# Uncertainty Quantification for Infrasound Propagation in the Atmospheric Environment
## 4 Quantitative Results Presentation and Analysis Based on the Actual Atmospheric Data  
### Sobol Sampling to Generate Atmospheric Profile Curves  
The Sobol index method is used to analyze the atmospheric physical parameters affecting infrasound propagation. After observing the infrasound propagation model, thermodynamic temperature, zonal wind, meridional wind, mean atmospheric molar mass, and specific heat ratio are identified as the physical parameters that may affect the atmospheric propagation. The atmospheric physical parameters are in the altitude range of 1-200 km, and 1000 points can be acquired at an altitude interval of \(\Delta z=0.2\) km. The baseline parameter curves for the above physical parameters are selected, i.e., the curves of the atmospheric physical parameter values with height (source of baseline parameter data: MSISE00 and HWM93 models). The format of the atmospheric background data is as follows:  
\[met=\left[z[\text{ km}],T(z)[\text{K}],w_{z}(z)\left[\frac{\text{m}}{\text{s }}\right],w_{m}(z)\left[\frac{\text{m}}{\text{s}}\right],m(z)\left[\frac{\text {g}}{\text{mol}}\right],\gamma(z)\right]_{1000\times 6}, \tag{21}\]  
where \(T\) is the temperature; \(w_{z}\) is the zonal wind speed; \(w_{m}\) is the meridional wind speed; \(m\) is the average molar mass of the atmosphere; \(\gamma\) is the specific heat ratio. The magnitudes of \(T\), \(w_{z}\), \(w_{m}\), \(m\) are (K), \(\text{m/s}\), (m/s), (g/mol), respectively. The baseline parameter curves are fitted using polynomial curve fitting in order to obtain the fitted coefficient values of the baseline curves. The sampling interval of the fitting coefficient is obtained by adjusting the sampling range of the fitting coefficient so that the distribution curve generated after sampling will conform to the actual atmospheric environment. The range of values of the curve fitting coefficients are uniformly distributed. The polynomial expansion of the five atmospheric parameters is shown in Equation (22). \[ \begin{array}{l}T(z)=n_{1}z^{5}+n_{2}z^{4}+n_{3}z^{3}+n_{4}z^{2}+n_{5}z+n_{6 }\\ w_{2}(z)=n_{1}z^{5}+n_{2}z^{4}+n_{3}z^{3}+n_{4}z^{2}+n_{5}z+n_{6}\\ w_{m}(z)=n_{1}z^{5}+n_{2}z^{4}+n_{3}z^{3}+n_{4}z^{2}+n_{5}z+n_{6}\\ m(z)=n_{1}z^{2}+n_{2}z^{1}+n_{3}\\ \gamma(z)=n_{1}z^{2}+n_{2}z^{1}+n_{3}\end{array}  \tag{22}\]  
where \(n_{i}\) (\(i=1,\ldots,6\)) is the polynomial fitting coefficient. The number of polynomial curve fitting coefficients for the five atmospheric parameters are 6, 6, 6, 3, and 3, respectively, and the range of their values is shown in Table 1.  
The number of samples \(N_{max}\) for the curve fitting coefficients is set to 1200. The sampled atmospheric parameter distribution curves are plotted in a graph, as shown in Figure 5. The black line represents the reference atmospheric parameter distribution  
\begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline  & & \(n_{1}\) & \(n_{2}\) & \(n_{3}\) & \(n_{4}\) & \(n_{5}\) & \(n_{6}\) \\ \hline \multirow{2}{*}{\(T\)} & min & \(-1.8359\times 10^{-9}\) & \(5.49439\times 10^{-6}\) & \(-0.00181\) & \(0.20020\) & \(-8.048\) & \(299.09\) \\  & max & \(-1.8363\times 10^{-9}\) & \(5.49549\times 10^{-6}\) & \(-0.00180\) & \(0.20060\) & \(-7.889\) & \(330.57\) \\ \hline \multirow{2}{*}{\(w_{z}\)} & min & \(9.880\times 10^{-9}\) & \(-4.377\times 10^{-6}\) & \(0.000655\) & \(-0.03674\) & \(0.511\) & \(-18.52\) \\  & max & \(9.882\times 10^{-9}\) & \(-4.376\times 10^{-6}\) & \(0.000657\) & \(-0.03666\) & \(0.521\) & \(6.17\) \\ \hline \multirow{2}{*}{\(w_{m}\)} & min & \(1.0201\times 10^{-8}\) & \(-4.410\times 10^{-6}\) & \(0.000639\) & \(-0.0355\) & \(0.665\) & \(-18.40\) \\  & max & \(1.0203\times 10^{-8}\) & \(-4.409\times 10^{-6}\) & \(0.000640\) & \(-0.0353\) & \(0.679\) & \(11.0 curve, and the red line represents the generated atmospheric parameter distribution curve after sampling. From Figure 5, it can be seen that the fitted parameter curves are well-distributed at a reasonable interval without exceeding the real atmospheric environment.