A-practical-guide-to-wavelet-analysis

61BulletinoftheAmericanMeteorologicalSociety1.IntroductionWaveletanalysisisbecomingacommontoolforanalyzinglocalizedvariationsofpowerwithinatimeseries.Bydecomposingatimeseriesintotime–fre-quencyspace,oneisabletodetermineboththedomi-nantmodesofvariabilityandhowthosemodesvaryintime.Thewavelettransformhasbeenusedfornu-merousstudiesingeophysics,includingtropicalcon-vection(WengandLau1994),theElNiño–SouthernOscillation(ENSO;GuandPhilander1995;WangandWang1996),atmosphericcoldfronts(GamageandBlumen1993),centralEnglandtemperature(Baliunasetal.1997),thedispersionofoceanwaves(Meyersetal.1993),wavegrowthandbreaking(Liu1994),andcoherentstructuresinturbulentflows(Farge1992).AcompletedescriptionofgeophysicalapplicationscanbefoundinFoufoula-GeorgiouandKumar(1995),whileatheoreticaltreatmentofwaveletanalysisisgiveninDaubechies(1992).Unfortunately,manystudiesusingwaveletanaly-sishavesufferedfromanapparentlackofquantita-tiveresults.Thewavelettransformhasbeenregardedbymanyasaninterestingdiversionthatproducescol-orfulpictures,yetpurelyqualitativeresults.Thismis-conceptionisinsomesensethefaultofwaveletanaly-sisitself,asitinvolvesatransformfromaone-dimen-sionaltimeseries(orfrequencyspectrum)toadiffusetwo-dimensionaltime–frequencyimage.Thisdiffuse-nesshasbeenexacerbatedbytheuseofarbitrarynor-malizationsandthelackofstatisticalsignificancetests.InLauandWeng(1995),anexcellentintroductiontowaveletanalysisisprovided.Theirpaper,however,didnotprovidealloftheessentialdetailsnecessaryforwaveletanalysisandavoidedtheissueofstatisti-calsignificance.Thepurposeofthispaperistoprovideaneasy-to-usewaveletanalysistoolkit,includingstatisticalsig-nificancetesting.TheconsistentuseofexamplesofAPracticalGuidetoWaveletAnalysisChristopherTorrenceandGilbertP.CompoPrograminAtmosphericandOceanicSciences,UniversityofColorado,Boulder,ColoradoABSTRACTApracticalstep-by-stepguidetowaveletanalysisisgiven,withexamplestakenfromtimeseriesoftheElNiño–SouthernOscillation(ENSO).TheguideincludesacomparisontothewindowedFouriertransform,thechoiceofanappropriatewaveletbasisfunction,edgeeffectsduetofinite-lengthtimeseries,andtherelationshipbetweenwaveletscaleandFourierfrequency.Newstatisticalsignificancetestsforwaveletpowerspectraaredevelopedbyderivingtheo-reticalwaveletspectraforwhiteandrednoiseprocessesandusingthesetoestablishsignificancelevelsandconfidenceintervals.Itisshownthatsmoothingintimeorscalecanbeusedtoincreasetheconfidenceofthewaveletspectrum.Empiricalformulasaregivenfortheeffectofsmoothingonsignificancelevelsandconfidenceintervals.Extensionstowaveletanalysissuchasfiltering,thepowerHovmöller,cross-waveletspectra,andcoherencearedescribed.ThestatisticalsignificancetestsareusedtogiveaquantitativemeasureofchangesinENSOvarianceoninterdecadaltimescales.Usingnewdatasetsthatextendbackto1871,theNiño3seasurfacetemperatureandtheSouthernOscilla-tionindexshowsignificantlyhigherpowerduring1880–1920and1960–90,andlowerpowerduring1920–60,aswellasapossible15-yrmodulationofvariance.ThepowerHovmöllerofsealevelpressureshowssignificantvariationsin2–8-yrwaveletpowerinbothlongitudeandtime.Correspondingauthoraddress:Dr.ChristopherTorrence,Ad-vancedStudyProgram,NationalCenterforAtmosphericRe-search,P.O.Box3000,Boulder,CO80307-3000.E-mail:[e-mail]20October1997.©1998AmericanMeteorologicalSociety62Vol.79,No.1,January1998ENSOprovidesasubstantiveadditiontotheENSOliterature.Inparticular,thestatisticalsignificancetestingallowsgreaterconfidenceinthepreviouswave-let-basedENSOresultsofWangandWang(1996).TheuseofnewdatasetswithlongertimeseriespermitsamorerobustclassificationofinterdecadalchangesinENSOvariance.Thefirstsectiondescribesthedatasetsusedfortheexamples.Section3de-scribesthemethodofwaveletanalysisusingdiscretenotation.ThisincludesadiscussionoftheinherentlimitationsofthewindowedFouriertransform(WFT),thedefinitionofthewavelettransform,thechoiceofawaveletbasisfunction,edgeeffectsduetofinite-lengthtimese-ries,therelationshipbetweenwaveletscaleandFourierperiod,andtimeseriesreconstruction.Section4presentsthetheoreticalwaveletspectraforbothwhite-noiseandred-noiseprocesses.ThesetheoreticalspectraarecomparedtoMonteCarloresultsandareusedtoes-tablishsignificancelevelsandconfi-denceintervalsforthewaveletpowerspectrum.Section5describestimeorscaleaveragingtoincreasesignificancelevelsandconfidenceintervals.Section6describesotherwaveletapplicationssuchasfiltering,thepowerHovmöller,cross-waveletspectra,andwaveletco-herence.Thesummarycontainsastep-by-stepguidetowaveletanalysis.2.DataSeveraltimeserieswillbeusedforexamplesofwaveletanalysis.TheseincludetheNiño3seasurfacetemperature(SST)usedasameasureoftheamplitudeoftheElNiño–SouthernOscillation(ENSO).TheNiño3SSTindexisdefinedastheseasonalSSTav-eragedoverthecentralPacific(5°S–5°N,90°–150°W).Datafor1871–1996arefromanareaaver-ageoftheU.K.MeteorologicalOfficeGISST2.3(Rayneretal.1996),whiledataforJanuary–June1997arefromtheClimatePredictionCenter(CPC)opti-mallyinterpolatedNiño3SSTindex(courtesyofD.GarrettatCPC,NOAA).Theseasonalmeansfortheentirerecor