## 3 Model Validation  
In this section, the computed values of the reflection coefficient are compared with the results calculated by ITM [6] to demonstrate the validation of the model. These two methods can be set with the same parameters for comparison, which overcomes the diffi  
Figure 3: Model of adjacent ice layer.  
culties of obtaining precise data from the changeable environment of the Arctic. The ITM is widely utilized to calculate the reflection coefficient of acoustic waves from multilayered materials. Therefore, the ITM was employed to verify the model. The principle of the ITM is to establish the relations of displacements and stresses between the first and the last layer. As for the displacement and stress field in the \(i\)th interface of ice, it can be described by the vector \(U(i)\):  
\[U(i)\ =\ [u_{xi},u_{zi},W_{zi},\tau_{zzi},\tau_{zxi},p_{fi}]^{\mathrm{T}} \tag{28}\]  
Furthermore, \(U(i)\) can be expressed by the amplitudes of displacement potentials:  
\[U(i)\ =\ A_{i}[\phi_{fi}^{+},\phi_{fi}^{-},\phi_{si}^{+},\phi_{si}^{-},\psi_{i}^ {+},\psi_{i}^{-}]^{\mathrm{T}} \tag{29}\]  
where \(A_{i}\) is the matrix of order 6, signifying the coefficient matrix of the \(i\)th interface of ice, and its elements can be deduced by Equations (8)-(13). In the paper, double-layered porous sea ice is chosen as the model to simplify the calculation. Then, from the bottom surface of ice to the top surface, the transfer matrix \(G\), which relates \(U(3)\) and \(U(1)\), is derived:  
\[U(1)\ =\ GU(3) \tag{30}\]  
\[G\ =\ A_{1}A_{2d}^{-1}A_{2u}^{-1}A_{3}^{-1} \tag{31}\]  
where \(A_{2d}\) and \(A_{2u}\) are the coefficient matrix of the second interface in the first layer and the second layer of ice. Based on the boundary condition at the ice-air half-space interface along the condition \(p_{f1}\ =\ -\beta p_{0}\) and \(\tau_{zx1}\ =\ 0\), the transfer matrix \(H\) from the air to the upper ice layer can be obtained:  
\[U(3){=}\ HU(a) \tag{32}\]  
where \(U(a)\ =\ [u_{a},p_{a}]^{\mathrm{T}}\). Therefore, the transfer matrix from the bottom surface of the ice to the ice-air half-space interface is deduced:  
\[U(1)\ =\ GU(3)\ =\ GHU(a) \tag{33}\]  
Furthermore, using the two remaining boundary conditions at the water-ice interface, the following can be derived:  
\[U(0){=}\ KU(1){=}\ KGHU(a) \tag{34}\]  
where \(U(0)\ =\ [u_{0},p_{0}]^{\mathrm{T}}\) and \(K\) is the transfer matrix. Solving Equation (34), the reflection can be calculated.  
For the double-layered model, the acoustic velocity and density are 1500 m/s and 1000 kg/m\({}^{3}\) in water and 343 m/s and 1.2 kg/m\({}^{3}\) in air, respectively. The density of the ice frame and the fluid in pores is 1.2 and 1000 kg/m\({}^{3}\), respectively. The porosity of the two layers is 0.1, and the other parameters of porous ice are given in Table 1. For long-range propagation of acoustic waves in the Arctic, the transmission loss increases dramatically with the increase in frequency. Therefore, we focused on the low-frequency region below 2000 Hz. Using the parameters in Table 1, reflection coefficients as a function of incident angle for sound frequencies of 500, 1000, 1500 and 2000 Hz were calculated by solving the system of equations method (EM) and the ITM. To better distinguish our method from the ITM in the following figures, the former method is described as the EM. Figure 4 shows the comparisons between the EM and the ITM on the basis of the same double-layered porous ice model.  
Curves of the reflection coefficient through EM and ITM nearly coincide with each other at different frequencies in Figure 4. The minor differences in those peaks are caused by the difference in accuracy of calculation between the two methods. Nevertheless, the similarity between the EM and ITM results is enough to confirm the validity of the model. Consequently, a conclusion can be drawn that the proposed double-layered porous model is valid, which simultaneously demonstrates the correctness of the multilayered and porous model.