Stability and the inverse gravimetry problem with minimal data

Abstract

The inverse problem in gravimetry is to find a domain ð • inside the reference domain ω from boundary measurements of gravitational force outside ω. We found that about five parameters of the unknown ð • can be stably determined given data noise in practical situations. An ellipse is uniquely determined by five parameters. We prove uniqueness and stability of recovering an ellipse for the inverse problem from minimal amount of data which are the gravitational force at three boundary points. In the proofs, we derive and use simple systems of linear and nonlinear algebraic equations for natural parameters of an ellipse. To illustrate the technique, we use these equations in numerical examples with various location of measurements points on\partialΩ. Similarly, a rectangular D is considered. We consider the problem in the plane as a model for the three-dimensional problem due to simplicity.