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The integration of experimental data into mathematical and computationalmodels is crucial for enhancing their predictive power in real-world scenarios.However, the performance of data assimilation algorithms can be significantlydegraded when measurements are corrupted by biased noise, altering the signalmagnitude, or when the system dynamics lack smoothness, such as in the presenceof fast oscillations or discontinuities. This paper focuses on variationalstate estimation using the so-called Parameterized Background Data Weak method,which relies on a parameterized background by a set of constraints, enablingstate estimation by solving a minimization problem on a reduced-orderbackground model, subject to constraints imposed by the input measurements. Toaddress biased noise in observations, a modified formulation is proposed,incorporating a correction mechanism to handle rapid oscillations by treatingthem as slow-decaying modes based on a two-scale splitting of the classicalreconstruction algorithm. The effectiveness of the proposed algorithms isdemonstrated through various examples, including discontinuous signals andsimulated Doppler ultrasound data.

Bias and Multiscale Correction Methods for Variational State Estimation Algorithms