In order to have physical meaning, a height system must have some relation to the Earth's gravity field. Of the height systems that do, orthometric heights match best with our intuitive understanding of height. The orthometric height of a point is the distance travelled along a plumbline from that point to the geoid, and can be arrived at either directly from leveling and gravity observations or indirectly by converting geodetic heights to orthometric heights using a geoid model. This dissertation investigates recent advances in orthometric height determination, to find out whether orthometric height determinations can meet modern centimetre-level accuracy requirements. Persistent barriers to improving orthometric height accuracy have been the impossibility of fully modeling topographical density effects, the lack of suitable numerical methods, and the lack of sufficient data.

The problem is addressed in six articles. The first two deal with direct calculation of orthometric heights, providing a practical implementation of a rigorous theory of orthometric heights able to deliver sub-centimetre accuracy in most cases, and showing that numerical errors in this process can be kept below the one centimetre level. The next two articles address the problem of the unknown density distribution in geoid determination, describing a framework for including the full three-dimensional effect of topographical density, and demonstrating that existing laterally-varying density models can provide sub-centimetre results in most areas. Vertical density variations neglected in such models are only expected to reduce accuracy to a few centimetres in mountainous areas. The fifth article demonstrates a new method for downward continuation of gravity anomalies, one of the largest numerical barriers to accurate geoid determination. The final article evaluates satellite altimetry as a source of gravity data over lakes, finding it promising but in need of further refinement.

The ultimate conclusion is that the physically meaningful system of orthometric heights can be now realized to about a centimetre in most areas, given suitable data, although in some especially challenging areas (e.g. mountain ranges) errors of several centimetres must be accepted.