Electromagnetic fields of a vertical magnetic dipole above and within a horizontal stratified earth in due consideration of the electrical permittivity

Our poster is motivated by the helicopter electromagnetic forward and inversion problem. In this context, we use the secondary field approach to evaluate the total electromagnetic fields of a vertical magnetic dipole (VMD) by a numerical discretization scheme. This approach requires the calculation of an analytic solution of the primary field at every receiver position, located within the air half-space. Furthermore, in order to calculate the Jacobian matrix using the sensitivity equation approach, these primary fields have to be calculated at every degree of freedom within the stratified media. The primary fields are the solution of the Helmholtz equation for a 1-D conductivity distribution.
We show the complete derivation of the electromagnetic fields of a VMD for the magnetic vector potential based on the spatial Hankel transformation. To avoid inaccuracies at high frequencies, varying electrical permittivities are incorporated. We furthermore explain the recursion algorithm that is used to find general expressions at arbitrary depths and which allows us to easily derive the different electromagnetic field components. The calculations start with the simple two layer case and are subsequently expanded to the more general N-layer case. Special attention is payed to a singularity problem arising at the derivation of the horizontal electrical and vertical magnetic fields in the air layer which is resolved by an integration by parts approach.