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

Misclosure — where it appears

The amount by which a survey loop fails to return to its starting point, which is the observations disagreeing with each other. It is the only evidence a closed figure gives about its own quality, and how it is spread is a choice with consequences.

Named by 10 essays across 4 fields — each of them below, with the objects they name alongside it.

A closed traverse cannot see a scale error. The same closed figure with two different errors in it. On the left every leg is 400 parts per million too long, which is roughly what forgetting the grid scale factor costs — and the figure closes to 2.3e-13 m, which is the last bit of a double rather than a measurement. Scaling every leg of a closed figure by the same factor produces a similar figure and a similar figure is still closed, so the check every specification leans on is blind to it. Every dimension in that figure is wrong: the perimeter is out by 1.17 m. On the right one angle is 20 seconds out — a far smaller disturbance in its own units — and the figure fails to close by 0.062 m. Closure tests the shape and says nothing about the size.

A traverse must close

The check every survey specification leans on cannot see the error this whole field is about. Scale every leg of a closed figure by four hundred parts per million and it closes to the last bit of a double, while every dimension in it is wrong.

practice · Reduction Two conventions, and neither of them finds the blunder. A blunder of 350 mm added to leg 3 of a closed traverse, producing a misclosure of 350 mm — one part in 8344, which most specifications would accept. Bowditch's rule shares the misclosure out in proportion to leg length; the Transit rule shares it in proportion to each leg's component along the axis being corrected. Both close the figure exactly, so both are valid; they disagree with each other by up to 14 mm; and neither puts more than 104 mm of correction on the leg that is actually wrong. A rule for distributing a misclosure is a convention for producing consistent numbers, and it is not an instrument for finding errors.

The two ways to spread a misclosure

Bowditch's rule and the Transit rule take the same closed figure and the same misclosure and disagree about which legs were wrong. Both close it exactly, neither puts the correction on the leg that actually carries the blunder, and no measurement can settle which is right.

practice · Reduction A published coordinate is a definition being quoted. Re-observing a control point perfectly, with an instrument good to 20 mm, and comparing the answer to what is published for it. The published coordinate was fixed on a datum realisation that has since been superseded, which accounts for 106 m, and the ground it marks has moved 750 mm in 30 years at 25 mm a year. Both numbers are larger than the observation and neither is an error in it: subtracting a published coordinate from an observed one measures the interval between two definitions.

A published coordinate is a result

Re-observe a control mark perfectly, with an instrument good to twenty millimetres, and the answer disagrees with the published value by a hundred metres. Neither number is wrong. The difference measures the interval between two definitions.

practice · Reduction What a 20 mm closure tolerance lets each station hide. A closed traverse of seven stations in plan, each labelled with the angle blunder that would leave the closure inside a 20 millimetre tolerance. An angle error at a station rotates everything downstream of it about that station, so the closing point moves by the distance from the station to the close — and the last station before the close stands 72 metres from it and can hide 57 arcseconds, against 2.1 at the worst-placed station. The check is not insensitive; it is unevenly sensitive, and nothing in the specification says so.

What a closed figure cannot see

A closed traverse imposes exactly two conditions on its observations, so everything else is free — and the freedom is not spread evenly. At a twenty-millimetre tolerance the worst-placed station in a seven-station loop hides an angle blunder of 2.1 arcseconds and the station standing seventy-two metres from the close hides 57, because a rotation about a point near the finish moves the finish hardly at all.

practice · Reduction 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 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 Adjusted to close on the second base, a sphere's drift of hundreds of ppm becomes a bow of under one. A strip of 40 thirty-kilometre equilateral triangles north from 45°, computed on the conformal sphere, between a base at each end. Unadjusted (dashed), the side error climbs to 315.0 ppm at the second base. Adjusted to close there, each triangle's share of the misclosure equal, what is left (solid) is the sphere's stretch less a straight line between the bases: 0.63 ppm at its largest, at triangle 23. Shaded: three standard deviations of what one-second angles leave in the adjusted strip, 37.6 ppm at the middle and nothing at either base.

An adjustment hides the sphere only in a strip of equal triangles

A strip computed on a substituted sphere drifts by the sphere's own stretch, 315 parts per million over forty triangles north from 45°, and an adjustment that closes it on a second base takes almost all of that away: in a strip of equal triangles what is left is the stretch less a straight line, 0.63 parts per million, a third-order bow sixty times below the noise of the old angles. A real chain is not made of equal triangles. The adjustment spreads the misclosure by how uncertain each triangle's angles are, the sphere put it in by how far north each triangle reaches, and the difference is a first-order residual that never clears the noise at any one side and that the old network's own geometry predicts exactly — so a re-observation matched against it reads the sphere at seven standard deviations.

wrong · Ellipsoid 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.

VerificationConventionInverse problemLevellingLeast-squaresToleranceAssertionBearingIdentifiabilityIsostasyNational GridNoise

All concepts