Synergy of inverse problems and data assimilation techniques

Abstract

This review article aims to provide a theoretical framework for data assimilation, a specific type of an inverse problem arising for example in numerical weather prediction, hydrology and geology. We consider the general mathematical theory for inverse problems and regularisation, before we consider Tikhonov regularisation as one of the most popular methods for solving inverse problems. We show that data assimilation techniques such as three-dimensional and fourdimensional variational data assimilation (3DVar and 4DVar) as well as the Kalman filter and Bayes’ data assimilation are, in the linear case, a form of cycled Tikhonov regularisation. We give an introduction to key data assimilation methods as currently used in practice, link them and show their similarities. We also give an overview of ensemble methods. Furthermore, we provide an error analysis for the data assimilation process in general, show research problems and give numerical examples for simple data assimilation problems. An extensive list of references is given for further reading.