Solving Vening Meinesz-Moritz inverse problem in isostasy

Vening Meinesz' inverse problem in isostasy deals with solving for the Moho depth from known Bouguer gravity anomalies and normal Moho depth (T-0, known, e. g. from seismic reflection data) using a flat Earth approximation. Moritz generalized the problem to the global case by assuming a spherical approximation of the Earth's surface, and this problem is also treated here. We show that T-0 has an exact physical meaning. The problem can be formulated mathematically as that of solving a non-linear Fredholm integral equation of the first kind, and we present an iterative procedure for its solution. Moreover, we prove the uniqueness of the solution. Second, the integral equation is modified to a more suitable form, and an iterative solution is presented also for this. Also, a second-order approximate formula is derived, which determines the Moho depth to first/linear order by an earth gravitational model (EGM), and the remaining short-wavelength/non-linear part, of the order of 2 km, can be determined by iteration. A direct, second-order formula, in principle accurate to the order of 25 m, combines the first-order solution from an EGM with a second-order term, which may include terrestrial Bouguer gravity anomalies around the computation point.