The Multi-Solution Problem in 4DVar and Solved in a Smooth Subspace

The solution of 4DVar may not be unique if observation information is not enough. In order to get the expected solution we can introduce a smooth operator to carry out transformation on the control variable. The numerical experiments with shallow-water-equation model and model data show that through the transformation the analysis increment is restricted in a smooth subspace and the analysis field is more close to the truth. The study explained the significance of background error covariance matrix in 4DVar from another side.