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.

Two-dimensional (2-D) models of electrical resistivity are improved by jointly inverting radiomagnetotelluric (RMT) and direct-current resistivity (DCR) data or by allowing for displacement currents in the inversion of RMT data collected on highly resistive bedrock. Uniqueness and stability of the 2-D models are assessed with a model variance and resolution analysis that allows for the non-linearity of the inverse problem. Model variance and resolution are estimated with a truncated singular value decomposition (TSVD) of the sensitivity matrix. In the computation of model errors, inverse singular values are replaced by non-linear semi-axes and the number of included eigenvectors is increased until a given error threshold is reached. Non-linear error estimates are verified with most-squares inversions. For the obtained truncation levels, model resolution matrices are computed. For RMT data, non-linear error appraisals are smaller than linearized ones. Hence, the consideration of the non-linearity in RMT data leads to reduced model errors or enhanced model resolution. The dielectric effect on RMT data is investigated with a new 2-D forward and inverse code that allows for displacement currents. As compared to the quasi-static approximation, apparent resistivities and phases of the impedance tensor elements are found to be significantly smaller and the vertical magnetic transfer function exhibits more distinct sign reversals. More reliable models of electrical resistivity are obtained from areas with highly resistive bedrock, if displacement currents are allowed for. In contrast, inversions with a quasi-static scheme introduce artefactual structures with extremely low or high resistivities. A smoothness-constrained 2-D joint inversion of RMT and DCR data is presented. The non-linear model variance and resolution analysis is applied to single and joint inverse models. For DCR data, the errors estimated by most-squares inversions are consistently larger than those estimated by the non-linear semi-axes, indicating that DCR models are poorly resolved. Certain areas of the joint inverse models are better resolved than in the single inverse models.