基于无导数优化方法的数值模式误差估计

摘要

Initial error and model error are key factors restricting the accuracy of numerical weather prediction (NWP). The purpose of the present study is to estimate the errors of spatiotemporal evolution model by using recent observations. By considering the continuous evolution of atmosphere, the observed data (ignoring the measurement error) can be viewed as a series of solutions of accurate model governing the actual atmosphere, and the model errors can be objectively assumed to be an unknown functional term (a missing forcing term) of the numerical model, thus the NWP can be considered as an inverse problem to uncover the unknown model error term by using the long periods of observed data. In this study, we first construct an inverse problem model with its optimization problem, which is constrained by the numerical model, to estimate the errors of spatiotemporal evolution model, then we present a derivative-free optimization (DFO) method to find the minimum solution of the optimization problem by running the numerical model with an external forcing term. The DFO method does not need to compute the gradient of the objective functional and the tangent linear model or adjoint model of the original numerical model. The numerical study of Burgers equation indicates that the presented methods can effectively uncover the model errors from the past data and evidently improve the numerical prediction. The precedures described in this paper open up possibilities for utilizing the past observation data to extract useful information about model errors and enhance the prediction efficiency in the operational models.%初始场误差和模式误差是制约数值预报准确率的两个关键因素，本文主要考虑利用历史观测资料实现时空演变的模式误差的估计问题。通过把模式误差综合考虑成为准确模式中的未知项，把历史资料看作是带有未知项的准确模式的特解，构造了求解时空演变的模式误差项的反问题及其最优控制问题。给出了一个解决最优控制问题的无导数优化方法，该方法的优点是不需要建立原数值模式的切线性模式与伴随模式，它只需在增加一个外强迫项的基础上运行原数值模式即可实现模式误差项的最优估计。关于Burgers方程的算例表明，无论模式的初始状态是否准确已知，无导数优化方法都能有效解决时空演变的模式误差的最优估计问题，它为实际业务模式利用历史数据提取模式误差信息并显著地改进预报效果提供了一种方便可行的数值方法与理论依据。