The magnetic gradient tensor of a right circular cylinder: theoretical considerations in the determination of magnetisation direction

Abstract

This paper investigates the use of the magnetic gradient tensor and its eigenvectors to derive expressions for estimating the direction of magnetisation of a uniformly magnetised right circular vertical cylinder. Expressions for the gradient tensor, the eigenvalues and eigenvectors, the normalised source strength (NSS), and the direction of magnetisation on the axis of a vertical cylinder are shown to be identical in form to those previously derived for a magnetised sphere or dipole. However, the off-axis gradient tensor field of a vertical cylinder is significantly different to that of a magnetic sphere or dipole particularly at low observation heights. These differences lessen with increasing observation height and in more compact pipes. The normalised source strength displays a directional asymmetry in the eigenvector field of the gradient tensor above a pipe which is related to the direction of magnetisation. This azimuthal asymmetry in the NSS results exclusively from the horizontal components of magnetisation, which are perpendicular to the longitudinal face of the vertical cylinder. In contrast, the vertical component of magnetisation is normal to the circular planar face and produces radial symmetry in the NSS. Additionally, the paper details some unusual properties of the NSS which are attributable to non-dipole field components in the gradient tensor field which cause pairs of eigenvectors to interchange directions when ordered by their eigenvalues, as is done when calculating the NSS. This results in spatial discontinuities in the eigenvector fields arising from crossovers in the ordered eigenvalue surfaces which vary with both magnetisation direction and observation height. This phenomenon is not present in a dipole field and thus provides direct evidence of non-dipole field components. Understanding this complexity in the eigenvector field is essential to designing methodologies which can successfully estimate the direction of magnetisation over a uniformly magnetised cylinder. One proposed method based on the peak NSS, uses airborne gradient tensor data to accurately estimate magnetisation direction over a reversely magnetised pipe-like body in the Diavik diamond field of north-western Canada. The results are verified using a full tensor inversion.