Abstract
It has been shown that some dynamic features hidden in the time series of complex systems can be uncovered if we analyze them in a time domain called natural time χ. The order parameter of seismicity introduced in this time domain is the variance of χ weighted for normalized energy of each earthquake. Here, we analyze the Japan seismic catalog in natural time from January 1, 1984 to March 11, 2011, the day of the M9 Tohoku earthquake, by considering a sliding natural time window of fixed length comprised of the number of events that would occur in a few months. We find that the fluctuations of the order parameter of seismicity exhibit distinct minima a few months before all of the shallow earthquakes of magnitude 7.6 or larger that occurred during this 27-y period in the Japanese area. Among the minima, the minimum before the M9 Tohoku earthquake was the deepest. It appears that there are two kinds of minima, namely precursory and nonprecursory, to large earthquakes.
Keywords: criticality, seismic electric signals
For a time series comprised of N events, we define the natural time for the occurrence of the kth event by 
(1), which means that we ignore the time intervals between consecutive events, but preserve their order. We also preserve their energy 
. We then study the evolution of the pair
, where 
is the normalized energy. We postulated that the approach of a dynamical system to criticality can be identified by the variance 
of natural time χ weighted for
, namely,
 |
Earthquakes (EQs hereafter) exhibit complex correlations in time, space, and magnitude, and the opinion prevails (e.g., ref. 2 and references therein) that the EQs are critical phenomena. In natural time analysis of seismicity, the quantity 
calculated from seismic catalogs serves as an order parameter (3, 4). Experiences have shown that the mainshock occurs in a few days to 1 wk after the 
value in the candidate epicentral area approaches 0.070 (5). This was found useful in narrowing the lead time of EQ prediction. However, to trace the time evolution of 
value, one needs to start the analysis of the seismic catalog at some time before the yet-to-occur mainshock. We chose, for the starting time for analysis, the initiation time of seismic electric signal (SES) activity. SESs are low-frequency (
1 Hz) electric signals that precede EQs (6). The reason for this choice was based on the consideration that SESs are emitted when the focal zone enters the critical stage (7). In the case of the lack of SES data, as in the Tohoku EQ, we cannot adopt this approach. In this study, therefore, we instead examine the fluctuations of 
near criticality, i.e., near the EQ occurrence. To compute the fluctuations, we apply the following procedure.
First, take an excerpt comprised of W (≥100) successive EQs from the seismic catalog. We then form its subexcerpts consisting of the nth to (n + 5)th EQs, (n = 1, 2,…,W-5) and compute 
for each of them. In so doing, we assign 
and the normalized energy 
, k = 1, 2,…,6 to the kth member of the subexcerpt. Note that at least 6 EQs are needed for obtaining reliable 
(3). We iterate the same process for new subexcerpts consisting of 7 members, 8 members, …, and finally W members. Then, we compute the average 
and the SD 
of the thus-obtained ensemble of (W − 4)(W − 5)/2 
values. The variability (4, 8) of 
for this excerpt W (≥100) is defined to be 
and is assigned to the (W + 1)th EQ, the target EQ.
The time evolution of the 
-value can be pursued by sliding the excerpt through the EQ catalog. Through the same process as explained above, the 
-values to be assigned to the (W + 2)th, (W + 3)th, EQs in the catalog will be obtained.
Data Analyzed
For our analysis, we used the Japan Meteorological Agency (JMA) seismic catalog and considered all of the EQs in the period from 1984 to the time of the M9 Tohoku EQ, within the area 25°–46°N, 125°–148°E, which covers the whole Japanese region (Fig. 1). The energy of EQs was obtained from MJMA after converting (9) to the moment magnitude Mw defined by Kanamori (10). Setting a threshold MJMA = 3.5 to assure the data completeness, we are left with 47,204 EQs in the concerned period of about 326 mo. Thus, we have on the average ∼102 EQs per month. We chose the values W = 200, 300, and 400, which would cover a period of a few months before each target EQ. This choice of a few months is based on the experience that the lead time of SES activities is of this order both in Japan (11) and Greece (5, 7, 12).
Fig. 1.
Epicenters (stars) of all major EQs with magnitude 7.6 or larger within the area 
since January 1, 1984 until the M9 Tohoku EQ (Table 1). The deep EQ on March 6, 1984 is depicted by a triangle.
Minimum of the Variability 
Before the M9 Tohoku EQ
Fig. 2A depicts about 47,200 
-values calculated for W = 300 versus the target EQ number from 1984 to the day of the Tohoku EQ, March 11, 2011. EQs with MJMA ≥ 6.9 (MJMA in the right scale) are shown by blue asterisks. One can see that 
-values fluctuate up and down so violently that it is hard to identify their correlations with EQs. However, one can notice that 
shows a deep minimum value just before the Tohoku EQ (rightmost side of Fig. 2A). This observation prompted us to investigate more about this 
minimum. Fig. 2B is an expanded version, in the conventional time, of the concerned part of Fig. 2A (the last 10-mo period shown in yellow). The red, blue, and green curves show what happened to 
for W = 200, 300, and 400. For brevity, we use hereafter the symbols 
and 
as needed. Putting the details aside, we observe that after around September 1, 2010 a decrease of 
became evident and 
went down to a minimum (
200,min ∼0.157, 
300,min ∼0.160, and 
400,min ∼0.150) in early January 2011, about 2 mo before the mainshock. (The abrupt increase of 
around December 22, 2010 was due to the M7.8 EQ on this date, e.g., ref. 3.)
Fig. 2.
Variability 
of 
(left scale) along with all MJMA ≥ 6.9 EQs (in blue, MJMA in the right scale). (A) Versus EQ number when a natural time window of length W = 300 events is sliding through the JMA catalog since 1984 until just before the M9 Tohoku EQ. (B) Versus the conventional time during the last 10-mo period (shown by yellow in A). Red for W = 200, blue for W = 300, and green for W = 400. Every tick is 10 d in the horizontal scale. (C) Variability 
for W = 2,000 (green) and W = 3,000 (blue).
Results of the computation on this minimum of 
are summarized as follows (Fig. 2 A and B, and Table 1):
i) Minimum of
with this depth was not observed at any other time during the whole period.ii)
300,min/
200,min = 0.160/0.157 = 1.02, i.e., nearly unity.iii) The dates of
for W = 200, 300, and 400 are January 5, January 5, and January 10, 2011, respectively, i.e., the dates of 
were almost the same.iv) The appearance of this minimum is less clear for greater W that would correspond to time intervals longer than a few months. It is almost invisible for W = 2,000 and 3,000. (Fig. 2C). The same applies to all other
as seen in Fig. 2 A and C. In what follows, for the sake of brevity we shall restrict ourselves to the cases of W = 200 and W = 300.
Table 1.
All shallow EQs with magnitude 7.6 or larger since January 1, 1984 until M9 Tohoku EQ within the area 
| Label | EQ date | EQ name | Lat.,°N | Long., °E | M |  |
 |
 |
Δt200 |
| a | 1993-07-12 | Southwest-Off Hokkaido EQ | 42.78 | 139.18 | 7.8 | 0.293 (1993-05-23) | 0.278 (1993-06-07) | 0.95 | 2 |
| b | 1994-10-04 | East-Off Hokkaido EQ | 43.38 | 147.67 | 8.2 | 0.295 (1994-06-30) | 0.319 (1994-07-22) | 1.08 | 3 |
| c | 1994-12-28 | Far-Off Sanriku EQ | 40.43 | 143.75 | 7.6 | 0.196 (1994-10-15) | 0.197 (1994-10-19) | 1.01 | 2–3 |
| d | 2003-09-26 | Off Tokachi EQ | 41.78 | 144.08 | 8.0 | 0.289 (2003-07-03) | 0.306 (2003-07-14) | 1.06 | 3 |
| e | 2010-12-22 | Near Chichi-jima EQ | 27.05 | 143.94 | 7.8 | 0.232 (2010-11-30) | 0.248 (2010-11-30) | 1.07 | 1 |
| f | 2011-03-11 | Tohoku EQ | 38.10 | 142.86 | 9.0 | 0.157 (2011-01-05) | 0.160 (2011-01-05) | 1.02 | 2 |
The symbols 
are the minima of the 
variability that preceded these EQs along with their dates. Δt200 is the difference in months between the dates of 
and EQ. Lat., latitude; Long., longitude.
Minima of the Variability 
Before Other Major EQs in Japan
During the 27-y study period, we had six shallow EQs with MJMA 7.6 or larger (Fig. 1 and Table 1). They are
EQa 1993-07-12: 1993 Southwest-Off Hokkaido EQ (MJMA = 7.8)
EQb 1994-10-04: 1994 East-Off Hokkaido EQ (MJMA = 8.2)
EQc 1994-12-28: 1994 Far-Off Sanriku EQ (MJMA = 7.6)
EQd 2003-09-26: 2003 Off Tokachi EQ (MJMA = 8.0)
EQe 2010-12-22: 2010 Near Chichi-jima EQ (MJMA = 7.8)
EQf 2011-03-11: 2011 Tohoku EQ (Mw = 9.0)
In the following, we examine if minimum of 
exists before these EQs also. Fig. 3 A–C are the expanded versions of Fig. 2A in the conventional time in three 10-y periods. EQs are marked by a–f. Because these figures are still too small, we expanded the time axis for each EQ as shown in Fig. 4 A–E, just as we did for in Fig. 2B for the Tohoku EQ. We can see minima of 
within 1–3 mo before all of the six mainshocks. In Table 1, these minima are listed along with the time-correlated EQs. As seen in this table, the values of the 
ratio and Δt200 of minima of 
before all of these EQs were very similar to those observed before the Tohoku EQ, i.e., they were in the narrow range 0.95–1.08. From these, it was considered that these minima may be precursory to the time-correlated EQs.
Fig. 3.
Variability 
versus the conventional time given in three consecutive 10-y periods in A, B, and C, respectively. No data are plotted in C after M9 Tohoku EQ. Every tick is 3 mo in the horizontal scale.
Fig. 4.
Excerpts of Fig. 3 plotted in expanded time scale. Every tick is 10 d in the horizontal scale. Red arrows show 
along with EQs (in black a–f, MJMA in the right scale). The figures show what happened for the variability 
before the occurrence of (A) EQa, (B) EQb, (C) EQb and EQc, (D) EQd, and (E) EQe and EQf (the M9 Tohoku EQ).
During the 27-y period, beyond the above-mentioned minima of 
before the six MJMA ≥7.6 EQs, there were many more minima as seen in Fig. 3. We therefore examined if they were also followed by EQs. For this purpose, we chose minima deeper than the shallowest one of the six 
in Table 1, which happened before EQb, the 1994 East-Off Hokkaido EQ (MJMA = 8.2), giving 0.295 for 
. It was found that, out of thus chosen 31 minima, nine (numbered 1–9 in Fig. 3) also exhibited 
ratio (Fig. 5, Table 2) similar to those in Table 1, i.e., in the range 0.95–1.08. These nine minima were followed by MJMA ≥6.4 EQs within 3 mo (Fig. 5 and Δt200 in Table 2). Naturally, such correspondences are less certain because of greater number of EQs. In fact, during the 27-y period, there were 139 MJMA ≥6.4 EQs. The cases mentioned here, however, may have some reason to be reliable as explained below (Table 2). For example, the EQ that apparently followed 
No. 6 in April 2000 was the largest EQ of the volcano-seismic activity in the Izu Island area (11), the largest EQ swarm ever recorded in Japan. For a second example, 
No. 7 in July 2000 was followed by MJMA 7.3 Western Tottori EQ. The period after this EQ in the year 2000 was completely free from shallow EQ greater than MJMA 6. Other examples are the minima Nos. 9, 2, 8, 3, and 4 which were followed by EQs of MJMA 7.2, 7.1, 7.0, 6.9, and 6.9. For brevity, each case is not described here, but it was inferred that these 
might have also been precursory to sizable EQs. In fact, there were only 43 MJMA ≥6.9 EQs during the 27-y period. Likewise, the 
Nos. 1 and 5 seemed followed by EQs of MJMA 6.6 and MJMA 6.4, respectively, although their correlations are even less certain. After handling these, we are still left with 22 minima unnumbered or unmarked in Fig. 3. We have checked the 
ratio of each of them. For example, 
(
0.213 and 
0.259) observed on December 4, 2008 (Fig. 3C) exhibited a ratio 
(=1.22), which lies outside the range 0.95–1.08. Fig. 6 is the histogram of the 
ratio for all of the 37 minima examined so far, consisting of the 6 in Table 1 marked a–f, 9 in Table 2 marked 1–9, and the 22 additionally chosen minima. From this figure, interestingly, none of the additional 22 minima exhibits the ratio within the range 0.95–1.08.
Fig. 5.
Excerpts of Fig. 3 but corresponding to each of the nine cases of 
given in Table 2: (A) 1, (B) 2, (C) 3 and 4, (D) 5, (E) 6 and 7, (F) 8, and (G) 9. Every tick is 10 d in the horizontal scale. Numbers 1–9 correspond to 
in Fig. 3 and Table 2. EQs time-correlated to minima 1, 5, and 6 are shown with the vertical black arrows.
Table 2.
Nine 
of the 
variability not included in Table 1 but chosen by the procedure described in the text
| No. |  |
 |
 |
EQ date | Lat., °N | Long., °E | M | Δt200, mo |
| 1 | 0.254 (1986-10-13) | 0.257 (1986-11-15) | 1.01 | 1987-01-14 | 42.45 | 142.93 | 6.6 | 3 |
| 2 | 0.278 (1989-08-08) | 0.292 (1989-09-15) | 1.05 | 1989-11-02 | 39.86 | 143.05 | 7.1 | 3 |
| 3 | 0.250 (1992-04-05) | 0.253 (1992-05-10) | 1.01 | 1992-07-18 | 39.37 | 143.67 | 6.9 | 3 |
| 4 | 0.188 (1993-07-13) | 0.182 (1993-07-15) | 0.97 | 1993-10-12 | 32.03 | 138.24 | 6.9 | 3 |
| 5 | 0.237 (1998-02-17) | 0.233 (1998-03-12) | 0.98 | 1998-05-31 | 39.03 | 143.85 | 6.4 | 3 |
| 6 | 0.229 (2000-04-12) | 0.219 (2000-05-06) | 0.96 | 2000-07-01 | 34.19 | 139.19 | 6.5 | 3 |
| 7 | 0.243 (2000-07-09) | 0.258 (2000-07-09) | 1.06 | 2000-10-06 | 35.27 | 133.35 | 7.3 | 3 |
| 8 | 0.244 (2002-05-12) | 0.252 (2002-06-03) | 1.03 | 2002-06-29 | 43.50 | 131.39 | 7.0 | 2 |
| 9 | 0.286 (2005-06-11) | 0.309 (2005-07-01) | 1.08 | 2005-08-16 | 38.15 | 142.28 | 7.2 | 2 |
The 
exhibit 
deeper than 0.295, which corresponds to the shallowest 
in Table 1. EQs are time-correlated to the 
.
Fig. 6.
Histogram of the 
ratio for the 37 minima in Fig. 3 which are deeper than the shallowest 
of Table 1. The minima marked a–f or numbered 1–9 in Fig. 3 are placed vertically in the corresponding column according to their 
values.
Summary and Conclusions
Analyzing in natural time the seismicity of Japan from January 1, 1984 to March 11 2011 (the time of M9 Tohoku EQ occurrence), using sliding natural time window of lengths W consisting of the number of events that would occur in a few months, the following results were obtained:
Almost 2 mo before the M9 Tohoku EQ, a minimum in the variability 
of the order parameter of seismicity 
is observed which is the deepest in the whole study period. Distinct minima of 
, but of shallower depth, were found also one month to a few months before the occurrence of all other Japanese major EQs (MJMA ≥ 7.6, depth <400 km) during 1984–2011. With less certitude, nine other minima of 
may have also been precursory to large EQs. The minima of 
which seem to be precursory to sizable EQ commonly show the 
ratio close to unity in the range of 0.95–1.08, whereas the other minima show the ratio outside this range. Thus, the phenomenon of minimum in the 
-value may play some role as a precursor in the EQ prediction in the future.
The approximate coincidence of the lead time of minima of 
with that of the SES activities may help in understanding the physics of both phenomena.
Footnotes
The authors declare no conflict of interest.
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