Filtering is concerned with online estimation of the state of a dynamical system from partial and noisy observations. In applications where the state is high dimensional, ensemble Kalman filters are often the method of choice. This paper establishes long-time accuracy of ensemble Kalman filters. We introduce conditions on the dynamics and the observations under which the estimation error remains small in the long-time horizon. Our theory covers a wide class of partially-observed chaotic dynamical systems, which includes the Navier-Stokes equations and Lorenz models. In addition, we prove long-time accuracy of ensemble Kalman filters with surrogate dynamics, thus validating the use of machine-learned forecast models in ensemble data assimilation.

翻译：滤波旨在从部分且含噪声的观测中在线估计动力系统的状态。在状态维度较高的应用中，集合卡尔曼滤波器通常是首选方法。本文确立了集合卡尔曼滤波器的长期精度。我们提出了关于动力学与观测的条件，在这些条件下，估计误差在长时间范围内保持较小。我们的理论涵盖了一类广泛的偏观测混沌动力系统，包括Navier-Stokes方程和Lorenz模型。此外，我们证明了使用替代动力学的集合卡尔曼滤波器的长期精度，从而验证了在集合数据同化中使用机器学习预报模型的可行性。