Nonlinear inversion in three-dimensional cross-well induction logging using full-vectorial multi-frequency data

A method for reconstructing the electrical conductivity from the cross-well induction logging measurements is presented. This work extends the method previously developed by Abubakar and van den Berg (see Radio Science, vol.33, no.4, p.989-1004, 1998) for the electrode logging (static) case to the more complicated induction logging (below 100 kHz) case using full-vectorial multi-frequency data. The target is assumed to be located in an object domain /spl Dscr/2 embedded in an unbounded homogeneous domain with conductivity /spl sigma//sub 0/. To reconstruct the conductivity of this object domain /spl Dscr/ from a knowledge of the secondary field, we assume that the object is illuminated successively by a number of divers point magnetic dipoles directed in the logging direction at the borehole axes (j=1,...,J), and at different frequencies (i=1,...,I) with time factor exp(-j/spl omega//sub i/t). For each excitation, we then have a full-vector three-dimensional problem where the primary electric field is denoted as E/sup p//sub ij/, and the total electric field as E/sub ij/. We assume that for the description of the material properties the Maxwell model for the diffusion problem holds. We then introduce the conductivity contrast /spl chi/=/spl sigma///spl sigma//sub 0/-1, where /spl sigma/ is the spatially dependent conductivity of the object domain /spl Dscr/.