## 5 Model Studies
### Pyramid Type Bathymetry Model  
Further verification of the new algorithm was performed by considering a pyramid-type bathymetry model (Figure 15) without a reservoir. The top area of the pyramid-type bathymetry model was 0.5 km \(\times\) 0.5 km in square, at 990 m depth. The bottom area was 1.0 km \(\times\) 1.0 km square, at 1000 m in depth. Air and sea water were 1000 m thick. In the domain of pyramid-type bathymetry, the depth of seawater ranges from 900 m to 1000 m. Figure 16 shows the pyramid-type bathymetry model with a reservoir. The domain of the reservoir was \(\Omega=\{-1000\) m \(\leq x,y\leq 1000\) m, 1500 m \(\leq z\leq 1600\) m}. The modeling domain was \(\Omega=\{-4\) km \(\leq x,y\leq 4\) km, \(-1\) km \(\leq z\leq 3\) km}. In these models, the conductivities of air, seawater, and sediment were the same as previously, whereas the reservoir's conductivity was 0.05 s/m. The electric dipole source was \(x\)-oriented at \((-3000,\ 0,\ 800)\) m at 1 Hz frequency, and the receiving station was located at \(z=800\) m.  
Figure 16: Section view of the grid used for the bathymetry model with reservoir. The reservoir occupies the blue area, and the current source is located at the red star. The dark blue, pink, blue, and green areas represent the air layer, sea water, oil and gas, and bedrock, respectively.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Element** & **Number of Elements** & **DOF** & **CPU Time (s)** \\ \hline SC & 560,000 & 6,679,300 & 141.7 \\ SN & 560,000 & 4,999,050 & 156.2 \\ LC & 560,000 & 9,998,100 & 311.4 \\ LN & 560,000 & 8,317,850 & 471.9 \\ \hline \hline \end{tabular}
\end{table}
Table 5: The cost-time of CPU when using different methods.  
Figure 15: Section view of the grid used for the bathymetry model without reservoir. The red star locates the dipole source. The dark blue, pink, blue, and green areas represent the air layer, sea water, oil and gas, and bedrock, respectively.  
Figures 17-20 show the secondary electric component in the \(x\)-direction of the bathymetry model, which has no reservoir or one reservoir in the isotropic sediment. Figures 21-24 show a comparison of the \(x\)-component of the secondary electric field of the bathymetry model with and without a reservoir in anisotropic sediment. Bathymetry and anisotropic background clearly affected the distribution of the marine controlled-source electromagnetic field (Figures 18-24), which should be considered in the processing of MCSEM data. Figures 25 and 26 show a comparison of the degrees of freedom (DOF) and CPU time using different elements, respectively. Clearly, these two new elements contributed significantly to reducing the CPU time in computing.  
Figure 21: No-reservoir bathymetry model in anisotropic sediment. Second electric component in the \(x\)-direction, which was calculated by using different quadratic edge elements.  
Figure 22: No-reservoir bathymetry model in anisotropic medium. Amplitude comparison between background field (blue line) and total field (dashed red line) for \(E_{x}\) component, at \(y=0\) m.  
Figure 23: Reservoir-embedded bathymetry model in anisotropic sediment. Second electric component in the \(x\)-direction, which was calculated by using different quadratic edge elements.  
Figure 24: Bathymetry model with a reservoir in anisotropic medium. Amplitude comparison between background field (blue line) and total field (dashed red line) for \(E_{x}\) component, at \(y=0\) m.  
Figure 25: A comparison of DOF using different elements. Model 1 = bathymetry + isotropic sediment; model 2 = bathymetry + isotropic sediment + reservoir; model 3 = bathymetry + anisotropic sediment; model 4 = bathymetry + anisotropic sediment + reservoir.