Data assimilation performance can be significantly impacted by biased noise in observations, altering the signal magnitude and introducing fast oscillations or discontinuities when the system lacks smoothness. To mitigate these issues, this paper employ variational state estimation using the so-called parametrized-background data-weak method. This approach relies on a background manifold parametrized by a set of constraints, enabling the state estimation by solving a minimization problem on a reduced-order background model, subject to constraints imposed by the input measurements. The proposed formulation incorporates a novel bias correction mechanism and a manifold decomposition that handles rapid oscillations by treating them as slow-decaying modes based on a two-scale splitting of the classical reconstruction algorithm. The method is validated in different examples, including the assimilation of biased synthetic data, discontinuous signals, and Doppler ultrasound data obtained from experimental measurements.

翻译：观测数据中的有偏噪声会显著影响数据同化的性能，改变信号幅值并在系统缺乏平滑性时引入快速振荡或不连续性。为缓解这些问题，本文采用参数化背景数据弱化方法进行变分状态估计。该方法依赖于由一组约束参数化的背景流形，通过求解降阶背景模型上的最小化问题实现状态估计，同时满足输入测量值施加的约束条件。所提出的公式包含新颖的偏差校正机制和流形分解技术，该技术通过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。该方法在不同案例中得到验证，包括有偏合成数据同化、不连续信号同化以及实验测量获得的超声多普勒数据同化。