GEOG 862
GPS and GNSS for Geospatial Professionals

The Geoid

photo of a small island surrounded by water
Sea Level is not the Surface of the Geoid
Source: NASA

An often-used description of the geoidal surface involves idealized oceans. Imagine the oceans of the world utterly still, completely free of currents, tides, friction, variations in temperature and all other physical forces, except gravity. Reacting to gravity alone, these unattainable calm waters would coincide with the figure known as the geoid. Admitted by small frictionless channels or tubes and allowed to migrate across the land, the water would then, theoretically, define the same geoidal surface across the continents, too. Of course, the 70% of the earth covered by oceans is not so cooperative, nor is there any such system of channels and tubes. In addition, the physical forces eliminated from the model cannot be avoided in reality. These unavoidable forces actually cause mean sea level to deviate from the geoid. This is one of the reasons that Mean Sea Level and the surface of the geoid are not the same. And it is a fact frequently mentioned to emphasize the inconsistency of the original definition of the geoid as it was offered by J.B. Listing in 1872. Listing thought of the geoidal surface as equivalent to mean sea level. Even though his idea does not stand up to scrutiny today, it can still be instructive.

The geoid is sometimes illustrated as the surface of the sea, sometimes known as mean sea level. I hasten to add, however, that mean sea level and the geoid are different. They are not the same. Close, yes, but the same, no.

One of the major reasons that mean sea level cannot be the same as the geoid is that the actual oceans of the Earth are affected by temperature, by wave motion, by salinity, and many other aspects that cause variation in their heights. So even with tidal monitoring, means sea level is not an indicator of gravity alone, and that is the concept of the geoid. The geoid is that surface which is affected only by gravity and it is an equipotential surface, where the gravity potential is always the same.

, the oceans of the earth can be thought of as an approximation of that idea, and if there was only gravity acting on them, and these waters were entirely calm, then of course the geoid and sea level would be exactly the same. But of course, the physical oceans are not nearly that cooperative.

An Equipotential Surface

graphic showing an exaggerated view of the Geoid of earth
And Exaggeration of the Geoid
Source: ESA

Gravity is not consistent across the topographic surface of the earth. At every point it has a magnitude and a direction. In other words, anywhere on the earth, gravity can be described by a mathematical vector. Along the solid earth, such vectors do not have all the same direction or magnitude, but one can imagine a surface of constant gravity potential. Such an equipotential surface would be level in the true sense. It would coincide with the top of the hypothetical water in the previous example. Mean sea level does not define such a figure, nevertheless the geoidal surface is not just a product of imagination. For example, the vertical axis of any properly leveled surveying instrument and the string of any stable plumb bob are perpendicular to the geoid. Just as pendulum clocks and earth-orbiting satellites, they clearly show that the geoid is a reality.

The geoid is an equipotential surface, and as we see on this image of what a lumpy surface Earth would be if we only considered gravity. This image is exaggerated, but you can see that gravity isn't consistent across the entire topographic surface of the Earth. At every place gravity has a direction and a magnitude. In other words, anywhere on the Earth, gravity can be described as a vector. And such vectors don't have all the same direction and magnitude, but you can imagine a surface that does have a constant gravity potential, and this would be the level surface, in the truest sense.

Think of an instrument, like a GPS receiver, being set up and leveled on the Earth. It is leveled in relation to gravity. It is leveled in relation to the surface that we're talking about. So, the vertical axis of any properly leveled surveying instrument and the string of any stable plumb bob are perpendicular to the geoid. In other words the geoid is a reality. It is a physical reality. I hasten to add that the ellipsoid, the nice, smooth, mathematical surface that we use as a reference in datums, is different than the geoid. They are two different surfaces. The geoid is defined entirely by gravity and is a physical reality. The ellipsoid is a purely mathematical imaginary surface.