Introduction
A fast, approximate 3D TEM inversion scheme has been developed, based on EM modeling at the resistive limit. The TEM data are converted to magnetic moments via time integration. In effect, the moment transformation converts the nonlinear 3D TEM inversion problem into a linear 3D magnetic inversion problem. The resistive limit response is realized as a linear combination of a discretized 3D target response and a continuous host response. A starting model is constructed from conductivity depth images of the TEM profiles. In addition, the inversion is constrained by geological information and by standard potential field inversion devices such as depth weights. The underlying model is both geological and petrophysical. The inverse problem is solved using a fast steepest-descent algorithm. The approximate inversion scheme has been successfully tested on synthetic fixed-loop TEM examples and on real fixed-loop TEM field data. Typically, for ground-TEM, the approximate 3D inversion completes in minutes on a Pentium 4 2.8-Ghz (from 2006).
A fast approximate 3D TEM modeling and inversion algorithm has been developed, based on the concept of TEM moments. The ground is divided into cubic cells, and electromagnetic interaction between volume elements is ignored. This is the first order TEM moment transform (Smith and Lee, 2002a). Application of the moment transform effectively converts the TEM inversion problem into a linear pseudo-magnetic inversion problem. The time-integration transforms each TEM decay into a single parameter. The time-depth resolution lost in the process is restored in three main ways: (i) a 3D starting model is constructed via interpolation of conductivity-depth imaging (CDI) sections; (ii) geological constraints are imposed if available; and (iii) the inversion is conditioned via weighting if desired, e.g. depth weighting (Li & Oldenburg, 1996). The 3D inversion algorithm was originally developed for potential fields data (Fullagar and Pears, 2007; Fullagar et al., 2008). The starting model is a geological model mapped onto the cubic 3D grid. Each cubic cell is assigned a rock type as well as a time constant. The geological significance of boundaries is preserved throughout the inversion process. Geological constraints ensure that the solution is consistent with all available geological information. The approximate inversion scheme is illustrated below, applied to both synthetic and real fixed loop TEM data.
The forward modeling scheme consists of superposition of the resistive limits of an anomalous volume, divided into cubic cells, together with a continuous host or background. Each cell is assigned a time constant, tk, proportional to its conductivity. Each induced magnetic dipole inherits the orientation of the primary field. Consequently, the induced vortex currents are static and current diffusion cannot be simulated. For this reason a continuous background is introduced. The resistive limit response of a continuous conducting half space is adopted as the background contribution. For a receiver on the surface of the half-space the Z-component TEM moment for a linear current segment is given by (Schaa, 2010). Summing the contributions from all four sides of the Tx-loop gives the net resistive limit response.