A plausible universal behaviour of earthquakes in the natural time-domain

In previous papers, we showed that in the new time-domain, called natural time: 1) the power spectrum of an SES-activity exhibits properties of critical phenomena and the power spectra of the local seismicity subsequent to an SES-activity become the same as that of the latter just before the main shock, 2) the power spectra of regional and long-term seismicity in both Greece and San Andreas fault system also exhibits features of critical phenomena. Here we show that the same can be found for the earthquakes in Japan. The “power spectra” in natural time of the seismicities in all the three areas fall on a universal curve.

Introduction. The analysis of seismicity as well as that of Seismic Electric Signal (SES) activities (e.g., refs. 1), 2)) has always been carried out in the conventional time-domain. It has recently been shown, 3)-7) however, that the use of a new time-domain, termed natural time χ, exhibits several distinct advantages. The use of this new time-domain was motivated by studies on complex systems and critical dynamics. 3) The natural time χ k = k/N serves as an index 3) for the occurrence of the kth event in a time series comprised of N events such as SES activity and associated seismicity. For the analysis of SES, we considered the time evolution of the pair of two quantities, i. e., (χ k , Q k ) where Q k denotes the duration of the kth signal. We introduced a continuous function F(ω) defined as follows: [1] where ω = 2πφ, and φ stands for the natural frequency. Dividing by F(0), we normalise F(ω): [2] where [3] We can regard, in analogy to the probability theory, p k and χ k = k/N a probability value and probability variable respectively.
Therefore, in what follows, we will be discussing on the properties of Π(ω) or Π(φ) in this range of φ. Motivated by the piezo-stimulated currents' model 1),2) for the SES generation, i.e., SES is generated by a cooperative reorientation of the electric dipoles within crystals of rock-forming minerals upon approaching a critical pressure, our theoretical analysis 3) based on the theory of critical phenomena (see also refs. 4) and 10) and references therein) showed that the

A plausible universal behaviour of earthquakes in the natural time-domain
following relations should hold for an SES-activity: [5] where ω(=2πφ) denotes the angular frequency in natural time. Expanding eq. [5] around ω = 0, we get This implies 3) that the variance of χ is: κ 1 = <χ 2 > -<χ> 2 = 0.07 . [7] According to the probability theory, the moments of a distribution and hence the distribution itself can be determined once the behaviour of the characteristic function of the distribution is known around zero. Our earlier studies 3),4) showed that actually all SES activities exhibit a universal behaviour obeying the relations [5] - [7], whereas these relations are not obeyed 5),6) by artifi-cial noises or ionic current fluctuations in biological membrane channels, even when they are indistinguishable in conventional time. This indicates that among those investigated only the observed SES have the properties compatible with those of the critical phenomena.
Natural time-domain analysis of seismicity. The analysis of the seismic activity in the natural timedomain can be made similarly. In this case, however, instead of Q k , we ascribed the corresponding seismic moment M 0 to the kth seismic event 3),8)-10) and the time evolution of the pair (χ k , M 0 k ) was investigated. Analysis of seismicity in the natural time-domain led to the following two major conclusions. It is the aim of the present study to investigate whether these two conclusions hold for Japanese earthquakes or not. If so, they may be regarded as universal. 1) Analysing the seismicity in a candidate area which can be estimated from the observed SES data, the following interesting property has been shown 3) : If we set Π(ω) = 18 5ω 2 -6cosω 5ω 2 -12sinω 5ω the natural time for the seismicity zero at the initiation time of the concerned SES-activity, we can form time series of seismic events in natural time for various time windows as the number of consecutive earthquakes N increases. When we compute Π(φ) for each of the time windows, we found that, in the range 0 < φ ≤ 0.5, it approaches, as N increases from 6 to some value less than 40, to that given by eq. [5], namely that of the SESactivity. Interestingly, this coincidence of Π(φ) of seismicity and SES-activity happens only a few days before the anticipated main shock, thus providing an effective way to make time prediction of a major earthquake much more precise than before. This investigation so far has been possible 3),8)-10) only for Greece because no SESactivity data has been available from other areas. When earthquakes occur in the way described above, we call the process "single correlated process". When the next main shock occurs within a short time, however, the situation becomes complex. As mentioned above, to observe the approach to the critical stage, i. e. the state of Π seis (φ) = Π SES (φ) = Π theoretical (φ), where the subscripts indicate seismicity, SES-activity and the theoretical, it was found necessary to sample seismic data before the main shock for an appropriate window length l which includes certain number of events (cf. it depends, of course, on the magnitude threshold) e.g. N = 40. After the main shock, there will be aftershocks for some time. If the time window to investigate the next main shock happens to overlap any part of the time windows and the aftershock period of the preceding earthquake, Π seis (φ) for the second earthquake would be seriously contaminated by un-related events.
2) The long period properties of the seismicity in wider regions also exhibit some characteristics associated with eqs.
3) and 8) for seismic areas in Greece and in San Andreas fault system and the results are summarized below.
Varotsos et al. 8) analyzed the seismic data of the whole Greek region, using the catalogue by the National Observatory of Athens, (NOA) for the period 1966-2001. First, calculation of the power spectra Π(φ) was made for an event taking time windows for 6 to 40 consecutive events, i. e., taking N = 6, 7, … , 39, 40. And second, this process was performed for all the events by scanning the whole catalogue. The lengths of the windows were chosen to cover the limits of the number of the earthquakes-after SES-activity-through which the coincidence of Π(φ) of seismicity and SES was reached as mentioned in 1). The choice of the precise value of the upper limit was not found decisive, because practically the same results were obtained 8) even if the number of consecutive events was changed from 6-40 to 6-100. Then, the values of the observed probability P[Π(φ)] versus Π(φ) have been plotted. An inspection of the results showed that the local maxima of the curves, each one of which was drawn for a certain φvalue, correspond to the Π(φ) values that lie very close to those predicted from eq. [5]. Essentially the same was found 3),8) for [1] each of the four smaller regions in Greece ( Fig. 1(b) and employed the following approximate formulae obtained from a fit to  Fig. 1a for their epicenters). Since no SES-activity data are available to us for these earthquakes, trial computation of Π(φ) was made for many time windows before them. Details will be published elsewhere, along with the results for the activity in 2000 in Izu Island region for which SES activity data have been published. 12      This could be understood in the context that the rupture length of the Off Tokachi earthquake (and hence the corresponding preparation area) was considerably larger than that of the Kobe. Long-term regional seismicity. In the following, we use, instead of P[Π(φ)] defined above, the corresponding probability density We find that the maximum value p max of p[Π(φ)] in each plot lies at a value of Π(φ)-hereafter called Π max (φ)-very close to that predicted from eq. [5]; as shown by the dotted vertical lines drawn in each plot. This proximity may be understood, as already mentioned, in the following context: when sliding, the time window through the whole catalogue, the most probable value of Π(φ) is the one predicted from the theory of critical phenomena, i.e., eq. [5]. It has been confirmed that the main feature of Fig.  2 (i.e., the proximity of Π max (φ)-values with those predicted from eq. [5]) remains practically invariant upon changing either the seismic region or the time-period (i.e., spatially and time invariant) as well as the magnitude threshold. This can be seen in Figs. 3, 4 and 5, respectively for the Japanese areas. More precisely, in Fig. 3, we show an example of p[Π(φ)] for φ = 0.1 for the three different regions indicated in Fig. 1a, i.e., Regions A (triangles), B (squares), and C (circles), along with the results for the whole area of Japan. In Fig. 4, we show, as an example, that the value of Π max (φ) for φ = 0.1 for all Japan is not practically affected if we consider -instead of the 37 year period-a shorter time period of 5 years. Note that the observed Π max (φ)-value again lies very close to that (vertical dotted line) predicted from eq. [5].
Finally, Fig. 5 shows that the value of Π max (φ) for all Japan is not practically affected if we change the magnitude threshold.
The values obtained for Π max (φ) are plotted in Fig. 6 versus φ. Actually, the results for all the areas studied, i.e., Japan, Greece and San Andreas, the points are on the theoretical line drawn in the figure according to the theory of critical phenomena, i.e., eq. [5]. It is striking that the independent analysis of SES-activities in the nat-ural time-domain led to the same theoretical universal curve. 3

),4)
The four cases depicted in Fig. 2 correspond to the p[Π(φ)] versus φ relation for different φ-values. These can be reduced to almost the same curve, for each area separately, if we plot in Fig. 7 p/p max versus p max [Π(φ) -Π max (φ)]. In Fig. 8, we plot, for all the seismic areas, the reduced probability density p/p max versus p max [Π(φ) -Π max, theoretical (φ)], where the subscript "max, theoretical" denotes the value calculated from eq. [5]. All the data seem to fall on the same curve (universal curve). Both panels of Fig. 8 refer to the same quantities: in the upper panel, a linear-linear plot is given, while in the lower we turn it to a log-linear plot. For the sake of brevity, we comment on the latter plot only. It clearly consists of two  segments: The upper right segment has an almost constant p/p max , while the segment to the left of the vertical line shows a decrease of p/p max by ~five orders of magnitude. The feature of the plot in Fig. 8 (lower) is strikingly reminiscent of the one obtained by Bak et al. 13) on different grounds within the frame of the "unified scaling law for earthquakes", using earthquakes in California only (see their relevant Fig. 4). The origin of this similarity is currently investigated in detail. Conclusion. By analysing the time-series of earthquakes in the natural time-domain, we find that the power spectrum of regional and long term seismicity in Japan, exhibits features of critical phenomena similar to those earlier found in Greece and the San Andreas faults system. All the power spectra in these three areas fall on a universal curve if an appropriate reduction has been made. Furthermore, a preliminary study in the natural time-domain of the seismicities that preceded the Kobe earthquake and Off Tokachi earthquake indicates that a critical state is approached just a few days before the occurrence of the corresponding mainshocks.