## 3 Results and Discussion
### Linear and Dynamic Coupling  
There is a change in the DFS as radio waves propagate through the ionosphere. We employed two approaches, linear and nonlinear, to investigate the relationship between the change in the DFS of a signal originating from one point on earth and its reflection by the ionosphere at another point. Table 2 shows the statistics for both correlation and transfer entropy measures for different time epochs considered in this study. We observed negative correlations between VTEC measurements and DFS at the two locations under consideration. We found similar correlation values for the two locations. The strongest negative correlation was observed in 2021, while the weakest correlation was reported in 2020. This implies that an increase in ionospheric activity at the two locations corresponds to a decrease in the change in the DFS at the receiving end. A similar analysis was carried out using the nonlinear transfer entropy. Our transfer entropy analysis showed that there was a strong relationship between the VTEC and DFS in 2020-2021 compared to the other two time frames. Across the years, a stronger connection was observed between VTEC and DFS in 2020 compared to 2021 in the two locations. Also, values of transfer entropy were found to be higher in Abuja compared to Lagos, whereas the correlation coefficient was slightly higher in Lagos compared to Abuja. While the values of transfer entropies were close in the two locations, there were sufficient differences to discriminate the coupling, which was not obvious in the correlation analysis. The year 2020, with a mean sunspot number (Rz) of 8.8, had a stronger transfer entropy than 2021, a year with high solar activity (Rz = 29.6). The relationship between VTEC and DFS was also considered on a monthly scale using the correlation (Figures 10 and 11) and transfer entropy (Figures 12 and 13) at the two locations. In Figure 10, correlation values showed a weaker relationship between VTEC and DFS during the months of April to August for the periods 2020, 2021, and 2020-2021. The values were largely negative at the two locations, except in the month of April, where a positive value was obtained. Furthermore, Figures 11 and 13, respectively, show the seasonal variation in the correlation values and transfer entropy between the VTEC and DFS. The summer (June) solstice has a higher correlation than the other seasons. This shows that an increase in \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Location** & **Period** & **Correlation** & **TE** \\ \hline Lagos & 2020 & \(-0.167\) & \(1.073\) \\ Lagos & 2021 & \(-0.279\) & \(0.385\) \\ Lagos & 2020–2021 & \(-0.219\) & \(1.338\) \\ \hline Abuja & 2020 & \(-0.164\) & \(1.114\) \\ Abuja & 2021 & \(-0.277\) & \(0.424\) \\ Abuja & 2020–2021 & \(-0.216\) & \(1.414\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Linear and nonlinear relationship between VTEC measurement and DFS at Lagos and Abuja using correlation measures and transfer entropy over different time periods. ionospheric activity corresponds to a decrease in the DFS. The correlation was observed to be stronger in the year with higher solar activity (2021), but weaker in 2020. Replicating the analysis using transfer entropy yielded another scenario, as presented in Figures 11 and 12. The influence of VTEC on DFS was different in 2020 compared to 2021 in both Lagos and Abuja. In 2020, there was little or no influence in most months of the year except June (Abuja), July (Lagos), September, and October. Figure 11 shows that the DFS of the propagated radio signal is more impacted by variability in the ionosphere, as captured by the VTEC in the solstitial months of June/July at Abuja/Lagos and the September equinox at the two locations. The seasonal effects in the ionospheric processes have been discussed by several authors using different parameters and techniques; see, for example, [31] and the references therein. Generally, the influence was weaker in 2020 compared to the other two periods under consideration in both Lagos and Abuja.