Theoretical and practical considerations for crosswell electromagnetic tomography assuming a cylindrical geometry

Abstract

An iterative Born imaging scheme is employed to analyze the resolution properties of crosswell electromagnetic tomography. The imaging scheme assumes a cylindrical symmetry about a vertical magnetic dipole source and employs approximate forward modeling at each iteration to update the internal electric fields. Estimation of the anomalous conductivity is accomplished through least‐squares inversion. Much of the mathematical formulation of this diffusion process appears similar to the analysis of wavefield solutions, but the attenuation implicit in the complex propagation constant invalidates many of the accepted wavefield criteria for resolution. Images of illustrative models show that vertical resolution improves with increasing frequency and with increased spatial sampling density. In addition, greater conductivity contrasts between the target and the background can result in better resolution. The horizontal resolution depends on the maximum aperture that is employed and with increasing frequency, larger apertures are needed to obtain optimal results. However, the maximum aperture that can be employed, and thus the horizontal resolution, is limited by the rate of attenuation and the noise present in the measurements. Weighting the long‐offset data equally with the zero‐offset data can improve the resolution if the noise is not a function of the dynamic range of the measurement system. At lower frequencies, the resolution can be improved by measuring both the horizontal and vertical components of the magnetic fields. In addition, multiple frequencies can be employed to improve the resolution for limited aperture measurements. The general applicability of the cylindrically symmetric geometry is examined by comparing the 2-D sensitivity functions to those produced by a 2.5-D model, and by imaging a 3-D body with the 2-D iterative Born scheme. For borehole separations greater than five skin depths it is demonstrated that the measurements, and thus the images, are not affected by the geometry of the conductive zone outside of the interwell plane. Thus the 2-D imaging scheme can be employed in these situations. For borehole separations less than five skin depths, artifacts are produced in the images which will lead to faulty interpretations.