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.

Abstract: During the past two decades, there have been significant efforts to better quantify emissions of environmentally important trace gases along with their trends. In particular, there has been a clear need for robust estimates of emissions on policy-relevant scales of trace gases that impact air quality and climate. This need has driven the expansion of the observing network to better monitor the changing composition of the atmosphere. This chapter will discuss the use of various data assimilation and inverse modelling approaches to quantify these emissions, with a focus on the use of satellite observations. It will discuss the inverse problem of retrieving the atmospheric trace gas information from the satellite measurements, and the subsequent use of these satellite data for quantifying sources and sinks of the trace gases.

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.