A joint parameter-state estimation framework using POD-Galerkin projected reduced order model and 3DVar data assimilation

Reduced-order models (ROMs) provide a fast and efficient approach to approximate full-order numerical models by capturing their essential dynamics. However, uncertainties in boundary conditions can lead to noticeable discrepancies between ROM simulations and real-world behavior. To address this challenge, this study introduces a novel framework that jointly corrects parameters and states by integrating ROMs with three-dimensional variational data assimilation (3D-Var). The framework operates in two sequential stages: a parameter estimation stage and a state reconstruction stage. Optimal parameter estimates are firstly derived to drive the ROM, generating optimal background information for the subsequent state reconstruction. A background error covariance matrix is computed from snapshots generating the ROM, significantly enhancing the reliability of the online data assimilation process. This study further explores the influence of sensor placement, measurement noise, and the number of measurement points on assimilation performance. Guidelines for optimal sensor placement are provided, along with an analysis of the prediction capability over extended parameter ranges. This framework offers a robust solution for enhancing the fidelity of ROMs in practical applications.