Run a precise level loop of a few hundred kilometers, mostly north and south, and it will fail to close by a measurable amount even if every single reading is perfect. Nothing is wrong with the instrument. The problem is that level surfaces are not parallel, so a sum of measured height differences is not a difference of heights.
The various height systems — orthometric, dynamic, normal — are three different ways of resolving that, each optimizing for something different. Which one you want depends on whether you care about geometric distance, about the direction water flows, or about computational rigour.
Why level surfaces converge
Gravity is stronger near the poles, because of both the Earth flattening and the reduced centrifugal effect of rotation. A level surface is defined by constant gravity potential, so where gravity is stronger the surfaces of equal potential lie closer together. Between two equipotential surfaces, the vertical separation shrinks as you move poleward.
The consequence is direct and slightly disturbing: two points on the same level surface — water would not flow between them — are not the same geometric distance above the geoid. Conversely, two points at the same geometric height above the geoid are not on the same level surface, and water would flow between them if you connected them.
A level loop run east and west feels almost none of this. A loop run north and south feels all of it, and the misclosure grows with the latitude range and the height of the terrain crossed.
Geopotential numbers: the quantity that actually behaves
- Geopotential number
- The difference in gravity potential between the geoid and a point, written C. It is obtained by summing the product of each measured height difference and the observed gravity along the levelling route, and it is independent of the path taken.
C = ∫ g dn (summed along the levelling route as Σ g × Δn)Geopotential numbers are path-independent, so a loop closes exactly on them, up to observational error. Every height system is then a different way of converting geopotential numbers into something with units of length. That is the cleanest way to see what distinguishes them: they all start from the same measurement and divide by a different gravity value.
The three height systems
| Height system | Divided by | Physical meaning | Where used |
|---|---|---|---|
| Orthometric H | Mean gravity along the plumb line at the point | Distance from the geoid to the point, along the curved plumb line | United States, Canada — NAVD 88 uses Helmert orthometric heights |
| Dynamic | Normal gravity at 45° latitude, a single constant | Not a distance; equal values mean the same level surface | Great Lakes and other large water bodies, IGLD 85 |
| Normal | Mean normal gravity along the normal plumb line | Distance above the quasigeoid; requires no assumption about crustal density | Much of Europe and the former Soviet Union |
Orthometric height is what surveyors normally mean by elevation. Its weakness is that mean gravity along the plumb line runs through rock you cannot measure, so it must be modelled — the Helmert approach uses a Poincaré-Prey reduction with an assumed crustal density. That assumption is where the theoretical impurity of orthometric heights lives.
Dynamic height gives up being a distance in exchange for being exactly what a hydrologist needs. Dividing every geopotential number by the same constant, normal gravity at 45° latitude, means equal dynamic heights lie on the same level surface, so water does not flow between them. That is why the International Great Lakes Datum of 1985 uses dynamic heights: a lake surface at rest is one level surface, and only dynamic heights give it one number.
Normal height sidesteps the density assumption entirely by using normal gravity, which is computable exactly from the reference ellipsoid. The price is that the reference surface is no longer the geoid but the quasigeoid, a computational surface that coincides with the geoid over the oceans and departs from it over high terrain.
The orthometric correction
In practice, precise levelling over long north-south lines is reduced by applying an orthometric correction to the raw sum of height differences. The correction accounts for the convergence of level surfaces along the route, and it depends on the latitudes of the endpoints, the height of the line, and the gravity field along it.
The magnitude is small over short lines and grows with distance and latitude range. Over a few kilometers it is negligible. Over a first-order line crossing several degrees of latitude at elevation it reaches centimetres, which is why it is standard in geodetic levelling and irrelevant to a construction site.
H_B − H_A = Σ Δn + orthometric correctionWhat matters in practice
- For ordinary construction and boundary work, orthometric heights on the national datum are the right choice and the corrections discussed here are below your noise.
- For precise levelling over long lines with a large latitude range, the orthometric correction is real and must be applied.
- For anything involving a large still water body — the Great Lakes above all — dynamic heights are the correct system and orthometric heights will mislead you.
- For work with European data, expect normal heights above a quasigeoid, and do not assume the reference surface is the geoid.
- The common thread is that raw levelled differences are observations, not heights. Something always has to be done to them before they become a height in a defined system.
Understanding this also explains why the modernized vertical datum is described as a geopotential datum rather than simply a new set of elevations. It is defined in terms of the potential field, which is the quantity that behaves properly, and heights in whatever system you need are derived from it.