Molodensky’s and Helmert’s theories: Two equivalent geodetic approaches to the determination of the gravity potential and the Earth surface

A fundamental problem of physical geodesy is the determination of the “Surface of the Earth” and its gravitational potential from various types of observations performed on the Earth surface S itself or in the outer space. When data are derived from gravimetry on S we speak of Molodensky’s problem. Since the gravity field depends linearly on its source, i.e. the mass distribution, it follows that we can manipulate the (unknown) internal density in a known way and still return to the same external solution once the effects of the manipulation have been eliminated (restored). This is used, in the frame of Molodensky’s theory, with the Residual Terrain Correction that is removed (and then restored) before approximating the solution by some regularized (collocation or other) approach. Differently, Helmert’s approach shifts the masses of the topographic layer, compressing them to some internal surface and substituting their effects on gravity by that of a single layer. Data are thus lowered to some internal ellipsoid or sphere and a solution is then easily computed. The effects of the internal changes are then inverted and added back to the solution. Despite the apparent completely different approach one can prove that the final solutions, when data are given continuously on the boundary and the errors are made to tend to zero, converge to the true potential on the surface S and then in the outer space. So the two solutions are geodetically equivalent and do not create any scientific conflict. Different is what happens inside S, down to the geoid level. Here Helmert’s approach, that introduces the density of the topographic layer as data, is certainly less erroneous in approximating the true potential. Yet due to the imperfect knowledge of the density and even more to the ill-posedness of the downward continuation operator, the internal potential can have large errors, unless the solution is duly regularized and an appropriate tuning is introduced between the regularization parameter and the size of data errors.