Implementations of a square-root ensemble analysis and a hybrid localisation into the POD-based ensemble 4DVar

The main purpose of this study is to propose a hybrid localisation technique for the proper orthogonal decomposition (POD)-based ensemble 4-D variational assimilation (PODEn4DVar) method with the aim of facilitating its implementation. This hybrid localisation scheme takes full advantages of both the implicit and explicit localisations and is relatively easy to be achieved by parallel programming. Besides, to strengthen its performance, we also incorporate a square root analysis scheme into the PODEn4DVar instead of its original one. The feasibility and effectiveness of the modified PODEn4DVar are demonstrated in a 2-D shallow-water equation model with simulated observations. It is found that the PODEn4DVar performs robust and is capable of outperforming the local ensemble transform Kalman filter (LETKF), its 4-D extension 4D-LETKF and another En4DVar method under both perfect-and imperfect-model scenarios.