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

The Back and Forth Nudging algorithm for data assimilation problems: theoretical results on transport equations