# Uncertainty Quantification for Infrasound Propagation in the Atmospheric Environment
## 4 Quantitative Results Presentation and Analysis Based on the Actual Atmospheric Data
### Uncertainty Quantification Analysis for the Travel Time of Infrasound Propagation on a Phase Loop  
The Sobol sensitivity analysis is used to quantify and analyze the main atmospheric physical parameters that affect the travel time on a phase loop. In this paper, atmospheric temperature, meridional wind, zonal wind, mean atmospheric molar mass, and specific heat ratio are selected as the atmospheric physical parameters to be quantified. The travel time of the tau-p model is chosen as the output. The source position of the tau-p model is set to the ground at an altitude of 0; the elevation angle of emission \(\theta\) is set to a fixed value of 30\({}^{\circ}\); the azimuth angle \(\phi\) is 20\({}^{\circ}\). The convergence curves of the first-order Sobol sensitivity index \(S\) and the total Sobol index \(ST\) are plotted in (a) and (b) of Figure 10, respectively. The range of Sobol sampling times \(N\) is from 200 to 2400. Figure 9: (**a**) First-order Sobol index pie charts and (**b**) total Sobol index pie charts of the five parameters for the infrasound propagation maximum height. The five uncertain atmospheric environmental parameters are independently and uniformly distributed. The number of Sobol samples \(N\) is 5000. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline  & \(T\) & \(w_{z}\) & \(w_{m}\) & \(m\) & \(\gamma\) \\ \hline \(S\) & 0.8496 & 0.0095 & 0.0727 & 0.0659 & 0.0033 \\ \(ST\) & 0.8563 & 0.0093 & 0.0847 & 0.0490 & 0.0007 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: First Sobol index \(S\) and total Sobol index \(ST\) of each atmospheric parameter for the infrasound propagation maximum height. It can be seen that the two indices converge as the number of Sobol samples increases. The results with a sampling number of 2400 were chosen, and pie charts of the first-order Sobol index and the total Sobol index were drawn separately, as shown in Figure 11. The sensitivity index values corresponding to the degree of influence of each atmospheric physical parameter on the infrasound propagation travel time in a phase loop are shown in Table 4.  
In Table 4, it can be seen that the parameters that have a large influence on the travel time of infrasound propagation have been ranked as follows:  
(1) The first-order influence indices in descending order are the mean atmospheric molar mass, specific heat ratio, thermodynamic temperature, meridional wind, and zonal wind;  
(2) The total impact indices in descending order are the mean atmospheric molar mass, specific heat ratio, thermodynamic temperature, meridional wind, and zonal wind. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline  & \(\mathbf{T}\) & \(\mathbf{w_{z}}\) & \(\mathbf{w_{m}}\) & \(\mathbf{m}\) & \(\mathbf{\gamma}\) \\ \hline \(S\) & 0.2685 & 9.7098 \(\times\) 10\({}^{-4}\) & 0.0045 & 0.6301 & 0.0960 \\ \(ST\) & 0.3396 & 2.6454 \(\times\) 10\({}^{-4}\) & 0.0036 & 0.5771 & 0.0795 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: First-order Sobol index \(S\) and total Sobol index \(ST\) of each atmospheric parameter for infrasound propagation travel time in a phase loop. Figure 11: (**a**) First-order Sobol index pie charts and (**b**) total Sobol index pie charts representing the five parameters for travel time \(t\). The five uncertain atmospheric environmental parameters are independently and uniformly distributed. The number of Sobol samples \(N\) is 2400. Figure 10: (**a**) Five atmospheric parameters’ first-order Sobol index convergence lines for the infrasound propagation travel time \(t\) in a phase loop; (**b**) Five atmospheric parameters’ total Sobol index convergence lines for the infrasound propagation travel time \(t\) in a phase loop (the blue, dark red, yellow, purple, and light green curves indicate temperature \(T\), zonal wind \(w_{z}\), meridional wind \(w_{m}\), mean atmospheric molar mass \(m\), and specific heat ratio \(\gamma\), respectively).