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 in a steady state system 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 data sets of ∼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. Plain Language Summary: Rivers transport solid and dissolved elements, chemicals, and pollutants from upstream to downstream. Knowing where these elements are coming from, and in what quantities, is useful in targeting environmental protection or learning about the geology and hydrology of an area. However, a single sample of a river can't provide much useful information because all of the different sources have been mixed together by the river. If multiple samples are taken upstream and downstream, we show that a computational technique called convex optimization can be used to "unmix" the river's waters or sediments revealing where the elements have come from. This approach is accurate and fast enough that we can run it thousands of times with small changes to the sampled values to get a sense of the uncertainty. Though we speak about our method in terms of rivers here, it could be useful in other contexts, such as determining where pollutants or tracers of disease outbreaks are entering sewage networks. Key Points: Unmixing sources and magnitudes of tracers from nested concentration measurements in a river basin is a convex optimisation problemWe provide new software that solves this problem very quickly given a D8 raster and some point observations of concentrationThis method can be used to apportion pollution in a river basin and in any geochemistry study that uses a nested sampling design [ABSTRACT FROM AUTHOR]