## 2 Materials and Methods
### STG Cycle Slip Detection Method  
We can make a satellite-difference combination to remove the second-order, time-difference receiver clock errors \(\Delta\Delta dt_{r}\). Thus, with Equation (4), the STG cycle slip detection values can be given as  
\[\begin{array}{ll}\
abla\Delta\Delta N_{1}+j\lambda_{2}/i\lambda_{1}\
abla \Delta\Delta N_{2}&=\
abla(i\Delta\Delta L_{1}+j\Delta\Delta L_{2}-(i+j)\cdot \Delta\Delta\rho)/i\lambda_{1}\\ &+\
abla(i+\alpha j)\Delta\Delta I/i\lambda_{1}\end{array}\;, \tag{9}\]  
where \(\
abla(\star)=\
abla(\star)^{G_{i}}-\
abla(\star)^{G_{j}}\) is the satellite-difference operator, \(G_{i}\) and \(G_{j}\) represent the different GPS satellites, and \(\Delta\Delta\rho=\Delta\Delta\rho_{0}+\overrightarrow{e}\cdot\Delta\Delta \overrightarrow{r}\).  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Sampling Interval (s)** & **1** & **5** & **10** & **30** \\ \hline Acceleration error (ms\({}^{-2}\)) & \(10^{-2}\) & \(4\times 10^{-4}\) & \(10^{-4}\) & \(10^{-5}\) \\ \hline \hline \end{tabular}
\end{table}
Table 1: Acceleration accuracy requirements for different sampling intervals with an \(\Delta\Delta\overrightarrow{r}(t)\) error at the 1-cm level.  
As stated above, the first part of the right side of Equation (9) can be precisely computed from the GPS phase observations and the known orbit dynamic models. Based on Equation (9), we can have several STG combinations, with different levels of ionospheric residuals as follows:  
\[\
abla\Delta\Delta N_{1}-\lambda_{1}/\lambda_{2}\
abla\Delta\Delta N_{2}=\
abla( \Delta\Delta L_{1}-\frac{f_{2}^{2}}{f_{1}^{2}}\Delta L_{2}-0.3928\Delta\Delta \rho)/\lambda_{1}, \tag{10}\]  
\[\begin{array}{ll}\
abla\Delta\Delta N_{1}-\
abla\Delta\Delta N_{2}&=\
abla( \Delta\Delta L_{1}-\frac{f_{2}}{f_{1}}\Delta L_{2}-0.2208\Delta\Delta\rho))/ \lambda_{1}\\ &-0.2833V\Delta\Delta I/\lambda_{1}\\ \
abla\Delta\Delta N_{1}-0.5\lambda_{2}/\lambda_{1}\
abla\Delta\Delta N_{2}&= \
abla(\Delta\Delta L_{1}-0.5\Delta\Delta L_{2}-0.5\Delta\Delta\rho)/\lambda_ {1}\\ &+0.1765\
abla\Delta I/\lambda_{1}\\ \
abla\Delta\Delta N_{1}=\
abla(\Delta\Delta L_{1}-\Delta\Delta\rho)/\lambda_ {1}+\
abla\Delta\Delta I/\lambda_{1}.\end{array}\, \tag{11}\]  
The coefficients \(i\) and \(j\) are shown in Table 2, corresponding to Equations (10)-(13).  
Equations (10)-(13) can be used to detect cycle slips, as Equation (10) is ionospheric-free, and Equations (11) and (12) have smaller ionospheric residuals (\(0.2833\Delta I\) and \(0.1765\Delta\Delta I\)) to STPIR (\(0.6469\Delta\Delta I\)). Furthermore, Equation (13) can be used for big cycle slip detection for single-frequency GPS carrier phase data, where even \(\Delta\Delta I\) cannot be practically ignored.  
If cycle slips \((dN_{1},dN_{2})\) occur on GPS \(L_{1}\) and \(L_{2}\) frequencies between time epochs \(t\) and \(t+\Delta t\), due to the second-order time-difference processing, by adding the time epoch mark \(t\) with Equation (9), we have the cycle slip values as follows:  
\[\begin{array}{ll}dN_{1}+j\lambda_{2}/i\lambda_{1}dN_{2}&=\
abla(i\Delta \Delta L_{1}(t)+j\Delta\Delta L_{2}(t)-(i+j)\cdot\Delta\Delta\rho(t))/i\lambda _{1}\\ &+\
abla(i+\alpha j)\Delta I(t)/i\lambda_{1}\end{array}\, \tag{14}\]  
\[\begin{array}{ll}-dN_{1}-j\lambda_{2}/i\lambda_{1}dN_{2}&=\
abla(i\Delta \Delta L_{1}(t+\Delta t)+j\Delta\Delta L_{2}(t+\Delta t)-(i+j)\cdot\Delta \Delta\rho(t+\Delta t))/i\lambda_{1}\\ &+\
abla(i+\alpha j)\Delta\Delta I(t+\Delta t)/i\lambda_{1}\end{array}. \tag{15}\]  
Equations (14) and (15) suggest that there will be two opposite peak values in the subsequent epochs when cycle slips occur, which can be used together to detect cycle slips.  
Moreover, it should be pointed out that \(\Delta\Delta dt_{r}\) for some LEO satellites (e.g., GOCE) can also be practically ignored, due to the high-stability atomic clocks usually equipped, similar to the GPS satellites. In that case, the satellite-difference operator \(\
abla(*)\) is not necessary for Equations (9)-(15) (presented in Section 3).