Geophys.J.Int.(2009)178,813 - PDF document

Geophys.J.Int.(2009)178,813±825doi:10.1111/j.1365-246X.2009.04177.x GJISeismologyTime-frequencymis®tandgoodness-of-®tcriteriaforquantitativecomparisonoftimesignalsMiriamKristekova,JozefKristekandPeterMoczoGeophysicalInstitute,SlovakAcademyofSciences,Dubravskacesta84528Bratislava,SlovakRepublicFacultyofMathematics,PhysicsandInformatics,ComeniusUniversityBratislava,MlynskadolinaF184248Bratislava,SlovakRepublic.E-mail:moczo@fmph.uniba.skAccepted2009March10.Received2009March5;inoriginalform2008October10SUMMARYWepresentanextensionofthetheoryofthetime-frequency(TF)mis®tcriteriaforquantitativecomparisonoftimesignals.Wede®neTFmis®tcriteriaforquanti®cationandcharacterizationofdisagreementbetweentwothree-componentsignals.WedistinguishtwocasesÐwithandwithouthavingonesignalasreference.Wede®nelocallyandgloballynormalizedTFcriteria.Thelocallynormalizedmis®tscanbeusedifitisimportanttoinvestigaterelativelysmallpartsofthesignal(e.g.wavegroups,pulses,transients,spikes,so-calledseismicphases)nomatterhowlargeamplitudesofthosepartsarewithrespecttothemaximumamplitudeofthesignal.TheyprovideadetailedTFanatomyofthedisagreementbetweentwoentiresignals.Thegloballynormalizedmis®tscanbeusedforquantifyinganoveralllevelofdisagreement.Theyallowaccountingforboththeenvelope/phasedifferenceataTFpointandthesigni®canceoftheenvelopeatthatpointwithrespecttothemaximumenvelopeofthesignal.WealsointroducetheTFenvelopeandphasegoodness-of-®tcriteriabasedonthecompletesignalrepresentation,andthussuitableforcomparingarbitrarytimesignalsintheirentireTFcomplexity.TheTFgoodness-of-®tcriteriaquantifythelevelofagreementandaremostsuitableinthecaseoflargerdifferencesbetweenthesignals.WenumericallydemonstratethecapabilityandimportantfeaturesoftheTFmis®tandgoodness-of-®tcriteriainthemethodologicallyimportantexamples.Keyword:Timeseriesanalysis.1INTRODUCTIONQuantitativecomparisonoftimesignals,timehistoriesofphysicalorchemicalquantitiesisoftennecessaryinmanyproblems.Devel-opingandtestinganewtheoreticalmethodofcalculationrequirescomparisonofatheoreticalsignalwithareferenceorexactsolution.Comparisonofatheoreticalsignalwithameasuredoneisnecessarytoverifythetheoreticalmodelofaninvestigatedprocess.Compar-isonoftwomeasuredsignalssigni®cantlyhelpsintheanalysisandinterpretationoftheprocessunderinvestigation.Asimplevisualcomparisonoftwosignalscanbeusefulinsomecases.Sometimesthesimplestpossiblemis®t,adifferencesr)betweenthetestedsignal)andreferencesignalsr),beingtime,isbetter.Asingle-valuedintegralquantityismoreappropriateifasetofsignalsistobecomparedwithanothersetofsignals.Asimplesingle-valuedmis®tbetweentwosignalscanbede®nedasMDsr.Probablytherootmeansquare(rms)mis®t,rms sristhemostcommonlyusedsingle-valuedmis®tcriterion.Althougheachofthethreeabovequantitiessomehowestimateadifferencebetweentwosignals,itisnotsodif®cultto®ndoutthatnoneofthemiscapabletocharacterizethenatureorreasonofthedifference.Consequentlyandeventuallyitisnotclearatallwhethertheyarecapabletoproperlyquantifythedifference.Therefore,Kristekovaetal.(2006)developedtime-frequency(TF)envelopeandphasemis®tcriteriaanddemonstratedtheircapabilitytoprop-erlyquantifyandcharacterizeadifferencebetweentwosignals.TheverybasicargumentsfordevelopingcriteriabasedontheTFrepresentationofsignalsthataretobecomparedare:1.Oneofthetwosignalscanbeviewedassomemodi®cationoftheothersignal.Itisclearthenthatsomemodi®cationsofthesignalcanbemorevisibleandunderstandableinthetimedomain,someinthefrequencydomain.Somemodi®cationscanchangemainlyoronlyamplitudesandanenvelope,someotheraphase.2.Themostcompleteandinformativecharacterizationofasig-nalcanbeobtainedbyitsTFrepresentation(inthissensewecanalsosaydecompositionintheTFplane).TheTFrepresentationen-ablesustoseeaspectralcontentatanytimeaswellastimehistoryatanyfrequency.TheTFcriteriaofKristekovaetal.(2006)wereapplied,forexamplebyPerez-Ruizetal.(2007),Moczoetal.(2007),Benjemaaetal.(2007),Kaseretal.(2008),Fichtner&Igel(2008)andSantoyo&Luzon(2008).Thecriteriawerealsousedfor2009TheAuthors813Journalcompilation2009RAS Thedefinitiveversionisavailableatwww.blackwell-synergy.comhttp://www3.interscience.wiley.com/journal/122443865/abstract 814M.Kristekova,J.KristekandP.MoczoevaluationoftheinternationalnumericalbenchmarkESG200forGrenoblevalley(Tsunoetal.2006;Chaljubetal.2009a,b),Bielaketal.(2008)appliedthecriteriatocomparethreenu-mericalsimulationsfortheShakeOutearthquakescenariover-sion1.1forthesouthernCalifornia.Thecriteriaserveforcom-parisonofsubmittedsolutionsintheframeworkoftheSPICECodeValidation(Igeletal.2005;Moczoetal.2006;Gallovicetal.2007,http://www.nuquake.eu/SPICECVal/).Therecentlyor-ganizednumericalbenchmarkforthegroundmotionsimulatiofortheEuroseistestsite,Mygdonianbasin,Greece(https://www-cashima.cea.fr/)willalsoapplytheTFmis®tcriteriaforevaluationofthesubmittedpredictions.ThepaperbyKristekovaetal.(2006)presentedTFmis®tcriteriaforone-componentsignalsinthecasewhenoneofthetwosignalscouldbeconsideredareference.Thepaperemphasizedtheglob-allynormalizedcriteriaandonlymarginallymentionedthelocallynormalizedcriteria.Clearly,thepaperpresentedbyKristekovaetal.(2006)didnotaddressallimportantaspectsandsituationsincomparingsignalsintheresearchpractice.Thiswasalsoclearfromfrequentlyaskedquestionsarisinginresponsetothepaper.Thequestionsbasicallyconcernthefollowing.(1)Thede®nitionoftheTFmis®tsinthecasewhennoneofsignalscanbeconsideredareference.(2)TheapplicationoftheTFmis®tstothree-componentsignals.(3)TheapplicabilityoftheTFmis®tsifthesignals,thataretobecompared,differ`toomuch',andconsequentlytherelationoftheTFmis®tcriteriatothegoodness-of-®tcriteria.(4)Theglobalversuslocalnormalization.(5)Theevaluationandinterpretationofthephasemis®ts,mainlytherelationtothephasejumps.Correspondingly,inthispaperwe®rstbrie¯ysummarizetheverybasicconceptsandrelationsnecessaryforthefurtherexposition.Wethenpayattentiontotheconceptsoftheenvelopeandphasedif-ferences,andstrategiesforde®ningTFmis®tcriteria.Wecontinuewithde®nitionsoftheTFmis®tcriteriaforthree-componentsig-nalsinbothsituationsÐwithandwithouthavingareferencesignal.Whereasthemis®tcriteriaaresupposedtoquantifyandcharacterizedifferencesbetweensignals,goodness-of-®tcriteriaaresupposedtoquantifythelevelofagreementbetweensignals.ForsuchsituationsweintroduceTFgoodness-of-®tcriteriaanddiscusstheirrelationtotheTFmis®tcriteria.EventuallywenumericallyillustratetheTFmis®tandgoodness-of-®tcriteriausingtwomethodologicallyimportantproblems.2CHARACTERIZATIONOFASIGNALHereweonlyverybrie¯ypresentconceptsandrelationsforchar-acterizationofasignalnecessaryinthefurtherexposition.2.1BasiccharacteristicsofasimplesignalInthesimplestcaseofamonochromaticsignalcosft(1)amplitude,phaseandfrequencyareunambiguouslyde®nedandveryeasytointerpret.Ifasignalismorecomplex,notionofamplitude,phaseandfrequencymaybenotsoobviousbecause,forexample,when)and)intheargumentofthecosinefunction,amplitudeandphaseareambiguous.Theanalyticalsignal(e.g.Flandrin1999)enablesustodevelopproperunambiguouscharacteristics.Theanalyticalsigna)withrespecttosignal)is(2)whereistheHilberttransformofsignal).RelationsDj;ÁArg[)](3)and dArg de®neenvelope,phaseand(so-calledinstantaneous)frequencyofthesignalattime.Althoughthesequantitiesareunambiguous,they,infact,representjustaveragedvalues.Forexample,Qian(2002)suggestsusingaterm`themeaninstantaneousfrequency'insteadoftheinstantaneousfrequency.Thenarrowerthespectralcontentattimeis,thebetteristheestimateofthedominantamplitude,phase,andfrequencybyrelations(3).Althoughtheconceptoftheanalyticalsignalcanbeappliedtosimplesignalsandserveasabasisforsimplemis®tcriteria(e.g.Kristeketal.2002;Kristek&Moczo2006),itclearlycannotbeappliedtosignalswithacomplexspectralcontentschangingwithtimeifthethreebasiccharacteristicsaretobedetermined.2.2Time-frequencyrepresentationofasignalAninstantaneousspectralcontentofasignaloratimeevolutionatanyfrequencyofthesignalcanbeobtainedusingtheTFrepresenta-tionofthesignal.TheTFrepresentationcanbeobtainedusing,forexample,thecontinuouswavelettransform.Thecontinuouswavelettransformofsignal)isde®nedbyCWT p ¡1 (4)withbeingtime,thescaleparameter,translationalparameter,andanalysingwavelet.Stardenotesthecomplexconjugatefunction.Thescaleparameterisinverselyproportionaltofrequency.Considerananalysingwaveletwithaspectrum,whichhaszeroamplitudesatnegativefrequencies.Suchawaveletisananalyticalsignalandiscalledtheprogressivewavelet.AMorletwaveleexp(i)exp(2)(5)with6isaproperchoiceforawideclassofsignalsandprob-lems.TheTFrepresentationofsignal)basedonthecontinuouswavelettransform,),canbethende®nedbychoosingare-lationbetweenthescaleparameterandfrequencyintheform=:,andreplacingby(becausethetranslationalparametercorrespondstotime).WeobtainCWT 2¼jfj ¡1 ¿:(6))representstheenergydistribution(energydensity)ofthesignalintheTFplane.Amoredetailedmathematicalback-groundonthecontinuouswavelettransformandMorletwavelecanbefound,forexample,inmonographsbyDaubechies(1992andHolschneider(1995),Kristekovaetal.(2006,2008b)numer-icallydemonstratedverygoodpropertiesoftheTFrepresentationde®nedabove. 2009TheAuthors,GJI178,813±825Journalcompilation2009RAS TFmis®tandgoodness-of-®tcriteria815HavingdeterminedtheTFrepresentation,anenvelopeandphase)atagivenpointoftheTFplanecanbede®ned;ÁArggW(t;f)]:(7)Holschneider(1995)showedthatif)isde®nedusingthecontinuouswavelettransformwiththeprogressivewavelet,theenvelope)andphase)areconsistentwiththosede®nedusingtheanalyticalsignal.NotethatthisTFrepresentationdoesnotsufferfromthewellknownproblemsandlimitationsofthewindowedFouriertransformduetothe®xedTFresolutionofthewindowedFouriertransform.ThesoftwarepackageSEIS-TFAdevelopedbyKristekova(2006)fornumericalcomputationoftheTFrepresentationusingthecon-tinuouswavelettransformandsixothermethodsisavailableahttp://www.nuquake.eu.3COMPARISONOFSIGNALS3.1TFenvelopeandphasedifferencesConsiderasignal)andareferencesignalsr).Given(6)and(7)itisclearthatArWr(8)de®nesthedifferencebetweentwoenvelopesateach()point.Similarly,1ÁArggW(t;f)]¡ArggWr(9)de®nesthedifferencebetweentwophasesateach()point.Theenvelopedifference)isanabsolutelocaldifferencethatcanattainanyvalue.Thephasedifferenceneedssomeexplana-tion.ThelittlecomplicationcomesfromthefactthatArg[]alwaysgivesthephaseofthecomplexvariableintherangeofh¡If,forexample,twophasesare170¼=180and160¼=180,eq.(9)formallygives330¼=180insteadofthecorrectvalue30¼=180.Itisclearthatde®nition(9)wouldneedanadditionalconditiontotreatsimilarsituations.Instead,however,wecanavoidthiscomplicationusingthefollowingequivalentde®nition1ÁArg Wr(10)Relation(10)alwaysgivesalocalphasedifferenceintherangeofh¡3.2Strategiesforde®ningTFmis®tcriteriaHavingtheenvelopeandphasedifferencesatagiven()point,wecande®neavarietyoftheTFmis®tcriteriatoquantitativelycomparetheentiresignals,importantpartsorcharacteristicsofthesignals.Inmanyproblemsitisimportanttoinvestigaterelativelysmallpartsofthesignal(e.g.wavegroups,pulses,transients,spikes,so-calledseismicphases)nomatterhowlargeamplitudesofthosepartsarewithrespecttothemaximumamplitudeoftheentiresignal.Asanexampleofanimportantseismicphasewecanmentiontheman-tlephasePcPÐtheseismicwavere¯ectedatthecore±mantleboundary.InsomeproblemsonemaybeinterestedinadetailedTFanatomyofthedisagreementbetweentwoentiresignals.Focomparingtwosignalsinsuchsituationsweneedtode®nelocamis®tcriteria±criteriawhosevaluesforone()pointwouldde-pendonlyonthecharacteristicsatthat()point.ConsideralocalTFmis®tcriterionfortheenvelope.Itisclearthatsuchcriterionshouldquantifytherelativedifferencebetweentwoenvelopesatagiven()point.Consequently,)givenbyeq.(8)shouldbenormalizedbyAr).Atthesametime,duetoitsnature,thephasedifference(10)itselfprovidestheproperquanti®cationforalocalTFphasemis®tcriterion.Wecanchoose,however,therangeh¡1,1insteadofh¡:wecandividethephasedifference(10)byTheprecedingconsiderationscanbetakenasargumentsandbasisforde®ningthelocallynormalizedTFmis®tcriteria.ThenTFEMLOC Ar(11)andTFPMLOC1Á (12)de®nethelocallynormalizedTFenvelope(TFEMLOC)andphaseTFPMLOC)mis®tcriteria,respectively.Insomeanalysesitmaybereasonabletogivethelargestweightstolocalenvelope/phasedifferencesforthosepartsofthereferencesignalinwhichtheenvelopereachesthelargestvalues.Forexample,itmaybereasonabletorequirethattheenvelopemis®tbeequaltotheabsolutelocalenvelopedifference)justatthat(pointatwhichenvelopeAr)ofthereferencesignalreachesitsmaximummaxAr.Attheother()pointswiththeenvelopesmallerthanmaxAr(andthereforealsowithsmallerenergycontent)suchamis®tcouldbeproportionaltotheratiobetweenAr)andmaxAr.Bothrequirementsaremetinthefollowingde®nitionsTFEMGLOBAr maxArTFEMLOC maxAr(13)TFPMGLOBAr maxArTFPMLOCAr maxAr1Á (14)Becausethede®nitionsapplythenormalizationbymaxArateach()point,wecanspeakofthegloballynormalizedTFenvelope(TFEMGLOB)andphaseTFPMGLOB)mis®tcriteria.Clearly,thevaluesofthegloballynormalizedTFmis®tcriteriaaccountforboththeenvelope/phasedifferenceata()pointandthesigni®canceoftheenvelopeatthatpointwithrespecttothemaximumenvelopeofthereferencesignal.Inthissensetheyquantifyanoveralllevelofdisagreementbetweentwosignals.Weapplytheglobalnormalizationinde®nitionoftheTFmis®tswhenwearenotmuchinterestedinadetailedanatomyofthesignalsandmis®tsinthosepartsofthesignalwhereitsamplitudesaretoosmallcomparedtothemaximumamplitudeofthereferencesignal.Thegloballynormalizedmis®tcriteriacanbeuseful,forexample,intheearthquakegroundmotionanalysesandearthquakeengineeringwhereweareusuallynotmuchinterestedinparticularwavegroupswithrelativelysmallamplitudes. 2009TheAuthors,GJI178,813±825Journalcompilation2009RAS 816M.Kristekova,J.KristekandP.Moczo4TFMISFITCRITERIAFORTHREE-COMPONENTSIGNALS4.1Three-componentsignals,onesignalbeingareferenceTheaboveconsiderationsonthegloballynormalizedmis®tcriteriaforone-componentsignalscanbeextendedalsotothemis®tsfothree-componentsignals.Ifamplitudesofonecomponentofthereferencesignalaresigni®cantlysmallerthanamplitudesoftwoothercomponents(acommonsituationwithapolarizedparticlemotion,forexample),theonlyreasonablechoicefortheglobalnormalizationistotakethemaximumTFenvelopevaluefromallthreecomponentsofthereferencesignal.Thischoicenaturallyquanti®esthemis®tswithrespecttothemeaningfulvaluesofthethree-componentreferencesignal.Italsopreventsobtainingtoolargemis®tvaluesduetopossibledivisionbyverysmallenvelopevaluescorrespondingtoinsigni®cantamplitudesofthesignalcom-ponents.Clearly,thischoiceisreasonablealsoiftheamplitudesofallthreecomponentsofthesignalsarecomparable.Aformalde®nitionofonelocalnormalizationfactorforallthreecomponentswouldclearlycontradicttothelocalcharacter.Eachcomponenthastobetreatedas,one-componentªsignalifoneisinterestedinthedetailedanatomyoftheTFmis®t.Nowwecande®neasetofthemis®tcriteriaforthethree-componentsignalswhenoneofthemcanbeconsideredarefer-Table1.LocallyandgloballynormalizedTFmis®tcriteriaforthree-componentsignals,onesignalbeingareference. );1,2,3Athree-componentsignalsr);3Athree-componentreferencesignalTFrepresentationofsignalWrWrTFrepresentationofthereferencesignalsrTime-frequencyenvelopeandphasemis®tsLocallynormalizedTFenvelopemis®tTFEMREFLOCWr WrLocallynormalizedTFphasemis®tTFPMREFLOC Arg WrGloballynormalizedTFenvelopephasemis®tTFEMREFGLOBTFPMREFGLOBWr maxWrTFEMREFLOCTFPMREFLOCTime-dependentenvelopeandphasemis®tsLocallynormalizedGloballynormalizedTEMREFLOCTPMREFLOCWrTFEMREFLOCTFPMREFLOC WrTEMREFGLOBTPMREFGLOBWrTFEMREFLOCTFPMREFLOC maxWrFrequency-dependentenvelopeandphasemis®tsLocallynormalizedGloballynormalizedFEMREFLOCFPMREFLOCWrTFEMREFLOCTFPMREFLOC WrFEMREFGLOBFPMREFGLOBWrTFEMREFLOCTFPMREFLOC maxWrSingle-valuedenvelopeandphasemis®tsLocallynormalizedGloballynormalizedEMREFLOCPMREFLOC WrTFEMREFLOCTFPMRELOC WrEMREFGLOBPMREFGLOB WrTFEMREFLOCTFPMREFLOC maxWr ence.ThelocallynormalizedandgloballynormalizedTFenvelopemis®tsTFEMREFLOC)andTFEMREFGLOB),andlocallynor-malizedandgloballynormalizedTFphasemis®tsTFPMREFLOCandTFPMREFGLOB)characterizehowtheenvelopesandphasesofthetwosignalsdifferateach()point.Theirprojectionontothetimedomaingivestime-dependentenvelopeandphasemis®ts,TEMREFLOC),TEMREFGLOB),TPMREFLOC)andTPMREFGLOB).Similarly,theprojectionoftheTFmis®tsontothefrequencydomaingivesfrequency-dependentenvelopeandphasemis®ts,FEMREFLOC),FEMREFGLOB),FPMREFLOC)andFPMREFGLOB).Finally,itisoftenveryusefultohavesingle-valuedenvelopeandphasemis®ts,EMREFLOCEMREFGLOBPMREFLOCandPMREFGLOB.Allthemis®tsaresummarizedinTables1and2.Theenvelopeandphasemis®tscanattainanyvalueintherangeh¡11iandh¡1,1respectively.4.2Three-componentsignals,nonebeingareferenceThemis®tcriteriaforthiscasecanbede®ned,infact,formallyinthesamewayascriteriainthecasewithareferencesignal.Theonlyquestioniswhichofthetwosignalsshouldbeformallytakenasareference.Thereasonablewayisto®ndamaximumenvelopeforeachofthetwosignals.Thenthesignalwithasmallermaximumcanbechosenasareferencesignal.Inthecaseofthegloballynormalizedcriteriaforthethree-componentsignalsthemaximum 2009TheAuthors,GJI178,813±825Journalcompilation2009RAS TFmis®tandgoodness-of-®tcriteria817Table2.LocallyandgloballynormalizedTFmis®tcriteriaforthree-componentsignals,nonebeingareference. );3two3-componentsignalsTFrepresentationsofsignals)andWrifmaxmaxifmaxmaxTime-frequencyenvelopeandphasemis®tsLocallynormalizedTFenvelopemis®tTFEMLOC WrLocallynormalizedTFphasemis®tTFPMLOC Arg GloballynormalizedTFenvelopephasemis®tTFEMGLOBTFPMGLOBWr maxWrTFEMLOCTFPMLOCTime-dependentenvelopeandphasemis®tsLocallynormalizedGloballynormalizedTEMLOCTPMLOCWrTFEMLOCTFPMLOC WrTEMGLOBTPMGLOBWrTFEMLOCTFPMLOC maxWrFrequency-dependentenvelopeandphasemis®tsLocallynormalizedGloballynormalizedFEMLOCFPMLOCWrTFEMLOCTFPMLOC WrFEMGLOBFPMGLOBWrTFEMLOCTFPMLOC maxWrSingle-valuedenvelopeandphasemis®tsLocallynormalizedGloballynormalizedEMLOCPMLOC WrTFEMLOCTFPMLOC WrEMGLOBPMGLOB WrTFEMLOCTFPMLOC maxWr istakenfromallcomponents.Inthecaseofthelocallynormalizedcriteriathereferencesignalshouldbechosenseparatelyforeachcomponent.NotethattheevaluationoftheTFmis®tsthemselvesdoesnotgiveareasontopreferthesmallerofthetwomaxima.Ourchoicecomesfromthepossiblelinktothegoodness-of-®tcriteriadevelopedbyAnderson(2004).WetakethesmallermaximumconsistentlywittheAnderson'scriteriadiscussedlater.5TFGOODNESS-OF-FITCRITERIATheenvelopeTFmis®ts,asde®nedinthepreviouschapter,quantifyandcharacterizehowmuchtwoenvelopesdifferfromeachother.Correspondingly,theenvelopemis®tcanattainanyvaluewithintherangeof(¡1)with0meaningtheagreement.Whileformallyapplicabletoanylevelofdisagreement,clearly,theenvelopemis®tsaremostusefulforcomparingrelativelycloseenvelopes.However,inpracticeitisoftennecessarytocomparesignalswhoseenvelopesdifferrelativelyconsiderably.Comparisonofrealrecordswithsyntheticsinsomeproblemscanbeagoodexample.Insuchacaseitisreasonabletolookforthelevelofagreementratherthandetailsofdisagreement.Thegoodness-of-®tcriteriaprovideasuitabletoolforthis.Thegoodness-of-®tcriteriaapproachzerovaluewithanincreas-inglevelofdisagreement.Ontheotherhand,some®nitevalueischosentoquantifytheagreement.TheTFenvelopegoodness-of-®tcriteriacanbeintroducedothebasisoftheTFenvelopemis®tsTFEGexpTFEMTEGexpTEMFEGexpFEMEGexpEM(15)Here,factorquanti®estheagreementbetweentwoenvelopesintermsofthechosenenvelopemis®t:Theenvelopegoodness-of-®tcriterionisequaltoiftheenvelopemis®tisequalto0.Choiceoftheexponentdeterminessensitivityofthegoodness-of-®tvaluewithrespecttothemis®tvalue.If10and1,theright-handsideofeq.(15)becomesformallysimilartoAnderson'sformula.Similarlywecande®neTFphasegoodness-of-®tcriteriaasthegoodness-of-®tequivalentstotheTFphasemis®tcriteria:TFPGTFPMTPGTPMFPGFPMPGPM(16)Fig.1showsthediscretegoodness-of-®tvaluesagainstthemis-®tvaluesfor10and1whatweconsiderapractically 2009TheAuthors,GJI178,813±825Journalcompilation2009RAS 818M.Kristekova,J.KristekandP.MoczoFigure1.Discretegoodness-of-®tvaluesagainstthemis®tvalues.reasonablechoiceforawideclassofproblems.Fig.1alsoincludesanexampleofapossibleverbalevaluationof®t.Thefourthcolumnofthetableassignsfourverbaldegreesorlevelstothegoodness-of-®tnumericalvalues.ThisexampleoftherelativelyrobustverbalevaluationistakenfromthepaperofAnderson(2004).Anderson'sgoodness-of-®tcriteriaarebasedoncharacteristicsrelevantintheearthquake-engineeringapplications.Hesplitthein-vestigatedfrequencyrangeintorelativelynarrowfrequencysubin-tervals.Thenhecomparedseismogramsthathadbeennarrow-band-pass®lteredforagivensubinterval.Heevaluatedgoodness-of-®tcriteriade®nedforthepeakacceleration,peakvelocity,peakdis-placement,Ariasintensity,theintegralofvelocitysquared,Fourierspectrumandaccelerationresponsespectrumonafrequency-by-frequencybasis,theshapeofthenormalizedintegralsofaccelera-tionandvelocitysquared,andthecrosscorrelation.Eachcharac-teristicwascomparedonascalefrom0to10,with10meaningagreement.Scoresforeachparameterwereaveragedtoyieldanoverallqualityof®t.BasedonthesystematiccomparisonofthehorizontalcomponentsofrecordedearthquakemotionsAnderson(2004)introducedthefollowingverbalscaleforgoodness-of-®t:Ascorebelow4isapoor®t,ascoreof4±6isafair®t,ascoreof6±8isagood®t,andascoreover8isanexcellent®t.WethinkthattheexampleoftheTFmis®ts,TFgoodness-of-®tsandverballevelsgiveninFig.1canbereasonablyappliedtoananalysisofearthquakerecordsandsimulationsandpossiblyalsotosomeotherproblems.Weshouldstress,however,thatthechoiceofthemappingbe-tweentheTFmis®tsandTFgoodness-of-®ts(inourcasethechoiceoftherangeofthegoodness-of-®tcriteria0,andexponent),andthechoiceoftheverbalclassi®cationshouldbeadjustedtotheproblemunderinvestigationandshouldbebasedonthenumeri-calexperience.Inotherwords,thechoiceshouldre¯ectarelevantaspectofthecomparativeanalysisorthecapabilityofaparticulartheorytomodelarealprocess.WethinkthattheconceptoftheTFmis®tsmakesitpossibletode®nepropergoodness-of-®tsandeventuallyalsotheverbalclassi®cationforthe®nal/overallrobustevaluation/comparisonofsignals.6NUMERICALEXAMPLESKristekovaetal.(2006)showeddetailednumericalexamplesoftheTFmis®tsforsignalsthatwererelativelyclose.ThechoiceoftheclosesignalsalloweddemonstratingthecapabilityoftheTFmis®tsnotonlytoquantifydifferencesbetweenthesignalsbutalsotocharacterizetheoriginornatureofthedifferences(e.g.pureamplitudemodi®cation,purephasemodi®cation,translationintime,frequencyshift).Herewefocusonverydifferentsituations.Inthe®rstexample,wecomparecomposeddispersivesignals.Inthesecondexample,wecomparesignals,whichdifferconsiderablyÐarecordedsignalwithasynthetic(numericallymodelled)signal.6.1DispersivesignalsDispersivesignalsareimportantandcommonbecausetheyarduetothewaveinterference.DispersivesignalsprovideagoodopportunitytoillustrateinterestingfeaturesoftheTFphasemis®t.Consideranexampleofasimpledispersivesignal,),¼=¼=50cos (17)Here,isaspatialcoordinate,istime.Aslightmodi®cationofthefrequencydependenceofthephasevelocityinthesignal(17)givesamodi®edsignal),¼=¼=50cos 9187(18)Bothsignalsfor1500kmareshowninFig.2(toppanel),)inred,)inblack.Consideralsoadispersivesignal),¼=¼=15cos (19)thatcouldbeconsideredasthe®rsthighermodeto)and);inthissenseandforthepurposeoffurtheranalysisletuscalled)and)fundamentalmodes,andthe®rsthighermode.Thesignalfor1500kmisshowninFig.2(middlepanel),inblue.Wecannowde®netwocomposeddispersivesignalsur(1500(1500(1500(1500(20)Bothsignalsaresuperpositionsofthefundamentalmodeand®rsthighermode.Thesignalsdifferinthefundamentalmodebuttheysharethesame®rsthighermode.ThesignalsareshowninFig.2(bottompanel)Ður)inred,)inblack.TheTFrepresentationsofthebothcomposedsignalsareshowninFig.3.ItisobviousfromtheTFrepresentationsthateachofthetwosignalscomprisestwomodes.Itisalsoobviousthatthesetwomodescannotbeseparated(withoutknowingtheirde®nitionformulas)onlyinthetimedomainoronlyinthefrequencydomainbecausethemodesoverlapinbothdomains.TheTFrepresentationisnecessarytorecognizethestructureofeachofthecomposedsignals.TheTFrepresentationsthemselves,however,arenotenoughforcomparingthetwocomposedsignals.AsimplevisualcomparisonoftheTFrepresentationsofur)and)onlypartlyallowsustorecognizebutdoesnotallowustoquantifydifferencesbetween 2009TheAuthors,GJI178,813±825Journalcompilation2009RASMisfitEnvelopeMisfitPhaseGoodness-of-FitNumericalvalueVerbalvalue ±0.00±0.010 ±0.11±0.19 excellent ±0.22±0.28 ±0.36±0.37 good ±0.51±0.46 ±0.69±0.55 fair ±0.92±0.64 ±1.20±0.73 ±1.61±0.82 ±2.30±0.91 poor ±1.00 TFmis®tandgoodness-of-®tcriteria819Figure2.Toppanel:dispersivesignals),inred,and),inblack.Middlepanel:dispersivesignal)consideredasthe®rsthighermodewithrespecttosignalsand.Bottompanel:composeddispersivesignalsur),inred,and),inblack.Allsignalsaredisplayedfor1500.Figure3.TFrepresentationsofthecomposeddispersivesignalsur),inred,and),inblack.Thesignalsaredisplayedfor1500.thesignals.TheTFmis®tcriteriaprovideareasonabletoolforthequantitativecomparison.ThegloballynormalizedTFmis®tsaredisplayedinFig.4.BoththeTFenvelopeandphasemis®tsclearlyshowthatur)and)differonlyinthefundamentalmodesÐthereisnomis®tintheTFregioncorrespondingtothe®rsthighermode).TFenvelopemis®tTFEMREFGLOB)andboththetime-andfrequency-dependentenvelopemis®tsTEMREFGLOB)andFEMREFGLOB 2009TheAuthors,GJI178,813±825Journalcompilation2009RAS 0 0 m u u 1u11 0 0mu uru u u u

Time[s] -0.2-0.2 820M.Kristekova,J.KristekandP.MoczoFigure4.GloballynormalizedTFmis®tsbetweenthecomposeddispersivesignalsur(1500,),takenasreference,and(1500,):TFEMandTFPM±TFenvelopeandphasemis®ts,TEMandTPM±time-dependentenvelopeandphasemis®ts,FEMandFPM±frequency-dependentenvelopeandphasemis®ts.)showtypicalsignaturesofthedifferencesbetweensignalcausedbyfrequencyshift;comparewiththesimplestcanonicalsituationsinKristekovaetal.(2006).ThesetypicalsignaturesofthefrequencyshiftincludemaximawithanalternatingsignalongthefrequencyaxisinboththeTFandfrequency-dependentenvelopemis®ts,andalsosigni®cantlylowervaluesofthetime-dependentenvelopemis®t.WecanclearlyrecognizedistinctfeaturesoftheTFphasemis®tTFPMREFGLOB):a.Thezeromis®t(whitecolour)showswhereur)and)areinphase.b.Thepositivemis®ts(warmcolours)showwhere)isphase-advancedwithrespecttour).c.Theneg-ativemis®ts(coldcolours)showwhere)isphase-delayedwithrespecttour).d.Linesofthediscontinuousmis®t-signchange(suddencolourchange)delineatesudden(discontinuous)changeofthephasedifferencebetweenur)and)fromto,ifwelookinthepositivedirectionalongthetimeaxis.Alongthelinesthesignalsareinantiphase.Notethatwhereasthephasedifferencebetweenur)and)jumpsfromto,theTFmis®tsvaluesfrombothsidesofthediscontinuitymaybesmallerinabsolutevaluethan100percent.Thereisnocontradictioninthis.Thisisjustasimpleconsequenceofthefactthatthedisplayedmis®tsaregloballynormalized.Timesoftheoccurrenceofthephasejumpsinthephasedifferencesbetweenthesignals(whenthesignalsareinantiphase)canbeclearlyidenti®edalsofromthetime-dependentphasemis®tTEMREFGLOB)asthetimesofasuddenchangeofthesignofthemis®tvalues.Again,duetotheglobalnormalization,theTPMREFGLOB)mis®tvaluesfrombothsidesofthediscontinuitymaybesmallerinabsolutevaluethan100percent.6.2RecordedandnumericallymodelledearthquakemotionTheobservedthree-componentsignalrepresentsthegroundmotionrecordedduringalocalsmallearthquakeatthetemporaryseismicstationintheMygdonianbasinnearThessaloniki,Greece.Thecom-putedthree-componentsignalrepresentsthenumericallysimulatedmotionforthepreliminarystructuralmodeloftheMygdonianbasin(Manakouetal.2004).BoththerecordedandcomputedsignalsareshowninFig.5.Therelativelylargedifferencesbetweentheob-servedandnumericallysimulatedsignalsmightmainlybeduetotheconsiderablysimpli®edvelocitymodelofthebasinsediments.Theobservedsignalistakenasareference.WepresentanddiscussherelocallyandgloballynormalizedTFrepresentationsofthesignals,locallyandgloballynormalizedTFenvelopeandphasemis®ts,andlocallyandgloballynormal-izedTFenvelopeandphasegoodness-of-®tcriteria.6.2.1LocallynormalizedTFrepresentationsThetoppanelofFig.6showsthreecomponentsoftherecordedsignal(red)andcomputedsignal(black)togetherwiththeirTFrepresentations,thatis).TheTFrepresentationofeach 2009TheAuthors,GJI178,813±825Journalcompilation2009RAS TFmis®tandgoodness-of-®tcriteria821Figure5.Recordedandcomputedthree-componentparticle-velocitysignalsatthetemporaryseismicstationsintheMygdonianbasin.N±north±southcomponent,E±east±westcomponent,Z±verticalcomponent.componentisnormalizedwithrespecttothemaximumvalueforthatcomponentandthenrepresentedusingthesamelog-arithmicred-colourorgreyscalecoveringtherangeofthreeordersofmagnitude.Thecombinationofthelocalnormalizationwiththelogarithmicscaleenablesustoseeverywellthedetaileddistribu-tionofthesignalenergylargerthan0.1percentofthemaximum.ItisobviousthatadirectvisualcomparisoncanneitherquantifynorproperlycharacterizedifferencesbetweentheTFrepresentationsofthesignals.6.2.2GloballynormalizedTFrepresentationsThetoppanelofFig.7showsthreecomponentsoftherecordedsignal(red)andcomputedsignal(black)togetherwiththeirTFrep-resentations,thatis).TheTFrepresentationofeachcom-ponentisnormalizedwithrespecttothemaximum)valuefromallthreecomponents;thesamelinearred-colourorgreyscaleisthenappliedtoeachnormalizedcomponent.ThecombinationoftheglobalnormalizationwiththelinearscaleshowsverywelltheTFstructure(pattern)ofeachcomponentrelativetothemaximum)value,thatistheenergeticallydominantTFcontentsofthesignal.AlthoughabiteasierthanwiththelocallynormalizedTFrepresentations,stilladirectvisualcomparisonprovidesnei-therquanti®cationnorapropercharacterizationofthedifferencesbetweentheTFstructuresoftheobservedandcomputedsignals.6.2.3LocallynormalizedTFmis®tcriteriaThemiddlepanelofFig.6displaysthedetailedanatomyofthTFenvelopeandphasemis®tsbetweencorrespondingcomponentsoftheobservedandcomputedsignalsintheentireconsideredTFrange.Duetorelativelyverylargeenvelope-mis®tvalues,allvaluesabove200percentareshowninthesamecolour(magenta),thatis,theyareclippedat200percent.Similarly,valuesbelow200percentareshownusingonecolour(lightgreen).AlthoughtheTFstructureoftheenvelopemis®tsremainsrelativelycom-plicatedtheTFenvelopemis®tsshowwheretheenvelopeofthcomputedsignalislargerandwhereitissmallercomparedtothatoftheobservedsignal.Becausethemis®tsarelocallynormalized,theycanattainverylargevaluesalsoatthose()pointsorpartsofthesignal,wheretheenvelopeitselfisrelativelyverysmallornegligiblecomparedtothemaximumenvelopevalue(thatis,wheretheenergyofthesignalisverysmallornegligible).Itisjustthisfeatureofthelocallynormalizedmis®tsthatmakestheirinterpre-tationrelativelydif®cult.Therefore,wheninterpretingthemis®ts,oneshouldalwayslookalsoattheTFrepresentationsofthesignalsthemselves.Thephasemis®tsshowwherethephaseofthecomputedsignalisadvancedandwhereitisdelayedcomparedtothatoftheobservedsignal.Note,however,thatduetothecomplexityofthesignalsandnatureofthephasemis®titsTFstructureismorecomplicatedandconsequentlyitsinterpretationmoredif®cultthanthoseoftheenvelopemis®t.6.2.4GloballynormalizedTFmis®tcriteriaThegloballynormalizedTFmis®tcriteriaareshowninthemiddlepanelofFig.7.Theenvelope/phasemis®tsclearlyre¯ectthoseTFpartsofthesignalswheretheenvelopes/phasesdifferandatleastoneofthesignalsisenergeticallysigni®cant.Inotherwords,thegloballynormalizedTFmis®tsaccountforboththeenvelope/phasedifferenceata()pointandthesigni®canceoftheenvelopeatthatpointwithrespecttothemaximumenvelopeofthereferencesignal.LookingatthemiddlepanelofFig.7wecanquitewell`sense'anoveralllevelofdisagreementbetweenthecomparedsignals6.2.5LocallynormalizedTFgoodness-of-®tcriteriaThebottompanelofFig.6showsthelocallynormalizedTFenve-lopeandphasegoodness-of-®ts.Althoughonlyfourdistinctcolourswereusedandassignedtointervalsofthegoodness-of-®tvalues,thepatternsoftheTFenvelopeandphasegoodness-of-®tsarecom-parablewiththoseoftheclippedlocallynormalizedTFmis®ts,thatis,theyarecomparablycomplicated(notethatnotclippedmis®tswouldbeobviouslyevenmorecomplicated).Thisisduetothelocalnormalizationinbothcases.Itisclearthatthelocalnormalization 2009TheAuthors,GJI178,813±825Journalcompilation2009RAS-0.040.000.04 computed recorded-0.040.000.04 computed recorded024681012141618202224262830-0.040.000.04 computed recordedN componentE componentZ componentTime[s]Particle velocity [m/s] 822M.Kristekova,J.KristekandP.MoczoFigure6.Toppanel:threecomponentsoftherecordedsignal(red)andcomputedsignal(black)togetherwiththeirTFrepresentations.TheTFrepresentationofeachcomponentisnormalizedwithrespecttothemaximum)valueforthatcomponent.Middlepanel:LocallynormalizedTFenvelope(TFEMandphase(TFPM)mis®tsbetweencorrespondingcomponentsoftheobservedandcomputedsignals.Theenvelope-mis®tvaluesabove200percentareshowninmagenta,valuesbelow200percentareshowninlightgreen.Bottompanel:locallynormalizedTFenvelopeandphasegoodness-of-®ts.makestheinterpretationoftheTFmis®tsandgoodness-of-®tsrela-tivelydif®cultifthesignalsthemselvesarenotsimple.Atthesametimethisrelativecomplexityisunavoidableiftheanalysisrequiresseeingdetailsofthedifference,forexample,relatedtospeci®cpartofthesignals.6.2.6GloballynormalizedTFgoodness-of-®tcriteriaThebottompanelofFig.7showsthegloballynormalizedTFenvelopeandphasegoodness-of-®ts.Comparedtoalltheprecedingmis®tsandgoodness-of-®tstheyprovidethesimplestandmostro-busttoolforvisualizingwhereintheTFplanetheamplitudesandphasesofthecomparedsignalsdifferandwheretheydonot.Thisisbecausetheyaregoodness-of®ts,theyaregloballynormalized,andtheyaredisplayedusingdistinctcoloursassignedtoonlyfourintervalsofthegoodness-of-®tvalues.Recallthatthefourcoloursrepresent,infact,thesimpleverbalevaluationofthelevelofagree-mentgiven(asanexample)inFig.1.Basedonthechosengloballynormalizedgoodness-of-®tswecansaythattheleveloftheoverallagreementinthe-componentisfair 2009TheAuthors,GJI178,813±825Journalcompilation2009RAS [Hz] [Hz] [Hz] [Hz] [Hz] [Hz] TFEM TFEM TFEM TFPM TFPM TFPM TFEG TFEG TFEG TFPG TFPG TFPG

 


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[s]EM= 124%PM= 41%EM= 194%PM= 53%EM= 122%PM= 56%EG= 2.9PG= 5.9EG= 1.4PG= 4.7EG= 3.0PG= 4.4N E Z TFmis®tandgoodness-of-®tcriteria823Figure7.Toppanel:Threecomponentsoftherecordedsignal(red)andcomputedsignal(black)togetherwiththeirTFrepresentations.TheTFrepresentationofeachcomponentisnormalizedwithrespecttothemaximum)valuefromallthreecomponents.Middlepanel:GloballynormalizedTFenvelopeTFEM)andphase(TFPM)mis®tsbetweencorrespondingcomponentsoftheobservedandcomputedsignals.Bottompanel:GloballynormalizedTFenvelopeandphasegoodness-of-®ts.inenvelopesandgoodinphases.Theleveloftheoverallagreementinthehorizontalcomponentsispoorinenvelopesandfair-to-goodinphases.Thedifferencebetweentheoverallagreementinthe-componentandhorizontalcomponentsisduetothepresenceofthewavegroupatlowerfrequenciesandlatertimesinthehorizontalcomponents,whichisnotpresentinthe-component.Thewavegroupisdistinctinthenumericallysimulatedmotion.Wecanjustnotethatitspresenceislikelyduetotheconsiderablysimpli®edvelocitystructureofthebasinsedimentsusedinthecomputationalmodel.7CONCLUSIONSWepresentedasystematicextensionandelaborationoftheconceptoftheTFmis®tcriteriaoriginallyintroducedbyKristekovaetal.(2006),Kristekovaetal.(2006)usedtheTFrepresentationofsignals 2009TheAuthors,GJI178,813±825Journalcompilation2009RAS [Hz] [Hz] TFEM TFEM TFEM TFPM TFPM TFPM TFEG TFEG TFEG TFPG TFPG TFPG    \n\n [Hz] [Hz] [Hz] [Hz]


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[s]EM= 124%PM= 41%EM= 140%PM= 38%EM= 68%PM= 31%EG= 2.9PG= 5.9EG= 2.5PG= 6.2EG= 5.1PG= 6.9N E Z 824M.Kristekova,J.KristekandP.Moczotode®neenvelopeandphasedifferencesatapointoftheTFplane,andthecorrespondingTFenvelopeandphasemis®tcriteria.Theyde®nedandnumericallytestedgloballynormalizedcriteriaforone-componentsignalsassumingthatoneofthecomparedsignalscanbeconsideredareference.Thelocallynormalizedcriteriawerede®nedbutnottestedandanalysed.Theextensionpresentedinthispapercanbesummarizedasfollows.Wefoundmoreproperde®nitionofthephasedifferenceatapointoftheTFplane.Wede®nedTFmis®tcriteriaforthree-componentsignals.Wedistinguishedtwobasicsituations:1.Itisreasonableandpossibletoconsideroneofthecomparedsignalsareference.2.Thereisnoreason,pertinentorattributabletotheinvestigatedproblem,tochooseonesignalareference.Wealsotreatedtwoprincipalnormalizationsofthemis®tsÐlocalandglobalnormalizationsÐinauni®edway.Thevaluesofthelocallynormalizedmis®tcriteriaforone)pointdependonlyonthecharacteristicsatthatpoint.Thelocallynormalizedmis®tcriteriashouldbeusedifitisimportanttoinvestigatethefollowing.(1)Relativelysmallpartsofthesignal(e.g.wavegroups,pulses,transients,spikes,so-calledseismicphases),nomatterhowlargeamplitudesofthosepartsarewithrespecttothemaximumampli-tudeofthesignal.(2)AdetailedTFanatomyofthedisagreementbetweensignalintheentireconsideredTFregion.Thegloballynormalizedmis®tcriteriagivethelargestweightstothelocalenvelope/phasedifferencesforthosepartsofthereferencesignal(trueorformallyalgorithmicallydetermined)inwhichtheenvelopereachesthelargestvalues.Thegloballynormalizedmis®tcriteriashouldbeusedifitisreasonable(1)toquantifyanoveralllevelofdisagreement,(2)toaccountforboththeenvelope/phasedifferenceata(pointandthesigni®canceoftheenvelopeatthatpointwithrespecttothemaximumenvelopeofthereferencesignal,forexample,intheearthquakegroundmotionanalysesandearthquakeengineering.WealsointroducedtheTFenvelopeandphasegoodness-of-®tcriteriaderivedfromtheTFmis®tcriteria.ThustheTFgoodness-of-®tcriteriaarebasedonthecompletesignalrepresentationandhavethesameTFstructureastheTFmis®ts.TheyaresuitableforcomparingarbitrarytimesignalsintheirentireTFcomplexity.TheTFgoodness-of-®tcriteriaquantifythelevelofagreementandaremostsuitableinthecaseoflargerdifferencesbetweenthsignals.Theycanbeusedwhenwelookfortheagreementratherthandetailsofdisagreement.Therobust`verbalquanti®cation'en-ablesustosee/®ndoutthe`essential'levelofagreementbetweenthecomparedsignals.Wenumericallydemonstratedthecapabilityandimportantfea-turesoftheTFmis®tandgoodness-of-®tcriteriaintwomethod-ologicallyimportantexamples.ProgrampackageTF_MISFIT_GOF_CRITERIA(Kristekovaetal.2008a)isavailableathttp://www.nuquake.eu/Computer Codes/.ACKNOWLEDGMENTSTheauthorsthankJacoboBielakandMartinKaserfortheirvaluablereviews.ThisstudywassupportedinpartbytheScienti®cGrantAgencyoftheMinistryofEducationoftheSlovakRepublicandtheSlovakAcademyofSciences(VEGA)Project1/4032/07andtheSlovakResearchandDevelopmentAgencyunderthecontracNo.APVV-0435±07(projectOPTIMODE).REFERENCESAnderson,J.G.,2004.Quantitativemeasureofthegoodness-of-®tofsyntheticseismograms,in13thWorldConferenceonEarthquakeEn-gineeringConferenceProceedings,Vancouver,Canada,Paper243,onCD-ROM.Benjemaa,M.,Glinsky-Olivier,N.,Cruz-Atienza,V.M.,Virieux,J.&Piperno,S.,2007.Dynamicnon-planarcrackrupturebya®nitevolumemethod,Geophys.J.Int.,171,271±285.Bielak,J.etal.2008.ShakeOutsimulations:veri®cationandcomparisonsinProceedings2008SCECAnnualMeetingandAbstracts,Vol.XVIII,SCEC,PalmSprings,CA,USA,p.92.Chaljub,E.,Cornou,C.&Bard,P.-Y.,2009a.Numericalbenchmarkof3DgroundmotionsimulationinthevaleyofGrenoble,FrenchAlps,PaperSB1,inProceedingsoftheThirdInternationalSymposiumontheEffectsofSurfaceGeologyonSeismicMotion,Grenoble,30August±1September2006,Vol.2,1365±1375(LCPCeditions).Chaljub,E.,Tsuno,S.&Bard,P.-Y.,2009b.Grenoblevalleysimulationbenchmark:comparisonofresultsandmainlearnings,PaperSB2,inPro-ceedingsoftheThirdInternationalSymposiumontheEffectsofSurfaceGeologyonSeismicMotion,Grenoble,30August±1September2006,Vol.2,1377±1436(LCPCeditions).Daubechies,I.,1992.TenLecturesonWavelets,SIAM,Philadelphia.Fichtner,A.&Igel,H.,2008.Ef®cientnumericalsurfacewavepropagationthroughtheoptimizationofdiscretecrustalmodelsÐatechniquebasedonnon-lineardispersioncurvematching(DCM),Geophys.J.Int.,173,519±533.Flandrin,P.,1999.Time-Frequency/Time-ScaleAnalysis,AcademicPress,SanDiego,CA,USA.Gallovic,F.,Barsch,R.,delaPuente,J.&Igel,H.,2007.Digitallibraryforcomputationalseismology,EOS,88,559.Holschneider,M.,1995.Wavelets:AnAnalysisTool,ClarendonPress,Oxford.Igel,H.,Barsch,R.,Moczo,P.,Vilotte,J.-P.,Capdeville,Y.&Vye,E.,2005.TheEUSPICEProject:adigitallibrarywithcodesandtrainingmaterialincomputationalseismology,EosTrans.AGU,86,FallMeet.Suppl.,AbstractS13A-0179.aser,M.,Hermann,V.&delaPuente,J.,2008.Quantitativeaccuracyanal-ysisofthediscontinuousGalerkinmethodforseismicwavepropagation,Geophys.J.Int.,173,990±999.Kristek,J.&Moczo,P.,2006.Ontheaccuracyofthe®nite-differenceschemes:the1Delasticproblem,Bull.seism.Soc.Am.,96,2398±2414.Kristek,J.,Moczo,P.&Archuleta,R.J.,2002.Ef®cientmethodstosimulateplanarfreesurfaceinthe3D4th-orderstaggered-grid®nite-differenceschemes,Stud.Geophys.Geod.,46,355±381.Kristekova,M.,2006.Time-frequencyanalysisofseismicsignals.Disser-tationthesis.GeophysicalInstituteoftheSlovakAcademyofSciences,Bratislava.(inSlovak)Kristekova,M.,Kristek,J.,Moczo,P.&Day,S.M.,2006.Mis®tcriteriaforquantitativecomparisonofseismograms,Bull.seism.Soc.Am.,96,1836±1850.Kristekova,M.,Kristek,J.&Moczo,P.,2008a.TheFortran95pro-grampackageTF_MISFIT_and_GOF_CRITERIAandUser'sguideathttp://www.nuquake.eu/Computer_Codes/Kristekova,M.,Moczo,P.,Labak,P.,Cipciar,A.,Fojtikova,L.,Madaras,J.&Kristek,J.,2008b.Time-frequencyanalysisofexplosionsintheammunitionfactoryinNovaky,Slovakia,Bull.seism.Soc.Am.,98,2507±2516.Manakou,M.,Raptakis,D.,Chavez-Garcõa,F.,Makra,K.,Apostolidis,P.&PitilakisK.,2004.Constructionofthe3DgeologicalstructureofMygdonianbasin(N.Greece),inProceedingsofthe5thInternationalSymposiumonEasternMediterraneanGeology,Thessaloniki,Greece,14±20April2004,Ref.S6-15.Moczo,P.,Ampuero,J.-P.,Kristek,J.,Day,S.M.,Kristekova,M.,Pazak,P.,Galis,M.&Igel,H.,2006.Comparisonofnumericalmethodsforseis-micwavepropagationandsourcedynamicsÐtheSPICEcodevalidation,inESG2006,ThirdInternationalSymposiumontheEffectsofSurface 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