A brief note on the computation of silent from nonsilent contributions of spatially localized magnetizations on a sphere
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(2020) 11:25
ORIGINAL PAPER
A brief note on the computation of silent from nonsilent contributions of spatially localized magnetizations on a sphere Christian Gerhards1 Received: 21 March 2020 / Accepted: 2 September 2020 © The Author(s) 2020
Abstract Any square-integrable vector field f over a sphere S can be decomposed into three unique contributions: one being the gradient of a function harmonic inside the sphere (denoted by f+ ), one being the gradient of a function harmonic in the exterior of the sphere (denoted by f− ), and one being tangential and divergence-free (denoted by fd f ). In geomagnetic applications this is of relevance because, if we consider f to be identified with a magnetization, only the contribution f+ can generate a non-vanishing magnetic field in the exterior of the sphere. Thus, we call f− and fd f “silent” and f+ “nonsilent”. If f is known to be spatially localized in a subregion of the sphere, then f+ and f− are coupled due to their potential field nature. In this short paper, we derive an approach that makes use of this coupling in order to compute the contribution f− from knowledge of the contribution f+ . Keywords Vector field decomposition · Inverse magnetization problem · Spatial localization · Uniqueness · Spherical harmonics Mathematics Subject Classification 31B20 · 41A30 · 65D15 · 65R32 · 86A22
1 Introduction 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
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Christian Gerhards [email protected] Geomathematics and Geoinformatics Group, Institute of Geophysics and Geoinformatics, TU Bergakademie Freiberg, Gustav-Zeuner-Str. 12, 09599 Freiberg, Germany 0123456789().: V,-vol
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GEM - International Journal on Geomathematics
(2020) 11:25
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 2016, 2019; Lima et al. 2013). We focus on spatially localized M but use computations in spectral domain. To be precise, throughout the course of the paper, we understand spatial localization as strict spatial localization, i.e., there exists a subregion ⊂ S such that M(x) = 0 for x ∈ S\. More precisely, let a square-integrable (vectorial) magnetization M be given on the unit sphere S. Its so-called Hardy–Hodge decomposition takes the form M = M+ +
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