Electromagnetic Source Equivalence and Extension of the Complex Image Method for Geophysical Applications

Abstract

Abstract-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 the 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 linear 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].

Complex Polar Coordinates in Electromagnetics

“…The analytical extension from real to complex coordinates -usually performed in a cartesian coordinate system -has been extensively used to model certain electromagnetic problems, in particular those related to the Laplacian operator, [1,2]. With regard to time-harmonic propagation problems, complex extensions of the coordinates have been used to study beam solutions, [3][4][5][6][7].…”

The Finite-Conducting Ground’s Effect on the Inductance of a Rectangular Loop

“…First of all, how the equivalence is built is explained here. The model in Figure 5 builds on the complex image theory [9,[11][12][13][14][15][16][17], which indicates that a wire above ground has an image at a complex depth . This is why effect of the grounds on the field of the wire is equivalent to the effect produced by the image wire.…”