  The use of model order reduction techniques in combination with
ensemble-based methods for estimating the state of systems described by
nonlinear partial differential equations has been of great interest in recent
years in the data assimilation community. Methods such as the multi-fidelity
ensemble Kalman filter (MF-EnKF) and the multi-level ensemble Kalman filter
(ML-EnKF) are recognized as state-of-the-art techniques. However, in many
cases, the construction of low-fidelity models in an offline stage, before
solving the data assimilation problem, prevents them from being both accurate
and computationally efficient. In our work, we investigate the use of
{\it{adaptive}} reduced basis techniques in which the approximation space is
modified online based on the information that is extracted from a limited
number of full order solutions and that is carried by the past models. This
allows to simultaneously ensure good accuracy and low cost for the employed
models and thus improve the performance of the multi-fidelity and multi-level
methods.
