## 2 Utilization of GNSS
### Multi-Frequency GPS Algorithm
The variance reduction term \(\sigma_{\rho,jono}^{2}\) describes the magnitude of the accuracy improvement due to the elimination of ionospheric errors. It is 0 for single-frequency measurements and is assigned according to (5) for ionosphere-free combinations. \[\sigma_{\rho,jono}^{2}= \begin{cases}25\,\mathrm{m}^{2},&\text{for }\theta\geq 20^{ \circ}\\ \frac{\sin 20^{\circ}}{\sin\theta}\cdot 25\,\mathrm{m}^{2},&\text{for }\theta<20^{ \circ}\end{cases}  \tag{5}\]  
This way, the ionosphere-free linear combinations get assigned a lower measurement variance than single-frequency pseudo-ranges. This contradicts (2), which specifies an increased standard deviation of ionosphere-free linear combinations compared to single-frequency pseudo-ranges. The contradiction is resolved by looking at the Kalman Filter's assumptions for measurement noise: zero-mean, Gaussian and uncorrelated in time. Noise of this type is fully specified by its covariance matrix. Unfortunately, none of these assumptions is true for pseudo-range measurements. Consequently, the measurement noise's variance is increased, especially to account for the nonzero mean of pseudo-range errors. The mean error's absolute value for ionosphere-free combinations is substantially smaller than for single-frequency measurements, because the ionospheric error is the largest contributor to the overall error. That is why the ionosphere-free combinations are modeled with smaller measurement noise variance as far as the Kalman Filter is concerned. While the measurement noise variance model turned out to be feasible, the idea to work with all possible ionosphere-free linear combinations as well as single-frequency pseudo-ranges without taking the DCBs into account did not. Firstly, the GNSS receiver used (a JAVAD TRIUMPH-LS) does not track the in-phase and quada-phase component on L5 independently, but uses a combined I + Q tracking. The resulting pseudo-ranges receive the observation code C5X in RINEX 3.03 [13]. No ISCs are provided in the CNAV data for this type of code tracking, so the satellite-specific DCBs cannot be eliminated in the way they could for independent I and Q tracking. In the following results, GPS C5X pseudo-ranges are processed with the ISCs for GPS L5I. Secondly, the receiver-specific DCBs proved to be too large to facilitate a composition of all available ionosphere-free combinations and single-frequency pseudo-ranges without DCB calibration (see Section 4). Table 2 summarizes the receiver clock biases obtained when working with different types of observations, all stemming from the same three data sets. \(c\delta t_{L1\ C/A}\), the receiver clock bias obtained from the EKF outputs when only processing L1 C/A pseudo-ranges, is treated as reference. The entries in the second column are gained by computing the mean and standard deviation of the differences \(c\delta t_{L1\ C/A}-c\delta t_{i}\), where the subscript \(i\) indicates the type of processed code observations and is specified in the first column. The first line in each cell specifies the mean, the second line specifies the standard deviation. The three entries in each line stem from three different sets of measurement data, collected between December 2016 and November 2017. All three data sets contain kinematic data and have durations of approximately \(120\,\mathrm{min}/60\,\mathrm{min}/30\,\mathrm{min}\) while covering distances of approximately \(100\,\mathrm{km}/13\,\mathrm{km}/8\,\mathrm{km}\). As the code biases in their non-differential form cannot be separated from the receiver clock bias, the difference in receiver clock bias is an indication for the DCBs. The last column ranks the positioning performance in each data set from best (1) to worst (7), based on the RMS position error. For two of the three data sets, it was not possible to compute a solution with L5-L1 C/A combinations alone, because there were not enough block IIF satellites in view.