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Original Research Papers

The role of model dynamics in ensemble Kalman filter performance for chaotic systems

Authors:

Gene-Hua Crystal Ng ,

Department of Civil and Environmental Engineering, Massachusetts Institute of Technology, Cambridge, MA; U.S. Geological Survey, Menlo Park, CA, US
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Dennis McLaughlin,

Department of Civil and Environmental Engineering, Massachusetts Institute of Technology, Cambridge, MA, US
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Dara Entekhabi,

Department of Civil and Environmental Engineering, Massachusetts Institute of Technology, Cambridge, MA, US
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Adel Ahanin

Department of Civil and Environmental Engineering, Massachusetts Institute of Technology, Cambridge, MA, US
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Abstract

The ensemble Kalman filter (EnKF) is susceptible to losing track of observations, or ‘diverging’, when applied to large chaotic systems such as atmospheric and ocean models. Past studies have demonstrated the adverse impact of sampling error during the filter’s update step. We examine how system dynamics affect EnKF performance, and whether the absence of certain dynamic features in the ensemble may lead to divergence. The EnKF is applied to a simple chaotic model, and ensembles are checked against singular vectors of the tangent linear model, corresponding to short-term growth and Lyapunov vectors, corresponding to long-term growth. Results show that the ensemble strongly aligns itself with the subspace spanned by unstable Lyapunov vectors. Furthermore, the filter avoids divergence only if the full linearized long-term unstable subspace is spanned. However, short-term dynamics also become important as nonlinearity in the system increases. Non-linear movement prevents errors in the long-term stable subspace from decaying indefinitely. If these errors then undergo linear intermittent growth, a small ensemble may fail to properly represent all important modes, causing filter divergence. A combination of long and short-term growth dynamics are thus critical to EnKF performance. These findings can help in developing practical robust filters based on model dynamics.

How to Cite: Ng, G.-H.C., McLaughlin, D., Entekhabi, D. and Ahanin, A., 2011. The role of model dynamics in ensemble Kalman filter performance for chaotic systems. Tellus A: Dynamic Meteorology and Oceanography, 63(5), pp.958–977. DOI: http://doi.org/10.1111/j.1600-0870.2011.00539.x
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  Published on 01 Jan 2011
 Accepted on 9 Jul 2011            Submitted on 7 Jun 2010

References

  1. Anderson , J. L. and Anderson , S. L . 1999 . A Monte Carlo implementa-tion of the nonlinear filtering problem to produce ensemble assimila-tions and forecasts. Mon . Wea. Re v . 127 , 2884 – 2903 .  

  2. Anderson , J. L . 2001 . An ensemble adjustment Kalman filter for data assimilation. Mon . Wea. Re v . 129 , 2884 – 2903 .  

  3. Annan , J. D . 2004 . On the orthogonality of bred vectors. Mon . Wea. Re v . 132 , 843 – 849 .  

  4. Bowler , N . 2006 . Comparison of error breeding, singular vectors, ran-dom perturbations and ensemble Kalman filter perturbation strategies on a simple model . Tellus 58A , 538 – 548 .  

  5. Carme , S. , Pham , D.-T. and Verron , J . 2001 . Improving the singular evolutive extended Kalman filter for strongly nonlinear models for use in ocean data assimilation . Inverse ProbL 17 , 1535 – 1559 .  

  6. Carrassi , A. , Vannitsem , S. , Zupanski , D. and Zupanski , M . 2009 . The maximum likelihood ensemble filter performances in chaotic systems . Tellus 61A , 587 – 600 .  

  7. Cohn , S. E. and Todling , R ., 1996 . Approximate data assimilation schemes for stable and unstable dynamics . J. Meteor Soc. Jpn , 74 , 63 – 75 .  

  8. Evensen , G . 1994 . Sequential data assimilation with a nonlinear quasi-geostrophic model using Monte Carlo menthods to forecast error statistics . J. Geophys. Res . 99 , 10143 – 10162 .  

  9. Evensen , G . 2003 . The ensemble Kalman filter: Theoretical formulation and practical implementation . Ocean Dyn . 53 , 343 – 367 .  

  10. Farrell , B. F. and Ioannou , P . J. 2001. State estimation using a reduced-order Kalman filter. J. Atmos. Sci . 58 , 3666-368 0 .  

  11. Fukumori , I. and Malanotte-Rizzoli , P . 1995 . An approximate Kalman filter for ocean data assimilation: an example with an idealized Gulf Stream model . J. Geophys. Res . 100 , 6777 – 6793 .  

  12. Furrer , R. and Bengtsson , T . 2007 . Estimation of high-dimensional prior and posterior covariance matrices in Kalman filter variants . J. Multi-variate Anal . 98 , 227 – 255 .  

  13. Hamill , T. M. , Whitaker , J. S. and Snyder , C. 2001. Distance-dependent filtering of background error covariance estimates in an ensemble Kalman filter. Mon. Wea. Rev. 129 , 2776 - 2790 .  

  14. Hansen , J. A. and Smith , L. A. 2000. The role of operational con-straints in selecting supplementary observations. J. Atmos. Sci. 57 , 2859 - 2871 .  

  15. Heemink , A. W. , Verlaan , M. and Segers , A . J. 2001. Variance reduced ensemble Kalman filtering . Mon. Wea. Rev . 129 , 1718– 1728 .  

  16. Houtekamer , P. L. and Mitchell , H. L . 1998 . Data assimilation using an ensemble Kalman filter technique. Mon . Wea. Re v . 126 , 796 – 811 .  

  17. Houtekamer , P. L. and Mitchell , H. L . 2001 . A sequential ensemble Kalman filter for atmospheric data assimilation. Mon . Wea. Re v . 129 , 123 – 137 .  

  18. Lawson , W. G. and Hansen , J. A. 2004. Implications of stochastic and deterministic filters as ensemble-based data assimilation methods in varying regimes of error growth . Mon. Wea. Rev . 132 , 1966– 1981 .  

  19. Legras , B. and Vautard , R . 1996 . A guide to Lyapunov vectors. In: Semi-nar on Predictability . ECMWF, Reading, United Kingdom, 143 - 156 .  

  20. Lermusiaux , P. E J. and Robinson , A. R. 1999. Data assimilation via error subspace statistical estimation. Part I: Theory and schemes. Mon. Wea. Rev. 127 , 1385 - 1407 .  

  21. Lorenz , E . 1996 . Predictability: A problem partly solved. In: Proc. Sem-inar on Predicability . Vol. 1, ECMWF, Reading, United Kingdom, 1 - 18 .  

  22. Lorenz , E. and Emanuel , K. A . 1998 . Optimal sites for supplementary weather observations: simulation with a small model. J. Atmos. Sc i . 55 , 399 – 414 .  

  23. Molteni , F. , Buizza , R. , Palmer , T. N. , and Petroliagis , T . 1996 . The ECMWF ensemble prediction system: Methodology and validation . Quart. J. Roy. Meteor Soc . 122 , 73 – 119 .  

  24. Oseledec , V. I . 1968 . A multiplicative ergodic theorem: Lyapunov char-acteristic numbers for dynamical systems . Trans. Moscow Math. Soc . 19 , 197 – 231 .  

  25. Ott , E. , Hunt , B. R. , Szunyogh , I. , Zimin , A. V. , Kostelich , E. J. , Corazza , M. , Kalnay , E. , Patil , D. J. and Yorke , J. A. 2004. A local ensemble Kalman filter for atmospheric data assimilation. Tellus 56A , 415 - 428 .  

  26. PazO , D. , Rodriguez , M. A. and Lopez , J. M. 2010. Spatio-temporal evolution of perturbations in ensembles intializaed by bred, Lya-punov and singular vectors. Tellus 62A , 10 - 23 . https://doi.org/10.1111/j.1600-0870.2009.00419.x .  

  27. Pham , D. T. , Verron , J. and Roubaud , M. C. 1998. A singular evolutive extended Kalman filter for data assimilation in oceanography. J. Mar SysL 16 , 323 - 340 .  

  28. Pires , C. , Vautard , R. and Talagrand , O. 1996. On extending the limits of variational assimilation in nonlinear chaotic systems. Tellus 48A , 96 - 121 .  

  29. Sacher , W. and Bartell° , P . 2008 . Sampling errors in ensemble Kalman filtering, Part I: Theory. Mon . Wea. Re v . 136 , 3035 – 3049 .  

  30. Sacher , W. and Bartell° , P . 2009 . Sampling errors in ensemble Kalman filtering, Part II: Application to a barotropic model. Mon . Wea. Re v . 137 , 1640 – 1654 .  

  31. Sakov , P. and Oke , P. R . 2008 . Implications of the form of the ensemble transformation in the ensemble square root filter. Mon . Wea. Re v . 136 , 1042 – 1053 .  

  32. Snyder , C. and Hamill , T. M . 2003 . Leading Lyapunov vectors of a turbulent baroclinic jet in a quasigeostrophic model. J. Atmos. Sc i . 60 , 683 – 688 .  

  33. Tippett , M. K. , Anderson , J. L. , Bishop , C. H. , Hamill , T. M. and Whitaker , J. S. 2003. Ensemble square root filters . Mon. Wea. Rev. 131 , 1485 - 1490 .  

  34. Toth , Z. and Kalnay , E . 1993 . Ensemble forecasting at NMC: The generation of perturbations . Bull. Amer. Meteor Soc . 74 , 2317 – 2330 .  

  35. Toth , Z. and Kalnay , E . 1997 . Ensemble forecasting at NCEP and the breeding method. Mon . Wea. Re v . 125 , 3297 – 3319 .  

  36. Trevisan , A. , D’Isidoro , M. and Talagrand , O. 2010. Four-dimensional variational assimilation in the unstable subspace and the optimal sub-space dimension. Q. J. R. Meteor. Soc . 136 , 487 - 496 .  

  37. Trevisan , A. and Palatella , L . 2011 . On the Kalman Filter error covari-ance collapse into the unstable subspace . Nonlin. Proc. Geophys . 18 , 243 – 250  

  38. Trevisan , A. and Pancotti , F . 1998 . Periodic orbits, Lyapunov vectors, and singular vectors in the Lorenz system. J. Atmos. Sc i . 55 , 390 – 398 .  

  39. Trevisan , A. and Uboldi , E 2004 . Assimilation of standard and targeted observations within the unstable subspace of the observation-analysis-forecast cycle system. J. Atmos. Sc i . 61 , 103 – 113 .  

  40. van Leeuwen , P . J. 1999. Comment on “Data assimilation using an ensemble Kalman filter technique.” Mon. Wea. Rev . 127 , 1374-137 7 .  

  41. Vannitsem , S. and Nicolis , C . 1997 . Lyapunov vectors and error growth patterns in a T21L3 quasigeostrophic model. J. Atmos. Sc i . 54 , 347 – 361 .  

  42. Verlaan , M. and Heemink , A. W . 1995 . Data assimilation schemes for non-linear shallow water flow models . Stoch. Hydro. HydrauL 11 , 349 – 368 .  

  43. Whitaker , J. S. and Hamill , T. M. 2002. Ensemble data assimilation without perturbed observations . Mon. Wea. Rev . 130 , 1913– 1924 .  

  44. Wolfe , C. L. and Samelson , R. S . 2007 . An efficient method for recover-ing Lyapunov vectors from singular vectors . Tellus 59A ( 3 ), 355 – 366 . https://doi.org/10.1111/j.1600-0870.2007.00234.x .  

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