GEOG 862
GPS and GNSS for Geospatial Professionals

Ellipsoidal Heights

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Ellipsoidal height aka Geodetic height
Ellipsoidal Height
Source: GPS/GNSS for Land Surveyors

As mentioned before, modern geodetic datums rely on the surfaces of geocentric ellipsoids to approximate the surface of the Earth. But the actual surface of the Earth does not coincide with these nice smooth surfaces, even though that is where the points represented by the coordinate pairs lay. Abstract points may be on the ellipsoid, but the physical features those coordinates intend to represent are on the Earth. Though the intention is for the Earth and the ellipsoid to have the same center, the surfaces of the two figures are certainly not in the same place. There is a distance between them. The distance represented by a coordinate pair on the reference ellipsoid to the point on the surface of the Earth can be measured along a line perpendicular to the ellipsoid -  which differs from the direction of gravity. This distance is known by more than one name. It is called the ellipsoidal height, and it is also called the geodetic height and is usually symbolized by h.  In the illustration, you see an ellipsoidal height of +2,644 meters.  The concept is straightforward. A reference ellipsoid may be above or below the surface of the Earth at a particular place. If the ellipsoid’s surface is below the surface of the Earth at the point, the ellipsoidal height has a positive sign; if the ellipsoid’s surface is above the surface of the Earth at the point, the ellipsoidal height has a negative sign. 

Diagram showing deflection of the vertical due to the ellipsoid, see text below
The Deflection of the Vertical
Source: GPS/GNSS for Land Surveyors

It is quite impossible to actually set up an instrument on the ellipsoid. That makes it tough to measure ellipsoidal height using surveying instruments. In other words, ellipsoidal height is not what most people think of as an elevation. Said another way, an ellipsoidal height is not measured in the direction of gravity. It is not measured in the conventional sense of down or up. In the illustration, there is an instrument level on the topographic surface of the Earth. The direction of gravity does not coincide with the perpendicular to the ellipsoid.  

Diagram showing the fundamentals of locating a point on the Earth's surface, see text below
A Few Fundamentals
Source: GPS for Land Surveyors

Nevertheless, the ellipsoidal height of a point is readily determined using a GPS/GNSS receiver. GPS/GNSS can be used to discover the distance from the geocenter of the Earth to any point on the Earth, or above it for that matter. In other words, it has the capability of determining three-dimensional coordinates of a point in a short time. It can provide latitude and longitude, and if the system has the parameters of the reference ellipsoid in its software, it can calculate the ellipsoidal height. The relationship between points can be further expressed in the ECEF coordinates, X, Y, and Z, or in a Local Geodetic Horizon System (LHGS) of north, east, and up. Actually, in a manner of speaking, ellipsoidal heights are new, at least in common usage, since they could not be easily determined until GPS/GNSS became a practical tool in the 1980s.

You may wonder why in the world spend any time at all on this bizarre idea of an ellipsoid height. Since you can't measure it directly, you can't set up on the ellipsoid, what's the point? Well, here's the point. As you see in this image, the ellipsoid height, not the orthometric height, is what is determined directly from a GPS/GNSS measurement. The GPS/GNSS satellites orbit around the center of mass of the Earth. They have no concept of where the surface of the Earth is. Your receiver, which is on the surface of the Earth, of course, could just as well be in outer space as far as the GPS/GNSS coordinate or Earth-Centered-Earth-Fixed XYZ is concerned. Starting from the 3D Cartesian coordinate and given a particular ellipsoid on which to work, it is possible for the computer in your GPS/GNSS receiver to calculate a latitude and longitude. It will base that latitude and longitude on the ellipsoidal parameters in its microcomputer. It follows that the height that is determined up to the station that you're interested in will also be based upon the ellipsoid.

Diagram showing a shift between the ITRF97 origin and the NAD83 origin.
A Shift to the Geocenter
Source: GPS for Land Surveyors

However, ellipsoidal heights are not all the same, because reference ellipsoids, or sometimes just their origins, can differ. For example, an ellipsoidal height expressed in ITRF might be based on an ellipsoid with exactly the same shape as the NAD83 ellipsoid, GRS80. Nevertheless, the heights would be different because the origin has a different relationship with the Earth’s surface. It's worthwhile to note that ellipsoidal heights vary as the ellipsoid changes. As the reference frame (datum) changes, the ellipsoid height changes. And that's what this image here is intended to represent. There is an approximately 2.24 meter separation between the origin of NAD83, the GRS80 ellipsoid, the ITRF ellipsoid. Therefore, the heights —the small h, the ellipsoid height— derived from each of these would be different.