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# 2: The shape of the earth

2.1 The first geodetic question

When you look at all the topographic and oceanographic details, the Earth is a very irregular and complex shape. If you want to map the positions of those details, you need a simpler model of the basic shape of the Earth, sometimes called the 'figure of the Earth', on which the coordinate system will be based. The details can then be added by determining their coordinates relative to the simplified shape, to build up the full picture.

The science of geodesy, on which all mapping and navigation is based, aims firstly to determine the shape and size of the simplified 'figure of the Earth' and goes on to determine the location of the features of the Earth's land surface - from tectonic plates, coastlines and mountain ranges down to the control marks used for surveying and making maps. Hence, geodesists provide the fundamental points of known coordinates which cartographers and navigators take as their starting point. The first question of geodesy, then, is 'What is the best basic, simplified shape of the Earth?' Having established this, we can use it as a reference surface, with respect to which we measure the topography.

Geodesists have two very useful answers to this question: ellipsoids and the Geoid. To really understand coordinate systems, you need to understand these concepts first.

2.2 Ellipsoids

The Earth is very nearly spherical. However it has a tiny equatorial bulge making the radius at the equator about one third of one percent bigger than the radius at the poles. Therefore, the simple geometric shape which most closely approximates the shape of the Earth is a biaxial ellipsoid, which is the three-dimensional figure generated by rotating an ellipse about its shorter axis (less exactly, it is the shape obtained by squashing a sphere slightly along one axis). The shorter axis of the ellipsoid approximately coincides with the rotation axis of the Earth.

Because the ellipsoid shape doesn't fit the Earth perfectly, there are lots of different ellipsoids in use, some of which are designed to best, fit the whole Earth and some to best fit just one region. For instance, the coordinate system used with the Global Positioning System (GPS) uses an ellipsoid called GRS80 (Geodetic Reference System 1980) which is designed to best-fit the whole Earth. The ellipsoid used for mapping in Britain, the Airy 1830 ellipsoid, is designed to best-fit Britain only, which it does better than GRS80, but; it is not useful in other parts of the world. So various ellipsoids used in different regions differ in size and shape, and also in orientation and position relative to each other and to the Earth. The modern trend is to use GRS80 everywhere for reasons of global compatibility. Hence, the local best-fitting ellipsoid is now rather an old fashioned idea, but it is still important because many such ellipsoids are built into national mapping coordinate systems.

Figure 2: Greatly exaggerated representation of a cross section through the Earth showing cross
sections of a globally best-fitting ellipsoid (black) and a regionally best-fitting ellipsoid (grey). The
regional ellipsoid is only intended for use in the region of best fit and does not fit the Earth in other
areas. Note that the ellipsoids differ in centre position and orientation as well as in size and shape.

2.3 The Geoid

If we want to measure heights, we need an imaginary surface of 'zero height' somewhere underneath us to which the measurements will be referred. The stated height of any point is the vertical distance above this imaginary surface. Even when we talk in casual, relative terms about height, we are implicitly assuming that this surface exists.

The fact that increasing heights on the map are taken to mean 'uphill' and decreasing heights on the map are taken to mean 'downhill' implies that the height reference surface must be a level surface - that is, everywhere at right angles to the direction of gravity. It is clear that if we want to talk about the heights of places over the whole world, the reference surface must be a closed shape, and it will be something like the shape of an ellipsoid. Its exact shape will be defined by the requirement to be at right angles to the direction of gravity everywhere on its surface. As was pointed out in section 1.2, level surfaces are not simple geometrical shapes. The direction of gravity, although generally towards the centre of the Earth, varies in a complex way on all scales from global to very local. This means that a level reference surface is not a simple geometric figure like the ellipsoid, but is bumpy and complex. We can determine level surfaces from physical observations such as precise gravity measurements. This is the scientific study known as gravimetry.

Depending on what height we choose as 'zero height', there are any number of closed level surfaces we could choose as our global height reference surface, and the choice is essentially arbitrary. We can think of these level surfaces like layers of an onion inside and outside the Earth's topographic surface. Each one corresponds to a different potential energy level of the Earth's gravitational field, and each one, although an irregular shape, is a surface of constant height. The one we choose as our height reference surface is that level surface which is closest to the average surface of all the world's oceans. This is a sensible choice since we are coastal creatures and we like to think of sea level as having a height of zero. We call this irregular three-dimensional shape the Geoid. Although it is both imaginary and difficult to measure, it is a single unique surface: it is the only level surface which best-fits the average surface of the oceans over the whole Earth. This is by contrast with ellipsoids, of which there are many fitting different regions of the Earth.

Figure 3: Why the gravity field is important in height measurement. On the left is a hill
in a uniform gravity field. On the right is a flat surface in a non-uniform gravity field. The white
dotted lines are level surfaces. The experience of the blindfolded stick-men is the same in both cases.
From this it is clear that the gravity field must be considered in our definition of height. This is why
the Geoid is the fundamental reference surface for vertical measurements, not an ellipsoid.

The Geoid is very nearly an ellipsoid shape-we can define a best-fitting ellipsoid which matches the Geoid to better than two hundred metres everywhere on its surface. However, that is the best we can do with an ellipsoid, and usually we want to know our height much better than that. The Geoid has the property that every point on it has exactly the same height, throughout the world, and it is never more than a couple of metres from local mean sea level. This makes it the ideal reference surface on which to base a global coordinate system for vertical positioning. The Geoid is in many ways the true 'figure of the Earth' that we introduced in 2.1, because a fundamental level surface is intrinsic to our view of the world, living as we do in a powerful gravity field. If you like, the next step on from understanding that the shape of the Earth is 'round' rather than 'flat', is to understand that actually it is the complex Geoid shape rather than simply 'round'.

For more explanation of this subject, see section 3.1 below. A colour image of the global Geoid relative to a best-fitting ellipsoid surface is available on the Internet - see the further information list in section 8.

Local geoids

Height measurements on maps are usually stated to be height above mean sea level. This means that a different level surface has been used as the 'zero height' reference surface - one based on a tide-gauge local to the mapping region rather than the average of global ocean levels. For most purposes, these local reference surfaces can be considered to be parallel to the Geoid but offset from it, sometimes by as much as two metres. For instance, heights in Britain are measured relative to the tide gauge in Newlyn, Cornwall, giving a reference surface which is about 80 centimetres below the Geoid.

What causes this discrepancy between average global ocean levels and local mean sea levels? We know that pure water left undisturbed does form a level surface, so the two should be in agreement. The problem is that the sea around our coasts is definitely not left undisturbed! The oceanic currents, effects of tides and winds on the coast, and variations in water temperature and purity all cause mean sea level to deviate slightly from the truly level Geoid surface. So the mean sea level surface contains very shallow hills and valleys which are described by the apt term 'sea surface topography'. The sea around Britain happens to form a 'valley' in the sea surface, so our mean sea level is about 80 centimetres below the Geoid. Different countries have adopted different local mean sea levels as their 'zero height' definition. Consequently there are many 'zero height' reference surfaces used in different parts of the world which are (almost) parallel to, but offset from the true global Geoid. These reference surfaces are sometimes called 'local geoids' - the capital G can be omitted in this case.

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