An improved forward modeling method for two-dimensional electromagnetic induction problems with bathymetry
© The Society of Geomagnetism and Earth, Planetary and Space Sciences (SGEPSS); The Seismological Society of Japan; The Volcanological Society of Japan; The Geodetic Society of Japan; The Japanese Society for Planetary Sciences; TERRAPUB. 2012
Received: 8 March 2012
Accepted: 28 April 2012
Published: 22 June 2012
Recently, electromagnetic observations have become common not only on land but also on the seafloor. In particular, thanks to the development of instruments for use in shallow seas, one can conduct observations along land-sea arrays. However, since there is a large contrast in conductivity between sea water and rocks in the crust and the mantle, we have to pay more attention to the accuracy of forward solvers used for electromagnetic induction problems, including bathymetry. In this paper, we develop a two-dimensional forward code using triangular finite elements, and confirm the accuracy of the new code by TM mode responses. The accuracy of the improved solver was tested by comparison with an analytical solution in a hemi-cylindrical geometry. We also show that triangular elements are more reliable than rectangular elements in determining conductivity structures beneath land-sea arrays. Our results indicate the importance of precisely discretized bathymetry and the accuracy of spatial derivatives of electromagnetic field components, especially in the vicinity of coastlines.
Recently, it has become quite common to carry out electromagnetic (EM) observations on the seafloor. For instance, the development of EM instruments applicable to shallow seas enables us to investigate subsurface conductivity structures near coastlines in more detail. However, magnetotelluric (MT) responses obtained in the vicinity of coastlines are known to be influenced strongly by large differences in conductivities between land and sea. Careful consideration, therefore, is necessary as to how electric currents flow within conductive sea water, which greatly depends on bathymetry, as well as whether this is reproduced accurately in forward modeling or not.
Many numerical approaches developed so far may be used in regions including coastlines theoretically. The finite element method (FEM) is one of very popular approaches because it is capable of including arbitrary bathymetry/topography by adopting various forms of elements. For example, Utada (1987) developed a two-dimensional (2-D) FEM code using triangular elements. On the other hand, Uchida (1993)/Ogawa and Uchida (1996) adopted rectangular elements for their FEM forward solver. There exist respective advantages for triangular and rectangular elements. Rectangular elements are conceptually simple to use, say, in generating numerical meshes and coding many desired mathematical/physical formulations. On the other hand, triangular elements are very useful in expressing complicated topography/bathymetry accurately, especially in regions around coastlines where at least one triangular element is indispensable at the very edge of the land-sea boundary in 2-D problems. In this study, we have adopted triangular elements in our 2-D FEM modeling and improved Utada’s (1987) forward solver, which we henceforth call UT, in order to develop a 2-D forward code that enables precise modeling of bathymetry. Expressions of bathymetry by triangular elements, however, may not be sufficient to obtain reliable MT responses near coastlines. To improve UT’s calculation algorithm itself, we also applied Li et al.’s (2008) differentiation method. In addition, we have extended the code so that one can use electric and magnetic fields at different observation sites to calculate the desired EM responses. This improvement helps us to calculate MT responses at sites where only electric variations were observed. In the following section, we will explain the improvements we have made in detail. However, we work with only 2-D problems because it is rather simple to show how MT responses near coastlines are affected by bathymetry, and the accuracy of the forward code, in two dimensions.
2. Improvement of the 2-D FEM Forward Code
Another essential point is that the extrapolation should be started from the resistive side. This is because the amplitude and phase of the along-strike fields vary more severely when the magnetic/electric fields propagate through conductive bodies. For instance, the auxiliary fields on the seafloor can be calculated more accurately by extrapolation from the sub-seafloor side than from the sea water side. From this point of view, we extrapolated the spatial derivatives from the sub-seafloor side in obtaining the auxiliary fields on the seafloor for both modes.
Furthermore, we modified the code so as to calculate MT responses using electric and magnetic fields observed at different sites. In marine and/or land EM observations, only electric fields are often observed at a significant number of sites and MT responses are calculated using magnetic fields obtained at other sites assuming a spatial uniformity of the inducing horizontal geomagnetic field. This modification enabled us to increase the number of observed MT responses for use in further forward modeling and/or inversion.
3. The Appraisal Method of the Improved Code
In this section, we test the accuracy of the improved 2-D FEM forward code using an analytical solution. Wannamaker et al. (1986) considered a hemi-cylindrical geometry, the analytical responses of which are identical at the dc limit to that of a cylinder excited by a time-varying horizontal electric field in the lower half-space. To test the accuracy of our new 2-D code for regions including coastlines, we considered the case where a cylinder full of a conductive (4 S/m) medium corresponding to seawater is embedded in a resistive (100 ohm.m) whole-space. Under this circumstance, a geometry whose upper half-space is replaced by an insulator can be regarded as a land-sea configuration where the hemi-cylinder, the lower half-space excluding the cylinder, and the upper half-space insulator correspond to the sea, land and the air, respectively. In addition, we did not apply dc electric currents but horizontal electric fields oscillating with an angular frequency, ω, as the inducing field for the TM mode.
4. Results and Discussion
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.
We have developed a new 2-D FEM forward code, which is useful especially for EM induction problems with bathymetry and coastlines. The FEM code adopts triangular elements which has an obvious advantage over rectangular elements. The improvements achieved in this study are two-fold: First, we applied Li et al.’s (2008) differentiation/extrapolation method in order to evaluate more accurate MT responses on the seafloor and in the vicinity of coastlines. Second, we enabled the code to calculate EM responses allowing any combination of observation sites and EM components.
We tested the accuracy of the new code by a comparison with the analytical solution in the hemi-cylindrical geometry. It was clearly shown that the new code yielded most reliable MT responses especially on the seafloor and at the coastline.
In conclusion, careful considerations are needed for 2-D EM FEM modeling on the seafloor and in the vicinity of coastlines. The determination of bathymetry near coastlines by numerical meshes is particularly critical because deflected electric currents can concentrate at the shore. We recommend avoiding rectangular grids to determine bathymetry, in which zigzag electric currents occur along bathymetry, especially in the vicinity of coastlines, in order to determine conductivity structures beneath coastal regions. It was found difficult, or very expensive, to achieve the same accuracy as by triangular grids by rectangular grids near coastlines. Without any tests of accuracy, the indiscriminate use of rectangular elements may cause critical errors in estimating theoretical EM response functions near coastlines. In addition, a differentiation method with a more precise extrapolation should be applied as well when seafloor EM observations are to be modeled accurately.
We were supported by Grants-in-Aid for Scientific Research of the Japan Society for the Promotion of Science (#19340127). We would like to express our sincere thanks to Profs. H. Utada and Y. Ogawa for kindly having made available their FEM forward codes. We are grateful to Prof. N. Oshiman and Drs. M. Uyeshima and R. Yoshimura for their many helpful suggestions.
- Ichihara, H., R. Honda, T. Mogi, H. Hase, H. Kamiyama, Y. Yamaya, and Y. Ogawa, Resistivity structure around the focal area of the 2004 Rumoi-Nanbu earthquake (M 6.1), northern Hokkaido, Japan, Earth Planets Space, 60, 883–888, 2008.View ArticleGoogle Scholar
- Li, S., J. R. Booker, and C. Aprea, Inversion of magnetotelluric data in the presence of strong bathymetry/topography, Geophys. Prospect., 56 (2), 259–268, 2008.View ArticleGoogle Scholar
- Ogawa, Y. and T. Uchida, A two-dimensional magnetotelluric inversion assuming Gaussian static shift, Geophys. J. Int., 126 (1), 69–76, 1996.View ArticleGoogle Scholar
- Uchida, T., Smooth 2-D inversion for magnetotelluric data based on statistical criterion ABIC, J. Geomag. Geoelectr., 45, 841–858, 1993.View ArticleGoogle Scholar
- Utada, H., A direct inversion method for two-dimensional modeling in the geomagnetic induction problem, PhD. Thesis, Univ. Tokyo, pp. 409, 1987.Google Scholar
- Wannamaker, P. E., J. A. Stodt, and L. Rijo, Two-dimensional topographic responses in magnetotellurics modeled using finite elements, Geophysics, 51 (11), 2131–2144, 1986.View ArticleGoogle Scholar
- Ward, S. H. and G. W. Hohmann, Electromagnetic theory for geophysical applications, in Electromagnetic Methods in Applied Geophysics Vol. 1, Theory, edited by Misac N. Nabighian, 529 pp., Soc. Explor. Geophys., Oklahoma, 1988.Google Scholar