1: Introduction

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1.1 Who should read this booklet?

This booklet is aimed at people whose expertise is in fields other than geodesy, who need to know the concepts of coordinate systems in order to deal with coordinate data, and who need information on using mapping coordinate systems in Great Britain. It explains:

• the basic concepts of terrestrial coordinate systems;
  
  
• the coordinate systems used with the Global Positioning System (GPS) and in Ordnance Survey mapping; and
  
  
• how these two relate to each other.

Although this booklet deals with the GPS system, the concepts and techniques can also be applied to other Global Navigation Satellite Systems (GNSS), for example GLONASS (russian), Galileo (European) and Compass (Chinese). 

The subject of geodesy deals, amongst other things, with the definition of terrestrial coordinate systems. Users of coordinates are often unaware that this subject exists, or that they need to know some fundamental geodetic concepts in order to use coordinates properly. This booklet explains these concepts. If you work with coordinates of points on the ground and would like to know the answers to any of the following questions, or if you don’t understand the questions, this booklet is a good place to start.

• How do geodesists define coordinate systems that are valid over large areas? What is difficult about this task, anyway? Why can’t we all just use one simple coordinate system for all positioning tasks?
  
  
• What exactly is WGS84? How accurate is it? How does WGS84 relate to map coordinates? Why are there other GPS coordinate systems that seem to be very similar to WGS84? Why are there so many acronyms used to describe GPS coordinate systems?
  
  
• How is the Ordnance Survey National Grid defined? How does OSGB36^® relate to the National Grid? Why does it seem to be difficult to relate the National Grid coordinates to GPS coordinates? How are grid references converted to latitude and longitude coordinates?
  
  
• Why do coordinate systems use ellipsoids? Why are there so many different ellipsoids? Why is it so difficult to convert coordinates from one ellipsoid to another? Is an ellipsoid the same thing as a datum? What is the difference between height above mean sea level and height above an ellipsoid?
  
  
• Why are transformations between different coordinate systems not exact? How can GPS coordinates be related precisely to the National Grid and mean sea level (orthometric) heights?

1.2 A few myths about coordinate systems

Myth 1: ‘A point on the ground has a unique latitude and longitude’

For reasons which are a mixture of valid science and historical accident, there is no one agreed latitude and longitude coordinate system. There are many different meridians of zero longitude (prime meridians) and many different circles of zero latitude (equators), although the former generally pass somewhere near Greenwich, and the latter is always somewhere near the rotational equator. There are also more subtle differences between different systems of latitude and longitude which are explained in this booklet.

The result is that different systems of latitude and longitude in common use today can disagree on the coordinates of a point by more than 200 metres. For any application where an error of this size would be significant, it’s important to know which system is being used and exactly how it is defined.

The figure below shows three points which all have the same latitude and longitude, in three different coordinate systems (OSGB36, WGS84 and ED50). Each one of these coordinate systems is widely used in Britain and fit for its purpose, and none of them is wrong. The differences between them are just a result of the fact that any system of absolute coordinates is always arbitrary. Standard conventions ensure only that different coordinate systems tend to agree to within half a kilometre or so, but there is no fundamental reason why they should agree at all.

Figure 1: Three points with the same latitude and longitude in three
different coordinate systems. The map extract is 200 metres square.

Myth 2: ‘A horizontal plane is a level surface’

Of course it cannot be, because the Earth is round - any gravitationally level surface (such as the surface of the wine in your glass, or the surface of the sea averaged over time) must curve as the Earth curves, so it cannot be flat (that is, it cannot be a geometrical plane). But more than this, a level surface has a complex shape - it is not a simple curved surface like a sphere. When we say ‘a level surface’ we mean a surface which is everywhere at right angles to the direction of gravity. The direction of gravity is generally towards the centre of the Earth as you would expect, but it varies in direction and magnitude from place to place in a complex way, even on a very local scale. These variations, which are too small for us to notice without specialist measuring equipment, are due to the irregular distribution of mass on the surface (hills and valleys) and also to the variable density of the Earth beneath us. Therefore all level surfaces are actually bumpy and complex.

This is very important to coordinate systems used to map the height of the ground, because the idea of quantified ‘height’ implies that there is a level surface somewhere below us which has zero height. Even statements about relative height imply extended level surfaces. When we casually say ‘Point A is higher than point B’, what we really mean is ‘The level surface passing through point A, if extended, would pass above point B’ So to accurately quantify the height difference between A and B, we would need to know the shape of the level surface passing through point A. In fact, we choose a general reference level surface of zero height covering the whole country to which we can refer all our measured heights. This reference level surface is not flat!

Myth 3: ‘The true coordinates of a ground point do not change’

They certainly do, due to the continuous deforming motions of the Earth. Relative to the centre of the Earth, a point on the ground can move as much as a metre up and down every day just because of the tidal influences of the sun and moon. The relative motion of two continents can be 10 centimetres a year, which is significant for mapping because it is constant year after year – after 50 years a region of the earth may have moved by 5 metres relative to a neighbouring continent. Many other small effects can be observed – the sinking of Britain when the tide comes in over the continental shelf (a few centimetres), the sinking of inland areas under a weather system high (about 5 millimetres), and the rising of the land in response to the melting of the last Ice Age (about 2 millimetres per year in Scotland, up to 1 centimetre per year in Scandinavia). Generally, as the size of the region of the Earth over which we want to use a single coordinate system increases, the more these dynamic Earth effects are significant.

The modern trend is to use global coordinate systems even for local applications. Therefore, it is important to realise that in a global coordinate system, the ground on which we stand is constantly moving. This leads to subtleties in coordinate system definition and use.

Myth 4: ‘There are exact mathematical formulae to change between coordinate systems’

Exact formulae only apply in the realm of perfect geometry – not in the real world of coordinated points on the ground. The known coordinates of a point in one coordinate system are obtained from a large number of observations which are averaged together using a whole raft of assumptions. Both the observations and the assumptions are only ever approximately correct and can be of dubious quality, particularly if the point was coordinated a long time ago. It will also have moved since it was coordinated, due to subsidence, continental plate motion and other effects.

The result is that the relationship between two coordinate systems at the present time must also be observed on the ground, and this observation too is subject to error. Therefore, only approximate models can ever exist to transform (convert) coordinates from one coordinate system to another. The first question to answer realistically is ‘What accuracy do I really require?’ In general, if the accuracy requirements are low (5 to 10 metres, say) then transforming a set of coordinates from one coordinate system to another is simple and easy. If the accuracy requirements are higher (anywhere from one centimetre to half a metre, say), a more involved transformation process will be required. In both cases, the transformation procedure should have a stated accuracy level.

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