A closed-form formula-based method for magnetic dipole localization in a constant magnetic field by measurement of its magnetic field vectors and magnetic gradient tensors

Abstract

The problem of dipole localization and its parameter estimation has been widely investigated. In the literature, various approaches have been proposed, including iterative methods and resolution schemes based on inverse formulas. This paper introduces the Closed-form Formula-based Method (CFM), which addresses the dipole localization problem when the source is placed in the Earth’s magnetic field, assumed to be constant within the region of interest. The CFM is based on a modified version of the Yin formula that is a closed-form expression originally developed to estimate the position of a dipole in the absence of a constant magnetic field. The proposed method utilizes this modified Yin formula in combination with a specific magnetometer array configuration designed to measure both gradients and magnetic field values, along with a selection algorithm. The concept was tested by simulations in which different dipole parameters and different positions are tested to verify all possible configurations. Both the Yin formula and the CFM were simulated in MATLAB by developing scenarios in which infinitesimal gradient schemes were considered. Additionally, the CFM was tested by taking into account a finite gradient scheme, the effect of white noise, different trajectories in space, and different strengths of the dipole. Results show that the CFM can be used in all configurations in which the Yin formula does not work, and that it localizes a 100 Am² dipole at a distance of 3 m, with a white noise of 0.01 nT as a standard deviation, providing a mean and maximum absolute error of 0.01 m and 0.03 m, respectively.