Sensitivity analysis and quantification of uncertainty for isotopic mixing relationships in carbon cycle research

Quantifying and understanding the uncertainty in isotopic mixing relationships is critical to isotopic applications in carbon cycle studies at all spatial and temporal scales. Studies that depend on stable isotope approaches must also address quantification of uncertainty for parameters derived from isotopic studies. An important application of isotopic mixing relationships is determination of the isotopic content of ecosystem respiration (δ13CS) via an inverse relationship (a Keeling plot) between atmospheric CO2 concentrations ([CO2]) and carbon isotope ratios of CO2 (δ13C). Alternatively, a linear relationship between [CO2] and the product of [CO2] and δ13C (a Miller/Tans plot) can also be applied. We used three datasets of [CO2] and δ13C in air to examine contrasting approaches to determine δ13CS and its uncertainty. These datasets were from the Niwot Ridge, Colorado, AmeriFlux site, the Biosphere-Atmosphere Stable Isotope Network (BASIN), and from the Grünschwaige Grassland Research Station in Germany. The analysis of this data included Keeling plots and Miller/Tans plots fit with both Model I (ordinary least squares) and Model II regressions (geometric mean regression and orthogonal distance regression). Our analysis confirms previous observations that increasing the range of the measurements ([CO2] range) used for a mixing line reduces the uncertainty associated with δ13CS. Using a Model II regression technique to determine δ13CS introduces a negatively skewed bias in δ13CS which is especially significant for small [CO2] ranges. This bias arises from comparatively greater variability in the dependent variable than the independent variable for a linear regression. For carbon isotope studies, uncertainty in the isotopic measurements has a greater effect on the uncertainty of δ13CS than the uncertainty in [CO2]. As a result, studies that estimate parameters via a Model II regression technique maybe biased in their conclusions. In contrast to earlier studies, we advocate Model I (ordinary least squares) regression to calculate δ13CS and its uncertainty. Reducing the uncertainty of isotopic measurements reduces the uncertainty of δ13CS, even when the [CO2] range of samples is small (<20 ppm). As a result, improvement in isotope (rather than [CO2]) measuring capability is presently needed to substantially reduce uncertainty in δ13CS. We find for carbon isotope studies no inherent advantage or disadvantage to using either a Keeling or Miller/Tans approach to determine δ13CS. We anticipate that the mathematical methods developed in this paper can be applied to other applications where linear regression is utilized.