Natural Time Analysis: Results Related to Two Earthquakes in Greece during 2019 †

: The following two earthquakes occurred in Greece during 2019: First, a Mw5.4 earthquake close to Preveza city in Western Greece on 5 February and a Mw5.3 earthquake 50 km East of Patras on 30 March. Here, we present the natural time analysis of the Seismic Electric Signals (SES) activities that have been recorded before these two earthquakes. In addition, we explain how the occurrence times of these two earthquakes can be identiﬁed by analyzing in natural time the seismicity subsequent to the SES activities.


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 22:55 UTC at an epicentral distance around 133 km SW of the city of Patras, Western Greece (see Figure 1). It was preceded by an anomalous geolectric signal that was recorded on 2 October 2018 at a measuring station 70km away from the epicenter [2]. Upon analyzing this signal in natural time, it was found [2] that it conforms to the conditions suggested (e.g., see [3][4][5])) for its clarification as precursory Seismic Electric Signal (SES) activity [4,6,7]. Notably, the observed lead time of 23 days lies within the range of values that has been very recently identified [8] 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 revealed [2] that critical 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 [4]. The application [2] of the recent method of nowcasting earthquakes [9][10][11][12][13], which is based on natural time, has revealed that an earthquake potential score of around 80% was observed just before the occurrence of this M w 6.8 earthquake. Here, we focus on the recording [14] of additional SES activities after the occurrence of the latter earthquake in the beginning of January 2019 (see below) that preceded the following two earthquakes in Greece during 2019: First, a M w 5.4 earthquake [15] close to Preveza city in Western Greece on 5 February 2019 and a M w 5.3 earthquake [16] on 30 March 2019 a few tens of km East of Patras SES measuring station (labeled PAT in Figure 1).

Results
Two SES activities were recorded [14] by the VAN telemetric network [3] operating in real time in Greece on 3 January 2019 and 9 January 2019 at the measuring stations PAT and PIR, respectively (see Figure 1).  [3] operating in Greece are shown by the blue circles. The blue square corresponds to the central station operating at Glyfada, Athens (ATH), where the data are collected. The thick black line depicts the Hellenic arc [17] while the gray shaded area and the black rectangle the selectivity map of Pirgos (PIR) measuring station (see Figure 1 in [2]) and Patras (PAT) measuring station (see the rectangle with solid lines in Figure 8 in [18]), respectively. After the recording of the SES activities on 3 January 2019 at PAT and on 9 January 2019 at PIR, the areas corresponding to the selectivity maps of these two measuring stations have been reported in [14] as probable to suffer a strong EQ. The red stars correspond to the epicenters of the M w 6. According to the VAN method of short-term earthquake prediction [3,4,6,7,[19][20][21], the electric signals that are emitted from the future focal area as the stress increases prior to the EQ due to the collective (re)orientation (cf. such a cooperativity is a hallmark showing that the region enters the critical stage) [22] of the pre-existing electric dipoles [23] in the ionic constituents of the rocks, e.g., see Figure 1 in [24], follow [7], conductive paths in the solid Earth crust and become detactable at certain (SES sensitive) sites on the Earth's surface giving rise to the so-called selectivity phenomenon [7,17,[25][26][27][28][29][30][31]. This means that an SES measuring station is capable of recording SESs emitted from certain EQ prone areas. After long experimentation (cf. SES research in Greece started in the 1980s, e.g., see [32,33]) for each measuring station, one may construct a selectivity map of this station by considering the EQs that have been preceded by SES recorded in the station as well as by using geological and geophysical data (since faults are usually highly more conductive than their surroundings, they consitute conductive paths, e.g., see [25]). The gray shaded area in Figure 1 depicts the selectivity map of the PIR measuring station as reported in [2] while the black rectangle in the same figure corresponds to the selectivity map of the PAT measuring station [14,18].

20˚E
The SES activity recorded on 3 January 2019 at PAT station can be seen in Figure 5 in [14]. The analysis in natural time has led [14] to values of κ 1 , S and S − which are compatible with those observed for SES (see Section 4.1). After applying the methodology suggested in [34] for the analysis of the SES activity recorded on 3 January 2019 at PAT we obtain κ 1 = 0.075 (22), S = 0.071 (22), and S − = 0.075 (30). More or less similar results are found for the SES recorded on 9 January 2019 at PIR.
After these observations and in order to estimate the occurrence time of the impending EQs, we started to analyze in natural time the seismic activity occurring after the SES within the respective selectivity maps of each measuring station, i.e., the gray shaded area of Figure 1 for PIR and the one shown by the black rectangle in Figure 1 for PAT. We observed (see Figure 7 in [35]) that when analyzing the seismicity within the PIR selectivity map, the criticality condition κ 1 = 0.070 has been fulfilled upon the occurrence of a ML(ATH) = 3.

Discussion
It is notable that the occurrence of the two EQs under study took place almost a week after the criticality condition κ 1 = 0.070 has been met for various magnitude thresholds. 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 [2,4,18,[36][37][38][39].

Natural Time Analysis (NTA)
In a time series consisting of N individual events(e.g., electric pulses or EQs), the natural time [4,[40][41][42] associated with the k-th event is given by χ k = k/N. In NTA [4,[40][41][42], the pair (χ k , Q k ) is studied, where Q k is proportional to the energy emitted during the k-th event. For example in the case of SES, Q k is proportional to the duration of each SES pulse [40,41], while for EQs it may be considered proportional to the seismic moment [40,42,43]. How the time series coming from a variety of complex systems are read in natural time can be seen in Figure 1 of [5].
The pair (χ k , Q k ) is studied by considering the normalized energy for the k-th event p k = Q k /∑ N n=1 Q n , where p k can be also considered as a probability distribution [5,44]. In view of the latter, the function [4,[40][41][42]44] provides information about the probability distribution p k when ω → 0. Expanding Equation (1) around ω = 0, we obtain that Π(ω) = 1 − κ 1 ω 2 + . . . , where κ 1 stands for the variance of natural time with respect to the distribution p k . When Q k are independent and identically distributed random variables, we have that p k → 1/N. This is the case of the so-called [4,45,46] 'uniform' distribution leading to a value of κ 1 equal to κ u = 1/12 ≈ 0.083. For critical systems, Varotsos et al. [47] have shown that for a variety of systems approaching criticality. Thus, κ 1 reaches the value of 0.070 for a critical system or 0.083 for a system exhibiting stationary or quasi-periodic behavior [5]. Apart from κ 1 , another useful quantity in NTA [4,5] is the entropy S given by [40,46,48] where the brackets . . . ≡ ∑ N k=1 . . . p k denote averages with respect to the distribution p k . The entropy S is a dynamic entropy that exhibits [49] positivity, concavity and Lesche [50,51] experimental stability. When Q k are independent and identically distributed random variables, S reaches [48] the value S u ≡ ln 2 2 − 1 4 ≈ 0.0966 that corresponds to the aforementioned 'uniform' distribution. For SES, it has been experimentally observed [4,49] that S SES < ∼ S u . Upon reversing the time arrow and hence applying the time reversal operator T to p k , i.e., T p k = p N−k+1 , the value of S changes to a value S − . Again, it has been experimentally observed [4,49] that for SES activities:

Conclusions
The two strongest earthquakes that occurred in Greece since 1 January 2019, i.e., the Mw5.4 earthquake close to Preveza city in Western Greece on 5 February and the Mw5.3 earthquake 50km East of Patras on 30 March, were preceded by Seismic Electric Signals (SES) activities that were identified as such before the earthquakes [14].
The occurrence times of these two earthquakes can be approached by analyzing in natural time the seismicity subsequent to the SES activities within the selectivity maps of the corresponding VAN stations that recorded the SES activities.

Conflicts of Interest:
The authors declare no conflict of interest.

Abbreviations
The following abbreviations are used in this manuscript:

ATH
Athens EQ Earthquake ML(ATH) Local EQ magnitude reported by the Institute of Geodynamics of the National Observatory of Athens Mw Moment magnitude NTA Natural time analysis PAT Patras SES measuring station PIR Pirgos SES measuring station SES Seismic Electric Signals VAN Varotsos Alexopoulos Nomikos