Seismology-A Statistical Vignette
Author(s): David Vere-Jones
Source: Journal of the American Statistical Association, Vol. 95, No. 451 (Sep., 2000), pp. 975978
Published by: American Statistical Association
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Statisticsin the Physicaland EngineeringSciences
975
lems also have a sequentialaspect and providethe oppor- plicatedobservationaldatasets.I hope thatit is clearthatno
tunityto use adaptivemethodsthatimproveover time.
single disciplinealone can approachthese problems:The
and
easy stuffhas been done! Progressin understanding
5. CLOSING REMARKS
forecasting
theearth'ssystemsrequirescollaborativeeffort
Popularviewsof statisticsin thegeophysicalsciencesof- amongteamsof scientists,includingstatisticians.
tenfocuson spatialmethods(e.g.,Kriging)and timeseries.
Of course,theseremainstandardmethodsfordata analysis,
REFERENCES
and non-Gaussianproand theirextensionto nonstationary
B. A., Berliner,L. M., Collins,W., and Kiehl,J. T. (1999), "Neucesses pose new researchproblems.Recently,Bayesian hi- Bailey,
in Studiesin theAtmospheric
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2000). One strength
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manuscript,
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I hope thatthisvignettebalances some coninterpretation.
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VersionOne," Journalof Climate,11, 1115-1130.
emergingareas in atmosphericscience.
Keihl, J. T., Hack, J. J., Bonan, G. B., Boville, B. A., Williamson,D.
Due to limitedspace, I have focusedsolelyon theatmoL., and Rasch, P. J. (1998), "The NationalCenterforAtmosphericReoftheearth'senviron- search CommunityClimate Model: CCM3," Journalof Climate, 11,
sphere.But a completeunderstanding
1131-1149.
mentmustincludechemicaland biologicalprocessesalong
withstudyof the sun. Overlaidon thisnaturalframework Salby, M. L. (1996), Fundamentalsof AtmosphericPhysics,San Diego:
AcademicPress.
is the influenceon physicalsystemsby humanactivities. Tebaldi,C., Nychka,D., Brown,B., and Sharman,R. (1999), "Predicting
These areas challengestatisticianswiththe need to work
Scienices,eds. L.
Clear-AirTurbulence,"in Stutdiesin theAtmospheric
M. Berliner,D. W. Nychka,and T. Hoar,New York:Springer-Verlag.
numericalmodelsand large,comcloselywithsubstantive
Seismology-A Statistical Vignette
David VERE-JONES
1. INTRODUCTION
Geophysics,and seismologyin particular,has a somewhatuneasyrelationship
withstatistics.On one side, geophysicistshave always been passionatecollectors,procesof observationaldata. FromHalley in
sors,and interpreters
in the 20thcentury,
its
the 17thcenturyto Harold Jeffreys
develleadingpractitioners
have also pioneeredimportant
opmentsin statistics-ingraphicaland numericalmethods,
in the treatment
of errors,in time series analysis,and in
manymorespecializedtopics.On theotherside,geophysitendto cling
cists,in commonwithotherphysicalscientists,
difto a view of theuniverseas governedby deterministic
ferentialequations.Probabilitymodelstendto be relegated
to therole of describingobservationalerrors,even where,
as in describingthe occurrencetimesof earthquakes,the
lie considerablydeeper.The upshot
sourcesof uncertainty
is thatthe general level of statisticalusage among geophysicists,and among seismologistsin particular,is very
importanceto
uneven,fromcontributions
of fundamental
disappointing
misunderstandings.
The role of seismologywithingeophysicsis greaterthan
its ratherspecial subject mattermightsuggest. This is
David Vere-Jonesis EmeritusProfessor,School of Mathematicaland
New Zealand.
Wellington,
ComputingSciences, VictoriaUniversity,
of thereflecchieflybecause formanyyears,measurement
of earthquakewaves passingthrough
tionsand refractions
the earthhas providedan importanttool for probingthe
earth's inner structure.At the same time,the subject is
keptin thepublic eye throughits applicationsto engineering, buildingcodes, insurance,and earthquakeprediction.
In all of theseapplications,and at theheartof the subject
itself,statisticalproblemsabound,few of themeasy and
some challengingthe limitsof currentstatisticalmethodology. In this vignetteI attemptto indicatethe natureof
theseproblems,firstby tracinga briefhistoryof seismology,and thenby selectinga few special issues of current
Bolt (1988) andBullen(1963) haveprovideduseful
interest.
to the subject.
generalintroductions
2. A BRIEF HISTORY OF SEISMOLOGY WITH A
STATISTICALBIAS
2.1
FirstStages: 1890-1920
Seismology has little statisticalhistorybefore the developmentof the Milne-Shaw seismographin the 1890s.
Somewhat earlier the theoryof wave propagationin an
elastic mediumhad been workedout, a qualitativeinten? 2000 American Statistical Association
Journal of the American Statistical Association
September 2000, Vol. 95, No. 451, Vignettes
976
Journalof the AmericanStatisticalAssociation,September2000
sityscale forassessinggroundmotionhad been developed, 2.2 The Classical Period: 1920-1950
to compilelistsof largehistor- The decades followingthe 1906 San Francisco earthand therehad been attempts
ical earthquakes.But the Milne-Shaw instrument
was the quake were markedby steadyimprovements
in instrumenfirstthatwas compactand accurateenoughto allow objec- tationand data collection.Networksof stationswere estivemeasurements
of earthquakewave motionin manydif- tablished,and information
began to be collected at both
ferentplaces to be made and compared.Early instruments global and local levels. The principalthemewas not the
were deployednot only in Europe and the United States, studyof the earthquakesthemselves,however,but rather
but also in Japan,New Zealand, and China.
the information
thattheyprovidedabout the earth'sinteWiththefirstreliableinstruments
came the firstreliable rior.As early as 1909, Mohorovicichad observedwaves
data and the beginningsof seismologyas a quantitative apparentlyreflectedfroman internalboundarysome kiloscientificdiscipline.As early as 1894, the Japaneseseis- metersbelow the earth's surface,and evidence for other
of largeJapanese boundaries,includingthatof a centralcore, accumulated.
mologistOmori,studyingtheaftershocks
earthquakes,
formulated
the firstempiricallaw of seismolThe disentanglement
of such data is a classical inversion
at time problem,the basic unknownbeing the velocitystructure
ogy,Omori'slaw: thefrequencyA(t) of aftershocks
t afterthe main shock decays hyperbolically.
This is most inside the earth.Many seismologistscontributed
to these
commonlyquotedin thegeneralizedform
were those of
issues, but the mostprofoundcontributions
laterassistedby K. E. Bullen.The notable
HaroldJeffreys,
feature
of
workwas its carefulattentionto staJeffreys's
A(t) = (1 + y)
tisticalprocedures.Like thatof Laplace and Gauss before
workon probabilityand statisfundamental
him,Jeffreys's
Remarkably,
to thisday,Omori's law remainswithoutany tical inference(Jeffreys
1939) was underlainby the better
clear physicalexplanation.It can be modeledto a firstap- partof a decade of experiencein thereductionof physical
proximationas a nonhomogeneousPoisson process (Jef- data, earthquaketravel-time
of
data and the establishment
freys1938), and it is used in Ogata's epidemic-typeaf- improvedproceduresfor epicenterand hypocenter(threetershocksequence (ETAS) model (Ogata 1988), whereev- dimensional)location.The workculminatedin 1940 with
eryeventis supposedto triggerits own aftershocks
("off- thepublicationof theJeffreys-Bullen
taglobal travel-time
spring"),butthemodelsremainat a descriptivelevel.
bles (Jeffreys
and Bullen 1940), which are still used for
The next major stimuluswas the 1906 San Francisco calculatingtraveltimesbased on theassumptionof a spherearthquake.ChallengingU.S. technical"know-how"at one ically symmetric
globe.
of its major centers,the earthquakedrewfortha massive
Anotherimportantstep taken duringthis period was
technicalreport(Lawson 1908) thatdocumentedtheextent Richter's(1935) developmentof an earthquakemagnitude
and characterof damage and of displacementsalong and scale. Based on the logarithmof the maximumamplitude
offtheSan Andreasfault,and therebylaid thefoundations recordedon a standardinstrument,
and adjustedto a stanforthe fieldof engineeringseismology.In a sequel to this dard distancefromthe source,thiswas the firstobjective
report,H. F. Reid (1911) set out his elastic reboundtheory measureof the size of an earthquake.Despite its limitaof earthquakes:On eitherside of a majorfault,large-scale tions,such a measureremainsan almostindispensibletool
forces operateto cause relativemotionbetweenthe two forthequantitative
analysisof earthquakecatalog data.
sides of the fault.Frictionopposes the motion.As time
Hard on theheels of the magnitudescale came the secpasses, therockmaterialdeformselasticallyand strain(de- ond empiricallaw of seismology,the Gutenberg-Richter
and hencestress(elasticforce),accumulate,un- frequency-magnitude
formation),
law. In statisticalterms,thislaw astil ultimatelythe strength
of the faultis exceeded and the sertsthatmagnitudesfollowan exponentialdistribution.
If
twosidesslipin an earthquake;thentheprocessstartsagain. the magnitudesare relatedback to physicalvariablessuch
fromsurveysbeforeand afterthe as theseismicenergyrelease,thenthistranslatesto a power
Detailed measurements
1906 earthquakestronglysupportedthishypothesis,
which law (Pareto'slaw) distribution
forthephysicalvariable.In
bothexplainedtheoriginof theearthquakewaves and gave particular,
it suggeststhatthetailsof theenergydistribution
somebasis forregarding
earthquakesas a recurrent
process. are of theform
In broad terms,Reid's hypothesishas dominatedthinking about earthquakemechanismsever since its formulapr(energy> E) oc E',
tion.It lies behindrecentstochasticmodelsforearthquake
suchas renewalor semi-Markovprocesseswith wheretheexponentaeis in therange.4-.8.
occurrence,
This law also remainswithoutany universallyaccepted
log-normalinterevent
times,or thestress-release
model,in
explanation,althoughthe problemhere is less an absence
whichtheconditionalintensity
at timet has theform
of models.It putsearthquakessquarely
thana proliferation
into the realm of phase-change-likephenomena,associA(t) exp[a + bX(t)],
ated with featuressuch as power law distributionsof
and selfwhereX(t) =X(O) + p(t- tYZt%<t Xi) iS a measureofthe size, long-rangespatialand temporalcorrelations,
currentstresslevel in theregion,theXi beingthe stresses similarity.But here thereare manypossible models, and
releasedin previousevents.Once again,however,themod- the Gutenberg-Richter
(G-R) law by itselfis not enough
els remainat a broadlydescriptivelevel.
to distinguishbetweenthem.Nevertheless,it and Omori's
Statisticsin the Physicaland EngineeringSciences
977
law provideconstraintsthatany successfulmodel of the Americanand Japanesescientists,consciousof the superiearthquakefractureprocessmustsatisfy.
orityof theirtechnicalequipment,were spurredto emulate
of additionalfunds.
theChinese,again assistedby offerings
however,theChineseprogramsufOnly2 yearsafterward,
2.3 Time Series Analysisand Explosions: 1945-1970
fereda severereversalwiththedevastating1978 Tangshan
Time seriesand geophysicshave grownup together,
each
No formalpredictionswere claimed,and masearthquake.
contributing
to the developmentof the other,and seismollosses
of
both life and propertywere incurred.Such
sive
ogy has been an integralpartof thisprocess.Early timeof earthquakepredictionever since;
has
been
the
progress
series work in seismologyrelatedto the largelyfruitless
has been matchedby an embarrassing
success
each
claimed
studyof hiddenperiodicitiesin earthquakeoccurrence.In
to
or a false alarm.The unpredictedearththe period followingthe Second World War, however,a failure predict
Californiaand Kobe, Japan,each in
quakes in Northridge,
numberof practicalproblemspushedtime-seriesmethods
of
did littleto help
researchterritory,
intothecenterof seismologicalresearch.The mostimpor- theheart earthquake
matters.Fundingstartedto dryup, the credibilityof scitantof these (not in the least because it led to substantial
and serious
entistsworkingon predictionwas threatened,
increasesin fundingfor seismology)was the problemof
doubtswere entertainedas to whetherearthquakepredicdetectingunderground
nuclearexplosionsand distinguishwas a feasibleor even a desirableaccomplishment.
ing themfromearthquakes.The analysisof data fromseis- tion
I
am
moreoptimistic
overthesemattersthanthelastparamic arrays(i.e., instruments
set up in a gridor otherstruc(Vere-Jones1995). The viewpointis
might
suggest
graph
turedpattern)
requiredthesolutionof further
problems.The
slowlygainingacceptancethatpredictionsmustbe couched
same periodsaw thegrowinguse of explosionseismology
in termsof probabilitiesof occurrence.Many embarrass(recordingof waves fromdeliberateexplosions)to invesmentsmighthavebeen avoidedhad thisviewpointprevailed
tigatesubsurfacestructuresfor oil explorationand other
thereseem to me two main stumbling
sooner.Currently,
purposes,and of spectralmethodsto analyze theresponse
blocks. The firstis in the physics,in the lack of an adeof buildingsand otherstructures
to earthquakewaves. As
of earthquakegenesis and growth.The secseismicnetworksbecamemorehighlyautomated,questions quate theory
ond is the lack of statisticalmodels for the highlyclusarose concerningthe automatictriggeringof unmanned
In
typesofdatafromearthquakepatterns.
tered,self-similar
equipmentand the effectiveanalysis and storageof data
is considerableroomforimprovement
and
both
areas
there
fromsuch equipment.All of theseissues requiredthe sosome indicationthatdespitecurrentpessimism,the proboftenhighlytechnicalproblemsin time
lutionof difficult,
to yield.Featuressuch as local activation,
lems are starting
seriesanalysisand engagedtheattention
of leadingexperts
and precursory
foreshocks
swarms,acceleratedmomentrein bothfields.New ideas arose,such as maximumentropy
lease, and precursoryquiescence do providesome degree
methods,and thelinksbetweenthedisciplinesremainvery
of enhancementof backgroundprobabilities,and suggest
close.
thattheaccumulationof stressbeforea largeeventmaybe
detectable.However,the factorsare not yet large enough,
2.4 Plate Tectonicsand EarthquakePrediction:
and the models are not well enoughestablished,forthem
1970-Present
to be usefulin directpracticalapplications.Improvements
Plate tectonicsis one of the scientificsuccess storiesof in data qualityand therangeof characteristics
studiedcan
the second half of the twentiethcentury.For seismology, onlylead to improvements
in thissituation.
it provideda unifyingprinciplethathelped explain many
In the meantime,seismologyoffersstatisticiansthe opresolvedissues. It gave meaningto thehighly portunity
incompletely
to collaboratein an extremelydiverserange of
of seismicallyactivezones aroundthe problems.Let me concludeby quotinga few examplesof
irregular
distribution
world,and indicatedthe natureof the "large-scaleforces" recentor currentwork which happen to have caughtmy
requiredby Reid's elastic reboundtheory-plate motions, interest(butare notclaimedto be representative).
impelledby convectionprocessesin theearth'smantle.Collisionand subductionzones,rigidplatesandfractured
plate3. SOME RECENT EXAMPLES
boundaries,mid-oceanridges,and heat flow and gravity
anomalies were conceptsilluminatedand coordinatedby
Dating of EventsAlong theNew Zealand Alpine Fault.
plate tectonics.
Althoughthis faultmarksa major plate boundary,it has
This somewhateuphoricperiod also saw the firststeps been a seismicallyquite zone ever since Europeansarrived
in what was to prove a salutoryreminderthatthe earth about 2 centuriesago. A centralissue was to determine
does not yield its secretscheaply.In the lull before the whetherlargeearthquakeshad occurredalongthefault,and
Chinese culturalrevolution,intriguing
rumorsof eccentric if so, when.A recentworkshopbroughttogetherscientists
animalbehaviorand anomolousphysicalmeasurements
be- who had been tacklingthis issue fromdifferent
pointsof
forelargeearthquakesemanatedfrombehindthe"bamboo view: carbondatingfrompeat residues,treeringdata,data
curtain."These culminatedin 1976 withtheclaimedpredic- fromlichensgrowingon theundersideof fallenrocks.As
tionof theHaichengearthquake,leadingto the evacuation theworkshopprogressed,
in one technique
theuncertainties
of residentsfromtheirhomes and the consequentsaving were resolvedby information
providedby another.By the
of manylives. Observersfromtheinternational
seismolog- timeit finished,a clear answerhad emerged-the last maical communityvisitedand confirmedmuch of the story. jor earthquakehad occurredin 1717. Before then,two or
978
Journalof the AmericanStatisticalAssociation,September2000
threefurther
dates were established,withless certainty,
at
StatisticalSeismologyand an S-PLUS-Based Software
intervalsof from100 to 300 years.
Environment. Recentworkof ourowngrouphas focussed
on reviewingthe applicabilityof catalog-basedprediction
A TimeSeries Problem:Identification
of Preseismicand methodsto New Zealand data.Workingjointlywitha group
Coseismic Changes in Water Well Levels. Water level headed by ProfessorMa Li fromthe Chinese Seismologichanges have long been toutedas an earthquakeprecur- cal Bureau,we have developedan S-PLUS-based software
(SSLIB; see Harte 1999) for subsettingand
sor. Kitagawa and Matsumoto(1996) finallymarrieddata environment
simulating,
predictof sufficient
qualityto noise-reduction
techniques(involving displayingcatalog data and forfitting,
models.
nonlinearfiltering)
of sufficient
sensitivity
to isolatecoseis- ing,and evaluatinga rangeof conditional-intensity
mic and some smallpreseismicsignals.
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the area on the
Matthews(1992) were able to reconstruct
Press.
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faultplane thatslipped,frommeasurements
takenon the Dahmen,K., Ertas,D., and Ben-Zion,Y. (1998), "GutenbergRichterand
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surfaceacrossand alongsidethefault.In somewhatrelated
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Harte,D. (1999), "DocumentationfortheStatisticalSeismologyLibrary,"
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geophysicalinversionproblems;one example is theiruse
VictoriaUniversity,
Wellington.
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arguments,suggesting"characteristic EarthquakeResearchInstitute,University
of Tokyo,61, 1-65.
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In a series of recentarticles Only From GravitationalData: An Objective Bayesian Approach,"J.
gestingextremeirregularity.
Geophys.Res., 12097-12109.
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Ogata,Y. (1988), "Statisticalmodelsforearthquakeoccurrencesandresidtheexiscolleaguesin theUnitedStateshave demonstrated
ual analysisforpointprocesses."JournalofAmericanStatisticalAssotenceof complexsystems,imitating
featuresof tectonically ciation,83, 9-27.
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and capable of existingin two pos- Omori,F. (1894), "On After-Shocksof Earthquakes,"Journalof the Col7, 111-200.
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Bulletin
Reid,H. F (1911), "The ElasticReboundTheoryofEarthquakes,"
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of California,6, 413-444.
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in Statistics,
PartA-Theory and Methods,A10, 1559Communications
of such behaviorin real faultsystems?
1585.