Abstract. The ensemble Kalman filter (EnKF) is widely used for state estimation in chaotic dynamical systems, including atmospheric and oceanic flows. One of the fundamental questions is how many samples are required for accurate long-term performance of the EnKF. In this study, we introduce a notion of time-asymptotic filter accuracy based on the scaling of the analysis error with respect to the observation noise level. This formulation provides a qualitative distinction between convergent and divergent filtering behavior, beyond standard criteria based on time-averaged RMSE at a fixed noise level. We investigate the minimum ensemble size m* required for this filter accuracy and relate it to intrinsic instability of dynamical systems. Using the Lyapunov exponents (LEs), which quantify asymptotic exponential growth rates of infinitesimal perturbations, we characterize degrees of instability by the number of positive exponents N+. Because spanning the unstable directions by a limited ensemble is essential for long-term accuracy, we propose an ensemble spin-up and downsizing strategy. Numerical experiments with the EnKF applied to the Lorenz 96 model indicate that the minimum ensemble size required for this filter accuracy satisfies m*=N++1. These results provide a practical guideline for ensemble-size selection based on a priori dynamical information and bridge idealized theoretical requirements with feasible numerical implementations via the ensemble downsizing method.
Takeda et al. (Mon,) studied this question.