A 4D-EnKF Method via a Modified Cholesky Decomposition and Line Search Optimization for Non-Linear Data Assimilation

This paper introduces an efficient approach for implementing the Four-Dimensional Variational Ensemble Kalman Filter (4D-EnKF) for non-linear data assimilation, leveraging a modified Cholesky decomposition (4D-EnKF-MC). In this method, control spaces at observation times are represented by full-rank square root approximations of background error covariance matrices, derived using the modified Cholesky decomposition. To ensure global convergence, we integrate line-search optimization into the filter formulation. The performance of the 4D-EnKF-MC is evaluated through experimental tests using the Lorenz 96 model, and its accuracy is compared to that of a 4D-Var extension of the Maximum-Likelihood Ensemble Filter (4D-MLEF). Through Root Mean Square Error (RMSE) analysis, we demonstrate that the proposed method outperforms the 4D-MLEF across a range of ensemble sizes and observational network configurations, providing a robust and scalable solution for non-linear data assimilation in complex systems.