Abstract. Following our earlier definition of the rigorous orthometric height (Tenzer et al. 2005, J Geod, 79), we present the derivation and calculation of the differences between this and the Helmert orthometric height, which is embedded in the vertical datums used in numerous countries. By way of comparison, we also consider Mader and Niethammer's refinements to the Helmert orthometric height. For a profile across the Canadian Rocky Mountains (maximum height of ~2,800 m), the rigorous correction to Helmert's height reaches ~13 cm, whereas the Mader and Niethammer corrections only reach ~3 cm. The discrepancy is due mostly to the rigorous correction's consideration of the geoid-generated gravity disturbance. We also point out that several of the terms derived here are the same as those used in regional gravimetric geoid models, thus simplifying their implementation. This will enable those who currently use Helmert orthometric heights to upgrade them to a more rigorous height system based on the Earth's gravity field and one that is more compatible with the regional geoid model.

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Poisson Downward Continuation Solution by the Jacobi MethodDownward continuation is a continuing problem in geodesy and geophysics. Inversion of the discrete form of the Poisson integration process provides a numerical solution to the problem, but because the B matrix that defines the discrete Poisson integration is not always well conditioned the solution may be noisy in situations where the discretization step is small and in areas containing large heights. We provide two remedies, both in the context of the Jacobi iterative solution to the Poisson downward continuation problem. First, we suggest testing according to the upward continued result from each solution, rather then testing between successive solutions on the geoid, so that choice of a tolerance for the convergence of the iterative method is more meaningful and intuitive. Second, we show how a tolerance that reflects the conditioning of the B matrix can regularize the solution, and suggest an approximate way of choosing such a tolerance. Using these methods, we are able to calculate a solution that appears regular in an area of Papua New Guinea having heights over 3200 m, over a grid with 1 arc-minute spacing, based on a very poorly conditioned B matrix.

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For decades now the geodetic community has been split down the middle over the question as to whether geoid or quasigeoid should be used as a reference surface for heights. The choice of the geoid implies that orthometric heights must be considered, the choice of the quasigeoid implies the use of the so-called normal heights. The problem with the geoid, a physically meaningful surface, is that it is sensitive to the density variations within the Earth. The problem with the quasigeoid, which is not a physically meaningful surface, is that it requires integration over the Earth's surface.Density variations that must be known for the geoid computation are those within topography and these are becoming known with an increasing accuracy. On the other hand, the surface of the Earth is not a surface over which we can integrate. Artificial "remedies" to this fatal problem exist but the effect of these remedies on the accuracy of quasigeoid are not known. We argue that using a specific technique, known as StokesHelmert's and using the increased knowledge of topographical density, the accuracy of the geoid can now be considered to be at least as good as the accuracy of the quasigeoid.