## 3 Proposed Self-Demodulation-Based Visualization Method
### Simulation Initialization  
The simulation conditions are set at the beginning of the initialization. As shown in Figure 3, we can divide the simulation initialization process into three steps: environment, sound source, and initial acoustic rays. Step 1: The initialization of the environment  
The initialization of the environment is almost same as other acoustic simulations. In this step, the simulation space and acoustic parameters are defined. The acoustic parameters include the speed of sound \(c\), the frequency-dependent attenuation rate \(\alpha\), the nonlinear factor of air \(\beta\), and the density of the air \(\rho\). In the case that the reflector exists in the simulation space, the reflection attenuation \(\eta\) should also be defined. In this method, we also define several parameters to keep the possibility of utilizing the practically measured results to define acoustic parameters, and their definitions will be given in the following description. Step 2: The initialization of the sound source  
In the \(xyz\)-coordinate system, the center of the PAL is set at  
\[\mathbf{u}_{\rm o}=\big{[}u_{\rm ox},u_{\rm oxy},u_{\rm ox}\big{]}^{T}. \tag{6}\]  
The propagation axis of the PAL is set as  
\[\mathbf{v}_{\rm o}=\big{[}v_{\rm ox},v_{\rm oxy},v_{\rm ox}\big{]}^{T}, \tag{7}\] \[\text{s.t.}\ v_{\rm ox}^{2}+v_{\rm oxy}^{2}+v_{\rm ox}^{2}=1. \tag{8}\]  
Other aspects of the sound source are also defined at the beginning, such as the shape, size, and directional angle of the target PAL. The carrier frequency \(f_{\rm c}\) and sideband frequency \(f_{\rm s}\) and their corresponding sound pressure are also defined. The above definitions will be used in the next step to set the initial acoustic rays. Step 3: The initialization of the initial acoustic rays  
The position vectors and direction vectors of acoustic rays for \(i=0\) are initialized based on the shape, size, and directional angle of the target PAL. When the target PAL has a rectangular shape, the position vectors are evenly distributed on the PAL surface. When the target PAL is circular (e.g., MEE or PS-60E), the position vectors are distributed as concentric circles on the PAL surface. In Figure 4, we give an example of initialization for a circular PAL. The position vectors are distributed on the \(yz\)-plane and then extended to three-dimensional space with affine transformation and Rodrigues' rotation formula. As shown in Figure 4, when the \(z\)-axis is the propagation axis (\(\mathbf{v}_{\rm o}=[0,0,1]^{T}\)), the position vectors on the \(yz\)-plane can be given as  
\[\mathbf{u}^{\prime}(\mu)=\big{[}u_{y}^{\prime}(\mu),u_{z}^{\prime}(\mu)\big{]}^{T}, \tag{9}\]where \(\mu=-M^{\prime},\cdots,-1,0,1,\cdots,M^{\prime}\) denotes an index of initial positions. In this example, the initial positions are equally distributed along \(y\)-axis; therefore, we have  
\[u^{\prime}_{y}(\mu) = y_{0}+\mu W/(2M^{\prime}+1), \tag{10}\] \[u^{\prime}_{z}(\mu) = z_{0}, \tag{11}\]  
where \(W\) denotes the diameter of PAL. The corresponding direction vectors can be given as  
\[\mathbf{v}^{\prime}(\mu)=\left[\sin(\varphi(\mu)),\cos(\varphi(\mu)) \right]^{\mathrm{T}}, \tag{12}\] \[\varphi(\mu)=\mu\Phi/M^{\prime}, \tag{13}\]  
where \(\Phi\) is the PAL directional angle. By rotating the \(yz\)-plane around the \(z\)-axis, we can obtain all the initial position vectors and direction vectors for acoustic rays, and we re-index them with index \(m=0,1,2,\cdots,M-1\), as shown in Figure 4c. Here, the acoustic ray indexed with \(m=0\) is located at the center of the target PAL, and the direction of the propagation axis remains constant. The initialization defines the initial position vector \(\mathbf{u}(m,0)\) and the initial direction vector \(\mathbf{v}(m,0)\) for each acoustic ray.