Characterising the shape and material properties of hidden targets from magnetic induction data

The purpose of this paper is to clear up a major mystery in metal detection and confirm that the engineering prediction of H T ¨ pM ind H M q, for the sensitivity of measurements of the perturbed magnetic field to the presence of a general conducting object placed in a low frequency background field, is correct. Explicitly, H T is the background field generated by the transmitter coil, H M is the background field generated by the receiving coil as if it were used as a transmitter and M ind is a rank 2 polarisation tensor, which describes the shape and material properties of the object. To show this, we apply a recently derived asymptotic formula for the perturbed magnetic field due to the presence of a conducting object, which is expressed in terms of a new class of rank 4 polarisation tensors (Target detection and characterization from electromagnetic induction data, Journal de Mathématiques Pures et Appliquées (2013) At first sight this appears to contradict the engineering prediction, however, contrary to this, we show that at most 9 rather than 81 coefficients are required to describe the rank 4 tensor for a conducting object and a further 9 are required if the object is magnetic. We then show that the rank 4 tensor does in fact reduce to a rank 2 tensor, thus providing a solid theoretical foundation for the engineering prediction. Furthermore, by combining the reduced conductivity and permeability tensors, we obtain a symmetric rank 2 tensor, which describes a general conducting object in terms of just 6 complex independent coefficients. For objects with rotational and mirror symmetries we show that the number of coefficients is still smaller. We include numerical examples to demonstrate that the new polarisation tensors can be accurately computed by solving a vector valued transmission problem by hp–finite elements and include evidence to confirm that the asymptotic formula describing the perturbed fields agrees with the numerically predictions.