Iterative Forms of the Ensemble Kalman Filter

Recently, the Ensemble Kalman filter (EnKF) has been proposed as an alternative to traditional history matching. Because of its computational efficiency, ease of implementation into a reservoir simulator and ability to provide an evaluation of uncertainty in the reservoir model and in production forecasts, EnKF appears to be a more attractive method for integrating essentially continuous streams of dynamic data to update reservoir models and characterize uncertainty than automatic history matching based on methods such as randomized maximum likelihood using the adjoint/LBFGS approach for optimization.<br><br>Although the standard theoretical underpinnings of EnKF rest on Bayesian updating with Gaussian priors, we show that the EnKF update equations can also be derived as an approximation to the Gauss-Newton method, which uses an ``average'' sensitivity matrix.<br>This suggests that for highly nonlinear, non-Gaussian problems, EnKF may not provide an appropriate characterization of uncertainty and that some form of iteration is required. By viewing EnKF through the lens of optimization, instead of Monte Carlo sampling, we derive an iterative EnKF procedure for nonlinear problems. Although the iterative scheme incorporates some of the main features of EnKF, the computational efficiency of the basic EnKF method is not preserved.<br>We show, however, that for some problems, iteration can provide an improved characterization of uncertainty.