Detection-Identification of multiple unknown time-dependent point sources in a 2D transport equation: application to accidental pollution

Abstract

We address the nonlinear inverse source problem of identifying multiple unknown time-dependent point sources occurring in a two-dimensional evolution advection–dispersion–reaction equation. Provided to be available within the monitored domain interfaces for recording the generated state and its flux crossing each suspected zone where a source could occur, we establish a constructive identifiability theorem based on an introduced dispersion-current function that yields uniqueness of the unknown elements defining all occurring sources. Then, the established theorem leads to develop a detection-identification method that goes throughout the monitored domain to detect in each suspected zone whether there exists or not an occurring source. Once a source is detected, the developed method determines lower and upper bounds of the mean value discharged by its unknown time-dependent intensity function. Thereafter, the method localizes the sought position of the detected source as the unique solution of an equation satisfied by the introduced dispersion-current function and identifies its unknown intensity function from solving an associated deconvolution problem. Ultimately, the unknown number of occurring sources is deduced as the sum of all detected-identified active sources. Some numerical experiments on a variant of the surface water BOD pollution model are presented.