Analytic formulas for complete and incomplete first-order TEM moments below ground in a conductive half-space

Abstract

In the resistive limit, discrete conductors give rise to magnetic dipole fields. Magnetostatic modeling of time integrals, or moments, of transient electromagnetic (TEM) data therefore offers a means for fast approximate 3D modeling and inversion of TEM data sets. In our approximate inversion scheme, the net TEM moment response is represented as the combination of discrete conductor and uniform host responses. The inversion algorithm first estimates a homogeneous host conductivity, and then it subtracts the host response and fits the residual moment data by adjusting the conductivities of cells comprising a 3D rectangular mesh. To expedite calculation of the host response, we have derived analytic formulas for first-order TEM moments produced on and under the ground by a horizontal electric dipole on the surface of a homogeneous conductive half-space. We present analytic expressions for idealized all-time “complete” moments, or resistive limits, as well as for realizable finite-time “incomplete” moments. The moments produced by an arbitrary horizontal polygonal loop are determined by combining contributions from appropriately oriented electric dipoles. Downhole TEM moments computed with the new expressions reveal substantial differences between incomplete and complete moments when early time data are excluded and between step and impulse response incomplete moments. The role of the formulas in the first stage of our moment-based 3D inversion scheme is illustrated via analysis of downhole TEM data recorded at Santander, Peru. The host conductivity of best fit for early time B-field moments is 2.40 mS/m, consistent with apparent conductivities derived from ground TEM data recorded in the same area.