[1] 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.

The En4DVar method is designed to combine the flow-dependent statistical covariance information of EnKF into the traditional 4DVar method. However, the En4DVar method is still hampered by its strong dependence on the adjoint model of the underlying forecast model and by its complexity, maintenance requirements, and the high cost of computer implementation and simulation. The primary goal of this paper is to propose an alternative approach to overcome the main difficulty of the En4DVar method caused by the use of adjoint models. The proposed approach, the nonlinear least squares En4DVar (NLS-En4DVar) method, begins with rewriting the standard En4DVar formulation into a nonlinear least squares problem, which is followed by solving the resulting NLS problem by a Gauss–Newton iterative method. To reduce the computational and implementation complexity of the proposed NLS-En4DVar method, a few variants of the new method are proposed; these modifications make the model cheaper and easier to use than the full NLS-En4DVar method at the expense of reduced accuracy. Furthermore, an improved iterative method based on the comprehensive analysis on the above NLSi-En4DVar family of methods is also proposed. These proposed NLSi-En4DVar methods provide more flexible choices of the computational capabilities for the broader and more realistic data assimilation problems arising from various applications. The pros and cons of the proposed NLSi-En4DVar family of methods are further examined in the paper and their relationships and performance are also evaluated by several sets of numerical experiments based on the Lorenz-96 model and the Advanced Research WRF (ARW) Model, respectively.

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A B S T R A C T This paper presents a novel non-linear least squares enhanced proper orthogonal decomposition (POD)-based 4DVar algorithm (referred as NLS-4DVar) for the non-linear ensemble-based 4DVar. In the algorithm, the GaussÁNewton iterative method is employed to handle the non-quadratic non-linearity of the 4DVar cost function while the overall structure of the algorithm still resembles the original POD-4DVar algorithm. It is proved that the original POD-4DVar algorithm is a special case of the proposed NLS-4DVar algorithm under the assumption of the linear relationship between the model perturbations (MPs) and the simulated observation perturbations (OPs). Under the assumption it is also shown that the solution of POD-4DVar algorithm coincides with the solution of the proposed NLS-4DVar algorithm. On the contrary, if the linear relationship assumption is dropped, the solution of the POD-4DVar algorithm is only the first iteration of the proposed NLS-4DVar algorithm. As a result, our analysis provides an explanation for the degraded and inaccurate performance of the POD-4DVar algorithm when the underlying forecast model or (and) the observation operator is strongly non-linear. The potential merits and advantages of the proposed NLS-4DVar are demonstrated by a group of Observing System Simulation Experiments with Advanced Research WRF (ARW) using accumulated rainfall-observations.