Identifiability of a pollution source: The distributed model and the semi-discretized differential model

This paper is devoted to the identification of a pollution source in a river. A simple mathematical model of such a problem is given by a one-dimensional linear advection-dispersion-reaction equation with a right hand side spatially supported in a point (the source) and a time variant intensity, both unknown. The identifiability of the distributed system was established for two points of observations one upstream, the other downstream from the source provided the pollutant flow rate is zero on an interval [T, T + deltaT] (T > 0). But the distributed system has to be discretized in order to do a numerical estimation of the unknown parameters. It is why this paper is devoted to the identifiability of the differential system obtained by using a semi-discretization scheme in space. The proof of the identifiability does not require the restrictive assumption about the pollutant flow rate but the knowledge of the initial condition and one observation located upstream from the source. Moreover, from this study, a numerical procedure is deduced for estimating the unknown parameters. It does not necessitate a priori knowledge about the parameters and the unknown function is not expanded on a basis of special functions. Both aspects play an important role in the real applications