Progress In Electromagnetics Research B
ISSN: 1937-6472
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By A. Pulkkinen, A. Viljanen, R. Pirjola, and L. Häkkinen

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In this work, source equivalence and computation of the reflected (induced) electromagnetic field in geophysical situations are studied. It is shown that the application of Huygens' principle allows for full generalization of Fukushima's equivalence theorem that applies only for magnetic field. The source equivalence is revisited for a vertical line current element, and it is shown that the equivalent charge required to replace the original source by a planar equivalent source together with the surface charge associated with the reflected field generates a purely vertical total electric field on the ground. Consequently, if the magnetic field and horizontal components of the total electric field on the ground are of interest, only equivalent currents need to be considered. The classical Complex Image Method (CIM) is derived from exact image theory for planar impedance surfaces. The classical CIM is extended by considering a divergence-free source current that may have components also perpendicular to the ground plane. The extension is seen to generate a complex image charge not present in the classical CIM. Further, a generalized application of the extended CIM to geophysical situations having divergence-free volume source currents is introduced. The application involves decomposition of the source into line current elements and rotations, translations and reflections of the electromagnetic field expressions associated with each element. The validity of the new approach is verified for an example of external current system and ground model setup by means of comparisons to results obtained from exact formulation by~[18].

A. Pulkkinen, A. Viljanen, R. Pirjola, and L. Häkkinen, "Electromagnetic Source Equivalence and Extension of the Complex Image Method for Geophysical Applications," Progress In Electromagnetics Research B, Vol. 16, 57-84, 2009.

1. Thomson, W., Reprint of Papers on Electrostatics and Magnetism, 2nd Ed., Macmillan & Co., 1884.

2. Taraldsen, G., "The complex image method," Wave Motion, Vol. 43, 91-97, 2005.

3. Wait, J. R., "Image theory of a quasi-static magnetic dipole over a dissipative half-space," Electronics Letters, Vol. 5, 281-282, 1969.

4. Wait, J. R. and K. P. Spies, "On the image representation of the quasistatic fields of a line current source above the ground," Can. J. Phys., Vol. 47, 2731-2733, 1969.

5. Wait, J. R., "Complex image theory-revisited," IEEE Antennas Propagat. Mag., Vol. 33, No. 4, 27-29, 1991.

6. Lindell, I. V. and E. Alanen, "Exact image theory for the Sommerfeld half-space problem, Part 1: Vertical magnetic dipole," IEEE Transactions on Antennas and Propagation, Vol. 32, 126-133, 1984.

7. Lindell, I. V. and E. Alanen, "Exact image theory for the Sommerfeld half-space problem, Part 2: Vertical electric dipole," IEEE Transactions on Antennas and Propagation, Vol. 32, 841-847, 1984.

8. Lindell, I. V. and E. Alanen, "Exact image theory for the Sommerfeld half-space problem, Part 3: General formulation," IEEE Transactions on Antennas and Propagation, Vol. 32, 1027-1032, 1984.

9. Deschamps, G. A., "Gaussian beam as a bundle of complex rays," Electronics Letters, Vol. 7, No. 23, 684-685, 1971.

10. Madden, T. M. and R. L. Mackie, Three-dimensional magnetotelluric modelling and inversion, Proceedings of the IEEE, Vol. 77, No. 2, 318-333, 1989.

11. Avdeev, D. B., A. V. Kuvshinov, O. V. Pankratov, and O. G. Newman, "Three-dimensional induction logging problems, Part I: An integral equation solution and model comparisons," Geophysics, Vol. 67, No. 2, 413-426, 2002.

12. Silva-Macedo, J. A., M. A. Romero, and B.-H. V. Borges, "An extended fdtd method for the analysis of electromagnetic field rotators and cloaking devices," Progress In Electromagnetics Research, Vol. 87, 183-196, 2008.

13. Soleimani, M., "Simultaneous reconstruction of permeability and conductivity in magnetic induction tomography," Journal of Electromagnetic Waves and Applications, Vol. 23, No. 5/6, 785-798, 2009.

14. Viljanen, A., O. Amm, and R. Pirjola, "Modeling geomagnetically induced currents during different ionospheric situations," J. Geophys. Res., Vol. 104, No. A12, 28059-28072, 10.1029/1999JA900337, 1999.

15. Pulkkinen, A., M. Hesse, M. Kuznetsova, and L. Rastatter, "First-principles modeling of geomagnetically induced electromagnetic fields and currents from upstream solar wind to the surface of the Earth," Annales Geophysicae, Vol. 25, 881-893, 2007.

16. Thomson, D. J. and J. T. Weaver, "The complex image approximation for induction in a multilayered Earth," J. Geophys. Res., Vol. 80, 123-129, 1975.

17. Pirjola, R. and A. Viljanen, "Complex image method for calculating electric and magnetic fields produced by an auroral electrojet of finite length," Annales Geophysicae, Vol. 16, 1434-1444, 1998.

18. Hakkinen, L. and R. Pirjola, "Calculation of electric and magnetic fields due to an electrojet current system above a layered earth," Geophysica, Vol. 22, No. 1--2, 31-44, 1986.

19. Fukushima, N., "Generalized theorem for no ground magnetic effect of vertical currents connected with Pedersen currents in the uniform conductivity ionosphere," Rep. Ionos. Space Res. Jpn., Vol. 30, 35-40, 1976.

20. Lindell, I. V., J. J. Hanninen, and R. Pirjola, "Wait's complex-image principle generalized to arbitrary sources," IEEE Transactions on Antennas and Propagation, Vol. 48, 1618-1624, 2000.

21. Pulkkinen, A., R. Pirjola, and A. Viljanen, "Determination of ground conductivity and system parameters for optimal modeling of geomagnetically induced current flow in technological systems," Earth, Planets and Space, Vol. 99, 999-1006, 2007.

22. Boteler, D. H. and R. J. Pirjola, "The complex-image method for calculating the magnetic and electric fields produced at the surface of the Earth by the auroral electrojet," Geophys. J. Int., Vol. 132, No. 1, 31-40, 1998.

23. Lindell, I., Huygens' principle in electromagnetics, IEE Proc.-Sci. Meas. Technol., Vol. 143, No. 2, 103-105, 1996.

24. Lindell, I., Methods for Electromagnetic Field Analysis, Oxford University Press, 1995.

25. Mayes, P. E., "The equivalence of electric and magnetic sources," IRE Transactions on Ant. Propag., Vol. 6, 295-296, 1958.

26. Dmitriev, V. I. and M. N. Berdichevsky, The fundamental model of magnetotelluric sounding, Proc. IEEE, Vol. 67, 1034-1044, 1979.

27. Sarabandi, K., M. D. Casciato, and Il-Sueh Koh, "Efficient calculation of the fields of a dipole radiating above an impedance plane," IEEE Transactions on Antennas and Propagation, Vol. 50, No. 9, 1222-1235, 2002.

28. Hanninen, J. J., R. J. Pirjola, and I. V. Lindell, "Application of the exact image theory to studies of ground effects of space weather," Geophys. J. Int., Vol. 151, 534-542, 2002.

29. Ward, A. J. and J. B. Pendry, "Refraction and geometry in Maxwell's equations," Journal of Modern Optics, Vol. 43, 773-793, 1996.

30. Post, E. J., Formal Structure of Electromagnetics, General Covariance and Electromagnetics, Interscience Publishers, 1962.

31. Kosmas, L. T. and O. Hess, "Optics: Watch your back," Nature, Vol. 451, 27, 2008.

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