The Spherical Harmonic Representation of the Geoid

Global Gravitational Models (GGMs) describe the Earth’s external gravitational field by a set of spherical harmonic (Stokes) coefficients. These coefficients are routinely used to compute the geoid model, while disregarding the upper continental crustal (i.e., topographic) masses above the geoid. Strictly speaking, however, these coefficients can describe only gravity field quantities at (or above) the Earth’s surface to satisfy Laplace’s equation. Consequently, the GGM coefficients cannot be used to define the geoid surface rigorously without accounting for the internal convergence domain and the gravitational effect of topographic masses. In most technical and scientific applications, the computation of the geoid model directly from the GGM coefficients has been accepted under the assumption that errors due to disregarding the internal convergence domain (inside the topographic masses) are typically less than a few centimeters (i.e., at the level of global geoid model uncertainties). In this study, we demonstrate that these errors reach several decimeters and even meters, with maxima in Tibet and Himalayas exceeding ~4 m. Moreover, relatively large errors, reaching decimeters, are already detected in regions with a moderately elevated topography. In scientific applications requiring a high accuracy, such errors cannot be ignored. Instead, GGM coefficients describing the Earth’s external gravitational field have to be corrected for the effect of (topographic) masses distributed above the geoid surface to obtain spherical harmonic coefficients that explicitly define the geoid globally. The explicit definition of the global geoid model in the spectral domain is derived in this study and used to compile spherical harmonic coefficients of the geoid up to degree/order 2160 from the EIGEN-6C4 global gravitational model.