An Ensemble Kalman Filter with Four-Dimensional Incremental Analysis Update to data assimilation

The analysis produced by the ensemble Kalman filter (EnKF) may be dynamically inconsistent and contain unbalanced gravity waves that are absent in the real atmosphere. These imbalances can be generated by the discontinuous nature of the EnKF, and exacerbated by covariance localization and inflation. One strategy to combat the imbalance in the analyses is the incremental analysis update (IAU), which uses the dynamic model to distribute the analyses increments over a time window. The IAU has been widely used in atmospheric and oceanic applications. However, the analysis increment that is gradually introduced along model integration is often computed once and is assumed to be constant for each assimilation window, which can be seen as a three-dimensional IAU (3DIAU). Thus the propagation of the analysis increment in the assimilation window is neglected, yet this propagation may be important, especially for moving weather systems.

To take into account the propagation of the analysis increment during an assimilation window, a four-dimensional IAU (4DIAU) update for the EnKF is proposed. It constructs time-varying analysis increments by applying all observations in an assimilation window to state variables at different times during the assimilation window. It then gradually applies these time-varying analysis increments through the assimilation window by use of IAU. The EnKF with 4DIAU (EnKF-4DIAU) reduces imbalances in the analysis and better retains the time-varying information in the analysis increments compared to the EnKF with 3DIAU (EnKF-3DIAU). The EnKF-4DIAU is tested in a dry two-layer primitive equation model and the NCEP Global Forecast System (GFS). Results from the two models show that the EnKF-4DIAU produces lower error than the EnKF without IAU and the EnKF-3DIAU.