Numerical weather prediction and many other geophysical problems are characterized by high-dimensional, nonlinear systems. The non-Gaussian distribution produced by nonlinearity combined with high dimensionality poses difficult challenges for data assimilation. We present a nonlinear, ensemble-based filtering scheme that builds on the techniques from the ensemble Kalman filter (EnKF) in order to be feasible in high dimensions.

Two filtering algorithms are presented which extend the ensemble Kalman filter by use of Gaussian mixtures. The first method, referred to as a mixture ensemble Kalman filter (XEnsF), adaptively represents local covariance structures using nearest neighbors. An efficient sampling algorithm is presented for XEnsF, and the filter is shown to be superior to existing methods in simulations on a three-dimensional model. A second algorithm, the local-local ensemble filter (LLEnsF), combines localizations in physical as well as phase space, allowing the update step in high dimensional systems to be decomposed into a sequence of lower-dimensional updates tractable by the XEnsF. Sequential XEnsF updates at different spatial locations are smoothly joined together using output from EnKF. Given the same ensemble in a 40-dimensional system, the LLEnsF update is shown to locally produce more accurate estimates of the state than the EnKF when the underlying distributions are strongly non-Gaussian. In the 40-dimensional system, a hybrid filter combining the output from LLEnsF with that of EnKF is shown to outperform the EnsKF by 5.7%.