An Iterative Ensemble Kalman Filter for Data Assimilation

Abstract The ensemble Kalman filter (EnKF) is a subject of intensive investigation for use as a reservoir management tool. For strongly nonlinear problems, however, EnKF can fail to achieve an acceptable data match at certain times in the assimilation process. Here, we provide iterative EnKF procedures to remedy this deficiency and explore the validity of these iterative methods compared to standard EnKF by considering two examples, one of which is pertains to a simple problem where the posterior probability density function has two modes. In both examples, we are able to obtain better data matches using iterative methods than with standard EnKF. In Appendix A, we enumerate the assumptions that must hold in order to show that EnKF provides a correct sampling of the probability distribution for the random variables. This derivation calls into question the common derivation in which one adds the data to the original combined state vector of model parameters and dynamical variables. In fact, it appears that there is no assurance that this trick for turning a nonlinear problem into a linear problem results in a correct sampling of the pdf one wishes to sample. However, we show that augmenting the state vector with the data results in a correct procedure for sampling the pdf if at every data assimilation step, the predicted data vector is a linear function of the combined (unaugmented) state vector and the average predicted data vector is equal to the predicted data evaluated at the average of the predicted combined state vector. Without these assumptions, we know of no way to show EnKF samples correctly. For completeness, in Appendix C, we show that each ensemble member of model parameters obtained at each step of EnKF is a linear combination of the initial ensemble, which emphasizes the importance of obtaining a sufficiently large initial ensemble.