# Evaluating the Effectiveness of Natural Carbon Sinks Through a Temperature-Dependent Model
## 2 Materials and Methods
### The Top-Down Carbon Sink Model from the Continuity Equation  
The continuity equation based on the conservation of mass implies that the atmospheric concentration growth \(G_{i}\) in year \(i\) results from the difference between the total emissions and all absorptions \(A_{i}\)[14], whereby the total emissions are the sum of the anthropogenic emissions \(E_{i}\) and the natural emissions \(N_{i}\), i.e.,  
\[G_{i}=E_{i}+N_{i}-A_{i} \tag{1}\]  
The effective sink capacity \(S_{i}\) is calculated as the difference between the anthropogenic emissions \(E_{i}\) and the concentration growth \(G_{i}\), i.e.,  
\[S_{i}=E_{i}-G_{i} \tag{2}\]  
Following the continuity equation above, the effective sink capacity \(S_{i}\), therefore, is the difference of ocean and plant absorptions \(A_{i}\) and natural emissions \(N_{i}\):  
\[S_{i}=A_{i}-N_{i} \tag{3}\]  
It is important to note that the sink effect consists of two components, the actual absorptions and the natural emissions. When measuring the sink effect, we do not know anything about the actual absorptions \(A_{i}\) nor about the natural emissions \(N_{i}\). All models for the sink effect \(S_{i}\) have this conceptual deficit. Nevertheless, they are the starting point of this investigation.  
A simple linear sink model for Equation (3) has been developed by several researchers [7; 14; 15; 16; 17; 18], where the sink effect of the yearly data model only depended on theCO\({}_{2}\) concentration of the previous year \(C_{i-1}\) with an offset \(C^{0}\), representing an equilibrium concentration:  
\[\hat{S}_{i}(C) = a^{\prime}\cdot(C_{i-1}-C^{0}) \tag{4}\] \[= a^{\prime}\cdot C_{i-1}-n \tag{5}\]  
The parameters \(a^{\prime}\) and \(n\) are determined by minimizing the sums of squares between measured data from Equation (2) and the sink model of Equation (5) over all known years \(i\):  
\[\min_{a^{\prime},n}\sum_{i}\left(E_{i}-G_{i}-(a^{\prime}\cdot C_{i-1}-n)\right) ^{2} \tag{6}\]  
All statistical evaluations, in particular, the ordinary least squares optimization, were conducted with the Python package Statmodels, version 0.13.5 [19], and the graphs were created with the Python package Matplotlib, version 3.7.0 [20].  
\(n=a^{\prime}\cdot C^{0}\) represents the assumed constant natural net emissions per year. Obviously, this is a very crude model because with the constant \(n\) it is not possible to represent any systematic variability.  
Questions about the principle validity of this model for multiple sinks have made it necessary to again clarify a point that has been discussed before [14; 16]. The total sink effect is composed of the sum of several linearized components, which may have different equilibrium concentrations. Let us pick ocean, phytoplankton, C\({}_{3}\) plants, and C\({}_{4}\) plants [21]:  
\[\hat{S}_{i}(C)=a_{ocean}\cdot(C_{i}-C^{0}_{ocean})+a_{phyto}\cdot(C_{i}-C^{0} _{phyto})+a_{C3}\cdot(C_{i}-C^{0}_{C3})+a_{C4}\cdot(C_{i}-C^{0}_{C4}) \tag{7}\]  
As this is a linear form, it maps perfectly into the form of Equation (5). The individual sink rates or their equilibrium value remain unknown, only their combined effect is determined.  
Land use change effects are not considered here as anthropogenic. They are assumed to be part of the unknown natural emissions. In a previous publication, land use change emissions have been included as part of the anthropogenic emissions [14], pulling down the resulting equilibrium concentration to 242 ppm. This contradicts the accepted consensus that the pre-industrial equilibrium concentration was between 2/70 ppm and 290 ppm. Furthermore, [10] demonstrated that prediction quality improves when land use change emissions are discarded as explicit anthropogenic emissions during the time since 1959. Land use change emissions might have had a significant role in the first half of the 20 th century [10], this will be subject to future research.