Uncertainty Quanti cation in the Ensemble Kalman Filter

The ensemble Kalman lter (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 nd 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 con rmed by a Gauss-linear simulation study. In order to improve on the EnKF, complementary shrinkage, transformation and resampling schemes are de ned, and quantitatively evaluated under the same Gauss-independent assumptions. The Gauss-linear simulation study is further used to empirically evaluate these complementary schemes. Shrinkage provides signi cant improvements of the EnKF, and additional use of transformation or resampling may also be bene cial.