## 3 Proposed Methods
### Multi-Epoch Method for Determining the Two-Dimensional Directions of Spoofing Signal Source  
After determining spoofed signals (Section 3.2), we track the same signal source in multiple epochs. First, an appropriate computational coordinate system is selected. Then, the dynamic multi-epoch inter-station single-difference carrier-phase measurements after integer ambiguity compensation and MIMU observation information are used to perform multi-epoch least-squares operation for the target spoofing signal source to obtain the initial position of the spoofing signal source. Finally, the two-dimensional direction information of the spoofing signal source is obtained through filtering in real-time. To determine the two-dimensional direction of the signal source, it is first necessary to select a suitable observation coordinate system. Because the direction of the signal source changes in real-time under the real-time body coordinate system for the carrier, the model constructed is characterised by high complexity. Additionally, geoid systems, especially CSCG2000, have larger value representations for positions, which is not beneficial to computation. Therefore, we choose the body coordinate system \(\mathbf{b}_{0}\) created for the epoch where the signal source points approximately to the \(X\)-axis in the motion as the observation coordinate system. In this coordinate system, the direction of the carrier-pointing signal source in this epoch is the value to be solved by the model. As shown in Figure 4, if the carrier has a hovering motion on a horizontal plane, then \(\rho_{s,i}^{b_{i}}\) is approximately stable, and the value of \(\theta_{s,i}^{b_{i}}\) is directly affected by \(a_{s,i}^{b_{i}}\). From the geometric relationship, we can obtain  
\[\cos\theta_{s,i}^{b_{i}}=\cos\beta_{s,i}^{b_{i}}\cos\alpha_{s,i}^{b_{i}} \tag{16}\]  
As shown in Equation (16), when the carrier phase reaches its maximum value, \(a_{s,i}^{b_{0}}\) approximates to 0, and \(\beta_{s,i}^{b_{i}}\approx\theta_{s,i}^{b_{i}}\). This moment is recorded as Moment 0, at which the body coordinate system is the \(b_{0}\) system. For a brief period, the body system error in the process of MIMU-assisted rotation does not significantly vary with time. The value of the signal source \(\mathbf{r}_{\text{s,t}}\) in the \(\mathbf{b}\) system at different moments can be expressed as \(\mathbf{r}_{\text{s,t}}^{b}\). The distance from the carrier to the signal source is much larger than the change in the position of the signal source, i.e., \(r_{\text{s,t}}^{b_{0},0}-\mathbf{r}_{\text{s,0}}^{b_{0},0}\approx 0\). As the MIMU can obtain the rotation matrix from the \(\mathbf{b}_{i}\) system (i.e., the body coordinate system for the current epoch) to the \(e\) system, we use the \(e\) system for the transfer. \(\mathbf{r}_{\text{s,i}}^{b}\) and \(\mathbf{r}_{\text{s,i}}^{b_{0}}\) are defined as the representations of the vector \(\mathbf{r}_{s}\) from the carrier to the signal source in the \(\mathbf{b}_{i}\) and \(\mathbf{b}_{0}\) systems, respectively; \(\mathcal{C}_{\text{e}}^{b_{0}}\) is the rotation matrix from the \(e\) system to the \(\mathbf{b}_{0}\) system; \(\mathcal{C}_{\text{b}_{i}}^{e}\) is the rotation matrix from the \(\mathbf{b}_{i}\) system to the \(e\) system. \[r_{\text{s,t}}^{b_{i}}=\mathcal{C}_{\text{e}}^{b_{i}}\cdot\mathcal{C}_{\text{ b}_{0}}^{e}\cdot\mathbf{r}_{\text{s,t}}^{b_{0}} \tag{17}\]A Taylor expansion is performed for Equation (17) at \( \begin{cases}\alpha_{s,0}^{b_{0}}=0\\ \beta_{s,0}^{b_{0}}=\theta\end{cases} \) to obtain  
\[r_{s,t}^{b_{0},0}\approx \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} - \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} \cdot