Continuous data assimilation with blurred-in-time measurements of the surface quasi-geostrophic equation

An intrinsic property of almost any physical measuring device is that it makes observations which are slightly blurred in time. We consider a nudging-based approach for data assimilation that constructs an approximate solution based on a feedback control mechanism that is designed to account for observations that have been blurred by a moving time average. Analysis of this nudging model in the context of the subcritical surface quasi-geostrophic equation shows, provided the time-averaging window is sufficiently small and the resolution of the observations sufficiently fine, that the approximating solution converges exponentially fast to the observed solution over time. In particular, we demonstrate that observational data with a small blur in time possess no significant obstructions to data assimilation provided that the nudging properly takes the time averaging into account. Two key ingredients in our analysis are additional boundedness properties for the relevant interpolant observation operators and a non-local Gronwall inequality.