Test a source estimate against a reacted-tracer counterexample

Connect the Earth-system mechanism

Endmember mixing uses distinct source compositions to infer contributions to a mixture. A two-source concentration relation is linear when volumes mix consistently, source concentrations are known and stable, and the tracer is conservative along the relevant flow paths. Those are causal assumptions, not consequences of solving the equation.

Build the declared budget

The original low and high sources have tracer A concentrations 10 and 50 mg/L. If the high-source water fraction is f, the unreacted mixture is 10+40 f. The inversion f=(C−10)/40 recovers the source fraction only if no process changes A. A second tracer B has endmembers 20 and 40 mg/L and is stipulated conservative.

Separate the competing quantities

The model deliberately removes an assigned concentration of A after mixing. Inverting the reacted concentration as if conservative underestimates the high-source fraction and can produce a negative value. A value outside 0–1 is evidence of inconsistent assumptions or data, not a physical negative source and not a reason to silently clamp the estimate.

Challenge the inference

Actual source interpretation must examine endmember variability, additional sources, reaction, measurement uncertainty and flow history. Several tracers can help discriminate hypotheses, but adding a second measurement does not automatically identify every source and reaction. Natural chemical contrasts can sometimes be used without adding tracers; no environmental release is requested by this exercise.

Worked original packet

At true f 0.5, tracer A starts 30 mg/L. Removing 4 leaves 26, which falsely implies f 0.4 under the conservative inversion. Tracer B remains 30 mg/L and recovers f 0.5. The disagreement tests the assumed observation model rather than proving the actual high source vanished.

Investigation

Compare removal 0 and 10 at true fraction 0.25. Retain both tracers, calculate the erroneous inversion and propose an independent observation that could distinguish reaction from a changed endmember.

Changed-case transfer

For true fraction 0 and A removal 6 mg/L, compute observed A and its naive inferred fraction. How should the negative estimate be reported?

Feedback

Observed A 4 mg/L; naive fraction(4−10)/40=−0.15. Report the incompatibility with the two-endmember conservative model and investigate reaction/endmembers/error rather than present−15% as a physical contribution.

Sources and evidence boundary

Original prose, numerical packets and exercises. Mechanism/method context checked 30 October 2026: primary source. These are conditional computational models, not measured field records, carbon credits, safe yields, treatment standards or independent mastery. No physical extraction, chemical addition or environmental release is requested.