Comparing different ensemble Kalman filter approaches

Over the last decade, the ensemble Kalman filter (EnKF) method has become an attractive tool for history matching reservoir simulation models and production forecasting. Recently, EnKF has been successfully applied to real field studies (e.g. Bianco et al., 2007). Nonetheless, Lorentzen et al. (2005) observed a consistency problem. They showed, using the Kolmogorov-Smirnov test, that for the PUNQ-S3 model, the posterior cdfs of the total cumulative oil production for 10 ensembles were not coming from the same distribution, as was expected since the 10 initial ensembles were generated using a common distribution. The forecasts from these initial ensembles gave consistent cdfs. We investigate if this issue is related to the inbreeding of the ensemble members when applying EnKF. Houtekamer and Mitchell (1998) proposed to use a paired EnKF with covariance localization to improve atmospheric data assimilation. This method was developed to hinder ensemble collapse due to inbreeding of ensemble members and spurious correlations problems far from the observation points. We show that using a paired EnKF, where each Kalman gain is used to update each other's ensemble, do not in fact prevent inbreeding. A new approach, coupled EnKF, that do prevent inbreeding is presented. The present work first review the issue of consistency encountered with EnKF and investigate the use of paired and coupled EnKFs without covariance localization as a possible remedy to this problem. The method is tested on a simple nonlinear model for which the posterior distribution can be analytically solved. The obtained results are compared with the traditional EnKF and the analytic solution. Next, a hierarchical ensemble filter approach inspired by Anderson (2007) acting as a covariance localization technique is proposed and tested on the PUNS-S3 model against the traditional EnKF. This approach seems better than paired or coupled EnKF to help solve the observed inconsistencies.