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An intrinsic property of almost any physical measuring device is that itmakes observations which are slightly blurred in time. We consider anudging-based approach for data assimilation that constructs an approximatesolution based on a feedback control mechanism that is designed to account forobservations that have been blurred by a moving time average. Analysis of thisnudging model in the context of the subcritical surface quasi-geostrophicequation shows, provided the time-averaging window is sufficiently small andthe resolution of the observations sufficiently fine, that the approximatingsolution converges exponentially fast to the observed solution over time. Inparticular, we demonstrate that observational data with a small blur in timepossess no significant obstructions to data assimilation provided that thenudging properly takes the time averaging into account. Two key ingredients inour analysis are additional boundedness properties for the relevant interpolantobservation operators and a non-local Gronwall inequality.

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