Local existence of the solution to a nonlinear inverse problem in gravitation

We consider the problem of modeling the external gravitational field of the earth from measurements of its intensity on the surface. The problem is formulated as a nonlinear oblique boundary value problem for the external gravitational potential. The equation for the potential is approximated for small anomalies and solved by a regular perturbation expansion. The convergence of the series is proved using Schauder estimates, and local existence for small anomalies is established. The technique used to prove the convergence of the series is in the same spirit as the Cauchy—Kowalewsky estimates. This study completes previous work done by Backus who proved uniqueness of the solution.