Variational Data Assimilation

1. Variational Data Assimilation Hans Huang NCAR Acknowledge: Dale Barker, The WRFDA team, MMM and DATC staff AFWA, NSF, KMA, CWB, BMB, AirDat

2. Outline • Introduction to data assimilation. • Basics of variational data assimilation. • Demonstration with a simple system. • The WRFDA approach.

3. Modern weather forecast (Bjerknes,1904) • A sufficiently accurate knowledge of the state of the atmosphere at the initial time • A sufficiently accurate knowledge of the laws according to which one state of the atmosphere develops from another. • Analysis: using observations and other information, we can specify the atmospheric state at a given initial time: “Today’s Weather” • Forecast: using the equations, we can calculate how this state will change over time: “Tomorrow’s Weather” Vilhelm Bjerknes (1862–1951) (Peter Lynch)

4. initial state forecast Model Analysis Model state x, ~107 Observations yo, ~105-106

6. Assimilation methods Empirical methods Successive Correction Method (SCM) Nudging Physical Initialisation (PI), Latent Heat Nudging (LHN) Statistical methods Optimal Interpolation (OI) 3-Dimensional VARiational data assimilation (3DVAR) 4-Dimensional VARiational data assimilation (4DVAR) Advanced methods Extended Kalman Filter (EKF) Ensemble Kalman Filter (EnFK)

7. Outline • Introduction to data assimilation. • Basics of variational data assimilation. • Demonstration with a simple system. • The WRFDA approach.

8. The analysis problem for a given time

9. BLUE: the Best Liner Unbiased Estimate

10. The analysis value should be between background and observation. If B is too small, observations are less useful. If R can be tuned, analysis can fit observations as close as one wants! Statistically, analyses are better than background. Statistically, analyses are better than observations!

11. 3D-Var

12. Sequential data assimilation (I)

13. Sequential data assimilation (II)

14. Sequential data assimilation (III)

15. Sequential data assimilation (IV) 4DVAR (continue)

16. Scalar to Vector Observation operator H

17. H - Observation operator H maps variables from “model space” to “observation space” x y • Interpolations from model grids to observation locations • Extrapolations using PBL schemes • Time integration using full NWP models (4D-Var in generalized form) • Transformations of model variables (u, v, T, q, ps, etc.) to “indirect” observations (e.g. satellite radiance, radar radial winds, etc.) • Simple relations like PW, radial wind, refractivity, … • Radar reflectivity • Radiative transfer models • Precipitation using simple or complex models • … !!! Need H, H and HT, not H-1 !!!

18. Level of preprocessing and the observation operator H=HFHGHRHN Raw data: Phase and amplitude Frequency relations Ionosphere corrected observables H=HGHRHN Geometry H=HRHN Bending angle profiles Abel trasform or ray tracing H=HN Refractivity profiles Hydrostatic equlibrium and equation of state H=I Temperature profiles

19. Sequential data assimilation

20. Issues on variational data assimilation

21. Outline • Introduction to data assimilation. • Basics of variational data assimilation. • Demonstration with a simple system. • The WRFDA approach.

22. The Lorenz 1964 equation

23. Issues on data assimilation for the system based on the Lorenz 64 equation

24. Too simple?Try: Lorenz63 modelTry: 1-D advection equationTry: 2-D shallow water equation…Try: ARW!!!

25. Outline • Introduction to data assimilation. • Basics of variational data assimilation. • Demonstration with a simple system. • The WRFDA approach. WRFDA is a Data Assimilation system built within the WRF software framework, used for application in both research and operational environments….

26. Goal: Community WRF DA system for • regional/global, • research/operations, and • deterministic/probabilistic applications. • Techniques: • 3D-Var • 4D-Var (regional) • Ensemble DA, • Hybrid Variational/Ensemble DA. • Model: WRF (ARW, NMM, Global) • Support: WRFDA team • NCAR/ESSL/MMM/DAG • NCAR/RAL/JNT/DATC • Observations: Conv.+Sat.+Radar WRF-Var (WRFDA) Data Assimilation Overview

27. KMA Pre-operational Verification: (with/without radar) Threat Score Bias WRFDA Observations • In-Situ: • Surface (SYNOP, METAR, SHIP, BUOY). • Upper air (TEMP, PIBAL, AIREP, ACARS, TAMDAR). • Remotely sensed retrievals: • Atmospheric Motion Vectors (geo/polar). • Ground-based GPS Total Precipitable Water. • SSM/I oceanic surface wind speed and TPW. • Scatterometer oceanic surface winds. • Wind Profiler. • Radar radial velocities and reflectivities. • Satellite temperature/humidities. • GPS refractivity (e.g. COSMIC). • Radiative Transfer: • RTTOVS (EUMETSAT). • CRTM (JCSDA).

28. Model-Based Estimation of Climatological Background Errors • Assume background error covariance estimated by model perturbations x’ : Two ways of defining x’: • The NMC-method (Parrish and Derber 1992): where e.g. t2=24hr, t1=12hr forecasts… • …or ensemble perturbations (Fisher 2003): • Tuning via innovation vector statistics and/or variational methods.

29. Single observation experiment- one way to view the structure of B The solution of 3D-Var should be Single observation

30. Example of B3D-Var response to a single ps observation Pressure, Temperature Wind Speed, Vector, v-wind component.

31. Examples of B T increments : T Observation (1 Deg , 0.001 error around 850 hPa) B from NMC method B from ensemble method

32. Sensitivity to Forecast Error Covariances in Antarctica(Rizvi et al 2006) 60km Verification (vs. radiosondes) T+24hr Temperature 60km Domain 1 GFS-based errors 20km Domain2 WRF-based errors

33. Incremental WRFDA Jb Preconditioning • Define preconditioned control variablev space transform where U transform CAREFULLY chosen to satisfy Bo = UUT. • Choose (at least assume) control variable components with uncorrelated errors: • where n~number pieces of independent information.

34. WRFDA Background Error Modeling Up: Change of variable, impose balance. Uv: Vertical correlations EOF Decomposition Uh: RF = Recursive Filter, e.g. Purser et al 2003

35. WRFDA Background Error Modeling Up: Change of variable, impose balance. Uv: Vertical correlations EOF Decomposition Uh: RF = Recursive Filter, e.g. Purser et al 2003

36. WRFDA Background Error Modeling Up: Change of variable, impose balance. Uv: Vertical correlations EOF Decomposition Uh: RF = Recursive Filter, e.g. Purser et al 2003

37. WRFDA Background Error Modeling Define control variables: Up: Change of variable, impose balance. Uv: Vertical correlations EOF Decomposition Uh: RF = Recursive Filter, e.g. Purser et al 2003

38. WRFDA Statistical Balance Constraints • Define statistical balance after Wu et al (2002): • How good are these balance constraints? Test on KMA global model data. Plot correlation between “Full” and balanced components of field:

39. WRF 4D-Var milestones 2003: WRF 4D-Var project. 2004: WRF SN (simplified nonlinear model). Modifications to WRF 3D-Var. 2005: TL and AD of WRF dynamics. WRF TL and AD framework. WRF 4D-Var framework. 2006: The WRF 4D-Var prototype. Single ob and real data experiments. Parallelization of WRF TL and AD. Simple physics TL and AD. JcDF 2007: The WRF 4D-Var basic system. 2008: Further optimization and testing

40. TL00 TLDF WRFINPUT WRF WRF_NL WRF_TL WRF_AD VAR Outerloop Mk dk Innerloop U UT call AF(K),…,AF(1) Basic system: 3 exes, disk I/O, parallel, full dyn, simple phys, JcDF I/O xb B WRFBDY call NL(1),…,NL(K) R y1 … yK WRF+ BS(0),…,BS(N) TL(1),…,TL(K) call xn AD00

41. Why 4D-Var? • Use observations over a time interval, which suits most asynoptic data and use tendency information from observations. • Use a forecast model as a constraint, which enhances the dynamic balance of the analysis. • Implicitly use flow-dependent background errors, which ensures the analysis quality for fast developing weather systems. • NOT easy to build and maintain!

42. Radiance Observation Forcing at 7 data slots

43. Single observation experiment The idea behind single ob tests: The solution of 3D-Var should be Single observation 3D-Var  4D-Var: HHM; HHM; HTMTHT The solution of 4D-Var should be Single observation, solution at observation time

44. Analysis increments of 500mb q from 3D-Var at 00h and from 4D-Var at 06h due to a 500mb T observation at 06h + + 3D-Var 4D-Var

45. + OBS 500mb q increments at 00,01,02,03,04,05,06h to a 500mb T ob at 06h

46. + OBS 500mb q difference at 00,01,02,03,04,05,06h from two nonlinear runs (one from background; one from 4D-Var)

47. + OBS 500mb q difference at 00,01,02,03,04,05,06h from two nonlinear runs (one from background; one from FGAT)

48. Real Case: Typhoon HaitangExperimental Design (Cold-Start) • Domain configuration: 91x73x17, 45km • Period: 00 UTC 16 July - 00 UTC 18 July, 2005 • Observations from Taiwan CWB operational database. • 5 experiments are conducted: • FGS – forecast from the background [The background fields are 6-h WRF forecasts from National Center for Environment Prediction (NCEP) GFS analysis.] • AVN- forecast from the NCEP AVN analysis • 3DVAR – forecast from WRF-Var3d using FGS as background • FGAT - forecast from WRF-Var3dFGAT using FGS as background • 4DVAR – forecastfrom WRF-Var4d using FGS as background

49. Typhoon Haitang 2005

50. A KMA Heavy Rain Case Period: 12 UTC 4 May - 00 UTC 7 May, 2006 Assimilation window: 6 hours Cycling (6h forecast from previous cycle as background for analysis) All KMA operational data Grid : 60x54x31 Resolution : 30km Domain size: the same as the KMA operational 10km domain.