Escaping the black box: Addressing the mathematical blindness of sediment fingerprinting models

In the last decade, sediment fingerprinting has evolved from a specialised geochemical technique to a widely used numerical tool for catchment management. However, as models become more complex (from Frequentist to Bayesian or Machine Learning approaches), a fundamental question arises: is the uncertainty in our results a product of environmental complexity or a consequence of the mathematical structures we use? This work advocates for a synergistic approach where mathematical rigor and field expertise are not just compatible, but inseparable.
Drawing on extensive research using virtual experiments and artificial laboratory mixtures, we demonstrate that unmixing models are inherently "blind" to any process not explicitly included in their underlying hypotheses. We show how common issues, such as high source variability, non-contributing sources, or particle size effects, often manifest as "model bias" when, in fact, they represent mathematical inconsistencies between the tracer signal and the model's assumptions.
We present the Consistent Tracer Selection (CTS) and the Linear Variability Propagation (LVP) methods as essential bridges between these two worlds. These tools allow researchers to test the mathematical consistency of their datasets before running any unmixing algorithm. Our findings, derived from comparing multiple model structures (including FingerPro, MixSIAR, and others), reveal a crucial reality: when tracers are selected following strict physical and mathematical criteria, the choice of the model becomes secondary.
The results show that different algorithms tend to converge on the same solution when the input data is consistent. Therefore, we argue that the future of sediment fingerprinting lies not in a "model war," but in a shift toward rigorous tracer validation. We conclude that understanding the mathematics behind the mixing process, such as the Conservative Balance (CB), is what allows us to interpret whether a model’s output represents a physical reality or merely a mathematical artifact.