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We investigate the numerical approximation of the nonlinear Molodenskyproblem, which reconstructs the surface of the earth from the gravitationalpotential and the gravity vector. The method, based on a smoothedNash-Hormander iteration, solves a sequence of exterior oblique Robin problemsand uses a regularization based on a higher-order heat equation to overcome theloss of derivatives in the surface update. In particular, we obtain aquantitative a priori estimate for the error after m steps, justify the use ofsmoothing operators based on the heat equation, and comment on the accurateevaluation of the Hessian of the gravitational potential on the surface, usinga representation in terms of a hypersingular integral. A boundary elementmethod is used to solve the exterior problem. Numerical results compare theerror between the approximation and the exact solution in a model problem.

A Nash-Hormander iteration and boundary elements for the Molodensky problem