5.3 Calculation of Meteoric Water Lines
A meteoric water line is a best fit line (i.e., a regression line) for a set of points in x-y space. The most common regression analysis is known as the least squares method. This assumes the x-variable is independent and accurately known, whereas the y-variable depends upon the x-value and has errors and random variations. An example of an independent x-variable would be time or distance, and a dependent y-variable could be temperature, precipitation or an isotope ratio. For analysis of one stable isotope ratio, say δ2H against one of these independent variables, like time, a least squares regression is a suitable representation of the relationship. However, where both variables are dependent, such as δ2H and δ18O, neither one should be treated as more certain than the other and so the reduced major axis form of a structural regression is suitable (Ma, 2019).
5.3.1 Least Squares Regression
The straight-line regression is of the form shown in Equation 9.
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y = mx + c |
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where:
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m |
= |
gradient, i.e., slope of the line (dimensions of y-axis values over dimensions of x-axis values) |
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c |
= |
y-intercept (dimensions of y-axis values) |
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x, y |
= |
regression variables |
the least squares regression approach produces Equation 10.
[latex] \begin{gathered} m=\underline{S P}_{x y} \\ S S_x \\ c=\bar{y}-m \bar{x} \\ S S_x=\Sigma_{i=1}^{n_i}(x i-\bar{x})^2 \\ S P_{x y}=\sum^{n_{i=1}}(x i-\bar{x})(y i-\bar{y}) \end{gathered}[/latex] (10)
where:
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SSx |
= |
sum of the squared deviations from the mean of x (squared dimensions of the x variable) |
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SPxy |
= |
sum of product of deviations from the mean of x and deviations from mean of y (dimensions are product of x and y dimensions) |
5.3.2 Reduced Major Axis Regression (RMA)
The RMA regression line is calculated in a manner similar to that shown in Section 5.3.1, with the single difference that the gradient, m, is calculated as shown in Equation 11.
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[latex]\begin{gathered} m_{R M A}=\sqrt{\frac{S S_y}{S S_x}} \\ S S_y=\sum_{i=1}^n\left(y_i-\bar{y}\right)^2 \end{gathered}[/latex] | |