## 4 Simulation Models in Marine Investigations
### Model 2: Rugged Seabed  
For further illustration, the seafloor topography in form of a rugged seabed is modeled in Figure 7 that had a cosine form with the period of 100 m and an amplitude of 10 m. The trench length \(l_{1}\) was 100 m, and the trench depth \(h_{2}\) was 20 m. The sea depth \(h_{1}\) was 15 m. The center of the loop source was at the origin of the axis, which was also just above the peak of the hump in the rugged seabed.  
Figure 8 shows the voltage response when the loop height is 10 m and the number of terms \(N\) for the Talbot algorithm is 8. Due to the rugged terrain of the seabed, different parts of the seabed produced different responses. At the location of the two trenches, two distinct humps in the curves were observed. However, the ridge part of the seabed yielded an apparent concavity in the curves. Since the conductivity of the ridge was smaller than that of the seawater in the trenches, this lead to a downward deviation between the response curves of the seawater and those of the ridge. Thus, the response generated by the ridge was smaller than that of the trenches. As time increased, the anomaly in the curves became gradually apparent. Other obvious anomalies were noted later, especially at 100 ms.  
Further, the value for the number of terms \(N\) was varied from 8 to 14, and the voltage response of the observation time of 100 ms is shown in Figure 9a. When the value of \(N\) was increased, the voltage response curve remained unchanged. However, its computation time considerably increased, which was consistent with the results obtained by the subsea tunnel model. Thus, the number of terms \(N\) was set as 8, which can meet the computing needing for this model. Although the rugged seabed was reflected directly when the loop source was placed at the height of 10 m, in practical TEM investigations, the loop source has a greater height; thus, the source height \(h_{0}\) was also analyzed for this model. In Figure 9b, when \(h_{0}\) was less than 20 m, there were two humps and a concavity corresponding to the location  
Figure 6: Voltage response at the observation time of 100 ms. (**a**) Voltage response for different values of \(N\); (**b**) Voltage response for different heights \(h_{0}\).  
Figure 7: 2.5D two-layered model with a rugged seabed.  
of the rugged terrain in the seabed. However, if \(h_{0}\) exceeded 20 m, the response curves became flat. In this modeling, it was noted that the rugged seabed was the anomaly. Considering the effect of the seabed terrain on the response, the maximum height of the source was different than that of the low-resistance body modeling. This demonstrates that in a TEM system a suitable detecting height of the source must be selected according to the subsea terrain and target. It was further validated that the Talbot algorithm provides favorable results for methods that require the inverse Laplace transform even when the Talbot algorithm has a smaller value of \(N\). In addition, the marine TEM method with a trapezoidal wave source works well for complex marine modeling.  
Figure 8: Profile map for the two-layered model with a rugged seabed when loop source height \(h_{0}\) was 10 m.  
Figure 9: Voltage response at the observation time of 100 ms. **(a)** Voltage response for different values of \(N\); **(b)** voltage response for different heights \(h_{0}\).