Reconstruction of axisymmetric media with an FFHT-enhanced extended Born approximation

We invert for the axisymmetric conductivity distribution from borehole electromagnetic induction measurements using a two-step linear inversion method based on a fast Fourier and Hankel transform (FFHT) enhanced extended Born approximation (EBA). In this method, the inverse problem is first cast as an under-determined linear least-norm problem for the induced electric current density; from the solution of this induced current density, the unknown conductivity distribution is then obtained by solving an over-determined linear problem using the newly developed, FFHT-enhanced EBA. Numerical results show that this inverse method is applicable to a very high conductivity contrast. It is a natural extension of the original two-step linear inversion method of Torres-Verdin and Habashy to axisymmetric media. In the first step, the CPU time costs O( N 2 ). In the second step, the CPU time costs O( N log 2 N ) where N is the number of unknowns. Because of the FFHT algorithm, this inverse method is actually more efficient than the conventional, brute-force first-order Born approximation.