The importance of measurement uncertainties in the regression of hydrogen and oxygen isotope values

The isotopic composition of hydrogen ( 2 H) and oxygen ( 18O) in water is an established and powerful tool in hydrological sensu lato investigations. Since Craig (1961) first documented the δ2 H-δ18O linear relationship in meteoric water across the globe, the Global Meteoric Water Line (GMWL − δ2 H = 8 * δ18O + 10), researchers have explored the variability in meteoric water lines both globally and locally. The isotopic composition of precipitation reflects fractionation processes during evaporation, condensation, and moisture transport, which are conditioned by the temperature, humidity, and morphological location of each site. Local Meteoric Water Lines (LMWLs) represent the relationship of δ2 H-δ18O at one specific site or area and are used for numerous hydrological, hydrogeological, and ecohydrological applications. A correct computation of LMWLs is, therefore, crucial. Several statistical regression methods have been employed to compute LMWLs: the Ordinary Least Squares regression (OLS), the Reduced Major Axis regression (RMA); the Major Axis regression (MA, also known as orthogonal regression), and the Error-In-Variables regression (Argiriou and Lykoudis, 2006 and Boschetti et al., 2019) which considers measurement uncertainties in δ2 H and δ18O. Some landmark papers in isotopes hydrology correctly argued that measurement uncertainties in δ2 H and δ18 O should be included in the regression methods, and consequently suggested the use of RMA and MA regression as the most suited for computing LMWLs. RMA and MA thus became the most common regression approaches. However, in those works, variables’ uncertainties are not inserted in RMA and MA equations; consequently, uncertainties are not used in the following works. A complete description of the implementation of δ2 H and δ18O measurement uncertainties in the computation of MWL regression is absent in the isotope hydrology literature. In this work, we illustrate how to correctly account for measurement uncertainties on both hydrogen and oxygen when performing their linear regression with the long established least-squares estimation (LSE). We discuss the assumptions to carry out the regression, the statistical methodology, and the evaluation of the goodness of fit. When computing the slope and intercept fit parameters the LSE regression is the least biased compared with OLS, RMA, and MA regressions. Four simulations are presented along to better illustrate the method application and evaluation of goodness of the fit statistic. Software tools, in R and Python, for computing the regression of water isotopes data (indeed any data) with uncertainties in both variables are provided. The use of the proposed method provides a better computation of MWLs (and of linear regressions of isotopic data in general) and the possibility of effectively evaluating the goodness of the computed linear fits. This method is not specific and thus applicable to any case in which the X and Y variables are characterised by measurement uncertainties