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We consider the problem of reconstruction of the overall magnetisation vector(net moment) of a sample from partial data of the magnetic field. Namely,motivated by a concrete experimental set-up, we deal with a situation when themagnetic field is measured on a portion of the plane in vicinity of the sampleand only one (normal to the plane) component of the field is available. Underassumption that the measurement area is a sufficiently large disk (lying in ahorizontal plane above the sample), we obtain a set of estimates for thecomponents of the net moment vector with the accuracy which improvesasymptotically with the increase of the measurement disk radius. Compared toour previous preliminary results, the asymptotic formulas are now rigorouslyjustified and higher-order estimates are derived. Moreover, the presentedapproach, based on an appropriate splitting in the Fourier domain and estimatesof oscillatory integrals (involving both small and large parameters),elucidates the derivation of asymptotic estimates of an arbitrary order, apossibility that was previously unclear. The obtained results are illustratednumerically and their robustness with respect to noise is discussed.

Magnetisation moment of a bounded 3D sample: asymptotic recovery from planar measurements on a large disk using Fourier analysis