Analysis of lithospheric magnetization in vector spherical harmonics

The lithospheric contribution to the geomagnetic field arises from magnetized rocks in a thin shell at the Earth's surface. The lithospheric field can be calculated as an integral of the distribution of magnetization using standard results from potential theory. Inversion of the magnetic field for the magnetization suffers from a fundamental non-uniqueness: many important distributions of magnetization yield no potential magnetic field outside the shell. We represent the vertically integrated magnetization (VIM) in terms of vector spherical harmonics that are new to geomagnetism. These vector functions are orthogonal and complete over the sphere: one subset (I) represents the part of the magnetization that produces a potential field outside the shell, the observed field; another subset (epsilon) produces a potential field exclusively inside the shell; and a third, toroidal, subset (T) produces no potential field at all. epsilon and T together span the null space of the inverse problem for magnetization with perfect, complete data. We apply the theory to a recent global model of VIM, give an efficient algorithm for finding the lithospheric field, and show that our model of magnetization is dominated by epsilon, the part producing a potential field inside the shell. This is largely because, to a first approximation, the model was formed by magnetizing a shell with a substantial uniform component by an potential field originating inside the shell. The null space for inversion of lithospheric magnetic anomaly data for VIM is therefore huge. It can be reduced if the magnetization is assumed to be induced by a known inducing field, but the null space for susceptibility is not so easily recovered.