Inversion of induction tool measurements using the distorted Born iterative method and CG-FFHT

The inversion of induction tool measurements using the distorted Born iterative method (DBIM) and the conjugate gradient-fast Fourier-Hankel transform (CG-FFHT) is described. The inverse problem is formulated in terms of an integral equation of scattering where the unknown to be sought is the conductivity in the rock formation, when the measurements along a borehole axis are performed. The nonlinear problem is linearized at each stage using the distorted Born approximation. The inhomogeneous medium Green´s function in the distorted Born approximation is found by solving a volume integral equation using the CG-FFHT method, which allows a rapid solution to a large problem with reduced computational complexity and memory requirement. In this manner, the inverse problem is solved with a computational complexity proportional to N_tlN log N where N_tl is the number of transmitter locations used in the data collection and N is the total number of pixels used to model the unknown formation. The memory requirement is of order NN_tl