# Ionospheric Error Models for Satellite-Based Navigation--Paving the Road towards LEO-PNT Solutions
## 4 Results
### Ionospheric Delay Results
#### 4.2.1 Data from Reference Stations--Static Conditions  
We acquired RINEX observation files from our reference base station at Tampere University (TAU). Note that the absolute location (or ground truth) of the TAU antenna was computed using a Precise Point Positioning (PPP) algorithm and based on five days of data collected from a Septentrio receiver, which is connected to a Talysman roof GNSS antenna. First, RINEX observation files were acquired from the Septentrio receiver. Next, a Weighted Least Squares (WLS) algorithm following [59] was applied to the observation data. Then, ionospheric corrections were applied following Methods 1-4. Note that we also applied a tropospheric correction model in all four methods (i.e., the dry/hydrostatic Saastamoinen ionospheric model [36]). Finally, we calculated the mean and standard deviation of the 3D positioning error for each method mentioned earlier. The mean and standard deviation of the 3D positioning error--denoted by \(\epsilon_{i}\) and \(\sigma_{i}\), respectively, with \(i=1,\ldots,N_{scen}\) being the scenario index and \(N_{scen}\) being the number of scenarios--was computed in \(x\), \(y\), \(z\) coordinates in a WGS84 coordinate system via  
\[\epsilon_{s}=\sqrt{(\bar{x}_{i}-x_{i})^{2}+(\bar{y}_{i}-y_{i})^{2}+(\bar{z}_{ i}-z_{i})^{2}} \tag{9}\]  
where \(\bar{x}_{i},\bar{y}_{i},\bar{z}_{i}\) are the estimated \(x\), \(y\), \(z\) coordinates after the atmospheric delay removal with a weighted least squares approach (following [59]) and \(x_{i},y_{i},z_{i}\) are the highly accurate reference positions (obtained via PPP) of the actual roof antenna's position. The results are presented in Table 4a. In order to diversify our data origins and further test the models, we also used FINPOS, a service provided by the National Land Survey of Finland (NLS). FINPOS offers various data, including raw observation files from several GNSS base stations across Finland. We collected data from different FINPOS stations at random dates and times and repeated the same process of calculating the 3D positioning error. The results are in Table 4b. The positioning error means \(\epsilon_{i}\) and standard deviations \(\sigma_{i}\) were computed over all tracks \(j=1:N_{i}\) of each scenario, where \(N_{i}\) is the total number of measurements for scenario \(i\). The number of scenarios for each of the three cases (i.e., measurements from TAU's base station, open-access measurements from FINPOS, and measurements collected with the GNSSLogger app from Android phones) was 8 per case, giving us a total of 24 scenarios. The total number of track points \(N_{i}\) per scenario varied, with an average of 28,800, 2883, and 13,699 track points from TAU data, FINPOS data, and Android data, respectively. \begin{table}
\begin{tabular}{c c} \hline \hline
**Number of Points Used in Statistics** & **Correlation Coefficient** \\ \hline
5183 & 0.9283 \\
15,549 & 0.9574 \\
31,098 & 0.9589 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Correlation coefficients between TEC values generated by IAMM and IGS. \(\epsilon_{i}\) is the mean 3D positioning error per scenario \(i\) over all tracks and is given by  
\[\epsilon_{i}=\frac{1}{N_{i}}\sum_{j=1}^{N_{i}}d_{i,j} \tag{10}\]  
where  
\[d_{i,j}=\sqrt{(x_{i,j}-\hat{x}_{i,j})^{2}+(y_{i,j}-\hat{y}_{i,j})^{2}+(z_{i,j}- \hat{z}_{i,j})^{2}} \tag{11}\]  
with \(x_{i,j},y_{i,j},z_{i,j}\) being the reference (or ground-truth) coordinates and \(\hat{x}_{i,j},\hat{y}_{i,j},\hat{z}_{i,j}\) being the estimated coordinates.