# A Shallow Seafloor Reverberation Simulation Method Based on Generative Adversarial Networks
## 2 Methodology  
### The Traditional Point-Scattering Method  
Traditional seafloor reverberation simulation theory is generally based on two types of scattering models, namely the point-scattering model and the element-scattering model. Based on a statistical approach, the point-scattering model assumes that scatterers are randomly distributed throughout the ocean. The reverberation is calculated by summing the echoes of each individual scatterer. The point-scattering model is applicable to high-frequency sonar and subglacial scattering [7]. In this study, the point-scattering model is used for simulation as a comparison. As an example, the model of the monostatic devices is shown in Figure 1.  
During a period \(T\), the scattering area corresponding to the transmitted signal can be roughly viewed as a fan ring. This region of the fan ring is called the sampling element. The azimuth of the sampling element is limited by the beam pattern and the distance is limited by the period of the transmitted signal. The area of the sector ring can be calculated based on the speed of sound and the period of the wavelength. After calculating the area of the fan ring, the number of scatterers in the fan ring can be calculated based on the scatterer density. Based on the above information, the reverberant signal can be modeled as follows [36]:  
Figure 1: Schematic diagram of the point-scattering model.  
\[R(t)=\mathcal{F}[\sum_{i=1}^{N(t)}S(r_{i},\varphi_{i})f^{2}(d(r_{i},\varphi_{i}))B_{ TR}(r_{i},\varphi_{i})x^{+}(t-\tau(r_{i},\varphi_{i}))e^{j\phi_{i}}] \tag{1}\]  
where \(r_{i}\) is the distance coordinate, \(\varphi_{i}\) is the azimuth coordinate, \((r_{i};\varphi_{i})\) is the coordinate vector of the \(i\)th scatterer, \(S\) is the scattering term, \(B_{TR}\) is the beam pattern of the monostatic sonar, \(f\) is the propagation term, \(d\) is the distance between the scatterer and the sonar, \(\tau\) is the two-way propagation time delay, \(x^{+}(t)\) is the analytic signal of the emitted signal \(x(t)\), and \(\phi_{i}\) is the random phase perturbation in the scattering process, uniformly distributed in \([0,2\pi]\).  
Usually, \(S,f,\) and \(B_{TR}\) exhibit slow fluctuation characteristics relative to \(x^{+}(t)\) in Equation (1). When the signal period \(T\) is small, \(x^{+}(t)\) also exhibits slow fluctuation characteristics within the sampling element. In this case, the reverberation model can be expressed as [37]  
\[R(t)\approx\mathcal{F}[\sum_{k=1}^{N_{a}(kT)}I(kT)x^{+}(t-kT)] \tag{2}\]  
where \(I(kT)\) denotes the result of the common action of the scatterers about \(Sf^{2}B_{TR}e^{j\phi}\) in the \(k\)th sampling element, which is calculated as follows:  
\[I(kT)=\sum_{i=1}^{N_{a}(kT)}S(r_{i})f^{2}(d(r_{i}))e^{j\phi_{i}} \tag{3}\]  
where \(N_{a}\) is the number of the scatterers in the sampling unit. In Equation (3), the scattering item \(S\) can be expressed as [37]  
\[S=\sqrt{10^{\frac{\mu}{\beta}}sin(\alpha_{i})sin(\alpha_{o})} \tag{4}\]  
where \(\mu\) is vertical scattering coefficient, \(\alpha_{i}\) is the incident grazing angle, and \(\alpha_{o}\) is the scattering grazing angle. In the case of monostatic sonar, \(\alpha_{i}\) is equal to \(\alpha_{o}\). The propagation term \(f\) can be expressed as follows [38]:  
\[f(d)=\frac{1}{d}10^{-\beta d}v \tag{5}\]  
where \(\beta\) represents the absorption coefficient of seawater.