Knowledge Bank

The spherical harmonic expansion of the external gravity field of the earth is divergent at the surface of the earth. From the practical point of view a truncated series can be used if the errors are small. The errors are of two types: the truncation error and the downward continuation error. In this report we investigate the latter, which is caused by the disturbing topography above the point where the series is developed. This error is studied for potentials, gravity anomalies and for the vertical gradient of gravity. It is shown that the usual relation between the spherical harmonics of T and ∂Δg/∂r is also valid for their errors, while the relation between Tn and Δgn does not hold for the downward continuation errors. The relative errors of Δg are found to be more serious than those for T. Some global RMS errors are estimated based on the degree variances of Tscherning and Rapp (1974). Furthermore, formulae are developed for a numerical integration of the errors over an approximately known topography. Finally, these formulae are tested for 1654 5° x 5° mean elevations. In the earth model "a level ellipsoid with topography of constant density" these computations gave the RMS error 0.13 m for global undulations in an expansion to degree 16. The gravity anomaly errors were generally within ± 5 mgal except at the edges of the continents and for rough areas inside the continents, where larger errors might be expected. [Some mathematical expressions are not fully represented in the metadata. Full text of abstract available in document.]