High Dimensional Ensemble Kalman Filter

The Ensemble Kalman Filter (EnKF), as a fundamental data assimilation approach, has been widely used in many fields of the sciences and engineering. When the state variable is of high dimensional accompanied with high resolution observations of physical models, some key theoretical aspects of the EnKF are open for investigation. This paper proposes several high dimensional EnKF (HD-EnKF) methods equipped with consistent estimators for the important forecast error covariance and Kalman Gain matrices. It then studies the theoretical properties of the EnKF under both fixed and high dimensional state variables, which provides one-step and multiple-step mean square errors of the analysis states to the underlying oracle states offered by the Kalman Filter and gives the much needed insight to the roles played by the forecast error covariance on the accuracy of the EnKF. The accuracy of the data assimilation under the misspecified physical model is also considered. Numerical studies on the Lorenz-96 and the Shallow Water Equation models illustrate that the proposed HD-EnKF algorithms outperform the standard EnKF and widely used inflation methods.