ENSEMBLE KALMAN FILTER IN THE PRESENCE OF MODEL ERRORS

1. ENSEMBLE KALMAN FILTER IN THE PRESENCE OF MODEL ERRORS Hong Li Eugenia Kalnay

2. If we assume a perfect model, we can grossly underestimate the errors imperfect model (obs from NCEP- NCAR Reanalysis NNR) ~Perfect model

3. We compare several methods to handle model errors imperfect model (obs from NCEP- NCAR Reanalysis NNR) ~perfect model

4. SPEEDY MODEL (Molteni 2003) • T30L7global spectral model • total 96x48 grid points on each level • State variables u,v,T,Ps,q • Data Assimilation: LETKF • Methods to handle model errors • Multiplicative/additive inflation • Dee & daSilva (1998) (DdS) • Low-dimensional method (LDM, Danforth et al, MWR, 2007) Dense Observations

5. Control run 100% inflation Dee & da Silva Low-order

6. Model error estimation schemes (1) 1a.Covariance inflation (multiplicative) (Ideal KF) (EnKF) 1b.Covariance inflation (additive)

7. Model error estimation schemes (DdS) 2. Dee and daSilva bias estimation scheme (1998) Do data assimilation twice: first for model error then for model state (expensive) and need to be tuned

8. Model error estimation schemes (LDM) 3. Low-dim method(Danforth et al, 2007: Estimating and correcting global weather model error. Mon. Wea. Rev) • Generate a long time series of model forecast minus reanalysis from the training period model NNR NNR NNR NNR t=0 NNR t=6hr We collect a large number of estimated errors and estimate bias, etc. Forecast error due to error in IC Time-mean model bias Diurnal model error State dependent model error

9. Further explore the Low-dimensional method Include Bias, Diurnal and State-Dependent model errors: Time-mean model bias

10. BIAS one month climatological debiased

11. Leading EOFs for 925 mb TEMP Diurnal model errors • Generate the leading EoFs from the forecast error anomalies fields for temperature. pc1 pc2 • Lack of diurnal forcing generates wavenumber 1 structure

12. 925hPa Temperature Black line: Blue line:

13. State-dependent model errors the localstate anomalies (Contour) and the forecasterror anomalies (Color) SVD2 SVD1 SVD4 SVD3

14. Correct state-dependent model errors 500hPa Uwind 500hPa Height Black line: Blue line: Univariate SVD (not account for the relations between different variables)

15. Impact of model error, and different approaches to handle it Perfect model imperfect model (obs from Reanalysis)

16. Simultaneous estimation of inflation and observation errors Hong Li Eugenia Kalnay University of Maryland

17. Motivation • Any data assimilation scheme requires accurate statistics for the observation and background errors. Unfortunately those statistics are not known and are usually tuned or adjusted by gut feeling. • Ensemble Kalman filters need inflation (additive or multiplicative) of the background error covariance, but 1)Tuning the inflation parameter is expensive especially if it is regionally dependent, and it may depend on time 2) Miyoshi and Kalnay 2005 (MK) proposed a technique to objectively estimate the covariance inflation parameter. 3) This method works, but only if the observation errors are known. • Here we introduce a method to simultaneously estimate observation errors and inflation.

18. MK method to estimate the inflation parameter (Miyoshi 2005, Miyoshi&Kalnay 2005) obs. increment in obs. space Assumption: R is known Should be satisfied if R, Pb andare correct (they are not!) So, at any given analysis time, and computing the inner product (1)

19. Diagnosis of observation error statistics (Desroziers et al, 2006, Navascues et al, 2006) Desroziers et al, 2006, introduced two new statistical relationships: if the R and B statistics are correct and errors are uncorrelated Writing their inner products we obtain two equations which we can use to “observe” R and: (2)

20. Simultaneous estimation of inflation and observation errors (1) (2) • Model : Lorenz-96 model / SPEEDY model • Perfect model scenario • Data assimilation scheme: Local ensemble transform Kalman filter(LETKF, Hunt et al. 2006) • We estimate both and R online at each analysis time

21. method method rms rms (1) 1 4.0 0.044 0.027 0.202 1.632 Tests within LETKF with Lorenz-96 model 40 observations with true Rt=1, 10 ensemble member. Optimally tuned rms=0.20 Perfect R, estimate inflation using (1) : it works Wrong R, estimate inflation using (1) : it fails (1)

22. R method method rms Tests within LETKF with L96 model (2) (1) 0.25 1.001 0.042 0.202 4.0 1.008 0.040 0.204 Now we estimate observation error and optimal inflation simultaneously using (1) and (2): it works! EstimatedR Estimated Rinit

23. Tests within LETKF with SPEEDY • SPEEDY MODEL (Molteni 2003) • Primitive equations, T30L7global spectral model • total 96x46 grid points on each level • State variables u,v,T,Ps,q

24. Tests within LETKF with SPEEDY • OBSERVATIONS • Generated from the ‘truth’ plus “random errors” with error standard deviations of 1 m/s (u), 1 m/s(v), 1K(T), 10-4 kg/kg (q) and 100Pa(Ps). • Dense observation network: 1 every 2 grid points in x and y direction • EXPERIMENTAL SETUP • Run SPEEDY with LETKF for two months ( January and February 1982) , starting from wrong (doubled) observational errors of 2 m/s (u), 2 m/s(v), 2K(T), 2*10-4 kg/kg (q) and 200Pa(Ps). • Estimate and correct the observational errors and inflation adaptively.

25. online estimated observational errors The original wrongly specified R converges to the correct R quickly (in about 5-10 days)

26. Estimation of the inflation Estimated Inflation Using an initially wrong R and but estimating them adaptively Using a perfect R and estimating adaptively After R converges, they give similar inflation factors (time dependent)

27. Global averaged analysis RMS 500hPa Temperature 500hPa Height Using an initially wrong R and but estimating them adaptively Using a perfect R and estimating adaptively

28. Summary • The online (adaptive) estimation of inflation parameter alone does not work without estimating the observational errors. • Estimating both of the observational errors and the inflation parameter simultaneously our approach works well on both the Lorenz-96 and the SPEEDY global model. It can also be applied to other ensemble based Kalman filters. • SPEEDY experiments show our approach can simultaneously estimate observational errors for different instruments. • Current work shows our method also works in the presence of random model errors.

29. A few more slides • Junjie Liu: Adaptive observations • Junjie Liu: Estimation of the impact of observations • Shu-Chih Yang: Comparison of EnKF, simple hybrid (3D-Var + Bred Vectors) and 4D-Var • Shu-Chih Yang: 4D-Var and initial and final SVs, EnKF and initial and final BVs • No cost smoother for reanalysis

30. Adaptive sampling with the LETKF-based ensemble spreadJunjie Liu • Purpose • Sample 10% adaptive DWL wind observations to get 90% improvement of full coverage • Compare ensemble spread method with other sampling strategies • How the results are sensitive to the data assimilation schemes (3D-Var and LETKF) • Note • same adaptive observations from ensemble spread method are assimilated by both 3D-Var and LETKF

31. 500hPa zonal wind RMS error Rawinsonde; climatology; uniform;random; ensemble spread; “ideal”; 100% 3D-Var LETKF RMSE • With 10% adaptive observations, the analysis accuracy is significantly improved for both 3D-Var and LETKF. • 3D-Var is more sensitive to adaptive strategies than LETKF. Ensemble spread strategy gets best result among operational possible strategies

32. 500hPa zonal wind RMS error (2% adaptive obs) Rawinsonde; climatology; uniform;random; ensemble spread; “ideal”; 100% 3D-Var LETKF • With fewer (2%) adaptive observations, ensemble spread sampling strategy outperforms the other methods in LETKF • For 3D-Var 2% adaptive observations are clearly not enough

33. Analysis sensitivity study within LETKF • The self sensitivity is the trace of the matrix S. • It can show the analysis sensitivity with respect to: • different types of observations (e.g., rawinsonde, satellite, adaptive observation and routine observations) • the observations in different area (e.g., SH, NH)

34. Analysis sensitivity of adaptive observation (one obs. selected from ensemble spread method over ocean) and routine observations (every grid point over land) in Lorenz-40 variable model 10-day forecast RMS error Analysis sensitivity • About 17% information of the analysis comes from observations over land. • About 85% information comes from observation for the adaptive observation (a single observation over ocean). • The single adaptive observation is more important than single observation over land.

35. Comparison of ensemble-based and variational-based data assimilation schemes in a Quasi-Geostrophic model.Shu-Chih Yang et al. 3D-Var Hybrid (3DVar+20 BVs) 12-hour 4D-Var LETKF (40 ensemble) 24-hour 4D-Var

36. Analysis increment (color shaded)vs. dynamically fast growing errors (contours) 12Z Day 24 00Z Day 25 Initial increment (smoother) vs. BV analysis increment vs. BV LETKF Initial increment vs. Initial SV analysis increment vs. Final SV 12-hour 4DVAR

37. 3D-LETKF time to t1 4D-LETKF No-cost LETKF smoother (cross): apply at t0 the same weights found optimal at t1, works for 3D- or 4D-LETKF

38. No-cost LETKF smoother LETKF analysis at time i LETKF Analysis Smoother analysis at time i-1 “Smoother” reanalysis

39. LETKF minimizes the errors of the day and thus provides an excellent first guess to the 3D-Var analysis 3DVar 3DVar with the background of the first 50 days provided from LETKF 3DVar with the background provided from LETKF (forecast mean) LETKF We conclude from this experiment that the errors of the day (and not just ensemble averaging) are important in LETKF and 3D-Var.