Estimating equivalent dipole polarizabilities for the inductive response of isolated conductive bodies

Away from a conductive body, secondary magnetic fields due to currents induced in the body by a time-varying external magnetic field are approximated by (equivalent) magnetic dipole fields. Approximating the external magnetic field by its value at the location of the equivalent magnetic dipoles, the equivalent magnetic dipoles' strengths are linearly proportional to the external magnetic field, for a given time dependence of external magnetic field, and are given by the equivalent dipole polarizability matrix. The polarizability matrix and its associated equivalent dipole location are estimated from magnetic field measurements made with at least three linearly independent polarizations of external magnetic fields at the body. Uncertainties in the polarizability matrix elements and its equivalent dipole location are obtained from analysis of a linearized inversion for polarizability and dipole location. Polarizability matrix uncertainties are independent of the scale of the polarizability matrix. Dipole location uncertainties scale inversely with the scale of the polarizability matrix. Uncertainties in principal polarizabilities and directions are obtained from the sensitivities of eigenvectors and eigenvalues to perturbations of a symmetric matrix. In application to synthetic data from a magnetic conducting sphere and to synthetic data from an axially symmetric elliptic conducting body, the estimated polarizability matrices, equivalent dipole locations, and principal polarizabilities and directions are consistent with their estimated uncertainties.