1. Space and Time Multiscale Analysis System A sequential variational approach Yuanfu Xie, Steven Koch Steve Albers and Huiling Yuan Global Systems Division Earth System Research Laboratory 5th Oceanic D-A Workshop9/22/2009

2. Outline Applications; Sequential variational analysis approach; Multigrid implementation; Numerical results; Summary. 5th Oceanic D-A Workshop9/22/2009

3. LAPS III Configuration Data Ingest Intermediate data files LAPSGSI Data STMAS3D Trans Post proc1Post proc2Post proc3 Model prep WRF-ARWMM5WRF-NMM Forecast Verification Error Covariance 5th Oceanic D-A Workshop9/22/2009

4. STMAS Applications FAA/MIT boundary detection; MDL boundary detection/nowcasting; Storm Prediction Center nowcasting; Central Weather Bureau reanalysis; AOML/ESRL hurricane data assimilation. National Marine Data Information Service for SST and height analysis; Tornado applications. 5th Oceanic D-A Workshop9/22/2009

5. It is derived from a statistical analysis assuming the error’s distribution is Gaussian. It solves a variational problem: subject to a model constraint for 4DVAR Single 3-4DVAR approach 5th Oceanic D-A Workshop9/22/2009

6. A Lognormal 1DVAR PDF: Cost function: 5th Oceanic D-A Workshop9/22/2009

7. Error Covariance  The critical issue is the accuracy of the background error covariance matrix B.  It is hard to obtain it.  It usually has a size of about 10 6 x 10 6 /2.  It is time, location, and flow dependent.  Model forecast differences are usually used to construct the covariance but model bias is missed from the covariance. 5th Oceanic D-A Workshop9/22/2009

8. Background Observation Longer wave Background Observation Longer wave Resolvable Information for a Given Observation Network Difference on longer waveDifference on shorter wave 5th Oceanic D-A Workshop9/22/2009

9. STMAS STMAS is implemented in two steps. 1.It retrieves the resolvable observation information. 2.After the resolvable information retrieved, STMAS is reduced to a standard statistical variational analysis With long waves retrieved, STMAS deals with a localized error covariance, a banded matrix, at its last phase of analysis. 5th Oceanic D-A Workshop9/22/2009

10. An idealized multiscale case Left: Mesonet surface stations; Right: An analysis function 5th Oceanic D-A Workshop9/22/2009

11. An idealized multiscale case (cont.) 5th Oceanic D-A Workshop9/22/2009

12. A Recursive Filter 3DVAR A single 3DVAR And B is approximated by a recursive filter (Hayden and Purser 95): 5th Oceanic D-A Workshop9/22/2009

13.  =0.5 0.7 0.9 A single 3DVAR with different  These analyses tend to approximate the truth: 5th Oceanic D-A Workshop9/22/2009

14. Recursive filter version of STMAS A sequential variational analysis implemented through a recursive filter. 1.Solve the VAR with large , e.g. 0.999; 2.Subtract the analysis from observation values used in previous VAR analysis; 3.Reduce  by a fraction, say  in (0.5,1); 4.Return to 1 if it is necessary; 5.Add the previous analyses together. 5th Oceanic D-A Workshop9/22/2009

15. Comparison: Single 3DVAR With  =0.5 or 0.9 STMAS-RF 5th Oceanic D-A Workshop9/22/2009

16. Approximation Theory Any smooth function can be decomposed by a series of base functions that is complete. where  k can be Fourier based functions, wavelets, or any smooth function with decreasing scales with k. STMAS uses a sequence of truncated series to approximate the the information contained in obs and background 5th Oceanic D-A Workshop9/22/2009

17. Advantages  STMAS is an iterative variational analysis;  It retrieves longer wave information that can be resolved by the observation network;  It is a variational generalization of a single 3-4DVAR and it can handle advanced data and observation errors globally;  Various balances can be applied at different STMAS levels, e.g., geostrophic on large scale analysis; hydrostatic over small ones. 5th Oceanic D-A Workshop9/22/2009

18. Multigrid Technique Using the number of gridpoints to control the base functions. 5th Oceanic D-A Workshop9/22/2009

19. STMAS Multigrid Implementation The number of grid points over a given domain determines the shortest wavelength allowed. A multigrid uses the number of grid points to control the wavelength. STMAS solves its variational problem over the coarsest grid and obtains observation information for longest waves. By gradually increasing the number of grid points, STMAS multigrid gains shorter waves by each iterations. 5th Oceanic D-A Workshop9/22/2009

20. An efficient analysis system  Since the multigrid determines the wavelength, there is no correlation involved in STMAS variational analysis over a given grid.  Only computation for the cost function is simple interpolations.  An STMAS 5 km surface analysis of 6 state variables over eastern US (two third of CONUS) using recursive filter spends 15 minutes; A multigrid STMAS analysis could take about 40 seconds. 5th Oceanic D-A Workshop9/22/2009

21. Different Implementations of STMAS Recursive filterWaveletMultigrid 5th Oceanic D-A Workshop9/22/2009

22. Real time analysis STMAS runs a real time analysis every 15 minutes over a domain over the east of US. It provides wind, temperature, pressure, dew-point, and many other analyzed fields. http://laps.noaa.gov/request/nph-laps.cgi 5th Oceanic D-A Workshop9/22/2009

23. FAA/MIT Boundary Detection Application 5th Oceanic D-A Workshop9/22/2009

24. STMAS for frontal boundary detection 5th Oceanic D-A Workshop9/22/2009

25. STMAS vs. HPC 5th Oceanic D-A Workshop9/22/2009

26. STMAS Dennis analysis (850mb) STMAS (V) Background (V) 5th Oceanic D-A Workshop9/22/2009

27. STMAS Dennis analysis (850mb) STMAS (U) STMAS wind 5th Oceanic D-A Workshop9/22/2009

28. STMAS: A typhoon test Analysis of radial+derived wind Substract (-) Derived wind (where it is available) uv Analysis of Derived wind Substract (-) Derived wind (where it is available) 5th Oceanic D-A Workshop9/22/2009

29. Intensity: WRF Katrina forecast by STMAS Wind Barb, Windspeed image, Pressure contour at 950mb Surface pressure j 5th Oceanic D-A Workshop9/22/2009

30. KATRINA LAPS / HRS(2005) 5th Oceanic D-A Workshop9/22/2009

31. Windsor tornado case, 22 May 2008 Tornado touched down at Windsor, Colorado around 17:40 UTC, 22 May 2008 STMAS initialization 1.67 km 301 x 313 background model: RUC 13km, 17 UTC hot start (cloud analysis) Boundary conditions: RUC 13km, 3-h RUC forecast (initialized at 15 UTC) WRF-ARW 1.67 km, 1-h forecast Thompson microphysics Postprocessing: Reflectivity 9/22/20095th Oceanic D-A Workshop

32. 00-01hr 800mb wind initialized at 17 UTC 22 May 2005, STMAS analysis vs. WRF forecast (STMAS) 9/22/20095th Oceanic D-A Workshop

33. 00-01hr wind cross-section initialized at 17 UTC 22 May 2005, STMAS analysis vs. WRF forecast (STMAS) 9/22/20095th Oceanic D-A Workshop

34. 00-01hr 800mb reflectivity initialized at 17 UTC 22 May 2005, mosiac radar vs. WRF forecast (STMAS) 9/22/20095th Oceanic D-A Workshop

35. 00-01hr reflectivity cross-section initialized at 17 UTC 22 May 2005, mosiac radar vs. WRF forecast (STMAS) 9/22/20095th Oceanic D-A Workshop

36. 00-01hr reflectivity cross-section initialized at 17 UTC 22 May 2005, WRF forecast, RUC vs. STMAS 9/22/20095th Oceanic D-A Workshop

37. Future Modeling Considerations Resolution (horizontal & vertical) Microphysics Scheme Reflectivity Calculation 9/22/20095th Oceanic D-A Workshop

38. Future of Multigrid STMAS At each multigrid level, STMAS solves a single 4DVAR just like other 4DVARs; At coarser grids, its adjoint of the 4DVARs exists and the adjoint is used; At finer grid where there is no adjoint, a non- differentiable optimization method will be used; At its finest grid, an EnKF can be applied. 5th Oceanic D-A Workshop9/22/2009

39. Summary  STMAS-MG (multigrid) is an efficient, multiscale analysis system;  It can use all possible data sources, including radar, satellite, SFMR and so on;  It can also impose balances or constraints to its analysis directly;  STMAS will become a sequential 4DVAR. 5th Oceanic D-A Workshop9/22/2009

40. Comparison of the responses For each response function of a single 3DVAR, a  can be found such that its response is close to the 3DVAR one. 5th Oceanic D-A Workshop9/22/2009

41. Composite of two corrections The responses of two corrections of both Barnes and 3DVAR: 5th Oceanic D-A Workshop9/22/2009

42. STMAS/LSI Boundary Detection 5th Oceanic D-A Workshop9/22/2009