Data assimilation and Uncertainty Quantification in the Geosciences*

In the statistics community “Big Data” science is meant to suggest the combining of inferential and computational thinking. We also speak of big data in the geosciences. However, the problems we pursue, e.g. Earth's climate, are often extreme in the number of degrees of freedom, and in many instances, non-stationary in their statistics.
This usually means that we are working with sparse observational data sets, even if the number of observations is large. The Bayesian framework is a natural inferential data assimilation strategy in geosciences, to some extent because the degrees of freedom in the problem vastly outnumber observations but more critically, because the models we use to represent nature have considerable predictive power. Data sparsity is thus mitigated through physics-informed models.

After presenting a review of this Bayesian estimation strategy we will summarize how this process has evolved to handle nonlinear and non-Gaussian processes. We will also suggest that it is possible to design estimators to highlight certain features or exploit structure in the dynamics or the physics. An example of an approximate Bayesian estimator informed by models and future data will be shown to lead to improvements in forecasts. Machine learning can be exploited to capture unknown or unresolved processes and made to work with these estimators.
We will conclude the presentation with a review of present challenges, encompassing multiscale dynamics and statistics, unresolved physics, and event forecasting.