Ensemble Updating of Binary State Vectors by Maximising the Expected Number of Unchanged Components

The main challenge in ensemble-based filtering methods is the updating of a prior ensemble to a posterior ensemble. In the ensemble Kalman filter (EnKF) the update is constructed based on a linear-Gaussian model assumption. In the present study we consider how the underlying ideas of the EnKF can be transferred to a situation where the components of the state vector are binary variables. Based on a generalised view of the EnKF, we formulate a class of possible updating procedures. We adopt a hidden Markov model for the state and observation vectors, and define an optimal update by maximising the expected number of binary variables that remain unchanged. The performance of our approach is demonstrated in a simulation example inspired by the movement of oil and water in a petroleum reservoir. In particular we empirically compare our results with corresponding results when using a naive updating strategy where the posterior ensemble is sampled independently of the prior ensemble, and with the results from a computationally intensive, but Bayesian optimal updating procedure. Our filter performs much better than the naive approach, but of course not as good as the Bayesian optimal filter.