# A Spatial Reconstruction Method of Ionospheric foF2 Based on High Accuracy Surface Modeling Theory
## 2 Data and Methods
### Modeling Methods  
This paper integrates HAS modeling with the characteristics of the ionospheric foF2 to perform spatial reconstruction. The initial surface function of the foF2 spans the modeling range and incorporates the IRI as the background field, being expressed as follows:  
\[f\text{oF2}=f(\varphi,\lambda) \tag{2}\]  
where f is the mapping satisfied by the surface, \(\varphi\) represents the longitude of the ionospheric observation station, \(\lambda\) represents the latitude of the ionospheric observation station, and (\(\varphi\),\(\lambda\)) represents the coordinates of the ionospheric observation station. During the numerical simulation of the function, the first and second fundamental quantities of the surface must first be interpolated using the sampled values [25]. According to the fundamental theorem of surfaces, the first and second fundamental quantities define many surface properties. The first fundamental quantities represent geometric properties, such as the length of the curves and the area of the surfaces [26]. In contrast, the second fundamental quantity represents macro information on the surface, such as local warping [27]. The first fundamental quantities \(B\), \(C\), and \(D\) can be expressed as follows:  
\[\left\{ \begin{array}{l}B=1+f_{\varphi}^{2}\\ C=f_{\varphi}f_{\lambda}\\ D=1+f_{\lambda}^{2}\end{array} \right. \tag{3}\]  
where f\({}_{\varphi}\) and f\({}_{\lambda}\) are denoted as the first-order derivatives of the function f(\(\varphi\),\(\lambda\)) in the \(\varphi\) and \(\lambda\) directions, respectively. Figure 2: Modeling flowchart of foF2 spatial interpolation model based on Kriging model and foF2 spatial interpolation model based on Kriging model. The second class of fundamental quantities E and F is denoted as  
\[\left\{ \begin{array}{l}E=\frac{f_{\varphi\varphi}}{\sqrt{1+f_{2}^{2}+f_{3}^{2}}} \\ F=\frac{f_{\lambda\lambda}}{\sqrt{1+f_{2}^{2}+f_{3}^{2}}}\end{array} \right. \tag{4}\]  
where f\({}_{\varphi\varphi}\) and f\({}_{\lambda\lambda}\) are the second-order derivatives of the function f(\(\varphi\),\(\lambda\)) in the \(\varphi\) and \(\lambda\) directions, respectively. Based on surface theory and space sampling, the surface can be expressed by the following set of Gaussian equations:  
\[\left\{ \begin{array}{l}f_{\varphi\varphi}=G_{11}^{1}f_{\varphi}+G_{11}^{2}f_ {\lambda}+\frac{E}{\sqrt{B+D-1}}\\ f_{\lambda\lambda}=G_{22}^{1}f_{\varphi}+G_{22}^{2}f_{\lambda}+\frac{F}{\sqrt {B+D-1}}\end{array} \right. \tag{5}\]  
where \(G_{11}^{1}\), \(G_{11}^{2}\), \(G_{22}^{1}\), and \(G_{22}^{2}\) are the coefficients of the Gaussian system of equations and depend only on the fundamental quantities of the first type and their derivatives, defined as  
\[\left\{ \begin{array}{l}G_{11}^{1}=\frac{DB_{\varphi}-2C\zeta_{\varphi}+CB_{ \lambda}}{2(BD-C^{2})}\\ G_{22}^{1}=\frac{2 DC_{\lambda}-DD_{\varphi}-CD_{\lambda}}{2(BD-C^{2})}\\ G_{11}^{2}=\frac{2 BC_{\varphi}-BB_{\lambda}-CB_{\varphi}}{2(BD-C^{2})}\\ G_{22}^{2}=\frac{BD_{\lambda}-2 CC_{\lambda}+CD_{\varphi}}{2(BD-C^{2})}\end{array} \right. \tag{6}\]  
The results of the numerical experimental analyses show that the inverse of the coefficient matrix needs to be reinverted at each iteration, which is highly computationally intensive. To reduce the computational effort, the finite difference form's expression of the Gaussian system of equations is updated as follows [28]:  
\[\left\{ \begin{array}{l}\frac{f_{i+1,j}^{m+1}-2f_{ij,i}^{m+1}+f_{i-1j}^{m+1}}{ h_{2}^{2}}=(G_{11}^{1})_{ij}^{n}\frac{f_{i+1,j}^{m}-f_{i-1j}^{m}}{2h_{\alpha}}+(G_{11}^{2})_{ ij}^{n}\frac{f_{i+1,j}^{m}-f_{i-1}^{m}}{2h_{\lambda}}+\frac{E_{ij}^{n}}{ \sqrt{B_{ij}^{n}+D_{ij}^{n}-1}}\\ \frac{f_{ij,i+1}^{m+1}-2f_{ij,i}^{m+1}+f_{i-1}^{m+1}}{{h_{\lambda}}^{2}}=(G_{2 2}^{1})_{ij}^{n}\frac{f_{i+1,j}^{m}-f_{i-1j}^{m}}{2h_{\lambda}}+(G_{22}^{2})_{ ij}^{n}\frac{f_{i,j+1}^{m}-f_{i-1}^{m}}{2h_{\lambda}}+\frac{F_{ij}^{n}}{ \sqrt{B_{ij}^{n}+D_{ij}^{n}-1}}\end{array} \right. \tag{7}\]  
where n denotes the number of iterations.