On the Reconstruction of Dipole Directions from Spherical Magnetic Field Measurements
This work addresses a fundamental non-uniqueness problem in geomagnetism and inverse problems for researchers studying magnetization reconstruction.
The paper investigates the uniqueness of reconstructing dipole directions from spherical magnetic field measurements, showing that under spatial localization or band-limited constraints, the dipole direction can be determined up to a polynomial condition, with examples of non-uniqueness and admissible candidates.
Reconstructing magnetizations from measurements of the generated magnetic potential is generally non-unique. The non-uniqueness still remains if one restricts the magnetization to those induced by an ambient magnetic dipole field (i.e., the magnetization is described by a scalar susceptibility and the dipole direction). Here, we investigate the situation under the additional constraint that the susceptibility is either spatially localized in a subregion of the sphere or that it is band-limited. If the dipole direction is known, then the susceptibility is uniquely determined under the spatial localization constraint while it is only determined up to a constant under the the assumption of band-limitedness. If the dipole direction is not known, uniqueness is lost again. However, we show that all dipole directions that could possibly generate the measured magnetic potential need to be zeros of a certain polynomial which can be computed from the given potential. We provide examples of non-uniqueness of the dipole direction and examples on how to find admissible candidates for the dipole direction under the spatial localization constraint.