Transient field solution for a layer of finite thickness on a resistive basement

Abstract

A closed-form asymptotic solution is derived for the magnetic field of the currents induced by a transient airborne magnetic source in a conductive layer of finite thickness. The conductive layer rests on top of a resistive half-space. Like the well-known solution found by J. C. Maxwell for a thin conductive sheet surrounded by an insulator, the secondary magnetic field is expressed in terms of an image source receding from the layer. However, the new solution also accounts for the layer thickness h and the conductivity of the half-space.One of the conclusions from the new solution is that the mirror plane that specifies the position of the image is located below the upper interface of the conductive layer at depth h/3. This indicates the correct position at which the equivalent thin sheet should be placed when Maxwell's solution is applied to a layer of finite thickness.If the basement that underlies the layer is highly resistive, Maxwell's solution becomes accurate when induced currents are almost uniformly spread across the layer. It remains accurate as long as currents induced in the basement can be neglected. Eventually, the secondary magnetic field of these currents will prevail over the field of currents in the layer. Maxwell's solution loses its accuracy long before this occurs. Depending on parameters of the model, the validity time range of Maxwell's solution may be narrow or even nonexistent.The generalized image solution is applicable in thewhere v s is the image recession speed, and σ s and σ b are the layer and basement conductivities, respectively. This range is significantly wider than that of Maxwell's solution. At early times, the secondary magnetic field is controlled by the position of the nearest interface of the conductive layer. Accounting for this observation, a simple modification of the new solution may be used to extend the applicability range towards even earlier times. The generalized solution is faster by several orders of magnitude than a numeric solution based on successive wavenumber-tospace and frequency-to-time domain fast Hankel transforms.

Generalized Tikhonov's algorithm for accurate calculation of one‐dimensional transient responses directly in time domain

“…The expression is simple, but mathematically is not strictly substantiated. Contrary, Singer and Green () showed more complicated, but strictly substantiated asymptotic solution for the same model.…”