The maximum likelihood ensemble filter performances in chaotic systems

Alberto Carrassi; Stephane Vannitsem; Dusanka Zupanski; Milija Zupanski

doi:10.1111/j.1600-0870.2009.00408.x

The maximum likelihood ensemble filter performances in chaotic systems

Tellus

Volume 61 (2009): Issue 5

By: Alberto Carrassi, Stephane Vannitsem, Dusanka Zupanski and Milija Zupanski

Open Access

|Jan 2009

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.

References

Anderson , J. L. 2001 . An ensemble adjustment Kalman filter for data assimilation . Mon. Wea. Rev . 129 , 2884 – 2903 .
Search in Google Scholar Back to article
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 .
Search in Google Scholar Back to article
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 .
Search in Google Scholar Back to article
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 .
Search in Google Scholar Back to article
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
Search in Google Scholar Back to article
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 .
Search in Google Scholar Back to article
Fletcher , S. J. and Zupanski , M. 2006a . A data assimilation method for lognormally distributed observation errors . Q. J. R. MeteoroL Soc . 132 , 2505 – 2520 .
Search in Google Scholar Back to article
Fletcher , S. J. and Zupanski , M. 2006b . A hybrid multivariate normal and lognormal distribution for data assimilation . Atmos. Sci. Let . 7 , 43 – 46 .
Search in Google Scholar Back to article
Fletcher , S. J. and Zupanski , M. 2008 . Implications and impacts of transforming lognormal variables into normal variables in VAR . Met. Zeit . 16 , 755 – 765 .
Search in Google Scholar Back to article
Ghil , M. 1997 . Advances in sequential estimation for atmospheric and oceanic flows . J. MeteoroL Soc. Jpn . 75 , 289 – 304 .
Search in Google Scholar Back to article
Jazwinslci , A. H. 1970 . Stochastic Processes and Filtering Theory , Academic Press , London .
Search in Google Scholar Back to article
Heemink , A. W. , Verlaan , M. and Segers , J. 2001 . Variance reduced ensemble Kalman filtering . Mon. Wea. Rev . 129 , 1718 – 1728 .
Search in Google Scholar Back to article
Houtelcamer , P. L. and Mitchell , H. L. 1998 . Data assimilation using an ensemble Kalman filter technique . Mon. Wea. Rev . 126 , 796 – 811 .
Search in Google Scholar Back to article
Kalman , R. 1960 . A new approach to linear filtering and prediction problems . Trans. ASME, J. Basic Eng . 82 , 35 – 45 .
Search in Google Scholar Back to article
Kalnay , E. 2003 . Atmospheric Modeling , Data Assimilation and Pre-dictability , Cambridge University Press , Cambridge .
Search in Google Scholar Back to article
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.
Search in Google Scholar Back to article
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 .
Search in Google Scholar Back to article
Lorene , A. C. 1986 . Analysis methods for numerical weather prediction . Q. J. R. MeteoroL Soc . 112 , 1177 – 1194 .
Search in Google Scholar Back to article
Lorenz , E. N. 1995 . ‘Predictability: a problem partly solved’. Proc. Seminar on Predictability , Vol. 1 , Reading, UK, ECMWF , 1 - 18 .
Search in Google Scholar Back to article
Nicolis , C. 2003 . Dynamics of model error: some generic features . J. Atmos. Sci . 60 , 2208 – 2218 .
Search in Google Scholar Back to article
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 .
Search in Google Scholar Back to article
Ott , E. 2002 . Chaos in Dynamical Systems, Cambridge University Press, Cambridge.
Search in Google Scholar Back to article
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 .
Search in Google Scholar Back to article
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 .
Search in Google Scholar Back to article
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 .
Search in Google Scholar Back to article
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 .
Search in Google Scholar Back to article
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 .
Search in Google Scholar Back to article
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 .
Search in Google Scholar Back to article
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 .
Search in Google Scholar Back to article
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 .
Search in Google Scholar Back to article
Whitaker , J. S. and Hamill , T. M. 2002 . Ensemble data assimilation without perturbed observations . Mon. Wea. Rev . 130 , 1913 – 1924 .
Search in Google Scholar Back to article
Zupanski , M. 2005 . Maximum likelihood ensemble filter: theoretical aspects . Mon. Wea. Rev . 133 , 1710 – 1726 .
Search in Google Scholar Back to article
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.
Search in Google Scholar Back to article
Zupanski , D. and Zupanski , M. 2006 . Model error estimation employing an ensemble data assimilation approach . Mon. Wea. Rev . 134 , 1337 – 1354 .
Search in Google Scholar Back to article
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 .
Search in Google Scholar Back to article
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 .
Search in Google Scholar Back to article
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.
Search in Google Scholar Back to article
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 .
Search in Google Scholar Back to article

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DOI: https://doi.org/10.1111/j.1600-0870.2009.00408.x | Journal eISSN: 3035-9554

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Language: English

Page range: 587 - 600

Submitted on: Jan 9, 2009

Accepted on: Jun 3, 2008

Published on: Jan 1, 2009

Published by: Stockholm University Press

In partnership with: Paradigm Publishing Services

© 2009 Alberto Carrassi, Stephane Vannitsem, Dusanka Zupanski, Milija Zupanski, published by Stockholm University Press
This work is licensed under the Creative Commons Attribution 4.0 License.

Volume 61 (2009): Issue 5