An ensemble-based explicit four-dimensional variational assimilation method

The adjoint and tangent linear models in the traditional four-dimensional variational data assimilation (4DVAR) are difficult to obtain if the forecast model is highly nonlinear or the model physics contains parameterized discontinuities. A new method (referred to as POD-E4DVAR) is proposed in this paper by merging the Monte Carlo method and the proper orthogonal decomposition (POD) technique into 4DVAR to transform an implicit optimization problem into an explicit one. The POD method is used to efficiently approximate a forecast ensemble produced by the Monte Carlo method in a 4-dimensional (4-D) space using a set of base vectors that span the ensemble and capture its spatial structure and temporal evolution. After the analysis variables are represented by a truncated expansion of the base vectors in the 4-D space, the control (state) variables in the cost function appear explicit so that the adjoint model, which is used to derive the gradient of the cost function with respect to the control variables in the traditional 4DVAR, is no longer needed. The application of this new technique significantly simplifies the data assimilation process and retains the two main advantages of the traditional 4DVAR method. Assimilation experiments show that this ensemble-based explicit 4DVAR method performs much better than the traditional 4DVAR and ensemble Kalman filter (EnKF) method. It is also superior to another explicit 4DVAR method, especially when the forecast model is imperfect and the forecast error comes from both the noise of the initial field and the uncertainty in the forecast model. Computational costs for the new POD-E4DVAR are about twice as the traditional 4DVAR method but 5% less than the other explicit 4DVAR and much lower than the EnKF method. Another assimilation experiment conducted within the Lorenz model indicates potential wider applications of this new POD-E4DVAR method.