## 2 Methods
### AMG Combined Cycle Slip Detection Model  
Most cycle slips processing methods are based on the linear combinations of pseudorange and the phase observation. The severe multipath in complex environments seriously affects the pseudorange accuracy, resulting in significant deviations in the MW combination cycle slips' estimation. In order to effectively utilize the pseudorange phase to detect and repair cycle slips, an AMG combined cycle slips detection method is proposed in this paper. First, an autoregressive (AR) model prediction method was used to predict pseudorange observations in a multipath environment. Then, an MW + GF combined cycle slips detection model was used to correct the cycle slips' estimation with the predicted pseudorange observations. Compared with the traditional MW method, the AMG method can effectively solve the cycle slips' detection problem in severe multipath conditions, and false detection rarely occurs. An AR model is a kind of widely used time series model. The prediction result at time \(t\) can be represented by a linear combination of the observations at time \(y\) before or after it, and its expression is as follows [32]:  
\[z_{t}=\sum_{i=1}^{p}\theta_{i}z_{t-i}+\varepsilon_{t} \tag{10}\]  
where \(z_{t}\) represents the predicted value, \(\theta_{i}\) represents the autoregressive coefficient, \(y\) represents the model order, and \(\varepsilon_{t}\) represents the system noise. The y-order AR model of the pseudorange \(P\) is established based on the sequential characteristic. In order to improve the performance of cycle slips' detection in a multipath environment, more accurate pseudorange observations are required; that is, the difference between the actual pseudorange observations and the predicted values is required to be as small as possible. Therefore, a sliding window is constructed based on the pseudorange observations without multipath contamination to predict the pseudorange observations in the multipath environment:  
\[ \begin{array}{l}p[1]=a_{1}p[0]+\varepsilon[n]\\ p[2]=a_{1}p[1]+a_{2}p[0]+\varepsilon[n]\\ \vdots\\ p[n]=a_{1}\ p[n-1]+a_{2}\ p[n-2]+\ldots+a_{y}\ p[n-y]+\varepsilon[n]\end{array}  \tag{11}\]  
Assuming that the different pseudoranges are of an equal accuracy, the accuracy \(\sigma_{z}\) predicted by the AR model can be inferred based on the error propagation law [33]:  
\[\sigma_{z}=\sqrt{{a_{1}}^{2}+{a_{2}}^{2}\cdots{a_{n}}^{2}}\cdot{\sigma_{p}}^{2} \tag{12}\]  
The order of the AR model has a great influence on the accuracy of its prediction. So the least squares and the final prediction error (FPE) criterion (\(FPE(n)=\frac{N+n}{N-n}\sigma_{a}^{2}\)), the information (AIC) criterion (\(FPE(n)=\frac{N+n}{N-n}\sigma_{a}^{2}\)), and the BIC criterion (\(FPE(n)=\frac{N+n}{N-n}\sigma_{a}^{2}\)) are used to jointly test the applicability in this paper. In order to save space, the results of the joint test are directly given, that is, the optimal model is taken when the model order is \(y=3\) by the joint test. Based on Equation (12), the MW combined observation equation assisted by the predicted value of the AR model can be obtained:  
\[{L^{\prime}}_{MW}=\frac{1}{f_{1}-f_{5}}(f_{1}L_{1}-f_{5}L_{5})-\frac{1}{f_{1}+f_ {5}}(f_{1}z_{t(P_{1})}+f_{5}z_{t(P_{5})}) \tag{13}\]  
By differencing the combined equations of two adjacent epochs, we can obtain:  
\[\Delta{N^{\prime}}_{MW}=\frac{L_{MW}^{\prime}(t+1)}{\lambda_{MW}}-\frac{L_{MW} ^{\prime}(t)}{\lambda_{MW}} \tag{14}\]  
Therefore, the standard deviation of the AR-assisted MW combination cycle slips' estimation is:  
\[\sigma_{\Delta{N_{MW}}}=\sqrt{(\alpha_{WL}\sigma_{\varphi})^{2}+(\beta_{WL} \sigma_{\varphi})^{2}+(\frac{\alpha_{NL}\sigma_{z_{p}}}{\lambda_{mw}})^{2}+( \frac{\beta_{NL}\sigma_{z_{p}}}{\lambda_{mw}})^{2}} \tag{15}\]  
where \(\sigma_{z}\) is the prediction accuracy of the AR model. Compared with the traditional MW combination observations, the MW observation values assisted by the AR model are not affected by the pseudorange multipath and pseudorange noise, but only by the prediction accuracy.