  We investigate the numerical approximation of the nonlinear Molodensky
problem, which reconstructs the surface of the earth from the gravitational
potential and the gravity vector. The method, based on a smoothed
Nash-Hormander iteration, solves a sequence of exterior oblique Robin problems
and uses a regularization based on a higher-order heat equation to overcome the
loss of derivatives in the surface update. In particular, we obtain a
quantitative a priori estimate for the error after m steps, justify the use of
smoothing operators based on the heat equation, and comment on the accurate
evaluation of the Hessian of the gravitational potential on the surface, using
a representation in terms of a hypersingular integral. A boundary element
method is used to solve the exterior problem. Numerical results compare the
error between the approximation and the exact solution in a model problem.
