  We present a boundary element method to compute numerical approximations to
the non-linear Molodensky problem, which reconstructs the surface of the earth
from the gravitational potential and the gravity vector. Our solution procedure
solves a sequence of exterior oblique Robin problems and is based on a
Nash-H\"{o}rmander iteration. We apply smoothing with the heat equation to
overcome a loss of derivatives in the surface update. Numerical results compare
the error between the approximation and the exact solution in a model problem.
