Uncertainty Quantification in the Ensemble Kalman Filter

The ensemble Kalman filter (EnKF) provides an approximate, sequential Monte Carlo solution to the recursive data assimilation algorithm for hidden Markov chains. The challenging conditioning step is approximated by a linear updating, and the updating weights, termed Kalman weights, are inferred from the ensemble members. The EnKF scheme is known to provide unstable predictions and to underestimate the prediction intervals, and even sometimes to diverge. The underlying cause for these shortcomings is poorly understood. We find that the ensemble members couple in the conditioning procedure and that the coupling increase multiplicatively in the recursive conditioning steps. Under reasonable Gauss-independence assumptions, exact expressions for this correlation are developed. Moreover, expressions for the precision of the predictions and the downward bias in the empirical variance introduced in one conditioning step are found. These results are confirmed by a Gauss-linear simulation study. Furthermore, we quantitatively evaluate an alternative, improved EnKF scheme on the basis of transformations of ensemble members under the same Gauss-independent assumptions. The scheme is compared with the frequently used ensemble inflation scheme.