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A Stochastic Ensemble Kalman Filter with Perturbation Ensemble Transformation

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Abstract

The Kalman filter is currently one of the most popular approaches to solving the data assimilation problem. A major line of the application of the Kalman filter to data assimilation is the ensemble approach. In this paper, a version of the stochastic ensemble Kalman filter is considered. In this algorithm, an ensemble of analysis errors is obtained by transforming an ensemble of forecast errors. The analysis step is made only for a mean value. Thus, the ensemble π-algorithm combines the advantages of stochastic filters and the efficiency and locality of square root filters. A numerical method of implementing the ensemble π-algorithm is proposed, and the validity of this method is proved. This algorithm is used for a test problem in a three-dimensional domain. The results of numerical experiments with model data for estimating the efficiency of the algorithm are presented. A comparative analysis of the behavior of the root-mean-square errors of the ensemble π-algorithm and the Kalman ensemble filter by means of numerical experiments with a one-dimensional Lorentz model is performed.

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References

  1. Bellman, R., Introduction to Matrix Analysis, New York:McGraw-Hill, 1960.

    MATH  Google Scholar 

  2. Zelenkov, G.A. and Zubov, N.V., On the Boundaries of the Spectrum of a Linear Operator Matrix in Unitary Space, in Mathematics. Computer. Education, Coll. Pap. of XIV Int. Conf., Izhevsk: Scientific Publishing Center “Regulyarnaya i Khaotichnaya Dinamika,” vol. 2, 2007, pp. 34–41.

    Google Scholar 

  3. Klimova, E.G., A Data Assimilation Technique Based on the pi-Algorithm, Russ. Meteorol. Hydrol., 2008, vol. 33, pp. 143–150.

    Article  Google Scholar 

  4. Lancaster, P., Theory of Matrices, New York: Academic Press, 1969.

    MATH  Google Scholar 

  5. Bjorck, A. and Hammarling, S., A Schur Method for the Square Root of a Matrix, Lin. Alg. Appl., 1983, vols. 52/53, pp. 127–140.

    Google Scholar 

  6. Burgers, G., Van Leeuwen, P.J., and Evensen, G., Analysis Scheme in the Ensemble Kalman Filter, Monthly Weather Rev., 1998, vol. 126, pp. 1719–1724.

    Article  Google Scholar 

  7. Evensen, G., Sequential Data Assimilation with a Nonlinear Quasi-Geostrophic Model Using Monte Carlo Methods to Forecast Error Statistics, J. Geophys. Res., 1994, vol. 99, pp. 10143–10162.

    Article  Google Scholar 

  8. Evensen, G., The Ensemble Kalman Filter: Theoretical Formulation and Practical Implementation, Ocean Dyn., 2003, vol. 53, pp. 343–367.

    Article  Google Scholar 

  9. Evensen, G., Data Assimilation. The Ensemble Kalman Filter, Berlin: Spriger-Verlag, 2009.

    MATH  Google Scholar 

  10. Higham, N.J., Computing Real Square Roots of Real Matrix, Lin. Alg. Appl., 1987, vols. 88/89, pp. 404–430.

    Book  MATH  Google Scholar 

  11. Hodyss, D. and Campbell, W.F., Square Root and Perturbed Observation Ensemble Generation Techniques in Kalman and Quadratic Ensemble Filtering Algorithms, Monthly Weather Rev., 2013, vol. 141, pp. 2561–2573.

    Article  Google Scholar 

  12. Houtekamer, P.L. and Mitchell, H.L., Ensemble Kalman Filtering, Quart. J. Royal Meteorol. Soc., 2005, vol. 131, pp. 1–23.

    Article  Google Scholar 

  13. Houtekamer, H.L. and Zhang, F., Review of the Ensemble Kalman Filter for Atmospheric Data Assimilation, Monthly Weather Rev., 2016, vol. 144, pp. 4489–4532.

    Article  Google Scholar 

  14. Hunt, B.R., Kostelich, E.J., and Szunyogh, I., EfficientData Assimilation for Spatiotemporal Chaos: A Local Ensemble Transform Kalman Filter, Phys. D, 2007, vol. 230, pp. 112–126.

    Article  MathSciNet  MATH  Google Scholar 

  15. Jazwinski, A.H., Stochastic Processes and Filtering Theory, New York: Academic Press, 1970.

    MATH  Google Scholar 

  16. Kalnay, E., AtmosphericModeling, Data Assimilation and Predictability, Cambridge Univ. Press, 2002.

    Book  Google Scholar 

  17. Klimova, E., A Suboptimal Data Assimilation Algorithm Based on the Ensemble Kalman Filter, Quart. J. RoyalMeteorol. Soc., 2012, vol. 138, pp. 2079–2085.

    Article  Google Scholar 

  18. Lawson, G.A. and Hanson, J.A., Implications of Stochastic and Deterministic Filters as Ensemble-Based Data Assimilation Methods in Varying Regimes of Error Growth, Monthly Weather Rev., 2004, vol. 132, pp. 1966–1981.

    Article  Google Scholar 

  19. Lei, J., Bickel,P., and Shyder,C., Comparison of Ensemble Kalman Filters under Non-Gaussianity,Monthly Weather Rev., 2010, vol. 138, pp. 1293–1306.

    Article  Google Scholar 

  20. Lorenz, E.N. and Emanuel, K.A., Optimal Sites for SupplementaryWeather Observations: Simulation with a Small Model, J. Atmosph. Sci., 1998, vol. 55, pp. 399–414.

    Article  Google Scholar 

  21. Sakov, P. and Oke, P.R., Implication of the Form of the Ensemble Transformation in the Ensemble Square Root Filters, Monthly Weather Rev., 2008, vol. 136, pp. 1042–1053.

    Article  Google Scholar 

  22. Whitaker, J.S. and Hamill, T.M., Ensemble Data Assimilation without Perturbed Observations, Monthly Weather Rev., 2002, vol. 130, pp. 1913–1924.

    Article  Google Scholar 

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Authors and Affiliations

  1. Institute of Computational Technologies, Siberian Branch, Russian Academy of Sciences, pr. Akad. Lavrent’eva 6, Novosibirsk, 630090, Russia

    E. G. Klimova

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  1. E. G. Klimova
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Correspondence to E. G. Klimova.

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Russian Text © E.G. Klimova, 2019, published in Sibirskii Zhurnal Vychislitel’noi Matematiki, 2019, Vol. 22, No. 1, pp. 27–39.

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Klimova, E.G. A Stochastic Ensemble Kalman Filter with Perturbation Ensemble Transformation. Numer. Analys. Appl. 12, 26–36 (2019). https://doi.org/10.1134/S1995423919010038

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  • DOI: https://doi.org/10.1134/S1995423919010038

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