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In many applications, a signal is deformed by well-understood dynamics beforeit can be measured. For example, when a pollutant enters a river, itimmediately begins dispersing, flowing, settling, and reacting. If thepollutant enters at a single point, its concentration can be measured before itenters the complex dynamics of the river system. However, in the case of anon-point source pollutant, it is not clear how to efficiently measure itssource. One possibility is to record concentration measurements in the river,but this signal is masked by the fluid dynamics of the river. Specifically,concentration is governed by the advection-diffusion-reaction PDE, with anunknown source term. We propose a method to statistically reconstruct a sourceterm from these PDE-deformed measurements. Our method is general and applies toany linear PDE. This method has important applications in the study ofenvironmental DNA and non-point source pollution.

Source Reconstruction for Spatio-Temporal Physical Statistical Models