# Evaluating the Effectiveness of Natural Carbon Sinks Through a Temperature-Dependent Model
## 2 Materials and Methods
### Separating Absorptions and Natural Emissions
#### 2.2.1 Estimating CO\({}_{2}\) Absorption by Means of the Bomb Test Data  
The nuclear bomb tests beginning in the 1950s stopped rather suddenly in 1963 with the nuclear test ban treaty. This provides a close-to-ideal identifiable carbon emission pulse of \({}^{14}\)C that has been thoroughly investigated for more than 40 years [13]. The data series is the global data sequence from 1950 to 2019 from the supplements of [13]. The relative deviation from the preindustrial zero level of \({}^{14}\)C concentration, \(\Delta^{14}\)C, is displayed as the blue graph in Figure 2.  
Why is this concentration decay representing \"pure\" absorptions? The CO\({}_{2}\) emissions from the oceans have the much lower \({}^{14}\)C concentration of the long-term equilibrium before the bomb tests; therefore, the upwelling \({}^{14}\)C can be neglected. The primary purpose of the publication from [PERSON] et al. [13] was to provide a data set usable for correcting carbon dating measurements. Therefore, the zero concentration reference level is the preindustrial level before 1900. The purpose here, however, is a completely different one. We want to determine the time constant of the atmospheric decay of \(\Delta^{14}\)C caused by ocean sinks and long-living plants. Therefore, we have to preprocess the data by treating the \({}^{14}\)C dilution in the atmosphere caused by fossil fuels that do not contain \({}^{14}\)C, the so-called Suess effect [26], in a different way than required by carbon dating. In order to deal with the Suess effect properly, we need to look at the definition of the provided data. They are calculated in the \(\Delta^{14}\)C convention [27; 28; 29], which is defined as follows:  
Figure 2: The decay of the global atmospheric \(\Delta^{14}\)C after 1965 [13]. Published data are blue, and adjusted data are orange. \[\Delta^{14}C(t)=\left(\frac{{}^{14}C_{Atm}(t)/^{12}C(t)}{{}^{14}C_{Std}/^{12}C_{Std}}- 1\right)\cdot 1000 \tag{18}\]  
\(\Delta^{14}C(t)\) are the relative deviations from the pre-industrial reference concentration level of \({}^{14}C\). This contains all deviations from the pre-industrial standard, both the Suess effect and the bomb test anomaly. But because we want to model the post-bomb test \({}^{14}C\) data as an exponential decay, the fact that the \(\Delta^{14}C(t)\) data are negative in the years before the bomb tests due to the Suess effect requires an adjustment. In the definition of \(\Delta^{14}C(t)\), the pre-industrial zero level had been fixed in order to meet the needs of carbon dating, but in order to measure the decay of the bomb test \({}^{14}C\) concentration, the zero level has to be the level just before the bomb test, that is, just before the sharp rise. The resulting decreasing atmospheric \({}^{14}C\) concentration shows, over a more than 40-year time period, that the contributing absorption sink processes exhibit an undistorted exponential decay of a first-order linear differential equation; see Figure 3.  
This decay includes both the decay of \({}^{14}C\) concentration into the sinks and the Suess effect due to atmospheric concentration change of \({}^{12}C\) by anthropogenic emissions. For the determination of the Suess effect, we take the 25 years from 1965 to 1990, where we have a good \({}^{14}C\) decay signal. The upper bound of the diluting Suess effect due to fossil fuels is obtained by pretending there is no sink effect, thus adding the cumulative emissions of 60 ppm (=127 GtC) between 1965 and 1990 to the 1965 CO\({}_{2}\) concentration of 320 ppm. The yearly diluting contribution is calculated by  
\[\Delta C_{Suess}=\sqrt[25]{\frac{320+60}{320}}=1.0069\]  
This means that the Suess effect contributes 0.69% per year to the decline of \(\Delta^{14}C\). The regression parameter for the slope in Figure 3 is \(-0.058\), with the 95% confidence interval [\(-0.054\), \(-0.063\)]. When this is reduced by 0.0069, the real absorption constant of \({}^{14}C\) is \(-0.051\), with a corresponding confidence interval.