# Evaluating the Effectiveness of Natural Carbon Sinks Through a Temperature-Dependent Model
## 3 Results and Discussion
### Evaluation of the Two Sink Models
#### 3.2.2 Evaluation with the Extended Concentration and Temperature-Dependent Sink Model  
The extended sink model \(\hat{S}(C,T)\), Equation (14), is replacing \(\hat{S}(C)\) in the least squares optimization (Equation (12)). Using the CO\({}_{2}\) concentration time shift of 16, the optimization is conducted for temperature time shifts from 0 to 12 months. Figure 9 shows that the explained variance ratio \(R^{2}\) has a rather clear maximum of more than 0.8 at a time shift of 4 months. This is a quite remarkable increase in the explained variance ratio, leaving only 20% of the total variance unexplained. Table 2 displays the dependence of \(R^{2}\), the absorption value \(a\), the temperature coefficient \(b\), the constant \(c\), and the equilibrium concentration \(\hat{C}^{0}\) on the temperature time shift.  
\begin{table}
\begin{tabular}{l c c c c c c c c} \hline \hline \multirow{2}{*}{**Metric**} & \multicolumn{8}{c}{**Shift**} \\ \cline{2-10}  & **0** & **1** & **2** & **3** & **4** & **5** & **6** & **7** & **8** \\ \hline \(R^{2}\) & 0.7321 & 0.7668 & 0.7902 & 0.8032 & 0.8078 & 0.8033 & 0.7943 & 0.7886 & 0.7755 \\ \(a\) & 0.0456 & 0.0484 & 0.0497 & 0.0500 & 0.0497 & 0.0492 & 0.0485 & 0.0481 & 0.0475 \\ \(b\) & \(-3.12\) & \(-3.45\) & \(-3.61\) & \(-3.66\) & \(-3.63\) & \(-3.59\) & \(-3.51\) & \(-3.49\) & \(-3.42\) \\ \(c\) & \(-14.20\) & \(-15.13\) & \(-15.58\) & \(-15.69\) & \(-15.57\) & \(-15.42\) & \(-15.18\) & \(-15.07\) & \(-14.85\) \\ \(\hat{C}^{0}\) & 311.8 & 312.9 & 313.5 & 313.6 & 313.5 & 313.4 & 313.2 & 313.1 & 312.8 \\ \hline \hline \end{tabular}
\end{table}
Table 2: Temperature time shift and corresponding metrics in extended sink model (\(R^{2}\), \(a\), \(b\), \(c\), and \(\hat{C}^{0}\)).  
Figure 9: Explained variance of the extended model as a function of temperature shift, measured in months.  
With this optimal timeshift, the final extended sink model \(\hat{S}_{i}(C,T)\) is determined:  
\[\hat{S}_{i}(C,T)=0.05\cdot(\sum_{k=12i}^{12i+11}\frac{1}{12}C_{k-16}^{\text{ month}}-313.5\text{ ppm})-3.63\frac{\text{ppm}}{\text{\tiny{$\circ$}}\text{C}}\cdot\sum_{k=12i}^{12i+11} \frac{1}{12}T_{k-4}^{\text{month}} \tag{20}\]  
where \(C_{k}^{\text{month}}\) are the measured monthly Mauna Loa CO\({}_{2}\) concentration data [11] and \(T_{k-4}^{\text{month}}\) are the timeshifted temperatures computed from monthly HadSST4 sea surface anomalies [12].  
The CO\({}_{2}\) dependent absorption with 5% is significantly larger than in the simple sink model, where it was 1.8%. At first sight, it is hard to understand how this discrepancy comes about. Therefore, it is visualized with the diagram, Figure 10.  
The measured sink effect from Equation (2) can be approximated directly with a linear function depending on the CO\({}_{2}\) concentration. This is represented by the simple sink model with the red arrow. It can also be represented by two processes: downwelling large absorption, represented by the green arrow, and upwelling natural emissions, represented by the blue arrow. The blue arrow partly compensates for the effect of the green arrow. The fact that these upwelling natural emissions are temperature-dependent makes them identifiable when including temperature in the model.  
Figure 11 shows the resulting sink effect of both the simple sink model and the extended model in comparison with the measured sink data. While the simple sink model represents the trend of the data, the extended model follows the short-term deviations of 1 to 4 years very well. There is one noteworthy exception. After 1991, there was a remarkable rise in the sink rate, which has no correspondence with the model variability caused by temperature. The reason has been described by [PERSON] [7], who writes, \"Significant model departures from observations occurred for three years after the 1991 eruption of Mt. Pinatubo\" and explains it as a consequence of increased photosynthesis due to more diffuse solar radiation.