## 3 Model Reconstruction
### Spatial Characteristics Reconstruction
\right], \tag{10}\]  
(2) In view of the facts that the ionosphere is controlled by both the orientation of the earth's rotation axis and the configuration of the geomagnetic field, the geomagnetic parameters play a significant role in mainly variations taking place in the ionosphere, most especially the F2 region. Simultaneously, the dependence of the tendency of changes in \(f_{0}\)F\({}_{2}\) on geomagnetic coordinate was proved [60]. Therefore, the geomagnetic functions of the geomagnetic longitude and dip latitude value are employed in the proposed model instead of simple geographic coordinates. Meanwhile, the modified radial basis function was defined in space reconstruction. Namely, the modified ionospheric distance between two different spatial locations in equation (7) is defined as following:  
\[r_{ij}=\left[\ln\left(\sqrt{\left(\phi_{i}-\phi_{j}\right)^{2}+\left(\beta_{i} -\beta_{j}\right)^{2}}\right)-1\right]\cdot\left[\left(\phi_{i}-\phi_{j} \right)^{2}+\left(\beta_{i}-\beta_{j}\right)^{2}\right]^{2}, \tag{11}\]  
where \(r_{ij}\) is the semivariogram value for the modified ionospheric distance between the \(i\)-th location (\(\delta_{i}\), \(\phi_{i}\)) and \(j\)-th location (\(\delta_{j}\), \(\phi_{j}\)), \(\delta\) is geomagnetic latitude, \(\phi\) is geomagnetic longitude. In this study, we tested three types of geomagnetic coordinates magnetic Apex coordinates [61], corrected geomagnetic coordinates (CGM) [62], and modified geomagnetic dip latitude coordinates (MGD) [63; 64]; see Figure 2. These coordinates are usually used in ionospheric physics and particularly suited for the organization of observations of plasma density in the ionospheric F2 layer [61]. From Figure 2, the trends for three types of coordinates obviously are different, and this circumstance becomes more serious for high-latitude regions. To determine the coordinates, the ionosonde data from 17 modeling stations (except the Jeju station because its data only cover the period 2013-2017) were chosen of which Irkutsk, Manzhouli, Khabarovsk, Wakkanai, Beijing, Akita, Seoul, Kokubunji, Yamagawa, Chongqing, Okinawa, Taipei, Guangzhou, and Haikou were selected for generalized cross-validation. From the root-mean-square error (RMSE), Figure 8 similar trends appear for the four different coordinates, and the RMSEs for the stations in the mid-latitude region were lower than those in low and high-latitude regions. Specifically, the RMSEs of the geographic coordinates, magnetic apex coordinates, CGM coordinates, and MGD coordinates are 1.0272 MHz, 0.96895 MHz, 0.90360 and 0.95144 MHz. In particular, the RMSE using geomagnetic dip coordinates is seen to have a lower minimum than those using the other three types of coordinates. The RMSE for the CGM coordinates is a minimum and lower than 0.1236 MHz (12.0327%) than for the geographic coordinates. Hence, the CGM coordinates were selected for the reconstruction of the spatial characteristics. Based on the above method, the map of weights \(W\) of equation (1) and (10) could be obtained, and the samples during four typical time period are shown in Figure 9. As shown in Figure 9, the weight variations are smooth with the solar activity parameters including the flux of solar radio waves at 10.7 cm and sunspot number. As well, there are different rules for different stations during different time period. Figure 8: Bubble chart of Root-mean-square error with four different coordinates. (**a**) Geographic coordinates, (**b**) Apex coordinates, (**c**) CGM coordinates, and (**d**) MGD coordinates. The radius of bubbles is set by 20\(\times\)2 RSME for better comparison.