Approximating conductive ellipsoid inductive responses using static quadrupole moments

Smith and Morrison (2006) developed an approximation for the inductive response of conducting magnetic (permeable) spheroids (e.g., steel spheroids) based on the inductive response of conducting magnetic spheres of related dimensions. Spheroids are axially symmetric objects with elliptical cross-sections along the axis of symmetry and circular cross sections perpendicular to the axis of symmetry. Spheroids are useful as an approximation to the shapes of unexploded ordnance (UXO) for approximating their responses. Ellipsoids are more general objects with three orthogonal principal axes, with elliptical cross sections along planes normal to the axes. Ellipsoids reduce to spheroids in the limiting case of ellipsoids with cross-sections that are in fact circles along planes normal to one axis. Parametrizing the inductive response of unknown objects in terms of the response of an ellipsoid is useful as it allows fitting responses of objects with no axis of symmetry, in addition to fitting the responses of axially symmetric objects. It is thus more appropriate for fitting the responses of metal scrap to be distinguished electromagnetically from unexploded ordnance. Here the method of Smith and Morrison (2006) is generalized to the case of conductive magnetic ellipsoids, and a simplified form used to parametrize the inductive response of isolated objects. The simplified form is developed for the case of non-uniform source fields, for the first eight terms in an ellipsoidal harmonic decomposition of the source fields, allowing limited corrections for source field geometry beyond the common assumption of uniform source fields