Reproducing-kernel-based splines for the regularization of the inverse ellipsoidal gravimetric problem

Abstract

To solve the inverse gravimetric problem, i.e. to reconstruct the Earth's mass density distribution by using the gravitational potential, we introduce a spline interpolation method for the ellipsoidal Earth model, where the ellipsoid has a rotational symmetry. This problem is ill-posed in the sense of Hadamard as the solution may not exist, it is not unique and it is not stable. Since the anharmonic part (orthogonal complement) of the density function produces a zero potential, we restrict our attention only to reconstruct the harmonic part of the density function by using the gravitational potential. This spline interpolation method gives the existence and uniqueness of the unknown solution. Moreover, this method represents a regularization, i.e. every spline continuously depends on the given gravitational potential. These splines are also combined with a multiresolution concept, i.e. we get closer and closer to the unknown solution by increasing the scale and adding more and more data at each step.

Keywords:

• inverse gravimetric problem
• ill-posed
• regularization
• scaling function
• multiresolution analysis

AMS Subject Classifications:

• 45Q05
• 65R30
• 45B05
• 41A15
• 46E22
• 86A22

Acknowledgements

NA gratefully acknowledges the support by the Higher Education Commission Pakistan and VM gratefully acknowledges the support by the German Research Foundation (projects MI 655/2-1 and MI 655/2-2).