A contribution of the multiquadric method: Interpolation of potential inside the earth

Abstract

The multiquadric (MQ) method, when converted into a proper integral equation, is a potential function. As a numerical method, it is capable of interpolation without the usual reciprocal distance singularities. An unusual result of a more detailed study is that the Newtonian attraction generated by each volume element inside a material body is a constant, proportional to the product of G and the density. Consequently, potential at the boundary is identical, whether computed internally with a proper integral for material space, or externally by the approach of unit masses from infinity in free space to the boundary points of a sphere. The capability of the proper integral form, and the corresponding MQ numerical approximation provides a basis for interpolation of potential and its functionals inside and on the boundary of the Earth.