EM fields of a VMD calculated at arbitrary points within a layered half-space

In this paper, we derive expressions for all electromagnetic (EM) field components which can be observed, when a vertical magnetic dipole (VMD) is located at z = −h, h > 0 over a stratified earth, i.e., when the electrical conductivity is a piecewise constant function of depth z. We further allow a non-vanishing but small electrical conductivity in the air layer, and let the electrical permittivity vary from its vacuum value. Apart from traditional approaches, we use a non-vanishing air conductivity to be consistent with our 3-D discretizations which would otherwise yield singular mass matrices. The basic ideas of the derivation within the following paragraphs emanate from Ward and Hohmann (1988) and Zhdanov (2009). While the first sections of this paper concern the two-layer (i.e., the nearly non-conductive air and the conductive homogeneous half-space) case we expand the concept to the general N-layer case in the last section. This work has been motivated by the one-dimensional forward and inverse problem of helicopter electromagnetics (HEM). To evaluate the observed total fields by a numerical discretization scheme, the secondary field approach requires the calculation of the analytical solution of the EM fields at the receiver positions within the air half-space. Furthermore, in order to calculate the Jacobian matrix, these fields are required at arbitrary points within the conductive layered half-space.