Height above what?

Two instruments are set up on the same benchmark. One is a satellite receiver, which reports a height of 84 metres. The other is a levelling staff whose readings have been carried from a tide gauge two hundred kilometres away, and it reports 39 metres.

Neither is broken. They are measuring heights above two different surfaces, and the surfaces are 45 metres apart.

The two heights #

Ellipsoidal height is the distance from the reference ellipsoid, along its normal. It is purely geometric: the ellipsoid is four constants, the normal is calculus, and the height is the third component that falls out of converting a Cartesian position back to latitude and longitude. Every satellite fix produces one, whether or not anybody asked.

Orthometric height is the distance from the geoid, measured along the plumb line. The geoid is the equipotential surface of the Earth’s gravity field that best coincides with mean sea level — the surface water would settle on if it could reach everywhere. It is not a shape anybody chose. It is where the rock happens to be.

The two differ by the geoid separation or undulation, $N$, and the relation is exact:

$$h = H + N$$

Globally $N$ runs from about $-106$ metres south of India to about $+85$ metres near New Guinea. It is not small, it is not a correction, and it does not average out over any region small enough to survey.

Why the difference is not academic #

Ellipsoidal height answers how far from the reference figure. Orthometric height answers which way will water flow, and those are different questions with different right answers.

Water flows down the gradient of the gravity potential, which is by definition perpendicular to the geoid. So two points at the same orthometric height are at the same potential and water will not flow between them. Two points at the same ellipsoidal height can differ in potential by whatever the geoid does between them, and water will flow from one to the other.

A drainage scheme, a canal, a sewer, a flood model and a runway approach are all governed by the second question. A satellite receiver answers the first. That mismatch is a real and recurring engineering failure, and its size is the local gradient of the geoid times the length of the works: geoid slopes of several parts per million are ordinary, which is metres over a hundred kilometres and centimetres over a kilometre.

Which works the geoid’s slope actually reaches #

Water flows the wrong way is the right criterion and it is not a uniform hazard, because a works has a design gradient of its own and the geoid’s slope only matters beside it.

The geoid’s slope is a few parts per million — five is an ordinary figure over ordinary ground — and a scheme is safe when its own gradient is much larger. Setting the two side by side:

The hazard is confined to the bottom of that table, and the bottom of the table is not a list of exotic cases: it is flood modelling, drainage over flat country, canal reaches, irrigation levelling and coastal inundation — precisely the works carried out over large areas of ground with almost no fall.

Two readings follow, and both sharpen the essay’s own claim.

A steep works is protected by its own steepness, not by care. A road designer using ellipsoidal heights is wrong by tens of metres in the absolute and by a thousandth of a per cent in the gradient, and nothing about the road will ever notice. That is why the mistake survives: most works are in the safe rows and the practitioners who make it never see a consequence.

And a flat works has no protection at all. A canal reach is level by definition, so its design gradient is zero and any geoid slope is infinitely larger than it. The whole of the works’ engineering is a statement about the potential, and the ellipsoidal height contains none of it.

So the criterion is not how long is the scheme but how flat. A hundred-kilometre motorway is safe and a ten-kilometre irrigation levelling is not, which inverts the intuition that a larger job carries more of the error.

The decision this site takes about the geoid #

This collection deferred a question early on and named it: does a published spherical-harmonic geoid model count as a measurement or as a citation?

The answer taken here is citation, for the same reason the coastline dataset was refused at the outset. A geoid model has a truncation degree. The separation it reports at a point is partly a measurement of that choice, and a reader handed the number cannot tell how much. The same site that will not compute Greenland’s area from somebody’s simplified polygons should not compute a height from somebody’s truncated harmonics and present the result as a measurement.

What is not a citation is the normal field — the gravity field of the level ellipsoid — which is determined exactly by four constants and has a closed form for everything anybody needs from it. So the arrangement in this collection is:

• the normal field is computed, in closed form, and checked against the published constants it is supposed to reproduce;
• the geoid separation is a stated input, left as the reader’s own number, with everything around it computed.

That costs almost nothing, because the results worth having are all in the normal field. The ellipsoid is a level surface derives equatorial and polar gravity, the potential of the ellipsoid and the Earth’s dominant gravity coefficient from $a$, $f$, $GM$ and $\omega$ — and reproduces every published value to ten significant figures.

What was computed, and how #

Three things in this essay are computed rather than asserted, and the third is the one that decides whether the distinction between the two heights matters in practice.

Normal gravity across latitude, by Somigliana’s formula:

$$\gamma(\varphi) = \frac{a\gamma_a\cos^2\varphi + b\gamma_p\sin^2\varphi}{\sqrt{a^2\cos^2\varphi + b^2\sin^2\varphi}}$$

exact rather than a series in the flattening, with $\gamma_a$ and $\gamma_p$ themselves derived from the four defining constants rather than quoted. It runs from 9.7803 metres per second squared at the equator to 9.8322 at the pole — half a per cent, and the reason a weighing machine calibrated in Quito reads differently in Oslo.

The convergence of the level surfaces. The surfaces of constant potential are not parallel to each other: a surface one kilometre above the ellipsoid at the equator is 994.7 metres above it at the pole. That is why a levelled height is not a distance, and it is computed here exactly, from the closed-form normal potential in ellipsoidal coordinates rather than to first order in the height.

The two reductions a measured distance needs. A tape on the ground is not a distance on the ellipsoid and is not a distance on the grid.

That is the working surveyor’s version of the whole question, and the ground is not the grid follows it through.

What the separation does to a levelling run #

That figure is the reason the vertical needs its own machinery rather than being a third number attached to the horizontal. A levelling instrument follows the local level surface exactly — that is what a bubble does — and the level surfaces are not parallel, so a chain of perfectly executed observations does not sum to a height difference. The correction is a property of the gravity field and is larger than the survey’s own error.

Where the vertical datum comes from #

A national height system is not the geoid either. It is a realisation of the geoid, in exactly the sense that a horizontal datum is a realisation of an ellipsoid: somebody chose a tide gauge, declared its mean sea level to be zero, and carried heights out from it by levelling.

Three consequences follow, and all three are the vertical analogue of something in the horizontal.

The starting point is arbitrary and local. Britain’s datum is Newlyn, Ireland’s is Malin Head, and the two differ by about two metres. Mean sea level is not the same surface everywhere — winds, currents, temperature and salinity hold the ocean’s surface up to a metre and a half off the geoid in places — so a tide gauge measures the geoid plus the ocean’s own topography.

The realisation carries the survey’s error. Levelling accumulates, and a national network’s error grows away from its origin. That is the vertical version of the network strain in where a fit leaves residuals.

Two neighbouring countries disagree, by a metre or two, at their shared border, and the disagreement is not resolvable by measuring better.

The response is the same as in the horizontal, and it is happening now: a global geoid model with a stated potential value $W_0$ replaces the tide gauge, and national systems are re-defined against it. A height then means this much potential below the global reference, and it means the same thing everywhere.

Where the model stops #

No geoid model is computed here. The separation in every figure is a stated input, and the arithmetic around it — that $h = H + N$, that a drainage gradient is a potential gradient, that a level surface converges polewards — is exact whatever $N$ is. What is not computed is any particular country’s $N$.

Mean sea level is not an equipotential surface. The ocean’s surface departs from the geoid by up to about 1.5 metres from steady currents and density differences, so the classical definition of the geoid — the equipotential that coincides with mean sea level — is self-inconsistent at the metre level. Modern practice defines the geoid by a stated potential value instead and treats the departure of the sea from it as a measurable quantity in its own right.

Orthometric height is itself model-dependent. Its definition divides the geopotential number by the mean gravity along the plumb line between the geoid and the point, and that mean is inside the rock. Getting it requires an assumption about density, which is why several countries use normal heights instead — defined against the normal field, which needs no such assumption. The difference between the two definitions is centimetres in flat country and decimetres in mountains.

The plumb line is not the ellipsoid normal, so “along the plumb line” and “along the normal” are different directions, by up to about half an arcminute in ordinary country and considerably more in mountains. That difference has its own consequences for what an astronomical observation means, taken up in the plumb line is not the normal.

Nothing here is about the tide. Sea level rises and falls by metres twice a day. A tide gauge’s mean sea level is an average over years, and which years, and whether the average is corrected for the solid Earth’s own tidal deformation, are decisions that move the answer by centimetres. The vertical has three separate conventions for that alone.

Reading a height critically #

Four questions, and a height that answers none of them is a number with a unit and no reference.

Above what surface? Ellipsoidal or orthometric — and if orthometric, which national datum, since they differ from each other by metres.

On which ellipsoid, if ellipsoidal? A height is measured from a specific figure, and the figures differ. Swapping WGS84 for a national ellipsoid changes an ellipsoidal height by tens of metres, which is the third coordinate moves too — the vertical component of a datum shift, routinely dropped because two-dimensional software has nowhere to put it.

Which geoid model produced the separation? Two models a decade apart differ by decimetres, and a dataset built by converting with one and extended by converting with another has a step in it at the boundary between them.

Orthometric or normal? The two definitions of “height above the geoid” differ by centimetres in flat country and decimetres in mountains, because one requires an assumption about the density of rock and the other does not.

None of the four is answerable from the number itself, which is why a height in a database without its metadata is in exactly the position of a latitude without its datum.

The generalisation #

The general point is about what a reference surface is for, and it is the same point which projection is best makes about projections one field over.

There is no such thing as the height of a point. There is height above a figure chosen for its mathematical convenience, and height above a surface defined by the physics that decides which way water runs. Asking which is correct is the wrong question; asking which one the work depends on is the right one, and the answer differs by tens of metres.

What makes the geodetic case a good example is that the two references are of genuinely different kinds:

• the ellipsoid is defined, by four numbers, and is therefore exactly computable everywhere and forever;
• the geoid is discovered, by measuring gravity, and is therefore known only as well as it has been surveyed and only where it has been.

That asymmetry is why the two heights have such different characters. Ellipsoidal height is available instantly, anywhere, at centimetre precision, and means almost nothing physically. Orthometric height requires a national levelling network or a geoid model, is known to a few centimetres at best, and is the one that determines what happens.

The failure mode is worth naming because it is now extremely common. Satellite positioning made the wrong height cheap. For most of surveying’s history the only affordable height was the orthometric one, so nobody had to state which was meant. A receiver hands out ellipsoidal heights by default, they are labelled “height”, and they are wrong for the purpose by tens of metres — with no error, no warning and a plausible number.

The two heights across a country #

The sign of the separation is not a detail. A conversion that assumes the geoid is above the ellipsoid is wrong by twice the separation wherever it is not, and there is no region of the world where the sign can be assumed.

Who found it, and when #

The idea that the Earth’s figure is a surface of equilibrium rather than a chosen shape goes back to Newton and Huygens, and to the demonstration that a rotating fluid body must be flattened. That line ends in the exact closed forms of the ellipsoid is a level surface.

The word geoid is Johann Benedict Listing’s, in 1873 — the same Listing who named topology — for the equipotential surface that continues the mean ocean surface under the continents. Carl Friedrich Gauss had described the concept fifty years earlier, in the course of the Hanover survey, as the surface a spirit level defines.

The measurement problem was solved twice. Stokes gave an integral in 1849 that recovers the geoid from gravity measured all over the Earth, on the assumption that no mass lies outside it — which requires knowing the density of the topography, and therefore an assumption about rock. Molodensky’s work in the 1940s and 1950s removed that assumption, at the cost of computing a slightly different surface, and it is his formulation that modern practice uses. The same Molodensky as the shortcut, and the transformation formulae are much the smaller contribution.

The satellite era inverted the difficulty. Determining $N$ used to require gravity measurements on the ground; now the ellipsoidal height is trivial and the geoid model is the hard part, so $N$ is what everything else waits on.

Where this goes next #

The next rung takes the half of the problem that is exactly computable and computes it. The ellipsoid is a level surface derives the whole normal gravity field from four constants — including the coefficient that describes the Earth’s own field — and checks every derived value against the published one it is supposed to reproduce.