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首页> 外文期刊>SIAM/ASA Journal on Uncertainty Quantification >Asymptotic Forecast Uncertainty and the Unstable Subspace in the Presence of Additive Model Error
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Asymptotic Forecast Uncertainty and the Unstable Subspace in the Presence of Additive Model Error

机译：渐近预测不确定性和不稳定子空间的存在加性模型错误

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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 phenomenon, 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 nonnegative 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. While forecast error in the stable subspace may not generically vanish, we show that even without filtering, uncertainty remains uniformly bounded due its dynamical dissipation. However, the continuous reinjection of uncertainty from model errors may be excited by transient instabilities in the stable modes of high variance, rendering forecast uncertainty impractically large. In the context of ensemble data assimilation, this requires rectifying the rank of the ensemble-based gain to account for the growth of uncertainty beyond the unstable and neutral subspace, additionally correcting stable modes with frequent occurrences of positive local Lyapunov exponents that excite model errors.
机译：这是很好理解的动态不稳定预测不确定性的主要驱动因素之一在混乱中,物理系统。技术设计开发现象,减少的有效尺寸数据同化问题的方向快速增长的错误。工作,此外,提供正式的证明中央假说的同化安娜Trevisan不稳定子空间方法和她的合作者:过滤器和流畅完美,线性、高斯模型渐近分布的预测错误符合unstable-neutral子空间。具体来说,预测和列空间后验误差协方差渐近一致与落后的李雅普诺夫向量张成的空间负的指数。研究集中在完美的模型,这一点现在目前的工作探讨了关系之间的动态不稳定,精度观察和预测误差的进化在线性模型相加模型错误。证明为渐近的不确定性范围,明确相关动态扩张,模型精度和观测准确性。提供一种新颖的、必要的标准有界性的预测错误。数值研究之间的关系观察设计,动力不稳定,过滤器有界性。详细的介绍了乘法遍历性定理和理论李雅普诺夫向量的建设。误差稳定子空间可能不一般消失,我们表明,即使没有过滤,不确定性仍然由于其一致有界动力耗散。回注模型误差的不确定性是兴奋的瞬态不稳定稳定模式的差异,使预测不确定性还大。整体数据同化,这要求整流的秩ensemble-based增益不确定性的增长超出了不稳定的和中性的子空间,此外纠正与频繁出现稳定的模式积极激发当地李雅普诺夫指数模型错误。

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