Ellipsoidal spline functions for gravity data interpolation and smoothing

Document Type : Original Article


1 School of Surveying and Geospatial Engineering, University of Tehran, Tehran, Iran

2 Department of Mathematics, Tafresh University, Tafresh, Iran


The aim of this paper is to study the theory of spline interpolation and smoothing problems on the surface of a triaxial ellipsoid for the Consecutive Iterated Helmholtz operator and a set of linearly independent evaluation functionals. Spline functions were introduced based on the minimization of a semi-norm in the context of a semi-Hilbert space whose domain was the surface of the ellipsoid. The semi-Hilbert space was decomposed into two different subspaces, a particular Hilbert space and the null space of the desired operator. Using surface Green’s functions for the Consecutive Iterated Helmholtz operator, the reproducing kernel for the Hilbert subspace was constructed. Spline and smoothing functions were explicitly represented based on the reproducing kernel and the evaluation functionals. An approximation formula was derived to facilitate the potential use in Earth’s gravity field data interpolation and smoothing. An application of this technique was presented to show the interpolation of potential fields over Iran. Ellipsoidal and spherical splines were compared as well. It revealed the ellipsoidal splines to be more accurate than the spherical counterparts.