## 2 Utilization of GNSS
### Multi-Frequency GPS Algorithm  
The overall pseudo-range error, often called _user equivalent range error_ (UERE), is usually decomposed into two error types which are considered statistically independent: The _signal in space ranging error_ (SISRE) and the _user equipment error_ (UEE) [9]. \[\text{UERE}=\sqrt{\text{SISRE}^{2}+\text{UEE}^{2}} \tag{1}\]  
Modern (circa 2016 and later) dual-frequency receivers provide a typical _root mean square_ (RMS) ionosphere error of 0.4 m, resulting in a UEE of 0.5 m and a UERE of 0.7 m (assuming no significant multipath error). In comparison, the typical RMS ionosphere error of modern single-frequency receivers is 5.0 m. As all other errors are significantly smaller, this results in both a UEE and a UERE of 5.0 m [9]. Hence, multi-frequency signals accessible to civil users provide the potential to drastically increase measurement accuracy. Out of a total of 31 active GPS satellites, the GPS L2C signal today is available from 19 satellites (7 block IIR-M and 12 block IIF), while the GPS L5 signal is available from the 12 block IIF satellites [10]. With the upcoming launches of block IIIA GPS satellites, both numbers will increase. For Galileo (see Section 2.3), availability of civil multi-frequency signals is not an issue as all satellites provide civil signals on E1, E5a and E5b. To input pseudo-range measurements on additional frequencies into the localization algorithm, the following alternatives were considered:  
1. Treat the new measurements exactly as the old ones, i.e. retain the assumption that all measurements are uncorrelated and assign similar variances to the new pseudo-range measurements as to the old ones. 2. Treat measurements stemming from the same satellite as a batch, i.e. assign similar variances to the new pseudo-range measurements as to the old ones but assume that pseudo-range measurements on different frequencies, but from the same satellite, are highly correlated. Pseudo-ranges from different satellites are still considered to be uncorrelated. 3. Introduce additional variables into the EKF's state vector to estimate the ionospheric refraction. The simplest version of this alternative adds a single additional state representing the zenith ionospheric delay on a reference carrier frequency. The ionospheric refraction's dependency on carrier frequency and elevation is represented in the pseudo-range measurement model. More complex versions add one state for each satellite in view, resulting in a variable-length state vector. 4. Work with ionosphere-free linear combinations of pseudo-ranges. Option (1) does not model the ionospheric refraction's dependency on the carrier frequency and is therefore clearly suboptimal. Option (2) does take this dependency into account. However, this is only done implicitly via the off-diagonal entries in the measurement noise covariance matrix. The result is an increased computational load as some of the measurements are now correlated to each other. Also, parametrizing the measurement noise covariance matrix becomes more difficult. While option (3) provides a high degree of flexibility, it also increases the algorithm's complexity significantly due to the additional variables in the state vector. Ionosphere-free linear combinations (option 4) offer the advantage of being able to keep many of the existing algorithm's features: Measurements from different satellites can still be considered uncorrelated and the EKF's state vector does not need to be changed. Also, the properties of ionosphere-free combinations are well known because they have been investigated since the launch of GPS, they are the preferred mode for ionospheric corrections according to the GPS interface specifications and the _civil navigation messages_ (CNAV) on L2C and L5 provide terms to correct for the effects of satellite-specific DCBs in the respective combinations [11; 12]. On the other hand, ionosphere-free linear combinations only support dual-frequency measurements and not three or even more measurement frequencies.