follow a power-law [8,18–25]. At this point, we note that, as we already have clarified, the identification of a valid (presenting critical characteristics) MHz EME anomaly is a necessary but not sufficient condition for the occurrence of a strong EQ [8,11,12]. This happens if and only if the EQ preparation process evolves up to the third stage of the abovementioned four-stage model during which valid kHz EME anomaly is emitted. In this work we attempt to establish a criterion, in terms of the organization of the critical state, for discriminating valid pre-EQ MHz EME into two categories: (i) those corresponding to geophysical processes that may evolve to strong EQs and (ii) those corresponding to processes that are not expected to evolve to strong EQs. The establishment of such a criterion is done through the MCF analysis of MHz EME time series, the mechanism of Lévy flight and its engagement with Gaussian stochastic processes according to Koponen [26], and correlation analysis. 2. Method of critical fluctuations The description of phase transition in equilibrium is a statistical description in the configuration space. In [15] it has been shown that a dynamical in time description in terms of the order parameter fluctuations can be obtained. This dynamics is expressed by an one-dimensional intermittent map. The invariant density ρ (φ ) of such a map is characterized by a plateau which decays in a super-exponential way. Importantly, the exact dynamics at the critical point can be determined analytically for a large class of critical systems introducing the so-called critical map. This critical map can be approximated as:
φn+1 = φn + uφnz + ϵn
(1)
The shift parameter ϵn introduces a non-universal stochastic noise which is necessary for the creation of ergodicity [16]. Each physical system has its characteristic ‘‘noise’’, which is expressed through the shift parameter ϵn . Notice, for thermal systems the exponent z is connected with the isothermal critical exponent δ as z = δ + 1. We emphasize that the plateau region of the invariant density ρ (φ ) corresponds to the laminar region of the critical map where fully correlated dynamics takes place. The laminar region ends when the second nonlinear term in Eq. (1) becomes dominant. However, due to the fact that this dynamical law changes continuously with φ , the end of the laminar region φl cannot be easily defined based on a strictly quantitative criterion. Thus, the end of the laminar region should be generally treated as a varying parameter. Based on the aforementioned description of the critical fluctuations, the Method of Critical Fluctuations (MCF) develops an algorithm permitting the extraction of the critical fluctuations, if any, in a measured time series. The crucial observation in this approach is the fact that the distribution P(l) of the suitable defined laminar lengths l (waiting times in the laminar region) of the above mentioned intermittent map of Eq. (1) in the limit ϵn → 0 is given by the power law [27]: P(l) ∼ l−pl
(2)
S.M. Potirakis, Y. Contoyiannis and K. Eftaxias / Physica A 528 (2019) 121360
3
where the exponent pl is connected with the exponent z by pl = z −z 1 . Therefore the exponent pl is connected with the isothermal exponent δ by pl = 1 + 1δ . Inversely, the existence of a power law such as in Eq. (2), accompanied by a plateau form of the corresponding density ρ (φ ), is a signature of underlying correlated dynamics similar to the critical behavior [28]. We emphasize the fact that it is possible, in the frame of universality characteristic for critical phenomena, to give meaning to the exponent pl beyond the thermal phase transitions [17]. Up to now, the MCF has been applied on a wide variety of complex systems such as numerical experiments of thermal systems [17], pre-EQ electromagnetic signals [8,18–25,29–31], electrocardiac signals from biological tissues [20,28], resistor-inductor-diode circuits [32], and economical systems [33]. We present in the following the general characteristics of MCF. The reader is referred to the above mentioned works for more details, while a step-by-step procedure for the application of MCF comprising six simple steps can be found in [30,31]. The main aim of the MCF is to estimate the exponent pl . In summary, we produce the laminar lengths l as the lengths that result from successive φ -values obeying the condition φo ≤ φ ≤ φl , where φo is the marginally stable fixed point and φl is the end of laminar lengths. In others words, the laminar lengths are the waiting times within the laminar region. The distribution of the laminar lengths of fluctuations is then fitted by the function: g(l) ∼ l−p2 e−p3 l
(3)
We focus on the exponents p2 and p3 . If the exponent p3 is zero, then, the exponent p2 is equal to the exponent pl . The relation pl = z −z 1 suggests that the exponent pl (or p2 ) should be higher than 1. On the other hand, from the theory of critical phenomena [34] it results that the isothermal exponent δ is higher than 1. Consequently, as results from pl = 1 + 1δ , pl (= p2 ) ∈ (1, 2). In conclusion, the critical profile of the temporal fluctuations is restored for the conditions: p2 > 1 and p3 ≈ 0. A time series excerpt satisfying the abovementioned criticality conditions is often referred to as ‘‘critical window’’. As the system departs from the critical state, then, the exponent p2 decreases while simultaneously the exponent p3 increases reinforcing, in this way, the exponential character of the laminar lengths distribution. It should be noted that Eq. (3) seems to be an appropriate choice for the fitting of the laminar lengths distribution of the order parameter fluctuations of possibly EQ-related phenomena. Its main feature is the combination of a power law with an exponential factor as happens with generalized gamma distributions that have been found to successfully describe the statistical properties of waiting-times arising from fracture systems, both at the laboratory scale (acoustic emission, AE, waiting-times distributions) and the geophysical scale (earthquake waiting-times distributions) [35–38]. 3. Lévy and Gauss statistics in fracture organization The Lévy statistics is applied to physical systems which develop stable processes with main characteristic the infinite variance and infinite moments of all orders. The characteristic function of this statistics has the form Pˆ LF (k) = e−c |k|a , where 0 < a < 2 [39]. This characteristic function is the Fourier transform of the jump probability PLF (x). The later function behaves for large |x| as [39]: PLF (x) ∼ |x|−(1+a) .
(4)
The random walk process, which has the above jump probability, is known as Lévy ‘‘flight’’. In [22] the strong relation between the Lévy statistics and critical phenomena has been shown. It has been shown that the Lévy random walk is a way to organize the fractal geometry of the critical state. In real world processes the Lévy walk obeys to the requirement of finite variance of the process. The topic has been faced by Mantegna and Stanley [40] introducing the truncated Lévy flight (TLF) where an upper cutoff is introduced to the values of random variables. The TLF introduces an exponential cutoff factor to the distribution of Eq. (4) and so the corresponding expression for TLF is written as: PTLF (x) ∼ e−λ|x| |x|−(1+a)
(5)
This form is similar to the function g (l) (Eq. (3)) proposed for the fitting of the distribution of the laminar lengths of fluctuations, where p2 corresponds to 1 + a and p3 corresponds to λ. However, there is a difference in the limits of the corresponding power law exponents. Specifically, in TLF the exponent 1 + a lies between 1 and 3 while the exponent p2 of critical phenomena lies between 1 and 2. This means that the critical phenomena is a subspace of the application of TLF in nature. As it is known [26] for values α ≥ 2 one obtains a Gaussian process. In [26] the TLF model has been modified in such way as to allow an analytical calculation of the characteristic function of the distribution PTLF (x). By using solely the characteristic function Pˆ TLF (k), the existence of a smooth but slow transition of a Lévy stable process towards a Gaussian process has been demonstrated. The transition of the density of the distribution from a Lévy form to a Gaussian form can be achieved by increasing the exponent a while keeping the exponent λ close to zero. Specifically, in shorter time-scales the Lévy distribution dominates while in longer timescales the convergence to a Gaussian distribution is accomplished. In [26] this transition is obtained numerically from the probability distribution (Eq. (5)) by increasing the exponent a starting from the value 0.5 for λ = 0.05. The value of the critical exponent p2 which corresponds to a = 0.5 is p2 = 1.5, while the condition λ ≈ 0 indicates proximity to the critical state. In critical phenomena, the specific value of p2 exponent (p2 = 1.5) has a particular
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importance. Generally, the exponent z of the non-linear term of the intermittent map of Eq. (1) determines the dynamics of intermittency. As we have already mentioned in Section 2, exponent z is connected with the critical exponent δ as z = δ + 1. So, since in critical phenomena δ > 1 it follows that z > 2. On the other hand, in terms of the power spectrum Sf ∼ f −β there is a relation of exponent β with the exponent z which is different for different ranges of the value of z [27]. The z value which separates two different power spectrum behaviors in critical phenomena is z = 3 [27], discriminating therefore two z value intervals: z ∈ (2, 3) and z ∈ (3, +∞). As we already mentioned in Section 2, pl = z −z 1 = 1 + z −1 1 , while at the critical state it is p2 (= pl ) = 1 + z −1 1 , which means that the corresponding intervals for p2 value are p2 ∈ (1.5, 2) and p2 ∈ (1, 1.5), respectively. Thereupon, the value p2 = 1.5 is dividing the range of p2 exponent values, (1, 2), into two intervals corresponding to different behaviors. In this way, it is confirmed that the value a = 0.5 discriminates two statistical behaviors corresponding to different dynamics, specifically Lévy dynamics from TLF (Lévy-Gaussian) dynamics. Thus, in the following we divide the p2 exponent range into two regions, namely p2 ∈ (1, 1.5) and p2 ∈ (1.5, 2). In the first region, the critical organization is achieved by the domination of the Lévy process while in the second range a slow transition from the Lévy to the Gaussian process appears. According to the four-stage model that has been proposed (see Section 1 and [12]), the preparation of a strong EQ begins when the heterogeneous focal area is at the critical state. At this state a micro-cracking procedure is performed during which both the times and lengths of the micro-cracking are organized according to the Lévy mechanism as has been shown in [22]. The micro-fracture organization is spread by Lévy steps across all space–time scales of the focal area. This mechanism, according to Lévy walk, begins to occur at a region where the resistance to rupture growth has the minimum value. Failure nucleation develops in this area with micro-cracking and then, after a great deal of such short steps, a local fracture occurs and the walk jumps to another point to repeat the process. This picture is compatible with the view that a major fault is activated after the activation of a population of local fractures (corresponding to small EQs) in the heterogeneous region that surrounds the major fault. Note that EQ triggering has been suggested to be driven by the smallest EQs at all scales, even for the largest EQs [12,41], while the small EQs are considered the agents by which longer stress correlations are established [12,42]. Note that EQ precursors in general are expected to be sourced from a wide area around the epicenter referred to as ‘‘EQ preparation zone’’ [43,44], while the MHz EME, that are in the focus of the present study and refer to the fractures in the heterogeneous region that surrounds the major fault during the critical phase, are expected to be sourced from the so called ‘‘critical region’’ [42]. In an attempt to describe, in terms of the MHz EME observable and using intermittency terminology, the organization to the critical state through Lévy walk mechanism, one gets the following picture: In the weaker section of the focal area local micro-cracking begins. This can be observed through MHz EME amplitude fluctuations, the waiting times of which within the laminar region are called laminar lengths. The short departures of the MHz EME amplitude fluctuations from the laminar region could be attributed to the creation of new fresh surfaces through ruptures occurring within the specific section. The waiting times constitute the observable Lévy flight (temporal) steps. At some point, when the stresses exceed the fracture threshold of the boundaries of the specific section, a local fracture occurs corresponding to a small EQ (which might be under the magnitude of completeness of the recorded EQ catalog), transmitting the stress to another section of the focal area. Such a local fracture is manifested by a relatively high amplitude MHz EME burst outside the laminar region. During the development of the cracking network within a specific section of the focal area, the rupture of the weaker parts of the heterogeneous material happens first and progressively, as the stress increases, the heterogeneity is reduced as the weaker parts are fractured and the stronger parts remain. In course of this process, the longer waiting times indicate higher fracture threshold differences. Under the hypothesis that the fracture threshold difference increases as the heterogeneity reduces, it is expected that the longer waiting time related to a specific section appears just before the local fracture, signifying a Lévy flight (temporal) jump. As long as the stress has been transferred to a new section the whole process is repeated until it spans over the whole focal area. Through this mechanism, the entire focal area is engaged in this cracking organization at the critical state, while, according to what we have previously outlined about the dynamics of the critical state, the distribution of the waiting times (laminar lengths) of the emitted MHz EME follows power-law. Note that the foreshock seismicity before a strong EQ has been found to reach criticality within the time frame during which MHz EME present critical characteristics [24,25,45]. Moreover, since at the critical state it is expected the appearance of scale-free structures both in time and space, one could imply that the observed temporal Lévy flight steps are expected to correspond to spatial steps the lengths of which also follow Lévy statistics. In the framework of this scenario, one understands that the critical organization we have described for the development of the cracking structure is a necessary condition for the final occurrence of a strong EQ. At the observation level, such a process is manifested by MHz EME recordings the laminar lengths distribution of which follow a power law according to what we have set out previously. According to the above mentioned about the discrimination of two regions for the observed p2 exponent range in the laminar phase, i.e., p2 ∈ (1, 1.5) and p2 ∈ (1.5, 2), we can consider that if the exponent p2 lies within the interval (1.5, 2), a slow transition from the Lévy to the Gaussian process appears where the long scales are cut-off due to the Gaussian mechanism, so the organization of the critical state is not completed according to the Lévy flight scenario since the correlations do not spread all over the focal area. On the contrary, if the exponent p2 lies within the interval (1, 1.5) then the organization of the critical state is completed according to the Lévy flight scenario. The results obtained after the analysis of the MHz EME observed before 15 strong EQs which occurred in Greece (see Section 4) are compatible with the above suggestion.
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Table 1 15 strong EQs, which have occurred on-land or near coastline in Greece during the time period 1995–2015 for which there are available MHz EME recordings, and the associated MCF analysis results. The reader is referred to [21,24] for more details on our EME telemetric stations network deployed throughout Greece. EQ details
Critical window details
Place of occurrence (near or at)
ML
Epicenter
Kozani-Grevena * Athens * Zante * Cephalonia * Methoni * Methoni Andravida * Lamia * Agios Nikolaos * Strofades * Karpathos Limnos Chania** Cephalonia*** Lefkada
6.1 5.4 5.4 5.5 6.2 6.0 6.5 5.2 5.2 5.5 6.2 5.8 6.2 5.8 6.0
(40.18 (38.15 (37.68 (38.34 (36.50 (36.18 (37.98 (38.72 (35.66 (37.13 (35.64 (39.67 (35.50 (38.22 (38.67
◦
N, N, ◦ N, ◦ N, ◦ N, ◦ N, ◦ N, ◦ N, ◦ N, ◦ N, ◦ N, ◦ N, ◦ N, ◦ N, ◦ N, ◦
21.71 23.62 20.91 20.42 21.78 21.72 21.51 22.57 26.39 20.78 26.56 25.56 23.28 20.53 20.60
◦
E) E) ◦ E) ◦ E) ◦ E) ◦ E) ◦ E) ◦ E) ◦ E) ◦ E) ◦ E) ◦ E) ◦ E) ◦ E) ◦ E) ◦
Depth (km)
Date
Window width (s)
# days before EQ
p2max
p3min
Recorded at the station
39 30 18 15 41 25 25 24 42 15 63 31 65 16 11
13-05-1995 07-09-1999 11-4-2006 25-03-2007 14-02-2008 20-02-2008 08-06-2008 13-12-2008 13-1-2009 16-02-2009 01-04-2011 08-01-2013 12-10-2013 26-1-2014 17-11-2015
11000 20000 17000 12000 19000 20000 18000 24000 20000 10000 16000 23000 27000 10000 12000
12 9 1 7 2 5 5 1 5 1 5 2 5.5 2 3
1.49±0.06 1.31±0.046 1.37+-0.044 1.43±0.065 1.37±0.044 1.35±0.07 1.41±0.047 1.48±0.04 1.48±0.01 1.37±0.049 1.48±0.05 1.38±0.04 1.51±0.026 1.35±0.047 1.28±0.05
0.009 0.005 0.001 0.009 0 0.002 0 0 0.04 0 0.048 0.001 0.035 0 0
Zakynthos Zakynthos Zakynthos Zakynthos Zakynthos Zakynthos Zakynthos Zakynthos Neapoli Zakynthos Vamos Lesvos Vamos Cephalonia Zakynthos
4. MHz fracture-induced electromagnetic emissions analysis results In the following, we present results concerning the 15 strong, relatively shallow, seismic events which have occurred on-land or near coastline in Greece during the time period 1995–2015 and for which we have available MHz EME recordings (see Table 1). Note that the MHz EME have been analyzed by MCF for 9 of them (see events with asterisk in Table 1) in [19], while the MHz EME for the 2013 Chania and 2014 Cephalonia EQ (see events with two and three asterisks in Table 1, respectively) have been analyzed in [24,25], respectively. All details concerning EQ have been retrieved from the EQ catalogs of the Institute of Geodynamics of the National Observatory of Athens (http://www.gein.noa.gr/en/ seismicity/earthquake-catalogs). As a representative example, we present in Figs. 1 and 2 the MCF analysis results for the 2013 Limnos and the 2015 Lefkada EQs (see also Table 1), respectively. Note that we have proved [8] that the electric field amplitude of MHz EME plays the role of an order parameter. It is noted that the critical windows (time series excerpts satisfying the MCF criticality conditions presented in Section 2) presented in Table 1 were not the only ones identified prior to the specific strong EQs. The criterion for the choice of the presented results was the lowest exponent p3 in the laminar phase which corresponds to the best effective laminar region, i.e., closer to critical state. As can be seen from Table 1, for all the events the estimated p2 max exponent values, within the margin of the estimation error, belong to the interval (1.1.5). According to the theoretical analysis presented in Section 3, this result signifies that the organization of these strong events to critical state was accomplished according to the Lévy scenario in which correlations are spread throughout the whole focal area, at all scales in agreement with the theory of critical phenomena. In a seismogenic country such as Greece many small EQs occur every day. So, a question effortlessly arises on whether the emergence of such a critical window could refer to the appearance of high seismic activity during which energy is released not through a strong EQ but through many weaker EQs within a specific area. The answer is positive. One such example of high seismic activity happened between the days from 22-5-2010 to 27-5-2010 around the neighboring islands of Zakynthos (Zante) and Cephalonia. Specifically, during the specific time period and within the area roughly defined by the triangle Cephalonia–Zakynthos–Western Greece coastline, a swarm of 55 seismic events with magnitudes from 2.5 to 3.5 were recorded within the specific area (35 of them occurred on a single day!). A few days before, on 20-5-2010, the Zakynthos station recorded a critical window of 12000 s duration, for which the MCF analysis resulted to p2 max value in the range (1, 1.5), specifically the analysis yielded p2 max = 1.17 and p3 min = 0.02. We suggest that it is possible, by means of the autocorrelation function, to discriminate the criticality detected in MHz EME which is related with an upcoming strong EQ from the criticality related with a high seismic activity of weaker events within a specific geographic area. According to our suggestion presented in Section 3, a swarm of weak EQs could correspond to local fractures. However, the appearance of local fractures per se is not enough to create a strong EQ. Local fractures must be correlated to lead to a strong EQ. Since we are interested in the correlations at the laminar phase where the critical dynamics have been developed, we herein estimate the ‘‘laminar’’ autocorrelation function, CL (m), where m denotes the time-lag (time-length), by using only the parts of the critical window of the MHz EME time series that belong to the laminar phase, i.e., by using the values φo ≤ φ ≤ φl . Fig. 3 depicts the plot of CL (m) vs. m on a log–log plot for the critical windows of the 7 stronger EQ events of Table 1 (with ML ≥ 6.0) and the abovementioned swarm of weaker EQs.
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Fig. 1. (a) The time series excerpt of the MHz EME recordings of 06-01-2013, at Lesvos station associated with the 2013 Limnos EQ; the time scale refers to the time in sec starting from 00:00:00 UT of the specific day. (b) The values’ distribution for the time-series excerpt of panel (a); the value φo = 450 was chosen as the start of laminar regions. (c) A representative example of laminar lengths distribution for the end point φl = 473, i.e., for the laminar region (450, 473). The continuous line corresponds to the fitted function of the form of Eq. (3). A clear power-law is identified. (d) The exponents p2 , p3 vs. the end point φl . The validity of criticality condition p2 > 1, p3 ≈ 0 for a wide range of end point values, and consequently for a wide range of laminar regions, is evident.
From Fig. 3, it is evident that there is a qualitative differentiation between the results obtained for the weak EQs swarm and the results obtained for the strong EQs in terms of the autocorrelation function. A ‘‘collapse’’ of autocorrelation is observed on the longer scales in the case of the weak EQs swarm, thus declaring the lack of long correlations among the local fractures (attributed to the specific seismic activity) which would be necessary for the preparation of a strong event. On the contrary, the behavior observed for the strong EQ cases, namely the fact that autocorrelation function indicates long-range correlations extending up to long scales (up to a different scale for each one of them), is compatible with a full deployment of critical dynamics throughout the whole focal area including not only short correlations between micro-fractures but also long correlations among the local fractures. Moving further with our investigation, we formulate the following questions: Have there been identified any MHz EME time series excerpts for which, by application of the MCF, power law can be found in the distribution of laminar lengths with exponent p2 in the range (1.5, 2)? Did a strong EQ followed such cases? In a country of high seismic activity such as Greece, it is difficult to find relatively long time periods without any strong EQ in order to check the answer to the above posed questions. However, we were able to find such a time period of 8 months duration (April to December 2009) during which no EQs with ML > 4.5 have been recorded in Greece. Nevertheless, during the specific time period 4 critical windows were identified among the MHz EME recordings of the telemetric stations of our network as shown in Table 2. According to the MCF analysis results presented in Table 2, for all cases p2 max was estimated in the range (1.5, 2), while no strong EQ or weaker EQ swarm did happen following any of the specific critical windows. According to the theoretical analysis presented in Section 3, p2 in the range (1.5, 2), signifies a slow transition from Lévy to Gaussian process, in which the later cuts-off the long spatial and temporal steps which are necessary to be developed in the micro-cracking network at the critical state, according to what we have explained in Section 3. 5. Conclusions In this work we focused on the analysis of MHz EME using the MCF time series analysis method which has been known to identify the approach of a system to critical state as well as its departure from it. Specifically, we further investigated the hypothesis that the MHz EME recorded prior to the occurrence of a strong EQ reflect the organization of the fracture of the highly heterogeneous system surrounding the fault to criticality according to the Lévy flight mechanism.
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Fig. 2. (a) The time series excerpt of the MHz EME recordings of 14-11-2015, at Zakynthos station associated with the 2015 Lefkada EQ; the time scale refers to the time in sec starting from 06:23:20 UT of the specific day. (b) The values’ distribution for the time-series excerpt of panel (a); the value φo = 590 was chosen as the start of laminar regions. (c) A representative example of laminar lengths distribution for the end point φl = 610, i.e., for the laminar region (590, 610). The continuous line corresponds to the fitted function of the form of Eq. (3). A clear power-law is identified. (d) The exponents p2 , p3 vs. the end point φl . The validity of criticality condition p2 > 1, p3 ≈ 0 for a wide range of end point values, and consequently for a wide range of laminar regions, is evident.
Fig. 3. The log–log plots of ‘‘laminar’’ autocorrelation function CL (m) vs. m of the MHz EME critical windows detected prior to the 7 stronger EQs events of Table 1 (ML ≥ 6.0) and a swarm of weaker EQs which occurred between 22-5-2010 and 27-5-2010 around the neighboring islands of Zakynthos (Zante) and Cephalonia (see text). Note that m denotes the time-lag in time units equal to the sampling period of the time series, here 1 s.
We theoretically proved that the MCF power law exponent p2 range can be divided into two regions, namely p2 ∈ (1, 1.5) and p2 ∈ (1.5, 2). In the first region, the critical organization is achieved by the domination of the Lévy process, while in the second region a slow transition from the Lévy to the Gaussian process appears. Based on this theoretic result, we checked the hypothesis that valid pre-EQ MHz EME, i.e., a MHz EME presenting critical characteristics, could be discriminating into two categories in terms of the organization of the critical state: (i) those corresponding to geophysical processes that may evolve to strong EQs and (ii) those corresponding to processes that are not expected to evolve to strong EQs. The MCF analysis of several MHz EME which have been recorded by our ground-based observatories indicates that such a discrimination is possible.
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S.M. Potirakis, Y. Contoyiannis and K. Eftaxias / Physica A 528 (2019) 121360 Table 2 MCF analysis results for the critical windows identified in the MHz EME recordings of our telemetric stations network from April to December 2009. Date
Window duration (s)
p2max
p3min
Recorded at the station
25-04-2009 08-07-2009 15-12-2009 25-12-2009
11000 18000 18000 19000
1.71 1.80 1.80 1.80
0 0 0 0.005
Zakynthos Zakynthos Zakynthos Komotini
Specifically, we presented the MCF analysis results for the pre-EQ MHz EME recorded a few days prior to each one of 15 strong EQ events which took place in Greece. The results show that in all these cases the values of p2max exponent (the p2 value corresponding to the best effective laminar region, i.e., closer to the critical state) lies within the interval (1, 1.5). In such cases, the organization of the critical state is considered to be completed according to the Lévy flight scenario spreading correlations all over the focal area during the fracture of the highly heterogeneous system surrounding the fault. Attention is also drawn to the fact that, as we demonstrated, the MCF analysis of a MHz EME time series yielding a p2max value in the interval (1, 1.5) may also be observed prior to a swarm of smaller events. A qualitative criterion which discerns a swarm of smaller events from a strong EQ is that in the first case a ‘‘collapse’’ of autocorrelation is observed on the longer scales of the critical MHz EME recorded prior to the EQ swarm appearance. On the other hand, we demonstrated that in one relatively long period during which no strong EQs (all recorded EQs had ML ≤ 4.5) or EQ swarms occurred in Greece, MHz EME time series excerpts with critical characteristics were identified by MCF, but in all cases p2 max was estimated in the range (1.5, 2). In such cases, it is considered that the organization of the critical state is not completed according to the Lévy flight scenario since the long scales are cut-off due to the Gaussian mechanism, and accordingly the correlations do not spread all over the focal area. Thereupon, in such a case a strong EQ should not be expected to follow. It has to be mentioned that long time periods without any strong EQ are rare in Greece, rendering very difficult to check more similar cases within the frame of the present work. We consider that the main outcome of the present work is that MHz EME time series excerpts presenting, by means of the MCF analysis, critical characteristics but with p2max values in the interval (1.5, 2) are not expected to be followed by a strong EQ. However, more MHz EME results in terms of the presented analysis would be necessary to reinforce the findings of this work. It should finally be underlined that a MHz anomaly for which critical characteristics with p2,max ∈ (1, 1.5) have been identified by MCF analysis, is considered a ‘‘necessary’’ condition for the occurrence of a strong EQ event, it is not ‘‘a necessary and sufficient’’ one. 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