## 2 Materials and Methods
### Weight Setting in Phase Pattern Fitting Process  
The PCC measurements constructed using the triple-differencing approach contain greater noise compared with the double-differencing approach, especially under a long time-differencing interval. In order to reduce the impact of the triple-differencing noise on the accuracy of the subsequent phase pattern fitting result, it is necessary to design a weighting method, giving more weight to the PCC measurements that are more accurate. This section first briefly describes the unweighted phase pattern fitting procedure and then proposes a new weight setting method. There are traditionally two approaches to model the elevation- and azimuth-dependent phase patterns: grid and spherical harmonics expansions [11; 12]. Since the direction of the satellite LOS vectors fully covers the antenna hemisphere, the approach with spherical harmonics is more compelling and meaningful. Moreover, most recent studies adopt the spherical harmonics of degree of 8 or 12 [6; 8; 11; 20; 34]. Using the spherical harmonics expansion, the relative PCC measurements can be expressed as  
\[ \begin{split}\Xi^{s}&=\Xi(azi^{s},ele^{s})=\sum \limits_{n=0}^{n_{\max}}\sum\limits_{m=0}^{m^{\prime}}\widetilde{P}_{nm}(\sin ele ^{s})(a_{nm}\cos(m\cdot azi^{s})+b_{nm}\sin(m\cdot azi^{s}))\\ m^{\prime}&=\max(m_{\max},n),m_{\max}\leq n_{\max} \end{split}  \tag{11}\]  
where \(n_{\max}\) and \(m_{\max}\) denote the maximum degree and maximum order of spherical harmonics, \(\widetilde{P}_{nm}\) denotes the normalized associated Legendre polynomial of degree n and order m, \(\widetilde{P}_{nm}\) is a function of the sine of the elevation angle \(ele^{s}\), and \(a_{nm}\) and \(b_{nm}\) denote the desired spherical harmonics coefficients that determine the phase pattern model. If we set \(n_{\max}\) and \(m_{\max}\) to 8 and 5, respectively, like the literature [6; 34] showed, there will be 38 coefficients to be resolved. The triple-differenced PCC measurement obtained in the last section is a combination of four relative PCCs:  
\[\Xi^{sr}\Big{(}t,t-T_{diff}\Big{)}=(\Xi^{s}(t)-\Xi^{r}(t))-\Big{(}\Xi^{s} \Big{(}t-T_{diff}\Big{)}-\Xi^{r}\Big{(}t-T_{diff}\Big{)}\Big{)}+\xi_{12}^{sr} \Big{(}t,t-T_{diff}\Big{)} \tag{12}\]  
By substituting Equation (11) into Equation (12), the triple-differenced PCC measurements can be expanded as a function of a set of azimuth and elevation angles:  
\[\Big{[}azi^{s}(t),ele^{s}(t),azi^{s}(t),ele^{s}(t),azi^{s}\Big{(}t-T_{diff} \Big{)},ele^{s}\Big{(}t-T_{diff}\Big{)},azi^{s}\Big{(}t-T_{diff}\Big{)},azi^{s }\Big{(}t-T_{diff}\Big{)},ele^{s}\Big{(}t-T_{diff}\Big{)}\Big{]} \tag{13}\]Once the spherical harmonics coefficients are resolved, the PCC values at grid points can be derived from the fitted function. The common grid spacing used in the file jigs14.atx is 5\({}^{\circ}\)[35]. The existing research usually sets a unitary weighting for all PCC measurements [6; 11; 9]. In the following, we present a weight setting method based on the area of spherical quadrilaterals. Each triple-differenced PCC measurement is associated with four directions--the directions of the LOS vectors of satellites \(r\) and \(s\) at epochs \(t\) and \(t-T_{diff}\). As we can see from Figure 3, the terminal points of the four corresponding unit LOS vectors lie on the antenna unit hemisphere, and these points enclose a spherical quadrilateral ACBD. Figure 2: Flowchart of constructing the triple-differenced PCC measurements. According to Equation (12), a triple-differenced PCC measurement contains two parts: (1) triple-differenced PCC and (2) the rest noise in triple-differenced carrier-phase observations. The magnitude of the second part is relatively stable once a reliable cycle-slip detection and rejection are completed, while the magnitude of the first part can fluctuate remarkably. The triple-differenced PCC reflects the differences between PCCs at four different directions.