Datum shifts dwarf projection errors

Everything else on this site measures a projection’s error and reports it. This essay is about a larger error that no projection is responsible for, that is present in almost every coordinate anybody handles, and that is essentially never mentioned in an argument about projections.

What a datum is #

An ellipsoid is a shape. A datum is that shape placed — positioned and oriented relative to the Earth, so that a latitude and longitude on it name a point on the ground.

The two are separate decisions and both are required. Airy 1830 is an ellipsoid; OSGB36 is that ellipsoid positioned to fit Britain, with its centre about seven hundred metres from the Earth’s centre of mass. WGS84 is a different ellipsoid positioned at the centre of mass, because that is where a satellite orbit is naturally referenced.

So a coordinate pair means nothing without three things: the angle convention (geodetic, which is safe to assume), the ellipsoid, and the datum. The first is universal. The other two are not, and they are not in the numbers.

The size of it #

The site computes the shift through the published seven-parameter transformations:

Set that against the projection errors argued about elsewhere. UTM’s worst scale error across a zone is 981 parts per million — under a metre per kilometre. At the scale of a survey site, a hundred metres across, the projection contributes about ten centimetres. The datum contributes a hundred and sixty metres.

The ratio is three orders of magnitude, in favour of the thing nobody discusses.

Why the shift is not a constant #

The obvious model of a datum shift is a translation: two ellipsoids offset from each other, so every coordinate moves by the same amount in the same direction. That is the three-parameter model, and it is what the NAD27 numbers above use.

It is not adequate, and the reason is that two ellipsoids fitted to different regions also differ in size, in shape and in orientation. The general transformation has seven parameters — three translations, three rotations and a scale change — and it acts on the geocentric Cartesian triple rather than on latitude and longitude:

$$\mathbf{X}_{\text{WGS84}} = \mathbf{T} + (1+s)\,\mathsf{R}\,\mathbf{X}_{\text{local}}$$

For OSGB36 the translation is 446, −125 and 542 metres, the rotations are under a second of arc, and the scale change is −20.5 parts per million. The rotations look negligible and are not: a second of arc across the Earth’s radius is thirty metres.

That is why the OSGB36 shift is 125 metres in London and 89 in Edinburgh. The transformation is not a translation, so it does different things in different places, and quoting one number for a country is already wrong by tens of metres.

The inverse is not the negative #

A detail that cost this site a centimetre and is worth recording, because it is the kind of error that survives.

The obvious inverse of the transformation above is to negate all seven parameters and apply the same formula. That is wrong, because the scale and the translation do not commute: the forward map scales the rotated point and then translates it, so the inverse must translate back before dividing by the scale.

Doing it the obvious way leaves an error of order $s\,|\mathbf{T}|$ — twenty parts per million of five hundred metres, which is about a centimetre. The site’s round-trip assertion caught it at 1.3 cm.

A centimetre is a good size of error to catch this way. It is far too small to notice in any map, far too large for a land registry, and it would have sat in the library indefinitely if forward-then-inverse had not been required to return the starting point.

With the inverse done in the right order the round trip closes to 1.1×10⁻⁴ metres, and the residual is the second-order term in the linearised rotation matrix — a tenth of a millimetre, which is what a first-order rotation of a body six thousand kilometres across is worth.

The transformation cannot be exact #

Worth being clear about, because a seven-parameter fit looks authoritative and is an approximation of a specific kind.

A Helmert transformation is a rigid motion plus a uniform scale. It has seven degrees of freedom for a whole continent’s worth of survey marks, and the two datums it connects were each fitted to their region by a network of triangulation with its own accumulated distortions.

Those distortions are not rigid. A nineteenth-century triangulation network drifts by metres across a continent in a pattern with no formula, so no seven-parameter transformation can fit better than the network’s own internal error.

The evidence is what the surveys actually publish. The United States does not use a Helmert transformation between NAD27 and NAD83 at all; it uses NADCON, a grid of interpolated corrections, precisely because no rigid transformation fits. Britain publishes OSTN, a similar grid, alongside the seven-parameter values.

So the numbers in the table above are themselves approximate — good to a few metres where the transformation was fitted, and worse elsewhere. The site’s NAD27 figures show this plainly: the three-parameter model gives 95 metres in Seattle and 49 in Denver, and the true shifts from the correction grid differ from both by several metres. A transformation quoting six significant figures is quoting the parameters, not the accuracy.

How a datum was actually realised #

Worth knowing, because it explains why the old datums are shaped the way they are and why no transformation fits them exactly.

Before satellites, a datum was defined by an origin point and an orientation. A single station — Meades Ranch in Kansas for NAD27, Herstmonceux for OSGB36’s predecessor, Potsdam for the German datum — had its latitude and longitude determined astronomically, its azimuth to a second station measured, and the ellipsoid was then declared to be tangent to the geoid there and oriented accordingly.

Everything else followed by triangulation. A network of measured angles was propagated outward from the origin, and every coordinate in the country was computed through that network. So a national datum is not a surface; it is one point plus a computation, and the computation carries the accumulated error of every angle in the chain.

That is why the old datums fit their own region beautifully and nothing else. The ellipsoid was chosen and placed so that the geoid separation was small near the origin, which is a local best fit by construction, and a hundred metres out at the far end of the continent.

It also explains why the transformations have rotations in them. Two datums oriented by astronomical observations at different origins are not merely offset; their axes point in slightly different directions, because each was aligned to a local vertical that is itself deflected by the mass distribution underneath it. A second of arc of deflection at the origin becomes thirty metres of discrepancy across the Earth’s radius, and the deflection of the vertical at a given station is routinely several seconds.

Height is a third datum #

The essay has been about horizontal position, and there is a separate and larger problem in the vertical.

Height above the ellipsoid is what a satellite receiver computes: a purely geometric quantity, the distance along the normal from the reference surface. Height above sea level is what everything else means: a quantity referred to the geoid, the equipotential surface gravity actually defines.

The two differ by the geoid separation, which ranges from about −106 metres near southern India to +85 metres near Iceland. Those are not errors. They are the amount by which the Earth’s gravity field departs from the smooth ellipsoid, and no formula produces them — the separation is given by a model with thousands of coefficients, fitted to gravity measurements.

So a receiver reporting “height 120 m” and a map contour reading “height 175 m” at the same point can both be right. The difference is that one is measured from an ellipsoid and one from the geoid, and the gap between them is a third piece of missing metadata alongside the ellipsoid and the datum.

The consistent lesson is the one this whole essay is about: a number describing a position is a measurement plus a reference, and the reference is never in the number.

Where it goes wrong in practice #

Three failure modes, and the order is by how quietly they fail.

Data from two sources overlaid. Two datasets on different datums line up almost perfectly — same shapes, same relative geometry — and are displaced from each other by a hundred metres. Nothing looks broken. Boundaries that should coincide do not, and the discrepancy is usually blamed on digitising error.

An old map georeferenced to a new one. A scanned sheet on a national datum, fitted to a modern basemap, absorbs the datum shift into whatever affine transformation the georeferencing used. The result is a map that fits at the corners and is out in the middle.

A coordinate transcribed without its datum. The most common of the three. A latitude and longitude in a spreadsheet, a report, a paper. There is nowhere in the notation to put the datum, so it is carried in prose or not at all, and in a decade it is gone.

Why the projection gets the attention #

The asymmetry is worth explaining rather than merely complaining about.

A projection’s error is visible. It shows in the shape of a continent, in the size of Greenland, in whether a route looks straight; it can be argued about from a picture, and it has been argued about from pictures for fifty years.

A datum’s error is invisible. It displaces everything on a map by the same amount, so the map looks entirely correct — every shape right, every relative position right — and the whole thing is a hundred and sixty metres from where it says it is. There is no picture in which that is visible, because the reference the eye would compare against is on the same sheet and has moved too.

Which is the same asymmetry as areal against angular distortion, one level up. The failure that looks like something gets the argument; the failure that looks like nothing gets the damage.

What GPS changed #

The situation has improved and it is worth saying how, because it is not the way most people assume.

Satellite positioning did not make datums unnecessary. It made one datum overwhelmingly common: a receiver reports WGS84 because that is the frame its orbits are computed in, so most new data is on a single global datum and the mismatch problem largely does not arise for anything collected since about 1995.

What it did not do is fix the old data. Every map, boundary, title deed and survey record from before then is on a national datum, and the conversion is still needed and still approximate.

It also introduced a new version of the problem. WGS84 is realised through a series of reference frames that have been revised several times, and it is aligned to the ITRF, which is fixed relative to the Earth as a whole rather than to any plate. Plates move at up to about ten centimetres a year, so a point in Australia has drifted about a metre and a half since 2000 relative to a globally fixed frame.

Australia’s response was to define a datum with an epoch and a plate motion model. A coordinate there is now incomplete without a date, which is the same lesson arriving again with a different missing field.

The rule #

Short, and it follows from everything above.

Record the datum with the coordinate, always. It costs a field. Without it a coordinate is uncertain by a hundred metres and there is no way to recover the information later.

Convert with a grid where one exists, and with the seven-parameter transformation only where one does not, knowing it is good to a few metres rather than to the six figures its parameters are quoted in.

Do not argue about projections in a dataset whose datum is unknown. The projection contributes centimetres and the unknown contributes a hundred metres, and improving the smaller one is not progress — the same order-of-magnitude reasoning that decides whether a projection choice can matter at all.

A datum shift is tens or hundreds of metres and a projection’s own corrections are centimetres. The second kind is worth knowing precisely because the first is so large: a correction nobody applies is indistinguishable from a datum nobody declared.

What was computed here #

The shifts are computed through the full seven-parameter Helmert transformation, applied to geocentric Cartesian coordinates, with the geodetic-to-Cartesian conversion and Bowring’s iteration back.

Three assertions hold it. The Cartesian conversion must round-trip across three ellipsoids and three heights, which it does to 1.9×10⁻⁹ metres — Bowring’s iteration is the piece most likely to be silently wrong, because it converges to something from almost any starting point. The Helmert transformation must round-trip, which it does to 1.1×10⁻⁴ metres once the inverse is done in the right order and did not before. And the resulting shifts must exceed twenty metres somewhere, which is the essay’s whole claim stated as a test that could fail.

The published transformation parameters are quoted, because they are definitions rather than derivations: a datum transformation is a fit somebody performed and published, and the numbers are the fit rather than a property of the Earth. The site does not attempt to re-derive them, and the essay says what they are worth.

What the pictures cannot show #

The shift itself, on a map. Displacing a whole map by a hundred metres produces a map that looks identical, which is the entire difficulty — the figure here is a bar chart because a bar chart is the only honest picture of an offset that has no visible signature.

The figures also cannot show the transformation’s own error. The bars are computed to millimetres and are accurate to a few metres, and the gap between those two numbers is the difference between a parameter and a measurement.

What the shift is made of #

The decomposition is worth having beside the total, because it separates two things the single number hides. The placement — the three translations — is a property of the datum’s definition and would be the same if the survey had been perfect. The rotations and the scale are a property of the realisation, and they are the residue of eighty years of triangulation. The seven parameters, and what each one does takes them apart; where a fit leaves residuals measures what even seven of them cannot carry.

The plausible size, which is why it survives #

This essay’s comparison is with projection error. The applied field makes a different comparison, with the other ways a coordinate can be misread, and the ordering explains why this failure outlives all of them.

Reading degrees as metres is wrong by five thousand kilometres and is fixed the day it happens. Reversing the axes is wrong by thousands of kilometres and is caught by a glance at a map. A datum confusion is wrong by 103 to 216 metres depending on where the point is — inside the range a reader would accept, self-consistent under every internal check, and detectable only against something external.

Eight hundred kilometres from where this essay measures them the same three shifts have reordered themselves, and OSGB36 has gone from the smallest to the largest at 216 metres — because a datum fitted to Britain has no obligation to behave in the Alps.

Even the loudest of those failures has one line where it hides. A five-kilometre band about the diagonal on which latitude and longitude are numerically equal is still a few hundredths of a degree wide, and inside it a swapped pair looks right — which is the point: obviousness is a property of most of the Earth rather than of the mistake.

Who found it, and when #

Every national survey of the nineteenth and early twentieth centuries defined its own datum, because it had to: a triangulation network has to be tied to the ellipsoid somewhere, and the only available tie was an astronomical observation at an origin point, with the ellipsoid then oriented to fit the network as well as possible.

OSGB36 dates from the retriangulation of Britain in 1936, on Airy’s 1830 ellipsoid; ED50 was assembled after the Second World War from the national networks of western Europe; NAD27 was adjusted in 1927 from an origin at Meades Ranch, Kansas.

The satellite era produced the first genuinely global datums, of which WGS84 in 1984 is the one that stuck. The transformations between the old and new are all later, all fitted, and all published with more digits than they are worth — which is the shape of the whole problem in one observation.

Where this goes next #

The angle the datum places is geodetic against geocentric latitude. The grid built on top of both is UTM and the zone system. And the body all of it references is the Earth is a sphere, and when it is not.