Unscented/ensemble transform-based variational filter

To deal with high dimensional nonlinear filtering problems, a hybrid scheme called Unscented/Ensemble transform Variational Filter (UEVF) is introduced. This is the combination of an Unscented Transform (UT), an Ensemble Transform (ET) and a rank-reduction method to compute the background covariance error matrices as well as a variational minimization to conduct the mean correction. The proposed UEVF is more efficient than the Unscented Kalman Filter (UKF) to estimate the ensemble mean and covariance by the blending of a variational optimization instead of a Kalman linear correction as well as the ET-like covariance estimation into the update step. Moreover, in order to tackle the high dimension dynamics, truncated singular value decomposition is applied to provide a size reduction of a sigma-points set with an adaptive fashion. For performance verifications, we present two numerical experiments with different dynamics. The first system is the chaotic and high dimensional Lorenz-95 model. We show the performance of different filters including the UEVF as the increasing of dimensionality or noise level. The second simulation is a model based on the 2D shallow water equation. The same tests are provided on this hydrodynamical system. All the numerical experiments confirm that the UEVF outperforms the widely applied Kalman-like filters explicitly.