On inversion for mass distribution from global (time-variable) gravity field

The well-known non-uniqueness of the gravitational inverse problem states that the external gravity field, even if completely and exactly known, cannot uniquely determine the density distribution of the body that produces the gravity field. In this paper, we provide conceptual insight by examining the problem in terms of spherical harmonic expansion of the global gravity field. By comparing the multipoles and the moments of the density function, we show that in 3-D the degree of knowledge deficiency in trying to inversely recover the density distribution from an external gravity field solution is (n + 1)(n + 2)/2 − (2n + 1) = n(n − 1)/2 for each harmonic degree n. On the other hand, on a 2-D spherical shell we show via a simple relationship that the inverse solution of the surface density distribution is unique. The latter applies quite readily in the inversion of time-variable gravity signals (such as those observed by the GRACE space mission) where the sources largely come from the Earth's surface over a wide range of timescales.