Concerning the Roundabout Method to Downward Continuation and the Evaluation of the Far-zone

English Abstract:

The Stokes-Helmert approach to precise geoid computation is used in the SHGeo software package at the University of New Brunswick. This method uses Stokes’s theorem which requires the knowledge of gravity anomalies on a boundary (the geoid); measurements on or above the surface of the Earth must therefore be downward continued to the geoid. The mathematical instability of the downward continuation is well-documented and its effect on the resulting geoid must be considered.

Helmert space differs from the real one by removing topography above the geoid and applying Helmert’s second condensation method. It has a similar gravity field to the real space while satisfying the harmonicity of the gravity potential above the geoid as required. Another space that can be used is the No-Topography (NT) space, defined by the Earth after the removal of topography; this causes the field to be much smoother than the real field. The NT anomaly can be downward continued and transformed to Helmert space by accounting for the condensation layer. We call this technique of evaluating Helmert’s anomalies on the geoid the ‘roundabout’ approach compared to continuing Helmert’s anomalies directly to the geoid (the ‘direct’ approach). Results from the roundabout technique should be more accurate due to the smoothness of the NT anomalies.

Applying this approach in Auvergne, France (44.5° < ϕ < 47.5°, 1.5° < λ < 4.5°), the results for the roundabout approach after downward continuation are worse than for the direct approach. After testing we found that the NT anomalies are about twice as sensitive to the far-zone contribution than the Helmert ones. By doubling the integration radius for the downward continuation of the NT anomalies, the geoid results from the roundabout approach are comparable to those obtained from the direct approach. The fit to GPS benchmarks in Auvergne is σ = 0.042m vs. 0.037m. This preliminary work shows that the NT field is worth further investigation.