Gravitational inversion for arbitrary surface mass distribution-application to earth's time-variable gravity with topography

The aim of this paper is gravitational inversion, that is, the determination of the mass density distribution of a central body when given the observed external gravitational field that is produced by the mass itself. This inversion for a 3-D mass distribution is known to be grossly non-unique; whereas that for a 2-D mass distribution on a spherical surface is known to be unique. The latter justifies the surface 'mascon', or equivalent-water-thickness, solutions when not considering any interior mass transports for the Earth's time-variable gravity as observed by the GRACE satellite. In this paper, using the gravitational multipole formalism cast in the framework of linear Hilbert space with the notion of inner product, we do two things further: (i) we prove mathematically that the 2-D gravitational inversion on an arbitrary surface is unique; and (ii) we derive the algorithm that leads to the unique exact 2-D mass distribution solution, up to a truncated finite spherical harmonic degree. This pertains directly to reaching refined GRACE mascon solutions that account for the actual Earth surface shape including the ellipsoidal figure and the topography. In the process, we also (re)formulate the exact equations, for spherical or arbitrary shape, unifying the formulae that have appeared in the GRACE-relevant literature.