## 3 Proposed Methods
### Multi-Epoch Method for Determining the Two-Dimensional Directions of Spoofing Signal Source
\begin{bmatrix}\delta\alpha_{s,t}^{ b_{0}}\\ \delta\beta_{s,t}^{b_{0}}\\ \delta\beta_{s,t}^{b_{0}}\end{bmatrix}  \tag{18}\]  
where \(r_{s,t}^{b_{0},0}\) is a unit vector for \(r_{s,t}^{b_{0}}\); \(\delta\alpha_{s,t}^{b_{0}}\) and \(\delta\beta_{s,t}^{b_{0}}\) are the yaw and pitch angle errors, respectively. By establishing a least-squares observation equation set, then Equation (7) can be transformed as follows:  
\[y_{t}=\mathbf{A}_{t}\cdot\delta\mathbf{X}+\varepsilon_{t} \tag{19}\]  
in which,  
\[\mathbf{A}_{t,1\times 2}=-\left(\mathbf{L}^{\mathrm{b}}\right)^{\mathrm{T}}\cdot\mathbf{C} _{\mathrm{b}_{0}}^{\mathrm{b}_{t}}\cdot \begin{bmatrix}-\cos\beta_{s,t}^{b_{0}} \sin\alpha_{s,t}^{b_{0}},&-\cos\alpha_{s,t}^{b_{0}}\sin\beta_{s,t}^{b_{0}}\\ \cos\beta_{s,t}^{b_{0}}\cos\alpha_{s,t}^{b_{0}},&-\sin\alpha_{s,t}^{b_{0}} \sin\beta_{s,t}^{b_{0}}\\ 0,&\cos\beta_{s,t}^{b_{0}}\end{bmatrix}  \tag{20}\]  
\[y_{t}=\lambda\cdot\left(\Delta\tilde{\varphi}_{12,t}^{\mathrm{i}}+N_{12,t}^{ \mathrm{i}}\right)- \begin{bmatrix}L\\ 0\\ 0\end{bmatrix} \cdot\mathbf{C}_{\mathrm{e}}^{\mathrm{b}_{t}}\cdot\mathbf{C}_{\mathrm{b }_{0}}^{\mathrm{e}_{t}}\cdot \begin{bmatrix}\cos\beta_{s,t}^{b_{0}}\cos\alpha_ {s,t}^{b_{0}}\\ \cos\beta_{s,t}^{b_{0}}\sin\alpha_{s,t}^{b_{0}}\\ \sin\beta_{s,t}^{b_{0}}\end{bmatrix}  \tag{21}\]  
\[\delta\mathbf{X}= \begin{bmatrix}\delta\alpha_{s}^{b_{0}}\\ \delta\beta_{s}^{b_{0}}\end{bmatrix}  \tag{22}\]  
The two-dimensional direction of the spoofing signal source is obtained after least-squares compensation as follows:  
\[ \begin{bmatrix}\alpha_{s}^{b_{0}}\\ \beta_{s}^{b_{0}}\end{bmatrix} = \begin{bmatrix}\alpha_{s,0}^{b_{0}}\\ \beta_{s,0}^{b_{0}}\end{bmatrix} - \begin{bmatrix}\delta\alpha_{s}^{b_{0}}\\ \delta\beta_{s}^{b_{0}}\end{bmatrix}  \tag{23}\]  
To obtain the real-time source position information, we construct an EKF model for the carrier-phase measurement information that is updated in real-time. We define the state parameters as the yaw angle (\(\alpha_{s}^{b_{0}}\)) and pitch angle (\(\beta_{s}^{b_{0}}\)) of the two-dimensional direction vector of the signal source in the \(\mathbf{b}_{0}\) system, and denote them in the following equation:  
\[\mathbf{X}= \begin{bmatrix}\alpha_{s}^{b_{0}}\\ \beta_{s}^{b_{0}}\end{bmatrix}  \tag{24}\]  
Given that the signal source is static, then  
\[ \begin{cases}\mathbf{X}_{t+1/t}=\mathbf{X}_{t/t}\\ \delta\mathbf{X}_{t+1/t}=\delta\mathbf{X}_{t/t}\\ \sum_{t+1/t}=\sum_{t/t}\end{cases}  \tag{25}\]  
The measurement equation at epoch (t+1) can be established as  
\[y_{t+1}=\mathbf{A}_{t+1}\cdot\delta\mathbf{X}_{t+1/t+1}+\varepsilon_{t+1} \tag{26}\]  
The error states and their variances can be estimated for epoch (t+1):  
\[ \begin{cases}\delta\mathbf{X}_{t+1/t+1}=\delta\mathbf{X}_{t/t}+K(y_{t+1}-A_{t+1}\delta \mathbf{X}_{t/t})\\ \sum_{t+1/t+1}=\left(I-KA_{t+1}\right)\sum_{t+1/t+1}(I-KA_{t+1})^{T}+\sigma_{ \Delta\phi}^{2}KK^{T}\end{cases}  \tag{27}\]\[K=\sum_{d/t}A_{t+1}\Big{(}A_{t+1}\sum_{d/t}A_{t+1}\mathrm{T}+\sigma_{\Delta\phi}^{2} \Big{)}^{-1} \tag{28}\]  
To reduce the linearisation error of the measurement equation, we calculate the initial state value and the estimated error state for epoch (t+1) as follows:  
\[ \begin{cases}\mathbf{X}_{t+1/t+1}=\mathbf{X}_{t/t}-\delta\mathbf{X}_{t+1/t+1}\\ \delta\mathbf{X}_{t+1/t+1}=\mathbf{0}\end{cases}  \tag{29}\]