  Four-dimensional weak-constraint variational data assimilation estimates a
state given partial noisy observations and dynamical model by minimizing a cost
function that takes into account both discrepancy between the state and
observations and model error over time. It can be formulated as a Gauss-Newton
iteration of an associated least-squares problem. In this paper, we introduce a
parameter in front of the observation mismatch and show analytically that this
parameter is crucial either for convergence to the true solution when
observations are noise-free or for boundness of the error when observations are
noisy with bounded observation noise. We also consider joint state-parameter
estimation. We illustrated theoretical results with numerical experiments using
the Lorenz 63 and Lorenz 96 models.
