A thin-sheet model for global electromagnetic induction

Abstract

We consider the frequency domain electromagnetic (EM) induction problem in a layered spherical Earth overlain by an inhomogeneous thin sheet of finite conductance, and derive an integral equation for the secondary electric fields resulting from surface conductivity variations. The Green′s function kernel of the integral equation is singular, and must in practice be regularized for a numerical treatment. We compare two approaches: truncation of the spherical harmonic representation of the Green′s function, and projection onto a discrete spatial grid, with elements of the resulting discrete system matrix evaluated as contour integrals. In either case rotational symmetry and the fast Fourier transform (FFT) may be exploited to reduce computational costs, and Krylov space methods may be used to solve the resulting discretized integral equation efficiently. We illustrate the methods for several source field geometries. The two regularization schemes result in solutions that agree well at large scales, but exhibit substantial differences as the nominal resolution is approached. The agreement improves with increased resolution. Although the level of solution errors for the two approaches is likely comparable in an L2 sense, the truncation approach results in Gibb′s oscillations near sharp conductivity contrasts. These may be reduced by tapering the truncated kernel, essentially equivalent to low-pass filtering the solution.