Accounting for Coarse Resolution Model Simulations in Data Assimilation for the Geosciences

In geophysical applications we are rarely if ever able to simulate the problem at hand at a resolution for which all important scales of motion are fully resolved. Almost universally we must truncate the continuous variables of interest to a discrete set and then concatenate those variables into a state-vector that does not fully describe the problem. Typically, we model this state-vector with a discretized partial differential equation (PDE) that coarsely models the entire physical system. The result of this coarsening of the simulation of the system is that the numerical model does not simulate the actual variables of interest, but simulates some (unknown) function of the variables of interest. For example, a coarse spatial mesh used to solve the typical hyperbolic PDEs of geophysical fluid dynamics delivers a solution that is smoother than a fine spatial mesh, and therefore the solution for each element of the coarse mesh model is some function of many elements of the fine mesh model. Therefore, observations of the actual physical system observe state variables on the high-resolution mesh that are not actually simulated by our coarse mesh forecast model, at least not directly. All of this has particular consequences on the data assimilation that has not been accounted for previously in a rigorous Bayesian framework. The questions to be examined in this work include: what does Bayes' rule mean in this context? What is the best way to make use of these kinds of observations of a high-resoloution world using a low-resolution model? And, what data assimilation system should be used in this situation? We will review some recent research towards a solution to this problem and illustrate its application in an example problem.