Gérald Desroziers Météo-France, Toulouse, France

1. Data assimilation schemes in numerical weather forecasting and their link with ensemble forecasting Gérald Desroziers Météo-France, Toulouse, France

2. Outline • Numerical weather prediction • Data assimilation • A posteriori diagnostics: optimizing error statistics • Ensemble assimilation • Impact of observations on analyses and forecasts • Conclusion and perspectives

3. Outline • Numerical weather prediction • Data assimilation • A posteriori diagnostics: optimizing error statistics • Ensemble assimilation • Impact of observations on analyses and forecasts • Conclusion and perspectives

4. Numerical Weather Prediction at Météo-France DX ~ 10 km Global Arpège model : DX ~ 15 km Arome : DX ~ 2,5 km

5. Ebauche xb = M(xa-) Observationsyo Initial condition problem Prévision état à t0 + 24h État atmosphérique à t0

6. Outline • Numerical weather prediction • Data assimilation • A posteriori diagnostics: optimizing error statistics • Ensemble assimilation • Impact of observations on analyses and forecasts • Conclusion and perspectives

7. Synops and ships Buoys Radiosondes Pilots and profilers Aircraft ATOVS Satobs Geo radiances SSM/I Scatterometer Ozone Data coverage 05/09/03 09–15 UTC (courtesy J-.N. Thépaut)

8. Satellites (EUMETSAT)

9. Satellite data sources (courtesy J-.N.Thépaut, ECMWF)

10. General formalism • Statistical linear estimation : xa = xb+ dx =xb+ K d = xb+ BHT (HBHT+R)-1 d, with d = yo – H(xb ), 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 » .

11. Statistical hypotheses • Observations are supposed un-biased: E(eo) = 0. • If not, they have to be preliminarly de-biased, • or de-biasing can be made along the minimization (Derber and Wu, 1998; Dee, 2005; Auligné, 2007). • Oservation error variances are supposed to be known ( diagonal elements of R = E(eoeoT) ). • Observation errors are supposed to be un-correlated : ( non-diagonal elements of E(eoeoT) = 0 ), • but, the representation of observation error correlations is also investigated (Fisher, 2006) .

12. Implementation • Variational formulation: minimization of J(dx)=dxT B-1dx + (d-H dx)T R-1(d-H dx) • Computation of J’: development and use of adjoint operators • 4D-Var : generalized observation operator H : addition of forecast model M. • Cost reduction : low resolution increment dx (Courtier, Thépaut et Hollingsworth, 1994)

13. obs Jo « old » forecast analysis Jo xb obs corrected forecast Jb Jo xa obs 9h 12h 15h Assimilation window 4D-Var : principle

14. Outline • Numerical weather prediction • Data assimilation • A posteriori diagnostics: optimizing error statistics • Ensemble assimilation • Impact of observations on analyses and forecasts • Conclusion and perspectives

15. A posteriori diagnostics • Is the system consistent? • We should have E[J(xa) ] = p, p = total number of observations, • but also E[Joi(xa) ] = pi – Tr(Ri-1/2 HiAHiT Ri-1/2 ), pi : number of observations associated with Joi (Talagrand, 1999) . • Computation of optimal E[Joi(xa) ] by a Monte-Carlo procedure is possible. (Desroziers et Ivanov, 2001) .

16. Application : optimisation of R ∙ ∙ One tries to obtain E[Joi (xa)] = (E[Joi (xa)])opt. by adjusting the soi ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ Optimisation of HIRS so ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ (Chapnik, et al, 2004; Buehner, 2005)

17. Outline • Numerical weather prediction • Data assimilation • A posteriori diagnostics: optimizing error statistics • Ensemble assimilation • Impact of observations on analyses and forecasts • Conclusion and perspectives

18. 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 day. (Evensen, 1997; Fisher, 2004; Berre et al, 2007)

19. Ensembles Based on a perturbation of observations The same analysis equation and (sub-optimal) operators K and H are involved in the equations of xa and ea: xa = (I – KH) xb+ K xo ea = (I – KH) eb + K eo The same equation also holds for the analysis perturbation: pa = (I – KH) pb+ K po

20. Background error standard-deviations Over a month Vorticity at 500 hPa For a particular date 08/12/2006 00H Vorticity at 500 hPa

21. Ensemble assimilation:errors 08/12/2006 06UTC 500 hPa vorticity error surface pressure

22. Ensemble assimilation:errors 15/02/2008 12UTC 850 hPa vorticity error (shaded) sea surface level pressure (isoligns) (Montroty, 2008)

23. Outline • Numerical weather prediction • Data assimilation • A posteriori diagnostics: optimizing error statistics • Ensemble assimilation • Impact of observations on analyses and forecasts • Conclusion and perspectives

24. Measure of the impact of observations • Total reduction of estimation error variance: r = Tr(K H B) • Reduction due to observation set i : ri = Tr(KiHiB) • Variance reduction normalized by B : riDFS = Tr(KiHi) • Reduction of error projected onto a variable/area: riP = Tr(PKiHi B PT) • Reduction of error evolved by a forecast model: riPM = Tr(PMKiHi B MTPT) = Tr(L KiHi B LT) (Cardinali, 2003; Fisher, 2003; Chapnik et al, 2006)

25. Randomized estimates of error reduction on analyses and forecasts It can be shown that This can be estimated by a randomization procedure: is a vector of observation perturbations and where the corresponding perturbation on the analysis. (Fisher, 2003; Desroziers et al, 2005)

26. Degree of Freedom for Signal (DFS) 01/06/2008 00H

27. Error variance reduction % of error variance reduction for T 850 hPa by area and observation type (Desroziers et al, 2005)

28. Outline • Numerical weather prediction • Data assimilation • A posteriori diagnostics: optimizing error statistics • Ensemble assimilation • Impact of observations on analyses and forecasts • Conclusion and perspectives

29. Conclusion and perspectives • Importance of the notion of « observation operator » : • most often explicit, • rarely statistical • Large size problems : • state vector : ~ 10^7 • observations : ~ 10^6 • Ensemble assimilation: • estimation error covariances • measure of the impact of observations • link with Ensemble forecasting (~ 40 members of +96h forecasts)