• \
    Start Submission Become a Reviewer

    Reading: The maximum likelihood ensemble filter performances in chaotic systems

    Download

     A- A+
    Alt. Display

    Original Research Papers

    The maximum likelihood ensemble filter performances in chaotic systems

    Authors:

    Alberto Carrassi ,

    Institut Royal Météorologique de Belgique, Av. Circulaire 3, 1180 Brussels, BE
    X close

    Stephane Vannitsem,

    Institut Royal Météorologique de Belgique, Av. Circulaire 3, 1180 Brussels, BE
    X close

    Dusanka Zupanski,

    Cooperative Institute for Research in the Atmosphere, Colorado State University, Fort Collins, CO, US
    X close

    Milija Zupanski

    Cooperative Institute for Research in the Atmosphere, Colorado State University, Fort Collins, CO, US
    X close

    Abstract

    The performance of the maximum likelihood ensemble filter (MLEF), is investigated in the context of generic systems featuring the essential ingredients of unstable dynamics and on a spatially extended system displaying chaos. The main objective is to clarify the response of the filter to different regimes of motion and highlighting features which may help its optimization in more realistic applications. It is found that, in view of the minimization procedure involved in the filter analysis update, the algorithm provides accurate estimates even in the presence of prominent non-linearities. Most importantly, the filter ensemble size can be designed in connection to the properties of the system attractor (Kaplan—Yorke dimension), thus facilitating the filter setup and limiting the computational cost by using an optimal ensemble. As a corollary, this latter finding indicates that the ensemble perturbations in the MLEF reflect the intrinsic system error dynamics rather than a sampling of realizations of an unknown error covariance.

    How to Cite: Carrassi, A., Vannitsem, S., Zupanski, D. and Zupanski, M., 2009. The maximum likelihood ensemble filter performances in chaotic systems. Tellus A: Dynamic Meteorology and Oceanography, 61(5), pp.587–600. DOI: http://doi.org/10.1111/j.1600-0870.2009.00408.x
    1
    Views
      Published on 01 Jan 2009
    Peer Reviewed
     CC BY 4.0
     Accepted on 3 Jun 2008            Submitted on 9 Jan 2009

    References

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

    2. Bishop , C. , Etherton , J. and Majumdar , S. J. 2001 . Adaptive sampling with the ensemble transform Kalman filter. Part I: theoretical aspects . Mon. Wea. Rev . 129 , 420 – 436 .  

    3. Carrassi , A. , Ghil , M. , Trevisan , A. and Uboldi , F. 2008a . Data assimilation as a nonlinear dynamical system problem: stability and convergence of the prediction-assimilation system . Chaos 18 , 023112 .  

    4. Carrassi , A. , Vannitsem , S. and Nicolis , C. 2008b . Model error and sequential data assimilation: a deterministic formulation . Q. J. R. Meterol. Soc . 134 , 1297 – 1313 .  

    5. Carrio , G. G. , Cotton , W. R. , Zupanski , D. and Zupanski , M. 2008 . De-velopment of an aerosol retrieval method: description and preliminary tests . J. AppL MeteoroL ClimatoL 47 , 2760 – 2776  

    6. Evensen , G. 1994 . Sequential data assimilation with a nonlinear quasi-geostrophic model using Monte-Carlo methods to forecast error statis-tics. J. Geophys. Res . 99 ( C5 ), 10 143 – 10 162 .  

    7. Fletcher , S. J. and Zupanski , M. 2006a . A data assimilation method for lognormally distributed observation errors . Q. J. R. MeteoroL Soc . 132 , 2505 – 2520 .  

    8. Fletcher , S. J. and Zupanski , M. 2006b . A hybrid multivariate normal and lognormal distribution for data assimilation . Atmos. Sci. Let . 7 , 43 – 46 .  

    9. Fletcher , S. J. and Zupanski , M. 2008 . Implications and impacts of transforming lognormal variables into normal variables in VAR . Met. Zeit . 16 , 755 – 765 .  

    10. Ghil , M. 1997 . Advances in sequential estimation for atmospheric and oceanic flows . J. MeteoroL Soc. Jpn . 75 , 289 – 304 .  

    11. Jazwinslci , A. H. 1970 . Stochastic Processes and Filtering Theory , Academic Press , London .  

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

    13. Houtelcamer , P. L. and Mitchell , H. L. 1998 . Data assimilation using an ensemble Kalman filter technique . Mon. Wea. Rev . 126 , 796 – 811 .  

    14. Kalman , R. 1960 . A new approach to linear filtering and prediction problems . Trans. ASME, J. Basic Eng . 82 , 35 – 45 .  

    15. Kalnay , E. 2003 . Atmospheric Modeling , Data Assimilation and Pre-dictability , Cambridge University Press , Cambridge .  

    16. Kaplan , J. L. and Yorke , J. A. 1979 . Chaotic behavior of multidimen-sional difference equations. In: Functional Differential Equations and Approximations of Fixed Points (eds H.-O. Peitgen and H.-O. Walter ). Lecture Notes in Mathematics , Vol. 730 . Springer, Berlin , p. 204.  

    17. 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 .  

    18. Lorene , A. C. 1986 . Analysis methods for numerical weather prediction . Q. J. R. MeteoroL Soc . 112 , 1177 – 1194 .  

    19. Lorenz , E. N. 1995 . ‘Predictability: a problem partly solved’. Proc. Seminar on Predictability , Vol. 1 , Reading, UK, ECMWF , 1 - 18 .  

    20. Nicolis , C. 2003 . Dynamics of model error: some generic features . J. Atmos. Sci . 60 , 2208 – 2218 .  

    21. Nicolis , C. , Perdigao , R. and Vannitsem , S. 2009 . Dynamics of prediction errors under the combined effect of initial condition and model errors . J. Atmos. Sci . 66 , 766 – 778 .  

    22. Ott , E. 2002 . Chaos in Dynamical Systems, Cambridge University Press, Cambridge.  

    23. Ott , E. , Hunt , B. R. , Szunyogh , I. , Zimin , A. V., Kostelich , E. J. and co-authors. 2004 . A Local Ensemble Kalman Filter for atmospheric data assimilation in oceanography. Tellus 56A , 273 - 277 .  

    24. Pham , D. T. , Verron , J. and Roubaud , M. C. 1998 . A singular evolu-five extended Kalman filter for data assimilation in oceanography . J. Marine Sys . 16 , 323 – 340 .  

    25. Reichle , R. H. , Walker , J. P. , Koster , R. D. and Houser , P. R. 2002 . Extended versus ensemble Kalman filtering to land data assimilation . J. Hydrometeor 3 , 728 – 740 .  

    26. Rozier , D. , Birol , F. , Cosme , E. , Brasseur , P. , Branlcart , J. M. and co-authors. 2007. A reduced-order Kalman filter for data assimila-tion in physical oceanography . SIAM Rev . 49 , 449 - 465 .  

    27. Sakov , P. and Oke , P. R. 2008 . Implications of the Form of the ensemble transformation in the ensemble square root filters . Mon. Wea. Rev . 136 , 1042 – 1053 .  

    28. Tippet , M. K. , Anderson , J. L. , Bishop , C. H. , Hamill , T. M. and Whitaker , J. S. 2003 . Ensemble square root filters . Mon. Wea. Rev . 131 , 1490 – 113 .  

    29. Trevisan , A. and Uboldi , F. 2004 . Assimilation of Standard and targeted observations within the unstable subspace of the observation-analysis-forecast cycle system . J. Atmos. Sci . 61 , 103 – 113 .  

    30. Uzunoglu , B. , Fletcher , S. J. , Zupanski , M. and Navon , I. M. 2007 . Adaptive ensemble size inflation and reduction. Q. J. R. MeteoroL Soc . 133 , 1281 - 1294 .  

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

    32. Zupanski , M. 2005 . Maximum likelihood ensemble filter: theoretical aspects . Mon. Wea. Rev . 133 , 1710 – 1726 .  

    33. Zupanski , D., 2009 . Information measures in ensemble data assimilation. Chapter in the book. In: Data Assimilation for Atmospheric, Oceanic, and Hydrologic Applications (eds S. K. Park and L. Xu ), Springer-Verlag, Berlin-Heidelberg . pp. 85 - 95 , 475pp.  

    34. Zupanski , D. and Zupanski , M. 2006 . Model error estimation employing an ensemble data assimilation approach . Mon. Wea. Rev . 134 , 1337 – 1354 .  

    35. Zupanski , M. , Fletcher , S. F. , Navon , I. M. , Uzunoglu , B. , Heikes , R. P. and co-authors . 2006. Initiation of ensemble data assimilation. Tellus 58A , 159 - 170 .  

    36. Zupanski , D. , Hou , A. Y. , Zangh , S. Q. , Zupanski , M. , Kummerow , C. D. and co-authors. 2007a. Application of information theory in ensemble data assimilation . Q. J. R. MeteoroL Soc . 133 , 1533 - 1545 .  

    37. Zupanski , D. , Denning , A. S. , Uliasz , M. , Zupanski , M. , Schuh , A. E. and co-authors. 2007b. Carbon flux bias estimation employing Maximum Likelihood Ensemble Filter (MLEF) . J. Geophys. Res . 112 , D17107.  

    38. Zupanski , M. , Navon , I. M. and Zupanski , D. 2008 . The Maximum Likelihood Ensemble Filter as a non-differentiable minimization al-gorithm . Quart. J. Roy. MeteoroL Soc . 134 , 1039 – 1050 .  

    Jump to Discussions Related content
    comments powered by Disqus