Continuation of Surface Anomalies to the Geoid in Regions of Negative Heights

English Abstract:

The Poisson integral is a spatial solution to the Dirichlet boundary value problem for the exterior of a sphere. In the context of geoid computation, the Poisson kernel is usually applied to gravity anomalies on the geoid (approximated locally by a spherical surface), to determine gravity values on topography. Conversely, a discrete form of the Poisson integration can be inverted to determine gravity anomalies on the geoid, from gravity values on topography. Direct Poisson integration attenuates high frequencies, while the inverse operation amplifies high frequencies, in accordance with the physics of potential fields.

The Poisson integral as usually implemented, choosing the geoid as the spherical boundary and applying it to gravity anomalies harmonic only above the geoid, cannot be applied to gravity measurements within (below) the geoid. Two methods are herewith proposed to resolve this issue, in the context of continuation of Helmert gravity anomalies. In the first, the Helmert anomalies on topography are converted to a space where they are harmonic above a reference sphere corresponding to the lowest elevation in the area. They are then downward continued to the reference sphere, upward continued to the geoid, and converted back to the Helmert space. In the second, the B matrix representing the Poisson integral kernel in its discreet form is partitioned into sub-matrices, separating areas of negative and positive heights. The sub-matrices associated with negative heights are applied to gravity anomalies at negative heights, while the sub-matrices of B^-1 associated with positive heights are applied to gravity anomalies at positive heights. Application in a test area over the Dead Sea shows agreement between geoid results from both methods of less than 1.5 cm in regions of negative height, and suggests that the second method is the more accurate.