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Analogue Correction Method of Errors by Combining Statistical and Dynamical Methods

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Analogue Correction Method of Errors by Combining Statistical and Dynamical Methods

    Analogue Correction Method of Errors by

    Combining Statistical and Dynamical

    Methods

    VOL.20NO.3ACTAMETEOROL0GICASINICA2006

    AnalogueCorrectionMethodofErrorsbyCombiningStatistical

    andDynamicalMethods

    RENHongli1,2t(任宏利)andCHOUJifan0(纪范)

    1Laboratory{0rClimateStudies}NationalClimateCenter,ChinaMeteorologicalAdministrationIBeijing100081

    2CollegeofAtmosphericSciences,LanzhouUniversity,Lanzhou73000

    (ReceivedApril30,2006)

    ABSTRACT

    Basedontheatmosphericanalogyprinciple.theinverseproblemthattheinformationofhistoricalana-

    loguedataisutilizedtoestimatemodelerrorsisputforwardandamethodofanaloguecorrectionoferrors

    (ACE1ofmodelisdevelopedinthispaper.TheACEcancombineeffectivelystatisticalanddynamical

    methods.andneednotchangethecurrentnumericalpredictionmodels.Thenewmethodnotonlyade

    quatelyutilizesdynamicalachievementsbutalsocanreasonablyabsorbtheinformationofagreatmany

    analoguesinhistoricaldatainordertoreducemodelerrorsandimproveforecastskil1. Furthermore.theACEmayidentifyspecifichistoricaldataforthesolutionoftheinverseproblemin

    termsoftheparticularityofcurrentforecast.ThequalitativeanalysesshowthattheACEisthe

oretically

    equivalenttotheprincipleofthepreviousanalogue

    dynamicalmodel,butneednotrebuildthecompli

    catedanalogue-deviationmodel,sohasbetterfeasibilityandoperationalforeground.Moreo

    ver,underthe

    idealsituations.whennumericalmodelsorhistoricalanaloguesareperfect,theforecastofthe

    ACEwould

    transformintotheforecastofdynamicalorstatisticalmethod,respectively.

    Keywords:combinationofstatisticalanddynamicalmethodsinverseproblemnumericalpre

    diction,

    analoguecorrectionoferrors(ACE)

    1.Introduction

    Ingeneral,therearetwobasicpredictivemeth

    odsincludingthedynamicalandstatisticalones,both ofwhichhavebenefitsandabsences.Thedynamical method,basedoninitialproblemofphysicalprinci

    ples,doesnotornotfullyutilizehistoricaldata.As acontrast.thestatisticalmethodcanusealotofin

    f0rmationofhistoricalobserveddata,butdoesnotor notfullyutilizephysicswehave(Chou,1986).Early in1958,Gu(1958)putforwardtheimportanceand feasibilityofintroducingpasthistoricaldataintothe numericalprediction.Thereafter,aseriesofinnovative andeffectivetheoriesandmethodswereputforward (Chou,1974;ZhengandDu,1973;Cao,1993;Zhang andChou,1997;Gu,1998;Gongeta1.,1999;Chen etal.,20031inordertoemcientlycombinestatistical methodwithnumericalmodelandfullyutilizeinfor

    mationofpastdatatoimprovedynamicalprediction. Manyresultsofnumericalexperimentshaveshown

    considerablepredictiveskil1.Especially,toeffectively combinenumericalpredictionmodelwithsubjective experiencesofforecastersinanalogueprediction,the dynamicalpredictionfieldmaybeassumedasasmall disturbanceoverhistoricalanaloguefieldsothatsyn

    opticexperiencesareintroducedtothenumericalpre

    diction(Chou,1979).Intermsofthisbasicprinciple, someanaloguedynamicalmodels(ADMs)wereestab

    lishedfortheweatherforecastingandseasonalpredic

    tion(QiuandChou,1989;HuangandWang,1991; Huangeta1.,1993).TheseADMshavehigheraccu

    racythanthetraditionalanalogueprediction,which wasdocumentedbypredictionexperiments(Schuur

    mans,1973;BarnettandPreisendorfer,1978;vanden Dool,1987).

    Itiswellknownthatthereinevitablyexister

    rorsinthenumericalmodel,thustheADMprinciple isjustproposedforreducingmodelerrors.However, itisquitedifficulttechnicallytodirectlybuildthe analoguedeviationversionofcomplicatednumerical SupportedjointlybytheNationalNaturalScienceFoundationofChinaunderGrantNos.402

    33031,40575036and40675039

    Correspondingauthor:renhl@cma.gov.cn.

368ACTAMETEOR0L0GICASINICA

    mode1.Actually,wehadbetterbaseonexistingdy

    namicalmodelsifwewouldfullyutilizephysicallaws. Atpresent,thereexistagreatmanydirectapproaches fordiminishingmodelerrors,butalongsuchroute,it isgettingmoredifficulttoheightenpredictionleve1.

Asacontrast,sometechniquessuchasmodeliden

    tificationwhichisregardedasthesecondkindofin. verseproblem(GaoandChou,1994;FanandChou, 1999)canfullyutilizeplentyofexistingrealobserved datatooptimizemodelparametersandimprovemodel withsmallercost(QiuandChou,1987,1988,1990). Itneedspointingoutthat,suchtheinverseproblem indicatesreimprovementofthenumericalmodeland thuscanimprovethenumericalpredictionalongwith thecontinualdevelopmentsofpositiveproblems.Re

    cently,therehasalreadybeensomeinnovativeworks fortherelatedattemptandexplorationbyusingcom

    plicatedmodel,andpreliminaryexperimentresults haveshownconsiderablevalidation(Baoeta1.,2004; Bao,2004).Thereforeinthepresentpaper,wewill baseontheatmosphericanalogyprincipleandputfor

    wardanewinverseproblemthattheinformationof historicalanaloguedataisutilizedtoestimatecurrent modelerrorsandthenamethodofanaloguecorrec

    tionoferrors(ACE),whichcancombineeffectively statisticalanddynamicalmethods.isfurtherdevel

    oped.

    2.Anewinverseproblem

    Becauseitisfocusedonhowhistoricaldataare utilizedtoreducemodelerrorsefficientlyandimprove currentprediction,theinfluencesofobserveddataer- rorsonlattertheoreticaldeviationarenotconsidered inthissituationillordertohighlightprimaryprob

    lems.Thatistosay,observeddataarecompletely proper.Aswehaveknown,numericalpredictionis

    mathematicallyputforwardintermsoftheinitial problemofpartialdifferentialequations.Ingeneral, numericalpredictionmodelcanbeexpressedasfol

    ows:

    .

    (r,0)=G(r),

    where(r,t)isthemodelstatevectortobepredicted V0L.20

    rthevectorinthespatialcoordinate,ttime,andL thedifferentialoperatorof,whichiscorresponding

    torealnumericalmodelandusuallynonlinear.When t>0.oritsfunctionalPmaybeobtainedbynumer

    icallyintegratinginitialvalues.Similarly,theexact modelthatrealatmospheresatisfiescanbewrittenas )=)

    whereEistheerrortermandstandsfortheprocess thatactuallyexistsbutisnotdescribedorexactlyde

    scribedinEq.(1),andjustreflectstheerrorsofrealnu

    mericalmode1.Thenhistoricaldatamayberegarded asaseriesofspecialsolutionsortheirfunctionalP ofEqs.(3)and(2).

    Atpresent,thereprimarilyexisttwomethodsfor thereductionofnumericalmodelerrors.Oneisdi

    rectlyimprovingeverysectionsofdynamicalmodel, whichhasbeenwidelyopera,tedinternationaIIy.But inthesedays,therealsoappearmanyproblemsinre

    latedstudiessuchasexpensivecost,longresearchpe

    riod,andslowadvancementonpredictiveleve1.For theothermethod,itisthemainideathattheun

knowntermEinEq.(3)callbeapproximatelyesti

    matedwhenaseriesofspecialsolutionsorPof Eqs.(3)and(2)aswellasLareknown,thatistosay, whichhasbecomethemodelidentificationproblem ofconfirmingunknownsectionsinequationsbyuti

    lizinghistoricalobserveddata(QiuandChou,1987, 1988,1990).Suchaproblembelongstothesecond kindofinverseproblem(GaoandChou,1994;Fanand Chou,1999),whichcanimprovemodelandheighten predictivelevelwithsmallercost.Thus,itwilljustbe investigatedhowcurrentmodelerrorsareestimated byeffectivelyusinginformationfromhistoricaldata, whichmayalsoberegardedasamodelidentification problem.

    Infact,foraspecialnumericalmodelL,onecan retrieveandobtainseveralverydifferentorevencom

    pletelydifferentestimationsofmodelerrorsbyusing observeddataindifferenttimes.Whjchestimationon earthshouldbechosentoimprovemodelandpredic

    tion?Whatoughttobebasedonforsuchachoice? Toanswertheseproblems,theparticularityofpre

    dictionobjectiveshouldbeconsideredinorderto \l,\l,

N03RENHongliandCHOUJifan

    usedistinguishinglyexistinghistoricaldata.Interms ofanalogyprinciple,estimatedmodelerrorsEbyuti

    lizinganalogicalatmosphericevolvingdatawouldbe closertoeachother,whichmaybeeasilyunderstood frompracticalexperiences.Forinstance,

thesame

    modeloftenmakesverysimilarfaultsforforecasting analogicalweatherprocess.Setasthehistorical analogicalstate(calledasanaloguereferencestate,or referencestateforshort,denotedasRS)of,which

    satisfies

    )=E()

    (r,0)=G(r).

    (4)

    (5)

    Becauseisquitecloseto,wecanmakethefirst

    orderTaylorexpansionofE()intermsofaround asfollows:

    E()E()+()Dl

    whereDrepresentsthesumofthepartialdifferentials ofEwithrespecttoeverycomponentof.Aswe cansee,whenDfisboundedandJJJJissmall

    enough,let=+anditisnotdifficulttoobtain E(+)E()l_<<JJE()

    Atthistime,providedthattheerrortermE() ontherighthandsideofEq.(3)isdirectlyestimated withtheerrortermE()ontherighthandsideof

    Eq.(4),wecanobtain

    )

    InEq.(6),thesmalltermE(+)E()has

    beenactuallyomitted.BecausetheRSsarefromhis

    toricaldata,thefirsttermontherighthandsideof

    Eq.(6)isknownandthesecondtermmaybecalcu

    latedbynumericalmodel(Bao,2004).Thus,Eq.(6)

Callbeconsideredtoappendananaloguecorrection

    termoferrorsintoEq.f11inordertobecloserto Eq.(3),whichisevidentlymoreexactthanthatby omittingE()ontherightthesideofEq.(3)com

    paredwithEq.(1).Here,wecallEq.(6)theanalogue

    correctionequationoferrors(denotedbyACEE1, 369

    whichstillreflectstheoriginalmodelbyonlyadding acorrectiontermsoastoreducemodelerrors. ItcanbeseenthatbyestimatingE()incurrent

    predictionfromE()onthebasisofexistingmodel andhistoricalanalogueRSs,wecanregardE()as

    correctionterminordertoreducemodelerrorsand improvecurrentprediction.Insuchasense,theprob. 1emimprovingthedynamicalpredictioninnumerical modelbyutilizinghistoricaldatahasbeenactually transformedintotheinverseproblemestimatingcur. rentunknownmodelerrorsbyusingknownhistorical analogicalinformation.

    3.Equivalentanaloguedynamicalmodel

    Ingeneral,thereexistmanyapproachesthatcan reducemodelerrorsandimprovepredictionbyuti 1izinginformationofhistoricaldata,suchasthesys

    tematiccorrectionmethodofmodelerrors.besidesthe modelidentificationtechniqueonthebasisofsolving inverseproblem(QiuandChou,1987,1988,1990). Theformercanbeemployedtoimprovepredictionby directlyutilizinghistoricalhindcasterrorstocorrect currentprediction,butthelatterisusedforimprov

ingmodelbasedonhistoricaldata.Underthepre

    requisitewithoutregardtoobservederrors,theerror

    correctionsectionscorrespondingtotheformermay justbeproducedbythemodelerrortermretrieved fromthelatter.Consequently,inthesenseofimprov

    ingprediction,thetwoapproachesareconsistentwith eachother,whichisveryimportantfortheapplica- tionsofEq.(6)tocomplicatedmode1.

    Asweknow,theinfluencesofmodelerrorson

    predictionarealterativewithtimeanddependenton flowpattern.Thus,itmaynotbethemosteffective thatallofdataorinthenearpastdataareusedfor

    theidentificationofmodelerrorsorsystematicerror correction.Generally,themoredataareusedin1inear system,thebettereffectforsolvingabovementioned

    inverseproblemwillbe.However,astheatmosphere isanonlinearsystem,theeffectforsolvingcorrespond

    inginverseproblemwillnotbedependentonquantity ofdatabutonparticularityofdata.Accordingtothe aforementionedanalyses,itcanbeknownthatforthe ChouJifan.Theanalogue

    dynamicalmodelofmonthlymeancirculation.personalcommunication.April25,2003

370ACTAMETEOROLOGICASINICA

    estimationproblemofmodelerrors,weshouldutilize specifichistoricaldataforthesolutionoftheinverse problemintermsoftheparticularityofcurrentfore

    castobjective,whichisalsothesameforsystematicer

    rotcorrection.Furthermore,intermsofthederivation ofEq.(6),itisnotdifficulttounderstandthatonlythe

    informationgeneratedfromhistoricalanaloguestates canbeeffectivelyusedforcorrectingmodelerrorsof currentforecast,butnonanaloguestatescouldpro

    videfaultcorrectinginformation.Here,"theparticu

    larityofforecastobjective''refersto''currentforecast" and"theparticularityofuseddata''refersto"histor

    icalanaloguestates".

    Actually,manypreviousstudiesonADMs(qiu andChou,1987;HuangandWang,1991;Huanget a1.,1993)havealreadyintegrallytakentheprediction objectiveandtheparticularityofuseddataintoac

    count.canbedividedintotheanaloguereference state(RS)andtheanaloguedisturbancestate(or disturbancestateforshort,denotedasDS),namely =+,whereisselectedfromhistoricaldata

    intermsofthesimilaritiesbetweentheRSsandthe currentinitialstateG(r).Substitute=+and intoEq.(1)respectively,andsubtractthelatterfrom theformer,andwecanobtaintheanaloguedeviation

    equation(ADE)asfollows:

    ))=.

    BycomparingEq.(7)withEq.(6),itcanbefound thattheyarecompletelyequivalent,buthavequite differentphysicalsense.Theformerexpressesthedy- namicalequationsatisfiedbyanaloguedeviationand needstorebuildadeviatedADM,whichisverydiffi

    cultforcomplicatedoperationalmode1.However,the latteronlycorrectsmodelerrorsincurrentprediction byutilizinghistoricalanalogicalinformationandneed

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