存在可加模型误差时的渐近预报不确定性和不稳定子空间
Asymptotic forecast uncertainty and the unstable subspace in the presence of additive model error
摘要：
众所周知，动态不稳定性是混沌物理系统中预测不确定性的主要驱动因素之一。数据同化技术的目的是利用这种现象，减少数据同化问题的有效维度的方向迅速增长的错误。此外，最近的数学工作为Anna Trevisan及其合作者的不稳定子空间方法中的同化的中心假设提供了正式的证明：对于完美的线性高斯模型中的滤波器和平滑器，预测误差的分布渐近符合不稳定中性子空间。具体而言，列跨度的预测和后验误差协方差渐近对齐的跨度后向李雅普诺夫向量与非负指数。早期的数学研究集中在完美的模型，目前的工作，探讨动态不稳定性，观测精度和预测误差的演变与添加剂的线性模型模型误差之间的关系。我们证明的渐近不确定性的界限，明确有关的动态膨胀率，模型精度和观测精度。形式化这种关系，我们提供了一个新的，必要的标准预测误差的有界性。此外，我们数值探讨观测设计，动力学不稳定性和过滤器有界性之间的关系。此外，我们还详细介绍了乘法遍历定理和李雅普诺夫向量的理论和建设。
英文摘要：
It is well understood that dynamic instability is among the primary drivers of forecast uncertainty in chaotic, physical systems. Data assimilation techniques have been designed to exploit this phenomena, reducing the effective dimension of the data assimilation problem to the directions of rapidly growing errors. Recent mathematical work has, moreover, provided formal proofs of the central hypothesis of the Assimilation in the Unstable Subspace methodology of Anna Trevisan and her collaborators: for filters and smoothers in perfect, linear, Gaussian models, the distribution of forecast errors asymptotically conforms to the unstable-neutral subspace. Specifically, the column span of the forecast and posterior error covariances asymptotically align with the span of backward Lyapunov vectors with non-negative exponents. Earlier mathematical studies have focused on perfect models, and this current work now explores the relationship between dynamical instability, the precision of observations and the evolution of forecast error in linear models with additive model error. We prove bounds for the asymptotic uncertainty, explicitly relating the rate of dynamical expansion, model precision and observational accuracy. Formalizing this relationship, we provide a novel, necessary criterion for the boundedness of forecast errors. Furthermore, we numerically explore the relationship between observational design, dynamical instability and filter boundedness. Additionally, we include a detailed introduction to the Multiplicative Ergodic Theorem and to the theory and construction of Lyapunov vectors.
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