Solution Exercise 1 – Meteoric Water Lines
Step-by-step calculation is shown in the tables and calculations that follow.
Calculation of a meteoric water line
|
Location |
Date |
Precipitation mm |
δ18O ‰ |
δD ‰ |
(δ18Oi– δ𝟏𝟖𝐎̅̅̅̅̅)2 |
(δDi − δ𝐃̅̅̅̅)2 |
(δ18Oi − δ𝟏𝟖̅̅̅̅𝐎̅)
(δDi − δ𝐃̅̅) |
|
UCT |
Jan 2010 |
9 |
0.75 |
9.7 |
8.619 |
342.87 |
54.36 |
|
UCT |
Feb 2010 |
12 |
-0.54 |
-5.8 |
2.709 |
9.10 |
4.96 |
|
UCT |
Mar 2010 |
7 |
-0.83 |
0.4 |
1.838 |
84.95 |
12.50 |
|
UCT |
Apr 2010 |
44 |
-2.25 |
-12.1 |
0.004 |
10.78 |
0.21 |
|
UCT |
May 2010 |
277 |
-2.78 |
-8.4 |
0.353 |
0.17 |
-0.25 |
|
UCT |
Jun 2010 |
222 |
-3.31 |
-17.0 |
1.264 |
66.97 |
9.20 |
|
UCT |
Jul 2010 |
166 |
-3.69 |
-15.0 |
2.263 |
38.23 |
9.30 |
|
UCT |
Aug 2010 |
121 |
-1.74 |
-4.9 |
0.199 |
15.34 |
1.75 |
|
UCT |
Sep 2010 |
100 |
-2.24 |
-3.3 |
0.003 |
30.43 |
-0.30 |
|
UCT |
Oct 2010 |
102 |
-2.15 |
-3.3 |
0.001 |
30.43 |
0.20 |
|
UCT |
Nov 2010 |
73 |
-2.39 |
-6.2 |
0.042 |
6.85 |
-0.53 |
|
UCT |
Dec 2010 |
17 |
-5.06 |
-39.9 |
8.261 |
966.17 |
89.34 |
|
|
|
Total |
Mean |
Mean |
SS-δ18O |
SS-δD |
SP-δ18O-δD |
|
|
|
1150 |
-2.19 |
-8.82 |
25.56 |
1602.30 |
180.74 |
SS = sum of squares; SP = sum of products.
For the best fit line by RMA (reduced major axis) regression:
[latex]\begin{aligned} & \text { the gradient }=\sqrt{\frac{S S \delta D}{S S \delta^{18} O}}=\sqrt{\frac{1602.30}{25.56}}=7.92 \\ & \text { the intercept }=\overline{\delta D}-\left(7.92 \overline{\delta^{18} O}\right)=-8.8-(7.92(-2.19))=8.49 \end{aligned}[/latex]
therefore, the LMWLUCT–2010 is D = 7.9218O + 8.49
Pearson’s r, the sample correlation coefficient
[latex]{=r}_{^{18}OD}=\frac{{SP}^{18}OD}{\sqrt{{SS}^{18}O}\sqrt{SSD}}=\frac{180.74}{\sqrt{25.56}\sqrt{1602.30}}=0.893[/latex]
Calculation of a precipitation–weighted meteoric water line
|
Location |
Date |
Precipitation mm |
δ18O ‰ |
δD ‰ |
Rain-fraction |
|
|
|
|
 |
|
UCT |
Jan 2010 |
9 |
0.75 |
9.7 |
0.0078 |
0.0059 |
0.0759 |
0.067 |
2.68 |
0.43 |
|
UCT |
Feb 2010 |
12 |
-0.54 |
-5.8 |
0.0104 |
-0.0056 |
-0.0605 |
0.028 |
0.09 |
0.05 |
|
UCT |
Mar 2010 |
7 |
-0.83 |
0.4 |
0.0061 |
-0.0051 |
0.0024 |
0.011 |
0.52 |
0.08 |
|
UCT |
Apr 2010 |
44 |
-2.25 |
-12.1 |
0.0383 |
-0.0861 |
-0.4630 |
0.000 |
0.41 |
0.01 |
|
UCT |
May 2010 |
277 |
-2.78 |
-8.4 |
0.2409 |
-0.6696 |
-2.0233 |
0.085 |
0.04 |
-0.06 |
|
UCT |
Jun 2010 |
222 |
-3.31 |
-17.0 |
0.1930 |
-0.6390 |
-3.2817 |
0.244 |
12.93 |
1.78 |
|
UCT |
Jul 2010 |
166 |
-3.69 |
-15.0 |
0.1443 |
-0.5326 |
-2.1652 |
0.327 |
5.52 |
1.34 |
|
UCT |
Aug 2010 |
121 |
-1.74 |
-4.9 |
0.1052 |
-0.1831 |
-0.5156 |
0.021 |
1.61 |
0.18 |
|
UCT |
Sep 2010 |
100 |
-2.24 |
-3.3 |
0.0870 |
-0.1948 |
-0.2870 |
0.000 |
2.65 |
-0.03 |
|
UCT |
Oct 2010 |
102 |
-2.15 |
-3.3 |
0.0887 |
-0.1907 |
-0.2927 |
0.000 |
2.70 |
0.02 |
|
UCT |
Nov 2010 |
73 |
-2.39 |
-6.2 |
0.0635 |
-0.1517 |
-0.3936 |
0.003 |
0.43 |
-0.03 |
|
UCT |
Dec 2010 |
17 |
-5.06 |
-39.9 |
0.0148 |
-0.0748 |
-0.5898 |
0.122 |
14.28 |
1.32 |
|
|
|
Total |
Mean |
Mean |
Sum |
Sum |
Sum |
|
|
 |
|
|
|
1150 |
-2.19 |
-8.82 |
1.00 |
-2.73 |
-9.99 |
0.909 |
43.98 |
5.08 |
SS = sum of squares; SP = sum of products; rf = rain fraction (i.e., weighted by precipitation).
For the best fit line by RMA (reduced major axis) regression:
[latex]\begin{aligned} & \text { the gradient }=\sqrt{\frac{S S \times r f \delta D}{S S \times r f \delta^{18} O}}=\sqrt{\frac{43.98}{0.909}}=6.95 \\ & \text { the intercept }=\sum r f . \delta D-6.95 \sum r f . \delta^{18} O=-9.99-(6.95(-2.73))=8.96 \end{aligned}[/latex]
the precipitation weighted LMWLUCT–2010 is δD = 6.95δ18O + 8.96
Pearson’s r, the sample correlation coefficient
[latex]r_{r f \delta^{18} O \delta D}=\frac{\operatorname{SPrf} \delta^{18} O \delta D}{\sqrt{\operatorname{SSrf} \delta^{18} O} \sqrt{\operatorname{SSrf} \delta D}}=\frac{5.08}{\sqrt{0.909} \sqrt{43.87}}=0.805[/latex]