# Assessment of IRNSS-Only Data Processing: Availability, Single-Frequency SPP and Short-Baseline RTK
## 3 Methods
\tag{13}\]  
\[X_{D}=\left[X_{r},Y_{r},Z_{r},\Delta\
abla N^{G_{1}},\cdots,\Delta\
abla N^{G_ {n-1}},\Delta\
abla N^{I_{1}},\cdots,\Delta\
abla N^{I_{m-1}}\right]^{\mathsf{T}} \tag{14}\]  
where \(O_{D}\) is the vector of dual-difference OMC code and carrier phase observables, \(v_{D}\) denotes the observation residual error vector in the short-baseline RTK positioning, \(H_{D}\) denotes the design matrix in the short-baseline RTK positioning, \(X_{D}\) is the vector of estimates in the short-baseline RTK positioning, \(\Delta\
abla p\) is the double-difference OMC code observables, \(\Delta\
abla\phi\) is the double-difference OMC carrier phase observables, \(n\) and \(m\) denote the satellite number of GPS and IRNSS, respectively, \(G_{b}\) and \(I_{b}\) denote the reference satellite of GPS and IRNSS, respectively, \(G_{k}(k=1,2,\cdots,n-1)\) and \(I_{q}(q=1,2,\cdots,m-1)\) denote a satellite of GPS and IRNSS, respectively, and \(\alpha\), \(\beta\) and \(\chi\) denote the linearization coefficients related to receiver coordinates in the design matrix \(H_{D}\). The meanings of other parameters are the same as the aforementioned ones, and thus we do not repeat them again. According to Equation (14), the parameters to be estimated in the short-baseline RTK positioning comprise the 3D receiver coordinates and the double-difference phase ambiguities of GPS and IRNSS. The Kalman filter is used for the parameter estimation in the short-baseline RTK processing. The receiver coordinates are estimated as white noise process, and the double-difference phase ambiguity parameters are estimated as integer constants. The least-squares ambiguity de-correlation adjustment (LAMBDA) method [21] is used for the ambiguity resolution (AR). In addition to the rigorous functional model, a suitable stochastic model can also improve the positioning performance. Usually, the observations can be influenced by the ionosphere, troposphere, and multipath effect during the propagation of satellite signals. All these influencing factors are related to the satellite elevation angles. Therefore, in this study, an elevation-dependent model is used to determine the weights of observations. Assuming that there is no correlation between the measurements of different types, from different satellites, or at different stations, the variances of undifferenced observations can be calculated as follows:  
\[\sigma^{2}(el)=\frac{\sigma_{0}^{2}}{(\sin{el})^{2}} \tag{15}\]  
where \(\sigma_{0}\) is the standard deviation (STD) of observations at zenith, \(el\) denotes the satellite elevation angle, and \(\sigma(el)\) is the STD of observations at the elevation angle \(el\). In this contribution, \(\sigma_{0}\) is set to 3 dm and 1.9 mm (one percent of chip length or wavelength) for the code and carrier phase observations from GPS satellites, respectively. It is considered that the accuracy of broadcast ephemeris and the quality of satellite signals of IRNSS are inferior to those of GPS, and thus the adopted \(\sigma_{0}\) of the code and carrier phase observations from IRNSS satellites is empirically 1.5 times and 2.0 times as large as that of GPS code and phase, respectively (i.e., 4.5 dm and 3.8 mm).