An Optimal Linear Transformation for Data Assimilation

Abstract

Linear transformations have wide application in data assimilation, for example, as the basis for computationally feasible models for the prior covariance (e.g., Derber & Bouttier, 1999). Here we consider how linear transformations of state and observation variables could improve covariance localization in ensemble data assimilation. In particular, we ask what is the optimal linear transformation to precede localization. As will be shown below, the optimal linear transformation diagonalizes the analysis (or update) step of the Kalman filter. While this diagonalization result is not novel, it is nevertheless fundamental to several aspects of data assimilation beyond ensemble-based techniques. A second purpose of this paper is to elucidate the insights provided by considering the update step of the Kalman filter in the transformed variables.At first glance, there is little to be gained by linear transformations of the analysis equations from data assimilation. For linear, Gaussian problems with known (prior) state and observation-error covariances, the result of the Kalman-filter update (or, equivalently, the minimizer of the cost function in a variational assimilation scheme) is independent of invertible linear transformations of both the state and the observations, in the sense that performing the linear transformation (with appropriate transformations of covariances), updating, and then applying the inverse transformation, gives results identical to updating in the original variables (see, e.g., Snyder, 2014). The estimate is neither improved nor degraded by choosing different state and observation variables for the analysis step.Ensemble data assimilation, however, requires estimation of state covariances, and observation-state covariances in some algorithms, from a finite-size ensemble of forecasts. Covariance localization (Hamill et al., 2001;