Two grids over the same ground

A survey mark at 1.9°W, 52.5°N has two coordinates in common use. On the British National Grid it is 406,788 east and 289,167 north. On UTM zone 31 it is 167,478 east and 5,827,940 north.

Subtracting them gives 5,544 kilometres, which is the sort of number that starts an argument about datums. It is almost entirely not about datums, and separating out what it is about produces a fact about transverse Mercators that this essay did not set out to find.

The three terms #

Where each grid puts zero, 5,544 kilometres. Four parameters between them: a different central meridian (2°W against 3°E), a different true origin parallel (49°N against the equator), and different false easting and northing constants. Every one of the four moves every coordinate and not one distance.

The datum, 106 metres. OSGB36 is fitted to Britain on the Airy 1830 ellipsoid; UTM zone 31 here is on WGS84. The same latitude and longitude, read against the two, name ground 106 metres apart. This is the only term in which anything on the ground has moved.

The scale factors, half a metre. 0.9996012717 against 0.9996 — a difference in the sixth decimal place, which at this point in the country is 50 centimetres.

And that third term is the whole of the geometry.

What a reader usually does with the number #

The failure this essay exists to prevent has a standard shape, and it is worth walking through because it is entirely reasonable at every step.

A dataset arrives with coordinates that do not line up with an existing one. The offsets are computed and come out in the thousands of kilometres, consistently. Somebody recognises that as far too large to be a datum problem and correctly concludes that the two files are in different coordinate systems.

So far so good. The next step is where it goes wrong: the second file is reprojected into the first system, the offsets drop to something small, and the job proceeds. Whether the reprojection handled the datum is now invisible, because the residual after a projection-only conversion is a hundred metres and the residual after a full conversion is a metre or two — and against an original discrepancy of thousands of kilometres, both look like success.

The large convention masks the small substance. Once the zero conventions are subtracted the remaining terms are 106 metres and half a metre, and only somebody who knows to expect the first will notice that it is still there.

Two orders of magnitude, in the wrong direction #

The largest term is the one that means least, and this is the finding worth carrying away.

Somebody comparing two coordinate systems by subtracting a coordinate gets thousands of kilometres and concludes the systems are wildly incompatible. What they have measured is the distance between two arbitrary decisions about where zero goes.

Somebody who has heard that the false origin is a convention subtracts that out and still has hundreds of kilometres, because the central meridian and the true origin’s parallel are two more conventions of the same kind and are much larger.

Only somebody who holds the parameters fixed one group at a time recovers the actual sizes, and the sizes carry the lesson: the term everybody argues about is not the largest and is not the smallest. The datum’s 106 metres sits between five thousand kilometres of bookkeeping above it and fifty centimetres of geometry below.

How the decomposition is done #

The three terms do not add to the total, and the reason is worth stating rather than hiding.

Each term is isolated by holding the others. The datum term is the published seven-parameter Helmert transformation applied to the geographic coordinates. The scale-factor term is the first grid’s projection given the second’s $k_0$ and nothing else. The origin term is the coordinate difference with both grids given the same scale factor, so that only the four zero-setting parameters differ.

Two earlier versions of that separation were wrong, and both were caught by the figure’s own assertion rather than by inspection. The first varied the ellipsoid under the label “projection” and reported zero for a pair of grids differing only in scale factor — the one case where the scale factor is the whole story. The second varied the true origin’s parallel as though it were geometry, and put 5,449 kilometres of meridian arc into a term meant to measure distortion.

The repair is the finding. In a transverse Mercator the latitude of the true origin sets where northing zero falls and changes nothing about the map, because the scale factor does not depend on the northing at all — so it belongs with the false constants and the central meridian as bookkeeping, and what is left is one number.

Those three quantities do not sum to the total, because a datum shift and a projection change do not commute. Projecting after shifting and shifting after projecting give different answers, by an amount that is small here and is not zero.

So what is being measured is not a partition. It is the size of each declaration’s contribution when the others are held — which is the question a practitioner asks when deciding what to fix first, and it is stated as such rather than presented as an accounting identity.

That non-commutation is the same structure as setting out runs the chain backwards and as the Helmert inverse in the seven parameters: operations whose arguments the other operations change do not commute, and the size of the gap is the difference between evaluating one at the other’s input and at its output.

What the datum term hides #

The 106 metres is the smooth part of the datum difference and is not the whole of it.

A published seven-parameter transformation is a global best fit to a set of common points, and its residuals are not noise — where a fit leaves residuals shows they are a field, with structure the seven parameters cannot express. National authorities therefore publish a correction grid on top of the parameters, worth metres, which is the subject of when a formula is not enough.

So a job that applies the seven parameters and stops has done most of the datum conversion and left a metre or two on the table. Whether that matters is a tolerance question and the answer is usually yes for anything cadastral.

Why the scale-factor term is as large as it is #

Half a metre from a difference in the sixth decimal place of a scale factor sounds implausible until the arithmetic is done.

The two grids’ scale factors differ by 1.3 parts per million. The point sits about 400 kilometres from the British grid’s own northing zero, and a scale factor multiplies the whole projected coordinate — so 1.3 parts per million of 400 kilometres is about half a metre.

That is the shape of every scale-factor effect on a coordinate: proportional to the coordinate, not to any distance in the work. It is the same structure as the units are part of the coordinate, where two parts per million of a state plane easting is 91 centimetres and two parts per million of a building’s dimension is a fifth of a millimetre. A scale factor and a unit are the same kind of parameter, and their errors have the same signature.

Against a tangent grid — the same British grid with $k_0$ left at unity — the difference is 400 parts per million rather than 1.3, and the term rises to 155 metres.

The comparison a job actually needs #

Comparing two grids at a point answers a question nobody has. The question a job has is whether this dataset and that one can be used together, and that is a question about distances rather than positions.

Two datasets in different grids, both converted correctly, agree in position to whatever the conversion is worth — a metre or two if the correction grid was used, a hundred metres if the datum was skipped. But two datasets in different grids, used without conversion, disagree in distance by the difference of their scale factors, which is a few hundred parts per million, and that disagreement is invisible in any overlay.

So there are two independent failure modes and the visible one is not the dangerous one. A hundred-metre positional offset is noticed the moment anything is drawn. A four-hundred-part-per-million scale disagreement produces two datasets that overlay convincingly and give different answers for the length of the same object — the similarity failure that a traverse must close shows no internal check can see.

The practical rule #

Three declarations, three sizes, and a rule for what to do about each.

Subtract the false origins first and always. They are known constants and they carry no information. Any comparison that has not removed them is measuring a convention.

Never assume the projection term is negligible. It is comparable to the datum term and is the one most often skipped, because “different projection” sounds like a formatting difference and “different datum” sounds like a physical one.

Treat the datum term as two parts. The seven parameters get most of it; the correction grid gets the rest. A job that needs metres uses the first, a job that needs centimetres uses both.

What was computed, and how #

Both coordinates, by projecting the same geographic position through each grid’s published parameters, with each grid’s true-origin meridian arc computed from its own ellipsoid rather than quoted.

The datum term, from the published OSGB36 seven-parameter transformation.

The projection term, by projecting through the second grid’s parameters on the first grid’s ellipsoid.

The false-origin term, as the difference of the constants.

Two assertions. The false-origin term must exceed the datum term by more than a factor of ten — which is the essay’s headline — and the datum term must exceed fifty metres. The second half is what stops the check from passing on a pair of grids that happened to share a datum, in which case the first would be true and would prove nothing about anything.

The numbers at another point #

The ranking is stable across Britain and the numbers are not, and it is worth seeing one more case before generalising from one.

At 5°W, 57°N — north-west Scotland — the datum term is larger, because a seven-parameter fit is a compromise and the fit’s residual grows toward the edges of the region it was fitted over. The projection term is larger still, because that point is further from both grids’ central meridians and the scale factors diverge quadratically. The false-origin term is exactly the same, because it is a constant.

So the ratio between the largest and the smallest term shrinks toward the edges of the country, and the essay’s headline — two orders of magnitude — is a statement about the middle of Britain rather than about grids in general. What survives everywhere is the ordering: convention largest, projection next, datum smallest, and both of the last two too large to drop.

Stating which of those is a general claim and which is a measurement of one point is the whole of the honesty here, and it is the same distinction the scale factor was chosen has to make about an optimum that depends on the region it was optimised over.

Where the model stops #

One point. All three terms vary across a country. The false-origin term is constant by construction; the datum term varies by tens of metres between the north and south of Britain, because a seven-parameter fit is a compromise over the whole area; and the projection term varies enormously, because it depends on the distance from each grid’s meridian and the two meridians are in different places. The ranking holds everywhere in Britain; the numbers do not.

Two grids of the same family. Both are transverse Mercators. Comparing a transverse Mercator against a Lambert conformal conic would put a differently-shaped term in the middle slot, and the decomposition into three declarations survives while the arithmetic for the second does not.

And the vertical is absent. A full comparison of two systems includes the height, where the two datums differ by tens of metres in a way that has nothing to do with either projection — the third coordinate moves too puts numbers on it.

The generalisation #

When a difference decomposes into terms of very different sizes, the size ordering and the information ordering need not agree — and where they disagree, every intuitive diagnosis is wrong.

That is a general hazard in any layered system where one layer contributes a large constant offset. The remedy is the one used here: hold the layers fixed one at a time and measure each contribution separately, rather than reasoning about the total.

What makes the cartographic case a good example is that the meaningless term is not merely large but three orders of magnitude larger, so it dominates any summary statistic anybody might compute. A mean, a maximum, a root-mean-square of the differences between two grids over a country are all essentially measurements of the false origins.

Who set the numbers, and when #

The two grids’ parameters were set thirty years and one war apart, by people with different problems.

Britain’s came from the retriangulation begun in 1936, designed for a country, computed on a datum fitted to that country, with an origin chosen to keep the coordinates positive over exactly that country. UTM’s came from a 1947 American military specification designed for everywhere, with a zoning scheme owing nothing to any national border and an origin convention chosen so that a coordinate has the same number of digits anywhere in the world.

Neither authority was designing against the other, which is why the difference between them decomposes so cleanly into terms with unrelated provenances — one term from a decision about a country, one from a decision about the world, and one from two independent decisions about a clerk’s handwriting.

A difference with three unrelated provenances #

The decomposition is clean because each term comes from a decision nobody was coordinating, and that is worth reading as more than an anecdote about two committees.

A difference between two systems is not a single quantity. It looks like one — a number of metres between two coordinates for the same mark — and the temptation is to treat it as an error, or a shift, or a correction. It is a sum of terms with different origins, different sizes and different behaviours across the country.

And the terms behave differently, which is what makes the decomposition useful. A datum term varies slowly and smoothly; a scale-factor term varies with distance from a central meridian; an origin term is a constant. A job that knows only the total cannot predict how it changes as the work moves, and a job that knows the three terms can.

The provenances explain the sizes. The largest term here comes from a decision about a country, made in 1936 for a survey of that country; the next from a decision about the world, made in 1947 for a system with no country in it; the smallest from two independent choices about keeping coordinates positive and legible. None of the three was made with the other two in view, so their relative sizes carry no information about which matters — they are three unrelated numbers that happen to be added together.

Which is the practical warning. The largest term in a difference is not necessarily the one a job should attend to. What a job needs is the term that varies most over the extent it works across, and that is a different question from which term is biggest — the constant term is often the largest and is always the least troublesome, because a constant offset is the one kind of error that cancels out of every difference.

Where this goes next #

Given all the corrections and all the declarations, a job still has to decide which of them it may leave out: the tolerance decides the model. And the coordinates the whole comparison rests on turn out not to be observations at all — a published coordinate is a result.