## 3 Model Reconstruction
### Temporal Characteristics Reconstruction
where  
\[f_{0}F_{2}{}^{F}(F_{12},m) =\sum_{k=0}^{K}\sum_{l=0}^{L}\left[c_{k,l}F_{12}{}^{l}\cdot\cos\left( \frac{2\pi km}{12}\right)+s_{k,l}F_{12}{}^{l}\cdot\sin\left(\frac{2\pi km}{12} \right)\right], \tag{5}\] \[f_{0}F_{2}{}^{R}(R_{12},m) =\sum_{k=0}^{K}\sum_{l=0}^{L}\left[c^{\prime}_{k,l}R_{12}{}^{l} \cdot\cos\left(\frac{2\pi km}{12}\right)+s^{\prime}_{k,l}R_{12}{}^{l}\cdot\sin \left(\frac{2\pi km}{12}\right)\right], \tag{6}\]  
and \(c_{k,l}\), \(s_{k,l}\), \(c^{\prime}_{k,l}\), and \(s^{\prime}_{k,l}\) are determined by the least square fitting approach. \(K\) is the index representing the annual, semi-annual, seasonal and monthly cycle, \(L\) is variations representing the solar cycle. The variations \(K=1\), \(2\), \(3\), and \(4\) of the trigonometric functions are expressed in terms of \(12\) months, \(6\) months, \(3\) months and \(1\) month respectively. The denominator \(12\) represent the maximum value of seasonal cycle, namely, twelve months. Moreover, the terms of solar activity can be expressed in terms of either a first-order, second-order, third-order or fourth-order function which is corresponding to \(L=1\), \(2\), \(3\), \(4\). Figure 5: Correlation coefficients between the monthly median values of critical frequency and the twelve-month running mean value of solar indices over Kokubunji, Japan. (**a**) The correlation coefficients between monthly median \(f_{0}\)F\({}_{2}\) and \(F_{12}\); (**b**) the correlation coefficients between monthly median \(f_{0}\)F\({}_{2}\) and \(R_{12}\); (**c**) the deviation of two type correlation coefficients. Figure 6 shows the root-mean-square error (RMSE) against different maximum regression order. As shown in Figure 6, it has better convergence and robustness with \(L=2\), which is better than that with \(L=1\). The root-mean-square error differences between measured value of \(f_{0}\)F\({}_{2}\) and regression values with \(L=2\) and \(K=2\) are already lower than that with \(L=1\) and \(K=4\). Furthermore, it is shown in Figure 6 that: the regress RMSE gradually decreased with considering the annual (\(K=1\)), semi-annual (\(K=2\)), seasonal (\(K=3\)) and monthly (\(K=4\)) cycle variation, when only using the maximum first-order form of solar activity parameter (\(L=1\)). It is also noticed that the reconstruction results can obtain better convergence and better accuracy only considering the annual (\(K=1\)) and semi-annual (\(K=2\)) cycle variation, when using the maximum second-order of solar activity parameter (\(L=2\)). Specially, the regression result based on the two solar activity index parameters is better than that based on the single parameter with 10.7 cm solar radio flux or sunspot. Considering the engineering simplification, the maximum orders of \(K\) and \(L\) was set as 2. Figure 7 shows sample of regression coefficients at Kokubunji station. The label number from 1 to 36 represent the coefficients \(c_{0,0}\), \(c_{0,1}\), \(c_{0,2}\), \(s_{0,0}\), \(s_{0,1}\), \(s_{0,2}\), \(c_{1,0}\), \(c_{1,1}\), \(c_{1,2}\), \(s_{1,0}\), \(s_{1,1}\), \(s_{1,2}\), \(c_{2,0}\), \(c_{2,1}\), \(c_{2,2}\), \(s_{2,0}\), \(s_{2,1}\), \(s_{2,2}\), \(c^{\prime}{}_{0,0}\), \(c^{\prime}{}_{0,1}\), \(c^{\prime}{}_{0,2}\), \(c^{\prime}{}_{1,0}\), \(c^{\prime}{}_{1,1}\), \(c^{\prime}{}_{1,2}\), \(s^{\prime}{}_{1,0}\), \(s^{\prime}{}_{1,1}\), \(s^{\prime}{}_{1,2}\), \(c^{\prime}{}_{2,0}\), \(c^{\prime}{}_{2,1}\), \(c^{\prime}{}_{2,2}\), \(s^{\prime}{}_{2,0}\), \(s^{\prime}{}_{2,1}\), and \(s^{\prime}{}_{2,2}\), respectively. As can be seen from Figure 7, the \(c^{\prime}{}_{0,0}\) is obviously more high than other coefficients in Kokubunji station. Moreover, the coefficients during the day are high than those during the night.