Scattering of electromagnetic waves by a perfectly conducting half plane below a stratified overburden and applications to geophysical prospecting by VLF

Sammanfattning: The present paper is an introduction to and a summary of a thesis consisting of the following papers:
1. Scattering of electromagnetic waves by a perfectly conducting half plane below a stratified overburden. Radio Sci. (1978), Vol. 13, No. 2, p. 391-397.
II. VLF anomalies from a perfectly conducting half plane below an overburden. Accepted for publication in Geophysical Prospecting.
The thesis also contains
III. VLF5 program documentation
IV. Program listing of VLF5-MO.
Electromagnetic methods have been used in geophysical prospecting since the beginning of the twentieth century. Since the first field tests there has been a tremendous development and refinement of field methods and instrumentation. The electromagnetic methods have proved to be an efficient tool for the localization of conductive zones in the earths crust, e.g. mineral deposits, shales and fracture zones. Even though the electromagnetic methods have been in extensive use for such a long time the intrepretation of the field data has been more an art than a science. This has to a large extent been due to lack of appropriate theoretical models amenable to computation that would explain the field data. For a long time the only theoretical models available were those where boundaries between media of different conductivity were defined by a constant on a coordinate axis such as a sphere, cylinder, half plane, or a horizontally stratified earth. Beginning in the nineteensixties when large digital computers came into more general use there has been an increased effort put into solving electromagnetic problems relevant to geophysics. Since about 1970 there has been a series of pertinent papers published by different workers on the subject. The problems have been attached by different analytical and numerical methods such as integral equation methods, finite element methods, and finite difference methods. The present thesis is a development of the integral-equation technique to deal with the problems often encountered in geophysical prospecting where a subsurface conductor in conducting host rock is situated below a stratified overburden. The electric and magnetic field is represented by the Hertz vectors. As has been discussed by Stratton (1941) it is sufficient with a single component of each Hertz vector. The fields are thus expressed through two scalar potentials satisfying the scalar Helmholtz equation. These scalar potentials may in certain cases be identified with the two polarizations of the electromagnetic field. To solve the boundary equations at the horizontal boundaries in the stratified overburden the potential functions are expanded into plane waves. In the plane wave representation the overburden is completely described through its reflection and transmission coefficients for each plane wave. For an arbitrary number of layers in the overburden the reflection and transmission coefficients are easily obtained through a recursion formula. When a plane wave propagating downward strikes an anomalous object situated in the halfspace below the overburden the secondary field generated by this object will consist only of upgoing waves at any horizontal surface above the anomalous object. The secondary field generated by the anomalous region may thus be completely represented by the scattering matrix for the plane waves. In certain cases the scattering matrix is fairly easy to calculate e.g. a sphere, cylinder, or halfplane (as will be shown below). In a series of papers Waterman (1965, 1971) has developed a method to calculate the scattering matrix for more complicated structures. The boundary equations at the bottom surface of the overburden gives an integral equation for the plane wave amplitudes where the kernel of the equation is the scattering matrix multiplied by the reflection coefficient of the overburden. To find the scattering matrix of the perfectly conducting halfplane the Hertz potentials are expanded into modified Bessel functions. After several manipulations the scattering matrix is found to be simple and its elements consist of elementary functions. The integral equation is approximated by a matrix equation which is solved by standard iterative methods. When the plane wave amplitudes at the bottom of the overburden have been obtained the wave amplitudes at the ground surface are readily obtained and the field components may be calculated. Numerical calculations have been carried out for the perfectly conducting halfplane when the incident field is a plane wave, with special application to VLF prospecting. The VLF method makes use of the electromagnetic field from radio transmitters in the frequency range 10-30 kHz. The VLF transmitter can usually be considered as a vertical electrical dipole situated at the ground surface. The transmitted field will thus have a magnetic field component which is approximately tangential to the surface of the earth. When the VLF field is considered in a limited region a few wavelengths from the transmitter the primary field is usually well approximated by a plane TM wave incident almost tangentially along the ground. The use of the VLF-method in mineral exploration started around 1960, since then the VLF-method has become an extensively used tool in mineral exploration and geological mapping. The method has proved to be an efficient tool in localizing structures of enhanced conductivity, e.g. mineralization, fracture zones, graphite shists, and shales. For many of these cases the halfplane can be considered to be an appropriate model. The VLF anomalies have been computed for different overburden conductivity and thickness and also for different dip angles of the halfplane. The anomaly curves, which are displayed as tilt and ellipticity of the polarization ellipse, show a fairly complicated behavior. However the qualitative behavior of the curves is mainly due to the phase shift and attenuation of the field caused by the conductivity of the overburden and the host rock. From the anomaly curves it is possible to define the apparent depth to the top of the conductor as the distance between the up-dip peak value and the cross-over of the real component. The apparent depth is usually larger than the actual depth, but it is possible to determine the actual depth to the conductor from the relation between the peak to peak anomaly and the apparent depth. When the peak to peak anomaly is fairly large it is also possible to make estimates of the dip angle. In a specific case the theoretical calculations are shown to be in good agreement with measured data.