# Research on Passive Ranging Technology of Moving Ship Based on Vertical Hydrophone Array
## 3 Simulation  
In this section, a simulation is carried out to verify the performance of the above-proposed method. The concrete simulation setting is as follows. The source radiates a broadband signal with a frequency band from 200 Hz to 1000 Hz and moves with a constant speed of 3.5 m/s. The source depth is 50 m. The range and time of the CPA point are 500 m and 500 s, respectively. The moving time of the ship is from 0 s to 1800 s, and the corresponding range between the sound source and the receiving hydrophone is from 500 m to 4568 m. Figure 4 shows the curve of the target range versus time. A vertical hydrophne array is adopted to collect the pressure information of the ship-radiated noise. It consists of 16 elements. The hydrophones at different depths receive different signals due to a variety of scattering and reflection in the sound field. The range estimation results obtained by each element are also different. In order to reflect the differences in the received signal, we designed different signal-to-noise ratios (SNR) for the hydrophones at different depths. The algorithm proposed in this paper makes no requirements on the sound field model and is not sensitive to environmental parameters. In order to prove that the method is adaptable to different sound field environments, we used a sound speed profile (SSP), which was measured at a sea depth of 2200 m. The concrete parameter settings such as the water density and sea floor attenuation coefficient are based on the experience value and presented in Figure 5.  
As described above, the high-frequency signal is modulated by a low-frequency signal. However, the low-frequency line spectrum in the wideband spectrum is unstable, so a DEMON spectrum analysis is used to extract the low-frequency line spectrum components. In this case, we set the modulation frequency to \(f=14\) Hz and employ the normal mode model to simulate the pressure signal. Figure 6 presents the wave of the broadband noise in the time domain and the DEMON spectrum result demodulated for a single element. It is clear that the wideband signal has a low-frequency envelope, and the demodulated line spectrum is stable at 14 Hz. Figure 4: The range from the moving source to the receiving point with time. Figure 5: Simulated sound speed profile in the deep-sea. The interference fringes with the changing time can be observed when the time interval \(\Delta t\) varies from 1 s to 200 s, as shown in Figure 7a. The period of the interference oscillations \(d_{t}\) can be obtained through calculating the Fourier transform of the sound intensity of the pressure difference. Then, the estimated radial velocity versus time is computed by Equation (16). As presented in Figure 7b, the yellow line represents the radial velocity estimation result, and the blue line stands for the actual radial velocity. For the CPA point, the estimated radial velocity, the range and the time are 3.7 m/s, 555 m and 496 s. Note that there is a widening of the estimated radial velocity near the CPA point because the radial component of the moving velocity in the region near the CPA point is quite small and even close to zero. Figure 7c compares the estimated range and actual range, which are represented by the solid red line and the blue dotted line, respectively. The comparison result shows that the two curves basically coincide. The two curves are relatively close, especially at the CPA point. However, the deviation increases over time. In order to describe the difference between the estimation value and true value at each time, we introduce a parameter, namely the relative error of the range estimation. It is defined as:  
\[\sigma_{r}=\left(\frac{r-R_{true}}{R_{true}}\right) \tag{27}\]  
where \(r\) is the range estimation, and \(R_{true}\) is the true range versus time. The corresponding calculation result is shown in Figure 7d. One can see that the relative error of the estimated result is stable at about 0.1 and changes little with time.