This survey paper is written with the intention of giving a mathematical introduction to filtering techniques for intermittent data assimilation, and to survey some recent advances in the field. The paper is divided into three parts. The first part introduces Bayesian statistics and its application to statistical inference and estimation. Basic aspects of Markov processes, as they typically arise from scientific models in the form of stochastic differential and/or difference equations, are covered in the second part. The third and final part describes the filtering approach to estimation of model states by assimilation of observational data into scientific models. While most of the material is of survey type, very recent advances in the field of nonlinear data assimilation covered in this paper include a discussion of Bayesian inference in the context of optimal transportation and coupling of random variables, as well as a discussion of recent advances in ensemble transform filters. References and sources for further reading material will be listed at the end of each section.
Regularity-informed data assimilation: A hierarchical Bayesian approach to ensemble Kalman filtering for hyperbolic conservation laws
Division of Applied Mathematics, Linköping University（林雪平大学应用数学系）；Center for Computational Science and Engineering, Massachusetts Institute of Technology（麻省理工学院计算科学与工程中心）
Random-Feature Kalman Filtering for Linear PDE Data Assimilation
Ceyear Technologies Co., Ltd（中国电子科技集团公司第五十四研究所）；School of Mathematics and Statistics, Xidian University（西安电子科技大学数学与统计学院）；Department of Mathematics, National University of Singapore（新加坡国立大学数学系）
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）
Amortized Filtering and Smoothing with Conditional Normalizing Flows
School of Mathematics and Statistics（数学与统计学学院）；The University of Sydney（悉尼大学）；Faculty of Science and Technology（科学与技术学院）；Beijing Normal-Hong Kong Baptist University（北京师范大学-香港 Baptist大学）；Institute of Computational Mathematics and Scientific/Engineering Computing（计算数学与科学/工程计算研究所）；Academy of Mathematics and Systems Science（数学与系统科学学院）；Chinese Academy of Sciences（中国科学院）；Department of Mathematics and Center for Computation and Technology（数学系与计算技术中心）；Louisiana State University（路易斯安那州立大学）
A Bayesian Approach to Multivariate Adaptive Localization in Ensemble-Based Data Assimilation with Time-Dependent Extensions
Virginia Polytechnic Institute and State University（弗吉尼亚理工大学）；Computational Science Laboratory, Department of Computer Science, Virginia Tech（弗吉尼亚理工学院计算机科学与计算科学实验室）
