## 3 Results and Discussion
### Analysis of the Second-Order Time-Difference Values for LEO GPS Observations  
Firstly, we evaluate the errors of the computed \(\Delta\Delta\overset{\rightarrow}{\tau}^{G}\) and \(\Delta\Delta\overset{\rightarrow}{\tau}_{r}\) via Equation (8), with the known orbit dynamic models, with different sampling intervals, with the final precise ephemeris products for GPS satellites and LEO satellites (GOCE, GRACE-A, and Jason-3 satellite) as \"true\" values. Based on one-day data, the errors of the computed \(\Delta\Delta\overset{\rightarrow}{\tau}^{G}\) and \(\Delta\Delta\overset{\rightarrow}{\tau}_{r}\) are shown in Figures 1 and 2. Here, only the Earth gravity model 2008 (EGM2008) up to 120 degrees and order is used as the known dynamic model to compute the \(\Delta\Delta\overset{\rightarrow}{\tau}^{G}\) and \(\Delta\Delta\overset{\rightarrow}{\tau}_{r}\) for the satellites, with \(\Delta t\)= 30s for GOCE and GRACE, and \(\Delta t=60\)s for Jason-3 (corresponding to Jason-3's orbit product). The results show that the computed \(\Delta\Delta\overset{\rightarrow}{\tau}^{G}\) errors all are less than 3 mm for all GPS satellites in three directions, while the RMSs of the computed \(\Delta\Delta\overset{\rightarrow}{\tau}_{r}\) error in the three directions are 1.8 mm, 1.7 mm, and 2.2 mm for GOCE, 0.8 mm, 0.8 mm, and 0.8 mm for GRACE-A, and 4.4 mm, 2.9 mm, and 2.9 mm for Jason-3. \begin{table}
\begin{tabular}{c c c c c c} \hline \multicolumn{2}{c}{**CSPs**} & \multicolumn{2}{c}{\(dN_{1}-A_{1}/A_{2}dN_{2}\)} & \(dN_{1}-dN_{2}\) & \(dN_{1}-0.5A_{2}/A_{1}dN_{2}\) & \(dN_{1}\) \\ \hline \(dN_{1}\) & \(dN_{2}\) & & & & \\ \hline \(1\) & \(1\) & \(0.2208\) & \(0\) & \(0.3583\) & \(1\) \\ \(3\) & \(4\) & \(-0.1169\) & \(-1\) & \(0.4333\) & \(3\) \\ \(4\) & \(5\) & \(0.1039\) & \(-1\) & \(0.7917\) & \(4\) \\ \(3\) & \(5\) & \(-0.8961\) & \(-2\) & \(-0.2083\) & \(3\) \\ \(7\) & \(9\) & \(-0.0130\) & \(-2\) & \(1.2250\) & \(7\) \\ \(7\) & \(11\) & \(-1.5714\) & \(4\) & \(-0.0583\) & \(7\) \\ \(9\) & \(14\) & \(-1.9091\) & \(-5\) & \(0.0167\) & \(9\) \\ \(60\) & \(77\) & \(0\) & \(-17\) & \(10.5917\) & \(60\) \\ \hline \end{tabular}
\end{table} 
Table 3: Several special cycle slip pairs (CSPs) and detected cycle slip values for the second-order time-difference geometry-free (STG) combinations. On one hand, due to GOCE's lower orbital altitude (about 260 km), the ignored gravitational accelerations (e.g., ocean tide, solid tide) are stronger than for GRACE. Thus, the errors of the computed STPs of GOCE are larger than those of GRACE. On the other hand, the errors of the computed STPs of Jason-3 are larger than those of GOCE and GRACE, because of the larger sample interval. The results are consistent with Equation (8). Moreover, the STP accuracies are still better than 0.01 m. Therefore, it can be concluded that the errors of the computed STPs \(\Delta\Delta\overrightarrow{r}^{G}\) and \(\Delta\Delta\overrightarrow{r}_{r}\) computed via Equation (8) are negligible for cycle slip detection and repair. Figure 1: The means (bottom) and RMSs (top) of the computed GPS satellite position (STP) errors, with one-day final 30-s International GNSS Service (IGS) GPS ephemeris products as the “true” value, with 30-s sampling intervals. Figure 2: The computed low-Earth orbit (LEO) satellite STP errors, with one-day precise reduced dynamic orbit products as the “true” value, with different sampling intervals: GOCE and GRACE (30-s intervals), and Jason-3 (60-s intervals). Furthermore, with one-day (18 November 2009) final 30-s clock products provided by IGS as the \"true\" values, \(\Delta\Delta dt^{G}\) is evaluated, and the means and RMSs of \(\Delta\Delta dt^{G}\) are statistically computed for a sampling interval of 30 s, as shown in Figure 3. The results clearly illustrate that \(\Delta\Delta dt^{G}\) values are as precise as 0.1 ns with nearly zero means for most GPS satellites, and are all smaller than 0.15 ns, which means that \(\Delta\Delta dt^{G}\) can also be practically ignored for cycle slip detection and repair.