Ensemble of Data Assimilations and uncertainty estimation

Transcription

1 Ensemle of Data Assimilations and uncertainty estimation Massimo Bonavita ECMWF Acnowledgments: Lars Isasen, Mats Hamrud, Elias Holm, Slide 1 Mie Fisher, Laure Raynaud, Loi Berre, A. Clayton

2 Outline Why do we need an Ensemle of Data Assimilations? Sequential DA methods and Non-Sequential DA methods Hyrids methods: the est of oth worlds? Use of EDA variances in a hyrid DA Use of EDA covariances in a hyrid DA Conclusions and perspectives Slide 2 Slide 2

3 Why do we need an EDA? A crash course in Data Assimilation! DA is the process through which all the availale information is used to estimate as accurately as possile the state of the atmospheric or oceanic flow (Talagrand, 1997) A Bayesian inference prolem (Lorenc, 1986; Wile and Berliner,2007) If X is the state and Y our data the full proaility model can e factored as: which can e written (Bayes Rule): p( x, y) p( x y) p( y) p( y x) p( x) p( x y) p( y x) p( x) p( y) Slide 3 Slide 3

4 Why do we need an EDA? p( x y) p( y x) p( x) p( y) i.e., in order to infer the distriution of the state given the data (posterior distriution, p(x y)), we need only form the product of the distriutions of measurement errors (data distriution, p(y x)) and our prior nowledge aout the state (prior distriution, p(x)). The marginal distriution p( y) p( y x) p( x) dx can e thought of as a normalising constant. Slide 4 Slide 4

5 Why do we need an EDA? Let us introduce the time dimension: we want to estimate a set of states X 0:t =[X 0,X 1,..,X t ] given all the oservations over the same time interval Y 1:t =[Y 1,Y 2,..,Y t ], i.e. p(x 0:t y 1:t ) p(y 1:t x 0:t )p(x 0:t ) Two hypotheses are commonly introduced: a) A Marov assumption on the prior distriution, i.e. p(x 0:t )=p(x 0 ) p(x 1 x 0 ) p(x t x t-1 ) ) Statistical independence of the oservations: This leads to: p(y 1:t x 0:t )=p(y 1 x 1 ) p(y t x t ) Slide 5 p(x 0:t y 1:t ) p(x 0 ) p(x 1 x 0 ) p(y 1 x 1 ) p(x t x t-1 ) p(y t x t-1 ) Slide 5

6 Why do we need an EDA? p(x 0:t y 1:t ) p(x 0 ) p(x 1 x 0 ) p(y 1 x 1 ) p(x t x t-1 ) p(y t x t-1 ) This form naturally leads to a sequential algorithm, i.e., when new oservations are availale the state is updated from the previously availale estimate. In real time applications we are mainly concerned with the Filtering prolem: Knowing p(x t-1 y 1:t-1 ) how does a new atch of oservations Y t change our estimate of the state? Two step procedure: 1. Compute the forecast distriution at time t (forecast step) : p(x t y 1:t-1 ) = p(x t x t-1 ) p(x t-1 y 1:t-1 )dx t-1 2. Compute the analysis distriution (analysis step) : Slide 6 p(x t y 1:t ) = p(x t y t,y 1:t-1 ) p(y t x t,y 1:t-1 )p(x t y 1:t-1 ) = p(y t x t ) p(x t y 1:t-1 ) Slide 6

7 Why do we need an EDA? For Gaussian error distriutions and linear model and oservation operators we recover the Kalman Filter (KF) equations: x t = M t-1,t x t-1 + η t-1,t η~ N(0,Q t-1,t ) (1) y t = H t x t + ε t ε t ~ N(0,R t ) (2) Models (1) and (2) give the prior p(x t x t-1 ) and data p(y t x t ) distriutions, so that the forecast distriution x t y 1:t-1 ~ N(x t t-1,p t t-1 ): x t t-1 = M t-1,t x t-1 (3) P t t-1 = M t-1 t P t t-1 M T t-1 t + Q t-1 t (4) The analysis distriution x t y 1:t ~ N(x t t,p t t ) is given y: x t t = x t t-1 + K t (y t - H t x t t-1 ) (5) P t t = (I - K t H t )P t t-1 (6) Slide 7 K t = P t t-1 H T t (R t + H t P t t-1 H T t) -1 (7) Slide 7

8 Why do we need an EDA? We may e interested in the distriution p(x t y 1:T ) for t=1,,t, i.e. we want to estimate the state using oservations oth efore and after time t (smoothing distriution). Under the same hypothesis used for the Kalman filter, a Kalman smoother (KS) can e derived (Cosme et al., 2011). Two aspects need to e emphasised: a) The Kalman smoother differs from the filter only y using crosscovariances in time to correct the state at time t using oservations at future times; ) At the end of the assimilation window (t=t) the KS and KF estimates are the same Slide 8 Slide 8

9 Why do we need an EDA? The KF is the optimal solution of the filtering prolem for linear, Gaussian systems. Unfortunately it is impractical for large systems: a current NWP system has N~10 8. In the KF we have to compute and evolve in time error covariances of NxN dimension! Two possile types of solutions: a) 4D Variational methods ) Reduced-ran Kalman Filters Slide 9 Slide 9

10 Why do we need an EDA? 4D Variational methods If we neglect model error (perfect model assumption) the smoothing prolem of finding the model trajectory that est fits the oservations over an assimilation interval (t=0,1,,t) given a acground state x and its error covariance P is the minimum of the cost function: J T 1 x x xo P x xo yt H tm 0t x0 T 1 R y H M 0 t t t 0t x0 t0 This is equivalent to the Kalman smoother solution over the assimilation interval for the same x, P and to the Kalman filter solution at the end of the interval (t=t). The 4D-Var solution implicitly evolves acground error covariances over the assimilation window (Thepaut et al.,1996), Slide 10 ut does not cycle them! Information from past oservations is only carried forward y x T Slide 10

11 Why do we need an EDA? 4D Variational methods What if we pushed ac the start of the assimilation window enough so that the smoothed solution (and the filter solution at the end of the window) would no longer depend on the specified P? Enough means 3-5 days for state of art NWP models: Time series curves 500hPa Geopotential Root mean square error forecast S.hem Lat to Lon to T+120 all os all os Slide AUGUST 2005 Slide 11

12 Slide 12 Why do we need an EDA? 4D Variational methods For assimilation windows > 12h we can not assume the model to e perfect any more. We have to add a model error term to our cost function (Wea-constraint 4D-Var): This is an elegant solution, ut: 1) Prolem is shifted from estimation of P to estimation of Q. Q can also have a non negligile flow-dependent component 2) It remains difficult in the 4D-Var framewor to have realistic estimates of P a Slide ,...,, t t t t T T t t t t t t t t t T T t t t t o T o T M M H H J x x Q x x x y R x y x x P x x x x x

13 Why do we need an EDA? Reduced-ran Kalman Filters In order to remain in the sequential paradigm we need to use the Kalman Filter analysis with a low-ran approximation of P /a In this framewor we loo for a low-ran approximation to P of the form P t =X (X ) T where X is Nxm and m<<n It then follows that K = P t H T [HP HT+R] -1 = X (HX ) T [(HX )(HX ) T + R] -1 P ta = X [I mxm + (HX ) T R(HX )] (X ) T P t+1 = M t->t+1 P ta M T t->t+1 + Q t->t+1 = M t->t+1 X [A mxm ] (M t->t+1 X ) T + Q t->t+1 Slide 13 i.e., dimension N has een replaced y m in the KF equations! However Slide 13

14 Why do we need an EDA? Reduced-ran Kalman Filters However x a -x = K (y H(x )) = X (HX ) T [(HX )(HX ) T + R] -1 (y H(x )) It then follows that the analysis increments are confined to the suspace spanned y X, which has ran m<<n Reduced-ran KF ecame popular only with the introduction of the Ensemle Kalman Filter (EnKF, Evensen, 1994; Burgers et al., 1998) EnKF is a Monte Carlo approx. of the KF which crucially does not require the Tangent Linear and Adjoints of M and H. But the suspace spanned y P ens= 1/ (N ens -1) X (X ) T, (X are Slide 14 the ensemle perturations to the ensemle mean) has still dimension N ens 1 << N Slide 14

15 Why do we need an EDA? Reduced-ran Kalman Filters P ens= 1/ (N ens -1) X (X ) T, N ens 1 << N There are ways to artificially increase the effective ensemle size (Shur product covariance localization, Hamill and Whitaer, 2001; Local analysis, Evensen, 2003; Ott et al., 2004; adaptive localization, Anderson 2007, Bishop and Hodyss, 2007,2009), ut they (too!) come at a price: a) Dynamical alance may e degraded; ) Asymptotic optimality of the EnKF lost; c) More difficult for non-local oservations, since usually applied in oservation space Slide 15 Slide 15

16 Why do we need an EDA? Results with the ECMWF EnKF Surface Pressure oservations only N.Hem. 500 hpa AC Slide 16 Slide 16

17 Why do we need an EDA? Results with the ECMWF EnKF Conventional oservations only N.Hem. 500 hpa AC Slide 17 Slide 17

18 Why do we need an EDA? Results with the ECMWF EnKF All oservations N.Hem. 500 hpa AC Slide 18 Slide 18

19 Why do we need an EDA? Quic recap: a) Kalman Filter is computationally unfeasile for realistic NWP; ) Non-sequential approx. (4D-Var) do not cycle state error estimates: wor well for short assimilation windows (6-12h), ut longer windows have proved more difficult; c) Sequential approx. (EnKF) cycle low-ran estimates of state error covariances, ut analysis increments are confined to perturations suspace;. Hyrid approach: Use flow-dependentstate errorestimates (from an EnKF/EDA system) in a 3/4D-Var analysis algorithm Slide 19 Slide 19

20 Hyrids Hyrid approx.: Use flow-dependent state error estimates (from an EnKF/EDA system) in a 3/4D-Var analysis algorithm This solution would: 1) Integrate flow-dependent state error covariance information into the variational analysis 2) Keep the full ran representation of B and its implicit evolution inside the assimilation window 3) More roust than pure EnKF for limited ensemle sizes and large model errors 4) Allow (eventual) localization of ensemle perturations to e performed in state space; 5) Allow for flow-dependent QC of oservations Slide 20 Slide 20

21 Hyrids In operational use (or in an advanced testing), there are currently two main approaches to doing an hyrid DA in a VAR context: 1. Alpha control variale method (Met Office, NCEP/GMAO) 2. Ensemle of Data Assimilations method (ECMWF, Meteo France) Slide 21 Slide 21

22 Hyrids: α control variale 1. Alpha control variale method (Met Office, NCEP/GMAO) Conceptually add a flow-dependent term to the climatological B matrix: B c is the static, climatological covariance P e C loc is the localised ensemle covariance B B In practice this is done through augmentation of control variale: x and introducing an additional term in the cost function: J 2 c c c 2 e 2 B 1 c v 1 v 2 T v 1 2 e T -1 α Cloc P e X C ' α α J loc Slide o 22 J c from: A.Clayton Slide 22

23 Slide 23 Hyrids: EDA The Ensemle of Data Assimilations (EDA,Isasen et al. 2010) can e considered a flow-dependent extension of the way the climatological acground error matrix is estimated (Fisher, 2003). For a linear system the data assimilation update is: In our system the sources of error are the oservations and the forecast model: x +1 = M x a + η η ~ N(0,Q ) y = H x + ζ ζ ~ N(0,R ) Slide 23 a a M x x x H y K x x 1 T a T T a Q M M P P K R K H K I P H K I P 1

24 Slide 24 Consider now the evolution of the same system where we pertur the oservations and the model evolution with random noise drawn from the respective error covariances: where η ~(0,R), ζ ~(0,Q). If we define the differences etween the pertured and unpertured state and, their evolution is otained y sutracting the unpertured state evolution equations from the pertured ones: Slide 24 a a ζ M x x x H η y K x x ~ ~ ~ ~ ~ 1 a a a x x ε ~ x x ε ~ a a ζ M ε ε ε H η K ε ε 1 Hyrids: EDA

25 Slide 25 i.e., the perturations evolve with the same update equations of the state What aout the errors? If we tae the statistical expectation of the outer product of the perturations: Slide 25 a a ζ M ε ε ε H η K ε ε 1 T T a a T T T T T a a Q M ε ε M ε ε K R K H K I ε ε H K I ε ε 1 1 Hyrids: EDA

26 Slide 26 These are the same equations for the evolution of the system error covariances: provided that: 1. The applied perturations η, ζ have the required covariances (R, Q); 2. At some stage in time (asymptotic convergence, Fisher et al., 2005) Slide 26 T T a a T T T T T a a Q M ε ε M ε ε K R K H K I ε ε H K I ε ε 1 1 T a T T a Q M M P P K R K H K I P H K I P 1 T P ε ε Hyrids: EDA

27 Hyrids: EDA What does all this mean in practice? 1. We can use an ensemle of pertured 4D-Var to simulate the errors of our reference high resolution 4D-Var; 2. The ensemle of pertured DAs should e as similar as possile to the reference DA (i.e., same or similar K matrix) 3. The applied perturations η, ζ must have the required error covariances (R, Q); however we do not need an explicit covariance model of Q Slide 27 Slide 27

28 Hyrids: EDA X 1 (t ) y+ε 1 o Boundary pert. 1 X 2 (t ) y+ε 2 o Boundary pert. 2 Analysis Analysis ε 1 m X 1a (t ) Forecast X 1 (t +1 ) ε 2 m X 2a (t ) Forecast X 2 (t +1 ). Slide 28 Slide 28

29 Hyrids: EDA 10 ensemle memers using 4D-Var assimilations T399 outer loop, T95/T159 inner loops. (Reference DA: T1279 outer loop, T159/T255/T255 inner loops) Oservations randomly pertured according to their specified R SST pertured with realistically scaled structures Model error represented y stochastic methods (SPPT, Leutecher, 2009) Slide 29 Slide 29

30 Hyrids: EDA The EDA system simulates the error evolution of the 4DVar analysis cycle. As such it has two main applications: 1. Provide a flow-dependent sample of analysis errors to use as initial perturations for the Ensemle Prediction system (EPS) 2. Provide a flow-dependent sample of acground errors at the initial time of the 4D-Var assimilation window Slide 30 Slide 30

31 Hyrids: EDA Improving Ensemle Prediction System y including EDA perturations for initial uncertainty The Ensemle Prediction System (EPS) enefits from using EDA ased perturations. Replacing evolved singular vector perturations y EDA ased perturations improve EPS spread, especially in the tropics. The Ensemle Mean has slightly lower error when EDA is used. N.-Hem. Tropics Slide 31 EVO-SVINI EDA-SVINI EVO-SVINI EDA-SVINI Slide 31 Ensemle spread and Ensemle mean RMSE for 850hPa T

32 Hyrids: EDA The EDA system simulates the error evolution of the 4DVar analysis cycle. As such it has two main applications: 1. Provide a flow-dependent sample of analysis errors to use as initial perturations for the Ensemle Prediction system (EPS) 2. Provide a flow-dependent sample of acground errors at the initial time of the 4D-Var assimilation window Slide 32 Slide 32

33 Hyrids: EDA In the ECMWF 4D-Var, the B matrix is defined implicitly in terms of a transformation from the acground departure (x-x ) to a control variale χ: So that the implied B=LL T. (x-x ) = Lχ In the current wavelet formulation (Fisher, 2003), the variale transform can e written as: T is the alance operator 1 1/ 2 2 x x T Σ C 1/, Σ is the gridpoint variance of acground errors C j (λ,φ) is the vertical covariance matrix for wavelet index j Slide 33 ψ j are the set of radial asis function that define the wavelet transform. j j j j Slide 33

34 Hyrids: EDA 1 1/ 2 2 x x T Σ C 1/, C j (λ,φ) are full vertical covariance matrices, function of (λ,φ). They determine oth the horizontal and vertical acground error correlation structures; In standard 4D-Var T and C j are computed off-line using a climatology of EDA perturations. Σ is computed y random sampling of the static B matrix (randomization procedure, Fisher and Courtier, 1995) How do we mae this error covariance model flow-dependent? j j j j We loo for flow-dependent EDA estimates of Σ and C j (λ,φ) Slide 34 Slide 34

35 EDA variances Σ is defined in grid space; it can e directly sampled from the EDA acground forecasts: Σ N 1 EDA, N 1 2 l i j, x i, j, x i, j, EDA However the sampled variance estimates are affected y two errors: a) Sampling Noise due to the small EDA dimensionality (N eda =10): ˆ 2 ) Systematic errors due to incorrect specification of error sources in Slide 35 the EDA (i.e., mis-specification of R, Q, uncertainties in the oundary conditions) l1 Σ Σ N EDA 1 Slide 35

36 EDA variances a) Sampling Noise due to the small EDA dimensionality (N eda =10) The ey insight is to recognise that sampling noise is small scale with respect to the error variance field (Raynaud et al., 2008) We may use a spectral filter to disentangle noise error from the signal Slide 36 Slide 36

37 EDA variances Raw Ensemle StDev VO ml64 Filtered Ensemle StDev VO ml64 Slide 37 Slide 37

38 EDA variances Operational StDev Random. Method (Fisher & Courtier, 1995) Filtered Ensemle StDev VO ml64 Slide 38 Slide 38

39 EDA variances: Ensemle Size The sampling noise effectively limits the scales that we can roustly estimate from the EDA. The effective spatial resolution of the diagnosed errors is much coarser than the nominal EDA resolution (T399) and is primarily determined y the ensemle size (Bonavita et al., 2010) Temperature ml 49 (~200 hpa) 10 memer EDA 20 memer EDA 30 memer EDA Slide 39 Slide 39

40 EDA variances: Ensemle Size A larger EDA effectively allows the sampling of errors at finer resolutions. This helps improve analysis and forecast sill! Slide 40 Slide 40

41 The current noise filter is spectral: EDA variances Vorticity ml=64 This means that there is full resolution in terms of scale ut none in physical space (i.e., the same filtering function is applied everywhere on the gloe). A wavelet filter would trade in some spectral resolution in exchange for Slide 41 spatial resolution: Slide 41

42 EDA variances A wavelet filter would trade in some spectral resolution in exchange for spatial resolution. Filter for Vorticity (ml=64), wavelet 14 (wavenumers ) Slide 42 Slide 42

43 EDA variances ) Systematic errors due to incorrect specification of error sources in the EDA (i.e., mis-specification of R, Q, uncertainties in the oundary conditions) A statistically consistent ensemle should satisfy: 1 N x N ens N N ens ens ens 2 * x x 1 1 x 2 i i1 <ensemle variance> <squared ensemle mean error> Slide 43 Slide 43

44 EDA variances Vorticity ml 78 (~850hPa) Ensemle Error Ensemle Spread Spread - Error Slide 44 Slide 44

45 EDA variances Slide 45 Conditional distriution of the EDA mean acground RMS error for given EDA acground standard deviation Slide 45

46 EDA variances Spread-Sill regressions of the type shown serve two purposes: 1. Diagnose the progress (or lac thereof!) in the modelling of system uncertainties in the EDA Slide 46 Slide 46

47 EDA variances Spread-Sill regressions of the type shown serve two purposes: 1. Diagnose the progress (or lac thereof!) in the modelling of system uncertainties in the EDA 2. Calirate on-line the EDA sample variances to otain realistic estimates of acground errors (Ensemle Variance Caliration, Kolczynsy et al., 2009, 2011; Bonavita et al., 2011) Slide 47 Slide 47

48 EDA variances Tropical Storm Aere, 9 May UTC: Operational Analysis E-suite Analysis Slide 48 Slide 48

49 EDA variances Tropical Storm Aere, 9 May UTC: Slide 49 Slide 49

50 EDA variances Tropical Storm Aere, 9 May UTC: Operational Analysis E-suite Analysis BG Error StDev Analysis Increments Slide 50 Slide 50

51 EDA variances What is the impact of flow-dependent EDA variances on the IFS scores? CY36R4, T1279L91 ffg (control: fezj): WINTER ffge (control: 0051): SUMMER Slide 51 Slide 51

52 Geopotential RMSE reduction Slide 52 winter Blue= Slide 52 summer

53 EDA variances CY37R2 (18 May 2011): Use of EDA Variances in 4D-Var Reduction of AMSU-A oservation errors Slide 53 Slide 53

54 Hyrids: EDA Covariances 1 1/ 2 2 x x T Σ C 1/, C j (λ,φ) are full vertical covariance matrices, function of (λ,φ). They determine oth the horizontal and vertical acground error correlation structures; How do we mae this error covariance model flow-dependent? We loo for flow-dependent EDA estimates of Σ and C j (λ,φ) j j j j Slide 54 Slide 54

55 Hyrids: EDA Covariances Diagnosing the Bacground Error Correlation Length-Scales Hurricane Fanele, 20 January 2009 Slide 55 Slide 55

56 Hyrids: EDA Covariances 20 memer EDA Surf. Press. Bacground Err. St.Dev. Surf. Press. BG Err. Correlation L. Scale Slide 56 Slide 56

57 Hyrids: EDA Covariances BG Error Length Scale field has more high frequency spatial structure than BG error StDev -> need for larger ensemle Off-line estimates of C j (λ,φ) are computed over a period of 2 months. Simplest approach to introduce flow-dependency in the correlation structures is through use of an evolving, on-line estimation of C j (λ,φ) over a short caliration period (Varella et al., 2011) Slide 57 Slide 57

58 Hyrids: EDA Covariances wavelet-implied length-scales of wind near 500 hpa 3-wee average, from 15/2 to 7/ Slide 58 from: L.Berre Slide 58

59 Hyrids: EDA Covariances wavelet-implied length-scales of wind near 500 hpa 4-day average, from 24/2 to 27/ Slide 59 from: L.Berre Slide 59

60 Hyrids: EDA Covariances Mean Geopotential Height at 500 hpa 4-day average, from 24/2 to 27/ Slide 60 Slide 60

61 Hyrids: EDA Covariances Regularization of the on-line correlation estimates through temporal averaging and the implicit spatial averaging of the wavelet representation Larger ensemle would allow for a larger flow-dependent component to e retained Other forms of regularization of the on-line correlation estimates can e envisaged (i.e., convex cominations of on-line and off-line C j (λ,φ) estimates) Slide 61 Slide 61

62 Conclusions and Perspectives The Kalman Filter/Smoother is still the gold standard of atmospheric gloal NWP data assimilation, ut practically unfeasile Non-sequential approx. to KF (4D-Var): 1. Keeps full-ran representation of B matrix and its implicit evolution during the assimilation window; 2. Unale to cycle B estimates; 3. Difficult to access realistic estimates of P a ; 4. Long-window wea-constraint 4D-Var would potentially solve issue 2. ut still to e demonstrated in realistic NWP settings Slide 62 Slide 62

63 Conclusions and Perspectives Low-ran, Monte Carlo, Sequential approx. to KF (EnKF): 1. Explicit evolution and cycling of low-ran approximation of B matrix (and P a ); 2. Computationally scalale and efficient; 3. The EnKF analysis is restricted to the error suspace spanned y the ensemle perturations. This is unrealistically small and requires covariance localization/inflation to eep the EnKF from diverging; 4. EnKF performance degrades with respect to 4D-Var when N os in the local analysis patch is >> N ens and oservations are non-local (satellite radiances). Can this prolem e cured with larger ut affordale ensemle size and more careful oservation selection? Slide 63 Slide 63

64 Conclusions and Perspectives Hyrid approx. to KF: try to comine the strengths of the sequential and non-sequential approaches a) Low-ran, Monte Carlo error representation through cycling EnKF/EDA system; ) State estimate from full-ran 4D-Var analysis where static B at the start of the window is (partially) replaced y EnKF/EDA flowdependent B Hyrid can e done y adding an ensemle, flow-dependent component to the static B used in 4D-Var (alpha control var.) Hyrid can also e done y using an EDA/EnKF to get an on-line, Slide 64 flow-dependent estimate of parameterised B (EDA approach) Slide 64

65 Conclusions and Perspectives Use of hyrids consistently improves deterministic analysis and forecast sill w.r.to pure sequential (EnKF) and non-sequential (4D-Var) solutions; EDA/EnKF, possily re-centred around deterministic analysis, provide improved sampling of initial errors for Ensemle Prediction We can expect growing ensemle use in 4D-Var: 1. A larger ensemle (oth in the EDA and EnKF) improves error characterization and ultimately sill scores; 2. 4D acground error covariances sampled from an EDA/EnKF could e used over the all 4D-Var assimilation window (not only at the start!): En-4D-Var (Liu et al., 2008; Buehner Slide 65et al., 2010). This would remove the need of developing and maintaining a TL and Adjoint version of the forecast model Slide 65

66 Conclusions and Perspectives We can expect growing ensemle use in 4D-Var: 3. Wea-constraint Long-window 4D-Var revolves around the estimation of Q: It is conceivale that an EDA will provide a way of effectively sampling Q 4. The EnKF is more computationally efficient than an ensemle of 4D-Var analysis (EDA): if it can e shown to e as accurate as standard 4D-Var with the full oserving system, then it will provide a relatively cheap and efficient way of cycling error estimates in a hyrid system Slide 66 Slide 66

67 Questions and Answers! Slide 67 Slide 67

68 Additional Slides Slide 68 Slide 68

69 Randomization Randomization procedure, Fisher and Courtier, 1995 Define N random vector in control-variale space, with independent elements drawn from a Gaussian distriution with zero mean and unit variance ξ i ~N(0,I). Then Lξ i will e drawn from the distriution N(0,B) A grid point estimate of acground error variances can then e computed from: Bˆ g 1 N N 1 1 S Li S Li Where S -1 denotes the inverse transform from Spectral space. i1 The variances are then rescaled ased on an estimate of analysis errors from the leading eigenvectors of the Hessian matrix. T Finally an error growth model (Savijärvi, 1995) Slide is 69applied to account for error growth over the short range forecast Slide 69

70 Use of EDA variances in 4DVar There is not much variaility on daily-weely scales ut seasonal variaility is important Slide 70 General solution: slowly varying adaptive caliration coefficients Slide 70

71 Temperature RMSE reduction Slide 71 winter Blue= Slide 71 summer

72 Wind Vector RMSE reduction Slide 72 winter Blue= Slide 72 summer

73 EDA Cycle x +ε i y+ε o i SST+ε SST i Analysis x a +ε i a Forecast x f +ε i f i=1,2,,10 Variance post-process ε if raw variances Variance Rescaling Variance Filtering EDA scaled variances 4DVar Cycle x EDA scaled Var Analysis x a Slide 73 Forecast x f Slide 73

74 EDA variances a) Sampling Noise due to the small EDA dimensionality (N eda =10) The ey insight is to recognise that sampling noise is small scale with respect to the error variance field (Raynaud et al., 2008) Define G e (i) as the sampling error in the estimated ensemle variance at e ~ gridpoint i: G ~ i B EB ii ii Then the covariance of the sampling noise can e shown to e a simple function of the expectation of the ensemle-ased covariance matrix: 2 e e 2 ~ ig j EB E G N 1 A consequence of (1) is that L G e(i)=l ε (i)/ 2, i.e., sampling noise is smaller scale than acground error. Slide 74 The variance field varies on larger scales then the acground error, so we may use a spectral filter to disentangle noise error from the variance field ij (1) Slide 74

75 EDA variances There is indeed scale separation etween signal and sampling noise! Truncation wavenumer is determined y maximizing signal-tonoise ratio of filtered variances (details in Raynaud et al., 2009; Bonavita et al., 2011) Optimal truncation wavenumer depends on parameter and model level Slide 75 Slide 75

76 EDA variances Vorticity ml 30 (~50hPa) Ensemle Error Ensemle Spread Spread - Error Slide 76 Slide 76

77 EDA variances How does a flow-dependent error variance estimate change the 4D-Var analysis? Z500 Geopotential (shaded) and MSLP Z Slide 77 Slide 77

78 EDA variances Vorticity Bacground Error ml=78 (850hPa) Randomization method EDA Slide 78 Slide 78

79 EDA variances Single os. experiment: T os -T fg =+1K, (34N,74W), 900 hpa Randomization method EDA Slide 79 Slide 79

80 Randomization EDA variances Single os. experiment: T os -T fg =+1K, (34N,74W), 900 hpa EDA T ana -T fg 900 hpa Slide 80 VO ana -VO fg 850 hpa Slide 80

81 EDA variances 1. The oservation weight in the analysis is increased in the area of large acground uncertainty: ΔT ΔVO EDA Randomiz K 0.37 K 5.E-5 s E-5 s The EDA analysis increments show a degree of flowdependency Slide 81 Slide 81

82 Why do we need an EDA? Results with the ECMWF EnKF All oservations S.Hem. 500 hpa AC Slide 82 Slide 82