The line a height is measured along

Every rung of this ladder has argued about the surface a height is measured from. Above the ellipsoid or above the geoid; above a level surface or above a mathematical one; the third coordinate moving when a datum shifts; a deflection being the slope of a mass.

Height above what? opened the ladder on that question and the plumb line is not the normal sharpened it. None of them has looked at the line the height is measured along. All five take it to be a straight vertical of known length, and it is neither.

The curvature costs nothing in length #

That last number is worth taking seriously before the rest. A line that bends by two seconds of arc over four kilometres is longer than a straight one by about three ten-millionths of a metre, because a path’s excess length over a chord is second order in its deflection and two arcseconds is 10⁻⁵ radians.

The asymmetry is the finding. A quantity that enters a length at second order and a position at first order will always behave this way, and it is the reason the curvature of the plumb line is absent from every treatment of orthometric heights and present in every treatment of astronomical position.

So the curvature of the plumb line is irrelevant to the length of the height and decisive for the position of its foot. An orthometric height is a distance along that curve, and to every precision anybody works to it is the same as the distance along the straight line. But the geoid point it starts from is 47 mm away from the point directly beneath the summit, and 47 mm is a great deal in a levelling network — larger than the closure a levelled height is not a distance treats as the network’s own error.

The control is the half that makes the measurement mean anything: with the offset mass removed, the line comes back straight to 10⁻¹⁴ metres. Nothing bends a plumb line except a mass beside it, which is a deflection being the slope of a mass integrated instead of differentiated.

The number that is not in doubt #

The geopotential number of a point is the difference in gravity potential between it and the geoid, and it is the honest quantity here for a reason that no assumption can undermine: a potential difference does not depend on the path. Any route from any point of the geoid to the summit gives the same number, because the geoid is a level surface and every point of it has the same potential.

For a point 4,000 m above the geoid at 46° north it is 39,204 m² s⁻². That is a measurement, it comes out of levelling and gravity observations along any route the surveyor pleases, and nothing in it is assumed.

Turning it into a length is where the difficulty is. H = C / ḡ, and ḡ is the mean gravity along the plumb line between the geoid and the point — which is inside the mountain, where no gravimeter has ever been.

Reconstructing what nobody can measure #

Gravity at the surface is observed. Getting from that to the mean along the line inside the rock is the Poincaré–Prey reduction, and it has three steps:

1. remove the attraction of the rock above the geoid — the Bouguer slab, 2πGρH;
2. apply the free-air gradient to move down to the midpoint of the line;
3. restore the slab.

Steps 1 and 3 do not cancel, because the point has moved between them, and what is left is

ḡ = g_observed + (0.15437 × 10⁻⁵ − 2πGρ) · H

At ρ = 2670 the bracket is 0.0423 × 10⁻⁵ m s⁻² per metre, which is Helmert’s published coefficient — and this is where that number comes from. It is not a property of the Earth. It is a property of an assumed density.

Crustal rock runs from about 2,400 kg/m³ for sedimentary material to about 2,900 for basalt, and what is under a given summit is not known. So the orthometric height of a 4,000-metre peak is uncertain by 342 millimetres, and no measurement of any kind reduces it — the uncertainty is in the model, not in the data.

It is quadratic, so it matters where mountains are #

At 500 m the ambiguity is five millimetres and nothing anybody does with heights notices it. At 8,000 m it is 1.37 metres, which is larger than the difference between any two published values for the height of a famous mountain, and larger than the accuracy those values are quoted to.

The quadratic law is the reason this appears where it does. Both terms are a fractional correction proportional to H applied to a height of H, so they grow as H², and a country whose highest ground is 500 m can ignore the whole subject while one whose highest ground is 5,000 m cannot.

What it does to a levelling network #

The consequence for practice is not that a summit is quoted wrongly — a summit’s height is a number in an almanac and nobody builds anything on it. It is that two points at different heights have their difference computed with different mean gravities, and a network spanning a mountain range therefore has a systematic error that depends on the terrain.

A benchmark at 2,000 m and one at 500 m differ, under the two extreme density assumptions, by 86 − 5 = 81 millimetres of their height difference. That is not a random error and no amount of redundancy in the network removes it: every observation in the network carries it in the same direction, because they all share the same assumed ρ. It is exactly the kind of error what a closed figure cannot see is about — a misclosure check passes, because the error is a property of the reduction rather than of the measurements.

What a height system can do instead #

Two of the alternatives this ladder has already met dispose of the problem entirely, and it is worth saying which and why.

The dynamic height divides C by a single constant, γ at 45°, chosen once and applied everywhere. It is constant on a level surface — which is the property a height that is not a length was about — and it involves no density at all, because there is no line to average along. It is unambiguous and it is not a distance from anything.

The normal height divides C by the mean normal gravity along the normal plumb line, which is a computed quantity in a stated field with no rock in it. It is unambiguous too, and it is a distance above a surface — the quasi-geoid — which is not the surface water settles on.

The orthometric height is the one that is a distance above the surface water settles on, and it is the one that needs the rock. That is the trade, and it is forced: a height above the geoid, measured along the real plumb line, is a statement about the interior of the Earth whether anybody wants it to be or not.

Which is why the countries that have changed height system in the last forty years have mostly changed to normal heights, and why the ones that have not are the ones with the flattest ground.

What was computed, and how #

The vertical structure is the Bouguer slab: a point at height z inside a slab of thickness H standing on the geoid feels the slab’s own attraction 4πGρ(z − H/2) as well as the free-air field. The total gradient inside the rock is −0.3086 + 4πGρ mGal per metre, which at ρ = 2670 is −0.0847, and the classical Poincaré–Prey gradient is −0.0848. That agreement is not an input; it is the check.

Gravity at the summit is computed once from the normal field plus the slab’s attraction at its own true density and then treated as an observation — so the reconstruction is genuinely blind to the truth, and assuming the true density returns the true height. It does, to 3.6 millimetres out of 4,000 metres, and the residual is the free-air gradient’s own linearisation.

The plumb line is integrated downward by following the direction of gravity — the slab’s field for the vertical, a buried sphere’s for the horizontal — with the sideways drift accumulated step by step. The bend is reported as the change in the line’s own tilt during the descent rather than as the tilt itself, because the tilt at the summit is a deflection of the vertical and is a different quantity.

The assertions carry four refusals. The height must move with the assumed density; the reduction must be worth more than the density ambiguity is, or it would not be worth doing; Helmert’s published coefficient must reproduce the slab model at 2670; and the plumb line must be straight when the offset mass is removed, which is what rules out an integration error masquerading as a physical effect.

What is safe to say about a summit #

Three statements about the height of a 4,000-metre peak, in order of how much they assume.

Its geopotential number is 39,204 m² s⁻². That is measured, it is path-independent, and no assumption about rock enters it. It is also not a length, and nobody outside geodesy has ever heard of it.

Its normal height is a length, and it is a length above a computed surface — the quasi-geoid — which is not the surface water settles on. It is unambiguous, it needs no density, and it is what most countries that have modernised their vertical datum now use.

Its orthometric height is a length above the surface water settles on, and it is uncertain by 342 millimetres because the rock beneath it has never been sampled. It is the number in the almanac, it is the one everybody means, and it is the only one of the three that requires knowing something nobody knows.

Where the model stops #

The slab is infinite and a mountain is not. A real summit has terrain falling away on every side, which reduces the attraction of the rock below the point and changes the reduction. That is the terrain correction, it is what separates Niethammer’s and Mader’s definitions of the orthometric height from Helmert’s, and computing it needs a digital elevation model — which is data, and is ruled out here for the same reason a geoid is.

The density is treated as uniform. It is not, and a summit made of a light sedimentary cap over a dense basement has a mean gravity along its plumb line that no single ρ reproduces. What is measured here is the sensitivity to the number, which is the right thing to measure, and it is a lower bound on the real ambiguity rather than an estimate of it.

And the horizontal deflection uses one buried mass at one offset. The 47 millimetres is a number about that model, and real deflections of the vertical run to tens of arcseconds in mountainous country — ten times what is used here, and therefore roughly half a metre of drift rather than five centimetres.

The density’s persistence is worth one more sentence, because it is the same shape as everything else this collection audits. 2670 kg/m³ was a reasonable mean for continental crust when it was chosen and it is still a reasonable mean. What changed is that it stopped being quoted as an assumption and started being a coefficient — 0.0424 — with no density visible in it at all, and a coefficient does not invite the question a density does.

Who found it, and when #

The problem is as old as levelling with gravity in it. Poincaré and Prey’s reduction dates from around 1900 and Helmert’s treatment from Die mathematischen und physikalischen Theorien der höheren Geodäsie of 1884; the density 2670 kg/m³ became the standard through Hayford and Bowie’s isostatic work in the 1910s, and has been the number in the formula ever since — chosen as a reasonable mean for continental crust, and inherited as though it were measured.

That is the shape of the finding and the reason it belongs in a collection about maps. A convention adopted for a good reason, carried for a century, and quoted as a property: it is the scale factor was chosen again, and a published coordinate is a result again, in the vertical.

What the density assumption is worth, as a number #

The complaint about 2670 kg/m³ is that it stopped being visible, and the useful form of that complaint is a figure for what its invisibility costs — which is available without any density model at all.

The correction is linear in the density. The mass column above the point is what bends the plumb line and what changes the gravity along it, and doubling the density doubles the effect. So a relative uncertainty in the density is the same relative uncertainty in the correction, exactly, with no modelling in between.

Continental rock runs from about 2,400 to 3,000 kg/m³ — sediments at the low end, mafic rock at the high — which is roughly ±11 per cent about the standard value. That is not a tail: it is the ordinary range of what a levelling line actually passes through.

So the orthometric correction carries an eleven per cent uncertainty that nothing in its statement reveals, and it is a systematic eleven per cent over whatever region shares a geology rather than a scatter that averages out along a line. A network run through a sedimentary basin and one run over a volcanic province are wrong in opposite directions, consistently, and the discrepancy appears when the two are joined.

Which puts a number on when it matters. The correction itself is small in gentle country and grows with the square of the height, so eleven per cent of it is negligible where the correction is negligible and is the dominant error where the correction is large. A mountain traverse is precisely the case where the correction is worth applying and the case where the assumption inside it is worth doubting, and those two facts are usually presented in different chapters.

There is a cheaper half-remedy available to anybody joining two networks, and it needs no geology. Recompute both with the density moved to each end of the plausible range and see how far the junction discrepancy moves. If it barely moves, the density is not what separates the two networks and the disagreement is somewhere else. If it moves by most of the discrepancy, the two networks were reduced through different rock and the standard constant is the thing to stop sharing.

And the remedy is not a better constant. It is quoting the correction with its density stated — 0.0424, at 2670 kg/m³ rather than 0.0424 — so that a user working over rock they know something about can substitute, and a user who cannot at least knows that a substitution exists. The coefficient hides an input; the input has a range; the range is the error bar nobody publishes.

That is a two-run experiment on data already adjusted, and it turns a suspicion about geology into a measurement of whether geology is the culprit.

Where the ladder goes next #

Six rungs have asked what a height is above, what it is measured along, and what it is not. What none of them has asked is what happens when the surface moves: the geoid is a snapshot of a mass distribution that changes with the seasons, the ice and the tides, and a height system whose zero is a tide gauge is a height system whose zero is drifting — which is the epoch being part of the coordinate applied to the one coordinate that was thought to be safe.