## 3 Model Reconstruction  
### Temporal Characteristics Reconstruction  
In describing solar activity, the flux of the solar radio waves at 10.7 cm (\(F_{10.7}\)) is affected by the corona and upper layer of the chromosphere, whereas sunspot activity is affected by the corona and the lower layer of the chromosphere [42]. A sample profile (shown in Figure 4) represents the correspondence between solar activity indices and monthly median values of \(f_{0}\)F\({}_{2}\) over Kokubunji station. There are monthly variations in the solar indices as well as monthly median values of \(f_{0}\)F\({}_{2}\) during the period from 1958 to 2005, covering four cycles in solar activity. As shown in Figure 4, the similar cycle behaviors are reflected by \(f_{0}\)F\({}_{2}\) and solar indices. They respectively reached its peak and valley at the same corresponding years. At the same time, it is easy to see that the two solar activity parameters are not completely consistent with the variation of \(f_{0}\)F\({}_{2}\). In particular, the correlation coefficients between the \(f_{0}\)F\({}_{2}\) monthly median values and the twelve-month running mean value of solar indices at sample station (Kokubunji) is fully confirmed in Figure 5. The same conclusion has been reported in the literature [43]. As shown in Figure 5, the variation trends of the two types of correlation coefficients are generally similar but are different in some minor detail. During certain times and months, the correlation coefficients between \(f_{0}\)F\({}_{2}\) and the  
Figure 3: The time of data collected from ionosonde station and the corresponding cycle variations of solar activity factor (\(F_{12}\) and \(R_{12}\)). (**a**) The time of data collected from ionosonde station; (**b**) the cycle variations of solar activity factor corresponding to the collected data. two solar activity parameters are obviously different, especially at sunrise (around 06:00 clock) and sunset (around 20:00 clock) in summer. Considering that the critical frequency is directly related to \(F_{10.7}\) and the sunspot dissimilarity in different time periods, an alternative numerical map function is therefore derived. Concurrently, the annual and semi-annual variations of the ionosphere or more subtle variations are considered [44; 45]. For given spatial location and time,\(f_{o}\)F\({}_{2}\) with the harmonic functions representing the seasonal (annual, semi-annual, quarter-annual, and monthly) and solar activity cycle variations (denoted by the variable \(F_{12}\), \(R_{12}\), \(m\)) can be formulated as the following:  
\[ \begin{array}{ll}f_{o}F_{2}(F_{12},R_{12},m)&=f_{o}F_{2}{}^{F}(F_{12},m)+f_{o }F_{2}{}^{R}(R_{12},m)\\ &=\sum\limits_{k=0}^{K}\sum\limits_{l=0}^{L}\left[C_{l}F_{12}\cdot\cos\left( \frac{2\pi km}{12}\right)+s_{{}_{k}l}F_{12}{}^{l}\cdot\sin\left(\frac{2\pi km}{ 12}\right)+\right.\\ &\left.c_{k,l}^{\prime}R_{12}\cdot\cos\left(\frac{2\pi km}{12}\right)+s_{{}_{ k}l}^{\prime}R_{12}\cdot\sin\left(\frac{2\pi km}{12}\right)\right]\end{array}  \tag{4}\]  
Figure 4: Behavior of the monthly median values of critical frequency and the twelve-month running mean value of solar indices at 4:00 UT during the years from 1958 to 2005 over Kokubunji, Japan. (**a**) Month median \(f_{o}\)F\({}_{2}\) (MHz) around midnight (Local time is 00:00 clock); (**b**) month median \(f_{o}\)F\({}_{2}\) (MHz) around noon (Local time is 12:00 clock); (**c**) the twelve-month running mean value of sunspot number (\(F_{12}\)); (**d**) the twelve-month running mean value of sunspot number (\(R_{12}\)).