Semi-Analytic Sensitivity of the MCIL Responses in Homogeneous Biaxial Anisotropic Media

We presented a semi-analytic solution for the sensitivity of the response tensor of the multicomponent induction logging (MCIL) tool in a homogeneous biaxial anisotropic (BA) medium. By using the perturbation principle and the spectral representation of electromagnetic (EM) fields, the sensitivity was formulated as a sevenfold integral whose integrand involves two sets of spectral electric fields. Fortunately, such a complicated integral can be reduced to a threefold integral, which consists of two parts: one is a definite integral over the spatial variable
$z$
that can be analytically integrated; the remaining is a twofold integral over the spectral variable
${\mathbf{k}}$
that is computed by numerical quadrature. We verified our formulation by comparing it with the exact solution and the finite-difference (FD) approximation.