We present an error analysis and further numerical investigations of the Parameterized-Background Data-Weak (PBDW) formulation to variational Data Assimilation (state estimation), proposed in [Y Maday, AT Patera, JD Penn, M Yano, Int J Numer Meth Eng, 102(5), 933-965]. The PBDW algorithm is a state estimation method involving reduced models. It aims at approximating an unknown function $u^{rm true}$ living in a high-dimensional Hilbert space from $M$ measurement observations given in the form $y_m = ell_m(u^{rm true}),, m=1,dots,M$, where $ell_m$ are linear functionals. The method approximates $u^{rm true}$ with $hat{u} = hat{z} + hat{eta}$. The emph{background} $hat{z}$ belongs to an $N$-dimensional linear space $mathcal{Z}_N$ built from reduced modelling of a parameterized mathematical model, and the emph{update} $hat{eta}$ belongs to the space $mathcal{U}_M$ spanned by the Riesz representers of $(ell_1,dots, ell_M)$. When the measurements are noisy {--- i.e., $y_m = ell_m(u^{rm true})+epsilon_m$ with $epsilon_m$ being a noise term --- } the classical PBDW formulation is not robust in the sense that, if $N$ increases, the reconstruction accuracy degrades. In this paper, we propose to address this issue with an extension of the classical formulation, {which consists in} searching for the background $hat{z}$ either on the whole $mathcal{Z}_N$ in the noise-free case, or on a well-chosen subset $mathcal{K}_N subset mathcal{Z}_N$ in presence of noise. The restriction to $mathcal{K}_N$ makes the reconstruction be nonlinear and is the key to make the algorithm significantly more robust against noise. We {further} present an emph{a priori} error and stability analysis, and we illustrate the efficiency of the approach on several numerical examples.
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