Pade Approximants for Geodesy

TL;DR

This paper explores the application of Padé approximants to extend spherical harmonic expansions beyond their convergence radius and to identify singularities in gravitational potential models.

Contribution

It provides an analysis of Padé approximants for geodetic problems, especially for downward continuation and singularity detection in gravitational models.

Findings

Padé approximants can extend the convergence domain of spherical harmonic expansions.

They are useful for identifying complex singularities in gravitational potential.

The convergence region may be larger for models with analytic topography and density.

Abstract

In this note we analyze the use of Pad\'e approximants for downward continuation beyond the radius of convergence of spherical harmonic expansions (SHEs), and for identifying the complex singularities of the gravitational potential. SHEs are, in essence, expansions in 1/r, i.e., expansions about the point at infinity. Their domain of convergence is generically the exterior of the Brillouin sphere. However, for synthetic models with analytic topography and density the region of convergence may be larger, with the deviation decreasing as the structural complexity of the planet increases.