Bretherton Winter 2014 Reference Hartmann Atm S 552 notes Chapter 612 111 White noise A common way to statistically assess the signi64257cance of a broad spec tral peak as in the Nino34 example is to compare with a simple noise process White noise h ID: 30463
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Lecture11:Whiteandrednoisec\rChristopherS.BrethertonWinter2014Reference:HartmannAtmS552notes,Chapter6.1-2.11.1WhitenoiseAcommonwaytostatisticallyassessthesigni canceofabroadspectralpeakasintheNino3.4exampleistocomparewithasimplenoiseprocess.Whitenoisehaszeromean,constantvariance,andisuncorrelatedintime.Asitsnamesuggests,whitenoisehasapowerspectrumwhichisuniformlyspreadacrossallallowablefrequencies.InMatlab,w=randn(N)generatesasequenceoflengthNofn(01)`Gaussian'whitenoise(i.e.withanormaldistributionofmean0andstd1).TheuppertwopanelsofFig.1showawhitenoisesequenceoflengthN=128anditsperiodogram,whichshowsthatthepowerspectrumisuniformlyspreadacrossfrequencieswithameanspectralpowerof1=Nperharmonic.11.2RednoiseRednoisehaszeromean,constantvariance,andisseriallycorrelatedintime,suchthatthelag-1autocorrelationbetweentwosuccessivetimesampleshascorrelationcoecient0r1.Aswewillshowshortly,rednoisehasapowerspectrumweightedtowardlowfrequencies,buthasnosinglepreferredperiod.Its`redness'dependsonr,whichcanbetunedtomatchtheobservedtimeseries.FortheNino3.4case,areasonablestatisticalnullhypothesiswouldbethattheobservedpowerspectrumcouldhavebeengeneratedpurelybyrednoise.Tosequentiallygeneratean(01)rednoisesequencexjfromawhitenoisesequencewj,wesetx1=w1xj+1=rxj+(1 r2)12wj+1;j1(11.2.1)Usingpropertiesofnormaldistributions,itiseasilyshownthatxj+1isn(01)(Gaussian)andthatthelag-1correlationcoecientofxj+1andxjisr.Itisalsoeasytoshowbyinductionthatthecorrelationcoecientofxj+pand1 Amath482/582Lecture11Bretherton-Winter20142 0 50 100 -3 -2 -1 0 1 2 3 jwjWhite noise 0 50 100 -3 -2 -1 0 1 2 3 jxjRed noise (r = 0.85) -50 0 50 0 0.01 0.02 0.03 0.04 0.05 MSpectral power -50 0 50 0 0.05 0.1 0.15 0.2 MSpectral power Figure1:Whiteandrednoisetimeseries(left)andtheirperiodograms(right)xjforp�1isrp=exp( plogr)=exp( Rpt),whereR= logr=tisthedecorrelationrate.Theautocovariancesequenceofrednoisethusdecaysexponentiallywithlag.Thelagatwhichtheautocorrelationdropsto1=eis =R 1Thefunctionrednoise.m(classwebpage)implementsthisalgorithm,ItwasusedwiththewhitenoisesequenceontheupperleftofFig.1andr=085togeneratetherednoisetimeseriesonthelowerleft.Theperiodogramofthissequence,showninthelowerright,nowhasapredominanceofspectralpowerinlowharmonicsM11.3TheoreticalpowerspectrumofrednoiseThetruepowerspectrumofn(01)rednoiseismosteasilydeducedastheDFTofitsautocovariancesequence.Ratherthangrindingthroughdiscretesums,itismorehelpfultointerprettheDFTasaRiemannsumthatapproximatesthecontinuousintegralforthecomplexFouriercoecientsofthecontinuous Amath482/582Lecture11Bretherton-Winter20143functiona(t)=e Rjtj;L-periodicallyextendedfort�L=2):Sm=N 1DFT(a)cMMa(t)]]=L 1ZL=2 L=2e Rjtj i!MtdtSofar,theapproximationisgoodifRt 1and! 1Mt 1,sothata(t)exp( i!mt)iswellresolvedbythegridofspacingt.IfinadditionRL=2 1,theintegrandbecomesverysmallfort�L=2.Then,withnegligi-bleerrorwecanextendtherangeofintegrationtoin nity:SmL 1Z1 1e Rjtj i!Mtdt=L 1Z0 1et[R i!M]dt+Z10e t[R+i!M]=L 11 R i!M+1 R+i!M=L 12R R2+!2M=! R R2+!2M(!=2 L)(11.3.1)Thepowerspectrumofrednoisehasamaximumvalueforlowfrequencies!M R,anddecreasesathighfrequencies-a'red'spectrum,asclaimed.Ifwesumthepowerspectrumacrosstheharmonics,andthinkofitasaRiemannsumapproximationtoacontinuousintegralNXm=1SmN=2 1XM= N=2! R R2+!2M1 Z(N=2 1)! N!=2d!R R2+!21 Z1 1d!R R2+!2=1 tan 1(!=R) 1 1=1ConsistentwithParseval'stheorem,wehavededucedthatthepowerspectrumsumsto1,thevariancethatweconstructedourrednoisetohave.Inthisderiva-tion,extendingthelimitsoftheintegraltoin nityisagoodapproximationifN!=2 R.SinceN!=2=N=L==t,thisisequivalenttoRt 1,whichwastheassumptionwemadeinderivingthediscreterednoisespectrum(thattherednoiseiswellresolved). Amath482/582Lecture11Bretherton-Winter2014411.4FittingrednoisetodataOnecommonwayof ttingtheautocorrelationsequenceisarednoise t,asanexponentiallydecreasingfunctionoflag.This tisshowninplotaschaindash,usingane-foldingtimeof =61months.Roughlyspeaking,measurementsclosertogetherthan willbesigni cantlycorrelatedandthosefurtherapartwillbeonlyweaklycorrelated.Thiscanbecastintermsofae ectivelag-1autocorrelationr=exp(t= )(=0.85inourcase).Becausetheactualautocorrelationisnotexactlyanexponentiallydecreasingfunctionoflag,risnotexactlythesameasthetruelag-1autocorrelationof0.9.Scriptnino2addsared-noise ttotheSSTApowerspectrumbasedon =61monthsandscaledtomatchtheobservedvarianceofSSTA.Therearefourharmonicsinthe0.2-0.4yr 1rangethatclearlystandabovetherednoisespectrum.Totesthowlikelythisistobeachanceoccurrence,wenowlookforalessnoisywaytoestimatethepowerspectrum.