# Accurate Collaborative Globally-Referenced Digital Mapping with Standard GNSS
## 3 GNSS Error Analysis
### Error Sources
#### 3.3.3 Ionospheric Modeling Errors
Figure 2: Azimuth and elevation dependence of post-fit IGS global ionospheric map (GIM) residuals. **(a)** A representative station from the northern hemisphere. **(b)** A representative station from the southern hemisphere. The average residual error (in TECU) is denoted by the color of the disc. The size of the disc indicates the number of samples of post-fit residuals available in each bin. From Figure 2, it is clear that the elevation gradients in the ionospheric residuals are pronounced. A subtle azimuthal gradient also exists, mainly along the north-south direction. Such spatial non-uniformity, coupled with the non-uniform satellite angular distribution, may be the reason for the observed persistent position biases. While the elevation gradients are consistent for stations at all locations, the azimuthal gradients appear to invert along the north-south direction between the northern and southern hemisphere. This is likely the reason for the opposite direction of the average horizontal position bias in the northern and southern hemispheres. Such persistent position-domain biases due to inaccurate ionospheric modeling have not been previously reported in the literature, and are a rather remarkable result. While some single-frequency PPP (SF-PPP) techniques eliminate the ionospheric delays based on the GRAPHIC combination [42], many other techniques that rely solely on ionospheric corrections from GIMs have been shown to achieve 30 cm 95% accuracy in the east-north plane after convergence, with sub-10-cm bias [43], seemingly contradicting the results here. The key difference is that the SF-PPP methods involve estimation of a float carrier ambiguity term for each satellite arc. A portion of the systematic biases in the GIM estimates is likely absorbed in these states of the estimator, thereby attenuating the position biases in the east-north plane. For instance, the SF-PPP technique in [43] is based on the phase-adjusted pseudorange algorithm [44], wherein the ambiguity term for each satellite, physically an unknown constant, is in fact iteratively estimated with small but non-zero process noise. In such an estimator, the ambiguity term can absorb slowly time-varying systematic biases. In other SF-PPP techniques, the ionospheric correction term is explicitly included as a state to be estimated, and the estimates from GIM are applied as pseudo-observations [45; 46]. Once again, decimeter-level biases in the GIM estimates of the ionospheric delay may not necessarily appear in the final reported position accuracy of the SF-PPP method. Of course, such absorption of biases in augmented states is not undesirable. However, for the case of urban vehicular positioning, convergence of SF-PPP is a concern due to carrier phase cycle slipping, as discussed earlier. In an enhanced code-phase-based receiver, the high variance of the code noise leads to poor observability of the decimeter-level horizontal position bias due to ionospheric modeling errors. Thus, ionospheric biases are not often estimated in a code-phase-based GNSS estimator. Another factor of note is that 2014 was a maximum in the 11-year solar activity cycle, and thus the IGS GIM accuracy may have been worse than usual over this period of time. In conclusion, persistent decimeter-level biases in the east-north plane and meter-level biases in the vertical direction can arise when ionospheric delay corrections are sourced from the IGS GIM, or similar, even under ideal open-sky conditions. More advanced models of the ionosphere with more accurate slant TEC measurements may achieve better results. Elimination of the ionospheric delay based on the ionosphere-free combination is another option, but tends to worsen multipath-induced position errors. If corrections from some ionosphere model lead to unbiased position errors, then for globally-referencing digital maps by averaging GNSS measurements over many sessions it is advisable to avoid the combination of multi-frequency signals.