This document discusses statistical physics models of seismogenesis and their implications for seismic hazard analysis. It reviews composite physical models that simulate earthquake populations during fault slip, fault growth, and fault nucleation. These models exhibit critical or near-critical behavior consistent with observed scaling laws of earthquakes, including the Gutenberg-Richter law. They also provide a scientific rationale for assumptions of stationarity used in probabilistic seismic hazard analysis. However, questions remain about other assumptions like Poissonian processes and zoning of potentially multifractal distributions. While models show some premonitory seismicity patterns, there is no consensus on reliable intermediate-term earthquake prediction.

1. STATISTICAL PHYSICS, SEiSMOGENESIS,
AND SEISMIC HAZARD
lan Main
Departmentof Geologyand Geophysics
Grant Institute,Universityof Edinburgh
Edinburgh,Scotland
Abstract. The scalingpropertiesof earthquakepopu-
lations show remarkable similarities to those observed at
or near the criticalpoint of other compositesystemsin
statisticalphysics.Thishasled.tothedevelopmentof a
varietyof differentphysicalmodelsof seismogenesisasa
criticalphenomenon,involvinglocallynonlineardynam-
ics,with simplifiedrheologiesexhibitinginstabilityor
avalanche-typebehavior,in a materialcomposedof a
largenumberof discreteelements.In particular,it has
been suggestedthat earthquakesare an e•ample of a
"self-organizedcritical phenomenon"analogousto a
sandpilethatspontaneouslyevolvesto a criticalangleof
reposein responseto the steadysupplyof newgrainsat
the summit.In thisstationarystateof marginalstability
the distributionof avalancheenergiesis a powerlaw,
equivalentto the Gutenberg-Richterfrequency-magni-
tudelaw,andthebehaviorisrelativelyinsensitiveto the
detailsof the dynamics.Here we reviewthe resultsof
someof the compositephysicalmodelsthat havebeen
developedto simulateseismogenesison differentscales
during(1) dynamicslipon a preexistingfault,(2) fault
growth,and(3) faultnucleation.The individualphysical
modelssharesomegenericfeatures,suchasa dynamic
energyflux applied by tectonicloading at a constant
strain rate, stronglocal interactions,and fluctuations
generatedeitherdynamicallyor by fixedmaterialheter-
ogeneity,buttheydiffersignificantlyin the detailsof the
assumeddynamicsand in the methods of numerical
solution. However, all exhibit critical or near-critical
behavior,with behaviorquantitativelyconsistentwith
manyof the observedfractalor multifractalscalinglaws
of brittlefaultingandearthquakes,includingthe Guten-
berg-Richterlaw.Someof the resultsare sensitiveto the
detailsofthedynamicsandhencearenotstrictexamples
of self-organizedcriticality.Nevertheless,the resultsof
thesedifferentphysicalmodelssharesomegenericsta-
tisticalpropertiessimilar to the "universal"behavior
seenin a widevarietyof criticalphenomena,with sig-
nificantimplicationsfor practicalproblemsin probabi-
listicseismichazardevaluation.In particular,thenotion
of self-organizedcriticality(or near-criticality)givesa
scientificrationalefor the a priori assumptionof "sta-
tionarity"usedas a first stepin the predictionof the
future levelof hazard.The Gutenberg-Richterlaw (a
powerlaw in energyor seismicmoment)is foundto
applyonlywithina finitescalerange,bothin modeland
natural seismicity.Accordingly,the frequency-magni-
tude distributioncanbe generalizedto a gammadistri-
butionin energyor seismicmoment(a powerlaw,with
an exponentialtail). This allowsextrapolationsof the
frequency-magnitudedistribution and the maximum
crediblemagnitudeto be constrainedby observedseis-
mic or tectonic moment release rates. The answers to
other questionsraisedare lessclear,for example,the
effectof the a priori assumptionof a Poissonprocessin
asystemwithstronglocalinteractions,andtheimpactof
zoninga potentiallymultifractaldistributionof epicen-
treswith smoothpolygons.The resultsof somemodels
showpremonitorypatternsof seismicitywhichcouldin
principlebe used as mainshockprecursors.However,
there remains no consensus,on both theoretical and
practical grounds,on the possibilityor otherwiseof
reliableintermediate-termearthquakeprediction.
1. INTRODUCTION
The predictionof individualearthquakeshas long
proven to be one of the "holy grails" of geophysics
[Macelwane,1946;Richter,1958;Suzuki,1982;Turcotte,
1991;Johnston,1996]. Of course,we would like to be
able to predict the exactlocation,size, and time of a
future event (i.e., before it happens),within narrow
limitsor errorbounds[WoodandGutenberg,1935],and
at alevelof statisticalsignificanceabovethenullhypoth-
esisof a randomPoissonprocess,but is it possible?
Scholz[1990a]haspointedout that there are practical
problemsin identifyingprecursorsat a usefullevel of
statisticalsignificance.For example,in recentexercises
in identifyingstatisticallysignificantprecursorsaccord-
ingto aprecisesetofcriteria,noclearpositiveexamples
werefound[Wyss,1991;Geller,1996].However,three
exampleswere accepted,on the balanceof evidence,
ontoa "preliminarylistof significantprecursors"[Wyss,
1991].The absenceof clear-cutprecursorsmaybedueto
eitherof tworeasons:(1) reliableearthquakeprecursors
do generallyexiston a timescaleusefulfor predictive
Copyright1996 by the AmericanGeophysicalUnion.
8755-1209/96/96 RG-02808515.00
ß 433 ß
Reviewsof Geophysics,34, 4 / November 1996
pages433-462
Papernumber96RG02808

2. 434 ß Main: STATISTICALPHYSICSAND SEISMIC HAZARD 34, 4 / REVIEWSOF GEOPHYSICS
purposes,but our instrumentationis currentlyinsuffi-
cientto recordthem, or (2) theydo not generallyexist
owingto the underlyingnonlinearphysics[e.g.,Brune,
1979;Kagan, 1994a].Nonlinear dynamicsimpliesex-
treme sensitivityto initial conditions,making accurate
long-termpredictionpotentiallydifficultor impossible,
becausewe can measuresuchconditionson potential
earthquakenucleationsitesonly indirectlyand incom-
pletely.Therefore it is possible,manywould sayeven
likely, that the reliablepredictionof individualearth-
quakes,on a timescaleof practicaluse for ordered
evacuation,mayproveto be an inherentlyunattainable
goal.
In contrast,therehasbeena greatdealof progressin
recentyearsin the analysisof the statisticalphysicsof
seismogenesisasaprocess,notablyin explainingtherole
of dynamiccomplexityin generatingthe richorderand
patternwe seein theoccurrenceof populationsof faults
andearthquakeson a varietyof scales.The aim of this
article is to reviewsomeof thisprogressand to assess,
evenat thisearlystage,theimpactof thisrelativelynew
paradigmonsomepracticalproblemsinseismology.The
emphasisisonprobabilisticseismichazard,whichrelies
on the statisticsof earthquakepopulations,rather than
on the predictionof individualearthquakes.
We beginwith a brief summaryof somebasiccon-
ceptsin statisticalphysicsand nonlineardynamicsand
then summarizesomeof the importantscalingproper-
ties to be addressedin the phenomenologyof earth-
quakes.The sectiononstatisticalphysicsandseismogen-
esisincludesa descriptionof recent resultsobtained
fromcompositephysicalmodelsof (1) earthquakepop-
ulationsin the planeof an existingfault, (2) the devel-
opmentof large-scaleearthquakefaults,and (3) the
nucleationandgrowthof smallerfaults.The influenceof
porefluidsasan extrasourceof complexityis discussed
with primary relevanceto fault nucleationand induced
seismicity.The interestedreaderis referredto the on-
goingdebateof the mechanicalrole of pore fluidsin
larger-scaleprocessesintroducedby Hickman et al.
[1995].In the finalsectionwe discusssomeof the prac-
ticalimplicationsof thestatisticalphysicsof earthquakes
for probabilisticseismichazardanalysis.For reference,a
glossaryof specializedtermsfollowsthe maintext.
1.1. SomeBasicConceptsin StatisticalPhysicsand
NonlinearDynamics
Statisticalphysics,or statisticalmechanics,is the
branch of condensedmatter physicsdealingwith the
physicalpropertiesof macroscopicsystemsconsistingof
a largenumberof elements(classically,atomsor mole-
cules).Thisapproachhasbeenappliedto variousprob-
lemswith analyticalsolutionsin theequilibriumthermo-
dynamics of composite systems,for example the
behaviorof idealgasesandparamagnetism[e.g.,Mandl,
1988].Examplesof analyticalmodelsfor the statistical
mechanicsof earthquakesaregivenbyMain andBurton
[1984,1986],Lomnitz-Adler[1985],Rundle[1988,1989a,
b, 1993],andSornetteand Sornette[1989].
More recently,the adventof powerfulcomputershas
permittedthe studyof a wider varietyof complexsys-
tems,involvinga competitionof local interactionsand
randomfluctuationsin a compositematerialcomposed
of a large,but finite, numberof discreteelements.Ex-
amplesof relevanceto this article include the Ising
model for magnetism[Bruceand Wallace,1989] and
resistornetwork modelsin electricalconduction[Van-
nesteand Sornette,1992].The applicationof this rela-
tivelynewbranchof computationalstatisticalphysicsto
problemsin seismogenesisis oneof the primetopicsof
this review.
Nonlinear dynamicsis the branch of mathematics
dealingwithdynamicaldifferential(or finitedifference)
equations,with nonlineartermsrepresentingthe feed-
backloopsoften seenin natural systems.The mathe-
maticalproblemisdeterministicinthesensethatweare
dealingwith dynamicalequationsthat are known ex-
actly.For linear dynamicssuchdeterminismresultsa
completelypredictablebehavior.However,evenforvery
simple nonlinear systems,with only a few degreesof
freedom (i.e., the number of independentdynamical
variables),we find a surprisingrangeof behavior,from
thecompletelyregularandpredictableto thecompletely
chaoticandunpredictable[e.g.,Tsonis,1992].Suchde-
terministicchaoshasbeenappliedin simplifiedmodels
of phenomenaas diverseas populationdynamicsin
biologyandearthquakedynamicsin geophysics[Huang
and Turcotte,1990a,b; Scholz,1990b;Turcotte,1992].
The idea of deterministicchaosis usuallyappliedto
simplesystemswith onlya few degreesof freedomand
has been used predominantlyas a forward modeling
tool. The main reasonfor this is that it has proven
extremelydifficultto invertfor the underlyingdynamics
from natural datawith a large randomcomponent,for
example,evenfor thebasicproblemof determiningthe
number of degreesof freedom in the system[Tsonis,
1992;Ruelle, 1994]. Similar difficultieshave alsobeen
foundin invertingfor thenumberof degreesof freedom
in model earthquakesequences[McCloskeyand Bean,
1992].More recently,attentionin thisfieldhasconcen-
trated on nonlinearsystemswith manydegreesof free-
dom:the studyof dynamiccomplexity[e.g.,Nicolisand
Prigogine,1989].
One of the propertiesof nonlinearsystemsis their
capacityto exhibitself-organization•thespontaneous
emergenceof configurationalorder or pattern•in a
varietyof naturalsystemsdrivenfar from equilibrium.
For example,Nicolis [1989] explainshow patternsof
thermal convectionrolls and chemicalzoning emerge
spontaneouslyfrom solutionsto the relevantnonlinear
dynamicalequationsfor convectionandreaction-advec-
tion-diffusionprocesses.The applicationof thesespe-
cific examplesto problemsin geophysics(mantle con-
vection) and geochemistry(zoning) are reviewed
respectivelyby Turcotte[1992]andOrtoleva[1994],who

3. 34, 4 /REVIEWS OF GEOPHYSICS Main: STATISTICALPHYSICSAND SEISMIC HAZARD ß 435
demonstratethat similar ordered patterns can evolve
under appropriateconditionsin the Earth. However,
much of the spatialorder in geophysicalsystemsdoes
nottaketheformofpatternswithacharacteristiclength.
For example,at high Rayleighnumber,convectionbe-
comesturbulent,andthe regularpatternof similar-sized
cellsgivesway to a more disorderedpattern involving
eddiesof motion on all scales.As an analogue,Kagan
[1992]refersto seismicityasthe "turbulenceof solids."
The geometricalpropertiesof suchhierarchicaldis-
tributions(turbulence,earthquakepopulations)are of-
ten scale-invariantor self-similar(the wordsare often
usedinterchangeablyin the literature,but seebelow).
Suchsystemshaveno characteristiclengthandhavethe
samebroadappearanceat all magnifications.This fun-
damentalscalingpropertyis the basisfor the definition
of the fractal geometryof natural systems.Mandelbrot
[1983,p. 15] definesa fractalsetstrictlyin termsof its
topology:"A fractal is a set for whichthe Haussdorff-
Besicovitchdimensionstrictlyexceedsthe topological
(Euclidean)dimension."This strictmathematicaldefi-
nition is not the one in common use in geologyand
geophysics.Instead,Turcotte[1992]definesa fractalset
asonein whichthe numberN of objectsor fragmentsof
lengthL scaleasa powerlaw of exponentD (equation
(1) below).Turcotte'sdefinitionincludesall of the frac-
tal setsdefinedby Mandelbrot but alsoincludesscale
invariance for sets in which there is no information on
the location of elements in the set, for instance, the
frequency-lengthdistributionfor seismicsources,or the
scalingof hierarchicalmodelsfor fragmentation.
Some fractal sets exhibit scale-invariantproperties
that are not isotropic.For example,the scalingproper-
tiesof a coastline(horizontalsection)aredifferentfrom
thoseof groundelevation(verticalsection),although
both exhibitscaleinvariancein their ownplane.Isotro-
pic scaleinvarianceis self-similar,meaningthat objects
appearsimilarin all directionsat all scales.In contrast,
self-arlineobjects,whenseenat differentmagnification,
appearself-similaronlywith someanisotropicmagnifi-
cation,for example,a verticalexaggerationof scale(illus-
tratedbyPowerandTullis[1991,Figure2]).Theroughness
of fault and fracturesurfaceshasbeen interpretedwith
methodsassumingbothself-similar[Avilesetal., 1987]and
self-affine[BrownandScholz,1985;ScholzandAviles,1986;
Schmittbuhletal., 1995]geometry.PowerandTullis[1991,
1995]concludedthat,for naturalfault andfracturesur-
faceson a scalerangefrom 10 !xmto 40 m, "approxi-
mate"self-similarityholdswhenthe dataareinterpreted
over the whole bandwidth, but self-aifine behavior is
more appropriatewithin smallerwavelengthbands.
Another complicationis the existenceof multifractal
scaling,whichhastwo relatedaspects.The firstinvolves
spatialvariability,sothat the overallscalingexponentof
the wholesetcanbe describedin termsof a superposi-
tion of smallersubsetswith differentindividualscaling
exponentsoti.Thisphenomenoniswell knownin turbu-
lenceandcanbe quantifiedbyplottingthe "spectrumof
singularities,"f(ot) [e.g.,Feder,1988,Figure6.5], equiv-
alent to the fractal dimensions of the subsets. Multifrac-
tal behaviorcannotbe establishedwithout a sufficiently
broad band of observation.For example,Lovejoyand
Schertzer[1991] establishedthe phenomenonfor cloud
patternsandrainfallovera bandwidthof nine ordersof
magnitude(10-3-106m).
The secondaspectof multifractalsis related to the
scalingof "mass"weightingsof objectsin the set at
differentmagnifications.For example,in a set of one-
dimensionallines,the massweightingin a generalized
"box-counting"algorithm is the product of the linear
densityand the total line length enclosedin a box in a
grid of scalelengthL [Grassberger,1983;Feder, 1988].
FractaldimensionsDqarethencalculatedbycounting
the numberof boxescontaininga line, weightedby the
massraisedto the powerq. Thisprocedureisequivalent
to estimatingthe different momentsq of a probability
distribution.For q : 0 there is no massweighting,so
Do, sometimesknown as the "capacity"dimension,is
equivalenttoMandelbrot's[1983]fractaldimension.The
higher-orderdimensionsinclude the "information" di-
mensionD 1andthe "correlation"dimensionD 2 [Feder,
1988].ThemultifractalspectrumDqcanberelatedmath-
ematicallytof(ot)by a Legendretransform[Halseyetal.,
1986;Geilikmanet al., 1990], so the massclusteringis
directlyrelatedto the spatialvariability(see,for example,
Feder[1988,Figures6.7,6.8,6.9]).The multifractalspec-
trumcanalsobecalculatedbyweightingaccordingto other
criteria.In describingtectonicdeformation,for example,it
is often useful to weight accordingto the cumulative
displacementon faultsrather than a massbasedon the
linear density[e.g.,Davyetal., 1992;Cowieetal., 1995].
The prime use of multifractalsis in quantifyingthe
degreeof concentration,clusteringproperties,and in-
termittency in the fractal set of interest. Multifractal
measureshave been found to describethe geometrical
propertiesof faults,fractures,and earthquakes,includ-
ing seismicityin Pamir, the Caucasus,and California
[Geilikmanetal., 1990];aftershocksof the 1992Erzincan
(Turkey) earthquake[Legrandet al., 1996];patternsof
faults in analoguelaboratorymodels of layered litho-
spheredeformation[Davyet al., 1992];the distribution
of openingdisplacementsin fracturesrecordedinwelllogs
[Belfield,1994];andthe concentrationof plasticdeforma-
tioninshearbandsinrocks[PoliakovandHerrmann,1994].
Anothermeasureof spatialclusteringisthetwo-point
correlationdimension,determinedbythe distributionof
spacingbetween points [Grassbergerand Procaccia,
1983]. This method has been applied to demonstrate
that the clusteringpropertiesof laboratoryhypocenters
[Hirataet al., 1987]and epicentraldata in naturalseis-
micity [Kaganand Knopoff,1980;Hiram, 1989;Hender-
sonet al., 1992]are fractal in nature.Hirata and Imoto
[1991] applieda massweightingto the distributionof
spacingsandshowedthat the two-pointspacingstatistics
in the Kanto region of Japancouldbe describedby a
multifractal set.Eneva [1996] applied this generalized

4. 436 ß Main: STATISTICALPHYSICSAND SEISMIC HAZARD 34, 4 / REVIEWSOF GEOPHYSICS
(a) California faults showingevidenceof activity in latest 15m.y.( Howard& others,
__--•'-.• , / .•.
{b) •osht-e8oyez eorthq.oke fo.lt, Iro. {Tcholenko, l•70}
-- m•/J • /'•1 lOOm
(c) Clay deformation in a Reidel shear experiment ( Tchalenko, 1970 )
I I
IOmm
(d) Detail of shear box experiment ( Tchalenko, 1970)
I Imm I
Figure1.Tracesoffaultpopulationsona rangeof scales,fromlaboratoryexperiments(Figureslc andld)
to plate-rupturingfaultssuchastheSanAndreas(Figurela) (afterMainetal. [1990];originaldatafrom
Tchalenko[1970],Howardetal. [1978],andShawandGartner[1986]).Thefaultpatternsarescale-invariant
in the sensethat without scalebarsit wouldbe dill]cultto tell them apart.
techniqueto mining-inducedseismicityin Canadabut
concludedthat incompletesamplingoften presentin
seismicitydatamayintroducesomespuriousmultifractal
effectsthat do not reflect the underlyingphysicalpro-
cess.Similarly,Onceletal. [1995]concludedthatsystem-
aticchangesin themeasuredcorrelationdimensionasa
functionof time in northernTurkey canbe assignedto
temporalchangesin instrumentaldeploymentrather
than an underlyingphysicalcause.
Theapplicabilityoffractalstogeologyandgeophysics
isthe subjectof a burgeoningliterature,includingtext-
books[e.g.,Turcotte,1992;Korvin,1992;Xie, 1993]as
well as specialvolumesin the form of collectionsof
papers[e.g.,SchertzerandLovejoy,1991;Kruhl,1994;
BartonandLa Pointe,1995].However,theprimeaimof
thisreviewisnotto examinethe detailedapplicabilityof
fractal measuresto fault nucleation,fault growth,and
seismogenesis.Insteadweevaluatetheimplicationsof a
new classof physicalmodelsthat haverecentlybeen
developedto describetheirscalingproperties.Thisnew
paradigminvolvesthestatisticalphysicsof a systemwith
manydegreesoffreedom,withpopulationsofindividual
elementsobeyingthe localnonlinearconstitutiverules
of fracture or friction. It is therefore an exampleof
dynamiccomplexityratherthanlow-dimensionalchaos.
Of particularinterestishowtheextraordinarilyordered
patternofseismogenicdeformationonEarthmighthave
been first developedand then maintained,usingcon-
ceptsdevelopednotin geologyor geophysicsbutin the
statisticalphysicsof criticalpointphenomena.Firstwe
brieflysummarizesomeofthephenomenologyof earth-
quakesandfaultpopulationstowhichthisclassofmodel
isprimarilyaddressed.
1.2. Phenomenologyof EarthquakesandFault
Populations
Anygeneraltheoryforthestatisticalphysicsoffault-
ingandearthquakeshasto explainthefollowingempir-
icalfactsor relationsconcerningtheirscalingproperties:
1. Faultpopulationsarebroadlyscale-invariantover

5. 34, 4 / REVIEWSOF GEOPHYSICS Main' STATISTICALPHYSICSAND SEISMIC HAZARD ß 437
severalordersof magnitude(e.g., Figure 1), and as a
consequencehavepowerlaw lengthdistributions(Fig-
ure 2) of the form
N(L) = AL-ø, (1)
whereN hereisthenumberof faultsof lengthexceeding
L, D is the scalingexponentof the lengthdistribution
(the slopeon a log-logplot), andA is a constant[King,
1983;Shawand Gartner,1986;Heiferand Bevan,1990;
Main et al., 1990; Sornetteand Davy, 1991; Turcotte,
1992].
2. Earthquake frequency-magnitudestatisticsalso
implypowerlaw scalingvia the Gutenberg-Richterlaw
for earthquakerecurrence[GutenbergandRichter,1954]
logN = a - bm, (2)
whereweshalltakeN to beanincrementalfrequencyof
occurrenceof earthquakesof magnitudein the range
rn _+ 8m/2 (usually 8m = 0.1) and a and b are
constants.The negativeslopeb is usuallyreferredto as
the seismic"b value."The Gutenberg-Richterlaw cor-
respondsto a power law distributionof fault length,
seismicmomentor seismicenergy.Figure3 [afterMain
andBurton,1984;Turcotte,1992]showsanexamplefrom
southern California.
3. Earthquakeshavea relativelyconstantandrela-
tivelysmallstressdropoverawiderangeof scalesduring
z
ß California fault system
-i- Dasht-e Bayezfault Iran
• oClaydeformationReidel
- shear experiment
• /•detailsofshearbox
experiment
Slope=-2•
-
ß ,-i-
SIo
I
0 I
log L
Figure2. Frequency-lengthdistribution(log-logplot) of the
faults determinedfrom Figure 1 (after Main et al. [1990];
originaldatafrom Tchalenko[1970],Howardetal. [1978],and
ShawandGartner[1986]).Thedatahavebeenrenormalizedso
thatdataondifferentscalescanbeplottedtogether.The solid
straightlinefor thesetwo-dimensionalmapsisconsistentwith
a powerlawsizedistributionof lengthof exponentD - 1. The
dashedline representsa slopeof D - 2, the usualcasefor
earthquakesourcedimensionswhensampledin threedimen-
sions[Turcotte,1992].
2.0
1.0
• 0.0-
o -1.0.
-3.0
4.0 9.05.0 6.0 7.0 8.0
MagnitudeMs
Figure 3. Discretefrequency-magnitudedistribution(log lin-
earplot)of earthquakesin southernCalifornia[afterMain and
Burton,1984].The solidlinerepresentsa maximumentropyfit
to the dataconstrainedbythe long-termmomentreleaserate.
The resultfitsboththe short-terminstrumentaldata (several
decades,openareas)andthelonger-termpalaeoseismicrecur-
rence(over2 millennia,shadedarea) in thiscase.
dynamicslip(3 MPa comparedwith tectonicand litho-
staticstressesin thecrustof the orderof 10-100 MPa).
Figure4 [afterAbercrombieandLeary,1993;Abercrom-
bie, 1995]showsa compositeof severaldatasetsexhib-
iting thispropertyon a wide rangeof scales,from 10 m
to 100 km. Althoughscaleinvarianceof stressdrop is
1021I' ' ' i•••'e,•'•CajonPassBorehole ,,,•.',,1019 ßVariousCombinedStudies
•' .• .-...ßZ•.1017 . ß ß1,.e- ß • eee ß
•0
10TM
Linesofmnstantstressdrop{bars)
109
5 10 102 10a 10• 10•
SourceDimension(m)
Figure 4. Scalingof seismicmoment as a function of source
size(log-logplot) afterAbercrombie[1995],whoalsogivesthe
original referencesfor the combinedstudiesindicated.The
solidlinesrepresentthe scalingpredictedat differentstress
drops(in bars)froma simpledislocationmodelfor theseismic
source.The high-resolutionboreholedatafromtheCajonPass
borehole(opentriangles)shownostrongsystematicdifference
fromearthquakedataat largerscales(solidcircles)withinthe
scatterof the data;i.e., there is no systematic"fmax"effect.

6. 438 ß Main: STATISTICAL PHYSICSAND SEISMIC HAZARD 34, 4 / REVIEWSOF GEOPHYSICS
100
101
b••"• ß Sec..ondary
[///•/ ßTertiary
104 I I I I
1 2 4 8 16 32
r, mm
Figure5. Cumulativeprobability(Pr) distributionthat hypo-
centersare separatedby a linear distanceR lessthan a char-
acteristicscaler (log-logplot) (after Main [1992a];original
datafromHirataetal. [1987]).The datacomprisehypocentral
locationsof acousticemissionsin rock samplesduring the
threephases(primary,secondary,andtertiary)of a creeptest
in initiallyintactcrystallinerock.The slopeof thesolidstraight
lines is equivalentto the correlationdimension,which de-
creasesasdeformationprogresses(asindicatedbythearrowin
the bottomleft corner).
indicatedby the data followingthe theoreticalstraight
lines on the figures,there remainsa considerablereal
scatterin the data abovethe levelof measuringuncer-
tainty [Abercrombie,1995].
4. Fault and fracturebreaksare rough,with self-
arline or self-similarscaling[Brownand Scholz,1985;
Scholz,1990a;Powerand Tullis, 1991, 1995;Schmittbuhl
et al., 1995].
5. Earthquakepopulationsin diversetectoniczones
exhibitspatialvariability,clustering,and intermittency,
quantitativelyconsistentwith multifractalscaling[Geil-
ikmanetal., 1990;Hirata andImoto, 1991].
6. The distributionof spacingsof hypocentralloca-
tionsof earthquakesand laboratoryacousticemissions
are powerlaw in both space[KaganandKnopoff,1980;
Hirata, 1989;Hendersonetal., 1992]andtime [Turcotte,
1992]. Figure 5 [after Hirata et al., 1987] showsan
examplefrom laboratoryhypocentraldata during the
three stagesof a creepexperiment.
7. Earthquakeshaveaftershocksequencesthat de-
cayat a rateR(t) determinedby Omori'slaw,general-
ized by Utsu[1961]in the form
= o/(t + to)" (3)
wherep isapowerlawindex,andR0andtoareconstants
to be determined from the data.
In additionto thesescalingpropertieswe make the
followingobservation:
8. seismicitycanbe inducedby stressperturbations
smallerthan the stressdrop in individualevents.These
maybe due to previousearthquakesoccurringat rela-
tivelygreatdistances[Hill etal., 1993;Steinet al., 1994;
Kagan, 1994b; Gombergand Davis, 1996; Stark and
Davis, 1996],or to changesin localpore fluid pressure
through man-made activity [Segall,1989]; i.e., earth-
quakescanbe "triggered"[Brune,1979].
In the two-dimensionalmap viewsof Figure 1 the
scalingexponentof the length distributionis D - 1
(Figure2). In contrast,naturalseismicsourcesoccurin
a three-dimensionalvolume, and the inferred distribu-
tion of faultlengthsimpliesD = 2 [King,1983;Turcotte,
1992]. Thus the dimensionalityof the samplingtech-
niqueexertsa stronginfluenceon the measuredfractal
dimension.The multifractal clusteringof faults and
earthquakelocationsis consistentwith two important
generalobservations,namely,(1) the observedconcen-
tration of seismicmoment or energyreleaseon large,
dominantfaultsand(2) thepresenceof someseismicity
everywhere,includingareasremotefromplateboundaries.
2. STATISTICAL PHYSICS AND SEISMOGENESIS
In this section we review some of the results of a
plethoraof physicalmodelsfor seismogenesisasa crit-
ical, or self-organizedcritical,phenomenon.Before do-
ing this it is worthwhileto discusswhat is meantby a
"model"in suchcases.There aretwostagesin reducing
what happensin a naturalcompositematerialto some-
thing that is computationallyor analyticallytractable.
The first is to developa generic"conceptualmodel,"
whichsimplifiesthebehaviorof thephysicalsystemto an
idealizedversionthat preservesthe essentialfeaturesof
theprocessto bemodeledbutiscapableof numericalor
analyticalsolution.This is the stagewhere mostof the
importantassumptionsregardingthe processare made.
The secondstepis to developa specificnumericalor
computationalmodel to solvethe set of equationsthat
describethe conceptualmodel,whichmay involvefur-
ther simplification.We thereforedistinguishin the text
between a conceptualmodel and a numericalmodel,
usingthe term "physicalmodel"or simply"model"to
implythe resultantof both steps.Firstwe examinehow
suchprocedureshave shed light on other systemsin
statisticalphysics.
2.1. Critical Point Phenomena
Oneof thebranchesof statisticalphysicswherepower
lawscalingwith long-rangecorrelationsisproducedisat
or nearthe criticalpoint in order-disorderphasetransi-
tions[Ma, 1976;Bruceand Wallace,1989].For example,
at the criticalpoint in the phasetransitionbetweena
liquid(moreorderedstate)anda gas(moredisordered
state),the densitydifferencebetweentwo phasesvan-
ishes. In the more ordered state the interactions between
molecules(vanderWaal'sforces)dominate,in contrast
to the thermal fluctuations,which dominate in the more
disorderedstate.The criticalpoint for thisphasetran-

7. 34, 4 / REVIEWSOF GEOPHYSICS Main: STATISTICALPHYSICSAND SEISMIC HAZARD ß 439
sitionoccursat a precisecombinationof pressureand
temperaturefor a givenmaterial.
Similarcriticalpoint behavioris seenwhen a mag-
netic material is cooled below its Curie temperature.
Here correlated "domains,"or clustersof large con-
nectedareaswith coherentmagnetisation,emergefrom
the chaosseenat highertemperatures.The Curie tem-
peraturealsorepresentsacriticalpointbetweenordered
behaviordominatedby spin-spininteractionsat lower
temperatures,and disorderedthermal fluctuationsat
highertemperatures.Similarbehaviorisseenin avariety
of criticalpointphenomena,givingriseto the notionof
"universality,"wherediversephysicalsystemssharesim-
ilar (in somecasesexactlythe same)scalingproperties
near their criticalpoints[Ma, 1976;Bruceand Wallace,
1989].It isthereforeinformativeto consideroneexam-
ple in a little detailbeforeprogressingto the physicsof
earthquakepopulations.
In so-called"Ising" modelsfor magnetism,the con-
ceptualmodel takes the form of a regular array of
individualmagnetizedelements,which interact solely
with their neighboringelements.The individualmag-
netic elementsmay be magnetizedeither parallel or
antiparallelto the externalgeneratingfield,generatinga
binary or Booleanfield. The physicalproblem is thus
reducedto a conceptualproblemin the form of a cellu-
lar automaton,whichsimilarin manyrespectsto a finite
elementmodelworkingon a regulargrid and whichis
capableof numericalsolution.The numericalsolution
involvesreplacingthe physicsof the spin exchange
forceswith local numerical "rules" that approximate
their behavior, for example,neglectingthe effect of
weaker long-rangeinteractions.Wolfram [1984, 1986]
describesseveralexamplesof computationalcellularau-
tomata,designedto model the behaviorof a varietyof
complexsystems.A commonfeatureof cellularautom-
ata,with differinglocalrules,is that theyproducelong-
range order that emergesspontaneouslythrough dy-
namicfeedbackand self-organization.
The resultsof the Isingmodel showthe spontaneous
developmentof organizedmagneticdomainsat the Cu-
rie temperature,with a powerlawsizedistribution,and
fractal scalingof the rough domainwalls [Bruceand
Wallace, 1989, Figure 8.2]. Above the critical point,
thermal fluctuations dominate, so the behavior is ran-
domor chaotic,andno orderedpatternisgenerated.As
the temperatureis reducedin the model,the local in-
teractionsbeginto take effect,andorganizedclustersof
alignedspinsbegin to emergein the results.At the
criticalpoint To,domainsof positiveandnegativemag-
netismpreciselycancel,andno additionalexternalmag-
neticfieldisgenerated.Belowthecriticalpoint(T < T0
increasinglyregular magneticdomainsare generated
parallel to the externalfield. As a consequence,the
"orderparameter"(definedasthe excessof areaswith
positiveovernegativemagnetization)increasesas the
material cools,resultingin a systematicincreasein the
externalfield.In thisexamplethe criticalpoint is bal-
ancedexactlyat the transitionfrom orderto disorderin
the competitionbetweenlocalinteractionsandthermal
fluctuations, a condition sometimes referred to as "on
theedgeof chaos."Nevertheless,thecriticalpointisalso
associatedwithlong-rangecorrelations[Bruceand Wal-
lace,1989,Figure8.3] andthe disappearanceof a char-
acteristiclengthscale.
2.2. Self-OrganizedCriticality
In the aboveexamplesthe criticalpointisreachedby
precisetuningof externalvariables(temperatureand/or
pressure).However,somecomplexphysicalsystemscan
organizethemselvesspontaneouslyto the criticalpoint,
giving rise to the notion of a state of self-organized
criticality.Thusthe systemorganizesitself,notjust into
apatternedstructure,butto theprecisestructureseenat
the critical point, and then remainsthere apart from
dynamicfluctuations.A stateof self-organizedcriticality
is thereforein somesensean attractorin the dynamics.
Baketal. [1988a,b] illustratedthe conceptbyapplyinga
cellular automaton model to the size distribution of
avalanchesin a sandpilemaintainedat a criticalangleof
reposebythe steadysupplyof newgrainsto the summit.
The modelsystemorganizesitselfspontaneouslyto the
critical point (the critical angle of repose) and then
remainsthere apartfrom dynamicfluctuations(the av-
alanches).Althoughdrivenfar from equilibriumby the
constantflux of sandgrains,the sandpileremainsin a
stationarystaterepresentedby a relativelyconstantan-
gleof reposeandpowerlaw scalingof the sizedistribu-
tion of avalanches[Bak et al., 1988b].However, minor
eventsin the dynamicscanstarta chainreactionthat can
affect any number of elementsin the system:i.e., the
systemexhibits"triggering"(observation8 in section
1.2). After an initial transient,the systemremainsin a
stationarybut metastablestate,balancedpreciselybe-
tween a more ordered and a more disordered state, "at
the edgeof chaos"[Baketal., 1988a,b].A keyproperty
of thisstateisthatthebehaviorof the systemisrelatively
insensitiveto the detailsof the dynamics,sothat thereis
no need to tune the systemexternally(to "twiddlethe
knobs" [Bak et al., 1988b]) as in other examplesof
criticalphenomena.Individualavalanchesmaybe hard
to predict becauseof the nonlinear physics,but the
averageproperties(angleof repose,sizedistributionof
avalanches,etc.) remainrelativelyconstantin time.
A compositesystemin a stateof self-organizedcriti-
cality cannot reach a stable equilibrium but instead
evolvesdynamicallyfrom one metastablestate to an-
other. In the sandpileanalogythe criticalangleof re-
pose,0•, representsa stationary,but potentiallyunsta-
ble, stateachievedin the dynamiccompetitionbetween
thesteadysupplyofpotentialenergy(byadditionof new
grains)and the intermittentenergydemand(from the
sand avalanches).Over long timescalesthe angle of
reposeaveragesout at a constantvalueto maintainthe
critical slope,but short-termdynamicfluctuationsare
fundamentalto maintainingthislong-termmetastability.

8. 440 •, Main: STATISTICALPHYSICSAND SEISMIC HAZARD 34, 4 / REVIEWSOF GEOPHYSICS
yv
•o•.•"'
Kc KC
Figure 6. Schematicdiagram of the compositespring-block
sliderconceptualmodel of Burridgeand Knopoff[1967],ex-
tendedto twodimensionsbyOtsuka[1972].l/is thevelocityof
the upperrigidplate,andKi• isthe stiffnessof the leaf springs
connectingthe plate to the compositefault elements.Kc isthe
stiffnessof thespringsconnectingtheneighboringfaultelements.
That is,weexpectsomestatisticalfluctuationinherentto
the processaswell as in any uncertaintyof measure-
ment. For a systemto remain near-critical, the ava-
lanchesmustreducethe slopeby only a smallamount
A0c,sothe notion dependson the presenceof a micro-
scopicmechanismthatresultsinthismacroscopicproperty.
The paradigm of self-organizedcriticalityhas also
beenappliedto a rangeof naturalcomplexphenomena,
includingecosystemsandstockmarketsaswell assand-
piles, avalanches,and earthquakes[Bak and Chen,
1991]. We now considerits applicationspecificallyto
earthquakes.
2.2.1. Earthquakesas critical (or self-organized
critical) phenomena. Earthquakepopulationshave
somepropertiesthat are strikinglysimilar to thoseof
self-organizedcriticalphenomena,includingavalanche
dynamics,powerlawscalingin the frequency-sizedistri-
bution (observation2 in section1.2), and the parallel
propertyof havingstressdropsthat are smallin com-
parisonwith the regionaltectonicstressfield (observa-
tion 3 in section1.2, analogousto small A0c). Self-
organizedcriticalityin the Earth is alsoconsistentwith
the observationof seismicityinducedor triggeredby
relativelysmallstressperturbations(observation8 in
section1.2;seealsothe discussionbyScholz[1990a]).
The notion that earthquakesmight alsobe an exam-
ple of self-organizedcriticalitywassuggestedmore or
less simultaneouslyby severalgroupsusing different
methodsas illustrations.Bak and Tang[1989]andIto
and Matsuzaki [1990] applied the genericconceptual
model of Burridgeand Knopoff[1967],extendedto two
dimensionsbyOtsuka[1972]asisillustratedin Figure6.
In the Burridge-Knopoffmodelthe lithosphereisrepre-
sentedby a compositearrangementof discretespring-
block slider elements,sandwichedbetweentwo rigid
plates that are driven past each other at a constant
velocityV. The twoplatesareconnectedbyleaf springs
to the discrete fault elements or blocks with constitutive
lawsthat representthe elastic-brittlefrictionalproper-
ties of a preexistingtwo-dimensionalfault. As in the
Isingmodelthe constitutiverulesare oftengreatlysim-
plifiedin thenumericalimplementationin orderto allow
a greater number of elementsin the calculationand
hence a greater bandwidthof observation.This is an
importantcompromiseif wewishto examinethe scaling
propertiesof the system.The blocksare alsoconnected
to oneanotherby elasticsprings(Figure6), sothat the
stressdrop causedby the failure of one element is
immediatelyredistributedto itsfournearestneighbors.
As in the Isingmodel,the weaker long-rangerinterac-
tionsare ignored.If we are near the criticalpoint, the
failureof oneelementcantriggeravalanchesof all sizes
without a significantchangein the strainat the bound-
ariesrepresentedbythetwoplates.Thisisanexampleof
a mechanismof unstablepositivefeedbackin theunder-
lyingdynamics,with nonlinearitybeingintroducedbe-
causeof stronglocal interactionsand the threshold
natureof an elastic-brittlerheology.The sizeof a single
avalanche event in the two-dimensionalBurridge-
Knopoffmodelisthe total areaof a clusterof connected
failed elements. This can be used to reconstruct the
frequency-magnituderelation,usingsomesimplescaling
rulesfrom a dislocationtheoryof the earthquakesource
and the knownpropertiesof seismicrecordinginstru-
ments[KanamoriandAnderson,1975].Despite its ac-
knowledgedsimplificationof the constitutiverules,the
numerical implementationof the Burridge-Knopoff
model by Bak and Tang [1989] andIto and Matsuzaki
[1990],usingthecellularautomatonapproach,neverthe-
lessspontaneouslyevolvesto a stateof stationarycritical
pointbehavior,with a frequency-magnitudedistribution
consistentwith the Gutenberg-Richterrelation(obser-
vation2 in section1.2).
At the sametime, CarlsonandLanger[1989]applied
a one-dimensionalversion of the Burridge-Knopoff
model but includedmore realisticdynamics,including
inertial terms and a simplifiedvelocity-weakeningfric-
tionalconstitutivelaw,ratherthanthesimplifiedcellular
automaton rules of Bak and Tang [1989]. The one-
dimensionalversioncorrectlyaccountsfor stressinten-
sity factors,an acknowledgeddrawbackof the two-di-
mensionalversionof Figure 6 (L. Knopoff,personal
communication,1992). Carlson and Langer's [1989]
model systemhad no initial mechanicalheterogeneity
but was neverthelesscapableof generatingand main-
taining its own dynamicheterogeneityby amplifying
smallperturbationsinevitablypresentin the computer-
generatedstarting'model.The resultscontainlargefluc-
tuationsin earthquakemagnitude,whichgrow,andthen
persist,becausethe systemisfirstattractedto, andthen
remainsin, a stationarystateof marginalstability.
Duringthesameperiod,SornetteandSornette[1989]
alsosuggestedthat the notionof self-organizedcritical-
ity could be applied to processesunderlyingearth-
quakes,buttheyuseda differentapproachto illustrating
its utility. Rather than derivingthe propertyof station-
arity or the Gutenberg-Richterlaw from a large com-

9. 34, 4 / REVIEWSOF GEOPHYSICS Main: STATISTICAL PHYSICS AND SEISMIC HAZARD ß 441
TABLE 1. Hallmarks of Self-Organized Criticality
Feature Sandpiles Earthquakes
Boundarycondition
Critical parameter
Dynamic fluctuation
Power law distribution
constant"grain" rate
reposeangle 0c
smallfluctuationsin angle
,50 << Oc
avalahchevolumeorenergy
constant strain rate
tectonicstresscrc
smallstressdrop
sourcelength,seismicmoment,or
energy(Gutenberg-Richterlaw)
puter-generatedmodelof fluctuationsand interactions,
theyassumedtheirexistenceaprioriasaconsequenceof
self-organizedcriticality.Using the assumptionof sta-
tionarity, they introducedthe constraintof a constant
energyfluxandhencederiveda probabilisticmeanfield
(analytical)solutionfor the returnperiodsof eventsof
differentenergiesin the form of a power law. (A dy-
namicenergyflux is crucialto the developmentof self-
organizingsystems,in contrastto the more staticbehav-
ior of equilibriumthermodynamics[Nicolis,1989]).
Somekey assumptionsare presentin all of the com-
putationalmodelsdescribedabove.For example,elastic
strain energyis assumedto be suppliedto the model
fault from a remote boundaryat a relativelyconstant
strain rate, with the intermittent release of this stored
energyresultingin earthquakes.The modelfault iscom-
posedof a regulargrid of individualdiscreteelements
that interactsolelythroughtheir nearestneighbors(al-
thoughlong-rangeinteractionspropagatedynamicallyin
the one-dimensionalmodel of Carlson and Langer
[1989]).The averagerate of energyinputandreleaseis
then maintainedrelativelyconstantin a stationarystate
of marginalstability,analogousto the sandpilemodel,
with powerlaw scalingin the sizedistributionof earth-
quakeenergies.Thesekeypropertiescanthenbe used,
in simpler analyticmodels,to derive some statistical
propertiesof earthquakerecurrence[e.g.,Sornetteand
Sornette,1989].Table 1 comparessomeof the essential
featuresof the earthquakeand sandpilemodels.
The different numerical variants of the Burridge-
Knopoffconceptualmodeldescribedaboveexhibitmany
of the empirical scalingrelationsobservedin natural
earthquakepopulations,but not all classesshow evi-
denceof trueself-organizedcriticality[RundleandKlein,
1993].In particular,the strengthof the permanentma-
terial heterogeneityand the plate-drivingvelocityboth
havesignificanteffectson the resultingfrequency-mag-
nitude distribution in different realizations of the two-
dimensionalversion of the Burridge-Knopoffmodel
[RundleandKlein, 1993].Permanentheterogeneity,or
"quencheddisorder" due to preexistingstructure,is
presentin all geologicalsystems,so it is reasonableto
examineits influenceon the dynamics.It is alsoreason-
able to examinethe effectof differentplate-drivingve-
locities,becausespatial variations in driving velocity
over 1.5to 2 ordersof magnitudeare a featureof plate
tectonicson Earth [e.g., DeMets, 1995]. The results
achievedby alteringthesevariablesare shownin Figure
7 [afterRundleandKlein, 1993].All exhibitfractalscal-
ing over a finite range,but only someshowevidenceof
true self-organizedcriticality.For example,when the
materialheterogeneityisstrong,the largestearthquakes
never cross the entire area of the model fault surface,
andtheprobabilityof occurrenceof the largesteventsis
reduced in comparisonwith an extrapolationof the
Gutenberg-Richtertrend (Figure7a). Under thesecon-
ditionsthe behavioris alsosensitiveto the plate-driving
velocity(Figure7a).Suchsystemsarenotstrictexamples
of self-organizedcriticalitybecausetheyare sensitiveto
externalconditions.We shallrefer to thisgenericclassof
models as "subcritical,"in the sensethat the largest
avalanches do not cross the entire area of the model
fault.An equivalentstatementisthatthe systemremains
belowthepercolationthreshold,definedasthepoint at
which the 'largestconnectedclusterof failed elements
just spansthe modelgrid [StaufferandAharony,1994].
For weaker heterogeneity,with intermediatedriving
velocities,the behavior is both preciselycritical and
insensitiveto the precisevalue of V (Figure 7b). This
representsa true state of self-organizedcriticality, in
which the systemcan generate and maintain its own
dynamicalheterogeneity,and whosescalingproperties
are relativelyinsensitiveto the detailsof the dynamics.
For evenlargerdrivingvelocitiesa "supercritical"state
mayoccurwith an elevatedprobabilityof occurrenceof
large"characteristic"events(Figure 7c), similarto be-
haviorseenabovethepercolationthreshold[Staufferand
Aharony,1994,pp. 72-73]. A characteristicpeak in the
sizedistributionat largelengthscalesisreminiscentof a
first-orderphasetransition[Lomnitz-Adleret al., 1992;
CevaandPerazzo,1993].
The range of behaviorillustratedin Figure 7 is not
specificto the detailsof the cellularautomatonusedby
Rundleand Klein [1993]. For exampleLomnitz-Adler
[1993] examined40 differentclassesof computational
model for the statisticalphysicsof earthquakepopula-
tions,basedon differentcombinationsof the following
assumptions:total stressdrop (crackmodel) or partial
stressdrop (frictional slip); homogeneousor random
loading;thepresenceor absenceof asperities;a shortor
long characteristictime for fracturing;and whether or
notthe dynamicstrainenergy(or stress)isconservedon
the fault plane.All of thesephysicalmodelswere found
togenerateresultswithinoneofthethreegenerictypes

10. 442 ß Main: STATISTICALPHYSICSAND SEISMIC HAZARD 34, 4 / REVIEWSOF GEOPHYSICS
(a) (b) (c)
).10 -2
Z
:::310-4
o
n-10-6
:• 0-8
V=8
V=
%• i t
I I i it • • t
10-1 10o 101 102 103 104
CLUSTER SIZE
V= 0
10-1 100 101 102 103 104
CLUSTER SIZE
% I
i i i % I I
10-1 10 0 101 102 103 104
CLUSTER SIZE
Figure7. Frequencydistributionof eventclustersizefromtheresultsof applyingthespring-blockslidermodel
ofFigure6,rununderdifferentinitialandboundaryconditions(log-logplot)(redraftedafterRundleandKlein
[1993]),The clustersizeS measuresthe connectedareaof failed elements,correspondingto an individual
modelearthquake.There is somestatisticalscatter,whichresultsin a broadeningof the linesencompassing
the datafor the rarer, largerevents.The numericalmodelwasrun with (a) strongpermanentheterogeneity
with differentplate velocityV (arbitraryunits),(b) weakheterogeneityand relativelylow or intermediate
velocityV, and(c)weakheterogeneityandhighV. TheinterestedreaderisreferredtoRundleandKlein[1993]
for more precisedetailsof the numericalmodel.All of the resultsexhibitcriticalbehavior,with power-law
scaling(straightline slope)overa finite scalerange.In Figure7b the scalingpropertiesof the systemare
powerlawrightup to thelargestevents,with exponentsthat arerelativelyinsensitiveto the degreeof external
forcing.Only Figure7b correspondsto a stateof self-organizedcriticality.The behaviorin Figure7a maybe
termed"subcritical,"and that in Figure7c, "supercritical."
shownin Figure 7, but only a few classesproduced
preciselycritical behavior (a Gutenberg-Richterlaw
over all magnitudes(e.g.,Figure 7b)). The samethree
generictypesseenin Figure 7 can alsobe seenin the
frequency-energyresultsof computationalmodelsfor
thedynamicsof sandpiles[CevaandPerazzo,1993;Carl-
son et al., 1993]. Lomnitz-Adler[1993] interpretsthis
genericbehaviorin termsof differentpositionsnearthe
critical point, Ceva and Perazzo[1993] interpret it in
terms of different positionsaround the percolation
threshold,andCarlsonetal. [1993]interpretit in terms
of different solutionsto the diffusionequation. This
similarityof behaviorin the resultsof suchdiversephys-
ical modelsis anotherexampleof the principleof uni-
versalityin suchsystems.
The resultsof Figure7 showthat the modelsmustbe
"tuned" after all (at leastto someextent) to produce
self-organizedcriticalityin the strictsense.That is,there
isa finiterangeof startingstates(i.e., dynamicvariables
andinitialheterogeneity)in numericalmodelsthatthen
evolvespontaneouslyto a true state of self-organized
criticality.Other combinationsevolveto a stationary,
near-critical state but are more sensitive to the details of
the dynamics.
One of the primevariablesthat isusedin practiceto
tune the behavior of these numerical models is the
degreeof conservationof energyfollowingthe failureof
an element.For example,in order to producea b value
near 1 (observation2 in section1.2), mostcellularau-
tomatonmodelsintroducean arbitrarydegreeof energy
dissipationto the localconstitutiverules.This is physi-
callyrealisticbecauseenergyis lost to effectssuchas
seismicradiation (radiation damping), deformation
around the fault zone, or the movement of fluids, in
additionto thatlostin thegenerationof heatonthefault
surfacedueto frictionin thecaseof a partialstressdrop.
If all of the storedelasticenergywere conserved,then
the failureof a singleelementresultingin a stressdrop
Air would result in the loading of the four nearest
neighborsin the cellularautomatonby an amountAir/4
[e.g.,Bak and Tang,1989].If the elasticenergyis not
conserved,the simplifiedtechniqueusedto represent
such damping involvesloading the neighborsby an
amount otAtr/4,where ot < 1. Models with ot = 1 are
termed "conservative,"and thosewith ot< 1 are termed
"nonconservative."Olami et al. [1992] examinedthe
statisticalpropertiesof suchnumericalmodelsand ob-
serveda systematicnegativecorrelationbetweenotand
the seismicb value.A b valuenear 1 isproducedwhen
ot = 0.8. In other words, the b value is sensitiveto the
inputparameters,sothat the resultsare sensitiveto the
detailsof the dynamics,in contrastto the strictrequire-
mentsof self-organizedcriticality[Kadanoffetal., 1989;
Socolaretal., 1993].
The resultsof applyingthe variousnumericalimple-
mentationsof the Burridge-Knopoffconceptualmodel
comparewell with many other aspectsof earthquake
phenomenology.Examplesincludethe spatiotemporal
form of patterns of seismicityprecedinglarge earth-
quakes[Shawet al., 1992],the phenomenonof seismic
quiescence[Brownet al., 1991],Omori'slaw (in a gen-
eralizedform) for aftershocksandforeshocks(observa-

11. 34, 4 / REVIEWSOF GEOPHYSICS Main: STATISTICAL PHYSICS AND SEISMIC HAZARD ß 443
tion 7 in section1.2 [Shaw, 1993a]), the power law
distributionof intereventtimes(observation6 in section
1.2 [Matsuzakiand Takayasu,1991], and the form of
earthquakesourcespectra [Shaw, 1993b].McCloskey
[1993] and McCloskeyet al. [1993] showedthat the
observedsystematicchangesin theb valueat highmag-
nitudecouldbe explainedby the effectof fault segmen-
tationonseismicity.Wang[1995]showed,byalteringthe
relativestiffnessof the connectingspringsand the leaf
springsin Figure6, thattheseismicb valueissensitiveto
the seismiccouplingcoefficientin a way qualitatively
consistentwiththe resultsof Olamietal. [1992].Shawet
al. [1992]found someevidencefor systematicprecur-
sors,in the form of an accelerated seismicevent rate, but
with smaller or nonexistent fluctuations in the seismic b
value, conclusionssimilar to those from a statistical
model based on the constitutive rules of fracture me-
chanics[YamashitaandKnopoff,1987,1989].Numerical
modelsthat includerate-dependent,velocity-weakening
friction introducea characteristiclength scaleto the
problemat highermagnitude[e.g.,Shaw,1993b,equa-
tion (2)], resultingin an elevatedprobabilityof occur-
rence of the largest magnitudescomparedwith the
Gutenberg-Richtertrend [seeShaw,1993b,Figure 3].
We havetermedsuchbehavior"supercritical,"although
in Shaw's [1993b] study, the dynamicsproduces a
smoothbumpat largemagnitudesratherthanthe sharp
truncationof Figure 7c.
In summary,there is generalagreementamongthe
seismologicalcommunityworking on these problems
that the genericclassof compositeearthquakemodels
basedon Figure6 that are drivento, andmaintainedin,
a stateof at leastnear-criticalityare consistent,to first
order at least,with mostof the phenomenologyof nat-
ural seismicity,includingthe Gutenberg-Richterlaw,
Omori'slaw,powerlaw scalingof intereventtimes,the
relativelysmallstressdrop, and earthquaketriggering.
Much remainsto be doneto comparethe resultsof the
numericalmodelswith natural seismicity,notablytheir
multifractalcharacteristicsandtheir spatialandtempo-
ral correlation.On the other hand,there is an ongoing
debate as to whether self-organizedcriticalityin the
strictsenseappliesat all, not onlyto earthquakepopu-
lations[Lomnitz-Adler,1993;Rundleand Klein, 1993],
but alsoto sandpilesand other natural dissipativesys-
tems [Kadanoffet al., 1989; Ceva and Perazzo,1993;
Socolar et al., 1993; Carlson et al., 1993; Frette et al.,
1996].
The degree of realisticcomplexityrequired in the
modelsisalsoa subjectof debate.Someadvocatekeep-
ing the model assimpleaspossiblein order to deter-
mine, and gain insightinto, the fundamentalstatistical
properties.Othersarguefor the inclusionof more real-
isticphysics(e.g.,rate- andstate-dependentfriction,the
effectof fluids,layeredlithosphererheology),whilestill
keepingthe modelassimpleasnecessaryto explainthe
observations.For examplethere is generalagreement
that the constitutiverules required in the numerical
modelsto reproducethe Gutenberg-Richterlaw over a
finite scalerange are simplerthan we know to be the
casein nature.However,the samegenericbehavior,in
the frequency-magnitudedistribution,emergesin the
results when the numerical models are made more real-
istic. If the resultsof even the simplestmodelsade-
quatelymatchthefirst-orderfeaturesof seismicityin this
way,thenwe canbe confidentthat theycapturein some
waythe generalpropertiesof earthquakegeneration.
2.2.2. Statisticalphysicsof faulting. The models
describedabove are useful for understandingearth-
quake populationsresultingfrom slip on an existing
modelfault. Thisbegsthe questionof howmajorfaults
evolvein the first place.In order to addressthis ques-
tion, variousaspectsof the statisticalphysicsof the
evolutionof organizedcrustal-scalefaulting, involving
the localizationof deformation,haverecentlybeen in-
vestigatedby a rangeof theoretical,computationaland
laboratoryanaloguetechniques[A. Sometteet al. 1990,
1993;D. Sometteetal., 1994a,b;SometteandDavy,1991;
Somette and Virieux, 1992; Cowie et al., 1993, 1995;
Miltenbergeret al., 1993].In this sectionwe summarize
someof the importantresultsfor crustal-scalefaulting.
The geometryof a generic,two-dimensional,concep-
tual model for the processof the localizationof defor-
mationduringlarge-scalefault growthis shownin Fig-
ure 8 [afterCowieetal., 1993].This modelisbasedon a
resistornetwork analogue,composedof a mosaicof
large (10 km) two-dimensionalcrustalblocks,interact-
ingthroughboth short-andlong-rangeelasticforcesin
responseto a constantdriving velocity at the model
boundary.A full elasticsolutionof the resultingnumer-
ical problemis determinedby a linear matrix inversion
techniqueon the failure of individual elements.The
failed elementsin the resultsrepresenta view of the
structurein a sectionthat cutsthe fault, for example,a
map view of a set of normal faults.This contrastswith
the"in-plane"viewshowninFigure6,i.e.,parallelto the
fault surface.
Althoughshort-rangeinteractionsdominatethe elas-
tic stresses,the inclusionof long-rangeelasticinterac-
tionsin the numericalmodel is importaritfor two rea-
sons.The firstisthatit isconsistentwith someempirical
observation,notablythe long-rangetriggeringof seis-
micity(observation8 in section1.2).The secondisthat
it allows more subtle effects to be examined more accu-
rately. This contrastswith the purely local "nearest
neighbor"interactionscommonlyused in the cellular
automationapproach(Figure6).
Initially,the calculatedstrainisdistributedbyslipon
smallfaults,moreor lessevenlyacrossthe modelspace,
resultingin a patternof distributeddamage(Figure9a).
Thesefaultsare assigneda permanentstrengthhetero-
geneitythat isfeatureless(random,uncorrelatednoise)
sothat the developmentof long-rangestructuresis not
preconditionedby the startingmodel.As the displace-
mentonthemovingboundaryisincreased,theinterplay
betweenthe long-rangeelasticity,thresholddynamics,

12. 444 ß Main' STATISTICAL PHYSICSAND SEISMIC HAZARD 34, 4 / REVIEWSOF GEOPHYSICS
(a) accumulated
displacement
(b)
'•,'-•.•.•••:•:...,•-•-.•":-•U'•-..•Iiiiß external
permanentstrain/ driving
displace•nent stress
Figure 8. Sketchof the geometryof a scalarresistor-network
conceptualmodelfor crustal-scalefaulting[afterCowieet al.,
1993].(a) A compositelithosphere(blocksof dimension10
km, indicatedby the regularmosaicof situareelements)is
strainedat a constantrate by a movingboundary.(The scalar
natureof the conceptualmodeldoesnot distinguishbetween
extensionaland shear strain at the boundary.) Permanent
displacementisaccumulatedasshownon theverticalgraphat
differenttimestoandt•. The presenceof a faultisindicatedby
a discontinuityin displacement(ontheverticalgraph)andthe
thicksolidlines(onthemapview).(b) Rheologicalmodelused
to representtheinteractionbetweenelements.The elementis
connectedto all of the otherelementsbyan elasticspringthat
holdselasticstrain,andpermanentdisplacementof anelement
isrepresentedbya ratchetthatmovesbyonetoothper event.
and permanent material heterogeneityresultsin the
spontaneousformationof fractalfault structuresby re-
peatedearthquakes,well describedby the Gutenberg-
Richterlaw [Cowieetal., 1993;Miltenbergeretal., 1993;
Sornetteet al., 1994b].
As timeprogresses,deformationbecomeincreasingly
concentratedon very large, dominant throughgoing
faults(Figure9b).The majorfaultsgrowalong"optimal
paths"of leastresistancethroughthe material, analo-
gousto a randomdirectedpolymerproblemin physics
[Miltenbergeret al., 1993].The concentrationof defor-
mation on the largestfaults occurswhile the stressis
simultaneouslyhighelsewherein themodelgrid,sothat
the systemremainscriticaleverywherebut is drivento
failurelessoftenin betweenthe majorfaultsbecauseof
the screeningeffectof thelargefaultsthatdominatethe
deformation[Sornetteet al., 1994b].By "screening"we
meanthat areasbetweenmajorfaultshavinglowerper-
manent strain,but retain a high (near critical) stress.
The near-critical stress,even in areas of low deforma-
tion, is a plausibleexplanationfor inducedor triggered
seismicityin areasremote from major faults or plate
boundaries(observation8 in section1.2).
The earthquakefaultswhichappearspontaneouslyin
the resultshaveself-arlinescalingof the fault morphol-
ogy(observation4 in section4.2 [Sornetteetal., 1994b])
with a roughnessexponentof 2/3 [Miltenbergeret al.,
1993],power-lawscalingof both their sizedistribution
andthat of the earthquakestheygenerate[observations
1 and 2 in section1.2 [Cowieet al., 1993]), and small
stressfluctuationsafter an initial transient(observation
3 in section1.2[Sornetteetal., 1994b]),but theyhavean
exponential[Cowieetal., 1993]ratherthanthe observed
Figure9. Faultpatternsfromtheresultsof Cowieetal. [1993]
attwodifferentstages:(a) earlyand(b) laterin themodelrun.
Darker tonesrepresenta greaterdegreeof accumulatedoffset
on the faults, normalized to take account of increasing
deformationon two main fault strands(labeled 1 and 2 in
Figure 9b).

13. 34, 4 / REVIEWSOF GEOPHYSICS Main: STATISTICALPHYSICSAND SEISMIC HAZARD ß 445
powerlawtemporalcorrelationin earthquakeinterevent
times(observation6 in section1.2 [Turcotte,1992]).
Cowieet al. [1995] alsoinvestigatedthe multifractal
scalingspectrumof higher-orderdimensionscalculated
byweightingthe observationof a fault in a box-counting
algorithmby the cumulativeslip on the fault. The ca-
pacitydimensionD ois dominatedby the initial random
distribution of small faults and remains constant at the
Euclideandimension,2, of the model space.However,
the higher-orderdimensionscorrespondingto higher-
order momentsof the distributiondecreasesystemati-
cally in the resultsas the deformationbecomesmore
concentrated.This is accompaniedby a decreasein the
exponentof the lengthdistribution[Cowieet al., 1995],
asseenin analogueexperimentaldata[Davyetal., 1992].
The resultingmultifractalspectrum(characterizedboth
byDqandf(ot)discussedabove)hasaqualitativeshape
similarto that found for the clusteringof natural seis-
micity (observation5 in section1.2 [Geilikmanet al.,
1990]). Thus a progressivelyorganized(multifractal)
pattern of faultingdevelopsevenwhenthe initial heter-
ogeneityof rock strengthis setto be random,uncorre-
lated noise.
The appearanceof a multifractalspectrum,indicative
of spatialclustering,is alsoaccompaniedby systematic
changesin the earthquakefrequency-magnitudedistri-
bution.After an initial fault nucleationstagedominated
by the correlationlengthof the backgroundnoise,the
syntheticearthquakepopulationsprogressduring the
"growth" phase of the faulting through distributions
similarto the subcriticalto criticalexamplesillustrated
in Figures 7a and 7b [Cowieet al., 1993]. Thus the
Gutenberg-Richter law emerges as the correlation
length in the model resultsincreasesto a value near
infinity(the largestfaultscrossthe modelspace).
The spontaneousemergenceof largefaultsis all the
more remarkablebecauseno materialweakeningis in-
vokedin the numericalmodelduringaccumulatedslip.
The highlyorganizedpatternof faultingisthereforedue
solelyto the statisticalphysicsof elastic-brittlefailure,
includingtheeffectsof long-rangeelasticinteractions,in
a heterogeneousgranularmedium.Thisprovidesa plau-
sible mechanismfor the spontaneousdevelopmentof
large-scalebrittle faulting on Earth, evenfrom an ini-
tially featurelessmaterialheterogeneityon the scaleof
t0 km or so.Thismechanismisimportantbecausesuch
organizedfaultingis a necessaryfirst stepto the devel-
opmentof newplateboundaries.However,Earth hasa
spatially-correlatedgeologicalheterogeneity,evenin its
earlyhistory,soit will be importantin future to investi-
gate the effectsof differentgeologicallyrealisticinitial
andboundaryconditionson the resultingdynamics.
2.2.•. Faultnucleation. The conceptualmodelof
Figure8 describesthe developmentof concentratedslip
in a preexistingcoarse-grainedmosaicof faultedblocks
of scalelength t0 km. In this sectionwe considerhow
suchlargefaultsmaygrowfrom evensmaller-scalepro-
cessesand how controlledlaboratoryexperimentscan
helpusdiscriminatebetweendifferentmechanismssug-
gestedfor their larger-scaleproperties.The greatadvan-
tagein laboratorystudiesisthat dynamicvariablessuch
asthe appliedstressesmaybe measuredindependently,
whereasin the Earth even the reported "stress"mea-
surementsare essentiallyscaledfrom observationsof
strain [Zoback, 1992]. The great disadvantageis the
mismatchof laboratorystrainrates(>10-8 S-I) com-
paredwithtectonicrates(<t0-12s-l), whereeventhe
quotedupperboundfor naturalEarth strainisonlyseen
by monitoringvery closeto activefaults.
One aspectof the nucleationof larger faults from
smalleronesis the scalingof displacementu with fault
length L. This problem has been studiedby various
groups,often using combineddata setsto attain the
necessarybandwidth.Dependingon the methodsused,
differentgroupshaveproposedself-similarscaling(u cr
L [CowieandScholz,t992a]) or a systematicincreasein
displacementwiththelargerfaults(u crL1.5[Marretand
Allmendinger,1991]or u •cL2 [WalshandWatterson,
1988]). Scale-invariantbehavioris consistentwith an
analytic elastic-plasticmodel developedfor postyield
fracture mechanics[Cowieand Scholz,t992b], rather
thanwith a purelyelasticfault growthmodelwhereu cr
Lø'5[Scholz,t990a],orakineticmodelwherethefault
growsby incrementswhosesizeis a constant,irrespec-
tiveof faultlength(u •cL2 [WalshandWatterson,
1992]).
The combineddata can be interpreted as showing
scaleinvariancein two separatedomainsof "small"and
"large"earthquakes[CowieandScholz,t992a],butwith
a transitionto a systematicallygreateroffsetat a scale
lengthcorrespondingto the seismogenicthickness(t0
km or so[Scholz,1982]).At thischaracteristicscalethe
implicationisthat faultsaccumulateslipfasterthanthey
grow.Thus there is someevidencethat the large-scale
granularityof the crust,on a scalelengthof t0 km or so,
assumedin the resistornetwork model describedabove,
is at leastplausible.
Cowieetal. [1996]reviewsomeof therecentprogress
in understandingthe scalingpropertiesof fault popula-
tions. One of the conclusionsthey come to is that the
debateon the precisevalueof the scalingexponentmay
be less important than an explanationfor the large
degreeof real scatterin the plots,which givesrise to
muchof the uncertaintyin the interpretations.A similar
real scattercanbe seenin the dynamicstressdrop asa
functionof earthquakesourcelengthin Figure4 [Aber-
crombie,t995], alsoreflectedin the scalingof seismic
moment with fault rupture area in the resultsof the
numericalexperimentsof Gross[1996].Thisistestament
to the dynamiccomplexityof the mechanismof fault
growthandislikelyto be relatedto the sameunderlying
physicsof fluctuationsandinteractionsdescribedabove.
One of the problemsof interpretationsbased on
compositedata setsis the difficultyof combiningthe
data setsobjectively.Figure 10 showsan exampleof a
singlebroadbandstudyof tensilefracturesdevelopedin

14. 446 ß Main' STATISTICALPHYSICSAND SEISMIC HAZARD 34, 4 / REVIEWSOF GEOPHYSICS
2-
• o
a I 2- b
, ,,/"+
4-.
* I
+$•*+, -•-
, ! , , , , , , ,0 1 2 3 4 -1
2-
log (length(m))
-1-
o 3 4 -1
Figure10.Plotof thewidth(openingdisplacement)asa functionof length(log-logplot) for tensilefissures
in northern Iceland, after Hatton et al. [1994]. (Reprinted with permissionfrom Nature; copyright1994
MacmillanMagazinesLtd.) The dataareplottedbothfor individualpopulationsat (a) Kraflaand(b) Myvatn
and(c) in combination.The dashedlinesindicateabreakin scalingat about3 m, andthesolidlinesrepresent
leastsquaresfitstothedataaboveandbelowthispoint.Thedataallshowasystematicchangeinthescaling
exponent(slope)from about 2, representing"characteristic"growthof the smallerfractures,to about 1,
representingscale-invariantgrowthof the larger fractures.
...
., .½,&.'.?.,..• .:-. .,....-..:,, --.:..:.
........;:•:i';;;:•':'•::.;"' "*;i: , ' ':;'..
::- •,,:.•.
:
%:.:i'
..
.:.
Figure11.Photographof anoutcropshowinga preservedsectionof the nucleationof a setof conjugatefaults
in Carboniferoussandstonein northernCornwallnear Bude (reproducedwith permissionfrom J. Dixon,
Universityof Edinburgh).Smallwhite quartzveinsrepresentingthe locationsof earliersmallcracksare cut
by the later fault offset.

15. 34, 4 / REVIEWSOF GEOPHYSICS Main' STATISTICALPHYSICSAND SEISMIC HAZARD ß 447
the Krafla and Myvatn fissureswarmsin northernIce-
land. The scalingis a power law, but with a marked
breakof slopefrom2 for smallfractures(<3 m) to 1for
larger ones.A slope near 2 is consistentwith crack
growth by a constantincrement AL, independentof
cracklengthL, anda slopenear 1 isconsistentwith AL
•cL [Hattonet al., 1994].The materialthroughwhich
these fractureshave grown is a homogeneousbasalt,
withjointsona scalelengthof theorderof 30 cm,sothe
transitionfrom "characteristic"growth(AL = const)to
scale-invariantgrowth(/XL • L) doesnot occurpre-
ciselyat the structural"grain" size provided by the
coolingjoints. Thus the mechanicalgrain need not be
equivalentto the mostevidentstructuralgrain.Paradox-
ically, suchfractal scalingof the structureabove this
grain size impliesa scale-dependentcrack extension
force, similar to that reported in ceramic materials,
possiblyasa resultof increasingenergydemandfrom a
processzoneof damagesurroundingthe crackandcon-
centratedat the tip [Hattonet al., 1994].Whateverthe
explanation,Figure 10 illustratesthe generalpoint that
scaleinvarianceis alwaysunderpinnedby granularpro-
cessesat smaller scales, also a feature of all of the
physicalmodelsdescribedabove (i.e., the inherently
discreteblocksin Figures6 and8).
Tensilecrackingmay alsobe importantin fault nu-
cleation at depth, even though the principal tectonic
stressesare compressive.An exampleof an outcrop
where an incipientprocessof fault nucleationhasbeen
preservedis shownin Figure 11. Here a small shear
offset cuts an organizedband of smaller and earlier
en-echelondilatantfractures,nowfilledwith quartzlo-
callyderivedby pressuresolutionfrom the hostrock.
Dilatant crackingmayoccurlocallyundercompression,
in the presenceof fluid overpressure,or stronglocal
stressgradientsin a mechanicallygranular material.
However,thegrowthof a tensilecrackisrapidlyarrested
'ina compressivestressfield,for example,bythemech-
anism of dilatant hardening [Scholz,1990a]. Thus a
negativefeedback(hardening)processinitiallyresultsin
a distributedarray of isolatedtensilecrackssimilar to
Figure 9a. The rate constantsof fluid flow and mineral
precipitationthereforeexert.a stronginfluenceon the
sizeandspacingof thesearraysof microfractures.Even-
tually,themicrofracturesbeginto interact,coalescingto
producea shearfaultona largerscale[Mainetal., 1993].
The phenomenonof crackcoalescencecanbe inves-
tigated by microseismictechniquesin the laboratory.
Locknetetal. [1991]usedservo-controlon the recorded
acoustic emissions to slow down the fault nucleation
processin granitespecimensand demonstratedconclu-
sivelythat shearfaulting is precededby progressively
more localizedmicrocrackingalongthe incipientfault
plane. It is this processthat hasbeen frozen into the
exposureseenon Figure 11.Suchcrackinteractionrep-
resentsa positivefeedbackmechanismwhich may be
quasi-staticor dynamic,dependingonthestrainrate and
Zone of Damage
ß...? ß . ....•.•., .:.;.:'.t•q•"'...:..:..
ß :. .(,,',[..:'...,. ,;.
:.. . ..•.-'•;.:",•..... :..,
..::• ß .. •....•..,;.
ß .:'•.:-.'•.,.'•::•;:.:.::.::•: .' ....... ß ..,,a•.::,.?'.
ß:-"' ,::i...:!'._':;i...'-;i•:'"':'. ..:...::....,¾i.::.:...',.. .....',••'?.."."'•':"' ' '..."•':;?.•.2:g'-" "' ' ,.•:.."."•:•"":;'
:' ':' ß •:i•'.'•".•..... :' ' ..,
ß .: ... •fi.•p:.... . ß , ..•.: ß
.;:. 5 :':.... ...,:
ß:'"'' :i•:'. ..:. ...../--.' .. ..11.-..•i,...,..:
ß.:.:' '-.... '•:r,:ß
!?-??.;.:";:',:
;'"'".. ß •i....":.':;'::'"'";.•': 'ii•.-;.",.:.':'..':."
.... ..":;?';..:i..
ß'•:•....;'".
Intact Rock Fracture 'ProcessZone'
Figure 12. A snapshotof failed elements(shownin black) in
the results of a cellular automaton model with local rules of
hardeningand softening[afterHendersonet al., 1994].White
areasrepresentintact rock, stippledareasrepresentzonesof
damagecontainingisolatedmicrofractures(generatedby a
hardeningrule), and largeblackareasrepresentareaswhere
the microcrackshave coalescedinto a macroscopicfracture
(generatedbya softeningrule). The tworulesarecombinedin
a singleoperatorf, whichmodifiesthefracturetoughnessfield
K'?-• f327afterthefailureofaneighboringelement.Theform
usedbyHendersonetal.[1994]isf = (1+ pe-(n-4)2/20)e-n2/16,
wheren isthe numberof failedneighboringelementsand p is
anempiricalnegativefeedbackor hardeningparameter.Large
fracturesare usuallysurroundedby a zoneof damagesimilar
to a "processzone" seen in ceramicsand other composite
materials.
the relativephysicochemicalpropertiesof the fault zone
andthe surroundingrock.
The cellularautomatonapproachhasalsobeenused
to examinethe processof fault nucleationbythe coales-
cenceof suchdilatant microcracks.For example,Hen-
dersonetal. [1994]showedthat the scalelengthof local
dilatancyhasa systematiceffecton the seismicb value,
byexaminingthe effectof hardeningandsofteningfeed-
back ruleson the form of fracturegrowthand coales-
cence,projectedontothe planeof a shearcrack[Wilson
et al., 1996].The conceptualmodelsof Hendersonet al.
[1994]andWilsonetal. [1996]describea singlecycleof
quasi-staticfracturegrowth,anddo not includehealing.
Figure12showsa snapshotof the evolutionof the crack
array obtained from the results of Hendersonet al.
[1994],whichshowmanyof thecharacteristicfeaturesof

16. 448 ß Main' STATISTICALPHYSICSAND SEISMIC HAZARD 34, 4 / REVIEWSOF GEOPHYSICS
15.0
10.0
5.0
0.0
0.0
p--1.0
•=3.0 p:2.0
i
5.0 10.0
magnitude,m
15.0
Figure 13. Equivalentcumulativefrequency-magni-
tude distribution(loglinearplot) for populationsof
syntheticcrackssuchasthoseshownin Figure12,for
differentvaluesof the hardeningparameterp, also
definedin the captionto Figure12 [afterHenderson
etal., 1994].Increasingp impliesgreaterlocalhard-
ening.N is the numberof earthquakeswith magni-
tude greater than m; the magnitudeis calculated
from the logarithmof the area of clustersof con-
nectedfailedelementssimilarto Figure7. The data
showthreeranges:a characteristicpeakat thesmall-
estmagnitudes,a scale-invariantrangeat intermedi-
ate magnitudes,and an exponentialdecayat high
magnitudes.
crackgrowthin the laboratory,includingisolateddila-
tant cracks,large connectedclusterssurroundedby a
zone of damage(processzone), and large correlated
areaswith no or relativelylittle damage.Similarbehav-
ior is seenwhenthe simplerulesof the cellularautom-
aton are replacedby a quasi-staticsolutionto the cou-
pled problemof deformationand fluid flow, usinga
latticegasto simulatethe effectof a fluidphase[Wilson
et al., 1996]. Wilsonet al. [1996] alsoshowedthat the
geometricalevolutionof crackarraysin the resultsis
sensitiveto the initial mechanicalheterogeneity,similar
to the subcritical class of models of Rundle and Klein
[1993],illustratedin Figure7a.
Cracksizedistributionsobtainedby Hendersonet al.
[1994] from differentsnapshotssuchas Figure 12 are
showninFigure13.Powerlawscalingappliesonlyin the
middlerangeof scales.A markedbreakin slopeisseen
in the results for cracks smaller than a characteristic
length,determinednot by an inherentmaterial grain
scale,but by the localmechanicalhardeningrule. This
WhinSillDolerite KI (NN.m-3/2)'b'
3 1 2.97-3-05 1.38
2 2.82- 2.87 1.60
3 2.t.1- 2.60 1.82
2.20- 2.'31 2-00
2-11- 2.19 2.86
S t. 3 2 1
I I I I I I
0 20 t.o 60
Amplitude(dB}
H20liquid20øC
ß
o granite
I I I I I I I
80 100 0 0.2 0-4 0'6 0'8
K/Kc
10
Figure14.(left) Cumulativefrequency-amplitudedistributionsfor acousticemissionsduringthequasi-static
tensilefailureof notcheddoubletorsionsamplesof Whin Sill doleritein waterat 20øC(afterMeredithetal.
[1990];reprintedwithkindpermissionof ElsevierScience-NL,SaraBurgerhartstraat25, 1055KV Amster-
dam,Netherlands).The differentdistributions1-5 correspondto differentstressintensityfactors(K/) andb
valuesmeasuredduringdifferentstagesof a singletest run. (The test runs are controlledby gradually
increasingthe stressintensity,sothe quotedvaluesfor K/are listedasrangesratherthan singlefigures).
(right)A compositeplotof measuredb valueandstressintensityfactorfor testrunsondifferentcrystalline
rocktypesandwatersaturation.For theseteststhe constantc (equation(5)) isequalto 1, sothe exponent
of the sourcelengthdistribution,D, is equal to b [Meredithet al., 1990].The stressintensityfactor is
normalizedbythefracturetoughnessKcto allowdirectcomparisonof thedifferenttests.Solidlinesrepresent
leastsquaresfitsthroughthe resultingcompositedata,showinga systematicdifferencebetweennominally
water-saturated("wet") andair-dried("dry")samples,whichdecreasesasthe stressintensityincreases.
ITI

17. 34, 4 / REVIEWSOF GEOPHYSICS Main' STATISTICAL PHYSICSAND SEISMIC HAZARD ß 449
1,5
b '''5. , , '1.0
1.o 3
05• ,111•I I, ,I I.1.I.!, ......
0.5
0.0
0.0 50.0 100.0 150.0 200.0
Remote stress
Figure15. Evolutionof the seismicb valuewith time (representedby increasingappliedstress)from the
resultsofHendersonetal. [1994],withdifferentvaluesof thehardeningparameterp.The b valuesdetermined
byHendersonet al. [1994],basedsolelyon the meanmagnitudeof the wholedatarange,havebeenrescaled
for the intermediaterangeof datashownin Figure 13 [seeMain et al., 1994].The insetshowsthe behavior
(includingtypicalerrorbars)observedin field-scaleinvestigationsof earthquakesequences(redraftedwith
permissionfromNature[afterSmith,1981];copyright1991MacmillanMagazinesLtd.). The verticallinesin
the insetcorrespondto individuallargeevents(with theirmagnitudescalegivenon the right-handside).
illustratesthe concept of a local mechanical"grain"
from the physicalprocess,alsorequiredto explainthe
break in slopeof the data in Figure 11 [Hatton et al.,
1994].The biggestcracksin the model snapshotshave
notyetcrossedthe entiregrid,soa subcritical(compare
Figure7a) sizedistributionresultsat largerscales.The
resultsalsoshowa systematicincreasein b value,asso-
ciatedwith a smallermaximummagnitude,asthe degree
of negative feedback or hardening increases.This is
similar to the behavior seen in the nonconservative
spring-block-slidermodel of Olami et al. [1992] and in
the scalingof seismicityinducedby hydraulicmining
[Main etal., 1994].
A positivecorrelationof increasednegativefeedback
with decreasedb valueis alsoseenin laboratoryexper-
imentsduringquasi-staticcrackgrowthby stresscorro-
sion(Figure14[afterMeredithetal., 1990]).Theimpor-
tant controllingvariablein the resultsof Meredithet al.
[1990]isthe stressintensity(proportionalto the square
root of the crack extensionforce), a measureof the
concentrationof elasticstressat the cracktip. At low
stressintensities,energy,whichmayhavebeenavailable
to drive the main crack,is dissipatedin local physico-
chemicalfluid-rock reactionsthat produce a zone of
damage(processzone), which screensthe main crack
from the remotestress,similarto that seenin Figure12.
The resultsof severalsimilarexperimentsondifferent
crystallinerocktypesare alsoshownin Figure14,which
plotsthe b value asa functionof the normalizedstress
intensity(K/Kc), whereKc is the fracturetoughnessof
the rocktypeused.The normalizedstressintensityand
the presenceof fluidsboth have systematicfirst-order
effectson the b value in these testson initially intact
rock.Stressintensityincreasesin proportionto boththe
appliedstressandthe squareroot of the cracklength,so
fracturegrowthitselfincreasesthestressintensity,which
in turn feedsbackinto fastercrackgrowth.Suchnon-
linear, local, physicsis just one example of how an
acceleratingor avalanche-typeprocesscan develop
throughlocalfeedbackrules.
In a true stateof self-organizedcriticalitywe might
expectthe b value to remain unchangedor to change
onlyslowlyin comparisonwiththe eventrate [e.g.,Shaw
et al., 1992].The predictedevolutionof the seismicb
valuein the numericalmodelof Hendersonetal. [1994]
isshownin Figure15.For zeronegativefeedback(p = 0)
the b value decreasesmonotonically,and failure is ap-
proachedrapidlybecauseof the absenceof local hard-

18. 450 ß Main- STATISTICALPHYSICSAND SEISMICHAZARD 34, 4 / REVIEWSOF GEOPHYSICS
u,,,, / s•,u•^• :IV
0 0 0 0 0 0
o o o o o o
o o o o o o
anloA-q
0 0 0 0
0 u• 0
½'4 •- •- • 00
] [ • [ ] [ I I [ I [ [ I ! [ I I I I I • I I I ' [ I I • I O
• O ß
• . _
.
o
8••
I', .........,.........,.........,........ •'•;;o
>(Odl//) aJnssaJdPinl-I aJOd
u,w / s•,ua^a:IV
anloA-q
lllilllll]l]1111
o • • •

19. 34, 4 / REVIEWS OF GEOPHYSICS Main: STATISTICAL PHYSICS AND SEISMIC HAZARD ß 451
ening.Thisbehaviorissimilarto'thatseenintheresults
of Huangand Turcotte[1988].However,for finite nega-
tivefeedback(p > 0), failureisdelayed,andatemporary
increasefollowed by a shorter decreasein b value is
seen,similarto somereported observationsof natural
seismicity(seeinset[afterSmith1981]).The resultsof
Hendersonet al. [1994], althoughnear-critical,do not
exhibitthe insensitivityto local dynamicsof strict self-
organizedcriticality.
Similar fluctuations in b value, associatedwith de-
layed failure, are seenin controlledlaboratoryexperi-
ments during fault nucleation(Figure 16 [after Sam-
mondset al., 1992]). In thesetests,experimentsdone
under"drained"conditions(wheredilatancyhardening
cannotoccur,sop = 0) showa constantporepressure
and a monotonic decrease in b value. However under
"undrained"conditionsthe pore fluid pressuredrops,
implyinglocal dilatancyhardeningaccordingto Figure
12(p > 0). The effectsof suchlocalnegativefeedbackon
the fracture processare both to delay failure and to
introducea temporaryincreaseand then a decreasein
the seismicb value. Thesequalitativefeaturesare also
seenin the resultsshownin Figure 15. In contrast,the
presenceof a fault is markedby a relativelyconstantb
value in the period of stable sliding(Figure 16). A
relativelyconstantb value during stableslidingis also
seenin the experimentalresultsofMeredithetal. [1990]
andLocknetet al. [1991].
Thus fault nucleation, although it does produce
power law scaling,is not an exampleof true self-orga-
nizedcriticalitybecausetheb valuedependsstronglyon
the externalvariables(notablystressintensityandpore
pressure).Instead,fault nucleationis more consistent
with the subcriticalclassof modelsillustratedin Figure
7a, implyinga strongsensitivityto materialheterogene-
ity, externalforcing,and the effectivedissipationin the
form of fluid-rock interactions. However, once a fault is
present, the b value remains relatively constant
[Meredithetal., 1990;Locknetetal., 1991;Sammondset
al., 1992;Liakopoulou-Morriset al., 1994].
The main outstandingquestionin fault nucleation
concernstheprecisenatureof the underlyingcharacter-
isticgranularityof the brittle crest:is this the mineral
grainof therocks(millimeterscale),or isit dueto some
mechanicalgrainassociatedfor examplewith localdila-
tancy hardening,or some other process,at a greater
lengthscale?The smallestnaturalearthquakesforwhich
stressdropshavebeeninvertedfrom a sufficientlybroad
bandof data[AbercrombieandLeafy,1993;Abercrombie
and Brune,1994;Abercrombie,1995]showscaleinvari-
ancein stressdrop andb valuedownto a sourcelength
of 10 m or so.Therefore this granularitymaylie some-
wherebetweenthemineralgrain(millimeters)and10m,
equivalentto the lowerscalerangeof fieldstudiesshown
in Figures4 and11.It isevidentthatmoredetailedwork
is requiredin the sub-10-mscalerange,both at outcrop
andin settingswheresuchsmall-scaleseismiceventscan
be recorded(e.g.,in minesandhydrocarbonfields).
3. APPLICATION TO SEISMIC HAZARD
So far we have concentrated on the fundamental
problemsof howto explainthephenomenologyof earth-
quakesandseismogenicfaultingin theEarth on a broad
range of scales.In this sectionwe now consider the
implicationsof a stateat or near self-organizedcritical-
ityma state borderingon deterministicchaos--for the
practicalproblemsof earthquakepredictionandseismic
hazard estimation.
3.1. EarthquakePrediction?
We have already addressedsome of the problems
associatedwith identifyingearthquakeprecursorsin the
introduction.Althoughnoprecursorssatisfyingall of the
criteria cited by Wyss[1991,box 3] have been found,
three were acceptedonto a "preliminarylist of signifi-
cant earthquakeprecursors"[Wyss,1991,p. 1], as fol-
lows:(1) seismicquiescencebefore strongaftershocks
(manycases);(2) foreshocks(February4, 1975,M = 7.3
Haicheng earthquake);(3) radon concentrationde-
crease(January14, 1978,M = 7.0 Izu-Oshima-Kinkai
earthquake).This list does not imply that these are
definiteprecursors,merelythat "the majorityof review-
ersandpaneliststhoughtit morelikelythannot that the
method might be useful for earthquake prediction"
[Wyss,1991,p. 1].If we assumethatthe initialsampleof
28 precursorssubmittedto this exerciseis alreadypre-
selectedby thosetakingpart to be the "bestexamples,"
then the whole questionof the general existenceof
precursorsis calledinto questionon empiricalgrounds.
We havealsoseen'thatreliablepredictionmaynot even
be possiblein principle,owingto a combinationof (1)
nonlinear dynamics,involvingtriggeringor avalanche-
type processes[e.g.,Brune,1979;Bak and Tang,1989];
(2) incompleteor imperfectdatasampling[e.g.,Scholz,
1990a;Wyss,1991];and (3) a systemexistingin a per-
petual stateof marginalstability,wherephysicalfluctu-
ationsresultingin apparentstatistical"noise"are inher-
ent to the process[Baket al., 1988a;Rundle,1988].
On theoptimisticside,Keilis-Borok[1990],in a review
of the implicationsof a conceptualmodel of the litho-
sphereof theEarth asa nonlinearsystem,concludesthat
earthquakepredictionmay at leastbe possibledespite
the aforementionedproblems.On the otherhand,Geller
[1991,1996]pointsoutthatthereisnoconsensusamong
seismologistson whether earthquakepredictionis pos-
sible, even in principle, and arguesthat seismologists
should instead concentrate their efforts more on achiev-
inga betterunderstandingof seismogenesisasa process.
In a recent concreteexample,Abercrombieand Mori
[1996]haveshownthat there is no systematicrelation-
shipbetweentinyaspectof foreshockoccurrenceandthe
magnitudeof the subsequentmainshockin the western
United States.Similarly,Mori andKanamori[1996]ex-
amined the nature of precursoryseismicradiation im-
mediately(i.e.,secondsor fractionsof a second)priorto
the main energyreleasein a range of earthquakesof

20. 452 ß Main- STATISTICAL PHYSICSAND SEISMIC HAZARD 34, 4 / REVIEWSOF GEOPHYSICS
different sizesin the 1995 Ridgecrest, California, se-
quenceand concludedthat the size of the nucleation
zone for the precursoryseismicradiation was of the
orderof 10m or less,independentof the eventualsizeof
the mainshock.They concludethat the sizeof the earth-
quake is determinedby the nature of rupture arrest,
rather than the nucleationprocess,consistentwith the
notionof earthquaketriggeringandavalanchedynamics.
Thus at least one of the crucial elements of a successful
earthquakeprediction(i.e., the magnitude),cannot,at
leastyet, be madefrom short-termprecursorydatacru-
cial to the triggeringprocess(foreshocksor immediate
seismicprecursors).This problem is potentiallyeven
more severefor intermediate-termprecursors.
In the meantime the numerical models that have been
developedto explainseismogenesisasa criticalphenom-
enon may be used at least to assessthe likelihood of
prediction in an ideal casewhere our knowledgeis
complete.That is, if we cannotpredictsyntheticearth-
quakesin a computationalsystem,whichisdeterministic
but simplified,we maybe forcedto concludethat inter-
mediate-termpredictionin the Earth, with incomplete
data samplingand more complicatedphysics,may be
inherentlyunattainable.In thisvein, Shawet al. [1992]
observesomepremonitoryintermediate-termandshort-
term changesin seismicityrate andpatternin the results
of their versionof the Burridge-Knopoffmodel. More
recently,Pepke et al. [1994] found that the heuristic
pattern-recognitionalgorithmsdevelopedby trial and
error from real earthquakesequences[Keilis-Borokand
Gossubov,1990;Keilis-Boroketal., 1990]canbe applied
successfullyto a dynamicmodelof a fault,atleastfor the
caseof a cycleinvolvingthe failure of the entire model
spacein a singledynamicevent.By comparingtheoret-
ical results from numerical models, run with different
startingassumptions,parameters,and boundarycondi-
tions, with observations,we may iterate to a better
understandingof the physicalbasisfor precursors.How-
ever,the true testof reliableearthquakepredictionasa
practicalpropositionwill remainwith the statisticalex-
aminationof real (and noisy)discretedata.
In summary,somepreliminarymodelingwork on the
dynamiccomplexityof earthquakepopulationsshows
thatwe cannotyet rejectthetheoreticalpossibilityof the
predictabilityof location, time, and size of individual
earthquakesin model earthquakesequences,at least
under certainconditions[e.g.,Shawet al., 1992].How-
ever, physicallyrealisticsolutionsin similar dynamical
modelscanalsobe obtainedwith no precursors,consis-
tent with recentobservationsof earthquaketriggering.
In terms of real data, there is still no conclusiveproof
that earthquakeprecursors,at leastasdefinedby Wyss
[1991], do existgenerallyin nature. Thus despiteover
100yearsof effort,we seemto be movingno closerto
havinga usefulearthquakepredictioncapability[Geller,
1996],althoughwork is continuingin this area.In con-
trast,jhe implicationsfor a probabilisticseismichazard
analysisbasedon a populationof earthquakesmay be
muchmoreimportantandi•mediately usefulbut have
received much less attention to date. We consider this
applicationin the followingsection.
3.2. Seismic Hazard
Earthquakesare a significantnaturalhazardthat an-
nuallybringdisasterto a significantandgrowingnumber
of the world'spopulation,predominantlyin areasnear
plateboundaries(e.g.,the greatMichoacanearthquake
of September19,1985,Ms - 8.1 [Pressetal., 1987,p. 5].
However seismicrisk can alsobe significantfar from
plate boundaries,asis illustratedby the recentKhillari
earthquakein centralIndia on September29, 1993(Ms
= 6.3), whichclaimedmanythousandsof lives,partly
because of the low seismic attenuation in cratonic litho-
spherebut alsoowingto a perceptionof lowhazardand
limitationson affordablelocalbuildingpractices.In fact,
allthreerecordedearthquakescausingthelargestlossof
human life occurredin an intraplate setting(China,
1556,1920,and 1976).Many other examplescouldbe
cited here.
The seismicriskthereforedependsboth on the seis-
michazard(source,path,andlocalsiteconditions)and
on engineeringand socialconditionssuchasthe vulner-
ability of individualbuildings,the population affected,
and the actualor achievablelevel of preparedness.The
backgroundseismichazard,whichultimatelydependson
the dynamicsand structure of the solid planet, is a
proper studyfor Earth scientists.Economicand engi-
neeringdecisionsare alsobasedon avulnerabilitywhich
issteadilyincreasingasmoreandmorecitieswith larger
populationsare built in earthquake-proneareas[Reiter,
1991].For exampleit is estimatedthat 290 million "su-
percity" dwellers,80% of them in developingnations,
will live in a regionof significantseismicriskbythe end
of the millennium[Bilham,1988].
Although the seismicrisk is increasing,the seismic
hazardremainsrelativelyconstant(at leastwhen aver-
aged over long time periods)becauseof the relative
stabilityof globaldynamicsthroughthe extremelylinear
andpredictablephenomenonof surfaceplate tectonics.
The regularityof plate tectonicsexistswhen a dynamic
equilibriumof drivingforces,strivingto loseheat from
the mantle,is achievedwith a constantterminalvelocity
in the individualfragmentsof a rigid outer shell(the
lithosphere).For example,thecurrentpatternof mantle
heterogeneity,basedon seismictomographyand obser-
vationsof the geoid,isreasonablywell matchedbyplate
tectonicreconstructionsof subductionextrapolatedback
overthe last 120m.y. [RichardsandEngebretson,1992].
On shorter(decadal)timescalesthe resultsof global
satellitegeodesyhave alsoshowna markedlinear con-
sistencywith the sameplatemodel[DeMets,1995].This
stationarityisreassuringbecausesinceitsinception,the
conceptof plate tectonicshashad a greatimpacton the
calculation of seismichazard through seismotectonic
considerations[Lomnitz,1974].
However,plate tectonicscannotalwaysbe usedsim-

21. 34, 4 / REVIEWSOF GEOPHYSICS Main: STATISTICALPHYSICSAND SEISMIC HAZARD ß 453
ply to predict seismichazard. For example,although
there is a closeassociationof earthquakeswith plate
boundaries,we have seen that damagingearthquakes
mayoccurvirtuallyanywhere.Thusalthoughtheseismic
hazardtendsto be higherat the edgesof plates,it isnot
zero elsewherebecauseplatesare not completelyrigid
elasticbodies. In particular, continental deformation
ofteninvolvessignificantstretchingor shorteningof the
lithosphere,andthepatternof shallowseismicityisoften
more diffusein back-arcprovincesin suchregions.Nev-
ertheless,the first-orderpatternof seismicityand focal
mechanismsis in goodagreementwith the first-order
featuresof globaltectonics[e.g.,Lomnitz,1974].
A stationarystate of self-organizedcriticality,or at
least a stationarystate maintained near criticality as
discussedabove,givesa goodrationalefor predictingthe
future hazardon the basisof past occurrence,a basic
tenet of all currentpracticesin seismichazardestima-
tion [e.g.,Reiter,1991].Althoughthe dynamicreleaseof
storedelasticenergyin individualeventsmay be inter-
mittent and unpredictable,the averagestrain energy
releasein the earthquakepopulationasa wholewould
be expectedto remain constantover long timescales.
This singleresultremovesmuchof the uncertaintyin-
volvedin seismichazardestimation,usingwhat has,up
to now,beenan a priori assumptionof "stationarity"in
the earthquakeprocess.However,the underlyingme-
chanicspredictssignificantdynamicfluctuationsabout
thismetastablestate,sothe applicationof thisassump-
tion dependscriticallyon havinga longenoughstatisti-
cal samplecomparedwith the recurrencerates of the
largestevents.If short-termdynamicfluctuationsare an
inevitableconsequenceof the underlyingphysics,then
we shouldlikewiseadopt a conservativeapproachto
seismic hazard estimation.
A secondkey assumptionin much seismichazard
analysisis that earthquakesare a randomPoissonpro-
cess[e.g.,Reiter,1991].However,thestronglyinteracting
natureof avalanche-typedynamicsis fundamentallyin-
consistentwith earthquakesas a Poissonprocess.For
example,Brown et al. [1991] applied the Burridge-
Knopoffmodel,with itsstronglocalinteractions(Figure
6), to the problemof earthquakerecurrencetimesand
their statisticalvariability.The resultsshowa lognormal
distributionof recurrencetimes that is quantitatively
inconsistentwith a Poissonprocess.Lomnitz [1996],by
examiningthecorrelationof eventsin the globalseismic
moment catalogueof Pachecoand Sykes[1992], con-
cludedthat eventhe largestshallowearthquakesoccur-
ring this century(of momentmagnitudeM•v > 7.0),
maynotbe independentontimescaleslessthan 1month
for lengthscalesup to severalhundredkilometers.
How canwe usethelong-termmetastabilityof a state
near self-organizedcriticalityto combinetectonicesti-
matesof seismicslipwith earthquakemagnitudecata-
loguesto obtainbetter estimatesof future hazard?One
answeris to use the approachof information theory
[Shen and Manshina, 1983; Main and Burton, 1984,
1986],whereweseeka maximumentropysolutionto the
available constraintsof mean catalogue magnitude
(availablesinceabout1900)andmeanseismicmoment
flux(potentiallyovermillionsof years).The answerhas
the form of a modifiedgammadistributionin the cumu-
lative frequencyof occurrence:
axN(Mo) = Nr M;-a-le-•ø' dM;. (4)
Nr is the total numberof seismiceventsin the magni-
tude catalogueper unit time;B and X are the distribu-
tion parameters,to be calculatedfrom the mean cata-
logue magnitude(m) and the mean moment (Me) =ß ß
Me/Nr, where Me is the measuredseismicmoment
releaserate;andM• axisthemaximumseismicmoment.
The staticseismicmomentMe for an individualevent
isthe productof the fault areaA, the averagedisplace-
mentu, andthe shearmodulusof the uppercrust(ix •
30 GPa) andisrelatedto the magnitudem (ML orMs)
by
logM0 = cm + d, (5)
where c and d are scalingconstants.The empirical
relation(5) isconsistentwitha simpledislocationmodel
for the seismicsource,wherec dependson the relative
frequencycontentof the seismiceventandthepassband
of therecordinginstrument,andd dependsonthemean
stressdrop [KanamoriandAnderson,1975;Hanks and
Kanamori,1979].The dislocationmodelpredictsthatfor
a seismometerresponseflat to velocity in the typical
frequencyrangeof thesourcespectrumc = 3/2, andfor
a stressdropof about3 MPa (30bars),d = 9.1 (forMe
in newtonmeters).The scalingconstantc determines
the relationshipbetweenthe seismicb value and the
exponentsB (equation(4)) orD (equation(1)). Usually,
c - 3/2, soD - 2b andB = 2b/3 [King,1983;Main and
Burton,1984;Turcotte,1992].
By adoptingthis procedure,the long-term seismic
hazardcanbe constrainedby a long-termseismicmo-
mentreleaserate3;/o,determinedfrompaleoseismic,
geodetic,or platetectonicstudiesandsynthesizedinto a
singledistributionwith the shorter-termseismicitycata-
logueviathemeanmagnitudeandtheseismiceventrate.
Nr [MainandBurton,1984,1986].Implicitin thispro-
cedureis the assumptionof stationarityin the flux of
energyor moment,alsopresentin the meanfield statis-
ticalmechanicalmodelof Sornetteand Sornette[1989].
The gammadistributionexhibitspowerlawscalingof
energyat smallmagnitudes,consistentwith the termb in
equation(2), orB in equation(4), butwitha tail at high
magnitudesthat mayeither continuethistrend rightup
to themaximummagnitude(X = 0), haveanexponential
tail (• > 0), or leadto a "characteristic"peaknearthe
largestmagnitude(• < 0). This genericbehavior is
similarto the subcritical(X > 0); critical(X = 0) and
supercritical(X < 0) classesof behaviorin Figures7a,

22. 454 ß Main' STATISTICALPHYSICSAND SEISMIC HAZARD 34, 4 / REVIEWSOF GEOPHYSICS
(a) Eastern Mediterranean
3.00,
2.00.
1.00
0-00•
- 1.00-
-2.00
4.0 9.05•0 6•0 7•0 8•0
MagnitudeMs
•.00,
1.00
"001
-1.001
-2-001-3.OOl
(b) Southern California
5.0 6.0 7.0 8.0
Magnitude Ms
(c) Mount St. Helens
3.0
2.4
1.8
1.2
0.6
0 ,
, i I
3.5 4.0 4.5 5.0
Magnitude ML
Figure17. Examplesof frequency-magnitudedistributions(loglinearplot) from threedifferenttectonic
settings:(a) a diffuseplateboundarywithdistributedseismicity(theeasternMediterranean[MainandBurton,
1984]), (b) a plate boundarywith deformationconcentratedon a singlethroughgoingfault (southern
California[MainandBurton,1984]),and (c) a zoneof highlylocalizedintraplatedeformation(Mount St.
Helens[Main, 1987],reprintedwith kindpermissionof ElsevierScience-NL,SaraBurgerhartstraat25, 1055
KV Amsterdam,Netherlands).Thesedistributionscorrespondrespectivelyto the genericclassificationof
subcritical,critical,and supercriticalbehaviorasshownin Figures7a, 7b, and 7c.The solidlinescorrespond
tolinefitstothedistribution,correspondingto agammadistributionof seismicmomentwithX2> 0 (equation
(4)) in Figure17a,a gammadistributionwith X2= 0 in Figure17b,anda breakin scalingof thedistribution
of seismicmomentfrom powerlaw to a characteristicGaussianpeak at highmagnitudein Figure 17c.
7b, and 7c. Negativevaluesof X maybe foundin other
examplesof closedsystemsin statisticalmechanics(e.g.,
paramagnetism[Mandl,1988]).For X -<0 themaximum
magnitudeis constrainedto be finite for a finite energy
flux,therebyintroducinganothernecessaryparameterto
the distribution.
The genericbehaviorof equation(4) wasinterpreted
in terms of a simplestatisticalmechanicalmodel for
seismogenesisby Main and Burton[1984],who postu-
lated that the term B is controlledby the geometrical
probabilityof a singleeventof sourceareaA occurring
in a fault of total areaAmax [KanamoriandAnderson,
1975]and that X is controlledby the strainenergyrate
(flux).Thissimpleanalyticmodelanticipatedthegeneric
behaviorof the dynamicmodelsshownin Figure 7.
Studiesof globalseismicity[Kagan,1993,Figure1]or
of large tectoniczonessuchas the easternMediterra-
nean (e.g., Figure 17a [after Main and Burton,1984])
commonlyreveal a distributionwith X > 0. Intermedi-
ate-sized areas such as the San Andreas, with a maxi-
mum magnitudecorrespondingto an earthquakethat
just transcendsthe studyarea, commonlyhave X = 0
(Figure 17b [afterMain and Burton,1984]).Individual
fault segments(analyzedasa closedsystem)sometimes
showa characteristicpeakin the distribution,whichcan
befittedto firstorderwith X < 0, similarto Figure7c.In
naturethe excessprobabilityat highmagnitudesis usu-
allyfoundto havea smootherpeak,moresimilarto the
characteristicearthquakemodel of Shawet al. [1992],
ratherthan the sharpcutofffor finiteM•0axinherentin
equation(4) andFigure7c.The broadpeakrequiresan
extra parameterto describethe tail of the distribution
(Figure 17c[afterMain, 1987]).
The importantdifferencefor determiningtheformof
the distribution(subcritical,critical,or supercritical)is
whether or not the area can sustaina throughgoing
seismicdislocation.An equivalentstatementishownear
the systemis to the percolationthreshold.In the labo-
ratorythe boundariesof the samplearefixedandfinite,
sothe sizeof the largesteventis alsofixed.However,in
seismichazardanalysisit isnecessaryto examinesource
zones of a size smaller than the whole Earth. Such
sourcezonesare commonlydefinedbysmoothpolygons
[e.g.,Reiter,1991]. However,the precedingparagraph
has highlightedthat this subjectivechoiceof seismic
zonecanhavea profoundinfluenceon the appropriate
form of the distributionat highmagnitudes.For a large
enoughseismiczone,X > 0. Thisformof thedistribution
hasalsobeenfoundfor a varietyof data setsin natural
seismicity[MainandBurton,1984;Kagan,1991],natural
fault exposureof fault lengthin the SanAndreassystem
[Davy, 1993], and syntheticfaults developedboth in
analogue laboratory models and a resistor network
modelwith a layeredrheology[Davyetal., 1995].
The type examplesof the frequency-magnitudedis-
tributionshownin Figure7 exhibitpowerlaw scalingat
smallmagnitude(linesof constantslope)and then ei-
ther a gradualroll-off,a continuationof thistrend,or a
characteristicpeak at high magnitudes.This generic
behaviorcanbeseenin realdata(Figure17),illustrating
the utility of constrainingthe hazard from long-term
seismicmoment release rates, whatever the exact form
of the distribution.The general utility of the gamma
distributionin describingthis range of behavior, to-
getherwith the need describedaboveto applyit to a
long-termdataset,highlightsthe importanceof obtain-