Inverse Gravimetry as an Ill-Posed Problem in Mathematical Geodesy

The gravitational potential of (a part of) the Earth is assumed to be available, the inverse gravimetry problem is to determine the density contrast function inside (the specified part of) the Earth from known potential values. This paper deals with the characteristic ill-posed features of transferring input gravitational information in the form of Newtonian volume integral values to geological output characteristics of the density contrast function. Some properties of the Newton volume integral are recapitulated. Different methodologies of the resolution of the inverse gravimetry problem and their numerical implementations are examined dependent on the data source. Three cases of input information may be distinguished, namely internal (borehole), terrestrial (surface), and/or external (spaceborne) gravitational data sets. Singular integral theory based inversion of the Newtonian integral equation such as Haar-type solutions are proposed in a multiscale framework to decorrelate specific geological signal signatures with respect to inherently available features. Reproducing kernel Hilbert space regularization techniques are studied (together with their transition to mollified variants) to provide geological contrast density distributions by “downward continuation” from terrestrial and/or spaceborne data. Finally, reproducing kernel Hilbert space solutions are formulated for use of gravimeter data, independent of a specifically chosen input area, i.e., in whole Euclidean space $$\mathbb{R}^3$$