This paper proposes two practical implementations of Four-Dimensional Variational (4D-Var) Ensemble Kalman Filter (4D-EnKF) methods for non-linear data assimilation. Our formulations' main idea is to avoid the intrinsic need for adjoint models in the context of 4D-Var optimization and, even more, to handle non-linear observation operators during the assimilation of observations. The proposed methods work as follows: snapshots of an ensemble of model realizations are taken at observation times, these snapshots are employed to build control spaces onto which analysis increments can be estimated. Via the linearization of observation operators at observation times, a line-search based optimization method is proposed to estimate optimal analysis increments. The convergence of this method is theoretically proven as long as the dimension of control-spaces equals model one. In the first formulation, control spaces are given by full-rank square root approximations of background error covariance matrices via the Bickel and Levina precision matrix estimator. In this context, we propose an iterative Woodbury matrix formula to perform the optimization steps efficiently. The last formulation can be considered as an extension of the Maximum Likelihood Ensemble Filter to the 4D-Var context. This employs pseudo-square root approximations of prior error covariance matrices to build control spaces. Experimental tests are performed by using the Lorenz 96 model. The results reveal that, in terms of Root-Mean-Square-Error values, both methods can obtain reasonable estimates of posterior error modes in the 4D-Var optimization problem. Moreover, the accuracies of the proposed filter implementations can be improved as the ensemble sizes are increased.

翻译：本文提出了两种实用的四维变分（4D-Var）集合卡尔曼滤波（4D-EnKF）方法的实施策略，用于非线性资料同化。我们方案的核心思想在于避免4D-Var优化过程中对伴随模型的内在需求，并进一步处理同化观测资料时的非线性观测算子。所提方法的工作流程如下：在观测时刻获取一组模型实现快照，利用这些快照构建控制空间，并据此估计分析增量。通过在观测时刻对观测算子进行线性化，提出了一种基于线性搜索的优化方法以估计最优分析增量。理论上证明了当控制空间维数等于模型维数时该方法的收敛性。第一种方案中，控制空间通过Bickel-Levina精度矩阵估计器对背景误差协方差矩阵进行满秩平方根逼近得到。在此框架下，我们提出迭代Woodbury矩阵公式以实现优化步骤的高效计算。最后一种方案可视为最大似然集合滤波在4D-Var框架下的推广，其采用先验误差协方差矩阵的伪平方根逼近构建控制空间。基于Lorenz 96模型的实验测试表明：就均方根误差指标而言，两种方法均能在4D-Var优化问题中获得后验误差模式的合理估计。此外，随着集合成员数量的增加，所提滤波实施策略的精度可进一步得到提升。