# Assessment of Quad-Frequency Long-Baseline Positioning with BeiDou-3 and Galileo Observations
## 2 Materials and Methods
### Theoretical Precision of Triple- and Quad-Frequency EWL/WL Signals of BDS-3 and Galileo  
Then, we analyzed the theoretical precision of the instantaneous ambiguity and range parameters at each step in the DD case. The precision of the GNSS positioning performance depends on the measurement noise of the observations and the strength of the geometric figures between the receivers and the GNSS satellites. The dilution of precision (DOP) factor is often used in positioning to quantitatively reflect the strength of the geometric figures [25]. The calculation of the DOP value and the theoretical precision of the baseline can be expressed as follows [26; 27]:  
\[ \begin{array}{l}DOP=\sqrt{trace\big{(}\mathbf{B}_{a}^{T}\mathbf{P}\mathbf{B}_{a}\big{)}^ {-1}}\\ \sigma_{3D}=\sigma_{b}\times DOP\end{array}  \tag{11}\]  
where \(\mathbf{B}_{a}\) is the design matrix of the entire equations; \(\sigma_{3D}\) is the three-dimensional (3D) theoretical precision of the baseline parameters (DD range); and \(\sigma_{b}\) is the normalized baseline noise variance, which is related to the noise of EWL/WL combinations. When the DOP (PDOP for absolute positioning and RDOP for relative positioning) for all satellites in view is equal to 1, the theoretical precision of station-satellite distance or baseline can be obtained [28]. Therefore, assuming that RDOP is equal to 1, the theoretical baseline precision can be calculated using Equation (11). Empirically, the DD pseudo-range and carrier phase precision are set as 50 cm and 5 mm, respectively. Various combinations can be formed based on the original observations. When the number of frequency increases, the number of optional combined signals also greatly increases, but most of them are unsuitable signals. Assuming that the coefficient value \(i,\ j,\ k,\ m\) ranges from \(-10\) to 10, some linearly independent EWL, second EWL, and WL combinations with a small noise amplitude factor and a small ionospheric scalar factor can be obtained in the two systems, respectively. Within this range, we first chose the combination with minimal theoretical DD ambiguity standard deviation (STD) as the first EWL, corresponding to the combination of BDS-3 (1,\(-\)1,0,0) and Galileo (0,\(-\)1,1,0), and the corresponding EWL ambiguity precisions were 0.041 cycles and 0.043 cycles (shown in Tables 2 and 3), respectively. The first choice of second EWL combination was \(\frac{\lambda_{\textit{{EWL}}}}{2.0}>\lambda_{\textit{{SEWL}}}>2.93\) m, and then, the combination with the minimum DD ambiguity STD within this wavelength range was selected. Ignoring the troposphere residuals and the orbital residuals, Tables 2 and 3 show the theoretical parameter precisions of BDS-3 and Galileo when the RDOP value is 1, respectively. When no ambiguities were fixed, the precision of the baseline parameter was decided by the DD pseudo-range noise. For the function model shown in Equations (7) and (10), when only observation noise is considered, the float ambiguities rounding success rates can be computed as follows [29]:  
\[P(-0.5<x<0.5)=\int_{-0.5}^{0.5}\frac{1}{\sigma\sqrt{2\pi}}exp\left(-\frac{x^{2}}{ 2\sigma^{2}}\right)dx \tag{12}\]  
where \(\sigma\) is the random bias caused by observation noises. It can be seen that the theoretical precisions of BDS-3 and Galileo are 1.012 m and 1.249 m, respectively. When only pseudo-range is used, DD range is more than 1.0 m and decimeter-level instantaneous positioning cannot be obtained. Additionally, after fixing the EWL, the DD range theoretical precisions of BDS-3 and Galileo are 0.949 m and 1.173 m, respectively, which have slight improvement. Similarly, if the second EWL is directly constrained by pseudo-ranges, the ambiguity STDs of BDS-3 and Galileo are 0.110 cycles and 0.070 cycles, respectively, and the a priori success rates are both greater than 99.9%, so they can be fixed reliably by rounding.