Ensemble transform Kalman filtering (ETKF) data assimilation is often used to combine available observations with numerical simulations to obtain statistically accurate and reliable state representations in dynamical systems. However, it is well known that the commonly used Gaussian distribution assumption introduces biases for state variables that admit discontinuous profiles, which are prevalent in nonlinear partial differential equations. This investigation designs a new structurally informed non-Gaussian prior that exploits statistical information from the simulated state variables. In particular, we construct a new weighting matrix based on the second moment of the gradient information of the state variable to replace the prior covariance matrix used for model/data compromise in the ETKF data assimilation framework. We further adapt our weighting matrix to include information in discontinuity regions via a clustering technique. Our numerical experiments demonstrate that this new approach yields more accurate estimates than those obtained using ETKF on shallow water equations, even when ETKF is enhanced with inflation and localization techniques.

翻译：集成变换卡尔曼滤波（ETKF）数据同化常被用于将可用观测数据与数值模拟相结合，以获取动力系统中统计上准确且可靠的状态表征。然而，众所周知，常用的高斯分布假设会为存在不连续剖面的状态变量引入偏差，这类变量在非线性偏微分方程中普遍存在。本研究设计了一种新的结构感知非高斯先验，通过利用模拟状态变量的统计信息。具体而言，我们基于状态变量梯度信息的二阶矩构建了一个新的权重矩阵，以替代ETKF数据同化框架中用于模型/数据折中的先验协方差矩阵。我们进一步通过聚类技术调整权重矩阵，使其包含不连续区域的信息。数值实验表明，即使ETKF采用膨胀和局地化技术进行增强，本方法在浅水方程上的估计精度仍优于ETKF。