This manuscript provides a mathematical underpinning of MMT inversions. It shows that it is theoretically possible to isolate the magnetic moments of individual grains inside a sample from measurements on the surface of a sample alone, if the locations of the magnetic sources are known.


Scanning magnetometers are increasingly used to characterize the magnetization of mineral grains in rock samples. Up-scaling this measurement technique to large numbers of individual particles is hampered by the intrinsic non-uniqueness of potential-field inversion. Here it is shown that this problem can be circumvented by adding tomographic information that determines the location of the possible field sources. Standard potential theory is used to prove a uniqueness theorem that completely characterizes the mathematical background of the corresponding source-localized inversion. It exactly resolves under which conditions a potential-field measurement on a surface can be uniquely decomposed into signals from the different source regions. The intrinsic non-uniqueness of potential-field inversion prevents that the source distribution inside the tomographically outlined regions can be recovered, but the potential field of each region is uniquely defined. For scanning magnetometers in rock magnetism, this result implies that magnetic dipole vectors of large numbers of individual magnetic particles can be reliably reconstructed from surface scans of the magnetic field, if the particle positions are independently determined. This provides an incentive to improve scanning methods for future palaeomagnetic applications.