A height that is not a length

Water settles on a level surface. That is the definition of level, and it is the only definition a height system can be built on that a reader will not immediately dispute.

The trouble is that a level surface is not at a constant height above anything. Level surfaces converge towards the poles, so the single surface a lake’s water settles on stands 5.28 metres closer to the ellipsoid at the pole than at the equator — and a height system that reports the distance to it reports two different numbers for one sheet of still water.

The problem, stated as a lake #

A levelled height is not a distance establishes the mechanism: a chain of perfectly executed levelling observations does not sum to a height difference, because each observation measures a step along the local plumb line and the plumb lines are not parallel.

This essay is about what a height system does with that, and the case that forced the issue is a large lake spanning latitude. The water surface is one equipotential; every point on it is at the same level in the only sense the word has; and its orthometric height — the distance from the geoid up the plumb line — is not the same at both ends.

For a lake spanning 3° of latitude at 47° north and standing 183 metres above the ellipsoid, the convergence is 50 millimetres over its length. That is a small number and it is not a small number for a lake: it is the difference between a shoreline gauge at one end agreeing with a gauge at the other and disagreeing by more than either instrument’s error.

The International Great Lakes Datum publishes dynamic heights for exactly this reason, and this essay’s arithmetic is what that decision costs.

The three quantities, and only one of them is a length #

The object that is genuinely constant on a level surface is the geopotential number: the difference in potential between the datum surface and the point, which has units of potential rather than of length and cannot be measured with a staff.

$$C = W_0 - W(\text{point})$$

Every height system in use is that number divided by some gravity value, chosen to give it the units and roughly the magnitude of a length:

• orthometric height divides by the mean gravity along the plumb line between the geoid and the point, which makes it exactly the distance travelled up that line;
• normal height divides by the mean normal gravity along the same line, which makes it a distance above a slightly different surface;
• dynamic height divides by one constant — normal gravity at 45° — which makes it exactly proportional to $C$ and therefore exactly constant on a level surface.

The divisor is the whole of the choice. Divide by something that varies with position and get a length that is not level; divide by a constant and get a level number that is not a length.

What the constant costs #

A dynamic height difference is not the distance climbed, and the discrepancy is the local gravity’s departure from the constant somebody chose.

A quarter of a per cent is a large discrepancy for a quantity called a height. A hundred-metre climb at the equator raises the dynamic height by 99.735 metres and the same climb at the pole by 100.263, and both numbers are correct: gravity is 0.53 per cent stronger at the pole, so the same climb costs more potential there.

Nothing is wrong with either system. One reports a distance and is not constant on a level surface; the other is constant on a level surface and does not report a distance. Both properties cannot be had, because the level surfaces are not parallel, and no choice of divisor changes that.

The residual at 45° is a second effect #

At 45° the latitude term vanishes by construction and the leftover is 1.57 millimetres in a hundred metres, which is not zero and is not noise.

It is the free-air gradient. The mean gravity over a climb is the gravity at half its height rather than at its foot, and normal gravity falls by about $3.086\times10^{-6}$ per metre of height, so over a 50-metre half-climb the mean is 15.4 milligal below the surface value. Dividing by the surface value at 45° rather than by the mean therefore leaves a relative residual of $1.573\times10^{-5}$ — and the measured leftover is $1.573\times10^{-5}$.

Two terms, then, and they scale differently. The latitude term is linear in the climb and the free-air term is quadratic, so at a 500-metre climb the residual at 45° is 39 millimetres rather than 1.6.

The zero of the system is not at 45°, and it moves #

The two terms have opposite behaviour in latitude — one changes sign at 45° and the other does not change sign at all — so the latitude at which a dynamic height increment actually equals the climb is not where the construction puts it.

Normal gravity varies with latitude at 9.23 parts per million per degree near 45°, and the free-air term is a fixed relative shift of 1.573 × 10⁻⁵ for a hundred-metre climb. Setting one against the other puts the true crossing 0.170° north of 45° — at 45° 10′, about nineteen kilometres of ground.

That displacement is not a constant either, because the free-air term grows with the climb while the latitude gradient does not. For a five-hundred-metre climb the relative shift is 7.868 × 10⁻⁵, five times as large, and the crossing moves to 45° 51′. A kilometre of climb would put it past 46°.

So a dynamic height system has no single latitude at which it reports lengths. It has a curve of them, one for each height climbed, running north from 45° as the climb increases — and the constant that was chosen to make 45° the reference latitude does not make 45° anything in particular once a real climb is involved.

Two things follow. The first is a small correction to how the system is usually described: dynamic heights are exact at 45° is true of the latitude term alone, and the residual there is 1.57 millimetres in a hundred metres rather than nothing, which the figure already reports and which this section identifies as the reason.

The second matters more and is about what a constant divisor can do. The divisor was chosen to cancel a variation in latitude, and it cancels it exactly on a surface. What it cannot cancel is a variation in height, because a constant has no height dependence to spend — and gravity varies with height as well as with latitude, at 3.086 × 10⁻⁶ per metre, which over the range of terrestrial topography is a fifth of the whole equator-to-pole variation. Choosing one number to stand for gravity fixes one of the two directions it varies in, and the remaining direction is the one the height itself runs along. That is a stronger statement of the trade this essay is about, and it says the residual is not a defect in the choice of 45° but a consequence of the divisor being a single number at all.

The system that is not visible here, and why its absence is honest #

Normal height should be a third curve on the first figure and there are only two, because in a normal field the normal height and the ellipsoidal height coincide — to 6 micrometres over the whole sweep.

That is not a defect in the model; it is what the model is. This site computes the normal gravity field, defined by four constants and nothing else, and the gap between normal and orthometric height in the real world is entirely a gravity-anomaly effect. Anomalies are a data product, and this collection has ruled that it does not carry one — height above what? says so, and the ruling has held since.

So the honest report is two curves and a stated absence rather than three curves, one of which would be a fabrication. Requiring the two to coincide is what keeps the absence visible: if the normal height ever departed from the ellipsoidal height in this model, something would have quietly acquired a gravity field it was not given.

What the reader is holding when a height is quoted #

A published height is a number with three conventions inside it, and nothing on the sheet says which three. The practical consequence is that two heights from two sources are not comparable until all three agree.

The four numbers in that stack do not add up unless every one of them is in the same system, and the failure mode is silent: a GNSS ellipsoidal height minus a geoid model gives an orthometric height, and subtracting that from a dynamic height published by a lake authority gives a number with no meaning at all. The difference is centimetres, which is the size of the answer the subtraction was performed to obtain.

The size, against the things it is compared with #

Five metres over ninety degrees of latitude is the headline, and the useful form is the rate: 0.39 millimetres per kilometre of northing at 500 metres of elevation, in mid-latitudes. A national levelling network 1,000 kilometres long at that elevation accumulates 392 millimetres of it if it is ignored, and the rate is proportional to the height, so the same network at 2,000 metres accumulates a metre and a half.

That is far larger than the network’s own closure. A levelled height is not a distance measures the correction over 400 kilometres of northing and finds it larger than the misclosure the same network would be rejected for — so the correction is not optional and never was.

Against the other vertical quantities the site measures: the geoid–ellipsoid separation is tens of metres, the correction here is decimetres to metres, the deflection of the vertical moves a horizontal position by hundreds of metres, and the free-air term is millimetres. The convergence of level surfaces sits in the middle of that list and is the only one of them that is pure geometry — no anomaly, no observation, no data product, just the shape of the normal field.

What each system is actually for #

Orthometric height is for anything involving a distance: a cutting’s depth, a building’s storey heights, an aircraft’s clearance, a contour line on a topographic map. It is a length and behaves like one.

Dynamic height is for anything involving flow. Water runs from a higher dynamic height to a lower one and never from a higher orthometric height to a lower one — the two differ, and the case where they differ is exactly the case a canal engineer meets. A lock system, a river gauge network, a lake datum: all dynamic.

Geopotential numbers are for the network itself. A national levelling network is adjusted in potential, because potential is what the observations sum to, and the heights are computed at the end by dividing.

The third of those is the one that makes the other two cheap. Once a network is held in potential, changing height system is a division rather than a re-observation, and a country can publish two systems from one adjustment. A country that adjusted in heights instead has baked one convention into its archive and can only change it by re-processing — which is the vertical version of the trap a published coordinate is a result describes for the horizontal.

That ordering is worth stating because it inverts the usual presentation, where orthometric height is the real one and the others are technical variants. In the observations there is only $C$; every height is a convention applied to it afterwards.

Where the convergence comes from #

The whole essay rests on level surfaces not being parallel, and that is a consequence of exactly one thing: gravity is stronger at the poles.

A level surface is where the potential is constant. Moving up from one surface to the next costs a potential difference $g\,\Delta h$, so where $g$ is larger the same potential difference is spanned in less distance — and $g$ is 9.7803 at the equator and 9.8322 at the pole, a difference of 0.53 per cent. Two surfaces a kilometre apart at the equator are 994.7 metres apart at the pole, and 1000 × (1 − 0.0053) is 994.7.

That arithmetic is the entire mechanism, and its inputs are the four constants defining the ellipsoid. The ellipsoid is a level surface shows that the two gravity values above are consequences of $a$, $f$, $GM$ and $\omega$ rather than measurements, and that the site’s implementation reproduces the published figures to the last digit anybody prints — so the 5.28 metres in this essay’s first figure is derivable from four numbers and nothing else.

What a datum has to declare #

A vertical datum that says only “heights above mean sea level at some tide gauge” has under-specified itself by all of the above. What it has to say is:

• which surface is zero, which is the tide gauge and an epoch;
• which divisor turns potential into height, which is the choice between orthometric, normal and dynamic;
• and which gravity model supplied the divisor, because two orthometric systems using different mean-gravity models differ.

A fourth is implied by the first and is worth separating out: which epoch the tide gauge’s mean was taken over, because sea level moves and a datum defined in 1921 is a statement about the sea in 1921. That is the vertical twin of the epoch is part of the coordinate, and it is the reason a national vertical datum is periodically re-realised rather than corrected.

Three declarations plus an epoch, then, of which the second and third are almost never in the phrase “above sea level”. That is the vertical half of what a grid is made of, which counts five declarations a horizontal coordinate does not carry, and the vertical case is worse because the shortfall is not even conventionally acknowledged.

The check the machinery has to pass #

Four clauses, and each rejects a different way this could have been a fact about the arithmetic.

The dynamic height must be constant along the surface, to 0.4 micrometres over ninety degrees. That is the system’s defining property and is what the divisor was chosen for.

The orthometric height must not be, by metres. A check that only demanded the first would be satisfied by a routine returning zero everywhere.

The dynamic height must fail to be a length, by more than two parts in a thousand. Without this clause the system would look free, and the essay’s whole subject is what it costs.

The normal height must coincide with the ellipsoidal height in a normal field. That keeps the missing third system visible rather than hidden, and would fail loudly if anything in the module quietly acquired a gravity anomaly.

The first two are stated relative to the surface’s own height rather than as fixed thresholds, because the spread is a fixed fraction — about half a per cent of the height — so a figure drawn for the surface a lake sits on rather than one a kilometre up would otherwise fail a threshold written for a kilometre. That is the same parameterisation discipline the ground is not the grid needed for its own crossover height.

Where this ladder goes #

The height ladder has taken the third coordinate from “above what” through the ellipsoid as a level surface, the levelled height that is not a distance, the plumb line that is not the normal, the ground that is not the grid, and the third coordinate that moves with a datum shift.

This adds the system that gives up being a length in order to be level, which completes the pair: the field now has both answers to the convergence of level surfaces and the measured price of each.

What is left is the surface that is neither the ellipsoid nor a level surface of the normal field — the geoid proper, defined by the Earth’s actual gravity rather than by four constants. It is a data product, this site does not own one, and the essays here have been careful to say so at every point where one would have been convenient.