The magnetic gradient tensor of a triaxial ellipsoid, its derivation and its application to the determination of magnetisation direction

Abstract

This paper presents a theory for the anomalous magnetic gradient tensor due to a uniformly magnetised general triaxial ellipsoid. Expressions for the magnetic field vector and its gradient tensor are derived from expressions for the gravitational field or the gravity gradient tensor via an application of Poisson’s theorem. This theory provides increased capability in forward modelling, inversion and equivalent source applications in both magnetic and gravimetric exploration. It provides an accurate and computationally efficient means of modelling the magnetic gradient tensor of ellipsoidal bodies which may possess isotropic or anisotropic magnetic susceptibility, remanent magnetisation and, in the case of highly magnetic ellipsoids, may be subject to the effect of self-demagnetisation. This paper presents a novel method based on the eigenvector decomposition of the magnetic gradient tensor to provide estimates of the magnetisation direction over an ellipsoidal source. This includes an investigation of the influence of shape detail, observation height and inclination of magnetisation on the positioning of global maxima in normalised source strength and how this affects the problem of estimating magnetisation direction. This study confirms that magnetisation directions may be accurately estimated for extremely elongated ellipsoidal bodies where the ratio of smallest observation height to maximum elongation (in plan view) is greater than 1.

KEYWORDS:

• Ellipsoid
• magnetic gradient tensor
• magnetisation
• self-demagnetisation
• eigenvector
• normalised source strength

Acknowledgements

First I would like to acknowledge my supervisor Mark Lackie. I should like to thank David Clark, David Pratt and Clive Foss for encouraging me to write this paper. I also acknowledge the efforts of David Pratt, Clive Foss and David Clark for having the foresight to support the development of theory and software for calculating the magnetic tensors well before these became of general interest in potential field geophysics.

Disclosure statement

No potential conflict of interest was reported by the author.