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Empirical Localization of Observation Impact in Ensemble Filters

Empirical Localization of Observation Impact in Ensemble Filters. Jeff Anderson IMAGe/DAReS Thanks to Lili Lei, Tim Hoar, Kevin Raeder, Nancy Collins, Glen Romine, Chris Snyder, Doug Nychka. Definition of Localization. For an observation y and state variable x ;

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Empirical Localization of Observation Impact in Ensemble Filters

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  1. Empirical Localization of Observation Impact in Ensemble Filters 5th EnKF Workshop, 23 May 2012 Jeff Anderson IMAGe/DAReS Thanks to Lili Lei, Tim Hoar, Kevin Raeder, Nancy Collins, Glen Romine, Chris Snyder, Doug Nychka

  2. Definition of Localization For an observation y and state variable x; Increments for N ensemble samples of x are: Where is a sample regression coefficient, and is a localization. Traditionally , but here there is no upper bound. 5th EnKF Workshop, 23 May 2012

  3. Empirical Localization Have output from an OSSE. Know prior ensemble and truth for each state variable. 5th EnKF Workshop, 23 May 2012

  4. Empirical Localization Have output from an OSSE. Know prior ensemble and truth for each state variable. Can get truth & prior ensemble for any potential observations. 5th EnKF Workshop, 23 May 2012

  5. Empirical Localization Estimate localization for set of observations and subset of state variables. e.g. state variables at various horizontal distances from observations. 5th EnKF Workshop, 23 May 2012

  6. Empirical Localization Example: how to localize impact of temperature observations (4 shown) on a U state variable that is between 600 and 800 km distant. 5th EnKF Workshop, 23 May 2012

  7. Empirical Localization Given observational error variance, can compute expected ensemble mean increment for state. Plot this vs prior state truth - ensemble mean. 5th EnKF Workshop, 23 May 2012

  8. Empirical Localization Do this for all state variables in subset. 5th EnKF Workshop, 23 May 2012

  9. Empirical Localization Do this for all state variables in subset. 5th EnKF Workshop, 23 May 2012

  10. Empirical Localization Do this for all state variables in subset. 5th EnKF Workshop, 23 May 2012

  11. Empirical Localization Find a least squares fit. Slope is . Least squares minimizes: Same as minimizing Posterior mean 5th EnKF Workshop, 23 May 2012

  12. Empirical Localization Define set of all pairs (y, x) of potential observations and state variable instances in an OSSE. (A state variable instance is defined by type, location and time). Choose subsets of this set. 5th EnKF Workshop, 23 May 2012

  13. Empirical Localization Find that minimizes the RMS difference between the posterior ensemble mean for x and the true value over this subset. This can be computed from the output of the OSSE. Can then use this localization in a new OSSE for all (y, x) in the subset. Call the values of localization for all subsets an Empirical Localization Function (ELF). 5th EnKF Workshop, 23 May 2012

  14. Lorenz-96 40-Variable Examples Assume all observations are located at a model grid point. (Easier but not necessary). Define 40 subsets of (y, x) pairs: x is 20 to the left, 19 to the left, … 1 to the left, colocated, 1 to the right, …, 19 to the right of y. 5th EnKF Workshop, 23 May 2012

  15. Computing ELFs Start with a climatological ensemble. Do set of 6000-step OSSEs. (only use last 5000 steps). First has no localization. Compute ELF from each. Use ELF for next OSSE. No Localization ELF1 ELF2 ELF3 ELF4 ELF5 5th EnKF Workshop, 23 May 2012

  16. Evaluation Experiments Start with a climatological ensemble. Do 110,000 step assimilation, discard first 10,000 steps. Adaptive inflation with 0.1 inflation standard deviation. Many fixed Gaspari-Cohn localizations tested for each case. Also five ELFs (or should it be ELVEs?). 5th EnKF Workshop, 23 May 2012

  17. Case 1: Frequent low-quality obs. Identity observations. Error variance 16. Assimilate every standard model timestep. 5th EnKF Workshop, 23 May 2012

  18. Case 1: Frequent low-quality obs. N=20 Gaspari Cohn (GC) function with smallest time mean prior RMSE. 5th EnKF Workshop, 23 May 2012

  19. Case 1: Frequent low-quality obs. N=20 first ELF is negative for many distances, but minimum localization is 0 when this ELF is used. 5th EnKF Workshop, 23 May 2012

  20. Case 1: Frequent low-quality obs. Subsequent N=20 ELFs are less negative, smoother, closer to best GC. 5th EnKF Workshop, 23 May 2012

  21. Case 1: Frequent low-quality obs. Subsequent N=20 ELFs are less negative, smoother, closer to best GC. 5th EnKF Workshop, 23 May 2012

  22. Case 1: Frequent low-quality obs. Subsequent N=20 ELFs are less negative, smoother, closer to best GC. 5th EnKF Workshop, 23 May 2012

  23. Case 1: Frequent low-quality obs. Subsequent N=20 ELFs are less negative, smoother, closer to best GC. 5th EnKF Workshop, 23 May 2012

  24. Case 1: Frequent low-quality obs. N=20, best GC has half-width 0.2, time mean RMSE of ~1.03. 5th EnKF Workshop, 23 May 2012

  25. Case 1: Frequent low-quality obs. N=20, best GC has half-width 0.2, time mean RMSE of ~1.03. ELFs give RMSE nearly as small as this. 5th EnKF Workshop, 23 May 2012

  26. Case 1: Frequent low-quality obs. N=20, best GC has half-width 0.2, time mean RMSE of ~1.03. ELFs give RMSE nearly as small as this. 5th EnKF Workshop, 23 May 2012

  27. Case 1: Frequent low-quality obs. Similar results for smaller ensemble, N=10. Note larger RMSE, narrower best GC half-width. 5th EnKF Workshop, 23 May 2012

  28. Case 1: Frequent low-quality obs. Similar results for larger ensemble, N=40. Note smaller RMSE, wider best GC half-width. 5th EnKF Workshop, 23 May 2012

  29. Case 1: Frequent low-quality obs. N=40 ELFs have smaller time mean RMSE than best GC. 5th EnKF Workshop, 23 May 2012

  30. Case 1: Frequent low-quality obs. ELFs are nearly symmetric so can ignore negative distances. 5th EnKF Workshop, 23 May 2012

  31. Case 1: Frequent low-quality obs. ELF for smaller ensemble is more compact. 5th EnKF Workshop, 23 May 2012

  32. Case 1: Frequent low-quality obs. ELF for larger ensemble less compact, consistent with GC results. 5th EnKF Workshop, 23 May 2012

  33. Case 1: Frequent low-quality obs. ELFs for even bigger ensembles are broader, but noisier at large distances. 5th EnKF Workshop, 23 May 2012

  34. Case 1: Frequent low-quality obs. ELFs for even bigger ensembles are broader, but noisier at large distances. 5th EnKF Workshop, 23 May 2012

  35. Case 1: Frequent low-quality obs. ELFs for even bigger ensembles are broader, but noisier at large distances. 5th EnKF Workshop, 23 May 2012

  36. Case 2: Infrequent high-quality obs. Identity observations. Error variance 1. Assimilate every 12th standard model timestep. 5th EnKF Workshop, 23 May 2012

  37. Case 2: Infrequent high-quality obs. For N=10, all ELF cases have smaller RMSE than best GC. 5th EnKF Workshop, 23 May 2012

  38. Case 2: Infrequent high-quality obs. For N=20, first ELF is worse than best GC; all others better. Best GC gets wider as ensemble size grows. 5th EnKF Workshop, 23 May 2012

  39. Case 2: Infrequent high-quality obs. For N=40, all ELFs have smaller RMSE. 5th EnKF Workshop, 23 May 2012

  40. Case 2: Infrequent high-quality obs. N=10 ELF is non-Gaussian. Has local minimum localization for distance 1. 5th EnKF Workshop, 23 May 2012

  41. Case 2: Infrequent high-quality obs. N=40 ELF is broader; also has local minimum for distance 1. Need a non-gaussian ELF to possibly do better than GC. 5th EnKF Workshop, 23 May 2012

  42. Case 3: Integral observations. Each observation is average of grid point plus its nearest 8 neighbors on both side; total of 17 points. (Something like a radiance observation.) 5th EnKF Workshop, 23 May 2012

  43. Case 3: Integral observations. Each observation is average of grid point plus its nearest 8 neighbors on both side; total of 17 points. (Something like a radiance observation.) Error variance 1. Assimilate every standard model timestep. Very low information content: Assimilate 8 of these observations for each grid point. Total of 320 observations per assimilation time. 5th EnKF Workshop, 23 May 2012

  44. Case 3: Integral observations. ELFs are not very Gaussian. No values close to 1, two peaks at distance +/- 7. 5th EnKF Workshop, 23 May 2012

  45. Case 3: Integral observations. ELFs are not very Gaussian. Best GC is much larger near the observation location. 5th EnKF Workshop, 23 May 2012

  46. Case 3: Integral observations. RMSE is a more complicated function of the GC half-width in this case. 5th EnKF Workshop, 23 May 2012

  47. Case 3: Integral observations. ELFs all have significantly smaller time mean RMSE than best GC. 5th EnKF Workshop, 23 May 2012

  48. Case 4: Frequent low-quality obs., imperfect model Identity observations. Error variance 16. Assimilate every standard model timestep. Truth has forcing F=8 (chaotic). Ensemble has forcing F=5 (not chaotic). 5th EnKF Workshop, 23 May 2012

  49. Case 4: Frequent low-quality obs., imperfect model These are the localizations for the Case 1 perfect model. 5th EnKF Workshop, 23 May 2012

  50. Case 4: Frequent low-quality obs., imperfect model Best GC is more compact for imperfect model case. Fifth ELF also more compact, but not as close to imperfect GC. 5th EnKF Workshop, 23 May 2012

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