Multi-Source Unmixing: Decomposing a Mixed Signature

The Quick Detection module, Multi-Source Unmixing tab, decomposes the isotopic signature of a sample resulting from the mixing of two to eight pure sources. It calculates the contribution fractions of each source with their associated uncertainties.

The Problem This Tab Solves

A confluence water sample shows an antimony isotopic signature that does not match any pure source in the database. It lies between the signature of an acid mine effluent and that of agricultural water contaminated by arsenic-antimony-based pesticides. What is the respective contribution of each source to the total contamination?

This question is essential for environmental impact studies, sizing of treatment units, and attribution of liability. Isotopic unmixing answers it with a precision that conventional hydrological and chemical mass balances cannot achieve alone, because the isotopic ratio is conservative upon mixing.

The Conservative Property of Isotopes in a Mixture

When two water streams mix, the isotopic signature of the mixture is the flow-weighted average of the two signatures. This is a strictly conservative property: it depends neither on dilution, nor on antimony concentration, nor on temperature. This property is the mathematical foundation of isotopic unmixing.

For a binary mixture (sources A and B):

IsoFind solves this system for N sources using the NNLS (Non-Negative Least Squares) algorithm with closure constraint (fractions sum to 1 and each fraction is non-negative). A Monte-Carlo bootstrap calculates 1σ uncertainties on each fraction (details per method below).

Accessing the Tab

Step-by-Step Workflow

1. Enter the Mixture Signature Enter the measured δ value in the mixture sample and its analytical uncertainty at 2σ. The sample can be selected from the database if the measurement is registered there.
2. Define the Pure Sources (End-Members) For each source contributing to the mixture, enter its name, δ value, and uncertainty. Use the Add a Source button to add up to 8 sources. Each source can be entered manually or selected from the IsoFind database.
   
   Sources must bracket the mixture value: if δ_mix = +0.64‰, the pure sources must have values on either side (e.g. +0.43‰ and +0.81‰). If all sources are on the same side, the NNLS residual will be high and the decomposition will be unreliable.
3. Choose the Calculation Method

   • NNLS: projected gradient resolution with closure constraint. Recommended for 3 or more sources.
   • Geometric: exact analytical solution for 2 sources. Faster, without Bootstrap uncertainty.
   • Bayesian: NNLS with 200-draw Monte-Carlo bootstrap for uncertainties. Recommended when uncertainties on pure sources are high.
4. Run the Unmixing and Read the Mixing Diagram Click Decompose Mixture. Results are displayed as fraction bars and in a binary mixing diagram (SVG mixing plot) showing the mixing line between the two dominant sources, with the observed point and its error bar.

Reading the Mixing Diagram

The mixing diagram (mixing plot) shows:

• The mixing line between the two dominant sources (blue line)
• The uncertainty band propagated around the line (grey zone)
• The end-members (coloured circles at the extremities)
• The observed point (red circle) with its analytical error bars

If the observed point falls on the mixing line within the uncertainty band, the two-source model is consistent with the data. If it departs significantly, a third end-member is likely needed, or the chosen pure sources are not representative.

Practical Case: Mining and Agricultural Confluence, Oruro Basin

Two pure sources available in the database:

• AMD Water (WAT-OR-003): δ = +0.698‰ ± 0.025
• Agricultural groundwater (WAT-OR-004): δ = +0.623‰ ± 0.022

Confluence sample (MIX-OR-001): δ = +0.641‰ ± 0.023

NNLS result: AMD fraction = 24% ± 8%, agricultural water fraction = 76% ± 8%. Residual = 0.002‰ (below analytical uncertainty). The mixture is dominated by the agricultural contribution, which directs corrective measures towards irrigation water management rather than mine drainage alone.

Limitations and Precautions

The mixing model assumes that end-members are stable and do not fractionate after mixing. This assumption is valid for rapid mixing in aqueous environments. It may break down in sediments, where secondary processes (adsorption, precipitation) can modify the signature after deposition.

For mixtures involving significant post-mixing processes, build a complete Nexus workflow from the advanced module. The Open in Nexus link at the bottom of the configuration panel transfers the current parameters to the Nexus canvas.