Universal Log-Periodic Correction to Renormalization Group Scaling for Rupture Stress Prediction From Acoustic Emissions

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Universal Log-Periodic Correction to Renormalization Group Scaling for Rupture Stress Prediction From

Acoustic Emissions

J.-C. Anifrani, C. Le Floc’H, D. Sornette, B. Souillard

To cite this version:

J.-C. Anifrani, C. Le Floc’H, D. Sornette, B. Souillard. Universal Log-Periodic Correction to Renor-

malization Group Scaling for Rupture Stress Prediction From Acoustic Emissions. Journal de Physique

I, EDP Sciences, 1995, 5 (6), pp.631-638. �10.1051/jp1:1995156�. �jpa-00247089�

Classification

Physics

Abstracts

64.60Ak 62.20Mk

Short Communication

Universal Log-Periodic Correction to Renormalization Group

Scaling for Rupture Stress Prediction From Acoustic Emissions

J.-C.

Anifrani(~),

C. Le

Floc'h(~),

_D.

Sornette(~>~)

and B.

Souillard(~)

(~)

Aérospatiale,

B-P. Il, 33165 St Médart _en

Jalles,

France

(~) Laboratoire de

Physique

de la Matière

Condensée,

CNRS URA 190 Université de

Nice-Sophia Antipolis,

B.P. 71, 06108 Nice Cedex 2, Fiance

(~)

X,RS, Parc-Club,

28 _rue Jean

Rostand,

91893

Orsay Cedex,

Fiance

(Received

20 December

1994,

revised 26

January 1995, accepted

31

January 1995)

Abstract. Based

on trie idea that trie rupture of

heterogeneous

_systems is similar to a

critical

point,

_we show how to

predict

the failure stress with

good reliability

and

precision (m

51~) ^{fro~n acoustic emission} measurements at constant stress rate up to _a maximum load 15-20i~ below the failure stress. Trie basis of _our

approach

is to fit the

experimental signais

to a mathematical

expression

deduced from _{a new}

scaling theory

for rupture _in terms of

complex

fractal exponents. ^{Trie method is tested}

successfully

_{on an} mdustrial

application, namely high

pressure

spherical

tanks made of _vanous liber-matrix composites. As _a

by,product,

_our results constitute the first observation in

a natural _context of the universal periodic corrections to

scaling

in the

renormalization-group

framework. Our method could be

applied usefully

to other similar

predicting problems

_m the raturai _sciences

(earthquakes,

volcanic

eruptions, etc.)

Acoustic e~nissions

(AE)

_are ~nechanical _waves

produced by

^sudden ~novement in stressed

systems, occurring

m a wide _range of

materials,

structures and _processes, from the

largest

scale

(seismic events)

_to the smallest _one

(motion

of

dislocations).

It is _a rather dehcate

technique

to _use and

interpret

since each

loading

is

unique,

tests the whole structure and _con

easily

be contaminated

by

noise.

Notwithstanding

trie

development

of _numerous AE structural

testing procedures _[1, 2], hopes

to _use AE _{as a} reliable

practical

method for failure

prediction

have trot beell borne out.

AE dilfers from _most other nondestructive

testing (NDT)

methods in that the AE

signal

has its ongm in the material

itself,

not in _an externat _source, and in that it detects

movement,

^while

most methods detect

existing geometrical

discontinuities.

Thus,

the

general

basis for

developing

a

prediction

scheme is _to first determine which detected AE _are relevant _precursory

phenomena (among

the various _causes such _as crack nucleation and

growth,

liber-matrix

delammation,

liber

rupture...)

and then

develop

_an

algorith~n

which allows _one to relate these _precursors to the final

rupture.

©

Les Editions de

Physique

1995

632 JOURNAL DE

PHYSIQUE

I N°6

Here,

_we address the

general

_case where

global

rupture ^is not controlled

by

_a

single

event

(nucleation

and

growth

of _a

single crack),

as occurs in _a

ho~nogeneous _system

with very few

defects,

but rather

by

_a succession of events, in

general corresponding

to diffuse

damage, possibly culminating

_m trie

catastrophic growth

of _a

dominating

crack. In this _{case, we} daim that failure _can

only

be

predicted by viewing rupture

_{as a}

cooperative phenomenon:

it is not

through

the

separated analysis

of the _succession of the recorded hits that

rupture

can

be

predicted

but

by

somehow

synthetizing

all trie information embedded in these events in _a

global

framework.

From _a formal

point

^of

view,

_our

approach

to

rupture prediction

consists in

viewing

the final

stage

of

rupture

_{as a} kind of critical

point,

which _can be described

by

_a fixed

point

in

a renormalization _group scheme

(see below).

This

description

is

inspired

from _a wealth of

recent work in trie statistical

physics community, mainly

based _on exact solutions of solvable

models and _on extensive numerical

simulations,

^which bave shown trie existence of certain

scaling

laws up ^to a time of total

rupture [3-6].

Dur method is thus at trie

opposite

^{of trie}

view often advocated in trie mechanical

literature,

which consists in

modelling

an

(evolving) heterogeneous system

as an effective medium characterized

by

_average

properties, including

an

increasing global damage parameter.

^These

approaches

_are suitable

only

far from

rupture

and _are unable _to

capture

the

specific N-body

nature of the

rupture

cntical

point,

and _are

therefore useless _as

predictors.

In order to fix

ideas,

_we shall consider _an industrial

application

_on which _our method bas been tested

extensively.

Trie

complexity

of such

systems

can be taken _{as a test} of the robustness of _our

technique.

The

systems

are

spherical

tanks made of carbon libers

pre-impregnated

in _a

resin matrix

wrapped

_up around _a metallic liner. We have tested dilferent materials

(carbon

or kevlar

libers,

dilferent metallic

(titanium

_or

steel) compounds

for the

liner),

_as well _as dilferent tank dimensions

(radius

of 0.2 to 0.42

m).

AE

signais

_are obtained from three to six acoustic transducers

(resonant frequency

of150

kHz) placed

at

equal

distances _on the

equator

of _a

given sphencal

tank

by mcreasing

at a constant rate

(3

to 6

bar/s)

the internal water pressure and thus the stress exerted _on the tank.

Figure

1 represents a

typical

data set of the

mstantaneous AE _{energy rate}

dE/dt

_{as a} function of the

applied

internal _{pressure p up} to the

rupture

threshold _pr, obtained

by simple

^addition at each time

step

of the intensities measured

on all

operating

transducers. Note the

complex

intermittent _structure of the

data,

^with quiet

periods separated by

bursts of

widely

dilferent

amplitudes,

and trie

large

increase of trie AE

energy ^rate on the

approach

to failure

(occurring

in the

present

_case at Pr = 713

bars).

In order _{to test} the cntical

point concept

_on this

particular

set of

systems,

we checked for the existence of _{a power}

la~A~,

dE/dt

₌

Eo(Pr P)~" (1)

where _a is _a sc-called critical

exponent.

^{In all} cases

explored,

_we found that for _p

sufficiently

close to Pr to within about less than

51~,

the dramatic increase of AE energy ^rate on the

approach

to failure _is mdeed fitted _very well

by

_{a power} law

(1),

with _an exponent a ₌ 1.5 +0.2

mdependent

of the

specific sample

_or

previous history

as

long

_as the critical _zone

(approximately

m trie interval

[o.95pr; pr]

^bas not been reached in

preceding

loads. Trie _power law

(1)

was

checked either

by representing Log(dE/dt)

as a function of

Log(pr p), using

trie measured Pr. This is illustrated in

Figure

2 for three dilferent tanks

(and

two dilferent

partitions

for _one

tank)

which exhibits

clearly

trie

asymptotic

linear

dependence

when

suiliciently

close to Pr.

We find that ail data tend to _a

straight

^{behavior whose}

slope

determines _o. We also studied

(dE/dt)~l/° _(with

_o

=

1.5)

_{as a} ^function of _p.

Agam,

_an

asymptotic straight

line is obtained whose intersect with the abscisse gave a very

good

estimate for Pr better than ll~. These two

fitting procedures

are not

independent

but put ^dilferent

weights

_on trie data

point. Therefore,

SYSTEM

6000

sooo

_o

o

4000

°

°

~~~~ ~

o

1o

u~ ^° ~

°

2000

~ o

]

~~~~

~ é o

~ _°

~ o o ^°

o o o

° °

o

~ o

° o

0

500

550 600 650 700

PRESSURE

Fig.

1. Instantaneous AE energy ^rate

dE/dt

in system ¹ as a function of the

applied

internai

pressure p ^{at constant} pressure rate of 6

bar/sec

_up ta the rupture ^threshold _pr = 713 bars.

Pr-P

~~~

i ~~

u

2G

'"'"'~,

g loo

^'" ~

~

~

ioooo

F-

x G

j~

_'~,~,~

C4 k,

ÎiÎ io

^~

looo

o o

w o o

10 100 1000 10000

ENERGY

Fig. 2. Upper

nght: dE/dt

in

log-scale

_{as a} function of _{pr p m}

log-scale,

_usmg the measured _pr for each system, m three dilferent systems: system 1

(O);

system 2

(0)

and

(+);

_system 3

(x),

_system

2

being represented

twice for two dilferent crame-graimng

(0):

11

intervals; (+):

27 intervals. The

straight

hne has _a

slope

_cY

= I.à. Lower left: dilferential distribution

N(Eb)

of burst _energy Eb _in

log-log

scale. The

straight

fine has _a

slope

of _{a =} -2.

634 JOURNAL DE

PHYSIQUE

I N°6

verifying

their

consistency

is

important

for

checking

the existence of the _power law

(1).

As

expected,

the

exportent

a is found "universal"and

equal

ta I.à and insensitive ta the

specific

tank realization. It could however

depend

_upon the tank

components

which central trie _nature of

damage. Figure

2 aise shows trie differential distribution of burst _energy

'Eb'

which is aise

a power law

N(Eb)

+~

Ej~~~~~,

with _an

exponent

^B

consistently equal

ta 1+ 0.2 for ail the

systems investigated.

This

law,

which is reminiscent of trie

Gutenberg-Richter

distribution of

earthquake

sizes

[7],

^{bas been studied in detail} in [8] in trie AE ^context.

Armed with these

empirical

tests of trie concept ^{of critical} rupture,

prediction

should in

principle

be

possible by extrapolation

of AE data

using expression (1).

Note that this scheme is similar ta that

proposed by Voight

_a few _{years ago ta} describe and

predict rate-dependent

material

failure,

based _on trie _use of _an

empirical

_power law relation

obeyed by

_a measurable

quantity [9,10]. However,

this

procedure

is

unpractical

due ta trie _narrowness of trie demain of

validity

of the law

(1) (critical region [0.95pr; pr]

in most

explored cases), preventing

a

prediction

at pressures less

than,

_say, 0.97 pr.

A useful scheme should allow _{one ta} make

predictions

at much lower _pressures. Consider for instance _an

attempt

ta

predict

_pr

by monitoring

trie AE _up

ta,

say, 0.85 _pr, 1-e- up ^ta about 610 bars in trie _case shown in

Figure

1.

Inspection

of

Figure

1 shows

that, apparently,

_very little of trie whole AE data is contained in trie AE set up ta 610 bars.

Furthermore,

trie

dependence

of trie rate of AE _energy release _{as a} function of trie instantaneous

applied

_{pressure up} ta 610 bars bas

apparently nothing

ta do with _a power law such _as

(1)

and trie

prediction

thus _seems

hopeless.

Dur fundarnental idea _is that trie

concept

^of

rupture criticality

embodies _more usable in- formation than

just

trie _power law

(1)

^valid in trie critical demain. We _argue that

specific

precursors outside trie critical

region

_can lead ta "universal"

recognizable signatures

in trie AE data of trie final rupture. ^{In order} ta

identify

these

signatures,

_we first _note that _an

expression

like

(1)

can be obtained from trie solution of _a suitable renormalization _group

(RG) _[11].

Trie RG

formalism,

introduced in field

theory

and in cntical

phase transitions,

arnounts ta view trie

N-body problem

_{as a} succession of

I.body problems

with effective

properties varying

with trie scale of observation. It is based _on trie existence of _a scale invanance of trie

underlying physics

which allows _one ta define _a

mapping

between

physical

scale and distance from trie critical

point

in trie central

parameter

axis: in _Dur

problem,

trie

damage

rate and therefore AE rate at _a

given

_{pressure p} are related ta those _at another _pressure

p' through

_a suitable non-linear

transformation

p'

= pr

#(x)

with _x

= pr p. The RG formalism then

provides

the

general

structure of the functional

equation

that trie

physical

observable must

obey (for

trie

simplest

case of _a

Due.parameter RG):

iLF(x)

=

F(4(z)) (2)

For trie sake of

simplicity,

_we bave introduced trie notation

F(z)

=

(dE/dt)~~,

trie inverse of trie AE _energy

rate,

^which _goes to _zero at trie critical

rupture point

_xr

=

0;

_{/J is} a real

positive

number. We _assume that trie function

F(x)

is continuous and that

#(z)

is dilferentiable. Trie connection between this formalism and trie cntical

point problem

stems from trie fact that trie cntical

point

is _on trie attractor of trie fixed

point

of trie RG flow.

Then,

trie function

#(z),

which

generates

trie so-called RG

flow,

is

usually

used _to extract trie

qualitative

behavior _as well _as trie

stability

of fixed

points

and to deduce trie

corresponding

critical

exponents.

Let _us

assume for

simplicity

that trie critical

point

is _net

only

_on trie attractor of _a fixed point but is

indistinguishable

from

it,

_{as can} be done

by

_a suitable

change

of variable.

Then,

if _z

= 0 denotes

a fixed

point (#(0)

=

0)

and

#(x)

₌ AT +.. is trie

corresponding

linearized

transformation,

then trie solution of

(2)

close to _{x =} 0 is

given by equation (1)

with _a

=Log/J/LogÀ.

To go

beyond

this local

analysis

_m trie critical region, we are interested in trie

general

solutions of

equation (2).

To

get them,

let _{us assume} that

Fo(x)(= x")

is _a

special solution,

then trie

general

solution

F(x)

is related to

Fo(x)

in terms of _a

periodic

function

p(x),

with _a

period Log/J,

_as:

F(x)

₌

Fo(x)p(logfo(x)) (3)

Since

logfo(x)

₌

aLogx,

this leads to _a

periodic (in Logx)

correction to trie

dominating scaling (1). Equivalently,

this

log-periodicity

_cari be

represented mathematically by

_a

complex

critical

exponent,

since

Re(z"'+~"")

=

x"' cos(a"Logz) gives

trie first term in _a Fourier series

expansion

^of

(3).

This

expression

thus introduces universal oscillations

decorating

trie

asymptotic powerlaw (1).

Trie mathematical existence of such corrections bas been identified

quite early _[12]

in RG solutions for trie statistical mechanics of critical

phase transitions,

but bas been

rejected

for

translationally

invariant

systems,

since _a

period (even

in _a

logarithmic scale) implies

the existence of _{one or} several characteristic scale which is forbidden in these

systems.

For the rupture of

quenched heterogeneous

systems, trie translational invariance does not hold due to the _presence of static

inhomogeneities [13]

^{and trie fact that} new

damage

occurs at

specific positions

^which are not

averaged

out

by

thermal fluctuations.

Hence,

such

log-periodic

corrections _are allowed and should be looked for in order to

embody

trie

physics

of

damage

in trie non-critical

region.

It is

interesting

to mention trie few known _cases where such _a behavior bas been observed.

Probably

the first theoretical

suggestion

of the relevance to

physics

of

log-penodic

corrections to

scaling

bas been

put

^forward

by

Novikov _to describe trie influence of

intermittency

in turbu- lent flows

[14]. Loosely speaking,

if _an unstable

eddy

in turbulent flow

typically

breaks _up into two _or three smaller

eddies,

but not into 10 _or 20

eddies,

then _{one can}

suspect

^trie existence of _a

preferable

scale

factor,

hence trie

log-periodic

oscillations. A clear-cut

experimental

_ver-

ification of trie

generation

of

log-penodic

oscillations

by

discrete scale invariant fractals bave been carried _{on on} man-made

Sierpmsky

networks of normal-metal

links,

in which trie normal-

superconductive

transition temperature

presents log-periodic

oscillations _{as a} function of trie

applied magnetic

^field

[15]. Complex

tractai

exponents

bave also been

argued

to describe trie

branching

architecture of trie mammalian

lung [16].

^To our

knowledge,

there is _no

experi-

mental verification of this fact but renormalization _group calculations of critical behavior of random

dipolar Ising systems il?]

^and _spm

glasses _[18]

bave shown trie existence of

complex

cntical

exponents.

These results taken very

cautiously by

their authors could be the

signature

of _a

spontaneous generation

of discrete scale invariance due to the

interplay

of the

physics (interaction)

and the

quenched heterogeneity.

Boolean

delay equations involving

two time

lags

with _an irrational

ratio,

used

recently

to model the climate

vanability,

aise exhibit

superdiffuse

behavior with

log-periodic

oscillations

[19].

^Here

again,

we have _an

example

of _a discrete scale invariance which is

spontaneously generated,

^{in trie} present case

by

the threshold

dynamics

and nonlinear feedback involved in trie Boolean

delay equations. Finally,

it bas been

pointed

Dut that vibration and _wave

properties

on discrete fractal _structures should be characterized

by log-periodic

corrections _to the

leading singular

behavior for trie

density

of _states close to the

band

edges _[20]. However,

_our results _on these

periodic corrections,

that _are

presented below,

are trie first _ones obtained in structures

containing

_an uncontrolled

heterogeneity and,

to _our

knowledge,

^bave not been observed

previously

m trie _context of

rupture.

Figure

3 shows _a fit

(continous fines)

obtained

using

trie solution of

equation (3) applied

_to

two AE data sets

(represented by

trie

symbols)

obtained _on two differents tanks

(systems

1 and

2). Here,

_as in

Figure 2,

we group trie AE bits in time intervals of _varying width in order to reduce trie noise. To _express trie solution of

equation (3), p(z)

bas been

expanded

in Fourier

series and _we bave

kept only

trie dominant term:

dE/dt

₌

Eo(pr p)~~ il

₊ C _cos

(#

+

2xLog(pr p) /LogÀ)]. (4)

636 JOURNAL DE

PHYSIQUE

I N°6

iooooo

ioooo sysTEM

i

~- jQ~~ ^Î

~

©~ f

iÎà ~~

~j 1ÙÙ ~~ ~

~ ^X

x

10

SYSTEM 2

500

550 600 650 700 750

PRESSURE

Fig.

3. Theoretical fit

using equation (3)

shown _in continous fines of two AE data sets

(represented by

the

symbols)

obtained _on two dilferents tanks

(systems

1 and

2).

For each system, the dilferent

symbols corresponds

to dilferent

coarse-graining.

The

resulting

mathematical

expression

^bas a

priori

six unknown

parameters:

_a

normalizing

factor

Eo,

the cntical exponent a, trie pressure pr at

rupture,

the

amplitude

C of trie

oscillatory

correction,

its

penod LogÀ

and

phase #.

To bave better

conditioning,

we

impose

_a

= I.à

obtained from _our

previous

^fit shown in

Figure 2,

^since we expect ^it to be "universal" within _a Mass of materials. This leaves _us with five unknown variables to determine from _a non-hnear fit.

We used the

Levenberg-Marquardt

method

[21],

^{which combines the}

steepest

^descent method with trie inverse Hessian method. One _can observe _m

Figure

3 trie

importance

of the

oscillatory

corrections to the

leading

_power law behavior for

correctly accounting

for the intermittent burst

structure of the AE data shown in

Figure

1. It is

important

to remark that trie

penodicity

m

Log(pr p) implies

_a subtle correlation between successive AE bits measured at constant pressure ^rate. This _seems to be borne _out

by

trie

data,

_as

presented

in

Figure

3. Note also that trie burst distribution _m time and

amplitude

_can be _very different from _one

system

to

another,

_{as seen} in

Figure

3

by

_companng

system

to

system

^2: trie distributions of bursts _m time and

amplitude

_are not

universal; however,

when

they exist,

on average

they

_are correlated

according

to trie

loganthmic oscillations,

whose mathematical structure is a universal

property.

For

system 2,

^trie

log-penodic

oscillations _are less obvious but _are mdeed _necessary to account for trie still

significative

distorsion from _a pure power law

(C

= 0 in

Eq. (4)).

We _now show that these results _can be

exploited

to improve

dramatically

the

prediction.

Using

this

theory

for

prediction

_is also _a crucial test to further establish its

validity,

_smce

one could argue that _a

good

fit with five

adjustable parameters

^bas too much freedom _to

provide

_a safe

proof

of _any

theory.

Trie basic

goal

of _any

theory

_is m its

predicting

_power,

which _{we now} test _as follows.

Using

_{a given} AE data set such _as those

presented previously,

we first coarse-grain trie data _as in

Figures

2 and 3. We then _construct another data set

by

erasmg ail data above _{an upper pressure} pmax. This mimics _an expenment

performed

under trie _same conditions but which would bave

stopped

at trie _{pressure pmax} without

reaching

trie _pressure for

global

_rupture. We then

apply

the non-linear fit to this truncated file and

deduce _{pf~~~~~~~~ as one} of trie five variables of this fit.

Figure

4 shows for

system

^with

16

14

ii ^fi

j/

¹²

.,/"'1

(

10 /:

O ~ Ii

(

~

Î

i

4 ~~.

j

à

2 <'~

i,, Q

iÎ

~

~ "~

àÎ

~ ^~

Îi

_-2

1, 'n

ÎÎ

-4 ^~

-6 -8

500 520 540 560 580 600 620 640 660 680 700

UPPER PRESSURE

Fig.

4. Relative

precision

of the

prediction,

namely

100~pr -pf~~~'~~~~)/pr,

_{as a} function

ofprrax

for system 1 with three different

coarse-graining represented by

the three dilferent

symbols,

where _prrax

is the upper pressure taken into account. The

straight

fine is _given

by 100(pr prrax)/pr enabling

_a

comparison between trie

precision

of trie

prediction

^and the _upper attained pressure.

three different

coarse-graining (hence

trie three sets of

symbols)

trie relative

precision

of the

prediction, namely 100(pr pf~~~~~~~~)/pr

as a function of _pmax.

Changing

_pmax allows _us to

explore

trie

reliability

of

"long-term" prediction,

1-e- far from

rupture. Comparing

different

coarse-graining

_is

important

to _assess trie robustness and

consistency

of trie

prediction

scheme.

We observe that _a

precision

better than

4Sl

is obtained for _pmax > 620

bars,

_I.e.

remarkably

far below trie

rupture threshold,

where _use of _a

single

oscillation _m trie data shown in

Figure

3

(which

has

apparently

little _to do with the _power law

(1)),

is sufficient to

get

the information

on pr. This remarkable _success, in _view of the

relatively tiny

arnount of

remaining

AE data used in trie

fit,

_can be attributed to trie _very

strong

constraint

imposed by

the mathematical

solution of

equation (3),

hence relies

fundamentally

_on trie

physical picture

of

rupture

as a

critical

point.

For lower values

(pmax

< 620

bars),

trie

prediction

deteriorates since

lowering

further _pmax

prevents

from

retrieving

trie information embedded in the first lobe of the data shown in

Figure

3. Note that a similar

procedure using only

the

asymptotic

_power law

(1) yields extremely

bad

predictions

_{as soon as pmax} is lower than 680 bars. We bave also tested

Dur method

"blind-folded",

1e. _on AE data obtained _on

systems

^{which have} net reached their

rupture

^threshold. The

comparison

between _Dur

prediction

and the actual _pressure threshold reached in _a

subsequent experiment

_was of similar

quality

_as shown in

Figure

4. We have aise tested different

loading procedures;

_Dur results remain robust in ail these _cases.

In summary, our main

point

is that mechanical breakdown of

heterogeneous media,

instead of

nucleating

_{on a}

single

crack _as would be trie _case in _a

homogeneous system,

involves collective

phenomena.

^Idealised

Ising

^nucleation models bave

recently

been _{proven to} exhibit _a similar behavior

[22]:

^for a range of parameters, nucleation of _a

single droplet

leads _to trie

decay

of

supersaturated

_vapor, whereas for other

parameter

combinations trie coalescence of clusters

arising

from many

separate

nucleation events drives trie

phase

transition. This mechanism bas been confirmed

by laboratory experiments

^and

computer

simulations.

Exploiting

these

ideas,

638 JOURNAL DE

PHYSIQUE

I N°6

we bave

described,

and tested _on industrial

systems,

_{a new}

scaling theory

of

rupture

viewed _as

a critical

point

endowed with _a

complex

cntical

exponent,

which allows _{one to}

get precise

and reliable

prediction

of

rupture

^thresholds from trie

analysis

of AE data obtained _{on pressure} load at constant pressure ^rates.

Applications

and extensions of these ideas for

earthquakes

will be

presented

elsewhere

[23].

Acknowledgments

We thank D. Stauffer _as trie editor for constructive comments. D. Sornette

acknowledges

useful discussions with J-P-

Bouchaud,

P.

Miltenberger,

H. Saleur and C. Vanneste. We also thank trie Centre National d'Etudes

Spatiales (CNES)

for their financial support.

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