# Assessment of IRNSS-Only Data Processing: Availability, Single-Frequency SPP and Short-Baseline RTK
## 3 Methods
The single-difference between stations can eliminate the satellite clock errors and satellite hardware delay, and weaken the ionospheric delay, tropospheric delay, and satellite orbit errors. The inter-satellite single-difference can remove the receiver clock errors and receiver hardware delay. The double-difference carrier phase measurements can be expressed as follows:  
\[\left\{ \begin{array}{ll}\Delta\
abla\Phi_{i,j}^{G,A,B}=\Delta\
abla p_{i,j}^ {G,A,B}-\Delta\
abla Ion_{i,j}^{G,A,B}+\Delta\
abla T_{i,j}^{G,A,B}+\lambda^{ G}\cdot\Delta\
abla N_{i,j}^{G,A,B}+\Delta\
abla\xi_{i,j}^{G,A,B}\\ \Delta\
abla\Phi_{i,j}^{I,A,B}=\Delta\
abla p_{i,j}^{I,A,B}-\Delta\
abla Ion_{i, j}^{I,A,B}+\Delta\
abla T_{i,j}^{I,A,B}+\lambda^{I}\cdot\Delta\
abla N_{i,j}^{I,A,B}+ \Delta\
abla\xi_{i,j}^{I,A,B}\end{array} \right. \tag{8}\]  
where \(\Delta\
abla\) denotes the double-difference operation, \(A\) and \(B\) denote two satellites from GPS or IRNSS, and \(i\) and \(j\) denote two stations. The effects of residual errors of ionospheric delay and tropospheric delay can be ignored during the processing of short-baseline RTK, and the double-difference carrier phase observation equation can be simplified as follows:  
\[\left\{ \begin{array}{ll}\Delta\
abla\Phi_{i,j}^{G,A,B}=\Delta\
abla\rho_{i,j }^{G,A,B}+\lambda^{G}\cdot\Delta\
abla N_{i,j}^{G,A,B}+\Delta\
abla\xi_{i,j}^{ G,A,B}\\ \Delta\
abla\Phi_{i,j}^{I,A,B}=\Delta\
abla\rho_{i,j}^{I,A,B}+\lambda^{I} \cdot\Delta\
abla N_{i,j}^{I,A,B}+\Delta\
abla\xi_{i,j}^{I,A,B}\end{array} \right. \tag{9}\]  
The linearized observation model of the GPS/IRNSS combined short-baseline RTK can be described as follows:  
\[O_{D}+v_{D}=H_{D}\cdot X_{D} \tag{10}\]\[\left\{ \begin{array}{l}O_{D}=\left[O_{D}^{G},O_{D}^{I}\right]^{\mathsf{T}}\\ O_{D}^{G}=\left[\Delta\
abla p_{i,i}^{\sigma_{1},\sigma_{b}},\Delta\
abla p_{i,j }^{\sigma_{1},\sigma_{b}},\ldots,\Delta\
abla p_{i,j}^{\sigma_{n-1},\sigma_{b} },\Delta\
abla\phi_{i,j}^{\sigma_{n-1},\sigma_{b}}\right]\\ O_{D}^{I}=\left[\Delta\
abla p_{i,i}^{I_{1},I_{2}},\Delta\
abla\phi_{i,j}^{I_ {1},I_{2}},\ldots,\Delta\
abla p_{i,j}^{I_{m-1},I_{2}},\Delta\
abla\phi_{i,j}^{ I_{m-1},I_{2}}\right]\\ \end{array} \right. \tag{11}\]  
\[H_{D}=\left[ \begin{array}{ccccccccc}\alpha^{G_{1}}&\beta^{G_{1}}&\chi^{G_{1} }&0&\cdots&0&0&\cdots&0\\ \alpha^{G_{1}}&\beta^{G_{1}}&\chi^{G_{1}}&\lambda^{G}&\cdots&0&0&\cdots&0\\ \vdots&\vdots&\vdots&\vdots&\ddots&\vdots&\vdots&\ddots&\vdots\\ \alpha^{G_{n-1}}&\beta^{G_{n-1}}&\chi^{G_{n-1}}&0&\cdots&0&0&\cdots&0\\ \alpha^{G_{n-1}}&\beta^{G_{n-1}}&\chi^{G_{n-1}}&0&\cdots&\lambda^{G}&0&\cdots&0 \\ \alpha^{I_{1}}&\beta^{I_{1}}&\chi^{I_{1}}&0&\cdots&0&0&\cdots&0\\ \alpha^{I_{1}}&\beta^{I_{1}}&\chi^{I_{1}}&0&\cdots&0&\lambda^{I}&\cdots&0\\ \vdots&\vdots&\vdots&\vdots&\ddots&\vdots&\vdots&\ddots&\vdots\\ \alpha^{I_{m-1}}&\beta^{I_{m-1}}&\chi^{I_{m-1}}&0&\cdots&0&0&\cdots&0\\ \alpha^{I_{m-1}}&\beta^{I_{m-1}}&\chi^{I_{m-1}}&0&\cdots&0&0&\cdots&\lambda^{I} \\ \end{array} \right] \tag{12}\]  
\[\left\{ \begin{array}{l}\alpha^{G_{k}}=\frac{\chi^{G_{k}}_{0}-\chi^{G_{k}}_{0 }}{\rho^{G_{k}}_{0}}-\frac{\chi^{G_{k}}_{0}-\chi^{G_{k}}_{0}}{\rho^{G_{k}}_{0}} \\ \beta^{G_{k}}=\frac{\chi^{G_{k}}_{0}-\chi^{G_{k}}_{0}}{\rho^{G_{k}}_{0}}-\frac{ \chi^{G_{k}}_{0}-\chi^{G_{k}}_{0}}{\rho^{G_{k}}_{0}}\\ \chi^{G_{k}}=\frac{Z^{G_{k}}_{0}-Z^{G_{k}}_{0}}{\rho^{G_{k}}_{0}}-\frac{Z^{G_{k }}_{0}-Z^{G_{k}}_{0}}{\rho^{G_{k}}_{0}}\\ \alpha^{I_{q}}=\frac{\chi^{G_{k}}_{0}-\chi^{I_{q}}_{0}}{\rho^{G_{k}}_{0}}-\frac {X^{G_{k}}_{0}-X^{I_{q}}_{0}}{\rho^{G_{k}}_{0}}\\ \beta^{I_{q}}=\frac{\chi^{G_{k}}_{0}-Y^{I_{q}}_{0}}{\rho^{G_{k}}_{0}}-\frac{Y^{ \prime}_{0}-Y^{I_{q}}_{0}}{\rho^{G_{k}}_{0}}\\ \chi^{I_{q}}=\frac{Z^{G_{k}}_{0}-Z^{I_{q}}_{0}}{\rho^{G_{k}}_{0}}-\frac{Z^{ \prime}_{0}-Z^{\prime}_{0}}{\rho^{G_{k}}_{0}}\\ \end{array} \right.