Evaluation of high-degree series expansions of the topographic potential to higher-order powers

Abstract

Mass associated with surface topography makes a significant contribution to the Earth’s gravitational potential at all spectral scales. Accurate computation in spherical harmonics to high degree requires calculations of multiple integer powers of the global topography. The purpose of this paper is to analyse the contributions of Earth’s topography to its potential to the tenth power of the topography, and quantify truncation errors resulting from neglecting higher-order powers. To account for the effect of gravity attenuation with height, we use series expansions for gravity upward-continuation to the Earth’s surface. With degree-2160 expansions, limitation to the first three powers of the topography, as often done in practice, may give rise to maximum truncation errors exceeding 100 mGal at a reference sphere, and ~25 mGal at the topography. Aiming for a maximum truncation error of 1 mGal we found that higher-order terms to the seventh power are required over the Himalaya Mountains as example of Earth’s most rugged land region. Upward-continuation of topographic gravity effects with mGal-precision from the sphere to the Earth’s surface is accomplished with a series expansion of fifth order. As a key finding, the accurate conversion of topography to gravity effects at the Earth’s surface is governed by two similar yet not identical series expansions. For degree-2160 expansions, we recommend that the powers of Earth’s topography be used up to seventh order to accurately evaluate the topographic potential to the mGal-level, as required, e.g., for the construction of high-resolution Bouguer gravity anomaly maps in spherical harmonics.