## 2 Mathematical Model and Inversion Method  
### Mathematical Model  
The model is established in a source-free conical domain denoted by \(\Omega\), which is bounded by a finite cone semi-angle of \(\theta_{0}\). The domain lower and upper boundaries along the radial \(r\) are defined with two spherical surfaces of radii \(a\) and \(b\), respectively, as shown in Figure 1. The boundary of cone \(\Omega\) was defined as follows: \(\partial\Omega=\partial_{\theta_{0}}\Omega\cup\partial_{\theta_{0}}\Omega \cup\partial_{\theta_{0}}\Omega\). \(\partial_{\theta_{0}}\Omega:\{\theta\leq\theta_{0},\ r=a\}\) and \(\partial_{b}\Omega:\{\theta\leq\theta_{0},\ r=b\}\) represent the lateral boundary, the lower and upper boundary of \(\Omega\), respectively [18; 30]. Suppose that the geomagnetic field meets the irrotational and solenoidal requirements in the closed region \(\Omega\). The geomagnetic field scalar potential meets the Laplace equation \(\
abla^{2}V=0\) under the condition of boundary constraint. By using the separation of variables method, the solution of [PERSON]'s equation can be written as a sum of infinite series in a cone coordinate system [30]:  
\[ \begin{array}{ll}V(r,\theta,\varphi)=&a\sum\limits_{k\geq 0}\sum\limits_{m \geq 0}\left(\frac{a}{r}\right)^{n_{k}+1}\left(G_{n_{k}}^{i,m}\cos(m\varphi)+H_{n _{k}}^{i,m}\sin(m\varphi)\right)\cdot P_{n_{k}}^{m}(\cos\theta)\\ &+a\sum\limits_{k\geq 0}\sum\limits_{m\geq 0}\left(\frac{r}{a}\right)^{n_{k}} \left(G_{n_{k}}^{o,m}\cos(m\varphi)+H_{n_{k}}^{e,m}\sin(m\varphi)\right)\cdot P _{n_{k}}^{m}(\cos\theta)\\ &+a\sum\limits_{p\geq 1}\sum\limits_{m\geq 0}R_{p}(r)\left(G_{p}^{m}\cos(m \varphi)+H_{p}^{m}\sin(m\varphi)\right)\cdot K_{p}^{m}(\cos\theta)\\ &+a\sum\limits_{m\geq 0}R_{0}\left(G_{0}^{m}\cos(m\varphi)+H_{0}^{m}\sin(m \varphi)\right)\cdot P_{0}^{m}(\theta)\end{array}  \tag{1}\]  
The corresponding expressions for the geomagnetic component are as follows [18; 31]:  
\[ \begin{array}{ll}X=&\sum\limits_{k\geq 0}\sum\limits_{m\geq 0}\left(\frac{a}{ r}\right)^{n_{k}+2}\frac{dP_{n_{k}}^{m}}{d\theta}\left(G_{n_{k}}^{i,m}\cos(m \varphi)+H_{n_{k}}^{i,m}\sin(m\varphi)\right)\\ &+\sum\limits_{k\geq 0}\sum\limits_{m\geq 0}\left(\frac{r}{a}\right)^{n_{k}-1} \frac{dP_{n_{k}}^{m}}{d\theta}\left(G_{n_{k}}^{o,m}\cos(m\varphi)+H_{n_{k}}^{e,m}\sin(m\varphi)\right)\\ &+\frac{a}{r}\sum\limits_{p\geq 1}\sum\limits_{m\geq 0}R_{p}(r)\frac{d_{n_{k}}^{m }}{d\theta}\left(G_{p}^{m}\cos(m\varphi)+H_{p}^{m}\sin(m\varphi)\right)\\ &+\frac{a}{r}\sum\limits_{m\geq 0}R_{0}\frac{dP_{n_{k}}^{m}}{d\theta}\left(G_{0}^{m }\cos(m\varphi)+H_{0}^{m}\sin(m\varphi)\right)\\ Y=&\sum\limits_{k\geq 0}\sum\limits_{m\geq 1}\left(\frac{r}{a}\right)^{n_{k}+2} \frac{m_{p_{k}}^{m}}{\sin(r)}\left(G_{n_{k}}^{i,m}\sin(m\varphi)-H_{n_{k}}^{i,m}\cos(m\varphi)\right)\\ &+\sum\limits_{k\geq 0}\sum\limits_{m\geq 1}\left(\frac{r}{a}\right)^{n_{k}-1} \frac{m_{p_{k}}^{m}}{\sin(\theta)}\left(G_{n_{k}}^{o,m}\sin(m\varphi)-H_{n_{k} }^{e,m}\cos(m\varphi)\right)\\ &+\frac{a}{r}\sum\limits_{p\geq 1}\sum\limits_{m\geq 1}R_{p}(r)\frac{m_{p_{k}}^{m }}{\sin(\theta)}\left(G_{p}^{m}\sin(m\varphi)-H_{p}^{m}\cos(m\varphi)\right)\\ &+\frac{a}{r}\sum\limits_{m\geq 1}R_{0}\frac{m_{p_{k}}^{m}}{\sin\theta}\left(G_{0}^{m }\sin(m\varphi)-H_{0}^{m}\cos(m\varphi)\right)\\ Z=&-\sum\limits_{k\geq 0}\sum\limits_{m\geq 0}\left(n_{k}+1\right)\left(\frac{r}{a} \right)^{n_{k}+2}P_{n_{k}}^{m}\left(G_{n_{k}}^{i,m}\cos(m\varphi)+H_{n_{k}}^{i,m }\sin(m\varphi)\right)\\ &+\sum\limits_{k\geq 0}\sum\limits_{m\geq 0}n_{k}\left(\frac{r}{a}\right)^{n_{k}-1} P_{n_{k}}^{m}\left(G_{n_{k}}^{e,m}\cos(m\varphi)+H_{n_{k}}^{e,m}\sin(m\varphi)\right)\\ &+\sum\limits_{p\geq 1}\sum\limits_{m\geq 0}\frac{dR_{p}(r)}{dr}K_{p}^{m}\left(G_{p}^{m }\cos(m\varphi)+H_{p}^{m}\sin(m\varphi)\right)\end{array}  \tag{2}\]In Equation (2), \(X,Y,Z\) represent the north, east, and vertical downward components of the geomagnetic field at an observation station, respectively. \(a\) denotes the mean radius of the earth, which is taken to be 6371.2 km.