  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 the
magnetic field is measured on a portion of the plane in vicinity of the sample
and only one (normal to the plane) component of the field is available. Under
assumption that the measurement area is a sufficiently large disk (lying in a
horizontal plane above the sample), we obtain a set of estimates for the
components of the net moment vector with the accuracy which improves
asymptotically with the increase of the measurement disk radius. Compared to
our previous preliminary results, the asymptotic formulas are now rigorously
justified and higher-order estimates are derived. Moreover, the presented
approach, based on an appropriate splitting in the Fourier domain and estimates
of oscillatory integrals (involving both small and large parameters),
elucidates the derivation of asymptotic estimates of an arbitrary order, a
possibility that was previously unclear. The obtained results are illustrated
numerically and their robustness with respect to noise is discussed.
