# Statistical Multipath Model Based on Experimental GNSS Data in Static Urban Canyon Environment
## 4 Statistical Model Analysis
### Multipath Fading Frequency  
Multipath fading frequency affects the periodic behavior of observation errors caused by multipath. There are three major factors that influence multipath fading frequency: the position and velocity of the satellite, the position and velocity of the receiver antenna, and the reflector position. In this paper, the statistical multipath fading frequency characteristics for MEO, IGSO and GEO orbits are discussed. Figure 11 gives the fading frequency distributions obtained from experimental data for different orbits in the form of histograms. It shows that the multipath fading frequency is axisymmetric approximately 0, so we take the absolute value of the multipath fading frequency for convenience in the following. The probability density is a monotonic decreasing function. Figure 12 shows the probability distribution of the multipath fading frequency of the MEO and IGSO satellite at different elevations. The exponential distribution is used to fit the experimental data. The statistical experimental data of IGSO are not very smooth, as the MEO data are, because there are only five IGSO satellites. Thus, the number of multipath detected from the IGSO satellite is smaller than that of MEO. Although the raw experimental data of the IGSO satellite are insufficient to obtain a smooth distribution, the conclusions are still reasonable. Considering the probability of the multipath fading frequency with a positive sign and negative sign is the same, the fitted probability distribution function is given as follows:  
\[f(x;\lambda)=\frac{1}{2}\lambda\cdot e^{-\lambda\cdot|x|} \tag{11}\]  
where \(x\) is the fading frequency. The parameter \(\lambda\) for different elevation angles is given in Table 7. Most of the values of the NGEO multipath fading frequency are smaller than 0.3 Hz, which shows \begin{table}
\begin{tabular}{c c c} \hline \hline
**Parameters** & \(a\) **(95\% Confidence)** & \(b\) **(95\% Confidence)** \\ \hline Mean of multipath power & \(-0.0081\)\((-0.0088,\)\(-0.0074)\) & 32.7 (32.3, 33.1) \\ Maximum of multipath power & \(-0.018\)\((-0.020,\)\(-0.016)\) & 45.7 (44.5, 46.5) \\ Mean of power attenuation & \(-0.0032\)\((-0.0039,\)\(-0.0025)\) & \(-12.3\)\((-12.7,\)\(-11.9)\) \\ \hline \hline \end{tabular}
\end{table} 
Table 7: Fitted parameters of multipath power. Figure 11: Histograms of multipath fading frequency distribution for three orbits. that the multipath fading frequency is small for a static receiver. The mean of each distribution can be calculated by \(1/\lambda\), and it can be concluded that mean period of NGEO multipath variation is less than one minute. The GEO satellite is almost static with respect to earth, and its elevation angle and azimuth angle change slowly. There are five GEO satellites in the BDS system, and the variation range of their elevation angle is less than 4 degrees. Figure 13 gives the elevation angle and variation velocity of elevation for five GEO satellites in one day. Figure 12: Statistical probability distribution of multipath fading frequency for the medium earth orbit (MEO) and inclined geosynchronous satellite orbit (IGSO) satellite in different elevations. Figure 13: Elevation and variation velocity of the five BeiDou GEO satellites in one day. The experimental statistical distribution of the multipath fading frequency for GEO satellite is given in Figure 14. However, it is difficult to analyze the relationship between the multipath fading frequency and elevation angle because the variation of the GEO elevation angle is very small. The exponential distribution is also used to match the experimental data of multipath fading frequency. The chi-square test is also used to test the goodness of fit in the fading frequency distribution in Table 8. The result shows that the parameters values matched the experimental data very well. Figure 15 shows the mean multipath fading frequencies for three orbit types under different elevation angles.