Convergence properties of a Gauss-Newton data-assimilation method

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.