La taille de l'ensemble d'ébauches détermine la précision avec laquelle cet en- semble représente la distribution de probabilité des ébauches. Elle in uence égale- ment la dispersion de cet ensemble. Pour les situations FOG et NEAR-FOG, la taille de l'ensemble a cependant peu d'in uence sur la qualité des conditions initiales de température et d'humidité spéci que. Les scores des expériences ENKF8, ENKF16 et ENKF32 ne montrent pas de tendance à l'amélioration avec la taille de l'ensemble. Ceci est dû au fait que l'algorithme d'in ation adaptative des variances/covariances compense e cacement le manque éventuel de dispersion de l'ensemble d'ébauches. En eet, le facteur d'in ation diminue lorsque la taille de l'ensemble augmente.
Lorsque l'on utilise des observations réelles, ce mécanisme de compensation fonc- tionne moins bien et la taille de l'ensemble a un impact important sur la qualité de la prévision des évènements LVP. Le taux de détection et le pseudo-taux de fausse alarme sont tous deux fortement corrélés avec la taille de l'ensemble, surtout pour les échéances inférieures à 4 heures.
II Article
Ensemble Kalman lter data assimilation in a 1D numerical model used for fog forecasting
Soumis le 22 juin 2009 au Monthly Weather Review, accepté le 13 novembre 2009.
Generatedusingversion3.0 oftheo ial AMSLT E
Xtemplate
Ensemble Kalman lter data assimilation in a 1D numeri al model
used for fog fore asting
Samuel Rémy
∗
and Thierry Bergot
∗
Corresponding author address: Samuel Rémy, CNRM/GAME, 42Av. Coriolis, 31057Toulouse Cedex
1,Fran e.
rateandupdatedfogandlow loudfore asts. COBEL-ISBA,aboundarylayer1Dnumeri almodel,
has been developed for the very short term fore ast of fog and low louds. This fore ast system
assimilates lo al observations to produ e initial pro lesof temperature and spe i humidity. The
initial onditions have agreat impa ton theskillofthefore ast.
In this work we rst estimated the ba kground error statisti s. They vary greatly with time,
and ross- orrelationsbetween temperature andhumidityintheba kground were signi ant. This
led to the implementation of an ensemble Kalman lter (EnKF) within COBEL-ISBA. The new
assimilationsystemwasevaluated withtemperature andspe i humiditys oresaswellasinterms
of itsimpa t onthequalityoffog fore asts. Simulated observations were used,and fo usedonthe
modeling oftheatmosphere beforefog formation, andalso onthesimulationofthelife- y le offog
and low louds. For both situations, the EnKF brought a signi ant improvement in the initial
onditionsand thefore asts. Thefore astoftheonsetandburnotimesoffogswasalsoimproved.
The EnKF was also tested withreal observations and gave goodresults. The size of theensemble
didnothavemu himpa twhensimulatedobservationswereused,thankstoanadaptive ovarian e
Low visibility onditions often ause problems at many international airports. Su h on-
ditions may redu e the landing/takeo tra by a fa tor of two, leading to delays or even
an ellations of ights. This is why a urate fore asts of these onditions have be ome an
important issue. Ea h airportde nes a set of visibility and eiling thresholds below whi h
safety pro edures, alledLowVisibilityPro edures (LVP),areapplied. AtParis-CharlesDe
Gaulle airport, the threshold values are set at 600m forvisibilityand 60m for the eiling.
Various approa hes are employed to fore ast low visibility onditions. For airports lo ated
in atterrain,1D modelsare suitableforthe now astingofradiationfogevents(Bergot and
Guédalia (1994a), Bergot and Guédalia (1994b)). They are urrently used in real time to
fore astfogatthelo als ale(e.g. Clark(2002),Clark(2006),Herzeghetal.(2003)). The1D
boundary layer modelCOBEL, oupledwith the land surfa e s heme ISBA asdo umented
in Bergot et al. (2005) has been in operational use sin e 2005 at Paris-Charles de Gaulle
airportin Fran eto provideestimated times forthe onset and liftingof LVP onditions.
Fog is a phenomenon that evolves at small spatial and time s ales. Its 1D modeling in-
volves intera tions between many parameterizations: turbulen e, mi rophysi s, radiative
s heme and surfa e-atmosphere ex hanges. This highlightsthe importan eof working with
a urate initial onditions: the quality of the COBEL-ISBA fore asts is dependant on the
initial onditions (Bergot and Guédalia(1994a), Roquelaure and Bergot(2007), Rémy and
Bergot(2009)). This paperaims to improveit by usingan ensembleKalmanFilter (EnKF,
Evensen(1994)andEvensen(2003)). Theoreti ally,ensemble ltersareanadequatemethod
for taking the atmosphere variability into a ount in the assimilations heme of non linear
systems, su h as boundary-layer 1D models. They have re ently been implemented invari-
ous o eani and atmospheri models (Houtekamer et al. (2005), Zhang (2005), Ha ker and
Snyder (2005),Ha ker and Rostkier-Edelstein (2007) amongothers). Here, animplementa-
tion of this method for 1D fog fore asts is presented, using both model simulated and real
observations.
Theframeworkofthisstudy isoutlinedinse tion2. Twosetsofsimulatedobservationswere
reated: one with mostly lear-sky onditions at the initialization, to study the formation
of fog, and the otherwith frequent o urren e of fog and low louds. Se tion3 presents the
setup of the EnKF and se tion 4 shows the results with the two sets of simulated observa-
tions. Next,inse tion 5,wefo us onresults obtained fromasystem usingreal observations
instead of simulatedones. In se tion 6,the impa t of the ensemble size onthe performan e
oftheEnKF,forsimulatedand realobservations,isdis ussed. Finally,se tion7summarizes
1) The model
COBEL-ISBA onsists of the ouplingof the high resolution atmospheri boundary layer
1DmodelCOBEL(Bergot(1993),BergotandGuédalia(1994a)andb)withthe7-layerland-
surfa e s hemeISBA(NoilhanandPlanton (1989),Boone(2000)). Tobeable toadequately
fore astradiativefogevents,itpossessesahighverti alresolution: 30levelsbetween0.5and
1360 m, with 20levels below 200 m. The physi al parameterizationsused in COBEL-ISBA
onsists of :
• a turbulent mixings heme with a 1.5 order turbulen e losure that uses a prognosti
turbulentkineti energy(TKE)equation. Themixinglengthdiers forstable(Estour-
nel(1988)) and for neutralor unstable onditions (Bougeault and La arrere (1989)),
• a warm mi rophysi als heme adapated tofog and low loudsin temperateregions,
• Detailed longwaveand shortwave radiationtransfer s hemes.
COBEL-ISBA is run atone hourintervals and providesup toeighthours of LVP fore asts.
Theinputsofthemodelaretheinitial onditionsandmesos ale for ings. Mesos alefor ings
(i.e. geostrophi wind, horizontal adve tion and loud over above the model olumn) are
given by the Numeri al Weather Predi tion (NWP) modelALADIN (AireLimitée Adapta-
tion Dynamique Développement INternational, http://www. nrm.meteo.fr/aladin).
2) The assimilation s heme
Theinitial onditionsare given byatwo-stepassimilations heme,usinglo alobservations
(Bergot et al. (2005)). The observation system used at Paris-Charles de Gaulle airport
is designed to provide up-to-date information on the state of the surfa e boundary layer
temperatureandmoisture. Itin ludesaweatherstationwhi hprovides2mtemperatureand
humidity, visibility and eiling, a measurement mast that gives temperature and humidity
observationsat1,5,10and30metres,radiative uxes(shortwaveandlongwave)observations
at 2and 45 metresand soil temperatureand water ontent between the surfa e and -40 m.
The assimilationsystem uses informationfroma rstguess orba kground (i.e. aprevious1
hourCOBEL-ISBAfore ast),lo alobservationsandpro lesfromtheALADINNWPmodel,
to generate aBest Linear Unbiased Estimator(BLUE) for initial onditions of temperature
and spe i humidity:
xa= xb+ K∗ (y o− Hx b) (1) K=BH T (HBH T +R) -1 (2) In this equation, x a is the analysis, x b
is the rst guess or ba kground, y
o
are the observa-
tions. Kis the Kalmangain that a omplishesthe observation weighting. B and R are the
grid to the observation grid. As the dimension of the system is low, matri es an be ex-
pli itly inverted and there is no need for a variational algorithm. The ALADIN data are
taken as observations for the upper levels of the modeldomain, soa part of R orresponds
to the error varian es and ovarian es of the ALADIN pro les. The ovarian es between
the other observations are zero. In the operational setup, the error statisti s are imposed
arbitrarily toallow the initialpro le tobe lose to observations near the surfa e and loser
tothe ALADIN pro lesabove. The ross- orrelationof temperatureand humidity errors in
the ba kgroundarezero : theoperationalassimilations hemeismonovariate. Ourobje tive
is to ompute owdependent ba kgrounderror statisti sand to builda multivariateassim-
ilation s heme by taking into a ount the ross- orrelations of temperature and humidity
errors in the ba kground.
When alayerof loudisdete ted,anadditionalstepusesaminimizationalgorithmtogether
withmeasurementofradiative uxes atthegroundandat45mtoestimate loudthi kness.
The radiations heme of COBELis used to ompute the modelled radiative uxes at2 and
45m, using dierent initialthi knesses of the fog layer. The best estimate of the initialfog
thi kness isthe onethatminimizestheerrorbetweenmodelledandobservedradiative uxes
(see Bergot et al. (2005) for more details). The relative humidity pro le is then modi ed
within the saturated layer.
Thesoiltemperatureandwater ontent pro lesusedtoinitializeISBAare obtaineddire tly
by interpolationof soilmeasurements.
b. Simulated observations
The Observing System SimulationExperiment (OSSE)is adequate tostudy the a ura y
of an assimilation s heme (e.g. Huang et al. (2007)). It onsists of generating pseudo-
observationsbyaddingperturbationstoareferen erunofthemodel. Thepseudo-observations
are thenassimilated,andtheinitialstateand fore ast anbe omparedtothereferen erun.
The advantages of this methodare :
• The same physi al pro esses are underlyingboth observations and simulations;whi h
leads to the fa t that there are no modelling errors. The only sour e of error when
using simulated observations are the initial onditions, whi h explains why they are
used often in data assimilation studies. The errors in the initial onditions originate
only in the observations and rst guess errors, themselves originating from errors in
initial onditions propagated by the previous fore ast. The la k of observations for
ertain parameters (e.g. the thi kness or water ontent of a loud layer) does not
allowthe assimilations heme toentirely orre tthe errorsof the rstguess eld. The
quality of initial onditions thus depends solely on the observations used and on the
assimilations heme.
• TheframeworkprovidessimulatedobservationfortheentiredomainofCOBEL-ISBA.
• Lastly,itispossibleto reatealargevarietyofobservationsetsthat a ommodateour needs for evaluationpurposes.
is thus entirely diagonal.
Two sets ofsimulatedobservations : one forthe study of lear-sky nightsandof shallow-fog
situations(NEAR-FOG), and the otherfor the study of frequent are deep fogs (FOG).
1) The NEAR-FOG situation
Simulated observations orresponding to lear-sky and shallow-fog situations were pro-
du ed. This observation set will be referred to as NEAR-FOG hereafter. 15 days of sim-
ulated observations were generated, during whi h no fog o urred for the rst 10 nights.
Shallow fog situations developed for the remaining ve nights. Their thi knesses did not
ex eed 10 m. 21 hours of Low Visibility Pro edure (LVP) onditions were observed for
this situation. The skies above the model olumn were entirely lear whi h ensured strong
night-time ooling. Figure 1 shows the true temperatureat 1m, and orresponding liquid
water path. Close to ground-level,the dailyhighs lay inthe 20-22 C rangewhile the lows
were around 8-9 C. Day and night relative humidity varied greatly from 30% to 100%,
orresponding to typi al onditions observed during autumnand winter overland.
Figure 2 show the mean Root Mean Square Error (RMSE) and the mean bias of the fore-
asted temperatureand spe i humidity versus fore ast time and altitude,when using the
operationalsetup. Thein uen eoftheobservations anbeseenbythelowervaluesofRMSE
at initializationtime below 50m, espe ially for temperature. For both temperature ( gure
2 ) and spe i humidity ( gure2a), most of the in reaseof the RMSE o uredduring the
rst two hours of fore ast time. For spe i humidity,the maximum of RMSE is always at
the surfa e, whereas fortemperature,the RMSEnolonger showed largedieren es between
the lower and upper part of the domain after 4h of fore ast time. The analysis is nearly
unbiasedforboth spe i humidityandtemperature( gures2bandd). The spe i humid-
ity bias be ame positive with fore ast time, with a maximum lose to the ground. A old
bias appeared rapidly for the fore asted temperature 2d) and in reased regularly with the
fore ast time, with maxima lose tothe ground-leveland above the top of the mast (30m).
2) The FOG situation
This situation was designed to study the fog and low loud life y le. Fogand low louds
o urred during many nights of the 15-day observation set, hereafter referred to as FOG,
be ause of high moisture ombined with strong night-time ooling due to lear skies above
the model olumn. Figure3shows thetrue temperatureobservationsat1mandthe true
liquidwater ontentintegrated overthe model olumn. In total,98hoursof LVP onditions
were observed in these 15 days, with fog o urren e on 11 nights. Stratus also o ured
in the upper part of the model olumn on days 7 and 8, whi h were not ounted as LVP.
Various fogsituationso urred, from shallow early-morningfog tofoglayers more than 200
m thi k.
Figure 4shows the mean RMSE and bias of temperature and spe i humidity when using
NEAR-FOGoverthewholedomain. This isduetoerrorsinthe initializationof fogandlow
louds. The in rease of RMSE with fore ast time is slower for FOG than for NEAR-FOG,
andaftertwohoursoffore ast,the values losetothesurfa eare similarforbothsituations.
The RMSE above 100m remain signi antly higher for FOG than for NEAR-FOG, for all
fore ast times. The spe i humidity bias ( gure 4b) is lose to zero for all fore asts time
below 50m whereas it is negative above that height. For all heights, the spe i humidity
bias doesn'tvarymu hwithfore ast time. The RMSEoffore astedtemperature( gure4 )
in reases mu h faster inthe lowerpart ofthe domainfor FOGthan forNEAR-FOG( gure
2 ) andrea hesamaximumof1Kafter7hoursof simulation. Amaximumappearsbetween
50and 150mof altitude,whi h orresponds tosituationswherethe fore astedheightof the
fog is dierent from the simulated observations. The inversion at the top of the fog layer
signi antly in reases theerror if the fore asted loud layerthi kness isnot the sameas the
observed one. The temperature bias ( gure 4d) also in reases with fore ast time, with a
maximum at the surfa e.
3. Setup of the ensemble Kalman lter
a. Diagnosis of ba kground error orrelations
In the operational setup, the ba kground error orrelations were xed in time and in the
verti aldire tion. Wediagnosedthese orrelations,usingtheNMC(NationalMeteorologi al
Center) method(Parrishand Derber (1992)). This methodapproximatedthe fore ast error
from aset of dieren es between several fore asts validatthe sametime. Figure5 presents
the temperature and temperature-spe i humidity orrelations at analysis times of 6 and
15UTC, averaged over the November 1st-January 31st period. The error statisti s followa
markeddiurnal y le,withhighervaluesduringtheday, orrespondingtothedevelopmentof
a mixed boundary layer. The ross- orrelations between temperature and spe i humidity
errors are higher during the nightthan during the day. At 6am and below100m, the ross-
orrelations are nearly symmetri ,i.e. the orrelationbetween the ba kground temperature
error at 10 m and humidity error at 50 m, for example, is lose to the orrelation between
the ba kground humidity and temperatureerrors at10 and 50m respe tively.
This analysis showed that we needed to build a more adaptive assimilationsystem able to
estimate the ow-dependent ba kground ovarian esfor ea h run of the model and to take
the ross- orrelations between temperature and humidity into a ount. The EnKF was a
simple method toa hieve that.
b. Constru tion of the ensemble
We implemented the perturbed observations on guration of the EnKF, as it in reases
the spread of the ensemble as ompared to other on gurations. For more details on this,
refer to Burgers et al. (1998). This version of the EnKF onsists in using an ensemble of
initial onditions that is built by adding white noise perturbations tothe observations and
i
weather stations. As ALADIN pro les were also used as observations, their error statisti s
were estimated using amethodproposed by Desroziers et al.(2005) with:
σ2 o = (Hx b− y o) ∗ ( Hx a− y o) (3)
Soilobservationsandmesos alefor ing(i.e. geostrophi wind,adve tionoftemperatureand
humidity) error statisti s were provided by a sensitivity study arried out by Roquelaure
and Bergot (2007).
The modelhas 30levelsfor two ontrolvariables temperatureand spe i humidity (liquid
water ontentisassimilatedseparately), whi hmakesthatthe dimensionofthe modelspa e
in the assimilations heme israther small: 60.
. Validation of the prior ensemble
In this se tion, the realism of an ensemble of 32 ba kgrounds or priors is assessed. This
assessment was arried out using rank histograms, also known as Talagrand histograms
(Talagrand et al. (1997), Hamill (2001)). It onsists of ranking the verifying data in the
sorted ensemble. Rank histograms are generated by repeatedly omparing the rank of the
veri ation (usually an observation) relative tovalues froman ensemble sorted fromlowest
tohighest. Theyprovidearapiddiagnosisoftheensemblereliability. Ala kofvariabilityin
the ensemble willgive a U-shaped rank-histogram, while a onvex shape indi atesthat the
observations are most of the time en ompassed ina subset of the a tual ensemble, i.e. that
theensemblespreadistoolarge. Skewed histogramsindi ateabiasintheensemble,i.e. that
the ensemblemean is biased. A at shape implies equiprobability of the verifying data and
of the ensemble, hen ethat the ensembleis reliable. A ording toHou etal. (2001),arank
histogram an be de ned as at if the adjusted missing-rate is lower than 10%. In order
to ompute the adjusted missing-rate, the missing-rate, whi h is the sum of the relative
frequen ies of the two extreme (the rst and the last) ategories, is rst omputed. The
adjusted missing rate is then de ned as the dieren e between the expe ted missing-rate