Ellipsoid – A mathematical surface created by rotating a 2-dimensional ellipse about its axis. Examples of ellipsoids include WGA-1984, International, Clarke 1866 and Bessel.

An ellipsoid is normally defined by its semi-major axis (a) and its semi-minor axis (b). The semi-major axis (a) and the flattening (f) also often define them. The flattening is a ratio of the difference between the two axes, divided by the semi-major axis.

   f = (a-b)/a

The semi-major and semi-minor axes are normally expressed in meters. The flattening is often expressed as the inverse (1/f) of the flattening.

Your position on the ellipsoid is defined with three variables:

•Latitude: Your Latitude is the angle that a line drawn from your position normal (perpendicular) to the ellipsoidal surface makes with the ellipsoidal equator.

•Longitude: Your longitude is the polar angle of your point, measured counter-clockwise from a user-defined reference. For many ellipsoids, this reference is the Greenwich meridian (0).

•Height: The height is the distance from your point to the surface, measured along a line, which is normal (perpendicular) to the ellipsoidal surface.

Your latitude, longitude, and height will differ, depending on the ellipsoid used as your reference surface. In other words, a single point can be described with a different latitude, longitude and height combination for each ellipsoid you create.

Ellipsoids are chosen so they conform to the shape of the geoid for your project area.

A geoid is an equipotential surface, meaning the pull of gravity measured anywhere on the surface is equal. Base on the sur­rounding mass (mountains, canyons, etc.), this surface rises and falls and is much more irregular than an ellipsoid, although much smoother than the earth’s surface.

One of the important features about a geoid is that a plumb bob always points normal (perpendicular) to the geoidal surface. It does not point directly to the center of the earth. This means that your local land measurements will be affected by the local geoidal sur­face. In order to reduce the errors caused in computing positions on the ellipsoid using measurements affected by the geoid, the ellipsoid is shifted so it closely matches the geoid in your local area. When this is done, it becomes a datum.

A datum is an ellipsoidal surface, which has been moved to closely match the geoidal surface for your project area.

When you move your ellipsoid to create a datum, you are also affecting the latitude, longitude and height above the ellipsoid of your point. When someone describes a position location to you with latitude, longitude and height, you don’t know anything until you know which datum was used to define the point and which ellipsoid the datum is based upon.

Example #1: Say that your friend confesses to you on his death­bed that he robbed a bank and buried the money at exactly 45N and 78W. You had better quickly ask him what his reference datum was, or you are going to be digging a long time!

Example #2: An oil company pays you a lot of money to survey between 26 00N and 72 00W and 26 01N and 72 01W. Since they are a modern survey company, you assume they are working on WGS-84 and go out and perform the survey. When you get back home, you find out they are working on the Everest ellipsoid and you should have been surveying an area 2 miles to the south.

Three-Parameter Datum Transformation

In geocentric methods, the latitude, longitude and height above the ellipsoid are converted to Cartesian XYZ coordinates using the center of the ellipsoid as the origin. These are referred to as “geo­centric coordinates”. Based upon the separation between the cen­ters of the two datums, an offset is added to each coordinate to “shift” them from the first datum to the second datum. The geocen­tric coordinates for the second datum are then converted back into latitude, longitude and height using the ellipsoidal constants for the second ellipsoid.

In summary:

Lat1/Long1/H1 à X1, Y1, Z1, à X2,Y2,Z2 à Lat2/Long2/H2

To go from X1, Y1, Z1 to X2, Y2, Z2, we added a dX, dY and dZ to each specific value. This is called a three-parameter shift and is typically used for only a local area (10km). The same process is performed in a technique known as the Molodensky Formulae. The Abridged Molodensky Formulae is very similar, but eliminates a few variables while giving almost the same result.

To obtain these dX, dY and dZ values, you can either look up pub­lished information (such as DMA TM 8511) or calculate them. To calculate them, you need to know the latitude, longitude and height for the same point in the two different datums. Calculate the geo­centric coordinates for both points, using the ellipsoidal parameters associated with each datum. Take the difference between the X1 and X2 values to determine the dX parameter. Repeat the same for the dY and dZ parameters. Voila, you have computed the exact datum transformation parameters for your area.

The dX, dY, and dZ values used in a datum transformation are typ­ically valid for a small area (10 km X 10km?). As you move further from your area, the values change as the relationship between the two ellipsoids change.

To cover a wider area, a seven-parameter datum transformation can be used. A seven-parameter transformation contains, the dX, dY and dZ values mentioned above, as well as ӨX, ӨY and ӨZ and dScale values. The Ө values represent the difference in alignment of the X-, Y-, and Z-axis of the two ellipsoidal geocentric axes. The dScale represents difference in scale measured between the two systems.

The advantage of a seven-parameter datum transformation over a three-parameter datum transformation is that it is valid for a much larger area. Many countries, such a Saudi Arabia publish a single seven-parameter datum transform, which is used for the entire country.

 NAD83 originally intended to be co-incident with WGS84

• Defined to be compatible with BIH Terrestrial System of 1984

 WGS84 doesn’t exist as a “control network”

• Only tracking stations

• Although CSRS points have positions published in ITRF epochs

 WGS84 origin has moved ~ 2 metres

• to reflect better determination of centre of mass  NAD83 did not move and now is no longer synonymous with WGS84  2 m shift between the origins of the WGS84 and NAD83

• 1.1 metres in Halifax

• 1.5 metres in St.John’s