Hidden Magnetizations and Localization Constraints

Recovering the full underlying magnetization from geomagnetic potential field measurements is known to be highly nonunique. Localization constraints on the magnetization can improve this to a certain extent. We present some analytic background as well as some examples on what is meant by 'to a certain extent'. In particular, if no constraints other than spatial localization are imposed, only two out of three components of the vectorial magnetization can be reconstructed uniquely. If it is additionally assumed that the magnetization is induced by an (unknown) ambient dipole field, then the susceptibility and the direction of the ambient dipole can be reconstructed.