The integration of experimental data into mathematical and computational models is crucial for enhancing their predictive power in real-world scenarios. However, the performance of data assimilation algorithms can be significantly degraded when measurements are corrupted by biased noise, altering the signal magnitude, or when the system dynamics lack smoothness, such as in the presence of fast oscillations or discontinuities. This paper focuses on variational state estimation using the so-called Parameterized Background Data Weak method, which relies on a parameterized background by a set of constraints, enabling state estimation by solving a minimization problem on a reduced-order background model, subject to constraints imposed by the input measurements. To address biased noise in observations, a modified formulation is proposed, incorporating a correction mechanism to handle rapid oscillations by treating them as slow-decaying modes based on a two-scale splitting of the classical reconstruction algorithm. The effectiveness of the proposed algorithms is demonstrated through various examples, including discontinuous signals and simulated Doppler ultrasound data.

翻译：将实验数据整合到数学模型和计算模型中，对于提升其在真实场景中的预测能力至关重要。然而，当测量数据被偏差噪声污染（导致信号幅度改变）或系统动力学缺乏光滑性（例如存在快速振荡或不连续性）时，数据同化算法的性能会显著下降。本文聚焦于基于所谓参数化背景数据弱方法的变分状态估计，该方法通过一组约束条件构建参数化背景，从而在满足输入测量约束的前提下，通过求解约化阶背景模型上的最小化问题实现状态估计。针对观测数据中的偏差噪声，提出了一种修正公式，引入校正机制，基于经典重构算法的双尺度分裂将快速振荡视为慢衰减模态进行处理。通过包括不连续信号和模拟多普勒超声数据在内的多种算例，验证了所提算法的有效性。