## 4 Simulation and Result Analysis  
### Equivalent Modeling in a Unified Spatiotemporal Reference Frame  
The calculation method for the contribution rate of the ELF magnetic field was established in the previous section. Next, an analysis and evaluation of the contribution rates are conducted using three typical ELF magnetic fields as examples: the wake magnetic field, shaft frequency magnetic field, and equipment magnetic field.  
Under a unified spatiotemporal reference, simulations of the shaft frequency magnetic field and the equipment magnetic field are performed using arrays of electric dipoles and magnetic dipoles [21,22]. The arrays are uniformly distributed along the length of the submarine, with different operating conditions corresponding to the simulated source strengths. The Kelvin wake model is used to calculate the wake Debye magnetic field [23]. According to relevant sources, the cruising speed of a conventional submarine is approximately 3 m/s, with a rotational speed of 1 r/s [24].  
The detection line at the stern selects detection points on a measuring line away from the stern of the submarine (with the stern of the submarine as the origin), taking  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Matrix Order** & **3** & **4** & **5** & **6** & **7** & **8** & **9** \\ \hline RI & 0.58 & 0.90 & 1.12 & 1.24 & 1.32 & 1.41 & 1.45 \\ \hline \hline \end{tabular}
\end{table}
Table 3: RI values for three matrix orders from 3 to 9.  
10 detection points along the longitudinal direction of the submarine on the measuring line, with 1 detection point placed every \(L\) meters (\(L\) is the length of the submarine) [24]. The radial detection line is chosen at a point 10 m from the middle point on the right side of the submarine, with one detection point placed every 50 m. A schematic diagram of the simulation scenario is shown in Figure 3, and the simulation parameter settings are presented in Table 4. In Table 4, the parameters presented are fixed values utilized within the simulation, representing both the submarine model parameters and the ocean environmental parameters.  
In accordance with the arrangement of the magnetic detection points mentioned above, this study analyzed the magnitude and signal characteristics of the magnetic field at the detection points. Through the simulation studies of the three typical ELF magnetic fields described, the simulated signals were obtained, and actual measured ocean noise from an experiment was added to the simulation signals. The experiment was conducted in Hainan, China, in August 2023, using a CS-VL magnetometer to measure real-world environmental noise.  
Figure 4 illustrates the time-domain simulation signal diagram at a measurement point 100 m from the submarine's stern.  
\begin{table}
\begin{tabular}{c c} \hline \hline
**Parameters** & **Numerical Value** \\ \hline Length \(L\) & 100 m \\ Breadth \(D\) & 16.67 m \\ Geomagnetic field strength \(F\) & \(5\times 104\) nT \\ Magnetic inclination \(I\) & \(\pi/4\) \\ Seawater conductivity \(\sigma\) & \(\sigma=4\) s/m \\ Relative dielectric constant \(\epsilon_{\mathrm{r}}\) & \(\epsilon_{\mathrm{r}}=81\) \\ \hline \hline \end{tabular}
\end{table}
Table 4: Simulation parameter settings.  
Figure 4: Original simulation signal diagrams for the three types of magnetic fields: (**a**) shaft frequency magnetic field signal; (**b**) wake magnetic field signal; (**c**) equipment magnetic field signal.  
Figure 3: Schematic diagram of simulation scene. (**a**) Simulation diagram of a dipole and wake magnetic field; (**b**) Schematic diagram of the lateral line position.