Comparison of different methods for estimating the geoid-to-quasi-geoid separation

In Molodensky's definition of the normal height, the mean normal gravity is computed explicitly. In definition of the orthometric height, however, the actual mean gravity along the plumbline within the topography could only be estimated approximately, because our knowledge about the actual topographic density distribution is still restricted. This practical limitation resulted in implementing various approximations to the orthometric height based on adopting a particular density hypothesis. Helmert's orthometric heights are the most commonly used for a vertical datum realization, because of their simple definition by assuming a planar topography of uniform density. To improve the accuracy, several different numerical models have been proposed by taking into consideration the terrain geometry, topographic density heterogeneities and implicitly also the mass density distributed below the geoid surface. Consequently, different formulae have been derived for conversion from the normal to orthometric heights (or vice versa), practically realized by computing the geoid-to-quasi-geoid separation (i.e. the difference between the geoid and the quasi-geoid). In this study, we review these methods and apply them at the regional study area of Himalaya and Tibet, characterized by the largest vertical displacement between the geoid and the quasi-geoid. We show that the maximum geoid-to-quasi-geoid separation reaches -4.06 m, when computed with a spectral resolution complete to the spherical harmonic degree of 2160. We further investigate the dependence of the geoid-to-quasi-geoid separation on the spectral resolution of used gravity and topographic models, showing that a higher degree harmonic spectrum (from 360 to 2160) modifies the geoid-to-quasi-geoid separation 2 m or even more, especially at the foothills of Himalaya. Our results also reveal that the topographic density variations within sediments and bedrock modify the geoid-to-quasi-geoid separation typically less than 5 per cent. This aspect is however still open for further analysis, because the upper continental crustal density models currently available have a low resolution and accuracy. We also discuss some aspects associated with the orthometric height definition, particularly addressing numerical schemes applied for computing the contribution of terrain geometry. We demonstrate that the mean spherical terrain correction has to be applied instead of a planar one (used in Mader, Niethammer, or Wirth's methods) for an accurate conversion between the normal and orthometric heights.