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We provide extended analyses and interpretations of the parametrized-background data-weak (PBDW) formulation, a real-time in situ data assimilation framework for physical systems modeled by parametrized partial differential equations. The new contributions are threefold. First, we conduct an a priori error analysis for imperfect observations: we provide a bound for the variance of the state error and identify distinct contributions to the noise-induced error. Second, we illustrate the elements of the PBDW formulation for a physical system, a raised-box acoustic resonator, and provide detailed interpretations of the data assimilation results in particular related to model and data contributions. Third, we present and demonstrate an adaptive PBDW formulation in which we incorporate unmodeled physics identified through data assimilation of a select few configurations. Keywords:variational data assimilation; parametrized partial differential equations; model order reduction; imperfect observations; acoustics.

PBDW State Estimation: Noisy Observations; Configuration-Adaptive Background Spaces; Physical Interpretations