Ensemble Square Root Filters

Michael K. Tippett International Research Institute for Climate Prediction, Palisades, New York

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Jeffrey L. Anderson GFDL, Princeton, New Jersey

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Craig H. Bishop Naval Research Laboratory, Monterey, California

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Thomas M. Hamill NOAA–CIRES Climate Diagnostics Center, Boulder, Colorado

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Jeffrey S. Whitaker NOAA–CIRES Climate Diagnostics Center, Boulder, Colorado

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Abstract

Ensemble data assimilation methods assimilate observations using state-space estimation methods and low-rank representations of forecast and analysis error covariances. A key element of such methods is the transformation of the forecast ensemble into an analysis ensemble with appropriate statistics. This transformation may be performed stochastically by treating observations as random variables, or deterministically by requiring that the updated analysis perturbations satisfy the Kalman filter analysis error covariance equation. Deterministic analysis ensemble updates are implementations of Kalman square root filters. The nonuniqueness of the deterministic transformation used in square root Kalman filters provides a framework to compare three recently proposed ensemble data assimilation methods.

Corresponding author address: Michael K. Tippett, IRI/LDEO, 223 Monell, P.O. Box 1000/61 Rt. 9W, Palisades, NY 10964-8000. Email: tippett@iri.columbia.edu

Abstract

Ensemble data assimilation methods assimilate observations using state-space estimation methods and low-rank representations of forecast and analysis error covariances. A key element of such methods is the transformation of the forecast ensemble into an analysis ensemble with appropriate statistics. This transformation may be performed stochastically by treating observations as random variables, or deterministically by requiring that the updated analysis perturbations satisfy the Kalman filter analysis error covariance equation. Deterministic analysis ensemble updates are implementations of Kalman square root filters. The nonuniqueness of the deterministic transformation used in square root Kalman filters provides a framework to compare three recently proposed ensemble data assimilation methods.

Corresponding author address: Michael K. Tippett, IRI/LDEO, 223 Monell, P.O. Box 1000/61 Rt. 9W, Palisades, NY 10964-8000. Email: tippett@iri.columbia.edu

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  • Anderson, J. L., 2001: An ensemble adjustment filter for data assimilation. Mon. Wea. Rev., 129 , 28842903.

  • Andrews, A., 1968: A square root formulation of the Kalman covariance equations. AIAA J., 6 , 11651168.

  • Bierman, G. J., 1977: Factorization Methods for Discrete Sequential Estimation. Vol. 128, Mathematics in Science and Engineering, Academic Press, 241 pp.

    • Search Google Scholar
    • Export Citation
  • Bishop, C. H., B. Etherton, and S. J. Majumdar, 2001: Adaptive sampling with the ensemble transform Kalman filter. Part I: Theoretical aspects. Mon. Wea. Rev., 129 , 420436.

    • Search Google Scholar
    • Export Citation
  • Burgers, G., P. J. van Leeuwen, and G. Evensen, 1998: On the analysis scheme in the ensemble Kalman filter. Mon. Wea. Rev., 126 , 17191724.

    • Search Google Scholar
    • Export Citation
  • Cohn, S. E., 1997: An introduction to estimation theory. J. Meteor. Soc. Japan, 75 , 257288.

  • Cohn, S. E., and R. Todling, 1996: Approximate data assimilation schemes for stable and unstable dynamics. J. Meteor. Soc. Japan, 74 , 6375.

    • Search Google Scholar
    • Export Citation
  • Cohn, S. E., A. M. da Silva, J. Guo, M. Sienkiewicz, and D. Lamich, 1998: Assessing the effect of data selection with the DAO Physical-space Statistical Analysis System. Mon. Wea. Rev., 126 , 29132926.

    • Search Google Scholar
    • Export Citation
  • Dee, D. P., 1995: On-line estimation of error covariance parameters for atmospheric data assimilation. Mon. Wea. Rev., 123 , 11281145.

    • Search Google Scholar
    • Export Citation
  • Evensen, G., 1994: Sequential data assimilation with a nonlinear quasi-geostrophic model using Monte Carlo methods to forecast error statistics. J. Geophys. Res., 99 , 10431062.

    • Search Google Scholar
    • Export Citation
  • Golub, G. H., and C. F. Van Loan, 1996: Matrix Computations. 3d ed. The Johns Hopkins University Press, 694 pp.

  • Hamill, T. M., and C. Snyder, 2000: A hybrid ensemble Kalman filter—3D variational analysis scheme. Mon. Wea. Rev., 128 , 29052919.

  • Hamill, T. M., J. S. Whitaker, and C. Snyder, 2001: Distance-dependent filtering of background error covariance estimates in an ensemble Kalman filter. Mon. Wea. Rev., 129 , 27762790.

    • Search Google Scholar
    • Export Citation
  • Heemink, A. W., M. Verlaan, and A. J. Segers, 2001: Variance reduced ensemble Kalman filtering. Mon. Wea. Rev., 129 , 17181728.

  • Houtekamer, P. L., and H. L. Mitchell, 1998: Data assimilation using an ensemble Kalman filter technique. Mon. Wea. Rev., 126 , 796811.

    • Search Google Scholar
    • Export Citation
  • Houtekamer, P. L., and H. L. Mitchell, 2001: A sequential ensemble Kalman filter for atmospheric data assimilation. Mon. Wea. Rev., 129 , 123137.

    • Search Google Scholar
    • Export Citation
  • Houtekamer, P. L., L. Lefaivre, J. Derome, H. Ritchie, and H. L. Mitchel, 1996: A system simulation approach to ensemble prediction. Mon. Wea. Rev., 124 , 12251242.

    • Search Google Scholar
    • Export Citation
  • Maybeck, P. S., 1982: Stochastic Models, Estimation, and Control. Vol. 1. Academic Press, 423 pp.

  • Mitchell, H. L., and P. L. Houtekamer, 2000: An adaptive ensemble Kalman filter. Mon. Wea. Rev., 128 , 416433.

  • Pham, D., 2001: Stochastic methods for sequential data assimilation in strongly nonlinear systems. Mon. Wea. Rev., 129 , 11941207.

  • Pham, D., J. Verron, and M. Roubaud, 1998: A singular evolutive extended Kalman filter for data assimilation in oceanography. J. Mar. Syst., 16 , 323340.

    • Search Google Scholar
    • Export Citation
  • Verlaan, M., and A. W. Heemink, 1997: Tidal flow forecasting using reduced rank square filters. Stochastic Hydrol. Hydraul., 11 , 349368.

    • Search Google Scholar
    • Export Citation
  • Whitaker, J., and T. M. Hamill, 2002: Ensemble data assimilation without perturbed observations. Mon. Wea. Rev., 130 , 19131924.

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