Downward continuation of Helmert's gravity

Abstract

. The aim of this contribution is to show that mean Helmert's gravity anomalies obtained at the earth surface on a grid of a `reasonable' step can be transferred to corresponding mean Helmert's anomalies on the geoid. To demonstrate this, we take the by mean Helmert's anomalies from a very rugged region, the south-western corner of Canada which contains the two main chains of the Canadian Rocky Mountains, and formulate the problem of downward continuation of Helmert's anomalies for this region. This can be done exactly because Helmert's disturbing potential is harmonic everywhere outside the geoid, therefore even within the topography. Then we solve the problem numerically by transforming the Poisson integral to a system of 53,856 linear algebraic equations. Since the matrix of this system is well conditioned, there is no theoretical obstacle to the solution. The correctness of the solution is then checked by back substitution and by evaluating the contribution of the downward continuation term to Helmert's co-geoid. This contribution comes out positive for all the points. We thus claim that the determination of the downward continuation of mean Helmert's gravity anomalies on a grid of a `reasonable' step is a well posed problem with a unique solution and can be done routinely to any accuracy desired in the geoid computaion.