We need a datum
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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.
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.
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.