Magnetisation moment of a bounded 3D sample: Asymptotic recovery from planar measurements on a large disc

Inverse magnetisation problem consists in inferring information about a magnetic source from measurements of its magnetic field. Unlike a general magnetisation distribution, the total magnetisation (net moment) of the source is a quantity that theoretically can be uniquely determined from the field. At the same time, it is often the most useful quantity for practical applications (on large and small scales) such as detection of a magnetic anomaly in magnetic prospection problem or finding the overall strength and mean direction of the magnetisation distribution of a magnetised rock sample. It is known that the net moment components can be explicitly estimated using the so-called Helbig’s integrals which involve integration of the magnetic field data on the plane against simple polynomials. Evaluation of these integrals requires knowledge of the magnetic field data on a large region or the use of ad hoc methods to compensate for the lack thereof. In this paper, we derive higher-order analogs of Helbig’s integrals which permit estimation of total magnetisation components in terms of measurement data available on a smaller region. Motivated by a concrete experimental setup for analysing remanent magnetisation of rock samples with a scanning microscope, we also extend Helbig’s integrals to the situation when knowledge of only one field component is necessary. Moreover, apart from derivation of these novel formulas, we rigorously prove their accuracy. 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 formulas for the net moment components to an arbitrary order, a possibility that was previously unclear. The obtained results are illustrated numerically and their robustness with respect to the noise is discussed.