# Three-Dimensional Wide-Band Electromagnetic Forward Modelling Using Potential Technique
## 3 Results and Discussion
### Model 2  
We consider a model consisting of two anomalies embedded in the two-layer background earth in a marine environment as indicated in Figure 4. The resistivity of the reservoir (hydrocarbon) is 100 \(\Omega\) m with a size of 5000 \(\times\) 5000 \(\times\) 100 m\({}^{3}\) while the resistivity of the salt dome is 33 \(\Omega\) m and its volume 3000 \(\times\) 3000 \(\times\) 5000 m\({}^{3}\). The incident electromagnetic wave is vertically propagated at frequency of 0.3 Hz.  
Figure 3: The plot of the (**a**) apparent resistivity (**b**) phase for the \(E^{x}\) polarization using the finite volume (FV) approach and the analytical (A) method. And the (**c**) apparent resistivity relative error (**d**) phase relative error of the finite volume method. The solid dash green lines indicate the analytical (A) solution while the red circle denotes the finite volume (FV) approach. And the solid dash red lines indicate the relative error for the FV method.  
Figure 2: Vertical cross-section view of the model \(1\) with the anomaly in a homogeneous half-space.  
A comparison of our finite volume techniques using potential formulation was carried out with the finite volume method (Direct E) from [40]. For this investigation, the number of cells for the computational domain implemented are 45 \(\times\) 45 \(\times\) 35 while the dimension of the grid is 15, 15, 35 km in the x, y, z direction. The plots of the real and imaginary components for the electric and magnetic field of the \(E^{x}\) polarization are shown in Figures 5 and 6. The comparison indicates that both numerical schemes show an excellent agreement.  
Furthermore, BiCGStab and QMR with block ILU and other preconditioners are examined to explore the best solver for this example. Figures 7 and 8 signify the respective comparison of the convergence for BiCGStab and QMR with different preconditioners such as block ILU, SSOR, and SGS for the marine model. As revealed in the Figures 7 and 8, the block ILU preconditioner shows the best convergence rate for both solvers among the tested preconditioners. With the same preconditioner, the BiCGStab solver outperforms the QMR, e.g., the block ILU for the BiCGStab converges at 45 iterations with computing time of 6 s while 140 iterations with a run time of 18 s for the corresponding QMR. Thus, the utilization of the block ILU preconditioner with BiCGStab iterative solver is favorable to simulate the EM responses for the marine model.  
Figure 4: The planar and vertical cross-section view of the marine model 2. The anomalous bodies are embedded in a homogenous layer below the sea water.  
Figure 5: The plot of the \(E^{x}\) polarization (real and imaginary) calculated using the Potential and Direct E formulation.  
Figure 8: The convergence plot of various preconditioners with QMR for marine model 2.  
Figure 6: The plot of the \(H^{y}\) for the \(E^{x}\) polarization (real and imaginary) calculated using the Potential and Direct E formulation.  
Figure 7: The convergence plot of various preconditioners with BiCGStab for marine model 2.