The Earth is not perfectly round, some thought it is even flat! Also, the Earth is not a perfect sphere because it is spinning around its poles. More material is trying to be ejected off the earth, and the best place to do it is as far as possible from the poles: around the equator. Fortunately the mattter doesn\'t get ejected because the gravity keeps it together. The gravity is based on the principle that matter atracts matter, but denser matter will atract less denser matter. The same way the earth attracted the apple, and Newton attracted the apple on his head. It means that seamounts will attract the water around them, which will create a hump. It is also true on land where rocks will attract soil around them. It means that the mean sea level around the earth is not always at the same distance from the earth centre.

Scientists have tried throughout the ages to have a good representation of the earth. In the begining the earth was flat, then later with the Greeks they realised that it wasn\'t possible. The best way to find your position is to look where you are going. The Romans were using this technique to build their roads. They were making fires and followed the direction of the smoke when building the pavement. We later developped better tools to measure direction and distance. At the same time people realised that the study of stars and their positioning the sky is giving your position on earth. The sextant is one of these tools that give your position by knowing the time of the day and the elevation of the sun or some stars in the sky. The position is not accurate, about a few km. You can also use radio waves to find your direction and the distance to the beacon. The precision is better: about a few 100 metres. And finally the latest developments brought us the GPS (Global Positionning System). The GPS gives accuracy of 100 metres when not differentially corrected and up to a few millimetres when properly corrected.

Instrument	Precision
Sextant	1-5km
Radio Beacons	100-1000m
GPS	0.01-100m

Well to come back to what is the earth, it is like this:

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But we didn't know it till we went in space.

The best mathematical representation of such object is called an ellipsoid?. The problem is to shape it and place it as close as possible from the true representation of the Earth. The true representation of the Earth is called the geoid?.

The geoid is not the surface of the earth but an equipotential surface based on the gravity field corrected by the centrifugal force of the earth\'s rotation. Basically this equipotential surface should be as close as possible from the mean sea level if the earth was made only of water. The centre of the geoid is also the earth centre, centre of the mass.

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As previously stated, the measuring instruments became better and better though the years that means that the representation of the Earth became better and better. The type of ellipsoid used is a biaxial ellipsoid or spheroid. A triaxial spheroid could have been used but this representation does not bring a better accuracy in comparison to the level of complexity of the geometric figure.

ImageImage?


A spheroid is characterised by its semi-major axis a, and its degree of flattening f. The semi minor axis is given by b=a*f.

Here is a list of a few spheroids and their year of adoption.

Spheroid	Year	a	1/f
Bessel	1841	6377397.155	299.1528128
Clarke	1880	1880	6378249.145	293.465
International	1924	6378388.0	297.0
WGS60	1960	6378165.0	298.3
WGS72	1972	6378135.0	298.26
WGS84	1984	6378137.0	298.257223563

Between WGS72 and WGS84 we gained a better understanding of the shape of the Earth through better measuring systems. It is also due to the instruments that allow us to locate our position compared to the spheroid. If we use a sextant to set up the spheroid we will mis-locate the spheroid by a few kilometres. It is not important as long as we use a sextant to find again our position, but with better instrumentation, problems start to begin.

First let\'s go back to the positionning of the Spheroid. Let\'s say Fiji is positioning the International spheroid and Solomon Islands is doing it too. As the two islands are far away from each other, there is no way to compute a distance or a direction between the two countries with enough precision. It means that if both countries use the same spheroid they will place it at two different locations. It is also important to note that the spheorid is not necessarly at the centre of the geoid

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The errors you can make are basically an error of location of the centre, an error of rotation, and an error of scale. This is expressed as a datum.

Before we speak of the datum, let\'s express our position on the earth. The first set of coordinates is defined by the major earth axis: the axis around which the earth is spinning. However we have to choose a second axis. This one is defined by the circle whose diameter is the earth rotation axis and whose perimeter passes through Greenwich. The third axis is easily given from the two others. This set of coordinates is called Terestrial Cartesian (x,y,z), because the axis are joined at the centre of the earth.

It is important to note that the axis of rotation is moving.

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Observed Polar Motion (Dr. S Yumi)


But as we have seen, we rarely use the geoid for the representation of the earth, but rather the spheroid. We can use a similar coordinate system based on the centre of the spheroid and its axis. This set of coordinates is called Geodetic Cartesian (x,y,z?) expressed in meters.

These cartesian coordinates are not pratical, and we prefer to use curvilinear coordinates (Latitude, Longitude, Altitude?) expressed in degrees and meters. Latitude and Longitudes are angles from the Meridian of Greenwich and the equator respectively. We have also two sets: Terestrial based on the geoid, and geodetic based on the spheroid. The later is the one found on maps, therefore it is the most used.

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If latitude and longitude are easy to understand, altitude is not well defined. The altitude in the geodetic curvilinear coordinate system is the height above the spheroid, it is not the height above mean sea level. A difference up to 60m can be observed.

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Geoid referred to best fitting ellipsoid (contours in meters)


The transformation from curvilinear coordinates to cartesian cordinates is given below knowing the semi-major axis a and the semi-minor axis b of the spheroid.

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The oposite transformation from cartesian coordinates to curvilinear coordinates is not trivial. Usually we use a a convergent method which is stopped as soon as a given precision is obtained.

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Height here is understood as height above the spheroid.

All this positionning is fine, but let\'s not forget that the earth is composed of plates floating on the earth magma. The movement of these plates could be up to 20cm a year in the South Pacific.

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Plates and continents as today - Plates and continents 10 million of years ago

From Loren W. Kroenke


From these pictures we notice that the east plate has moved on some points of more than 100km in 10 million of years which is about 10cm a year.

This was all the theory about the spheroid, and placing it as close as possible as from the true representation of the earth. As different people had different interpretations, the spheroid was placed differently. The fact to place the spheroid in relation to a few points on the earth is called the datum. We will see now what eactly is a datum...

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Franck Martin 1998