Abstract This paper introduces a computationally efficient data assimilation scheme based on Gaussian quadrature filtering that potentially outperforms current methods in data assimilation for moderately nonlinear systems. Moderately nonlinear systems, in this case, are systems with numerical models with small fourth and higher derivative terms. Gaussian quadrature filters are a family of filters that make simplifying Gaussian assumptions about filtering pdfs in order to numerically evaluate the integrals found in Bayesian data assimilation. These filters are differentiated by the varying quadrature rules to evaluate the arising integrals. The approach we present, denoted by Assumed Gaussian Reduced (AGR) filter, uses a reduced order version of the polynomial quadrature first proposed in Ito and Xiong [2000. Gaussian filters for nonlinear filtering problems. IEEE Trans. Automat. Control. 45, 910–927]. This quadrature uses the properties of Gaussian distributions to form an effectively higher order method increasing its efficiency. To construct the AGR filter, this quadrature is used to form a reduced order square-root filter, which will reduce computational costs and improve numerical robustness. For cases of sufficiently small fourth derivatives of the nonlinear model, we demonstrate that the AGR filter outperforms ensemble Kalman filters (EnKFs) for a Korteweg-de Vries model and a Boussinesq model.

中文翻译：

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一种用于中等非线性系统的基于数值积分的卡尔曼滤波器

摘要 本文介绍了一种基于高斯正交滤波的计算效率高的数据同化方案，该方案在中等非线性系统的数据同化中可能优于当前方法。在这种情况下，中等非线性系统是具有四阶和更高阶导数项的数值模型的系统。高斯正交滤波器是一系列滤波器，它们简化了关于过滤 pdf 的高斯假设，以便对贝叶斯数据同化中的积分进行数值评估。这些滤波器通过不同的正交规则进行区分，以评估产生的积分。我们提出的方法由假设高斯减少 (AGR) 滤波器表示，它使用 Ito 和 Xiong [2000] 中首次提出的多项式正交的降阶版本。用于非线性滤波问题的高斯滤波器。IEEE 翻译 自动机。控制。45, 910–927]。该正交使用高斯分布的特性来形成有效的高阶方法，从而提高其效率。为了构建 AGR 滤波器，该正交用于形成降阶平方根滤波器，这将降低计算成本并提高数值鲁棒性。对于非线性模型的四阶导数足够小的情况，我们证明 AGR 滤波器优于 Korteweg-de Vries 模型和 Boussinesq 模型的集成卡尔曼滤波器 (EnKFs)。该正交用于形成降阶平方根滤波器，这将降低计算成本并提高数值鲁棒性。对于非线性模型的四阶导数足够小的情况，我们证明 AGR 滤波器优于 Korteweg-de Vries 模型和 Boussinesq 模型的集成卡尔曼滤波器 (EnKFs)。该正交用于形成降阶平方根滤波器，这将降低计算成本并提高数值鲁棒性。对于非线性模型的四阶导数足够小的情况，我们证明 AGR 滤波器优于 Korteweg-de Vries 模型和 Boussinesq 模型的集成卡尔曼滤波器 (EnKFs)。