# Uncertainty Quantification for Infrasound Propagation in the Atmospheric Environment
## 3 Sobol Index for Sensitivity Analysis of the Tau-p Model
### Mathematical Principles of Sobol Sensitivity Analysis  
The theory of variance-based Sobol sensitivity indices [15] is introduced in this section. The Sobol sensitivity analysis is achieved based on the quasi-Monte Carlo (QMC) [17]. The first-order Sobol index \(S_{m}\) and the total Sobol index \(S_{T_{m}}\) are calculated by this method to estimate the effect of individual variables or groups of variables on the model output. Given \(N_{\text{QMC}}\) samples of \(\mathbf{k}\), let \(h_{n}\big{(}n=1,\ldots,N_{\text{QMC}}\big{)}\) represent the output computed from the infrasound physical model (tau-p model):  
\[h(\mathbf{k})_{n}=h(\mathbf{k}_{1},\ldots,\mathbf{k}_{M})_{n}, \tag{12}\]  
where \(\mathbf{k}_{m}\) (\(m=1,\ldots,M\)) denotes the \(m\)-th atmospheric environmental parameter. The mean value \(\mu(h)\) of the tau-p model output is as follows:  
\[\mu(h)=\frac{1}{N_{\text{QMC}}}\sum_{n=1}^{N_{\text{QMC}}}h_{n}. \tag{13}\]  
Figure 3: Graphs (**a**–**d**) represent the sampling numbers of 10, 50, 100, 500 two-dimensional Sobol sampling schematics, respectively. \(u_{1}\), \(u_{2}\) are two parameters in the range [0–1].  
The total variance \(\sigma^{2}(h)\) can be calculated by the following equation:  
\[\sigma^{2}(h)=\frac{1}{N_{QMC}}\sum_{n=1}^{N_{QMC}}(h_{n}-\mu(h))^{2}. \tag{14}\]  
Similarly, the first-order Sobol index for the \(m\)-th uncertain atmospheric environmental parameter is calculated as follows:  
\[S_{m}=\sigma_{m}^{2}/\sigma^{2}, \tag{15}\]  
where \(\sigma_{m}^{2}\) is the partial variance. It can be obtained by the following equation:  
\[\sigma_{m}^{2}=\text{Var}_{u_{m}}(\mathbb{E}_{\mathbf{k}_{\sim m}}(h\mid u_{m }))=\frac{1}{N_{QMC}}\sum_{n=1}^{N_{QMC}}h_{n}(\Psi_{2})(\text{h}_{n}(\Psi_{1} ^{m})-h_{n}(\Psi_{1})), \tag{16}\]  
where \(\Psi_{1}\) and \(\Psi_{2}\) are two independent Sobol sample sets from \(\mathbf{k}\). The format of the sample set is a matrix of \(N_{QMC}\times M\), where \(M\) denotes the number of uncertain atmospheric environmental parameter types. \(\Psi_{1}^{m}\) is the m-th column of the first sample set \(\Psi_{1}\) replaced by the mth column of the second sample set \(\Psi_{2}\). \(\mathbf{k}_{\sim m}\) denotes the set of all variables except \(\mathbf{k}_{m}\), and Var denotes the variance symbol. \(\mathbb{E}\) denotes the mathematical expectation symbol. The second-order Sobol index can be obtained by the following equation:  
\[S_{m_{1}m_{2}}=\sigma_{m_{1}m_{2}}^{2}/\sigma^{2}, \tag{17}\]  
which describes the mutual cross-effects of uncertain atmospheric parameters \(m_{1}\)-th and \(m_{2}\)-th on the tau-p model output \((m_{1},m_{2}\in\{1,\ldots,M\})\), in which  
\[\sigma_{m_{1}m_{2}}^{2}=\text{Var}_{u_{m_{1}}u_{m_{2}}}\Big{(}\mathbb{E}_{ \mathbf{k}_{\sim m_{1}m_{2}}}(h\mid u_{m_{1}},u_{m_{2}})\Big{)}=\frac{1}{N_{ QMC}}\sum_{n=1}^{N_{QMC}}h_{n}(\Psi_{2})\big{(}h_{n}\big{(}\Psi_{1}^{m_{1}m_{2}} \big{)}-h_{n}(\Psi_{1})\big{)}. \tag{18}\]  
Then, the total Sobol index [16; 17] is calculated as follows:  
\[\begin{split} S_{T_{m}}&=\frac{\mathbb{E}_{\mathbf{ k}_{\sim m}}(\text{Var}_{u_{m}}(h\mid\mathbf{k}_{\sim m}))}{\sigma^{2}}\\ &=S_{m}+\sum_{m_{1}\
eq m}S_{mm_{1}}+\sum_{m_{1}\
eq m,m_{2}\
eq m,m_{2}>m_{1}}S_{mm_{1}m_{2}}+\cdots(m=1,\ldots,M),\end{split} \tag{19}\]  
which describes the extent to which the cross-effect of the uncertain atmospheric parameter \(m\)-th with other atmospheric parameters affects the output of the tau-p model. With the increase of uncertain parameters \(M\), it is necessary to calculate the \(2^{M}-1\) Sobol index. To simplify the calculation, the total Sobol index is approximated as follows [17]:  
\[S_{T_{m}}\approx\frac{1}{2\sigma^{2}N_{QMC}}\sum_{n=1}^{N_{QMC}}(h_{n}(\Psi_{1 }^{m})-h_{n}(\Psi_{1}))^{2}. \tag{20}\]