found anomalous behavior of geomagnetic diurnal variations mainly in the vertical component at the Esashi station (ESA) located at about 135 km from the M 9 Tōhoku EQ focal zone for about 10 d, i.e., 4–14 January 2011. They analyzed geomagnetic data of a 3-year period, i.e., from 1 January 2010 to 31 December 2012, by computing ratios of diurnal variation range between the target station ESA and the remote reference station Kakioka about 300 km from the EQ epicenter. To validate this finding, further investigations were reported by after analyzing geomagnetic data of 16-year long-term observations in Japanese stations. They again found that the mean values of the ratios of the diurnal variations in the vertical component showed a clear anomaly exceeding the statistical threshold during the aforementioned period, i.e., 4–14 January 2011, and in addition they emphasized that this anomaly is unique in over 16 years. This has been further validated by the most recent study of , who analyzed geomagnetic data of long-term observations at 17 stations in Japan. They found that the above unique anomaly in the vertical component has also been observed at a second station at Mizusawa (MIZ) in the Tōhoku region, which is about 20 km southwest of the ESA station. This fact that both ESA and MIZ show clear anomalies at the same time suggests that the anomaly cannot be the result of observation system error or artificial noises . Furthermore, this anomaly cannot be attributed to magnetic storms since it has been observed during a period in which no moderate–strong magnetic storms were recorded.

Daily resolution data retrieved from the 1243 GPS stations in Japan were utilized by to expose surface displacements before the M 9 Tōhoku EQ. They applied the method proposed by on filtering long-term plate movements, short-term noise, and frequency-dependent (i.e., semiannual and annual) variations from the three-component GPS data for all stations. The N–S and E–W components were utilized to compute the orientations of the horizontal azimuths, termed GPS azimuths.

In general, the residual surface displacements are random . However, as depicted by black text in Fig. 3, southward movements became evident on 5 January 2011, i.e., 65 d before the Tōhoku EQ. Other changes before and after 5 January 2011 have also been observed as follows: while during the period 12–22 December 2010 random orientations of GPS azimuths prevailed, a gradual alignment toward the southern direction started on 22 December 2010 and continued until around 5 January 2011, accompanied with a gradual uplift of the crust. The most intense crust uplift was observed approximately on 5 January 2011 together with the full alignment of GPS azimuths southwards (more details are given in Sect. 4).

reported that anomalous groundwater changes started 3 months before the Tōhoku EQ. In particular, groundwater level and temperature decreased almost simultaneously in a 2000 m well at a spa, Goyo-onsen, in Iwate Prefecture, 155 km northwest of the epicenter. This simultaneous decrease occurred only once in the 3.5-year records when considering that the recordings started in this source since October 2007. emphasized that Tōhoku EQ is the only EQ that was preceded by anomalous changes in both water level and temperature. The exact date of the initiation of this phenomenon is not mentioned explicitly by , because the measurements were not made continuously but were taken intermittently and irregularly (the average interval between them being 8 d). They plotted, however, the consecutive measurements versus conventional time in their Fig. 1 (period 2007–2012) in which one can read that the phenomenon initiated around 20 December 2010, which agrees with what they state, i.e., around 3 months before Tōhoku EQ.

In addition, reported that, according to radon concentration measurements in the groundwater in the Izu Peninsula (at a distance about 500 km from the epicenter), an increase started almost 3 months before Tōhoku EQ. This increase occurred only this time during a 35-year observation.

In a time series comprising N EQs, the natural time for the occurrence of the kth EQ of energy Qk is defined as χk=k/N. In natural time analysis, we study the evolution of the pair (χk,pk), where 1pk=Qk/∑n=1NQn denotes the normalized energy released during the kth EQ. Qk and hence pk for earthquakes are estimated through the relation 2Qk∝101.5Mk.

It is widely accepted that the observed earthquake scaling laws indicate the existence of phenomena closely associated with the proximity of the system to a critical point (the mainshock is the new phase). In particular, it has been indicated by that it seems possible that systems that operate persistently near a threshold of instability are in some way like thermodynamic systems near critical points (EQ can be regarded as a stick-slip frictional instability of a preexisting fault). The order parameter of seismicity is the quantity by which one can identify the approach of the dynamical system to a critical point (the mainshock is the new phase). It was argued by that the variance 3κ1=〈χ2〉-〈χ〉2 of natural time χ weighted for pk, given by 4κ1=∑k=1Npk(χk)2-∑k=1Npkχk2, may serve as an order parameter of seismicity.

The entropy S in natural time is defined by 5S≡〈χln⁡χ〉-〈χ〉ln⁡〈χ〉, where the brackets 〈…〉≡∑(…)pk denote averages with respect to the distribution pk, i.e., 〈f(χ)〉≡∑f(χk)pk. Notably, the functional given by Eq. () has been shown to exhibit positivity, concavity, and experimental stability, which are the three requirements in order to be characterized as entropic functional. Furthermore, note that the entropy S is a dynamic entropy depending on the sequential order of the events and not simply a statistical entropy (e.g., Shannon entropy) see. Upon considering time reversal T^, i.e., T^pk=pN-k+1, the value S changes to a value S-: S-=∑k=1NpN-k+1kNln⁡kN6-∑k=1NkNpN-k+1ln⁡∑l=1NlNpN-l+1. The physical meaning of the entropy change ΔS≡S-S- in natural time under time reversal is discussed in and .

Using a moving window of length i (number of events) sliding through the time series of L consecutive events, the entropy in natural time is determined for each position j=1,2,…,L-i of the sliding window. Thus, a time series of Si is obtained. By considering the standard deviation σ(ΔSi) of the time series of ΔSi≡Si-(S-)i, we define the complexity measure Λi: 7Λi=σ(ΔSi)σ(ΔS100), where the denominator has been selected to correspond to the standard deviation σ(ΔS100) of the time series of ΔSi of i=100 events e.g.,.

ΔS constitutes a key measure that may identify when the system approaches the critical point (dynamic phase transition). For example, ΔS has been applied for the identification of the time of an impending sudden cardiac death risk . Furthermore, it has been used for the study of the predictability of the Olami–Feder–Christensen (OFC) model for earthquakes , which is probably the most studied non-conservative self-organized criticality (SOC) model. The OFC model originated by a simplification of the Burridge and Knopoff spring–block model by mapping it into a non-conservative cellular automaton simulating the earthquake's behavior and introducing dissipation in the family of SOC systems. In particular, it was found that ΔS exhibits a clear minimum (or maximum if we define ΔS≡S--S instead of ΔS≡S-S-; e.g., see ) before a large avalanche in the OFC model, which corresponds to a large earthquake. For example, by analyzing the seismicity during the period 2012–2017 in natural time in the Chiapas region of Mexico where the M 8.2 earthquake occurred on 7 September 2017, we observed that the entropy change ΔS of seismicity under time reversal was minimized almost 3 months before and in particular on 14 June 2017.

Interesting results have been recently obtained upon analyzing the Japan seismic catalog in natural time and computing the fluctuations of κ1. To compute the κ1 fluctuations, we use a sliding natural time window comprising the number i of EQs that would occur on average in a few months or so . We then calculate the average value μ(κ1) and the standard deviation σ(κ1) of the ensemble of κ1 obtained. The quantity 8βi≡σ(κ1)/μ(κ1) is defined as the variability of κ1 . The time evolution of the β value can then be pursued by sliding the excerpt i through the EQ catalog, and the corresponding minimum value is labeled βmin. The following key results have been obtained.

found that the fluctuations β of κ1 of seismicity exhibited a clearly detectable minimum approximately at the time of the initiation of a pronounced SES activity recorded by around 2 months before the volcanic–seismic swarm activity in 2000 in the Izu Islands region, Japan.

analyzed the Japan seismic catalog in natural time from 1 January 1984 to 11 March 2011, and their results showed that the fluctuations β of κ1 of seismicity exhibited distinct minima βmin a few months before all the shallow earthquakes of magnitude 7.6 or larger that occurred during this 27-year period in the Japanese area N2546E125148. Among these minima, the minimum before the M 9 Tōhoku EQ observed at around 5 January 2011 was the deepest. Subsequently, found that the spatiotemporal variations of βmin enable the estimation of the epicentral area of the impending mainshock for all these EQs of magnitude 7.6 or larger.

In addition, focused on the minima βmin preceding all magnitude 8 (and 9) class EQs in the Japanese area from 1 January 1984 to 11 March 2011 and applied detrended fluctuation analysis (DFA) to the earthquake magnitude time series. DFA has been established as a standard method to investigate long-range correlations in non-stationary time series in diverse fields e.g., see including the study of geomagnetic data associated with the M 9.0 Tōhoku EQ . The results of DFA are described in terms of the so-called DFA exponent , hereafter labeled α (α=0.5 means random, α greater than 0.5 means long-range correlations, and α less than 0.5 means anti-correlations). The following three main features have been identified : the minima βmin are observed during periods when long-range temporal correlations between EQ magnitudes have been developed since the corresponding DFA exponent is α>0.5. Before (bef) the minima βmin there exists a stage in which an evident anti-correlated behavior appears showing a minimum αmin,bef in the DFA exponent markedly smaller than 0.5. Finally, after (aft) the minima βmin, the long-range correlations break down to an almost random behavior (α=0.5), possibly turning to anti-correlation exhibiting a minimum αmin,aft with αmin,aft<0.5. These three main features of temporal correlations between EQ magnitudes can be visualized in Fig. 3 for the case of the M 9 Tōhoku EQ. The deepest minimum βmin was observed around 5 January 2011 during a period in which long-range correlations between EQ magnitudes prevail with a DFA exponent α>0.5. Before this minimum an anti-correlated behavior was identified on 22 December 2010 with αmin,bef=0.35. After βmin the long-range correlations break down on 13 January 2011 to an almost random behavior with α=0.5, and subsequently the behavior turned to anti-correlation on 23 January 2011 with αmin,aft=0.42. As explained in the next section, the main features of the temporal correlations between EQ magnitudes obtained by DFA appeared simultaneously with distinct phases of crustal deformation identified by GPS measurements described in the previous section.

Recently we have shown that almost 3 months before the M 9 Tōhoku earthquake, i.e., on 22 December 2010, the following additional facts have been observed: first, the complexity measure Λi associated with the fluctuations of the entropy change of seismicity under time reversal exhibited an abrupt increase which conforms to the seminal work by Lifshitz and Slyozov and independently by for phase transitions showing that the characteristic size of the minority phase droplets exhibits a scaling behavior in which time growth has the form A(t-t0)1/3 . It was also found that the increase ΔΛi of Λi follows the latter form and that the prefactors A are proportional to the scale i, while the exponent (1/3) is independent of i . Second, the Tsallis entropic index q shows a simultaneous increase which interestingly exhibits the same exponent (1/3) . Third, a minimum ΔSmin of the change ΔS of the entropy of seismicity in the entire Japanese region under time reversal was found by , who also demonstrated that the probability of obtaining such a minimum by chance is approximately 3 %, thus showing that it is statistically significant. In addition, the robustness of the appearance of this minimum on 22 December 2010 upon changing the EQ depth, the EQ magnitude threshold, and the size of the area investigated has been documented . Such a minimum is of precursory nature, signaling that a large EQ is impending according to the natural time analysis of the OFC model as mentioned in Sect. 3.1. Fourth, studying the fluctuations β of κ1 of seismicity in the entire Japanese region N2546E125148 versus the conventional time from 1 January 1984 until the Tōhoku EQ occurrence on 11 March 2011, we find a large fluctuation of β upon the occurrence of the M 7.8 earthquake near Chichijima on 22 December 2010. This finding has also been checked for several lengths from i=150 to 500 events, which also revealed the following : upon increasing i it is observed see Figs. 2b and 4e of that the increase Δβi of the βi fluctuation on 22 December 2010 becomes distinctly larger – obeying the interrelation Δβi=0.5ln⁡(i/114.3) – which does not happen see Fig. 4a–d of for the increases in the β fluctuations upon the occurrences of all other shallow EQs in Japan of magnitude 7.6 or larger during the period from 1 January 1984 to the time of the M 9 Tōhoku EQ. Hence, the β fluctuation on 22 December 2010 accompanying the minimum ΔSmin is unique. Its presence is of paramount importance for the validity of the physical model that will be discussed in the next section.