Error analysis of the projected PO method with additive inflation for the partially observed Lorenz 96 model

We consider the filtering problem with the partially observed Lorenz 96 model. Although the accuracy of the 3DVar filter in this problem has been established, the theoretical guarantee for the ensemble Kalman filter (EnKF) remains limited due to the analytical difficulty of handling non-symmetric matrices that emerge in the partial observation setting. This study establishes uniform-in-time error bounds of a stochastic variant of the EnKF, known as the perturbed observation (PO) method. By utilizing additive covariance inflation, we successfully obtain the bounds both with and without projecting the background covariance onto the observation space. Our analysis with the projection complements existing results for the deterministic variant of the EnKF, while our approach without the projection offers an extended mathematical framework to handle the non-symmetric matrix products directly. A numerical example validates the theoretical findings and shows comparable accuracies between the two settings.