Plane Wave Propagation Problems in Electrically Anisotropic and Inhomogeneous Media with Geophysical Applications

Abstract

Boundary value problems required for modelling plane wave propagation in electrically anisotropic and inhomogeneous media relevant to the surface impedance methods in electromagnetic geophysics are formally posed and treated. For a homogeneous TM-type wave propagating in a half space with both vertical and horizontal inhomogeneities where the TM-type wave is aligned with one of the elements of the conductivity tensor, it is shown using exact solutions that the shearing term in the homogeneous Helmholtz equation for inclined anisotropic media: [Equation 1], unequivocally vanishes and solutions need only be sought to the homogeneous Helmholtz equation for biaxial media: [Equation 2]. This implies that those problems posed with an inclined uniaxial conductivity tensor can be identically stated with a fundamental biaxial conductivity tensor, provided that the conductivity values are the reciprocal of the diagonal terms from the Euler rotated resistivity tensor: [Equation 3], [Equation 4], [Equation 5]. The applications of this consequence for numerical methods of solving arbitrary two-dimensional problems for a homogeneous TM-type wave is that they need only to approximate the homogeneous Helmholtz equation and neglect the corresponding shearing term. The self-consistent impedance method, a two-dimensional finite-difference approximation based on a network analogy, is demonstrated to accurately solve for problems with inclined uniaxial anisotropy using the fundamental biaxial anisotropy equivalence. The problem of a homogeneous plane wave at skew incidence upon an inclined anisotropic half space is then formally treated. In the half space, both TM- and TE-type waves are coupled and the linearly polarised incident TM- and TE-type waves reflect TE- and TM-type components. Equations for all elements of the impedance tensor are derived for both TM- and TE-type incidence. This offers potential as a method of predicting the direction of anisotropic strike from tensor impedance measurements in sedimentary environments.