Closed-form solution of magnetic field and magnetic gradient tensor for magnetic minerals

Abstract

Magnetization is a natural property of magnetic materials which is widely used to study the ancient climate change whose footprint was recorded in rocks by palaeomagnetic data, uncover the location of unexploded ordnance, mineral deposits and gas and oil resources, and image regional crust and planet structure. An essential but very challenging task is to find analytical solutions of induced magnetic signals for general magnetic targets. In this study, for the first time, we demonstrate the existence of such solutions and explicitly give the analytical expressions of magnetic potential, magnetic field and magnetic gradient tensor for magnetic materials with polyhedral shapes and polynomial magnetization vectors. The order of magnetization polynomials can vary from linear order to arbitrary integer orders along both horizontal and vertical directions. Several synthetic and realistic iron deposit models are used to validate the accuracy and efficiency of our analytical solutions. Due to its inherent importance, our findings would have an immediate impact on magnetic explorations in earth science and have potentials in different applications.