Meteoric Water Lines in isotope hydrology: a guide to Error-In-Variables regression

The stable isotopic composition of hydrogen (δ²H) and oxygen (δ¹⁸O) in precipitation is an established and powerful tool in water cycle investigations. Since Craig (1961) first documented the δ²H-δ¹⁸O linear relationship in meteoric water across the globe, the Global Meteoric Water Line (GMWL - δ²H = 8 ∗ δ¹⁸O + 10), researchers have explored the variability in MWLs 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 δ²H-δ¹⁸O at one specific site or area and are used for numerous hydrological, hydrogeological, and eco-hydrological 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), the Error-In-Variables regression (EIV) considering measurement uncertainties in δ²H and δ¹⁸O. Some landmark papers correctly argued that, since both δ²H and δ¹⁸O are characterized by measurement uncertainties, the RMA and MA regression must be considered 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; and consequently, uncertainties are not used in the following works. A more complete description of the possibility of using EIV regression method to isotopic composition in precipitation analysis is absent in the literature. In this work we present a method, based on a particular case of the EIV, in which the uncertainties on both variables can be different from each other and from one measurement to another. It is a generalization of the MA method (in which the uncertainties on the two variables are equal to each other) and, in its simplest version, it is called the Deming method. We discuss the assumptions to carry out the regression, the statistical methodology, and the evaluation of the goodness of fit. In addition, software tools for computing Deming regression of isotopes in precipitation data will be presented. Finally, three examples will be presented along with LMWLs evaluations. The use of the proposed method provides a better evaluation both of LMWLs parameters and of the associated uncertainties, allowing a better comparison within and between the years of the monitoring periods. This could improve, from that point onward, the analysis of δ²H-δ¹⁸O linear relationship in meteoric water.  We believe that this is a robust method to compute LMWL and suggest its use in isotope hydrological analyses.