The Vertical Equation
Ellipsoid height, orthometric height, and geoid height are connected by a simple relationship: h = H + N. The equation is not an arbitrary rule to memorize. It is a compact description of the three vertical distances we have already identified.
Build the Equation from the Picture
Return once more to the monument and picture the two reference surfaces beneath the antenna. Begin at the ellipsoid and move upward toward the antenna. The total distance traveled is the ellipsoid height, h. If the geoid lies between the ellipsoid and the antenna, that same trip can be divided into two parts: first the distance from the ellipsoid to the geoid, N, and then the distance from the geoid to the antenna, H.
The full distance must equal the sum of those two parts. That is all the vertical equation says. Ellipsoid height equals orthometric height plus geoid height. Written symbolically, the relationship is h = H + N. The equation follows directly from the geometry of the three surfaces; it does not depend on a particular receiver brand, software package, or survey method.
Think of h as the complete vertical path from the ellipsoid to the antenna. The quantities H and N divide that same path into two segments. The equation h = H + N simply states that the whole equals the sum of its parts.
Solving for the Elevation We Usually Want
In practice, the GNSS receiver has already supplied h, while a geoid model supplies N. The quantity most users want is H, the orthometric height. Rearranging the same relationship gives H = h - N. This is the conversion performed when a receiver or data collector turns an ellipsoid height into a gravity-based elevation.
Suppose the receiver reports an ellipsoid height of 42.60 meters and the geoid model reports a geoid height of -28.35 meters at that location. Substituting those values gives H = 42.60 - (-28.35), so the orthometric height is 70.95 meters. The negative sign does not indicate an error. It means the geoid lies below the ellipsoid at that location, so subtracting the negative separation increases the reported orthometric height.
A negative geoid height does not mean the geoid model is wrong or that the elevation must be negative. The sign only describes whether the geoid lies above or below the ellipsoid at that location.
Why the Sign Causes So Much Confusion
Many elevation mistakes are not caused by poor satellite observations. They begin when the sign convention in the vertical equation is misunderstood. A user sees a geoid value of -28.35 meters, assumes that a negative number must always be subtracted in magnitude, and applies the correction in the wrong direction. The resulting elevation can be wrong by roughly twice the geoid separation while the RTK solution itself remains perfectly sound.
The safest approach is not to memorize a regional shortcut such as 'always add about thirty meters.' That shortcut may appear to work in one area and fail somewhere else. Instead, write the relationship h = H + N, preserve the sign provided by the geoid model, and solve algebraically for the unknown quantity. The equation works anywhere because it describes the reference surfaces rather than a local numerical habit.
If N is negative, then H = h - N becomes subtraction of a negative number. The orthometric height will therefore be larger than the ellipsoid height. Did the antenna move upward? No. The reference surface lies lower.
The Equation Does Not Create Accuracy
The vertical equation is exact as a relationship, but the values placed into it still carry uncertainty. The ellipsoid height depends on the quality of the GNSS solution. The geoid height depends on the selected geoid model and how accurately that model represents the gravity field at the location. The computed orthometric height inherits uncertainty from both sources.
This matters because a precise-looking result can conceal a reference problem. A receiver may display H to the nearest millimeter even though the geoid model, datum realization, antenna setup, or local control introduces much larger uncertainty. The number of digits on the screen does not establish that the elevation is compatible with the benchmark being checked.
Follow the Number Through the Conversion
We can now describe the final vertical step in the processing chain. The satellites contribute observations. The RTK engine solves the rover position. The ECEF coordinates are converted to latitude, longitude, and ellipsoid height. A geoid model is then evaluated at the horizontal position to obtain N. Finally, the software computes H = h - N and displays the result as an orthometric elevation.
Only the last step changes the reported height value. It does not alter the satellite observations, the ECEF coordinate, the latitude and longitude, or the physical location of the antenna. The software has changed the reference from which the vertical distance is measured. Once that distinction is clear, the equation becomes a diagnostic tool rather than a source of confusion.
The vertical relationship is h = H + N, or equivalently H = h - N. Keep the sign of N exactly as provided by the geoid model. The calculation changes the reference surface used to report height; it does not change the coordinate itself.
The equation is simple, but its result depends on the geoid value placed into it. In the next section, we will examine what a geoid model actually contains, why it must be selected deliberately, and how two different models can produce different elevations from the same ellipsoid height.