Limitations and strengths of 4D-Var and Possible use of a variational analysis in the EnKF

1. Limitations and strengths of 4D-Var and Possible use of a variational analysis in the EnKF EnKF Internal Workshop CMC, Dorval Mark Buehner ASTD/MRD/ARMA February 2, 2011

2. Contents • Limitations of 4D-Var approach • limitations related to use of GEM TL/AD • other limitations • Strengths of 4D-Var approach • Results from deterministic experiments that motivate possible use of a variational analysis within EnKF • variational analysis with 4D ensemble covariances, similar to how EnKF does analysis • some important differences with EnKF sequential analysis approach (talk/discussion led by Hersh)

3. Limitations of 4D-Var approach • Computational cost of GEM TL/AD: • integrations of TL/AD can only start after the obs cut-off time • integrations must be done sequentially, not in parallel • difficult to make efficient use of high number of processors for low resolution TL/AD • difficult to increase resolution of the analysis increment in currently operational 4D-Var (wall-clock time constraint) • Development cost of GEM TL/AD: • model formulation and/or optimization strategy of high-resolution NLM may not be appropriate for lower resolution TL/AD • theoretically/practically difficult to linearize highly non-linear physical parameterizations  must be simplified • major changes to NLM require changes to TL/AD • development time spent on improving/optimizing TL/AD could be spent on improving high resolution NLM

4. Limitations of 4D-Var approach • Limitations of any variational approach: • Background-error covariances not estimated as part of approach, currently use (time consuming) ad-hoc method to estimate static covariances for winter/summer • Analysis-error covariances not easily obtained • Approach not designed for initializing ensemble forecasts, other centers add perturbations to deterministic analysis (using e.g. singular vectors or ensemble of data assimilation systems) • TL/AD of observation operators required in addition to original nonlinear operators

5. Strengths of 4D-Var approach • Implicit evolution of covariances through assimilation window in 4D-Var: • allows evolution of full-rank covariance matrix: e.g. operational covariances or localized EnKF covariances or combination of both • covariances mostly evolved with linearized model, but outer loop allows inclusion of non-linearity • allows use of temporal penalty term in cost function: e.g. weak constraint digital filter • Variational analysis approach • common to all variational flavors: 3D-FGAT, 4D-Var, En-4D-Var • global solution, spatial localization of B directly, var QC, computational cost may scale better with respect to Nobs

6. Experiments that motivate use of a variational analysis within EnKF • Experiments performed in context of EnKF—4D-Var intercomparison project: • same observations used in all cases, 58 levels, model top at 10hPa • spatial resolution of variational analysis increment equal to EnKF resolution, EnKF uses 96 members • experiments over February 2007 • Deterministic analysis in variational system using 4D EnKF ensemble covariances: En-4D-Var • could be used to perform analysis step within EnKF: one analysis for each ensemble member • allows for flexible approaches to model covariances, such as combining spatially localized ensemble covariances with more filtered covariances (similar to Bnmc)

7. Analysis and Forecast Verification Results En-4D-Var vs. standard approaches En-4D-Varvs. EnKF and En-4D-Var vs. 4D-Var-Bnmc

8. 4D error covariancesTemporal covariance evolution (explicit vs. implicit evolution) EnKF and En-4D-Var: 96 NLM integrations 4D-Var: 55 TL/AD integrations, 2 outer loop iterations -3h 0h +3h

9. Forecast Results:En-4D-Var vs. EnKF north tropics south Difference in stddev relative to radiosondes: Positive En-4D-Var better Negative  EnKF better En-4D-Var uses incremental approach, deterministic analysis zonal wind temp. height

10. Forecast Results:En-4D-Var vs. EnKF north tropics south Significance level of difference in stddev relative to radiosondes: Positive En-4D-Var better Negative  EnKF better zonal wind temp. Computed using bootstrap resampling of the individual scores for 48-hour non-overlapping periods. height Shading for 90% and 95% confidence levels

11. Forecast Results:En-4D-Var vs. 4D-Var-Bnmc north tropics south Difference in stddev relative to radiosondes: Positive En-4D-Var better Negative  4D-Var-Bnmc better zonal wind temp. height

12. Forecast Results:En-4D-Var vs. 4D-Var-Bnmc north tropics south Significance level of difference in stddev relative to radiosondes: Positive En-4D-Var better Negative  4D-Var-Bnmc better zonal wind temp. Computed using bootstrap resampling of the individual scores for 48-hour non-overlapping periods. height Shading for 90% and 95% confidence levels

13. Analysis and Forecast Verification Results Averaged covariances vs. NMC and EnKF Bavg = ½ Bnmc + ½ Benkf 3D-Var-Bavg vs. 3D-Var-Bnmc and 3D-Var-Bavg vs. 3D-Var-Benkf

14. Forecast Results:3D-Var-Bavg vs. 3D-Var-Bnmc north tropics south Difference in stddev relative to radiosondes: Positive 3D-Var-Bavg better Negative  3D-Var-Bnmc better zonal wind temp. height

15. Forecast Results:3D-Var-Bavg vs. 3D-Var-Benkf north tropics south Difference in stddev relative to radiosondes: Positive 3D-Var-Bavg better Negative  3D-Var-Benkf better zonal wind temp. height

16. Analysis and Forecast Verification Results En-4D-Var vs. combined 4D-Var – EnKF approach En-4D-Varvs. 4D-Var-Benkf

17. Forecast Results:En-4D-Var vs. 4D-Var-Benkf north tropics south Difference in stddev relative to radiosondes: Positive En-4D-Var better Negative  4D-Var-Benkf better zonal wind temp. height

18. Forecast Results:En-4D-Var vs. 4D-Var-Benkf north tropics south Significance level of difference in stddev relative to radiosondes: Positive En-4D-Var better Negative  4D-Var-Benkf better zonal wind temp. Computed using bootstrap resampling of the individual scores for the 56 cases (28 days, twice per day). height Shading for 90% and 95% confidence levels

19. Summary • Major future improvements of 4D-Var would require significant effort: • optimization/reformulation of GEM TL/AD and development of linearized physics • improved background-error covariances by using EnKF ensemble  requires synchronized development of 4D-Var and EnKF • significant redesign of variational code to facilitate major future changes to model (vertical co-ord, yin-yang, icosahedral etc.) • Use of En-4D-Var (without GEM TL/AD): • advantages of a variational analysis could be preserved by using a variational solver within EnKF (e.g., QC-var) • allows use of some alternative approaches for modelling covariances: e.g. averaged covariances • allows use of var QC • requires further research to determine if it can be made sufficiently computationally efficient (in progress)

20. Radiance assimilation and bias correction: EnKF issues L. Garand, S. MacPherson, A. Beaulne February 2-3, 2011

21. Assimilated radiances: major input in Strato-2b from new data and increased thinning Number of radiance observations assimilated February 1st, 2009 (4 analyses):

22. Issues for implementation in EnKF • Which trial to use for Bias-Cor, ensemble mean? Do different members have significant different BC characteristics? Answer: output offline cardiograms from EnKF trials • Is current vertical localization optimal for all channels? Answer: 1-ob testing of radiance assimilation under various B conditions • How to go about cloudy radiance assimilation? Partial answer: link cloud water to other variables in B Value of ensemble mean for cloud variables?

23. Assessing impact of radiance assimilation • Impact of AIRS and IASI was shown to be very significant at the MSC and other centers • No such impact noted yet in EnKF (AIRS was tested) • It would be good to compare the impact in EnKF and 4Dvar at same analysis resolution • Need tools to analyse relative impact in both systems • Ideal channel selection may differ in each system

24. Conclusion/Discussion • EnKF lags behind 4Dvar in terms of volume of radiances assimilated. In particular no IR radiances yet. • Best vehicle to rapidly increase number of assimilated data is through current system (need better computer and optimization of minimization). • No significant issue with implementing radiance Bias Correction in EnKF system (no VarBC is used at MSC) • No technical difficulty in EnKF to take into account interchannel correlations. • Open avenue of research for cloudy radiance assimilation.

25. Strengths and weaknesses of the EnKF Peter Houtekamer and Herschel Mitchell February 2-3, 2011

26. Historical review of the development of the EnKF 1994: experiment with an hemispheric barotropic model. 1997: experiment with the Marshall and Molteni quasi-geostrophic model. 2001: import of large portions of code from the 3D-Var for observation processing 2005: first operational implementation of the EnKF in the Canadian EPS 2008: inter comparison of EnKF and 4D-Var (Buehner et al., 2010-a,b), 2011: experimental configuration for a new version of the GEM stratospheric model with a new job sequencer.

27. Scalability With operations in decreasing order of importance: • GEM model integrations: 192 times more parallel than the model itself. No problem until 192 x 16 = 3072 CPU (the task is completed ~ 50 times faster than needed). • Computation of : independent per grid-point.. • Matrix inversion for each batch of observations: does not scale with more than 24 CPUs. • However, smaller regions can be considered to reduce the relative importance of this operation. • Computation of H(X) scales well up to192 CPU.

28. Sequential algorithm • Schematic illustration of the strategy used to form batches of observations. • At each assimilation step, • the circles represent the observations to be assimilated at this step, while • the x's denote observations that have not yet been assimilated.

29. Sequential algorithm • In the EnKF, batches of pmax (~1000) neighbouring observations are assimilated using a sequential algorithm. • Allows use of a direct solution method (Cholesky decomposition) for solving the analysis equation. • Computational cost increases as pmax3 and approximately linearly with number of batches. • In practice, then, more observations implies more batches.

30. Efficiency of the ensemble Kalman filter EnKF uses a sequential algorithm to solve This approach would have to be changed if the volume of data is to be doubled

31. Impact of altering the order of observations in the processing Where there are lots of observations, changing the order of the observation processing can significantly alter the result Results from one extreme case

32. Impact of having larger volumes of data • The EnKF algorithm behaves poorly when the number of observations exceeds the number of degrees of freedom of the model state • The sequential algorithm then shows a large dependence to the order in the observation processing and the ensemble then lacks dispersion • To allow for small scale structures, with the current algorithm, it would be necessary to localize even more (at the expense of the larger scales) or increase the number of members.

33. Conclusion (from Houtekamer and Mitchell) • The EnKF is by nature simple, modular and generally easy to parallelize • The B matrix estimated with the EnKF provides information about the evolution of errors during the assimilation cycle • However, to assimilate larger volumes of data, numerical and statistical considerations demands modifications or replacement of the sequential algorithm. • New avenues are being explored (variational solver)

34. Extra Slides