Bias and Multiscale Correction Methods for Variational State Estimation

TL;DR

Variational state estimation using the so-called parametrized-background data-weak method, which 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.

Abstract

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.

Figures (13)

• Figure 1: Setting for Example 1. The plot (a) shows the noise-free measurements, whereas the plot (b) shows the noisy measurements according to the observation model \ref{['eq:test_bias']}. Plot (c) and (d) shows the basis for $W_m$ and $V_n$, respectively.
• Figure 2: Reconstruction with PBDW and bPBDW for the case of noise parameters $\sigma=A_{gt}/10$ and $\alpha = 0.2$, considering 5 POD modes (optimal dimension for $V_n$, see also Figure \ref{['fig:simple_experiment_bench']}).
• Figure 3: Left: $\ell^2$ relative error $e(n)$ for $\alpha = 0.1$ and $\sigma = A_{gt}/100$ The 64 lighter curves correspond to the error \ref{['eq:error']} for every reconstruction, whereas the thicker black curve shows the mean error among the 64 test cases. Right: Average reconstruction errors for the standard PBDW and for the bias-correction method bPBDW.
• Figure 4: Maximal $e(n)$ for bPBDW with respect to the number of measurements $m$ considered, i.e., the dimension of the space $W_m$ ($\alpha = 0.1$ and $\sigma = A_{gt}/100$).
• Figure 5: Average $e(n)$ with bPBDW for different bias intensities $\alpha$ (see equation \ref{['eq:test_bias']})

Theorems & Definitions (7)

• Remark 1: Parametrized-background as solution of a PDE
• Remark 2: Application to ultrasound imaging
• Remark 3
• Remark 4: Well-posedness
• Remark 5: Computational cost of the bPBDW method vs. PBDW
• Remark 6: Computational cost associated with additional measurements
• Remark 7: Computational cost of the sPBDW vs. PBDW