The 4D-Var method for filtering partially observed nonlinear chaotic dynamical systems consists of finding the maximum a-posteriori (MAP) estimator of the initial condition of the system given observations over a time window, and propagating it forward to the current time via the model dynamics. This method forms the basis of most currently operational weather forecasting systems. In practice the optimisation becomes infeasible if the time window is too long due to the non-convexity of the cost function, the effect of model errors, and the limited precision of the ODE solvers. Hence the window has to be kept sufficiently short, and the observations in the previous windows can be taken into account via a Gaussian background (prior) distribution. The choice of the background covariance matrix is an important question that has received much attention in the literature. In this paper, we define the background covariances in a principled manner, based on observations in the previous b assimilation windows, for a parameter \(b\ge 1\). The method is at most b times more computationally expensive than using fixed background covariances, requires little tuning, and greatly improves the accuracy of 4D-Var. As a concrete example, we focus on the shallow-water equations. The proposed method is compared against state-of-the-art approaches in data assimilation and is shown to perform favourably on simulated data. We also illustrate our approach on data from the recent tsunami of 2011 in Fukushima, Japan.

用于过滤部分观察到的非线性混沌动力系统的 4D-Var 方法包​​括在给定时间窗口内的观察中找到系统初始条件的最大后验 (MAP) 估计量，并通过模型将其向前传播到当前时间动力学。这种方法构成了目前大多数可操作的天气预报系统的基础。在实践中，如果时间窗口过长，由于成本函数的非凸性、模型误差的影响以及 ODE 求解器的精度有限，优化变得不可行。因此，窗口必须保持足够短，并且可以通过高斯背景（先验）分布考虑先前窗口中的观察结果。背景协方差矩阵的选择是文献中备受关注的一个重要问题。在本文中，我们根据前面的观察，以原则的方式定义背景协方差b同化窗口，用于参数\(b\ge 1\)。该方法最多比使用固定背景协方差的计算成本高出b倍，几乎不需要调整，并且大大提高了 4D-Var 的准确性。作为一个具体的例子，我们关注浅水方程。将所提出的方法与数据同化中的最新方法进行了比较，并显示出在模拟数据上表现良好。我们还说明了我们对 2011 年日本福岛最近发生的海啸数据的处理方法。