Physics Models of Earthquake

0.76 MB·English

by  Srutarshi Pradhan

Physi
s Models of Earthquake
∗
Srutarshi Pradhan
Department of Physi
s, Norwegian University of S
ien
e and Te
hnology, N(cid:21)7491 Trondheim, Norway
7 Sin
elongba
k,s
ientistshavebeenputtingenormouse(cid:27)orttounderstandearthquakedynami
s
0 -thegoalistodevelopasu

essfulpredi
tions
hemewhi
h
anprovidereliablealarmthatanearth-
0 quakeisimminent. Modelstudiessometimeshelptounderstandinsomeextendthebasi
dynami
s
2 of the real systems and therefore is an important part of earthquake resear
h. In this report, we
n reviewseveralphysi
smodelswhi
h
apturesomeessentialfeaturesofearthquakephenomenonand
a also suggest methods to predi
t
atastrophi
eventsbeing within the range of model parameters.
J

0
100000
1 Introdu
tion entire Japan 1985-1998

10000
] y
h c
c `Earthquake' is a long standing problem to the s
ientist uen 1000 slope = 0.90
e
ommunity as it is a subje
t of many unknowns. Strong q
e
m earthquakes often
auses huge devastation in terms of nt fr 100
- livesandpropertiesandthereforethissubje
talwaysde- e
v
at mands high priority. But so far, little advan
ement has e 10
t beenmadetounderstandtheentireearthquake-dynami
s
s
. and the
hallenge still remains -to develop a su

essful 1
at predi
tion s
heme whi
h
an provide exa
t spa
e-time 2 3 4 5 6 7 8
m information of future earthquakes and their expe
ted m
magnitudes. However, several models and hypothesis
-
d [1, 2, 3, 4, 5, 6℄ have been proposed and some re
ent
n studiessuggestpotentialmethodstopredi
t
atastrophi
(froFmiguJrUe.N1E:CT
haetamloagg)n[i2tu7℄d.eTdihsetrisbtruatiigohntolfineearwthitqhuaskloepsein−J0a.p9ains
o m>3
events in some earthquake models. In this short report thebest(cid:28)ttothedatapointsfor andsupportsGutenberg-
c
wedis
usssu
hphysi
smodelsofearthquake,givingim- Ri
hterlaw.
[
portan
e to their
apability of predi
tion.

1
Physi
s models
v
4 Geologi
al fa
ts
0 The overall frequen
y distribution of earthquakes in-
2

luding foresho
ks, mainsho
ks and aftersho
ks, seems
1
0 Plate-te
toni
theoryexplainstheoriginofearthquake to follow the empiri
al Gutenberg-Ri
hter law [5℄:
7 through a sti
k-slip dynami
s: Earth's solid outer
rust lnN(m)=constant−bm,
0 (about 20 km thi
k) rests on a te
toni
shell whi
h is
/ divided into numbers (about 12) of mobile plates, hav- N(m)
t where isthenumberofearthquakeshavingmagni-
a ing relativevelo
ities of the orderof few
entimeters per m b
m year. This motion of the plates arises due to the power- tude(inRi
hters
ale)greaterthanoreqbualto , isthe
powerexponent. Theobservedvalueof rangesbetween
- ful
onve
tive(cid:29)owoftheearth'smantleattheinner
ore 0.7 1.0 ǫ
d and (Fig.1). The amount of energy released in
of earth. On the other hand solid-solid fri
tional for
e
n an earthquake is related to the magnitude as
arises at the
rust-plate boundary and it sti
ks them to-
o
lnǫ=constant+am.
c gether. This kind of sti
king developselasti
strains and
: the strain energy graduallyin
reases be
ause of the uni-
v
i form motion of the te
toni
plates. Therefore a
ompe- ThereforeGutenberg-Ri
hterlaw
anbe expressedasan
X tition
omes to play between the sti
king fri
tional for
e alternative form:
ar and the restoring elasti
for
e (stress). When the a

u- N(ǫ)=ǫ−α,
mulated stress ex
eeds the fri
tional for
e, a slip (earth-
quake) o

ursand it releasesthe stored elasti
energy in N(ǫ)
the form of sound, heat and me
hani
al vibrations. It where isthenumberǫofearαth=quba/kaesreleasingenergy
has been observed that generally a series of small earth- greaterthan or equal to and . The Gutenberg-
quakesappearbefore(foresho
ks)andafter(aftersho
ks) Ri
hter law is being
onsidered as one of the most fun-
a big quake (main sho
k). damental observations by the physi
ists.
Severalmodelshavebeenproposedtostudythenature
oftheearthquakephenomenon. Themainintensionisto
∗
apture the Gutenberg-Ri
hter type power law for the
Ele
troni
address: pradhan.srutarshintnu.no frequen
y distribution of failures (quakes) by modelling
2
di(cid:27)erentaspe
tsoffaults(cid:21)stru
ture,materialproperties, and dynami
phase begins. Due to the presen
e of fri
-
geometry et
. These models
an be
lassi(cid:28)ed into four tional for
e this dynami
phase is de(cid:28)nitely dissipative
groups a

ording to their basi
assumptions: (A) Fri
- in nature. This dissipation redu
es the relative velo
ity
tion models, in
orporatethe sti
k-slipdynami
sthrough oftheblo
kandthesystemagaingoesba
kto thestati
the
olle
tivemotionof anassemblyoflo
ally
onne
ted phase. The dynami
fri
tion is assumed as a velo
ity
f
elementssubje
ttoslowdrivingfor
e(B) Fra
turemod- weakening fun
tion (shown in the Fig.2 ). Presen
e of
els, look at earthquake phenomena as a fra
ture-failure spring for
e and velo
ity weakening fri
tion for
e leads
pro
ess of deformable materials that break under ex- the system to a
omplex dynami
al state. At the initial
ternal loading through slow build-up of stress (C) Self- stage, individual small slip o

urs. But due to
onstant
organised
riti
al (SOC) models,
onsider earthquake as pulling, the springs are gradually stret
hed and attain
a self-driven slow pro
ess and (D) Fra
tal models, give the limit of total fri
tional stability of the blo
ks where
importan
e to fra
tal natureof the
rust-plateinterfa
es the
olle
tiveslipsofalmostalltheblo
kso

ur. Clearly
and address the phenomenon as a two-fra
tal overlap this is a big event and a large amount of elastij
thenergy
problem. isreleased here. The equationofmotion of the blo
k
of the system is
(A) Fri
tion models
mx¨j =kc(xj+1−2xj +xj−1)−kpxj −f(x˙j −V)
t m
In 1967 Burridge and Knopo(cid:27) [7℄ introdu
ed model
where dots denote di(cid:27)erentiation with respe
t to , is
k k
studies in earthquakeresear
h. They proposed a spring- p c
themassofablo
k, and arespring
onstantsofthe
f
blo
k model to mimi
the typi
al sti
k-slip dynami
s of

onne
ting springs and represents the nonlinear velo
-
x
earthquake phenomena, whi
h has been extended later
ity weakening fri
tion for
e. The position
oordinate
V
by Carlson and Langer [8℄. The Burridge-Knopo(cid:27) type
m ismeasured alongthe
hainlength and the velo
ity of
x
model
ontainsalineararrayofblo
ksofmass
oupled
the platform is in the in
reasing dire
tion. The total
t
to ea
h other by identi
al harmoni
springs of strength
k energyofthe system at time
anbe
al
ulatedby solv-
c
and also atta
hed to a (cid:28)xed surfa
e at the top by a
k ing the above equation. A sudden drops in the energy
p
di(cid:27)erent set of identi
al springshaving strength . The
of the system is identi(cid:28)ed as the released energy during
blo
ks are kept on a horizontal platform (rough surfa
e)
V a slip event and the distribution of su
h energies follows
whi
h moves with a uniform velo
ity (Fig.2). Here
power-law:
qualitativelytheblo
ks
anbethoughtofasthepointsVof N(ǫ)∼ǫ−c,c∼1,

onta
tbetweentwoplatesmovingatarelativespeed ,
k k
c p
wherethespring
onstants and representsthelinear N(ǫ)
where isthenumberofslipsreleasingenergygreater
elasti
responseofthe
onta
tregionto
ompressionand ǫ
than or equal to .
shear respe
tively.
kp Predi
tion possibility of major events (slips)
k
c Re
ently Dey et al. [9℄ developed a method to predi
t
m
the major slip event in Burridge-Knopo(cid:27) type spring-
V f
blo
k model. Introdu
ing an additional dissipative for
e
in the spring-blo
k arrangement they identi(cid:28)ed the dis-
R(t)
sipative fun
tional as energy bursts similar to the
v v
a
ousti
emission signals [10, 11℄ observed in experi-
0 R(t)
ments. The distribution of shows power laws if one
re
ords all slip events in
luding the major slips.

Figure 2: TheBurridge-Knopo(cid:27) model(above) and twodi(cid:27)er-
entformsofvelo
ityweakening fri
tionfor
e(below)[9℄.
-4
Starting from unstrained
ondition, when the system is 1)0
t
(
pulled slowly (at
onstant rate), the blo
ks initially re- R
main stu
k to the surfa
e due to fri
tion between sur-
fa
es. A slip of a blo
k o

urs when the
orrespond-
-8
ing spring for
e over
omes the threshold value (maxi- 10
mum stati
fri
tional for
e between that blo
k and the
rough surfa
e). The dynami
s of ea
h blo
k is basi
ally 5240 t 5255 5270
the resultant of two phases: a stati
phase and a dy- R(t) t
Figure 3: The dissipative fun
tional vs. time ( ) in
nami
phase. Duringstati
phasetheblo
kremainsstu
k
Burridge-Knopo(cid:27)model[9℄.
on the rough surfa
e and the elasti
strain
ontinuously
R(t) t
growsup. The stati
phase
omes to a sudden end when The plot of vs. shows a gradual in
rease in
driving for
e on that blo
k attains the threshold value a
tivity (Fig.3) prior to the o

urren
e of a major slip.
3
R(t)
But as is noisy it
an not help mu
h to predi
t the dynami
sandtheavalan
hestatisti
sareexpe
ted to be
majorslipetvent,′rat′herthe
umulativeenergydissipated extremely useful in analysingfra
ture and breakdownin
E (t) ∼ R R(t )dt
ae 0 grows in steps and it seems to di- real materials, in
luding earthquakes [15, 16, 17, 18℄.
verge as a major slip event is approa
hed (Fig.4). From
t
c
su
h divergen
e one
anpredi
t the o

urren
e time ( )
of a major slip through proper extrapolation.
−8 −2
10
R (t) δ
−12 −8 F
10
e 4250 t 4450
a
E
g
o Figure 5: The(cid:28)berbundlemodel.
l
−16
When external load is applied, the surviving fra
tion
0 25 50 t 75 100 of total (cid:28)bers follows a simple re
ursion relation
Eae t Ut+1(σ)=1−P(σ/Ut),
Figure 4: Cumulative energy versus time ( ) plot. Inset
R(t)
showsthetimeseriesof in
ludingtwomajorslipevents[9℄.
Ut = Nt/N0 σ P
where , is the external stress and is
(B) Fra
ture models
umulative probability fun
tion. The re
ursion relation
Ut+1 =Y(Ut)
hastheformofaniterativemap and(cid:28)nally
Ut+1 =Ut
the dynami
s stops at a (cid:28)xed point where .
Earthquake
anbe
onsideredasafra
ture-failurephe-
nomenon through slow build up of stress at the plate- ∆FIf external load is in
reased in steps by equal amount

rustboundarywith themovementofthe plateasexter- , then the entire failure pro
ess
an be formulated
naldrivingfor
e. The deformationpropertiesof the ma- throughthere
ursivedynami
s[19℄mentionedaboveand
terialssittingattheboundaryplay
ru
ialroleonspatial tσhe (cid:28)xed point solution gives the value of
riti
al stress
c
redistribution of the stress around a broken region and above whi
h the bundle
ollapses. It
an be shown
this a
tually de
ides in whi
h dire
tion the
ra
k front that the bundle undergoes a phase transition from par-
should propagate. Fiber bundle model and random fuse tiallybrokenOstateto
ompletelχybrokenstate. Theordτer
model address su
h s
enario in material breakdown and parameter( ), sus
eptibility( )andrelaxationtime( )
sometimestheyaretreatedasmodelsofearthquakesin
e follow robust power laws with universal exponent values
theyprodu
etimeseriesofavalan
heswhi
hfollowpower [20℄.
law distribution similar to Gutenberg-Ri
hter law. In
aseofquasi-stati
loadin
rement,onlytheweakest
(cid:28)ber(amongtheinta
t(cid:28)bers)failsafterloadingandthen
thebundleundergoesloadredistributiontilla(cid:28)xedpoint
Fiber bundle model isrea
hed. The(cid:29)u
tuationsinstrengthdistributionspro-
du
esdi(cid:27)erentsizeofavalan
hesduringtheentirefailure
Fiber bundle (RFB) model
onsists of many (N0) (cid:28)bers pro
essand theavalan
−he5/d2istributionfollowspowerlaw

onne
ted in parallel to ea
h other and
lamped at their (Fig.6) with exponent . Hemmer and Hansen [14℄
two ends and having randomly distributed strengths. has analyti
al proved that this exponent is universal un-
Themodelexhibitsatypi
alrelaxationaldynami
swhen der mild restri
tion on strength distributions.
external load is applied uniformly at the bottom end
10000 100
(Fig.5). In the global load-sharing approximation [12,
13, 14℄, surviving (cid:28)bersFshare equally the external load. 1000 10-2 ∆−5/2
Initially, after theload isapplied on the bundle, (cid:28)bers
having strength less than the applied stress σ = F/N0 ∆ 10-4
100 D(∆)
fail immediately. After this, the total load on the bun-
N 10-6
dleredistributesgloballyasthestressistransferredfrom
10
broken (cid:28)bers to the remaining unbroken ones. This re- 10-8
distribution
auses se
ondary failures whi
h in general
1 10-10

ausesfurtherfailuresandprodu
esan(cid:16)avalan
he(cid:17) whi
h 50000 s15t0e00p0 300000 1 10 1∆00 1000 10000
denotes simultaneous failure of several elements. With
steady in
rease of external load, avalan
hes of di(cid:27)erent
size appear before the global breakdown where the bun- Figure 6: Avalan
he time series in a (cid:28)ber bundle model (left)
dle
ollapses. The s
aling properties of su
h mean-(cid:28)eld andthe
orresponding avalan
hedistribution(right).
4
106
Random fuse model 100
105
D(∆) 10-1 x0 = 0.9 xc 104 ∆−3
I D(1) 10-2 ∆) 103
(
D ∆−2
V 10-3 x0 = 0 102
101
10-4
100
1 ∆ 10 1 ∆10 100
Figure 9: Crossover signature inavalan
he power law. In (cid:28)ber
x0
Figure 7: Therandomfusemodel. bundxlce model: is the starting position of re
ording avalan
5h/e2s
and istheglobalfailurepoint;exponentvalue
hangesfrom
3/2
Thefusemodel[21,22,23℄
onsistsofalatti
einwhi
h to 3(left2) and in random fuse model: exponent value
hanges
from to when the system
omes
loser to breakdown point
ea
h bond is a fuse, i.e., an ohmi
resistoras long as the
(right).
ele
tri

urrentit
arriesisbelowathresholdvalue. Ifthe
threshold its passed, the fuse burns out irreversibly. The Re
ently Kawamura [27℄ observed similar
rossover
threshold opf(eta)
h bond is drawn from an u45n◦
orrelated behavior for the lo
al magnitude distribution of earth-
distribution . The latti
e is pla
ed at with re- quakesinJapan(Fig.10). Thisobservationhasstrength-
gardsto the ele
tri
albusbars (Fig.7) and an in
reasing ened the possibility of using
rossoversignal as a tool of

urrent is passed through it. Numeri
ally, the Kir
hho(cid:27) predi
ting
atastrophi
events.
equations are solved at ea
h node.
4.5
When a bond breaks,
urrent value on the neighbor- entire Japan 1985-1998
ing bonds in
reases and sometimes it triggers se
ondary 4 rmax=30km, mc=4
3.5
y
failures. Finally the system
omes to a state where no nc 3 slope = 0.60
e slope = 0.88

urrent passes through the latti
e; that means there is qu 2.5
e
agr
ardau
aklwinh
ir
ehasseepoafr
autrersentth/evloaltttaig
eeainsetrwieospoifei
netse.rmWeidtih- ent fr 1 .25 0-100 days
ate avalan
hes appear before the (cid:28)nal breakdown. The ev 1 100-200 days
200-300 days
distribution of su
3h avalan
hes follows power law with 0.5 all period
exponent
lose to (Fig.8). 0
2 3 4 5 6 7
10000 106 m
105
1000 ∆−3 Figure 10: Crossoversignatureinthemagnitudedistributionof
∆ D(∆)104 earthquakes withinJapan[27℄.
100 103
102 (C) Self-organised
riti
al models
10
101
In various thermodynami
s systems, there is a (cid:16)
riti-
1 100
0 1000 2000 30t00 4000 5000 6000 1 ∆10 100
al(cid:17) point where the systems be
ome totally
orrelated
and show s
ale free (power law) behaviour. Apart from
the
riti
al point the average mi
ros
opi
quantities of
Figure 8: Avalan
he time series in a fuse model (left) and the thesystemsfollows
alingbehaviour. Generallysu
h
rit-

orresponding avalan
he distribution(right).
i
al states are a
hieved through the (cid:28)ne-tuning of phys-
i
al parameters, su
h as temperature, pressure et
. and
thepowerlawbehavioris
onsideredto beasignatureof
Crossover behavior: signature of imminent breakdown
the (cid:16)
riti
al(cid:17) state of the system. However, it has been
observed that some
omplex systems evolve
olle
tively
Arobust
rossoverbehaviorhasbeenobserved[24,25, to su
h
riti
al state only through mutual intera
tions
26℄ re
ently in the two very di(cid:27)erent models des
ribed and show power law behavior there. These systems do
abovewhere the system graduallyapproa
hesthe global notneedany(cid:28)netuningofphysi
alparameterandthere-
failure through several intermediate failure events. If in- foreare
onsideredas(cid:16)self-organised
riti
al(cid:17) (SOC)sys-
termediate avalan
hes are re
orded, avalan
he distribu- tems. The term SOC was (cid:28)rst introdu
ed by Bak, Tang
tion follows a power law with an exponent that
rosses and Wiesenfeld in 1987.
over from one value to a very di(cid:27)erent value when the The magnitude distribution of earthquake shows
system is
lose to the global failure or breakdown point power-law(Gutenberg-Ri
hterlaw),thereforeitistempt-
(Fig.9). Therefore, this
rossoveris a signature of immi- ing to assume that earthquake happens through a self-
nent breakdown. organiseddynami
s: thebuildupofstressduetote
toni
5
h
c
motionisaselforganisedslowpro
ess;graduallythe
rit- ultimatelysettlesat a
riti
alstatehavingheight and
i
al state is a
hieved where the stress releases in bursts exhibits s
ale free behavior in terms of avalan
he and
of various sizes. Extensive resear
h have been going on life time distributions. But the power law exponents are
to establish relation between earthquake and SOC sys- di(cid:27)erent
ompared to those in BTW model and there-
tems, for whi
h several models have been proposed. So foreMannamodelbelongstoadi(cid:27)erentuniversality
lass
far,sandpilemodelsarethebestexampleofSOCsystem. [30℄.
BTW sandpile model
100000 10000
Critical state
Critical state
The(cid:28)rstattempttostudySOCthroughmodelsystems
10000
was made by Bak, Tang and Wiesenfeld [28℄. This is 1000
∆
a model of sandpile whose natural dynami
s drives it
1000
towards the
riti
al state. The model
an be des
ribed ∆ 100
(oin,ja)two dimensional square latti
eh. At ea
h latti
e site 100
i,j
, there is an integer variable whi
h represents
10
the height of the sand
olumn at that site. A unit of 10
height (one sand grain) is added at a randomly
hosen
site at ea
h time step and the system evolves in dis
rete 1 1
(i,j) 5000 15000 30000 0 500 1000 1500 2000 2500 3000
time. The dynami
s starts as soon as any site has t t
h 4
th
got a height equal to the threshold value ( = ): that
h
i,j
sitetopples,i.e., be
omeszerothere,andtheheights
of the four neighbouring sites in
rease by one unit
hi,j →hi,j −4,hi±1,j →hi±1,j +1,hi,j±1 →hi,j±1+1. Figure 12: Avalan
hetimeseriesinaBTWmodel(left)andin
aMannamodel(right).
1
3 3 3 3 3 3 3 4 3 5 0 4 1 2 0 1 2 1
3 3 2 3 4 2 4 0 3 0 2 3 1 2 4 1 3 0 Pre
ursory a
tivities
2 0 1 2 0 1 2 1 1 3 1 1 3 1 1 3 1 2
On
onstant adding of grains, sandpile models gradu-
Figure 11: BTWmodelonsquarelatti
e[29℄. ally attains
riti
al state from sub-
riti
al states. Now
the question is:
an one predi
t the
riti
al state from
The pro
ess
ontinues till all sites be
ome stable the sub-
riti
al response of the pile? A pulse perturba-
(Fig.11).4 In
ase of toppling at the boundary of the tion method [31, 32℄ gives the answer:
latti
e ( nearest neighbours are not available), grains h
av
At an average height , a (cid:28)xed number of height
falling outside the latti
e are removed and
onsidered h
p
units (pulse of sand grains) is added at any
entral
to be absorbed/
olle
ted at the boundary. Gradually
h h
av c point of the system. Just after this addition, the lo
al
the average height attains a
riti
al value , be-
dynami
s starts and it takes a (cid:28)nite time or iterations
yond whi
h it does not in
rease at all - on an average
to return ba
k to the stable state after several toppling
the additional sand grains are (cid:29)own away from the lat- ∆ →
events. One
anmeasurethe response parameters:
ti
e . Total number of toppling between two su

essive τ → ξ →
number of toppling number of iteration and
stable states, determines the size of an avalan
he and at

orrelation length whi
h is the distan
e of the furthest
the
rinti
a∼lstsa−tΓeavalan
hnesizedistributionfollowspsower hp
s s toppled site from the site where has been dropped.
laws: , where denotes the density of size
Γ Γ≃1.15±0.10
avalan
hes. Theexponent hasthevalue To ensure toppling at the target site, the pulse hight
h = 4 h =
in 2D [29℄. has been
hosen as p for BTW model and p
2
for Manna model. Simulation studies
on
lude [32℄
h
c
that all the resp∆ons∝e p(hara−mehter)−foλlloτw∝pow(her−lawh a)s−µ
Manna model iξs∝ap(phroa−
hhed:)−ν λ ∼= 2c.0 µav∼= 1.,2 ν ∼=c 1.0av ,
c av
∆−1/λ τ−1/µ ; ξ−1/ν, and h. Now
av
Manna proposed the sto
hasti
sand-pile model [29℄ if , and are plotted against , all
by introdu
ing randomness in the dynami
s of sand-pile the
urve follow straight line and they should tou
h the
2 h = h
av c
growth. Here, the
riti
al height is . Therefore at ea
h x axis at . A proper extrapolation estimates
h = 2.13±.01
c
toppling,thetworeje
tedgrains
hoosetheirhostamong the
riti
al hight for BTW model and
h =0.72±.01
c
the four availableneighbours randomlywith equalprob- for Manna model, whi
h agree well with
ability. After
onstantadding ofsand grains,the system dire
t estimates [29℄.
6
Self-a(cid:30)ne asperity model
30
0.5 0.4 V. De Rubies et. al. [34℄ has proposed a new model for
earthquakeswherethes
aleinvarian
eoftheGutenberg-
0.4 20
0.3
Ri
hter law is
laimed to
ome from the fra
tal geome-
20 0.3
−1.0ξ 0.2 −1.0ξ 0.2 try of the fault surfa
es. In this model the sliding fault
ξ ξ 0.1 surfa
es have been represented b|hy(xfr+a
tri)o−nahl(Bxr)o|w∼niraζn
0.1 surfa
es, whose height s
ales as .
10
10 0 0 The roughness of the surfa
es are determined from the
1.8 1.9 2 2.1 2.2 0.55 0.6 0.65 0.7 ζ 0 1
h h value ofthe exponent whi
h liesbetween and . Two
av av
surfa
esaresimulatedby twostatisti
ally self-a(cid:30)nepro-
h1(x) h2(x)
(cid:28)les, say, and , one drifting overotherwith a
01 1.2 1.4 1.6 1.8 2 2.2 00.4 0.5 0.6 0.7
onstant speed v su
h that h1(x,t)=h2(x−vt). An in-
h h
av av tera
tionbetweentwopro(cid:28)lesrepresentsasingle seismi
event and the energy released is assumed to be propor-
ξ tional to the breaking area of the asperities. At the
on-
Figure 13: Sub-
riti
al response: Correlation length ( )versus
hav ta
tpointofthesurfa
esthe`surfa
eroughness'prevents
average hight ( ) in BTW model (left) and in Manna model
(right). Inset shows the plot of inverse ξ versus hav and predi
ts slipping and the stress is a

umulated there. When the
the
riti
alhightthrough extrapolation. stressex
eedsa
ertainthresholdvalue,breaking(earth-
quake) o

urs. TPhi(sEm)o∼deEl p−rβo−d1u
es GutPen(bEe)rdgE-Ri
hter
Therefore, although BTW and Manna models belong
type power law: , where is the
todi(cid:27)erentuniversality
lasseswithrespe
ttotheirprop-
probability that an earthquake releases energy between
E E + dE β
erties at the
riti
al state, both the models show similar
and and the exponent is dire
tly related
ζ β = 1−ζ/(d−1) =
sub-
riti
al response or pre
ursors. A proper extrapola-
to the roughness exponent as :
(D −1)/(d−1) D d
tion method
an estimate the respe
tive
riti
al heights f f
, where and are respe
tively the
of the models quite a

urately.
fra
tal dimension and the embedding dimension of the
surfa
es.
(D) Fra
tal models
Chakrabarti-Stin
h
ombe model
The surfa
es of earth's
rust and te
toni
plate at the
fault zone are not
ompa
t rather fra
tal in nature. In Thisisananalyti
almodel,proposedbyChakrabartiand
fa
t, these surfa
esarethe resultsof the larges
alefra
- Stin
h
ombe[35℄, whi
h in
orporatesthe self-similarna-
tureseparatingthe
rustfromthemovingte
toni
plate. ture of both the
rust and the te
toni
plate. They used
It has been observed that these surfa
es are self sim- self-similar fra
tals to represent fault surfa
es (Fig.15).
ihla(λrxf,rλa
yt)a∼ls λ[3ζ3h℄(xh,ayv)ing the she(lfx-)a(cid:30)ne s
aling property Crust
, where denotes the height of
x ζ
the
ra
k surfa
e at the point and is the roughness
exponent. It has been
laimed re
ently that sin
e the
fra
tured surfa
es have got well-
hara
terized self-a(cid:30)ne (cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)
(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1) (cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1) (cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1) (cid:0)(cid:1)(cid:0)(cid:1) (cid:0)(cid:1)(cid:0)(cid:1)
properties,thedistributionoftheelasti
energiesreleased (cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)
(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1) (cid:0)(cid:1)(cid:0)(cid:1) (cid:0)(cid:1)(cid:0)(cid:1) (cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1) (cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)
during the slips (earthquake events) between two rough (cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)
(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)
(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)
surfa
es (
rust and plate) may follow the overlap distri- (cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)T(cid:0)(cid:0)(cid:1)(cid:1)ec(cid:0)(cid:0)(cid:1)(cid:1)to(cid:0)(cid:0)(cid:1)(cid:1)nic(cid:0)(cid:0)(cid:1)(cid:1) P(cid:0)(cid:0)(cid:1)(cid:1)la(cid:0)(cid:0)(cid:1)(cid:1)te(cid:0)(cid:0)(cid:1)(cid:1)
(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)
bution of two fra
tal surfa
es [34, 35℄ (cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)
(a) (b)
Figure 15: S
hemati
representation of the rough surfa
es of
earth's
rustandmovingte
toni
plate[35℄.
The total
onta
t area between the surfa
es is assumed
to be proportional to the elasti
strain energy that
an
be grown during the sti
king period, as the solid-solid
fri
tion for
e arises from the elasti
strain at the
on-
ta
ts between the asperities. This energy is
onsidered
to be released as one surfa
e slips over the other and
sti
ks again to the next
onta
t between the rough sur-
2+1
Figure14: Atypi
alself-a(cid:30)nefra
turesurfa
ein( )dimen- fa
es. Chakrabarti and Sti
h
ombe have shown analyti-
ζ=0.8
sionwithroughnessexponent .
allythroughrenormalizationgroup
al
ulationsthatfor
7
regular fra
tals (Cantor sets and
arpets) the
onta
t 00000.....22222
9999000000000000
area follows power law distribution:
ρ(s)∼s−γ;γ =1,
00000.....1111155555 ))))
mmmm5555000000000000
((((
PPPP
)))))
’’’’’
whi
h is
omparable to that of Gutenberg-Ri
hter law. mmmmm
((((( 00000.....11111
’’’’’ 00000000 0000....0000000000003333 0000....0000000000006666
PPPPP mmmm
00000.....0000055555
Numeri
al veri(cid:28)
ation
The
laim of Chakrabarti-Sti
h
ombemodel has been 00000
00000 22222 44444 66666 mmmmm’’’’’88888 1111100000 1111122222 1111144444
veri(cid:28)ed by extensive numeri
al simulations [36℄ taking
di(cid:27)erent type of syntheti
fra
tals: regular or non- P′(m′) m′
Figure 17: The plotof against for Cantor sets with
random Cantor sets, random Cantor sets (in one dimen- df =ln2/ln3at(cid:28)nitegenerations: n=10(square),n=11(plus),
n=12 n=13
sion), regular and random gaskets on square latti
e and (
ross) and (as(txar−) abn)−ddthedottedd=lin1esindi
atethe
per
olating
lusters embedded in two dimensions. bPe(smt)(cid:28)t
umrves of the form ; where . Inset shows
vs. plots.
(a) (b)
Predi
tion possibility of large events
n=1
Ifone
antorsetmovesuniformlyoverother,theover-
lapbetweenthetwofra
tals
hangequasi-randomlywith
m(t)
n=2 timeand produ
es atime seriesofoverlaps . Su
h a
time series is shown in Fig.18, for Cantor sets of dimen-
ln2/ln3
sions . While most of the overlaps are small in
n=3 n=3 magnitude, some areretally big, where as the
umulative
Q(t) = R mdt
overlap size o `on average' grows linearly
with time. Is it possible to predi
t a large future over-
lap analysing the time series data? A re
ent study [38℄
ln2/ln3
Figure 16: (a)AregularCantorsetofdimension ;only suggests a method:
three(cid:28)nitegenerationsareshown. (b)Theoverlapoftwoidenti
al
n = 3
(regular) Cantor sets, at , when one slips over other; the 200
overlap sets are indi
ated withintheverti
al lines,where periodi
boundary
onditionhasbeenused.
)150
(t
m
P(m,L) e
The
onta
t areadistributions seem to follow z
a universal s
aling: p si100
a
P(m,L)∼LαP′(m′);m′ =mLα, verl
O
50
L α = 2(d−
where denotes the size of the fra
tal and
d ) d d
f ; f being the mass dimension of the fra
tal and is 00 1000 2000 3000 4000 5000 6000 7000
Time (t)
the embedding dimension. Also the overlap′dist′ribution
P(m) P (m)
m, and mhe′n
e the s
aled distribution , de
ay t m
with or following a power law (Fig.17) for both Figure 18: The time( )series dlna2ta/lonf3overl8apsize ( )for
regular and random Cantor sets and gaskets: regularCantorsets: ofdimension ,at thgeneration.
t
P(m)=m−β;β =d. One
anidentifythe`largeevents'o

urringat time i
m(t) m(t )≥M
i
inthe series,where ,apre-assignednum-
Q(t) =
Rbetir+,1tmhednt
al
ulate the
umulative overlap size
A very re
ent report [37℄ analyti
ally explains the ori- ti , where the su

essive large events o

ur at
ti ti+1 Q(t) 0
gin of su
h asymptoti
power laws in
ase of Cantor set times and . Obviously is reset to value
Q
i
overlap. after every large event. The behavior of with time is
8
shown in Fig.19 for regular
antor sets . It appears that
haoti
-thereforete
hni
ally unpredi
tablewhi
hagrees
Q
i
therearedis
retevaluesuptowhi
h growswithtime. well with the o

urren
e of earthquakes. However, the
method proposed by Dey et. al [9℄ to predi
t a major
18000 1200
e Q(t)i1146000000 (a) e Q(t)i1000 (b) smliapyeovpenentbuypmaowniidteorsi
nogptehoef
fuumtuurlearteivseeaern
ehrginytfhuins
(cid:28)tieolnd,.
ap siz12000 ap siz800 hOanvethbeeeonthdeervhelaonpde,d(cid:28)bbaesri
baullnydtloe mstuoddyelbarnedakfduoswenmpohdee-l
verl10000 verl600 nomena in
omposite materials. As earthquake is a ma-
o8000 o
e e jor breakdown phenomenon, these models
an be used
ativ6000 ativ400 as earthquake models due to their inherent mean-(cid:28)eld
umul4000 umul200 nature. Avalan
hedistributionsshowpowerlawsinboth
C2000 C themodelsanda
rossoverinexponentvalueappears[25℄
00 1000 2000 3000 4000 5000 6000 7000 00 1000 2000 3000 4000 5000 6000 7000 near breakdown point, whi
h
an be treated as a
rite-
Time (t) Time (t)
rionforimminentbreakdown. SOCmodelsassumeaself-
drivenslowdynami
sandreprodu
esGutenberg-Ri
hter
law at the
riti
al point. Although the
riti
al state
an
Figure19: The
umulativeoverlaplsniz2e/vlnar3iation8withtime
be predi
ted a

urately from the sub-
riti
al response of
(forregularCantorsetsofdimension ,at thg0enera-
the systems, the behavior remains unpredi
table at the
tion), wherethe
umulativeoverlapha≥sbMeenreset tMo =va1l2u8e

riti
al state. Fra
tal overlap models are di(cid:27)erent (from
after32every big event (of overlap size where
the other three types) modelling approa
hesin the sense
and respe
tively).
that they fo
us on the fra
tal nature of the fault inter-
fa
es and not on the dynami
s. The
onta
t area distri-
bution follows asymptoti
power law and this suggests a
possibility that fra
tal geometry of the faults might be
M
Therefore, if one (cid:28)xes a magnitude of the overlap the true origin of Gutenberg-Ri
hter law.
m m≥M
sizes , so that overlapswith are
alled `events' There are several di(cid:30)
ulties in studying earthquake
Q
i
(or earthquake), then the
umulative overlapQi ∼=grloQw0s phenomenon,asiti3sa(cid:16)Nbody(cid:17)
omplexproblem. While
linearly with time up to some dis
rete levels , exa
tsolutionofa -bodyproblemneeds rigorousmath-
Q0 M
where is the minimal overlap size, dependent on emati
al
al
ulations and it does not work for a mutu-
l 3
and is an integer. This information
ertainly does not ally intera
ting system having more than bodies, nat-
helptopredi
tafuturelargeeventa

urately,butitgives urally, theoreti
al physi
s
an not help to formulate the
Q
i
some hints by identifying dis
rete levels of where a entire earthquake dynami
s. Again, as the dynami
s of
20
large overlap is likely to happen. earthquake happens at a depth more than km from
earth surfa
e -experimental observation of su
h dynam-
Dis
ussions and
on
luding remarks i
sisalmostimpossible. Moreover,earthquakedynami
s
involves
rust and plates whi
harehighly heterogeneous
at many s
ales from the atomi
s
ale to the s
ale of te
-
TheGutenberg-Ri
hterlawisawellestablishedlawin
toni
plates,with thepresen
eofdislo
ation, impurities,
earthquakeresear
h. ItshouldbementionedthatGuten-
grains, water et
and the s
enario be
omes very mu
h
berg and Ri
hter obtained this law from the statisti
s of

ompli
ated. Inthissituation,modelstudiesareveryim-
earthquake events observed throughout the world. Also,
portantinearthquakeresear
hinthesensethatthey
an
earthquakes within a te
toni
ally a
tive region (Japan,
produ
esyntheti
earthquakeevents,allowustomonitor
California et
) follow similar power law. It is still a
on-
thedynami
sandanalysetheeventstatisti
sto
ompare
troversial issue whether the exponent of the Gutenberg-
with the real earthquake data. Moreover, su
h studies
Ri
hter power law is an universal
onstant or it varies
0.8 1.2
suggest potential methods to predi
t a major event. It
in a narrow range ( to ) depending upon the na-
will be a real breakthrough if any of su
h methods
an
ture of the fault zone. As the motion of the te
toni
help a little to predi
t a future earthquake.
plates is surely an observed fa
t, the sti
k-slip pro
ess
should be a major ingredient of earthquake models. Al-
though Burridge-Knopo(cid:27) type spring-blo
k model su
- A
knowledgment:

essfully
apturesthesti
k-slipdynami
sandreprodu
es
Gutenberg-Ri
hter type power law in the size distribu-
tionofevents,itisfarfromthea

uraterepresentationof We are grateful to Bikas. K. Chakrabarti for impor-
earthquakedynami
s-asitdoesnot
ontainame
hanism tant
ollaborations and useful suggestions. Thanks to
for aftersho
kswhi
h areobserved fa
ts. But this model Resear
hCoun
ilofNorway(NFR) for(cid:28)nan
ialsupport
hassomeimportantfeatures: Thedynami
sisinherently through grant No. 166720/V30.
9
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