Phillips, D.L. and J.W. Gregg. 2003. Source partitioning using stable isotopes: coping with too many sources. Oecologia 136:261-269. WED-02-184

Stable isotopes are increasingly being used as tracers in environmental studies. One
application is to use isotopic ratios to quantitatively determine the proportional
contribution of several sources to a mixture, such as the proportion of various pollution
sources in a waste stream. In general, the proportional contributions of n+ 1 different
sources can be uniquely determined by the use of n different isotope system tracers
(e.g., δ^{13}C, δ^{15}N, δ^{18}O) with linear
mixing models based on mass balance equations. Often, however, the number of potential
sources exceeds n+1, which prevents finding a unique solution of source proportions. What
can be done in these situations? While no definitive solution exists, we propose a method
that is informative in determining bounds for the contributions of each source. In this
method, all possible combinations of each source contribution (0-100%) are examined in
small increments (e.g., 1%). Combinations that sum to the observed mixture isotopic
signatures within a small tolerance (e.g., ±0.1‰ ) are considered to be feasible solutions,
from which the frequency and range of potential source contributions can be determined.
To avoid misrepresenting the results, users of this procedure should report the
distribution of feasible solutions rather than focusing on a single value such as the
mean. We applied this method to a variety of environmental studies in which stable isotope
tracers were used to quantify the relative magnitude of multiple sources, including (1)
plant water use, (2) geochemistry, (3) air pollution, and (4) dietary analysis. This
method gives the range of isotopically determined source contributions; additional
non-isotopic constraints specific to each study may be used to further restrict this
range. The breadth of the isotopically determined ranges depends on the geometry of the
mixing space and the similarity of source and mixture isotopic signatures. A sensitivity
analysis indicated that the estimated ranges vary only modestly with different choices of
source increment and mass balance tolerance parameter values. A computer program (IsoSource)
to perform these calculations for user-specified data is available at
http://www.epa.gov/wed/pages/models.htm.