On the inverse gravimetry problem with minimal data

The inverse problem in gravimetry is to find a domain D inside the reference domain Omega from measurements of gravitational force outside Omega. We consider the problem in three dimensions where we found that a few parameters of the unknown D can be stably determined given data noise in practical situations. An ellipsoid is the best approximation of D. We prove uniqueness of recovering an ellipsoid in a particular case for the inverse problem from minimal amount of data which are the approximated gravitational force at nine boundary points. In the proofs, we derive and use simple systems of linear and nonlinear algebraic equations for the parameters of an ellipsoid. Similarly, a rectangular parallelepiped D is considered. To support our theory, we use numerical examples with different location of measurements points on partial derivative Omega .