(ARW) Model in Geopotential Height and Temperature

12 Pages·2012·0.79 MB·English

DECEMBER2012 WEE ET AL. 3907
Two Overlooked Biases of the Advanced Research WRF (ARW) Model
in Geopotential Height and Temperature
TAE-KWONWEE
UniversityCorporationforAtmosphericResearch,Boulder,Colorado
YING-HWAKUO
UniversityCorporationforAtmosphericResearch,andNationalCenterforAtmosphericResearch,Boulder,Colorado
DONG-KYOULEE
AtmosphericSciencesProgram,SchoolofEarthandEnvironmentalSciences,DepartmentofAtmosphericSciences,
SeoulNationalUniversity,Seoul,SouthKorea
ZHIQUANLIUANDWEIWANG
NationalCenterforAtmosphericResearch,Boulder,Colorado
SHU-YACHEN
UniversityCorporationforAtmosphericResearch,Boulder,Colorado
(Manuscriptreceived8February2012,infinalform17April2012)
ABSTRACT
TheauthorshavediscoveredtwosizeablebiasesintheWeatherResearchandForecasting(WRF)model:
anegativebiasingeopotentialandawarmbiasintemperature,appearingbothintheinitialconditionandthe
forecast.Thebiasesincreasewithheightandthusmanifestthemselvesattheupperpartofthemodeldomain.
Both biases stem from a common root, which is that verticalstructures of specific volume and potential
temperature are convex functions. The geopotential bias is caused by the particular discrete hydrostatic
equationusedinWRFandisproportionaltothesquareofthethicknessofmodellayers.Forthevertical
levelsusedinthisstudy,thebiasfarexceedsthegross1-dayforecastbiascombiningallothersources.Thebias
isfixedbyrevisingthediscretehydrostaticequation.WRFinterpolatespotentialtemperaturefromthegrids
of an external dataset to the WRF grids in generating the initial condition. Associated with the Exner
function,thisleadstothemarkedbiasintemperature.ByinterpolatingtemperaturetotheWRFgridsand
thencomputingpotentialtemperature,thebiasisremoved.Thebiascorrectionsdevelopedinthisstudyare
expectedtoreducethedisparitybetweentheforecastandobservations,andeventuallytoimprovethequality
ofanalysisandforecastinthesubsequentdataassimilation.Thebiascorrectionsmightbeespeciallybeneficial
toassimilatingheight-basedobservations(e.g.,radiooccultationdata).
1. Introduction use of the assumption that short-term forecast of the
modelisfreeofsystematicerrors,whicharecollectively
Most of the data assimilation methods being used
referred to as bias. However, as noted by Mass et al.
nowadays to produce the best possible initial condition
(2008), virtually all NWP models possess substantial
for numerical weather prediction (NWP) models make
systematicerrors.Biasinthemodelmakesthedataas-
similationineffectiveasitviolatesthebasicassumption
ofzeromeanandrandomerrors.Thebiasedmodelalso
Corresponding author address: Tae-Kwon Wee, University
leads to a biased analysis even with unbiased observa-
CorporationforAtmosphericResearch,P.O.Box3000,Boulder,
tions(DeeandDaSilva1998).Sinceobservationsalso
CO80307.
E-mail:[email protected] tend to hold their own biases (e.g., Tenenbaum 1996;
DOI:10.1175/MWR-D-12-00045.1
(cid:1)2012AmericanMeteorologicalSociety
3908 MONTHLY WEATHER REVIEW VOLUME140
Wangetal.2002;DeeandUppala2009),correctionof possess the same biases even against the very global
the model’s bias through data assimilation is difficult analysisthatwasusedtogeneratetheinitialcondition.
even with a fair amount of observations. Therefore, Sincecreatingtheinitialconditionfromothermodels’
lesseningthemodel’sbiasisofparamountimportance dataismerelyviewedasasetofinterpolationprocedures,
forrobustdataassimilation. theinitialbiaswasquitesurprisingandledustospeculate
Inpracticaldataassimilationapplications,observation the source of the disparity. Tracing further back to the
biasrelatestomodelbiaseventhoughtheyareirrelevant sources,welearnedthatthegeopotentialbiasisduetoa
innature.Agoodexampleinthisregardisthevariational particulardiscretizationofthehydrostaticequationused
biascorrectionforsatelliteradiancedata(DerberandWu in the model. The hydrostatic equation is used in the
1998;Dee2005).Inthemethod,themodel’sownstateis modeltodiagnosethespecificvolume(fornonhydrostatic
used as an unbiased reference to uncover bias in obser- runs)orgeopotential(forhydrostaticruns).Accuracyin
vations.Biasinthesatelliteradiances,forinstance,canbe thediagnosedvariablesdependsonhowtheintegration
introducedthroughmanyprocessesincludinginstrument ofthehydrostaticequationiscarriedout.Inthisstudy,we
characterization;sensorcalibration;andcollection,pro- willshowhowthediscretehydrostaticequationleadsto
cessing, and archiving of the data. If uncorrected, the asubstantial modelbias.Themodel usespotentialtem-
observation bias depreciates value of the observation perature as a prognostic variable. Accordingly, the po-
andhasthepotentialtodamagethequalityofanalysis tentialtemperatureisinterpolatedfromglobalanalysesto
andforecast.Therefore,theestimatedobservationbias thegridsofthelimited-areamodeltoprovidetheinitial
is strongly affected by the model bias. Regarding this condition, leading to a marked bias in the temperature.
fact,thebiascorrectionagainstthemodelstateissome- Giventhatmodelerrorsathighaltitudehavelongbeena
timesrestrictedtoareasnearradiosondeobservations problemforglobalmodels(e.g.,TrenberthandStepaniak
(e.g.,HarrisandKelly2001)soastoreducethecontami- 2002;Kobayashietal.2009)andforlimited-areamodels
nationduetomodelbias.AsaddressedbyKobayashietal. (e.g., Wee and Kuo 2004), the two biases in the model
(2009),itisthepresenceoflargebiasesintheassimilating mayhavebeenconsideredassuch.Aswillbedescribed
modelthatcausesspuriousshiftsandotherartifactsinthe later,theabove-mentionedsourcesarehardlyviewedas
analysis,evenifallassimilatedobservationsareunbiased problematic since the procedure and assumption that
andcorrectlyrepresentedbytheassimilationsystem. leadtothesystematicerrorsareseeminglylegitimate
Systematic model errors are attributed to diverse andreasonable.Section2offersabriefdescriptionof
sources.Besidesthebiasintheinitialcondition,thenu- the numerical model and data used in this study. The
merical model is imperfect in the governing equations, biases in geopotential height and in temperature will
numerics, surface forcing, and parameterizations of un- bedescribedalongwithsuggestedremediesinsections
resolvedphysicalprocesses.Despitetheextensiveefforts 3 and 4, respectively. The summary and conclusions
to understand and reduce model bias in the literature, willbepresentedinsection5.
dealingwiththemodelbiasisstillanenduringchallenge
formodeldevelopment.Thisisprobablyduetothelarge
2. Numericalmodelanddata
variety of possible sources of modeling errors and non-
linearinteractionamongthem.Ontheotherhand,there The numerical model used in this study is the Ad-
could alsobe some tractable sources of model bias. If vanced Research WRF (ARW; Skamarock et al. 2008;
suchsourcesexist,eradicatingthemmakesitalotless hereafter referred to as WRF model) version 3.2. The
complicated to track down remaining sources. There- WRF is a fully compressible and nonhydrostatic model
fore,understanding,identifying,andeliminatingobvi- andisthussuitableforabroadspectrumofapplications
oussourcesofmodelbiasarevitalindealingwiththe acrossabroadrangeofscales.Theprognosticvariables
imperfectmodel. are column mass of dry air, three-dimensional compo-
While making efforts to optimize a version of the nents of wind, potential temperature, and geopotential.
WeatherResearchandForecasting(WRF)modelthat Diagnostic variables (e.g., temperature, pressure, and
is to be used for the Arctic System Reanalysis (ASR; density) are derived from the prognostic variables. The
Bromwich et al. 2010), we have noticed inexplicable modelalsopermitshydrostaticruns.Inthiscase,thege-
model biases in the forecast compared with verifying opotentialisdiagnosedthroughthehydrostaticequation.
global analyses, which are persistent over the period AlthoughtheWRFmodelallowsforecaststobemadeon
andhorizontallyunvarying.Thebiasesintemperature aglobalscale(moreinformationcanbefoundonlineat
andgeopotentialheightappearattheupperpartofthe http://www.ncar.ucar.edu/feature/articles/wrf_global.php),
modeldomainandincreasewiththealtitude.Wetracked inthisstudyitisusedasalimited-areamodel.Accordingly,
down the biases and found that the initial conditions theWRFneedstoobtainitsinitialandlateralboundary
DECEMBER2012 WEE ET AL. 3909
FIG.1.Verticaldistributionofhdepthofmodellayersalong(a)thelevelindexand(b)theapproximatepressure.
Thehdepthisthedifferenceinhbetweentwoadjacentfullhlevels.Dashedlinesareforthemodelwith28vertical
layers(L28)andsolidlinesarethemodelwith55layers(L55).
conditions from other models that cover an area wider sigma-pressure vertical coordinate with the model top
thantheWRFdomain,usuallyaglobalanalysisorfore- locatedat0.1 hPa.TheERA-Interimdataalsoprovided
cast. This involves interpolating another model’s data to lateral boundary conditions forthe WRF,updatedwith
WRFgrids,andthesystematicmodelerrorswefoundare 6-h intervals. The model domain used here is a coarse-
pertinent to but not limited to this procedure. Hereafter resolution replica of the ASR domain, centered at the
theprocedureisreferredtoas‘‘initialization’’forthesake SouthPolewith30-kmhorizontalgridspacing.Wealso
ofconvenience,anditdiffersfromtheinitializationtech- usedmeansoundingsthatwereobtainedfromaveraging
niques that adjust the initial condition of a numerical the ERA-Interim for 10 Decembers during the years
modelinordertoreduceinitialimbalance. 2000–09andovertheASRdomain(hereafterreferredto
The vertical coordinate used in WRF is a terrain- asclimatology).TheERA-Interimdatausedinthisstudy
followinghydrostatic-pressurecoordinateh5(p2p)/m, isrepresentativeofexternaldatasets(XDS)fromwhich
t
wherem5p 2p isproportionaltothemassoftheairin theinitialandlateralboundaryconditionsofWRFareto
s t
averticalcolumnofthemodeldomainwithpbeingthe be obtained. A description on the other details of the
hydrostaticpressure,andp andp refertothepressures modelsettingirrelevanttothisstudyisomitted.
s t
atthemodel’ssurfaceandtop,respectively.Inthisstudy
10 hPaisusedforthepressureatthetopofmodel.The
3. Negativebiasingeopotentialheight
result we are presenting in this study is sensitive to
verticalresolution;hence,weendeavortotakeatypical ThehydrostaticequationusedinWRFis
example. As the baseline, we pick up a set of h values
thathas28fulllevels(hereafterL28).Itistheonepro- ›F
52a*m*, (1)
vided with the WRF source code as an example of h ›h
specification for tutorial purposes. To demonstrate the
dependencyofmodelbiasonresolution,wealsouse55 where F 5 gz is the geopotential with g and z being
full levels (hereafter L55), where each layer of L28 is accelerationduetogravityandgeopotentialheight,re-
halved in depth. The depth of each layer in h for both spectively;aisthespecificvolume(orinversedensity).
L28andL55isshowninFig.1. Othervariablesaredescribedinsection2.Thisequation
ToprovidetheinitialconditionfortheWRF,weused describeshydrostaticbalanceamongthemassvariables
theEuropeanCentreforMedium-RangeWeatherFore- and can be used to diagnose one mass variable from
casts(ECMWF)Re-AnalysisInterim(ERA-Interim;Dee others.ItmustbementionedthatinWRF,(1)isapplied
etal.2011)foramonth:December2006.Thedatahas only to the dry hydrostatic component of a and p, de-
aspectralT255horizontalresolution,whichcorresponds notedbysuperscript*.Therefore,forhydrostaticruns,
toapproximately79-kmspacingonareducedGaussian moisture-related perturbations are in the hydrostatic
grid.Theverticalresolutionis60modellayersonahybrid balanceseparately.Keepingthatinmind,thesuperscript
3910 MONTHLY WEATHER REVIEW VOLUME140
FIG.2.StructureoftheverticalgridoftheWRFmodelinwhich
solid(dashed)linesrepresentfull(half)hlevels.Geopotentialis
defined on full levels, while potential temperature and specific
volumearedefinedonhalflevels.Pressureisavailableonbothfull
andhalflevels.
is omitted hereafter for the sake of simplicity. The
hydrostatic equation can be rewritten in a form more
widelyused: FIG.3.Differencebetweengeopotentialheights(m)computed
ð usingtwodiscretehydrostaticequations:onmodellevelsofL28
Fk112Fk52 pk11adp, (2) (wdhaesrheedz3)aannddLz655a(rseotlhide).gTeohpeovteernttiicaallhperiogfihltesocfozm3p2utezd6iusssinhgow(3n),
pk and (6), respectively. The hydrostatic equations are integrated
upwardstartingfromthelowestleveltothemodeltop.Amean
wheresubscriptskandk11representverticalindicesof soundingoftemperature(monthlymeananddomainaveraged)is
model levels. Inthe vertical grid of WRF model that is interpolatedtothemodellevelsandthenusedastheinputforthe
hydrostaticintegrations.
staggeredasshowninFig.2,allmassvariablesarelocated
athalflevelsexceptforthegeopotential.Uponthedefi-
nitionofverticalcoordinateh,thedryhydrostaticpres- whereTisthetemperatureandR isthegasconstantof
d
sureisavailableatbothhalfandfulllevels,andpk11/25 thedryair.Thediscreteformof(5)mightbeasfollows:
(pk1pk11)/2.Seeingthewaythatthegridisstaggered,
anaturalchoiceofdiscretizationfor(2)wouldbe DF5RdTk11/2ln(pk/pk11). (6)
DF52a Dp, (3) The atmospheric temperature in normal conditions
k11/2
varieslinearlywithheight,andthenaturallogarithmof
where Dj 5 jk11 2 jk for an arbitrary variable j. As pressure is largely proportional to the height. An ex-
amatteroffact,(3)istheexactformusedinWRF.Inusual ample of this is the standard temperature lapse rate of
atmosphericconditions,atendstobeininverseproportion the U.S. Committee on Extension to the Standard At-
top(i.e.,a}p21)andthusaasfunctionofpisconvex, mosphere(COESA1976).Therefore,Tk11/2iscloseto
meaningthata(p)#lak1(12l)ak11foranyl2[0,1], thelayer-meantemperaturetakeninheightorlnp:
wherep5lpk1(12l)pk11.Thisagainmeansthatak11/2 ð
issmallerthanthemeanspecificvolumeofthelayer: T~5 1 pk Td(lnp). (7)
a5 1 ðpk adp. (4) ln(pk/pk11) pk11
p 2p
k k11 pk11
This again implies that (6) is superior to (3) in esti-
Theuseof(3)henceleadstoanunderestimatedthick-
mating the thickness. In summary, although (2) and (5)
ness. To improve accuracy of the computed thickness
are equivalent with each other and have no problem
andalsotodoawaywiththenonobservationalvariable
mathematically,discreteforms(3)and(6)differintheir
(i.e.,thespecificvolumeordensity),analternativeform
accuracy.Todemonstratethedifferencebetween(3)and
of(2)hashistoricallybeenusedinstead(e.g.,Bjerknes
(6),aclimatologyoftemperature(describedinsection2)
andSandstro¨m1910):
isinterpolatedtoaverticalgridoftheWRFmodel.We
ð thenintegrate(3)and(6)upwardstartingfromthelowest
p
F 2F 52R k11Td(lnp), (5) leveltocalculatetheheightoffullmodellevelsfromthe
k11 k d
p interpolated temperature on half model levels. Figure 3
k
DECEMBER2012 WEE ET AL. 3911
showsz32z6,wherez3andz6arethegeopotentialheights
computed using (3) and (6), respectively. The difference
betweenthegeopotentialheightsexplainsthat(3)leadsto
a significant negative bias that increases with the height.
Comparing results for L28 (dashed) and L55 (solid), we
observeastrongdependencyontheverticalresolution.
For thedry atmosphere, (6) can berewritten via the
idealgaslaw:
DF5a p ln(p /p )
k11/2 k11/2 k k11
!
120:5Dp/p
5a p ln k11/2 . (8)
k11/2 k11/2 110:5Dp/p
k11/2
FIG.4.Permillageerror(mkm21)ofDF3againstDF8,defined
ExpandingintheTaylorseries,ifjDpj/(2pk11/2),1, aasre10th0i0c(kDnFes3se2sDofFa8m)/DoFde3l,laasyfeurninctgioenopooftdepn/pti.aHlceorme,pDuFte3dawndithDF(38)
oneobtains and(8),respectively.Seetextfordetails.
" ! !
3
Dp 1 Dp
DF522a p 1 formforanoceanmodel.Equation(11)explainswhythe
k11/2 k11/2 2pk11/2 3 2pk11/2 increasedresolutionreducestheheightbiasconsiderably.
!5 !7 # Meanwhile, thehdepth used in thisstudy isshallow
1 Dp 1 Dp
1 1 1 (cid:2)(cid:2)(cid:2) . (9Þ nearthesurfaceandatthemodeltop,anddeeperinthe
5 2pk11/2 7 2pk11/2 middle troposphere (Fig. 1). This differs from typical
practicesinthepast.Forinstance,intheeraofthefifth-
FactoringDp/(2pk11/2)fromtheseries,itreads: generation Pennsylvania State University–National Cen-
" ! ter for Atmospheric Research (PSU–NCAR) Mesoscale
1 Dp 2 Model(MM5;Dudhia1993),betterverticalresolutionis
DF52ak11=2Dp 11 3 2p usuallyonlyusedforthelowerpartofthemodelsoasto
k11/2 betterresolvetheplanetaryboundarylayer,butarela-
! ! #
1 Dp 4 1 Dp 6 tivelycoarseresolutionisusedelsewhere(e.g.,Weeand
1 1 1 (cid:2)(cid:2)(cid:2) . (10) Kuo 2004). As more attention is paid to the benefit of
5 2p 7 2p
k11/2 k11/2 adding more layers to the upper part of the model at-
mosphere (e.g., Kimball and Dougherty2006), the gen-
Therefore,thefractionaldifferenceingeopotentialis
eraltrendatpresentistousedenserlayersforthehigh
" ! !
2 4 altitudes.However,thereisnotheoreticexplanationyet
DF32DF8 1 Dp 1 Dp
52 1 thatsupportsthistrend.Whileearlierstudies(e.g.,Zhang
DF3 3 2pk11/2 5 2pk11/2 andWang2003;KimballandDougherty2006)accentu-
! #
ate forecast sensitivity to the vertical resolution, (11)
6
1 Dp
1 1 (cid:2)(cid:2)(cid:2) , (11Þ entailsthat the poor resolution has adverse effects as it
7 2pk11/2 causesasignificantmodelbias.AtestwithanMM5-type
h specification showed an enormous bias in the geo-
whereF3andF8arethegeopotentialscomputedusing potentialheight(notshown).
(3)and(8),respectively.Thismeansthenegativebiasof We now consider the problem in practical WRF ini-
F3withrespecttoF8increasesathighaltitudesandas tialization and forecast. In creating initial conditions for
themodel’sverticalresolutiondecreases.Figure4shows WRF,(3)isusedtodiagnosegeopotentialheight.Unless
thepermillage(i.e.,proportionperthousand)biasofF3 any other information is added to enhance the initial
,
asfunctionofDp/pk11/2thatisagaininproportiontothe condition (e.g., data assimilation, or any physical or dy-
depthoflayerinheight.Asimilaranalysiswasdoneby namical initialization techniques to improve balances
Duthie(1946)forradiosondeobservations.Adcroftetal. among the model state variables), the diagnosed geo-
(2008) also discussed errors arising from the use of the potentialheightissupposedtobeidenticalwiththegeo-
finite-difference form of the hydrostatic equation, and potential in XDS. However, when the two geopotential
howtheycanberelievedbyinsteadusingthefinite-volume heightsarecomparedwitheachotherbyinterpolatingthe
3912 MONTHLY WEATHER REVIEW VOLUME140
FIG.5.Comparisonofbiasingeopotentialheight(m)forthe(a)initialconditionsand(b)1-dayforecasts.The
modellevelsusedherearefromL28.ThebiasisagainstERA-Interimanalysiswith(solid)andwithout(dashed)the
biascorrection.Profilesaremonthlymeananddomainaveraged.
WRF initial condition back to the grids of XDS or vice insteadof(3),the24-hforecastsshowpracticallynobias
versa, a systematic difference appears. Figure 5a shows in the predicted geopotential height (solid line in Fig.
the bias of the WRF initial condition against the ERA- 5b).The WRF modelhas a tendencytoproduce slight
InterimanalysisforL28averagedoverthedomainofthe negativebiasatthelowerpartofthemodeldomainand
WRFmodelandoverthemonth-longperiod.Theinitial slight positive bias at stratospheric altitudes. However,
conditions are generated with a 12-h interval, at every their magnitudes are negligible compared with the ini-
0000and1200UTC.Beingsolelydeterminedbythedry tialbiasduetotheuseof(3).
hydrostaticpressureonthemodellevelasshownin(11), ThebiasesassessedforL55areshowninFig.6.Asthe
the bias is horizontally uniform if the undulation of h results without the bias correction are compared
surfaceduetothemodel’stopographyisneglected.Thisis (dashed lines in Figs. 5 and 6), the doubled vertical
whythearea-andtime-averagedbiasofz3showninFig. resolutionreducesthebiastoone-fourthofL28inboth
5a(dashedline)isaboutthesameasthatshowninFig.3, theinitialconditionandforecast[notethedifferencein
forwhichasingleclimaticsoundingisconsidered.Forthis scaleofthemodelbias(xaxis)betweenFigs.5and6].
reason, the bias is by and large insensitive to the geo- This confirms the resolution dependency of the bias
graphicallocationofthemodeldomain.When(8)isused presentedin (11).However,theincreaseintheresolu-
instead of (3), the bias is almost completely removed tion is ineffective in addressing the bias. On the con-
(solidlineinFig.5a). trary, the correction (i.e., use of the revised discrete
The bias correction made in the initial condition by hydrostatic equation) eliminates the still-existing con-
using(8)doesnoteradicatethebiasintheforecast.The siderable bias in L55 almost completely. The small re-
WRF model allows the forecast to be made either in maining bias in the forecast of L55 with the bias
hydrostaticornonhydrostaticmodeatruntime.Inboth correctionreflectsthetotalforecastbiaswithcombined
cases,themodeluses(3)torelateFandawhilemaking contributions from all other dynamical and physical
the forecast. For the hydrostatic run, (3) is used to di- sourcesaswellasuncertaintiesintheverifyinganalysis.
agnose F from a in the forecast, via other prognostic When the remaining bias is compared with the bias
variables.Onthecontrary,(3)isusedtodiagnoseafrom presentedinthecasewherethecorrectionisnotmade,
predictedFforthenonhydrostaticrun.Aslongas(3)is the latter exceeds the former. Therefore, if such an
used for the model in one way or another, the geo- overwhelming bias exists, it would be impossible to
potentialheightwithrespecttothepressureisbiasedin identify,understand,andcorrectothersubtlesourcesof
thesameway.Even aftertheinitialbiaswasremoved, bias. In other words, rectifying one source of bias pro-
the bias comes back quickly in the forecast and soon videsasolidfoundationtostudyothersourcesaswell.
reaches the same level as in the case where correction Inanidealizedsimulationofahurricane,Kimballand
was not made (dashed line in Fig. 5b). By using (8) Dougherty (2006) showed for the MM5 model that the
DECEMBER2012 WEE ET AL. 3913
FIG.6.AsinFig.5,butformodellevelsofL55.
storm’sintensityissensitivetothedistributionofvertical wherePistheExnerfunction;p isareferencepressure,
0
levels.Theyemphasizedtheimportanceofwell-resolved usuallytakenas1000 hPa;andk5R /C ’0.286,with
d p
stratosphericoutflowinthestormdevelopmentandad- C being the heat capacity of the dry air at constant
p
dressed strong need for well-established guidelines for pressure. After that, the potential temperature on the
verticalresolutioninthemodel.Theirstudysuggeststhat gridsofXDSisinterpolatedtotheWRFmodel’sgrids.
anadequate numberofmodellevelshastobeused for Fromatechnicalpointofview,thepotentialtemperature
the(otherwiseassumedlessimportant)stratospherical- isnothingbut apressure-weightedtemperature.Iftem-
titudestoachievearealisticsimulationofthehurricane. peratureislaterrecoveredfromtheinterpolatedpoten-
Ourstudyindicatesthatanincreasedverticalresolution tial temperature, one observes a systematicerror inthe
partlyreducesthebiasintheheightoftheWRFmodelas temperature,whichiscausedbytheExnerfunction.
well.TheresultshowninFig.5alsosuggeststhattheuse Toexplainthis,onemayconsideralinearinterpolation
ofthereviseddiscretehydrostaticequationin(8)isvery of potential temperature from the XDS’ grids to the
effective in reducing the model bias even for a lower WRFmodel’s,withpressureastheindependentvariable,
verticalresolution. as illustrated in Fig. 7. Suppose we have a level of the
4. Warmbiasintemperature
Likeothernonhydrostaticmodels,WRFusespotential
temperature, instead of temperature, as a prognostic
variable.Themainadvantageforthisisthatthedensity,
anonobservationalvariable,doesnotexplicitlyappearin
thegoverningequations.Therefore,onemightbetemp-
tedtointerpolatethepotentialtemperaturedirectlyfrom
the grids of XDS to the WRF model’s grids in the ini-
tialization procedure. Otherwise, one has to interpolate
bothtemperatureandpressureindividually(asthepres-
sureusedforthecoordinateofWRFisdryhydrostatic) FIG.7.Geometryassociatedwithverticalinterpolationofpo-
tentialtemperature.Solidlinesrepresentlevelsofglobalanalysis,
and then must compute the potential temperature on
whilethedashedlineinthemiddledenotestheleveloftheWRF
WRFgridsfromtheinterpolatedvariables.IntheWRF modeltowhichpotentialtemperatureistobeinterpolated.Sub-
initialization, the potential temperature is precomputed scripts‘‘u’’and‘‘l’’denoteupperandlowerlevelsofthelayerofthe
ontheglobalmodel’sgrids: global analysis, respectively. The relative distances in pressure
fromtheupperandlowerlevelstotheWRFlevelaredenotedasd
(cid:2) (cid:3) u
u5TP215T p0 k, (12) taondthdel,lreevseplesc.tNivoeltye.tLhiaktewthiesew,weuigahntidnwglisstainnvdefrosrelwyerieglhattiendgstgoivthene
p
distanceinthelinearinterpolation.
3914 MONTHLY WEATHER REVIEW VOLUME140
WRFmodel(themiddledashedlevel)towhichtheXDS relative distance of the WRF level from the lower and
hastobeinterpolated.Thelevelresidesinalayerofthe upperboundsofthelayer).Aswithw 512w andp5
u l
XDS whose lower and upper bounds are designated by p 1wdp,bytakingderivativewithrespecttow foran
u l l
subscripts l and u, respectively. Therefore, the interpo- isothermal layer, it can be shown that the second term
latedpotentialtemperatureisasfollows: hasitsmaximumvalueat
u5w u 1wu , (13) 1
u u l l w 5 . (17)
l 21dp/p
u
wherew andw standfortheweightgiventoeachbound,
l u
involvedwiththeinterpolation.Notethatinthelinearin- This means that if dp/p is negligibly small then the
u
terpolation,theweightisininverseproportiontotherel- maximum error appears at half pressure level of the
ativedistanceinpressure:w 5(p2p)/dp,w 5(p2p )/dp, layer.Otherwise, the location approachesto theupper
u l l u
anddp5 p 2 p .Thetemperaturerecoveredfromthe boundfromthehalfpressurelevel.Obviously,theerror
l u
interpolatedpotentialtemperatureTuisthen due to the interpolation vanishes when the WRF level
(cid:2) (cid:3)k (cid:2) (cid:3)2k (cid:2) (cid:3)2k coincideswithoneoftheboundssinceeitherwl50or
T 5u p 5w T pu 1wT pl . (14) wu50in(16).
u p u u p l l p As the weights make it difficult to appreciate each
0
term in (16), we consider a simple case that the WRF
As pu/p 5 1 2 wldp/p and pl/p 5 1 1 wudp/p, the levelislocatedattheexactmiddleofalayer(i.e.,wl5
Taylorseriesexpansionof(14)readsas w 5 0.5). This can be accomplished by interpolating
u
! ! potential temperature from full h levels of the WRF
T 5 w T 1wT 1kw wdp T 2T modeltohalfhlevelsandthenbyrecoveringtempera-
u u u l l u lp u l tureatthelocationfromtheinterpolateduandp.Todo
(cid:2) (cid:3) (cid:2) (cid:3) so, the climatological temperature, used earlier to
k(k11) dp 2 T T
1 w w u1 l demonstratethegeopotentialbias,isinterpolatedtofull
2 u l(cid:2)p wu(cid:3) wl ! hlevelsoftheWRFmodelandthenpotentialtemper-
k(k11)(k12) dp 3 T T ature is computed there. The dp/pu for 28L, shown in
1 w w u2 l 1 (cid:2)(cid:2)(cid:2) . Fig.8a,increaseswithheightaspressurebecomeslower.
6 u lp w2 w2
u l Figure 8b shows that the second term of rhs in (16)
(15) evaluatedforthiscaseismuchlargerthanthefirstterm.
Also shown in Fig. 8b is Tu 2 T, where Tu is the re-
In the above equation, the sum of first two terms of
covered temperature from interpolated potential tem-
rhs, w T 1 wT, is the temperature that one would
u u l l peratureandTisthetemperaturedirectlyinterpolated
obtainifthetemperaturewasdirectlyinterpolatedfrom tohalfhlevels.TheTu2Tisverycloseto«T.Figure9
theXDS.Therefore,inthecontextoflinearinterpolation,
shows dependency of the temperature bias on the res-
theremainingtermsareerrorsduetointerpolatingpo-
olutionofXDS.Here,thesameclimatologicalsounding
tentialtemperatureinsteadofinterpolatingtemperature
of temperature is interpolated to levels of L55 (solid)
directly.Thetemperatureerrorcanbeapproximatedas
andL28(dashed)inordertomimictwodifferentreso-
follows:
lutionsoftheXDS.TheresultforL28isthesameasthat
(cid:2) (cid:3)
dp showninFig.8,butduplicatedforeaseofcomparison.
« ffikw w T 2T It is clear from the comparison that use of a higher-
T u lp u l
resolutionXDSreducesthebiasasshownin(16).
(cid:2) (cid:3) (cid:2) (cid:3)
k(k11) dp 2 T T Notethatthetemperatureerrorshownhereiscloseto
1 w w u1 l . (16)
2 u lp w w the largest possible value because the interpolation is
u l
donetothehalflevelofeachlayer.InactualWRFini-
In(16),higher-ordertermsaretruncated,astheyare tializations, theerror diminishes in size and its vertical
small compared with the first two lower-order terms. structuregetsintricate.Figure10showsthetemperature
Betweenthetwotermsretained,thefirsttermchanges biasesintheinitialconditionsofL28andL55duetothe
its sign depending on the vertical thermal gradient. u interpolation, which are against the ERA-Interim
However, it will later be shown that the first term is analysis used for initialization and averaged over the
much smallerthanthesecond, whichispositive allthe modeldomainandoverthemonth-longperiod.Differ-
time.Thesizeofthesecondtermismainlydetermined entfromwhatisshowninFigs.8and9,thebiasinthe
by the position of the WRF level in the layer (i.e., the initial conditions does not increase monotonically with
DECEMBER2012 WEE ET AL. 3915
FIG.8.(a)Verticalprofileofdp/puonlevelsofL28.(b)Comparisonofthefirst(solid)andsecond(dashed–dotted)
termsinrhsof(16).Heavygraylinerepresentsthetotaltemperatureerrorthatincludesallerrorterms(exceptforthe
firstterm)inrhsof(15),whichisagainequivalentwithTu2T(seetextfordetails).Themeansoundingoftem-
perature(monthlymeananddomainaveraged)interpolatedtothefulllevelsofL28isusedtoevaluatetheterms.
ThetotalerrorrepresentsthemaximumpossibletemperaturebiasforL28.
the height. Instead noticeable zigzags appear in both removedandthepotentialtemperaturedoesnotpossess
resolutions,causedbyalternatingthedistanceofWRF anysystematicerrors.
modellevelstothelevelsoftheglobalanalysis.Thatis
to say, if a certain level of WRF is close to one of the
5. Summaryandconclusions
levels of the global analysis, the bias is small. On the
contrary,iftheWRFmodellevelapproachesthemiddle Inthecourseoftheworkthatmotivatedthisstudy,we
of a layer of the global analysis, the bias gets bigger. noticed considerable systematic differences between a
Anotherfeatureworthmentioningisthatthebiasesof WRF forecast and the verifying global analysis. How-
L28andL55areaboutthesameinsizeregardlessofthe ever,wewerenotsureabouttheexistenceofthebiases
WRFresolution.Thereasonforthisisthatdp/p in(17) presented in this study until they were also found in
u
referstotheparameteroftheglobalanalysis.Thisagain the validation of initial conditions. Since generating a
indicates that the bias is irrelevant to the resolution of WRF initial condition from another model’s data is
the WRF model and can be reduced by using higher- considered to be a set of interpolation procedures, the
resolution XDS. The zigzags in the bias suggest that apparentbiasesintheinitialconditionagainstthevery
the interpolation of the potential temperature brings input data used for the initialization were surprising
artificial oscillations in the initialized temperature. enoughtoleadustolookintothedisparity.Evenafter
Differently from the bias in geopotential height, the theirexistencewasacknowledged,itwasnoteasyforus
temperature bias is caused only in the initialization. torelatetheperceivedbiasestotheactualerrorsources
The bias in the initial condition remains the same in since the procedure and assumption behind the biases
the forecast (not shown). The bias is caused by inter- [i.e.,thediscretehydrostaticequation(3)andthedirect
polating potential temperature despite it being a con- interpolation of potential temperature to WRF grids]
served variable in adiabatic flow and thus preferred in areseeminglyproperandbarelyconsideredtobeerro-
manymodelformulationsthantemperature.Thebiasis neous.Eventually,wehaveidentifiedandcorrectedtwo
remedied simply by interpolating temperature and sizeablebiasesintheWRFmodel.Theyareanegative
pressureindividually, and then by computingpotential bias in the geopotential and a warm bias in the tem-
temperature from the interpolated variables. By doing perature,appearingbothintheinitialconditionandin
so, the warm bias in the initial condition is completely the forecast. The biases increase with height and thus
3916 MONTHLY WEATHER REVIEW VOLUME140
FIG.9.Dependencyofthetemperaturebiasontheresolutionofglobalanalysis.Here,theclimaticsoundingof
temperatureisinterpolatedtolevelsofL55(solid)andL28(dashed)inordertomimictwodifferentresolutionsof
theglobalanalysis.ResultsforL28areasinFig.8,butduplicatedheretohighlighttheresolutiondependency.(a)As
inFig.8a,butfortheresultforL55isincluded.(b)Themaximumpossiblebiasintemperature,Tu2T.
manifest themselves at the upper part of the model weighting(i.e.,theExnerfunction).Interestingly,through
domain. ananalyticalderivation,itisfoundthatthemagnitudeof
The geopotential bias is caused by the discrete hy- thetemperaturebiasisdependentontheverticalresolu-
drostatic equation used in WRF, which disregards the tion of the XDS rather than on that of the WRF model
innate vertical structure of the specific volume (or in- itself.Thismeansthatuseofahigher-resolutionXDSfor
versedensity)thatisunequivocallyconvexwhenviewed theinitializationreducesthebias.Thesizeofthebiasfor
inthemodel(orpressure)coordinate.Astheequationis eachhalflevelofWRFisalsodependentontherelative
used not only for generation of the initial and lateral distancebetweentheWRFleveltothetwonearestlevels
boundary conditions but also for model integration as oftheXDS.ThebiasgrowsastheWRFlevelapproaches
a part of the governing equations, the bias quickly re- the middle of the layer between the two levels, while it
vivesintheforecastevenafteritwasremovedfromthe vanishes when the WRF level coincides with one of the
initial condition. Hydrostatic and nonhydrostatic runs levels. The distance dependency introduces artificial os-
areverysimilarwitheachotherinthebias.Magnitude cillations in the model temperature. By interpolating
ofthebiasisanalyticallyfoundtobeproportionaltothe temperatureintheinitializationandthencomputingpo-
squareofthethickness(inheight)ofthemodellayers. tential temperature on the WRF grids, the temperature
For the vertical levels used in this study, the negative biasisremoved.
bias is foundto be about 450 m for L28 and 120 m for As to explaining and correcting the model bias, our
L55 at the model’s top. The bias is fixed by using an approach is analytical. While the validation conducted
alternative form of the discrete hydrostatic equation. in thisstudy bymeansof comparingcorrectedand un-
ThevalidationagainstERA-Interimanalysisillustrates corrected model states with verifying global analyses
thatthecorrectionalmostcompletelyremovesthebias offersademonstration,ourfindingsinthisstudyarenot
intheinitialconditionandintheforecastregardlessof restrictedbytheuncertaintiesassociatedwitherrorsin
theverticalresolutionusedfortheWRFmodel. theverificationdatathataremoreorlessunavoidable.
The WRF model uses the potential temperature as Previousstudiesdonotprovideaconsensusastowhether
a prognostic variable. Accordingly, the potential tem- abiascorrectioncanimproveforecastskillinmeasures
perature is interpolated from the grids of XDS to the otherthanthebias.Someshowedpositiveimpactofbias
WRFgridstoprovidetheinitialcondition.Thisleadsto correctiononrandomcomponentsofforecasterror(e.g.,
a marked bias in the temperature due to the pressure Johansson and Saha 1989; Yang and Anderson 2000;