displayedwinds

InternationalJournalofGeosciences,2010,1,70-78doi:10.
4236/ijg.
2010.
12010PublishedOnlineAugust2010(http://www.
SciRP.
org/journal/ijg)Copyright2010SciRes.
IJGABasisforImprovingNumericalForecastingintheGulfAreabyAssimilatingDopplerRadarRadialWindsFathallaA.
Rihan1,ChrisG.
Collier21DepartmentofMathematicalSciences,CollegeofScience,UnitedArabEmiratesUniversity,Al-Ain,UAE(Permanentaddress:FacultyofSceince,HelwanUniversity,Cairo,EGYPT)2NationalCenterforAtmosphericScience,SchoolofEarthandEnvironment,UniversityofLeeds,Leeds,UKE-mail:frihan@uaeu.
ac.
ae,C.
G.
Collier@leeds.
ac.
ukReceivedJune7,2010;revisedJuly1,2010;acceptedJuly26,2010AbstractAnapproachtoassimilateDopplerradarradialwindsintoahighresolutionNumericalWeatherPrediction(NWP)modelusing3D-Varsystemisdescribed.
Wediscussthetypesoferrorsthatoccurinradarradialwinds.
Somerelatedproblemssuchasnonlinearityandsensitivityoftheforecasttopossiblesmallerrorsininitialconditions,randomobservationerrors,andthebackgroundstatesarealsoconsidered.
Thetechniquecanbeusedtoimprovethemodelforecasts,intheGulfarea,atthelocalscaleandunderhighaerosol(dust/sand/pollution)conditions.
Keywords:3D-Var,DataAssimilation,DopplerWinds,Errors,NWP,Nonlinearity,Sensitivity1.
IntroductionNumericalWeatherPrediction(NWP)isconsideredasaninitial-boundaryvalueproblem:givenanestimateofthepresentstateoftheatmosphere,themodelsimulates(forecasts)itsevolution.
Specificationofproperinitialconditionsandboundaryconditionsfornumericaldy-namicalmodelsisessentialinordertohaveawell-posedproblemandsubsequentlyagoodforecastmodel(Awell-posedinitial/boundaryproblemhasauniquesolu-tionthatdependscontinuouslyontheinitial/boundaryconditions).
Thegoalofdataassimilationistoconstructthebestpossibleinitialandboundaryconditions,knownastheanalysis,fromwhichtointegratetheNWPmodelforwardintime.
AssimilationofDopplerradarwinddataintoatmos-phericmodelshasrecentlyreceivedincreasingattentionduetodevelopmentsintheuseoflimitedareahighreso-lutionnumericalmodelsforweatherprediction[1].
Themodelsrequireobservationswithhighspatialandtem-poralresolutiontodeterminetheinitialconditions,forwhichpurposeradardataareparticularlyappealing.
However,theresolutionofDopplerradarobservationsismuchhigherthanthatofthemesoscaleNWPmodel.
Beforetheassimilation,thesedatamustbepreprocessedtoberepresentativeofthecharacteristicscaleofthemodel.
Toreducetherepresentativenesserrorandcorre-spondthedatamorecloselytothemodelresolutionsthandotherawobservations,onemayspatiallyinterpolatefromtherawdatatogeneratethesocalledsuper-ob-servations;seeSection8.
Overthelastthirtyyearsorsonetworksofweatherradars,providingmeasurementsofradarreflectivity,fromwhichrainfallhasbeenestimated,havebeenestab-lishedwithinoperationalobservingsystems.
Initiallytheradars,operatingatS-band(10cm)orC-band(5-6cm)wavelengthsdidnothavethecapabilitytomeasurethemotionofthetargets(mainlyhydrometeorsbutalsoin-sectsandbirds,andforhighpowersystems,refractiveindexinhomogeneities)towardsorawayfromtheradarsite.
DuringthelasttwentyyearsorsoweatherradarshavingDopplercapabilitymeasuringradialmotionofthetargetshavebecomestandardsuchthatnowinEuropewelloverhalfoftheoperationalradarsareDopplersys-tems(see[2,3]).
Recently,DopplerradarradialwindshavebeenassimilatedintoNWPmodelsasverticalwindprofilesderivedfromVelocityAzimuthDisplay(VAD)analysis[4,5],andusingvariationaltechniques[6-8].
InordertoassimilateDopplerradialvelocityobserva-tions,theobservationerrorswhichcomefromseveralsourcesareestimatedforinclusioninthevariationalsys-tem.
InthispaperweoutlinethelikelyerrorsinestimatesofDopplerradarradialwinds,andhowtheymightberepresentedmathematically.
Toillustratetheresults,weapplythemethodologytoartificialdata.
WedescribetheradialwindanderrorrepresentationaspartofasystemF.
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IJG71forgeneratingsimulateddataforuseinthe3D-Vari-ational(3D-Var)system.
InSection2,wediscussthetypesoferrorsinradarradialwinds.
Section3describesasimulationmodelwhichisusedtoanalyseactualradialwinds.
Weoutlinehowsuchobservationsareassimilatedin3D-Variational(3D-Var)systeminSection4.
SomeassociatedissuesduetothenonlinearityaredescribedinSection5.
Aproposedmethodologyisdescribedtode-rivethevariationoftheerrorswithrangeinSection6.
DirectassimilationofradialwindsinPPIformat,andpreprocessingthedataarediscussedinSections7&8.
StepsofNWPandconclusionsaregiveninSections9&10.
ThisworkismainlybasedontheworkdescribedinRihanetal.
[1].
2.
ErrorsinDopplerRadialVelocityTargetsmovingawayfromortowardsaradarproduceaDopplershiftbetweenthefrequencyofthetransmittedsignal(pulse),andthesignalreflectedfromthetargetsandreceivedbackattheradar.
However,ambiguitiesmayariseinthesemeasurementsduetorangefoldingandvelocityaliasing[9].
Fortunatelyprocedureshavebeendevelopedtominimizetheseproblems[10].
Otherproblemsremain,namelytheexistenceofdataholes(wheretherearenotargets),andirregularcoverage,instrumentalnoiseandsamplingerrors.
Varioustypesofinterpolationschemeshavebeenusedtofillindataholesandpoorcoverage[11],althoughsuchschemesareun-necessarywhenthreedimensionalassimilationschemesareimplemented.
However,theimpactsofinstrumentalnoiseandsamplingaremoreproblematic.
Mayetal.
(1989)discuss,andassess,anumberoftechniquesusedtoestimatetheDopplershiftinthereceivedsignals.
TheDopplershiftisproportionaltotheslopeofthephaseoftheautocorrelationfunction(atzerolag)ofthereturnedsignals.
Anestimatoroftheshiftisthephaseatthefirstlagdividedbythevalueofthelagintimeunits.
Thisisknownaspulsepairprocessing,andmaybeimprovedbyaveragingmorethanonevalueofthephasedividedbythelag(polypulsepair).
Samplingerrorsdependuponthesizeofthepulsevo-lumecorrespondingtoeachdatapoint.
Inpracticethesamplingerrorscouldbeweaklycorrelatedfrompointtopoint,butonlyaverysmalladditionalerrorwillbein-troducedifthisisignored.
Practically,samplingerrorsdominatesinceinstrumentalerrorsareusuallyminimizedinoperationalsystems.
Inthefollowingweoutlineasystemforcreatingartificialradarradialwinddatasetswithinwhichdifferenttypesoferrormaybeincluded.
Figure1showsschematicsoftheimpactuponaGaus-sianDopplerspectrumofvariouseffectsofstrongwindshearalongthepulsevolume,andinstrumentallyinduc-edeffects.
SeveraloftheseeffectsupontheDopplerspe-ctrummaybepresentinthesameradarimage,and,inthecaseofgeophysically-inducedeffects,theirmagni-tudemayvarywithrangeandazimuth.
Theheightandsizeofthepulsevolumeswillincreasewithincreasingdistancefromtheradar.
3.
SimulationofDopplerRadialWindsTheconstructionofartificialradardatasetshasbeencarriedoutforseveralstudiesoverthelasttwentyyears[12-14].
Consideraconicalradarscan(seeFigure2)inaCartesiancoordinatesystem(x,y,z).
Thecomponentsofthewindfieldcorrespondingtothesecoordinatesareu,vandwrespectively.
Itisassumedthatvelocity-rangefoldinghasbeenremoved.
ThewindisassumedtovarywithheightaccordingtoanEkmanspiralwithvariablesurfacefriction.
Thewinddirectionatthetopoftheboundarylayerisparalleltotheisobars,whilstthewinddirectionatthesurfaceisinthedirectionofthelowerpressureduetothesurfacefriction,thecoriolisforceandpressuregradientforce.
--sin,gazuUeaz(1)-(1-cos),gazvUeaz(2)wheregUisthegeostrophicwind,/2afk,fisthecoriolisparameterandkistheeddyexchangecoeffi-cients(≈5*104cm2sec-1)inmiddlelatitudes.
Thesimulateddataareassumedtobeavailableonthemeasurementpoints(seeFigure2).
Theradialvelocityiscalculatedfromsincoscoscossinrvuvw(3)whereu,v,warethewindcomponents;and,aretheazimuthangle,andtheelevationangleoftheradarbeam.
Figure1:Distributionofwindspeederror.
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GeometryforscanofvelocitiesonaVelocityAzi-muthDisplay(VAD)circle(top)andthevariationoftheradial.
AteachmeasurementpointaGaussian(oramodifica-tionofaGaussian)distributionisintroduced,themagni-tudeandspatialvariationofwhichmayrepresentthedifferenterrortypesshowninFigure1.
ExamplesofthetypeofartificialradialvelocityfieldproducedusingFormula(3)aredisplayedinFigure3(top).
SuchdatacanbeusedtotesttherepresentationoftheerrorsintheDopplerradialwinds,andvariationalanalysisschemes.
Airmovementis3-dimensionalandvariesovertimeandspace.
However,Dopplerradarallowsthemeasure-mentofonlyone(radial)componentofthevelocityofthetargetsataspecificrangeandazimuth.
SinceweonlytakethedatafromsingleradarinthepresentstudyratherthansimultaneousmeasurementswiththreeDopplerradars,weareforcedtomakeasimplifyingassumptiontothestructureoftheobservedwindfieldduringthecreationofDopplerproducts.
Thesimplestcaseistoconsiderahorizontallyuniformwindfieldforboth,horizontalandvertical(precipitationfallvelocity)components.
Insuchacase,ifwemakemeasurementsofthevelocityalongcirclescentredattheradarbyazimuthalscanningataconstantelevationangle(PPI),weget,foraconstantdistancefromtheradar,asinusoidaldependencyofthemeasuredradialvelocityontheazimuthalangle.
Assumingthatthehorizontalwindvelocityvhandhydrometeorfallspeedwareuniformovertheareabeingobserved,thenthemeanDopplervelocityvrvariedsinusoidallywithmaximaandminimaoccurringwhenthebeamazimuthpassestheupwind(θ=0)anddownwind(θ=π)directions,thatiswhenr1hr1h=cos+sin,when=0(4)=-cos+sin,when=vvwvvwHence1212=,(5)2cos2sinrrrrhvvvvvwThenthehorizontaldivergenceisgivenbytheformula2012tandiv,(6)coshrwvvdRRwhereRistheradiusoftheradarsamplingcircleatheightl.
However,Equation(6)isonlyvalidforlowele-vationangles.
UsingFormula(3)toderivethewindvelocity,wecompareaplotderivedfromFigure3(top)withaper-fectsinewavedisplayedinFigure3(bottom).
Theim-pactofthesimulatederrorsistocausethedifferencesbetweenthedata(dots)andthenoerrorsinecurve(solidline)shown.
Itispossibletousethesimulatortoinvesti-gate,inmoredetail,theimpactofvariouserrorsontheradialwinds.
Thisisthesubjectofcontinuingstudy.
4.
3D-VarDataAssimilation3D-Varsystemsuseanincrementalformulation(forareviewsee,forexample,[6]).
UndertheassumptionthatthebackgroundandobservationerrorsareGaussian,randomandindependentofeachother,theoptimalesti-mateoftheCartesianwindabXXXintheanalysisspaceisgivenbytheincrementalcostfunction,1-11[]217)2TTbbJXXBXHXYHXEHXYHXwhereabXXXisthestatevectoroftheanalysisincrements(theestimatedradialwindsisgivenbybaXXHH,bXthestatevariableofthebackgroundCartesianwinds,andYdenotestheobservedradialwindsintheobservationspace.
Histhenonlinearob-servationoperatorthatrelatesthemodelvariablestotheobservationvariableandatransformationbetweenthedifferentgridmeshes,andHisthelinearobservationoperatorwithelements/jhXijiH.
Someconstruc-tionsofthebackgroundandobservationerrorcovarianceF.
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IJG73matricesBandEaregivenin[15].
MillerandSun[16],andXuandGong[14]assumedthattheobservationerrorcovariancematrixEisdiagonalwithconstantdiagonalelementsgivenbytheestimatedobservationerror,whichwastakenas1m/sfortypicalradarobservations.
InthenextSection,weprovideadifferentapproachtotherep-resentationoftheerrorofobservedradialwinds.
ToavoidthecomputationallyoverwhelmingproblemofinvertingthecovariancematrixBintheminimizationofthecostfunction(7),andtoacceleratetheconver-genceoftheminimizationalgorithm,apre-conditioningoftheminimizationproblemisneeded[15].
ThiscanbeachievedbydefiningavariableUtobeappliedtotheassimilationincrementX(UXX)suchthatittransformstheforecasterrorinthemodelspaceintoavariableofanidentitycovariancematrix(i.
e.
,,IT,where.
,.
isaninnerproduct).
Thischangeofvariablecanbewrittenas1U.
Thus118)TTTBUUorBUUFigure3.
ArtificialradialvelocitywithGaussiannoise.
Thisleadstoanewrepresentationoftheincrementalcostfunctionoftheform1-111122[-].
(9)TTbbJHUYXEHUYXXXXXHXHWiththiscostfunction,noinversionofBisneeded.
ThecontrolvariablesXarevelocitypotential,streamfunction,unbalancedpressureandrelativehumidity.
Here,weassumethatthematrixEincludestheerrorsfromtheobservations(originalmeasurements),observa-tionoperator,andsuper-obbingprocedure1.
The3D-Varanalysisisthenperformedusingcontinuouscyclingpro-cedure.
Thelengthoftheassimilationwindowineachanalysisisdeterminedaccordingtothemodelresolution.
Ineachanalysiscycle,theoptimalanalysisisobtainedbyminimizingthecostfunction(9)usingiterativepro-cedure.
Thematrix1Uin(9)mayberealizedas1UDF(10)whereDisadiagonalmatrixofstandarddeviationofthebackgrounderrorspecifiedbytheerrorestimationofnumericalexperiments,andFisthesquarerootofama-trixwhosediagonalelementsareequaltoone,andoff-diagonalelementsarethebackgrounderrorcorrelationcoefficients.
InpracticaldataassimilationforNWP,thefullmatrixFistoolargetocomputeexplicitlyorstoreintocomputermemory.
Assumptionsandapproxima-tionsaremadesuchthattheeffectofFonthecontrolvariableXinEquation(9)isachievedthroughtheuseofequivalentspatialfilter.
FollowingtheworkofPurserandMcQuigg[17],Lorenc[18],andHaydenandPurser[19],theeffectofFusingarecursivefilterisdefinedby1i1i=+(+1)for=1,2,.
.
.
,Ζ=+(+1);for=,-1,.
.
.
,1;iiiiXinYinn(11)whereiXistheinitialvalueatgridpointi,i(1,.
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.
,in)isthevalueafterfiltering,andiistheinitialvalueafteronepassofthefilterineachdirection.
isthefiltercoefficientsgivenin[15]by221(2),2/(4)NxL(12)whereListhehorizontalcorrelationscale,xisthegridspacing,andNisthenumberoffilterpassestobeapplied.
Thisisafirst-orderrecursivefilter,appliedinbothdirectionstoensurezerophasechange.
Multipassfilters(Ngreaterthanunity)arebuiltupbyrepeatingapplicationof(11).
Thisfiltercanbeconstructedinall1Super-obbingprocedureisatechniquetocombine(re-scale)theradarobservations,usingstatisticalinterpolation,atalargerspatialscalewhichiscompatiblewiththemodel.
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IJG74threedirections.
Wenextprovideanapproachtoesti-matingtheerrorcovarianceofradialvelocityobserva-tionscanbeconstructedinallthreedirectionsupbyre-peatingapplicationof(11).
Thisfiltercanbeconstructedinallthreedirections.
Wenextprovidesomerelatedissuesduetononlinear-ityofthedynamicsystem.
5.
NonlinearityoftheAtmosphericDynamicsIngeneral,forecastskillincreasesnotonlybyincreasingmodelresolution,butalsobyimprovingthenumericalmodelsandthemethodofsolution.
However,evenifwehadaforecastmodelthatrepresentedatmosphereproc-essesperfectly,wewouldneverbeabletopredictthestateofatmosphereaccuratelyforlongleadtimes.
Achaoticbehavioroccurs(andleadstoanunpredictablelong-termevolution)whensolvingdeterministic,non-linear,dynamicalsystemsthatexhibitsensitivitytoini-tialconditions.
Thisoccurs,becausethenonlineardy-namicalsystemsthatdescribetheatmosphericbehaviouraresensitivetosmallchangesininitialconditions.
Inthissectionweconsider,twoissuesassociatedwithdataas-similation,sensitivityanalysisandbiasesduetononlin-earity.
Now,towhatleadtimeforecastsremainskillfulde-pendsonhowsmallerrorsintheinitialconditions,boundaryconditions,ormodelspecificationsgrowtoaffectthestateoutputortheforecast.
Becauseerrorstendtogrowrapidlyinprocessesthatoccuratsmallerspa-tial-scales,thenforecastsforsmallscaleprocessesmaybepredictableonlyforfewhours.
However,forecastsoflargescaleprocessescanbepredictedforperhapstwoweeksahead.
Thus,whensolvingtheforwardproblem,itisveryimportanttoassessthesensitivityofthestateoutputvariablesofthedynamicsystemtosmallchangesintheinitialconditions.
Aknowledgeofhowthestatevariablescanvarywithrespecttosmallchangesintheinitialdatacanyieldinsightsintothebehaviourofthemodelandassistthemodellingprocesstodetermine(forexample)themostsensitivearea.
Thesensitivityanalysis,ofthedynamicsystem,entailsfindingthepartialderiva-tiveofthestatevariable(ortheanalysis)withrespecttotheparameters,whichisabigchallengeinalargenon-linearsystem.
Forfurtherstudyofsensitivityanalysisduetothenonlinearity,werefertosee[6,20,21].
Itshouldalsobenotedthatthepredictabilityoftheatmosphericstatedependsmainlyontheaccuracyoftheparameterestimates(thecontrolvariables),whensolvingtheinverseproblem.
Sincetheultimategoalistopro-duceananalysisthatgivesthebestforecast,itisdesir-abletohaveinformationabouttheeffectontheanalysissystem(ortheestimates)duetoperturbingtheobserva-tions(ornoisydata),orsmallchangesinthebackground;See[6].
ThenonlinearbehaviorofthewindfieldhasbeendiscussedinLovejoyetal.
[22].
Verticallypropagatinggravitywavesarebrokenupbyunstablelayers(seeforexampleBrowningetal,2009[23])whichhavefractalstructures.
Lovejoyetal.
[22]showthatonecanreadilymakestronglynonlinearmodelsbasedonlocalizedtur-bulencefluxeswhichhavewavelikeunlocalizedvelocityfields,andthisrespectingtheobservedhorizontalandverticalscaling.
Thisturbulentanisotropicscalingcangiveriseto(nonlinear)dispersionrelationsnotsodif-ferentthanthosepredictedbylineartheorysoitmaybesufficienttoreinterprettheempiricalstudiesofwavesinthisanisotropicscalingframework.
Wenextprovideanapproachtoestimatingtheerrorcovarianceofradialvelocityobservations.
6.
ErrorsinObservationRadialWindsInordertooptimallyassimilateDopplerradarradialve-locityobservationsintoNWPmodel,itisnecessarytoknowtheirerrorcovariances.
Weassumethattheobser-vationalerrorsareuncorrelatedinspaceandtime.
Underthisassumption,theobservationerrorcovariancematrix(E)inthecostfunction(9)canbereducedtoadiagonalmatrix.
ThenthematrixE,inEquation(9),isregardedasaweightingcoefficientthatreflectstherela-tiveprecisionofthedata(measurementuncertaintyandrepresentativenesserror).
Thematrixcanbeex-pressedas:2[()],diag(13)where2()istheerrorvarianceoftheradialvelocityrv.
Themostcommonerrorinradarradialwindsare1)thenoiseintheradialvelocityinducedbythevelocitygradientacrossthepulsevolumewithvariance2()v,and2)theinstrumentalerrorduetohardwaredegrada-tionofvariance2()i.
MillerandSun[16]statethatthesemeasurementerrorsneedtobespecifiedsothatradarobservationscanbeproperlyassimilatedforNWP.
However,theynotethatthemeanradialvelocityandspectralwidthestimatorsareproportionaltotheradarwavelengthandthetimespectralwidth[24],andthere-foreareratherimpracticalasestimatesofthemeasure-menterrors.
Theythereforenoteaneedforerrorestima-torsofradialvelocitythatcanbeobtainedfromthemeasurementsthemselves.
6.
1.
ErrorDuetotheVelocityGradientThelocalsamplingoftheradialvelocityisemployedtoapproximatetheerrorvariance2rv,sincenoisydataareusuallyassociatedwithhighvaluesofradialvelocityF.
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Errorsintheoriginalmeasurementsoftheradialve-locitywithineachradarpulsevolumedependonthestrengthofthereturnedsignalandthespread(orwidth)oftheDopplervelocityspectrumthatdependsonthevelocitygradients.
Sincetheradarscatterersinthepulsevolumemoverandomly,weassumethattheerrorsofthevelocitygradientoftheradarbackscatterersaregivenbyanormal(Gaussian)distribution,where,2/21().
2vvvpdfed(14)Theerrorvariance2ismodifiedbythevelocitygra-dient(whichvarieswithtime)alongthepulsevolume.
Thevariationsofthisvelocitydifferencealongthepulsevolumecausethekineticenergy(KE)ofthemovingscattererstochange.
Wewillassumehereforsimplicitythatthisvelocitydifferenceistakenintheradialdirec-tiononly.
Here,therateofchangeinthe=,rKEv(15)whereisanarbitraryforceappliedtothescatterersandrvisthevelocitydifferencealongthepulsevol-ume.
Thustheincreaseinthekineticenergyintimein-tervaldtduringwhichthescatterermovesadistancedr,is()=,dKEdW(16)whereWbyistheworkdonetheforcetheinfini-tesimaldisplacementdrtakenasthepulselengthandovertimedt.
RogersandTrips[24]showthatthechangeintheki-neticenergyperunitmasscanbeexpressedas2rdKEv(17)where2()rvisthevarianceofthemeanDopplerve-locity,whichcanbeexpressedby(see[24])2()=.
8rvMT(18)Hereistheradarwavelength,isthetruespec-tralwidth,Misthenumberofequallyspacedpulses,andTisthetimebetweenpulses.
(Themaximumun-ambiguous(Nyquist)velocityis/4nyqvT).
Nastom[25]investigatedthefactorsimpactingonthespectralwidthofDopplerradarmeasurements.
Forverysmallbandwidthsitwasfoundthatthevariancewasdominatedbytheeffectsofwindspeedchangesalongtheradarbeam.
Theexpressionforthevariancewasde-rivedasafunctionofthebeamelevationandtheverticalwindshear.
Wethereforechoseheretoexpresstheerrorvariance2()vofradialwindsmoresimplyintermsofthegradientvariancealongthepulsevolumeinara-dialdirectionasfollows|/|22()(1)(),vrvvvrev(19)wherervisthegradientoftheradialvelocity,meas-uredasacentreddifferenceacrossthepulsevolume.
TheerrorinradarradialwindsduetothevelocitygradientalongthepulsevolumevarieswiththerangeR.
Figure4showsaproposedS-functionfortheobservationerrorsasafunctionoftherange,whichisacceptabletorepre-senttheerrorsoftheradialvelocity.
Notethatastherangeincreasestheerrorincreases.
Thisistobeexpectedastheradarbeamgetswiderandthepulsevolumegreat-erthekineticenergyvariationofthescattersinthepulsevolumeincreases,withincreasingrange.
6.
2.
ErrorDuetoHardwareDegradationAlthoughtheinstrumentalerrorcanhaveasignificantimpactontheretrieval,inpracticeitisdifficulttodeter-minehowthiserrorvarieswithtime.
Inthiscaseweas-sumethattheinstrumentalerrordoesnotvarytemporally,andtaketheinstrumentalerrorvarianceas2()ias-sumingthereisnohardwaredegradationwithtime.
Thereforethetotalerrorvarianceoftheradialwindsisgivenby222vi(20)ateach(,)r.
Theinstrumentalerrormaynotbeafunctionof(,)r,butisafunctionoftime.
Inthiscase,weassumethatthiserrorisrepresentedbya"skewed"distributionsuchasaChi-Squareddistributionwithprobabilitydensityfunc-tion(fordegreesoffreedom)givenby.
Figure4.
Theproposeds-functionfortheobservationer-rorsasafunctionoftherange.
Thevariabilityoftheerrorincreasesastherange.
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2(/2)iiepdf(21)Here(.
)isthegammafunctionandiisthein-strumentalerror.
Thustheerrorvarianceofthedegrada-tionisdefinedby22iiiiipdfd(22)whereiisthemeanoftheinstrumentalerrors.
7.
DirectAssimilationofPPIDataDuetothepoorverticalresolutionofradardata,averti-calinterpolationofradardatafromconstantelevationlevelstomodelCartesianlevelscanresultinlargeerrors.
ForthisreasonadirectassimilationofPPIdatawithnoverticalinterpolationwasrecommendedin[1,7,8].
However,radardatahasbetterhorizontalresolutionthanthatofthemodel(thepoorestpolarradardataisap-proximately0.
5kmatthefarthestrangedistance).
Anobservationoperatormustbeformulatedtomapthemodelvariablesfrommodelgridintotheobservationlocationssuchthatthedistancebetweentheobservationsandmodelsolutionisestimatedinthecostfunction.
Thus,wetakeadvantageoftheverticalresolutionofthemodelbeingmuchbetterthanthoseofradardata.
Theobservationoperation,eH,formapping(andaveraging)thedatafromthemodelverticallevelstotheelevationanglelevelsisformulatedas(),rrGvzvGzr,ee=H(23)wherer,eistheradialvelocityonanelevationanglelevel,rvisthemodelradialvelocity,andzisthemodelverticalgridspacing.
Thefunction22/Gerepre-sentsthepowergainoftheradarbeam,(inradians)isthebeamhalf-widthandisthedistancefromthecentreofradarbeam(inradians).
Thesummationisoverthemodelgridpointsthatlieinaradarbeam.
WenextdiscusstheobservationoperatortoconverttheCartesianmodelcomponentstotheradialcompo-nents.
7.
1.
ObservationOperatorTherearetwotypesofobservationoperators.
Oneisusedtointerpolateandtransfertheradardatafromob-servationlocationstothemodelgrids.
Thesecondisusedtomapthemodeldataintotheobservationlocations.
Inthecaseofadirectassimilationofradarradialwindatconstantelevationangles,whichisnotamodelvariable,theobservationoperatorinvolves:1)abilinearinterpola-tionoftheNWPmodelhorizontalandverticalwindcomponents,,uvwtotheobservationlocation;2)aprojectionoftheinterpolatedNWPmodelhorizontalwind,atthepointofmeasurement,towardstheradarbeamusingtheformula=sin+cos,hvuv(24)whereisthetheazimuthangle(clockwisefromdueNorth).
Theelevationangleshouldincludeacorrectionwhichtakesaccountofearthsurfacecurvatureandradarbeamrefraction(see[26]);thenthethirdstep3)involvestheprojectionofhvintheslantwisedirectionoftheradarbeamas1cos()sin().
cos()tansin()rhvvwrrdh(25)Hereistheelevationangleoftheradarbeam.
TheformulaforrepresentsapproximatelythecurvatureoftheEarth.
Intheterm,ristherange,distheradiusoftheEarthandhistheheightoftheradarabovethesealevel;see[27].
8.
Pre-ProcessingofDopplerRadialWindDataThemajorstepsinprocessingthedatabeforetheassimi-lationareinterpolationofdatafrom/toaCartesiangrid,removingthenoisydata,andfiltering.
Dataqualitycon-trol(QC)isatechnique,whichshouldbeperformedforeachscan,toremoveundesiredradarechoes,suchasgroundclutterandanomalouslypropagatedclutter(APclutter),seaclutter,velocityfolding,andnoiseusingthethreshold,thatanyvelocitydatawithvalueslessthan,say,0.
25m/sandtheircorrespondingreflectivityareremoved.
UnfoldedDopplervelocity,inDopplerradar,ismeasuredusingbothhorizontallyandverticallypolar-isedpulses.
Thisparameteristhecomponentofthetar-getvelocitytowardstheradar(positivevelocitiesaretowardstheradar).
Noiseareremovedonthebasisofthevarianceofthevelocityateachpixelwithitsneighbours,whichcanoccasionallyremovegooddatawithgenuinelyhighvariance;see[7].
8.
1.
Super-ObbingRadialWindDataDopplerradarsproducerawradialwinddatawithhightemporalandspatialdensity.
Thehorizontalresolutionofthedataisaround300m(thatistoohightobeusedintheassimilationscheme)whereasthetypicalresolutionofanoperationalmesoscaleNWPmodelisoftheorderofseveralkilometers.
Toreducetherepresentativenesserror,andcorrespondtheobservationstothehorizontalF.
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IJG77modelresolution,onemayusespatialaveragesoftherawdata,calledsuper-observations.
Thedesiredresolu-tionforthesuper-observationscanbegeneratedbyde-finingparameters(whichcanbefreelychosen)fortherangespacingandtheanglebetweentheoutputazimuthgates.
Aswehavementionedpreviously,adirectassimila-tionofPPIdatawithnoverticalinterpolationisrecom-mended.
MoreoverassimilationusingradardatadirectlyatobservationlocationsavoidsinterpolationfromanirregularradarcoordinatesystemtoaregularCartesiansystem,whichcanoftenbeasourceoferrorespeciallyinthepresenceofdatavoids(see[6]).
Wehavedevelopedasoftwarepackage,whichisbasedonspatialinterpola-tioninpolarspace,forprocessingofrawvolumedataofradialvelocityinPPIformattosuper-obbthedatatotherequiredresolutionsforthe3D-VarsystemintheMetOffice(UK);see[1].
9.
StepsofNWPNWPisaninitial-valueproblem00(),()XFXXtXt(26)forwhichweshouldprovidetheinitialconditions(ICs).
Theseequationsareingeneral"partialdifferentialequa-tions"ofwhichthemostimportantareequationsofmo-tion,thefirstlawofthermodynamics,andthemassandhumidityconservationequations.
NWPcanbesumma-rizedinthefollowingthreesteps:Thefirstistocollectallatmosphericobservationsforagiventime.
Second,thoseobservationsarediagnosedandanalyzed(thatrep-resentedindataassimilation)toproducearegular,co-herentspatialrepresentationoftheatmosphereatthattime.
ThisanalysisbecomestheinitialconditionfortimeintegrationofNWPmodelthatbasedonthegoverningdifferentialequationsoftheatmosphere.
Finally,theseequationsaresolvednumericallytopredictthefuturestatesoftheatmosphere.
Forfurtherinformationaboutthegoverningequationsoftheevolutionoftheatmos-phere,wereferto[28-30].
10.
ConcludingRemarksThebenefitsofassimilatingDopplerradarwindsinNWParethat:1)theLimitedAreaModelsrequireob-servationswithhighspatial-temporalresolutiontofore-castthedetailsofweatheranditsdevelopment;2)Dop-plerradarsareabletoscanlargevolumesoftheatmos-phere,andprovidehighresolutionmeasurementsofre-flectivityandradialvelocityinforecastingofquicklydevelopingmesoscalesystems.
Weshouldmentionherethatthepredictabilityoftheatmosphericstatedependsmainlyontheaccuracyoftheparameterestimates(thecontrolvariables),whensolvingtheinverseproblem.
Sincetheultimategoalistoproduceananalysisthatgivesthebestforecast.
Aparticularchallengeinthefo-recastingofthetimeevolutionofatmosphericsystemisthenonlinearityofthesystemandthecorrespondingsensitivityoftheinitialconditions.
ThemajorimpactofassimilatingDopplerradialwindsuponmodelperform-anceislikelytoarisefromtheimpactofthesedatauponmoistconvectiveprocesses,throughmoisturerelatedvariablessuchasverticalvelocity.
TheaimofthispaperwasmainlytoinvestigateissuesconcernedwiththeassimilationofDopplerradialwindsintoaNWPmodelusinga3D-VarsystemtoimprovethenumericalforecastingintheGulfArea.
Thisisduetothefactthatthereisawitnessingrapideconomicalandtechnologicaldevelopmentassociatedwithconsiderableinvestmentsininfrastructureanddevelopmentalprojects.
ThetechniquedisplayedinthispapercanbeappliedindatacollectedintheGulfarea,especiallyinUAE:BothwindandvisibilitydatainferredfromtheRadarmeas-urementsaretheassimilatedusing3D-VarsystemandMM5mesoscalemodelareusedtofortheforecasts.
Themodelexperimentscanbeperformedunderdifferentweatherconditions,withparticularemphasisonimprov-ingbehaviorofthemodelatthelocalscaleandunderhighaerosol(dust/sand/pollution)conditions.
11.
AcknowledgementTheauthorsthankProfM.
MaimAnwarforhisvaluablecommentsinthispaper.
12.
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