Abstract
Abstract. In ensemble Kalman filtering (EnKF), the small number of ensemble members that is feasible to use in a practical data assimilation application leads to sampling variability of the estimates of the background error covariances. The standard approach to reducing the effects of this sampling variability, which has also been found to be highly efficient in improving the performance of EnKF, is the localization of the estimates of the covariances. One family of localization techniques is based on taking the Schur (element-wise) product of the ensemble-based sample covariance matrix and a correlation matrix whose entries are obtained by the discretization of a distance-dependent correlation function. While the proper definition of the localization function for a single state variable has been extensively investigated, a rigorous definition of the localization function for multiple state variables that exist at the same locations has been seldom considered. This paper introduces two strategies for the construction of localization functions for multiple state variables. The proposed localization functions are tested by assimilating simulated observations experiments into the bivariate Lorenz 95 model with their help.
Highlights
The components of the finite-dimensional state vector of a numerical model of the atmosphere are defined by the spatial discretization of the state variables considered in the model
In the light of this, we suggested two different approaches for the construction of positivedefinite filtered estimates of the background covariance matrix
The results of our numerical experiments show that a mathematically proper localization function often leads to improved state estimates
Summary
The components of the finite-dimensional state vector of a numerical model of the atmosphere are defined by the spatial discretization of the state variables considered in the model. An ensemble-based Kalman filter (EnKF) data assimilation scheme treats the finite-dimensional state vector as a multivariate random variable and estimates its probability distribution by an ensemble of samples from the distribution. An EnKF scheme assumes that the probability distribution of the state is described by a multivariate normal distribution, and it estimates the mean and the covariance matrix of that distribution by the ensemble (sample) mean and the ensemble (sample) covariance matrix. The estimate of the mean and the estimate of the covariance matrix of the analysis distribution are obtained by updating the mean and the covariance matrix of a background (prior) distribution based on the latest observations. The background distribution is represented by an ensemble of short-term forecasts from the previous analysis time. This ensemble is called the background ensemble
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