A published coordinate is a result

A survey crew sets up over a national control mark and observes its position with equipment good to a couple of centimetres. The answer disagrees with the published coordinate by something over a hundred metres.

The instrument is fine. The observation is fine. The published value is fine. What the crew has measured is the gap between two definitions of where the mark is, and definitions do not have error bars.

What a control coordinate actually is #

A published control coordinate is not an observation of a mark. It is the output of an adjustment — a computation that took thousands of observations across a network, imposed the condition that they be mutually consistent, and produced one coordinate per station.

That has two consequences and both are usually forgotten.

No individual station’s coordinate is what anybody observed. Each is the network’s compromise, and the compromise was struck under whatever constraints the adjustment imposed — which stations were held fixed, which observations were weighted how, and what model of the Earth was in use.

Once published, it is held fixed by convention. A national datum is defined by its published coordinates. OSGB36 is not a set of parameters from which coordinates can be derived; it is the set of coordinates the 1936–62 retriangulation produced, and the parameters are a description fitted to them afterwards. That is what a realisation means, and it is the subject of what a coordinate refers to.

The adjustment is where the coordinates come from #

It is worth being concrete about what an adjustment does, because the word suggests a tidying-up and the operation is nothing of the kind.

A national network has thousands of stations and tens of thousands of observations — angles, distances, azimuths — and the observations are mutually inconsistent, because every measurement is. The adjustment finds the set of coordinates most consistent with all of them at once, subject to whatever was held fixed, and publishes that set.

So the published coordinate of any one station depends on every other station in the network. Re-observe one mark and the answer differs from its published value not only because of the two terms above but because the published value was pulled by observations at stations hundreds of kilometres away. The coordinate is a property of the network, not of the mark.

That is also why a network cannot be improved piecemeal. Better coordinates for a few stations, computed independently, would be inconsistent with their neighbours — so an improvement means readjusting the whole thing and republishing everything, which is the expense the next section is about.

The adjustment itself is a least-squares computation and this collection does not own it: numerical-linear-algebra.com owns least squares as a numerical procedure and normaldistribution.xyz owns it as an estimator. What is owned here is the consequence — that a published coordinate is an output rather than an observation — which is a fact about coordinate systems rather than about estimation.

What the network’s own inconsistency leaves behind is a residual field the parameters cannot express — structure rather than noise, which is why a transformation between two realisations needs a correction grid on top of its formula.

The two terms in the disagreement #

The realisation, 106 metres. The published value is on a national datum fitted to one country in the 1930s; the observation is on a global datum realised by satellites. The mark has not moved. The reference surface underneath it has been replaced.

The plate motion, 750 millimetres over thirty years at 25 millimetres a year. Here the mark genuinely has moved, along with the continent it is set in, and the published coordinate records where it was at the epoch of its adjustment. That is the epoch is part of the coordinate.

Both are far larger than the observation. The assertion the figure carries says so in both directions: the realisation term must exceed the drift by more than a factor of ten, and the drift must itself exceed fifty millimetres — so that neither term can be dismissed as noise and the larger one is identified as a change of definition rather than a physical movement.

Which of the two is a measurement #

Neither, and the distinction between them is still worth making.

The plate motion is a fact about the world that the coordinate system fails to track. The mark is somewhere different from where it was; a perfect coordinate system would say so, and the published one does not because it was fixed at an epoch.

The realisation change is a fact about the coordinate system with no counterpart in the world at all. Nothing moved. The same ground, the same mark, the same monument, and a different set of numbers because a different reference surface was adopted.

Reporting them as one number — “the published value is 106 metres out” — merges a real displacement with a change of convention, and the merged quantity means nothing. That is the same failure two grids over the same ground identifies when the false origin is folded into a datum comparison, one level up the stack.

Why the coordinates are not simply corrected #

If the published values are known to disagree with reality by a hundred metres, the obvious remedy is to republish them.

That has been done, in most countries, and doing it is enormously expensive in a way that has nothing to do with the surveying. Every legal boundary, every utility record, every planning consent, every as-built drawing and every land registry entry expressed in the old coordinates would have to be reinterpreted. The alternative — publishing a transformation and leaving the old values alone — costs nothing and is what almost everybody does.

So a country typically maintains two systems: the historical one that all the records are in, and the modern one that the satellites produce, with a published transformation between them. The transformation is a seven-parameter fit plus a correction grid, and the fit’s residuals are a field rather than noise — where a fit leaves residuals measures that, and when a formula is not enough is why the grid exists.

The historical coordinates are not wrong and are not going to be replaced. They are the definition of a datum that a century of records depends on, and the modern system’s job is to be relatable to them rather than to supersede them.

What the disagreement looks like at other places #

Britain’s hundred metres is not a universal figure and the shape of the disagreement varies more than its size.

A country whose historical datum was fitted well and whose plate moves slowly — most of western Europe — has a large realisation term and a small drift term, which is the case computed here. A country on a fast-moving plate has the opposite balance over a long enough interval: Australia moves about 70 millimetres a year, so a coordinate fixed forty years ago is nearly three metres out from ground motion alone, and the country has redefined its datum twice in response.

The extreme case is a country astride a plate boundary, where the two halves move relative to each other and no single rigid-plate correction can be right for both. New Zealand’s answer is a deformation model rather than a datum: the published coordinates come with a field describing how the ground moves, so a coordinate and a date can be converted to a coordinate at another date.

That is the logical end of this essay’s argument. Once it is accepted that a published coordinate is a definition attached to an epoch, the natural next step is to publish the rule for moving between epochs — which turns a datum from a set of numbers into a model, and turns the question “where is this mark?” into “where was it, and when?”

The balance between the two terms differs by country for a reason that has nothing to do with either practice. Each plate carries its markers at its own rate and in its own direction, so the drift term over a working career runs from centimetres to metres depending on where the work is.

The consequence for a job #

A job that ties to control has to decide what it is tying to, and the decision is not about accuracy.

Tie to the historical system if the work has to agree with historical records — a boundary, an easement, an extension to an existing structure. The coordinates will disagree with a satellite fix by a hundred metres and that is correct.

Tie to the modern system if the work has to agree with satellite positioning, with other modern work, or with anything international. The coordinates will disagree with the old maps and that is correct too.

Never mix them, because the difference is a hundred metres and is invisible in the numbers — the argument of what a grid is made of, and the reason the declarations have to travel with the data.

The digits, which claim more than they know #

A published coordinate is quoted to a millimetre or better, and the precision of the digits is not a statement about the accuracy of the position.

It is a statement about the adjustment. The computation produced a number, the number has digits, and truncating them would introduce inconsistency between stations — so the digits are published because they are needed for the coordinates to remain mutually consistent, not because anybody believes the mark is located to a millimetre in any absolute sense.

That distinction is easy to state and almost never made in practice, and it is the same one a grid reference makes honestly and a decimal coordinate does not. SU 387 145 is a hundred-metre square and says so; 406,788.312 asserts a millimetre whether or not one was ever observed. The convention that makes a coordinate easy to compute with is the convention that strips out its own claim about precision, which is a grid has an origin that is not there noticed from a different angle.

So there are two independent things a reader might want from a published coordinate — how consistent it is with its neighbours, and how close it is to the ground — and the digits answer the first while appearing to answer the second.

What this does to the checks #

Every check in this field compares something against something else, and this essay is about what happens when the reference is itself a convention.

A traverse closing on published control is checked in position, orientation and scale — the remedy a traverse must close recommends for errors a loop cannot see. But the check is against a definition, so what it verifies is agreement with the definition rather than agreement with the ground. A survey that closes perfectly on OSGB36 control has demonstrated that it is consistent with a 1930s adjustment, which is exactly what is wanted for a boundary and is not the same as being right.

That is not a criticism. It is the point of a datum: a shared convention that everybody’s work agrees with is more useful than everybody’s work being independently more accurate, because the purpose of a coordinate is to be compared.

What was computed, and how #

The realisation term, from the published seven-parameter transformation between the national and global datums at the stated point.

The drift term, from the plate’s rotation vector at that point, integrated over the stated interval — the machinery of the epoch is part of the coordinate.

The observation, as a stated instrument capability rather than a computed quantity, because it is the one number here that is a property of equipment.

Two assertions, described above, one bounding the ratio and one putting a floor under the smaller term.

Where the model stops #

The drift is a rigid-plate model. Real deformation includes strain near plate boundaries, post-glacial rebound and local subsidence, none of which a rotation vector carries. Britain is rising in the north and sinking in the south at a few millimetres a year from rebound alone, which is comparable to the horizontal term over a career and is not in this computation.

The realisation term is the smooth part. The full difference includes the correction grid, worth a metre or two, and that part varies across the country in a way the seven parameters cannot express.

And the observation is idealised. A real re-observation carries centring error, antenna phase-centre variation, atmospheric modelling error and the accumulated uncertainty of whatever reference stations it was tied to. Twenty millimetres is a good figure for a careful job and it is not a lower bound on anything.

One asymmetry between the two terms is worth keeping, because it decides which of them a country can do anything about. The plate motion accumulates whatever anybody does: it is a rate, so waiting makes it larger and no adjustment removes it. The realisation gap does not accumulate — it was fixed on the day the modern datum was adopted and it has stayed the same size ever since. So a country that republishes has converted a permanent hundred-metre offset into a growing millimetres-a-year one, and a country that does not has kept the offset constant. Neither choice makes the disagreement go away; they choose which of the two shapes it has.

The generalisation #

A published reference value is a definition being quoted, and the difference between it and a fresh measurement is not an error in either.

That structure appears wherever a field maintains a standard: a reference mass, a reference frequency, a reference epoch. In every case the published value is right by construction — it is what the standard says — and a better measurement does not make it wrong but instead raises the question of when to redefine, which is a decision about disruption rather than about accuracy.

Cartography’s version is unusually stark because the disagreement is a hundred metres, which is large enough that everybody meets it, and because the records depending on the old definition are legal rather than merely scientific. A physicist redefining a unit inconveniences some laboratories. A country redefining its national grid touches every property boundary in it.

What a picture can and cannot witness #

A control coordinate is held fixed by convention and re-observing disagrees. A related question is what a drawn coordinate can be evidence of, and the answer is a floor set by the medium rather than by the observation.

A screen map’s pixel is 23.8 metres of ground at zoom 12 and 1.5 at zoom 16, and every vertex is moved to the nearest one. A centimetre-level coordinate survives being drawn only at zoom 23 — four levels past the bottom of any scheme in service, and 70 trillion tiles. So a picture cannot witness a position at survey precision at any zoom anybody serves, whatever the survey did.

Two things a picture cannot witness are worth naming, because they bound what any figure here could have shown. The medium has a floor: at zoom 16 a pixel is 1.5 metres of ground and rounding moves a vertex by up to a metre, which is two orders of magnitude above the coordinate this essay is about. And a published coordinate carries a realisation, which no drawing of it can display — the same numbers on two datums are a hundred metres apart, and that disagreement is definitional rather than an error.

Who realised it, and when #

The problem arrived with satellites, and arrived very suddenly.

Before the 1980s a national datum was the only thing there was: a country’s coordinates were defined by its own network, there was no independent way to check them, and the question of whether they were “right” had no operational meaning. Doppler positioning in the 1970s and then GPS in the 1980s produced, for the first time, coordinates on a globally consistent frame that could be compared against every national datum simultaneously — and the comparison revealed offsets of tens to hundreds of metres everywhere.

None of those offsets was a discovery of error. Every national network had been internally excellent; what the satellites revealed was that internal excellence and global consistency are different properties, and that a century of surveying had optimised the first because the second was not measurable.

That is the whole of this field in one observation, and it is where this ladder ends: the thing a survey checks itself against determines what its checking can find, and for most of the history of the subject the only available reference was itself.

Where this goes next #

The practice field’s two ladders end here — the grid as an object with declarations, and the chain that turns a measurement into one of its coordinates. What both rest on is the datum underneath, which is what a coordinate refers to, and the reason a datum can never be finished is a datum is fitted to a region.