# A Shallow Seafloor Reverberation Simulation Method Based on Generative Adversarial Networks
## 3 Simulation
### The Simulation Based on the Point-Scattering Method
#### 3.1.3 Time Domain Correlation Characteristics of the Reverberation Simulation  
Although the statistical characteristics of the reverberation signal depend on time, it can still be considered as a stationary random process locally or in a short time. Since the time-varying characteristics of the signal is much slower than itself, some random process analysis can still be used here.  
The time dependence of a reverberation signal is described by the autocorrelation function, if a reverberation signal is  
\[s(t)=s_{0}(t)cos[\omega_{0}(t)+\mathbf{\Phi}(t)] \tag{13}\]  
Figure 7: Amplitude statistical characteristics of LFM reverberation signal.  
then the autocorrelation function [44] is  
\[R(\tau)\approx\frac{1}{\delta}\bigg{\{}\int_{-\infty}^{\infty}s_{0}(t)s_{0}(t+ \tau)cos[\mathbf{\Phi}(t+\tau)-\mathbf{\Phi}(t)]dt\bigg{\}}cos\omega_{0}\tau \tag{14}\]  
After a little analysis of the above equation, it can be seen that the autocorrelation function of a reverberation signal is similar to that of its transmitted signal, and its autocorrelation function decays with the time. The duration of a reverberation signal is much longer than that of the transmitted pulse width. It has relevant characteristics similar to that of the transmitted signal within the duration. With the increase in duration, the amplitude gradually decreases. When the time correlation is close to zero, it means that the reverberation signal is basically over.  
Figure 8a shows the autocorrelation function of the CW signal, and Figure 8b shows the autocorrelation function of the CW reverberation signal. Comparing the two figures, we can find that the autocorrelation function of reverberation and signal is similar. The autocorrelation function is the largest in the middle moment, and the middle moment corresponds to the moment of 0. After that, the amplitude gradually decreases with the increase in time, and finally tends to 0, which is consistent with the theoretical analysis. Figure 9 is similar to Figure 8.  
Figure 8: Autocorrelation functions of CW signal and CW reverberation signal.  
In this study, the time domain correlation characteristics of the reverberation signal are described by its autocorrelation function. The time domain correlation functions of the CW signal, the CW reverberation signal, the LFM signal, and the CW reverberation signal are plotted as follows.  
From a theoretical point of view, the calculation amount of the point-scattering model is influenced by the period of the emitted signal, the density of the ocean scatterer, and the sampling frequency. The area of the scattering element in the reverberant zone is first determined according to the emission signal period, then the number of sampling points in its next period is determined according to the sampling frequency. Finally, the calculation amount of each sampling point is determined according to the density of the scatterer. In the calculation process, when any one of the above three factors is amplified, the overall calculation cost tends to increase by hundreds or thousands of times. In this case, it is necessary to wait a long time for each reverberation simulation. In addition, the nonuniversality of some parameters in the scattering model, for example, the scatterer density which changes with the specific sea area, leads to a reduction in the generalizability of the reverberation model.  
Figure 9: Autocorrelation functions of the LFM signal and the LFM reverberation signal.