Use of observations in data assimilation Gérald Desroziers Météo-France, Toulouse, France Outline Introduction � Optimizing observation error statistics � Ensembles based on a perturbation of observations � Impact of observations on analyses and forecasts � Conclusion and perspectives � 1
Synops and ships Buoys Data coverage 05/09/03 09–15 UTC (courtesy J-.N. Thépaut) Radiosondes Pilots and profilers Aircraft ATOVS Satobs Geo radiances SSM/I Scatterometer Ozone Satellites (EUMETSAT) 2
In 2007, ECMWF uses ~ 40 different satellite data sources Number of satellite sources used at ECMWF 55 50 45 AEOLUS SMOS 40 TRMM CHAMP/GRACE COSMIC 35 METOP MTSAT rad MTSAT winds 30 JASON GOES rad No. of sources METEOSAT rad 25 GMS winds GOES winds METEOSAT winds 20 AQUA TERRA QSCAT 15 ENVISAT ERS DMSP 10 NOAA 5 0 1996 1997 1998 1999 2000 2001 2002 2003 2004 2005 2006 2007 2008 2009 Year (courtesy J-.N.Thépaut, ECMWF) General formalism Statistical linear estimation : � x a = x b + δ x = x b + K d = x b + BH T ( HBH T +R ) -1 d , with d = y o – H ( x b ), innovation, K, gain matrix , B et R, covariances of background and observation errors, H is called « observation operator » (Lorenc, 1986), � It is most often explicit , � It can be non-linear (satellite observations) � It can include an error , � Variational schemes require linearized and adjoint observation operators, � 4D-Var generalizes the notion of « observation operator » . � 3
Statistical hypotheses Observations are supposed un-biased: E( ε o ) = 0. � If not, they have to be preliminarily de-biased, � or de-biasing can be made along the minimization � (Derber and Wu, 1998; Dee, 2005; Auligné, 2007). Observation error variances are supposed to be known � ( diagonal elements of R = E( ε o ε oT ) ). Observation errors are supposed to be un-correlated : � ( non-diagonal elements of E( ε o ε oT ) = 0 ), but, the representation of observation error correlations is also � investigated (Fisher, 2006) . Outline Introduction � Optimizing observation error statistics � Ensembles based on a perturbation of observations � Impact of observations on analyses and forecasts � Conclusion and perspectives � 4
A posteriori diagnostics Is the system consistent? � We should have � E[J( x a ) ] = p, p = total number of observations, but also � T R i E[J o i ( x a ) ] = p i – Tr( R i -1/2 H i A H i -1/2 ), p i : number of observations associated with J o i (Talagrand, 1999) . Computation of optimal E[J o i ( x a ) ] by a Monte-Carlo procedure is � possible. (Desroziers and Ivanov, 2001) . Application : optimization of R ∙ ∙ One tries to obtain ∙ ∙ ∙ ∙ E[J o i ( x a )] = (E[J o i ( x a )]) opt . ∙ ∙ by adjusting the σ o i ∙ ∙ Optimization of HIRS σ o ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ (Chapnik, et al, 2004; Buehner, 2005) 5
Diagnostics / observations y o d = y o – H ( x b ) d oa = y o – H ( x a ) = ( I - HK ) d d oa d ab = H ( x a ) – H ( x b ) = HK d d ε o E[ d oa d T ] = ( I-HK) E[ d d T ] = R x a E[ d ab d T ] = HK E[ d d T ] = HBH T ε a d ab < ε,ε ’ > = E[ ε ε ’ ] x b x t ε b (Desroziers et al, 2005) Implementation in 4D-Var For any subset i with p i observations, simply compute = 2 ab T d d σ b ( / ) ( ) p i i i i pi ∑ = − − a b o b ( y y )( y y ) / p j j j j i = j 1 and = 2 d oa T d o σ ( ) ( ) / p i i i i pi ∑ = − − o a o b ( y y )( y y ) / p j j j j i = j 1 This is nearly cost-free and can be computed, a posteriori, over one or several analyses. 6
Implementation in 4D-Var U(TEMP) Q(TEMP) Computation over four 6h 4D-Var analyses (one day) Outline Introduction � Optimizing observation error statistics � Ensembles based on a perturbation of observations � Impact of observations on analyses and forecasts � Conclusion and perspectives � 7
Ensemble of perturbed analyses � Simulation of the estimation errors along analyses and forecasts. � Documentation of error covariances – over a long period (a month/ a season), – for a particular date. (Ehrendorfer, 2006) (Evensen, 1997; Fisher, 2004; Berre et al, 2007) Ensembles based on a perturbation of observations The same analysis equation and (sub-optimal) operators K and H are involved in the equations of x a and ε a : x a = ( I – KH) x b + K x o ε a = ( I – KH) ε b + K ε o The same equation also holds for the analysis perturbation: e a = ( I – KH) e b + K e o , with e o = R 1/2 η and η a vector of random numbers Covariance matrix of analysis error: P a = E( e a e aT ) = ( I – KH) E( e b e bT ) ( I – KH) T + K E( e o e oT ) K T 8
Background error standard-deviations Over a month Vorticity at 500 hPa For a particular date 08/12/2006 00H 6 member ensemble Vorticity at 500 hPa Validation/tuning of ensemble variances � Background errors e b can be projected to observation space by applying the observation operator: H ε b . � Using several e b , ensemble variances in observation space can be computed and compared to - variances of innovations (d) minus variances of observation errors - or background error variances given by statistics of d ab x d cross products 9
Ensemble / diagnosed bg error std-dev HIRS 5 (28/8/6 00h) Ensemble (6 members) Diagnosed from d ab x d cross-products Outline Introduction � Optimizing observation error statistics � Ensembles based on a perturbation of observations � Impact of observations on analyses and forecasts � Conclusion and perspectives � 10
Measure of the impact of observations Total reduction of estimation error variance: � r = Tr( K H B ) Reduction due to observation set i : � r i = Tr( K i H i B ) Variance reduction normalized by B : � DFS = Tr( K i H i ) = E[J o r i i ( x a )] Reduction of error projected onto a variable/area: � P = Tr( P K i H i B P T ) r i Reduction of error evolved by a forecast model: � PM = Tr( P M K i H i B M T P T ) = Tr( L K i H i B L T ) r i (Cardinali, 2003; Fisher, 2003; Chapnik et al, 2006) Randomized estimates of error reduction on analyses and forecasts It can be shown that r = L K H B L T tr ( ) i i = H B L T L K tr ( ). i This can be estimated by a randomization procedure: ∑ − ≈ δ δ B T L K o y o T R 1 H L y r ( ) ( ) j j i i i i j ∑ − ≈ δ δ o T 1 * ' a y x 1 / 2 T / 2 ( ) R L L ( ) H B B i j j i j δ y o ) is a vector of observation perturbations and where ( j δ a ) x the corresponding perturbation on the analysis. ( j (Fisher, 2003; Desroziers et al, 2005) 11
Degree of Freedom for Signal (DFS) TEMP_U lat>20° AMSU -20°<lat<20° SATOB lat<-20° 3000 TEMP_T AIREP_U 2000 TEMP_Q PILOT AIREP_T 1000 0 (Chapnik et al, 2006) Error variance reduction % of error variance reduction for T 850 hPa by area and observation type 12
Conclusion and perspectives Observation operators allow the use of a wide range of observations � Statistics on observation errors: � – variances not perfectly known, but may be tuned by using optimality criteria Simulation of analysis and background errors: � - can be obtained by ensembles based on a perturbation of observations - variance of ensembles might be validated in the space of observations Measure of the impact of observations: � – how observations contribute to the reduction of the uncertainty in the analyzed and forecast state? – which are the most useful observations? – observation impacts are by-products of ensembles based on a perturbation of observations 13
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