214 Physicsofthe Earth andPlanetaryInteriors, 30 (1982) 214—227ElsevierScientificPublishingCompany,Amsterdam— Printedin The Netherlands

Seismicsourcespectraandmomenttensors*

D.J. Doornbos

NTNF/NORSAR,PostBox51, 2007 Kjeller (Norway)

(ReceivedDecember16, 1981; revisionacceptedApril 6, 1982)

Doornbos,D.J., 1982. Seismicsourcespectraandmomenttensors.Phys.EarthPlanet. Inter.. 30: 2 14—227.

When thesourcespectrumis expandedin powersof frequency,thecoefficientof then th poweris a linearfunctionofthen th momenttensor.This showsexplicitly thelow-frequencyapproximationby momentsof low degree,andsuggestssuitableways of extrapolation. In extrapolatingto higher frequenciesthe spectralassumptionimplied restricts thesourcemodel to a particularclass,andseveralpossibilities,includingw

2 andGaussianmodels,arehereconsidered.Themodel has20 parameters,involvingmomentsof degreezero,one,andtwo. Model-dependentconstraintscan reducethisnumberto that employedin other methods.Deterministicandstochasticinterpretationsaregiven, togetherwith therelations to source dimension and correlation length. The models consideredlead to simple expressionsfor thedisplacementfield and the total radiatedseismic energy, and can be made to satisfy certaingeneralpropertiesofobservedfar-field spectra,including the corner-frequencyshift of P waveswith respectto S waves.Momentsof firstdegreedeterminethephasespectrumof themodel,andtheirdeterminationis equivalentto theclassicalsourcelocationproblem by travel-time analysis.Momentsof second degreecontrol the spectralbandwidthand its variation withtake-off angle and wave velocity, and their determinationis comparable(but not equivalent) to corner-frequencymethodsof determiningsourcedimensions.A preliminary investigationsuggeststhat in particulartheclassof Gaussianmodels satisfiesa numberof criteriasetforth in this paper.

A practical inversionmethod andits restrictionsare discussedand illustrated by the analysisof SRO/ASROdatafrom a deep-focusevent. It is observedthat the digital SRO responsepresentlyemployedcan resolveonly poorly theexcitationspectraof this source,and that frequency-dependentQ andanomalousshort-periodamplitudevariationscansignificantly affect theestimates.The numericalresultsin this exampleshould be regardedas tentative.

1. Introduction The seismic-momenttensorrepresentsbothscalarmomentandradiationpattern,but its applications

The quantificationof seismic sources(e.g., by are restrictedto what may be regardedas pointscalarmomentor magnitude), the determination sources.In principle it is possible to removethisof sourcesize (e.g., using corner-frequencymeth- restrictionby extendingthe representationto mo-ods), and the determinationof radiation pattern ment tensorsof higherdegree(Backus,1977),and(e.g., from fault-plane solutions) are procedures inpracticeit is possibletoestimatemomenttensorswhich usuallyproceedseparatelyandusedifferent up to degreetwo (Doornbos, 1982).At this stage,partsof the seismic spectrum.Of course,the van- however, a long-periodapproximationis still im-oussourceparametersare not independentof one plied; it may be consideredthat momenttensorsanother,and are often related through semi-em- of degreeup to two can summaniseinformationpirical rules (Kanamori and Anderson, 1975). from sourceswhose rise-time and spatial extentThere would be obvious advantagesif source are smaller respectively than wave period andparameterscould be determinedsimultaneously. wavelength.In otherwords, spectral information

abovethe corner frequencycannotbe.usedin the* NORSARContributionNo. 308. presentprocedure.

0031-920l/82/0000—0000/$02.75 © 1982 ElsevierScientific PublishingCompany

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In this paperwe discussthe possibility of using wherep is the ray parameteror slowness,c theall of the spectralinformationto determinesource wave velocity, and Yk the direction cosineof theparameterssimultaneously,through the moment wave in ~.

tensorapproach.We will be guidedby the consid- Theexpansionof thfk in momentsof degreeuperationthat, for practicalpurposes,the numberof to two requires90 parameters,but if a commonsource parametersmust be kept limited; more spaceand time history is assumedfor the compo-specifically, we retain the parametersinvolving nentsof mfk, i.e.,momenttensorsof degreeup to two, as discussed th (~~)= M f(~T) (2)in previouswork (Doornbos,1981, 1982). A spec- j~’~ jk

tral assumptionis thenneededto allow for theuse then the scalarfunctionf(~,i-) canbe expandedinof high-frequency information, although there moments and the representationup to secondshould be no needto model the high-frequency degreerequires 20 parameters,six of which de-part of individual sourcespectraprecisely.In the termineMJk, the moment tensorof zero degree.following Sectionswe discussspectralmodels, in- The assumption(2) is introduced to reduce thecluding a class of ~ and Gaussianmodels. It numberof source parametersfor practical pun-remainsto testtheir consequencesagainstobserva- poses,but the equationis satisfiedfor mostof thetional evidence, but a preliminary investigation “classical” sourcemodels(Haskell, 1964; Savage,suggeststhat, in particular, the classof Gaussian 1966). The functionf(~,i~)may be assumednor-models satisfies the criteria set forth here. We malised,andthe expansionin momentsrelativetodiscuss the implications and limitations of this a referencepoint(~,r,~)canbe writtenapproach,andthen illustratethe performanceof a j~,T) = (1 — F,8,— F0 + ~F, 8,8 + F, 84~8

method basedon such modelsby inverting long- r m

andshort-perioddatafrom the SROnetwork. ++1~8r8~+...) ~~—~0)&(T—T0) (3)

where the summationconventionis assumedforthe indices I andm. F, andF.~are respectivelythe

2. Moment tensorsand low-frequencyapproxima- spatial and temporalcomponentsof Ff1,, the mo-tions ment tensorof first degree.F,,,,, F,~and.1~are the

spatial—temporalcomponentsof F~2),the momentA representationby spatial and temporalmo- tensorof seconddegree.It may be notedthat the

mentsrequires the source function to be a tran- n th-degreemomenttensorof f(~,T) is of order n,sientin bothspaceand time. The following mod- whereasthe generaln th-degreemomenttensorofification of a conventionalGreen’sfunctionrepre- mJk(~,‘r) is of order n + 2:sentation(e.g.,StumpandJohnson,1977;Aki and Mk, M.kF,Richards,1980,p. 59) is thenuseful:

An interpretationof thesemomentsin termsofu,(x, t) ff G~k(~~x, t — r)rnJk(~,r)drdV kinematicsourceparametershasbeengivenprevi-

V —~ ‘ ously (Backus, 1977; Doornbos, 1981). The mo-(I) mentsof first degreedeterminethe “mislocation”,

i.e., the differencebetweenthe referencepoint(~whereu is displacementat positionx andat time . ,, “

•1 . . ‘r ) andthe source’s centreof gravity (~, T ) for1, m is the temporaldenvativeof the moment- . . ,, ° 0

J . . . . . , which ~= 0 (called the centroid by Backus):tensor density, and GJk is a modified Greensfunction(Doornbos,1982).In integral form, F~1) = (~— ~ ‘~ — (4)

ç . The momentsof seconddegreeare determinedbyG(~,x,t—r)= g (E,x,t—r)dp . . .

-‘ Jr. ‘ the sources fimtenessin spaceand time, and therupturevelocity. They alsodependon the misloca-

and, for asymptoticwave functions, .

G]k(~,x, t — T) = — f (yk/c)g~(~, x, t — T)dp 1~2)= ~2) + 1~i)F(T) (5)

216

where is measuredwith respect to (~,f0). tion u( r) canbe written in a similar form:Only F(2) representsintrinsic sourceparameters. U(w) = L~O)(l— i~L~1)— +

In the frequency—wavenumberdomain,eq. (3)becomes Xexp(—i~T0) (9)

F(.c, to) = (i + K1F1 itoF~~~sc1iç,F,m

+ ic,coF,~— ~w2F~+ ...) 3. Source spectrawith equivalent low-frequency

X exp[i( K . — ~~)] (6) approximations

which shows the momentsas coefficients of a Fromeqs.(5) and (8), the low-frequency excita-Taylor expansionabout (it, to) = (0,0), and the tion spectrumcanbe written asfirst few terms given here form a low-frequency P( ~,~) = 1 — ~ I) — ~ .ç T(~

2) + ~ ~i~)~approximationof the spectrum.An integral repre-sentationof the displacementspectrumbecomes (10)

+00

U1(x, to) = MJk( G)k(Ic, x, ~ whereit is recalledthat the termswith F~I)merely— reflect the choiceof referencepoint. The aim is to

XF(—,c,co)dic/(2ir)3 (7) improve the source representationin the high-

frequencyrangewithout increasingthe numberofIn many cases,the Green’sfunction is strongly parameters,while satisfyingeq. (10) in the low-peakedin wavenumberspace,andthe integral in frequencylimit. On the b~sisof thesecriteria weeq. (7) is trivial. For notational convenience,we considerthe following functions:drop theintegral in mostof what follows. Wenote (a) a particularclassof to2 models(AU, 1967):that representingthe Green’sfunctionin eq.(7) bythe delta-function H(~,co)= (i ~

(11)

x [exp(i.c0. ~0)]~(~— ic0) (b) a rationalapproximation,of the form

would be equivalentto the Fraunhoferapproxima- [i — (1/4)~

2 ,~.T ~2) ~]tion often used in source analysis (AU and K(.~,to)=[1 +(l/4)w2~T~

2)~]

Richards,1980,p. 804).If wavenumbercomponentsare interpretedin xexp(_jw~T1~1)) (12)

termsof propagatingwavesit, = to~y,/c,wherec isthe wave velocity and y, a directioncosineof the (c) a classof Gaussiansourcespectra:wave(possiblygeneralisedin thesenseof Richards, N( ~,to) = exp(— ~ ~ ~ — ~ ~. T~1~) (13)1976).To be consistentwith our notationand useof the Green’sfunction, the wave direction defi- All of the models(11), (12) and (13) approachning ‘y is from the receiver to the source.Then a eq. (10) at low frequencies.Their relative perfor-slownessvector canbe formed: manceat higherfrequenciesdependson the char-

acter of the actualsource pulses,and the conse-= (~/~Y2/” y3/c, 1) quencesshould be tested against observational

and an expressionobtainedfor the displacement evidence;however,a preliminary investigationisspectrumin termsof momenttensors: conductedhereon thebasisof a few typical earth-

quake source pulses: a triangular pulse and aU1(x, to) = MJkGJk(~o,x, to)(l — ito~TF~i1 rectangularpulse.Of these,the triangularmodel

— ~ T ~2) ~ + ... )exp(— itoT0) (8) seemsto be more commonlyused,sometimesinconnectionwith a boxcarmodel. A commonap-

Notethat the spectrumof a one-dimensionalfunc- proximationto thesetwo differentfunctionsshould

217

tendto favourtheclassof spectralmodelswhich is 1.0 I I I

sufficiently general to be applicable to a wide - ‘IT

rangeof sourcefunctions.The results are given in Figs. I and 2. It is

shown that the low-degree moment-tensorap-proximationP(~,to) is reasonablebelowthecorner . T/2 l~J T/2

frequency, as are all the other equivalent low- ‘

frequencyapproximations.The rational function ~‘ 11(w)

K(~’,to)is a good approximationup to at leasttwice the corner frequency, although its high- ~ .

frequencybehaviourillustrates that it does not ~ w)

really representa physical sourcepulse.The spec-tral bandwidthof the particularto

2 model adopted K _____

is relatively large, as is the total energy, anda fit o -

by this model would give second-degreemomentswhich are correspondinglylarge. The Gaussian P(Wl F(w)

model leadsto areasonableapproximationat both -

low and high frequencies,and the total energy isgiven to within 4%. It may be noted that on alogarithmic scale,the asymptoticGaussianspec- -0.5 I I

0 W~ 5 10FREQUENCY (rad/s)in wilts of ‘/T —~

1.0 ‘ l/T Fig. 2. Spectralapproximationsto rectangularsourcepulsef(t)

(seeinset).Functionsandmodelsasin Fig. 1.

_T 0 T trum will be quite different from a conventional

I 0.5 - high-frequencyasymptote(appropriate,for exam-1 ple, for an to2 or to3 model). However, for many

purposes,including least-squaresinversion, it is• the absolutedifferenceor differencesquaredthat

• is important,andthis differencegoesto zeroin theN(w) high-frequencylimit.

0F(w)

4. Someimplications

P1*4 ICIw) Having availablea boundedsourcefunction, itis possibleto computethe total radiatedseismicenergy.Equation(8) can be rewritten in the form

0 ‘ u’C —FREQUENCY (rad/s)inunitsuf l/T—~-- ‘ U1(x, to) = M’JkG) k(~o’x, to)F(~,w)exp(—ito’~0)

Fig. I. Spectralapproximationsto triangularsourcepulsef(t) (14)(seeinset)with amplitudespectrumF(w) andcornerfrequency . .

w~:P(~),parabolicapproximation(eq. (10)); H(t.~),~2 model Thereis no loss of generalityin makinguseof the(eq. (11)); K(w), rational approximation(eq. (12)); N(w), far-field Green’s function for a homogeneousGaussianmodel (eq. (13)). mediumhaving propertiesequivalentto thoseof

218

the region surroundingthesource,andan expres- that F~,—. 0 andE, —~ ~, emphasisingthe fact thatsion for the seismicenergyis then usually a point-sourcemodel can serve only as a

low-frequencyapproximation.More generally,eqs.

E = 1 j ~ (15)and(16) imply that, for afixed staticmoment,161T2p 12 • a smaller source radiates more seismic energy.

+$5B2(~5)E(~s)[y,ykMJ,Mk, Thus eq. (17) will usually overestimatethe actual

energyradiated.

— (y,y,,M,,,)2]} (15) Thequadraticform in eq.(16) canbe rewrittenin the principal-axessystemof 1.,,,,the spatialpart

whereaand$ are the PandSvelocities,~ and~ of F(2). Let p~be the unit eigenvectorsof 1~,,,,andare the associatedslownessvectors, and we have X~, the associated(positive)eigenvalues.The spa-introducedthe energyE andspectralbandwidthB tial—temporal componentsP,., are determinedby

the coupling betweenspaceandtimethrough rup-of a pulse accordingtoture velocity. Averageruptureover the source,if

+00

E(fl=f (dto/2,~-)IF(~,to)I2 nonzero,is expectedto be in the direction of the

—00 major principal axis, say p1. Then

+00

B2(~)=f (dto/2~)to2IF(~,to)I2/E(~)

Thus the seismic energy dependsonly on two + (2/c)i,,.~(7.Pt) + i~ (18)parametersof the sourcespectrum,and any spec- The reciprocalof this form determinesthe spectraltral model which fits theseparametersshouldgive bandwidth in all of the source models(l0)—(13),a good estimate,irrespectiveof the implied sec- eventhoughthe P(,~,to) andK(~,to) modelsareond-degreemoments.For the models(11) and(13) invalid at high frequenciesandonly H( ~‘, to) has aof Section3,

well-definedcornerfrequency.Sincec is~the P- orS-wavevelocity, it is seenthat the modelsusuallypredicta wider frequencyband for P than for 5,

x = (1/4) — I / 2 (Gaussian) except in a range of directionsfor sourceswhose= l/~ to2 averagerupture velocity is close to the S-wave

velocity (unidirectional Haskell type of model).(16 ) Note that the spectral bandwidthdifferencebe-

tweenP and S in theseapproximatemodelsmayIn generalit doesnot seempossibleto evaluate . -arise irrespectiveof the propertiesof the actual

analytically the integral over solid angle~2in eq.source. For example,a Haskell-typemodel with

(15), except for the caseof a point source.In ‘the bidirectional faulting is known to havea cornerlatter caseeq. (15) reducesto an equivalenttime- frequencyfor S that is usually higher than for Pdomainresultof Rudnicki andFreund(1981),and (Savage, 1972), whereasits moment tensor and~ — F’r’r. Theresult is

associatedlow-frequencyapproximations(l0)—(13)

E~—(XP_3/2/60irp){css[2MMk+(M)2] do not. It is thereforeimportantthat this particu-lar spectralpropertyof thesemodelsis consistent

— M)2]} (17) with what seemsto be a consensusof observa-tionalopinion, namely,that cornerfrequenciesfor

However,it shouldbe realisedthat, as pointedout P areusually(thoughnotalways)higherthan for Spreviously(Backus,1977), a point sourcein space (Hanks, 1981).but not in timeis usuallynot a physicallyaccepta- Thesourcemodelsin Section3 were introducedble model, except perhapsfor explosions.On the in the frequency—wavenumberdomain,but it isotherhand,a point sourcein bothspaceandtime alsoof interesttoconsidertheinterpretationin the(i.e., with a step-functiontimedependence)implies space—timedomain. In eqs. (11)—(13), the phase

219

factor involving (~,T0) cancelsa similar factor in beendiscussedby Haskell (1966) andAki (1967),

thedisplacementresponse(eq.(8)), showingmerely although the assumedcorrelation functions werethat the result doesnot dependon the choice of different; for example,Aki’s to

2 model assumesanreferencepoint. The remaining source spectrum exponentialautocorrelationof the form exp(— ar).

canbe transformedto the space—timedomain,and It may be notedthat a model describedby Gaus-the Gaussianmodel gives the well-known expres- sian spatial correlation functions has been used,sionfor a normaldistribution: for similar reasons,to accountfor wave-scattering

sourcesin the Earth (e.g., Chernov,1960). In theN( ~ T) = (2~ —2 I~

2)I —1/2 Gaussiancorrelation model, scale lengths have

beencommonlyidentifiedwith the e~correlation

xexP[ — ~ — ~)~‘(~— ~o) value. From eqs. (18) and (21), this would implyTT0 scalelengthsof ~,= 2A, and ‘~,= 2F,~/

2,and the

(19) associatedvolume,surfaceandtime length are

Provided that this function is properly truncated, = (32/3)~ I

2A3it canbe interpreteddeterministicallyin termsof S~= 4~-A1A2the region occupied by the sourcein space and T = 4fr’/

2 (22)time. The spatial momentsfi,,,, canbe consideredas representinga uniform distribution in space.Knowledge of the correspondingequivalentuni-form sourceregionwould be useful for somepur- 5. Inversiontechniquesposes,including the determinationof stressdrop,although its value will underestimatethe total In previouswork, the time-domainversion ofregionfor a nonuniformsourcedistribution. Simi- eq. (8) was used as a basis for the inversion oflarly, the temporalmoment1~.,can be interpreted long-perioddata(Doornbos,1982):in terms of a uniform region in time. Thus, the u((x,t)M~kG]k(~O,x,t—TO)*F(.~,r) (23)equivalentuniform sourceregion is

= (20/3)~rA1A2A3 and for the Gaussianmodel, the sourcefunction

F( ~‘, T) becomesS~,=

= ~ (20) N(~,T)= (2~T~2)~)~2

where V~is a volume, S~is a surfaceappropriate xexp[ (T — ~ (24)for a planefault, and T., is a time length. At this point it shouldbe recalledthat a corn-

An alternativeview is obtainedby forming thefrequency—wavenumberspectrumI F(~,to)I 2~ For pleterepresentationrequiresintegrating the forms(8) or (23) over the ray parameteror slowness.Inthe Gaussianmodel, the transformto spaceand an asymptoticapproximation,the Green’sfunc-time gives tion G] is usually strongly peaked in slowness

IN(~,to)I2—~r(~,T)= ~ spaceand the integrationis trivial, giving a nurn-

— I \ 1 ber of terms representingfor exampleP and S(21) waves. In the moment-tensorapproach(eq. (8))x exp[— ~ ~ T (2~2~ ~T)] the integralis not a realprobleman~ay,sincethe

andthis leadsto correlationfunctions,which may moment tensors are constantsand the Green’sbe interpretedin a statisticalsense.In this view, functions and slownessvectorscan be takento-the source surfacebreakscoherentlyon a spatial getherto form new, modified Green’s functions.and temporal scalegiven by the componentsof With a Gaussianmodel in eq. (23) this is not

which may be small comparedto the total possible.In the following we usedatawhich aresourceregion. Suchstatisticalsourcemodelshave represented sufficiently by the asymptotic ap-

220

proximation.Should it happenthat this is suffi- Observationsin at least two frequencybandscientfor thehigh-frequencypartof a spectrumbut (e.g., long- and short-perioddata) areneededtonot for the low-frequencypart, then thetwo repre- estimateA. and B.; then the linear systems(29)sentations(23) and(8) canbe usedfor alternative and(30) can be inverted. The second-degreemo-partsof the spectrum. ment tensor~2) is determinedby ten parameters,

With thesereservations,it is found that the but constraintson the orientation of the sourcemoment tensors of the spectral model can be regionandon the averagerupturedirectionreduceobtainedin a way which is very similar to the this numberto six. They are mosteasilyobtainedmoment-tensorinversion presentedby Doornbos by meansof eq. (18). Here the minor axis, say p3,(1982). The linearisedequationsfor perturbations is inferredfrom the zero-degreemomenttensorMof the moment-tensorcomponentsare (apart from the so-calledfault-plane ambiguity).

One other parameter,~, is then requiredto de-— = ~ — JkGf.k~

8’~I) terminethe orientationsof p1 and p2. Thus, solu-

~ *N(~,T) (25) tions for A~,A~,X~,P andP canbe obtainedPi.”

as a function of 4. It shouldbe noted,however,where i2, is givenby eq. (23), (MJk+

8MJk) is taken that a least-squaressolution of the system(30) isto be an updatedversionof MJk, andsimilarly for actuallydeterminedby log U.; it is thereforemosttheothermoments.In practice,the solutionshould sensitiveto the very small amplitudes,where thebe constrainedfor it to be consistentwith our effect of noise is relativelysevere,and it is not aconceptionof themechanismof faulting. good measureof spectralbandwidthand energy.

It is often possible to treat the problem of For thesereasonsit is suggestedthat the least-determiningtherelocationvector1~separately.If squarescriterion be appliedto the dataset itself,Green’s functions are determined by only one and that the six parametersrelated to F(

2) beasymptoticwave,the form of eq. (24) showsthat estimateddirectly using eq. (28), or its equivalentan estimateof time-domainexpression.

(26)

canbe obtainedby standardtravel-timeanalysis,6. An exampleand solving the linear system (26) would then

amountto the usualprocedureof estimatingsourceWe haveapplied the techniqueoutlined abovelocation (e.g.,Buland, 1976). In the following we

assumez~T,to be eliminated by “aligning” ob- in an inversion of long- and short-perioddatafrom the SRO/ASROnetwork. Partly to investi-servedandsyntheticseismograms.

Simple Green’sfunctionsin the abovesensecan gate its potential in applications with lessbedecomposedas sophisticateddata, we hereusedinformationcon-

cerning body-wave polarity, long-period ampli-GJk(~O,x, ~) (sJ~k).G,(~O,x, to) (27) tudes,andshort-periodenergyin theband0.5—2.0

where and ~k are componentsof the unit dis- Hz. Amplitude datamay be relevant for sourceplacementvector and the slownessvector of the pulses which are simple in the passbandof thewave in ~. Thissuggestsa practicalprocedurefor observations,andsourceparametersrelatedto sec-obtaining initial estimatesof the momenttensors ond-degreemomentscan be interpretedin termsM and F~2).Substitutingthe Gaussianmodel, eq. of whole-sourcedimensions.If the sourcepulse is(8) canbe rewritten as complicated, energy in one or more frequency

bands should replacethe amplitude information,U,(x,to)=A,G,(~0,x,to)exp(—~oi

2B,—itoTo)(28)and the interpretationof the second-degreemo-

with mentsis in termsof correlationdistanceandtime.

A, = (s~k) ,~k (29) Wecomparedlong-periodamplitudeinversionwithcompletewaveforminversionfor the Bali Seaevent

B, = ~ (30) used in previous work. The resultsas shown in

221

TableI areverysimilar, suggestingthat at leastfor eigenvaluesof ~2)’ no otherconstraintswere ap-simpleGreen’sfunctions, the datain a waveform plied in the inversion. Therefore the parameterare highly correlated,owing to their almost identi- valuesobtaineddo not necessarilycorrespondto acalsamplingof the focal sphere.The implications particulartypeof faulting. Also, the uncertaintyinof the short-perioddataare discussedfurther at the resultsmight be larger than the formal stan-the end of this Section. dard errors indicate. This is due partly to the

We havealso applied the simultaneouslong- rather small number of stations used and theirand short-periodinversionto a morerecentevent, poorazimuthaldistribution. Figure4 clearly showsfor which more short-period data from the that the nodal planesare poorly constrainedbySRO/ASRO network were available. Figure 3 the polarities, and the nodal-plane solution isshows the vertical componentsof the body waves determinedto a largeextentby the relatively large(including P, PKP, SKP) that were used; at pre- PKP amplitudesin the N—E quadrant.The resultsent no horizontal-componentshort-perioddata shouldthereforeberegardedas tentative,althoughare available.In the inversionthe PREM velocity it is not unusualin that it indicatesdown—dipand density model (Dziewonski and Anderson, compression (cf. Isacks and Molnar, 1971).1981) wasused,togetherwith the PREM Q-model Another source of uncertainty is the scatter infor long-period data and the Q-model of particular of the short-perioddata. Although weArchambeauet a!. (1969) for short-perioddata, do not hereinvestigatethis questionin detail,weDifferencesin Q betweenlong- and short-period speculatethat systematicshort-periodamplitudedata have been suggestedrecently by several anomaliesmay lead to source models having aauthors(e.g., Lundquistand Cormier, 1980).The relativelypronounceddirectivity pattern.For eachinversion results are summarisedin TableII and phase,a Gaussianspectrumwasfitted to the long-Fig.4. Here the scalarmomentsM and N corre- andshort-perioddataafterdeconvolutionof Earthspondto a (nonunique)decompositionof the as- and instrumentresponses.Figure 5 shows thesesumeddeviatorictensorMJk into a major and a spectranormalisedto a unit DC level, and Fig. 6minordoublecouple;M correspondsto thelargest, gives the secondmoment(B, in eq.(28)), which isN to the smallest eigenvalueof MJk. Standard simply related to the spectralbandwidth and todeviationsof the scalarmomentswere estimated the pulse width. The theoretical values for theby linearising the relationship with M~.k,whose sourcesolutionin TableII are also givenin Fig. 6.standarddeviationswere obtainedas a usualby- The variationwith take-offangleis determinedbyproduct of the linear least-squaresprocedure. the spatial and spatial—temporalcomponentsofStandarddeviationsof the second-degreemoments ~2)’ and eq. (18) providesinsight concerningthewereestimatedby linearising the relationshipbe- maximumallowablevariation. Thequadraticformtweenthe solution andthe data. has a minimum for waves in the propagation

Apart from thezero-traceconstrainton M and direction of a unilateral propagatingsource (thethe application of positivity constraintson the Doppler effect), and it goes to zero only in the

TABLE!

Comparisonof amplitudeinversionandwaveforminversionfor moment tensorof Bali Seaeventof 1978, June10 a

M22 M33 M12 ~“I3

Amplitude 4.13 —1.77 —2.36 0.67 —0,48 0.28(±0.51) (±0.54) (±0.90) (±0,37) (±0.27) (±0.25)

Waveform 3.87 —0.77 —3.11 0.70 —0,58 0.31(±0.88) (±0.94) (±1.56) (±0.64) (±0.46) (±044)

a Standarddeviationsin parentheses.Dataandotherconditionsasin caseNo. 4, tables2 and 3 of Doornbos(1982).

222

RNNO 87.5 DES 8. 53.24.D—~~-f\-----------—-..’- ~ 8. 54,42.1

SNZO 21.6 DES 8. 49.46.6~~~~ P 8. 47.4.2

BCRO 157.2 DEC 9. 1. 4.2 —‘-—------.-—-“~-~—‘~‘-—-—‘\._.—-‘- PKP 9. 2. 20.2 —--“v---’------—-—.————-—--

CTRO 33.i DES 8. 47.26.8~ P 8.

Cl-ITO 89.7 DES 8. 93.33.3 .\,‘\fVtN,,../~—~...t\j\/\p ~. ~

TRIO 73.6 DEe 8. 52.9.8 ~ ~ 8. 53.29.7

ANTO 146.8 DES 9. 0. 11.8~ PKP ~

1IAJO 69,7 DEC 8, 5i.52.6 ..~,.y.....~.~.1\f\J\~l\,JVV\J\,l\V~ 8. 53.6.8

GRFO 149.5 DEC 9. 0. 19.8 ._—.~_.__.J’\~—-_._.__._____-. pkp g, ~~ ~

KONO 140.1 DES 9. 2. 43.6 ~ SKP 9. 4. 0.1 —\./~‘-—-—--—---—--.------

L-P S—P

Fig. 3. SROandASRO recordsfromFiji Islandsevent(seeTableII for details)with vertical componentsof P. PKPor SKP.Note thedifferent amplitude scalesfor the long- and short-period sections;the long-periodrecord length is 4 mm, the short-periodrecordlength, 12 s.

unlikely caseof sonicrupturevelocity. Forsources an upper bound. Finally, the assumptionof awith Fk = 0, theminimumis F~in thedirectionof particularsourcemodel would predictspatial mo-the minor principal axis. On the otherhand, the mentsconsistentwith F~,and the seismicenergymaximum is less than 5F~for several fault for that model can be computed.For example,modelswith subsonicrupturepropagation.A final case(c) of TableII was obtainedby approachingsourceof uncertaintyis due to the fact that, for in the low-frequencylimit the model of Molnar etthe data from this event, the sensitivity to source a!. (1973), for which spatial and temporal mo-finitenessand also to take-off angleappearsto be ments have been given previously (Doornbos,most importantin the frequencyrangewhere the 1982).Thus,disregardingfor the momentthe pos-digital SROresponseis small.Theresponsecurves sibility of systematicerrors,it appearsthat seismicin Fig. 5 serveto emphasisethis fact. This may be energyis estimatedto within a factorof two.a generalproblem with eventsin the magnitude In the contextof a sheardislocationmodel, therange5—6. apparent stress 775 may be obtained from the

Thetotalradiatedseismicenergy(TableII) was scalarmomentM andthe seismicenergy E,, andcomputedusingeq.(15) or (17). We also inverted the stressdrop ~a maybeobtainedfrom the scalarthe data in terms of a point sourcein spacebut momentandthe fault shapeandsurfacearea(e.g.,with a temporalmoment1~(TableII, (b)). The AU, 1972).The fault surfaceareain TableII wascorrespondingseismicenergyshouldbeconsidered computedusingeq. (20), and the stressdrop was

223

a;

CC0~5

a‘C

a

00 00 00

‘Ca —o- E ~ ,_ ~ ~ ,‘, ‘, ~a —~~a’~— _~eaa ,,

o ~ X ~ ~m 0. ~ ~- ‘C ~ ~- i~’I 00 ‘C Q ~

r~H~ a~ ~

r- ‘5 II o II o c ‘~ ~

—o ~ ~ ~

r-

1~~:,_,‘C m i-i n r’~ a

a ~, ~ 0 Q ‘ C~ a a

a~- ~-‘ :~‘~

a —

C 0.00

C a 0,’-

,-‘iT~ r~0 0 0 a+1

< •~ a00

“6 0 0

—V

C anV ‘- 0C ,,

— N,.., 0C n

— TI a‘C ‘‘ ‘‘ ‘‘

oCCE

00~ ~ 0 III

I ~‘C r-’n r-~i ~ I~C 0 — 0 — 0 —

~ cL:~~ ~. ~J.

2 a~

- fr~J’ ~ -~ -~‘

‘a _,a

224

N

P2

.P

W - P3 E

1.

S

Fig. 4. Fault-planesolution for major doublecouple, with observedpolaritiesand reducedlong-periodamplitudesof P(±)andSH(—.). PandTare thecompressionand tensionaxes;Pi, p~,p3 aretheprincipalaxesof thesourceellipsoid(in decreasingorder).

, I I I I I I

i calculatedsimply for a circular fault. Despitethe/ ratherdifferentconstraintsthe resultsfor cases(a)/ and(c) arereasonablyclose,suggestingthat useful

\~\ / • estimatesof seismicenergyandstressdrop canbe/ obtainedwithout knowing the fault geometryand/ • rupture history in detail. The estimated fault/ surfaceareaseemslargecomparedto estimatesfor/ similar earthquakesby Chung and Kanamori/ (1980),but smallcomparedto the resultsof Wyss

~ Lp • and Molnar (1972). Consequently,the estimated\‘~~‘~\ \‘\ / stressdrop also lies betweenthe resultsof these

~ \‘\ / • authors.‘\‘~~‘\\ \ \ / We now considerthe implicationsof the short-

‘I. • perioddatain connectionwith the Q-modelbeing\‘.‘~\~ ‘\ used.We repeatedthe inversionwith short-period

‘\)(“\. \ • energy in the frequency bands0—1.0 and 0—0.5

~ ~ long short penodL responses.The digital long. (Lp) and short-period(Sp) SRO

— FREQUENCY (Hz)—- responses are alsoindicated(different scales).

225

11~0o 1.0 ~ (7ç~c)~O.5 : : :

—FREQUENCY (Hz)—-

Fig. 7. Gaussianexcitationspectrafor spatialpointsourcewith0 finite temporalmomentobtainedby fitting long-periodampli-

—STATION SEOUENCE— tudes and short-period energy: (I) short-period bandwidth0.5—2,0 Hz, temporal moment /~=0.33s

2 (2) bandwidthFig. 6. Secondmomentof Gaussiansourcespectrum(B, in eq. 0—1.0 Hz, Pr., =082 s2 (3) bandwidth0—0.5 Hz, P~,= 1.47 ~2•

(28)): •, observed;0, excitationfor sourcemodel in TableII. Curve (a) indicates the digital short-period SRO response,Thestationsequenceis asin Fig. 3. Curves(b) and (c) indicate a typical Green’s function in a

standardEarth model, for P at teleseismicdistance:(b) PREMQ-model (usedwith the long-perioddata); (c) Q-model fromArchambeauet al. (usedwith theshort-perioddata).Note thedifferentamplitudescalesfor thedifferent functions.

Hz, respectively.For a spatial point source thisgaveF,.,= 0.82 s2 and1.47 s2, ascomparedto 0.33s2 for the frequencyband 0.5—2.0Hz. Thesere-sults are illustrated by the excitation spectrain 7. ConclusionFig. 7. This figure also shows the spectrumof atypical P-waveGreen’sfunction at teleseismicdis- The general moment-tensorrepresentationoftancein a standardEarthmodel andat theoutput seismic sources becomes a low-frequency ap-of the digital short-periodSROsystem.Using the proximation when only the moments of lowerQ-model of Archambeauet a!. (1969) which was degree are retained. A complete representationusedto deconvolvethe short-perioddata,thepeak above the corner frequency requires momentis near 1.4 Hz. But using the PREM Q-model tensorsof degreehigher than two, but a usefulapplied in the deconvolutionof the long-period approximation is given by the low-degreemo-data, the peakis near0.8 Hz. The conclusionis ments togetherwith a sp~ctralassumption;suchthat a frequency-independentQ-modelaffects the an assumptionrestricts tjhe source models to aestimateof sourcepulselength; it similarly affects particular class.Thesem~delscan be interpretedestimatesof sourcesize,seismicenergyandrelated in a deterministicor a stc~chasticsense,they leadquantities.This corroboratesa conclusionreached to simple expressionsfor~the displacementfieldby Choy and Boatwright (1981), also in a source andthetotal radiatedseismicenergy,andtheycananalysisstudy.Der (1981)ratherextensivelydocu- be made to satisfy certain generalpropertiesofmentedthe presentdiscrepancybetweencurrent observedfar-field spectra,including the corner-Q-modelsandhigh-frequencydata,and Lundquist frequency shift for P waves with respect to Sand Cormier (1980) have made an attempt to waves.It remainsto testthe resultsof thesemod-estimatethe frequencydependenceof Q. Quantita- els against observational evidence, but a pre-tive sourceanalysiscan be expectedto improve liminary investigationusing synthetic waveformssignificantly if adequatecorrectionscan be oh- suggeststhat, of the modelsconsidered,the classtamedfor this effect, andfor the effect of anoma- of Gaussianmodels in particular satisfies thebusamplitudevariations, criteria set forth in this paper. Although on a

226

logarithmic scale the high-frequencypart of the phases.Sinceno attemptwas madeto correctforGaussianspectrummay differ significantly from the above-mentionedeffects,the numericalresultsconventionalto2 or ~ asymptotes,the absolute obtainedshouldbe regardedas tentative.amplitude differenceis small and would not sig-nificantly affect least-squaresestimates of thesourceparameters. Acknowledgement

A misbocationof the sourcegives rise to mo-mentsof first degree.They determinethe phase This researchwas supportedby the Advancedspectrumin the spectralmodel, andcan often be ResearchProjectsAgency of the Departmentofanalysedseparatelyby standardtravel-timeanaly- Defenseand was monitoredby AFTAC, Patricksis. The source’sspatial and temporalextentgive AFB, FL 32925, underContractNo. F08606-79-rise to momentsof seconddegree,which control C-000l.the spectralbandwidthandits variationwith take-off angleand wave velocity. In the past, muchemphasishasbeenplacedon thecornerfrequency Referencesas a measureof the spectralbandwidth and as ameansto determinethe sourcedimension.Brune’s Aki, K., 1967. Sealing law of seismic spectrum.J. Geophys.

(1970) model is often used,partly becauseit pro- Res.,72: 1217—1231.

vides an estimateof sourcedimensionevenfrom a Aki, K., 1972. Earthquakemechanism.Tectonophysics,13:

singlestationrecord.It is possibleto do this in the 423446.Aki, K. and Richards, P.G., 1980. QuantitativeSeismology.

contextof a moment-tensorspectralmodel too: if W.H. Freeman,SanFrancisco,932 pp.

sourceshapeand rupturevelocity are prescribed, Archambeau,C.B., Flinn, E.A. andLambert, D.G., 1969. Fine

the number of independentsecond-degreemo- structureof theupper mantle.J. Geophys.Res., 74: 5825—mentsis two; if the fault is circular, this number 5865.

Backus, G.E., 1977. Interpretingthe seismicglut momentsofreducesto one.total degreetwo or less. Geophys.J. R. Astron.Soc., 51:Long- andshort-perioddata,or moreprecisely, 1-25.

data in at least two frequency bands, can be Brune. J.N., 1970. Tectonicstressand thespectraof seismic

inverted by a perturbationmethod analogousto shearwavesfrom earthquakes.J. Geophys.Res.,75: 4997—

that used in previouswork. However, it is often 5009.Buland, R., 1976. Themechanicsof locatingearthquakes.Bull.possibleto obtaininitial estimatesof the moments Seismol.Soc. Am., 66: 173—187.of degreezero andtwo separately,wherethe zero- Chernov,L.A., 1960. Wavepropagationin a RandomMedium.

degreemomentsare obtainedby linearinversion, McGraw-Hill, New York.

and the second-degreemomentsby nonlinearin- Choy, G.L. and Boatwright,J., 1981. The rupturecharactens-version.In an applicationof this techniqueto the tics of two deep earthquakesinferred from broadband

analysisof a deepeventin the Fiji Islands region GDSNdata. Bull. Seismol.Soc. Am., 71: 691—711.we observedthat (1) the obtainableresolution of Chung, W. and Kanamori, H., 1980. Variation of seismicsourceparametersandstressdropswithin a descendingslabthespatialandtemporalparametersof this source andits implications in plate mechanics.Phys.Earth Planet.would be maximum in a frequencyrangewhere Inter., 23: 134-159.

the digital SROresponseis small,(2) knowledgeof Der, Z.A., 1981. Q of the earth in the 0.5—10 Hz bandthe frequencydependenceof Q is neededto recon- (unpublisheddata).Doornbos,D.J., 1981. Seismicmoment tensors.In: E.S. Huse-cue long- and short-perioddata, or even data bye and S. Mykkeltveit (Editors),Identificationof Seismicwithin the short-periodband, and (3) amplitude Sources.D. Reidel,pp. 207—232.

variationsaresometimeslarger than explainedby Doornbos,D.J., 1982. Seismicmoment tensorsand kinematic

sourceradiationand standardpropagationeffects. sourceparameters.Geophys.J.R. Astron.Soc.,69: 235—251.Points(2) and(3) corroboratethe resultsof other Dziewonski, A.M. and Anderson,D.L., 1981. Preliminaryref-erenceEarthmodel.Phys.EarthPlanet.Inter., 25: 297—356.workers. The dataset used in our examplewas Hanks, T.L., 1981. The corner frequencyshift, earthquake

ratherlimited, consistingof the polarity andlong- sourcemodels,andQ. Bull. Seismol.Soc.Am., 71: 597—612.and short-periodamplitudesof ten body-wave Haskell, N.A., 1964. Total energyand energyspectraldensity

227

of elastic wave radiation from propagatingfaults. Bull. quenciesof P and S waves and models of earthquakeSeismol.Soc. Am., 54: 1811—1841. sources.Bull. Seismol.Soc. Am., 63: 2091—2104.

Haskell,NA., 1966. Total energyand energyspectraldensity Richards,P.G., 1976. On the adequacyof plane-wavereflec-of elasticwaveradiationfrom propagatingfaults.PartII. A tion/transmissioncoefficients in the analysis of seismicstatistical sourcemodel. Bull. Seismol.Soc. Am., 56: 125— bodywaves.Bull. Seismol.Soc. Am., 66: 701—717.140. Rudnicki, J.W. and Freund,LB., 1981. On energyradiation

Isacks,B. and Molnar, P., 1971. Distributionof stressesin the from seismicsources.Bull. Seismol.Soc. Am., 71: 583—595.descendinglithospherefroma global surveyof focalmecha- Savage,J.C., 1966. Radiationfroma realisticmodel of faulting.nismsolutionsof mantleearthquakes.Rev. Geophys.Space Bull. Seismol.Soc.Am., 56: 577—592.Phys.,9: 103—174. Savage,J.C., 1972. Relationof corner frequencyto fault dimen-

Kanamori,H. and Anderson,D.L., 1975. Theoreticalbasis of sions.J. Geophys.Res.,77: 3788—3795.someempirical relationsin seismology.Bull. Seismol. Soc. Stump, B.W. and Johnson,L.R., 1977. The determinationofAm., 65: 1073—1095. sourcepropertiesby the linear inversionof seismograms.

Lundquist,G.M. andCormier, V.C., 1980.Constraintson the Bull. Seismol.Soc. Am., 67: 1489—1502.absorptionbandmodel of Q. J. Geophys.Res.,85: 5244— Wyss, M. and Molnar, P., 1972. Sourceparametersof inter-5256. mediate and deep focus earthquakesin the Tonga Arc.

Molnar, P., Tucker, B.E. and Brune,J.N., 1973. Corner fre- Phys.Earth Planet. Inter., 6: 279—292.