On the reconstruction of inducing dipole directions and susceptibilities from knowledge of the magnetic field on a sphere

Abstract

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 assumption of bandlimitedness. 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. ARTICLE HISTORY

Unveiling Earth's Hidden Magnetization

“…However, retrieving the magnetization distribution associated with a given magnetic field set of measurements in a unique way has been a long‐standing open question in geophysics that has severely restricted the use of magnetic data for probing solid Earth's upper layers. Decomposing the magnetization into three parts, from which only the one contributes to the observed magnetic field and is therefore directly recoverable from magnetic field measurements, constituted an important step forward (Baratchart et al, ; Gerhards, , ; Gubbins et al, ).…”

A brief note on the computation of silent from nonsilent contributions of spatially localized magnetizations on a sphere

“…The vector spherical harmonics (2.3)-(2.5) correspond (up to a possibly varying sign or normalization factor) to Gubbins et al (2011), Vervelidou and Lesur (2018) and Lesur and Vervelidou (2020) Freeden and Gerhards (2012), Freeden and Schreiner (2009), Gerhards (2016) and Gerhards (2019). Some Sobolev spaces that are required later on are gathered in the following definition.…”

“…The non-uniqueness of the reconstruction of a magnetization M from magnetic field data B is well-known (e.g., Backus et al 1996;Blakely 1995). Recently this has been investigated in more detail in several publications: for the Euclidean setup with applications in SQUID microscopy in Baratchart et al (2013) and Lima et al (2013), for spherical geometries with applications in geomagnetism and planetary magnetism in Baratchart and Gerhards (2017), Gerhards (2016), Gerhards (2019), Gubbins et al (2011), Vervelidou and Lesur (2018), Vervelidou et al (2017) and Lesur and Vervelidou (2020). The studies above can be divided into those that focus on dealing with M in spectral domain (cf.…”

“…Gubbins et al 2011;Vervelidou and Lesur 2018;Vervelidou et al 2017;Lesur and Vervelidou 2020) and those dealing with spatially localized M (cf. Baratchart and Gerhards 2017;Baratchart et al 2013;Gerhards 2016Gerhards , 2019Lima et al 2013). We focus on spatially localized M but use computations in spectral domain.…”

“…If we assume that M is spatially localized, then both M + and M − are uniquely determined by knowledge of B in the exterior. Inverting B subject to localization constraints on M might be one way to obtain the contributions M + and M − (as has been done, e.g., in Baratchart et al (2013), Gerhards (2016) and Gerhards (2019)). However, the short paper at hand more generally aims at obtaining M − from knowledge of M + under the assumption that (an otherwise unknown) M is spatially localized, without considering any connection to a magnetic field B.…”