## 2 Materials and Methods
### Carrier Single Difference Solution  
As can be seen from the previous section, the carrier single difference value is the premise of solving the DOAs. This section describes the solution method of the carrier single difference. The carrier phase observation equation obtained by low-cost commercial receiver measurement is as follows [31]:  
\[\widetilde{\varphi}_{i}^{s}\lambda=\rho_{i}^{s}+c(dt_{i}-dt^{s})+d_{trop,i}^{s} -d_{ion,i}^{s}+\lambda N_{i}^{s}+\lambda b^{s}+\lambda b_{i}+d_{mul,i}+d_{ apc,i} \tag{11}\]  
where \(i\) and \(s\) denote the index of the station and satellite, respectively. \(\widetilde{\varphi}_{i}^{s}\) is the actual observed value of the carrier phase measurement of the receiver in cycles. \(\rho_{i}^{s}\) is the distance between the satellite \(s\) and the station \(i\), c is the speed of light, \(dt_{i}\) is the receiver clock offset, \(d^{s}\) is the satellite clock offset, \(d_{trop,i}^{s}\) is the troposphere delay error and \(d_{ion,i}^{s}\) is the ionosphere delay error. \(N_{i}^{s}\) is the integer ambiguity, \(b^{s}\) is the instrumental biases of the satellite, \(b_{i}\) is the cable delay between the receiver and the antenna, \(d_{mul,i}\) is the equivalent distance error caused by the multipath effect, and \(d_{apc,i}\) is the equivalent distance error caused by the antenna phase center error. When two stations \(i\) and \(j\) observe a satellite \(s\) at the same time, the inter-station carrier single difference model can be established:  
\[ \begin{array}{l}\Delta\widetilde{\varphi}_{ij}^{s}=\widetilde{\varphi}_{i}^ {s}-\widetilde{\varphi}_{j}^{s}\\ =\Delta\rho_{ij}^{s}+c\Delta dt_{ij}+\lambda\Delta N_{ij}^{s}+\lambda\Delta b _{ij}+\Delta d_{mul,j}+\Delta d_{apc,ij}\end{array}  \tag{12}\]  
where \(d_{ion,i}^{s}\) and \(d_{trop,i}^{s}\) are eliminated due to the shorter baseline distance. The inter-station carrier single difference eliminates the error terms \(dt^{s}\) and \(\lambda b^{s}\). The following equation can be further obtained:  
\[\Delta\widetilde{\varphi}_{ij}^{s}-\lambda\Delta N_{ij}^{s}-c\Delta dt_{ij}- \lambda\Delta b_{ij}-\Delta d_{mul,ij}-\Delta d_{apc,ij}=\Delta\rho_{ij}^{s} \tag{13}\]  
\(\Delta\rho_{ij}^{s}\) is consistent with the meaning of \(\Delta\varphi_{AB}\lambda\) in Equation (3), which is the real distance difference between the satellite \(s\) and the station \(i\),\(j\). After solving \(\Delta\rho_{ij}^{s}\), the signal DOAs can be solved. \(\Delta\widetilde{\varphi}_{ij}^{s}\) is given by the receiver and is a known value. In order to facilitate the subsequent solution, the three errors in Equation (13) are combined and expressed as follows:  
\[\eta=\lambda\Delta b_{ij}+\Delta d_{mul,ij}+\Delta d_{apc,ij} \tag{14}\]  
\(\Delta N_{ij}^{s}\), \(\Delta dt_{ij}\) and \(\eta\) are three unknowns. First, consider the solution of \(\Delta N_{ij}^{s}\). At present, the integer ambiguity resolution method is mainly based on the search algorithm of the search domain. The LAMBDA algorithm is a widely used method at present, but its solution requires multiple epochs to converge. There are also methods that use baseline information to solve ambiguities, such as the long-short baseline method. This method requires that the distance between two antennas is less than half a wavelength. The half wavelength of GPS L1 is 0.0952m. Because of the manufacturing process of the receiving antenna, it is difficult for the distance to be less than half the signal wavelength. Even if the carrier single difference ambiguity problem is solved successfully, low-priced receiver boards may still face the problem of frequent cycle slips, making data processing more difficult. Based on the consideration of the above problems, this paper innovatively proposes a method of calculating the integer and decimals of the carrier single difference value separately. The carrier single difference is modeled as the addition of the integer part and the fractional part.