2D Linear inversion of ground-based controlled-source electromagnetic data under a low induction number condition

Low-induction-number frequency-domain geoelectromagnetic (LIN-GeoFEM) instruments are ground conductivity meters that use a small coil transmitter (Tx) and one coil receiver (Rx). This coil–coil system is designed to propagate alternating electromagnetic fields through the earth at small Tx–Rx separations and low frequency and receive the EM field coupling in the shallow subsurface to provide direct measures of the apparent electrical conductivity. This measured property is a complicated average of spatially distributed localized electrical conductivities in the subsurface. Instruments capable of operating as LIN FEM instruments include the EM38, EM31, and EM34 (Geonics Ltd., Mississauga, ON), the DUALEM instruments series (DUALEM, Inc., Milton, ON), the GEM instrument series (Geophex Ltd., Raleigh, NC) and CMD series (GF Instruments, s.r.o.). The Tx and Rx coils can be oriented relative to each other and the earth's surface. Orientations considered in this study are horizontal coplanar (HCP) (both coils lie flat on the ground) and vertical coplanar (VCP) (coils are upright and coplanar). The range of LIN FEM instruments applications for environmental and hydrologic characterization and monitoring is large and increasing.The LIN-GeoFEM applications are industrially feasible as long as there is a reasonably fast algorithm that is accurate enough to invert the survey data. Furthermore, forward modeling plays a key role in the inversion procedure. The linear integral equation (IE) method is a powerful tool in EM forward modeling for geophysical applications, especially for simple background conductivity structures. The main advantage of the IE method in comparison with the finite difference (FD) and finite element (FE) methods is its fast and accurate simulation of the response for models with compact 2-D or 3-D bodies in a layered background. The main limitation of the IE method is that the background conductivity model must have a simple structure to allow for an efficient Green’s function calculation. Fortunately, the most widely used background models in LIN-GeoFEM explorations are those formed by horizontally homogeneous layers. A main issue is that the EM field integral equation is nonlinear. However, an approximate linear equation is obtained for the electromagnetic induction at low induction numbers using the Born approximation. A 2D forward modeling code for LIN-GeoFEM is developed based on the integral equation (IE) method. Here, a linear relation between model parameters and apparent conductivity values is proposed. The 2D problem is obtained from 3D using numerical integration along the y-axis (strike direction) from minus infinity to infinity. So, the linear approximation is applied to the 2D inversion of LIN measurements. We use a damped minimum length solution using depth weighting to solve this problem iteratively. Thus, we obtain a better estimate of conductivity in a few iterations. Using this 2D linear inversion or imaging technique, we can produce reasonably good results of inverting jointly and individually VCP and HCP for low and moderate conductivity contrasts.To validate the algorithm, we consider two 2D synthetic scenarios and field data acquired on a thick conductive dyke in the Bloemfontein Nature Reserve region in South Africa. The first synthetic scenario consists of one 3 W.m conductive horizontal or vertical prism immersed in a 100 W.m resistive host. In this example, the recovered models from the inversion of the HCP (VMD) and VCP (HMD) data show good results for the vertical and horizontal prism, respectively. The second scenario simulates four 20 W.m conductive vertical and horizontal prisms in a 100 W.m resistive background. The recovered conductivity from the inversion of the VCP data has the weakest results, especially in the case of vertical prisms. In the conductivity section from the inversion of HCP data, the existence of the four anomalous bodies is evident. However, the image obtained from the joint inversion of HCP and VCP data has generated useful information about the true model in all recovered models. The result of jointly inverting VCP and HCP field data confirms the presence of the dyke as a zone of low conductivities.