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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)

2CollegeofAtmosphericSciences,LanzhouUniversity,Lanzhou73000

ABSTRACT

Basedontheatmosphericanalogyprinciple.theinverseproblemthattheinformationofhistoricalana-

loguedataisutilizedtoestimatemodelerrorsisputforwardandamethodofanaloguecorrectionoferrors

(ACE1ofmodelisdevelopedinthispaper.TheACEcancombineeffectivelystatisticalanddynamical

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

lishedfortheweatherforecastingandseasonalpredic

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

Itiswellknownthatthereinevitablyexister

33031,40575036and40675039

Correspondingauthor:renhl@cma.gov.cn.

368ACTAMETEOR0L0GICASINICA

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

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

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

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

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

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

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".

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