On building the state error covariance from a state estimate

Abstract: It was recently found with the aid of machine learning that for a variety of toy data assimilation systems with chaotic Lorenz-96 model it is possible to achieve a nearly-optimal data assimilation without carrying the state error covariance between cycles. This result does not look surprising on its own because not carrying covariance is the approach taken by standard 4D-Var, but it was found ``astonishing'' in the context of the machine learning-based system trained on the ensemble Kalman filter. This note proposes two algorithms for building the state error covariance from a state estimate that yield good performance and could be worked out by the deep learning-based system.