New analytical expression of the magnetic gradient tensor for homogeneous polyhedrons

Abstract

We have developed a new analytical expression for the magnetic-gradient tensor for polyhedrons with homogeneous magnetization vectors. Instead of performing the direct derivative on the closed-form solutions of the magnetic field, it is obtained by first transforming the volume integrals of the magnetic-field tensor into surface integrals over polyhedral facets, in terms of the gradient theorem. Second, the surface divergence theorem transforms the surface integrals over polyhedral facets into edge integrals and structure-simplified surface integrals. Third, we develop analytical expressions for these edge integrals and simplified surface integrals. We use a synthetic prismatic target to verify the accuracies of the new analytical expression. Excellent agreements are obtained between our results and those calculated by other published formulas. The new analytical expression of the magnetic-gradient tensor can play a fundamental role in advancing magnetic mineral explorations, environmental surveys, unexploded ordnance and submarine detection, aeromagnetic and marine magnetic surveys because more and more magnetic tensor data have been collected by magnetic-tensor gradiometry instruments.