1. Toward a Real Time Mesoscale Ensemble Kalman Filter Gregory J. Hakim Dept. of Atmospheric Sciences, University of Washington Collaborators: Ryan Torn (UW) Sebastien Dirren (UW) Chris Snyder (NCAR) “Analysis PDF of Record” http://www.atmos.washington.edu/~hakim

2. Two Distinct AOR Priorities 1)NDFD forecast verification. Nationwide analyses; no critical delivery time? An a posteriori approach could use max data. Better centralized for uniformity? No distribution costs; no hard deadlines. 2) Real-time mesoscale analyses & forecasts. Regional analyses & short (<12 h) forecasts. Delivery time critical; use available data. A distributed/regional approach is helpful? DA community resource (cf. MM5, WRF, etc.) EnKF appears well suited.

3. Summary of Ensemble Kalman Filter (EnKF) Algorithm (1)Ensemble forecast provides background estimate & statistics (P b ) for new analyses. (2)Ensemble analysis with new observations. (3) Ensemble forecast to arbitrary future time.

4. Strengths & Weakness of EnKF Probabilistic analyses & probabilistic forecasts. –prob. forecasts widely embraced. –prob. analyses don’t yet exist. –account for ensemble variance in NDFD verification? Straightforward implementation; ~parallelization. Do not need –background error covariance models. –adjoint models (cf. 4DVAR). Weakness: Rank deficient covariance matrices. –ensemble may need to be very large. –Many ways to boost rank for small ensembles O(100).

5. Synoptic Scale Example Weather Research and Forecasting Model (WRF). –100 km grid spacing; 28 vertical levels. Assimilate 250 surface pressure obs ONLY. Perfect model assumption. –Observations = truth run + noise.

6. Surface Pressure Errors

7. ~500 mb Height Errors

8. Boundary Conditions (Ryan Torn)

9. Ensemble Surface Pressure & 

10. Ensemble 500 hPa Height & PV

11. Surface Cov(P, P low ) & Cov (V, P low )

12. Cov(Z 500, P low ) & Cov (V 500, P low )

13. Covariance Convergence

14. 500 mb  Covariance, N e = 20

15. 500 mb  Covariance, N e = 40

16. 500 mb  Covariance, N e = 60

17. 500 mb  Covariance, N e = 80

18. 500 mb  Covariance, N e = 100

19. Mesoscale Examples 12 km grid spacing, 38 vertical levels. 3-class microphysics. TKE boundary layer scheme. 60 ensemble members. Assimilate surface pressure observations. –Hourly observations. –Drawn from truth run plus noise. –Realistic surface station distribution.

20. Observation Network

21. Surface Pressure Error Snapshot

22. Mesoscale Covariances Camano Island Radar|V 950 |-q r covariance 12 Z January 24, 2004

23. Surface Pressure Covariance OceanLand

24. Toward a Real-Time Mesoscale EnKF Prototype Surface observations (U,V,T,RH, green) Radiosondes (U,V,T,RH) Scatterometer winds (U,V over ocean, red) ACARS (U,V,T)

25. Summary Ensemble Kalman filter AOR opportunities: –Ensemble mesoscale analyses & short-term forecasts. –Lowers barriers-to-entry for DA. –Regional DA (prototype in progress at UW). –Community DA resource (cf. MM5, WRF). Background Error Covariances: –Automatic & flow dependent with EnKF. Cloud field analyses no more difficult than, e.g., 500 hPa height. Optimal ensemble size? –Vary strongly in space & time. Difficult to assume mesoscale covariances, unlike synoptic scale.

26. THE END

27. EnKF Sampling Issues Problem #1: “under-dispersive” ensembles. overweight background relative to observations. Solution: Inflate K by a scalar constant. Problem #2: spurious far-field covariances. affect analysis far from observation. Solution: Localize K with a window function.

28. Computational & Plotting Domains

29. Analysis-update Equation analysis = prior + weighted observations

30. Traditional Kalman Filter Problem A forecast of P b is needed for next analysis. Problem : P b is huge (N x N) and cannot be evolved directly. Solution : estimate P b from an ensemble forecast. “Ensemble” KF (EnKF).

31. Current DA and Ensemble Forecasting 3D/4Dvar: P b is ~ flow independent. –Assumed spatial influence of observations. –Assumed field relationships (e.g. wind—pressure balance). –P b assumptions for mesoscale are less clear. –Deterministic: a single analysis is produced. Ensemble forecasts –perturbed deterministic analyses (SVs, bred modes). EnKF: unifies DA & ensemble forecasting.

32. Kalman Gain

33. Application of Localization

34. Ensemble Tropopause 

35. Cov(  trop, P low ) & Cov (V trop, P low )

36. Synoptic Observation Network

37. Application to an Extratropical Cyclone 23 March 2003.

38. Motivation Probabilistic forecasts well accepted. –e.g. forecast ensembles. Genuine probabilistic analyses are lacking. –singular vectors & bred modes are proxies. Solving this problem creates opportunities. –probabilities: structural & dynamical information. –old: dynamics  data assimilation. –new: data assimilation  dynamics.

39. Ensemble Statistics Ensemble-estimated covariance between x and y: cov(x, y)  (x – x) (y – y) T. Here, we normalize y by  (y). cov(x, y) has units of x. linear response in x given one-  change in y. take y = surface pressure in the low center.

40. Mesoscale Challenges Cloud fields & precipitation. –No time for “spin up.” Complex topography. Background covariances vary strongly in space & time. –can’t rely on geostrophic or hydrostatic balance. Boundary conditions on limited-area domains.