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.

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