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.
Comparing an Ensemble Kalman Filter to a 4DVAR Data Assimilation System in Chaotic Dynamics
Universidade Federal de Pelotas – Faculdade de Meteorologia（佩洛塔斯联邦大学气象学院）；Departamento de Ciência e Tecnologia Aeroespacial – Instituto de Aeronáutica e Espaço – Divisão de Ciências Atmosféricas（航空航天科学与技术系 – 航空与航天研究所 – 大气科学部）
Flow Matching for Efficient and Scalable Data Assimilation
The Computing and Mathematical Sciences Department, California Institute of Technology（加州理工学院计算与数学科学系）；Department of Mathematics and Scientific Computing and Imaging Institute, University of Utah（犹他大学数学与科学计算系和成像研究所）
