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

Geoid — where it appears

The equipotential surface of the Earth's gravity field that best matches mean sea level, and the surface heights above sea level are measured from. It departs from the ellipsoid by up to about a hundred metres and its slope is the deflection of the vertical.

Named by 16 essays across 3 fields — each of them below, with the objects they name alongside it.

Three surfaces, and the two heights between them. The ellipsoid, the geoid and the ground, with the two heights a coordinate can carry. Ellipsoidal height h is what a satellite fix returns and is measured from a surface defined by four constants. Orthometric height H is what a level and a staff measure and is referred to the geoid — the equipotential surface that best fits mean sea level. They differ by the separation N, drawn here as 45 metres because that is a stated input rather than a computed one: a geoid model is a data product with a truncation degree in it, and this site computes rather than downloads. The arithmetic h = H + N is exact whatever N is.

Height above what?

A satellite reports a height above a mathematical surface. A level and a staff report a height above the surface water settles on. The two disagree by tens of metres, both are correct, and only one of them decides which way a pipe drains.

datums · Height Normal gravity, derived from four constants. Gravity on the surface of the level ellipsoid, by Somigliana's closed form, for WGS84. Nothing here is measured: a, f, GM and ω go in and the whole curve comes out, rising 5186 milligal — 0.53 per cent — from equator to pole. The two open marks are the published values of equatorial and polar gravity for WGS84, which the derivation reproduces to ten significant figures rather than borrowing.

The ellipsoid is a level surface

WGS84 publishes two dozen constants and defines four of them. The other twenty are consequences — polar gravity, the potential of the ellipsoid, the coefficient that dominates the Earth's gravity field — and every one comes back here from a, f, GM and ω to the last digit published.

datums · Height Four steps between a tape and a drawing. A slope distance of 76.895 km measured at 3.2° on ground 220 m above sea level, reduced to the British National Grid, with every step drawn on a logarithmic scale in millimetres. The slope reduction is the largest by a wide margin at 119.90 m and is the one everybody applies. The grid reduction is 30.23 m. The fourth bar is not a step at all: it is what using the levelled height where the height above the ellipsoid is wanted costs, with a geoid separation of 48.5 m — 583 mm, which is 1.9% of the grid reduction it sits beside and 29 times the tolerance the job closes to. Reading a correction's importance off its share of the chain is how it gets dropped.

What a tape measures

Four steps stand between an instrument reading and a coordinate, and their sizes are not in the order anybody expects. On a 77-kilometre line the slope reduction is 120 metres, the grid reduction 30, and a step nobody names — using the height above sea level where the height above the ellipsoid is wanted — is 583 millimetres.

practice · Reduction 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 A deflection of 10 arcseconds, and what it hides. The ellipsoid normal and the plumb line at one point, with the geoid tilted against the ellipsoid by 10 arcseconds — drawn 3000× steeper than life, because at true scale the two lines are indistinguishable. The relation is exact and linear: an arcsecond of deflection is the geoid rising 4.85 millimetres in a kilometre, so 10 arcseconds is 48.5 millimetres per kilometre. A star sight measures the plumb line's direction, so astronomic latitude differs from geodetic by exactly this angle — 309 metres of ground, at a point where the coordinate itself is correct.

The plumb line is not the normal

A latitude measured from the stars and a latitude that means a position on the ellipsoid are different angles, because a plumb bob hangs along gravity and gravity is not perpendicular to a mathematical surface. Ten arcseconds of difference is 309 metres of ground.

datums · Height The two reductions, at a grid factor of 1.0004. A measurement made on the ground has to be brought to the ellipsoid and then to the grid, and the two corrections have opposite signs. The elevation factor is R/(R+h) and always shrinks; the grid factor here is 1.0004, which stretches. Their product is the only number a surveyor can use. They cancel exactly at 2551 metres. At 1500 metres the combined factor is 165 parts per million, which is 1.65 metres on a 10 kilometre baseline.

The ground is not the grid

A tape measure on a hillside has to be brought down to the ellipsoid and then out onto the map, and the two corrections have opposite signs. On a grid whose scale factor exceeds one there is exactly one elevation where they cancel — 2,551 metres, for a factor of 1.0004.

datums · Height The same coordinate on four datums. One pair of numbers — 2.0° west, 54.5° north — read as a coordinate on OSGB36, ED50, NAD27 and on WGS84, and plotted where each reading puts the mark on the ground. The spread runs to 195 metres, and the height changes by 51, 44, 47 metres, which is the component nobody quotes.

The third coordinate moves too

A datum shift is quoted as a horizontal displacement because horizontal is what people look at. The transformation acts on a three-dimensional point, and its vertical component is between a quarter and a half of the horizontal one — 51 metres, for a British coordinate.

datums · Height The equator, with its ellipticity exaggerated forty thousand times. The dashed circle is the equator every projection formula on this site assumes. The solid curve is the equator satellite geodesy reports, drawn with its departure multiplied by 40,000 so that seventy metres on a six-thousand-kilometre radius can be seen at all. The long axis is at 14.9° west and the short one ninety degrees from it, and the difference between them is 70.0 metres — a real quantity, about the height of a twenty-storey building, on a body every geodetic computation treats as a surface of revolution.

The equator is not a circle either

Eleven rungs price what pretending the Earth is a sphere costs, and every one of them replaces the sphere with a surface of revolution — a body whose equator is a circle. It is not. The two equatorial radii differ by seventy metres, the two surfaces part by thirty-five, and the auxiliary latitudes every ellipsoidal formula is written in stop existing.

wrong · Ellipsoid Shape predicted from field, against shape as published. Clairaut's theorem gives a body's flattening from two numbers of its gravity field: J₂, which is how its mass is arranged, and m = ω²a³/GM, which is how fast it spins. For a body in hydrostatic equilibrium the prediction is the shape, and the diagonal is where such a body sits. Earth is on it to 0.05 per cent — 12 metres at the pole, out of twenty-one kilometres of flattening. Mars is 12.5 per cent off it, which is 2.23 kilometres, and the excess is Tharsis: a body carrying a continent-sized volcanic load is not a fluid figure, so its ellipsoid is not one of its own level surfaces, and its zero of height has to be chosen rather than found.

A body with no sea level

On Earth the zero of height is found rather than chosen — water settles onto the equipotential surface by itself. Nowhere else has one, and the difference is measurable: Clairaut's theorem predicts the Earth's flattening from its own gravity field to twelve metres at the pole and misses Mars's by 2.2 kilometres.

datums · Bodies 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. That is the number this site has been able to relate and unable to compute since its practice phase.

A deflection is the slope of a mass

The site has computed the exact relation between a geoid slope and a deflection of the vertical — one arcsecond is 4.85 millimetres in a kilometre — and computed no deflection anywhere, because a deflection is the gradient of a geoid and the geoid is a citation. 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 ladder 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

The previous rung 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 baseline, turned. The part of a 20 km height difference a degree-360 geoid model omits, against the direction the baseline runs, at four anisotropy ratios. A ratio of one is the model rung 9 used and is a flat line — the isotropic covariance cannot depend on a direction, by construction, which is the whole of the objection. At a ratio of two the same baseline omits 165 millimetres along the grain and 241 across it.

The correlation is not the same in every direction

Every number in the previous rung came out of Σ cₙ Pₙ(cos ψ) — a covariance that depends on the angular distance and nothing else. Ground has grain: at a modest anisotropy the same 20 km baseline omits 165 millimetres along it and 241 across, and the isotropic answer understates the worse direction by 18.4 per cent.

datums · Height 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 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

Named alongside it

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

Orthometric heightEquipotentialVertical datumLevellingRealisationToleranceDeflection of the verticalDatumEllipsoidal heightGeopotential numberVerificationConvention

All concepts