  In this paper, we provide conditions, \emph{based solely on the observed
velocity data}, for the global well-posedness, regularity and convergence of
the Azouni-Olson-Titi data assimilation algorithm (AOT algorithm) for a
Leray-Hopf weak solutions of the three dimensional Boussinesq system. This
condition also guarantees the construction of the {\it determining map}. The
aforementioned conditions on the (finite-dimensional) velocity observations,
which in this case comprise either of a finite-dimensional \emph{modal}
projection or finitely many \emph{volume element observations}, are
automatically satisfied for solutions that are globally regular and are
uniformly bounded in the $H^1$-norm. However, neither regularity nor uniqueness
is {\it a priori} assumed on the solutions. To the best of our knowledge, this
is the first such rigorous analysis of the AOT data assimilation algorithm for
the three-dimensional Boussinesq system. As a corollary, we obtain that the
condition that we imposed is in fact {\it a new observable regularity criterion
on the weak global attractor.} The proof of this fact proceeds through the
construction of the determining map.
