## 2 Quality Control Algorithms for GPS Observations
### Cycle Slip Effects  
Cycle slip is an error resulting from the instantaneous loss of carrier-phases in GPS observations. Cycle slip is associated with a failure of the carrier-phase tracking loop either because the signal is blocked physically or because the signal is weak [9]. The detection of cycle slip is important for precise geodetic positioning using carrier-phase data. Geodetic quality receivers usually flag the user when a cycle slip has occurred. To repair the cycle slip in GPS observations, the exact characteristics of cycle slips need to be known.  
A cycle slip can be detected if double and triple differences are formed using more than two points in GPS carrier-phase observations, which are observed simultaneously. In the case of a single GPS observation, however, a different methods for cycle slip detection based on undifferenced carrier-phase observations should be applied. In this study, cycle slip detection is performed using the TurboEdit algorithm developed by [PERSON] [8]. TurboEdit is an algorithm for cycle slip identification and repair as well as outlier removal using undifferenced, dual-frequency GPS observation at single point. In this algorithm, the detection of cycle slip is based on Melbourne-Wubenna (M-W) linear combination [20] and a geometry-free combination.  
The M-W linear combination has been widely applied to cycle slip detection in undifferenced observation and double differences. A major advantage of M-W combination is that it is not only geometry-free but also ionospheric-free. Therefore, it can be used even if the GPS receiver undergoes high levels of dynamics and/or ionospheric variation. For ionospheric combinations, the M-W combination was used in the TurboEdit algorithm to detect and repair cycle slip. The M-W combination uses pseudo-range observations that have greater noise than carrier-phase observations. Under some observation conditions, the pseudo-range noise may be much larger than usual, for example, in the presence of multi-path, increased ionospheric delay, and low _DOP_. In such circumstances, the recursive averaging algorithm used in TurboEdit may fail to detect small cycle slips of 1-2 cycles [20].  
For observations from a dual-frequency GPS receiver, the M-W combination can be expressed as follows,  
\[L_{MW}=\frac{f_{1}\cdot\lambda_{1}\Phi_{1}-f_{2}\cdot\lambda_{2}\Phi_{2}}{f_{ 1}-f_{2}}-\frac{f_{1}\cdot P_{1}+f_{2}\cdot P_{2}}{f_{1}+f_{2}}=\lambda_{W_{2 }L}N_{W_{2}L} \tag{16}\]  
where \(\lambda_{MW}=c/(f_{1}-f_{2})\simeq 0.86\) m and \(N_{MW}=N_{1}-N_{2}\) are the wide-lane wavelength and wide-lane ambiguity, respectively.  
The wide-lane ambiguity can be easily obtained from Equation (16) as follows,  
\[N_{WL}=\frac{L_{MW}}{\lambda_{WL}}=\Phi_{1}-\Phi_{2}-\frac{f_{1}\cdot P_{1}+f _{2}\cdot P_{2}}{\lambda_{WL}(f_{1}+f_{2})} \tag{17}\]As long as the phase observations are free of cycle slips, the wide-lane ambiguity remains quite stable over time. In utilizing the M-W combination to detect cycle slips, a recursive averaging filter is used in TurboEdit as follows,  
\[\overline{N}_{\mu_{L}}(k)=\overline{N}_{\mu_{L}}(k-1)+\frac{1}{k}[N_{\mu_{L}}(k )-\overline{N}_{\mu_{L}}(k-1)] \tag{18}\]  
\[\sigma^{2}(k)=\sigma^{2}(k-1)+\frac{1}{k}[(N_{\mu_{L}}(k)-\overline{N}_{\mu_{ L}}(k-1))^{2}-\sigma^{2}(k-1)] \tag{19}\]  
where \(\overline{N}_{\mu_{L}}\) is the mean value of \(N_{\mu_{L}}\), \(k\) and \(k-1\) represent the current and previous observation epochs, respectively. The standard deviation of \(\overline{N}_{\mu_{L}}\) at epoch (\(k\)) is \(\sigma(k)\). When the following conditions are satisfied, it is considered that a cycle slip (\(cyc\)) occurs,  
\[\left|N_{\mu_{L}}(k)-\overline{N}_{\mu_{L}}(k-1)\right|\geq 4\sigma(k) \tag{20}\]  
and  
\[\left|N_{\mu_{L}}(k+1)-N_{\mu_{L}}(k)\right|\leq 1 \tag{21}\]  
where \(\sigma(k)\) was considered as configuration parameter and set at 0.5 cycles as a default value.