A. Lyubushin / Natural Science 5 (2013) 1-7

6

2000 2004 2008 2012

0.0

0.4

0.8

1.2

1.6

2.0

2000 2004 2008 2012

0.0

0.2

0.4

0.6

0.8

2000200420082012

-0.4

-0.2

0.0

2000 2004 2008 2012

0.0

0.2

0.4

0.6

0.8

1.0

min

En

1998 2000 2002 2004 2006 2008 2010 2012 2014

8

16

24

32

40 2011.03.11, M=9.0

Right-hand end of 365 days moving time window with mutual shift 3 days

Number of clusters providing maximum

of pseudo-F-statistics.

(a) (b)

(c) (d)

(e)

Figure 5. (a)-(d)—median values of 4 daily parameters of

seismic noise

,

*,

min—multi-fractal singularity spectra

parameters; En—minimum normalized entropy of squared or-

thogonal wavelet coefficients, green lines—running average

within 57 days moving time window. (e)—result of clustering

of 3 first principal components of medians of 4 daily seismic

noise parameters (a)-(d) within moving windows of the length

365 days with mutual shift 3 days.

0

1t

—mean vector of the whole cloud of

principal components

Lt

zL

t

.

The rule is not working if we try to dis-

tinguish cases q* = 1 and q* = 1 because the value

is not defined for q = 1. These cases could be

distinguished by existing of break point of the monoto-

nous function J(q) at the argument q = 2 [11]. Let

maxPFS

2

1q

q

be the deflection of the dependence of

q

n

2

0

ln

(a,b)

miq

on

from linear approximation:

, where coefficients (a,

b) are found by least squares: 1q. The

final rule for selecting q* is the following.

ln q

2

0

ln

lnqa

qb q

S. If

40 2

Let then q* = q0. Else

0240

arg max

q

qPF

q

02q

if

240

1max 1

qq

then q* = 1 else q* = 2.

Graphic at Figure 5(e) presents evolution of the esti-

mates of the best number of clusters q* in dependence on

the right-hand end of moving time window of the length

1 year. This plot contains the most intrigue characteris-

tics of the data: now we observe the same unstable be-

havior of q* which was observed before 2011.03.11 and

during some time immediately after Tohoku mega-ear-

thquake.

A question arises: does the Figure 5(e) mean that the

next mega-earthquake is already prepared and waits for

its trigger?

4. CONCLUSION

The averaged maps of singularity spectra support

width and minimum normalized entropy of squared or-

thogonal wavelet coefficients of low-frequency seismic

noise could be regarded as a new tool of dynamic esti-

mate of seismic danger. These maps give a possibility to

inspect the origin and evolution of the SSD—“spots of

seismic danger”. Analysis of seismic noise at Japan is-

lands from broad-band seismic network F-net gave a

possibility for prediction of Tohoku mega-earthquake

2011.03.11, which was published in advance of the event.

According to the analysis of seismic noise (correlation

2

between 2 multi-fractal parameters and the “best”

number q* of clusters for annual clouds of properties of

median values of 4 daily statistics) after 2011.03.11 the

next mega-earthquake with magnitude near 9 could occur

at the region of Nankai Trough during peri od 20 1 3-2 01 4.

5. ACKNOWLEDGEMENTS

This work was supported by the Russian Foundation for Basic Re-

search (project No. 12-05-00146) and Russian Ministry of Science

(project No. 11.519.11.5024). The author is grateful to National Insti-

tute for Earth Science and Disaster Prevention (NIED), Japan, for pro-

viding free access to the source of broadband seismic noise waveforms

registered at the F-net stations.

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