Cartographic Projection
every map is wrong, and by exactly how much
Concept

Levelling — where it appears

Carrying a height difference along a chain of instrument readings, each measured along the local plumb line rather than along a fixed direction. Because each reading is along the local plumb line, and the level surfaces are not parallel, the sum of the readings is not a height difference until it is corrected.

Named by 13 essays across 2 fields — each of them below, with the objects they name alongside it.

Why a levelled height is not a distance. The correction between raw levelling and orthometric height, for lines at 200, 500, 1000, 2000 metres above the geoid running north from 50°. It is not instrument error: level surfaces converge towards the pole, so a run that stays on one of them gains height relative to another. A line 2000 metres up reaches 647 millimetres over 400 kilometres. The dashed line is 10 millimetres, which is about what a first-order levelling network closes to over that distance — so this is not a refinement, it is the larger of the two numbers.

A levelled height is not a distance

Level surfaces converge towards the poles by five metres in a thousand, so a chain of perfectly executed levelling observations does not sum to a height difference. The correction over four hundred kilometres of northing is larger than the network's own closure.

datums · Height One level surface, 1000 m up at the equator, in two height systems. A single equipotential surface — the shape a body of water takes — with the number each height system gives it, all the way from the equator to the pole. The orthometric height, which is the distance up the plumb line and therefore a length, falls by 5.28 metres along it, because level surfaces converge polewards. The dynamic height, which is the geopotential number divided by one constant gravity value, is flat to 0.000 millimetres — it is the same number everywhere on the surface, and it is not a distance from anything.

A height that is not a length

Level surfaces converge polewards, so the surface a lake sits on is 5.28 metres lower at the pole than at the equator and a height system that reports lengths says a lake runs downhill. The fix reports a number that is constant on the surface and is not a distance from anything: a hundred-metre climb raises it by 99.73 metres at the equator and 100.26 at the pole.

datums · Height A mass, the geoid it raises, and the plumb lines that lean towards it. A sphere of 5 km radius buried 8 km down, denser than its surroundings by 500 kg per cubic metre — a salt dome, or an ore body. Its mass is 261.8 × 10¹² kg, and outside itself its field is a point mass's exactly, so everything above is a closed form. The geoid rises 22 centimetres over it, drawn at 20,000× the true slope; the plumb line leans by at most 2.21 arcseconds, and it does so 5.7 km to the side rather than above the body, because the deflection is the geoid's SLOPE and a slope is zero at a summit. Without a stated mass the relation between slope and deflection can convert that number but never produce it.

A deflection is the slope of a mass

One arcsecond of deflection of the vertical is 4.85 millimetres of geoid slope in a kilometre, and that exact relation produces no deflection anywhere by itself, because a deflection is the gradient of a geoid and a geoid model is cited rather than computed. State a mass instead, and every quantity is a closed form: a salt dome five kilometres across bends the plumb line by 2.21 arcseconds.

datums · Height The height of a 4000-metre summit, against the density assumed beneath it. The geopotential number is 39204 m² s⁻² and is not in doubt. Turning it into a length divides it by the mean gravity along the plumb line, which is inside the mountain — and reconstructing that from the gravity measured at the surface needs a density. Taking the rock to be 2400 rather than the 2670 it actually is puts the summit 185 mm low; taking it to be 2900 puts it 157 mm high. Skipping the reduction entirely puts it 691 mm high, which is why the reduction exists.

The line a height is measured along

Five rungs have argued about the surface a height is measured *from* and every one of them took the line it is measured *along* to be straight and known. It is neither: through a stated buried mass a plumb line arrives 47 millimetres from the point below the summit, and the height it gives depends on the density of rock nobody has seen — 342 millimetres of spread at 4,000 metres and 1.37 metres at 8,000.

datums · Height The drift is a straight line in the deflection. The horizontal distance between where a plumb line hangs at the top of a column of rock and where it hangs at the bottom, against the deflection of the vertical the mass produces at the surface. Four heights of column. Every line is straight through the origin: 12.54 mm of drift per arcsecond of deflection over a 4,000 m line, to two parts in ten thousand across a fortyfold range of deflection. Which is what makes the number transferable — the 47 mm the first calculation started from was a statement about one buried sphere, and this is a statement about any mass that produces the same deflection.

How far the plumb line bends

An earlier measurement dropped a plumb line down a four-kilometre column of rock beside one buried mass and found it arrived 47 millimetres from the point below the summit. That is a number about that mass. Parameterising by the deflection of the vertical instead — the quantity surveyors actually measure — gives 12.54 mm per arcsecond, exactly linear across a fortyfold range.

datums · Height What a geoid model leaves out, against the degree it stops at. The RMS of everything above the model's highest degree, from Kaula's rule — the statement that the normalised coefficients at degree n are about 10⁻⁵/n². The line is R × 10⁻⁵ ÷ n, so a model to degree 360 omits 17.7 centimetres and one to 2190 omits 2.9. Every orthometric height derived from such a model carries that as an error, and it is not quoted with the height.

The geoid model stops at a degree

Eleven essays treat the geoid as a surface that exists. Every geoid anybody uses is a series truncated at a degree, so every orthometric height derived from one carries an omission error nobody quotes with the height — eighteen centimetres at degree 360 — and the same truncation removes two thirds of the slope, which does not converge at all.

datums · Height The same deflection, from a compact mass and from a broad root. Each curve is the deflection of the vertical across a mass buried 8 km down, with the mass solved so that all of them peak at 10″. The narrow one is the buried sphere the earlier measurement used; the broad ones are crustal roots 40 and 160 km wide, modelled as that mass spread along a line. They agree where it matters most and disagree everywhere else: the signal is 30 km wide for the compact body and 188 km for the widest, which is the difference the plumb line feels as it descends.

A mountain is not a buried sphere

The plumb line's drift was measured over a compact buried body and grows as the 0.69 power of the column's height — an exponent that is a statement about how quickly a buried sphere's field weakens with distance rather than about mountains. Spread the same mass into a crustal root and the exponent climbs to 0.84, while the proportionality to the deflection survives exactly.

datums · Height Round a triangle the geoid's deflection has a flux and no circulation. A fifty-kilometre triangle on a stated patch of geoid whose undulations follow Kaula's spectrum between 10 and 400 km, with the deflection of the vertical drawn as strokes down the geoid's slope. Beside each side, the deflection's component across it and along it, each integrated along the side, in metres: across −0.689, −0.538, +0.499; along −0.235, +0.275, −0.040. The sum across is −0.727 m, which over the Earth's radius is −0.0235″ of the triangle's excess. The sum along is +0.000003 m — the geoid's height carried round the loop and returned to where it started.

A levelling loop reads the circulation, and the geoid has none

A triangle of angles reads the geoid as the flux of the deflection of the vertical through its sides. A levelling loop is the other integral round the same curve, the circulation, and at the geoid that one is identically zero, because the deflection is the slope of a height. What a levelling loop run on the ground misses by is the Jacobian of gravity and height over the loop: 31 mm round a fifty-kilometre loop in a stated mountain range, none of it from the free-air gradient that changes gravity by 1,333 mGal there, uncorrelated with what the triangle reads, and two and a half times the noise of the levelling where the triangle's reading is a fiftieth of its own.

impossibility · Curvature A compacting basin, and the tilt it produces. The stated vertical velocity field — a bowl of subsidence 120 km across, with no mass leaving — drawn as circles proportional to the rate, with the tilt of the ground surface as the arrows. The subsidence is largest at the centre and the tilt is exactly zero there, because a smooth bowl has no gradient at its own bottom. The largest tilt is 125.5 nanoradians a year, on a ring at the bowl's own scale length over root two, and it is the quantity a levelling network measures.

A vertical rate needs a height system

Four rungs of this anchor measure the two-by-two horizontal tensor, because that is what a tangent chart returns. The larger signal in a subsiding basin is vertical — 126 nanoradians a year of tilt against 0.76 nanostrain a year of horizontal strain — and it is not a measurement at all until the surface it is measured against is named, because that surface is moving too.

datums · Strain The mountain's gravity follows the ground and the root's does not. A profile 260 km long across the stated mountain patch. Top: the height of the ground. Bottom: the gravity at the ground from the terrain's own rock, a sheet at 2,670 kg/m³ (solid), and from a root at 30 km carrying the same mass with the opposite sign (dashed). Over the whole patch the rock's gravity correlates with the height at 0.99, so of its 54.0 mGal rms only 7.8 is not a function of the height under it. The root's, smoothed over thirty kilometres, correlates at −0.85, and of its 10.0 mGal 5.3 is free of the local height. A levelling loop misses by exactly that free part, which crosses the ground's contours.

A levelling loop weighs the root, and a triangle weighs the mountain

A mountain range is carried partly by a root of light crust beneath it, and a survey that knows the ground can ask how much. Put a root thirty kilometres down under a stated range and it moves a levelling loop's misclosure by 85 per cent as much as the mountains' own rock does, and a triangle's excess by 16 per cent: the root's gravity is a smoothed copy of the ground, and a smoothed copy crosses the contours. Read against their noise, pinning the root's share to a tenth takes about 2,700 fifty-kilometre loops, a million triangles or 550 astronomic stations — and if the rock's density is unknown too, a triangle cannot tell a compensated mountain from a lighter one.

impossibility · Curvature The sea surface, against the surface heights are measured from. The stated mean dynamic topography — how far the sea stands above the geoid — drawn on Mollweide, with the eight tide gauges marked. It ranges over 2.09 metres, it is smooth, and it is what every national datum's zero is sitting on. Nothing here is a model's output: it is a stated closed form, chosen to have the observed sign structure and the observed size.

Every country's zero is a different surface

Ten rungs measure a height against a geoid and treat the geoid as one object. No national datum is on it: each is pinned to the mean sea level at one tide gauge, the sea surface stands up to two metres from the geoid, and the eight European zeros measured here spread over 462 millimetres — a step no levelling can remove and nobody's error.

datums · Height A triangle never sees the root against its own noise; a levelling loop sees it until it is deep. For a root at each depth carrying all of the terrain's mass, the rms of each reading's response to it over 160 placements, divided by that reading's noise — the background the root sits in plus the observing error. A 25 km loop: 34.2 per cent at 10 km, 15.6 at 30, 3.9 at 60. A 50 km loop: 37.6 per cent at 10 km, 20.1 at 30, 5.5 at 60. A 100 km loop: 34.9 per cent at 10 km, 20.2 at 30, 6.1 at 60. A 50 km triangle: 3.1 per cent at 10 km, 1.0 at 30, 0.2 at 60. The triangle's response falls with depth more steeply than the loop's, which is what made their ratio look like a gauge of depth, but at every depth it is a few per cent of its noise or less.

A loop reads a root's depth from its own pattern, and a triangle reads none of it

A levelling loop and a geodetic triangle respond to a buried root in proportions that change with its depth, so their ratio looked like a gauge of how deep a range is compensated. It is not, because a triangle's response to a root at 30 km is a hundredth of its own noise. The depth is there in the loops alone: a root changes the pattern of misclosures from loop to loop as it deepens, and 1,600 loops of 50 km place a 10 km root within a factor of 1.5 three times in four and a 30 km root about half the time. A 60 km root is not placed even by 6,400, and when the depth is fitted rather than stated, such a root is read as a light shallow one carrying 25 per cent of the mass where it carries 60.

impossibility · Curvature A gravity station sees a buried root against its own noise three or four times as well as a loop near the surface, and half again as well at sixty kilometres. For a root at each depth carrying all of the terrain's mass, the rms of each reading's response to it over 160 placements on the stated range, divided by that reading's noise — the background the root sits in plus the observing error. A gravity station (noise 28.6 mGal): 111.5 per cent at 10 km, 38.5 at 30, 9.0 at 60, 1.4 at 100. A 50 km loop (noise 25 mm): 37.6 per cent at 10 km, 20.1 at 30, 5.5 at 60, 0.9 at 100. A 100 km loop (noise 39 mm): 34.9 per cent at 10 km, 20.2 at 30, 6.1 at 60, 1.1 at 100. A 50 km triangle (noise 1.7″): 3.1 per cent at 10 km, 1.0 at 30, 0.2 at 60, 0.0 at 100.

Gravity is worth four loops a station, and the background still hides a deep root

Levelling loops alone place a range's compensating root at ten kilometres three times in four from 1,600 loops, and do not place one at sixty from 6,400. Gravity measured on the ground reads the root's attraction directly, and a station is worth about four loops for a shallow root and two for one at thirty kilometres. It does not place the deep root either: 1,600 stations put a sixty-kilometre root within a factor of 1.5 of its depth in fewer than one survey in five. With the background field taken away, a hundred stations place it every time. What hides the deep root is not the reading but the geoid's own mass that the range does not explain, and either half of its spectrum is enough to do it. Fitted for depth, gravity reads the deep root as too heavy, where the loops read it as too light.

impossibility · Curvature

Named alongside it

The objects these essays reach for when they reach for this one.

Orthometric heightGeoidGeopotential numberDeflection of the verticalEquipotentialVertical datumVerificationGravity anomalyMisclosureIsostasyNormal gravityConvention

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