## 3 Results and Discussion
### Using the Proposed STG Method to Detect and Repair Cycle Slips  
In order to investigate the performance of the proposed STG cycle slip detection and repair method, the GPS observation data of GOCE satellite on 21 November 2009 were collected with a sampling interval of 1 s. Figure 6 shows GOCE satellite latitude, elevation, and STPIR (1 s) for GPS PRN 9 and PRN 22 (max elevation 2.1\({}^{\circ}\)). Due to the lower elevation, the quality of the carrier phase observations corresponding to PRN 22 is poor. Firstly, simulated special CSPs were added to the GPS \(L_{1}\) and \(L_{2}\) carrier phase data every 5 min, from 5:20 to 5:45 p.m. (GPS time) for PRN 9, and from 5:35 to 5:45 p.m. (GPS time) for PRN 22, as shown in Table 4.  
Figures 7 and 8 show the cycle slip detection results for the testing data with special CSPs using the STG method, without the satellite-difference processing. For convenience of display, the detected cycle slip values larger than 1.5 cycles were set as 1.5 in the figures below. In each plot, time series of the combinations from Equations (10)-(13) are shown. The time series of the ionosphere-free STG combination in Equation (10) shown in Figures 7 and 8 are stable at zero cycles, which indicates that the geometric variations are removed clearly, and GOCE's \(\Delta\Delta dt_{r}\) can be practically ignored, similar to the GPS satellites, due to the high-stability atomic clocks equipped [3]. The STG combinations in Equations (11)-(13) are also stable at zero cycles, which illustrates that the ionospheric residuals \(\Delta\Delta I\) for GOCE are significantly small and negligible in our work. Moreover, as Equations (14) and (15) suggested, there are two opposite peak values in the adjacent epochs when the cycle slip is simulated. \begin{table}
\begin{tabular}{c c c} \hline \hline
**GPS Time** & **CSPs on PRN 9** & **CSPs on PRN 22** \\ \hline
5:20 p.m. & (5, 4) & – \\
5:25 p.m. & (3, 4) & – \\
5:30 p.m. & (3, 2) & – \\
5:35 p.m. & (1, 1) & (1, 1) \\
5:40 p.m. & (9, 7) & (3, 4) \\
5:45 p.m. & (4, 5) & (5, 4) \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Simulated special CSPs at GOCE satellite GPS observations. Figure 6: GOCE satellite latitude, elevation, and second-order time-difference PIR (STPIR) (1 s) for GPS PRN 9 and PRN 22 on 21 November 2009. As expected, as shown in Figures 7 and 8, each combination of Equations (10)-(13) has some insensitive CSPs. For instance, Equation (10) is insensitive to the CSPs (3, 4) and (4, 5), but they can be distinguished using the other combinations. Thus, simultaneous application of all four STG combinations is necessary to detect and repair all unknown CSPs. As Equation (13) suggests, it can be used to detect big cycle slips. Thus, only random small CSPs between \(-10\) and \(10\) added to the GPS \(L_{1}\) and \(L_{2}\) carrier phase data at GPS PRN 9 and PRN 22 at random epochs are tested in our experiments. The test is repeated over 1000 independent runs. Considering the fact that the raw LEO satellite GPS sampling interval is usually less than 10 s, such as for GOCE [34], SWARM [5], FY-3C [37], and Sentinel-3A [38], the max interval is set to 10 s.  
Figure 8: Cycle slip detection results for GPS PRN 22 using STG combinations in Equations (10)–(13), denoted by (**a**–**d**), respectively, with simulated cycle slip pairs at GOCE on 21 November 2009. Figure 7: Cycle slip detection results for GPS PRN 9 using STG combinations in Equations (10)–(13), denoted by (**a**–**d**), respectively, with simulated cycle slip pairs at GOCE on 21 November 2009. Table 5 shows the percentage of correct instances of detection and repair of the simulated cycle slips on observations of GPS PRN 9 and PRN 22 for datasets with sampling intervals of 1, 5, and 10 s. The data with sampling intervals of 5 and 10 s are extracted from the original dataset.