5.4 Weighted Regression Line Equations
As noted by Hughes and Crawford (2012), weighting of isotopic values for monthly cumulative precipitation by the precipitation amount produces regression lines (meteoric water lines) with higher gradients, as a result of minimizing the influence of evaporated samples from low precipitation events. The difference in gradient and intercept between weighted and unweighted regression lines depends on the dataset and the regression method, and although the differences are generally minor, they can be significant (Boschetti et al., 2019). These meteoric water lines better characterize the average precipitation and especially heavier events that are more likely to play an important role in hydrological processes such as groundwater recharge (Li et al., 2018). These weighted regressions use methods similar to those presented in Section 5.3, but include a precipitation term in the statistical quantities as shown in Equation 12.
| [latex]\begin{gathered} S S_x=\sum_{i=1}^n\left(\operatorname{rain}_i\right)\left(x_i-\bar{x}\right)^2 \\ S S_y=\sum_{i=1}^n\left(\operatorname{rain}_i\right)(y-\bar{y})^2 \\ S P_{x y}=\sum_{i=1}^n\left(\operatorname{rain}_i\right)\left(x_i-\bar{x}\right)\left(y_i-\bar{y}\right) \end{gathered}[/latex] | ||
It is useful to be aware that some computer statistical programs will perform least squares regression as their standard method to produce a best fit line. Thus, it is essential to either: 1) know what the program is doing; or, 2) set up the calculations in a spreadsheet using the formulae of Equations 10 – 12.