## 3 Mapping Performances of the Four Models over the Arctic and Antarctic
### Establishment of Four Ionospheric TEC Models  
In this section, four models were considered to map the polar ionospheric TEC, including the POLY model, the GTSF model, the SH model, and the SCH model. A brief introduction of these four models is given in the following paragraphs. The POLY model has a simple structure and is mainly used for regional TEC mapping. In this model, the TEC value is regarded as a series function of various factors, such as sun angle and latitude [14]. Its expression is as follows:  
\[\text{VTEC}(\varphi,\lambda,t)=\sum\limits_{i=0}^{n}\sum\limits_{j=0}^{m}A_{ ik}(\varphi-\varphi_{0})^{i}(\lambda-\lambda_{0}+t-t_{0})^{j}, \tag{2}\]  
where \(\text{VTEC}(\varphi,\lambda,t)\) is the vertical TEC (VTEC) at the ionosphere pierce point (IPP) (\(\varphi,\lambda\)) at time \(t\); \(n\) and \(m\) are the degree and order of the model, respectively; \(A_{ik}\) are the unknown model coefficients; \((\varphi_{0},\lambda_{0})\) are the geographic latitude and longitude at the regional center; \(t_{0}\) is the sun angle corresponding to the middle time of the mapping period. According to the spatiotemporal characteristics of regional ionospheric TEC, [PERSON] and [PERSON] [11] used both the polynomial function related to latitude and LT and the trigonometric series function related to LT to map the diurnal variation of VTEC, which improved the simulation capabilities of regional TEC. This model is developed on the basis of a trigonometric series function model and is called GTSF model. The expression is as follows:  
\[\left\{ \begin{array}{c}\text{VTEC}(\varphi,\lambda,t)=\sum\limits_{i=0}^{N_{ I}}\sum\limits_{j=0}^{N_{I}}A_{ij}\varphi_{m}{}^{i}h^{j}+\sum\limits_{k=0}^{N_{ K}}(B_{k}\cos(kh)+C_{k}\cos(kh))\\ \varphi_{m}=\varphi+0.064\cos(\lambda-1.617)\\ h=2\pi(t-14)/T,\ T=24\ h\end{array} \right., \tag{3}\]  
where \(N_{I}\) and \(N_{J}\) are the maximum degree and order of the polynomial function, respectively; \(N_{K}\) is the maximum degree of the trigonometric series function; \(A_{ij}\), \(B_{k}\), \(C_{k}\) are the model coefficients to be estimated; \(\varphi_{m}\) is the geomagnetic latitude at IPP; \(h\) is the variable related to LT. The SH model can precisely map the global ionospheric TEC [3]. In regional mapping, the basis functions of SH do not have orthogonality. However, the lower-order SH model still can be used for regional TEC mapping with high accuracy [16]. The function model is as follows:  
\[VTEC(\varphi,\lambda)=\sum\limits_{n=0}^{n_{\max}}\sum\limits_{m=0}^{m_{\max }}P_{nm}(\sin\varphi)\cdot(A_{nm}\cos(m\lambda)+B_{nm}\sin(m\lambda)), \tag{4}\]  
where \(\text{VTEC}(\varphi,\lambda)\) is the VTEC at the geomagnetic coordinate \((\varphi,\lambda)\) of the IPP; \(n_{\max}\) and \(m_{\max}\) are the maximum degree and order of the SH function, respectively; \(P(\sin\varphi)\) is the Legendre function; \(A_{nm}\) and \(B_{nm}\) are the model coefficients to be estimated. Because the basis functions of the SH model are not orthogonal in regional mapping, the SCH functions were introduced. The SCH model consists of a set of functions obtained by solving a Laplace's equation on a specific spherical cap [12]. For regional modeling, the zero-order term indicates the average TEC within the local area. The function expression of the SCH model is similar to that of the SH model; the main difference between them is that the SCH model replaces the integer-order Legendre function with a non-integer Legendre function.