# Comparison of \({}^{210}\)Pb Age Models of Peat Cores Derived from the Arkhangelsk Region
## 3 Results and Discussion
### Chronology \({}^{210}\)Pb and Rate of Peat Accumulation
#### 3.3.4 Application of the CF Model Alone and via Monte Carlo Simulation  
The constant flux model (the CF model) is known as the CRS model (CRS) [4; 5; 25; 26] (note that we are sticking to these authors' views [21] in the title of the model). In a Monte Carlo simulation, it is assumed that repetition of the experiment will produce a frequency distribution of results. Assuming that the final distribution is Gaussian, we can express the uncertainty in terms of the mean and its standard deviation. The values used in constant flux model were defined as follows:  
* The accumulated \({}^{210}\)Pbex deposits under horizon (\(i\)) (Bq\(\cdot\)m\({}^{-2}\)) were computed as follows: \(A(i)=\sum_{j=i+1}^{j=\infty}\Delta A_{i}\). As there is no layer 41 \({}^{210}\)Pb\({}_{\text{ex}}\) below, the calculation starts with \(A(41)\) = 0 Bq\(\cdot\)m\({}^{-2}\). For upper layer, \(A(39)\) = \(\Delta A_{39}\). In addition, \(u(A(39))\) = \(u(\Delta A_{39})\). For layer 37, \(A(37)\) = \(A(39)\) + \(\Delta A_{37}\), and this pattern continued towards the top. If certain intervals were not analysed, the value of the mass concentration (\(C_{i}\)) (and \(\Delta m_{i}\) if its corresponding value is unknown) were estimated via interpolation from neighbouring intervals, and \(\Delta A_{i}\) was computed for the missing section. Relevant values were defined as follows:  
* Uncertainty \(u(A(37))\): \(u(A(37))=\sqrt{u((A(39)))^{2}+(u(\Delta A_{37}))^{2}}\). This pattern continued towards the top. * \(A(0)\) = 1682 \(\pm\) 117 Bq\(\cdot\)m\({}^{-2}\)--reserve \({}^{210}\)Pbex, on the basis of which we determined its flux to the sediment surface to be 52 \(\pm\) 4 Bq\(\cdot\)m\({}^{-2}\cdot\)year\({}^{-1}\). * The age of CF was determined via the expression \(t(i)=\frac{1}{\lambda}ln\frac{A(0)}{A(i)}\). The extension of uncertainty was performed with care (see [29]). * The expression for the uncertainty of age was defined as follows: \[u(t(i))=\frac{1}{\lambda}\sqrt{(u(\lambda)t(i))^{2}+\left(\frac{u(A(0))}{A(0) }\right)^{2}+\left(1-\frac{2A(i)}{A(0)}\right)\left(\frac{u(A(i))}{A(i)} \right)^{2}}\]
* To calculate the calendar year \(T(i)\), we subtracted the age of the CF from the sampling date. * MAR (\(r(i)\), kg\(\cdot\)m\({}^{-2}\cdot\)year\({}^{-1}\)) was determined via the formula \(r(i)=\frac{\lambda A(i)}{C(i)}\), and its uncertainty was \(u(r(i))=r(i)\sqrt{\left(\frac{u(\lambda)}{\lambda}\right)^{2}+\left(\frac{u( A(i))}{A(i)}\right)^{2}+\left(\frac{u(C(i))}{C(i)}\right)^{2}}\). MAR was shown in Figure 6.
* SAR (\(s(i)\), cm\(\cdot\)year\({}^{-1}\)) was computed using the bulk density of dry sediment \(\rho\), as \(s(i)=\left(\frac{r(i)}{\rho(i)}\right)\times 100\). SAR is shown in Figure 6.
* Uncertainty \(u(s(i))\): \(u(s(i))=s(i)\sqrt{\left(\frac{u(r(i))}{r(i)}\right)^{2}+\left(\frac{u(\rho(i) )}{\rho(i)}\right)^{2}}\)[21]. The value SAR (\(s_{i}\)) for the upper core layers 0-9 cm varied little, and the value range was 0.44 to 0.53 cm\(\cdot\)year\({}^{-1}\), as can be seen in Table 2; from 9 to 33 cm, we observed a relatively uniform and gradual decrease in the mean sediment accumulation rate to 0.03 cm\(\cdot\)year\({}^{-1}\). We also noted the increased \(s_{i}\) values for the 35-39-cm interval. The values of MAR (\(r_{i}\)) for all core layers were almost constant and fluctuated very slightly from to 0 and 0.01 to 0.02 g\(\cdot\)cm\({}^{-2}\cdot\)year\({}^{-1}\). In Table 5, the CF dating model used in the ISNO-1 core for the 19-21-cm layer shows an age of 1965. The Monte Carlo method was employed to provide a refined age of 1962, which slightly improved the agreement with the \({}^{137}\)Cs reference horizon, considering the error. The CF/CS and CF models, in combination with the Monte Carlo method, were used to calculate the linear accumulation rate \(s\) and the mass accumulation rate \(r\) in the sampled peat core, as illustrated in Figure 7. The \(s\) values for CF ranged between 0.09 \(\pm\) 0.02 and 1.3 \(\pm\) 0.05 cm-year\({}^{-1}\), with an average of 0.48 \(\pm\) 0.08 cm-year\({}^{-1}\).