Physicsof theEarth andPlanetaryInteriors, 10 (1975)12—48©ElsevierScientificPublishingCompany,Amsterdam— Printedin TheNetherlands

PARAMETRICALLY SIMPLE EARTH MODELS CONSISTENT WITH GEOPHYSICAL DATA

A.M. DZIEWONSKI*, A.L. HALES andE.R. LAPWOOD**

ResearchSchoolof EarthSciences,AustralianNational University, Canberra,A.C. T. (Australia)

SubmittedSeptember3, 1974; revisedversionacceptedDecember3, 1974

We presenta setof threeparametricearthmodels(PEM) in which radialvariationsof thedensityandveloc-ities arerepresentedby piecewisecontinuousanalyticalfunctionsof radius(polynomialsof ordernothigherthanthethird). Whileall threemodelsareidentical belowa depthof 420km, modelsPEM-OandPEM-Caredesignedto reflect the differentpropertiesof theoceanicand continentaluppermantles,respectively.The thirdmodelPEM-A is a representationof anaverageearth.

Thedatausedin inversionconsistof observationsof eigenperiodsfor 1064 normalmodes,246 travel timesofbody wavesfor five different phasesandregional surface-wavedispersiondataextendingto periodsasshortas 20seconds.Agreementof thefunctionalsderivedfor thePEM modelswith theappropriateobservationsis satisfactory.In particular,the fit of free-oscillationdatais comparableto that obtainedin inversionstudiesin which constraintsimposedon the smoothnessof structurewerenot assevereasin our study.

Our density distributionfor all depthsgreaterthan670 km is consistentwith theAdams-Williamsonequationtowithin 0.2%maximumdeviation,and theseminutedeparturesresultonly fromthelimitations imposedby theparam-etric simplicity of our models.We alsoshowthat thevelocitiesin the lowermantleareconsistentwith the coin—pletethird-orderfinite-strain theory to within 0.2%for Vp and0.4%for VS (r.m.s.relativedeviations).The derivedpressurederivativesof thevelocitiesarevery similar to thoseobtainedfor corundumstructuresin laboratoryex-periments.

We concludethat anydeparturesfromhomogeneityandadiabaticitywithin the innercore,outer coreor lowermantlemustbevery small,andthatintroduction of suchdeviationsis not necessaryon thebasisof the availableobservationalevidence.

1. Introduction If sucha modelwere to be usedasabasisfor com-

parisonwithotherearthmodels,or their functionals,Haleset al. (1974)suggestedthe constructionof a thenrepresentationof the seismicparametersby

sphericallysymmetricalearthmodelin which the piecewisecontinuousanalyticalfunctionswouldhaveradialvariationsof seismicparametersareexpressed severaladvantagesovera modeldefinedat a numberby piecewisecontinuousanalyticalfunctionssuchas of discretepoints.Theseadvantagesareprincipallylow-order polynomialsin theearthradius.This sugges- relatedto the fact thatthe seismicparametersandtion was relatedto theactivitiesof the StandardEarth their derivativescouldthenbecalculatedexactlyforModel Committeeof theI.U.G.G. underthe chairman- any desiredvalueof the radiuswithout resorttoshipof ProfessorK.E. Bullen. This representationis numericalinterpolationwhichalwaysinvolvescertainconsistentwith theparametricsimplicity requiredin subjectiveassumptions.Also, the functionalsof theanyreferenceearthmodel. modelsuchas thetravel timesof body wavesand their

derivativeswould alwaysvary smoothlyas a functionof distanceon a particularbranchof a travel-time

* Permanentaddress:Departmentof GeologicalSciences, curveHarvardUniversity,Cambridge,Massachusetts,U.S.A. .

** Onleavefrom: Departmentof Applied Mathematicsand The remarksabovedescnbethegenesisof our ap-TheoreticalPhysics,Universityof Cambridge,Cambridge, proachto constructionof an earthmodel,but theEngland. moreimportantreasonsfor undertakingthis effort

PARAMETRICALLY SIMPLE EARTH MODELS 13

are closelyrelatedto basicproblemsof the structure Gilbert (1971b) hasshownthat if we averageob-of the deepinteriorof the earth,suchasthe evidence servationsof thenormal-modeeigenfrequenciesob-for radialinhornogeneitiesin the chemicalor mineral- tamedfor a well-distributednetworkof stationsandogical compositionof thematterin the lowermantle sources,the effectof lateralheterogeneitieswould beandthe cores.To be detectedby seismologists,such eliminated,andthat theaverageeigenfrequencyob-inhomogeneitiesmustbe expressedby a changein tamedin this waywould correspondto the degener-seismicvelocitiesor in density,andcouldbe represent- ateeigenfrequencyof a radiallysymmetric,non-edby a perturbationin thederivativeof a seismicparam- rotating,averageearth.However,the seismicstationseterwith respectto radiusor by a first-orderdisconti- are certainlynotevenlydistributedon thesurfaceofnuity, dependingon themethodof inversion.If, on the the earth,andthebulk of our normal-modedataotherhand,thematerialwithin the lowermantle,or comesfrom only two events(theAlaskanearthquakethecores,werehomogeneous,the absolutevaluesof of 1964 andthe Colombianearthquakeof 1970).Thusderivativesof the velocitiesanddensitywith respect a possibility of biasexists.Recognizingthis,Gilbertto the radiusshoulddecreasesmoothlyandmonotoni- and Dziewonski(1975)assigneda minimumrelativecally with increasingdepth.Furthermore,if the temper- standarderrorof 0.05%,and they checkedthat theaturegradientbelow a depthof 600 — 700 km were overall distributionof residualsfollowed closely theadiabatic,thenthe densitygradientshouldsatisfythe normal distribution.Adams-Williamsonequation. However,it is assumedin the inversionthat the

It hasbeensuggestedin severalstudiesbasedon covariancematrix is diagonal,i.e. thaterrorsarenotdT/dz~measurementsusingseismicarraysthat there correlated.If biasexiststhis maynot be true and thearenumerousperturbationsin thegradientof corn- precisionof the significantearthdatamay be over-pressionalvelocitiesin the lowermantle (for reviews estimated.Someindicationcanbe foundin table 7seeHalesandHerrin, 1972;Wright andCleary, 1972 of Gilbert and Dziewonski(1975)wheredeviationsor Wiggins et al., 1973).The discrepanciesbetween betweenthe calculatedandobservedperiodsshowthe resultsobtainedat different sitessuggest,how- the samesign formoreconsecutivemodesthanstatisti-ever,that theperturbationsin thedT/d~curvesmight cally probable.result from regionalandlocal variationsin the crustal Theeffect of the estimatesof the precisionof theandupper-mantlestructures, dataon the measureof theperturbationaddedto the

Freeoscillations,or normalmodes,of the earthcan startingmodel is particularly clearin theformulationbe assumedto be free of suchlocal effectsbecauseof of Gilbert (1971a,eqs.8—10). For a particularsetoftheir largewavelengths.Recently,JordanandAnder- dataanartificial decreasein theestimatesof standardson(1974)reportedan earthmodel (Bl) consistent errorswill, in general,resultin an increasedmeasureofwith free-oscillationandtravel-timedata;Gilbert and perturbationin themodel,unlessthestartingmodelDziewonski(1975)derivedearthmodels(1066Aand fits the dataperfectly.1066B) usingover one thousandnormal-modedata. The significanceof the featuresin anearthmodel,All of thesemodelsshowminorperturbationsin the obtainedby applicationof an inversionmethodingradientsof seismicparametersthatareinconsistent which the wavelengthof perturbationsare dependentwith thehypothesisof homogeneityof the lower on theadoptederrorsof observations,is questionablemantleandthe cores.The perturbationsareparticularly oncethe possibilitythat errorsare correlatedhasnoticeablein theoutercoreof model Bl and for the arisen.compressionalvelocity in the lowermantleof the In this paperwe adopta certainhypothesiswithmodelsof Gilbert andDziewonski.The inversion regardto thesmoothnessof thedistribution of themethodsused,althoughdifferentin eachstudy, re- seismicparameterswith theradius;the observeddataquirethat theperturbationaddedto the startingmodel are invertedusingall thedegreesof freedomallowedbe assmoothaspossible.Thus, it would appearthat by thehypothesis;following the inversion,theresidualsthesevariationsin gradientmustbereal, are examined— if they are acceptable,the hypothesis

Thisconclusionmaybe invalid if the precisionof mustbe consideredplausible— if they exceedthe al’thedata,or subsetsof data,usedin the inversionwere lowablelimits, thehypothesismustbe rejected.overestimated. Our hypothesis,consistentwith the approachpro-

14 A.M. DZIEWONSKI, A.L. HALES AND E.R. LAPWOOD

posedby Haleset al. (1974),is that distributionof the uousand discontinuousstructuresin the uppermantle,velocitiesanddensityin the lowermantleand thecores which is notsurprisingif oneexaminestheaveragingcanbe adequatelydescribedby a setof second-and lengthsfor this regionof the earthcomputedbythird-degreepolynomialsin radius,the absolutevalues Gilbert et al. (1973).Anotherconclusionis that as-of their first derivativeswithrespectto the radiusde- sumptionsmadewith respectto thedetailsof thecreasingwithin eachregionsmoothlyand monotoni- structureof the uppermantledo notintroducea de-cally with increasingdepth.The requirementthat the tectablebias in the resultsof inversionat greaterdensitydistributionshouldfollow theAdams-William- depths.This allows usto considercertain aspectsofsonequationis consideredasanadditionaloption in theearth’sstructureat depthslessthan 420kin simul-further constrainingthenumberof degreesof freedom. taneouslywith the alreadyoutlinedproblemrelatedtoThe densitydistributionsin many of the recentearth the deepinterior of the earth.modelsarevery closeto satisfyingthis equation(cf. Hales(1974a)hasquestionedthevalidity of the ap-Dziewonski,1971a;JordanandAnderson,1974; proachesto derivationof a structurein theoutermostGilbert andDziewonski,1975). 100—200km usedby DziewonskiandGilbert (1972)

Themostsignificant differencesbetweentwo earth andJordanandAnderson(1974)in their constructionsmodelspresentedby Gilbert andDziewonski(1975) of an averageearthmodel.He proposedthat one shouldoccurin theuppermantle;model 1066Ais described considertwo earthmodels— onewith anoceaniccrustby a set of functionsthat are continuousfrom the and uppermantle (including a layerof waterat thebottom of the crustto thecore—mantleboundary; surface)andtheotherwith an averagecontinentalmodel l066B hastwo first-orderdiscontinuitiesat structure.The observedperiodsof free oscillationsdepthsof 420 and 670 km, asdoesmodel Bl of shouldthen be comparedwith a weightedaverageofJordanand Anderson(1974)which wasusedasa the periodscomputedfor eachof the structures.Al-startingmode in derivationof 1066B.Yet both 1066- thoughthereexistsa satisfactoryapproachto deriva-modelsgive equally good fits to the data,and their tion of an averageearthmodel,aswe shall showlater,parametersbelowa depthof 1000km are practically thesuggestionof Halesof derivationof two earthidentical.Oneobviousconclusionis That the dataused modelsis worth following up for at least two reasons:in the inversionfail to distinguishbetweenthe contin- Fig. I showscomputedgroup-velocitydispersion

GROUP VELOCITIES OF RAYLEIGH WAVES

~3.5 / A 00 0

— MODEL I-$BI

—— MODEL 5081E0

MODEL RI0

0 AR A OBSERVED OCEANIC0

0 000 OBSERVED CONTINENTAL3-0 ooo°°°

15 20 30 40 50 100 150 200 300 400

PERIOD (s)

Fig. 1. Groupvelocitiesof Rayleighwavescomputedfor model HB1 of Haddonand Bullen(1969), 5.08Mof KanamoriandPress(1970)andBl of JordanandAnderson(1974)arecomparedwith typical dispersioncurvesobservedfor oceanicandcontinentalpaths.

PARAMETRICALLY SIMPLE EARTH MODELS 15

curvesto periodsasshortas 15 secondsfor three by thoseauthorscouldbe biased,becauseof thestruc-widely quotedaverageearthmodels:HB1 of Haddon ture adoptedin theuppermostregionof their averageandBullen (1969),5.08Mof KanamoriandPress earthmodels.As themeasurementsandthe reduction(1970)andBi of JordanandAnderson(1974).The of the travel-timedataarealmostentirely for land-group velocitiesof Rayleighwavesfall, for all these basedstations,it is logical that the travel-timedatamodels,outsidetherangeof observedvaluesfor any shouldbe invertedfor an earthmodelwith continentaltype of terrestrialstructure.Forexample,the group crustanduppermantle.The base-linecorrectionsvelocity for model 5.08Mreachesa valueof 4.20 km/ determinedfor this type of model shouldbebothsecat a periodof 37 sec,which is 4—6%higher than more objectiveand also of moreimmediateapplica-themaximumobservedin this periodrangefor ocean- tion, asit would benotunreasonableto suggestthatic paths.It should,of course,be recognizedthat short- suchcorrectionsbeintroducedto existing travel-perioddatawerenotusedin theconstructionof timetables.model 5.08Mandbecauseof thebasicdifferencesbe- . -

2. Inversionprocedureanddataselectiontweenthe oceanicand continentaldispersioncurvesat shortperiodsan averageearthmodel would be of As usual,the first step in attemptingto solve a non-doubtful usefulnessin this period rangeanyway.On linearinverseproblemby the linearestimationmethodthe otherhand,themodelsof thecrustandupper is to createa satisfactorystartingmodel.Sinceour hy-mantle that are obtainedby inversionof the short- pothesisrequiresthat thefinal modelshouldbe re-perioddataare usually terminatedat a depthof sev- presentedby a set of low-order polynomialsin theeralhundredkilometers,andlittle considerationis radius,it is only naturalto demandthatboth thegiven to therequirementthat if combinedwith a startingmodel andtheperturbationsdeterminedmodel of thedeepinteriorof the earth,suchmodels throughinversionbe expressedby functiOnsof theshouldproduceperiodsof free oscillationsthat are sametype.consistentwith observations. Haleset al. (1974)madespecific recommendations

We presentin this paperaverageoceanicandcon- with respectto parameterizationof thestartingandtinentalmodelsthat areconsistent,in thesenseof final models.While their suggestionthat the lowerweightedaverages,with the free-oscillationdataand mantleand the coresbe representedby smoothfunc-“pure path” dispersiondatafrom 150 to 300 sec,and tionsshouldbe considereduncontroversialin viewalso reproducethe characteristicpatternof dispersed of the objectivesof this work, the reasonsfor rep-surfacewavesin the periodrangefrom 20 to 100 sec. resentingtheuppermantleby a seriesof regionsSuchreferencemodelscouldbeused,for example,in separatedby abruptdiscontinuitiesare notequallycalculationof excitationparametersby earthquake compelling.Onecanproducea numberof argumentssources,or in applicationof the “residual dispersion bothfor andagainstthis form of representationofmethod”of Dziewonskiet al. (1972).But, aboveall, thepropertiesof theuppermantle.Our final deci-the requirementthat the modelsshouldbe consistent sion in this respectwasbasedon conceptualgroundswith the datain a periodrangefrom 20 secto over ratherthan on directevidence,whichis difficult to3000secmay be expectedtoimposeadditionalcon- obtain.We believethat our representationclearlystraintson thepropertiesof eachtype of upper-mantle identifiesthe particularregionsof theuppermantlestructure, suchasthe lithosphere,the low-velocity zoneand re-

Another problem that canbe investigatedby con- gionsof phasetransformations.Eachof theseregionsisstructionof distinctmodelsfor the continentsand of distinctsignificanceinourpresentunderstandingofoceansis that of thebase-linesto the travel timesof large-scalegeologicalprocesses,and for this reasonwebody waves.Gilbert et al. (1973)andJordanand think that they shouldbe clearly identified.At theAnderson(1974)havepointedout that it is neces- sametime, we mustcautionthe readeragainstattach-saryto introducecorrectionsto thebase-linevalues ing toomuchsignificanceto thespecific valuesofof thetravel times so as to assuretheir compatibility depthsassignedto thediscontinuities,or of thegra-with the normal-modedata. Hales(l974a)notedthat dientswithinparticularregions.the estimatesof thebase-linecorrectionssuggested Ourstartingmodelwasbasedon model 1066Bof

16 A.M. DZIEWONSKI, A.L. HALES AND E.R. LAPWOOD

Gilbert andDziewonski(1975),which resultedfrom 220km, andthevelocitiesin the LVZ (constant)andinversionof model Bi of JordanandAnderson(1974) theregionbetween220and420km (linearvariation)with anaugmentedset of normal-modedata.In the werechosensoasto representasclosely aspossiblelowermantleandthe cores,polynomialsof theorder thepropertiesof model 1066Bin this range.specifiedin table I of Haleset al. (1974)were fitted Calculationsof Gilbert et a!. (1973)indicatethatby leastsquaresto the specificparametersof model attemptsto determineindependentlydensitiesfor each1066B. The fit wasquiteclose;in the lowermantle of thethreeregionsbetweenthe crustandadepthofthemaximumdeviationwas of theorder0.5%. We 420km would notbejustifiedby the resolvingpowerhavefound that the densityin the lowermantlecan of thedataset.Accordingly, we haverepresentedbe adequatelydescribedby a polynomialof only densitiesfor this entiredepthrangeby a first-ordersecondorderand,also, that the densitygradientin polynomialfitted to the densitydistributionof modelthe lowermantleandthe outercorefollows very 1066B.Thisrepresentationhasenoughdegreesof free-closely the Adams-Williamsonequation.Consequent- dom that if a resultsuchasthat presentedby Pressly, we havechangedtheparametersof densitydistribu- (1969)wererequiredby thedata,it couldbe accomo-tion of our startingmodel so that it follows exactly datedby anincreasein densitiesimmediatelybelowtheAdams-Williamsonequationfor all regionsbelow thecrust.670 km depth.The densitygradientin the outermost Once thestartingmodelsareestablished,it is possi-partof the innercore of model 1066B is significantly ble to computetheir functionalssuchaseigenfrequen-greaterthan adiabatic,butwe haveforcedit to follow ciesor travel timesof bodywavesandtheir first-ordertheAdams-Williamsonequation,as the local deviations variationalparameters.of the densityfrom valuespredictedby theAdams- The dataconsideredin the initial stageof our studyWilliamsonequationare of the orderof 0.5%and consistedof asetof 1064normal-modedatapresentedbeyondtheresolvingpowerof the normal-modedata in table 7 of Gilbert andDziewonski(1975)andasetfor thedensityin this depthrange.Thecoreradii r1 set of travel timesof bodywaves.We havechosentoandr2 (Hales et a!., 1974)of the startingmodel were include in our inversiontravel-timedatafor P, S, SKSthoseof model 1 066Bandtheseparameterswere andPKIKP anddifferential travel timesof SKKS—SKS.allowedto vary in the inversion. Thenormal-modedataareconsideredto represent

Forthe radii r~andr~of theupper-mantlediscon- theaveragepropertiesof theearthandwill bereferredtinuities(Hales et al., 1974) we haveadoptedvalues to as grossearthdata(GED),following the termfirstof 5701 km and5951 km (depthsof 670 and420km, usedby BackusandGilbert (1967).The travel-timerespectively).Thesetwo radiiwere heldfixed in the data,which accordingto our assumptionsareto beinversion.Linear relationshipswere assumedfor the consistentwith thecontinentaluppermantle,repre-parametersin thedepthrangefrom 420 to 670km. sent “regionalearthdata” (RED). The subsetof REDThe zonefrom the surfaceto a depthof 420 km is the will be expandedlaterby incorporationof regionalregion in which oceanicandcontinental-typemodels surface-wavedispersiondata.may,accordingto our assumptions,havedifferent If anobserveddatum‘y~is of GED type,thentheproperties.Ourcrustalmodelswere adoptedaccording computedaveragefunctional is definedas:to recommendationsof Hales(1974b);theoceanicmodelhaving 4 km of waterat surfaceand 7 km g~= ~ + ~g,~crust;the continentalcrustbeing representedby twolayers— uppercrustof thickness20km, andlower whereg~andg~are thefunctionalscomputedfor ourcrust 15 km. The top of the low-velocity zone(LVZ) oceanicandcontinentalmodels,respectively,andthewasassumedfor bothmodelsto be at 120 km depth weighting factors2/3 and1/3are meantto approxi-(but was reconsideredlater andchangedfor oceanic matethe arealproportionsof theearth’ssurfacecover-structuretu be consistentwith the surface-waveevi- edby oceansandcontinents.dence).The velocitiesin theoverlyinglid were assumed We developtwo types of observationalequations:to be8.1 km/secfor Vp and4.65 km/secfor V~.The onefor GED andthe other for RED data.For GEDbottomof the LVZ wasassumedto be at a depthof datawe have:

PARAMETRICALLY SIMPLE EARTH MODELS 17

N N0 long-periodsurface-wavedatacanbeexpectedto con-

Ap~jXn + ~ A~°1X~+ tam informationthat representsa betterarealaveragen1 n 1 of thepropertiesof theearth.To reconcilethesetwo

typesof datait maybe necessaryto introducecorrec-Nc tionsto the base-linelevel of particularsubsetsof the

+ ~ ~ A~X~= travel-timedata,andsomespecificvalueshavebeenn’l recommendedby Gilbert et al. (1973)andJordanand

Anderson(1974).JordanandAndersonusedin theirwhere6gf~= — g~Nrepresentsthe numberof un- inversiondifferential traveltimeswhich shouldbe toknownparameters(coefficientsof polynomials)that agreatextentfree of thenear-surfaceregionaleffects.are commonto bothmodels,N0andNc the numberof However,the observationsof differentialtravel timesparametersthatare distinctfor oceanicandcontinen- suchasPcP—P,ScS—S,etc.arenotnearlyasnumer-tal models,respectively. ousasthosefor phasessuchasP, S, SKS. Also, the

If X,~is a coefficientof anm-thpowertermin a assumptionthat the differentialtravel timesarefreepolynomialdescribingthedistribution of a seismic from the effectsof lateralheterogeneitiesmaynotparameterbetweenradii and then: be justified. Unpublishedobservationsof a nuclear

explosionby HalesandNation indicatea differencer1~1 of 1 secin thedifferential traveltime of PcP—Pob-

= J” G1(r) . r°~dr servedatlocationslessthan 100 km apartafterarj correctionfor theepicentraldistancehasbeenapplied.

whereG~(r)is the differentialkernelfor a particular Forthesereasonswe decidedthat it would be bestseismicparameter;for example,shearvelocity.More to usethe travel times(P, S, SKS,PKIKP) thatarespecificallyG7 for thecommonparametersXn is morelikely to representaveragepropertiesof thedeterminedasa weightedaverageof the differential earth’sinterior and to allow in theprocessof inver-kernelsobtainedfor oceanicandcontinentalstruc- sion for an adjustmentin base-linevaluessuchasistures. requiredto assurecompatibilityof traveltime and

An bbservationalequationfor a regionaldatum, free-oscillationdata.is: In selectinga setof observedP-wavetravel times

we haveconsideredthe resultsof ClearyandHalesN NR (1966)andHerrin et al. (1968).The principal differ-

R~ AniXn + ~ A’~’.X~— V1 ‘f~= g1 encebetweenthesetwo setsis in the slope of then1 n1

respectivetravel-timecurves,which amountsto travel-NR beingthenumberof parametersfor the relevant timedifferencesof approximately1 .5 secovertheregion. TheunknownY~representsa correctionto epicentraldistancerangefrom 30 to 90°.We decidedthej-th subsetof RED. Forexample,it maybe a to usebothsetsof datain separateinversionrunsandbase-linecorrectionto the setof observedP-wave to retainfor furtheranalysisthe setthat shouldprovetravel times; in this casef~,= 1. Later whenwe intro- morecompatiblewith the normal-modedata.duceregionalsurfacewave datawe shallhavef~j= The S and SKS traveltimeswere takenfrom the[(Ii + ~)w~]’,wherew, is the eigenfrequencyof the studyof HalesandRoberts(1970),andin additionwei-th datumand1, theangularordernumber, havealsousedSKKS—SKSdifferentialtravel timesof

Becausethe uppermantleof theearthis heteroge- HalesandRoberts(1971).Theinversionstudyofneous,the traveltimesmeasuredat particularlocations Gilbert and Dziewonski(1975)in which only normal-showsystematicdifferencesin comparisonwith other modedatawereusedshowedthat their free-oscilla-locations.To removethis effect a conceptof “station tion dataare morecompatible,regardlessof thestart-corrections”hasbeenintroduced.As theproper ing model,with the PKIKP dataof Clearyand.Halesworld-wideaveragesof thetravel times are notknown, (1971)thanwith thoseof Bolt (1968).Accordingly,thechoiceof “zero-correction”or thebase-linelevel we haveadoptedtheset of ClearyandHales.is to somedegreearbitrary.The free oscillationsand ThebranchesAB andBC of the PKP phaseoccur

18 A.M. DZIEWONSKI, A.L. HALES AND E.R. LAPWOOD

on seismogramsaslaterarrivals,and for this reason pathcompositionshouldapproachtheglobal area!maybe knownless accurately.Also thereis some composition.questionasto the extentof errorsdue to theapproxi- Therehasbeensomeconcernoverthe discrepancymationsinherentin geometricalray theory. The dis- betweenthe resultsof Rayleighwave “pure path”coveryby Haddon(1972),later elaboratedby Cleary analysesby Kanamori(1970)andDziewonski(1971b).and Haddon(1972), that the so-calledGH branchis Wu (1972)divided the earthinto four typesof regionrelatedto scatteringat the core—mantleboundary — oceans,continents,arcs andridges— andfound thatputs in doubt the interpretationof publishedobserva- thetwo setsof datagive similar results.As thediscre-tions of the BC branch.Rays belongingto theAB pancybetweenthe resultsof Kanamoriand Dziewonskibranchbottom in thedepthrangecoveredby theob- is clearly dueto inadequatesampling,it would appearservationsof SKS.Forthesereasonswe decidedthat that in a searchfor consistencyoneshouldreduceincorporationof PKP travel times is unnecessary,if ratherthanincreasethenumberof typesof region;fornot undesirable. a single “region” — theaverageearth,the resultsof

The initial inversionruns showedthat withoutaddi- KanamoriandDziewonskiarepracticallyidentical.tionalconstraintstheproblemof base-linesof travel We havefound that if theearthis divided into two re-timescouldnotbe resolvedsuccessfully.We found gions— oceansandcontinents,thesetsof dataofit possibleto satisfy the travel-timedatawithout base- Içanamoriand Dziewonskialso give similar results.line corrections,thenecessaryincrease(model l066B From the analysisof combinedsetsof dataofwould haverequiredpositivebase-linecorrectionsfor KanamoriandDziewonskiwe obtainedperiodsofbothPand S travel times) of thevelocitiesof thecon- spheroidaloscillationsfrom 0S21to 0S61for averagetinental uppermantlebeingcompensatedby a decrease oceanicandcontinentalregions,and periodsofof velocitiesof the oceanicuppermantle.However, toroidaloscillationsfrom 0T23to 0T67from thewe could notverify whethertheresulting differences Love wave dataof Kanamori.in thevelocitieswerenotartificial. Nevertheless,as the After thesedatawereincludedin our inversion,itresultof this experimentwe were ableto determine becameclear that theproblemof base-linecorrectionsthat the P-wavetravel-timedataof Herrin et al. (1968) to thetravel timescannotbe eliminatedby merelyin-were morecompatiblewith the normal-modedatathan creasingaveragevelocitiesin theuppermantleunderthoseof ClearyandHales(1966)andthustheformer continentsanddecreasingthemundertheoceans.set wasretainedfor further analysis. The base-linecorrectionsnecessary,at this pointof

The datathat might providean effectivecontrol our study,to satisfy thenormal-mode,travel-timeandover distributionof shearvelocitiesand,to someex- regional-dispersiondata,were+3.5 secfor S traveltent,compressionalvelocitiesfor oceanicandconti- timesof HalesandRoberts(1970)and+1.1 secfor Pnentalregionsare theso-called“pure path” dispersion travel timesof Herrin et al. (1968).curves.The conceptandthe methodof “pure path” The averager.m.s.error for 1064normal-modeanalysiswasintroducedby ToksozandAnderson data,246 travel-timedataand32 regional-dispersion(1966). In their original formulation the approachin- datawas 0.183%,which liesbetweentheoverall r.m.s.volvedratherformidableassumptionsthat the disper- errorsobtainedby Gilbert and Dziewonski(1975)sionis thesamefor eachparticularregion (oceanic, for their models1066AandB being0.177and0.186,shieldor tectonic)and thatgeometricalray theoryis respectively.applicableto long-periodwaves.Theseassumptions It appearedthat we had reacheda satisfactorywerenecessaryin thestudyof ToksOzandAnderson, agreementwith observations.However,at this pointasthey derivedthree“pure path” dispersioncurves we calculateddispersioncurvesfor our modelsforfrom four great-circlepathmeasurements. periodsfrom 150 to 15 sec.Our oceanicgroup-veloc-

As the numberof availableobservationsincreases, ity dispersioncurvehas a maximumof 4.3 km/sectheconceptof “pure path” dispersioncanbe replaced at 50 sec;thisis evenfurther from thenormally ob-by “averageregionaldispersion”.Also, the dependence servedvaluesthan is the casefor any of thedisper-on applicability of the raytheory is less,as for a large sion curvesshownin Fig. 1.numberof randomlydistributedpaths theaverage It becameclearthat majormodificationsin our

PARAMETRICALLY SIMPLE EARTH MODELS 19

oceanicstructurewerenecessaryto achieveat least curvesis achieved.Translatingtheequationaboveintoa qualitativeagreementwith thetypical shapeof termsof freeoscillations,thegroup velocitieswill re-group-velocitydispersioncurvesobservedfor oceanic main unchangedif the periodsin a setunderconsidera-paths. tion are offsetby 6,1T1 A’~~T1/(l+ ~),whereA’ is

The principalproblemin introducingshort-period an arbitraryconstant.dispersiondataintoan inversionstudysuchas the Startingvaluesof periodsof free oscillationsfor thecurrentone is that theproperspatialaveragesare not continentalregionswere obtainedby integratingtheknown. Biasedshort-perioddatawould not be corn- group-velocitycurve in theleft part of fig. 16 ofpatible with eitherregionalaveragesfor long-period Landismanet al. (1969).The integrationconstantwaswavesor thegrossearthdata.Yet constructionof a adjustedso that the averagephasevelocitiesin themodelthat would predictrealisticvaluesof group and periodrangefrom 18 to 85 secare thesameasthephacevelocities is desirablefor thereasonsspecified averagephasevelocitiesfor thethreecurvesin thein the introduction. figure of Blochet al. (1969).The rangeof angular

Fig. 5 of Blochet al. (1969)comparesa numberof ordernumbersof spheroidaloscillationsextendsfromphase-velocitycurvesfor differentcontinentalregions: ~1O9 at 87.29secto 0S613at 18.44sec.Particularthreecurvesobtainedfor the centralUnited States modeswereselectedfor inversionwith approximately(McEvilly, 1964),easternAustralia (Landismanet al., equalstepson a logarithmic frequencyscale.1969) and southernAfrica (Bloch et al., 1969).The Startingvaluesfor short-periodfree oscillationsthreecurveshavenearlyidenticalphasevelocitiesfor for the oceanswereobtainedby combiningthe groupthesethreewidely separatedregions.Comparisonby velocitiesobservedby Landismanet al. (1969)for aKnopoff (1972,seefig. 7) of Rayleighwave space long (12790km) oceanicpath from an epicentrevelocitiesindicatesthat thedispersioncurvesfor con- near the west coastof Mexico to Riverview,Australia,tinental aseismicregionshavevaluesbetweenthosefor with the regionalgroupvelocitiesfrom 150 to 300 secshieldsandtectonicallyactiveregions(rifts) andagree obtainedfrom a two-regionanalysisof resultsofcloselywith thethreeof Bloch et al. (1969). It maybe Dziewonski(1971b).Groupvelocitiesfrom 18 to 300assumed,therefore,that thephaseandgroup veloci- secwere derivedby the least-squaresfit of an 8-th de-ties for thesepathsarecloseto an averagefor conti- gree polynomial to thetwo setsof datadescribeda-nentalregions. bove,and theintegrationconstantwas chosensothat

Landismanet al. (1969)havealsomeasuredinter- the phasevelocitiesobservedin a periodrangefromstationgroupvelocity,usingthe cross-correlation 150 to 300 secwere satisfied.The resultingphasemethod,for thesamepath acrosseasternAustralia re- velocitiesat shortperiodsarerelatively high;betweenferredto above. 30 and 60 secphasevelocitiesarenearlyconstantand

Theuseof observationsof group velocitiesasthe are closeto 4 km/sec,slightly abovethe rangeof valuesbasisfor derivationof thedatafor inversionhas the indicatedby Knopoff (1972).following advantage.All phasevelocity curves—C(w), The group-velocitycurve usedaboveappearsto bethat satisfy theequation: quite typical; it may be compared,for example,with

the resultsfor the pathRat Island—ChartersTowers,Australia (fig. 10 in Landismanet al., 1969).Numer-

= -~— + ~— ~ + A ousotherexamples(cf. Goncz,1974) support,inC(co) C0 w J u w qualitativeterms,thevalidity of our choice.

Incorporationof short-perioddatarequiredsubstan-

tial modificationsin the structureof the oceanichitho-havethesamegroup velocities—u(w).Sincewe may sphere.Preliminaryinvestigationrevealedthat twoexpectthat a selectedphase-velocitycurve might de- typesof modelsof the oceaniclithosphereare con-viatefrom thetrue averagefor a particularregion,the sistentwith the data.The first is characterizedby aarbitraryconstantA couldbe allowedto float,so relatively thin (60 km, includingthe crustand water)that theoptimumlevel of consistencywith the world- lithospherewith a shearvelocity in the lid of 4.55wide averagesandlong-periodregional-dispersion km/sec.Theothertype allowsfor a somewhatgreater

20 A.M. DZIEWONSKI, A.L. HALES AND E.R. LAPWOOD

total thicknessof the lithosphere(80 km), with the Anderson(1974)wereable to satisfy theseobserva-mantleshearvelocitiesbeing4.5km/secto a depth tions,but their dataset did not include the largenum-of 50 kin, and4.65 km/secin the depthrangefrom ber of toroidalovertonedatawhichhaverecentlybe-50 to 80 km;shearvelocitiesin the low-velocity zone come available(BruneandGilbert, 1974;Gilbert andaresomewhatlower in this case. Dziewonski,1975).Although theovertonedataare

The appealingfeatureof themodelwith a two- not sensitiveto thedetailsof theupper-mantlestruc-layerlithosphereis that it explainsthe observedveloci- ture, they aresensitiveto the averagepropertiesof thistiesof the oceanicSn-waves(MolnarandOliver, 1969; region. Becauseof the obviousbiasof the modesfromHartand Press,1973).Existenceof a zoneof increased 0T34 to 0T46we decidedto removethesedatafromvelocitieswould be consistentwith anincreaseof the the dataset to be usedin thefinal inversion.The sameabundanceof garnetin that depthrange:Observations decisionwasmadein respectof the regionalLove waveof Ptravel timesfor an oceanicpath acrossthe Gulf data.Obviously,becauseof greatersensitivity of Loveof Mexico by Haleset al. (1970)indicatean abrupt wave datato lateralheterogeneities,a significantin-increasein velocitiesat a depthof 57 km. A depthof creasein thenumberof observationsis neededbefore80 km to thetop of the low-velocity zoneis more thederivedaveragescould be consideredrepresentativeconsistentwith thecurrentmineralogicalandpetro- of the averageearthand evenmore,of anaveragecon-logical theories(cf. Green,1973, fig. 3). A depthof tinentalor oceanicstructure.60 km would requirea substantialincreasein thewatercontentof theuppermantle,perhapslargerthan is possibleaccordingto thecurrentevidence. 3. Inversionandparametersof regionalizedearthmodels

Despitetheaboveargumentsin favourof the two-layerhithosphericmantlewe adopta model of the Thefinal datasetusedin inversionconsistsof 1051first type in this studyfor thesakeof parametric grossearthdatarepresentingtheobservedfree-oscilla-simplicity becausewe believethat the introduction tion periodsfrom table 7 of Gilbert andDziewonskiof additionalcomplicationsin the structureshould (1975)after 13 observationsfor 0T34 — 0T46havebe supportedby a largerandmorerepresentative beenculledfor reasonsexplainedin theprevioussec-bodyof data. tion; theregionalizedearthdatafor continentalstruc-

Our final decisionregardingthedatato be usedin ture consistingof 246 travel-timeobservationsandthe inversionconcernsobservationsof the fundamen- periodsfor 43 spheroidalmodes~ with 1 rangingtal toroidalmode.Observationsof Mendiguren(1973) from 24 to 613;andthe datafor oceanicstructureandGilbert and Dziewonski(1975)indicatedifferences consistingof 43 periodsof 0S1with a rangeof 1 fromof theorderof 0.5%between.theperiodsof ~7’~ob- 24 to 554. The regional-dispersiondatawerederivedservedfor the Colombianearthquakeandthe Alaskan from group and/orphase-velocitymeasurements,asearthquake(DziewonskiandGilbert, 1972).Inconsis- explainedpreviously.tenciesamongtheaverageperiodsof toroidalmodes Ourstartingmodel definedat thebeginningof theandLove wavephasevelocitieshavebeennotedby previoussectionhasundergonechangesat all levels,asKanamori(1970)and DziewonskiandGilbert (1972). in the processof testingthe varioussubsetsof dataweIt is clear that we do nothaveat thepresenttime haveperturbedthemodelsto improveagreementwithsatisfactoryworld-wide averagesfor theperiodsof observations.However,thechangesfor depthsgreaterthefundamentaltoroidalmodeswith angularorder than670 km representedin all casesonly a fractionofnumbersgreaterthan 33—35.The factthat the “all 1%.data” averagesof DziewonskiandGilbert (1972)are We haveperformedanumberof experimentsrelatedbiasedwith respectto the remainderof the presently to determinationof thebase-linecorrectionsto theavailabledataset of free oscillationsis evidentfrom subsetsof travel-timedatause.Despiterelativelytable 7 of Gilbert andDziewonski(1975); theresi- numerousobservationsfor spheroidalovertonesdomi-duals for themodes0T35 — 0T46, whereall observa- natedby compressionalenergyit doesnot seempossi-tionscomefrom thepaperby DziewonskiandGilbert ble to determineuniquelythebase-linevalue for P(1972),arelargeandall of thesamesign.Jordanand traveltimes. Correctionsto thetravel timesof Herrin

PARAMETRICALLY SIMPLE EARTH MODELS 21

et a!. (1968)rangedfrom approximately1 secto 2 sec sion datahasimposedconstraintson the degreetodependingon theset of dataincluded.Fixing the base- which velocitiescanbe changedin the oceanicupperline correctionat,for example,1 sec,doesnotchange mantle to compensatefor thehighervelocitiesof thesubstantiallytheoverall fit to the datain comparison continentaluppermantle.with the casein which thebase-lineis allowedto as- With thebase-linesfixed as describedabove,wesumean arbitraryvalue. In general,thepresentset of performedthefinal inversion.Before presentingthefree-oscillationdataseemsto preferabase-linecorrec- resultswe definethe third typeof earthmodel — thetion of the orderof 2 sec.Thereis oneimportantex- averageearth.ception,however.The residualsfor thefundamental Realizingthata modelof anaverageearthmayberadial mode0S0,which is notassubjectto a biasdue usefulin someapplications,we definesucha modelto anunevenandincompletedistribution of sources asa weightedaverageof theoceanicandcontinentalandreceiversasotherspheroidalmodes,seemto corre- models.Theparametersfor the startingaverageearthlatewith themagnitudeof thebase-linecorrection. modelwere derivedfrom thefinal regionalizedmodels.Forbase-linecorrectionsof the order of 2 secthe resi- Parametersfor depthsgreaterthan420km, assumeddual is of theorder of 0.2%;it canbe reducedto less identicalfor bothregionalmodels,remainthesamethan0.1%for base-linecorrectionsof 1 sec.Model for the averagemodel and arenot allowedto change.1066Arequiresa2.4-seccorrectionand the residual The densitydistribution at shallowerdepthswas alsofor ~ is 0.23%;thecorrectionfor model 1066Bis not allowedto changeasits gradientand the valueof1.6 secandthe ~ residualis 0.19%.With the base- densityat420km wereassumedto be the sameasforline correctionof 1.25 sec,finally adoptedin this both regionalmodels.paper,we were able to reducetheresidualfor ~ to If thethicknessof a zonesuchas,for example,the0.09%. If a correctionof +1.25 secis adoptedfor P lid or LVZ is d0 for the oceanicmodelandd~for thetraveltimes, thePKIKP traveltimesof Cleary and continentalmodel thenthe thicknessdA for the aver-Hales(1971)requirea correctionof +0.19 sec. agemodelis:

The questionof base-linecorrectionsfor the Stravel timesis relativelylesscomplicated.The data in-

d -2d+1dcluding the regionalfree-oscillationperiods,clearly A 3 o ~

prefera base-linecorrectionof +2.6 secto the travel .

timesof HalesandRoberts(1970),for epicentraldis- the averagevertical travel time is:tancesfrom 30°to 82°.The correctionincreasesto — d~, d~+3 secif the datafor distancesup to 98°are included. tA — T + 1

Themagnitudeof this correction(for continentalstructure)appearsto be quite reasonable. and theaveragevelocity:

Although the S andSKS travel timesin the study —

of HalesandRoberts(1970)shouldhavea common 1”A — dA/tA

base-linecorrection,our resultsindicatethat thecor-rectionfor SKS shouldbe only +0.85 sec. Thisdis- This definition of anaveragemodel is consistentcrepancycould be partly relatedto thedifficulties in with thenormal-mode—body-waveanalogyin whichdeterminationof S traveltimesat distancesgreater the overtonesof normal modescanbe comparedtothan82°,beyondwhich theS phaseis a secondary multiply reflectedbody waves(cf. Singli andBen-arrival afterSKS. The differentialtraveltimesSKKS— Menahem,1969;DziewonskiandGilbert, 1972andSKSwere assumednot to requireany base-linecorrec- 1973).After a largenumberof reflections,the averagetion (Jordanand Anderson,1974). traveltime throughtheuppermostregions of the earth

It wasnotedearlierthat without theconstraints shouldbe consistentwith the representationproposedimposedby theregional-dispersiondatait waspossi- above.ble to achievesatisfactoryfits to the travel-timedata In thefinal inversionfor an averageearthmodelwewithoutbase-linecorrectionsandalso to satisfy the useonly the grossearthdata(free-oscillationperiodsgrossearthdata.Incorporationof the regional-disper- of Gilbert andDziewonski, 1975)andonly parameters

22 A.M. DZIEWONSKI, A.L. HALES AND E.R. LAPWOOD

in theuppermantleareallowedto vary, to compen- coefficientsof Table I. Table II also containsothersatefor possibleerrorsintroducedby our averaging parametersimportantin the studiesof the propertiesmethod, of theearth’s interior suchasthe incompressibility,

TableI lists the coefficientsof polynomialsdescrib- Lameconstants,Poisson’sratio,pressureandgravity.ing thedistribution of densityandvelocitieswithin Parametersfor depthswithin the uppermost420kmtheappropriaterangesof radiusfor the oceanic,con- are specifiedseparatelyfor the oceanic,continentaltinental andaveragemodelscalledPEM-O, PEM-Cand andaveragemodels.PEM-A (parametricearthmodel)respectively.The ra- The continentalmodel is plottedin Fig. 2 and thedius in thepolynomialexpressionsis normalized(R = uppermantlesof the oceanic,continentalandaveragena, wherea is the radiusof theearth— 6371 km). modelsare comparedin Fig. 3.

Table II gives the parametersof theearthmodels In TableIII we comparer.m.s. residualsfor groupscomputedfor particularvaluesof radiususingthe of modeswith the sameradialordernumbercalculated

ioóo 2000 aobo

3000 4000 5000 6000DEPTH ~km)

Fig. 2. Earthmodel PEM-C computedusingthe parametersof Table I.

PARAMETRICALLY SIMPLEEARTh MO1)ELS 23

TABLE I

Coefficientsof thepolynomialsdescribingtheparametrizedearthmodels (PEM).ThevariableR is t1~enormalizedradius:R = na,wherea is theearthradius— 6371km

Region Radiusrange(km) Density (g/cm3) Vp (km/sec) V~(km/sec)

Innercore 0 —1217.1 13.01219 11.24094 3.56454— 8.45292*R2 — 4.09689*R2 — 3.45241*R2

Outercore 1217.1 — 3485.7 12.58416 10.03904— 1.69929*R + 3.75665*R 0— 1.94128*R2 _13.67046*R2— 7.11215*R3

Lowermantle 3458.7 — 5701.0 6.81430 16.69287 9.20501— 1.66273*R — 6.38826*R — 6.85512*R— 1.18531*R2 + 4.68676*R2 + 9.39892*R2

— 5.30512*R3 — 6.25575*R3

Transitionzone 5701.0—5951.0 11.11978 21.05692 15.04371— 7.87054*R _12.31433*R _10.69726*R

Oceanicstructure

Be1owLVZ 5951.0—6151.0 7.15855 22.53683R _ii.86483*R

LVZ 6151.0—6311.0 — 3.85999*R 7.87320 4.33450

AboveLVZ 6311.0—6360.0 7.90000 4.55000

Crust 6360.0 — 6366.0 2.85000 6.40000 3.70000

Sediments 6366.0 — 6367.0 1.50000 2.00000 1.00000

Ocean 6367.0—6371.0 1.03000 1.50000 0

Continentalstructure

Be1owLVZ 5951.0—6151.0 7.15855 17:63609R _9.32106*R

LVZ 6151.0 — 6251.0 — 3.85999*R 7.84750 4.45860

Above LVZ 6251.0 — 6336.0 8.02000 4.69000

Lower crust 6336.0 — 6351.0 2.92000 6.50000 3.75000

Uppercrust 6351.0—6371.0 2.72000 5.80000 3.45000

Average structure

28.48832 15.09536BelowLVZ 5951.0—6151.0 _20.90003*R _11.01544*R

7.15855LVZ 6151.0—6291.0 — 3.85999*R 7.89520 4.34060

Above LVZ 6291.0 — 6352.0 7.93420 4.65400

Lower crust 6352.0 — 6357.0 2.90200 6.50000 3.75000

Uppercrust 6357.0 — 6368.0 2.80200 6.00000 3.55000

Ocean 6368.0 — 6371.0 1.03000 1.50000 0

24 A.M. DZIEWONSKI, A.L. HALES AND E.R. LAPWOOD

~01 ~ ~ ~

~0 PS * 0•••SU• W*•*•• •0000S*•0 P *0000**• SOPS*~•~•S * P0 ~0 PP•

.~ ~.:——————————

SB -EJ ~ N~RC~’~—RE.-—.tV--.P.,Q’C~o ,‘.RIs .00* S**0**..S*.00060t*..,00000 .......0 ~.Rn ~.~ ~A~SB

a ~p * •0.p000p. IPSO P S * P0* *0*5 •****• SO..,... *

— eva ~.i~ u~,g ~

(.J P’~C~ .-‘~O4E~C~V~V ~t~a~t%c~———--—-A———

a~$~— Io ~RE ..0

Rfl S •*SS** 0**S*S*• S •5000•5005 •P500*000 •* S. 0050*0500S* .......... S

0.

~ W ~ ~ ~ ~r..I..P.iCIc.0 ~. ~o ~ z ~ ~ r~p,j.-,,. ~

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a~ ~ ~0 O~

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‘- E a.—. ~I- a~ a.a~aa.aca~oaa~a-.,-..,c~oic

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RE ~ LB ~0. ttl ~‘5 ,~ 0.t1) ~

4_._s >—~-. 05050*000 555.050..*. P. 0*o S~c~OR0,0Ba CvevvCMO. ..-.———— .alo,o,a,a,o,RE .~RE ‘0

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.4 ~ ~°“~ ~ ~0

k~ o_~ -‘U E ~ °

RE ~ — ev~i-~”,~ ‘~~ ~4 i~ ~ ~~A0A:: _,

PARAMETRICALLY SIMPLE EARTH MODELS 25

~ ~

‘—LB ~ ~‘‘IL .

.4”. ~

0.1I4~’ttP’. ~ IL .-~ ~P’.,CRC ~ ~ ‘~U~O~ ~

-o . .

— —I-.. —ais.

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— ~H ~ ~ ~

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0 ~ C) ~—‘It’~~—~’I ~ ~~4)

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~DI~flevCVev-.-~O~

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0 s—~~~u’a~~.-———.Inr Cr) ~010.~N. ~~“..N.’GCC’4(SJ N.

to, ., I—I 5*is..’ ~ ~—Cq~ev0~

C’. ~F)tI1t~** Z

z1.1 I.B .4,,~F.e-O.a,a.a.Ia.a.a.S.~ ~ ~ ~ r~a.v)p...a.0.1 ~ 5.’ I~.”S-I

~ ~ C!) ~ H— *0 SeS5S * SO~ c ~

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I C.- I

I ~ ~u~-,fl’.a.s~ ~ -~CL) I~I~ ~ a.~

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a.‘-41 .... ..OC t5 15’Ø S S. OrQf.-p.-..S ~C —C ‘-s——CMOS,.’.N. t t —

0 ~CI~)N.-~C~CCCCP.’.0 ~ ~a.ERE4.ACMCM3~W2’5).’~S~) aa~a.——~

— ~CC~CC~C

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~-B ~J

26 A.M. DZIEWONSKI, A.L. HALES AND ER. LAPWOOD

~-‘::~~

0 100 200 300 400 500 600

DEPTh kn,)

Fig. 3. Upper-mantlemodelsPEM-C (continental),PEM-O(oceanic),andPEM-A (averageearth).For depthsgreaterthan420 kmparametersfor all threemodelsare identical;seeFig. 2.

for model 1066Bof Gilbert andDziewonski(1975), PEM-O model,whereit is measuredfrom the bottomtheweightedaveragesfor our oceanicand continental of theocean(4 km).models,andour averageearthmodel.The overall The correspondingtraveltimes for P-wavesarestandarddeviationof our modelsis only slightly greater 103.4and 102.50sec,but if theoceanicP travel times(2 and 5%) than thecorrespondingvalue for model were measuredat the surfacethelatter estimatewouldl066B. The r.m.s.residualsof theweightedaverage increaseto 107.8sec.However,thereflection coeffi-periodsfor thefundamentalmodes0T1 and~ are cientat the oceanbottomis —0.85 andthe effectiveconsiderablygreaterthanthose for models1066Band (in the senseof the body-wave—normal-modeanal-PEM-O;undoubtedlybecauseof the biasintroducedby ogy) Ptraveltimes are only slightly greaterfot thetheregionalthort-periodsurface-wavedata.The over- oceanicstructure.Thismay be inferredfrom thetones10S1, ~~S1and14S1 showa markedlypoorer comparisonof eigenperiodsof modesdominatedbyoverall fit to thedataascomparedwithmodel 1066B. compressionalenergy,suchas theradialmodes.As theseseriescontain a numberof modessensitiveto The calculatedphaseand group velocitiesfor modesthecompressionalvelocity in the lowermantleit may PEM-O and PEM-Care comparedwith the observationsbe that this particulargroup of the observedmodes in Figs. 4A and B. It will be notedin Fig. 4A thatshouldbe closelyscrutinizedwithrespectto a possi- thereare systematicdifferencesbetweentheobservedble biasin observations, velocity and thosecomputedfor thecontinentalmod-

Table IV containsa list of theobservedeigenperiods el. Thesearisebecausewe allowedthe constantA’ toandthecorrespondingvaluescomputedfor the ocean- float in orderto preserveconsistencywith thegrossic, continentalandaverageearthmodels.The eigen- earthdata.periodsaregreater,in general,for theoceanicstructure Comparisonof the computedand observedtravelthanfor thecontinentalone.This is patticularlyclear timesis madein Figs. 5A—E. While the overalltrendfor thetoroidalmodes,and is undoubtedlyassociated of theP traveltimescomputedfor modelPEM-Cwith the fact that theverticaltwo-way travel time for agreeswell with thatof the 1968-tablesof Herrin etthe S-wavesbetweenthe surfaceand420km depthis al. (1968),it maybe observedthat the detailsof the186.3secfor thePEM-C modeland 187.5secfor the deviationscoincidevery closely with those for the

PARAMETRICALLY SIMPLE EARTH MODELS 27

TABLE III

Comparisonof r.m.s. relativeresidualsof theeigenperiodsofradial modesandparticularovertonesof toroidalandspheroidalmodescomputedfor model 1066Bof Gilbert andDziewonski (1975),weightedaveragesfor PEM-OandPEM-C modelsand PEM-Amodel

Overtone Number of Relativeerror (%) Overtone Number of Relativeerror (%)modes modes

1066B regional PEM-A 1066B regional PEM-A

Allmodes 1049 0.182 0.191 0.186 ,S1 39 0.106 0.139 0.118

8S1 19 0.098 0.107 0.096

n5o 12 0.129 0.123 0.120

9S1 19 0.099 0.131 0.14931 0.103 0.170 0.130 ioSi 22 0.120 0.187 0.16358 0.064 0.079 0.102 uSl 31 0.102 0.112 0.106

2T1 38 0.062 0.086 0.072 1251 15 0.065 0.131 0.1103T1 39 0.110 0.101 0.103 i3Si 21 0.164 0.146 0.148

34 0.212 0.210 0.223 14S1 12 0.094 0.202 0.18732 0.210 0.202 0.199 15SI 13 0.098 0.126 0.13632 0.356 0.353 0.327 16S1 13 0.116 0.117 0.123

7T1 34 0.332 0.351 0.344 17S1 13 0.103 0.114 0.116

8T1 25 0.403 0.400 0.399 18S1 12 0.175 0.173 0.1859Tj 28 0.436 0.437 0.444 19S1 7 0.065 0.091 0.089

ioTl 23 0.461 0.452 0.469 2051 12 0.138 0.108 0.15711T1 2 0.383 0.378 0.382 2151 9 0.105 0.120 0.11812T1 2 0.315 0.347 0.328 22S1 7 0.123 0.134 0.129

2 0.197 0.238 0.209 23S1 9 0.090 0.093 0.092147’! 2 0.243 0.223 0.232 24S1 8 0.105 0.090 0.097

2 0.232 0.252 0.226 25S1 7 0.060 0.062 0.0642 0.131 0.134 0.131 26S1 6 0.188 0.119 0.120

i,T/ 2 0.242 0.199 0.272 27S1 5 0.177 0.164 0.16318T1 2 0.198 0.244 0.192 ~S1 4 0.139 0.156 0.152

65 0.057 0.131 0.069 2951 2 0.102 0.139 0.13855 0.105 0.126 0.115 3o~l 3 0.050 0.091 0.08851 0.174 0.112 0.091 31S1 1 0.258 0.329 0.32744 0.073 0.098 0.071 32~l 1 0.072 0.032 0.033

4~1 30 0.063 0.082 0.112 ~3S~ 1 0.221 0.289 0.28742 0.065 0.082 0.089 34S1 1 0.091 0.095 0.095

48 0.089 0.113 0.112

travel timesof Cleary and Hales(1966).TheP-waves timesof Halesand Roberts(1970)is 0.75secfor thefor model PEM-Creachthecore—mantleboundary distancerangebetween30°and82°while eq.2 ofat a distanceof 94.5°.This is somewhatearlier than HalesandRobertsgives an r.m.s.errorof 0.73 sec.is commonly accepted,but is in generalagreement Thevertical line in Fig. SC separatesthe distancewith observationsof amplitudesby Sacks(1966,fig.l). rangesin which the S-waveprecedesand then follows

The computedtraveltimes for thePKIKP phase theSKSphase.The increasein the scatterof observa-(Fig. SB) follow very closely observationsof Cleary tionsfor distancesgreaterfrom 82°is evident;it alsoand Hales(1971); the r.m.s.error within the epicentral coincideswith the occurrenceof systematicdifferencesdistancerangefrom 130 to 180°is below 0.2 sec.The betweenthe computedandobservedtraveltimes.differencesaregreaterfor distanceslessthan 130°, The computedtravel timesfor the SKS phase(Fig.but the amplitudesin this distancerangeare smalland SD) agreewith the observationsof HalesandRobertsobservationsare difficult. (1970)nearlyaswell asthepolynomial representation

The r.m.s.error of our fit of the S-wavetravel givenby thoseauthorsin their eq.5. Also the devia-

I S ~ O~OQR~~~ORRU.A.4— 0~.

I .5.154Z’— • I I I I I

4.1.4

15.11SrisJC.’- ~.4a.0s..13_ ~0.1 S Sro 3_RI) .AItO~’.a,V’)S35OR~s—’U~.~N.a.~,’.’Uc,.> ,5~

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PARAMETRICALLY SIMPLE EARTH MODELS 39

PHASE VELOCITIES OF RAYLEIGH WAVES /5~0

>- 4.5—oO —54 -

>

I — — — 0 0O~

40 — 000

_~ o~

0 — PEM-OCEANIC

.‘ 0 ___ PEM-CON4TINENTAL

“0 000 OBSERVED CONTINENTAL‘0

~0

3~ A -,

15 20 30 40 50 100 150 200 300

PERIOD (s)

Fig. 4A. ComputedRayleigh wavephasevelocitiesfor modelsPEM-OandPEM-C. The systematicdifferencesbetweentheob-servedand computedcontinentalphasevelocities resultfrom anadjustmentrequiredto satisfythegrossearthdata;seethetextfor details.

4.5GROUP VELOCITIES OF RAYLEIGH WAVES

00 0 OBSERVED CONTINENTAL0~

30 QQQo000~

B -,

I ~ I I I 4 4 I

15 20 30 40 50 100 150 200 300 400

PERIOD (s)

Fig. 4B. Comparisonof groupvelocitiesof Rayleighwavesobservedfor continentaland oceanicpathswith thosecomputedformodelsPEM-C and PEM-O.

40 A.M. DZIEWONSKI, A.L. HALES AND E.R. LAPWOOD

P — P04-C

~ I’ iii ~ ::~/

:E~/ 10 20 300060 70 80 90 ~0

EPICENTRAL DISTANCE (DEOREES)

Fig. SA.Deviationsof the P-wavetraveltimescomputedfor ModelPEM~Cfrom the “1968 TravelTimeTables”of Herrinetal.Deviationsfor JeffreysandBullen (1940)andClearyandHales(1966)travel timesareaddedfor comparison.Note thesimilaritybetweenthe short wavelengthfeaturesin thecurvefor PEM-C andthedataof Clearyand Hales.

PKIKP

— — — — — — MODEL PEM-C

— ——— CLEARY & HALES (1971)

I I I120 130 140 150 160 170 180

EPICENTRAL DISTANCE (DEGREES)

Fig. SB.Deviationsof thePKIKP traveltimescomputedfor modelPEM-C andobservationsofCleary andHales(1971)from thedataof Bolt (1968).

C11~:~oS

74P~.s

EPICENTRAL DISTANCE (DEGREES)

Fig. SC. Deviationsof theS-wavetravel timescomputedfor modelPEM-C from theSeismologicalTablesof JeffreysandBuilen(1940).Observationsandpolynomialrepresentation(eq. 2) ofHalesandRoberts(1970)areaddedfor comparison;a base-linecorrectionof +2.60sechasbeenappliedto thevaluesof HalesandRoberts.Theverticalline designatesthe epicentraldistancebeyondwhichS-wavearrivalsfollow thoseof theSKS phase.

PARAMETRICALLY SIMPLE EARTH MODELS 41

SKS

8-

• — MODEL PEM-C

— —— HALES & ROBERTS 1970 EQ 5)0 - ••• HALES & ROBERTS 1970 OBSERVATIONS

.~ D

80 90 100 110 120 130

EPICENTRAL DISTANCE (DEGREES)

Fig. SD. SKS travel timescomputedfor modelPEM-CplottedagainsttheJeffreys-Bullentraveltimes. Observationsandpoly-nomial representation(eq. 5) of HalesandRoberts(1970)areaddedfor comparison;abase-linecorrectionof +0.85 sechasbeenappliedto thevaluesofHalesandRoberts.

1 SKKS - SKS

8 E

EPICENTRAL DISTANCE (DEGREES)

Fig. 5E. Deviationsof the differentialtravel timesSKKS—SKScomputedfor modelPEM-C from thevaluespredictedby eq. 3 of

HalesandRoberts(1971).

tionsof thecomputeddifferential travel times SKKS— Dziewonski(1975). It is our beliefthat modelswithSKSfrom thevaluespredictedby eq.3 of Halesand this type of parameterizationareparticularlywellRoberts(1971) arenot excessive,in viewof thescatter suitedfor useasreferenceor comparisonmodels.of theoriginal observationsshownin their fig. 1. However, we must emphasisethat this model,like all

othermodelsobtainedby generalizedinversionproce-

dures,is dependenton thecharacteristicsof thestart-4. Discussion ing modelon which theinversionis based,theaver-

agingor smoothingprocedureusedin the inversionThe comparisonsin Table III showthat ourparam- and, of course,theobservationaldatasetselected

etrically simplespherically symmetricearthmodels for inversion.We havebeencarefulto describein de-fit theobservationaldataaboutaswell asthemodels tail thechoiceswith regardto the observationaldataderivedby other inversionprocedures,for example setandthestartingmodel for our inversion.We be-thoseof JordanandAnderson(1974)or Gilbert and lieve theuseof simplepolynomial-typeparameteriza-

42 AM. DZIEWONSKI, AL. HALES AND E.R. LAPWOOD

tion exposesasfully asanyother procedurethelimita- cantdeparturefrom the linearrelation.Supportfortionson the model whichhavebeenimposedby the this deviationcomesfrom theamplituderelationsneedfor averagingor smoothing.We think, therefore, (cf. HalesandHerrin, 1972,fig. 13). It is possiblethatthat thismodel is anadequaterepresentationof the thePEM model doesnot adequatelyreflect thebeha-real earth. viour of dT/di~beyond 85°.

It should berecognized,however,that the aver- The otherregionof dT/d~wherea geophysicallyagingprocessmay haveobscuredcertainminor devia- significantdeviationmight occuris between670 andtionswhichcouldconceivablyhaveconsiderablegeo- 1000km correspondingto body-wavetravel-timephysical significance.Suchdeviationsmay be revealed arrivalsfrom 25°to 40°.Thereis someindication bothby specialstudiesof the travel timesor theapparent from S travel times(HalesandRoberts,1970)andPslownessof particularphases. travel timesthat dT/d~doesnot follow thelinearre-

In theinnercore of ourmodeltheshearvelocity lation in this rangeof i~. Carefulstudiesof thesecondvariesfrom 3.44to 3.56km/secconfirming the earlier arrival phasesandcomparisonwith syntheticseismo-estimateof 3.5 km/secby DziewonskiandGilbert grams(cf. HelmbergerandWiggins, 1971) should clar-

(1971).We arenot awareof anydatawhichsuggest ify thevelocitydistributionbetween670 and1000possiblegeophysicallysignificantdeviationsfrom the km.PEM modelsof thecores.The differencesof theveloci- The choiceof a startingmodelhasits most signifi-ties andradii of the PEM coremodel from thoseof cant effectbetween220 and670 km. Therewe haveMasse’et al. (1974)seemto us to be within the limits chosento usesharpdiscontinuitiesratherthanaof observationalerror, smoothvelocity distribution. It is clearfrom table 7

In thelower mantletherearetwo regionswhere of Gilbert andDziewonski(1975) thatmodels1066A,

deviationsfrom the PEM modelmay begeophysically which is relativelysmooth,and1066B,which hassignificant. We showin Fig. 6 calculatedvaluesof sharpdiscontinuities,fit theobservationaldataequallydT/d~for themodel PEM-C.It is known (Haleset al., well. Thus free-oscillationdatado not permit discrimi-1968;HalesandHerrin, 1972; Wiggins et al., 1973; nationbetweenthesetwo radically differentmodelsWright andCleary, 1972),that dT/d~varieslinearly of the transition zonein theuppermantle.It is possi-with ~ up to about850. Thereafterthereis asignifi- ble to showalso thatthesurface-wavedispersionfor

8

7~

6 — PLO-C

—— — HERRIN 21 .1 (I9RB)

N5 ~1~4%

~ 30 40 50 60 70 80 90

EPICENTRAL DISTANCE (DEGREES)

Fig. 6. Comparisonof thedT/d~valuescomputedfor modelPEM-C with thecorrespondingvaluesof Herrin et a!. (1968).

PARAMETRICALLY SIMPLE EARTH MODELS 43

these two modelsdoesnot lie sufficiently far outside TABLE Vtheexperimentalerrorof both thegroupandphase- Deviationsof the densitiesin the coresandlowermantleofvelocity determinationsto makediscriminationsbe- thePEM models(p) from the densitiespredictedby the

tweenthemodels1066A and 1066Bpossibleon the Adams-Williamsonequation(PAW)basisof dispersionmeasurements. _____________________________

The choiceof asharpdiscontinuityat 670km is Radius(km) p (g/cm3) p — PAW (g/cm3)supportedby theevidenceof EngdahlandFlinn (1969) _____ ______

that thereareclearshort-periodreflectionsof the 0 13.012 0.002PKIKP phasefrom theundersideof this discontinuity. 300 12.993 0.000

As waspointed out by Hales(1972)andRichards 600 12.937 0.000

(1972),thesesharpreflectionsfrom theundersideof 1217 121704 —0.001

thediscontinuityimply that themajorpart of the 1217 12.139 —0.015

changeof thevelocity musttakeplacewithin a rela- 1500 11.984 0.000tively shortdepthrange,estimatedby Richardsat 4 1800 11.789 0.006

km. Petrologicalconsiderationssuggestthat theremay 2100 11.558 0.005

bemorethanonephasetransitionin the600—700 2700 101974 —01006km depthrange. The seismologicalevidenceimplies, 3000 10.611 —0.007however,thatat leastoneof theseis sharpandthata 3300 10.195 0.000

majorpartof thevelocitychangeis discontinuousor 3486 9.909 0.017nearly so. Forthe420km discontinuity theevidence 3486 5.550 —0.008

for or againstsharpnessis not clear.Thechoiceof a 4100 5253 0.002

single sharpdiscontinuitywasmadein the interestof 4400 s.ioi 0.000simplicity of themodel.Thus againprogresstowards 4700 4.943 —0.001

moreprecisemodelsof thevelocitydistributionsbe- 5000 4.779 —0.003tweeñ 220 and670km mustwait for further precise 5300 4.611 0.000

travel-timestudiesespeciallyof thesecond-arrival 5600 4.437 0.005phasescoupledwith synthetic-seismogramcomparisons. . -________

It should,of course,beclearthat thevelocity dis-tributions so derivedmustbe in accordwith the free-oscillation data.The lack of control of thedetailsof We areconfident,however,that anynewmodelsthevelocity distribution in theuppermantledoesnot foundby travel-timestudies,or in other ways,will

imply that thefree-oscillationdatacanbe fitted by haveaveragesof thedensitiesandvelocitiesoverdepthanyvelocity distributionwhatsoever.The average rangesof 200—300km in substantialagreementwithvelocitiesareconstrainedwithin relatively closelimits similar averagesfound from the PEM model.(Gilbert et al., 1973),andthus the free-oscillation We remarkedearlierin thispaperthatour starting

dataserveasacontrol on thevalidity of thevelocity modelfor density followed the Adams-Williamsondistributionsderivedin otherways.It is probable equationvery closely.Sincethechangesin densityalso that oncethevelocitydistributions in theupper duringtheinversionwere small,thedeparturesfrommantlehavebeenderivedusingothermethods,it will uniformity in the final PEM model,i.e. from theAdam-be possibleto obtain tighterconstraintson theden- Williamson densities,would be expectedto be small.sity distribution in this regionfrom thefree-oscilla- This is confirmedby Table V, which showsthedevia-tion data. lions of the PEMmodel from theAdams-Williamson

In theuppermostuppermantlethePEM modelfits predicteddensities.The maximumdeviationis 0.2%.thedatareasonablywell. It is clear,however,that it The questionof uniformity can also be examinedmay be somewhattoo simple in this regionto fit the in termsof equationsinvolving thevelocities.Bullendataexactly.A combinationof travel-timeandsurface- (1949)introduced1 —g1(d~/dr)asa measureofwavedispersionstudiesmayleadto more detailed uniformity. Birch (1952)showedthat 1 g1(dØ/dr)modelsfor specificregionsandultimately to better dK

5/dP+ a~r/gwherea= coefficientof thermalexpan-knowledgeof upper-mantlestructure. sion,r theexcessof theactualtemperaturegradient

44 A.M. DZIEWONSKI, A.L. HALES AND E.R. LAPWOOD

overtheadiabatic,K5 theadiabaticincompressibilit1 TABLE VIand0 is theseismicparameter(0 = K5/p = V~— ~ l’~). Zero-pressureparametersderivedfrom a least-squaresfit of

Usingfinite straintheoryhe showed: finite-strain theory to thePEM models

1dØ (aK~\ l2—49e . . —_________

1 —g —~ J = +aterminvolving~,dr ‘ 3P i T 3(1—7e) Parameter CaseI CaseII3 3486~r~57013831~r~5359

e beingthestrainandgivenby p/p0= (1 — 2e)~and~a temperature-dependentcoefficientarisingin the p(g/cm

3) 3.991 3.986finite-strainequationof state. ?t(Mbar) 1.305 1.293

This equationshowsthat for ~= 0, [1 —g~(dØ/dr)] ~2(Mbar) 1.332 1.346should be4 for zero strainanddecreaseas ci increases. K (Mbar) 2.193 2:190

Valuesof [1 —g1(d~/dr)]for the PEM modelsare Vp (km/sec) 9.972 9.999shownin thelast columnof Table II. They decrease VS (km/sec) 5.777 5.811

from 3.388 at 670 km to 2.808at 2886km. Reference ~I (km2/sec2) 54.95 54.94

to Birch’s table 5 showsthat thesevaluesaretoo low ..L ~! (Mbar—1) 0.483 0.477to fit thecase~ = 0 andmustcorrespondto asmall Vp a

1°positivevalueof ~ 1 ~VS

Later,Sammiset al. (1970)derivedcompletethird- ~ -~-(Mbar’) 0.324 0.314orderEulerianequationsfor thevelocitiesin Birch-

Murnaghanfinite-straintheory(Birch, 1939). For 0.239 0.247

hydrostaticcompressionthese.equationsare: r.m.s.error (%) Vp 0.232 0.107V8 0.390 0.182

pV~= (1 — 2e)512 [X0+ 2/10 — e(11k + 10/10 — ~ 1.244 0.413

pV~= (1 — 2C)5/2 [~o— e (o + ~ + ii)] to r = 5359km alone.The deviationsfor this calcula-tion arealsoshownin Fig. 7. The shapeof thecurve

P = —3K0e(1 — 2e)SI’

2 (1 + 2e~) doesnot change,but theerrorsarereducedbecauseinthefirst caseconsideredthelargestdeviationswere

where~, i~aredependenton temperatureand: towardstheendsof thecurves.The parametersK0,

3~+ 4 p0, etc.werenot changedappreciablyas canbeseen

= 12K from Table VI. The systematicnatureof thedevia-0 tions mayarisebecausethethird-orderstrain theory

We havefittedby leastsquaresthesethreeequations is not adequate,or becauseof thetemperature-de-to thelower-mantlevaluesof V13~V~,p andPgiven pendenceof theconstants~, 17, and~ in third-order

in Table II to find k~~ ~ andK0 for fixedvalues theory, or by reasonof minor inhomogeneityof theof p0. Analogouscalculationsusing theSammiset al. materialof the lower mantle.It will be studiedfurther(1970)equationswerecarriedout by Andersonet al. in aseparatepaper.

(1971). TheconstantsK0, ~ [(1/V~)(aV5/aP)] andThe deviationsof thecalculatedvaluesfrom the [(1/ V~)(aVp/al’)] areimportantfor theidentification

valuesgivenin TableII area minimum for p0 = 3.991. of thematerialsof thelower mantle.In so far asThe parametersfor this casearegiven in Table VI. andp0 areconcernedourvaluesconfirm Birch’s(Note that theseparametersarefor lower-mantle (1952)conclusionthat theconstantsfor “periclase,temperaturesandthat density shouldbe increasedby corundumandrutile showthat oxidestructurescan4%and0 by 10%,approximately,for comparison possesstherequiredtightnessof binding, combinedwith laboratorymeasurements.)The deviationsof with asuitabledensity”.thevelocitiesareplotted in Fig. 7. The deviationsare TableVII showsvaluesof theseparametersfor asmall,0.23%for Vp,0.38 for V~,but aresystematic. numberof minerals;it waspreparedfor us by Dr.We, therefore,fitted the tabularvaluesfrom r = 3831 RobertC. Liebermann.It is clear that thelogarithmic

PARAMETRICALLY SIMPLE EARTH MODELS 45

DEVIATIONS OF FiNITE STRAIN MODELFROM PEM MODEL

I \\~,_~ I

VFS - v~M CASE 1

-0~O5’ 5

I I

600 1000 2000 3000DEPTH )km)

Fig. 7. Deviationsof thecompressionalandshearvelocitiescomputedaccordingto finite-straintheoryfrom the appropriatevaluesfor thelower mantleof thePEM models.For CaseI thefinite-straintheoryequationswereappliedto theentirelowermantle;CaseII correspondsto aradiusrangefrom 3831 to 5359 km.

TABLE VII

Pressurederivativesfor variousminerals

Structure Compound 1/V[(a V)/(aP)TI, (Mbaft) Reference

Compressional Shear

Olivine Mg2SiO4 1.249— 0.714 KamazawaandAnderson(1969)

(Mg0 93Fe0 07)2SiO4 1.211 0.736 KamazawaandAnderson(1969)Fe2SiO4 1.325 0.172 Chung (1971)

Pyroxene (Mg0 8Fe0 2)Si03 2.644 1.093 Friliblo and Barsch (1972)

Garnet almandite-pyrope 0.919 0.456 Anderson et al. (1968)spessartite-almandine 0.842 0.467 Wang and Simmons(1974)grussularite 0.5 13 —0.022 Halleck (1973)

Spinel MgAI2O4 0.563 —0.008 Chang and Barsch (1973)Mg0 75Fe036Al190O4 0.559 —0.093 WangandSimmons(1972)MgO~2.6 A12O3 0.494 0.076 Andersonet al. (1968)NiFe2O4 0.610 —0.008 Liebermann (1972)

RutileS Ti02 0.825 0.101 Manghnani (1969)Ge02 0.652 0.206 Wang and Simmons (1973)Sn02 0.4.65 0.026 Chang and Graham (1974)

Rocksalt MgO 0.740 0.621 Spetzler (1970)CaO 1.308 0.602 Anderson et al. (1968)

Corundum A12O3 0.478 0.347 Anderson et al. (1968)

Fe203 0.591 0.151 Andersonet al. (1968)

a-quartz Si02 1.325 0.172 Andersonetal. (1968)

46 AM. DZIEWONSKI,A.L. HALES AND E.R. LAPWOOD

derivativesof thevelocities,[(l/V~)(aV~/3P)Jand S. Conclusions[(l/V~)(aV5/aP)],may providea tight control on thelattice structuresof thematerialpredominantin the It hasbeenshownthat it is possibleto constructalower mantle.The only single mineral in TableVII parametricallysimple earthmodel(PEM-A) which fitswhich comesclose to fitting theseparametersis theobservedfree-oscillationdatasatisfactorily.Thiscorundum(A1203). Andersonet al. (1971)observed averageearthmodelwasderivedfrom aninversioninthat thelaboratorymeasurementsof thelogarithmic which separatemodels,PEM-O andPEM-C, of thedeviationsof thevelocitiesfor Al203 andFe203give uppermost420km wereusedfor theoceansandcon-theclosestmatchto thevaluesderivedfrom finite- tinents.Below a depthof 420km all threemodelsarestraintheory. Our valuesof [(l/V~)(3V5/aP)] aretoo thesame.Theseregionalmodelsareconsistentwithhigh to be compatiblewith Fe203. principal featuresof oceanicandcontinentaldisper-

In generaloneexpectsthat thematerialsof the sion curvesto periodsof about20 sec.lower mantlewill consistof high-pressuretransforms The seismic-bodytravel-timedataareregardedasof olivine (Mg,Fe)25i04andpyroxene(Mg,Fe)SiO3. applyingprincipally to thecontinentsbecauseof the

Oneof thepossibilitiesis disproportionationinto pen- preponderanceof land-basedstations.The continentalclaseandstishovite.The logarithmic derivativesof the structurewasthereforeusedfor theinversionof thevelocitieshavenot yet beendeterminedfo~stishovite, travel-timedata. If theregionalsurface-dispersiondatabut would beexpectedto be similar to thosefor the arenot includedin theinversion,satisfactoryfits torutile structuresin Table VII (R. Liebermann,personal all datacanbe obtainedwithout introducingbase-communication,1974). Clearlyit wouldbedifficult to line corrections.In thefinal modelthe incorporationmatchøo’ p0 andthelogarithmic derivativeswith such of regional-dispersiondatamadeintroduction of base-amixturebecauseof theconsiderabledifferencesin line correctionsnecessary.Thesecorrectionsare+ 1.25[(l/v~)(av~/aP)]0and [(I/V5)(aV5/3P)]0. for the Herrin et al. (1968)P traveltimes,+3.0 sec

Although logarithmic derivativesof thevelocities for the HalesandRoberts(1970)S-times,+0.19 forfor stishovitearenotavailable,Mizutani et al. (l972a) ClearyandHales(1971)PKIKP times and+0.85to thehavedeterminedthe velocitiesfor stishoviteat pres- SKS traveltimesof HalesandRoberts(1970). Thesureslessthan 10 kbarandroomtemperature.Their differencein thetwo-wayvertical travel timesfor S-valuefor theshearmodulus,p, is 1.30±0.07Mb al- wavesof PEM-O andPEM-C to 420km depthis onlymost exactly the sameasthevaluefor theshear 1.1 sec.modulusof p for periclasegivenby Andersonet al. The densitiesandvelocitiesin thelower mantle

(1968), namely,1.288 Mbar.The valuesof p for andcoreswere found to be consistentwith theAdams-wtistite (FeO)quotedby Mizutani et al. (l972b),and Williamson equationto less than0.2%.Thusit can beAkimoto (1972)are0.51 and0.55,respectively. Thus inferred that anydeparturesfrom homogeneityand

anydisproportionationof olivine orpyroxeneinto adiabaticitywithin eachof theseregionsmustbeMgO, SiO2,FeOwill inevitably result in a zero-pressure very small. It wasshownalso that the velocitiesin theroom-temperaturevalue for p lessthan 1.30Mbar. lower mantlewereconsistentwith thecompletefirst-This is lower than thevalue of 1.332 Mbar given in orderfinite-straintheoryto within 0.2%for V~andTable VI whichis for lower-mantletemperatures. 0.4%for V5. Thelogarithmicderivativesof theveloci-Thus it seemsthat unlesstheproportion of FeO to tieswith respectto pressureobtainedfrom theapplica-MgO is improbablycloseto zero thematerialof the tion of finite-straintheorywerevery similar to thoselower mantlemustconsistat leastin part of denser for corundumstructures.structures(e.g.,perovskite,calcium ferrite) though

admixturewith oxidestructuresis possible(Ringwood.1970). Similar conclusionswith regardto theimplica- Acknowledgementstionsof theshearvelocitiesfoundfor thelower mantlewere reachedby Mizutani et al. (1972a). This paperwaswritten while two of theauthors

47

(A.M. DziewonskiandE.R. Lapwood)were Visiting Gilbert, F., 1971a. Geophys. J., 23: 125.

Fellowsat theAustralianNational University. Gilbert, F., 1971b. Geophys. J., 23: 119.Gilbert, F., andDziewonski,A.M., 1975. Philos,Trans.R.Dr. RobertC. Liebermannhasreadthemanuscript

Soc. London, Ser. A, 278: 187.critically andmademanyhelpful suggestions.We wish Gilbert, F., Dziewonski, A. andBrune,J., 1973. Proc. Nati.

to thankDr. Liebermann,ProfessorA.E. Ringwoodand Amd. Sci. (U.S.A.), 70: 1410.

Dr. D.H. Greenfor their interestandhelpful discussion. Goncz,J.H.,1974.Surfacewavestudiesof the AustralianWe havepleasurein acknowledgingour indebtedness Upper Mantle,Thesis,AustralianNationalUniversity.

Green,D.H., 1973. Earth Planet.Sci. Lett., 19: 37.to ProfessorFreemanGilbert for the useof his program Haddon, R.A.W., 1972. Trans. Am. Geophys.Union, 53:for thenumericalsolutionof thenormal-modepro- 600.

blem andto Dr. BruceJulian for theuseof his travel- Haddon,R.A.W. andBullen, K.E., 1969. Phys.EarthPlanet.

Inter., 2: 35.time program.Oneof us (A.M. Dziewonski)wishesto acknowledge Hales, A.L., 1972. The traveltimesof P seismicwavesand

their relevance to the upper-mantle velocity distribution.supportunderNationalScienceFoundationGrant In: A.R. Ritsema(Editor),TheUpperMantle.Elsevier,GA-32320andtheCommitteefor ExperimentalGeo- Amsterdam.

logy andGeophysics,HarvardUniversity. Hales,A.L., 1974a.J. Geophys.Res., 79: 422.Hales,A.L., 1974b.Phys.Earth Planet.Inter., 9: 7.Hales,A.L. andHerrin, E., 1972. Travel timesof seismic

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