Data assimilation provides algorithms for widespread applications in various fields. It is of practical use to deal with a large amount of information in the complex system that is hard to estimate. Weather forecasting is one of the applications, where the prediction of meteorological data are corrected given the observations. Numerous approaches are contained in data assimilation. One specific sequential method is the Kalman Filter. The core is to estimate unknown information with the new data that is measured and the prior data that is predicted. As a matter of fact, there are different improved methods in the Kalman Filter. In this project, the Ensemble Kalman Filter with perturbed observations is considered. It is achieved by Monte Carlo simulation. In this method, the ensemble is involved in the calculation instead of the state vectors. In addition, the measurement with perturbation is viewed as the suitable observation. These changes compared with the Linear Kalman Filter make it more advantageous in that applications are not restricted in linear systems any more and less time is taken when the data are calculated by computers. The thesis seeks to develop the Ensemble Kalman Filter with perturbed observation gradually. With the Mathematical preliminaries including the introduction of dynamical systems, the Linear Kalman Filter is built. Meanwhile, the prediction and analysis processes are derived. After that, we use the analogy thoughts to lead in the non-linear Ensemble Kalman Filter with perturbed observations. Lastly, a classic Lorenz 63 model is illustrated by MATLAB. In the example, we experiment on the number of ensemble members and seek to investigate the relationships between the error of variance and the number of ensemble members. We reach the conclusion that on a limited scale the larger number of ensemble members indicates the smaller error of prediction.

翻译：数据同化为各领域广泛应用提供算法，其在处理难以估计的复杂系统大量信息时具有实际价值。天气预报是该应用之一，即通过观测数据修正气象预测结果。数据同化包含多种方法，其中一种特定序列方法为卡尔曼滤波。其核心思想是利用新测量数据与先验预测数据估计未知信息。事实上，卡尔曼滤波存在多种改进方法。本研究考虑添加扰动观测的集合卡尔曼滤波，通过蒙特卡洛模拟实现。该方法以集合代替状态向量参与计算，并将含扰动的测量值作为合理的观测数据。相较于线性卡尔曼滤波，这些改进使该方法更具优势：应用不再局限于线性系统，且计算机处理数据时耗时更少。本文旨在逐步构建添加扰动观测的集合卡尔曼滤波。在包含动力系统介绍的数学基础部分，首先建立线性卡尔曼滤波，推导预测与分析过程；随后通过类比思想引入非线性添加扰动观测的集合卡尔曼滤波。最后，利用MATLAB对经典Lorenz 63模型进行仿真实验。实验中我们对集合成员数量进行试验，探究误差方差与集合成员数量的关系。我们得出结论：在有限范围内，集合成员数量越大，预测误差越小。