  In this paper, we consider the back and forth nudging algorithm that has been
introduced for data assimilation purposes. It consists of iteratively and
alternately solving forward and backward in time the model equation, with a
feedback term to the observations. We consider the case of 1-dimensional
transport equations, either viscous or inviscid, linear or not (B\"urgers'
equation). Our aim is to prove some theoretical results on the convergence, and
convergence properties, of this algorithm. We show that for non viscous
equations (both linear transport and Burgers), the convergence of the algorithm
holds under observability conditions. Convergence can also be proven for
viscous linear transport equations under some strong hypothesis, but not for
viscous Burgers' equation. Moreover, the convergence rate is always exponential
in time. We also notice that the forward and backward system of equations is
well posed when no nudging term is considered.
