Electromagnetic induction in a conductive strip in a medium of contrasting conductivity: application to VLF and MT above molten dykes

Abstract

Very low frequency (VLF) electromagnetic waves that penetrate conductive magma-filled dikes generate secondary fields on the surface that can be used to invert for dike properties. The model used for the interpretation calculates currents induced in a conductive strip by an inducing field that decays exponentially with depth due to the conductivity of the surrounding medium. The differential equations are integrated to give an inhomogeneous Fredholm equation of the second kind with a kernel consisting of a modified Bessel function of the second kind. Numerical methods are typically used to solve for the induced currents in the strip. In this paper we apply a modified Galerkin-Chebyshev method, which involves separating the kernel into source and field spectra and integrating the source terms to obtain a matrix equation for the unknown coefficients. The incident wave is expressed as a Chebyshev series. The Modified Bessel function is separated into a logarithmic singularity and a non-singular remainder, both of which are expanded in complex Chebyshev polynomials. The Chebyshev coefficients for the remainder are evaluated using a fast Fourier transform, while the logarithmic term and incident field have analytic series. The deconvolution then involves a matrix inversion. The results depend on the ratio of strip-size to skin-depth. For infinite skin-depth and a singular conductivity distribution given by τ 0 a/ √ a 2 − z 2 (where τ 0 is the conductance, a is the half-length, and z the distance from the center), Parker (2011) gives an analytic solution. We present a similar analytic series solution for the finite skin-depth case, where the size to skin depth ratio is small. Results are presented for different ratios of size to skin depth that can be compared with numerical solutions. We compare full-space and half-space solutions. A fit of the model to VLF data taken above a magma filled dikes in Hawaii 1 and Mt Etna demonstrates that while properties such as depth to top, conductivity ratio, tilt, and dip can be determined, the depth to bottom is indeterminate due to the exponential decay of the inducing field.2