The Geoid - An Equipotential Description with Gravity

The Geoid is a surface that is not often talked about on blogs or mentioned on the web - so this may be a first.

The Geoid surface is irregular, unlike reference ellipsoids (such as Clarke 1866, Bessel, Hayford, etc.) which have been used to approximate the shape of the physical Earth at a local point. The geoid is considerably smoother than Earth's physical surface.

In looking for a good description that makes sense to many people, I stumbled upon this one on Wikipedia:

"In geodetic surveying, the computation of the geodetic coordinates of points is commonly performed on a reference ellipsoid closely approximating the size and shape of the Earth in the area of the survey. The actual measurements made on the surface of the Earth with certain instruments are however referred to the geoid. The ellipsoid is a mathematically defined regular surface with specific dimensions. The geoid, on the other hand, coincides with that surface to which the oceans would conform over the entire Earth if free to adjust to the combined effect of the Earth's mass attraction (gravitation) and the centrifugal force of the Earth's rotation. As a result of the uneven distribution of the Earth's mass, the geoidal surface is irregular and, since the ellipsoid is a regular surface, the separations between the two, referred to as geoid undulations, geoid heights, or geoid separations, will be irregular as well."

and further it states:

"The geoid is a surface along which the gravity potential is everywhere equal and to which the direction of gravity is always perpendicular. The latter is particularly important because optical instruments containing levelling devices are commonly used to make geodetic measurements. When properly adjusted, the vertical axis of the instrument coincides with the direction of gravity and is, therefore, perpendicular to the geoid. The angle between the plumb line which is perpendicular to the geoid (sometimes called "the vertical") and the perpendicular to the ellipsoid (sometimes called "the ellipsoidal normal") is defined as the deflection of the vertical. It has two components: an east-west and a north-south component."

The reference surface for heights is traditionally taken as Mean Sea Level (MSL).

The geoid, as described above, is a surface of equal gravity potential which closely approximates mean sea level.

With GPS becoming more and more relevant in our daily lives, what is the height measurement we get?

The heights derived from GPS are relative to the GPS reference ellipsoid (WGS84). The separation between the geoid and an ellipsoid is known as the geoid-ellipsoid separation, or N value.

In a mathematical sense, we have the following then:

H = h - N

where H = Orthometric Height

h = Ellipsoidal Height (for example, the height above the ellipsoid WGS84)

N = Geoid-Ellipsoid Height (this is also called the Geoid Undulation)

Note that with N, that if the geoid is above the ellipsoid, N is positive. If the geoid is below the ellipsoid, N is negative.

How does mass effect the geoid and the ellipsoid?

Where a mass deficiency exists, the geoid will dip below the mean ellipsoid and where a mass surplus exists, the geoid will rise above the mean ellipsoid.

Where are the largest undulations?

Well, the largest undulations known, with the minimum in the Indian Ocean at a value of N = -100 metres and the maximum in the northern part of the Atlantic Ocean with N = +70 metres.

So how do we describe the shape and size of the Earth?

There are three surfaces to be considered:

• The topography - the physical surface of the earth.



• The Geoid - the level surface (also a physical reality).



• The Ellipsoid - the mathematical surface for computations.
  

Mean Sea Level (MSL) points, an approximation to the geoid, and can be used as reference surfaces for height measurements (i.e. orthometric heights).

Ellipsoidal heights (such as those derived by GPS) have to be adjusted before they can be compared to the orthometric heights given on topographic maps.

The deviation between the geoid and an reference ellipsoid is called Geoid undulation (N). Geoid undulations can be used to adjust the ellipsoidal heights (H = h +/- N).

This is an introduction to the Geoid and the science of Geodesy. It hopefully clears up some questions about this surface.

I'll explain in more detail some of the formulations of the Geoid and how geodesists over time have tried to model it and some of the efforts being conducted presently to come up with a global gravity model to aid in height determination at a later date.

There are some very interesting projects going on in Africa, South America, and Canada.

The Robinson Projection: Not shaped like an Egg

A Pleasing View of the World

The Robinson Projection came into being in 1963 and was introduced by Dr. Arthur H. Robinson.

This projection can be classified as a pseudo-cylindrical projection because of its straight parallels, along each of which the meridians are spaced evenly. The central meridian is also a straight line and all other meridians are curved.

The projection is neither equal-area nor conformal, therefore abandoning both for a compromise for creating what Dr. Robinson felt produced a better overall view of the world. This was the first map projection to be developed for commercial interests. Rand McNally felt that many of the map projections in use did not present the earth as a whole very well. With Mercator the poles were distorted. Robinson, was essentially contracted to develop a map projection that did not maintain angle, direction, or limit distortion, but was sanctioned to produce a map projection that "looked good" for books and atlases.

Remember maps are designed for one of the following 4 reasons:

• Conformality - the shapes of places are accurate
• Distance - measured distances are accurate
• Area/Equivalence - the areas represented on the map are proportional to their area on the earth
• Direction - angles of direction are portrayed accurately

Dr. Robinson specified the projection to be constructed by referring to a table of cartesian coordinate values at specific intersections of latitude and longitude. The intermediate locations are to be found by interpolation. Dr. Robinson developed the projection through a series of trials, continually iterating till he settled upon the meridian shapes and parallel spacing most pleasing to the eye. In comparison with other Map Projections, they are mainly developed or are formulated as mathematical equations.

Parallels are straight parallel lines, equally spaced between latitudes 38 degrees north and south. Space decreases beyond these limits. The Equator is 0.8487 times as long as the circumference of a sphere of equal area. The central meridian is a straight line 0.5072 as long as the Equator. Other meridians are equally spaced elliptical arcs and concave toward the central meridian. The scale is true along latitudes 38 degrees north and south, constant along any given latitude, and the same for the latitude of opposite sign (Robinson 1974; Snyder and Voxland 1989).

This map projection is also based on a sphere, not an ellipsoid. This is an important point to remember, as the earth is being modelled differently.

What is the true shape of the Earth?

The Earth, in actual fact, is shaped more like an egg, hence, even an ellipsoid is not the best model. In the past, when datum's were more local (such as NAD27, SAD69, Cape Datum, etc.), various ellipsoids were tied to local points on Earth. For NAD27, this was Meades Ranch, Kansas.

But back to the Earth's shape; it is very close to an oblate spheroid — a rounded shape with a bulge around the equator — although the precise shape (the geoid) varies from this by up to 100 metres.

The rotation of the Earth creates the equatorial bulge so that the equatorial diameter is 43 km larger than the pole to pole diameter.

What about height? A whole other story.

Interesting, eh? There are so many different ways to see the Earth. When we look at height systems and describe the geoid, we are describing an equipotential surface. But height is a whole other story, as gravity is involved and it is not as simple as getting the "Zed" or "Zee" measurement from your GPS.

I'll explain height later and how we can determine MSL (and where the "mean" actually comes from).