What is a geodetic transformation?

A geodetic transformation is a mathematical operation that takes the coordinates of a point in one coordinate system and returns the coordinates of the same point in a second coordinate system. Hopefully we will also have an ‘inverse transformation’ to get back to the first coordinate system from given coordinates in the second. Many types of mathematical operation are used to accomplish this task. The term ‘geodetic transformation’ does not indicate a particular type of formula.

To properly understand geodetic transformations, it is essential to understand the relationship between a datum and the TRF that realises the datum. Having looked at several examples of real TRFs in Modern GNSS coordinate system and Ordnance Survey coordinate systems, we know that they are of widely variable quality. Unless a TRF is perfect, the datum it realises is ambiguous. We can think about this in the following way: the given three-dimensional coordinates of any three ground points in the TRF imply the location of the coordinate origin, and the orientation of the coordinate axes, relative to those three points. If we select a different set of three ground points, the origin and orientation implied will be slightly different. This is due to observational errors in the determination of the coordinates of the ground points. Taking the TRF as a whole, the datum that it implies has some degree of uncertainty, because the origin positions implied by different selections of TRF ground points are never in perfect agreement with each other.

The usual situation in geodesy is that we have two TRFs realising two different datums, with a large number of points in common between the two TRFs. That is, a particular ground point will be included in both TRFs, but it will have different coordinates in each. Because the TRFs are not perfect realisations of their respective datums, it is impossible to say exactly what the transformation between the two datums is. We can only give a best estimate of the transformation with a statistical measure of the quality of the estimate.

To clarify this idea, let’s look at an example. To keep things simple, we’ll consider one‑dimensional coordinate systems (The Ordnance Datum Newlyn page is an instance of a real one-dimensional coordinate system). Imagine we have two coordinate systems: A and B, each of which state coordinates for three points called White, Grey and Black. So here we have 100% overlap between the TRFs of the two coordinate systems – all three points appear in both TRFs.

An example of two one-dimensional coordinate systems with three points appearing on both TRFs

Point	Coordinate system A	Coordinate system B
White	4.1	7.3
Grey	9.8	12.7
Black	12.2	15.0

We want to work out a transformation from the datum of coordinate system A to the datum of coordinate system B. This transformation will be a single number t that can be added to any coordinate in system A to give the equivalent coordinate in system B. We don’t have any theoretical datum definitions of these two systems to work from, and even if we did we would not use them (in this example, we don’t even know what the coordinates represent – it doesn’t matter). We want a practical transformation that will convert any coordinate in coordinate system A to coordinate system B. So, we use points that have known coordinates in both systems – in other words, we relate the two datums by using their TRFs.

In this example, we have three points that are coordinated in both TRFs – White, Grey and Black. The best estimate of the transformation t between the frames is simply the average of the transformation at each of these points:

$$t_{white}=white_B-white_A=3.2\\ t_{grey}=grey_B-grey_A=2.9\\ t_{black}=black_B-black_A=2.8\\ so, t = \frac{t_{white}+t_{grey}+t_{black}}{3} = 2.97$$

However, this ‘best estimate’ value for t doesn’t actually agree with the translation at any of the three known points. We need to compute a statistic that represents the ‘amount of disagreement’ with this estimate among the known points. This is the standard deviation of the three individual t values from the average t value:

$$σ_t= \sqrt{\frac{(t_{white}-t)^2+(t_{grey}-t)^2+(t_{black}-t)^2}{3}} = 0.17$$

So, our estimate of t would be written t = 2.97 ±0.17, that is, t could quite probably be anything from 2.80 to 3.14 based on the information we have available. It is common to state a range of three standard deviations as a likely range within which the true figure lies, that is, between 2.46 and 3.48.

So, what is the transformation between datum A and datum B? We don’t know exactly, and we can never know exactly. But we have a best estimate and a measure of the likely error in the estimate.

The point of all this is that since TRFs aren’t perfect, neither are transformations between the datums they realise. In practice, this means that no exact transformation exists between two geodetic coordinate systems. People often assume, quite reasonably, that the transformation between two ellipsoidal datums can be exactly specified, in the way that a map projection can. In theory, this is true, but what use would it be? What you really want to know is what the transformation is for real points on the ground – so you are actually asking a question about TRFs. And the answer to such a question is always an approximation.

The degree of error in a geodetic transformation will depend on the patterns of errors present in the TRFs A and B (which are often characteristic of the methods used to establish the TRF in the first place), and also on how carefully the transformation has been designed to take account of those errors. For example, TRFs established by triangulation generally contain significant errors in the overall scale of the network and often, this scale error varies in different parts of the network. Therefore, a real transformation is likely to represent not only the difference between geodetic datums, but also the difference between the TRFs that realise those datums due to errors in the original observations.