# Statistical Multipath Model Based on Experimental GNSS Data in Static Urban Canyon Environment
## 4 Statistical Model Analysis
### Multipath Time Delay  
We build the statistical distribution models for different elevation angle ranges because the multipath characteristics are closely related to the satellite elevation angle. Figure 6 shows the probability density distribution of multipath time delay with different satellite elevations. In Figure 6, the histogram is obtained from statistical experimental data, and the Gamma distribution is adopted to fit the data. The probability density function of the Gamma distribution is as follows:  
\[f(x;\gamma,\varsigma)=\tfrac{1}{\Gamma(\gamma)\cdot\varsigma^{\gamma}},x^{ \gamma-1}\cdot e^{-\tfrac{x}{\varsigma}}\quad x\in(0,\infty) \tag{7}\]  
where \(x\) is the time delay. \(\Gamma(\cdot)\) is the Gamma function. \(\gamma\) and \(\varsigma\) are the parameters which is different for different elevation angles. The multipath time delay is shorter when the satellite elevation is higher. The distribution parameters under different elevations are given in Table 4. The table shows that the shape parameters \(\gamma\) are similar under different elevation angles but that the scale parameters \(\varsigma\) are inversely proportional to the elevation. Figure 5: The multipath characteristics of one geosynchronous orbit (GEO) satellite and one non-geosynchronous orbit (NGEO) satellite at collection spot 1. To verify that the fitted distribution is acceptable, the chi-square test is used in every probability distribution model [26]. The formula of the chi-square test is as follows:  
\[\chi^{2}=\sum_{i=1}^{k}\frac{(f_{i}-n\cdot p_{i})^{2}}{n\cdot p_{i}}, \tag{8}\]  
where \(f_{i}\) is the sample number of the \(i\)th interval and \(p_{i}\) is the theoretical probability of \(i\)th interval. \(n\) is the sample size, and \(k\) is the number of intervals. The result should follow a chi-square distribution with \(k-1\) degrees of freedom. Since this paper mainly focus on the empirical model based on the experiment measurements, a common method of studying distribution function is to use all potential functions with similar trend to fit the statistical histogram. The chi-square test result is used to represent the fitting consistency of each function. It can be found that the overall probability distribution of time delay first rise and then drop, so the following six distribution functions are chosen to fit the data. The test result of time delay distribution fitted by these functions are shown in Table 5. According to the value of Table 5, it can be seen that the Gamma distribution is the most suitable for the time delay. In addition, the gamma distribution is equal to the sum of multiple independent and identically distributed exponential distribution, and the exponential distribution is commonly used to describe the waiting time of Poisson distributed events. Thus, the Gamma distribution \(Gamma(\gamma,\xi)\) can represent the waiting time of \(\gamma\)-th event. To interpret the gamma distribution of multipath time delay from a physical standpoint, it can be suggested that the multipath signal may be reflected \(\gamma\) times in statistical analysis. Table 4 indicates that the statistical reflecting times are between 2 and 3 times for different elevations. The chi-square test of time delay with different elevation fitted by the Gamma distribution are shown in Table 6. It shows that most of the test results are less than \(\chi^{2}_{0.01}(k-1)\), which confirmed that \begin{table}
\begin{tabular}{c c c} \hline
**Elevation** & \(\gamma\) & \(\xi\) \\ \hline \((0,15)\) & 2.62 & 129.83 \\ \((15,30)\) & 2.77 & 105.52 \\ \((30,45)\) & 2.81 & 80.93 \\ \((45,60)\) & 2.56 & 65.12 \\ \((60,75)\) & 2.47 & 53.22 \\ \((75,90)\) & 2.40 & 43.24 \\ \hline \end{tabular}
\end{table} 
Table 4: Distribution parameters of time delay with different elevations. Figure 6: Statistical distribution of time delay in different elevation angle ranges.