Proper determination of the δ180-δD relationship for ice and water by least-squares cubic regression

The δ180-δD relationship for ice and water is frequently summarized with a line fitted by least-squares linear regression. This technique assumes that one variable is known exactly and all error can be ascribed to the other. Unfortunately, determinations by mass spectrometry of both δ180 and δD are subject to experimental error. Often a blanket laboratory precision is provided for δ180 and δD, in which case functional analysis, accounting for the relative error in the variables, is appropriate. Properly, however, each sample has an individual analytical error in both variables, defined by the variance in estimates of isotope concentration provided by the mass spectrometer. Where individual errors are known, the least-squares cubic method, which assigns a weight to each sample and generates the summary line by an iterative method, may be used. An algorithm sufficient to determine both the functional fit and the least-squares cubic regression line is presented. Illustrations are provided, one of which demonstrates that if the plot of δ180 versus δD is scattered (r2 < 0.9), both the functional fit and the least-squares cubic regression line may be significantly different from the least-squares linear regression lines