# A Spatial Reconstruction Method of Ionospheric foF2 Based on High Accuracy Surface Modeling Theory
## 2 Data and Methods
### Modeling Methods
From Equation (7), it can be found that the left-hand side's coefficient matrices of the solved system of linear algebraic equations are the same in each iteration such that the inverse matrices are required only once, thus reducing the computational effort considerably. In this case, the finite difference form of the fundamental quantities of the first category and the second category is shown below:  
\[\left\{ \begin{array}{l}B_{i,j}=1+(\frac{\tilde{f}_{i+1,j}-\tilde{f}_{i-1,j} }{2h_{\varphi}})^{2}\\ C_{i,j}=(\frac{\tilde{f}_{i+1,j}-\tilde{f}_{i-1,j}}{2h_{\alpha}})(\frac{\tilde{ f}_{i+1,j}-\tilde{f}_{i-1}}{2h_{\lambda}})\\ D_{i,j}=1+(\frac{\tilde{f}_{i,j+1}-\tilde{f}_{i,j-1}}{2h_{\lambda}})^{2}\end{array} \right. \tag{8}\]\[\left\{ \begin{array}{l}E_{ij}=\frac{\frac{\tilde{I}_{i+1,j}-\tilde{J}_{i,j}\tilde{I }_{j,l}+\tilde{I}_{i-1,j}}{h_{\varphi}^{2}}}{\sqrt{1+(\frac{\tilde{I}_{i+1,j} -\tilde{I}_{i-1,j}}{2h_{\varphi}})^{2}+(\frac{\tilde{I}_{i+1}-\tilde{I}_{i,j-1 }}{2h_{\lambda}})^{2}}}\\ F_{ij}=\frac{\tilde{I}_{j,i+1}-\tilde{I}_{i,j}\tilde{I}_{j,l}+\tilde{I}_{i-1 }}{\frac{h_{\lambda}^{2}}{2h_{\varphi}}}\\ \sqrt{1+(\frac{\tilde{I}_{i+1,j}-\tilde{I}_{i-1,j}}{2h_{\varphi}})^{2}+(\frac{ \tilde{I}_{i,j+1}-\tilde{I}_{i,j-1}}{2h_{\lambda}})^{2}}}\end{array} \right. \tag{9}\]  
where h\({}_{\varphi}\) and h\({}_{\lambda}\) are the calculation steps in the longitude and latitude directions and \(\widetilde{f_{i,j}}\) is the predicted value of the foF2 calculated by the IRI model at the grid position (\(\varphi_{i}\),\(\lambda\)). The finite difference formulation of the Gaussian system of equations with coefficients \(G_{11}^{1}\), \(G_{11}^{2}\), \(G_{21}^{1}\), and \(G_{22}^{2}\) are  
\[\left\{ \begin{array}{l}\left(G_{11}^{1}\right)_{ij}=\frac{D_{ij}(B_{i+1,j}-B _{i-1,j})h_{\lambda}-2C_{ij}(C_{i+1,j}-C_{i-1,j})h_{\lambda}+C_{ij}(B_{i,j+1}- B_{i,j-1})h_{\varphi}}{4(B_{ij}D_{ij}-C_{ij}^{2})h_{\lambda}h_{\lambda}}\\ \\ \left(G_{22}^{1}\right)_{ij}=\frac{2D_{ij}(C_{i,j+1}-C_{i,j-1})h_{\varphi}-D_{ ij}(D_{i+1,j}-D_{i-1,j})h_{\lambda}+C_{ij}(D_{i,j+1}-D_{ij-1})h_{\varphi}}{4(B_{ij}D _{ij}-C_{ij}^{2})h_{\theta}h_{\lambda}}\\ \\ \left(G_{11}^{2}\right)_{ij}=\frac{2B_{ij}(C_{i+1,j}-C_{i-1,j})h_{\lambda}-B_{ ij}(B_{i,j+1}-B_{i,j-1})h_{\varphi}+C_{ij}(B_{i,j+1}-B_{i,j-1})h_{\varphi}}{4(B_{ij}D _{ij}-C_{ij}^{2})d_{\varphi}d_{\lambda}}\\ \\ \left(G_{22}^{2}\right)_{ij}=\frac{B_{ij}(D_{ij,i+1}-D_{ij-1})h_{\varphi}-2C_{ ij}(C_{i+1,j}-C_{ij-1})h_{\varphi}+C_{ij}(D_{i+1,j}-D_{i-1,j})h_{\lambda}}{4(B_{ij}D _{ij}-C_{ij}^{2})h_{\theta}h_{\lambda}}\\ \end{array} \right. \tag{10}\]  
The difference steps in the longitude and latitude directions are expressed as follows:  
\[\left\{ \begin{array}{l}h_{\varphi}=\frac{1}{\varphi+1}\\ h_{\lambda}=\frac{1}{\lambda+1}\\ \end{array} \right. \tag{11}\]  
where \(\varphi\) denotes the number of grid points along the longitude direction and \(\lambda\) represents the number of grid points along the latitude direction. The algebraic equations are then reformulated into matrix form. The matrix expression's form of the numerical simulation equations for the HAS method is  
\[\left\{ \begin{array}{l}Lf^{n+1}=M^{n}\\ Jf^{n+1}=N^{n}\\ \end{array} \right. \tag{12}\]  
where \(f^{n+1}=[f_{1,1}^{n+1},\ldots f_{1,\lambda}^{n+1},f_{2,1}^{n+1},\ldots f_{2, \lambda}^{n+1},\ldots f_{\varphi-1,1}^{n+1},\ldots f_{\varphi+1,\lambda}^{n+1},f_{\varphi,1}^{n+1},\ldots f_{\varphi,\lambda}^{n+1}]\), T\({}_{0}\) is the interpolation result of the sampling point value, L and Mn are the left end's coefficient matrix and the right end's constant term matrix of the first equation in the HAS equation system, respectively, and J and N\({}^{n}\) are the left end's coefficient matrix and the right end's constant term matrix of the second equation in the HAS equation system, respectively.