# Assessment of IRNSS-Only Data Processing: Availability, Single-Frequency SPP and Short-Baseline RTK
## 3 Methods  
In this section, the mathematical model for the GPS/IRNSS integrated position determination is developed, while the corresponding model for the IRNSS-only and GPS-only data processing can be easily derived with the developed mathematical model. The positioning model can be divided into absolute and relative positioning based on different data processing strategies. The absolute positioning includes the SPP and the precise point positioning (PPP). This contribution only analyzes the positioning results of SPP and short-baseline RTK, and does not cover the PPP due to the absence of IRNSS precise ephemeris. Only the pseudorange measurements on the L5 band are used for SPP processing as the receivers at MGEX stations that support IRNSS satellites only receive the L5 signal [15]. The code observations from the IRNSS and GPS satellites can be expressed as follows:  
\[\left\{ \begin{array}{l}P^{G}=\rho^{G}+t_{r}^{G}-t_{s}^{G}+Ion^{G}+T^{G}+ \varepsilon^{G}\\ P^{I}=\rho^{I}+t_{r}^{I}-t_{s}^{I}+Ion^{I}+T^{I}+\varepsilon^{I}\end{array} \right. \tag{1}\] \begin{table}
\begin{tabular}{c c c} \hline \hline
**Signal** & **Carrier Frequency (MHz)** & **Bandwidth (MHz)** \\ \hline L5 & 1176.450 & 24.0 (1164.45–1188.45) \\ S & 2492.028 & 16.5 (2483.50–2500.00) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Information about IRNSS signals. Figure 1: Ground tracks of eight IRNSS satellites on 13 July 2019.  
where \(G\) and \(I\) denote satellites of GPS and IRNSS, respectively, \(P\) is the measured pseudorange, \(\rho=\sqrt{\left(X_{s}-X_{r}\right)^{2}+\left(Y_{s}-Y_{r}\right)^{2}+\left(Z_{s} -Z_{r}\right)^{2}}\) is the geometric distance, where \(\left(X_{s},Y_{s},Z_{s}\right)\) denotes the satellite coordinates in three dimensions, which can be calculated by the broadcast ephemeris, and \(\left(X_{r},Y_{r},Z_{r}\right)\) denotes the receiver coordinates in three dimensions, \(t\), denotes the receiver clock offset (in meters) that absorbs the receiver code hardware delay, \(t_{s}\) denotes the satellite clock offset (in meters) after applying the time group delay (TGD) corrections, \(Ion\) denotes the ionospheric delay, \(T\) denotes the tropospheric delay, and \(\varepsilon\) denotes the code measurement noises including the multipath errors. The ionospheric delay and tropospheric delay are corrected using the GPS Klobuchar model [16] (can also use the Galileo Nequik-G [17] or the BDS-3 BDGIM [18]) and the Saastamoinen model in this paper, respectively. When using different observations to estimate the satellite clock offset, the obtained satellite clock estimates are not consistent with each other as the frequency-related satellite code hardware delay is absorbed. Therefore, a bias correction is necessary when the code observations used for SPP processing are different from those adopted by satellite clock estimation. The TGD parameter provided in the broadcast ephemeris (generated with dual-frequency observations) can be employed to derive the consistent satellite clock offset for the single-frequency SPP users. The satellite clock offset with TGD corrections can be formulated as follows:  
\[\left\{ \begin{array}{l}t_{s}^{G}=t_{SV}^{G}-\gamma^{G}\cdot T_{GD}^{G}\\ t_{s}^{I}=t_{SV}^{I}-\gamma^{I}\cdot T_{GD}^{I}\end{array} \right. \tag{2}\]  
with \(\gamma^{G}=\left(f_{1}/f_{15}\right)^{2}\) and \(\gamma^{I}=\left(f_{S}/f_{L5}\right)^{2}\) where \(t_{SV}\) denotes the satellite clock offset calculated by the broadcast ephemeris, \(f_{1}\), \(f_{L5}\) and \(f_{S}\) denote the carrier frequency on the L1, L5 and S bands, respectively, and \(T_{GD}\) denotes the correction of the TGD. The IRNSS System Time (IRNSST) started at 00:00:00 on 22 August 1999, which corresponds to the Coordinated Universal Time (UTC) 23:59:47 on 21 August 1999 (same time as the first GPS week with a roll-over). IRNSST is a continuous time without leap seconds, and it is determined by the IRNSS System Precise Timing Facility (IRNPT) with an ensemble of Caesium and Hydrogen maser standard atomic clocks, which is steered to UTC.