# 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 Infrasound Propagation Distance  
The sampled parameter profiles are imported into the tau-p model as atmospheric background parameters to calculate the corresponding outputs. In this paper, the infrasound propagation distance \(R\) is selected as the output of the infrasound propagation physical model. The source position of the tau-p model is set to the ground at an altitude of 0; the elevation angle of the emission angle \(\theta\) is set to a fixed value of 30\({}^{\circ}\); the azimuth angle \(\phi\) is 45\({}^{\circ}\). The first-order sensitivity index and the total Sobol index of each parameter are solved by the Sobol index method. These two indices can reflect the degree of influence of the atmospheric parameters on the physical model of infrasound propagation. The first-order Sobol index convergence curves and the total Sobol index convergence curves for the five atmospheric parameters are shown in (a) and (b) of Figure 6, respectively. The range of Sobol sampling times \(N\) is from 500 to 3000. Figure 5: Distribution curves of each atmospheric parameter generated after Sobol sampling (the number of sampling is 1200). The black line represents the reference atmospheric parameter distribution curve, and the red line represents the atmospheric parameter distribution curve generated after sampling. Graphs (**a–f**) illustrate temperature \(T\), zonal wind speed \(w_{z}\), meridional wind speed \(w_{m}\), average atmospheric molar mass \(m\), specific heat ratio \(\gamma\), and adiabatic sound speed \(c\), respectively (Data from: [[https://kauai.ccmc.gsfc.nasa.gov/instantrun/msis](https://kauai.ccmc.gsfc.nasa.gov/instantrun/msis)]([https://kauai.ccmc.gsfc.nasa.gov/instantrun/msis](https://kauai.ccmc.gsfc.nasa.gov/instantrun/msis)), accessed on 15 May 2021). It can be seen from Figure 6 that with the increase of sampling times, the first-order Sobol index and total Sobol index of each atmospheric parameter converge. The first-order Sobol index and the total Sobol index when the number of Sobol samples \(N_{max}\) is 2000 are shown in Figure 7. The specific values of the two indices are shown in Table 2.  
It can be seen in Table 2 that the parameters that have a great impact on infrasound propagation distance have been sorted as follows:  
(1) The first-order impact indices in descending order are meridional wind, thermodynamic temperature, zonal wind, mean atmospheric molar mass, and specific heat ratio;  
(2) The order of the total impact indices from the largest to the smallest is as follows: meridional wind, thermodynamic temperature, zonal wind, mean atmospheric molar mass, \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 \(S\) & 0.3768 & 0.2684 & 0.3528 & 0.0016 & 0.0004 \\ \(ST\) & 0.3533 & 0.2649 & 0.3814 & 0.0003 & 0.0001 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: First Sobol index \(S\) and Total Sobol index \(ST\) of each atmospheric parameter for the infrasound propagation distance \(R\). Figure 6: (**a**) Five atmospheric parameters’ first-order Sobol index convergence lines for the infrasound propagation distance \(R\); (**b**) Five atmospheric parameters’ total Sobol index convergence lines for the infrasound propagation distance \(R\) (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). Figure 7: (**a**) First-order Sobol index pie charts and (**b**) total Sobol index pie charts of the five parameters for the infrasound propagation distance \(R\). The five uncertain atmospheric environmental parameters are independently and uniformly distributed. The number of Sobol samples \(N\) is 2000.  
and specific heat ratio.