## 3 Proposed Methods
### Method for Determining the AFM Carrier-Phase Ambiguity at Dual Frequencies  
In this method, the short-baseline inter-station single-difference integer ambiguity is effectively determined after real and spoofed GNSS signals are successfully received, thus realising subsequent positioning of spoofed signals. This method utilises the inter-frequency information difference to fuse the inter-station single-difference carrier-phase information at dual frequencies so as to obtain a strong correlation constraint between dual-frequency carrier phases and their ambiguities. Additionally, a new method for resolving the ambiguities of short-baseline inter-station carrier phases is proposed. As the effect of the inter-station clock difference between the two antennas still exists in Equation (2) given in Section 2, we can obtain the following equation by subtracting the upper and lower equations of Equation (2):  
\[ \begin{split}\lambda_{B_{1}}\Big{(}\Delta\phi_{12,B_{1}}^{i}+N_{1 2,B_{1}}^{i}\Big{)}-\lambda_{B_{3}}\Big{(}\Delta\phi_{12,B_{3}}^{i}+N_{12,B_{ 3}}^{i}\Big{)}\\ =c\cdot\Big{(}\delta t_{12,B_{1}}^{i}-\delta t_{12,B_{3}}^{i} \Big{)}+\varepsilon_{12}^{i}\end{split}  \tag{8}\]  
where \(\Delta\phi_{12,B_{1}}^{i}\) and \(\Delta\phi_{12,B_{3}}^{i}\) are the inter-station single-difference carrier-phase measurements for satellite \(i\) at two frequency points, \(B_{1}\) and \(B_{3}\), respectively; \(\varepsilon_{12,B_{1}}^{i}\) and \(\varepsilon_{12,B_{3}}^{i}\) are the inter-station single-difference carrier-phase measurement errors for satellite \(i\) at two frequency points, \(B_{1}\) and \(B_{3}\), respectively. By analysing the clock difference part in Equation (8), we obtain  
\[ \begin{split}\delta t_{12,B_{1}}^{i}-\delta t_{12,B_{3}}^{i}& =\big{(}\delta t_{1,B_{1}}-\delta t_{2,B_{1}}\big{)}-\big{(} \delta t_{1,B_{3}}-\delta t_{2,B_{3}}\big{)}\\ &=\big{(}\delta t_{1,B_{1}}-\delta t_{1,B_{3}}\big{)}-\big{(} \delta t_{2,B_{1}}+\delta t_{2,B_{3}}\big{)}\end{split}  \tag{9}\]  
The clock difference between two frequency points, \(B_{1}\) and \(B_{3}\), of the same receiver can be ignored because it is considerably smaller than the noise (random error). Moreover, the inter-station single-difference carrier phase at \(B_{1}\) and \(B_{3}\) represents the same distance, and this distance does not exceed the baseline length. Therefore, we obtain  
\[ \begin{split}\left\{\begin{aligned} \left|\lambda_{B_{1}}\Big{(}\Delta\phi_{12,B_{1}}^{i}+N_{1 2,B_{1}}^{i}\Big{)}\right|&\leq L\\ \left|\lambda_{B_{3}}\Big{(}\Delta\phi_{12,B_{3}}^{i}+N_{12,B_{3}}^{i} \Big{)}\right|&\leq L\end{aligned}\right.\\ \lambda_{B_{1}}\Big{(}\Delta\phi_{12,B_{1}}^{i}+N_{12,B_{1}}^{i} \Big{)}-\lambda_{B_{3}}\Big{(}\Delta\phi_{12,B_{3}}^{i}+N_{12,B_{3}}^{i} \Big{)}=0\end{split}  \tag{10}\]  
As depicted in Figure 5, the points in the range of Equation (10) represent the integer ambiguity group \(\Big{(}N_{12,B_{1}}^{i},N_{12,B_{3}}^{i}\Big{)}\) at \(B_{1}\) and \(B_{3}\). Considering the stochastic measurement errors \(\varepsilon_{1}\) and \(\varepsilon_{3}\) with the same standard deviation \(\sigma\), Equations (10) and (11) can be transformed as follows:  
\[\left\{ \begin{matrix}N_{12,B_{3}}^{i}=\frac{\lambda_{B_{1}}}{\lambda_{B_{3}}} \Big{(}\Delta\phi_{12,B_{1}}^{i}+\varepsilon_{12,B_{1}}^{i}+N_{12,B_{1}}^{i} \Big{)}-\Big{(}\Delta\phi_{12,B_{3}}^{i}+\varepsilon_{12,B_{3}}^{i}\Big{)}\\ -\frac{L}{\lambda_{B_{1}}}-\Delta\phi_{12,B_{1}}^{i}-3\sigma\leq N_{12,B_{1}}^ {i}\leq\frac{L}{\lambda_{B_{1}}}-\Delta\phi_{12,B_{1}}^{i}+3\sigma\\ -\frac{L}{\lambda_{B_{3}}}-\Delta\phi_{12,B_{1}}^{i}-3\sigma\leq N_{12,B_{3}} ^{i}\leq\frac{L}{\lambda_{B_{3}}}-\Delta\phi_{12,B_{1}}^{i}+3\sigma\end{matrix} \right.