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

geosciences

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

Filippos Vallianatos 1,2,*, Georgios Michas 1,2 and George Hloupis 2,3

1 Institute of Physics of the Earth’s Interior and Geohazards, UNESCO Chair on Solid Earth Physics and Geohazards Risk Reduction, Hellenic Mediterranean University Research Center, 73133 Chania, Greece; [email protected] 2 Section of Geophysics—Geothermics, Department of Geology and Geoenvironment, National and Kapodistrian University of Athens, 15784 Athens, Greece; [email protected] 3 Department of Surveying and Geoinformatics Engineering, University of West Attica, Egaleo Campus, 12244 Athens, Greece * Correspondence: [email protected] or [email protected]

Abstract: 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 Citation: Vallianatos, F.; Michas, G.; Hloupis, G. Seismicity Patterns Prior new universal principles that describe the generation processes of strong earthquakes. In the present to the Thessaly (Mw6.3) Strong work, we investigate this new framework in the seismicity prior to the Mw6.3 Thessaly earthquake. Earthquake on 3 March 2021 in Terms Initially, we apply MRWA to the interevent time series of the successive regional earthquakes in order of Multiresolution Wavelets and to investigate the approach of the regional seismicity at critical stages and to deﬁne the starting point Natural Time Analysis. Geosciences of the natural time domain. Then, we apply the NT analysis, showing that the regional seismicity 2021, 11, 379. https://doi.org/ approached criticality a few days before the occurrence of the Mw6.3 earthquake, when the κ1 natural 10.3390/geosciences11090379 time parameter reached the critical value of κ1 = 0.070.

Academic Editors: Ioannis Keywords: Thessaly earthquake; seismicity patterns; natural time; multiresolution wavelet analysis; Koukouvelas, Riccardo Caputo, criticality Tejpal Singh and Jesus Martinez-Frias

Received: 9 August 2021 Accepted: 8 September 2021 1. Introduction Published: 9 September 2021 On 3 March 2021, an MW6.3 earthquake occurred in northern Thessaly (Central Publisher’s Note: MDPI stays neutral Greece), close to the cities of Tyrnavos, Elassona and Larisa (Figure1). The earthquake with regard to jurisdictional claims in occurred in a region that is one of the most seismically active in Greece, mainly character- published maps and institutional afﬁl- ized by normal faulting along NW–SE striking faults, which belong to the Thessaly fault iations. zone [1–10]. Based on the provided focal plane solutions [11], the mainshock was generated by the activation of an NW–SE striking normal fault (Figure1)[ 12]. The mainshock was widely felt in the Thessaly basin and in the surrounding areas, from Athens in the south to the northern borders of Greece.

Copyright: © 2021 by the authors. The Thessaly basin has a well-known history of large earthquakes, with mainshocks Licensee MDPI, Basel, Switzerland. presenting typical magnitudes between 6.0 and 7.0 [3–15]. During the last century, eight This article is an open access article major earthquakes, with magnitudes equal to or larger than 6.0, have occurred in this area. distributed under the terms and The M = 7.0, 1954 Sofades earthquake was the most destructive event, resulting in heavy conditions of the Creative Commons damages in the broader southern Thessaly region (Papazachos and Papazachou, 2002; [15]). Attribution (CC BY) license (https:// The M = 6.8, 1957 Velestino earthquake and the M = 6.5, 1980 Almyros earthquake, are two creativecommons.org/licenses/by/ more signiﬁcant events, with similarly destructive consequences. 4.0/).

Geosciences 2021, 11, 379. https://doi.org/10.3390/geosciences11090379 https://www.mdpi.com/journal/geosciences Geosciences 2021, 11, 379 2 of 14

Strong arguments show that the earthquake generation process can be considered Geosciences 2021, 11, x FOR PEER REVIEW 2 of 14 as a critical point phenomenon that culminates with a large event as some critical point is approached [16–25]. New ﬁndings regarding the complex dynamics that characterize various geodynamic phenomena illustrate stimulating features in the framework of new earthquake,concepts, asare that two of more non-extensive significant events statistical, with physics similarly [22 –destructive25], multiresolution consequences. wavelets analysis [26–28] and of the novel time domain, termed as natural time [29–39].

Figure 1. Earthquake epicenters of the aftershock sequence of the 3 March 2021, Thessaly earthquake Figure(yellow 1. star),Earthquake from 3 Marchepicenters 2021 of to the 18 Aprilaftershock 2021 (M sequence≥ 2.0). Theof the focal 3 March mechanisms 2021, Thessaly of the mainshock earth- quakeand the(yellow largest star), aftershock from 3 areMarch also 2021 indicated, to 18 alongApril with2021 the(M regional≥ 2.0). The faults focal shown mechanisms with black of solidthe mainshocklines (for detailsand the see largest the text). aftershock are also indicated, along with the regional faults shown with black solid lines (for details see the text). The concept of natural time (NT) has been introduced recently to analyze possible pre-seismicStrong arguments signals [29 show,30,35 that]. The the analysis earthquake of various generation complex process systems can be in theconsidered NT domain as a enablescritical point the optimal phenomenon extraction that of culminates signal information with a large by reducing event as the some uncertainties critical point related is approachedto the conventional [16–25]. New time, findings as well asregarding the identification the complex of long-range dynamics correlationsthat characterize in the variousevolution geodynamic of the system, phenomena even in illustrate the presence stim ofulating “heavy features tails” [in40 the]. The framework usefulness of ofnew NT concepts,analysis as has that been of discussednon-extensive in a numberstatistical of physics applications [22–25], to knownmultiresolution critical phenomena, wavelets analysissuch as [26–28] fracturing, and earthquakes,of the novel time the 2-Ddomain, Ising termed model andas natural 3-D turbulent time [29–39]. flow [24,35], and referencesThe concept therein, of andnatural it has time been (NT) tested has experimentallybeen introduced in recently fracturing to experimentsanalyze possible in the pre-seismiclaboratory signals by analyzing [29,30,35]. acoustic The analysis emissions of timevarious series complex [23,41]. systems in the NT domain enablesFurthermore, the optimal extraction wavelet-based of signal methods information have been by reducing introduced the touncertainties characterize related fractal tosignals the conventional [42–45] and time, to overcome as well effectsas the iden associatedtification with of non-stationaritieslong-range correlations [46,47], in a verythe evolutionfrequent of effect the insystem, the time even dynamics in the presence of an earthquake of “heavy sequence. tails” [40]. The usefulness of NT analysisThe has goal been of thediscussed present in work a number is to test of andapplications evaluate to the known seismicity critical patterns phenomena, in terms suchof MRWA as fracturing, and NT earthquakes, analyses, as the applied 2-D Ising in the model evolution and 3-D of seismicity turbulent priorflow [24,35], to the recent and referencesMw6.3 Thessaly therein, strongand it has event. been The tested recent experimentally upgrading ofin thefracturing regional experiments seismological in the net- laboratoryworks [48 by] provides analyzing an acoustic accurate emissions catalogue time of microseismicity series [23,41]. in the area and enables the applicationFurthermore, of such wavelet-based methodologies. methods The earthquake have been catalogs introduced used herein to characterize are taken fromfractal the signalsHellenic [42–45] Unified and Seismological to overcome Networkeffects associated (HUSN) (withhttp://eida.gein.noa.gr/ non-stationarities [46,47],, last accessed a very frequenton 27 May effect 2021 in), the where time instruments dynamics of belonging an earthquake to the sequence. HL (National Observatory of Athens, The goal of the present work is to test and evaluate the seismicity patterns in terms of MRWA and NT analyses, as applied in the evolution of seismicity prior to the recent Mw6.3 Thessaly strong event. The recent upgrading of the regional seismological networks [48] provides an accurate catalogue of microseismicity in the area and enables the

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application of such methodologies. The earthquake catalogs used herein are taken from Geosciences 2021, 11, 379 the Hellenic Unified Seismological Network (HUSN) (http://eida.gein.noa.gr/, last3 ofac- 14 cessed on 27 May 2021), where instruments belonging to the HL (National Observatory of Athens, Institute of Geodynamics, 1997) [49] and HT (Aristotle University of Thessa- lonikiInstitute Seismological of Geodynamics, Network, 1997) 1981) [49] [50] and networks HT (Aristotle provide University a complete of Thessaloniki spatial coverage Seismo- in thelogical broader Network, area of 1981) Greece, [50] with networks a magnitude provide of a completecompleteness spatial (Mc) coverage down to in 2.0. the Figure broader 2 presentsarea of Greece, the seismic with aactivity magnitude observed of completeness in the region (Mc) of downThessaly to 2.0. forFigure a period2 presents starting the in Januaryseismic activity2016, approximately observed in the1900 region days before of Thessaly the 3 March for a period 2021 mainshock starting in and January within 2016, an areaapproximately of radius, 1900R = 80 days km, before around the its 3 Marchepicenter. 2021 The mainshock main objective and within of this an study area of is radius, to in- vestigateR = 80 km, the around applicability its epicenter. of NT analysis, The main as objective presented of in this the study regional is to seismic investigate activity the priorapplicability to the M ofw6.3 NT Thessaly analysis, earthquake, as presented integrated in the regional with seismic the results activity of MRWA prior to applied the Mw6.3 to theThessaly interevent earthquake, time series integrated of the withsuccessive the results events, of MRWAin order applied to define, to the with interevent an objective time technique,series of the the successive starting point events, for in the order analysis to deﬁne, in the with NT an domain. objective The technique, description the startingof seis- micitypoint for evolution the analysis with inthe the NT NT parameters, domain. The integrated description with of the seismicity results evolutionof MRWA, with repre- the sentsNT parameters, a novel framework integrated that with may the resultslead to of a MRWA,better understanding represents a novel of the framework evolution that of earthquakemay lead to generation a better understanding processes. of the evolution of earthquake generation processes.

Figure 2. The observed seismicity in the Thessaly region between January 2016 and 3 March 2021, in Figurean area 2. of The radius observed R = 80 seismicity km around in the the mainshock Thessaly region (star). between January 2016 and 3 March 2021, in an area of radius R = 80 km around the mainshock (star). 2. Multiresolution Wavelets Analysis in the Seismicity of Thessaly Region 2. MultiresolutionThe temporal Wavelets evolution Analysis of seismicity in the and Seismicity the time-scaling of Thessaly properties Region are of crucial importanceThe temporal [51–53 ]evolution for understanding of seismicity the and correlation the time-scaling properties properties of seismicity are of [54 crucial]. The importanceanalysis of time [51–53] intervals for understanding between successive the corre seismiclation events properties can be groupedof seismicity in exponential [54]. The analysisor power of laws time revealing intervals similarbetween behaviours successive over seismic different events scales can [be55 ].grouped in exponential orA power Wavelet laws Transform revealing (WT)similar involves behaviours the decompositionover different scales of a [55]. signal function into simpler,A Wavelet ﬁxed building Transform blocks (WT) at different involves scales the decomposition and positions. Inof a a similar signal way function to Fourier into simpler,transform fixed (FT), building WT operates blocks at on different a signal scales and transformsand positions. it from In a similar the time way domain to Fourier to a transformdifferent domain, (FT), WT offering operates a new on a representation. signal and transforms In Fourier it analysis,from the the time sine domain and cosine to a differentfunctions domain, are used offering as a basis a new and representation localized in the. In frequency Fourier analysis, domain the with sine a difﬁculty and cosine to functionsprocess a functionare used havingas a basis components and localized thatare in localizedthe frequency in the domain time domain. with a Asdifficulty a result, to a processsmall frequency a function change having in components FT produces that changes are localized everywhere in the in time this domain. As On a the result, other a smallhand, frequency wavelet functions change in are FT localized produces both changes in frequency everywhere or scale, in this and domain. in time, On via the dilations other and translations of the mother wavelet, respectively. This leads to compact representation of large classes of functions in the wavelet domain. This is one of the major advantages of WT: an event can be simultaneously described in the frequency domain as well as in the time domain, unlike the usual Fourier transform where an event is accurately described Geosciences 2021, 11, x FOR PEER REVIEW 4 of 14

hand, wavelet functions are localized both in frequency or scale, and in time, via dilations and translations of the mother wavelet, respectively. This leads to compact representation of large classes of functions in the wavelet domain. This is one of the major advantages of WT: an event can be simultaneously described in the frequency domain as

Geosciences 2021, 11, 379 well as in the time domain, unlike the usual Fourier transform where an event is accu-4 of 14 rately described either in the frequency or in the time domain. As a consequence, MRWA of data with different behaviour on different scales can greatly benefit from the use of WT. eitherThe in discrete the frequency wavelet ortransform in the time (DWT) domain. transf Asorms a consequence, a data vector MRWA of length of M data into with a differentdifferent vector behaviour of the on same different length. scales A wavelet can greatly basis benefitis characterized from the useby a of particular WT. set of parameters,The discrete called wavelet wavelet filter transform coefficien (DWT)ts. In transforms practice, DWT a data is commonly vector of length implemented M into a usingdifferent dyadic vector multirate of the samefilter length.banks, Awhich wavelet divide basis the is characterizedsignal frequency by a band particular into setsub of bandsparameters, [28]. For called a point wavelet process filter such coefficients. as that of Inthe practice, interevent DWT times is commonly sequence, implementedthe wavelet coefficientsusing dyadic can multiratebe derived filter from banks, which divide the signal frequency band into sub bands [28]. For a point process such as that of the interevent times sequence, the wavelet coefficients can be derived from / 𝑊, = 2 𝑡𝜓(2 𝑖 −𝑛) (1) L wav −m/2 −m Wm,n = 2 ∑ tiψ(2 i − n) (1) where the scale variable m and the translationi=1 variable n are integers, L represents the totalwhere number the scale of interevent variable mtimesand t thei analysed translation and variableψ is the waveletn are integers, function.L represents The DWT theis evaluated at the points (m, n) in the scale-interval-number plane. Smaller scales corre- total number of interevent times ti analysed and ψ is the wavelet function. The DWT is spondevaluated to more at the rapid points variations (m, n) in and, the scale-interval-numbertherefore, to higher frequencies. plane. Smaller scales correspond to moreIn the rapid current variations study, and,we perform therefore, MRWA to higher by frequencies.examining the standard deviation of waveletIn coefficients the current as study, a function we perform of scale, MRWA as described by examining from the standard deviation of wavelet coefficients as a function of scale, as described from v 1 u N 𝜎(𝑚) = u 1 (𝑊, −< 𝑊, 2 >) (2) σ (m) = t𝑁−1 Wwav − Wwav (2) wav − ∑ m,n m,n N 1n=1

where N is the number of wavelet coefﬁcients at a given scale m and the brackets indicate wherethe average N is the number among of the wavelet coefﬁcients coefficients at a scaleat a givenm. Figurescale m3 andpresents the brackets the interevent indicate the times av- eragebetween among two the successivecoefficients eventsat a scale versus m. Figure the occurrence3 presents the time interevent of the times second between event two until suc- the cessive events versus the occurrence time of the second event until the major seismic event, in a major seismic event, in a region within a circle of radius, R = 80 km, around the epicenter region within a circle of radius, R = 80 km, around the epicenter and a magnitude threshold, Mth = 2.0.and The a time magnitude period that threshold, was covered Mth =for 2.0. MRWA The timeof interevent period times that wasspanned covered from forJanuary MRWA 2016 of untilinterevent 3 March times 2021, spannedwhen the frommain Januaryevent of 2016Mw6.3 until occurred. 3 March In 2021,Figure when4 the thetime– main earthquake event of magnitudeMw6.3 occurred. plot for a Inradius Figure R =4 80 the km time– around earthquake the epicenter magnitude and a magnitude plot for threshold, a radius RM =th = 80 2.0 km is presented.around the epicenter and a magnitude threshold, Mth = 2.0 is presented.

Figure 3. Interevent times between two successive events versus occurrence time of each event for a

radius R = 80 km around the epicenter and a magnitude threshold, Mth = 2.0.

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Geosciences 2021, 11, 379 5 of 14 Figure 3. Interevent times between two successive events versus occurrence time of each event for a radius R = 80 km around the epicenter and a magnitude threshold, Mth = 2.0.

Figure 4. Time–Magnitude plot for a radius R = 80 km around the epicenter and a magnitude Figure 4. Time–Magnitude plot for a radius R = 80 km around the epicenter and a magnitude threshold, Mth = 2.0. threshold, Mth = 2.0. The wavelet choice is dictated by the requirement to identify a rather sharp change in a The wavelet choice is dictated by the requirement to identify a rather sharp change possible cyclic sequence. Following the same pre-processing approach as in [28], we tested in a possible cyclic sequence. Following the same pre-processing approach as in [28], we several candidate wavelets at small scales up to m = 4 and received quite similar results. tested several candidate wavelets at small scales up to m = 4 and received quite similar Thus, in the current work, we present results from the analysis using the db4 wavelet. results. Thus, in the current work, we present results from the analysis using the db4 We investigated the time evolution of the σwav(m), using fixed event number windows ofwavelet. 16 events shifting through the entire series. The shift between successive windows was set inWe two investigated events. Consistently the time withevolution the length of the of σ thewav(m time), using window, fixed we event analysed number the timewin- dows of 16 events shifting through the entire series. The shift between successive win- variation of the σwav(m) for lower scales (m = 1 to 4) since the number of available events is limited.dows was Each set calculatedin two events. value Consistently is associated with with the the length time of of the the last time event window, in the window.we analysed the time variation of the σwav(m) for lower scales (m = 1 to 4) since the number of Figure5 shows a representative set of results for the time evolution of the σwav(m) using theavailable db4 wavelet events withis limited. four scalesEach calculated for MRWA, va forlue the is associated seismicity observedwith the time in three of the circles last aroundevent in thethe epicenterwindow. ofFigure the mainshock5 shows a represen and withintative a radiusset of results of 30 km, for the 50 kmtime and evolution 80 km, respectively.of the σwav(m) using the db4 wavelet with four scales for MRWA, for the seismicity ob- servedAn in initial three comment circles around from Figure the epicenter5 is the significant of the mainshock temporal and variability within in a theradius strength of 30 ofkm, the 50 multiscale km and 80 properties km, respectively. of the interevent times. As observed in previous studies [26,27], beforeAn the initial major comment event of thefrom seismic Figure sequence 5 is the a traceablesignificant decrease temporal in thevariability temporal in evo-the lutionstrength of of the theσwav multiscale, m(t) appeared, properties especially of the atinterevent lower scales. times. Plots As atobserved Figure5 indictate previous the searchstudies for [26,27], a time before marker the beginning major event several of the months seismic before sequence the major a traceable event for decrease all the scalesin the analyzed.temporal evolution The sharp of decrease the σwav,at m( lowert) appeared, scales (especiallym = 1 and mat =lower 2), which scales. is Plots observed at Figure before 5 thedictate major the event, search can for bea time qualified marker as beginning such a time several marker months since thebefore decrease the major is evident event forfor severalall the scales days andanalyzed. is clearly The identifiable. sharp decrease at lower scales (m = 1 and m = 2), which is observedTranslating before thethe result major from event, lower can scales be qual in anified alternative as such way, a time we proposemarker since the use the of thede- observedcrease is evident time markers, for several which days appear and ais few clearly months identifiable. before the major event, as the initiation point for the natural time analysis that follows. The main purpose is to combine the two independent methods (MRWA and NT analysis) that have been successfully used for the identification of critical stages in earthquake preparation processes, in a joint approach that will maximize the advantages of each one. More specifically, the initial application of MRWA in a broader time period reveals time segments where the NT analysis is then used to investigate for indicators suggesting the entrance to the critical stage.

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. Figure 5. Cont.

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Figure 5. Time variation of σwav(m) with scale m ranging from 1 up to 4, for moving windows Figure with5. Time length variation of 16 eventsof σwav(m and) with a shift scale of m 2 events.ranging Interevent from 1 up timesto 4, for were moving estimated windows using with the HUSN length of 16 events and a shift of 2 events. Interevent times were estimated using the HUSN cata- catalogue (from January 2016 until 3 March 2021), for events within a radius R = 30 km (top 4 plots), logue (from January 2016 until 3 March 2021), for events within a radius R = 30 km (top 4 plots), 50 50 km (middle 4 plots) and 80 km (bottom 4 plots) around the Mw = 6.3 epicenter and a magnitude km (middle 4 plots) and 80 km (bottom 4 plots) around the Mw = 6.3 epicenter and a magnitude threshold, M = 2.0. Red vertical line indicates the day of minimum in variance, observed at each threshold, Mth = 2.0.th Red vertical line indicates the day of minimum in variance, observed at each scale. scale. 3. Natural Time Analysis of Seismicity before the Thessaly Mw6.3, March Translating2021 Earthquake the result from lower scales in an alternative way, we propose the use of the observed time markers, which appear a few months before the major event, as the The analysis of a complex system in the NT domain has been introduced in [29,35]. initiation point for the natural time analysis that follows. The main purpose is to combine In the case of seismicity, the natural time χ, deﬁned as χ = k/N, serves as an index for the two independent methods (MRWA and NT analysis) thatk have been successfully the occurrence of the kth event out of N total events. The seismic moment released during used for theth identification of critical stages in earthquake preparation processes, in a joint the k event is then considered, forming the pair (χk, Mk) for further analysis (see [28]). approach that will maximize the advantages of each one. More specifically, the initial The evolution of (χk,Mk) is further described by the continuous function F(ω), deﬁned as: application of MRWA in a broader time period reveals time segments where the NT F(ω) = N M exp iω k (3) where ω = 2πφ and φ stands for the natural frequency. analysis is then∑ usedk=1 tok investigateN for indicators suggesting the entrance to the critical stage. F(ω) is normalized by division with F(0) N k N 3. Natural Time Analysis of Seismicity∑k=1 beforeMkexp theiω NThessaly Mw6.3, Marchk 2021 Φ(ω) = = p exp iω (3) Earthquake N ∑ k N ∑n=1 Mn k=1 The analysis of a complex system in the NT domain has been introduced in [29,35]. N In the wherecase ofp seismicity,k = Mk/ ∑ then=1 naturalMn. The time quantity χ, definedpk describes as 𝜒 =𝑘/𝑁 the probability, serves as to an observe index for an earth- the occurrencequake event of the at naturalkth event time out χofk. N The total normalized events. The power seismic spectrum moment can released then be obtained during from 2 the kth (4),event as Πis (thenω) = considered,|Φ(ω)| . In theforming context the of pair probability (χk, Mk) theory,for further and foranalysis natural (see frequencies [28]). of The evolutionφ less than of 0.5,(χk, ΠM(kω) )isreduces further to described a characteristic by the functioncontinuous for thefunction probability F(ω), distributiondefined pk. as: 𝐹(𝜔It) has= ∑ been 𝑀 shown𝑒𝑥𝑝 that 𝑖𝜔 the (3) following where 𝜔=2𝜋𝜙 relation holds and [𝜙29 ,stands56] for the natural frequency. 18 6 cos ω 12 sin ω Π(ω) = − − (4) F(ω) is normalized by division with F(0)5ω 2 5ω2 5ω3

According to probability theory, once𝑘 the behavior of the characteristic function of the ∑ 𝑀 𝑒𝑥𝑝 𝑖𝜔 𝑘 distribution is known around zero, then the𝑁 moments and, hence, the distribution(3) itself can Φ(𝜔) = =𝑝 𝑒𝑥𝑝 𝑖𝜔 ∑ 𝑀 𝑁 be approximately determined. For ω → 0, (4) leads to

Π(ω) ≈ 1 − κ ω2 (5) where 𝑝 =𝑀⁄∑ 𝑀. The quantity pk describes the 1probability to observe an earthquake event at natural time χk. The normalized power spectrum can then be obtained where κ is the variance in natural time, given as from (4), as Π1(𝜔) =|Φ(𝜔)|. In the context of probability theory, and for natural frequencies of 𝜙 less than 0.5, Π(ω) reduces to a characteristic function for the probability N N !2 distribution pk. It has been shown that2 the following2 relation2 holds [29, 56] κ1 = hχ i − hχi = ∑ pkχk − ∑ pkχk (6) k=1 k=1

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18 6 cos 𝜔 12 sin 𝜔 Π(𝜔) = − − (4) 5𝜔 5𝜔 5𝜔

According to probability theory, once the behavior of the characteristic function of the distribution is known around zero, then the moments and, hence, the distribution itself can be approximately determined. For ω → 0, (4) leads to

Π(ω) ≈1−κ𝜔 (5)

where κ1 is the variance in natural time, given as

Geosciences 2021, 11, 379 𝜅 = 〈𝜒 〉 − 〈𝜒〉 =𝑝 𝜒 −𝑝 𝜒 8 of(6) 14

It has been shown that the properties of Π(ω) at ω → 0, i.e., the values of κ1 = 0.07, can signifyIt has been the approach shown that of thea complex properties system of Π (towardsω) at ω →some0, i.e., critical the valuespoint [35], of κ1 such= 0.07, as canthat signifyof an impending the approach large of aearthquake complex system (see [24,31,37] towards and some references critical point therein). [35], Figure such as 6 thatshows of an earthquake impending timeseries large earthquake in conventional (see [24,31 (Figure,37] and 6a) references and natural therein). time (Figure Figure 6b)6 showsdomains. an earthquakeFigure 6c shows timeseries the corresponding in conventional power (Figure spectrum6a) and Π natural(ω) for timethe critical (Figure stage6b) domains.with κ1 = Figure0.070, 6basedc shows onthe Equation corresponding (6), while power the two spectrum other curvesΠ(ω) for are the for critical non-critical stage withstages.κ1 =Theoretically, 0.070, based onit has Equation been shown (6), while that the κ two1 approaches other curves 0.083 are as for N non-critical → ∞, when stages. there Theoretically,are no long-ranged it has correlations been shown in that theκ system1 approaches [35]. 0.083 as N → ∞, when there are no long-ranged correlations in the system [35].

(a)

(b)

Figure 6. Cont.

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(c)

Figure 6. Time series of seismic events (a) in conventional time t and (b) in the natural time χ. Figure 6. Time series of seismic events (a) in conventional time t and (b) in the natural time χ. (c) (Schematicc) Schematic diagramdiagram showing showing the the power power spectrum spectrum ΠΠ(ϕ(φ) )in in natural natural time. time. Solid Solid line line is Π(φϕ)) obtainedobtained fromfrom EquationEquation (4) for for the the critical critical stage stage (κ(κ1 1= =0.070), 0.070) whereas, whereas two two other other lines lines are are for for κ1 >κ 10.07> 0.07 andand κ1 < κ0.07.1 < 0.07.

AsAs aa newnew eventevent occurs,occurs, thethe pairpair ((χχkk,, ppkk)) isis rescaledrescaled andand κκ11 varies.varies. ItIt hashas beenbeen veriﬁedverified thatthat when the the parameter parameter κκ1 1convergesconverges to tothe the value value 0.070, 0.070, the thesystem system enters enters a critical a critical state state[33,35,56]. [33,35 ,56]. Furthermore, the entropy in the NT domain, S , is deﬁned as [35] Furthermore, the entropy in the NT domain, Sntnt, is defined as [35]

N N N = h i − h i h i = − ( ) ( ) 𝑆 =<𝜒𝑙𝑛𝜒>−<𝜒>𝑙𝑛<𝜒>=𝑝Snt χlnχ χ ln χ𝜒𝑙𝑛𝜒∑ −pk χ(𝑝klnχκ𝜒) 𝑙𝑛(∑ 𝑝pkχk 𝜒ln) ∑ pkχk = κ=1 κ=1 k 1

N wherewhere h<𝑓f (χ)(𝜒i )=>=∑κ=∑1pk 𝑝f (χ𝑓(𝜒k).). TheThe entropy,entropy,S ntS,nt is, is a dynamica dynamic quantity quantity that that depends depends on the on sequential the sequential order oforder events. of Moreover,events. Moreover, upon the upon time the reversal time reversalT, i.e., TpT, mi.e.,= Tp pNm− = mpN + −1 m, + the 1, the entropy, entropy,Snt S−nt,−, is is furtherfurther defined.defined. When the analysedanalysed seismicityseismicity approachesapproaches aa “true”“true” criticalcritical state,state, thethe followingfollowing conditionsconditions shouldshould bebe fulfilledfulfilled [[35,56]:35,56]: (i).(i). The “average” distance D, defineddefined byby thethe normalizednormalized powerpower spectraspectra ΠΠ((ωω)) ofof thethe ω evolving seismicity and by thethe theoreticaltheoretical estimationestimation ofof ΠΠ((ω)) forfor κκ11 == 0.070,0.070, shouldshould −2 be less than 10 −2. . (ii). The parameter κ should approach the critical value of κ = 0.070 by “descending (ii). The parameter κ1 should approach the critical value of κ11 = 0.070 by “descending from above”. (iii). Both natural time entropies, Snt and Snt−, should be lower than the entropy of uniform (iii). Both natural time entropies, Snt and Snt−, should be lower than the entropy of uniform noise Su = (ln2/2) − 1/4 when κ approaches 0.070. noise Su = (ln2/2) − 1/4 when κ1 approaches1 0.070. (iv).(iv). Since the dynamic evolution of the system is expectedexpected toto bebe self-similarself-similar inin thethe criticalcritical state, the time of the true coincidence should not vary upon changing (within reason- state, the time of the true coincidence should not vary upon changing (within rea- able limits) either the magnitude threshold, Mth, or the area used in the calculation. sonable limits) either the magnitude threshold, Mth, or the area used in the calcula- Intion. [28 ], the authors proposed the use of the time marker indicated by MRWA in the seismicity evolution before the major event as the initiation point for the NT analysis. In [28], the authors proposed the use of the time marker indicated by MRWA in the In the frame of this approach, the two independent methods (MRWA and NT analysis) seismicity evolution before the major event as the initiation point for the NT analysis. In were integrated to identify the approach to the critical stage in the earthquake preparation the frame of this approach, the two independent methods (MRWA and NT analysis) process. In particular, the initial application of MRWA in a broader time period of the were integrated to identify the approach to the critical stage in the earthquake prepara- regional seismicity before the major event reveals time segments where the NT analysis is tion process. In particular, the initial application of MRWA in a broader time period of going to investigate for indicators suggesting the entrance to the critical stage. the regional seismicity before the major event reveals time segments where the NT anal- In Figure7, the computed Π(φ) curves are shown as they approach the critical Π(φ) ysis is going to investigate for indicators suggesting the entrance to the critical stage. in the regional seismicity, for a threshold magnitude of Mth = 2.0 and for areas of radius R = 30In km Figure, R = 7, 50 the km computed and R = 80 Π km,(ϕ) curves respectively, are shown around as thethey epicenter approach of the the critical main event. Π(ϕ) Thisin the analysis regional clearly seismicity, demonstrates for a threshold that, from magnitude 28 February of Mth to= 2.0 2 March and for 2021, areas one of radius to three R = 30 km, R = 50 km and R = 80 km, respectively, around the epicenter of the main event. days before the Mw6.3 earthquake of 3 March 2021, the critical Π(φ) was approached. In all

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This analysis clearly demonstrates that, from 28 February to 2 March 2021, one to three days before the Mw6.3 earthquake of 3 March 2021, the critical Π(ϕ) was approached. In all cases, for R = 30 km, R = 50 km and R = 80 km, the NT analysis starts at approximately one to two months around the corresponding time markers indicated by MRWA (see Figure 5). It may, thus, be considered that the critical point for the regional seismicity was approached around that time. What happened during the last time period before the main event can be seen in Figure 7, which depicts the time evolution of Π(ϕ), for 0 ≤ ϕ ≤ 0.5, for Mth ≥ 2.0 and R = 30 km, 50 km and 80 km, when calculations started on 6 August 2019 for R = 30 km and R = 50 km and on 7 November 2019 for R = 80 km. It also becomes interesting that around that time, when the critical point was reached, seismicity started Geosciences 2021, 11, 379 10 of 14 to occur in the epicentral region registering a few shallow, weak earthquakes prior to the mainshock. These events, which may be considered as foreshocks, do not affect the analysis, as their occurrence coincides with the approach to the critical point. cases,Applying for R = 30 the km, NT R = analysis 50 km and to the R = seismicity 80 km, the that NT analysisoccurred starts prior at to approximately the Mw6.3 event one in tothe two Thessaly months region, around starting the corresponding the analysis time from markers approximately indicated bythe MRWA time markers (see Figure in 5the). Itlower may, thus,scales be indicated considered by that MRWA the critical and up point to the for thetime regional of the seismicitymainshock was occurrence, approached we aroundobserve that that time. all criticality What happened requirements during theare last fulfilled. time period The latter before is the more main clearly event demon- can be seenstrated in Figure by the7 ,parameters which depicts D, theκ1, timeSnt and evolution Snt−, as oftheyΠ (evolvedφ), for 0 ≤eventφ ≤ 0.5,by event, for M thand≥ 2.0are andcomputed R = 30 km, and 50 plotted km and in 80 the km, natural when calculations time domain started and on in 6 the August conventional 2019 for R time, = 30 kmap- andproximately R = 50 km 100 and days on before 7 November the mainshock 2019 for (Figure R = 80 km.8). In It Figure also becomes 8, we observe interesting that all that the aroundrequirements that time, are whenfulfilled the a critical few days point before was reached,the mainshock seismicity for startedall three to cases occur that in the we epicentralstudy, i.e., region for R = registering30 km, R = 50 a fewkm and shallow, R = 80 weak km around earthquakes the epicenter prior to of the the mainshock. mainshock. TheseThe results, events, thus, which indicate may be that considered the regional as foreshocks, seismicity dopresented not affect criticality the analysis, characteristics as their occurrencea few days coincides before the with main the event. approach to the critical point.

R=30km; M =2.0; 06/08/2019 - 09/01/2021 R=30km; M =2.0; 06/08/2019 - 28/02/2021 th th 1 1

0.9 0.9

0.8 0.8

0.7 0.7 0.6

0.6 0.5

0.5 0.4

0.3 0.4 0 0.1 0.2 0.3 0.4 0.5 0 0.1 0.2 0.3 0.4 0.5

R=50km; M =2.0; 06/08/2019 - 23/01/2021 R=50km; M =2.0; 06/08/2019 - 02/03/2021 th th 1 1

0.9 0.9

0.8 0.8

0.7 0.7

0.6 0.6

Geosciences 20210.5, 11, x FOR PEER REVIEW 0.5 11 of 14

0.4 0.4 0 0.1 0.2 0.3 0.4 0.5 0 0.1 0.2 0.3 0.4 0.5

R=80km; M =2.0; 07/11/2019 - 09/01/2021 R=80km; M =2.0; 07/11/2019 - 28/02/2021 th th 1 1

0.9 0.9

0.8 0.8

0.7 0.7

0.6 0.6

0.5 0.5

0.4 0.4 0 0.1 0.2 0.3 0.4 0.5 0 0.1 0.2 0.3 0.4 0.5

Figure 7. Time evolution of Π(φ), for 0 ≤ φ ≤ 0.5, of the seismic activity, for M ≥ 2.0 and R = 30 km (top), R = 50 km Figure 7. Time evolution of Π(ϕ), for 0 ≤ ϕ ≤ 0.5, of the seismic activity, for Mthth ≥ 2.0 and R = 30 km (top), R = 50 km (middle(middle),), and and RR == 80 km km ( (bottombottom),), when when calculations calculations start start on on 06/08/2019 6 August for 2019 R = for30 km R = and 30 kmR = and50 km R and = 50 on km 07/11/2019 and on 7for November R = 80 km. 2019 Π(forϕ) curves R = 80 (dashed km. Π(φ lines)) curves fall (dashed on the theoretical lines) fall onΠ(ϕ the) curve theoretical (red solidΠ(φ )lines), curve calc (redulated solid from lines), Equation calculated (6), fromas the Equation critical stage (6), as is the approached. critical stage is approached.

Figure 8. Time evolution of the NT analysis parameters κ1, D, Snt and Snt−, in natural time (left) and conventional time (right), as they evolve event by event prior to the Thessaly Mw6.3 mainshock, considering an area with radius of R = 30 km (top), R = 50 km (middle), and R = 80 km (bottom) around the epicenter and for a magnitude threshold Mth = 2.0. The

Geosciences 2021, 11, x FOR PEER REVIEW 11 of 14

R=80km; M =2.0; 07/11/2019 - 09/01/2021 R=80km; M =2.0; 07/11/2019 - 28/02/2021 th th 1 1

Geosciences 2021, 11, 379 11 of 14 0.9 0.9

0.8 0.8

0.7 0.7 Applying the NT analysis to the seismicity that occurred prior to the Mw6.3 event in the Thessaly region, starting the analysis from approximately the time markers in the lower 0.6 0.6 scales indicated by MRWA and up to the time of the mainshock occurrence, we observe 0.5 that all criticality requirements are fulﬁlled.0.5 The latter is more clearly demonstrated by the parameters D, κ1, Snt and Snt−, as they evolved event by event, and are computed and 0.4 0.4 0 0.1 0.2plotted 0.3 inthe 0.4 natural 0.5 time domain and in the0 conventional 0.1 0.2 time, 0.3 approximately 0.4 0.5 100 days

before the mainshock (Figure8). In Figure8, we observe that all the requirements are

Figure 7. Time evolution offulﬁlled Π(ϕ), for a few 0 ≤ daysϕ ≤ 0.5, before of the the seismic mainshock activity, for for all M threeth ≥ cases2.0 and that R = we 30 study, km (top i.e.,), R for = R50 = km 30 km, (middle), and R = 80 km (bottomR = 50), km when and calculations R = 80 km start around on 06/08/2019 the epicenter for R of = the30 km mainshock. and R = 50 The km results,and on 07/11/2019 thus, indicate for R = 80 km. Π(ϕ) curves that(dashed the lines) regional fall on seismicity the theoretical presented Π(ϕ) criticalitycurve (red characteristicssolid lines), calculated a few daysfrom Equation before the (6), main as the critical stage is approached.event.

Figure 8. Time evolution of the NT analysis parameters κ1, D, Snt and Snt−, in natural time (left) and conventional time Figure 8. Time evolution of the NT analysis parameters κ1, D, Snt and Snt−, in natural time (left) and conventional time (right), as they evolve event by event prior to the Thessaly M 6.3 mainshock, considering an area with radius of R = 30 km (right), as they evolve event by event prior to the Thessaly Mww6.3 mainshock, considering an area with radius of R = 30 km ((toptop),), RR = = 5050 kmkm (middle(middle),), and and R R = = 80 80 km km ( bottom(bottom)) around around the the epicenter epicenter and and for for a a magnitude magnitude threshold threshold M Mthth= =2.0. 2.0. TheThe analysis was started on 6 August 2019 for R = 30 km and R = 50 km and on 7 November 2019 for R = 80 km. The dashed

horizontal lines indicate the entropy limit of Su = 0.0966 and the value κ1= 0.070. The shaded rectangle marks the time when the critical stage is approached. Geosciences 2021, 11, 379 12 of 14

4. Concluding Remarks In the present work, we investigated the regional patterns of seismicity in the area of the Thessaly (Mw6.3) strong earthquake on 3 March 2021, by applying MRWA and NT analysis, two methods that have been used for the identification of critical stages in the preparation process of major earthquakes. The analysis was performed in the natural time domain, with an approximate starting point indicated by MRWA. The latter showed a decrease in the standard deviation of the wavelet coefficients σwav(m) at much lower scales, similar to the observations in [26,27,54] prior to the occurrence of major events. Within this joint approach, the initial application of MRWA in regional seismicity around the epicenter, and for a wide time period before the mainshock, indicated a time segment where the NT analysis was applied in order to explore possible indicators that suggested the entrance to a critical stage. The results demonstrated that regional seismicity approached criticality a few days before the Mw6.3 earthquake that occurred on 3 March 2021 in the Thessaly region, in agreement with the results in [57]. In other words, the curve of the power spectrum, Π(φ), in the natural time domain that characterizes the evolution of the regional seismicity, coincided with the theoretical curve of critical point phenomena a few days before the Mw6.3 mainshock, in a similar way to that of non-equilibrium critical systems. Hence, the analysis of the regional seismicity in the natural time domain, initiated at approximately the time marks indicated by the results of MRWA, pointed to an approximate date of the impending large Mw6.3 earthquake of 3 March 2021, within a narrow time window in the order of a few days. These results lay further support to the methodology introduced in [28] regarding the combination of MRWA and NT analyses for the identification of critical stages of regional seismicity prior to strong earthquakes, providing a novel and promising framework for better understanding the evolution of earthquake generation processes.

Author Contributions: Conceptualization, F.V.; methodology, F.V., G.M. and G.H.; validation, F.V., G.M. and G.H.; resources, F.V., G.M. and G.H.; data curation, F.V.; writing—original draft preparation, F.V.; writing—review and editing, F.V., G.M. and G.H. All authors have read and agreed to the published version of the manuscript. Funding: We acknowledge support of this study by the project “HELPOS: Hellenic Plate Observing System” (MIS 5002697) which is implemented under the action “Reinforcement of the Research and In- novation Infrastructure”, funded by the Operational Programme “Competitiveness, Entrepreneurship and Innovation” (NSRF 2014–2020) and co-financed by Greece and the European Union European Regional Development Fund. Data Availability Statement: Data are available at http://eida.gein.noa.gr/, at the National Obser- vatory of Athens, Institute of Geodynamics, doi:10.7914/SN/HL, and the Aristotle University of Thessaloniki Seismological Network, doi:10.7914/SN/HT (last accessed on 1 June 2021). Acknowledgments: The authors are grateful to the personnel of the Geodynamic Institute of the National Observatory of Athens and the Department of Geophysics of the Aristotle University of Thessaloniki involved in installing, operating and maintaining the stations used in this study, as well as all staff involved in the Hellenic Unified Seismic Network. We thank K. Pavlou for her help in the creation of Figure2. Conflicts of Interest: The authors declare no conflict of interest.

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