  Inversion of potential field data is a central technique of remote sensing in
physics, geophysics, neuroscience and medical imaging. In spite of intense
research, uniqueness theorems for potential-field inversion are scarce. Applied
studies successfully improve potential-field inversion results by including
constraints from independent measurements, but so far no mathematical theorem
guarantees that source localization improves the inversion in terms of
uniqueness of the achieved assignment. Empirical inversion techniques therefore
use numerical and statistical approaches to assess the reliability of their
results. Especially when inverting magnetic field surface measurements, even
seemingly advanced mathematical approaches require substantial additional
assumptions about the source magnetization to achieve a useful reconstruction.
Here, standard potential field theory is used to prove a uniqueness theorem
which completely characterizes the mathematical background of source-localized
inversion. It guarantees for an astonishingly large class of source
localizations that it is possible by potential field measurements on a surface
to differentiate between signals from different source regions. Non-uniqueness
of potential field inversion only prevents that the source distribution within
the individual regions can be uniquely recovered.
