The datum hides inside the projection's parameters

Every essay in this ladder has identified a projection. A map arrives with no statement of how it was made, a set of control points is read off it, each candidate is fitted, and the residuals rank them. The answer may not be in the library; a residual has more than one explanation; the control points’ arrangement decides what is recoverable. Six rungs, and every one of them takes the body as given.

It is not given. A map is drawn on an ellipsoid — and the flattening is not a free parameter — and which ellipsoid is a second unknown that nobody has asked about — because the answer is that it cannot be found.

Why the two are confounded #

A datum shift moves every coordinate in a region by nearly the same amount. Over Britain OSGB36’s displacement from WGS84 is about 99 metres and varies across the country by only a few — a shift far larger than the projection errors it is mixed with, and far more uniform.

A plane fit contains a translation. The similarity that every candidate is scored with has four parameters — a scale, a rotation and two offsets — and a nearly-uniform displacement is exactly what an offset removes.

So the two are not merely hard to separate; the fit is designed to remove the thing the datum does. Every rung of this ladder has been quietly discarding the datum in order to discard the reproduction’s own scale and rotation, and the two go out together.

What survives, and how little it is #

The part a translation cannot absorb is second order in the extent of the sheet. A datum shift is not exactly uniform: it varies across a region because the two ellipsoids differ in shape as well as in position, and the residual variation is what a fit leaves behind.

Measured, that residual is 1.1 parts per million of the map’s own size on a two-degree sheet, rising to 11 parts per million across sixty degrees of latitude. On a printed sheet half a metre across, eleven parts per million is five microns.

Nothing recovers a signal at that level from a paper map. Digitising a scanned sheet is good to a few tenths of a millimetre at best, which is a coordinate with a width in the most literal sense, which is a few hundred parts per million; paper itself moves more than that with humidity. The datum is not merely hard to see — it is four orders of magnitude below the noise of the medium.

The plan for this rung expected otherwise #

The expectation going in was that the residual would clear a digitising tolerance once the sheet was large enough, and that the interesting number would be the extent at which it did. That is not what happens, and the correction is worth recording because it makes the result stronger rather than weaker.

The residual does grow with the extent — by a factor of ten between two degrees and sixty, exactly as the second-order argument predicts. It grows from undetectable to undetectable. There is no crossing, because the starting point is six orders of magnitude below the medium’s noise and ten times six orders is still five.

So the honest statement is not “a datum needs a large sheet to be recoverable”. It is a datum is not recoverable from the geometry of a map at all, at any extent, and the growth of the residual is a measurement of how far from recoverable it stays.

The arithmetic of the absorption #

It is worth seeing why the residual is second order, because the reason says which cases could ever be different.

Write the datum’s effect on a coordinate as a vector field over the region. Expand it about the region’s centre: a constant term, a linear term, and the rest. The constant term is a translation and the fit removes it exactly. The linear term is a scale, a rotation and a shear; a similarity fit removes the first two of those exactly and an affine fit removes all three. What is left begins at the quadratic term, and a quadratic term over a region of half-width $h$ is of order $h^2$ relative to a linear one of order $h$.

So the surviving fraction scales as the extent, and the surviving absolute residual scales as the extent squared while the map’s own size scales as the extent — which is why the ratio grows only linearly, from 1.1 to 11 parts per million across a factor of thirty in extent.

That is also the answer to when could this be different. It could be different for a datum whose effect is not smooth — a regional distortion, a warped historical network — because such a field has no small quadratic term and is not approximated by anything the fit removes. Where a fit leaves residuals is about exactly that case, one field over, and it is the case in which a datum does leave a trace.

The parameter does not move either #

The second thing worth checking is whether the wrong body corrupts the answer — whether a projection fitted on the wrong ellipsoid comes back with displaced parameters, so that the cone constant or the standard parallel is quietly wrong even when the projection’s name is right.

It does not. The recovered cone constant is identical, digit for digit, between the right body and the wrong one — and the search demonstrably can resolve that parameter, because two maps drawn at 0.70 and 0.45 come back at their own values.

Exaggerating the datum shift sixty times moves nothing either, which was the first refusal tried and which failed to fire. The refusal that works is the one above: show that the search resolves the parameter, and then show that the datum does not move it.

What this does to the ladder’s earlier results #

It validates them, which is a pleasant result for a rung that set out to complicate things.

Every identification in the previous six rungs assumed a known body. That assumption turns out to be free: the answers those rungs produced would be identical on any of the ellipsoids in common use, because the fit removes the difference before it scores anything. Two projections that cannot be told apart remain exactly as indistinguishable, and the ones that could be separated remain separable.

So the ladder’s results are robust to an unknown this ladder never modelled, which is the best possible outcome of asking about it — and it is only knowable by asking.

And what it does to the reader who wanted the datum #

It is bad news, and it is worth being direct about it.

A common practical question about an old map is what datum is this on, and it is usually asked in order to overlay the map on modern data. This rung says the map does not contain the answer. Whatever else the sheet carries — a graticule, a projection, a grid, a scale — none of it constrains the ellipsoid to better than parts per million of the sheet, and the sheet is not that good.

The answer has to come from somewhere else: the sheet’s own legend, the series it belongs to, the survey organisation’s records, or a known point whose modern coordinates are available. That last one works and it is the only one that does, because a known point is a datum measurement rather than a geometric one.

An absorption that is not quite complete #

The absorption falls from 99.68 per cent to 85.16 per cent across the sweep, and the falling is the part that is easy to over-read.

Fifteen per cent of a 99-metre shift is fifteen metres, which sounds recoverable. It is not, because the fifteen metres are not in one place: they are the map-wide root-mean-square of a field whose largest component has already been removed, spread over sixty degrees of latitude, and expressed as a fraction of a sheet that is now enormous. The same fifteen metres on a hemispheric sheet is eleven parts per million of the drawing.

That is the trap in reading an absorption percentage on its own. What matters to a reader of a picture is the residual relative to the picture, and both numerator and denominator grow with the sheet. The percentage falls and the visibility does not rise.

The size of the thing being hidden #

The absorbed quantity is not small in the world; it is small on the page, and the difference between those two statements is the whole point.

A 99-metre horizontal shift is enormous by any survey standard. It is the difference between two fields, two buildings, two sides of a road. Applied to a map at 1:50,000 it is two millimetres, which is a line width; applied at 1:1,000,000 it is a tenth of a millimetre, which is nothing.

So the same error is fatal to the data and invisible on the picture, and which of the two a reader is looking at decides everything. This ladder reads pictures, and pictures do not carry it.

Where the fit could see it #

There is one configuration in which the body is recoverable, and it is not a map.

If the control points come with ground distances — measured, not scaled off the sheet — then the ellipsoid enters through the metric rather than through the drawing, and the two ellipsoids give different distances between the same pair of coordinates. That is a survey rather than a georeferencing exercise, and it is how the ellipsoids were determined in the first place.

The distinction is exactly the one this field’s separating test makes: identification from a drawing is limited by what a plane fit cannot absorb, and identification from observations is not, because observations carry a metric and a drawing does not.

Where the model stops #

The measurement here uses one datum pair and one projection family, and the numbers would move a little for others. The structure would not: any datum shift is dominated by its translation, any plane fit contains a translation, and the residual is second order in the extent for all of them.

What is not treated is a datum with a large rotation rather than a large translation, which is a different shape of error — it is not absorbed by an offset and is partly absorbed by the fit’s rotation instead. DHDN’s rotations are nearly three arcseconds and a fit on a large sheet might see something. That is a rung this ladder has not written and the machinery for it is one line away.

What a gate can check here #

The site’s own assertion for this rung had to be rewritten twice and the final form is worth stating, because it is a case where the interesting claim is a negative.

A negative claim needs two supports. The first is that the effect is genuinely absent: the recovered parameter is identical between the right body and the wrong one, to every digit printed. The second is that the instrument could have detected it: the same search, run on two maps drawn at different cone constants, returns their own values a quarter apart.

Without the second, the first is consistent with a search that resolves nothing. With it, the two together say that the search is sharp and the signal is absent, which is the only way to establish an absence at all.

The refusal tried first was to exaggerate the datum shift and check that the parameter then moved. It does not move, at sixty times the real displacement, because the exaggerated shift is still a translation and a translation is still removed exactly. That failure is itself informative and is the reason the final assertion takes the form it does.

Who found it, and when #

The confounding is well known in georeferencing practice and is usually expressed as advice rather than as a measurement: do not attempt to determine a datum by fitting, use documentary evidence. The advice is right and the reason given for it is usually vague — “the effects are similar” — where the actual statement is that one of them is a translation and the fit removes translations exactly.

The nearest thing to a published number is in the rubber-sheeting literature, where the residual after an affine fit is used as a quality measure for a scanned sheet and is reported at a few tenths of a millimetre. That number is the digitising noise, and the datum’s contribution to it is five microns.

What to do with a map whose datum is unknown #

The practical advice follows from the measurement rather than from taste, and it is short.

Do not fit for it. Any procedure that scores candidate ellipsoids by residual is scoring noise, and it will return an answer with an apparently convincing margin because a search over a small set always does.

Use the sheet’s own metadata. A published series states its datum in the marginalia, and the marginalia is the only part of the sheet that carries the information.

Failing that, use one known point. A single feature whose modern coordinates are available converts the problem from a geometric one to a datum one, and a 99-metre discrepancy is unmissable in that comparison while being invisible in the fit.

And record the uncertainty rather than resolving it. A map georeferenced without a known datum is good to whatever the datum ambiguity is — a hundred metres, for the common European and North American cases — and a product that quotes a fit residual of half a millimetre as its accuracy is quoting the wrong number by three orders of magnitude.

The test that says whether a fit can see a parameter #

The finding is specific — the datum is absorbed by the projection’s parameters — and the way it was established is general enough to be worth stating on its own, because any fit reporting a parameter owes the same test.

Perturb the parameter and refit everything else. Hold the candidate value away from its optimum, let every other parameter re-optimise against it, and look at the best residual achievable. If that residual is essentially unchanged, the data cannot see the parameter: whatever it was set to, the rest of the model absorbed it, and the value the original fit reported is an artefact of where the optimiser started.

The test is cheap and its verdict is sharp. It is one refit per perturbation, and it needs no theory about which parameters are confounded with which — the absorption shows up as a flat curve whatever its mechanism.

And it is the diagnostic the identification literature skips. A fit is run, a residual is reported, and the parameters are printed to several decimals; the question of whether a different value would have fitted just as well is not usually asked, and when it is asked informally the answer is taken from the covariance matrix, which reports the local curvature and can look reassuring in a valley that runs for a hundred metres.

A flat curve is not a failed fit, which is the reading that matters here. It is a correct statement that the quantity is not in the data, and reporting it as such is more useful than reporting a number — because the reader then knows to go and find the datum somewhere else, which is exactly what the recommendations above tell them to do.

Where the ladder goes next #

Seven rungs have asked what a map can be made to confess. The pattern of the answers is consistent: what survives a plane fit is recoverable and what does not is gone, and almost everything anybody wants to know is in the second category. The unasked question is the one about time — whether a map’s date leaves a trace, through the survey conventions of its period rather than through its geometry.