Data assimilation methods for highly non-linear models via efficient and practical covariance matrix estimators (2020)
In this research, efficient and practical implementation methods for data assimilation will be proposed using covariance matrix estimators, like Ledoit and Wolf (LW), and using an hybrid method based on modified Cholesky decomposition. The main idea is explote the rank-deficiency of the ensemble covariance matrix, and some properties of the trace of a matrix, in order to develop a tractable implementation of a covariance matrix estimator in high-dimensional probability spaces, such as those found in the context of operational data assimilation. In this manner, the ensemble covariance matrix is replaced by a well-conditioned, full-rank estimator wherein the impact of spurious correlations can be mitigated during assimilation steps. The work was organized as follow, first, an efficient and practical implementation of the ensemble Kalman filter (EnKF) via the distribution-free Ledoit and Wolf (LW) covariance matrix estimator is proposed (EnKF-LW). Second, the intrinsic needed of adjoint models in the four-dimensional context is avoided using an efficient and practical implementation of a Four-Dimensional Variational Ensemble Kalman Filter (4D-EnKF) via a modified Cholesky decomposition (4D-EnKF-MC). Last a framework for wind energy potential estimation is proposed using Four-Dimensional Variational (4D-Var) data assimilation. Experimental tests are performed by using an Atmospheric General Circulation Model. The results reveal that EnKF-RBLW by employing Gaussian relaxation on prior ensemble members during assimilation steps and that the proposed 4D-EnKF-MC method outperforms traditional filter formulations (4D-ENKF) in terms of L--2 error norms and RMSE values.