Estimation of high-dimensional prior and posterior covariance matrices in Kalman filter variants
Journal of Multivariate Analysis
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The ensemble Kalman filter (EnKF) is an ensemble-based Monte Carlo formulation of the Kalman filter. In most practical cases it is based on a low-rank approximation of a covariance matrix from a moderately sized ensemble. Sampling errors lead to artificial effects, such as spurious correlations, deteriorating the estimates and the forecasts of the system states. Using random matrix theory, we derive the distribution of an energy norm of the EnKF sampling error for the estimate of the mean, assuming noiseless data. Despite this restriction, the obtained distribution should improve the understanding of the EnKF reliability. The distribution depends on ensemble size, model dimension, and observation locations. We demonstrate the use of the distribution on an example.