{"id":126,"date":"2023-08-23T20:46:48","date_gmt":"2023-08-23T20:46:48","guid":{"rendered":"https:\/\/books.gw-project.org\/stable-isotope-hydrology\/chapter\/solution-exercise-2-recharge-altitude\/"},"modified":"2024-01-15T14:48:09","modified_gmt":"2024-01-15T14:48:09","slug":"solution-exercise-2-recharge-altitude","status":"publish","type":"chapter","link":"https:\/\/books.gw-project.org\/stable-isotope-hydrology\/chapter\/solution-exercise-2-recharge-altitude\/","title":{"raw":"Solution Exercise\u00a02\u00a0-\u00a0Recharge Altitude","rendered":"Solution Exercise\u00a02\u00a0–\u00a0Recharge Altitude"},"content":{"raw":"
\r\n

1. The mean stable isotope composition for the Kirstenbosch spring is:<\/p>\r\n\r\n\r\n\r\n\r\n\r\n
\r\n

<\/p>\r\n<\/td>\r\n

\r\n

[latex]\\bar{\\delta}=\\frac{\\sum_i^n \\delta_i}{n}[\/latex]<\/p>\r\n<\/td>\r\n

\r\n

<\/p>\r\n<\/td>\r\n<\/tr>\r\n

<\/td>\r\n<\/td>\r\n<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n

where:<\/p>\r\n\r\n\r\n\r\n\r\n\r\n\r\n
\r\n

\u03b4i<\/sub><\/p>\r\n<\/td>\r\n

\r\n

=<\/p>\r\n<\/td>\r\n

\r\n

value for a given sample<\/p>\r\n<\/td>\r\n<\/tr>\r\n

\r\n

n<\/em><\/p>\r\n<\/td>\r\n

\r\n

=<\/p>\r\n<\/td>\r\n

\r\n

total number\u00a0of samples<\/p>\r\n<\/td>\r\n<\/tr>\r\n

<\/td>\r\n<\/td>\r\n<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n

the means are -11.45\u2030 and -3.11\u2030 for \u03b42<\/sup>H and d18<\/sup>O respectively.<\/p>\r\n

2. The weighted means for the precipitation stations are:<\/p>\r\n\r\n\r\n\r\n\r\n\r\n
\r\n

<\/p>\r\n<\/td>\r\n

\r\n

[latex]\\bar{\\delta}=\\sum_i^n \\delta_i \\times r f_i[\/latex]<\/span><\/p>\r\n<\/td>\r\n

\r\n

<\/p>\r\n<\/td>\r\n<\/tr>\r\n

<\/td>\r\n<\/td>\r\n<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n

where:<\/p>\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n
\r\n

\u03b4<\/p>\r\n<\/td>\r\n

\r\n

=<\/p>\r\n<\/td>\r\n

\r\n

denotes a year<\/p>\r\n<\/td>\r\n<\/tr>\r\n

\r\n

\u03b4i<\/sub><\/p>\r\n<\/td>\r\n

\r\n

=<\/p>\r\n<\/td>\r\n

\r\n

value for a given month<\/p>\r\n<\/td>\r\n<\/tr>\r\n

\r\n

rf<\/em>i<\/em><\/sub><\/p>\r\n<\/td>\r\n

\r\n

=<\/p>\r\n<\/td>\r\n

\r\n

fraction of the total precipitation<\/p>\r\n<\/td>\r\n<\/tr>\r\n

<\/td>\r\n<\/td>\r\n<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n

(over 3 years) for that month, calculated as:<\/p>\r\n\r\n\r\n\r\n\r\n\r\n
\r\n

<\/p>\r\n<\/td>\r\n

\r\n

[latex] r f_i=\\frac{r_i}{r_t} [\/latex]<\/span><\/p>\r\n<\/td>\r\n

\r\n

<\/p>\r\n<\/td>\r\n<\/tr>\r\n

<\/td>\r\n<\/td>\r\n<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n

where:<\/p>\r\n\r\n\r\n\r\n\r\n\r\n\r\n
\r\n

r<\/em>i<\/em><\/sub><\/p>\r\n<\/td>\r\n

\r\n

=<\/p>\r\n<\/td>\r\n

\r\n

rain amount for a month<\/p>\r\n<\/td>\r\n<\/tr>\r\n

\r\n

r<\/em>t<\/em><\/sub><\/p>\r\n<\/td>\r\n

\r\n

=<\/p>\r\n<\/td>\r\n

\r\n

total rain for the period of the study<\/p>\r\n<\/td>\r\n<\/tr>\r\n

<\/td>\r\n<\/td>\r\n<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n

For months with missing data, the total precipitation amount must exclude those months. Final answers of -9.15\u2030 and -2.90\u2030 for \u03b42<\/sup>H and \u03b418<\/sup>O at University of Cape Town and -14.34\u2030 and -3.73\u2030 similarly for the Table Mountain Cableway.<\/p>\r\n

3. The altitude effect per isotope pair is calculated as:<\/p>\r\n\r\n\r\n\r\n\r\n\r\n
\r\n

<\/p>\r\n<\/td>\r\n

\r\n

[latex] \\frac{\\Delta \\delta}{m}=\\frac{\\bar{\\delta}_{T M C}-\\bar{\\delta}_{U C T}}{h_{T M C}-h_{U C T}} [\/latex]<\/span><\/p>\r\n<\/td>\r\n

\r\n

<\/p>\r\n<\/td>\r\n<\/tr>\r\n

<\/td>\r\n<\/td>\r\n<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n

where:<\/p>\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n
\r\n
\u0394\ud835\udeff\u2044\ud835\udc5a<\/span><\/div><\/td>\r\n
\r\n

=<\/p>\r\n<\/td>\r\n

\r\n

change in values per meter of altitude<\/p>\r\n<\/td>\r\n<\/tr>\r\n

\r\n

\ud835\udeffx<\/sub><\/p>\r\n<\/td>\r\n

\r\n

=<\/p>\r\n<\/td>\r\n

\r\n

mean stable isotope composition for a station<\/p>\r\n<\/td>\r\n<\/tr>\r\n

\r\n

h<\/em>x<\/em><\/em><\/sub><\/p>\r\n<\/td>\r\n

\r\n

=<\/p>\r\n<\/td>\r\n

\r\n

elevation of that station<\/p>\r\n<\/td>\r\n<\/tr>\r\n

<\/td>\r\n<\/td>\r\n<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n

giving answers of -0.0055\u2030\/m for \u03b42<\/sup>H and -0.00089\u2030\/m for \u03b418<\/sup>O. These answers are more reasonably displayed for \u03b42<\/sup>H and \u03b418<\/sup>O as -0.55\u2030\/100\u00a0m and -0.089\u2030\/100\u00a0m.<\/p>\r\n

4. The recharge elevation for Kirstenbosch can be calculated by addition of the isotopic altitude effect to the starting elevation at UCT, as follows:<\/p>\r\n\r\n\r\n\r\n\r\n\r\n
\r\n

<\/p>\r\n<\/td>\r\n

\r\n

[latex] h_{K b}=h_{U C T}+\\frac{\\left(\\bar{\\delta}_{K b}-\\bar{\\delta}_{U C T}\\right)}{(\\Delta \\delta \/ m)} [\/latex]\r\n<\/span><\/p>\r\n<\/td>\r\n

\r\n

<\/p>\r\n<\/td>\r\n<\/tr>\r\n

<\/td>\r\n<\/td>\r\n<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n

or by using the same formula with the hTMC<\/sub> and TMC<\/sub> values, which will produce a subtraction, both of which should give the same answer. Using \u03b42<\/sup>H gives 591\u00a0m and \u03b418<\/sup>O gives 368\u00a0m, the average of which is 479\u00a0m. This means the recharge area is 319\u00a0m above the spring at 160\u00a0m. This is obviously an average elevation as recharge occurs over an area and not at one point.<\/p>\r\n

Return to Exercise\u00a02<\/span><\/a><\/strong><\/p>\r\n

<\/p>\r\n\r\n<\/div>","rendered":"

\n

1. The mean stable isotope composition for the Kirstenbosch spring is:<\/p>\n\n\n\n\n
\n

\n<\/td>\n

\n

[latex]\\bar{\\delta}=\\frac{\\sum_i^n \\delta_i}{n}[\/latex]<\/p>\n<\/td>\n

\n

\n<\/td>\n<\/tr>\n

<\/td>\n<\/td>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

where:<\/p>\n\n\n\n\n\n
\n

\u03b4i<\/sub><\/p>\n<\/td>\n

\n

=<\/p>\n<\/td>\n

\n

value for a given sample<\/p>\n<\/td>\n<\/tr>\n

\n

n<\/em><\/p>\n<\/td>\n

\n

=<\/p>\n<\/td>\n

\n

total number\u00a0of samples<\/p>\n<\/td>\n<\/tr>\n

<\/td>\n<\/td>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

the means are -11.45\u2030 and -3.11\u2030 for \u03b42<\/sup>H and d18<\/sup>O respectively.<\/p>\n

2. The weighted means for the precipitation stations are:<\/p>\n\n\n\n\n
\n

\n<\/td>\n

\n

[latex]\\bar{\\delta}=\\sum_i^n \\delta_i \\times r f_i[\/latex]<\/span><\/p>\n<\/td>\n

\n

\n<\/td>\n<\/tr>\n

<\/td>\n<\/td>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

where:<\/p>\n\n\n\n\n\n\n
\n

\u03b4<\/p>\n<\/td>\n

\n

=<\/p>\n<\/td>\n

\n

denotes a year<\/p>\n<\/td>\n<\/tr>\n

\n

\u03b4i<\/sub><\/p>\n<\/td>\n

\n

=<\/p>\n<\/td>\n

\n

value for a given month<\/p>\n<\/td>\n<\/tr>\n

\n

rf<\/em>i<\/em><\/sub><\/p>\n<\/td>\n

\n

=<\/p>\n<\/td>\n

\n

fraction of the total precipitation<\/p>\n<\/td>\n<\/tr>\n

<\/td>\n<\/td>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

(over 3 years) for that month, calculated as:<\/p>\n\n\n\n\n
\n

\n<\/td>\n

\n

[latex]r f_i=\\frac{r_i}{r_t}[\/latex]<\/span><\/p>\n<\/td>\n

\n

\n<\/td>\n<\/tr>\n

<\/td>\n<\/td>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

where:<\/p>\n\n\n\n\n\n
\n

r<\/em>i<\/em><\/sub><\/p>\n<\/td>\n

\n

=<\/p>\n<\/td>\n

\n

rain amount for a month<\/p>\n<\/td>\n<\/tr>\n

\n

r<\/em>t<\/em><\/sub><\/p>\n<\/td>\n

\n

=<\/p>\n<\/td>\n

\n

total rain for the period of the study<\/p>\n<\/td>\n<\/tr>\n

<\/td>\n<\/td>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

For months with missing data, the total precipitation amount must exclude those months. Final answers of -9.15\u2030 and -2.90\u2030 for \u03b42<\/sup>H and \u03b418<\/sup>O at University of Cape Town and -14.34\u2030 and -3.73\u2030 similarly for the Table Mountain Cableway.<\/p>\n

3. The altitude effect per isotope pair is calculated as:<\/p>\n\n\n\n\n
\n

\n<\/td>\n

\n

[latex]\\frac{\\Delta \\delta}{m}=\\frac{\\bar{\\delta}_{T M C}-\\bar{\\delta}_{U C T}}{h_{T M C}-h_{U C T}}[\/latex]<\/span><\/p>\n<\/td>\n

\n

\n<\/td>\n<\/tr>\n

<\/td>\n<\/td>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

where:<\/p>\n\n\n\n\n\n\n
\n
\u0394\ud835\udeff\u2044\ud835\udc5a<\/span><\/div>\n<\/td>\n
\n

=<\/p>\n<\/td>\n

\n

change in values per meter of altitude<\/p>\n<\/td>\n<\/tr>\n

\n

\ud835\udeffx<\/sub><\/p>\n<\/td>\n

\n

=<\/p>\n<\/td>\n

\n

mean stable isotope composition for a station<\/p>\n<\/td>\n<\/tr>\n

\n

h<\/em>x<\/em><\/em><\/sub><\/p>\n<\/td>\n

\n

=<\/p>\n<\/td>\n

\n

elevation of that station<\/p>\n<\/td>\n<\/tr>\n

<\/td>\n<\/td>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

giving answers of -0.0055\u2030\/m for \u03b42<\/sup>H and -0.00089\u2030\/m for \u03b418<\/sup>O. These answers are more reasonably displayed for \u03b42<\/sup>H and \u03b418<\/sup>O as -0.55\u2030\/100\u00a0m and -0.089\u2030\/100\u00a0m.<\/p>\n

4. The recharge elevation for Kirstenbosch can be calculated by addition of the isotopic altitude effect to the starting elevation at UCT, as follows:<\/p>\n\n\n\n\n
\n

\n<\/td>\n

\n

[latex]h_{K b}=h_{U C T}+\\frac{\\left(\\bar{\\delta}_{K b}-\\bar{\\delta}_{U C T}\\right)}{(\\Delta \\delta \/ m)}[\/latex]
\n<\/span><\/p>\n<\/td>\n

\n

\n<\/td>\n<\/tr>\n

<\/td>\n<\/td>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

or by using the same formula with the hTMC<\/sub> and TMC<\/sub> values, which will produce a subtraction, both of which should give the same answer. Using \u03b42<\/sup>H gives 591\u00a0m and \u03b418<\/sup>O gives 368\u00a0m, the average of which is 479\u00a0m. This means the recharge area is 319\u00a0m above the spring at 160\u00a0m. This is obviously an average elevation as recharge occurs over an area and not at one point.<\/p>\n

Return to Exercise\u00a02<\/span><\/a><\/strong><\/p>\n

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