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.

翻译：四维弱约束变分数据同化通过最小化一个同时考虑状态与观测偏差以及随时间变化的模型误差的成本函数，在给定部分含噪声观测和动力学模型的情况下估计状态。该方法可表述为关联最小二乘问题的高斯-牛顿迭代。本文在观测失配项前引入一个参数，并通过解析方法证明：当观测无噪声时，该参数对收敛到真实解至关重要；当观测存在有界噪声且含噪声时，该参数对误差有界性至关重要。我们还考虑了状态-参数联合估计。使用洛伦兹63和洛伦兹96模型的数值实验验证了理论结果。