Figure 4 shows the results of the appraisal in terms of the apparent resistivity and phase in the TM mode. We henceforth call the calculated result by Utada’s (1987) original 2-D triangular FEM code, that with Li et al.s (2008) method, and that by Ogawa and Uchida’s (1996) 2-D rectangular FEM code, as T0, T1 and R0, respectively. The numerical responses were calculated for the periods 8, 16 and 32 s. The skin depths in seawater are approximately 0.7, 1.0 and 1.4 km, respectively, which are much larger than the radius of the hemi-cylinder, 50 m. This means that the condition of the non-decaying horizontal electric field was fulfilled.
It is evident from the figure that the MT responses of R0 are very different from those of the analytical solution. In particular, the biggest discrepancies between them, both in apparent resistivity and phase, occur at the edge of the hemi-cylinder. This means that one should be very careful in applying the rectangular FEM code, especially in the vicinity of coastlines. On the other hand, both the T0 and T1 responses fit the analytical solution at the coastline very well. The reason why only R0 failed to reproduce the analytical solution is because the simulation of bathymetric slopes using rectangular elements are much inferior to that using triangular elements. This can be attributed to the presence of rectangular steps along the seafloor and at the coastline. In the rectangular grid, vertical walls arising from of the steps, even if they are small, cause zigzag electric currents at each small step. For plane wave sources, electric currents tend to flow in the horizontal direction basically. If they encounter a resistive wall in seawater, they will be deflected to flow vertically. As a result, the deflected electric currents finally concentrate at the wedge of seawater near the coastline. This implies that discretization of bathymetry, especially in the vicinity of coastlines, is very important for the accurate evaluation of MT responses on the seafloor and at the coast. The fact that the largest discrepancy in the calculated responses is present at the coast supports this conjecture.
Furthermore, in our numerical experiments, the rectangular grid has more than four times as many elements as the triangular grid. Therefore, Fig. 4 also illustrates that, regarding rectangular elements, a large number of elements are not sufficient to achieve the same accuracy as in the case of triangular elements, and a much finer discretization of the hemi-cylinder (i.e., bathymetry) is needed especially in the vicinity of the coastline. This implies that it is very critical in 2-D EM FEM modeling near coastlines that appropriate numerical grids are employed that allow smooth and continuous tangential components of the electric field with respect to bathymetry.
As for the two numerical solutions using triangular elements, T1 becomes superior to T0 with regard to the apparent resistivity close to the bottom of the hemi-cylinder, while there are almost no differences between the two from the coastline to landward. This suggests that the accuracy of the spatial derivatives may greatly affect MT responses on the deep seafloor. It can be stressed that the improvements we achieved on the 2-D FEM forward code for EM induction in the Earth is necessary in regions including bathymetry and coastlines.