基于子空间方法的四维变分同化系统及其在气候预测中的应用

Decadal prediction is not only a frontier science, but also a national great demand. Advanced data assimilation method is one of the key factors for decadal predicting success. Mathematically, it can be formulated by an inverse initial-value problem for partial differential equations, with unknown random errors in both observational data and mathematical model, aiming at providing better initial condition by continuously incorporating observational information. The optimization model corresponding to the four-dimensional variational (4DVar) data assimilation problem is not only of high nonlinearity, huge scale of variables, but also faced with an inevitable difficulty that its gradient can only be evaluated by solving adjoint operators of complicated climate system models, which is intractable. Thus, to find an effective algorithm to solve this nonlinear optimization model is extremely challengeable. In this project, we plan to propose a new assimilation scheme suitable for decadal climate prediction through merging the advantages of both 4DVar and ensemble Kalman Filter (EnKF), based on advanced mathematical methods, especially optimization approaches. The scheme is hopefully able to solve the difficulties of EnKF and 4DVa, including the representative problem of ensemble members and the flow-dependent problem in both explicit and implicit ways to match the real atmospheric motion feature better. It will be applied to establishment of a new ensemble-based four-dimensional variational data assimilation system for initialization of coupled climate system model before decadal climate prediction. The project is expected to achieve breakthrough in theory and methodology of data assimilation for coupled climate system models.

年代际气候预测不仅是国际科学前沿问题，也是重大国家需求问题。先进的资料同化方法是年代际气候预测不可或缺的关键环节之一，其在数学上可表示为观测数据和数学模型存在未知随机误差的发展型偏微分系统的初值反演问题，其目的在于通过不断吸收观测信息，产生更加准确可靠的模式初始场。由于气候系统模式的高度复杂性和非线性，四维变分同化方法的主要困难在于求解具有超大规模变量的非线性优化问题和估计具有显式流依赖特征的背景误差协方差矩阵，这成为其在气候预测中成功应用的瓶颈。本项目拟基于先进的数学方法，特别是优化算法，通过四维变分同化和集合卡曼滤波的有机结合，提出适用于年代际气候预测的新同化方案，解决集合样本代表性问题，实现背景误差协方差矩阵的显式和隐式流依赖特征，使其更符合大气运动演变规律，以此建立新的集合四维变分资料同化系统，并将其应用于年代际气候预测中，可望实现我国耦合模式资料同化领域内的理论和方法上的重大突破

资料同化是求解一种带算子约束的大规模非线性最优化问题，它是提高天气（气候）预报（预测）的关键技术之一。针对现有资料同化方案存在维数高、计算量巨大或样本代表性差等瓶颈，本项目通过数学求解算法的创新，重点解决了集合样本代表性及背景误差协方差矩阵的流依赖等问题，发展了基于子空间方法的集合四维变分同化方法。项目提出一种正交展开的滤波函数来简化滤波矩阵与高维协方差矩阵之间的局地化算法，数值试验表明，当选取的滤波函数主特征向量个数到达20时，新滤波函数与原滤波矩阵几乎一致，而计算代价要小得多。在此基础上，本项目吸取已有子空间方法的内核，发展快速构造相似样本的数学方法,数值试验结果表明在集合变分资料同化中加入代表性样本能有效地改进同化效果。