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不规则地形和随机土壤属性对地震地面运动空间的影响Influence of irregular topography and random soil

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Influence󰀈of󰀈irregular󰀈topography󰀈and󰀈randomsoil󰀈properties󰀈on󰀈coherency󰀈loss󰀈of󰀈spatialseismic󰀈ground󰀈motions

Article󰀃󰀃in󰀃󰀃Earthquake󰀃Engineering󰀃&󰀃Structural󰀃Dynamics󰀃·󰀃July󰀃2011

DOI:󰀃10.1002/eqe.1077

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EARTHQUAKEENGINEERINGANDSTRUCTURALDYNAMICSEarthquakeEngngStruct.Dyn.2011;40:1045–1061

Publishedonline22November2010inWileyOnlineLibrary(wileyonlinelibrary.com).DOI:10.1002/eqe.1077

Inﬂuenceofirregulartopographyandrandomsoilproperties

oncoherencylossofspatialseismicgroundmotions

KaimingBi∗,†andHongHao

SchoolofCivilandResourceEngineering,UniversityofWesternAustralia,35StirlingHighway,

Crawley,WA6009,Australia

SUMMARY

Coherencyfunctionsareusedtodescribethespatialvariationofseismicgroundmotionsatmultiplesupportsoflongspanstructures.Manycoherencyfunctionmodelshavebeenproposedbasedontheoreticalderivationormeasuredspatialgroundmotiontimehistoriesatdenseseismographicarrays.Mostofthemaresuitableformodellingspatialgroundmotionsonﬂat-lyingalluvialsites.Ithasbeenfoundthatthesecoherencyfunctionsarenotappropriateformodellingspatialvariationsofgroundmotionsatsiteswithirregulartopography(Struct.Saf.1991;10(1):1–13).Thispaperinvestigatestheinﬂuenceoflayeredirregularsitesandrandomsoilpropertiesoncoherencyfunctionsofspatialgroundmotionsongroundsurface.Groundmotiontimehistoriesatdifferentlocationsongroundsurfaceoftheirregularsitearegeneratedbasedonthecombinedspectralrepresentationmethodandone-dimensionalwavepropagationtheory.Randomsoilproperties,includingshearmodulus,densityanddampingratioofeachlayer,areassumedtofollownormaldistributions,andaremodelledbytheindependentone-dimensionalrandomﬁeldsintheverticaldirection.Monte-Carlosimulationsareemployedtomodeltheeffectofrandomvariationsofsoilpropertiesonthesimulatedsurfacegroundmotiontimehistories.Thecoherencyfunctionisestimatedfromthesimulatedgroundmotiontimehistories.Numericalexamplesarepresentedtoillustratetheproposedmethod.Numericalresultsshowthatcoherencyfunctiondirectlyrelatestothespectralratiooftwolocalsites,andtheinﬂuenceofrandomlyvaryingsoilpropertiesatacanyonsiteoncoherencyfunctionsofspatialsurfacegroundmotionscannotbeneglected.Copyright᭧2010JohnWiley&Sons,Ltd.

Received21January2010;Revised4July2010;Accepted27September2010KEYWORDS:

coherencylossfunction;irregulartopography;randomsoilproperties;Monte-Carlosimulation

1.INTRODUCTION

Forlargedimensionalstructures,suchaslong-spanbridges,pipelinesandcommunicationtrans-missionsystems,theirsupportsinevitablyundergodifferentseismicmotionsduringanearthquakeowingtothegroundmotionspatialvariation.Pastinvestigationsindicatethattheeffectofthespatialvariationofseismicmotionsonthestructuralresponsescannotbeneglected,andcanbe,incases,detrimental[1].Groundmotionspatialvariationeffecthasbeenextensivelystudiedbymanyresearchersespeciallyaftertheinstallationofstrongmotionarrays(e.g.theSMART-1arrayinLotung,Taiwan).Manyempirical[2–6]andsemi-empirical[7,8]modelshavebeenproposedmostlyforﬂat-lyingalluvialsites.Thesecoherencyfunctionsusuallyconsistoftwoparts,themodulusorcalledlaggedcoherency,whichmeasuresthesimilarityoftheseismicmotionsbetween

∗Correspondence

to:KaimingBi,SchoolofCivilandResourceEngineering,TheUniversityofWesternAustralia,

35StirlingHighway,Crawley,WA6009,Australia.†E-mail:bkm@civil.uwa.edu.au

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K.BIANDH.HAO

thetwostations,andthephase,whichdescribesthewavepassageeffect,i.e.thedelayinthearrivalofthewaveformsatthefurtherawaystationcausedbythepropagationoftheseismicwave.Itisgenerallyfoundthatthelaggedcoherencydecreasessmoothlyasafunctionofstationseparationandwavefrequency.Toconsiderlocalsiteeffect,DerKiureghian[9]proposedatheoreticalmodeltodescribecoherencyfunctionofmotionsonthegroundsurface,inwhichheassumedthatsiteeffectinﬂuencesthephaseofthecoherencyfunctiononly,whileitdoesnotaffectthelaggedcoherency.

Contrasttotheobservationsontheﬂat-lyingsites,Somervilleetal.[10]investigatedthecoherencyfunctionofgroundmotionsonasitelocatedonfoldedsedimentaryrocks(theCoalingaanticline),andfoundthatthelaggedcoherencydoesnotshowastrongdependenceonstationseparationandwavefrequency,andtheincoherencyisgenerallyhigherthanthatontheﬂat-lyingsites.Theyattributedthechaoticbehaviourtothewavepropagationinamediumhavingstronglateralheterogeneitiesinseismicvelocity.Liaoetal.[11],basedontheseismicdatarecordedattheParkwayarrayinWainuiomataValley,NewZealand,comparedthelaggedcoherencyfunctionsofdifferentstationcombinations,i.e.fourgroupswithstationpairslocatedonthesediments,onegroupwithonesedimentarystationandonerockstation.Theyconcludedthatthelaggedcoherencybetweenthesedimentandrockstationsexhibitlargevariabilityandfollownoconsistentpattern.Theseobservationssuggestthatthespatialcoherencyfunctionmeasuredonﬂat-lyingsedimentarysitesmaynotprovideagooddescriptionofspatialgroundmotioncoherenciesonsiteswithirregulartopography.TheseobservationsalsoindicatethatthetheoreticalmodelproposedbyDerKiureghian[9]mightnotbeabletoreliablydescribetheinﬂuenceoflocalsiteeffectonthecoherencyfunction.

Althoughitwasobservedthattheheterogeneityofsiteconditionsstronglyaffectthegroundmotionspatialvariations[10,11],allthepreviousstudiesandtheoreticalandempiricalcoherencymodelsmentionedaboveassumedthatthesitecharacteristicsarefullydeterministicandhomoge-neous.However,inreality,therealwaysexistspatialvariationsofsoilpropertiesanduncertaintiesindeﬁningthepropertiesofsoils.Thisresultsfromthenaturalheterogeneityorvariabilityofsoils,thelimitedavailabilityofinformationaboutinternalconditionsandsometimesthemeasurementerrors.Theseuncertaintiesassociatedwithsystemparametersarealsolikelytohaveinﬂuenceonthecoherencyfunction.ZervaandHarada[12]modelledhorizontalsoillayersatasiteasa1-DOFsystemwithrandomcharacteristicstostudytheeffectofuncertainsoilpropertiesonthecoherencyfunction.Theypointedoutthatthespatialcoherencyofmotionsonthegroundsurfaceissimilartothatoftheincidentmotionatthebaserockexceptatthepredominantfrequencyofthelayer,whereitdecreasesconsiderably.Theeffectofuncertainsoilpropertiesshouldalsobeincorporatedinspatialvariationmodelofgroundmotions.Theirexplanationforthisphenomenonwasthatforinputmotionfrequenciesclosetothemeannaturalfrequencyofthe‘oscillators’,theresponseofthesystemswasaffectedbythevariabilityinthevalueofthisnaturalfrequency,andresultedinlossofcorrelation[12].However,itshouldbenotedthata1-DOFsystemcannotrealisticallyrepresentthemultiplepredominatefrequenciesthatmayexistatasitewithmultiplelayersandmultiplemodes.LiaoandLi[13]developedananalyticalstochasticmethodtoevaluatetheseismiccoherencyfunction,inwhichanumericalapproachtocomputecoherencyfunctionisdevelopedbycombiningthepseudo-excitationmethodwithwavemotionﬁniteelementsimulationtechniques.Anorthogonalexpansionmethodisintroducedtostudytheeffectofuncertainsoilpropertiesonthecoherencyfunction.Theresultsalsodemonstratethatthelaggedcoherencyvaluestendtodecreaseinthevicinityoftheresonantfrequenciesofthesite.Thismethodis,however,difﬁculttobeimplementedandsometimesalittlearbitrarytoselecttheabsorbingboundaryconditions,andisdifﬁculttoexplainwhythelaggedcoherencyfunctionvariessigniﬁcantlyoverrelativelyshortdistancesowingtotheinherentlimitationsofusingﬁniteelementmethodtomodelwavemotioninaunboundedmedium[14].

Itisobviousthattheeffectsofirregulartopographyandrandomsoilpropertiesofasiteonthecoherencyfunctionofspatialgroundmotionscannotbeneglected.However,atpresent,onlyverylimitedrecordedspatialgroundmotiondataonsitesofdifferentconditionsareavailable.Theyarenotsufﬁcienttodeterminethegeneralspatialincoherencecharacteristicsofgroundmotions

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andderiveempiricalrelationstomodelspatialgroundmotionvariationsatasitewithvaryingsiteconditions.Ontheotherhand,totheauthors’bestknowledge,nomoretheoretical/analyticalanalysisinthisﬁeldcanbefoundexceptforthestudiesmentionedabove[9,12,13].

Thepresentstudyinvestigatestheinﬂuenceofalayeredcanyonsiteandrandomlyvaryingsoilpropertiesoncoherencyfunctionofspatialgroundmotions.Thesiteisassumedconsistingofhorizontallyextendedmultiplesoillayersonahalf-space(baserock).Thebaserockmotionsatdifferentlocationsareassumedtohavethesameintensity,andaremodeledbyaﬁlteredTajimi-Kanaipowerspectraldensityfunction.Thespatialvariationofgroundmotionsonthebaserockisaccountedforbyanempiricalcoherencyfunctionforspatialgroundmotionsonaﬂat-lyingsite.Usingtheone-dimensionalwavepropagationtheory[15],thepowerspectraldensityfunctionsofspatialgroundmotionsatvariouslocationsonsurfaceofthecanyonsitecanbederivedbyassumingthebaserockmotionsconsistingofout-of-planeSHwaveorin-planecombinedPandSVwavespropagatingintothesitewithanassumedincidentangle.Thespatiallyvaryinggroundmotiontimehistoriescanthenbegeneratedbasedonthespectralrepresentationmethod.Inordertotakeintoconsiderationtherandomsoilproperties,Monte-Carlosimulationmethodisusedinthestudy.Therandomsoilpropertiesconsideredincludetheshearmodulus,densityanddampingratioofeachlayer,andtheyareallassumedtohavenormaldistributionsintheverticaldirectionandaremodelledasindependentone-dimensionalrandomﬁelds[16].Innumericalcalculations,foreachrealizationoftherandomsoilproperties,spatialgroundmotiontimehistoriesaregenerated.Thesetimehistoriesarethenusedtocalculatethelaggedcoherencybetweenanytwogroundmotiontimehistories.Thenumericalcalculationsincludethefollowingsteps:(1)randomgenerationofsoilproperties;(2)estimationofgroundmotionpowerspectraldensityfunctionsatvariouspointsonthecanyonsurface;(3)simulationsofspatialgroundmotiontimehistoriesand(4)calculationsofcoherencyfunctions.Thesestepsarerepeateduntiltheestimatedmeanandstandarddeviationofthelaggedcoherencybetweengroundmotionsatanytwopointsconverge.Numericalexamplesarepresentedtodemonstratetheproposedmethodandtostudytheeffectsofirregulartopographyandrandomsoilpropertiesoncoherencyfunctionofspatialgroundmotions.

2.THEORETICALBASIS

2.1.Estimationofcoherencyfunction

Letuj(t)anduk(t)betherecorded(simulated)accelerationtimehistoriesatlocationsjandkofasite,andthecorrespondingFouriertransformofthetimehistoriesareUj(󰀏)andUk(󰀏),respectively.Thesmoothedautospectraldensityfunctionofgroundmotionatlocationjorkisthen

Sii(󰀏n)=

M󰀉m=−M

W(m󰀂󰀏)|Ui(󰀏n+m󰀂󰀏)|2

i=jork(1)

andthecrosspowerspectraldensityfunctionbetweenmotionsatstationsjandkis

Sjk(󰀏n)=

M󰀉m=−M

∗

W(m󰀂󰀏)Uj(󰀏n+m󰀂󰀏)Uk(󰀏n+m󰀂󰀏)

(2)

whereW(󰀏)isthespectralsmoothingwindow,󰀂󰀏isthefrequencystep,󰀏n=n󰀂󰀏isthenth

discretefrequencyand∗denotesthecomplexconjugate.

Thecoherencyfunctionofthespatialgroundmotionscanbeobtainedas[3]

󰀄jk(󰀏)=󰀋Sjk(󰀏)Sjj(󰀏)Skk(󰀏)(3)

ThecoherencyfunctioninEquation(3)isgenerallyacomplexfunctionandcanbewrittenas

󰀄jk(󰀏)=|󰀄jk(󰀏)|exp[i󰀇jk(󰀏)]

(4)

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inwhich|󰀄jk(󰀏)|isthelaggedcoherency,󰀇jk(󰀏)=tan−1[Im(Sjk(󰀏))/Re(Sjk(󰀏))]isthephaseangle,‘Im’and‘Re’denotetheimaginaryandrealpartsofacomplexnumber.

Basedontheanalysisabove,thecoherencyfunctioncanbereadilyestimatediftheaccelerationtimehistoriesateachlocationareavailable.Thesimulationofgroundmotiontimehistoriesisbasedontheone-dimensionalwavepropagationtheory[17]andthespectralrepresentationmethod.ThesetwopartsarebrieﬂyintroducedinSections2.2and2.3,moredetailedinformationcanbefoundinReference[15].

2.2.One-dimensionalwavepropagationtheory

Forasitewithhorizontallyextendedmultiplesoillayersonahalfspace(baserock),thebaserockmotionscanbeassumedtoconsistofout-of-planeSHwaveorin-planecombinedPandSVwavespropagatingintoasitewithanassumedincidentangle.Foraharmonicexcitationwithfrequency󰀏,thedynamicequilibriumequationscanbewrittenas[17]󰀏2

∇e=−2e

󰀏2

or∇{󰀃}=−2{󰀃}

(5)

where∇2eand∇2{󰀃}aretheLaplaceoperatorofthevolumetricstrainamplitudeeandrotational-strain-vector{󰀃}.cpandcsaretheP-andS-wavevelocities,respectively.ThisequationcanbesolvedbyusingtheP-andS-wavetrialfunctions.Theout-of-planedisplacementswiththeamplitudeviscausedbytheincidentSHwave,whereasthein-planedisplacementswiththeamplitudeuandwinthehorizontalandverticaldirectionsdependonthecombinedPandSVwaves.Theamplitudevisindependentofuandw,hence,thetwo-dimensionaldynamicstiffness

L]and[SLmatrixofeachsoillayerfortheout-of-planeandin-planemotions,[SSHP−SV],canbe

formulatedindependentlybyanalysingtherelationsofshearstressesanddisplacementsattheboundaryofeachsoillayer.Assemblingthematricesofeachsoillayerandthebaserock,thedynamicstiffnessofthetotalsystemisobtainedanddenotedby[SSH]and[SP−SV],respectively.Thedynamicequilibriumequationofthesiteinthefrequencydomainisthus[17]

[SSH]{uSH}={PSH}or

[SP−SV]{uP−SV}={PP−SV}

(6)

where{uSH}and{PSH}aretheout-of-planedisplacementsandloadvectorcorrespondingtotheincidentSHwave,{uP−SV}and{PP−SV}arethein-planedisplacementsandloadvectorofthecombinedPandSVwaves.Thestiffnessmatrices[SSH]and[SP−SV]dependonsoilproperties,incidentwavetype,incidentangleandcircularfrequency󰀏.Thedynamicload{PSH}and{PP−SV}dependonthebaserockproperties,incidentwavetype,incidentwavefrequencyandamplitude.BysolvingEquation(6)inthefrequencydomainateverydiscretefrequency,therelationshipoftheamplitudesbetweenthebaserockandeachsoillayercanbeformed,andthesitetransferfunction[H(󰀏)]intheout-of-planeandin-planedirectionscanbeestimated.

2.3.Groundmotiongeneration

Consideracanyonsitewithhorizontallyextendedmultiplesoillayersrestingonanelastichalf-spaceasshowninFigure1,inwhichhm,Gm,󰀋m,󰀉mand󰀍misthedepth,shearmodulus,massdensity,dampingratioandPoisson’sratiooflayerm.Thespatiallyvaryingbaserockmotionsareassumedtoconsistofout-of-planeSHwaveorin-planecombinedPandSVwavesandpropagatingintothelayeredsoilsitewithanassumedincidentangleasdiscussedabove.Theincidentmotionsatdifferentlocationsonthebaserockareassumedtohavethesamepowerspectraldensity,andaremodelledbyaﬁlteredTajimi-Kanai[18]powerspectraldensityfunction.Thespatialvariationofgroundmotionsatbaserockismodelledbyanempiricalcoherencyfunctionforspatialground

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kLayer l:hl,Gl,ρl,ξl,υljLayer m-1:

Layer m:hm,Gm,ρm,ξm,υmhm−1,Gm−1,ρm−1,ξm−1,υm−1

j’

Layer 1:

h1,G1,ρ1,ξ1,υ1GB,ρB,ξB,υB

k’

Base rock:

Figure1.Schematicviewofalayeredcanyonsite.

motionsonaﬂatsite.Thecrosspowerspectraldensityfunctionsofsurfacemotionsatnlocationsofthelayeredsitecanbewrittenas

⎤⎡

S11(󰀏)S12(i󰀏)···S1n(i󰀏)

⎥⎢

⎢S21(i󰀏)S22(󰀏)···S2n(i󰀏)⎥

⎥(7)S(i󰀏)=⎢⎥⎢

············⎦⎣

Sn1(i󰀏)

where

Sjj(󰀏)=|Hj(i󰀏)|2Sg(󰀏),Sjk(i󰀏)=

j=1,2,...,n

j,k=1,2,...,n

∗

Hj(i󰀏)Hk(i󰀏)Sg(󰀏)󰀄j󰀅k󰀅(dj󰀅k󰀅,i󰀏),

Sn2(i󰀏)···Snn(󰀏)

(8)

aretheautoandcrosspowerspectraldensityfunctions,respectively.InwhichSg(󰀏)istheground

motionpowerspectraldensityonthebaserock;󰀄j󰀅k󰀅(dj󰀅k󰀅,i󰀏)isthecoherencyfunctionbetweenlocationsj󰀅andk󰀅onthebaserock;Hj(i󰀏),Hk(i󰀏)arethesitetransferfunctionsatlocationsjandkonthegroundsurface,whichcanbeformulatedbasedonone-dimensionalwavepropagationtheorydiscussedinSection2.2.

DecomposingtheHermitian,positive-deﬁnitematrix,S(i󰀏),intothemultiplicationofacomplexlowertriangularmatrix,L(i󰀏),anditsHermitian,LH(i󰀏),

S(i󰀏)=L(i󰀏)LH(i󰀏)

(9)

thestationarytimeseriesuj(t),j=1,2,...,n,canbesimulatedinthetimedomaindirectlyas[5]uj(t)=

where

j󰀉N󰀉m=1n=1

Ajm(󰀏n)cos[󰀏nt+󰀃jm(󰀏n)+󰀎mn(󰀏n)]

(10)

Ajm(󰀏)=

√

4󰀂󰀏|Ljm(i󰀏)|,0󰀁󰀏󰀁󰀏N

󰀂󰀁

Im[L(i󰀏)]jm

,0󰀁󰀏󰀁󰀏N󰀃jm(󰀏)=tan−1

Re[Ljm(i󰀏)]

(11)

aretheamplitudesandphaseanglesofthesimulatedtimehistories,whichensurethespectraofthesimulatedtimehistoriescompatiblewiththosegiveninEquation(8);󰀎mn(󰀏n)istherandomphaseanglesuniformlydistributedovertherangeof[0,2󰀊],󰀎mnand󰀎rsshouldbestatisticallyindependentunlessm=randn=s;󰀏Nrepresentsanuppercut-offfrequencybeyondwhichtheelementsofthecrosspowerspectraldensitymatrixgiveninEquation(7)isassumedtobezero.

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ThegeneratedtimeseriesbyEquation(10)arestationaryprocesses.Inordertoobtainthenon-stationarytimehistories,anenvelopefunction󰀆(t)isappliedtouj(t).Thenon-stationarytimehistoriesatdifferentlocationsarethenobtainedby

fj(t)=󰀆(t)uj(t),

2.4.Randomﬁeldtheory

Inengineeringpracticetherearealwayssomeuncertaintiesinthesoilpropertiesbecauseofthereasonsmentionedabove.Therandomﬁeldtheory[16]iswidelyusedtodescribethevariabilityofsoilproperties.Inthistheorytherandomsoilproperty,u(z),ischaracterizedbythemeanvalue,u¯,standarddeviation,󰀌uandthecorrelationdistance,󰀅u.󰀌umeasurestheintensityofﬂuctuationordegreetowhichactualvalueofu(z)maydeviatefrom.󰀅umeasuresthecorrelationlevelorpersistenceofthepropertyfromonepointtoanotherinasite,smallvaluesof󰀅usuggestrapidﬂuctuationabouttheaverage,whereaslargevaluesof󰀅uimplythataslowlyvaryingcomponentissuperimposedontheaveragevalueofu¯.

Considerthataone-dimensionalrandomﬁeld,u(z),withmeanvalue,u¯(z),andstandarddevi-ation,󰀌u,itslocalaverageprocessuZ(z)ofu(z)overtheintervalZcenteredatzisdeﬁnedas

󰀊

1z+Z/2󰀅󰀅

uZ(z)=u(z)dz(13)

Zz−Z/2Itcanbeseenthatthelocalaverage,uZ(z),dependsonthespeciﬁclocationoftheintervalzwithinthestatisticallyhomogeneoussoillayer.ThemeanandvarianceofuZ(z)are[16]

E[uZ(z)]=E[u(z)]=u¯(z)Var[uZ(z)]=󰀌2u󰀈(Z)

(14)

j=1,2,...,n

(12)

where󰀈(Z)isavariancereductionfunctionofu(z)thatmeasuresthereductionofpointvariance,

󰀌2u,underlocalaverage.Thevariancefunction,󰀈(Z),canbederivedfromauto-correlationfunction,󰀋u(󰀂z),inthefollowingform:

󰀂󰀊󰀁

2Z󰀂z

󰀈(Z)=󰀋u(󰀂z)d(󰀂z)(15)1−

Z0ZByusingtheexponentialauto-correlationfunction[19]

󰀋u(󰀂z)=exp(−2|󰀂z|/󰀅u)

thevariancereductionfunctioncanbederivedas[19]

󰀈(Z)=

[2(Z/󰀅u)+e−2(Z/󰀅u)−1]22(Z/󰀅u)

(17)(16)

Inthisstudy,theshearmodulus,densityanddampingratioofeachsoillayerofthesiteare

regardedasrandomﬁelds,andareassumedtofollownormaldistributionsintheverticaldirection.Theserandomﬁeldscanbemodelledbyintroducingthemeanvalue,standarddeviationandcorrelationdistanceofeachparameterasmentionedabove.Takeshearmodulusasanexample

󰀋󰀋

¯+󰀌G󰀈(Z)󰀏=G¯(1+COV×󰀈(Z)󰀏)G=G(18)¯and󰀌Garethemeanvalueandstandarddeviationofshearmodulus,󰀈(Z)isthevariancewhereG

reductionfunctionand󰀏isanormaldistributedrandomprocesswithzeromeanandunityvariance.

¯isthecoefﬁcientofvariation.COV=󰀌G/G

2.5.Monte-Carlosimulation

Monte-Carlosimulationshavebeenextensivelyusedinmanyscientiﬁcﬁeldswithrandomparam-eters.Itwasfoundthatfortherangeofvariabilityusuallypresentinsoilproperties,Monte-Carlo

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basedmethod,thoughcomputationallyintensive,mightbethesimplestandmostdirectmethod.Othermethods,whicharebasicallyexpansionbased,donotprovideaccurateresultswhenthecoefﬁcientsofvariationofsoilpropertiesarelarge[20].Inthisstudy,Monte-Carlosimulationsarealsoemployedtoaccountfortheinﬂuenceofrandomsoilpropertiesonspatialgroundmotions.InMonte-Carlosimulations,soilpropertiesarerandomlygeneratedaccordingtotheirdistributions.Eachsetofrandomsoilpropertiesareconsideredasdeterministicinestimatingthepowerspectraldensitiesofgroundmotions.Thenspatialgroundmotiontimehistoriesaresimulatedaccordingtotheproceduresdescribedabove.

3.NUMERICALEXAMPLE

Tostudytheinﬂuenceofirregulartopographyandrandomsoilpropertiesonthecoherencyfunctionbetweendifferentmotionsonthegroundsurface,afour-layercanyonsiterestingonthebaserockisselectedasanexampleasshowninFigure2.Themeanvaluesofthecorrespondingsoilpropertiesofeachsoillayerandbaserockarealsogivenintheﬁgure.

ThemotionsonthebaserockareassumedtohavethesameintensitiesandfrequencycontentsandaremodelledbytheﬁlteredTajimi-Kanaipowerspectraldensityfunctioninthefollowingform:

221+4󰀉2󰀏4g󰀏g󰀏

Sg(󰀏)=|HP(󰀏)|S0(󰀏)=2󰀁222222222(󰀏f−󰀏)+(2󰀏f󰀏󰀉f)(󰀏g−󰀏)+4󰀉g󰀏g󰀏

(19)

where|HP(󰀏)|isahighpassﬁlterfunction[21],whichisappliedtoﬁlteroutenergyatzeroand

verylowfrequenciestocorrectthesingularityingroundvelocityanddisplacementpowerspectraldensityfunctions.S0(󰀏)istheTajimi-Kanaipowerspectraldensityfunction[17],󰀏gand󰀉garethecentralfrequencyanddampingratiooftheTajimi-Kanaipowerspectraldensityfunction,󰀏fand󰀉farethecorrespondingcentralfrequencyanddampingratioofthehighpassﬁlter.󰀁isascalingfactordependingonthegroundmotionintensity.Intheanalysis,theout-of-planehorizontalmotionisassumedtoconsistofSHwaveonly,whereasthein-planehorizontalandverticalmotionsareassumedtobecombinedPandSVwaves.Theparametersforthehorizontalmotionareassumedas󰀏g=10󰀊rad/s,󰀉g=0.6,󰀏f=0.5󰀊,󰀉f=0.6and󰀁=0.0034m2/s3.TheseparameterscorrespondtoagroundmotiontimehistorywithdurationT=20sandpeakgroundacceleration(PGA)0.2gbasedonthestandardrandomvibrationmethod[22].TheverticalmotiononthebaserockisalsomodelledwiththesameﬁlteredTajimi-Kanaipowerspectraldensityfunction,buttheamplitudeisassumedtobe2/3ofthehorizontalcomponentofPGA0.2g.

kNo.4 Sandy fill, h=5m, G=30MPa, ρ=1900kg/m3,ξ=5%,υ=0.40No.3 Soft Clay, h=15m, G=20MPa, ρ=1600kg/m3,ξ=5%,υ=0.40jNo.2 Silt sand, h=16m, G=220MPa, ρ=2000kg/m3,ξ=5%,υ=0.33

No.1 Firm clay, h=12m, G=30MPa, ρ=1600kg/m3,ξ=5%,υ=0.40j’ k’ Base rock: G=1800MPa, ρ=2300kg/m3,ξ=5%,υ=0.33

Figure2.Afour-layercanyonsitewithdeterministicsoilproperties(nottoscale).

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Figure3.Simulatedaccelerationtimehistories:(a)baserockmotion;(b)horizontalout-of-planemotion;

(c)horizontalin-planemotion;and(d)verticalin-planemotion.

TheSobczykmodel[23]isselectedtodescribethecoherencylossbetweenthegroundmotionsatpointsj󰀅andk󰀅onthebaserock:

󰀄j󰀅k󰀅(i󰀏)=|󰀄j󰀅k󰀅(i󰀏)|exp(−i󰀏dj󰀅k󰀅cos󰀂/vapp)=exp(−󰀃󰀏d2j󰀅k󰀅/vapp)·exp(−i󰀏dj󰀅k󰀅cos󰀂/vapp)

(20)

where󰀃isacoefﬁcientreﬂectingthelevelofcoherencyloss;󰀃=0.001thatisusedinthepresentpaperrepresentsintermediatelycorrelatedmotions;dj󰀅k󰀅isthedistancebetweenthepointsj󰀅andk󰀅,anddj󰀅k󰀅=100misassumed;󰀂istheincidentangleoftheincomingwavetothesite,andisassumedtobe60◦;vappistheapparentwavevelocityonthebaserock,whichis1768m/saccordingtothebaserockpropertyandthespeciﬁedincidentangle.Seismicwavesareassumedpropagatingverticallyfromthebaserocktothegroundsurface.

Takethecanyonsitewithdeterministicsoilpropertiesasanexample.Assumingthesoilprop-ertiesofeachsoillayerequaltotheirmeanvaluesasgiveninFigure2,theaccelerationtimehistoriesonthebaserockandthegroundsurfacearesimulatedbasedontheprocedurespresentedinSections2.2and2.3.Thesamplingfrequencyandtheuppercut-offfrequencyaresettobe100Hzand󰀏N=20Hz,respectively.2048samplingpointsareusedineachsetofgroundmotiontimehistories.As󰀎mninEquation(10)isarandomvariableuniformlydistributedovertherangeof[0,2󰀊],anyrealizationofarandomangle,󰀎mn,willresultinagenerationofasetofspatialgroundaccelerationtimehistoriesthatarecompatiblewiththespectraldensityfunctioninEquation(8).Figure3showsonesetofthesimulatedaccelerationtimehistories.

Thecoherencyfunctionbetweendifferentmotionsonthegroundsurfacecanbeestimatedafterthegenerationofaccelerationtimehistories.However,itneedstobeemphasizedthatcoherencyestimatesdependstronglyonthetypeofthesmoothingwindowandtheamountofsmoothingperformedontherawdata.Abrahamsonetal.[24]notedthatthechoiceofthesmoothingwindowshouldbedirectednotonlyfromthestatisticpropertiesofthegroundmotiontimehistories,butalsofromtheproblemforwhichitisanalysing,sothattherequiredresolutionisnotlost.Theysuggestedan11-pointHammingwindow,ifthecoherencyestimatesistobeusedinstructuralanalysis,fortimewindowslessthanapproximately2000samplesandforstructuraldampingcoefﬁcient5%ofcritical[24].Itshouldalsobenotedthatifnosmoothingisperformedonthe

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Figure4.Meanvaluesandstandarddeviationsofthelaggedcoherencyofthehorizontal

out-of-planemotionat0.2,2.0,5.0and9.0Hz.

Figure5.Comparisonofthemeanlaggedcoherencyonthebaserockfrom

600simulationswiththetargetmodel.

rawdata,thelaggedcoherencywillalwaysbeunityforeachfrequency,andnoinformationaboutthecoherencycanbeextractedfromthedata.

Toobtainthemeanlaggedcoherencyfunctionsonthebaserockandgroundsurface,Monte-CarlosimulationmethodisusedasdiscussedinSection2.5.ConvergencetestneedstobeconductedtocheckthenumberofMonte-Carlosimulationsrequiredtoobtainconvergedsimulationresults.SincealargernumberofMonte-Carlosimulationsisrequiredforthesimulationtoconvergeiftherandomvariablesunderconsiderationhavelargercoefﬁcientsofvariation(COV),thecasewiththelargestCOVconsideredinthisstudy,i.e.aCOVof60%forshearmodulusanddampingratioofeachsoillayerand5%forsoildensity,whichwillbefurtherdiscussedinSection3.2,isusedtoperformtheconvergencetest.Themeanvaluesandstandarddeviationsofthelaggedcoherencyfunctionofthehorizontalout-of-planemotionat0.2,2.0,5.0and9.0Hzareusedasthequantityforconvergencetest.AsshowninFigure4,thecorrespondingvaluesvirtuallyunchangedafter600simulations,indicatingthesimulationsconvergedwith600simulations.Resultsofthesimulatedin-planemotions,whicharenotshown,alsoconvergeafter600simulations.Therefore,600simulationsareperformedforeachcaseinthesubsequentcalculations.Figure5showsthecomparisonbetweenthemeanlaggedcoherencyfunctionsfromthe600simulatedspatialgroundmotiontimehistoriesonthebaserocksmoothedbythe11-pointHammingwindowwiththetargetmodel.Itisevidentthatverygoodagreementcanbeobtainedexceptforthefrequenciesnearzero.Infact,theoretically,coherencyshouldtendtobeunityasfrequencytendstozero,however,coherencyestimatesfromgroundmotiontimehistories,duetosmoothing,canrarelyreachthisvalue.

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Figure6.Comparisonofthemeanlaggedcoherencybetweenthesurfacemotions(j,k)withthatoftheincidentmotiononthebaserock:(a)horizontalout-of-planemotion;

(b)horizontalin-planemotion;and(c)verticalin-planemotion.

Figure7.Standarddeviationsofthelaggedcoherencyonthegroundsurface:(a)horizontalout-of-plane

motion;(b)horizontalin-planemotion;and(c)verticalin-planemotion.

3.1.Inﬂuenceofirregulartopography

Assumingthatallthesoilpropertiesaredeterministicandequaltotheirmeanvalues,theinﬂuenceofirregulartopographyisstudiedﬁrst.Figure6showsthemeanvaluesofthelaggedcoherencyfunctionsbetweenthespatialgroundmotionsofpointsjandkonthegroundsurfaceofthecanyonsite.Forcomparisonpurpose,thelaggedcoherencybetweenincidentmotiononthebaserockatj󰀅andk󰀅isalsoplotted.Figure7showsthecorrespondingstandarddeviations.Asshown,thestandarddeviationshaveageneraltrendofincreasingwithfrequency,butarerelativelysmall,alllessthan0.13.Thisindicatesthatthelaggedcoherencyismoredifﬁculttobeaccuratelymodelledathighfrequencies.Nonetheless,asthestandarddeviationsarerelativelysmallascomparedtothemeanlaggedcoherencyvalues,includingthemwillchangethelaggedcoherencyvalue,butnottheoveralltrend.Figure6showsthatthecoherencyfunctionbetweensurfacegroundmotionsdiffersfromthatbetweenbaserockmotionssigniﬁcantly.Atallfrequencies,thecoherencylossfunctionsonthegroundsurfacearesmallerthanthoseonthebaserock,i.e.thecoherencyfunctiononthebaserockistheupperboundofthecoherencyofspatialgroundmotionsonthesurfaceofacanyonsite.ThisconclusionisinagreementwiththatofLouandZerva[25],andLiaoetal.[11].Itindicatesthatwavepropagationthroughalocalsiteevenwithdeterministicsitepropertiesfurtherreducesthecrosscorrelationbetweenspatialgroundmotionsonthebaserock.Asshown,therearemanyobviouspeaksandtroughsinthecoherencyfunctionofsurfacemotions.Thesepeaksandtroughsdirectlyrelatetothemodulusofthespectralratiooftwolocalsites,namely|Hk(i󰀏)/Hj(i󰀏)|,asshowninFigure8.Hj(i󰀏)andHk(i󰀏)arethetransferfunctionsofsitesjandk,respectively.Theyarethespectralratioofthesurfacemotionatjorktothecorrespondingbedrockmotionatj󰀅ork󰀅,whichcanbecalculatedbasedontheone-dimensionalwavepropagationtheoryasdiscussedinSection2.2.Figure9showsthemodulusofthetransferfunctionsatsitesjandk.

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Figure8.Modulusofthesiteampliﬁcationspectralratiooftwolocalsites:(a)horizontalout-of-plane

motion;(b)horizontalin-planemotion;and(c)verticalin-planemotion.

Figure9.Amplitudesofthesiteampliﬁcationspectraoftwolocalsites:(a)horizontalout-of-planemotion;

(b)horizontalin-planemotion;and(c)verticalin-planemotion.

Itisobviousthatsiteampliﬁesthemotionsonthebaserocksigniﬁcantly,whichmakestheenergyofsurfacegroundmotionsconcentrateatafewfrequenciescorrespondingtothevariousvibrationmodesofthesite.Thisresultindicatestheimportanceofconsideringthemultiplemodesofalocalsoilsitewhenestimatingtheseismicwavepropagationandsiteampliﬁcation.Thepresentresultisanextensionofthoseobtainedwitha1-DOFmodel[12].Witha1-DOFmodel,theinﬂuenceofthehighervibrationmodesofthesiteonsiteampliﬁcationandhencethespatialgroundmotioncoherencycannotbeincluded.ComparingFigures6and8,itcanbenotedthatwhenthespectralratiosdifferfromeachother,thespatialgroundmotionsonthegroundsurfaceareleastcorrelatedwithaminimumlaggedcoherencyvalue.Takingthehorizontalout-of-planemotionasanexample,fourobviousminimacanbeobservedaroundthefrequencies0.78,1.90,4.20and7.10Hz,whichcorrespondtothefourevidentpeaksinthespectralratioasshowninFigure8(a).Similarconclusionscanbeobtainedforthein-planemotions.Thisisexpectedbecausethelaggedcoherencymeasuresthesimilarityofthemotionsattwodifferentlocations.Iftwositesamplifythegroundmotionstothesameextentatcertainfrequencies,thecoherencylossismainlycausedbytheincoherenceeffectandwavepassageeffect,localsiteeffecthaslittleinﬂuenceonthelaggedcoherency.However,ifthesiteampliﬁcationspectraaredifferentfromeachotheratcertainfrequencies,thelocalsiteeffectonwavepropagationisdifferent.Thereforesurfacegroundmotionswillbedifferentatthesefrequencies,whichresultsinspatialsurfacegroundmotionslesscorrelated.TheseobservationscoincidewiththerecordeddatafromtheCoalingaanticlineinCalifornia[10]andtheWainuiomataValleyinNewZealand[11].Theseobservationsalsoindicatethatsiteeffectwillnotonlycausephasedifferenceofthecoherencyfunction[9],butwillalsoaffectitsmodulus.

LiaoandLi[13]usedtheauto-powerspectraldensityofgroundmotionatonelocationofthesitetoidentifythelaggedcoherencyfunctiononthegroundsurface,andconcludedthat

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thesurfacelayerirregularityofasitecanreducethelaggedcoherencyfunctionvaluesinthevicinityoftheresonantfrequenciesofthesite.Toexaminetheirobservation,thehorizontalout-of-planemotionofsitejisusedasanexample.Thefundamentalvibrationfrequencyofthesiteisabout1.25HzasshowninFigure9(a).AccordingtoLiaoandLi’sconclusion,thelaggedcoherencyshouldhaveaminimumvalueatthisfrequency.However,thepresentresultsactuallydisplayapeakvalueinthelaggedfrequencyatthisfrequencyasshowninFigure6(a).ThiscontradictswithLiaoandLi’sconclusion.Thisisbecauseinthepresentexample,bothwavepathsfromj󰀅tojandk󰀅tokorbothsitesamplifythebedrockmotionaroundthisfrequency,althoughtoadifferentextent.Thereforewavepropagationthroughthetwositesdoesnotsigniﬁ-cantlyreducethecrosscorrelationofspatialbedrockmotionsatthisfrequency.Thisobservationdemonstratesthatusingtheamplitudeofthepowerspectraldensityofgroundmotionatjustonelocationtoassesstheinﬂuenceofwavepropagationinacanyonsiteandhencethecoherencyfunctionofspatialsurfacegroundmotionsmaynotleadtoareliablecoherencyestimation.Thespectralratiobetweenthetwoconsideredsitesortwowavepathsisamorereliableandappro-priateparametertomeasurethelocalsiteeffectoncrosscorrelationofspatialsurfacegroundmotions.

3.2.Inﬂuenceofrandomsoilproperties

Theinﬂuenceofrandomlyvaryingsoilpropertiesonthecoherencylossfunctionsbetweenthesurfacemotionsisstudiedinthissection.Withoutlosinggenerality,assumingshearmodulus,dampingratioandsoildensityarerandomﬁeldsinallsoillayers,andallfollowanormaldistribution.ThemeanvaluesofsoilpropertiesineverylayeraregiveninFigure2.Accordingtoamorespeciﬁcreviewandsummary[26],inmostcommonﬁeldmeasurements,theCOVforthecohesionandundrainedstrengthofclayandsandareinarangeof10–100%.Thestatisticalvariationofthesoildensityis,however,relativelysmallascomparedwithothersoilparameters.Therefore,inthepresentstudy,itisassumedthattheshearmodulusanddampingratiohaveCOVof20,40and60%forallsoillayers,whereastheCOVofsoildensityisassumedtobe5%inallthecases.Vanmarcke[16]studiedthescaleofsoilﬂuctuation,andconcludedthatthecorrelationdistanceofvarioussoilsvaryfrom0.16to46m.Fortypicalclay,itisabout5m.Thecorrelationdistanceof4misusedinthepresentpaper.Itshouldbenotedthatinthepresentstudy,onlytherandomﬂuctuationsofsoilpropertiesintheverticaldirectionareconsidered,thoseinthehorizontaldirectionareneglectedbecauseseismicwavesareassumedpropagatingverticallyandmodeledwiththeone-dimensionalwavepropagationtheory.

Figures10and11showtheinﬂuenceofrandomvariationsofsoilpropertiesonthemeanvaluesandstandarddeviationsofthelaggedcoherenciesofspatialsurfacemotions.Forcomparisonpurpose,thecorrespondingvalueswithdeterministicsoilproperties(COV=0),andthatoftheincidentmotiononthebaserockarealsoplotted.Asshown,theinﬂuenceofrandomsoilpropertiesonthelaggedcoherencybetweenthemotionsonthegroundsurfaceshouldnotbeneglected,especiallyforhorizontalmotions.Thelaggedcoherencybetweenthemotionsonthegroundsurfaceissmallerthantheincidentmotiononthebaserockasobservedabove.WhentheCOVofsoilpropertiesis0.2,themeanlaggedcoherenciesaresimilartothoseobtainedbydeterministicanalysis.IncreasingCOVofsoilpropertiesingeneralleadstosmallerlaggedcoherenciesbetweenthemotionsonthegroundsurface,butcouldresultinlargercoherencyvaluesatcertainfrequencieswherethespectralratiosofthetwositesdifferfromeachothersigniﬁcantlyasshowninFigure12.Inthiscase,largerCOVleadstosmallerspectralratiosthatresultsintherelativelylargerlaggedcoherencyvalues.AsshowninFigure11,largerCOVofsoilpropertiesresultsinlargervariationsofthelaggedcoherencyfunctiononthegroundsurface,asexpected.Itshouldbenotedthattheseobservationsarebasedonthesimulateddatafromacanyonsite.Ifaﬂatsiteisunderconsideration,andtherandomnessofsoilpropertiesinthehorizontaldirectionisneglected,thetwolocalsitesamplifygroundmotionsonthebaserocktothesameextentalthoughrandomnessintheverticaldirectionisconsidered.Inthiscase,thespectralratiosoftwolocalsitesequalunity,andthecoherencyfunctiononthegroundsurfaceisthenthesameasthatonthebaserock(incidentmotion).Therandomsoilpropertieshavenoinﬂuenceonthecoherencyfunctionontheground

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Figure10.Inﬂuenceofuncertainsoilpropertiesonthemeanvaluesoflaggedcoherencyfunctions:(a)horizontalout-of-planemotion;(b)horizontalin-planemotion;and(c)verticalin-planemotion.

Figure11.Inﬂuenceofuncertainsoilpropertiesonthestandarddeviationsoflaggedcoherencyfunctions:(a)horizontalout-of-planemotion;(b)horizontalin-planemotion;and(c)verticalin-planemotion.

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Figure12.Inﬂuenceofuncertainsoilpropertiesonthemeanspectralratiosoftwolocalsites:(a)horizontal

out-of-planemotion;(b)horizontalin-planemotion;and(c)verticalin-planemotion.

surfaceinthiscase.Thisobservationprovesagainthattheinﬂuencesoflocalsiteonsurfacegroundmotionspatialvariationsdependonthesimilarityofthetwowavepaths.Ifthetwowavepathsarethesame,localsitewillnotaffectthesurfacegroundmotionspatialvariations.3.3.Inﬂuenceofrandomvariationofeachsoilparameter

Toinvestigatetheeffectofrandomvariationofeachsoilparameteronthelaggedcoherencyfunctionbetweendifferentmotionsonthegroundsurface,assumingonlyonesoilparameter,namelyeithershearmodulus,soildensityordampingratio,israndom,whereastheothertwoparametersareassumedtobedeterministicinthecalculation.TheCOVsforshearmodulusanddampingratioareassumedtobe40%andtheCOVforsoildensityisassumedtobe5%.Figures13and14showthemeanvaluesandthecorrespondingstandarddeviationsofthelaggedcoherency,respectively.Thecorrespondingvalueswithdeterministicsoilproperties,andthatbetweentheincidentmotionsonthebaserockareplottedagainforcomparisonpurpose.Asshown,meanvaluesandstandarddeviationsofthelaggedcoherencyobtainedbyconsideringonlythedampingratioorsoildensityasrandomparameterarealmostthesameasthosewithdeterministicsoilpropertyassumption,indicatingthattheinﬂuenceofrandomdampingratioandsoildensityonlaggedcoherencyisinsigniﬁcantandcanbeneglected.Ontheotherhand,theinﬂuenceoftherandomvariationsofshearmodulusisobviousespeciallyforthehorizontalmotions.TheseresultscanbeexplainedbythespectralratiosofthetwolocalsitesasshowninFigure15,inwhichtheinﬂuenceofrandomdampingratioandsoildensityonthespectralratiosisinsigniﬁcantwhiletheinﬂuenceoftheshearmodulusispronounced.Becausethelaggedcoherencyfunctiondirectlyrelatestothespectralratiosoftwolocalsitesasdiscussedabove,thisleadstotheobservationsoflaggedcoherencyfunctionsinFigures13and14.

Itshouldbenotedthatalltheresultsobtainedabovearebasedontheassumptionofacorrelationdistance,󰀅u,of4mforatypicalclaysite.Infact,thecorrelationdistancevariesinarelativelywide

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Figure13.Inﬂuenceofeachrandomsoilpropertyonthemeanvaluesoflaggedcoherencyfunctions:(a)horizontalout-of-planemotion;(b)horizontalin-planemotion;and(c)verticalin-planemotion.

Figure14.Inﬂuenceofeachrandomsoilpropertyonthestandarddeviationsoflaggedcoherencyfunctions:(a)horizontalout-of-planemotion;(b)horizontalin-planemotion;and(c)verticalin-planemotion.

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Figure15.Inﬂuenceofeachrandomsoilpropertyonthemeanspectralratiosoftwolocalsites:(a)horizontalout-of-planemotion;(b)horizontalin-planemotion;and(c)verticalin-planemotion.

range[16],whenlargercorrelationdistanceisconsidered,similarconclusionscanbeobtainedbutmoreprominentvariationwillbeobserved.Theseresultsarenotshowninthecurrentpaperowingtothepagelimit.

4.CONCLUSIONS

Thispaperevaluatestheinﬂuenceoflocalsiteirregulartopographyandrandomsoilpropertiesonthecoherencyfunctionbetweenspatialsurfacemotions.Followingconclusionsaredrawn:1.Thecoherencyfunctionbetweensurfacegroundmotionsonacanyonsiteisdifferentfromthatbetweenbaserockmotions.Thelaggedcoherencyfunctiononthebaserockistheupperboundofthatonthegroundsurface.

2.Foracanyonsite,thecoherencyfunctionofspatialsurfacegroundmotionsoscillateswithfrequency.Themaximumandminimumcoherencyvaluesarerelatedtothespectralratiosoftwolocalsitesortwowavepaths.Whenthespectralratiosoftwolocalsitesdifferfromeachothersigniﬁcantly,thespatialgroundmotionsonthegroundsurfaceareleastcorrelated.Thecoherencyfunctionmodelsformotionsonaﬂat-lyingsitecannotbeusedtomodelthatofmotionsonacanyonsite.

3.Theinﬂuenceofrandomsoilpropertiesonthelaggedcoherencyfunctiondependsonthelevelofvariationsofsoilproperties.Ingeneral,themoresigniﬁcantaretherandomvariationsofsoilproperties,thelargeristhelocalsiteeffectonspatialsurfacegroundmotionvariations.Therandomvariationsofsoildampingratioanddensityhaveinsigniﬁcanteffectonthelaggedcoherencywhencomparedwiththerandomvariationsofshearmodulus.Itshouldbenotedthatthesoilnonlinearitiesalsoaffectthesurfacemotionspatialvariations,butarenotconsideredinthepresentpaper.Itissuggestedtomonitorsomecanyonsitestocheck

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theresultsobtainedinthepresentpaper.Furtherstudyisalsoneededtodevelopanalyticalorempiricalrelationoflocalsitecharacteristicswithgroundmotionspatialvariationsforeasyuseinengineeringapplication.

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