The quality of water in a river can be analysed by measuring the amount of chemicals in it. If the level of chemicals from human, agricultural and industrial sources exceeds a certain threshold, this reduces the amount of oxygen in the water which, in turn, has adverse consequences for aquatic life.

The extent of pollution in a river can be quantified by two parameters: the biological oxygen demand and the chemical oxygen demand. Engineers frequently model these quantities, which obey a one-dimensional linear advection-dispersion-reaction equation. The coefficients in this equation depend on the flow of water in the river and on how the pollutants travel through the stream. Now, Abdellatif El Badia and colleagues at the University of Compiègne have looked at the inverse of this problem, which involves determining the source of the pollution by observing the effect it has on the river.

The new technique measures the concentration of pollutants at two points in the river, one upstream and one downstream of the stretch of river being studied. El Badia and colleagues derive a formula that yields the exact position of the source of pollution, and then expand this solution into a Fourier series to reconstruct how the intensity of the pollution has evolved over time.

“We have solved a mathematical problem that corresponds to a model that is used by engineers for the surveillance of pollution sources in rivers, whether they are of urban, agricultural or industrial origin,” says El Badia. “Our algorithm is simple enough to be implemented into software, which would allow scientists to detect accidental, or intentional, pollution spills.” The team is now working on a two-dimensional model that could be applied to estuaries.