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Data Assimilation comprehensively covers data assimilation and inverse methods, including both traditional state estimation and parameter estimation. This text and reference focuses on various popular data assimilation methods, such as weak and strong constraint variational methods and ensemble filters and smoothers. It is demonstrated how the different methods can be derived from a common theoretical basis, as well as how they differ and/or are related to each other, and which properties characterize them, using several examples. It presents the mathematical framework and derivations in a way which is common for any discipline where dynamics is merged with measurements. The mathematics level is modest, although it requires knowledge of basic spatial statistics, Bayesian statistics, and calculus of variations. Readers will also appreciate the introduction to the mathematical methods used and detailed derivations, which should be easy to follow, are given throughout the book. The codes used in several of the data assimilation experiments are available on a web page. The focus on ensemble methods, such as the ensemble Kalman filter and smoother, also makes it a solid reference to the derivation, implementation and application of such techniques. Much new material, in particular related to the formulation and solution of combined parameter and state estimation problems and the general properties of the ensemble algorithms, is available here for the first time.

Stochastic processes --- Kalman filtering. --- Numerical weather forecasting. --- Prediction theory. --- Stochastic processes. --- Kalman filtering --- Mathematical Statistics --- Mathematics --- Physical Sciences & Mathematics --- Earth sciences. --- Mathematical models. --- Probabilities. --- Physics. --- Applied mathematics. --- Engineering mathematics. --- Earth Sciences. --- Earth Sciences, general. --- Probability Theory and Stochastic Processes. --- Theoretical, Mathematical and Computational Physics. --- Mathematical Modeling and Industrial Mathematics. --- Appl.Mathematics/Computational Methods of Engineering. --- Random processes --- Probabilities --- Filtering, Kalman --- Control theory --- Estimation theory --- Prediction theory --- Kalman, filtrage de --- Processus stochastiques --- EPUB-LIV-FT LIVTERRE SPRINGER-B --- Geography. --- Distribution (Probability theory. --- Mathematical and Computational Engineering. --- Engineering --- Engineering analysis --- Mathematical analysis --- Distribution functions --- Frequency distribution --- Characteristic functions --- Cosmography --- Earth sciences --- World history --- Mathematical physics. --- Probability --- Statistical inference --- Combinations --- Chance --- Least squares --- Mathematical statistics --- Risk --- Geosciences --- Environmental sciences --- Physical sciences --- Models, Mathematical --- Simulation methods --- Physical mathematics --- Physics

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This open-access textbook's significant contribution is the unified derivation of data-assimilation techniques from a common fundamental and optimal starting point, namely Bayes' theorem. Unique for this book is the "top-down" derivation of the assimilation methods. It starts from Bayes theorem and gradually introduces the assumptions and approximations needed to arrive at today's popular data-assimilation methods. This strategy is the opposite of most textbooks and reviews on data assimilation that typically take a bottom-up approach to derive a particular assimilation method. E.g., the derivation of the Kalman Filter from control theory and the derivation of the ensemble Kalman Filter as a low-rank approximation of the standard Kalman Filter. The bottom-up approach derives the assimilation methods from different mathematical principles, making it difficult to compare them. Thus, it is unclear which assumptions are made to derive an assimilation method and sometimes even which problem it aspires to solve. The book's top-down approach allows categorizing data-assimilation methods based on the approximations used. This approach enables the user to choose the most suitable method for a particular problem or application. Have you ever wondered about the difference between the ensemble 4DVar and the "ensemble randomized likelihood" (EnRML) methods? Do you know the differences between the ensemble smoother and the ensemble-Kalman smoother? Would you like to understand how a particle flow is related to a particle filter? In this book, we will provide clear answers to several such questions. The book provides the basis for an advanced course in data assimilation. It focuses on the unified derivation of the methods and illustrates their properties on multiple examples. It is suitable for graduate students, post-docs, scientists, and practitioners working in data assimilation.

Earth sciences --- Probability & statistics --- Bayesian inference --- Data Assimilation --- Parameter Estimation --- Ensemble Kalman Filter --- 4DVar --- Representer Method --- Ensemble Methods --- Particle Filter --- Particle Flow