A Method to Extrapolate the Data for the Inverse Magnetisation Problem with a Planar Sample

A particular instance of the inverse magnetisation problem is considered. It is assumed that the support of a magnetic sample (a source term in the Poisson equation in $\mathbb{R}^3$) is contained in a bounded planar set parallel to the measurement plane. Moreover, only one component of the magnetic field is assumed to be known (measured) over the same planar region in the measurement plane. We propose a method to extrapolate the measurement data to the whole plane relying on the knowledge of the forward operator and the geometry of the problem. The method is based on the spectral decomposition of an auxiliary matrix-function operator. The results are illustrated numerically.