# GNSS Real-Time Precise Point Positioning in Arctic Northeast Passage
## 4 Discussion  
Ionospheric refraction error is one of the main error sources in GNSS positioning, and the first-order ionospheric term can reach several meters or even tens of meters because it is frequency dependent and can be eliminated using an IF-LC model at different frequencies [25; 26]. However, for the SF PPP, ionospheric models need to be used for correcting ionospheric propagation errors. In Section 3.2.2, we used the Klobuchar model [27] because it is simple and parameters can be obtained in the broadcast ephemerides; therefore, we can calculate the ionospheric propagation errors in real time. However, this model only offers 50-75% accuracy for ionospheric corrections, and the accuracy is lower at high latitudes [28]. The IGS developed and published RTCM-SSR messages for vertical total-electron-content (VTEC) ionospheric messages in October 2020 [29], which means that the grid ionospheric model (GIM) information will be released in real time, effectively improving the positioning accuracy of single-frequency receiver users in the NEP. However, IGS does not provide real-time GIM products in SSR formats; therefore, we used the post-processing GIM products provided by the Chinese Academy of Sciences [30] to simulate real-time VTEC ionospheric messages. \begin{table}
\begin{tabular}{l l l l l l} \hline \hline  & **E (m)** & **N (m)** & **2D (m)** & **U (m)** & **3D (m)** \\ \hline GPS & 0.153 & 0.223 & 0.300 & 0.257 & 0.434 \\ G+E+C & 0.043 & 0.058 & 0.080 & 0.057 & 0.107 \\ \hline \hline \end{tabular}
\end{table} 
Table 7: Mean RMS values of DF RT–PPP performance in the east, north, horizontal, up, and 3D directions for GPS and multi–GNSS from 10 September to 20 September 2019. Figure 12: RMS values of DF RT–PPP in the east, north, horizontal (2D), up, and 3D directions for GPS and multi–GNSS from 10 September to 20 September 2019. After obtaining the ionospheric grid file, the interpolation method can be used to obtain the TEC values at time \(t\):  
\[TEC(\lambda_{IPP},\varphi_{IPP},t)=\frac{T_{i+1}-t}{T_{i+1}-T_{i}}TEC_{i}(\lambda_ {IPP},\varphi_{IPP})+\frac{t-T_{i}}{T_{i+1}-T_{i}}TEC_{i+1}(\lambda_{IPP},\varphi _{IPP}) \tag{12}\]  
where \(\lambda_{IPP}\) and \(\varphi_{IPP}\) are the longitude and latitude of the satellite signal ionospheric pierce point (IPP), and \(TEC_{i}\) and \(TEC_{i+1}\) are the provided TEC data at time \(T_{i}\) and \(T_{i+1}\). Then, the vertical ionospheric delay \(I_{v}\) can be calculated as follows:  
\[I_{v}=-\frac{40.3}{f^{2}}TEC(\lambda_{IPP},\varphi_{IPP},t) \tag{13}\]  
Finally, the slant ionospheric delay \(I\) can be expressed as:  
\[I=\frac{1}{z^{\prime}}I_{v} \tag{14}\]  
where \(z^{\prime}\) is the zenith at IPP. Substituting Equation (14) into the SF-PPP model (Equation (1)), the user's real-time coordinates can be obtained. We performed the two aforementioned SF RT-PPP scenarios, including the Klobuchar- and GIM-based models, to examine and compare the effectiveness of ionospheric products. Figure 13 shows the east/north/up-coordinate solutions of the two scenarios utilizing the GPS/Galileo/BDS combination from 10:00 to 22:00 on 12 September 2019, while the GIM- and Klobuchar-based SF RT-PPP models are shown in green and red, respectively. We determined that the positioning accuracy can be considerably improved by using GIM products, especially in the north component. Table 8 summarizes the RMS values in each direction of the two aforementioned SF RT-PPP methods. Compared with the accuracy of 4.365 m and 2.630 m for the horizontal and vertical components derived from Klobuchar-based RT-PPP solutions, the results of the GIM-based solutions show improvements of 1.440 m and 0.863 m, which is approximately 67.0% and 67.2%. The positioning accuracy for the 3D component improves from 5.096 m with the Klobuchar-based solutions to 1.649 m with the GIM-based solutions--an improvement of 66.7%. Figure 13: Klobuchar–based and GIM–based SF RT–PPP solutions in the east, north, and up directions from 10:00 to 22:00 on 12 September 2019.