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
Bellman, R., Introduction to Matrix Analysis, New York:McGraw-Hill, 1960.
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.
Klimova, E.G., A Data Assimilation Technique Based on the pi-Algorithm, Russ. Meteorol. Hydrol., 2008, vol. 33, pp. 143–150.
Lancaster, P., Theory of Matrices, New York: Academic Press, 1969.
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.
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.
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.
Evensen, G., The Ensemble Kalman Filter: Theoretical Formulation and Practical Implementation, Ocean Dyn., 2003, vol. 53, pp. 343–367.
Evensen, G., Data Assimilation. The Ensemble Kalman Filter, Berlin: Spriger-Verlag, 2009.
Higham, N.J., Computing Real Square Roots of Real Matrix, Lin. Alg. Appl., 1987, vols. 88/89, pp. 404–430.
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.
Houtekamer, P.L. and Mitchell, H.L., Ensemble Kalman Filtering, Quart. J. Royal Meteorol. Soc., 2005, vol. 131, pp. 1–23.
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.
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.
Jazwinski, A.H., Stochastic Processes and Filtering Theory, New York: Academic Press, 1970.
Kalnay, E., AtmosphericModeling, Data Assimilation and Predictability, Cambridge Univ. Press, 2002.
Klimova, E., A Suboptimal Data Assimilation Algorithm Based on the Ensemble Kalman Filter, Quart. J. RoyalMeteorol. Soc., 2012, vol. 138, pp. 2079–2085.
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.
Lei, J., Bickel,P., and Shyder,C., Comparison of Ensemble Kalman Filters under Non-Gaussianity,Monthly Weather Rev., 2010, vol. 138, pp. 1293–1306.
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.
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.
Whitaker, J.S. and Hamill, T.M., Ensemble Data Assimilation without Perturbed Observations, Monthly Weather Rev., 2002, vol. 130, pp. 1913–1924.
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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