## 4 Simulation Models in Marine Investigations
### Model 1: Subsea Tunnel  
As shown in Figure 4, an immersed tunnel (similar to that of the Hong Kong-Zhuhai-Macau Bridge) was modeled. To simplify the analysis, the tunnel was regarded as a rectangle having a size of \(10\,\mathrm{m}\times 40\,\mathrm{m}\) (height and width) and a buried depth of \(10\,\mathrm{m}\) beneath the seabed. The sea depth \(h_{1}\) was \(30\,\mathrm{m}\), and the conductivities \(\sigma_{0}\), \(\sigma_{1}\), \(\sigma_{2}\) were set to \(0\), \(4\), and \(0.5\,\mathrm{S/m}\), respectively. The tunnel equivalent conductivity \(\sigma_{a}\) was assumed to be \(10^{3}\,\mathrm{S/m}\). The loop source was moved from \(-100\,\mathrm{m}\) to \(100\,\mathrm{m}\) on the x-axis, and the center of the rectangle was on the y-axis, which was just above the origin of the axis. The loop height \(h_{0}\) was set to \(10\,\mathrm{m}\).  
The number of terms \(N\) for the Talbot algorithm was first set as \(8\). In Figure 5, the response curves are flat with high amplitudes at early times. However, an abnormal hump becomes evident near \(25\,\mathrm{m}\)s. As the conductivity of the tunnel is greater than that of the seabed, the field encountering the tunnel yielded stronger responses than that springing from the seabed. In addition, because the secondary field decayed slowly in seawater, the hump became gradually more apparent at later stages. Moreover, the peak of the hump appeared at the sounding point of \(0\,\mathrm{m}\). It can be thought that the TEM response reflects the subsea tunnel directly. Furthermore, it was shown that the Talbot algorithm matched with a trapezoidal wave source in the airborne TEM can reflect the low-resistance body distinctly.  
Figure 4: 2.5D model for a subsea tunnel buried in layered medium.  
In Figure 5(a), the number of terms \(N\) is changed from 8 to 14, and the voltage response is selected at the observation time of 100 ms because the anomaly in the curve is more apparent at this time. As the value of \(N\) increased, the voltage response curve remained almost constant. When \(N\) had a smaller value, the Talbot algorithm with the trapezoidal wave in the FEM method yielded a high accuracy for the subsea tunnel model. However, with increasing \(N\), the computation time also increased. Thus, the number of terms \(N\) was set as 8 for this model. The impact of the loop source height on this model was also analyzed; the voltage response is presented in Figure 5(b). The response was also chosen at the observation time of 100 ms. As the source height \(h_{0}\) was increased, the voltage response curve gradually became flatter. At greater height, the hump gradually vanished, and flat curves were recovered. Along with the increase of \(h_{0}\), the field diffusing into the seawater was reduced. Due to the strong attenuation of the field in seawater, the detecting depth decreased, and the tunnel had little impact on the response. This demonstrates that the Talbot algorithm with a trapezoidal wave source has good performance for the sounding of a conductive target in the seabed within the range allowed by the detecting depth.  
Figure 5: Profile map for the subsea tunnel model in the layered medium when the loop source height \(h_{0}\) is 10 m.  
Figure 6: _Cont._