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Slide 1 ECMWF Data Assimilation Training Course – May 2010 Ensemble Data Assimilation Massimo Bonavita ECMWF Acknowledgments: Lars Isaksen, Elias Holm, Mike Fisher, Laure Raynaud

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Slide 2 ECMWF Data Assimilation Training Course – May 2010 Outline The Ensemble Data Assimilation method What do we do with the EDA? Use of EDA variances in ECMWF 4DVar 4DVar assimilation experiments Conclusions and plans Slide 2

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Slide 3 ECMWF Data Assimilation Training Course – May 2010 Outline The Ensemble Data Assimilation method What do we do with the EDA? Use of EDA variances in 4DVar 4DVar assimilation experiments Conclusions and plans Slide 3

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Slide 4 ECMWF Data Assimilation Training Course – May 2010 When simulating the error evolution of the reference system one should use the reference gain matrix K ( Berre et al ) The EDA method Slide 4

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Slide 5 ECMWF Data Assimilation Training Course – May 2010 This is what an ensemble of 4DVar analyses with random observation and SST perturbations does! ε a/f/o = perturbations w.r.to ensemble mean The EDA method Analysis x b +ε b y+ε o SST+ε SST (etc.) x a +ε a Forecast x f +ε f Slide 5

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Slide 6 ECMWF Data Assimilation Training Course – May 2010 After a few cycles the EDA system will have forgotten the initial background perturbations We are diagnosing the background error statistics of the actual analysis system However: We assume that observation and SST errors are correctly specified We need to account for the model error component of the background error The EDA method Slide 6

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Slide 7 ECMWF Data Assimilation Training Course – May 2010 Stochastic Kinetic Energy Backscatter SKEB (Berner et al., 2009) ΔX perturbed physics = f(KE,ΔX physics,n) A fraction of the dissipated energy is backscattered upscale and acts as stream function forcing on resolved-scale flow. Spectral Markov chain: temporal and spatial correlations prescribed Only vorticity perturbed, so only wind field directly affected All levels are perturbed. Either randomly or partially correlated. Stochastically Perturbed Parameterisation Tendencies SPPT (Leutbecher, 2009) ΔX perturbed physics = ( 1+μr) ΔX physics Random pattern r varies smoothly in space and time, with de-correlation scales 500 km and 6 hours. Gaussian distribution with no bias and stdev 0.5 (limited to ±3stdev) Same random number r for X=T, q, u, v No perturbations in lowest 300 m and above 50 hPa (0 μ 1). from: Lars Isaksen The EDA method Slide 7

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Slide 8 ECMWF Data Assimilation Training Course – May 2010 The EDA method Comparing the impact of various stochastic perturbation methods: zonal mean u wind ensemble background forecast spread NS Slide 8 Impact of SKEB on U spread Additional Impact of SPPT on U spread from: Lars Isaksen

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Slide 9 ECMWF Data Assimilation Training Course – May 2010 The EDA method Comparing the impact of various stochastic perturbation methods: zonal mean Temperature ensemble background forecast spread NS Slide 9 Impact of SKEB on T spread Additional Impact of SPPT on T spread from: Lars Isaksen

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Slide 10 ECMWF Data Assimilation Training Course – May 2010 In the EDA context model error parametrizations should increase spread where/when errors are larger => should increase spread-error correlation Are they doing this? Slide 10 The EDA method

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Slide 11 ECMWF Data Assimilation Training Course – May 2010 EDA (baseline) - EDA+SKEB EDA+SPPT - EDA+SPPT+SKEB Slide 11 The EDA method

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Slide 12 ECMWF Data Assimilation Training Course – May 2010 EDA (baseline) - EDA+SKEB EDA+SPPT - EDA+SPPT+SKEB Slide 12 The EDA method

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Slide 13 ECMWF Data Assimilation Training Course – May 2010 Model error parametrizations do increase spread- error correlations Large effect in the lower troposphere Tropics Small effect in the Extratropics Similar behaviour-performance of the two schemes and their combination Slide 13 The EDA method

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Slide 14 ECMWF Data Assimilation Training Course – May ensemble members using 4D-Var assimilations T399 outer loop, T95/T159 inner loop (reduced number of iterations) Observations randomly perturbed Cloud track wind (AMV) correlations taken into account SST perturbed with realistically scaled structures Model error represented by stochastic methods ( SPPT, Leutbecher, 2009 ) All 107 conventional and satellite observations used The EDA method from: Lars Isaksen Slide 14

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Slide 15 ECMWF Data Assimilation Training Course – May 2010 Outline The Ensemble Data Assimilation method What do we do with the EDA? Use of EDA variances in ECMWF 4DVar 4DVar assimilation experiments Conclusions and plans Slide 15

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Slide 16 ECMWF Data Assimilation Training Course – May 2010 The EDA simulates the error evolution of the 4DVar analysis cycle. As such it can be applied to: 1. Compute climatology of B matrix for use at the initial time of 4DVar (Analysis-Ensemble method, Fisher 2003; see talk on B matrix modelling) 2. Provide initial conditions for ensemble forecasts (EPS) 3. Provide a flow-dependent sample of background errors at the initial time of 4DVar What do we do with the EDA? Slide 16

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Slide 17 ECMWF Data Assimilation Training Course – May 2010 Improving Ensemble Prediction System by including EDA perturbations for initial uncertainty The Ensemble Prediction System (EPS) benefits from using EDA based perturbations. Replacing evolved singular vector perturbations by EDA based perturbations improve EPS spread, especially in the tropics. The Ensemble Mean has slightly lower error when EDA is used. What do we do with the EDA? Slide 17 EVO-SVINI EDA-SVINI EVO-SVINI EDA-SVINI N.-Hem.Tropics Ensemble spread and Ensemble mean RMSE for 850hPa T

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Slide 18 ECMWF Data Assimilation Training Course – May 2010 The EDA simulates the error evolution of the 4DVar analysis cycle. As such it can be applied to: 1. Compute climatology of B matrix for use at the initial time of 4DVar (Analysis-Ensemble method, see talk on B matrix modelling) 2. Provide initial conditions for ensemble forecasts (EPS) 3. Provide a flow-dependent sample of background errors at the initial time of 4DVar What do we do with the EDA? Slide 18

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Slide 19 ECMWF Data Assimilation Training Course – May DVAr does not cycle error information (B), only the state estimate 4DVar behaves like a Kalman filter in which the covariance matrix is reset to some static matrix B every few hours (12h in ECMWF implement.) Two possible ways around this: 1. Run 4DVar over a long enough window so that the influence of the initial state and errors on the final state analysis is negligible Slide 19 What do we do with the EDA?

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Slide 20 ECMWF Data Assimilation Training Course – May 2010 Analysis experiment started with satellite data reintroduced on 15 th August 2005 From Mike Fisher Memory of the initial state disappears after approx. 3 days Slide 20

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Slide 21 ECMWF Data Assimilation Training Course – May 2010 An analysis window of 3 days would allow 4DVar analysis at final time to be independent of initial state and error estimates However: 1.An effective model error representation must be applied to reconcile model and measurements over a long analysis time window (i.e., weak constraint 4DVar) 2.We would still lack an estimate of analysis errors Slide 21 What do we do with the EDA?

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Slide 22 ECMWF Data Assimilation Training Course – May 2010 Two possible ways around this: 1. Run 4DVar over a long enough window so that the influence of the initial state and errors on the final state analysis is negligible 2. Use a sequential method (EDA, EnKF) to cycle error covariance information Slide 22 What do we do with the EDA?

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Slide 23 ECMWF Data Assimilation Training Course – May 2010 Hybrid methods Use of ensemble perturbations in a 3-4DVar analysis Cycle error information through ensemble of Data Assimilations Retain the implicit full rank error representation of 3-4DVar µεσον τε και αριστον Aristotle, Nic. Ethics 2.6 Slide 23

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Slide 24 ECMWF Data Assimilation Training Course – May 2010 Hybrid systems Ensemble perturbations can be used in a 3-4DVar analysis in a number of different ways: 1.Use ensemble variances for observation QC 2.Use ensemble (co)variances as starting B matrix of minimization (often in linear combination with climatological B, extra control variable) 3.Use of ensemble covariances inside 4DVar minimization (En4DVAR) Slide 24 What do we do with the EDA?

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Slide 25 ECMWF Data Assimilation Training Course – May 2010 Outline The Ensemble Data Assimilation method What do we do with the EDA? Use of EDA variances in ECMWF 4DVar 4DVar assimilation experiments Conclusions and plans Slide 25

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Slide 26 ECMWF Data Assimilation Training Course – May 2010 We want to use EDA perturbations to simulate 4DVar flow-dependent error covariance evolution We start with the diagonal of the P f matrix, i.e.: Estimate the first guess error variances with the variance of the EDA short range forecasts This has been tried before (Kucukkaraca and Fisher, 2006, Fisher 2007, Isaksen et al., 2007) but results have been inconclusive Use of EDA variances in 4DVar Slide 26

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Slide 27 ECMWF Data Assimilation Training Course – May 2010 Define: V eda = EDA sampled variance V* = True error variance Then the estimation error can be decomposed: V eda -V* = E[V eda -V*] + (V eda -E[V eda ]) systematic random Similarly for mean square of the estimation error: E[(V eda -V*) 2 ] = (E[V eda -V*]) 2 + E[(V eda -E[V eda ]) 2 ] systematic random We should try to minimize both! Slide 27 Use of EDA variances in 4DVar

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Slide 28 ECMWF Data Assimilation Training Course – May 2010 What raw ensemble variances look like? Spread of Vorticity FG t+9h ml=64 Slide 28 Use of EDA variances in 4DVar

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Slide 29 ECMWF Data Assimilation Training Course – May 2010 Noise level is due to sampling errors: 10 member ensemble EDA is a stochastic system: variance of variance estimator ~ 1/N ens We need a system to effectively filter out noise from first guess ensemble forecast variances: Reduce the random component of the estimation error Slide 29 Use of EDA variances in 4DVar

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Slide 30 ECMWF Data Assimilation Training Course – May 2010 Mallat et al.: 1998, Annals of Statistics, 26,1-47 Define G e (i) as the random component of the sampling error in the estimated ensemble variance at gridpoint i: Then the covariance of the sampling noise can be shown to be a simple function of the expectation of the ensemble-based covariance matrix: (1) Slide 30 Use of EDA variances in 4DVar

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Slide 31 ECMWF Data Assimilation Training Course – May 2010 Mallat et al.: 1998, Annals of Statistics, 26,1-47 A consequence of (1) is that: (2) i.e., sampling noise is smaller scale than background error. If the variance field varies on larger scales then the background error => we can use a spectral filter to disentangle noise error from the sampled variance field Slide 31 Use of EDA variances in 4DVar

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Slide 32 ECMWF Data Assimilation Training Course – May 2010 Slide 32 Use of EDA variances in 4DVar

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Slide 33 ECMWF Data Assimilation Training Course – May 2010 There is indeed a scale separation between signal and sampling noise! Truncation wavenumber is determined by maximizing signal-to-noise ratio of filtered variances (details in Raynaud et al., 2009, and forthcoming Tech. Memo) Optimal truncation wavenumber depends on parameter and model level Slide 33 Use of EDA variances in 4DVar

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Slide 34 ECMWF Data Assimilation Training Course – May 2010 Raw Ensemble StDev VO ml64 Filtered Ensemble StDev VO ml64 Slide 34 Use of EDA variances in 4DVar

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Slide 35 ECMWF Data Assimilation Training Course – May 2010 Raw Ensemble StDev Spec. Hum. ml64 Filtered Ensemble StDev Spec. Hum. ml64 Slide 35 Use of EDA variances in 4DVar

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Slide 36 ECMWF Data Assimilation Training Course – May 2010 Operational StDev Random. Method (Fisher & Courtier, 1995 Filtered Ensemble StDev VO ml64 Slide 36 Use of EDA variances in 4DVar

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Slide 37 ECMWF Data Assimilation Training Course – May 2010 Is Filtering the Ensemble Variances enough to improve the analysis? Not quite… N.Hem. Z 500hPA AC S.Hem. Z 500hPA AC Slide 37 Use of EDA variances in 4DVar

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Slide 38 ECMWF Data Assimilation Training Course – May 2010 Should we also do something about the systematic component of the estimation error? E[(V eda -V*) 2 ] = (E[V eda -V*]) 2 + E[(V eda -[V eda ]) 2 ] systematic random A statistically reliable ensemble satisfies: Slide 38 Use of EDA variances in 4DVar

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Slide 39 ECMWF Data Assimilation Training Course – May 2010 Vorticity ml 30 (~50hPa) Ensemble Error Ensemble Spread Spread - Error Slide 39 Use of EDA variances in 4DVar

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Slide 40 ECMWF Data Assimilation Training Course – May 2010 Vorticity ml 78 (~850hPa) Ensemble Error Ensemble Spread Spread - Error Slide 40 Use of EDA variances in 4DVar

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Slide 41 ECMWF Data Assimilation Training Course – May 2010 Is the ensemble fg statistically calibrated? Calibration factors needs to be model level, latitude and parameter dependent Calibration factors seems also to be flow- dependent, i.e. depend on the size of the expected error Slide 41 Use of EDA variances in 4DVar

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Slide 42 ECMWF Data Assimilation Training Course – May 2010 Slide 42 Use of EDA variances in 4DVar

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Slide 43 ECMWF Data Assimilation Training Course – May 2010 Calibration factors need to be flow-dependent, too! Do they also change in time? Slide 43 Use of EDA variances in 4DVar

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Slide 44 ECMWF Data Assimilation Training Course – May 2010 Slide 44 Use of EDA variances in 4DVar

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Slide 45 ECMWF Data Assimilation Training Course – May 2010 Slide 45 Use of EDA variances in 4DVar

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Slide 46 ECMWF Data Assimilation Training Course – May 2010 There is not a large day-to-day variability but seasonal variability is important General solution: slowly varying adaptive calibration coefficients Slide 46 Use of EDA variances in 4DVar

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Slide 47 ECMWF Data Assimilation Training Course – May 2010 Variance post-process x a +ε i a AnalysisForecast SST+ε i SST y+ε i o x b +ε i b xf+εifxf+εif i=1,2,…,10 EDA Cycle ε i f raw variances Variance Recalibration Variance Filtering EDA scaled variances 4DVar Cycle xaxa AnalysisForecast EDA scaled Var xbxb xfxf Slide 47

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Slide 48 ECMWF Data Assimilation Training Course – May 2010 Slide 48

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Slide 49 ECMWF Data Assimilation Training Course – May 2010 Outline The Ensemble Data Assimilation method What do we do with the EDA? Use of EDA variances in ECMWF 4DVar 4DVar assimilation experiments Conclusions and plans Slide 49

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Slide 50 ECMWF Data Assimilation Training Course – May DVar assimilation with EDA variances Deterministic DA experiments with EDA variances 1.CY35r3_esuite, T799L91, 7/01 – 16/ Control f8a4 3.Experiment fb4k with ensemble DA variances: a)Calibration step: adaptive, flow-dependent, regionally varying, for each parameter and model level b) Filtering step: Optimal spectral filtering c)EDA variances are used both in observation QC and start of 4DVar minimization Slide 50

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Slide 51 ECMWF Data Assimilation Training Course – May DVar assimilation with EDA variances Slide 51

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Slide 52 ECMWF Data Assimilation Training Course – May DVar assimilation with EDA variances N.HEM Z ac Jan-Feb Slide 52

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Slide 53 ECMWF Data Assimilation Training Course – May DVar assimilation with EDA variances S.HEM Z ac Jan-Feb Slide 53

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Slide 54 ECMWF Data Assimilation Training Course – May DVar assimilation with EDA variances TROP. VW RMSE Jan-Feb Slide 54

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Slide 55 ECMWF Data Assimilation Training Course – May DVar assimilation with EDA variances Deterministic DA experiments with EDA variances 1.CY36r2_esuite, T1279L91, 7/08 – 16/ Control far9 3.Experiment fb9x with ensemble DA variances: a)Calibration step: adaptive, flow-dependent, regionally varying, for each parameter and model level b) Filtering step: Optimal spectral filtering c)EDA variances are used both in observation QC and start of 4DVar minimization Slide 55

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Slide 56 ECMWF Data Assimilation Training Course – May DVar assimilation with EDA variances N.HEM Z ac Aug-Sep Slide 56

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Slide 57 ECMWF Data Assimilation Training Course – May DVar assimilation with EDA variances S.HEM Z ac Aug-Sep Slide 57

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Slide 58 ECMWF Data Assimilation Training Course – May DVar assimilation with EDA variances TROP VW RMSE Aug-Sep Slide 58

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Slide 59 ECMWF Data Assimilation Training Course – May DVar assimilation with EDA variances Deterministic DA experiments with EDA variances 1.Results are generally positive in the Extra- Tropics (more so in the NH) 2.Small positive impact in the Tropics Slide 59

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Slide 60 ECMWF Data Assimilation Training Course – May DVar assimilation with EDA variances Slide 60 Observation departures statistics support the idea that (slightly) different observation usage is the result of smarter OBS QC decisions

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Slide 61 ECMWF Data Assimilation Training Course – May 2010 Outline The Ensemble Data Assimilation method What do we do with the EDA? Use of EDA variances in ECMWF 4DVar 4DVar assimilation experiments Conclusions and plans Slide 61

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Slide 62 ECMWF Data Assimilation Training Course – May 2010 First step towards an error cycling DA Use of flow dependent EDA variances has the potential to improve the deterministic scores A careful post-processing step of the raw ensemble first guess forecast is necessary to: a)Filter sampling noise b)Adaptively calibrate the ensemble Conclusions and plans Slide 62

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Slide 63 ECMWF Data Assimilation Training Course – May 2010 Better OBS QC decisions seems to be playing a part Further improvements in model error parameterizations will directly benefit the system Increase in ensemble size will benefit the system Conclusions and plans Slide 63

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Slide 64 ECMWF Data Assimilation Training Course – May 2010 WHERE NEXT Operational implementation and testing Further tuning of system at full operational resolution (T1279L91) Conclusions and plans Slide 64

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Slide 65 ECMWF Data Assimilation Training Course – May 2010 Generalize the use of EDA variances to unbalanced components of control vector Relax the assumption: analysis=truth in the calibration step Conclusions and plans Slide 65

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Slide 66 ECMWF Data Assimilation Training Course – May 2010 Medium term: Refine the representation of initial uncertainties (correlated perturbations, surface fields uncertainties) in stochastic EDA Evaluate EnKF covariances Further develop the hybridization of 4DVar with EDA (investigate the use of EDA covariances) Conclusions and plans Slide 66

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Slide 67 ECMWF Data Assimilation Training Course – May 2010 Thanks for your attention! I welcome your questions/comments… Slide 67

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