Data assimilation is a method of uncertainty quantification to estimate the hidden true state by updating the prediction owing to model dynamics with observation data. As a prediction model, we consider a class of nonlinear dynamical systems on Hilbert spaces including the two-dimensional Navier-Stokes equations and the Lorenz '63 and '96 equations. For nonlinear model dynamics, the ensemble Kalman filter (EnKF) is often used to approximate the mean and covariance of the probability distribution with a set of particles called an ensemble. In this paper, we consider a deterministic version of the EnKF known as the ensemble transform Kalman filter (ETKF), performing well even with limited ensemble sizes in comparison to other stochastic implementations of the EnKF. When the ETKF is applied to large-scale systems, an ad-hoc numerical technique called a covariance inflation is often employed to reduce approximation errors. Despite the practical effectiveness of the ETKF, little is theoretically known. The present study aims to establish the theoretical analysis of the ETKF. We obtain that the estimation error of the ETKF with and without the covariance inflation is bounded for any finite time. In particular, the uniform-in-time error bound is obtained when an inflation parameter is chosen appropriately, justifying the effectiveness of the covariance inflation in the ETKF.

翻译：数据同化是一种不确定性量化方法，通过利用模型动力学预测与观测数据更新，来估计隐藏的真实状态。作为预测模型，我们考虑希尔伯特空间上的一类非线性动力系统，包括二维纳维-斯托克斯方程以及Lorenz '63和'96方程。对于非线性模型动力学，通常使用集合卡尔曼滤波（EnKF）通过一组称为集合的粒子来逼近概率分布的均值和协方差。本文研究一种确定性的EnKF版本，即集成变换卡尔曼滤波（ETKF），与EnKF的其他随机实现相比，该滤波器即使在有限集合规模下也能表现出良好性能。当ETKF应用于大规模系统时，常采用一种称为协方差膨胀的临时数值技术来降低近似误差。尽管ETKF在实践中有效，但其理论分析尚不充分。本研究旨在建立ETKF的理论分析框架。我们证明了无论是否使用协方差膨胀，ETKF的估计误差在任何有限时间内都是有界的。特别地，当适当选择膨胀参数时，可得到时间均匀误差界，从而验证了ETKF中协方差膨胀的有效性。