## 2 Materials and Methods
### Natural Time Analysis: Background
Finally, how \(\beta_{W}\) varies in conventional time can be pursued by sliding event by event the window of \(W\) consecutive EQs in the EQ catalog; the occurrence time of the EQ that followed the last one of the window studied is attributed to each \(\beta_{W}\) obtained. It has been experimentally observed that appropriately selected minima of \(\beta_{W}\) labeled \(\beta_{W,min}\) in regional EQ catalogs [51; 99] coincide with the initiation of the emission of SES activities signaling that a strong EQ is going to occur within a maximum lead time of \(5\frac{1}{2}\) months [23]. The entropy \(S\) in natural time [20; 54] is given by  
\[S=\langle\chi\ln\chi\rangle-\langle\chi\rangle\ln\langle\chi\rangle=\sum_{k=1} ^{N}p_{k}\frac{k}{N}\ln\left(\frac{k}{N}\right)-\left(\sum_{k=1}^{N}p_{k} \frac{k}{N}\right)\ln\left(\sum_{k=1}^{N}p_{k}\frac{k}{N}\right). \tag{9}\]  
In the case of a \"uniform\" distribution [54], \(Q_{k}\) are considered independent and identically distributed (IID) positive random variables and \(S\) becomes \(S_{u}\equiv\frac{\ln 2}{2}-\frac{1}{4}\approx 0.0966\). The application [23; 38] of time-reversal \(\hat{T}\), i.e., \(\hat{T}p_{k}=p_{N-k+1}\), in Equation (9) leads to  
\[S_{-}=\sum_{k=1}^{N}p_{N-k+1}\frac{k}{N}\ln\left(\frac{k}{N}\right)-\left( \sum_{k=1}^{N}p_{N-k+1}\frac{k}{N}\right)\ln\left(\sum_{k=1}^{N}p_{N-k+1} \frac{k}{N}\right). \tag{10}\]  
The quantity \(S_{-}\) differs from \(S\), leading to an entropy change  
\[\Delta S\equiv S-S_{-} \tag{11}\]  
upon time reversal. Thus, \(S\) is time-reversal asymmetric [23]. Moreover, \(S\) is a _dynamic_ entropy, different from static ones, e.g., Shannon entropy, that is positive, concave, and experimentally ([PERSON]) stable [100; 101] (see Chapter 3 of [23]). The entropy change \(\Delta S\) under time reversal is also crucial for the identification of the approach of the system to a dynamic phase transition [5; 38; 74]. For this purpose, \(S\) and \(S_{-}\) are calculated for a natural time window of length \(i\) (=number of consecutive events) sliding event by event through the whole time-series. Equation (11) is then used for estimating \(\Delta S\), thus resulting in a time-series of consecutive \(\Delta S_{i}\) values. NTA can provide an estimate of the occurrence time of the impending EQ when analyzing seismicity in the area candidate to suffer the strong EQ after the initiation of an SES activity or after the observation of \(\beta_{W,min}\). Such examples for Greece can be found in Chapter 7 of [23] while for Japan, Mexico, and California, they can be found in Chapter 5 of [38]. Usually [20; 22; 44; 55; 56], the strong EQ occurs a few days up to one week or so after a true coincidence of \(\Pi(\omega)\) of the evolving seismicity to that of the critical state, \(\Pi_{crit}(\omega)\) of Equation (6). The following criteria have been used (see, e.g., [23; 38]) to assure such a coincidence:  
1. The \"average\" distance \(\langle D\rangle\) between the curves of \(\Pi(\omega)\) and \(\Pi_{crit}(\omega)\) for \(\omega\leq\pi\) should be \(\langle D\rangle<10^{-2}\). Details on this calculation can be found in [20; 23]. 2. The final approach of the evolving \(\Pi(\omega)\) to \(\Pi_{crit}(\omega)\) must be from below, see, e.g., Figure 5 of [77], which alternatively means that \(\kappa_{1}\) gradually decreases with time before reaching the critical value 0.070 of Equation (7). 3. At the coincidence, both entropies \(S\) and \(S_{-}\) in natural time must be smaller than \(S_{u}\). 4. Since this process (critical dynamics) is considered to be self-similar, the occurrence time of the true coincidence should not markedly vary upon changing the magnitude threshold \(M_{thres}\) which is used for NTA of seismicity. 5. The final approach described by the second criterion starts to be obeyed after an EQ for which \(\Delta S\) of seismicity in the candidate epicentral area under time reversal exhibits a local minimum. In simple words, after \(S_{-}\) exceeds \(S\) by a large amount, \(\kappa_{1}\) starts decreasing to finally approach 0.070.