Ensemble Data Assimilation with the NCEP Global Forecasting
Ensemble Data Assimilation with the NCEP Global Forecasting System Jeff Whitaker, Tom Hamill, Xue Wei (ESRL/PSD) Zoltan Toth, Yucheng Song, Richard Wobus (NCEP)
Advantages of EDA • Asymptotes to optimal solution (KF) when all sources of error are accurately sampled (large ensemble, perfect model, known R). • Provides a distribution - automatic initialization of ensemble forecasts. • Algorithmically simple and easy to code (compared to 4 DVar) - but computationally expensive (but no more than 4 DVar).
Example: Sparse Network (Ps obs only) Whitaker et al 2004, MWR p. 1190 Full NCEP-NCAR Reanalysis (3 DVar) (200, 000+ obs) Ensemble Filter (214 surface pressure obs) Black dots show pressure ob locations RMS = 39. 8 m Climatological covariances (214 surface pressure obs) RMS = 82. 4 m (3 DVar is worse!)
Motivation • How do ensemble-based data assimilation algorithms compare with existing NCEP 3 DVar with full current observational data set? • Problem: Too expensive for us to assimilate radiances, and run at operational (T 254) resolution. • Compromise: compare against 3 D-Var in reduced-resolution (T 62) model with all observations except satellite radiances.
Bottom Line - T 62 En. SRF beats T 62 3 DVar Z 500 T 62 Forecast Skill 36. 4 31. 2 29. 4 27. 5 10. 4 7. 9
Bottom Line - T 126 En. SRF without radiances close to T 254 Oper 3 DVar with radiances in NH 36. 4 31. 2 26. 7 26. 4 27. 5 30. 3 29. 4 23. 5 10. 4 8. 9 8. 3 7. 9
En. SRF cycle (serial processing) Loop over analysis times: • run 6 h forecast for each ensemble member from the previous analysis Loop over observations: • Do we need this ob? (will it significantly reduce ensemble uncertainty estimate? ) If not continue to Next observation • Update the ensemble mean and the ensemble perturbations using the KF update equations. Next observation • Add variance to account for errors unaccounted for. Next analysis time
Sources of error in EDA • Sampling error (not a large enough ensemble) - covariance localization. • Errors in forward operator (H - interpolates from model state vector space to observation space) • Don’t know model error and/or observation error perfectly. In practice, all of these (especially last two) are convolved and difficult to separate
Implementation details • Accounting for sampling error (cov local) – – • Horizontal: Montonically decreasing function that tapers to zero at 2800 km. Vertical: Zero at 2 scale heights [ such that -ln( )=2]; surface observation has no influence above ~ 135 h. Pa Accounting for model error 1) Covariance inflation (multiplicative, in ensemble subspace) – 25% in NH tapering to 15% in SH. Tapers vertically to zero at 6 scale heights. . 2) Additive covariance inflation (like standard KF formulation) – add random samples of 6 -h tendencies of reanalysis. Balanced, look like synoptic scale growing structures. – Relaxation to prior (Snyder and Zhang, MWR 2003) – relax analysis ensemble deviations back toward prior at each grid point x’a = cx’a + (1 -c)x’b
Simple Model Test: Lorenz 1996 ‘Double Ring’
Lorenz 96 results • Forcing parameterized as F = a + b*X (linear regression) • Stochastic param: Account for ‘scatter’ with AR 1 (Wilks 05) • Ob error variance 0. 25, assimilate every 0. 1 time units. exp En. SRF (P) 3 DVar covinf additive relaxprior stochastic prior err 0. 268 0. 596 0. 792 0. 652 0. 596 0. 972 0. 592 prior sprd 0. 26 0. 74 0. 544 2. 936 0. 668 post err 0. 16 0. 364 0. 436 0. 336 0. 288 0. 544 0. 292 post sprd 0. 152 0. 26 0. 284 1. 888 0. 292
Lorenz 96 results • Forcing parameterized as F = a + b*X (linear regression) • Stochastic param: Account for ‘scatter’ with AR 1 (Wilks 05) • Ob error variance 0. 25, assimilate every 0. 1 time units. exp prior err prior sprd post err post sprd En. SRF (P) 3 DVar covinf additive relaxprior 0. 268 0. 596 0. 792 0. 652 0. 596 0. 972 0. 26 0. 152 0. 74 0. 544 2. 936 0. 16 0. 364 0. 436 0. 336 0. 288 0. 544 stochastic 0. 592 0. 668 0. 292 0. 26 0. 284 1. 888
Comparison of model error parameterizations
LETKF vs En. SRF • En. SRF/serial processing - for each ob increment part of state vector influenced by that ob. No matrix inversions, can intelligently thin obs. • LETKF - for each element of state vector, compute increment based on nearby obs. Update in space of ensemble (Nx. N matrices), scales better on parallel machines when k > N.
LETKF vs serial En. SRF
Conclusions • Experimental En. SRF outperforms operational 3 DVar run at same resolution with same subset of observations. The sparser the network, the bigger the advantage for EDA (SH, historical reanalysis). • En. SRF at T 126 without radiances is comparable to operational SSI at T 254 with radiances. • Additive model error parameterization works best. • Next: – – – Compare with LETKF (underway) Assimilate radiances Techniques for ‘super-obbing’ Model error/bias correction (never done!) Parallel testing on NCEP machine?
Radiosonde Fits (6 -h forecast)
Radiosonde Fits (6 -h forecast)
Ensemble Square-Root Filter (En. SRF; Whitaker and Hamill, MWR ‘ 02) background-error covariances estimated from ensemble, with covariance localization. Mean state updated, correcting background to new observations, weighted by K, the Kalman gain “reduced” Kalman gain calculated to update perturbations around mean Forecast forward to the next time when data is available. Optionally, add noise or inflate to simulate model error.
Effect of Flow-Dependent Covariances 3 D-Var Ensemble Filter
Covariance localization: a way of dealing with inappropriate covariance estimates due to small ensemble size. from Hamill review paper, to appear in upcoming Cambridge Press book. See also Houtekamer and Mitchell, MWR, March 1998
500 h. Pa geopotential height Air Weather Service analysis valid 0400 GMT 27 December 1947 Ensemble Filter analysis has comparable quality to NCEP 3 D-VAR 5500 m (18000 ft) contour is thickened Ensemble Filter T 62 analysis valid 0600 GMT NCEP T 254 3 hr forecast valid 0600 GMT
Additive “model” errors from 6 -h tendencies spread mostly added in storm tracks.
Benefits of Flow-Dependent Background Errors Example 1: Fronts Example 2: Hurricanes
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