On the Consistency between a Classical Definition of the Geoid-to-Quasigeoid Separation and Helmert Orthometric Heights

It is acknowledged that a classical definition of the geoid-to-quasigeoid separation as a function of the simple planar Bouguer gravity anomaly is compatible with Helmert’s definition of orthometric heights. According to Helmert, the mean actual gravity along the plumbline between the geoid and the topographic surface in the definition of orthometric height is computed approximately from the measured surface gravity by applying the Poincaré-Prey gravity reduction. This study provides theoretical proof and numerical evidence that this assumption is valid. We demonstrate that differences between the normal and (Helmert) orthometric corrections are equivalent to the geoid-to-quasigeoid separation differences computed for individual levelling segments. According to our theoretical estimates, maximum differences between these 2 quantities should be less than ±1 mm. By analogy, differences between the Molodensky normal and Helmert orthometric heights at levelling benchmarks should be equivalent to the geoid-to-quasigeoid separation computed from the Bouguer gravity data. Both theoretical findings are inspected numerically by using levelling and gravity data along selected closed levelling loops of the vertical control network in Hong Kong. Results show that values of the geoid-to-quasigeoid separation at levelling benchmarks differ less than ±0.1 mm from differences between the normal and orthometric corrections. Relatively large differences (slightly exceeding 2 mm) between values of the geoid-to-quasigeoid separation and differences between the normal and (Helmert) orthometric heights at levelling benchmarks are explained by errors in levelling measurements rather than by inconsistencies in computed values of the geoid-to-quasigeoid separation and (Helmert) orthometric correction.