In the following pages, you will be introduced to the concept of geodetic transformations, essential for converting coordinates between different systems. You will learn about the mathematical operations involved and the importance of an inverse transformation to revert to the original coordinate system.

The transformations between ETRS89 (European TRS 1989, the Europe-fixed version of WGS84) and the various versions of the ITRS (International Terrestrial Reference System) are published by IERS (International Earth Rotation Service) via their Internet site. The easiest way to carry out these transformations is using the ETRS89/ITRS transformation tool on the EUREF Permanent Network (EPN) website – http://www.epncb.oma.be/_productsservices/coord_trans/.

The usual application of these transformations is in dealing with GNSS coordinates determined in a ‘fiducial network’ using IGS permanent stations as fixed control points and using IGS products such as satellite ephemerides, satellite clock solutions and Earth orientation parameters. In this application, the coordinates adopted for the fixed IGS stations should be ITRS coordinates in the same realisation of ITRS as used to determine the IGS precise ephemerides and other products. The ITRF coordinates of the IGS reference stations at the epoch of observation are obtained by multiplying the ITRF station velocity by the time difference between the ITRF reference epoch and epoch of observation and adding this position update to the ITRF reference epoch station positions.

The resulting coordinates of the unknown stations are in ITRS at the epoch of observation. The EPN transformation tool referenced above can then be used to obtain coordinates in a frame of the ETRS89.

A vertical transformation between ETRS89 ellipsoid height and orthometric height is also available from Ordnance Survey. This transformation is based on a gravimetric Geoid model covering Great Britain, Ireland and Northern Ireland aligned with their respective vertical datums. In mainland Britain, OSGM15 therefore gives Ordnance Datum Newlyn (ODN) orthometric heights directly from GNSS survey for direct compatibility with OS mapping, without the surveyor having to visit Ordnance Survey bench marks.

OSGM15 also covers areas of Britain not related to ODN such as the Orkney Islands, Shetland Isles, Outer Hebrides, Scilly Isles and the Isle of Man. Orthometric heights in these areas are related to specific local vertical datums.

In the previous model (OSGM02) all the datums were extended from land to a distance of 10 km offshore and beyond this limit transformation parameters were set to zero. For OSGM15 datums are extended to just 2 km offshore. All areas beyond this are populated with a version of ODN named “ODN Offshore” to signify its lower quality from ODN on land. This approach avoids the arbitrary “cliff” in the transformation data at 10 km but signifies the extended nature (and therefore lower quality) of the datum when used offshore. Away from land “ODN Offshore” values are more closely aligned to the gravimetric geoid rather than the one fitted to the FBMs on land.

The National Geoid Model OSGM15 is a grid of geoid-ellipsoid separation values on the same 1 kilometre resolution grid as the OSTN15 transformation. In the same way as for OSTN15 a geoid-ellipsoid separation value is obtained for any location by bi-linear interpolation. The nominal accuracy of the geoid model is 8 mm rms (root mean square) in mainland UK, better than 2 cm rms for other areas and 3 cm rms for Isle of Man. This transformation is freely available from the OS website. OS recommends the use of the National Geoid Model OSGM15 and OS Net for high-accuracy orthometric heighting, in preference to Ordnance Survey bench marks.

As detailed in section 5.1 the OS Net ETRS89 realisation was originally related to ITRF97 at epoch 2001.553. This original realisation is now designated “OS Net v2001”. In 2016 it was updated to a realisation related to ITRF97 at epoch 2009.756, designated “OS Net v2009”. The average shift between the two sets of coordinates is small (rms of all components ~18mm). A file of the shifts for individual OS Net stations is available on the OS website. The transformation parameters below can also be used (with equation 3 from the 'Helmert datum transformations' page) to move any coordinates related to OS Net v2001 to be compatible with OS Net v2009.

OS Net v2001 to OS Net v2009 Helmert transformation parameters.

There is a range of options for transforming coordinates computed under OS Net v2001 to be more compatible with OS Net v2009. In increasing order of accuracy they are:

1. Apply the 3 parameter transformation given in the table above.

2. Apply a simple mean shift in X, Y and Z based on the nearest n OS Net stations with n based on the most likely number of OS Net stations used in the original processing. The OS Net station coordinate shifts (in X, Y, Z) are in a file on the OS web site. Both v2001 and v2009 coordinates are given in the OS Net coordinate file on the OS web site so shifts can also be computed from there.

3. (Equal accuracy to 4 below). Apply weighted mean shifts based on the nearest n OS Net stations with n based on the most likely number of OS Net stations used in the original processing. Weighting can be based on the distance to the OS Net stations.

4. (Equal accuracy to 3 above). Apply a simple mean shift in X, Y and Z based on the OS Net stations actually used in the processing.

5. Apply weighted mean shifts based on the OS Nets actually used in the processing. Weighting can be based on the distance to the OS Net stations.

6. Re-process the raw GNSS data using the OS Net v2009 coordinates.

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 sections 4 and 5, 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.

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:

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:

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.

The following Helmert parameters transform WGS84 (or ETRS89 or ITRS, the differences are negligible here) coordinates to ‘something like’ OSGB36 and ‘something like’ ODN heights. The error is up to 3.5 metres (95%) both horizontally and vertically. This transformation is for use with equation (3). Note the remarks made about Helmert transformations in section 6.2.

ETRS89 (WGS84) to OSGB36/ODN Helmert transformation parameters.

To cope with the distortions in the OSGB36 TRF, different transformations are needed in different parts of the country. For this reason, the national standard datum transformation between OSGB36 and ETRS89 is not a simple Helmert datum transformation. Instead, Ordnance Survey has developed a ‘rubber-sheet’ style transformation that works with a transformation grid expressed in easting and northing coordinates. The grids of northing and easting shifts between ETRS89 and OSGB36 cover Britain at a resolution of one kilometre. From these grids, a northing and easting shift for each point to be transformed is obtained by a bilinear interpolation. This is called the National Grid Transformation OSTN15, and it is freely available, as an online service, a software package or as raw data for developers, from the Ordnance Survey website.

The National Grid Transformation copes not only with the change of datum between the two coordinate systems, but also with the TRF distortions in the OSGB36 triangulation network, which make a simple datum transformation of the Helmert type limited to applications requiring an accuracy level of only about 3 metres. OSTN15 removes the need to estimate local Helmert transformations between ETRS89 and OSGB36 for particular locations.

Because the National Grid Transformation works with easting and northing coordinates, other ETRS89 coordinate types (3D Cartesian or latitude and longitude) must first be converted to eastings and northings. This is done using the same map projection as is used for the National Grid (see 'Transverse Mercator map projections'), except that the GRS80 ellipsoid rather than the Airy 1830 ellipsoid is used. The parameters and formulae required to obtain these ETRS89 eastings and northings are given in 'Datum, ellipsoid and projection information' ,'Converting between 3D Cartesian and ellipsoidal latitude, longitude and height coordinates', and 'Converting between grid eastings and northings and ellipsoidal latitude and longitude'. After the transformation, the resulting National Grid eastings and northings can be converted back to latitude and longitude (this time using the Airy ellipsoid) if required.

OSTN15 is the definitive OSGB36/ETRS89 transformation. OSTN15 in combination with the ETRS89 coordinates of the OS Net stations, rather than the old triangulation network, define the National Grid. This means that, for example, the National Grid coordinates of an existing OSGB36 point, refixed using GNSS from OS Net and OSTN15, will be the correct ones. The original archived OSGB36 National Grid coordinates of the point (if different) will be wrong, by definition, but the two coordinates (new and archived) will agree on average to better than 0.1 m (0.1 m rmse, 68% probability).

The ways in which two ellipsoidal datums can differ are (i) position of the origin of coordinates; (ii) orientation of the coordinate axes (and hence of the reference ellipsoid) and (iii) size and shape of the reference ellipsoid. If we work in 3D Cartesian coordinates, item (iii) is not relevant (and any position given in latitude, longitude and ellipsoid height coordinates can be converted to 3D Cartesian coordinates using the formulae in 'Converting between 3D cartesian and ellipsoidal latitude, longitude and heigh coordinates'). Working with 3D Cartesian coordinates, therefore, we use six parameters to describe the difference between two datums, which are three parameters to describe a 3D translation between the coordinate origins, and three parameters to describe a 3D rotation between the orientations of the coordinate axes.

It has become traditional to add a seventh parameter known as the ‘scale factor’, which allows the scale of the axes to vary between the two coordinate systems. This can be confusing because the scale factor is nothing to do with the conventional definition of the datum expressed by the six parameters mentioned above. It was introduced to cope with transformations between TRFs measured by theodolite triangulation, which was the standard method before the latter half of the 20th century. These TRFs have the characteristic that network shape (which depends on angle measurement) is well measured, but network scale (which depends on distance measurements) is poorly measured, because distances were very hard to measure accurately before electronic techniques arrived in the 1950s. Strictly speaking, a Cartesian datum transformation has only six parameters, not seven. However, the scale factor is included in the usual formulation of the datum transformation, which is variously known as the ‘Helmert transformation’, ‘3D conformal transformation’, ‘3D similarity transformation’, or ‘seven parameter transformation’. All these terms refer to the same thing.

When this transformation is applied to a TRF, it has the effect of rotating and translating the network of points relative to the Cartesian axes and applying an overall scale factor (that is, a change of size), while leaving the shape of the figure unchanged.

The Helmert transformation can only provide a perfect transformation between perfect mathematical reference systems. In practice, we determine and use transformations between sets of real ground points, that is, between TRFs that are subject to the errors of the real world. The more distortion, subsidence, and so on present in the TRF, the worse the Helmert transformation will perform. Both the parameters of the transformation and the output coordinates will contain errors. Bear this in mind especially when working with pre‑GNSS coordinate sets and orthometric heights over large areas.

The usual mathematical form of the transformation is a linear formula which assumes that the rotation parameters are ‘small’. Rotation parameters between geodetic Cartesian systems are usually less than five seconds of arc, because the axes are conventionally aligned to the Greenwich Meridian and the Pole. Do not use this formula for larger angles – instead apply the full rotation matrices, which can be found in most photogrammetry and geodesy textbooks.

In matrix notation, the coordinates of each point in reference system B are computed from those in reference system A by:

where tX, tY and tZ, are the translations along the X, Y and Z axes respectively in metres rX, rY and rZ are the rotations about the X, Y and Z axes respectively in radians, and s is the scale factor (unitless) minus one. This definition of s is convenient, since its deviation from unity is usually very small. s is often stated in parts per million (ppm) – remember to divide this by a million before using equation 3. Rotations are often given in seconds of arc but must be converted to radians for use in equation (3).

Beware that two opposite conventions are in use regarding the rotation parameters of the Helmert transformation. The form of transformation as given in equation (3) is referred to as the “Position Vector transformation”. It is the form in most common use in Europe (particularly in the oil and gas industry), is used by the International Association of Geodesy (IAG) and recommended by ISO 19111 and is EPSG dataset coordinate operation method code 1033. The form of transformation that uses opposite rotation signs to those in equation (3) is referred to as the “Coordinate Frame Rotation transformation” and is common in the USA oil and gas industry and is EPSG dataset coordinate operation method code 1032. If there is any ambiguity about which convention applies it is best, for safety, to always include a test point with coordinates in systems A and B when stating a transformation.

Sometimes, three-parameter (translation only) or six-parameter (translation and rotation only) transformations are encountered, which omit some of the seven parameters used here. The same formula can be used to apply these transformations, setting parameters not used to zero.

Any ‘small’ Helmert transformation can be reversed by simply changing the signs of all the parameters and applying equation (3). This is valid only when the changes in coordinates due to the transformation are very small compared to the size of the network.

The Helmert transformation is designed to transform between two datums, and it cannot really cope with distortions in TRFs. As we’ve seen, in practice, we want an accurate transformation between the coordinates of points on the ground in the two coordinate systems for the area we’re interested in. If there are large regional distortions present in one or both of the TRFs, a single set of Helmert parameters for the whole coordinate system will not accomplish this. For the transformation from ETRS89 to OSGB36 in Britain, using a single Helmert transformation will give errors of up to 3 m (95%) in plan and 3.5 m (95%) plan and height, depending where in the country the points of interest are. One solution to this is to compute a special set of Helmert parameters for a particular region – this is known as a ‘local transformation’. Alternatively, more complex transformation types that model TRF distortion can be used. An example of this is the National Grid Transformation OSTN15 discussed in 'National Grid Transformations'.

To summarise: For a simple datum change of latitude and longitude coordinates from datum A to datum B, first convert to Cartesian coordinates (formulae in 'Converting between 3D Cartesian and ellipsoidal latitude, longitude and height coordinates'), taking all ellipsoid heights as zero and using the ellipsoid parameters of datum A; then apply a Helmert transformation from datum A to datum B using equation (3); finally convert back to latitude and longitude using the ellipsoid parameters of datum B (formulae in 'Converting between grid eastings and northings and ellipsoidal latitude and longitude') discarding the datum B ellipsoid height. There is a worked example Helmert transformation in the 'Helmert transformation worked example' page.