## 2 Materials and Methods
### Second-Order, Time-Difference Carrier Phase Combination  
The dual-frequency (subscript 1/2 for carriers \(L_{1}\)/\(L_{2}\)) observations between the LEO satellite receiver (subscript \(r\)) and GPS satellites (superscript \(G\)) are commonly described as  
\[ \begin{array}{l}L_{1,r}^{G}=c\cdot\Phi_{1,r}^{G}/f_{1}=\rho_{r}^{G}+c(dt^{G} -dt_{r})+\lambda_{1}N_{1,r}^{G}-I_{r}^{G}+\varepsilon_{1,r}^{G}\\ L_{2,r}^{G}=c\cdot\Phi_{2,r}^{G}/f_{2}=\rho_{r}^{G}+c(dt^{G}-dt_{r})+\lambda_{ 2}N_{2,r}^{G}-\alpha I_{r}^{G}+\varepsilon_{2,r}^{G}\end{array} \,, \tag{1}\]  
where \(c\) is the velocity of light, \(\rho_{r}^{G}\) is the geometric distance between the GPS satellite and receiver, \(L_{1,r}^{G}\), \(L_{2,r}^{G}\) are the dual-frequency carrier phase observations, \(\Phi_{1,r}^{G}\), \(\Phi_{2,r}^{G}\), \(\lambda_{1}\), \(\lambda_{2}\), \(N_{1,r}^{G}\), \(N_{2,r}^{G}\) are the carrier phases, wavelengths, and ambiguities at frequencies \(f_{1}\), \(f_{2}\), respectively, \(dt^{G}\), \(dt_{r}\) are the clock offset of the GPS satellite and receiver, respectively, \(\alpha=f_{1}{}^{2}/f_{2}^{2}\), \(I_{r}^{G}\) is the ionospheric delay, and \(\varepsilon_{1,r}^{G}\), \(\varepsilon_{2,r}^{G}\) are the noise of the dual-frequency carrier phase observations. The antenna phase center correction, relativistic effect, and wind-up effect, which can be calculated with the known correction models, are eliminated in Equation (1). For simplicity of expression, the time epoch \((t)\) of some symbols is omitted, such as \(\rho_{r}^{G}\) for \(\rho_{r}^{G}(t)\), whereas subscript \(r\) and superscript \(G\) are also omitted in some of the following equations. Given Equation (1), the linear combination of carrier phase \(Lc\) can be described as follows:  
\[ \begin{array}{ll}Lc&=iL_{1}+jL_{2}\\ &=(i+j)\cdot(\rho+c(dt^{G}-dt_{r}))-(i+\alpha j)I+i\lambda_{1}N_{1}+j\lambda_{ 2}N_{2}\end{array} \,, \tag{2}\]  
where \(i\) and \(j\) are coefficients of the observations \(L_{1}\) and \(L_{2}\), respectively. Assuming the true position of the GPS satellite and receiver's mass center are \(\overrightarrow{r}^{G}\) and \(\overrightarrow{r}_{r}\), respectively, and the approximate positions are \(\overrightarrow{r}_{0}^{G}\) and \(\overrightarrow{r}_{r,0}\), respectively, then we have  
\[\rho=\rho_{0}+\overrightarrow{e}\cdot d\overrightarrow{r}, \tag{3}\]  
where \(\rho_{0}\) is the geometric distance between \(\overrightarrow{r}_{0}^{G}\) and \(\overrightarrow{r}_{r,0}\), \(\overrightarrow{e}=\frac{\overrightarrow{r}_{0}^{G}-\overrightarrow{r}_{r,0}} {\rho_{0}}\) is unit direction vector between the GPS satellite and receiver, and \(d\overrightarrow{r}=(\overrightarrow{r}^{G}-\overrightarrow{r}_{r})-( \overrightarrow{r}_{0}^{G}-\overrightarrow{r}_{r,0})\). With Equations (2) and (3), the second-order, time-difference carrier phase combination \(\Delta\Delta Lc\) can be represented as  
\[ \begin{array}{ll}\Delta\Delta Lc&=(i+j)\Delta(\rho_{0}+\overrightarrow{e} \cdot d\overrightarrow{r}+dt^{G}-dt_{r})-(i+\alpha j)\Delta\Delta I\\ &+i\lambda_{1}\Delta\Delta N_{1}+j\lambda_{2}\Delta N_{2}\end{array} \,. \tag{4}\]Let us analyze the terms in Equation (4). \(\overset{\rightarrow}{r}_{0}^{G}\) is available from the GPS broadcast ephemeris in the real-time application or IGS final precise ephemeris products in the post-processing applications, and \(\overset{\rightarrow}{r}_{r,0}\) is available from a LEO a priori orbit, e.g., a GPS code solution [32]. Thus, \(\Delta\Delta\rho_{0}\) can be treated as known. Moreover, the term \(\Delta\Delta(\overset{\rightarrow}{e}\cdot d\overset{\rightarrow}{r})\) can be given as  
\[