Bayesian Reduced-Resolution Data Assimilation

The numerical simulation of geophysical problems invariably leads to using a mesh that is coarser than what is required to resolve all of the important physical processes being described by the set of governing partial differential equations. This coarse mesh will therefore miss important physical phenomena that the observational instruments used for data assimilation will see. The performance of a data assimilation algorithm can be improved by accounting for these missing physical processes. We briefly review recent work describing how to properly use Bayes' rule when the model is attempting to predict a truncated version of a much higher resolution state-vector and the observations that are being assimilated are observing the elements of this high-resolution state-vector. Then, we go on to describe a practical ensemble (Monte Carlo) data assimilation system that makes use of this theory in a simple problem which has the property that data assimilation at low-resolution works very poorly unless the aforementioned theory is properly accounted for.