## 4 Simulation Verification  
The echo of any target is composed of several sub-echoes iteratively; each sub-echo can be regarded as emitted from a scattering point, which is the bright spot. Therefore, we can equate any target to a combination of several bright spots. The acoustic decoys are usually modeled as multi-highlight echoes to simulate scale targets [20,21]. When the echo simulation is performed by suspended acoustic decoys, the virtual-scale target simulation is achieved by simulating the echoes of the bright spots of the suspended acoustic decoys at both ends of the torpedo course plumb line; in addition, the time delay simulation amount of the rest of the acoustic decoys is determined according to their distance from the bright spots of their corresponding virtual-scale targets. When the echo simulation is performed by the mobile acoustic decoy line array, each array element performs delay superposition and echo time broadening on the torpedo-seeking signal, according to the incoming torpedo posture, respectively, to realize the virtual-scale simulation of the submarine target. Among them, the bright spot echo model of a single suspended acoustic decoy and a single array element of a homing acoustic decoy array can be determined based on the three parameters of amplitude factor, time delay, and phase jump. The transfer function can be expressed as:  
\[H_{i}(r,\theta,\psi,\omega)=A_{i}(r,\theta,\psi)e^{i\omega(r)\tau_{i}}e^{i\varphi} \tag{10}\]  
where \(r\) is the distance from the torpedo to the target bright spot; \(A_{i}(r,\theta,\Psi)\) is the amplitude of the echo of this bright spot, which is related to the distance \(r\) of the target and the direction of acoustic wave incidence, i.e., the incidence angle \(\theta\) and the pitch angle \(\Psi\). \(\tau_{i}\) denotes the time delay of this bright spot, determined by the acoustic range of the equivalent acoustic center concerning some reference point, and is a function of \(\theta\). \(\omega(r)\) denotes the difference between the echo's center frequency and the incident wave's center frequency, due to the target's relative motion by a Doppler frequency shift; \(\varphi\) is the phase jump of the return wave. Therefore, the total transfer function of the target is:  
\[H_{i}(r,\theta,\psi,\omega)=\sum_{i=1}^{N}A_{i}(r,\theta,\psi)e^{i\omega(r) \tau_{i}}e^{i\varphi} \tag{11}\]  
where \(N\) is the number of target highlights. As the plate element analysis method is capable of accurate prediction of underwater complex target geometry backscattering, we built a benchmark submarine scattering model by plate element method [22,23]. Based on the high-frequency approximation of [PERSON]'s  
Figure 8: The correlation between apparent scale and echo broadening. integral formula, we can divide the target surface into plate elements, calculate the scattered sound field for each plate element separately, and obtain the whole scattered sound field after superposition. The Kirchhoff formula for the acoustic scattering problem is [24]:  
\[\varphi_{s}(r_{2})=\frac{1}{4\pi}\iint\limits_{S}\left[\varphi_{s}\frac{\partial }{\partial n}\left(\frac{\mathrm{e}^{\mathrm{i}kr_{2}}}{r_{2}}\right)-\frac{ \partial\varphi_{s}}{\partial n}\frac{\mathrm{e}^{\mathrm{i}kr_{2}}}{r_{2}} \right]ds \tag{12}\]  
where \(s\) is the scatterer surface, \(n\) is the surface outer normal, \(r_{2}\) is the scattering point vector diameter, \(\varphi_{s}\) is the scattering potential function, and \(k\) is the wave vector. The Kirchhoff approximation is a high-frequency approximation that typically assumes that:  
The contribution of the geometric shadow region to the scattered field is neglected. The object surface satisfies the rigid boundary conditions. Specifically, the scattering potential function under rigid boundary conditions is equal in size to the incident potential function, and the angle is symmetric about the target:  
\[\left\{ \begin{array}{l}|\varphi_{s}|=|\varphi_{i}|\\ \frac{\partial(\varphi_{s}+\varphi_{i})}{\partial n}=0\end{array} \right.