## 2 Analysis of Signal Power and Noise
### Code Measurement Noise  
The precision of code observations is generally limited by the combined influence of receiver noise and site-specific multipath effect. The aggregate impact of both these factors is usually evaluated using multipath combinations, which are formed from dual-frequency code and phase observations [47]. Under the assumption \(f_{1}>f_{2}\), the multipath combinations for satellite \(i\) and receiver \(k\) can be written as follows:  
\[MP1=P_{k,f_{1}}^{i}-\left(1+\frac{2}{\mu-1}\right)\lambda_{f_{1}}\varphi_{k,f_{ 1}}^{i}+\left(\frac{2}{\mu-1}\right)\lambda_{f_{2}}\varphi_{k,f_{2}}^{i} \approx M_{k,P,f_{1}}^{i}+\epsilon_{k,P,f_{1}}^{i}+B_{k,MP1}^{i} \tag{1}\]  
\[MP2=P_{k,f_{2}}^{i}-\left(\frac{2\mu}{\mu-1}\right)\lambda_{f_{1}}\varphi_{k,f_ {1}}^{i}+\left(\frac{2\mu}{\mu-1}-1\right)\lambda_{f_{2}}\varphi_{k,f_{2}}^{i} \approx M_{k,P,f_{2}}^{i}+\epsilon_{k,P,f_{2}}^{i}+B_{k,MP2}^{i} \tag{2}\]  
where \(P\) and \(\varphi\) are dual-frequency code and phase observables, respectively; \(\lambda\) is the signal wavelength; \(M\) and \(\epsilon\) denote multipath and noise effects affected code measurements; \(B\) is a constant factor including ambiguity terms and hardware delay biases (eliminated by the subtracting the mean values of \(MP1\) or \(MP2\) time series for entire arc; and \(\mu\) is the constant factor applied for conversion of the ionosphere delay on first frequency into that on second frequency according to the formula:  
\[\mu=f_{1}^{2}/f_{2}^{2}. \tag{3}\]  
Figure 2: Carrier-to-noise density ratio of the Galileo E14 and E18 signals on different frequencies as function of satellite elevation and altitude (two top panels—IAWA, two bottom panels—OLSZ). Black lines correspond to average C/N0 for Galileo satellites signals (E1 or E5a) at particular station.  
The above equations were applied to generate time series of dual-frequency multipath combinations for all observational arcs. For this purpose we used code observations C1C, C2W (GPS), C1X, C5X (Galileo) and the corresponding phase data. Subsequently, their RMSs in 10\({}^{\circ}\) bins of elevations were computed separately for GPS, common Galileo, and satellites with eccentric orbits. The results obtained for two selected stations are presented in Figure 3.  
With regard to results obtained for E14 and E18 satellites, one can conclude that the precision of their code observations is at the same level as for other Galileo satellites. Comparing only European satellites, the highest differences in elevation patterns did not exceed 3 cm and seem to be related to site-specific multipath effect. On the other hand, the results demonstrated significant distinctions between stations and systems. The former issue is a consequence of choke ring antenna installed at OLSZ station, which allows improved multipath mitigation. Thus, errors of code observables are significantly lower than at the IAWA site. The improvement at high elevations reaches about 10 cm for both Galileo (C1X) and GPS (C1C) code signals. Only in the case of C2W observations is the decrease of code observation errors for OLSZ site much lower, not exceeding 5 cm. Furthermore, the results basically confirm that the precision of Galileo code observations outperforms those of the GPS ones. RMS values of MP1 and MP2 combinations at high elevations for the European system are at the level of 0.19 m, 0.16 m (IAWA) and 0.08 m, 0.08 m (OLSZ). The corresponding values for GPS are 0.22 m, 0.23 m and 0.13 m, 0.19 m, respectively. The analysis executed for the remaining sites (BYCE and CHLE) showed a similar precision of code observations as for IAWA station (see Table 2). Analysing these results, one can also conclude that the precision of E5a code measurements at sites with common antennas (BYCE, CHLE, IAWA) is higher than for the E1 signal. This improvement, observed for all three cases, varies between 0.02 m and 0.06 m.  
Figure 3: Variation of code noise and multipath with elevation at stations IAWA (top panel) and OLSZ (bottom panel).