On the Renormalization of the Covariance Operators

Abstract

Many background error correlation (BEC) models in data assimilation are formulated in terms of a smoothing operator [Formula: see text], which simulates the action of the correlation matrix on a state vector normalized by respective BE variances. Under such formulation, [Formula: see text] has to have a unit diagonal and requires appropriate renormalization by rescaling. The exact computation of the rescaling factors (diagonal elements of [Formula: see text]) 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 [Formula: see text] 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.