Using convex optimization to efficiently apportion tracer and pollutant sources from point concentration observations

Rivers transport elements, minerals, chemicals, and pollutants produced in their upstream basins. A sample from a river is a mixture of all of its upstream sources, making it challenging to pinpoint the contribution from each individual source. Here, we show how a nested sample design and convex optimization can be used to efficiently unmix downstream samples of a well-mixed, conservative tracer into the contributions of their upstream sources. Our approach is significantly faster than previous methods. We represent the river's sub-catchments, defined by sampling sites, using a directed acyclic graph. This graph is used to build a convex optimization problem which, thanks to its convexity, can be quickly solved to global optimality---in under a second on desktop hardware for datasets of $\sim$100 samples or fewer. Uncertainties in the upstream predictions can be generated using Monte Carlo resampling. We provide an open-source implementation of this approach in Python. The inputs required are straightforward: a table containing sample locations and observed tracer concentrations, along with a D8 flow-direction raster map. As a case study, we use this method to map the elemental geochemistry of sediment sources for rivers draining the Cairngorms mountains, UK. This method could be extended to non-conservative and non-steady state tracers. We also show, theoretically, how multiple tracers could be simultaneously inverted to recover upstream run-off or erosion rates as well as source concentrations. Overall, this approach can provide valuable insights to researchers in various fields, including water quality, geochemical exploration, geochemistry, hydrology, and wastewater epidemiology.