An Efficient Algorithm of Both Fréchet Derivative and Inversion of MCIL Data in a Deviated Well in a Horizontally Layered TI Formation Based on TLM Modeling

In this paper, we set up an efficient Fréchet derivative algorithm of inversion of multicomponent induction logging (MCIL) data in horizontal layered transversely isotropic (TI) formations based on transmission line method (TLM) in order to simultaneously reconstruct the model vector including both horizontal and vertical conductivities, horizontal interfaces, borehole dipping angle, and tool azimuth in deviated well from the MCIL data. First, MCIL responses in the TI model are efficiently determined by both the spectrum EM fields obtained by TLM and the semianalytical computation of Sommerfeld integrals based on the cubic spline interpolation. Then, the Born approximations are executed to derive the MCIL Fréchet derivatives as the sixfold integrals of the products of two spectrum EM fields in infinite domains. By using the integral characters of the 2-D Dirac function, the sixfold integrals of Fréchet derivatives are further simplified into twofold integrals: One is the Sommerfeld integral in the radial spectrum domain, and the other is the definite integral in the vertical spatial domain. They are efficiently computed by the semianalytical approach similar to the MCIL simulation. Therefore, we obtain the linear equations between changes in the MCIL responses and small perturbations in the model vector. After that, we iteratively modify all of the model parameters to realize the best fit between the input data and the synthetic data of the inverted model by the normalization of the Fréchet derivative and singular value decomposition technique. Finally, we apply numerical results to investigate the characteristics of the Fréchet derivatives and to validate the inversion method and its antinoise ability.