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

A strong earthquake of magnitude Mw6.8 struck Western Greece on 25 October 2018 with an epicenter at 37.515∘ N 20.564∘ E. It was preceded by an anomalous geolectric signal that was recorded on 2 October 2018 at a measuring station 70 km away from the epicenter. Upon analyzing this signal in natural time, we find that it conforms to the conditions suggested for its identification as precursory Seismic Electric Signal (SES) activity. Notably, the observed lead time of 23 days lies within the range of values that has been very recently identified as being statistically significant for the precursory variations of the electric field of the Earth. Moreover, the analysis in natural time of the seismicity subsequent to the SES activity in the area candidate to suffer this strong earthquake reveals that the criticality conditions were obeyed early in the morning of 18 October 2018, i.e., almost a week before the strong earthquake occurrence, in agreement with earlier findings. Finally, when employing the recent method of nowcasting earthquakes, which is based on natural time, we find an earthquake potential score around 80%.


Introduction
According to the United States Geological Survey (USGS) [1], a strong earthquake (EQ) of moment magnitude M w 6.8 occurred on 25 October 2018 10:55 p.m. Coordinated Universal Time (UTC) at an epicentral distance around 133 km SW of the city of Patras, Western Greece. Patras has a metropolitan area inhabited by a quarter of a million persons and fatal casualties have been probably avoided because, among others, at 10:23 p.m. UTC, almost half an hour before the strong EQ, a moderate EQ of magnitude M = 5.0 occurred approximately at the same area as the strong EQ [2] (see Figure 1).  Figure 1. Map of the larger area N 47 29 E 35 12 in which the EQs of magnitude greater than or equal to 6.0 are shown by the green circles. The locations of the measuring stations operating in Greece of the VAN telemetric network are shown by the blue circles. The thick black line depicts the Hellenic arc [3] while the gray shaded area the selectivity map of Pirgos (PIR) measuring station. The red star corresponds to the epicenter of the M w 6.8 EQ on 25 October 2018 and the red circle delimits a circular region with radius R = 225 km around the city of Patras.
Geoelectric field continuous monitoring is operating in Greece by the Solid Earth Physics Institute [4][5][6] at nine measuring field stations (see the blue circles in Figure 1) aiming at detecting Seismic Electric Signals (SES). SES are low frequency (≤1 Hz) variations of the electric field of the Earth that have been found to precede strong EQs in Greece [7][8][9][10][11], Japan [12][13][14], China [15][16][17][18], Mexico [19,20], and elsewhere [21]. They are emitted due to the cooperative orientation of the electric dipoles [5,22,23] (that exist anyhow due to defects [24,25] in the rocks) of the future focal area when the gradually increasing stress before the strong EQ reaches a critical value [10]. SES may appear either as single pulses or in the form of SES activities, i.e., many pulses within a relatively short time period ([9], e.g., see Figure 2). The lead time of single SES is less than or equal to 11 days, while, for SES activities, it varies from a few weeks up to 5 1 2 months [6,9]. SES are recorded [9,10] at sensitive points [26] on the Earth's surface that have been selected after long experimentation in Greece during the 1980s and 1990s that led [27,28] to the construction of the so-called VAN telemetric network (from the acronym of the scientists Varotsos, Alexopoulos and Nomicos who pioneered this research) comprising the nine measuring field stations depicted in Figure 1. Each measuring station records SES from specific EQ prone areas which constitute the so-called selectivity map of the station [3,10,29]. The gray shaded area of Figure 1 depicts the selectivity map for the Pirgos (PIR) measuring station as it resulted after the recording of SES from various epicentral areas [30]. A basic criterion for distinguishing SES from noise is that the recorded signal should [9] exhibit properties compatible with the fact that it was emitted far away from the recording station. This is usually called [9] ∆V/L criterion (where ∆V stands for the potential difference between two electrodes that constitute a measuring electric dipole and L for the distance between them) and has been found [31-34] to be compatible with the aforementioned SES generation model if we take into account that EQs occur in faults (where resistivity is usually orders of magnitude smaller than that of the surrounding rocks, e.g., see [5] and references therein). The SES research has been greatly advanced after the introduction of the concept of natural time in 2001 [35][36][37]. Firstly, the criticality properties of SES activities (like the existence of long-range correlations and unique entropic properties) has been revealed by natural time analysis and hence new possibilities have been provided for the identification of SES and their distinction from man-made noise [11,[38][39][40][41][42][43]. Secondly, natural time analysis allowed the introduction of an order parameter for seismicity the study of which allows the determination of the occurrence time of the strong EQ within a few days up to one week [6,30,40,[44][45][46][47]. Thirdly, minima of the fluctuations of the order parameter of seismicity have been identified before all shallow EQs with M ≥ 7.6 in Japan during the 27 year period from 1 January 1984 to 11 March 2011, the date of the M9.0 Tohoku EQ occurrence [48,49]. Finally, the interrelation of SES activities and seismicity has been further clarified because, when studying the EQ magnitude time series in Japan, it was found that the minimum of the fluctuations of the order parameter of seismicity, which is observed simultaneously with the initiation of an SES activity [50], appears when long range correlations prevail [51].
The scope of this paper is twofold. First, we report the geoelectrical field changes (SES) observed before the M w 6.8 EQ that occurred on 25 October 2018. Second, we present the natural time analysis of both the SES activity and the seismicity preceding this EQ. The paper is structured as follows: the background of natural time analysis is presented in the next section. In the subsequent section, we give the results obtained. A discussion follows in Section 4 and a summary of our results and the main conclusions are presented in the final section.

Natural Time Analysis Background
Natural time analysis, introduced in the beginning of the 2000s [35-39], uncovers unique dynamic features hidden behind the time series of complex systems. In a time series comprising N events, the natural time χ k = k/N serves as an index for the occurrence of the k-th event. This index together with the energy Q k released during the k-th event, i.e., the pair (χ k , Q k ), is studied in natural time analysis. Alternatively, one studies the pair (χ k , p k ), where stands for the normalized energy released during the k-th event. As is obvious from Equation (1), the correct estimation of p k simply demands that Q k should be proportional to the energy emitted during the k-th event. Thus, for SES activities, Q k is proportional to the duration of the k-th pulse while, for EQs, it is proportional to the energy emitted [52] during the k-th EQ of magnitude M k , i.e., Q k ∝ 10 1.5M k (see also [6,53]). The variance of χ weighted for p k , labeled by κ 1 , is given by [6,35,[38][39][40] For the case of seismicity, the quantity κ 1 has been proposed to be an order parameter since κ 1 changes abruptly when a mainshock (the new phase) occurs, and, in addition, the statistical properties of its fluctuations are similar to those in other non-equilibrium and equilibrium critical systems ([40], see also pp. 249-253 of Reference [6]). It has been also found that κ 1 is a key parameter that enables recognition of the complex dynamical system under study entering the critical stage [6,[35][36][37]. This occurs when κ 1 becomes equal to 0.070 {( [54], see also page 343 of Reference [6]). In Table 8.1 of Reference [6], one can find a compilation of 14 cases including a variety of dynamical models in which the condition κ 1 = 0.070 has been ascertained (cf. this has been also later confirmed in the analyses of very recent experimental results in Japan by Hayakawa and coworkers [55][56][57]). Especially for the case of SES activities, it has been found that, when they are analyzed in natural time, we find κ 1 values close to 0.070 ([35,36,38,39], e.g., see Table 4.6 on p. 227 of Reference [6]), i.e., when analyzing in natural time the small EQs with magnitudes greater than or equal to a threshold magnitude M thres that occur after the initiation of an SES activity within the selectivity map of the measuring station that recorded the SES activity, the condition κ 1 = 0.070 is found to hold for a variety of M thres a few days up to one week before the strong EQ occurrence [6,30,40,41,[44][45][46][47]54,58] . This is very important from a practical point of view because it enables the estimation of the occurrence time of a strong EQ with an accuracy of one week or so. The entropy S in natural time is defined [6,35,39,53,59] by the relation It is a dynamic entropy showing [60] positivity, concavity and Lesche [61,62] experimental stability. When Q k are independent and identically-distributed random variables, S approaches [59] the value S u ≡ ln 2 2 − 1 4 ≈ 0.0966 that corresponds to the case Q k = 1/N, which within the context of natural time is usually termed "uniform" distribution [6,39,53]. Notably, S changes its value to S − upon time-reversal, i.e., when the first event becomes last (Q 1 → Q N ), the second last but one (Q 2 → Q N−1 ), etc., and hence it gives us the possibility to observe the (true) time-arrow [60]. Interestingly, it has been established [6,53] that both S and S − for SES activities are smaller than S u , On the other hand, these conditions are violated for a variety of similar looking electrical noises (e.g., see Table 4.6 on p. 228 of Reference [6]).
Natural time has been recently employed by Turcotte and coworkers [63][64][65][66] as a basis for a new method to estimate the current level of seismic risk called "earthquake nowcasting". This will be explained in the next section.

Natural Time Analysis of Geoelectrical Signals. Criteria for Distinguishing SES
Apart from the aforementioned ∆V/L criterion suggested long ago for the distinction of SES from man-made noise [9], natural time analysis has provided, as mentioned, three additional criteria for the classification of an electric signal as SES activity. These criteria are Equation (3) and the conditions (6). The analysis in natural time of the dichotomous representation shown in Figure 2 results in κ 1 = 0.072(2), S = 0.066(2) and S − = 0.079(3), which are obviously compatible with the criteria for distinguishing SES from noise. This leads us to support that the anomalous variation of the electric field of the Earth observed on 2 October 2018 is indeed an SES activity.

Estimation of the Occurrence Time of the Impending EQ
We now follow the method suggested in Reference [30] for the estimation of the occurrence time of the impending strong EQ by analyzing in natural time all the small EQs of magnitude greater than or equal to M thres that occurred after the initiation of the SES activity recorded on 2 October 2018 within the selectivity map of PIR measuring station shown by the gray shaded area in Figure 1. The EQ catalog [67] of the Institute of Geodynamics of the National Observatory of Athens has been used and, each time a new small EQ takes place, we calculate the κ 1 values corresponding to the events that occurred within all the possible subareas of the PIR selectivity map that include this EQ [30]. This procedure leads to an ensemble of κ 1 values from which we can calculate the probability Prob(κ 1 ) of κ 1 to lie within κ 1 ± 0.025. Figure 3a-d depict the histograms of Prob(κ 1 ) obtained after the occurrence of each small EQ with magnitude greater than or equal to 2.7, 2.8, 2.9, and 3.0, respectively. We observe that, within a period of 5 h around 18 October 2018 00:30 a.m. UTC, all four of the distributions Prob(κ 1 ) exhibit a maximum at κ 1 = 0.070. This behavior has been found, as already mentioned, to occur a few days up to one week or so before the strong EQ occurrence [6,30,[45][46][47]. Actually, one week later, i.e., on 25 October 2018, a strong M w 6.8 EQ occurred [1] within the selectivity map of the PIR measuring station (see the red star in Figure 1). Interestingly, as it is written in the legends of the panels of Figure 3, two of the three small EQs that led to the fulfillment of the criticality condition κ 1 = 0.070 originated from epicentral areas located only 20 or 25 km south that of the strong EQ.

Estimation of the Current Level of Risk by Applying EQ Nowcasting
Nowcasting EQs is a recent method for the determination of the current state of a fault system and the estimation of the current progress in the EQ cycle [63]. It uses a global EQ catalog to calculate from "small" EQs the level of hazard for "large" EQs. This is achieved by employing the natural time concept and counting the number n s of "small" EQs that occur after a "strong" EQ. The current value n(t) of n s since the occurrence of the last "strong" EQ is compared with the cumulative distribution function (cdf) P(n s < n(t)) of n s obtained when ensuring that we have enough data to span at least 20 or more "large" EQ cycles. The EQ potential score (EPS) which equals the "current" cdf value, EPS = P(n s < n(t)) is therefore a unique measure of the current level of hazard and assigns a number between 0% and 100% to every region so defined. Nowcasting EQs has already found many useful applications [64][65][66] among which is the estimation of seismic risk to Global Megacities. For this application [64], the EQs with depths smaller than a certain value D within a larger area are studied in order to obtain the cdf P(n s < n(t)). Then, the numberñ s of "small" EQs around a Megacity, e.g., EQs in a circular region of epicentral distances smaller than a radius R with hypocenters shallower than D, is counted since the occurrence of the last "strong" EQ in this region. Based on the ergodicity of EQs that has been proven [68][69][70] by using the metric published in References [71,72], Rundle et al. [64] suggested that the seismic risk around a Megacity can be estimated by using the EPS corresponding to the currentñ s estimated in the circular region. Especially in their Figure 2, they used the large area N 47 29 E 35 12 in order to estimate the EPS for EQs of magnitude greater than or equal to 6.5 at an area of radius R = 400 km around the capital of Athens in Greece. Figure 4 shows the results of a similar calculation based on the United States National EQ Information Center PDE catalog (the data of which are available from Reference [73]), which we performed focusing on the city of Patras, Greece, for EQs of magnitude greater than or equal to 6.0. Notably, before the occurrence of the M w 6.8 EQ on 25 October 2018, EPS was found to be as high as 80%.  Figure 1 between the occurrence of two strong (M ≥ 6.0) EQs. The numberñ s of small EQs that occurred within R ≤ 225 km with depths D ≤ 200 km from the city of Patras since the M w 6.5 EQ on 17 November 2015 07:11 a.m. UTC [74] and before the occurrence of the M w 6.8 EQ on 25 October 2018 was 212.

Discussion
Recently, the statistical significance of the Earth's electric and magnetic field variations preceding EQs has been studied [75] on the basis of the modern tools of event coincidence analysis [76][77][78] and receiver operating characteristics [79,80]. Using an SES dataset [9,10,81] from the 1980s, it was found that SES are statistically significant precursors to EQs for lead times in the following four distinct time periods: 3 to 9 days, 18 to 24 days, 43 to 47 days, and 58 to 62 days (the first one corresponds to single SES, while the latter to three SES activities [75]). Since the SES activity, shown in Figure 2, was recorded on 2 October 2018, the SES lead time for the present case of the M w 6.8 EQ on 25 October 2018, which is 23 days, falls favorably within the second time period of 18 to 24 days. Moreover, the analysis of the seismicity subsequent to the initiation of the SES activity in the selectivity area of the PIR station has led to the conclusion that the criticality condition κ 1 = 0.070 has been satisfied early in the morning on 18 October 2018. This compares favorably with the time window of a few days up to one week already found from various SES activities in Greece, Japan and United States [6,30,[45][46][47]58].
Let us now turn to the results concerning the entropy of the SES activity of Figure 2 in natural time. As it was reported, both S and S − are well below S u in accordance with the findings (e.g., see Reference [53]) so far for SES activities. Based on the critical properties that characterize the emission of signals that precede rupture (i.e., infinite range correlations compatible with a detrended fluctuation analysis (DFA) [82][83][84] exponent α DFA = 1), a fractional Brownian motion [85,86]  Finally, the successful results (i.e., the 80% EPS found before the occurrence of the M w 6.8 EQ on 25 October 2018) from the EQ nowcasting method, which is based on natural time, are very promising. Nowcasting does not involve any model and there are no free parameters to be fit to the data [63].

Conclusions
The strong EQ of magnitude M w 6.8 that occurred in Western Greece on 25 October 2018 was preceded by an SES activity on 2 October 2018 recorded at the PIR measuring station of the VAN telemetric network. The EQ epicenter was located within the selectivity map of PIR depicted by the gray shaded area in Figure 1.
The lead time of 23 days between the precursory SES activity and the strong EQ is statistically significant as recently found by the recent methods of event coincidence analysis and receiver operating characteristics. Both the entropy S and the entropy S − under time reversal in natural time are compatible with previous observation for SES activities as well as agree with a model for SES activities based on fractional Brownian motion. The analysis in natural time of the seismicity subsequent to the SES activity by considering the events occurring within the selectivity area of PIR shows that criticality has been reached early in the morning on 18 October 2018, almost a week before the strong EQ occurrence, in accordance with the earlier findings. Finally, EQ nowcasting has revealed an 80% EPS. In general, here we showed that the strong EQ under discussion provides an excellent validation of the methods developed so far in natural time for EQ prediction.