We have introduced an irregular, dynamic Earth and the concepts of ellipsoid and Geoid that are used to describe its basic shape. Now we want to describe with certainty where we are on that Earth, or where any feature is, in a simple numerical way. So, the challenge is to define a coordinate system with which we can uniquely and accurately state the position of any topographic feature as an unambiguous set of numbers. In the fields of geodesy, mapping and navigation, a ‘position’ means a set of coordinates in a clearly defined coordinate system, along with a statement of the likely error in those coordinates. How do we obtain this?
The answer to this question is the subject of the whole of this section. In 'Types of coordinates', we review the different types of coordinates we commonly need to work with. In 'We need a datum' and 'Realising the datum definition with a Terrestrial Reference Frame' we will look at the two essential concepts in creating a terrestrial coordinate system that gives us a detailed insight into what a set of coordinates (a geodetic position) really tells us.
We have come some way in answering the question ‘What is position?’ by introducing various types of coordinates: we use one or more of these coordinate types to state the positions of points and features on the surface of the Earth.
No matter what type of coordinates we are using, we will require a suitable origin with respect to which the coordinates are stated. For instance, we cannot use Cartesian coordinates unless we have defined an origin point of the coordinate axes and defined the directions of the axes in relation to the Earth we are measuring. This is an example of a set of conventions necessary to define the spatial relationship of the coordinate system to the Earth. The general name for this concept is the Terrestrial Reference System (TRS) or geodetic datum. Datum is the most familiar term amongst surveyors, and we will use it throughout this booklet. TRS is a more modern term for the same thing.
(This term is often misused. The term datum refers only to the arbitrarily chosen elements of a coordinate system necessary to define the origin of coordinates – not to any control network based on this.)
To use 3-D Cartesian coordinates, a 3-D datum definition is required, in order to set up the three axes, X, Y and Z. The datum definition must somehow state where the origin point of the three axes lies and in what directions the axes point, all in relation to the surface of the Earth. Each point on the Earth will then have a unique set of Cartesian coordinates in the new coordinate system. The datum definition is the link between the ‘abstract’ coordinates and the real physical world.
To use latitude, longitude and ellipsoid height coordinates, we start with the same type of datum used for 3-D Cartesian coordinates. To this, we add a reference ellipsoid centred on the Cartesian origin, the shape and size of which is added to the datum definition. The size is usually defined by stating the distance from the origin to the ellipsoid equator, which is called the semi-major axis a. The shape is defined by any one of several parameters: the semi-minor axis length b (the distance from the origin to the ellipsoid pole), the squared eccentricity e², or the inverse flattening . Exactly what these parameters represent is not important here. Each conveys the same information: the shape of the chosen reference ellipsoid.
The term geodetic datum is usually taken to mean the ellipsoidal type of datum just described: a set of 3-D Cartesian axes plus an ellipsoid, which allows positions to be equivalently described in 3-D Cartesian coordinates or as latitude, longitude and ellipsoid height. The datum definition consists of eight parameters: the 3-D location of the origin (three parameters), the 3-D orientation of the axes (three parameters), the size of the ellipsoid (one parameter) and the shape of the ellipsoid (one parameter).
There are, however, other types of geodetic datum. For instance, a local datum for orthometric height measurement is very simple: it consists of the stated height of a single fundamental bench mark.
Modern height datums are becoming more and more integrated with ellipsoidal datums through the use of Geoid models. The ideal is a single datum definition for horizontal and vertical measurements
What all datums have in common is that they are specified a priori – they are in essence, arbitrary conventions, although they will be chosen to make things as easy as possible for users and to make sense in the physical world. Because the datum is just a convention, a set of coordinates can in theory be transformed from one datum to another and back again exactly. In practice, this might not be very useful, as we shall see in later pages
How do we specify the position and orientation of a set of Cartesian axes in relation to the ground, when the origin of the axes, and the surface of the associated ellipsoid, are within the Earth? The way it used to be done before the days of satellite positioning was to use a particular ground mark as the initial point of the coordinate system. This ground mark is assigned coordinates that are essentially arbitrary, but fit for the purpose of the coordinate system.
Also, the direction towards the origin of the Cartesian axes from that point was chosen. This was expressed as the difference between the direction of gravity at that point and the direction towards the origin of the coordinate system. Because that is a three-dimensional direction, three parameters were required to define it. So, we have six conventional parameters of the initial point in all, which correspond to choosing the centre (in three dimensions) and the orientation (in three dimensions) of the Cartesian axes. To enable us to use the latitude, longitude and ellipsoid height coordinate type, we also choose the ellipsoid shape and size, which are a further two parameters.
Once the initial point is assigned arbitrary parameters in this way, we have defined a coordinate system in which all other points on the Earth have unique coordinates – we just need a way to measure them! We will look at the principles of doing this in the next section.
Although it is a good example of defining a datum, these days, a single initial point would never be used for this purpose. Instead, we define the datum ‘implicitly’ by applying certain conditions to the computed coordinates of a whole set of points, no one of which has special importance. This method has become common in global GNSS coordinate systems since the 1980s. Avoiding reliance on a single point gives practical robustness to the datum definition and makes error analysis more straightforward.
With our datum definition we have located the origin, axes and ellipsoid of the coordinate system with respect to the Earth’s surface, on which are the features we want to measure and describe. We now come to the problem of making that coordinate system available for use in practice. If the coordinate system is going to be used consistently over a large area, this is a big task. It involves setting up some infrastructure of points to which users can have access, the coordinates of which are known at the time of measurement. These reference points are typically either on the ground, or on satellites orbiting the Earth. All positioning methods rely on line-of-sight from an observing instrument to reference points of known coordinates. Putting some of the reference points on orbiting satellites has the advantage that any one satellite is visible to a large area of the Earth’s surface at any one time. This is the idea of satellite positioning.
The network of reference points with known coordinates is called the coordinate Terrestrial Reference Frame (TRF), and its purpose is to realise the coordinate system by providing accessible points of known coordinates. Examples of TRFs are the network of Ordnance Survey triangulation pillars seen on hilltops across Britain, and the constellation of 24 GPS satellites operated by the United States Department of Defense. Both these TRFs serve exactly the same purpose: they are highly visible points of known position in particular coordinate systems (in the case of satellites, the points move so the ‘known position’ changes as a function of time). Users can observe these TRF points using a positioning tool (a theodolite or a GNSS receiver in these examples) and hence obtain new positions of previously unknown points in the coordinate system.
A vital conceptual difference between a datum and a TRF is that the former is errorless while the latter is subject to error. A datum might be unsuitable for a certain application, but it cannot contain errors because it is simply a set of conventionally adopted parameters – they are correct by definition! A TRF, on the other hand, involves the physical observation of the coordinates of many points – and wherever physical observations are involved, errors are inevitably introduced. Therefore it is quite wrong to talk of errors in a datum: the errors occur in the realisation of that datum by a TRF to make it accessible to users.
Most land surveyors do not speak of TRFs – instead we often misuse the term ‘datum’ to cover both. This leads to a lot of misunderstandings, especially when comparing the discrepancies between two coordinate systems. To understand the relationship between two coordinate systems properly, we need to understand that the difference in their datums can be given by some exact set of parameters, although we might not know what they are. On the other hand, the difference in their TRFs (which is what we generally really want to know) can only be described in approximate terms, with a statistical accuracy statement attached to the description.
In the next few pages, we look at real geodetic coordinate systems in terms of their datums and TRFs in some detail. These case studies provide some examples to illustrate the concepts introduced here.
With the three concepts summarised in table, we can set up and use a coordinate system.
| Coordinate system concept | Alternative name | Role in positioning |
|---|---|---|
We have answered the question ‘What is position?’ in a way that is useful for positioning in geodesy, surveying and navigation. A position is a set of coordinates, hopefully with an accuracy statement, together with a clear understanding of the coordinate system to which it refers in terms of the three items in the table above.
The following two sections are case studies of two coordinate systems in common use in Britain – that used for GNSS positioning, and that used for OS mapping. As we shall see, a close look at either of these examples shows that even within one coordinate system, there are alternative datums and TRFs in use, sometimes under the same name.
The most common way of stating terrestrial position is with two angles, latitude and longitude. These define a point on the globe. More correctly, they define a point on the surface of an ellipsoid that approximately fits the globe. Therefore, to use latitudes and longitudes with any degree of certainty, you must know which ellipsoid you are dealing with.
The relationship between the ellipsoid and latitude and longitude is simple (see diagram below). North-south lines of constant longitude are known as meridians, and east-west lines of constant latitude are parallels. One meridian of the ellipsoid is chosen as the prime meridian and assigned zero longitude. The longitude of a point on the ellipsoid is the angle between the meridian passing through that point and the prime meridian. Usually, the scale of longitude is divided into eastern and western hemispheres (hemiellipsoids, actually!) from 0 to 180 degrees West and 0 to 180 degrees East. The equator of the ellipsoid is chosen as the circle of zero latitude. The latitude of a point is the angle between the equatorial plane and the line perpendicular to the ellipsoid at that point. Latitudes are reckoned as 0 to 90 degrees North and 0 to 90 degrees South, where 90 degrees either North or South is a single point – the pole of the ellipsoid.
So latitude and longitude give a position on the surface of the stated ellipsoid. Since real points on the ground are actually above (or possibly below) the ellipsoid surface, we need a third coordinate, the so called ellipsoid height, which is simply the distance from the point to the ellipsoid surface along a straight line perpendicular to the ellipsoid surface. The term ‘ellipsoid height’ is actually a misnomer, because although this is an approximately vertical measurement, it does not give true height because it is not related to a level surface. It does, however, unambiguously identify a point in space above or below the ellipsoid surface in a simple geometrical way, which is its purpose.
With the coordinate triplet of latitude, longitude and ellipsoid height, we can unambiguously position a point with respect to a stated ellipsoid. To translate this into an unambiguous position on the ground, we need to know accurately where the ellipsoid is relative to the piece of ground we are interested in. How we do this is discussed in 'We need a datum' and 'Realising the datum definition with a Terrestrial Reference Frame' below.
Rectangular Cartesian coordinates are a very simple system of describing position in three dimensions, using three perpendicular axes X, Y and Z. Three coordinates unambiguously locate any point in this system. We can use it as a very useful alternative to latitude, longitude and ellipsoid height to convey exactly the same information.
We use three Cartesian axes aligned with the latitude and longitude system (see diagram above). The origin (centre) of the Cartesian system is at the centre of the ellipsoid. The X axis lies in the equator of the ellipsoid and passes through the prime meridian (0 degrees longitude). The negative side of the X axis passes through 180 degrees longitude. The Y axis also lies in the equator but passes through the meridian of 90 degrees East, and hence is at right angles to the X axis. Obviously, the negative side of the Y axis passes through 90 degrees West. The Z axis coincides with the polar axis of the ellipsoid; the positive side passes through the North Pole and the negative side through the South Pole. Hence it is at right angles to both X and Y axes.
It is clear that any position uniquely described by latitude, longitude and ellipsoid height can also be described by a unique triplet of 3-D Cartesian coordinates, and vice versa. The formulae for converting between these two equivalent systems are given in 'Converting between 3D cartesian and ellipsoidal latitude, longitude and height coordinates'.
It is important to remember that having converted latitude and longitude to Cartesian coordinates, the resulting coordinates are relative to a set of Cartesian axes that are unique to the coordinate system concerned. They cannot be mixed with Cartesian coordinates associated with any other coordinate system without first applying a transformation between the two systems (see 'We need a datum', 'Realising the datum definition with a terrestrial reference frame' and 'Geodetic transformations'). When considering using coordinates from different sources together, beware that one named coordinate system can have several different realisations (see 'We need a datum'), which are not necessarily compatible with each other.
Similarly, having converted Cartesian coordinates to latitude, longitude and ellipsoid height, the resulting coordinates are relative to the ellipsoid chosen, and also to the Cartesian reference system of the input coordinates. They cannot be used together with latitudes, longitudes and ellipsoid heights associated with any other ellipsoid or coordinate system, without first applying a suitable transformation (see 'Myths about coordinate systems').
The term ellipsoid height is misleading because a distance above a reference ellipsoid does not necessarily indicate height – point A can have a greater ellipsoid height than point B while being downhill of B. As we saw in the 'Shape of the Earth' page, this is because the ellipsoid surface is not level – therefore a distance above the ellipsoid is not really a height at all. The reference surface that is everywhere level is the Geoid. To ensure that the relative height of points A and B correctly indicates the gradient between them, we must measure height as the distance between the ground and the Geoid, not the ellipsoid. This measurement is called ‘orthometric height’ or simply ‘Geoid height’.
(We are making some reasonable simplifications here. We assume that the level surfaces passing through our points of interest are parallel to the Geoid. This is not actually true, but the difference is negligible.)
The relationship between ellipsoid height H and Geoid height (orthometric height) h is:
where N is (reasonably enough) the ‘Geoid-ellipsoid separation’[4]. Because the Geoid is a complex surface, N varies in a complex way depending on latitude and longitude. A lookup table of N for any particular latitude and longitude is called a Geoid model. Therefore, you need a Geoid model to convert ellipsoid height to Geoid height and vice-versa. The figure below shows these quantities for two points A and B. The orthometric height difference between A and B is:
Because both GNSS determination of H and Geoid model determination of N are more accurate in a relative sense (differenced between two nearby points) than in a global ‘absolute’ sense, values of will always be more accurate than either or . This is because most of the error in is also present in and is removed by differencing these quantities. Fortunately, it is usually that we are really interested in: we want to know the height differences between pairs of points.
The development of precise Geoid models is very important to increasing the accuracy of height coordinate systems. A good Geoid model allows us to determine orthometric heights using GNSS (which yields ellipsoid height) and equation (1) (to convert to orthometric height). The GNSS ellipsoid height alone gives us the geometric information we need, but does not give real height because it tells us nothing about the gravity field. Different Geoid models will give different orthometric heights for a point, even though the ellipsoid height (determined by GNSS) might be very accurate. Therefore, orthometric height should never be given without also stating the Geoid model used. As we will now see, even height coordinate systems set up and used exclusively by the method of spirit levelling involve a Geoid model, although this might not be stated explicitly.
We will now take a look at the Geoid model used in OS mapping on the British mainland, although most surveyors might not immediately think of it as such. This is the Ordnance Datum Newlyn (ODN) vertical coordinate system. Ordnance Survey maps state that heights are given above ‘mean sea level’. If we’re looking for sub-metre accuracy in heighting this is a vague statement, since mean sea level (MSL) varies over time and from place to place, as we noted in the previous section.
ODN corresponds to the average sea level measured by the tide‑gauge at Newlyn, Cornwall between 1915 and 1921. Heights that refer to this particular MSL as the point of zero height are called ODN heights. ODN is therefore a ‘local geoid’ definition as discussed in the previous section. ODN heights are used for all British mainland Ordnance Survey contours, spot heights and bench mark heights. ODN heights are unavailable on many offshore islands, which have their own MSL based on a local tide‑gauge.
A simple picture of MSL heights compared to ellipsoid heights is shown in the figure below. It shows ODN height as a vertical distance above a mean sea level surface continued under the land.
What does it mean to continue mean sea level under the land? The answer is that the surface shown as a dotted line in the figure above is actually a local geoid model. The lower-case ‘g’ indicates a local geoid model as opposed to the global Geoid, as discussed in 'The shape of the earth'. It was measured in the first half of the 20th century by the technique of spirit levelling from the Newlyn reference point, which resulted in the ODN heights of about seven hundred thousand Ordnance Survey bench marks (OSBMs) across Britain, the most important of which are the two hundred fundamental bench marks (FBMs). Hence, we know the distance beneath each bench mark that, according to the ‘ODN geoid model’, the geoid lies. By measuring the ellipsoid height of an Ordnance Survey bench mark by GNSS, we can discover the geoid‑ellipsoid separation N according to the ODN model at that point.
ODN orthometric heights have become a national standard in Britain, and are likely to remain so for some considerable time. It is important to understand the reasons for the differences that might arise when comparing ODN orthometric heights with those obtained from modern gravimetric geoid models. These discrepancies might be as much as 1 metre, although the disagreement in in equation (2) is unlikely to exceed a few centimetres. There are three reasons for this:
The ODN model assumes that mean sea level at Newlyn coincided with the Geoid at that point at the time of measurement. This is not true – the true Geoid is the level surface that best fits global mean sea level, not MSL at any particular place and time. MSL deviates from the Geoid due to water currents and variations in temperature, pressure and density. This phenomenon is known as sea surface topography (SST). This effect is important only in applications that require correct relationships between orthometric heights in more than one country; for all applications confined to Britain, it is irrelevant. The ODN reference surface is a local geoid model optimised for Britain, and as such it is the most suitable reference surface for use in Britain.
Due to the ODN geoid model only being tied to MSL at one point, it is susceptible to ‘slope error’ as the lines of spirit levelling progressed a long distance from this point. It has long been suspected that the whole model has a very slight slope error, that is, it may be tilted with respect to the true Geoid. This error probably amounts to no more than twenty centimetres across the whole 1000 km extent of the model. This error might conceivably be important for applications that require very precise relative heights of points over the whole of Britain. For any application restricted to a region 500 km or less in extent, it is very unlikely to be apparent.
Most importantly, we have the errors that can be incurred when using bench marks to obtain ODN heights. Some OSBMs were surveyed as long ago as 1912 and the majority have not been rigorously checked since the 1970s. Therefore, we must beware of errors due to the limitations in the original computations, and due to possible movement of the bench mark since it was observed. There is occasional anecdotal evidence of bench mark subsidence errors of several metres where mining has caused collapse of the ground. This type of error can affect even local height surveys, and individual bench marks should not be trusted for high-precision work. However, ODN can now be used entirely without reference to bench marks, by precise GNSS survey using OS Net® in conjunction with the National Geoid Model OSGM15™. This is the method recommended by Ordnance Survey for the establishment of all high-precision height control. See the Ordnance Survey coordinate systems page for more details.
The last type of coordinates we need to consider is eastings and northings, also called plane coordinates, grid coordinates or map coordinates. These coordinates are used to locate position with respect to a map, which is a two-dimensional plane surface depicting features on the curved surface of the Earth. The ‘map’ might be a computerised geographical information system (GIS) but the principle is exactly the same. Map coordinates use a simple 2-D Cartesian system, in which the two axes are known as eastings and northings. Map coordinates of a point are computed from its ellipsoidal latitude and longitude by standard formulae known collectively as a map projection. This is the coordinate type most often associated with the Ordnance Survey National Grid.
A map projection cannot be a perfect representation, because it is not possible to show a curved surface on a flat map without creating distortions and discontinuities. Therefore, different map projections are used for different applications. The map projections commonly used in Britain are the Ordnance Survey National Grid projection, and the Universal Transverse Mercator projection. These are both projections of the Transverse Mercator type. Any coordinates stated as eastings and northings should be accompanied by an exact statement of the map projection used to create them. The formulae for the Transverse Mercator projection are given on the Transverse Mercator projections page and the parameters used in Britain are on the 'Datum, ellipsoid, and projection information page'.
In geodesy, map coordinates tend only to be used for visual display purposes. When we need to do computations with coordinates, we use latitude and longitude or Cartesian coordinates, then convert the results to map coordinates as a final step if needed. This working procedure is in contrast to the practice in geographical information systems, where map coordinates are used directly for many computational tasks. The Ordnance Survey transformation between the GNSS coordinate system and the National Grid works directly with map coordinates – more about this on the geodetic transformations page.
datum
Terrestrial Reference System (TRS)
The set of parameters which defines the coordinate system and states its position with respect to the Earth’s surface.
datum realisation
Terrestrial Reference Frame (TRF)
The infrastructure of ‘known points’ that makes the coordinate system accessible to users
Type of coordinates
The way we describe positions in the coordinate system
