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On error covariances in variational data assimilation. (English) Zbl 1120.93056
Summary: The problem of variational data assimilation for a nonlinear evolution model is formulated as an optimal control problem to find the initial condition function. The equation for the error of the optimal solution (analysis) is derived through the statistical errors of the input data (background and observation errors). The numerical algorithm is developed to construct the covariance operator of the analysis error using the covariance operators of the input errors. Numerical examples are presented.

93E20 Optimal stochastic control
93E25 Computational methods in stochastic control (MSC2010)
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[1] Agoshkov V. I., Russ. J. Numer. Anal. Math. Modelling 8 (1) pp 1016– (1993)
[2] E. Blayo, J. Blum, and J. Verron, Assimilation variationnelle de donnees en oceanographie et reduction de la dimension de l’espace de contraole. In: Equations aux Derivees Partielles et Applications. Elsevier, Paris, 1998, pp. 2050219. · Zbl 0915.35106
[3] Math. Appl. Comp. 2 pp 3022– (1983)
[4] A. L. Dontchev, Perturbations, Approximations and Sensitivity Analysis of Optimal Control Systems. Lecture Notes in Control and Information Sciences, Vol. 52. Springer, Berlin, 1983.
[5] M. Fisher and P. Courtier, Estimating the covariance matrices of analysis and forecast error in variational data assimilation. ECMWF Research Department Tech. Memo. 220, Reading, 1995.
[6] P. E. Gill, W. Murray, and M. H. Wright, Practical Optimization. Academic, San Diego, Calif. 1981. · Zbl 0503.90062
[7] Glowinski R., Acta Numerica 1 pp 2690378– (1994)
[8] Math. Comp. 24 pp 1022– (1970)
[9] Gunson J. R., J. Geophys. Res. 101 pp 28473028488– (1996)
[10] L. Hascoet and V. Pascual, TAPENADE 2.1 user’s guide. INRIA Tech. Report No. 0300, 2004.
[11] F.X. Le Dimet and I. Charpentier, Methodes du second ordre en assimilation de donnees. Equations aux Derivees Partielles et Applications. Articles dedies a J.L. Lions. Elsevier, Paris, 1998. · Zbl 0953.35019
[12] DOI: 10.1175/1520-0493(2002)130<0629:SOIIDA>2.0.CO;2
[13] Le Dimet F.-X., Russ. J. Numer. Anal. Math. Modelling 17 (1) pp 71097– (2002)
[14] Le Dimet F.-X., J. Meteorol. Soc. Japan 75 (1) pp 2450255– (1997)
[15] Tellus 38 pp 970110– (1986)
[16] Nonlinear Processes in Geophysics 14 pp 1010– (2005)
[17] J.L. Lions, Contraole optimal des systemes gouvernes par des equations aux derivees partielles. Dunod, Paris, 1968.
[18] J.L. Lions, Contraolabilite Exacte Perturbations et Stabilisation de Systemes Distribues. Masson, Paris, 1988.
[19] DOI: 10.1007/BF01589116 · Zbl 0696.90048
[20] G. I. Marchuk and V. V. Penenko, Application of optimization methods to the problem of mathematical simulation of atmospheric processes and environment. In: Modelling and Optimization of Complex Systems. Proc. of the IFIP-TC7 Working Conf. Springer, New York, 1978, pp. 2400 252.
[21] G. I. Marchuk, V. I. Agoshkov, and V. P. Shutyaev, Adjoint Equations and Perturbation Algorithms in Nonlinear Problems. CRC Press Inc., New York, 1996. · Zbl 0828.47053
[22] Navon I. M., New York 1 pp 7400746– (1995)
[23] S. V. Patankar, Numerical Heat Transfer and Fluid Flow. Hemisphere Publishing Corporation, New York, 1980. · Zbl 0521.76003
[24] E. Polak, Optimization: Algorithms and Consistent Approximations. Applied Mathematical Sciences, Vol. 124. Springer, New York, 1997. · Zbl 0899.90148
[25] L. S. Pontryagin, V. G. Boltyanskii, R. V. Gamkrelidze, and E. F. Mischenko, The Mathematical Theory of Optimal Processes. John Wiley, New York, 1962.
[26] W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in Fortran 77: the Art of Scientific Computing. Cambridge Press, 1992. · Zbl 0778.65002
[27] Rabier F., Quart. J. Roy. Meteorol. Soc. 118 pp 6490672– (1992)
[28] Y., Monthly Weather Review 98 pp 8570883– (1970)
[29] V., Russ. J. Numer. Anal. Math. Modelling 10 (4) pp 3570371– (1995)
[30] W., J. Geophys. Res. 94 pp 617706196– (1989)
[31] DOI: 10.1002/qj.49711750206
[32] F. Veerse, Variable-storage quasi-Newton operators as inverse forecast/analysis error covariance matrices in variational data assimilation. INRIA Tech. Report No. 3685, 1999.
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