volume | PIER Journals

Abstract

I present an algorithm to simulate low-frequency electromagnetic propagation in an anisotropic earth, described by a general (non-diagonal) conductivity tensor. I solve the electric formulation by explicitly imposing an approximate form of the condition ∇·J = 0, where J is the current density vector, which includes the source and the induced current. The numerical algorithm consists of a fully spectral explicit scheme for solving linear, periodic parabolic equations. It is based on a Chebyshev expansion of the evolution operator and the Fourier and Chebyshev pseudospectral methods to compute the spatial derivatives. The latter is used to implement the air/ocean boundary conditions. The results of the simulations are verified by comparison to analytical solutions obtained from the Green function. Examples of the electromagnetic field generated by a source located at the bottom of the ocean illustrate the practical uses of the algorithm.

1. Abramowitz, M. and I. A. Stegun, Handbook of mathematical functions, Dover, 1972.

2. Adhidjaja, J. I., G. W. Hohmann, and M. L. Oristaglio, "Two-dimensional transient electromagnetic responses," Geophysics, Vol. 50, 2849-2861, 1985.
doi:10.1190/1.1441904 Google Scholar

3. Al-Garni, M. and M. E. Everett, "The paradox of anisotropy in electromagnetic loop-loop responses over a uniaxial half-space," Geophysics, Vol. 68, 892-899, 2003.
doi:10.1190/1.1581041 Google Scholar

4. Anderson, B., T. Barber, and S. Gianzero, "The effect of crossbed-ding anisotropy on induction tool response," Petrophysics, Vol. 42, 137-149, 2001. Google Scholar

5. Badea, E. A., M. E. Everett, G. A. Newman, and O. Biro, "Finite-element analysis of controlled-source electromagnetic induction using Coulomb-gauged potentials," Geophysics, Vol. 66, 786-799, 2001.
doi:10.1190/1.1444968 Google Scholar

6. Carcione, J. M., "Ground penetrating radar: wave theory and numerical simulation in conducting anisotropic media," Geophysics, Vol. 61, 1664-1677, 1996.
doi:10.1190/1.1444085 Google Scholar

7. Carcione, J. M., "A spectral numerical method for electromagnetic diffusion," Geophysics, Vol. 71, I1-I9, 2006.
doi:10.1190/1.2159050 Google Scholar

8. Carcione, J. M., Wave Fields in Real Media. Theory and Numerical Simulation of Wave Propagation in Anisotropic, Anelastic, Porous and Electromagnetic Media, 2nd Ed., Elsevier, 2007.

9. Carcione, J. M. and M. Schoenberg, "3-D ground-penetrating radar simulation and plane wave theory," Geophysics, Vol. 65, 1527-1541, 2000.
doi:10.1190/1.1444841 Google Scholar

10. Carslaw, H. S. and J. C. Jaeger, Conduction of Heat in Solids, Clarendom Press, 1984.

11. Davydycheva, S., V. Druskin, and T. Habashy, "An efficient finite-difference scheme for electromagnetic logging in 3D anisotropic inhomogeneous media," Geophysics, Vol. 68, 1525-1536, 2003.
doi:10.1190/1.1620626 Google Scholar

12. Druskin, V. L. and L. A. Knizhnerman, "Spectral approach to solving three-dimensional Maxwells diffusion equations in the time and frequency domains," Radio Science, Vol. 29, 937-953, 1994.
doi:10.1029/94RS00747 Google Scholar

13. Druskin, V. L., L. A. Knizhnerman, and P. Lee, "New spectral Lanczos decomposition method for induction modeling in arbitrary 3-D geometry," Geophysics, Vol. 64, 701-706, 1999.
doi:10.1190/1.1444579 Google Scholar

14. Eidesmo, T., S. Ellingsrud, L. M. MacGregor, S. Constable, M. C. Sinha, S. Johansen, F. N. Kong, and H. Westerdahl, "Sea bed logging (SBL), a new method for remote and direct identification of hydrocarbon filled layers in deepwaters areas," First Break, Vol. 20, 144-151, 2002. Google Scholar

15. Everett, M. E. and S. Constable, "Electric dipole fields over an anisotropic seafloor: Theory and application to the structure of 40Ma Pacific Ocean lithosphere," Geophys. J. Int., Vol. 136, 41-56, 1999.
doi:10.1046/j.1365-246X.1999.00725.x Google Scholar

16. Jiang, B.-N., J. Wu, and L. A. Povinelli, "The origin of spurious solutions in computational electromagnetics," Journal of Computational Physics, Vol. 125, 104-123, 1996.
doi:10.1006/jcph.1996.0082 Google Scholar

17. Kong, F. N., S. E. Johnstad, T. Røsten, and H. Westerdahl, "A 2.5D finite-element-modeling difference method for marine CSEM modeling in stratified anisotropic media," Geophysics, Vol. 73, F9-F19, 2008.
doi:10.1190/1.2819691 Google Scholar

18. Lee, K. H., G. Liu, and H. F. Morrison, "A new approach to modeling the electromagnetic response of conductive media," Geophysics, Vol. 54, 1180-1192, 1989.
doi:10.1190/1.1442753 Google Scholar

19. Leppin, M., "Electromagnetic modeling of 3-D sources over 2-D inhomogeneities in the time domain," Geophysics, Vol. 57, 994-1003, 1992.
doi:10.1190/1.1443325 Google Scholar

20. Maaø, F., "Fast finite-difference time-domain modeling for marine-subsurface electromagnetic problems," Geophysics, Vol. 72, A19-A23, 2007.
doi:10.1190/1.2434781 Google Scholar

21. Mackie, R. L., T. R. Madden, and P. E. Wannamaker, "Three-dimensional magnetotelluric modeling using finite difference equations --- Theory and comparisons to integral equation solutions," Geophysics, Vol. 58, 215-226, 1993.
doi:10.1190/1.1443407 Google Scholar

22. Mittet, R., "High-order finite-difference simulations of marine CSEM surveys using a correspondence principle for wave and diffusion fields," Geophysics, Vol. 75, F33-F50, 2010.
doi:10.1190/1.3278525 Google Scholar

23. Negi, J. G. and P. D. Saraf, Anisotropy in Geoelectromagnetism, Elsevier, 1989.

24. Olhoeft, G. R. and D. E. Capron, "Petrophysical causes of electromagnetic dispersion," Proceedings of the Fifth Internat. Conf. on Ground Penetrating Radar, 145-152, University of Waterloo, 1994.

25. Oristaglio, M. L. and G. W. Hohmann, "Diffusion of electromag-netic fields into a two-dimensional earth: A finite-difference approach," Geophysics, Vol. 49, 870-894, 1984.
doi:10.1190/1.1441733 Google Scholar

26. Pellerin, L., J. M. Johnston, and G. W. Hohmann, "A numerical evaluation of electromagnetic methods in geothermal exploration," Geophysics, Vol. 61, 121-130, 1996.
doi:10.1190/1.1443931 Google Scholar

27. Sasaki, Y. and M. A. Meju, "Useful characteristics of shallow and deep marine CSEM responses inferred from 3D finite-difference modeling," Geophysics, Vol. 74, F67-F76, 2009.
doi:10.1190/1.3168616 Google Scholar

28. Smith, J. T., "Conservative modeling of 3-D electromagnetic fields: Part 2 --- Biconjugate gradient solution and an accelerator," Geophysics, Vol. 61, 1319-1324, 1996.
doi:10.1190/1.1444055 Google Scholar

29. Stalnaker, J. L., "A finite element approach to the 3D CSEM modeling problem and applications to the study of the effect of target interaction and topography,", PHD Thesis, Texas A&M University, 2004. Google Scholar

30. Tal-Ezer, H., "Spectral methods in time for parabolic problems," SIAM Journal of Numerical Analysis, Vol. 26, 1-11, 1989.
doi:10.1137/0726001 Google Scholar

31. Wang, T. and S. Fang, "3D electromagnetic anisotropy modeling using finite differences," Geophysics, Vol. 66, 1386-1398, 2001.
doi:10.1190/1.1486779 Google Scholar

32. Wang, T. and G. W. Hohmann, "A finite-difference, time-domain solution for three-dimensional electromagnetic modeling," Geophysics, Vol. 58, 797-809, 1993.
doi:10.1190/1.1443465 Google Scholar

33. Wang, T. and J. Signorelli, "Finite-difference modeling of electromagnetic tool response for logging while drilling," Geophysics, Vol. 69, 152-160, 2004.
doi:10.1190/1.1649383 Google Scholar

34. Weiss, C. J. and S. Constable, "Mapping thin resistors and hydrocarbons with marine EM methods, Part II --- Modeling and analysis in 3D," Geophysics, Vol. 71, G321-G332, 2006.
doi:10.1190/1.2356908 Google Scholar

35. Weiss, C. J. and G. A. Newman, "Electromagnetic induction in a fully 3D anisotropic earth," Geophysics, Vol. 67, 1104-1114, 2002.
doi:10.1190/1.1500371 Google Scholar

36. Yee, K. S., "Numerical solution of initial boundary value problems involving Maxwells equations in isotropic media," IEEE Transactions on Antennas and Propagation, Vol. 14, 302-307, 1966.
doi:10.1109/TAP.1966.1138693 Google Scholar

37. Yin, C. and H. M. Maurer, "Electromagnetic induction in a layered earth with arbitrary anisotropy," Geophysics, Vol. 66, 1405-1416, 2001.
doi:10.1190/1.1487086 Google Scholar

38. Yu, L., R. L. Evans, and R. N. Edwards, "Transient electromagnetic responses in seafloor with triaxial anisotropy," Geophys. J. Int., Vol. 129, 292-304, 1997.
doi:10.1111/j.1365-246X.1997.tb01582.x Google Scholar

39. Zhdanov, M., "Generalized effective-medium theory of induced polarization," Geophysics, Vol. 73, F197-F211, 2008.
doi:10.1190/1.2973462 Google Scholar

Dr. Jose M. Carcione