The third coordinate moves too

Every published summary of a datum shift is a horizontal number. OSGB36 to WGS84 moves a British coordinate about a hundred metres; ED50 moves a European one about 130; NAD27 moves an American one about 190. Those figures are in every textbook, every specification and every migration note.

The transformation is a rigid motion of a three-dimensional point. It has a vertical component, and nobody quotes it.

The claim #

The vertical component of a datum shift is not small. For the three datums this collection carries it runs between 43 and 51 metres, against horizontal shifts of 99 to 195. It is dropped for a structural reason rather than because it is negligible: the software that consumes coordinates is usually two-dimensional, and a national horizontal datum has a separate vertical datum which the seven parameters know nothing about.

So the height in a converted coordinate is the worst-determined of its three numbers, and it is the one nobody checks.

What was computed, and how #

The computation is the same one everything in this field runs on, with the third component kept rather than discarded.

A geographic coordinate on the source datum, with a stated ellipsoidal height, is converted to Cartesian coordinates on that datum’s ellipsoid; the seven parameters are applied; the result is converted back to geographic coordinates on WGS84. The inverse conversion returns three numbers, and the third is the ellipsoidal height. Differencing it against the height that went in gives the vertical component:

Every one of them is above forty metres. For OSGB36 the vertical is more than half the horizontal, and the horizontal is the number that gets a paragraph in every migration guide.

There is nothing surprising in the arithmetic — a translation of 715 metres in Cartesian space has a radial component wherever the point is not perpendicular to it — and the surprise is entirely in the reporting.

Why it is dropped #

Three separate reasons, and only the first is defensible.

Most coordinates have no height in them. A two-dimensional coordinate cannot have its height transformed, so the transformation is applied with $h = 0$ assumed and the resulting height thrown away. That is correct if the horizontal result is all that is wanted — the horizontal components are barely affected by the assumed height, since $h/R$ is a part in $10^4$ — and it is a real and sensible practice.

The vertical datum is a different object. A country’s heights come from a levelling network tied to a tide gauge, and that network has no relationship to the horizontal datum’s ellipsoid. So converting the horizontal datum does not convert the heights, because the heights were never expressed against the thing being converted. The seven parameters relate two ellipsoids; a national height is referred to a geoid, which the parameters cannot see.

The number looks wrong. Fifty metres of vertical shift on a country whose terrain is measured in hundreds looks like an error, and it is not — it is the ellipsoid moving under the heights, and the orthometric heights do not move at all, because they were never measured from the ellipsoid.

That figure is the resolution. The ellipsoidal height changes and the orthometric height does not, because the geoid separation changes by exactly the same amount in the opposite direction. $h = H + N$, with $h$ moving by 51 metres, $H$ fixed, and $N$ moving by −51. That is why a geoid model is always stated with the datum it belongs to: a separation computed against one ellipsoid is meaningless against another.

The vertical is the worst-determined component anyway #

Set aside the transformation for a moment. Even within one datum, the height is the number to distrust, and there are three independent reasons that all point the same way.

Satellite geometry is worse in the vertical. A receiver sees satellites above it and none below, so the geometry that fixes the vertical is one-sided in a way the horizontal geometry is not. The standard consequence is that vertical precision is two to three times worse than horizontal from the same observations — a rule of thumb that has survived every generation of the technology because it is about where the satellites can be rather than about the equipment.

The atmosphere delays the signal along the vertical. Tropospheric delay is estimated as part of the solution and is highly correlated with the height, so an error in the delay model appears almost entirely in the height.

The reference surface is worse. The horizontal reference is an ellipsoid known to millimetres by definition. The vertical reference that anybody actually wants is the geoid, which is known to a few centimetres at best and to decimetres in poorly surveyed country.

Three independent mechanisms, all making the vertical the weak component. The datum shift’s fifty metres sits on top of that, and it is by far the largest of the four.

Where the vertical actually hurts #

Given all that, the fifty metres is bookkeeping — until it is not. Three cases where it becomes an error rather than a change of reference.

A dataset with heights converted horizontally. If a file of three-dimensional coordinates has its latitude and longitude transformed and its heights passed through unchanged, the heights are now referred to nothing. They were ellipsoidal heights on the old datum, they are being read as ellipsoidal heights on the new one, and they are fifty metres out. Nothing in the file records it.

A geoid model applied to the wrong datum. A national geoid model gives $N$ relative to a specific ellipsoid. Using it with coordinates on another datum introduces the full vertical shift — tens of metres — in a calculation whose output is a height above sea level, which is exactly the number a flood model or a drainage design consumes.

A survey combining sources. This is the case the ground is not the grid meets from the other side: the elevation factor wants an ellipsoidal height, the job has an orthometric one, and the seven parts per million that costs is the smallest of the height problems in the room. Points from a satellite receiver, points from a national network, and points from an older archive have three different height references, and the differences between them are the vertical shift, the geoid separation, and the vertical datum’s own offset. All three are tens of metres and none of them is visible in the numbers.

Putting the two beside each other is the fair way to rank the things that move a coordinate. A change of datum moves it a hundred metres horizontally and fifty vertically, at once, when somebody decides. Plate motion moves it a metre or two horizontally over decades. Both are silent, and only the second has a mechanism — the epoch — that has been built into the way modern coordinates are quoted.

What a height’s metadata has to say #

A coordinate’s horizontal part needs a datum and an epoch, as what a coordinate refers to sets out. Its vertical part needs four more things, and a height that does not carry them is a number with a unit and no reference.

Which surface. Ellipsoidal or orthometric. This is the one that decides whether the fifty metres in this essay is an error or a change of bookkeeping.

Which ellipsoid, if ellipsoidal. An ellipsoidal height is measured from a specific figure, and swapping the figure moves it by the vertical component of the datum shift.

Which vertical datum, if orthometric. National height systems differ by a metre or two, because each was tied to its own tide gauge and each carries its own levelling network’s error — the vertical version of the realisation problem in where a fit leaves residuals.

Which geoid model, if the two have been converted between. Models a decade apart differ by decimetres, and a dataset built with one and extended with another has a step at the join.

That figure is the mechanical reason the vertical component goes unreported. It is not in the parameters. It emerges from applying them at a particular place, so there is no row in any table where somebody could have written it down — which is a good illustration of how a quantity becomes invisible without anybody deciding to hide it.

Where the model stops #

The heights here are ellipsoidal throughout. Everything computed is a change in $h$, and the orthometric height is unaffected by construction. Whether a real dataset’s heights are ellipsoidal or orthometric is a metadata question that the numbers cannot answer, and it is the question that decides whether the fifty metres is a change of reference or an error.

The transformation’s own vertical residual is not modelled. As with the horizontal, seven parameters carry a rigid motion exactly and network distortion not at all. Vertical network distortion — a levelling network’s accumulated tilt — is a separate field again, larger in relative terms than the horizontal one because levelling accumulates along a route.

No geoid model is used. The separation appears in the figures as a stated input, following the decision in height above what?. What is computed is the change in $h$; what is asserted is that $H$ is unaffected, which follows from the definitions and not from any model.

Nothing here uses a real height. The stated ellipsoidal height in the computation is an input; what is measured is the change, which is almost independent of it. A height a kilometre different changes the vertical component by parts in $10^4$ — which is why the effect can be quoted for a datum rather than for a point, and why Molodensky’s shortcut can drop the height terms and still get the horizontal right.

The vertical also moves with time. Plate motion is essentially horizontal, but the ground rises and falls for other reasons — glacial rebound at up to ten millimetres a year in Scandinavia and Canada, subsidence where fluids are extracted — at rates comparable to the horizontal plate rates in the epoch is part of the coordinate. Those are not in any plate model.

The generalisation #

A component that is never displayed is never checked. That is the whole of it, and the geodetic case is a clean example because the arithmetic is unambiguous and the omission is universal.

The mechanism is worth naming precisely, because it is not laziness. The vertical component is dropped at a boundary between two systems — a three-dimensional transformation feeding two-dimensional software — and at such a boundary the dropped quantity has no owner. The transformation computed it correctly. The consumer never asked for it. Neither is at fault, and the number is gone.

Two consequences follow for any system with this shape:

The dropped component’s error is unbounded, because nothing ever compares it against anything. A quantity that is displayed is checked by everybody who looks at it; a quantity that is computed and discarded accumulates whatever error it likes.

The failure appears far away. It appears when the discarded component is needed again — in the flood model, in the drainage design, in the merge of two datasets — at which point its provenance is several systems back.

The remedy is the same one this collection keeps arriving at. Carry the metadata with the number. A coordinate needs its datum, its epoch and, if it has a height, the surface that height is measured from. Each of those is a few bytes, each is omitted routinely, and each omission is worth between a metre and a hundred.

The arithmetic, done once #

It is worth working one conversion through in three dimensions, because the fifty metres becomes obvious the moment the third component is written down and is invisible when it is not.

Take a mark at 2° west, 54.5° north, at 100 metres of ellipsoidal height on OSGB36.

1. To Cartesian on Airy 1830. The prime vertical radius at that latitude is 6,391 kilometres; the three components come out at about $(3.71, -0.13, 5.17) imes 10^6$ metres.
2. Apply the seven parameters. Translations of 446, −125 and 542 metres, rotations under an arcsecond, a scale change of −20.5 parts per million. The point moves by 715 metres in a direction fixed by the translation vector, less about 130 metres inward from the scale.
3. Back to geographic on WGS84. Latitude and longitude move by fractions of an arcsecond — 99 metres on the ground — and the ellipsoidal height comes back as 150.7 metres.

The height changed by 50.7 metres and nothing anywhere in the process announced it. Steps 1 and 3 are conversions nobody looks at; step 2 is seven numbers in a table with no vertical among them. The change appears only if the third output is compared with the third input, and the third input was usually zero because the coordinate was two-dimensional.

That is the whole mechanism, and it is why the seven parameters insists on doing the round trip: an operation with three outputs should be checked on all three.

The vertical component across a spread of ground #

Comparing that with the British figure above is the argument against quoting a single vertical shift for a datum. It is a field, like the horizontal one, and it is a field that nobody plots.

The horizontal component of the same three transformations is the quantity every migration note quotes, and it varies across a spread of latitudes in exactly the way the vertical does. The vertical runs between a quarter and a half of it everywhere, and appears in none of them.

Who found it, and when #

Nobody discovered this, which is part of the point. The vertical component has been in the transformation since the transformation was written down, and every geodesist has always known it is there.

What happened is that heights and horizontal positions were separate trades for a century and a half, and their separation was institutional as well as technical. A triangulation network and a levelling network were observed by different parties with different instruments to different specifications, adjusted separately, and published separately. The horizontal datum and the vertical datum of a country are different objects with different names and different origin dates, and the software written to handle each inherited that division.

Satellite positioning collapsed the distinction at the measurement end — a receiver produces three numbers at once, from one observation, in one frame — without collapsing it at the reference end. So the modern situation is that the instrument is three-dimensional, the reference systems are two plus one, and the conversion between them is where the fifty metres lives.

The direction of travel is toward genuinely three-dimensional datums with the geoid model as an official component rather than an accessory, so that a height’s reference is part of the datum’s definition. That is what the newer national reference systems are doing, and it is the same move as putting the epoch in the definition: taking something everybody knew and nobody recorded, and making it part of the object.

The pattern is worth naming because it recurs whenever an institutional division outlives its technical reason. The separation of horizontal from vertical was not an error — it was forced by the instruments, since a theodolite and a level measure different things by different means — and it produced two bodies of practice, two sets of standards and two kinds of specialist. When the instrument stopped enforcing the division, everything built on top of it carried on enforcing it, because the software, the formats, the training and the job descriptions had all been shaped to fit.

Where this goes next #

That closes the vertical ladder, which has run from the two heights to the field that defines them to the reduction a job actually applies. What is left is the ellipsoid itself, which this collection has taken as given since the second phase. Two essays go under it: where its shape came from, and why its flattening was never free to be chosen — the flattening is not a free parameter.