  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.
