中文摘要：

针对变分资料同化中目标泛函梯度计算精度不高且复杂等问题，提出了一种基于对偶数理论的资料同化新方法，主要优点是：能避免复杂的伴随模式开发及其逆向积分，只需在对偶数空间通过正向积分就能同时计算出目标泛函和梯度向量的值。首先利用对偶数理论把梯度分析过程转换为对偶数空间中目标泛函计算过程，简单、高效和高精度地获得梯度向量值；其次结合典型的最优化方法，给出了非线性物理系统资料同化问题的新求解算法；最后对Lorenz 63混沌系统、包含开关的不可微物理模型和抛物型偏微分方程分别进行了资料同化数值实验，结果表明：新方法能有效和准确地估计出预报模式的初始条件或物理参数值。

英文摘要：

In gradient computations of the variational data assimilation （VDA） by the adjoint method, in order to overcome a lot of shortcomings such as low accuracy, di？cult implementation, and great complexity, etc., a novel data assimilation method is proposed based on the dual-number theory. The important advantages are that the coding of adjoint models and reverse integrations are not necessary any more, and the values of cost functional and its corresponding gradient vectors can be attained simultaneously only by one forward computation in dual-number space. Furthermore, the accuracy of gradient can be close to the computer machine precision without other error sources. The paper is organised as follows. Firstly, the dual-number theory and algorithm rules are introduced. Then, the issues of gradient analysis and computation in VDA are transformed into the processes of calculating the cost functional numerically in dual-number space, and the gradient vectors can be obtained at the same time in an easy, e？cient and accurate way. Secondly, the new algorithm for data assimilation in nonlinear physical systems is developed by combining accurate gradient information from the dual-number method with classical optimization algorithm. Thirdly, numerical experiments on sensitivity analysis for an ENSO nonlinear air-sea coupled oscillator are implemented, and the results are presented to demonstrate the important advantages of the dual-number method in the calculation of derivative information. Finally, numerical simulations for data assimilation are carried out respectively for the typical Lorenz 63 chaotic systems, the specific humidity evolving equation with physical “on-off” process at a single grid point, and a parabolic partial differential equation. Some conclusions can be drawn from the numerical experiments. The newly proposed method may be suited to many kinds of optimization problems with ordinary or partial differential equations as constraints, such as data assimilation, parameter estimation, inverse pro