Abstract
In the present work, we consider a magnetization moment recovery problem, that is finding integral of the vector function (over its compact support) whose divergence constitutes a source term in the Poisson equation. We outline derivation of explicit asymptotic formulas for estimation of the net magnetization moment vector of the sample in terms of partial data for the vertical component of the magnetic field measured in the plane above it. For this purpose, two methods have been developed: the first one is based on approximate projections onto spherical harmonics in Kelvin domain while the second stems from analysis in Fourier domain following asymptotic continuation of the data. Recovery results obtained by both methods agree and are illustrated numerically by plotting formulas for net moment components with respect to the size of the measurement area.
Highlights
Kelvin transformationIm z > 0 onto the unit disk |z| < 1 preserving harmonicity
Introduction and problem formulationEarth rocks and meteorites may preserve invaluable records of ancient planetary and solar nebula magnetic fields in the form of remanent magnetization
We focus on (iii), the last of the formulated issues, namely, dealing with incomplete data
Summary
Im z > 0 onto the unit disk |z| < 1 preserving harmonicity. Kelvin transformation is a generalization of this concept to higher dimensions [1]. We consider a one-parameter (R0 > 0) family of transforms. F Rξ , ξ−s where ξ := (ξ1, ξ2, ξ3)T , s := (0, 0, −R0)T , e0 := 2R0 (R0 + h), and. Which map functions on horizontal plane x3 = h onto those defined on the sphere of radius R0 centered at the origin. ∆f (x, x3) = 0, x3 > h ⇐⇒ ∆f ξ = 0, ξ < R0 Application of this transform to the potential followed by restriction to the sphere SR0 gives. Because of a more complicated angular dependence, the simple link between the net moment components and projections onto the first three spherical harmonics is broken:.
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