geosystemics and entropy of earthquakes - new

GEOSYSTEMICS and

ENTROPY of EARTHQUAKES

Angelo De Santis 1,2

Geosystemics

De Santis, Geosystemics & Entropy, Samarkand (Uzbekistan), 21 May, 2013

1 Istituto Nazionale di Geofisica e Vulcanologia (INGV), Rome, Italy 2University G. D’Annunzio, Chieti, Italy

[email protected]

A vision of our planet and a key of access

mailto:[email protected]

Acknowledgements

Thanks to some friends & colleagues:

Gianfranco Cianchini & Leda Qamili by INGV for their

discussions and enthusiasm between a coffee and another.

Paolo Favali & Laura Beranzoli by INGV for their support

between a meeting and another.

Patrizio Signanini & Bruno Di Sabatino by Chieti University

for exchanging ruminations between a lecture and another.

Giovanni Gregori by CNR for his incredible knowledge about

everything as wide as his modesty.

David Barraclough by BGS for his endless capability to

encode my Italian to a better English.

… to NATO and the Organisers of this Workshop for the

support and the invitation here.

INGV Geosystemics

3

SAGA-4-EPR: Satellite, Seafloor and Ground data

Analyses for (“4”) Earthquake Pattern Recognition

Geosystemics

DS3F: Deep Sea & Subseafloor Frontier

Project of Excellence Italy-China 2010-2012

Funded by: Italian Foreign Office, INGV (Italy), Northeastern

University (China)

European Project 2010- 2012

Funded by the European Commission

Projects and Funding

Institutions Project of Exchange Research Visits Italy-Albania 2012-2014

Funded by: Italian Foreign Office & Ministry of Education

and Science (Albania)

Earth Magnetic Field measurements, Modelling

and Seismicity – EM3S EM3S


4

Outline

1. Introduction

2. Earth as a Complex system

3. Geosystemics

4. Universal Tools

5. Entropy of Earthquakes

6. Focalisation

7. Other applications

8. Conclusions

9. Selected bibliography

INGV Geosystemics


1. INTRODUCTION:

WORLD IS CHANGING

• growing population

• global warming

• pollution

• biodiversity reduction

• geohazards

& greater weakness against disasters

• reduction of resources

Man is both affecting and affected

Geosystemics


EARTH: Interconnected system

of systems


magma chambers

geodetic

spreading

hydrothermalism

1 mm 1 cm 10 cm 1 m 100 m 1 km 10 km 100 km 10 3 km 10 4 km 10 5 km

vertical turbulent

mixing

capillary

waves

molecular

processes

surface gravity

waves

internal waves &

inertial motions

internal tides surface tides

eddies &

fronts

Seasonal

cycles

Rossby

waves

basin

scale

variability

El Ni ño &

NAO

climate

change

= mesoscale

physical and

biological

interactions

mantle

convection

EQ fault

coastal

upwelling

barotropic

variability

10 4 yr

1000 yr

100 yr

10 yr

1 yr

1 mon

1 wk

1 h

1 d

1 min

1 s

0.1 s

Spatial scale

Geohazards

Organisms

Temporal and Spatial Scales

Ruhl et al., 2011

8

2. Earth as a Complex System

1. A complex system contains many constituents interacting nonlinearly,

2. and interdependent;

This is the specific case of our planet, composed by an enormous number of sub-

systems and elements and sub-elements, placed into around 1012 km3 of solid

volume and much more larger volume of its oceans, coversphere, biosphere and

gaseous atmosphere.

Geosystemics

Points in bold from Baranger, 2001

3. A complex system possesses a structure spanning several scales.

Earth phenomena range from atomic scale to thousand - km scale, from almost

instant processes to billion – year timescale;

5. It is characterised by an interplay between chaos and non-chaos,

This point indicates that chaos can emerge sporadically, due to some change of

boundary conditions under which the phenomenon is occurring.

4. and it is capable of emerging behaviour. Concepts of : surprise, capability of

change, self-reproduction (autopoiesis) and self-organization, mixing. For

instance, Wilson cycle and cybertectonics of Earth.

6. and between cooperation and competition. Positive and negative feedbacks

in Earth processes (e.g. see Gaia Hypothesis)


9

The aim of science is not things themselves, but the relations between things; […] outside those

relations there is no reality knowable.

Henri Poincaré in “Science and Hypothesis”, 1905

Geosystemics


10

3. Geosystemics: definition

1 Systemics is the science of complex systems studied from a holistic point of view (in their

wholeness) (e.g. Klir, 1991). 2 Cybernetics is the science that studies phenomena of self-regulations and

communications among natural and artificial systems (Wiener, 1948).

Transfer of knowledge from “classic” disciplines

MATHEMATICS, PHYSICS (GEOPHYSICS), CHEMISTRY (GEO CHEMISTRY),

BIOLOGY, GEOLOGY, INFORMATICS (GEOINFORMATICS)

& more recent disciplines

SYSTEMICS1 and CYBERNETICS2

to Geosystemics

Geosystemics studies Earth system from the holistic point of view,

looking with particular attention at self-regulation phenomena and relations

among the parts composing Earth (De Santis, WSEAS, 2009)

classic

systemics

cybernetics

INGV Geosystemics

(trans-disciplinary approach)

11

Geosystemics: Science /1

INGV

Ask anybody what the physical world is made of, and you are likely to be told “matter and energy”. Yet if we have learned anything from engineering, biology and physics, information is just as crucial an ingredient.

Jacob D. Bekenstein

in Scientific American (2003)

Geosystemics


http://images.google.it/imgres?imgurl=http://www.physics4u.gr/articles/images6/bekenstein.jpg&imgrefurl=http://www.physics4u.gr/articles/2006/mond.html&usg=__Y2SpKfAGSezq881OIi_Ns76hCf8=&h=316&w=241&sz=27&hl=it&start=10&um=1&tbnid=6D8iSkoZU77S0M:&tbnh=117&tbnw=89&prev=/images?q=jacob+Bekenstein&um=1&hl=it&sa=G

12

Geosystemics: Science /2

Geosystemics puts the emphasis on contextuality

and interactions among the elements of Earth

System, on the cause-effect relationships, on

various sub-systems couplings and on both

production and transfer of Information (Shannon,

1948) from a sub-system to another.

INGV Geosystemics

Not only energy and matter are important, but also

(sometime even more) Information, self-regulation,

nonlinear coupling, emergent behaviour,

irreversibility which are decisive ingredients of

Geoscience, and matter of study for Geosystemics.

13

Science /3

Geosystemics overcomes traditional boundaries

between science, mathematics and philosophy,

between harmony and diversity, invariance and

variability, simplicity and complexity, symmetry and

asymmetry, uniformity and diversity, order and

disorder, reversibility and irreversibility, which all

together characterise Earth’s evolution.

Geosystemics

Universal tools (e.g. fractal dimension, phase space,

degrees of freedom, information and entropy) +

Multi-scale/parameter/platform observation help in

this challenge. De Santis, Geosystemics & Entropy, Samarkand (Uzbekistan), 21 May, 2013

14

4. Universal tools

Fractal dimension

In a fractal with N=N() elements with size covering the

whole structure, the fractal dimension D is defined as:

0

log ( )lim

log1

ND

This figure shows an example of fractal

interpretation that has been given for

the core-mantle boundary of Earth,

from the study of the geomagnetic field

over the last 400 years (De Santis &

Barraclough, PEPI, 1997).

Geosystemics


15

Universal tools

Phase Space The phase space of a dynamical system is the ideal space where each

state of the system can be represented by a single point. The minimum

number E of phase space axes, which contain all orbits of the dynamics,

defines the degrees of freedom of the system, i.e. the number of

variables that are required to describe that system. E is also said

embedding dimension.

This figure shows an example of phase

space reconstruction that has been

given for the geomagnetic field in 3

observatories for the last 150 years

(De Santis et al., Fractals, 2002).

INGV Geosystemics

Theorem for Reconstruction of

pseudo space phase (Takens,

1981)

Ha

rtla

nd

Irkutsk

Alibag


16

Shannon Entropy INGV Geosystemics

1

( ) ( ) log ( )N

i i

i

H t p t p t

Shannon Entropy (Shannon, 1948) of a “system” characterized by N independent states and a probability distribution pi(t)

Some Physical Interpretations

1. Measure of disorder 2. Measure of the average information content missing to the

knowledge about the state of the system; 3. Measure of unpredictability of the state of the system among

many alternatives; 4. Measure of the degree of dispersion of an observable among the

system’s parts

if pi(t)=0 then we impose log pi(t)=0

Claude E. Shannon


http://www.acorn.net.au/telecoms/infotheory/shannon.gif

17

Shannon Information and Entropy

over a sphere

If B(t) is a physical quantity defined over a sphere, we can write it as

sum of orthonormal spherical harmonics nm with maximum degree N

(which defines the smallest detail of the representation), i.e.

where pn(t) is the probability to have a certain n-degree spherical

harmonic power contribution instead of another (De Santis et al.

2004):

1

( ) ( ) ln ( )N

n n

n

I t H p t p t

2

2

0

'22

'

' 1 0

( )

( )

nm

n

n mn N n

m

n

n m

cB

pB

c

INGV

1 0

( ) ( )n

N nm m

n

n m

B t c t

then

Geosystemics Other Universal tools


18

> Any attack to the problem of

earthquake is worth doing

> Our approach (geosystemics) is

holistic (not against reductionism but in

parallel/complementary):

- earth as a whole,

- phenomenon in its most important

macroscopic features

--> Here we will see some cases

INGV Geosystemics

5. Entropy of Earthquakes


Some empirical statistical laws

On all earthquakes:

The rate of earthquakes occurrence (number of earthquakes N in a certain time interval) in a given region follows an exponential law of the magnitude M. (small earthquakes are many more than larger ones). log N = a – bM (b 1) Case of M5 in Italy: 1/year

Case ofM6 in Italy : 0.1/year, i.e. 1/10 yrs. M7 0.01/year, 1/100 yrs

1. Gutenberg-Richter Law (1944)


Some empirical statistical laws On the aftershocks: 2. Omori Law (1894; modified by Utsu in 1961): inverse power law of the rate n of aftershosks occurrence n(t) = K/(c+t) p with p 1

3. Båth Law (1965):

DM= Mmain-maxMafter 1.2±0.2

For 2009 M6.2 L’Aquila Eq.: DM = 1.0

For 2012 M5.9 Emilia it does not fit: DM = 0.1 !!

Date (day/month/year)

n(t

) e

art

hquakes/d

ay

L’Aquila earthquakes rate after mainshock


Some empirical statistical laws On the aftershocks:

4. Felzer & Brodsky (2006): inverse power law of the probability P of having an aftershosk at distance r from the mainshock epicenter (at least up to 100 km) P(r) = K/r s with s 1.4-1.8


P(r)

r , distance (km) from epicenter

1

0 0 100 50

22

Entropy of Earthquakes INGV Geosystemics

Gutenberg-Richter Law log ( )n M a bM

1

( ) ( ) log ( )N

i i

i

H t p t p t

Shannon Entropy

max 1.2

10 10H H

bb

b-value and Entropy

(De Santis et al., BSSA, 2011)

( ) log( log ) log ' logH t e e b k b

n(M) EQs magnitude ≥ M

a, b several interpretation

pi probability of the i-th level

of seismicity defined by a

range of magnitudes

0 1 2 3 4 5 6

-0.5

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

4.5-600 -400 -200 0 200 400

1

2

3

4

5

6

b)

b=0.89±0.03lo

g n

(M)

number of earthquakes

cumulative

Magnitude

22 June 2009 (ML=4.6)

6 Apr.2009 (ML=5.9)

Ma

gn

itu

de

Day w.r.t. main shock (6 Apr. 2009)

a)

So, this is a new insight into the b-value meaning!

0 ≤ H ≤1 if we divide by logN


23

Entropy of Earthquakes Two Cases in Central Italy

-600 -400 -200 0 200 400

-0,05

0,00

0,05

0,10

0,15

0,20

0,25

1.5 days after

6 days before

Increasing (cumulative) windowsEn

tro

py, H

(i

ncre

asin

g w

ind

ow

s)

Day w.r.t. main shock (6 Apr. 2009)

Main Shock

-150 -120 -90 -60 -30 0 30 60 90 120 150

0.20

0.25

0.30

0.35

0.5 days before

22 days before

2 days after

En

tro

py, H

Days w.r.t. main Shock (26 Sep. 1997)

Main Shock

Colfiorito (Umbria-Marche, Central Italy) 1997

Entropy - L’Aquila sequence

Central Italy, mainshock M6.2 2009

Entropy - Colfiorito sequence,

Central Italy, mainshock M6.1

1997

(De Santis et al., BSSA, 2011)

In general, some major features can be noticed:

• preparation phase a generalized increase

(months before main-shock)

• concentration phase a sudden jump from days to hours

before main-shock

• diffusive phase a decrease “after” the main-shock

main-shock is not a singularity

it is a part of a population of events

Focalization

24

Magnetic Transfer Function Entropy

(Cianchini et al., NPG, 2012)

Z(w) = A(w) X(w) + B(w) Y(w)

In the frequency domain the time variations of the components X,Y,Z of the geomagnetic field

observed at Earth surface are each other coupled:

A(w) and B(w) are the Magnetic Transfer Functions which are related with the conductivity at a

certain depth inversely proportional to the square root of frequency w.

The (normalised) entropy contribution of the harmonic wi is given by :

( , ) log ( , )( )

log

i ii

p t p tE t

N

w w

2

2

1

( , )( , )

( , )

r ii N

r i

i

K tp t

K t

ww

w

where:

- E

30-4

0s(t

)

Migration of fluids from below to the hypocentral zone

(10km) activated the seismic sequence that culminated

with the M6.3 mainshock of 6 Apr. 2009.

Transfer Function Entropy for periods of 30-40 sec.

(15-20 km depth)- L’Aquila (Central Italy), 2007-2009

Kr=real parts of A

or B


Geosystemics

From Fourier analysis → the spectral entropy (Powell and Percival, 1979)

lni i

i

S P P 2

2

'

'

i

i

i

i

fP

f

where for frequency

if

It measures how energy “concentrates” or “spreads” in frequency.

Wavelet analysis decomposes a signal ( )f t both in time and scale (or frequency)

, 2 ,logW s sS p p

,

( , )

( , )s

s

E sp

E s

( )f t ( , )W s Wavelet Transform

2( , ) ( , )E s W s

Wavelet Entropy

where

Probability

Energy

Champ filtered magnetic field modulus

Wavelet Entropy

Wavelet Spectrum

Example 26th Dec, 2004

(Sumatra EQ, M9)

25 Entropy of Earthquakes: Wavelet Entropy of satellite magnetic signal

The case of magnetic

signal from CHAMP

satellite

(in orbit 2000-2010)

Cianchini et al.,ESA, 2009

6. SPACE-TIME

FOCALIZATION

Geosystemics

THE CASES OF M6.3 L’Aquila (2009) and M5.9 Emilia (2012) EARTHQUAKES

An attempt to solve an important conundrum of seismology:

How does the stress at the tectonic plates transfer to an individual fault?


27

Lessons learnt from L’Aquila /1

INGV Geosystemics

FAULT GAP & SEISMICITY CONCENTRATION (FOCALIZATION)

De Santis et al., BSSA, 2011

M>1.5


28


Geosystemics

FOCALIZATION IN TIME: ACCELERATING STRAIN

De Santis et al., Tectonophysics, 2010

Mpred= 5.3

ML= 5.9

0.5

1.5 4.8

( ) ;

( ) 10

i

i

M

i

s t E

E Joule

s(t

)


29


Geosystemics

CHAOTIC PROCESS WITH TIME SCALE OF 10 DAYS

De Santis et al., Tectonophysics, 2010


30

Focalization of earthquakes:

Emilia 2012

Geosystemics

9.5 10.0 10.5 11.0 11.5 12.0 12.5

43.8

44.0

44.2

44.4

44.6

44.8

45.0

45.2

45.4

45.6

45.8

46.0

15 Jul 2005

M5.8 29 May 2012

Longitude

La

titu

de

Earthquakes R<200km , M4, 2005-2012

M5.9 20 May 2012

A partial Mogi doughnut (Mogi, 1969):

1. Former seismicity at the periphery 2. the present seismicity filled the gap


31

Focalization of earthquakes

in time: ASR

(Accelerated Strain Release)

Geosystemics

( ) ( )m

fs t A B t t

0 500 1000 1500 2000 2500 3000

0.0

5.0x106

1.0x107

1.5x107

2.0x107

2.5x107

R=200km, M>4

Mpred

=6.1+/-0.6

tf(pred.)=day 2577+/-12 days

C-factor=0.6

Cu

mu

lative

Be

nio

ff S

tra

in (

Jo

ule

0.5)

Day from 1 May 2005

*day 2576 (20 May 2012) M5.9

exponent m =0.25

as some theoretical

critical models

0.5

1.5 4.8

( ) ;

( ) 10

i

i

M

i

s t E

E Joule

s(t

)

De Santis et al., Tectonophysics, under revision


32


in time: Chaos

INGV Geosystemics

CHAOTIC PROCESS WITH 10- DAY TIME SCALE?

-300 -250 -200 -150 -100 -50 0

0

200

400

600

800

1000

1200

1400

Data: Data5_D

Model: y = A1*exp[(t-tf)/t1] + y0

Weighting:

y No weighting

Chi^2/DoF = 54172.87493

R^2 = 0.89525415

y0 56 ±120

A1 3.1319E-6 ±0.00005

t1 13 ±11

Ab

so

lute

Err

or

in p

red

ictio

n (

da

ys)

Days to Mainshock (20 May, 2012)

M>=2; R<100 km

De Santis et al., Tectonophysics, under revision


33


Distances from epicenter

INGV Geosystemics

0 500 1000 1500 2000 2500 3000

0

20

40

60

80

100

120

140

160

180

Day from 1 May 2005

Dis

tan

ce

fro

m m

ain

sh

ock e

pic

en

tre

(km

)

M4.1 day 2573.97

19 May 2012 23.13 UTC

( ) ( )m

r fr t D t t R<200 km; M>=4

De Santis et al.,

Tectonophysics, under revision


Negative slope

35


in time: Inter-events

INGV Geosystemics

-500 0 500 1000 1500 2000 2500 3000

0

100

200

300

400

500

M4.1 day 2573.97

19 May 2012 23.13 UTC

Day from 1 May 2005

Inte

r-e

ve

nts

(d

ays)

( ) ( )m

ft D t t R<200 km; M>=4


Negative slope

36

Multiscale Focalization of

earthquakes in space & time:

INGV Geosystemics

Stress from tectonic

plates to fault diffuses

at the minimum velocity

of 1-2 km/day?

100 80 60 40 20 0

0

10

20

30

40

50

60

0

10

20

30

40

50

60

Inte

r-e

ven

ts (

da

ys)

Distances (km)

1/slope = 1.6 km/day

De Santis et al.,

Tectonophysics, under revision


37

Focalization

confirmed

from space:

IR thermal

anomaly

from

satellite

INGV Geosystemics

Qin et al., Annals Geoph., 2012

Satellite

Ground


38

Laboratory fracturing-

model INGV Geosystemics

Infrared Detection Experiment of Compressed-shearing Loaded Rock

Settings rotated by 90o Original Settings

Experiments made in the frame of SAGA-4-EPR by Proff Liu and Wu of NEU (China)


t=001s

Experimental result of sample 1-9


t=040s



t=080s



t=086s



t=090s



t=095s



t=100s



t=106s



t=108s



t=109s



t=110s



t=140s

Experimental result of a sample 1-9


51

Conclusions on Emilia

Focalisation

INGV Geosystemics

•Seismicity that happened before the recent EMILIA earthquakes

is compatible with a partial Mogi doughnut model characterized

by a precursory space-time focalization process.

•The time evolution has behaved as a critical point process both

in space and in time.

• A simple laboratory model was able to grasp some the most

important features of the focalization process.


52

A systemic approach to

investigate Deep Seafloor

Beranzoli et al., PEPI, 1998

INGV Geosystemics

Seafloor Station deployed

several times

down to 3000 m depth

in Tyrrhenian Sea

Under the European

Projects:

GEOSTAR

ORION

EMSO

Instruments:

Seismometer

Magnetometers

Gravimeter

Currentmeter

…

From

Multiparameter

monitoring to

Trans-disciplinary

analysis A recent effort: Deep sea

and subseafloor frontier

DS3F Project

http://www.deep-sea-frontier.eu/

7. OTHER APPLICATIONS


The Deep Sea White Paper

54

Geomagnetism

Normalised Entropy

over a sphere

INGV

H*=0.0

H*=0.5 H*=1.0

H*=0.3

order

disorder

Geosystemics

Adapted from De Santis & Qamili, 2008

The present geomagnetic field


55

Geomagnetism and climate SAA and mean sea level

E

|

Geosystemics INGV

(De Santis et al., JASTP, 2012)

1 2 1 1 2( , ) ( ) ln[ ( ) / ( )]LK p p p s p s p s ds

Kullback-Leibler (1951) Relative Entropy

KL =0.05-0.06 (0.20-0.40 for surrogate data)

rspearman =0.94 1600 1700 1800 1900 2000 2100

-200

-150

-100

-50

0

50

100

150

1600 1700 1800 1900 2000 2100

0

1x107

2x107

3x107

4x107

5x107

6x107

7x107

8x107

1600 1700 1800 1900 2000 2100

-0.8

-0.6

-0.4

-0.2

0.0

0.2

0.4

0.6

Se

a L

eve

l (m

m)

Year

Sea level

SA

A A

rea (k

m2)

Year

SAA GUFM1

SAA IGRF

SAA CHAOS

Te

mp

era

ture

an

om

alie

s (

°C)

Year

SAA= South Atlantic Anomaly


56

8. Conclusions

1. We defined a new systemic approach (vision) to Earth

system study called GEOSYSTEMICS

where multi-platform/parameter/scale observations are

fundamental to take a whole picture of our planet

2. Fundamental tools (keys) have been proposed, mainly based

on - Entropy and Information

measuring the whole & the relationships among components

3. We showed an important application to Seismology

(disclosing the relationship between b-value and entropy)

4. and then something about focalisation of the process

(cases of 2009 L’Aquila and 2012 Emilia earthquakes)

5. Together with some other applications

6. Future can provide other cases of application in other fields of

Earth sciences.

Geosystemics


57

9. Selected bibliography

Baranger, M. (2001) Chaos, Complexity and Entropy: a Physics Talk for Non- Physicists, Wesleyan University

Physics Dept. Colloquium.

Bekenstein J. D. (2003), Information in the Holographic Universe. Scientific American, Vol. 289, No. 2, August 2003, p. 61.

Beranzoli L., De Santis A., Etiope G., Favali P., Frugoni F., Smriglio G., Gasparoni F. and Marigo A., (1998) GEOSTAR: a

Geophysical and Oceanographic Station for Abyssal Research, Phys. Earth Plan. Inter., 108 (2), 175-183.

Cianchini G. , De Santis A., Balasis G., Mandea M. , Qamili E. (2009) Entropy based analysis of satellite magnetic data for

searching possible electromagnetic signatures due to big earthquakes, Proceedings of the 3rd IASME/WSEAS

International Conference on Geology and Seismology (GES’09), pp. 29-35.

Cianchini G., De Santis A., Barraclough D.R., Wu L.X., Qin K. (2012), Magnetic Transfer Function Entropy and the 2009

Mw6.3 L’Aquila eaerthquake (Central Italy), Nonlinear Processes in Geoph., 19, 401-409.

De Santis A. (2009), Geosystemics, WSEAS, GES’09, Feb. 2009, Cambridge, 36-40.

De Santis A. & Qamili E. (2008), Are we going toward a global planetary magnetic change?, Proceedings of 1st WSEAS

International Conference on Environmental and Geological Science and Engineering (EG’08), Malta Spt. 2008, 149-

152.

De Santis A., Tozzi R., Gaya-Piqué L., (2004) Information Content and K-Entropy of the present geomagnetic field, Earth

Plan. Sci. Lett.,218, No.3, 269-275.

De Santis A., Cianchini G., Favali P., Beranzoli L., Boschi E. (2011), The Gutenberg–Richter Law and Entropy of

Earthquakes: Two Case Studies in Central Italy, BSSA, v. 101; no. 3, p. 1386-1395.

De Santis A., Qamili E., Spada G., Gasperini P. (2011) Geomagnetic South Atlantic Anomaly and global sea level rise: a

direct connection? JASTP, 2012.

Klir G.J. (1991), Facets of Systems Science, Plenum Press, London.

Powell G., and I. Percival (1979), A spectral entropy method for distinguishing regular and irregular motion of Hamiltonian

systems. J.Phys. A, 2053-2071.

Qin K., Wu L.X., De Santis A. Cianchini G., (2012) Preliminary analysis of surface temperature anomalies preceding the two

major 2012 Emilia (Italy) earthquakes, Annals Geoph., 55, 4, 823-828.

Shannon C. (1948). A mathematical theory of communication, The Bell Syst. Tech. J., 27, 379-423, 623-656.

Wiener N., (1948) Cybernetics: or Control and Communication in the Animal and the Machine, Cambridge: MIT Press.

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