Every country's zero is a different surface

Ten rungs of this ladder measure a height against a geoid. The geoid is treated throughout as one object: the equipotential surface that best fits mean sea level, the thing a levelled height is measured from, the surface height above what? names as the answer.

No national vertical datum is on it, and none ever has been.

What a datum’s zero actually is #

A national vertical datum is realised the same way everywhere: pick a tide gauge, observe the sea for years, average out the tides, and call the result zero. Britain’s is Newlyn, the Netherlands’ is Amsterdam, France’s is Marseille, the Baltic states use Kronstadt.

That definition contains an assumption which is stated nowhere: that mean sea level is an equipotential surface.

It is not, and the reason is ordinary physics rather than measurement error. Wind piles water against coasts. Temperature and salinity change its density, so a warm column stands taller than a cold one of the same mass. The ocean circulates, and a current in geostrophic balance has a slope across it. The result — the mean dynamic topography — holds the sea surface up to about a metre and a half from the geoid, and it varies from coast to coast.

So each country’s zero is the geoid plus whatever the dynamic topography is at its own gauge, and no two are the same surface.

What was computed, and how #

The dynamic topography here is a stated closed form rather than a published model, for the reason this collection’s coastline decision gives: a number computed off a model is partly a measurement of that model, and the reader cannot tell how much.

$$N_{\text{dyn}} = 0.62\sin\varphi\cos(\lambda - 2.4) + 0.55\cos 2\varphi - 0.38\sin 2\varphi \sin(\lambda + 0.6) + 0.24\cos 3\varphi\cos 2\lambda - 0.35$$

in metres. It is smooth, it ranges over 2.09 metres between ±76°, and it has the observed sign structure. Every number below is recomputable from that line.

The absolute column is a property of the stated field’s own offset and is not the interesting number. The differences are: Amsterdam’s zero and Cascais’s are 462 millimetres apart, so the same mountain has two published heights differing by nearly half a metre depending on which country published it.

Why it is not a misclosure #

The obvious objection is that half a metre across a continent is what a levelling network’s own errors look like, and the objection is testable.

A first-order levelling loop closes at about four millimetres times the square root of its length in kilometres. That is the standard against which any discrepancy has to be judged.

Six of the twenty-eight pairs are beyond what levelling error can account for. Nine are under it, and the second fact is the more interesting one.

Why nobody noticed for a century #

The pairs that hide are mostly the neighbours. Amsterdam and Newlyn differ by 60 millimetres and their levelling connection has a closure of 111. Genoa and Marseille differ by 30 against a closure of 71.

That is not an accident of this field. The dynamic topography is smooth, so nearby coasts share most of it, and the step between two adjacent countries is small — while the distance between them is small too, so the closure that would reveal it is also small. The two shrink together, and the ratio stays under one for most neighbouring pairs.

A border crossing therefore cannot find the effect. What finds it is a continental adjustment that connects the far ends of the network, where the step has accumulated and the loop closure has only grown as a square root. That is the historical shape of the discovery exactly: the European levelling networks were connected across borders throughout the twentieth century without the datum question becoming urgent, and it became urgent when a single continental adjustment had to reconcile them all.

Distance predicts the closure exactly and the step not at all #

The claim that the two shrink together deserves testing, because the table refutes it in both directions and what replaces it is more useful.

The closure is a clean function of distance: four millimetres times the square root of the length in kilometres, by definition of a first-order specification. The step is not. Taking the sea surface’s own slope of about 0.2 millimetres a kilometre would predict a step proportional to the separation, and a ratio growing as 0.05√d — which crosses one at about four hundred kilometres and reaches 2.5 at two and a half thousand.

Against the measured pairs that law is right in order of magnitude and wrong in every particular:

Newlyn and Kronstadt are two and a half thousand kilometres apart and their zeros differ by six millimetres. Amsterdam and Genoa are a third of that distance apart and differ by two hundred and forty-seven.

The step is not a function of separation at all. It is the difference between two values of a smooth field, so it depends on which contour of the dynamic topography each gauge happens to sit on — and two far-apart gauges on the same contour agree exactly while two near ones across a gradient do not. Distance sets the closure and the geography sets the step, and the two are independent.

That changes what a network has to contain to find the effect. Not a long baseline: a long baseline between two gauges that happen to agree finds nothing, and Newlyn–Kronstadt is precisely that case at a ratio of 0.03. What is needed is a pair straddling a gradient, and whether a network contains one is a matter of how many pairs it has.

Which makes the discovery grow as the square of the number of countries connected. Six of the twenty-eight pairs here exceed their own closure — a rate of just over a fifth — so two countries joining have about a one-in-five chance of noticing anything, four countries have six pairs and would expect to find one, and eight have twenty-eight and find six.

That is the historical shape exactly, and it explains the timing better than any account of improving instruments. The networks were connected across borders throughout the twentieth century one pair at a time, and each connection was overwhelmingly likely to close within its own tolerance; the problem became undeniable when a single adjustment held all of them at once, because the number of chances had grown quadratically while the individual chance had not changed at all.

One country’s own gauges disagree too #

The sea has a slope along a coast as well as across an ocean: about 0.14 to 0.25 millimetres a kilometre on the stated field, which is 56 to 99 millimetres over four hundred kilometres.

So a country with two tide gauges has two candidate zeros, differing by decimetres, and which one it adopted is a nineteenth-century decision about which port had the longer record. The choice is a convention with a metric consequence, which is what a datum is fitted to a region says about the horizontal and what makes both of them conventions rather than measurements. Britain’s use of Newlyn rather than Liverpool, France’s of Marseille rather than Brest, are choices of that kind and each one fixed a national zero for a century.

What a step actually breaks #

It is worth being concrete about what goes wrong, because a constant offset between two datums sounds like the mildest possible error.

A height difference within one country is unaffected. Both ends are counted from the same zero, so the offset cancels, and every gradient, every cut-and-fill volume and every flood level inside a national network is correct.

A height difference across a border is wrong by the step. A river crossing a frontier has a fall computed from two zeros, and the step is added to or subtracted from it. At 60 millimetres between Amsterdam and Newlyn that is not much for a river; at 462 between Amsterdam and Cascais it is more than the fall of many navigable channels over their last hundred kilometres.

And a continental model is wrong everywhere. A geoid model fitted to national heights inherits every step as a discontinuity at every border, so the model has jumps in a quantity that is smooth by construction — which is exactly the symptom that made the problem urgent rather than academic.

The last of the three is the one that decides the practice. A datum step is harmless to the country that has it and destructive to anything computed across several, which is why the pressure to fix it came from continental and global work rather than from any national office. It is the same asymmetry a grid stops fitting the ground it was laid on reports in the horizontal: a national system can be internally excellent for as long as nobody asks it to join up with anything.

The repairs, and what each one leaves #

Average several gauges. The obvious fix, and it is what several national datums actually do. Averaging Newlyn, Amsterdam, Marseille, Genoa and Trieste moves the zero by 102 millimetres from Newlyn’s alone — and lands 1,111 millimetres from the geoid rather than on it. A mean of points on a non-flat surface is a point on it, so this makes the zero more stable and no more correct.

Use geopotential numbers. A levelling network measures potential differences, and those are consistent whichever country observes them. Publishing a geopotential number rather than a height sidesteps the whole problem: it is the quantity a height that is not a length prices, it is what the physics gives, and it needs no zero at all until somebody wants metres.

Adopt one global equipotential as the zero. This is what an international height reference system does, and it converts the problem from a datum question into a conversion question: every national datum gets one number, the offset above, and every published height gets corrected by it. The offsets then have to be determined, which needs the dynamic topography, which needs an ocean model — so the correction is only as good as a model this collection has deliberately not used.

Where the model stops #

The topography is stated, not observed. Its structure is chosen to have the right size and the right signs, and no number here is a measurement of the real ocean. What is being demonstrated is the mechanism and the arithmetic — a spread of this size produces steps of that size against closures of that size — and the real spread is of the same order.

The gauges are placed correctly and the datums are simplified. Several of the countries named use datums with more history than one gauge, and several have changed datum since. The list is a set of plausible realisations rather than a register.

And the epoch is ignored. Mean sea level is a mean over a period, sea level is rising, and a datum realised in 1915 has a zero that referred to the sea of 1915. That is a second, independent offset, it grows with time, and it is the epoch is part of the coordinate applied to the vertical — which is a rung this ladder has not written.

Nor is the geoid model’s own error here. The geoid model stops at a degree prices the truncation, which is a separate contribution of its own size, and it would have to be added to anything above before a real conversion could be published.

What the correction would look like #

The repair that is actually being carried out worldwide is the third one, and its practical form is worth stating because it is unusually simple.

Every national datum gets one number: the offset of its zero from a stated global equipotential. A published height in that datum is converted by adding that number, once, everywhere in the country. There is no field, no interpolation and no model of the local terrain — the offset is a constant because a datum’s zero is a single surface, however wrong.

So the entire correction for a country is a scalar, and the difficulty is not in applying it but in determining it. That needs the dynamic topography at the country’s own gauge, which needs an ocean model, or a gravimetric geoid accurate enough to compare against levelled heights, or both. Those are the hard part, and they are hard for reasons that have nothing to do with this rung’s arithmetic.

The measurement here says how large the numbers are and therefore what accuracy the determination needs: decimetres to distinguish the countries, centimetres to be useful, and millimetres to be worth publishing beside a modern levelling network.

It is worth adding what the rung does not claim. None of the eight datums is badly realised, none of the tide gauges was read carelessly, and no country would improve its own network by moving its zero. Each is an accurate measurement of the sea at one place, and the sea at one place is not the sea at another. What is wrong is only the sentence everybody says about them — that a height is measured from mean sea level — which is true of each datum separately and false of any two together.

It also says what a new connection is worth. Adding one country to a network of n adds n pairs, so the marginal chance of exposing the problem grows with the size of the network already assembled — the last country to join a continental adjustment is the one most likely to be the one that reveals it, and is the one least likely to be responsible for anything.

The generalisation #

A datum is a convention realised at a point, and the point is not on the surface the convention names.

That is the horizontal story exactly, one field over. A datum is fitted to a region and fits it well and its neighbours badly; two parameter sets give one transformation because a realisation is not a definition; a chain of transformations does not close because each link is a fit.

What makes the vertical version starker is that the discrepancy is not an error in anything. A horizontal datum’s departure from the geocentre is a consequence of the observations available when it was fitted, and a better fit would reduce it. A vertical datum’s departure from the geoid is a consequence of the sea not being flat, and no amount of observing improves it — the gauge is measuring the right thing, the average is correct, and the surface it defines is simply not the surface everybody assumes it is.

Who found it, and when #

That mean sea level is not an equipotential has been understood since the nineteenth century, and the discrepancies between European datums were being measured as soon as the networks were connected. The United European Levelling Network and its successors exist to reconcile them, and the standard reference is Amsterdam’s zero — chosen as a convention rather than as a correction.

The modern form is the International Height Reference System, adopted in 2015, which defines the zero as a stated value of the gravity potential rather than as a sea level anywhere. That is the third repair above, made international, and it is the one that removes the tide gauge from the definition entirely.

What is worth carrying is the size and the shape: decimetres, invisible between neighbours, and visible only to a network large enough that its own closure has stopped growing as fast as the step.

Where the ladder goes next #

Eleven rungs price a height’s surface, its path, its instrument, its model, its correlation and now its zero. What none of them prices is the height’s epoch: a datum realised at one date refers to the sea of that date, the sea has risen since, and the ground has moved too — so a published height is a statement about a moment as much as about a place, and nothing in the number says which moment.