## 2 Theoretical Modeling
### Calculation of Displacements, Pressures and Stresses  
To calculate the reflection coefficient of the plane wave from the model, first, the relations between displacement potentials and displacements, stresses or pressures in different media are deduced. Then, a set of simultaneous equations for the amplitudes of displacement potentials is derived using the boundary conditions of the displacements, stresses or pressures at each interface. Eventually, the reflection coefficient can be calculated by solving these equations. When a plane wave with the angular frequency \(\omega\) propagates from water into sea ice, the sound field in water contains the incident and reflected waves. However, in air, only the propagated upward wave exists because of its half-space. Therefore, the displacement potentials in water and air are as follows:  
\[\phi_{0}\ =\ e^{i(\omega t+a_{0}z-k_{x}x)}+Ve^{i(\omega t-a_{0}z-k_{x}x)} \tag{1}\]  
\[\phi_{a}\ =\ \phi_{a}^{+}e^{i(\omega t+a_{a}-k_{x}x)} \tag{2}\]  
where \(\alpha_{0}\ =\ \sqrt{k_{0}^{2}-k_{x}^{2}}\), \(k_{x}\) and \(k_{0}\) refer to the vertical wavenumber, horizontal wavenumber and wavenumber in water, respectively; \(V\) is the reflection coefficient of the plane wave; \(\alpha_{a}\ =\ \sqrt{k_{a}^{2}-k_{x}^{2}}\) is the vertical wavenumber in air and \(k_{a}\) is the wavenumber in air. Then, the normal displacement \(u_{z0}\), \(u_{za}\) and the pressure \(p_{0}\), \(p_{a}\)[19] in water and air are given by the following:  
\[u_{z0}\ =\ \frac{\partial\phi_{0}}{\partial z},\ u_{za}\ =\ \frac{\partial\phi_{a}}{ \partial z} \tag{3}\]  
\[p_{0}\ =\ \rho_{0}\omega^{2}\phi_{0},\ p_{a}\ =\ \rho_{a}\omega^{2}\phi_{0} \tag{4}\]  
where \(\rho_{0}\) and \(\rho_{a}\) are the density of water and air. Taking the \(i\)th layer of sea ice as an example, the sound field contains three reflected and three refracted waves. Thus, the displacement potentials of the fast longitudinal wave, slow longitudinal wave and transverse wave can be written as follows:  
\[\phi_{fi}\ =\ \phi_{fl}^{+}e^{i(\omega t+a_{fl}z-k_{x}x)}+\phi_{fi}^{-}e^{i( \omega t-a_{fl}z-k_{x}x)} \tag{5}\]  
\[\phi_{si}\ =\ \phi_{si}^{+}e^{i(\omega t+a_{si}z-k_{x}x)}+\phi_{si}^{-}e^{i( \omega t-a_{si}z-k_{x}x)} \tag{6}\]  
\[\psi_{i}\ =\ \psi_{i}^{+}e^{i(\omega t+\sigma_{i}z-k_{x}x)}+\psi_{i}^{-}e^{i( \omega t-\sigma_{i}z-k_{x}x)} \tag{7}\]  
where \(\alpha_{fi}\ =\ \sqrt{k_{fl,i}^{2}-k_{x}^{2}}\), \(\alpha_{si}\ =\ \sqrt{k_{sl,i}^{2}-k_{x}^{2}}\) and \(\sigma_{i}\ =\ \sqrt{k_{l,i}^{2}-k_{x}^{2}}\) and \(k_{fl,i}\), \(k_{sl,i}\) and \(k_{l,i}\) are the vertical wavenumbers and wavenumbers of the three kinds of waves, respectively; \(\phi_{fi}^{+}\phi_{fi}^{-}\phi_{fi}^{+}\phi_{si}^{-}\), \(\psi_{si}^{+}\) and \(\psi_{i}^{-}\) are the amplitudes of the three kinds of waves to be determined and the subscripts \(+\) and \(-\) signify waves propagating upward (i.e., in the positive \(z\)-axis) and downward, respectively. Figure 2: Cross-section of the \(i\)th ice layer. The normal displacements of the ice frame and the fluid in pores can be represented by displacement potentials as follows:  
\[u_{xi} = \frac{\partial(\phi_{fi}+\phi_{si})}{\partial x}-\frac{\partial\psi_{ i}}{\partial z} \tag{8}\] \[u_{zi} = \frac{\partial(\phi_{fi}+\phi_{si})}{\partial z}+\frac{\partial \psi_{i}}{\partial x}\] (9) \[\phi_{a}\ =\ \phi_{a}^{+}e^{i(\omega t+a_{a}-k_{x}x)} \tag{2}\]0  
where \(r_{i1}\) and \(r_{i2}\) represent the ratio of the velocity of the fluid in pores over the velocity of the frame for the fast longitudinal and slow longitudinal waves, and \(r_{i3}\) is the ratio of the amplitude of displacement of the fluid and of the frame. Then, according to the stress-strain relations in [PERSON]'s theory [20], the normal and tangential stresses of the ice frame and the pressure of fluid in pores can be obtained:  
\[