# Analysis of the Impact of Multipath on Galileo System Measurements
## 3 Methodology
### Code Multipath
For this purpose, the combination equations for the two frequencies, omitting the satellite and receiver indices, can be formulated as follows:  
\[\mathcal{CM}_{f}=P_{f}-L_{f}-2\delta I_{f}+\lambda_{f}N_{f} \tag{5}\]  
Next, using the relation between the magnitude of the ionospheric delay for different carrier frequencies,  
\[\delta I_{2}=\gamma\delta I_{1}\quad\text{where:}\quad\gamma=\left(\frac{f_{ 1}}{f_{2}}\right)^{2}, \tag{6}\]  
the Geometry-Free combination can be formulated, omitting the time constant values of the carrier wave ambiguity, as follows:  
\[L_{\text{GF}}=L_{1}-L_{2}=\delta I_{2}-\delta I_{1}=\delta I_{1}(\gamma-1) \tag{7}\]  
The value of the ionospheric delay for L1 can be determined using  
\[\delta I_{1}=\frac{L_{1}-L_{2}}{1}\gamma-1 \tag{8}\]  
The \(\delta I_{1}\) determined in this way can be substituted into Equation (5), obtaining  
\[\mathcal{CM}_{1} =P_{1}-\frac{\gamma+1}{\gamma-1}L_{1}+\frac{2}{\gamma-1}L_{2} \tag{9}\] \[\mathcal{CM}_{2} =P_{2}-\frac{2\gamma}{\gamma-1}L_{1}+\frac{\gamma+1}{\gamma-1}L_{2} \tag{10}\]  
Using a moving average, the impact of time constant values of ambiguity was eliminated, averaged for \(2k\) epochs:  
\[\overline{\mathcal{CM}} =\frac{\sum_{j=-k}^{k}\mathcal{CM}_{j}}{2k+1} \tag{11}\] \[\widetilde{\mathcal{MC}}_{j} =\mathcal{CM}_{j}-\overline{\mathcal{CM}}_{j} \tag{12}\]  
where \(\overline{\mathcal{CMC}}\) denotes the average CMC observable for \(2k\) epochs and \(\widetilde{\mathcal{MC}}_{j}\) denotes the CMC residuals. Such an approach assumes constancy over time of the \(N\) parameter. In the event of cycle slips, the filtering of observations using a moving average should be reset. The CMC residuals determined in this way only contain a component of the multipath effect and measurement noise of the code observations:  
\[\mathcal{CM}_{a,f}^{i}=M_{a,f}^{i}+E_{a,f}=P_{a,f}^{i}-L_{a,f}^{i}-2\delta I_{ a,f}^{i}+\lambda_{f}N_{a,f}^{i} \tag{4}\]0According to the error propagation law, the standard deviation of CMC can be expressed as follows:  
\[\mathcal{CM}_{a,f}^{i}=M_{a,f}^{i}+E_{a,f}=P_{a,f}^{i}-L_{a,f}^{i}-2\delta I_{ a,f}^{i}+\lambda_{f}N_{a,f}^{i} \tag{4}\]1  
In order to determine the STD of the multipath effect, the STD of the measurement noise must therefore be taken into account. It can be determined using the differences of observation \(\widehat{\alpha}\widehat{\alpha}\) over time. Since multipath is a that parameter slowly varies over time, the value of this error in adjacent time epochs (with observations registered every 1 second) are almost identical. Hence, differentiation of the observations over time eliminate this component:  
\[\mathcal{CM}_{a,f}^{i}=M_{a,f}^{i}+E_{a,f}=P_{a,f}^{i}-L_{a,f}^{i}-2\delta I_{ a,f}^{i}+\lambda_{f}N_{a,f}^{i} \tag{4}\]2  
Assuming \(M(t_{1})\simeq M(t_{2})\), the final form of the CMC can be represented as follows:  
\[\mathcal{CM}_{a,f}^{i}=M_{a,f}^{i}+E_{a,f}=P_{a,f}^{i}-L_{a,f}^{i}-2\delta I_{ a,f}^{i}+\lambda_{f}N_{a,f}^{i} \tag{4}\]3  
Making the stochastic assumption that the statistical characteristics of the measurement noise in adjacent epochs is identical, the STD of the noise for these epochs is also identical. Taking into account the time correlation of the measurement noise for adjacent epochs [34], the STD of the time difference of CMC can be expressed as follows:  
\[\mathcal{CM}_{a,f}^{i}=M_{a,f}^{i}+E_{a,f}=P_{a,f}^{i}-L_{a,f}^{i}-2\delta I_{ a,f}^{i}+\lambda_{f}N_{a,f}^{i} \tag{4}\]4  
\[\mathcal{CM}_{a,f}^{i}=M_{a,f}^{i}+E_{a,f}=P_{a,f}^{i}-L_{a,f}^{i}-2\delta I_{ a,f}^{i}+\lambda_{f}N_{a,f}^{i} \tag{4}\]5  
\[\mathcal{CM}_{a,f}^{i}=M_{a,f}^{i}+E_{a,f}=P_{a,f}^{i}-L_{a,f}^{i}-2\delta I_{ a,f}^{i}+\lambda_{f}N_{a,f}^{i} \tag{4}\]6  
where \(q_{t}\) denotes the observation time correlation coefficient. The time correlation coefficient was determined on the basis of the combinations of double- and triple-differences (TD) of observation for the zero-length baseline.