Iterative Ensemble Kalman Filters for Data Assimilation

Summary The ensemble Kalman filter (EnKF) is a subject of intensive investigation for use as a reservoir management tool. For strongly nonlinear problems, however, the EnKF can fail to achieve an acceptable data match at certain times in the data assimilation process. Here, we provide two iterative EnKF procedures to remedy this deficiency and explore the validity of these iterative methods compared to the standard EnKF by considering two examples. In both examples, we are able to obtain better data matches using iterative methods than with the standard EnKF. The simplest derivation of the EnKF analysis equation "linearizes" the objective function by adding the vector of predicted data to the original combined state vector of model parameters and dynamical variables. We show that there is no assurance that this trick for turning a nonlinear problem into a linear problem results in a correct sampling of the probability density function (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. Without this assumption, we know of no way to show EnKF samples correctly.