Bouguer anomalies and the geoid: a reassessment of Stokes’ method

Abstract

The classical gravimetric approach to determining the form of the geoid conventionally involves a global surface integral of free‐air anomalies weighted by Stokes’ Function. While this approach is conceptually satisfactory at sea, where gravity observations are actually made on (or at least very close to) the geoid and there is consequently no topographic effect, it is only an approximation on land, where the precise definition of the geoid itself is somewhat vague. This paper discusses what is meant by the geoid on land and introduces a more precise definition which is practically realizable. Although this paper describes two small modifications of Helmert's condensation reduction (actually the procedure described by Stokes) so that it provides an acceptably rigorous determination of the geoid on land, such a method makes very inefficient use of the available information. The Stokes‐Helmert method condenses the topographic masses to a surface density and then adds their resulting gravitational attraction to the Bouguer anomaly. The result, similar to a terrain‐corrected free air anomaly, is then integrated to transform the gravity anomaly into its equivalent equipotential undulation. Here, a more efficient method is proposed whereby the two independent contributions to the geoid from subsurface anomalous density and from the topographic masses are computed in separate surface integrals and then added, thus finding the geoid as a correcting undulation on the Bouguer co‐geoid. Gravity observations, reduced to Bouguer anomalies, are only required for the component of the geoid due to anomalous density, whereas most local detail and usually the larger contribution is due to the topographic masses and can be computed directly from topographic maps. The classical approach using free‐air anomalies is conceptually equivalent to using a gravity meter as an altimeter to find the geometrical form of topography. A consequence of the ‘new’ approach (actually suggested by Stokes) is its ability to generate a satisfactory geoid from a much poorer coverage of gravity observations or, alternatively, it can give greatly increased detail with the same gravity data. Ibis is illustrated by a local geoid map computed for the highland region of northern Britain. Even greater local resolution can be achieved using new contour integrals to evaluate the potential due to local topography, but at the expense of much greater computation time. The contour integral approach is also probably the most efficient way to determine the small indirect effect of the Stokes‐Helmert condensation reduction, that is, the difference in potential between the condensed topographic masses and those in situ. Unlike the classical approach it is shown that the indirect effect must be evaluated at the observation point rather than on the geoid. Copyright © 1988, Wiley Blackwell. All rights reserved