The relation between rigorous and Helmert's definitions of orthometric heights

Following our earlier definition of the rigorous orthometric height [J Geod 79(1-3):82-92 (2005)] 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 a regional geoid model.