A spectral method for the inverse electromagnetic logging problem

Abstract

This paper presents a new inverse method for electromagnetic logging. The analysis assumes a two‐coil sonde operated in a CW mode. The geometric model allows for a piecewise continuous, axially inhomogeneous formation intersected at a right angle by a circular borehole. The inversion uses both in‐phase and quadrature‐phase log data. These data are Fourier transformed with respect to the axial coordinate z. An optimized fast Fourier transform technique, which selects the numerical transform parameters for both the sampling interval and window size, provides an accurate parallel computation of the theory’s integrals and yields the entire log segment in one computation. The inversion of the borehole voltage data to determine the formation conductivity profile requires no matrices and is ideally suitable for combinations on a minicomputer array processor. Central to the method is a transfer function relationship between the spectra of the anomalous borehole voltage and the formation conductivity. Typically, 1 024 point transforms are adequate. When the complex transfer function sequence is stored on disc, log data can be inverted in real time. Numerical simulations show that the constrained inversion procedure tolerates the zeros in the transfer function and is robust in the presence of additive noise. Both formation conductivity and dielectric constant estimates are computed. Results are given for a piecewise constant formation model.