## 2 Methods
### Function Model  
The functional model relative positioning establishes the mathematical relationship between the unknown parameters and the double-difference observations. Developing an accurate functional model requires a comprehensive understanding of the various physical phenomena occurring during signal transmission from GNSS satellites and on the receiver's side. GPS and BDS navigation systems utilize code division multiple access (CDMA) to distinguish satellites. The original observation equation for the carrier and pseudo-rangeof the observation data corresponding to any satellite's ephemeris can be expressed [34] (unit: meter):  
\[\left\{ \begin{aligned} P_{k,i}^{j}=\rho_{k,i}^{j}+I_{k,i}^{j}+T_{k,i}^{ j}+M_{k,ip}^{j}+c\big{(}\varepsilon_{t}-\varepsilon t^{j}\big{)}+\varepsilon_{p}^{j} \\ \lambda_{i}\varphi_{k,i}^{j}=\rho_{k,i}^{j}+\lambda_{i}N_{k,i}^{ j}-I_{k,i}^{j}+T_{k,i}^{j}+M_{k,i\varphi}^{j}+c\big{(}\varepsilon_{t}- \varepsilon^{j}\big{)}+\varepsilon_{\varphi}^{j}\end{aligned} \right. \tag{5}\]  
where \(k\) is the ground receiver number, \(j\) is the satellite number, \(i\) is the frequency of the observed data, \(P\) is the pseudo-distance observation, \(\rho\) is the distance from the satellite to the receiver, \(I\) is ionospheric delay, \(T\) is the tropospheric delay, \(M\) is the multipath error, \(c\) is the speed of light, \(\varepsilon t^{j}\) is the clock difference of the receiver \(i\), \(\varepsilon t_{i}\) is the satellite clock difference, \(\varepsilon_{p}\) is the chance error of the pseudo-distance observation, \(\varepsilon_{\varphi}\) is the chance error of the carrier observation, \(\lambda\) is the corresponding carrier wavelength, \(\varphi\) is the carrier phase observation, and \(N\) is the nondifferential raw ambiguity of the corresponding satellite for this receiver. After processing the original observation equation for inter-station and inter-star differences, we obtained the following:  
\[\left\{ \begin{aligned} \Delta\
abla P_{kr,i}^{lh}=\Delta\
abla\rho_{kr,i }^{lh}+\Delta\
abla I_{kr,i}^{lh}+\Delta\
abla T_{kr,i}^{lh}+\Delta\
abla \varepsilon_{kr,ip}^{lh}\\ \lambda_{i}\Delta\
abla\varphi_{kr,i}^{lh}=\Delta\
abla\rho_{kr,i }^{lh}+\lambda_{i}\Delta\
abla N_{kr,i}^{lh}-\Delta\
abla I_{kr,i}^{lh}+ \Delta\
abla T_{kr,i}^{lh}+\Delta\
abla\varepsilon_{kr,ip}^{j\text{th}}\end{aligned} \right. \tag{6}\]  
where \(\Delta\
abla\) is the double difference operator, \(h\) is the reference satellite number, and \(r\) is the reference station receiver number. Inter-station differencing is effective in mitigating satellite orbit errors and satellite clock differences. On the other hand, inter-satellite differencing helps eliminate receiver-side errors and partially attenuates the effects of ionospheric and tropospheric delays. When the baseline between stations is short, double-difference processing can significantly reduce the impact of atmospheric delay. However, for longer baselines, residual errors are typically compensated for by employing empirical models or introducing additional parameters. Through double-difference processing, common error terms in relative positioning are eliminated or mitigated, making this method a preferred choice for high-precision positioning purposes. In relative positioning, the coordinates of the reference point (assumed to be A) are generally precisely determined. Thus, the vector of unknown parameters contains mainly the coordinates of the flow station (assumed to be B) and the double-difference ambiguity:  
\[x=\Big{(}X_{B},Y_{B},Z_{B},N_{AB,f}^{i}\Big{)}^{T},i=1,\cdots,m_{sp} \tag{7}\]  
where \(m_{sp}\) refers to the number of double-differential satellite pairs (\(m_{sp}\) should be \(\geq\)4 in relative positioning).