An Inverse Source Problem in a Coupled BOD--DO Model with Partial DO Observations

Abstract

This work addresses an inverse source problem for a coupled reaction–diffusion system that models the interaction between biochemical oxygen demand (BOD) and dissolved oxygen deficit (DO) in water-quality monitoring. The objective is to reconstruct an unknown pollution source acting in the (BOD) equation from partial observations of the (DO) deficit, a difficult task due to the indirect coupling between the source and the observations. We propose a non-iterative reconstruction strategy based on topological sensitivity analysis. The method evaluates the first-order variation of a quadratic misfit functional induced by the nucleation of an infinitesimal source inclusion, thereby yielding an explicit topological gradient whose negative values indicate likely source locations. We establish well-posedness of the forward coupled system, prove an identifiability result for the admissible class of sources under exact data, and derive the associated topological sensitivity formula. Numerical experiments with noisy boundary data illustrate accurate localization and robustness with respect to moderate modelling uncertainties.