变分连续同化中优化步长的推导方法及数值试验

一个好的变分同化方法,必须考虑到预报模式具有误差,并对模式误差进行纠正.变分连续同化(VCA)通过在模式方程中引进一个修正项,连续地调整模式变量,从而达到以上目的.在变分连续同化的迭代过程中,优化步长的选取方法对于同化迭代是否收敛、收敛速度快慢与否都至关重要,文中推导了一种有效利用观测资料计算优化步长的新方法,并以大气运动方程中一维非线性平流波和二维惯性波为例,进行了四维变分资料同化数值试验.数值结果表明,VCA方法和计算优化步长的公式都是有效的.     

基于非线性滤波的风廓线雷达数据处理

针对实际工程中风廓线雷达风向、风速随高度分布取值的非线性特性以及非气象干扰因素,基于非线性化方法——扩展卡尔曼滤波法,对风廓线雷达探测数据进行滤波处理.先利用泰勒展开式的一次项对非线性方程作线性化处理,再结合经典的卡尔曼滤波进行滤波估计,将非线性滤波问题转化为一个近似的线性滤波问题.仿真实验结果表明,该方法可以有效去除风场数据中掺杂的噪声干扰,很好地发挥了其非线性特性,滤波效果优于传统的卡尔曼滤波,具有一定的工程应用前景.     

基于NLS-4DVar方法的雷达资料同化及其在暴雨预报中的应用

在基于本征正交分解POD（Proper Orthogonal Decomposition）的集合四维变分同化方法（POD4DEnVar）建立的雷达资料同化系统（PRAS）的基础上,本文利用非线性最小二乘法的集合四维变分同化方法（NLS-4DVar）对PRAS进行改进,解决PRAS在高度非线性情况下的适应性问题,建立了新的雷达资料同化系统（NRAS）。通过观测系统模拟试验OSSEs（Observing System Simulation Experiments）和两次实际暴雨同化试验（2010年7月8日,中国中部地区;2014年3月30日,中国华南地区）对NRAS进行检验,并与PRAS的同化结果进行了对比。结果表明:无论是OSSEs还是实际雷达资料的同化,相对于PRAS,NRAS能够进一步提高同化效果。通过增加迭代的次数,NRAS能够有效地调整初始场的风场和水汽场,进一步提高了降水强度和位置的预报精度。但随着迭代次数的增加,对初始场的调整变小,进而对降水预报效果的改进也减小。试验结果表明NRAS能够有效解决PRAS在高度非线性情况下的应用问题,通过有限次数的迭代,即可得到近似收敛的结果。因而NRAS有望在数值预报中更有效地同化雷达资料,提高中小尺度天气的预报水平。     

一种基于无约束总变分模型的阴影检测方法

阴影检测是进行阴影处理前的重要步骤,总变分模型可以用于影像阴影检测.通过对总变分模型进行改进,提出了一种基于无约束总变分模型的阴影检测方法.经实验及统计分析证明,在合适的迭代条件下,该方法对于单一阴影影像的处理效果理想.     

层结大气中重力惯性波的非线性周期解

本文在动量无辐散近似下的层结大气中,考虑密度空间分布的不均匀性引入非线性项,并假设解为行波形式,得到了一个二阶非线性系统。应用本系统不仅可以得到刘式适等(1984)所得到的一切结果,而且可以证明一次近似系统存在周期解时,非线性系统也存在周期解,且避免了应用级数展开时所带来的一些数学问题。文中还讨论了非线性系统的一系列近似解。     

正压大气中地转偏差引起的非线性Rossby波

正压大气中的非线性波动的研究业已在60年代开始,70年代到80年代取得了丰硕的成果。在我国巢纪平等(1980)首先讨论了旋转大气中的正压Rossby椭圆余弦波。之后刘式适、黄思训等也做了大量的工作。特别是刘式适等,鉴于以前的工作多采用多尺度方法,比如约化摄动法,而这种方法中的G-M变换本身就意味着将要导出的结果。考虑到长波的弱非线性近似,将非线性项在平衡点附近展开为Taylor级数,求得了方程的非线性近似解。黄思训等认为应用展开法只能求得原方程的近似解,问题的解析解尚未找到。于是利用类似的模式导出了方程的解析解,且给出了存在非线性     

应用变分原理积分正压原始方程

将正压原始方程组无量纲化以后,把量级近似相当的动能与有效位能之和守恒作为变分约束条件,有效地控制了非线性不稳定,使长时间积分中能保持总能量守恒。其结果可和其它积分方法相比。     

正压大气非线性波的二级近似(二)——位涡度方程的求解

本文是用摄动法求取正压原始方程一类非线性大尺度慢波解的二级近似的第二部分,提出了一种在电子计算机上进行某些初等函数非数值运算的方法,由此求得了二级问题中位涡度方程的解.数值试验的结果表明,所求得的二级近似是合理的.     

受迫广义Ｋｌｅｉｎ Ｇｏｒｄｏｎ方程的孤子近似解

利用同伦映射方法和理论讨论了一类受迫广义非线性Ｋｌｅｉｎ Ｇｏｒｄｏｎ方程,在适当的条件下,较简捷地得到孤波的任意次精度的近似解．     

系统辨识(7)：递阶辨识原理与方法

递阶辨识是系统辨识的一个重要分支.递阶辨识原理是在大系统递阶控制的"分解-协调原理"基础上发展起来的,它不仅能够解决参数数目多、维数高、大规模系统辨识算法计算量大的问题,而且能够解决结构复杂的双线性参数系统、多线性参数系统以及非线性系统的辨识问题.首先介绍递阶辨识原理和线性方程组Ax=b的著名雅可比迭代和高斯-赛德尔迭代,给出了线性方程组的迭代方法族;其次将雅可比迭代思想和递阶辨识原理用于研究一般矩阵方程和耦合矩阵方程的递阶梯度迭代求解方法和递阶最小二乘迭代求解方法;再次介绍了方程误差模型的两阶段最小二乘辨识方法(一个简单的递阶辨识方法)和线性回归模型的递阶最小二乘辨识方法;最后研究了类多变量CARMA系统的递阶辨识方法.     

An Ensemble-Based Three-Dimensional Variational Assimilation Method for Land Data Assimilation

Land surface models are often highly nonlinear with model physics that contain parameterized discontinuities. These model attributes severely limit the application of advanced variational data assimilation methods into land data assimilation. The ensemble Kalman filter （EnKF） has been widely employed for land data assimilation because of its simple conceptual formulation and relative ease of implementation. An updated ensemble-based three-dimensional variational assimilation （En3-DVar） method is proposed for land data assimilation This new method incorporates Monte Carlo sampling strategies into the 3-D variational data assimilation framework. The proper orthogonal decomposition （POD） technique is used to efficiently approximate a forecast ensemble produced by the Monte Carlo method in a 3-D space that uses a set of base vectors that span the ensemble. The data assimilation process is thus significantly simplified. Our assimilation experiments indicate that this new En3-DVar method considerably outperforms the EnKF method by increasing assimilation precision. Furthermore, computational costs for the new En3-DVar method are much lower than for the EnKF method.     

Retrieval of eddy thermal conductivity in the weakly nonlinear Prandtl model for katabatic flows

Because the nonlinearity of actual physical processes can be expressed more precisely by the introduction of a nonlinear term, the weakly nonlinear Prandtl model is one of the most effective ways to describe the pure katabatic flow (no background flow). Features of the weak nonlinearity are reflected by two factors: the small parameter ε and the gradually varying eddy thermal conductivity. This paper first shows how to apply the Wentzel–Kramers–Brillouin (WKB) method for the approximate solution of the weakly nonlinear Prandtl model, and then describes the retrieval of gradually varying eddy thermal conductivity from observed wind speed and perturbed potential temperature. Gradually varying eddy thermal conductivity is generally derived from an empirical parameterization scheme. We utilize wind speed and potential temperature measurements, along with the variational assimilation technique, to derive this parameter. The objective function is constructed by the square of the differences between the observation and model value. The new method is validated by numerical experiments with simulated measurements, revealing that the order of the root mean squre error is 10^–2 and thus confirming the method’s robustness. In addition, this method is capable of anti-interference, as it effectively reduces the influence of observation error.     

Effect of nonlinear dynamic process on formation and breakdown of blocking

With the L-P approximate method (variation of parameter method), a barotropic channel model in β-plane is used to study the effect of nonlinear interaction between two waves with different scales on the formation of blocking. The approximate analytical solution, which can describe the process of the blocking formation, maintenance and breakdown, has been obtained by using the method of approximate expansion. The importance of nonlinear interaction between two waves with different scales is stressed in the solution. The result suggests that the nonlinear interaction is the main dynamic process of the blocking formation. Some required conditions of blocking formation are also discussed.     

对称不稳定的非线性问题和对称型重力惯性波的非线性周期解

本文利用密度的空间分布不均匀性引入非线性项，从准动量无辐散模式出发导出了一个自治的二阶非线性系统，应用这个系统讨论了非线性项对对称不稳定以及对称型重力惯性波非线性周期解的作用。从本系统的一次近似系统可得到类似Hoskins于1974年得到的结论，同样可导出对称不稳定的位涡判据。由于本系统是一有限次的非线性系统，故应用Poincare形式级数法可证明非线性周期解的存在性，并可求得周期解的一系列近似解。     

用一种新的同化方法同化降水量资料

Observations of accumulated precipitation are extremely valuable for effectively improving rainfall analysis and forecast. It is, however, difficult to use such observations directly through sequential assimilation methods, such as three-dimensional variational data assimilation or an Ensemble Kalman Filter. In this study, the authors illustrate a new approach that makes effective use of precipitation data to improve rainfall forecast. The new method directly obtains an optimal solution in a reduced space by fitting observations with historical time series generated by the model; it also avoids the implementation of tangent linear model and its adjoint. A lot of historical samples are produced as the ensemble of precipitation observations with the fully nonlinear forecast model. The results show that the new approach is capable of extracting information from precipitation observations to improve the analysis and forecast. This method provides comparable performance with the standard four- dimensional variational data assimilation at a much lower computational cost.     

THE INVERSE PROBLEM OF KLEIN-GORDON EQUATION BOUNDARY VALUE PROBLEM AND ITS APPLICATION IN DATA ASSIMILATION

Since the solution of elliptic partial differential equations continuously depends on the boundary condition, the Euler equation derived from variational method cannot be solved without boundary condition. It is often difficult to provide the exact boundary condition in the practical use of variational method. However, in some application problems such as the remote sensing data assimilation, the values can be easily obtained in the inner region of the domain. In this paper, the boundary condition is tried to be retrieved by using part solutions in the inner area. Firstly, the variational problem of remote sensing data assimilation within a circular area is established. The Klein-Gordon elliptic equation is derived from the Euler method of variational problems with assumed boundary condition. Secondly, a computer-friendly Green function is constructed for the Dirichlet problem of two-dimensional Klein-Gordon equation, with the formal solution according to Green formula. Thirdly, boundary values are retrieved by solving the optimal problem which is constructed according to the best approximation between formal solutions and high-accuracy measurements in the interior of the domain. Finally, the assimilation problem is solved on substituting the retrieved boundary values into the Klein-Gordon equation. It is a type of inverse problem in mathematics. The advantage of this method lies in that it overcomes the inherent instability of the inverse problem of Fredholm integral equation and alleviates the error introduced by artificial boundary condition in data fusion using variational method in the past.     

均匀基流中非线性行波解的稳定性及其存在条件

本文利用地转动量近似＾「１」并结合行波解的方法,研究了均匀基本气流中的层结大气非线性波动,首先由绝热无磨擦的闭合方程组导得关于垂直Ｐ速度的单一变量的非线性方程,然后讨论了非线性方程解的稳定性,并求得近似的非线性方程的椭圆余弦波和孤立波解及其存在条件。     

Criteria for the nonlinear stability of three—dimensional quasi—geostrophic motions

Some more proper criteria for the nonlinear stability of three-dimensional quasi-geostrophic motions are given by combining variational principle with a prior estimates method. The criteria are suitable for perturbations of initial condition as well as parameters in the model. The basic flow can be steady or unsteady. Particularly the difficulty due to the nonlinear boundary condition is completely overcome by the use of our method.     

基于进化方向遗传算法的四维变分资料同化方法

数值天气预报模式初始时刻要素场的变分同化问题是一个非线性最优化问题。利用进化方向遗传算法（EDGA）求解该最优化问题，并对理想初始场作数值模拟，结果表明模拟的效果较好。