## 2 Materials and Methods
### DOA Calculation  
The schematic diagram of the elevation angle \(\alpha\) and azimuth angle \(\beta\) of the satellite \(S\) in the \(ENU\) coordinate system is shown in Figure 4.  
Figure 3: Schematic diagram of algorithm process. The following will take the \(A\) and \(B\) antennas as an example to illustrate the process of solving \(\alpha\) and \(\beta\). In the \(ENU\) coordinate system, the unit vector of the satellite signal direction can be expressed as:  
\[\hat{d}_{em}=[\cos\alpha\sin\beta,\cos\alpha\cos\beta,\sin\alpha]^{T} \tag{1}\]  
The precise relative coordinates of the four antennas in \(ENU\) can be calculated by RTK (Real Time Kinematic). Take antenna \(A\) as a reference and set it as the origin of \(ENU\). Then we can solve the relative coordinates between the remaining three antennas and antenna \(A\). The coordinates of \(A\) and \(B\) antennas in \(ENU\) are prior values, set as:  
\[\Delta X_{AB}=[\Delta x_{AB},\Delta y_{AB},\Delta z_{AB}] \tag{2}\]  
The geometric diagram of carrier single difference between antennas \(A\) and \(B\) is shown in Figure 5.  
Figure 4: Schematic diagram of DOA in \(ENU\) coordinate system. \(A\) and \(B\) respectively represent two GNSS receiving antennas. Figure 5: Geometrical diagram of carrier single difference between receivers. The carrier difference of the two real signals is expressed as \(\Delta\varphi_{AB}\). The \(\Delta\varphi_{AB}\) is the projection of \(\overset{\rightharpoonup}{AB}\) on the direction vector of the signal between the \(A\) antenna and the satellite, which can be expressed as:  
\[ \begin{array}{ll}\Delta\varphi_{AB}\lambda&=\Delta X_{AB}\cdot\hat{d}_{em}\\ &=\cos\alpha\sin\beta\Delta x_{AB}+\cos\alpha\cos\beta\Delta y_{AB}+\sin\alpha \Delta z_{AB}\end{array}  \tag{3}\]  
where \(\alpha\in[0,\pi/2]\), \(\beta\in[0,2\pi]\), \(\lambda\) is the carrier wavelength. When the coordinates of the baseline are known, the carrier single difference is completely determined by the elevation and azimuth angles. If the antenna C is considered, the relationship can be obtained as follows:  
\[\left\{ \begin{array}{ll}\Delta\varphi_{AB}\lambda=cos\alpha\sin\beta\Delta x _{AB}+cos\alpha\cos\beta\Delta y_{AB}+sin\alpha\Delta z_{AB}\\ \Delta\varphi_{AC}\lambda=cos\alpha\sin\beta\Delta x_{AC}+cos\alpha\cos\beta \Delta y_{AC}+sin\alpha\Delta z_{AC}\end{array} \right. \tag{4}\]  
It can be seen from Equation (4) that if \(\Delta\varphi_{AB}\), \(\Delta\varphi_{AC}\), \(\Delta X_{AB}\) and \(\Delta X_{AC}\) can be obtained, \(\alpha\) and \(\beta\) can be solved. The relative coordinates between the antennas can be accurately calculated by RTK, and the solution of the carrier single difference will be deduced in detail in the next section. However, the equation similar to Equation (4) is difficult to be solved. Focusing on this problem, this paper proposes a double rotation method to solve it by eliminating some items containing unknowns through graphic rotation. The process of solving the equation is simplified. The schematic diagram before and after rotation in \(ENU\) coordinate system is shown in Figure 6.  
In Figure 6, \(\bigtriangleup AB_{1}C_{1}\) can be obtained through the rotation matrix \(R_{1}(r,p,y)\). Since the coordinates of \(A\), \(B\), and \(C\) are known, \(R_{1}(r,p,y)\) can be solved easily. The coordinate relationship before and after rotation is as follows:  
\[\left\{ \begin{array}{ll}\Delta X_{AB_{1}}=R_{1}(r,p,y)(X_{AB}{}^{T})\\ \Delta X_{AC_{1}}=R_{1}(r,p,y)(X_{AC})^{T}\end{array} \right. \tag{5}\]  
The purpose of the rotation is to make \(\overset{\rightharpoonup}{AB}\) coincide with the \(E\) axis and \(\bigtriangleup AB_{1}C_{1}\) to be in the same plane as the plane \(\circ EAN\). Then \(\Delta X_{AB_{1}}\) and \(\Delta X_{AC_{1}}\) can be expressed as:  
\[\left\{ \begin{array}{ll}\Delta X_{AB_{1}}=[\Delta x_{AB_{1}},0,0]^{T}\\ \Delta X_{AC_{1}}=[\Delta x_{AC_{1}},\Delta y_{AC_{1}},0]^{T}\end{array} \right. \tag{6}\]  
Figure 6: Three-dimensional schematic diagram of \(\bigtriangleup ABC\) before and after rotation.