Simultaneous reconstruction of geometric parameter and resistivity around borehole in horizontally stratified formation from multiarray induction logging data

The new multiarray induction logging tool consists of eight different spacing three-coil arrays, operates at three frequencies, and measures both in-phase and quadrature signals to provide more information on the distribution of conductivity around the borehole than the conventional induction tool. A horizontally layered medium with a step-profile invasion per bed is used to describe the formation model. In this paper, we will establish a regularized iterative inversion algorithm to simultaneously reconstruct geometric parameter and resistivity per bed from the multiarray induction logging data. During the inversion, the Frechet derivative matrix with respect to the model parameters is efficiently calculated in terms of the perturbation principle, and the hybrid approach of Maxwell´s equations and the two-order Langrange function is used to determine the eigenmode solution in the radial direction in order to further enhance efficiency of forward modeling and calculation of the Frechet derivative matrix. Normalizations are used to transform the model vector, log data, and the Frechet matrix into dimensionless variables. Regularization and exponential damping factors are used to enhance inversion stability. We will also analyze and reduce the inversion errors originating from the noise in input data and the errors in bed thicknesses.