In this work we propose and evaluate two variational data assimilation techniques for the estimation of low order surrogate experimental dynamical models for fluid flows. Both methods are built from optimal control recipes and rely on proper orthogonal decomposition and a Galerkin projection of the Navier Stokes equation. The techniques proposed di er in the control variables they involve. The first one introduces a weak dynamical model defined only up to an additional uncertainty time-dependent function whereas the second one, handles a strong dynamical constraint in which the dynamical system's coe cients constitute the control variables. Both choices correspond to di erent approximations of the relation between the reduced basis on which is expressed the motion field and the basis components that have been neglected in the reduced order model construction. The techniques have been assessed on numerical data and for real experimental conditions with noisy Image Velocimetry data.