Closed form transient solution for a layer of finite thickness on resistive half-space

Abstract

The time-domain thin sheet solution derived by J. C. Maxwell more than a century ago remains an important tool for fast interpretation of airborne electromagnetic data. The solution is exact if a conductive layer is thin and surrounded by an isolator. As shown by Maxwell, the magnetic field of the currents induced by a turned-on magnetic dipole coincides with the field of an auxiliary dipole receding at a constant speed, which depends only on the layer conductance. The solution allows for an easy determination of the layer conductance from the measured magnetic field. The layer conductivity and thickness cannot be directly determined. Maxwell's solution becomes accurate at the late stage of the decay of induced currents provided that the basement that underlies the layer is non-conductive. The moment when the thin sheet solution becomes applicable is specified by the time necessary for the receding dipole to cross the layer. On the other hand, the secondary magnetic field of the currents induced in a basement of finite resistivity eventually prevails over the magnetic field of currents in the conductive layer. Depending on the layer thickness and layer-to-basement conductivity contrast, the time applicability range of the thin sheet solution narrows down or even disappears. We derive an asymptotic solution that accounts for the layer thickness as well as the basement resistivity. A special correction makes the solution applicable immediately after the source is turned on. The new solution is by orders of magnitude faster than a numeric solution based on successive wave number-to-space and frequency-to-time domain Fast Hankel Transforms.