## 2 Mathematical Model and Inversion Method
### Inversion Method  
The observation vector of the field \(L=\{l_{1},l_{2},\cdots l_{n}\}\) and parameter vector to be estimated (revised spherical cap harmonic coefficient) \(X=\{x_{1},x_{2},\cdots,x_{k}\},\ n>>k\), satisfy the following linear relation:  
\[L=AX+\Delta. \tag{3}\]  
In Equation (3), \(\Delta\) represents the observation noise. Its stochastic characteristic is \(E(\Delta)=0,\ D(\Delta)=\sigma_{0}^{2}{P_{L}}^{-1}\) where \(P_{L}\) represents the observation vector weight matrix and \(\sigma_{0}\) denotes the mean square error of unit weight. A target function is constructed according to the estimation criterion of the least square principle of compensation to provide a unique stable resolution for Equation (3):  
\[J_{\alpha}(L,X)=\left\|AX-L\right\|_{P_{L}}^{2}+\alpha\cdot\Omega(X). \tag{4}\]  
Parameter vector \(\hat{X}\) that meets the minimization of Equation (4) is the solution of the linear model by Equation (3). \(\Omega(X)\) is called a stable function. It functions to transfer ill-posed problems to well-posed problems. \(\alpha\) represents the regularization parameter (smoothing factor). It functions to balance the weights of two items of target function \(J_{\alpha}(L,X)\). \(\|\cdot\|_{P_{L}}^{2}\) represents the weighted Euclidean two-norm [38]. In practice, stable functional \(\Omega(X)\) typically uses the Tikhonov regularization method with the following form [39; 40; 41; 42]:  
\[\Omega(X)=\left\|X\right\|_{P_{X}}^{2}=X^{T}P_{X}X. \tag{5}\]  
In Equation (5), \(P_{X}\) represents the parameter weight matrix. For the above stable functional, the minimum value of target function \(J_{\alpha}(L,X)\) is calculated as follows:  
\[J_{\alpha}(L,X)=\left\|AX-L\right\|_{P_{L}}^{2}+\alpha\cdot X^{T}P_{X}X=\text {min}. \tag{6}\]If \(\partial J_{\alpha}(L,X)/\partial X=0\), then a parametric solution can be obtained as follows:  
\[X=\left(A^{T}P_{L}A+\alpha P_{X}\right)^{-1}A^{T}P_{L}L. \tag{7}\]  
Based on the abovementioned analysis, the selection of reasonable parameter weight function \(P_{X}\) is the key to solving ill-posed problems. Given that the Legendre and Mehler functions are mutually orthogonal, parameter weight function \(P_{X}\) can be set as a diagonal matrix. The elements in the diagonal line of matrix \(P_{X}\) can be determined by geomagnetic energy per coefficient. Besides the additional constraint or a stable functional \(\Omega(X)\), regularization parameter \(\alpha\) also plays an important role in ill-posed problems [43]. \(\alpha\) balances the influence of data fitting degree and regularized error for parameter estimation [43]. The optimal regularization parameter should minimize the comprehensive error caused by both factors. When \(\alpha\) is excessively small, the inversion model tends to fit the observation data effectively and with a small constraint effect, which may induce over-fitting of high-frequent noise in the observation data. As a result, the solution of target function loses physical meaning because of shock. If \(\alpha\) is excessively large, then the inversion model will be inclined to constraints, thereby leading to excessive smoothing of the model parameters. As a result, high-frequency information is lost, and the fitting effect of observation data is poor. To estimate the optimal regularization parameter \(a\), [PERSON] et al. used CM4 synthetic data to calculate the correlation coefficient between the R-SCHA prediction and the expected magnetic field and in the data gap layer at a height of 100-300 km. After testing, set \(\alpha=1e^{-6}\) was identified for the internal Legender functions. Set \(\alpha=9e^{-7}\) was determined for the Mehler functions. The R-SCHA algorithm can provide the best balance between the fit and the smoothness of the edges, as well as the reliability of the data gap layer [31]. By using multiple simulation data, [PERSON] and [PERSON] (2008) found that the model can also provide a stable space continuation in equal weight without considering regularization conditions.