A GNSS receiver produces a height above the ellipsoid. A benchmark carries a height above the geoid. The equation that connects them is short enough to memorize and subtle enough to get wrong in the field on a regular basis.
h = H + NRearranged for the case you actually meet — you have a GNSS height and you want an elevation the client can use — it becomes H = h − N. Since N is negative throughout the conterminous United States, subtracting it makes the orthometric height larger than the ellipsoid height, typically by twenty to fifty meters.
The three terms
- Ellipsoid height (h)
- The distance from the reference ellipsoid to the point, measured along the ellipsoid normal. Purely geometric. It is what a GNSS solution natively produces and it has no physical meaning for drainage or flow.
- Orthometric height (H)
- The distance from the geoid to the point, measured along the curved plumb line. This is elevation in the ordinary engineering sense, the number on a benchmark and the number a level run carries.
- Geoid height (N)
- The separation between the ellipsoid and the geoid at a point, positive where the geoid is above the ellipsoid. Also called geoid undulation or geoid separation. It comes from a published geoid model, never from a measurement you make.
A worked example
Suppose a receiver on a control point returns an ellipsoid height of 384.126 m on NAD 83 (2011), and the geoid model interpolated at that latitude and longitude gives N = −29.512 m.
H = h − N = 384.126 − (−29.512) = 413.638 mIn feet that is about 1357.1 international feet. The sign is where crews lose points: N is negative, so the minus sign in H = h − N becomes an addition. Anyone who mechanically subtracts 29.512 lands 59 m low, which is an error large enough to be caught immediately — the dangerous version of this mistake is a small sign error in a geoid model applied to only part of a project.
Why the equation is an approximation
Strictly, h is measured along the straight ellipsoid normal and H along the curved plumb line, and those two lines are not collinear. The angle between them is the deflection of the vertical. Because the deflection is small — usually a few arcseconds — the resulting error in h = H + N is well under a millimeter in ordinary terrain, and the equation is treated as exact for all practical survey work.
There is a second and much larger source of inexactness, and it is not geometric. It is that H and N have to be consistent with each other. A hybrid geoid model is built specifically so that h from GNSS minus N from the model reproduces the published H on the existing benchmark network. Mix a geoid model from one generation with a datum realization from another and the equation stops balancing.
Where the error in H actually comes from
| Source | Typical magnitude | Can you reduce it in the field? |
|---|---|---|
| GNSS vertical positioning error | 2 to 4 times the horizontal error | Yes — longer occupations, better geometry, redundant sessions |
| Geoid model error | A few centimetres, worse in rough terrain | No — but you can constrain it by holding local benchmarks |
| Local distortion in the vertical datum | Centimetres to a decimetre regionally | Partly — by tying to several benchmarks and checking agreement |
| Antenna height blunders | Whole decimetres or meters | Yes — measure twice, record the antenna reference point |
Notice that the geoid model is rarely the dominant term. Vertical GNSS error and antenna measurement blunders usually are. The standard defence is to occupy at least two published benchmarks bracketing the project, compare the modelled H against the published H, and apply the residual as a local shift or an inclined plane. That procedure absorbs geoid model bias and local datum distortion at the same time.
Practical rules
- Record and report h, H and N separately. A single elevation with no provenance cannot be checked later.
- Name the geoid model on the drawing, not just the vertical datum.
- Hold benchmarks, do not merely check them, when the project has any tolerance tighter than a tenth of a foot.
- Never mix an ellipsoid height from one realization with a geoid model built for another.
- Remember that h has no hydraulic meaning. Nothing drains relative to the ellipsoid.
The modernized vertical datum is designed to make this workflow the primary one rather than a workaround. Instead of a national levelling network defining heights and a geoid model bridging to GNSS, the gravimetric geoid becomes the definition, and an orthometric height is obtained from a GNSS position plus the model directly. The equation h = H + N does not change. What changes is which term is considered the authority.