Electromagnetic induction in a conductive strip in a medium of contrasting conductivity: Application to VLF andMT above molten dykes

Abstract

Very low frequency (VLF) electromagnetic waves that penetrate conductive magma-filleddykes generate secondary fields on the surface that can be used to invert for dyke properties.The model used for the interpretation calculates currents induced in a conductive strip by aninducing field that decays exponentially with depth due to the conductivity of the surroundingmedium. The differential equations are integrated to give an inhomogeneous Fredholm equationof the second kind with a kernel consisting of a modified Bessel function of the secondkind. Numerical methods are typically used to solve for the induced currents in the strip. In thispaper, we apply a modified Galerkin-Chebyshev method, which involves separating the kernelinto source and field spectra and integrating the source terms to obtain a matrix equation forthe unknown coefficients. The incident wave is expressed as a Chebyshev series. The modifiedBessel function is separated into a logarithmic singularity and a non-singular remainder, bothof which are expanded in complex Chebyshev polynomials. The Chebyshev coefficients forthe remainder are evaluated using a fast Fourier transform, while the logarithmic term andincident field have analytic series. The deconvolution then involves a matrix inversion. Theresults depend on the ratio of strip-size to skin-depth. For infinite skin-depth and a singularconductivity distribution given by τ0a/va2 - z2 (where τ0 is the conductance, a is the halflengthand z the distance from the centre), Parker gives an analytic solution. We present asimilar analytic series solution for the finite skin-depth case, where the size to skin depth ratiois small. Results are presented for different ratios of size to skin depth that can be comparedwith numerical solutions. We compare full-space and half-space solutions. A fit of the modelto VLF data taken above a magma filled dykes in Hawaii and Mt Etna demonstrates that whileproperties such as depth to top, conductivity ratio, tilt and dip can be determined, the depth tobottom is indeterminate due to the exponential decay of the inducing field.