An Investigation of Strong and Weak Constraints to Improve Variational Surface Analyses

1. Daniel Paul Tyndall 4 March 2010 Department of Atmospheric Sciences University of Utah Salt Lake City, UT An Investigation of Strong and Weak Constraints to Improve Variational Surface Analyses

2. Outline • Introduction • Literature Review • 2DVar/3DVar Analysis Methodologies • Strong and Weak Constraints • Current Progress • Analysis Equation Solution • Modifications to 2DVar analysis system • Computer Independent Analysis System • Comparison to INCA • Research Goals • Research Timeline

3. Introduction • High resolution analysis needs: • Operational weather forecasting • Wildfire management • Road maintenance operations • Air pollution management • Typical data assimilation techniques: • Cressman method • 2D variational (2DVar) and 3D variational (3DVar) methods • 4D variational (4DVar) and ensemble methods

4. Literature Review

5. Data Assimilation • 2DVar/3DVar ingredients • Observations • Background field • Background and observation error covariance matrices • Typical undersampling problem • Observation to grid point ratios: • 1.5:100 for Real-Time Mesoscale Analysis (RTMA; de Pondeca 2007) • 1.7:1000 for Integrated Nowcasting through Comprehensive Analysis (INCA)

6. The Cost Function • 2DVar and 3DVar analyses depend on the cost function: • Expanded to: background observations

7. Constraints • Goal: adding data to undersampled analysis equation • Understood balances or correlations between meteorological fields can help constrain the analysis equation • Constraints can be formulated as: • Weak constraints • Strong constraints

8. Weak Constraints • Implemented as 3rd term in cost function: • Usually takes form: • Does not force analysis to fit constraint • Sometimes constraint is an approximation • Multiple constraints can be combined into a single term • Makes solution of analysis equation more complicated

9. Strong Constraints • Implemented into cost function through: • Modification of Pb • Modification of background field • Assumes constraint is perfect • May add: • Balanced coupling between 2 assimilated fields • Error correlation to metrological parameter or topography field • Fundamental law or impose limit to analysis

10. Strong Constraint Implementations • Protat and Zawadzki (1999) • Utilized continuity equation as strong constraint • Trying to form 3D wind field through assimilation of Doppler velocities from multiple radar receivers • Gustafsson et al. (2001) • Geostrophic approximation as a strong constraint in new version of HIRLAM model • New version believed to out perform old version because of constraints

11. Strong Constraint Implementations (continued) • Žagar et al. (2004); Žagar et al. (2005) • Implemented shallow water equation model as strong constraint • Attempting to assimilate wind information in tropics

12. Weak Constraint Implementations • Protat and Zawadzki (1999) • Also used Doppler velocities from receivers as weak constraint (in addition to continuity equation strong constraint) • Analysis problem would become oversampled otherwise • Analysis method resulted in unrepresentative wind velocities • Probably due to integration technique of strong constraint

13. Weak Constraint Implementations (continued) • Xie et al. (2002) • Tested geostrophic constraints between uand vwind components and ψ and χ • Analyzing constraint impacts on mesoscale analyses • Found that constraint helpeduand vwind assimilation, but degraded mesoscale features when using ψ and χassimilation

14. Literature Review Conclusions • Poorly implemented constraints can degrade analysis • Where is all the research on mesoscale constraints? • Xie et al. (2002) and Protat and Zawadzki (1999) only ones here to look at mesoscale problems • Other mesoscale research looks at radar assimilation, but not conventional surface observation assimilation • Doesn’t seem to be a lot of research on this particular topic

15. Current Progress

16. Solving the Analysis Equation • Analysis space (used by Tyndall 2008, local analysis system [LSA]) • Observation space (Lorenc 1986, da Silva et al. 1995, to be used in this research)

17. Modified 2DVar Analysis System • Modified analysis system written in MATLAB • Like Tyndall (2008), uses Generalized Minimum Residual (GMRES) method to solve analysis equation • Why MATLAB? • Easy parallelization • Easy vectorization • Easy post processing of graphics • Intuitive debugger

18. Analysis System Improvements • Sparse matrices/covariance localization • Vectorization and parallelization • Precomputation of pbht for data denial experiments

19. Sparse Matrices and Covariance Localization • Using built-in sparse matrix data type • Test domain of 39,817 grid points and 588 observations (5-km resolution) • H is mathematically sparse • Reduction in memory: 187 MB → 0.3 MB • Pb is not mathematically sparse • Requires covariance localization (300 km) to make it sparse • PbHT reduction in memory: 187 MB → 83 MB • Optimal computation time when PbHTis converted to sparse after computation

20. Vectorization and Parallelization • Vectorization adds an order of magnitude increase in computation speed • MATLAB has easy for loop parallelization for k=1:numxb; pb_row = zeros(1,numxb); dx = radius .* cos(pi .* xb_lat ./180.) .* pi .* .. (xb_lon - xb_lon(k)) ./ 180.; dy = radius .* pi .* (xb_lat - xb_lat(k)) ./ 180.; dz = xb_felv - xb_felv(k); r2 = dx .* dx + dy .* dy; z2 = (dz .* dz); pb_row(1,:) = sigb .* (exp(-r2./rad2).*exp(-z2/radz2)); pbht(k,:) = pb_row * ht; end;

21. Pre-computation of pbht • pbht does not need to be recomputed unless: • Matrix Pb changes • Observation locations change • Optimizations decreased pbht computation time: 7 h → 7 min on 6 2-GHz cores • Data denial data set easily created by: • Single observation innovation = 0 • Particular observation error = 109

22. Operating System Independent Analysis System • MATLAB can create compiled executables • Executables can be run in UNIX, Windows, or Mac OS • Computer running executables does not need MATLAB license • Analysis system easily ported to this framework when GUI is completed • Is it worth it? • Kochanski seminar – analyses too complex

23. Analysis Domain • Proposing to investigate impacts of constraints over Austria • Why Austria? • High resolution background fields already computed and used for different analysis system (INCA) • Approximate spatially uniform observation dataset • Can compare 2DVar analyses to INCA analyses as a baseline

24. Comparison to INCA • 2DVar and INCA temperature analyses tested during 4 day Föhn period • Period selected because of high INCA errors • 2DVar found to have similar RMSE to INCA (0.1-0.2°C agreement)

25. Difference between 2DVar and INCA Temperature Analyses (0500 UTC 21 November 2007)

26. 2DVar Analysis Increments (0500 UTC 21 November 2007)

27. 2DVar Integrated Data Influence 1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0

28. Larger Differences between 2DVar and INCA… • Certain times where 2DVar does poorly compared to INCA • Why is this the case?

29. Cross Validation Results1100 UTC 23 November 2007

30. Difference between 2DVar and INCA Temperature Analyses1100 UTC 23 November 2007

31. Research Goals and Timeline

32. Research Goals • Test various strong and weak analysis constraints • Current hypotheses: • Specifying Pb using both spatial distances and potential temperature gradients will improve 2-m temperature analyses • 10-m wind analyses can be improved by added terrain-channeling constraint • Need accurate estimates of background error correlation • Using method by Lönnberg and Hollingsworth (1986); also used by Tyndall (2008) • Test hypotheses through data denial experiments and RMSE and sensitivity statistics (see Tyndall 2008)

33. Research Timeline • Project will be composed of two journal publications • First publication to be submitted summer 2010 • Comparison between INCA and 2DVar systems • Second publication to be submitted summer 2011 • Investigation of strong and weak constraints on surface variational analyses

34. Questions?