## 4 Conclusions  
In this paper, a second-order, time-difference geometry-free (STG) cycle slip detection and repair approach was proposed for dual-frequency GPS carrier phase observations. Based on a reformulation of [PERSON]'s equation of motion, the second-order time difference of the LEO satellite's position (STP) is only related to the satellite's acceleration and the sampling interval. Then, with the GPS/LEO satellite acceleration derived from known orbit dynamic models, the geometric variations in GPS carrier phase observations can be removed. Several kinds of STG combinations were presented, one of which was ionospheric-free, while some were more sensitive to cycle slips than STPIR. Moreover, one STG combination can be used to detect cycle slips on single-frequency carrier phase observations, if STPIR can be practically ignored. The combination of STG linear combinations can be used to uniquely determine cycle slips on both GPS \(L_{1}\) and \(L_{2}\) frequencies. To investigate the performance of the proposed method, extensive experiments were carried out with GPS datasets from the GOCE satellite, with simulated cycle slip pairs. The results indicated that the proposed method can detect and repair cycle slips with a high success rate, even at low elevation angles, especially for LEO GPS datasets with a sampling rate of 1 Hz. Moreover, because one of the STG combinations, Equation (10), was ionospheric-free, the ionospheric-free carrier phase combination can be used as an observation for LEO satellite orbit determination. The undetected special cycle slip pairs for the STG approach will have a slight influence on LEO satellite orbit determination. This is, thus, an effective method for cycle slip detection and repair for GPS carrier phase observation. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**GPS Satellite** & **Interval(s)** & **Detection** & **Repair** \\ \hline \multirow{3}{*}{PRN 9} & 1 & 100 & 100 \\  & 5 & 100 & 99.9 \\  & 10 & 99.8 & 98.5 \\ \hline \multirow{3}{*}{PRN 22} & 1 & 100 & 100 \\  & 5 & 100 & 99.4 \\ \cline{1-1}  & 10 & 99.7 & 97.6 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Correct instances of detection and repair of cycle slips (in percentage) using STG methods for the GOCE GPS dataset above with sampling rates of 1-s, 5-s, and 10-s intervals for PRN 9 and PRN 22. Unlike the traditional TurboEdit method, the proposed method is almost independent of the pseudorange. Benefiting from the high-stability atomic clock equipped on the GPS satellites and the known high-precision satellite orbit dynamic models, the geometric variations between the satellites can be derived precisely and ultimately removed, even with GPS broadcast products. Thus, the STG approach can also be applied in real-time application. Moreover, similar to previous work [39; 40], the performance of the proposed method is expected to be further tested for multi-frequency GNSS observations in the future. Furthermore, as stated, only the GPS receiver's acceleration information is necessary, the proposed method is suitable for ground-based receivers whose acceleration information is known, such as the static IGS station, and kinematic receivers equipped with an inertial navigation system (INS). What is more, as [PERSON] et al. [11] mentioned that STPIR can be practically ignored for ground GPS data, Equation (13) can be used alone to detect and repair cycle slips for single-frequency ground-based GPS receivers, when the acceleration of the receivers is available, even with 30-s sampling intervals. Data curation, H.W. ; formal analysis, H.W. and J.L. ; funding acquisition, J.L., S.Z., and X.X. ; investigation, S.Z. ; methodology, H.W. ; software, H.W. ; writing--original draft, H.W. ; writing--review and editing, S.Z. and X.X.