Simultaneous reconstruction of permittivity and conductivity for crosshole geometries

Abstract

We develop the theory and algorithm for the simultaneous inversion of permittivity and conductivity maps in the region between two boreholes in the earth. The three‐dimensional region is modeled as an inhomogeneous annulus of known thickness and height with radial and vertical variations of the electrical properties. The medium of interest is probed by a vertical magnetic dipole located on the axis of the cylindrical geometry and the receivers can be placed anywhere outside the inhomogeneous annulus. The inversion procedure is formulated in terms of a source‐type integral equation using monochromatic data. The integral equation is solved using an iterative approach where a Born approximation is applied at each iteration step. The nonuniqueness of the problem is overcome by imposing additional constraints on the solution using the method of regularization of Tikhonov. The distribution of electrical properties is obtained directly by this method. Numerical simulations, including multisource inversions, show that good results can be obtained for smoothly varying electrical properties, even for large contrast cases. In the case of rapid local variations of such properties, convergence can still be reached but at a slower rate and the reconstructions are smoothed versions of the original properties. In the presence of noise, the permittivity reconstruction is more robust than the conductivity reconstruction for the model considered here.