On Derivations and Properties of Stokes' Gravity Formula

Abstract

Several independent derivations of Stokes' gravity formula have been critically analysed. It is shown that a ‘revised’ Stokes formula derived by R. A. Hirvonen is not valid for gravity data given on an arbitrary surface as claimed, but only for data given on a sphere as in the original Stokes formula. This demonstration invalidates the entire foundation of Hirvonen's New Theory of Gravimetric Geodesy. In derivations based on Poisson's integral theorem a more rigorous treatment of the elimination of zeroth and first order terms of anomalous gravity has been presented. The influence of remote zones in the applications of Stokes' formula has been evaluated by numerical integration. It is shown that tremendous truncation errors occur unless the integration is extended over the entire globe. The differences in truncation error behaviour for different geographical locations are so large that data representing world-wide average behaviour may be very misleading when applied to individual locations. The lack of gravity data coverage over very large parts of the globe causes significant errors in geoidal height determinations and vertical extensions of anomalous gravity by Stokes' formula. In most cases these errors are so large as to render the results virtually meaningless. It is shown that similar drawbacks exist for the calculation of deflexions of the vertical from formulae derived as partial derivatives of the Stokes expression. Significant reduction in the truncation errors is obtained if, following A. H. Cook, a reliable third or higher order spherical harmonic reference model is used in the definition of Δg.