Resampling the ensemble Kalman filter

Ensemble Kalman filters (EnKF) based on a small ensemble tend to provide collapse of the ensemble over time. It is shown that this collapse is caused by positive coupling of the ensemble members due to use of one common estimate of the Kalman gain for the update of all ensemble members at each time step. This coupling can be avoided by resampling the Kalman gain from its sampling distribution in the conditioning step. In the analytically tractable Gausslinear model finite sample distributions for all covariance matrix estimates involved in the Kalman gain estimate are known and hence exact Kalman gain resampling can be done. For the general nonlinear case we introduce the resampling ensemble Kalman filter (ResEnKF) algorithm. The resampling strategy in the algorithm is based on bootstrapping of the ensemble and Monte Carlo simulation of the likelihood model. An empirical study demonstrates that ResEnKF provides more reliable prediction intervals than traditional EnKF, on the cost of somewhat less accuracy in the point predictions. In a synthetic reservoir study, it is shown the the hierarchical ensemble Kalman filter (HEnKF) provides more reliable predictions and prediction intervals than both ResEnKF and traditional EnKF. HEnKF requires additional modeling, however.