# Assessment of IRNSS-Only Data Processing: Availability, Single-Frequency SPP and Short-Baseline RTK
## 3 Methods
The time offset between IRNSST and GPS time could be several nanoseconds [19], and thus cannot be ignored in the IRNSS/GPS integrated data processing. IRNSS takes WGS-84 coordinate system as its space reference. Therefore, the transformation of satellite coordinates is unnecessary in the integrated data processing with IRNSS and GPS. Although the center frequency of L5 band is identical for IRNSS and GPS, there may still be difference in the receiver code hardware delay between the two satellite systems. This contribution uses GPS as the reference system, and introduces the ISB parameter into the code observation equation of IRNSS to account for the distinct time scale and hardware delay [20]. The linearized observation model for the GPS/IRNSS combined SPP can be expressed as follows:  
\[O_{U}+v=H\cdot\Delta X \tag{3}\]  
where \(O_{U}\) denotes the vector of observed-minus-computed (OMC) code observables, \(v\) denotes the observation residual error vector, \(H\) denotes the design matrix, and \(\Delta X\) denotes the unknown parameter residual error vector between the true value and the approximate value. \(H\) and \(\Delta X\) can be formulated as follows:  
\[O_{U}=\left[p^{G_{1}},\cdots,p^{G_{m}},p^{I_{1}},\cdots,p^{I_{m}}\right]^{T} \tag{4}\]\[H=\left[ \begin{array}{ccccc}\frac{X_{0}^{\prime}-X_{s}^{C_{1}}}{c_{1}}&\frac{ Y_{0}^{\prime}-Y_{s}^{C_{1}}}{c_{1}^{\prime}}&\frac{Z_{0}^{\prime}-Z_{s}^{C_{1}}}{c_{1} ^{\prime}}&1&0\\ \rho_{0}^{\prime}&\rho_{0}^{\prime}&\rho_{0}^{\prime}&\rho_{0}^{\prime}&1&0\\ \vdots&\vdots&\vdots&\vdots&\vdots\\ \frac{X_{0}^{\prime}-X_{s}^{C_{1}}}{\rho_{0}^{\prime\prime}}&\frac{Y_{0}^{ \prime}-Y_{s}^{C_{n}}}{\rho_{0}^{\prime\prime}}&\frac{Z_{0}^{\prime}-Z_{s}^{C_ {n}}}{\rho_{0}^{\prime\prime}}&1&0\\ \frac{X_{0}^{\prime}-X_{s}^{\prime\prime}}{P_{0}^{\prime}}&\frac{Y_{0}-Y_{s}^{ \prime\prime}}{P_{0}^{\prime}}&\frac{Z_{0}^{\prime}-Z_{s}^{\prime\prime}}{P_{0 }^{\prime\prime}}&1&1\\ \vdots&\vdots&\vdots&\vdots&\vdots\\ \frac{X_{0}^{\prime}-X_{s}^{\prime\prime}}{\rho_{0}^{\prime\prime}}&\frac{Y_ {0}^{\prime}-Y_{s}^{\prime\prime}}{\rho_{0}^{\prime\prime}}&\frac{Z_{0}^{ \prime}-Z_{s}^{\prime\prime}}{\rho_{0}^{\prime\prime}}&1&1\\ \end{array} \right] \tag{5}\]  
\[\Delta X=\left[ \begin{array}{ccccc}\Delta x&\Delta y&\Delta z&\Delta t_{r}^ {G}&\Delta t_{ISB}\end{array} \right]^{\rm T} \tag{6}\]  
where \(p\) denotes the OMC code observables, \((X^{\prime}_{0},Y^{\prime}_{0},Z^{\prime}_{0})\) denotes the approximate 3D coordinates of the receiver, \(\rho_{0}\) is the geometric distance between the receiver approximate position and the satellite position, \(n\) and \(m\) are the satellite numbers of GPS and IRNSS, respectively, \((\Delta x,\Delta y,\Delta z)\) denotes the receiver coordinate residual errors (i.e., corrections) with respect to the approximate position, \(\Delta t_{r}^{G}\) denotes the residual error of the GPS receiver clock offset, and \(\Delta t_{ISB}\) is the residual error of the ISB parameter \(t_{ISB}\) (with \(t_{ISB}=t_{r}^{I}-t_{r}^{G}\)). The least squares adjustment is used for the parameter estimation in SPP processing. In addition to the code observations, the RTK positioning includes the carrier phase observations, which can be expressed as follows:  
\[\left\{ \begin{array}{ll}\Phi^{G}=\rho_{r}^{G}+t_{r}^{G}-t_{s}^{G}-Ion^{G}+T ^{G}+d_{r}^{G}+d_{s}^{G}+\lambda^{G}\cdot N^{G}+\xi^{G}\\ \Phi^{I}=\rho^{I}+t_{r}^{I}-t_{s}^{I}-Ion^{I}+T^{I}+d_{r}^{I}+d_{s}^{I}+\lambda ^{I}\cdot N^{I}+\xi^{I}\end{array} \right. \tag{7}\]  
where \(\Phi\) is the phase observation in meters, \(\lambda\) is the wavelength, \(N\) is the integer phase ambiguity in cycles, \(\xi\) is the phase measurement noises including multipath errors, \(d_{r}\) is the grouped receiver phase and code hardware delay, and \(d_{s}\) is the grouped satellite phase and code hardware delay. Regarding the relative positioning, the difference between different observations can form a combined observable, so as to weaken the influence of various errors and biases.