  This paper considers a nudging-based scheme for data assimilation for the
two-dimensional (2D) Navier-Stokes equations (NSE) with periodic boundary
conditions and studies the synchronization of the signal produced by this
algorithm with the true signal, to which the observations correspond, in all
higher-order Sobolev topologies. This work complements previous results in the
literature where conditions were identified under which synchronization is
guaranteed either with respect to only the $H^1$--topology, in the case of
general observables, or to the analytic Gevrey topology, in the case of
spectral observables. To accommodate the property of synchronization in the
stronger topologies, the framework of general interpolant observable operators,
originally introduced by Azouani, Olson, and Titi, is expanded to a far richer
class of operators. A significant effort is dedicated to the development of
this more expanded framework, specifically, their basic approximation
properties, the identification of subclasses of such operators relevant to
obtaining synchronization, as well as the detailed relation between the
structure of these operators and the system regarding the syncrhonization
property. One of the main features of this framework is its "mesh-free" aspect,
which allows the observational data itself to dictate the subdivision of the
domain. Lastly, estimates for the radius of the absorbing ball of the 2D NSE in
all higher-order Sobolev norms are obtained, thus properly generalizing
previously known bounds; such estimates are required for establishing the
synchronization property of the algorithm in the higher-order topologies.
