State estimation theory in geosciences is commonly referred to as data assimilation. This term encompasses the entire sequence of operations that, starting from the observations of a system, and from additional statistical and/or dynamical information (such as an evolution model), provides the best possible estimate of its state. Data assimilation is common practice in numerical weather prediction but its application is becoming widespread in many other areas of climate, atmosphere, ocean and environment modelling; in all those circumstances where one intends to estimate the state of a large dynamical system based on limited information. While the complexity of data assimilation, and of the methods thereof, stands on its interdisciplinary nature across statistics, dynamical systems and numerical optimisation, when applied to geosciences an additional difficulty arises by the, constantly increasing, sophistication of the environmental models.

This overview course is aimed at geoscientists, who are confronted with the model-to-data fusion issue and would benefit from the application of data assimilation techniques, but so far have not delved into their conceptual and methodological complexities.

The course will provide first the formulation of the problem from a Bayesian perspective and will then present the two popular families of Gaussian based approaches, the Kalman-filter/-smoother and the variational methods. Ensemble based methods will then be considered, starting from the well known Ensemble Kalman filter, in its stochastic or deterministic formulation, and then the state-of-the-art ensemble-variational methods. The course will then conclude by presenting some of the nowadays active lines of development and current challenges, including coupled data assimilation and the particle filters.
The course will focus on the specific challenges that data assimilation has encountered to deal with high-dimensional chaotic systems, such as the atmosphere and ocean, and the countermeasures that have been taken and which have driven the dramatic development of the field experienced in the last decades.