# Accurate Collaborative Globally-Referenced Digital Mapping with Standard GNSS
## 3 GNSS Error Analysis
### Error Sources
#### 3.3.3 Ionospheric Modeling Errors
A numerical summary of the IGS GIM position bias is presented in Table 1, along with a similar analysis for the Wide Area Augmentation System (WAAS) ionospheric corrections available for the contiguous United States (CONUS) region. As reported in [15], the WAAS model was found to exhibit a significantly smaller RMS error in ionosphere TEC estimates when compared to the IGS GIM; however the long-term position bias due to WAAS corrections is similar to or worse than those for the IGS model. Another global ionospheric model, the Fast PPP model [41], was also studied as above. Fast PPP natively models the ionosphere as a two-layered shell, but is also made available in the standard one-layer IONEX (ionosphere-map exchange) format [15] for dissemination. The results presented in Table 1 represent the IONEX version of Fast PPP. In comparison with the IGS corrections, it is clear that the Fast PPP IONEX GIM corrections result in substantially unbiased long-term position errors at the global test locations. However, it must be conceded that the results in Table 1 are best-case results, as they are based on data from the same permanent reference stations used to constrain the model. Figure 1: Direction and magnitude (the latter represented by color, in meters) of the long-term average horizontal position error due to errors in the delay estimates provided by the IGS GIM. Note that the meridians are curved outwards due to projection of the spherical map, and that arrows parallel to the curved meridians point directly south or north. To understand the reason behind the systematic biases with IGS corrections, note that any ionospheric modeling bias that identically affects all satellites does not have any impact on the accuracy of the GNSS position solution, as this common error is absorbed in \(\delta t_{\rm R}\). Rather, position-domain biases arise from the azimuthal- and elevation-dependence of ionosphere model errors. From analysis of the spatial distribution of post-fit residuals, it was found that appreciable azimuthal and elevation residual gradients persist in the IGS ionospheric corrections. These gradients are represented graphically in Figure 2 for one representative station from the northern hemisphere (station code: EUSK, latitude: \(50^{\circ}40^{\prime}26.87^{\prime\prime}\), longitude: \(6^{\circ}45^{\prime}48.72^{\prime\prime}\)) and one representative station from the southern hemisphere (station code: VACS, latitude: -\(20^{\circ}17^{\prime}48.47^{\prime\prime}\), longitude: \(57^{\circ}29^{\prime}13.79^{\prime\prime}\)). The post-fit residuals are binned in azimuth and elevation and the average value in each bin is denoted by the color of the representing disc. The size of the disc denotes the number of samples of post-fit residuals available in each bin. Due to the inclination angle of the GPS satellite orbits, the angular distribution of satellites at any given latitude is non-uniform. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Ionosphere Model** & **Region** & **East (m)** & **North (m)** & **Up (m)** \\ \hline \multirow{3}{*}{IGS} & \(\phi\geq 30^{\circ}\) & \(0.0107\) & \(-0.2129\) & \(0.6733\) \\  & \(30^{\circ}>\phi>-30^{\circ}\) & \(-0.0651\) & \(-0.0692\) & \(1.5467\) \\  & \(\phi\leq-30^{\circ}\) & \(0.0237\) & \(0.2450\) & \(0.3355\) \\ \hline WAAS & CONUS & \(-0.0048\) & \(-0.2916\) & \(-0.1248\) \\ \hline \multirow{3}{*}{Fast PPP IONEX} & \(\phi\geq 30^{\circ}\) & \(-0.0042\) & \(-0.0099\) & \(-0.0122\) \\  & \(30^{\circ}>\phi>-30^{\circ}\) & \(-0.0390\) & \(0.0013\) & \(-0.3053\) \\ \cline{1-1}  & \(\phi\leq-30^{\circ}\) & \(-0.0325\) & \(-0.0087\) & \(0.0309\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Long-term average position error due to ionospheric model errors (\(\phi\) denotes station latitude). IGS: International Global Navigation Satellite System (GNSS) Service; PPP: precise point positioning; WAAS: Wide Area Augmentation System; CONUS: contiguous United States; IONEX: ionosphere-map exchange format.