Fast direct conductivity transforms for TEM systems

Abstract

NEKUT'S METHOD When current I flows around a circular loop of radius R, the magnitude of the magnetic field H produced at distance D along the central axis of the loop is expressed as     2 0 3/2 2 2 2 I R H D R D    , (1) where μ 0 is the permittivity of free space. We imagine this loop placed on the surface of the earth as in a traditional electromagnetic sounding. When current I is abruptly terminated at time t = 0, a secondary magnetic field is induced in the earth beneath the loop. A magnetometer placed in the centre of the loop measures the decay of the vertical component of this secondary magnetic field. The approximation technique used by Nekut (1987) for the direct inversion of (current step-down) time-domain data relies on a comparison of the vertical magnetic fields produced at the centre of the circular loop. The magnetic field amplitude ratio, H(D)/H 0 , is found using equation (1), where H 0 is equal to μ 0 I/2 :     2 3/2 2 2 0 H D R H R D  . (2) Nekut uses the approximation that the secondary field induced in the earth can be replaced by a receding image source identical to the ground loop of radius R. The image descends through a homogeneous half-space as t advances. In his formulation, Nekut replaces D by the variable 2δ, in analogy with the image method of solution for a circular loop distance δ above an infinitely conductive thin sheet. When above a homogeneous ground that has conductivity σ, the secondary field can be defined to penetrate to depthˆdepthˆ depthˆ , a time domain analogue of the frequency domain 'skin depth' and which is described by the diffusion depth formula in the time domain, ie 0 2 ˆ t    . (3) Equations (2) and (3) are combined with D = 2δ to produce an equation that provides an approximate description of the time-decay of the secondary magnetic field H(t). Nekut uses this simple calculation is a comparison to the exact long-wavelength approximation solution (Ward and Hohmann, 1988) provided in the literature. The vertical magnetic field amplitude ratios compared to normalised half-space diffusion depth for both the approximate and exact solutions are shown in Figure 1. Both curves exhibit a t-3/2 fall-off in amplitude ratio, typical of a homogeneous half-space. In order to correct the differences between the exact central-loop solution of the literature and the approximate receding-image solution provided by the vertical magnetic field component at the centre of a circular loop, Nekut applied a correction factor that corrects the mirror depth δ to the diffusion depthˆdepthˆ depthˆ. In essence, Nekut used measured amplitudes to predict a mirror depth δ, then used a correction factor to predict a diffusion depth. Presumably, he did this because the correction factor curve is close to unity and slowly varying and, hence, easily interpolated. It is far simpler and more intuitive to us to simply assume that the amplitude measured is for a half-space and therefore find from the analytic solution a diffusion depthˆ/depthˆdepthˆ/ R  that matches each data point. SUMMARY In 1987, Nekut published in Geophysics a method that used the receding-image approximation of the time domain electromagnetic (TEM) response of a concentric loop system above a half-space to derive a simple, fast, direct transform that calculates resistivity as a function of depth. This method is by far the fastest of published transforms from TEM data to resistivity. Following this example, we make a further simplification that completely eliminates one intermediate step required by Nekut. His intermediate step was used to resolve differences between mirror depth (half the image depth) and the half-space diffusion depth. We simply use the half-space diffusion depth directly in Nekut's receding image method without requiring a mirror-depth calculation and a further calculation of its associated correction. The result is an even faster direct resistivity transform method that exactly matches the published results of Nekut. A further conceptual advance is immediately clear: the fast direct resistivity transform can be expanded to other common survey geometries such as coincident square-and circular-loop TEM systems. This is achieved through use of the diffusion depth with either direct forward modelling of the half-space or the mirror approximation. We explore this conceptual advantage and give an example of direct resistivity transforms for the Slingram geometry commonly used in electromagnetic surveys.