# Uncertainty Quantification for Infrasound Propagation in the Atmospheric Environment
## 2 Physical Model of Infrasound Propagation  
In the real atmospheric environment, the propagation speed of infrasound is affected by the wind speed \(\mathbf{u}\). The type of wind \(\mathbf{u}\) affecting infrasound propagation can be classified into meridional wind \(w_{m}\) and zonal wind \(w_{z}\). The effective sound velocity \(c_{eff}\) can be obtained by superimposing the wind velocity onto the sound speed \(c\)[30] at the receivingpoint. The effective sound velocity \(c_{eff}\) under the influence of wind speed \(\mathbf{u}\) is shown in Figure 1, and the effective sound velocity \(c_{eff}\) can be modeled as follows:  
\[c_{eff}=c+\mathbf{\hat{n}}\cdot\mathbf{u}=\sqrt{\frac{\gamma RT}{m}}+w_{u}\sin \phi+w_{v}\cos\phi, \tag{1}\]  
where \(\gamma\) is the specific heat ratio; \(T\) is the thermodynamic temperature; \(m\) is the atmospheric molar mass; \(R\) is the the universal gas constant equal to \(8.314\)\(J/(\mathrm{mol}\cdot\mathrm{K})\); \(\phi\) is the azimuth angle; \(\mathbf{\hat{n}}\) is the unit vector along the direction of the wind speed \(\mathbf{u}\). In this paper, the tau-p model [28] is used to simulate the propagation process of infrasound in a moving atmospheric environment (effect of wind speed). The infrasound propagation distance, maximum height of infrasound propagation, and infrasound propagation time in a phase loop can be derived from the tau-p model. The infrasound propagation distance \(R\) is the distance propagated from the bottom to the top of the atmospheric waveguide and then down the phase cycle. The specific formula is as follows:  
\[R(z,p)=2\int_{z_{0}}^{z(p)}\xi(z,p)\bigg{[}\frac{p}{(1-u(z)p)}+\frac{u(z)}{c^{ 2}}\bigg{]}dz, \tag{2}\]  
while the corresponding travel time \(t(z,p)\) of a phase loop is  
\[t(z,p)=2\int_{z_{0}}^{z(p)}\frac{1}{c^{2}}\xi(z,p)dz, \tag{3}\]  
where \(z_{0}\) is the surface altitude; \(z(p)\) is the maximum altitude of infrasound propagation. The transverse offset \(Q\) of the infrasound propagation model is calculated using the following equation:  
\[Q(z,p)=\int_{z_{0}}^{z(p)}\frac{1}{c^{2}}\xi(z,p)v(z)dz, \tag{4}\]  
where \(v(z)\) is the horizontal wind speed in the vertical propagation direction. The diagram of the infrasound propagation model is shown in part (a) of Figure 2. The model of infrasound propagation in the atmospheric environment is derived from the classical WKBJ (Wentzel-Kramers-Brillouin-Jeffreys) [31] ray theory. When the horizontal phase velocity \(V_{theta}=1/p\) matches the effective sound velocity \(c(z)+u(z)\), the infrasound propagation ray will turn. Different infrasound propagation ray trajectories can be obtained by changing the elevation angle of the emission, as shown in part (b) of Figure 2.  
Figure 1: Schematic diagram of effective sound velocity under wind speed. The \(X\)-axis indicates the meridional direction; \(Y\)-axis indicates the zonal direction. \(w_{z}\) is the zonal wind; \(w_{m}\) is the meridional wind. \(u_{1}\), \(u_{2}\) and \(v_{1}\), \(v_{2}\) are respectively the components of \(w_{z}\) and \(w_{m}\) along the direction of infrasound propagation and the components perpendicular to the direction of infrasound propagation. Each ray is represented by an invariant ray parameter \(p\).