4. Electromagnetic Theory for Geophysical Applications

Abstract

To comprehend the bases and the interpretational techniques of electrical prospecting methods, requires first a knowledge of the tools of electromagnetic theory. The ability to solve a boundary-value problem in electromagnetic theory then becomes the objective. All electromagnetic phenomena are governed by the empirical Maxwell's equations; we must start with them. Maxwell's equations are uncoupled first-order linear differential equations but can be coupled by the empirical constitutive relations which reduce the number of basic vector field functions from five to two. Care must be taken in selecting the form of the constitutive relations pertinent to the earth. In particular, for most earth problems, we assume isotropy, homogeneity, linearity, and temperature-time-pressure independence of the electrical parameters of local regions of the earth. A more complicated earth model is formed by juxtaposition of several such regions.A concept fundamental in electricity and magnetism is that of magnetic and electric polarization. We introduce the polarization vector in a symmetrical sense which demands recognition of, at least, the theoretical existence of magnetic current density and magnetic monopoles, as well as electric current density and electric monopoles. Maxwell's equations are generalized and made more symmetric in the process with a conceptual advantage afforded when we come to deal with scalar and vector potentials. Total vector polarization functions, consisting of the algebraic sums of induced and source parts, frequently facilitate description of a physical problem.Many boundary-value problems can be solved in terms of the vector electric and magneticfield intensity functions. In an earth composed of several juxtaposed homogeneous isotropic linear regions, a solution of a wave equation is postulated for each region. These solutions must be matched at every boundary according to prescribed boundary conditions on two vector field functions or potentials. The wave equation used for each region is derived directly 132 Ward and Hohmann from Maxwell's equations and really represents a compact form of these equations. If the wave equation is homogeneous, no sources are present in that region. If the wave equation is inhomogeneous, sources exist in that region. Often a boundary-value problem is difficult to solve in terms of vector field functions andis easier to solve in terms of vector and/or scalar potential functions from which the vector field functions may be derived. Several different sets of potential functions appear in the literature; we use the Schelkunoff potentials because of their symmetry and because of the ease of relating them to the TE and TM modes of excitation. Any electromagnetic field, in a homogeneous, source-free region, may be decomposed into a part for which the electric field is transverse to some axis (TE mode) and a part for which the magnetic field is transverse to this same axis (TM mode). Mode decomposition simplifies the solution of boundary-value problems. The development of a se...

On the Radiation From a Short Current-Carrying Straight Wire Oriented Perpendicular to a Stratified Medium

“….. Notice that the integral along the upper infinite semi-circle has been neglected in Eq. (14). In fact, this contribution cannot but be identically zero, because the Hankel function in the integrand decays exponentially with increasing |λ| in the upper half-plane.…”

“…The study of the radiation characteristics of dipole antennas situated in proximity to stratified media has attracted the interest of several scientists in the past years [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20]. This is because electric dipoles are extensively used in a variety of engineering applications, especially in the areas of close-to-thesurface radio communication, geophysical prospecting, radio remote sensing, and hypethermia [2-7, 11-15, 22, 23].…”

Two-Dimensional Magnetotelluric Modelling of Ore Deposits: Improvements in Model Constraints by Inclusion of Borehole Measurements

“…In West and Ward's (1988) synthetic 3D AMT forward modelling studies and in Sasaki et al's (1992) synthetic 2D AMT inverse modelling studies, these axial components were normalised by horizontal magnetic fields simulated at the Earth's surface. In vertical boreholes, such data have no sensitivity to the resistivity distribution of planar layered models (Ward and Hohmann 1987). However, for 2D and 3D models, there is limited sensitivity to the normal (layered background) resistivity distribution, mostly in the form of sensitivity to the anomalous resistivity contrast (Bahr 1983;West and Ward 1988).…”

A flow‐through Hankel transform technique for rapid, accurate Green's function computation

“…One of the most straightforward derivations of the spectral GTE for multilayered structures [Ward and Hohmann, 1988] begins by constructing the Schelkunoff [1945] potentials at the surface for vertical electric and vertical magnetic dipolcs in the source layer, corresponding to the transverse electric (TIE) and transverse magnetic (TM) modes, respectively. One then calculates the impedance and admittance of the layered Earth structure from the surface looking downward and applies propagator matrices to compute these potentials in any layer.…”