The epoch is part of the coordinate

Three of a datum’s four commitments are about geometry: a shape, a placement, a set of marked points. The fourth is not. It is a date, and it became necessary the moment measurement got good enough to see that the marked points are not where they were.

The claim #

Plate motion is not a correction to a coordinate. It is a property of the ground the coordinate names, and it runs at between ten and seventy millimetres a year, which is above the tolerance of ordinary survey work within about two to five years and above the tolerance of any legal boundary description within a decade.

So a coordinate with no epoch on it is not slightly imprecise. It is a claim about a moving object with the time left out.

What was computed, and how #

Rigid-plate kinematics is one line of vector algebra and the rest is arithmetic.

A plate is modelled as a rigid cap rotating about an axis through the centre of the Earth. Its motion is fully described by one vector $\boldsymbol{\Omega}$ — the rotation vector, whose direction is the axis and whose magnitude is the rate. The velocity of a point at position $\mathbf{r}$ is then

$$\mathbf{v} = \boldsymbol{\Omega} \times \mathbf{r}$$

and that is the whole model. The rotation vectors are published inputs — a plate model is a fit to observed station velocities, and this site does not fit one — and they are treated the way an ellipsoid’s $a$ and $f$ are treated everywhere here: quoted, with everything downstream computed. They arrive in milliarcseconds a year, which converts to radians a year by a factor of $4.848 \times 10^{-9}$.

Resolving the result into the local horizontal frame gives an east and a north component, and their magnitude is the speed:

Six places, six plates, and not one of them is stationary. The slowest of them moves a quarter of a metre in a career.

The claim that can fail, and does not #

A model is worth what its refusals are worth, so the rigid rotation is given something it must satisfy: a rigid body deforms nothing.

The check runs each plate’s rotation forward by a million years — using the exact Rodrigues rotation rather than the first-order velocity, so that a long baseline in time cannot be mistaken for a deformation — and requires the chord between two markers on the plate to be unchanged to a part in $10^9$. It is, for all six.

That is not a test of geology. It is a test of the arithmetic: an error in the cross product, or in the construction of the local east–north–up frame, would show up as strain that is not in the model. Requiring the strain to be zero is how the velocity field is known to be a rotation rather than something that merely looks like one on a map.

The vertical is small and is not zero #

One number in the computation came out unexpectedly, and it turned out to be right for a reason worth keeping.

A rigid rotation about the Earth’s centre gives a velocity perpendicular to the position vector, so the motion should be purely horizontal and the vertical component should be exactly zero. It is not: it comes out at about 0.17 millimetres a year at Sydney, which is a thousandth of the horizontal speed.

The reason is that “perpendicular to the position vector” and “horizontal” are different things on an ellipsoid. Local up is the ellipsoid normal, and the normal does not pass through the centre — that is the entire difference between geodetic and geocentric latitude, which is up to 11.5 arcminutes. So a perfectly rigid rotation of a perfectly rigid plate shows an apparent vertical velocity of the horizontal speed times the sine of that angle, which is the tenth of a millimetre a year that appeared.

A check demanding exactly zero would have been demanding the wrong thing, and would have been “fixed” by moving to a spherical Earth — which would have hidden a real property of the coordinate system inside a simplification. The assertion instead requires the vertical to be under one per cent of the horizontal and above a floor, so that it cannot silently become zero either.

The velocity is a field, and its shape is a rotation #

That caution is not pedantry here. A velocity field is a set of directions, and what survives a change of coordinates is the standing question of this collection: the speed of a point is a property of the plate, and the bearing shown on the paper is a property of the plate and the projection together. Two maps of the same plate motion can show arrows pointing measurably differently, and neither is wrong.

The bands the arrows are drawn along are computed rather than outlined. A rigid rotation is fastest on the great circle ninety degrees from its own pole, so the band’s position is a property of the rotation vector. Drawing plate outlines instead would mean holding a boundary dataset, which this site does not — for the same reason it holds no coastline, set out in measuring instead of naming: the outline would be somebody else’s generalisation and the reader could not tell how much of the picture was it.

What a country does about it #

There are three strategies in use and they are genuinely different, rather than three names for the same thing.

Freeze the datum at an epoch. Australia’s GDA94 fixed its coordinates at 1 January 1994. The advantage is that a coordinate never changes: a boundary surveyed in 1996 has the same numbers in 2026. The cost is that the datum drifts away from the global frame at 57 millimetres a year, so by 2020 the offset was 1.8 metres — at which point GDA2020 was adopted and every coordinate in the country changed.

Track the global frame. North America’s newer frames tie to the global reference frame and quote coordinates at a stated epoch with a velocity, so a user computes the position at the date they need. The advantage is that satellite positioning agrees with the published coordinates directly. The cost is that a coordinate is now a function rather than a number.

Use a plate-fixed frame. Europe’s ETRS89 is defined to rotate with the Eurasian plate, so coordinates within Europe are nearly static while the frame as a whole moves. This is the compromise most large plates have adopted, and its cost is at the plate boundaries, where the ground does not move with the plate the frame is tied to.

Setting the two side by side keeps the sizes honest. Choosing the wrong datum costs hundreds of metres and is a one-off; ignoring the epoch costs a metre or two and grows. Both are silent, and the second is the one that gets worse while a dataset sits in a database being trusted.

Where the model stops #

A plate is not rigid, and the model says it is. Deformation is concentrated near boundaries, and within a few hundred kilometres of an active margin the rigid-rotation velocity can be wrong by a large fraction of itself. Japan, New Zealand, the western United States and the Mediterranean are all places where a plate model is the wrong instrument and a deformation model is needed.

Earthquakes are not in this at all. The 2011 Tōhoku earthquake moved parts of Japan by more than five metres in minutes, and the 2016 Kaikōura earthquake moved parts of New Zealand by six. No amount of steady-rate modelling anticipates that; the response is a new realisation of the datum, which is why several countries have datum versions named after earthquakes rather than after decades.

The rotation vectors are published, not derived. Different plate models — geological ones averaged over millions of years, geodetic ones fitted to a decade of station velocities — give different vectors, and they disagree by a few per cent. What is computed here is everything downstream of a stated vector, and the rigidity check is a check on that arithmetic rather than on the vector.

Nothing here is about the realisation’s own error. The velocities are what a rigid model predicts; the published coordinates of a marker also move when the network is re-adjusted, by amounts that have nothing to do with geology, and that is where a fit leaves residuals again — the difference between two realisations is a rigid motion plus a field.

Vertical motion is real and is not this. Land rises where ice sheets have gone, at up to ten millimetres a year in Scandinavia and Canada, and subsides where water or oil has been extracted. Those are vertical rates comparable to the horizontal ones here and they come from an entirely different mechanism — which is one more reason the third coordinate moves too is a separate essay.

What it does to a grid and to a boundary #

The abstraction has two concrete consequences, and they pull in opposite directions.

A national grid does not move. A projected coordinate — an easting and a northing on a national grid — is computed from a latitude and a longitude by a formula, and the formula has no opinion about time. So freezing the datum freezes the grid, and every map, every title deed and every asset database stays consistent with itself indefinitely. That is worth a great deal and is why the freeze strategy is popular.

Satellite positioning does move. A receiver computes its position in the global frame at the current instant, which is not the frozen datum’s epoch. The gap grows at the plate rate, so a surveyor with a frozen national datum and a modern receiver has to apply a correction that gets larger every year — and the correction is a rigid motion, so it looks exactly like the datum shifts of the seven parameters, with the difference that this one has a time in it.

The two together explain the shape of every national decision here. A country freezes until the accumulated gap becomes larger than the cost of re-coordinating everything, then re-freezes. Australia’s interval was twenty-six years and 1.8 metres. The next one will be shorter, because the tolerance people work to keeps falling while the plate keeps moving at the same speed.

The generalisation #

A measurement of a moving thing needs a time as well as a value, and the reason this is worth saying is that whether something counts as moving depends entirely on the precision available.

For two thousand years the ground was stationary, because nothing could measure it well enough to be wrong. Triangulation could not see 50 millimetres a year across a continent; the systematic errors of the network were larger than the signal by an order of magnitude, so the assumption of a static Earth cost nothing and was therefore correct in the only sense that matters to an engineer. Space geodesy in the 1980s changed the precision by two orders of magnitude and the assumption became the largest term in the error budget. The same thing happened one layer down when the ellipsoid stopped being optional: the Earth is a sphere, and when it is not is the record of a simplification that was adequate for centuries and then was not.

The general shape: a simplification is not true or false, it is adequate or inadequate, and improving the instrument can make a long-standing simplification into the dominant error without anything about the world having changed.

That has a practical corollary worth stating. When a measurement technique improves by an order of magnitude, the thing that then limits it is usually not a refinement of what was already being modelled — it is something that was not being modelled at all, because it had never been visible. Plate motion is the clean example: it did not become the limiting error by growing.

The failure mode is the one this whole field keeps producing. A coordinate with no epoch does not report its own staleness. It is a number of the right shape, in the right units, with the right number of digits, and it silently accumulates error at a rate its own metadata does not record — which is exactly the shape of the datum problem in what a coordinate refers to, one dimension over.

The same arithmetic over a longer interval #

That last figure is the reason the plate map takes its projection as a parameter. A velocity field drawn on a conformal projection has its directions right everywhere and its lengths wrong by the scale factor, which for Mercator at 60° is a factor of two.

Who found it, and when #

Continental drift was proposed by Alfred Wegener in 1912 and rejected for half a century, largely because he had no mechanism. Sea-floor spreading and the magnetic stripe evidence settled it in the 1960s, and by the late 1970s the geological plate models — reconstructions averaged over millions of years from magnetic anomalies and fault geometry — were mature enough to predict velocities at any point.

Measuring the motion directly took longer. Very long baseline interferometry and satellite laser ranging in the 1980s got the first direct geodetic velocities, and the global positioning system made them routine in the 1990s. The striking result was that the geodetic rates and the geological rates agree, to a few per cent, over intervals differing by six orders of magnitude — the same plates, moving at the same speeds, measured over a decade and over ten million years.

The consequences for coordinates arrived on the same schedule. The International Terrestrial Reference Frame, first published in 1988 and re-realised every few years since, is defined with an epoch and a velocity field because it could not have been defined any other way once the velocities were observable. National datums followed, each choosing one of the three strategies above, and the choices are still being revisited: GDA2020 in 2020, and continuing argument about whether Australia’s next frame should be static at all.

The vocabulary is a good place to end. A frame’s epoch is not when it was published. It is the date at which its coordinates are stated to be correct, and the two can be decades apart — GDA94 was published in 1995 and is correct in 1994, and a modern realisation of it is correct in 1994 too, by construction, for all time.

One consequence is worth stating before the vertical, because it is the way the epoch actually causes trouble. Two datasets are joined — a survey from one decade against a base layer from another — and nothing in either file objects, because both hold coordinates in the same frame and the frame’s name is the same. The discrepancy is the plate velocity times the gap between the epochs: five centimetres a year over twenty years is a metre, uniformly, in one direction, across the whole overlap. It looks like a systematic survey error and it is a date. The tell is that it is uniform: a survey error varies across a job and a plate velocity does not, so a discrepancy that is the same magnitude and the same bearing everywhere in the overlap is evidence about time rather than about measurement. That is a diagnosis anybody can make from the residuals they already have, and it is the one thing that distinguishes this failure from the dozen ordinary ones it resembles.It looks like a systematic survey error and it is a date.

Where this goes next #

That closes the four commitments of a datum: a shape, a placement, a realisation and a date. What has been assumed throughout is that a coordinate has two components and the third is a detail. It is not. Height above what? opens the vertical, where the reference surface is defined by gravity rather than by geometry and cannot be written down as an equation at all.