## 3 Model Reconstruction
### Spatial Characteristics Reconstruction  
Ionospheric parameters such as \(f_{0}\)F\({}_{2}\) have an irregular non-homogeneous spatial distribution and therefore studies mapping its spatial characteristics have been conducted without interruption over several years. At present, numerous techniques based on different empirical or mathematical methods have been proposed to describe ionospheric spatial characteristics, including inverse distance weighted interpolation [46; 47], spline interpolation methods [48; 49; 50], spherical harmonic (Legendre) functions [12; 13], the SIRM [26; 51; 52] and improved versions [25] in Europe, along with statistical approaches such as kriging [53]. Among the above methods, inverse distance weighted interpolation is lack of geophysical mechanism and low accuracy where long distance from measured stations. The spherical Legendre functions recommended by the ITU [14; 54] are only orthogonal over the whole sphere and do not efficiently represent a signal from a restricted region on the sphere [55]. Figure 6: Bubble chart of Root-mean-square error of different analysis order. The size of bubble is set by \(10^{7(\text{REMS}-0.5)}\) for better comparison viewing. On the other hand, surface spline interpolation based on radial basis functions (such as [PERSON]'s functions) has excellent approximation properties and has become a mainstream method and a very powerful tool in multivariate approximations because of its high precision, simplicity, and flexibility [50; 56]. Simultaneously, [PERSON]'s functions have been tested and evaluated for analytical interpolation tests and shown to be the most robust, cost effective and accurate [57; 58]. Furthermore, the Kriging method is an interpolation algorithm widely used in ionospheric parameter reconstruction and can offer the best linear unbiased estimates with an accurate description of the spatial structure of the ionospheric data. It uses known sample values and variogram to determine unknown values at different spatial locations. The variogram function of Kriging method describes the spatial correlation among the measured samples used in the interpolation and are calculated by taking the difference between pairs of measurements for a given distance [4; 24; 53]. Splines and kriging are two methods that should be used alternately, depending on what one wants to obtain [59]. To find the extreme accuracy limit of the regional model for \(f_{0}\)F\({}_{2}\), we implement two processes in developing an algorithm to describe the \(f_{0}\)F\({}_{2}\) spatial characteristics that provides high speed and high accuracy. (1) Based on modified ionospheric distance from surface spline interpolation theory, the weights \(W_{n}\) of equation (1) could be obtained from linear kriging equations:  
\[\left\{ \begin{array}{l}\sum\limits_{j=1}^{N}r_{ij}\cdot w_{j}=r_{i0}-\mu,\ i=1,2, \cdots,N\\ \sum\limits_{j=1}^{N}w_{j}=1\end{array} \right., \tag{7}\]  
where \(N\) is maximum number of measured stations used to space interpolation, \(\mu\) is the Lagrange factor, \(r_{ij}\) is one of radial basis functions for modified ionospheric distance between two known measured stations based on surface spline interpolation with modified radial basis function. It is expressed in matrix notation:  
\[R=\left[ \begin{array}{cccc}0&r_{12}&\cdots&r_{1N}\\ r_{21}&0&\cdots&r_{2N}\\ \vdots&\vdots&\ddots&\vdots\\ r_{N1}&r_{N2}&\cdots&0\end{array} \right], \tag{8}\]  
On this condition, we can calculate the ionospheric distance vector between unknown location (\(\beta_{i}\), \(\phi_{i}\)) and known measured station (\(\delta_{0}\), \(\phi_{0}\)), which contains the semivariogram estimations (\(R_{0}\)), namely,  
\[R_{0}=\left[ \begin{array}{c}r_{10}\\ r_{20}\\ \vdots\\ r_{N0}\end{array} \right], \tag{9}\]  
Additionally, the weights \(W\) of equation (1) could be obtained, which is expressed in vector notation:  
\[W=\left[ \begin{array}{c}w_{1}\\ w_{2}\\ \vdots\\ w_{N}\end{array}