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On the Renormalization of the Covariance Operators
NAVAL RESEARCH LAB STENNIS DETACHMENT STENNIS SPACE CENTER MS OCEANOGRAPHY DIV
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Many background error correlation BEC models in data assimilation are formulated in terms of a smoothing operator B, which simulates the action of the correlation matrix on a state vector normalized by respective BE variances. Under such formulation, B has to have a unit diagonal and requires appropriate renormalization by rescaling. The exact computation of the rescaling factors diagonal elements of B is a computationally expensive procedure, which needs an efficient numerical approximation. In this study approximate renormalization techniques based on the Monte Carlo MC and Hadamard matrix HM methods and on the analytic approximations derived under the assumption of the local homogeneity LHA of B are compared using realistic BEC models designed for oceanographic applications. It is shown that although the accuracy of the MC and HM methods can be improved by additional smoothing, their computational cost remains significantly higher than the LHA method, which is shown to be effective even in the zeroth-order approximation. The next approximation improves the accuracy 1.5-2 times at a moderate increase of CPU time. A heuristic relationship for the smoothing scale in two and three dimensions is proposed for the first-order LHA approximation.
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