# Accurate Collaborative Globally-Referenced Digital Mapping with Standard GNSS
## 3 GNSS Error Analysis
### Error Sources
#### 3.3.3 Ionospheric Modeling Errors  
The code-modulated GNSS signal propagates slower through the ionosphere as compared to vacuum due to the slightly-greater-than-unity _group_ index of refraction for this atmospheric layer. The excess group delay is given as  
\[\Delta\tau_{g}=\frac{40.3\cdot TEC}{cf^{2}},\]  
where \(TEC\) is the total electron content in electrons/m\({}^{2}\) and \(f\) is the frequency of the propagating signal. At the GPS L1 frequency centered at 1575.42 MHz, the excess ionospheric group delay is roughly 16.24 cm per TECU (1 TECU \(\triangleq\) 10\({}^{16}\) electrons/m\({}^{2}\)). If not modeled, the ionospheric delay can lead to ranging errors greater than 15 m.  
The ionospheric delay can be estimated via an ionosphere model or, in case of a multi-frequency receiver, eliminated via a combination of multiple-frequency pseudorange measurements. The latter technique does not require any external aiding, but the formation of the ionosphere-free combination exacerbates pseudorange noise, including any biases due to tracking of multipath signals. Compensating for ionospheric delay with the aid of an ionosphere model is applicable to both single- and multi-frequency receivers. It relies on accurate delay modeling based on ionospheric measurements at permanent GNSS reference stations, such as those that form the IGS network. While both methods have their merits, the analysis in this section considers corrections from an ionospheric model, and thus will not be relevant to applications where the ionosphere-free combination is applied. Note that those applications would likely experience worse multipath errors than the ones presented later, requiring a separate multipath analysis along the lines of Section 3.3.5. Ionospheric model accuracy was studied comprehensively in [15]. The method in [15] generates unambiguous carrier-phase measurements from a global distribution of permanent receivers to compute the true slant total electronic content (STEC) for each satellite, and compares the model prediction for a number of models with the ground truth. In [41], the same authors compared PPP convergence times when applying different ionospheric correction models. This section extends the analysis in [15; 41] to examine whether there exist long-term position-domain biases in enhanced code-phase positioning. The post-fit residuals for multiple regional and global ionospheric models, computed as described in [15], were graciously made available by the same authors for the year 2014. These residuals were computed for GPS signals as observed at about 150 reference stations around the globe at 5 min intervals. To observe the position-domain effect of the ionospheric modeling errors in isolation, this section neglects all other error sources, reducing the linearized measurement model in (3) to  
\[\mathbf{z}=H\mathbf{x}+\tilde{\mathbf{I}}.\]  
Historical GPS satellite almanacs can be combined with the timestamps from the residuals data to obtain the measurement sensitivity matrix \(H\) at each epoch for each station. With an elevation-dependent measurement covariance matrix \(R\), the error in the weighted least-squares solution due to errors in ionospheric modeling is \[\mathbf{\hat{x}}-\mathbf{x}=\left(H^{T}R^{-1}H\right)^{-1}H^{T}R^{-1}\mathbf{\tilde{I}}.\]  
Figure 1 summarizes the results for ionospheric corrections obtained from the IGS global ionospheric map (GIM). Each of the arrows in Figure 1 points in the direction of the position bias in the east-north plane, as estimated over 12 months of data from 2014 (more than 800,000 samples per station). The magnitude of the horizontal position bias is depicted by the color of the arrow according to the scale shown on the right. Interestingly, there is a clear trend of southward bias in the position error for most stations in the northern hemisphere, and a mild trend of northward bias in the position error for stations in the southern hemisphere.