Eartkquake Death Tolls

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Eartkquake Death Tolls

Leon Knopoff, Didier Sornette

To cite this version:

Leon Knopoff, Didier Sornette. Eartkquake Death Tolls. Journal de Physique I, EDP Sciences, 1995,

5 (12), pp.1681-1668. �10.1051/jp1:1995223�. �jpa-00247166�

Classification

Physics

Abstracts

02.50-1 91.30Px

Earthquake Death Tolls

Leon

Knopoff(1)

and Didier

Somette(~)

(~)

Department

of

Physics

and Institute of

Geophysics

and

Planetary Physics, University

of Califomia, Los

Angeles,

California 90095, USA

(~) Laboratoire de

Physique

de la Matière Condensée, CNRS URA 190,

Université de

Nice-Sophia Antipolis,

B-P. 71, Parc Valrose, 06108 Nice Cedex 2, France

(Received12 August1995, accepted

31

August 1995)

Abstract. In the risk and insurance literature, the

(one-point)

distributions of fosses in raturai disasters have been

proposed

to be characterized

by

"fat tait" _po~ver laws, 1.e. very

large

destruction _{may occur} with

a

non-vanishing

rate. A _naive

hypothesis

of uncorrelated Poissonian _occurrence would suggest ^{that the fosses} are

solely

characterized by the

properties

of the

underlying

_power la~v distributions, _i.e. the

longer

_we watt, the _more dramatic ~vill be trie

largest

disaster, which coula be _as much _{as a} finite fraction of the total

population

_or the

total wealth of _a country. ^{We find indeed that the numbers Z of deaths in the} very

largest

earthquakes of this Century can be described

by

_{a power} law distribution

P(Z)

_m

^Z~(~+~~

with à = I.o + 0.3,

implying

_an unbounded behavior for trie _most

devastating earthquakes. However,

the distribution of the number of deaths per capita in each country m this Century has _a well- defined _maximum value,

suggesting

that the naive

extrapolation

of the power law distribution

is incorrect and that the

understanding

of correlations is necessary ^to ascertain the level of

nsk from natural disasters. The

one-point

distributions

only provide

_{an upper} bound of the

expected

risk. We propose a

speculative

mortel _to explain the correlations bet~veen deaths in

large earthquakes

and their countries of occurrence: ~ve suggest ^that

large

ancrent civihzations

that have matured into

large present-day populations

_were the beneficianes of isolation from

marauders due to the relative

geographic protection by

tectonic processes

largely

of _{an orogenic} nature.

1.

Earthquake

Size Distribution

The estimation of risk due to natural disasters is of

increasing importance

in _view of the

growth

of the world's

population

and its concentration _{in areas} prone ^to

earthquakes,

volcanic

eruptions, typhoons

_or floods. In the risk and insurance

hterature,

the

(one-point)

distributions of losses bave been

proposed

to be characterized

by

"fat tail" _power

laws,

_{i e. very}

large

destruction may occur M.ith _a

non-vanishing

rate. How do these distributions evolve _over

long

times and what _are the

possible

consequences of correlations between events _on

long-

term _msurance

policies?

A _naive

hypothesis

of uncorrelated Poissoman _occurrence would

suggest

that the losses _are

solely

characterized

by

the

properties

of the

underlymg

_power la~K=

Q

Les Editions de

Physique

1995

1682 JOURNAL DE

PHYSIQUE

I N°12

distributions,

1-e- the

longer

_we

_wait,

the _more dramatic will be the

largest

disaster, which could be _as much _{as a} finite fraction of the total

population

_or the total wealth of _a

country.

Is this _a valid

assumption?

We address this

question using rank-ordering

statistics _on the distribution of

earthquake

fatalities Z derived from historical

data,

which has been

proposed

to be _a robust _measure of losses.

It is well-kno~vn that "small"

earthquakes

are distributed in size

according

to the

Gutenberg-

Richter _power

law, P(E)dE

~-

E~(1+~~/~),

where E is the _energy released in the

earthquakes.

For

example,

_a detailed

analysis Ill

of the World-Wide Harvard

earthquake catalog (19îî 1995)

shows that b

= 1.00 + 0.02 for shallow

earthquakes

with

magnitudes

less than _a value

in the 7.1 to 7.6 range.

Laiger earthquakes

_can be also described

by

_{a power} law distribution but with _a

larger exponent

b

= 2.3 + 0.3. The _cross-over between these two

populations

_is

ill-defined and _we caution that there is _no direct evidence to confirm the

hypothesis

that the

large-earthquake

branch is indeed _{a power} law. In

fact,

a gamma distribution

(power

law truncated

by

_an

exponential roll-off)

also fits the entire suite of

earthquake

moments from the smallest to the

largest satisfactorily _[2j.

Another well-documented system is that of the

earthquakes

of the Southern California

catalog (1932 -1995):

in this _case, b

= 1.00 + 0.02

up to the

largest magnitude observed,

J£I~v CÎ 7.5

Ill.

We caution that ail

existing earthquake catalogs

_{span a} time scale which is

significantly

shorter than the time interval between the

earthquakes

with

magnitudes

in the _range

Mw

_=f 8.5 9.5 known to occur.

2. Distribution of Death Tolls

A first

step

^{toward the} assessment of the level ofrisk associated with

earthquakesis

to determine the distribution function of the _sizes of these disasters. The destructive _power of _an

earthquake

results from _a

conjunction

of several

factors, including

the

magnitude

and moment of the

earthquake,

the

depth

of

focus,

the orientation of the

earthquake fault,

the distribution of

population

_m the

neighborhood

of the

earthquake,

^site effects such _as

liquefaction, landslides,

the nature of the

soil,

the

quality

of the infrastructure

including

failure of

utility lifelines,

etc.

In

addition,

_one should take into account collateral

damage

due to

fire,

_tsunami, environmental

contamination,

etc.

The

physical

sizes of

earthquakes,

the _economic losses and the deaths that result from earth-

quakes

_are each _measures of

magnitude

of risk. As _an introduction to the

problem,

consider first the number of human deaths in each

earthquake

worldwide _m this

century [3j.

^Let

Zi

>

22

> >

Zn

> >

ZN

be the rank-ordered number of deaths in each

earthquake.

For

example, Zi corresponds

to the

Tangshan,

^China

earthquake

of 1976.

Although

deaths due _to

earthquakes

are rare occurrences, the observation of _a few _tens of the

largest

_{cases is} sufficient to determine the distribution with

good precision using rank-ordering techniques;

these meth- ods _were

originally

introduced in the

study

of the statistics of

languages

_[4j and then afterward

m a

variety

of fields

[5j, including earthquakes

_[1]. The

rank-ordering

distribution _is the _same

as theicumulative distribution but with _an

interchange

of _axes.

However,

the statistical

analy-

ses of the two are different: the statistics of the

rank-ordering procedure

is concerned with the uncertainties in trie values of Z while the cumulative distribution is concerned with the _uncer- tainties of the order numbers, which _are integers.

Here,

_we focus attention _on those _extreme

events which characterize the distribution at the

large

end of the tail. In the _case of power law

distributions of the

largest

events, the tail of the distribution is

dramatically

constramed, and

a

surprisingly good

_recovery of the

exponent

con be derived from _an

extremely

small data set.

The

reliability

of the retrieval of the

exponent

bas been

explored

in detail _m [1].

Z~

o

i

i io ioo iooo

n

Fig.

1.

Log-log rank-ordering plot

of the distribution of deaths per

earthquake:

Zn _is the nth

largest

number of deaths per

earthquake,

in

decreasing

order. The best fit to the entries

having

ranks 1 to 25 is shown

by

the

straight line,

which defines _a power law for the extreme tail of the distribution

(see text).

In

Figure

1 _we show the

log-log

rank-order distribution of Z. The

equation logZn

-~

=

logn

+

C,

with à

= 1-o + 0.3 and C _a

constant,

^1-e- that

Zn

_~-

n~l/~

_is

a

good

fit

à

up ^to rank around 25. We _recover the distribution

P(Z)

_~-

^Z~ll+~) characterizing

the _very

+m

largest

death tolls~ This is obtained

using

the

equation

N

Î~ P(Z)dZ

_{m n,} which expresses that there _are

typically

_n values of Z

larger

than _or

equal

to

Zn

out of _a set of N entries

(see Ill

for _{a moi-e precise} derivation and the

appendix).

The _error bar +0.3 _on the determination of

the

exponent

^à results from trie usual _square root

uncertainty

^of the maximum likelihood esti-

mator

[fil,

^from a

systematic

bias

resulting

from the estimator

itself,

_as well _as the existence of

a crossover to a different distribution for the

larger ranks,

which

correspond

to smaller death

tolls

(see Ill

for

discussion).

The above result

suggests

that the

longer

we

wait,

the _more dramatic will be the

largest disaster,

which could be _a

significant

fraction of the

population

of _a

country.

If à _<

1,

then

zi

the _average number < Z _> of deaths per

earthquake

_is

given by

< Z _>=

Î ZP(Z)dZ

_=f

~

~Z)~~,

^where

^Zi

^{is the maximum observed size. Since}

^Zi

^{grows as}

^N~/~

^with

^N,

^then

1-

< Z _>

diverges

_as < Z _>~-

Nl~l. Practically,

this _means that the number of deaths

Zi

due to the

smgle

most disastrous

earthquake

_among the total of N

earthquakes

accounts for

a finite fraction of the total deaths _in all

earthquakes,

~vhich is ,àT < Z >.

Indeed,

the above estimate

gives

the finite fraction

~

~~

=f ~

~, confirming

_our statement; a more

rigorous

< Z _>

derivation is given in

[7j, p.169. Thus,

for

example

if à

= 2

/3,

the correct formula

predicts

that the

largest

event

corresponds

_{on average} to

1/3

of the total.

1684 JOURNAL DE

PHYSIQUE

I N°12

If à _{tutus out to} be

larger

thon 1, the average number _< Z _> of deaths _per

earthquake

is

well-defined.

Ho~vever,

its standard deviation <

Z~

_{> <} Z

>~ diverges, meaning

that _as time goes on there _are

larger

and

larger

fluctuations. The

practical

consequence is that the

probability

of human extinction

(or

the

probability

of ~'ruin" remains

important.

These

ideas,

put ⁱⁿ

quantitative

terms in

[8j, rely

_on the

assumption

that the _power law distribution is limited

only by

the finite size of the

population,

and hence that the

possibility

of _an extinction

is finite _eiren if

quite

small.

3. Per

Capita

Death Toll

We show that the above estimates may be

pessimistic

because

they

overlook the existence of correlations between different _measures of destruction _m

earthquakes.

We propose that _{a more} sensible _measure of the impact ^of

earthquake

destruction _on human

activity

is _a irariable that

mcreases with time _{over a}

long time,

normahzed

by

_a relevant _measure of wealth.

Indeed,

for

governments

^{and insurance} companies

[9],

^the fundamental

question

is the

adequate

_coverage

of the risk. One

strategy

is to divide the region ^at risk into districts and

apply

_an estimate for each districts

separately. Thus,

in

principle,

_one could focus the _resources of the

insuring agencies

_on trie

regions

at

greater

^{risks. For the} purposes of

constructing

an

example.

_suppose

the

region

at risk _to be the entire

earth,

and the districts to be the individual countries. On the basis of _our assumed

local,

_i-e-

national, strategy,

we would then

expect

a

greater

risk for

an

earthquake

_prone

country,

thon in _a

tectonically

stable part ^of the vTorld. ~vhile this is _a trivial statement, and is

easily

extended to smaller

regions

at risk with smaller subdivisions _into

districts,

the _moi-e

important question

is to estimate the risk to be

applied

to each district

[10].

As _an introduction to the

problem

of the

development

of _a risk

strategy

that takes into

accotint the cumulative value of

loss,

_we take _{as our} estimate of risk the _per

capita

number of

human deaths due to

earthquakes

_m this

century

^{for each}

country,

^referred to the

population

in 1973

[11,12].

Our

justification

for this

procedure

is that what is

important economically

is not the absolute value of the loss but its

magnitude

relative to the

surrounding

wealth.

Also,

the

analysis

of

single

events _can tell _us

something

_over

long times, only

if _we make _reason- able

assumptions

_on their coi-relations in time and _space. ive

attempt

to obtain information

regarding

the

important

correlations between losses _{over a}

relatively long

time.

Let

Di

>

D2

> >

Dn

> >

DN

be trie rank-ordered per

capita

deaths for each

country.

For

example, Dl

" 9.36 _x

10~~, D2

" 8.39

x10~3

_and

D3

" 4.95

x10~~ correspond

to

Aimenia,

Turkmenia and Guatemala.

Figure

2 shows three types ^of

plots: a) logDn

_{as a} function of

log

n

b) Dn

_{as a} function of

log

_n;

c) log Dn

_{as a} function of _n. Fioni examination of

Figure 2a,

it is clear that the

log-log plot

is not

linear,

which would haine defined _a power la~v distribution, If1ve insist _on

fitting

a power law distribution. _~ve find that

only

about the first

largest

10

entries _con be fitted with

exponent

^à = 0.7 + 0.4.

Figure

2b shows _a

slightly

better fit: _a linear relation between

Dn

and

log

_n is a

reasonably good description

of the data _up to rauk about là. In this case, we

gel

^the distribution

P(D)

~-

e~~/~°

_where

_Do

"

(8 +1)

x

10~3 Finally,

Figure

2c shows _a remarkable linear

dependence

of

logDn

_{as a} function of _n for the first 3î countries out of the 47 countries with deaths due _to

earthquakes

_in this

century. Beyond entry 37,

there _{is a} roll-off ~vhich may be attributable to finite size effects, smce the last ten points

correspond

to countries

having

about 200 _or less

deaths;

a

figure

that is _very small

compared

to the number of deaths for the

largest

entries _in the

list,

~vhich _are

roughlj<

4 _x 10~ _for China.

1.5

x10~

_for

Japan,

10~ for

Itàly,

etc. Trie best fit is

log Dn

=

log Dm~x

_an, with Dn~~x =f

10~~

and _a ci 0.18. Thus trie distribution _is

P(D)

~-

D~l

for _{D <}

Dm~x

and is 0 for D >

D,i,ax.

Let _{us stress} that this distribution _is

completely

different from _{a pure}

powerlaw ^D~(1+~)

lvith

à _~ 0 and with _no cut-off. Ei~en for à _as small _as

10~~ (this

_is all trie _more true for

larger à).

a

o o

° ° o o

o

°a oo

ooo

c ~~°°~

~ %b

~$~« ~~W

~ ^~

îJ

0,0 0,2 0,4 0,6 0,8 1,0 1,2 1,4 1,6 1,8 2.0

log

_n

1°

a)

D~

o o

u

o~ oo

0,0 0,2 0,4 0,6 0,8 1,0 1,2 1,4 16 18 2,0

'°gio"

b)

~c

ço~ ^°

1

oo

o

~

0 5 10 15 20 25 30 35 40 45 50

n ~)

Fig.

2. The DN _are Îhe rank-ordered pet capita ^{deaths for} each country.

a) log

Dn _{as a} function

of

logn; b)

Dn _{as a} function of

logn; c) log

D~ _{as a} function of _{n: m} this _case, the

straight

hne fit defines the distribution

P(D)

+~

D~~ _{for D <} Dma~ and 0 for D > D,»ax.

1686 JOURNAL DE

PHYSIQUE

I N°12

~~

ÎÔJo

o

i io ioo iooo

n

Fig.

3.

Log-log rank-ordering

of the distribution of national

populations

_pN in 1994

(pi corresponds

to

China, etc.).

The linear fit defines _{a power} law distribution with exponent _~t = 1+ 0.3 for the 50 most

populous

countries.

trie

representation (a) si~owing IogDn

_{as a} function of

Iogn qualifies

_as trie best fit to

D~(~+ô),

as have been checked with

synthetic

tests

[13]. Therefore,

the

representation (c) showing

_a

linear

dependence

of

logDn

_{as a} function of _{n is} characteristic of _a distribution with _a limite upper cut-cif. These results shows that power la~v behavior with _nonzero à for the per capita distribution _is trie poorest ^{fit of trie three}

attempts. Despite

trie existence of the power law distribution

D~(~+~)

_{for trie death toit per}

earthquake,

the

important

result of this

analysis

^is

that there is _a well-defined maximum number of deati~s _per

capita Dmax

that _seems to have been

adequately sampled

even for the limited

earthquake history

of this

century.

The value of

Dmax

is

mainly

constrained

by

the entries

having

trie

large

ranks _{up to} 37 whose deaths _per

capita

range frein

10~~

_to

10~~;

nevertheless the entries with the lowest

ranks,

1-e- those with the

largest

numbers of deaths _per

capita,

are

quite compatible

with this determination. While

we bave _no

exploration

for the

D~~ dependence

for D <

Dmax,

we stress that the

important

result is trie existence of the cut-cif.

4. Discussion

TO

summarize,

the distribution of the number of deaths _per

earthquake

is _a power law with

exportent

^à ci 1. The _same law halas for tl~e distribution of trie total number of deaths due to

earthquakes

in each country, wl~ich is reasonable in _view of tl~e

stability properties

of _sums of variables distributed witl~ power laws

(Lévy

laws _are stable laws witl~ respect ^to

addition).

However,

tl~e distribution of deatl~ _per capita for countries _in this

century

^bas a well-defined upper

limit,

_le- there is _a

sharp

cut-off _to tl~e distribution. How _{con we} rationalize tl~is

paradox?

A

rank-ordenng analysis

of tl~e distribution of national

populations

sl~own _in

Figure

3 shows tl~at the

probability P(p)

that _a

country

^has _p inl~abitants is given

by P(p)

_ci

p~(~+P)

witl~

p ci 1 for tanks _up to

50,

1-e- for the 50 most

populated

countries. If the death toit _pet

eartl~quake (or

tl~e total number of deatl~s _pet

country)

and tl~e national

population

_were

uncorrelated,

_we coula

simply

deduce the distribution of the death toit D _pet

capita, using

D

mw

Z/p.

Tl~e distribution of _a variable which is tl~e ratio of _two

variables,

eacl~ with _power law

distributions,

is itself _{a power} law with _an exportent

equal

ta tl~at of tl~e variable in tl~e

numerator, ^{since in} an uncorrelated universe tl~e

largest

value of tl~e ratio _occurs wl~en tl~e

numerator

explores

tl~e tait of its distribution while the denominator remains in the center of

its _own distribution

(this argument

is correct if _{p is} bounded from below and is

nonzero).

Dur result is in

disagreement

with this

simple _argument,

but instead

suggests

^{that there is}

a

strong

correlation between tl~e deatl~ toi] in

eartl~quakes

and

population.

This

might

_{seem a}

posteriori

_a trivial statement but it is

interesting

ta _see it confirmed

bjr

a statistical

analysis.

Tl~is

example provides

_a non-trivial _case wl~ere

important

correlations

destroy

_power

laws, showing

a

possible

outcome of the

interplay

between _power law toits and correlation. The

simplest

correlation between

country

size and

eartl~quake

deatl~ toi] is tl~e obvious fact tl~at deatl~ toit increases witl~

population.

A second

type

^of correlations is tl~at destruction and death _{are more}

probable

in poor countries _or in countries witl~ _poor construction standards. A

more subtle correlation tl~at _we would like ta

suggest

is tl~at

large populations

_are often found in

tectonically active, eartl~quake

_{proue areas.} We

speculate

that tl~e

development

of ancient

civilizations,

1-e- the

development

of

large

communal

populations,

teck

place

because of _geo-

graphicaI isolation,

induced

by

tectonic processes of mountain

building

that

presented

raturai defenses.

Examples

of sucl~ ancient civilizations _are to be found in Anatolia

(Turkey), China, Greece, India, Japan,

Java

(Indonesia), Mexico, Persia, Peru,

Rome. Tl~e absence of tectonic

defences in tl~e

plains promoted

^diffusion

by

marauders and l~ence tl~e incubation _processes necessary to

develop large populations

coula trot emerge. Tl~e

growtl~

of tl~ese

populations

has continued to this

day.

It _was much

later,

in the middle _âges, that

global

trade

emerged

_{as an}

even more

patent

^{force that allowed for tl~e}

development

of

large populations

in tl~e

plains,

for

example,

of nortl~ern and western

Europe.

And the

exploration

and

development

of North America teck

place

because of these _more patent forces _as well. We note tl~at otl~er studies bave

proposed

correlations between

earthquake

active

regions

and the

development

of l~uman

populations _[14]

and between

eartl~quakes

and

emerging

infections

ils]

and it remains to be

seen if these issues _are components of the correlations that must be

quantified

for _a

global

insurance

policy.

In this

regard,

_we find that the distribution of

population

sizes of countries

in tl~e death toit

earthquake

list is very diiferent frein that of tl~e

complete

set. As

pointed

eut, tl~is

suggests

tl~at tl~e _occurrence of

eartl~quakes

_may influence

population growtl~.

In sum, our results stress tl~e importance of

taking

into account correlations between tl~e number of casualties _in

eartl~quake

events in order to define _{a more} relevant

global

insurance

policy

tl~an bas been considered to tl~is time.

Acknowledgments

We _are

grateful

to C. Vanneste and Y.Y.

Kagan

for useful discussions. Tl~is work bas been

partially supported by

tl~e CNRS-NSF International

Cooperation

_program, and is Publication

no. 223 of tl~e Soutl~em Califomia

Eartl~quake

Center and Publication _no. 4305 of tl~e Institute of

Geophysics

^and

Planetary Physics, University

^of

Califomia,

Los

Angeles.

Appendix

A

Consider _a stochastic process m which the outcome

E,

which _we call trie energy

generically,

has the distribution

P(E)dE

=

)dE, _il

~

1688 JOURNAL DE

PHYSIQUE

I N°12

with

Em;n

= 1 < E < cc; then C

= p.

Suppose

that N events _occur within _a

given

time interval. Let

Ei

>

E2

> >

En

> >

EN

be the energies of the events listed in

descending

order. The

probability F(En)dEn

that the _energy for the nth order _e,rent be

equal

to

En

1&>ithin

dEn

is

[si

F(En)dEn

=

(~) (l /~ _{P(E)dE) (~)} /~ _P(E)dE) P(En)dEn (2)

n

~~

^{~ ~}

l

~,~

^{~ ~}

for _an

arbitrary

distribution

P(E).

For the _power

law, F(En

has _a

peak

at

Ez~~

₌

fi] ~/~,

wl~icl~ _recovers and makes

precise

tl~e

scaling

law for tl~e rank ordered values stated in tl~e _text, If

P(E)

is

given by

the power law

(Il, F(En)

_con be

expanded

around tl~is maximum _as

à2

_~n(~

FlEn)

"

FlEll~~) _~

~~2

~

(En Ell~~)~

+

, n ~>iiax

whicl~ allows _{us to}

get

an estimate of tl~e standard deviation of

En tl~rough

tl~e calculation of

AEn

_e<

(En Ez~~)~ >~/~,

wl~ere tl~e brackets indicate tl~at tl~e statistical average is taken,

We obtain

m

-

l~jn/+ ^{~1~~~. 13)}

References

iii

Sornette

D., Knopoff L,, Kagan

Y,Y. and Vanneste

C.,

J.

Geophys. Res.,

submitted.

[2]

Kagan

Y.Y,, Seismic

moment-frequency

relation for shallow earthquakes:

regional

comparison, preprint

(1995).

[3j ^Lomnitz C., Fundamentals of

Earthquake

Prediction

(Wiley,

New York,

1994).

[4]

Zipf

G-K-, Human behavior and trie

principle

of least-effort

(Addison-Wesley, Cambridge~ 1949).

[5] Gumbel E.J.~ Statistics of extremes

(Columbia University

Press,

New-York, 1960).

[6] ^Hill

B.M.,

Ann. Statistics 3

(1975)

l163-l174.

[7] ^Feller W., An introduction _ta

probability theory

and its

applications,

Vol. 2 (J.

Wiley,

New

York, 1966).

[8]

Kagan Y.Y., Earthquake

_size distribution and

earthquake

_msurance, preprint

(1995).

[9] Abbott A., Nature 372

(1994)

212-213.

[loi

Johnston

A.C.,

Seismotectonic

interpretations

and conclusions from stable continental region

seismicity database, _m: The

earthquake

of stable continental regions, Electric Power Research

Institute,

Palo Alto

(Cahfornia)

A.C. Johnston et ai.

Eds.,

Vol. l, _pp. 4-1 4-103.

iii]

The Times Atlas of trie

World,

sth ed.

(Times Books,

London,

1975).

[12] ^Because the

political

boundaries of the world change with _a

relatively

short time constant, ^the definition of _a nation

is

ephemeral.

We bave considered trie nations that

emerged

from trie

breakup

of trie Soviet Union _as

mdependent

entities for these purposes.

Yugoslavia,

Indonesia, New Guinea and others _are

single

entries

m our tabulation.

[13] We thank C. Vanneste for these simulations.

[14]

King

G.,

Bailey

G. and

Sturdy

D., J.

Geopàys.

Res. 99

(1994)

20 063.

[15] Earthquakes and

Emergence,

EOS, Trans. Amer.

Geophys.

Un. 75

(1994)

474.