Abstract: This chapter provides a broad introduction to Bayesian data assimilation that will be useful to practitioners in interpreting algorithms and results, and for theoretical studies developing novel schemes with an understanding of the rich history of geophysical data assimilation and its current directions. The simple case of data assimilation in a ‘perfect’ model is primarily discussed for pedagogical purposes. Some mathematical results are derived at a high level in order to illustrate key ideas about different estimators. However, the focus of this chapter is on the intuition behind these methods, where more formal and detailed treatments of the data assimilation problem can be found in the various references. In surveying a variety of widely used data assimilation schemes, the key message of this chapter is how the Bayesian analysis provides a consistent framework for the estimation problem and how this allows one to formulate its solution in a variety of ways to exploit the operational challenges in the geosciences.

Three areas where machine learning (ML) and physics have been merging: (a) Physical models can have computationally expensive components replaced by inexpensive ML models, giving rise to hybrid models. (b) In physics-informed machine learning, ML models can be solved satisfying the laws of physics (e.g. conservation of energy, mass, etc.) either approximately or exactly. (c) In forecasting, ML models can be combined with numerical/dynamical models under data assimilation.