A minimum energy condition for the inverse gravimetric problem

Abstract

The problem of finding the density distribution of the Earth from gravity data is called the inverse gravimetric problem. It is well known that this problem has not a unique solution. A possible approach to force the uniqueness of the solution is to impose the solution to realize a minimum of energy, that is to be an equilibrium distribution.

Several authors have considered this possibility by setting the energy to the sole energy of the potential field. In this paper, we show that such approaches correspond to the maximization of a quadratic form in the set of the density distributions considered as an Hilbert space. As a consequence, they cannot deliver realistic results without a constraint that reduces the set of possible density distributions to a bounded domain of the considered Hilbert space. Moreover, under such constraints, the result will necessarily be located at the boundary of the searched domain.

In order to express a steady state condition leading to a freer solution, we propose to introduce a compression energy. We choose the state equation of Murnaghan that is known to apply reasonably in the upper mantle. We give an integral form of the compression energy and show that the resulting total energy admits a lower bound, which assess the existence of a solution. Finally, we derive a differential equation verified by that the absolute minima from the equilibrium equation under its local form.