A levelled height is not a distance

Levelling is the most reliable measurement in surveying. A level is set between two staffs, the two readings are differenced, and the result is a height difference of a fraction of a millimetre. Repeat along a line for four hundred kilometres, accumulate, and the total is good to about ten millimetres.

Then run the same line back the other way, or around a loop, and it does not close. Not because of the instrument.

The claim #

A level bubble defines the local equipotential surface exactly: that is what gravity does to a fluid. So a levelling run measures increments perpendicular to whatever level surface it is standing on, faithfully, at every setup.

The trouble is that the level surfaces are not parallel to each other. They converge towards the poles, by five metres in every thousand, so a run that goes north at altitude and comes back at sea level is following surfaces that have closed up between the two legs. The sum of the observed increments is not a difference of anything, because there is nothing it is a difference of.

Raw levelling is not path-independent. That is the whole content of the essay, and it has a precise remedy.

What was computed, and how #

The convergence is a property of the gravity field, and the normal field is available exactly — from the ellipsoid is a level surface, where the potential is written in closed form and checked against every published constant it should reproduce.

The computation is then direct. A surface of constant potential is defined by a stated potential difference $\Delta U$ below the ellipsoid’s own. Its height above the ellipsoid at latitude $\varphi$ is found by solving

$$U(\varphi, h) = U_0 - \Delta U$$

by Newton’s method on the exact potential — not by the first-order shortcut $h = \Delta U/\gamma$, which is 15.8 centimetres low at a kilometre.

The convergence comes out at $5.276 \times 10^{-3}$ of the separation, and the gravity flattening $f^* = (\gamma_p - \gamma_a)/\gamma_a$ is $5.302 \times 10^{-3}$. Those agree to within second-order terms, and the agreement is asserted rather than noticed: a quantity defined as a fractional difference in a force is also the geometric flattening of the surfaces a bubble follows.

The correction to a levelling run then follows by evaluating the same solution at the two ends of the line. For a line at 1,000 metres running 100 kilometres north from 50°, the two level surfaces differ by 82 millimetres, or 0.82 millimetres per kilometre of northing. At two kilometres of elevation it is twice that.

The loop that does not close #

The clearest form of the failure is a circuit, and it can be constructed on paper.

Take a rectangular loop: run 100 kilometres north along a ridge at 1,000 metres, drop to sea level, run 100 kilometres south, and climb back. Every leg is levelled perfectly. The two vertical legs measure the same 1,000 metres — the climb and the descent are at the same latitude, so the surfaces there are spaced identically. The two horizontal legs each measure zero, because each follows a level surface.

The sum is zero. And the loop has not closed, because the two horizontal legs were not on the same surface: the northern one is 82 millimetres closer to the ellipsoid than the southern one, and that discrepancy is real, is not in the observations, and does not appear in the arithmetic.

That is the trap. The misclosure does not show up as a misclosure. The observations sum to zero because each one is correct about the surface it was taken on, and the fact that a height difference was never being measured is invisible from inside the traverse. It emerges only when the same benchmark is reached by two different routes at two different elevations — which is why the effect was found by national networks and not by individual jobs.

The remedy is a change of variable #

The fix is not a correction bolted onto levelling. It is a change in what is being measured.

Instead of accumulating height increments, accumulate potential increments: multiply each observed increment $\delta n$ by the gravity $g$ measured at that setup, and sum. What comes out is the geopotential number

$$C = \int_{\text{geoid}}^{P} g \,\mathrm{d}n$$

which is path-independent by construction, because it is the difference of a potential and a potential is a function of position. A loop closes exactly, and the closure of a real loop is then a measure of the survey’s error and of nothing else.

The geopotential number is the honest quantity. It has units of $\text{m}^2\text{s}^{-2}$, which nobody can build a road with, so it is converted back into a length by dividing by a gravity value — and which gravity value is what separates the height systems in use:

• orthometric height divides by the mean gravity along the plumb line between the geoid and the point, which is inside the rock and requires an assumption about its density;
• normal height divides by the mean normal gravity along the corresponding segment, which needs no assumption because the normal field is defined everywhere;
• dynamic height divides every point’s geopotential number by one constant, usually normal gravity at 45°, so points at the same potential get the same number everywhere.

Each is a different trade and each is used somewhere. Dynamic heights are what a lake’s surface has — the Great Lakes’ levels are quoted this way, because two points on one lake ought to have the same height and under an orthometric system they do not.

What survives, and what does not #

This collection’s standing question is which quantities are properties of the thing and which are properties of the description, and the vertical answers it sharply.

The geopotential number is the invariant. It is a difference of a scalar field evaluated at two points, so it does not depend on the route taken, the instrument, or the latitude of the work.

Every height is a derived quantity with a convention in it. Orthometric, normal and dynamic heights are three different functions of the same geopotential number, and they disagree — by centimetres in flat country and by decimetres in mountains — while all three are correct.

That is the same shape as the distinction in what survives a change of coordinates: the scale along the meridian and the scale along the parallel depend on how the sphere was parameterised, while the principal scales do not. Here, the geopotential number is what does not depend on the convention and the height is what does.

The figure is a reminder of where the correction sits in the stack. Everything in this essay is about getting from raw observations to $H$, the middle quantity. Getting from $H$ to $h$ needs the geoid separation, which is a different problem with a different answer, and getting from either to a distance on a map needs the projection, which is a third. Three reference surfaces, three conversions, and every one of them is a place where a number can be correct about the wrong thing.

What it costs a job, and when it does not #

The correction is proportional to the height of the line and to how far north it runs, so most work never meets it, and the boundary is sharp enough to state.

Setting the two figures side by side is the reason the generator takes the heights and the distance as parameters. A correction that matters everywhere would have been applied everywhere; this one matters for national networks, for long north–south routes, and for anything at altitude, and it is genuinely negligible for a building site.

Naming the boundary is more useful than naming the effect. At 0.8 millimetres per kilometre of northing per kilometre of elevation, the correction reaches a millimetre when the product of the elevation in kilometres and the northing in kilometres reaches about 1.2. A site 200 metres up and 5 kilometres across gives 0.001 — a micrometre. A railway at 1,500 metres running 300 kilometres north gives 0.45 — 370 millimetres, and unignorable.

The rule, with all three factors in it #

The two figures and the boundary above give the correction at particular latitudes, and the three dependences can be collected into one expression that a reader can apply without either figure.

The correction is proportional to the elevation of the line, to the northing it covers, and to the rate at which gravity changes with latitude, which is sin 2φ. Fitting the constant against the numbers this essay computes gives

$$\Delta H \;\approx\; 0.83\; H_{\text{km}} \; \Delta N_{\text{km}} \; \sin 2\varphi \quad\text{millimetres.}$$

Against the essay’s own worked cases: a line at 1,000 metres running 100 kilometres north from 50° gives 0.83 × 1 × 100 × 0.985 = 82 millimetres, which is the figure computed by Newton’s method on the exact potential; the railway at 1,500 metres over 300 kilometres gives 368 against the 370 quoted; a site 200 metres up and 5 kilometres across gives 0.0008, a micrometre.

Two things in that expression are worth reading rather than evaluating.

Only the northing enters. A levelling line running due east accumulates nothing at all, whatever its length and whatever its elevation, because it stays on one level surface — so a network’s exposure is decided by the shape of its routes and not by their total length. Two national networks of identical extent, one built along a coast running east and one along a valley running north, meet this correction to completely different degrees.

And the latitude factor has a maximum in the middle. sin 2φ is zero at the equator and at the pole and one at 45°, so a country at either extreme is nearly exempt and a country at mid-latitude pays the most. That is the opposite of the usual expectation that polar work is harder, and it is the reason the nineteenth-century campaigns that found this were French, German and American rather than tropical or Arctic.

Where the model stops #

This is the normal field’s convergence, not the Earth’s. The level surfaces computed here belong to the level ellipsoid. The real field’s surfaces are bumpy at the scale of the geoid, so a real levelling run has an additional correction that depends on the actual gravity along the route — which is why serious levelling is accompanied by gravity measurements at the setups rather than computed from latitude alone. What is computed here is the systematic part, which is the part that accumulates in one direction and therefore does not average away.

The orthometric correction depends on the height of the line, not on the height difference. A line at two kilometres accumulates twice what a line at one kilometre does, even if both are level. That is counter-intuitive and it is what the figure shows: the correction is proportional to the separation between the two surfaces, which is the elevation.

Nothing here says which height system a country uses. Most of Europe uses normal heights, North America orthometric, and the differences at a border are decimetres in mountains. That is a third source of disagreement on top of the two in height above what? — different tide gauge, different network, different definition of the word.

Gravity is assumed to follow the normal field. Real gravity departs from it by tens of milligal over ordinary topography, which is a few parts in $10^5$ — so the correction computed from latitude alone is itself approximate at the level of a few per cent of itself. That is why precise levelling is accompanied by gravimetry, and why the quantity a national network actually stores is the geopotential number rather than any height.

The tide is excluded. The solid Earth deforms twice a day and the potential changes with it, and there are three conventions for whether a published height includes that. They differ by up to a few centimetres, which is the same size as the corrections in this essay.

The generalisation #

The lesson is about path dependence, and it is one of the more portable things in this collection.

An accumulated measurement is trustworthy only if what is being accumulated is the increment of a function of state. Height increments are not: they are increments along a path, and the path matters. Potential increments are, and their sum is a difference of a scalar field.

The test is exactly the one this essay opened with. Go around a loop. If the sum is not zero to within the instrument’s error, the quantity being accumulated is not a state function, and no amount of instrumental care will fix it — the discrepancy is in the definition rather than in the measurement.

That test is doing a lot of work across this collection. Total curvature and the scale rule integrates Gaussian curvature over a closed surface and gets a topological invariant; what a coordinate refers to requires a datum transformation to compose with its own inverse and catches a 1.3-centimetre error in the order of operations. Loop closure is the same instrument each time: a claimed state function is required to behave like one.

And the practical corollary is worth stating plainly: when a loop does not close, the first suspect is the quantity, not the instrument. Surveyors spent a long time looking for the instrument.

Why the vertical resisted a global answer for so long #

The horizontal got a global datum in the 1980s and the vertical did not, and the reason is in this essay rather than in politics.

A horizontal network can be re-observed from orbit: a satellite measures a position in a geocentric frame directly, and the old network’s distortion becomes measurable in one campaign. There is no equivalent for heights, because a satellite measures ellipsoidal height, and converting that to a physically meaningful height needs the geoid — which is the thing that has to be surveyed on the ground.

So the vertical’s global answer had to wait for satellite gravimetry, which arrived in the 2000s, and for the agreement that a height should be defined by a stated potential value rather than by a tide gauge — which is an international decision rather than a measurement and was taken in 2015. The ground is not the grid works through the reduction in that figure, which is where a height and a horizontal distance meet on a job sheet.

The correction at another latitude #

That latitude dependence is the sin 2φ of the classical formula, and it has a practical consequence worth naming: a levelling network in the tropics is easier than one at mid-latitudes, by a factor that reaches two, for reasons that have nothing to do with the terrain.

Who found it, and when #

The non-parallelism of level surfaces follows from Clairaut’s work in 1743 and was understood in principle from then. The practical consequence took much longer to matter, because it is invisible until levelling networks are both long and precise.

The nineteenth century’s great levelling campaigns are where it arrived. Runs of hundreds of kilometres across France, across the German states and across the United States produced misclosures that were too large and too systematic to be instrument error, and the pattern — worse in mountainous country, worse in the north–south direction — pointed at the field rather than the instrument.

Helmert, who also gave his name to the seven-parameter transformation, set out the orthometric correction and the definition of orthometric height in the 1880s. Molodensky introduced normal heights in the 1940s and 1950s, specifically to remove the assumption about rock density that orthometric height smuggles in — the same motivation as his work on determining the Earth’s surface from surface measurements alone.

The choice between the two is still live, and it is a good example of a technical decision made on grounds that are not purely technical: normal heights are cleaner and orthometric heights were already in every archive.

The naming is unhelpful, as usual. Orthometric means “measured along the correct line”, which suggests that the alternatives are wrong. They are differently defined, and the one that is genuinely well-defined without assumptions is the one whose name sounds like an approximation.

Where this goes next #

The gravity field bends the vertical as well as spacing the level surfaces. The plumb line is not the normal takes the angle between the direction a bubble finds and the direction an ellipsoid defines, and shows what it does to an astronomical position — 309 metres of ground, at a point whose coordinate is perfectly correct.