## 2 Methods
### Cycle Slips Detection Model of Melbourne-Wubbena Combination  
The MW combination is a linear model obtained by subtracting the narrow-lane combination of pseudorange from the wide-lane combination of the carrier phase, which can eliminate the influence of the geometric distance, atmosphere, and clock difference on the detection of cycle slips. The MW combination can be expressed as [29]:  
\[L_{MW}=\frac{1}{f_{1}-f_{5}}(f_{1}L_{1}-f_{5}L_{5})-\frac{1}{f_{1}+f_{5}}(f_{1 }P_{1}+f_{5}P_{5}) \tag{4}\]  
where \(L_{MW}\) are the MW combination observations and \(P_{1}\), \(P_{5}\) are the pseudorange observations of B1c and B2a, respectively. The rest of the symbols are the same as formula (1). The wide-lane wavelength is \(\lambda_{MW}=c/(f_{1}-f_{5})\). \(L_{MW}\) can eliminate the influence of factors such as the inter-station distance, atmospheric delay error, satellite clock error, etc., and simply retain the influence caused by the multipath effects and observation noise, which can be weakened or eliminated by making differences between epochs. Its expression is:  
\[\Delta N_{MW}=\frac{L_{MW}(t+1)}{\lambda_{MW}}-\frac{L_{MW}(t)}{\lambda_{MW}} \tag{5}\]  
where \(\Delta N_{MW}\) is the detection value of the cycle slips of the MW combination after the difference between epochs, that is, the cycle slips' estimation.  
Equation (4) can be simplified as \(L_{MW}=L_{WL}-P_{NL}\), where \(L_{WL}\) is the wide-lane phase combination (WL) and \(P_{NL}\) is the narrow-lane pseudorange combination (NL),  
Therefore, \(L_{WL}\), the carrier phase wide-lane phase combination of BDS B1c and B2a, can be expressed as:  
\[L_{WL}=\alpha_{WL}L_{1}+\beta_{WL}L_{5} \tag{6}\]  
Its combination coefficient can be described as: \(\alpha_{WL}=f_{1}/(f_{1}-f_{5})\) and \(\beta_{WL}=-f_{5}/(f_{1}-f_{5})\)[30].  
Similarly, the narrow-lane combination \(P_{NL}\) code combination is given as:  
\[P_{NL}=\alpha_{NL}P_{1}+\beta_{NL}P_{5} \tag{7}\]  
The combination coefficient in Equation (7) is: \(\alpha_{NL}=f_{1}/(f_{1}+f_{5})\) and \(\beta_{NL}=f_{5}/(f_{1}+f_{5})\).  
It is assumed that the carrier phase observations and the pseudorange observations are independent of each other and have the same accuracy. Then, the standard deviation of the MW combination cycle slips' estimation can be approximately expressed as:  
\[\sigma_{\Delta N_{MW}}=\sqrt{\left(\alpha_{WL}\sigma_{\varphi}\right)^{2}+ \left(\beta_{WL}\sigma_{\varphi}\right)^{2}+\left(\frac{\alpha_{NL}\sigma_{ \varphi}}{\lambda_{mw}}\right)^{2}+\left(\frac{\beta_{NL}\sigma_{\varphi}}{ \lambda_{mw}}\right)^{2}} \tag{8}\]  
where \(\sigma_{\varphi}\) is the precision of the carrier phase observations, which is 0.01 cycles in this paper, and \(\sigma_{\varphi}\) is the precision of the pseudorange observations, which is 0.3 m [31] in this paper.  
Under normal circumstances, the influence of the multipath on the pseudorange observations is much greater than that of the carrier phase observations. If the GF combination does not adopt the pseudorange observations, it will not alarm when the multipath conditions occur. When the multipath effect is quite serious, the multipath residual between epochs should be regarded as white noise, then the standard deviation equation of the MW combination cycle slips' estimation should be rewritten as:  
\[\sigma_{\Delta N_{MW},M}=\sqrt{(\alpha_{WL}\sigma_{\varphi})^{2}+(\beta_{WL} \sigma_{\varphi})^{2}+(\frac{\alpha_{NL}\sigma_{p}}{\lambda_{mm}})^{2}+(\frac{ \beta_{NL}\sigma_{p}}{\lambda_{mm}})^{2}+(\frac{{M_{1}}^{2}+{M_{2}}^{2}}{ \lambda_{mm}})} \tag{9}\]