Quantifying the minimum ensemble size for asymptotic accuracy of the ensemble Kalman filter using the degrees of instability

Abstract. The ensemble Kalman filter (EnKF) is widely used for state estimation in chaotic dynamical systems, including the atmosphere and ocean. However, the required ensemble size for accurate state estimation remains unclear. In this study, we define filter accuracy based on its time-asymptotic performance relative to the observation noise. We then investigate the minimum ensemble size, m*, required to achieve this accuracy, linking it to the degrees of instability in the chaotic dynamics. Since the well-defined characteristic numbers of dynamical systems called the Lyapunov exponents (LEs) quantify the timeasymptotic exponential growth or decay rates of infinitesimal perturbations, we define the degrees of instability N₊ by the number of positive LEs. In the EnKF, capturing such instabilities with limited ensemble is crucial for achieving long-term filter accuracy. Therefore, we propose an ensemble spin-up and downsizing method within data assimilation cycles. Numerical experiments applying the EnKF to the Lorenz 96 model show that the minimum ensemble size required for filter accuracy is estimated by m* = N₊ +1. This study provides a practical estimate for the minimum ensemble size based on a priori information about the target dynamics, along with a method to achieve long-term accuracy.