Concept

Geodetic datum — where it appears

An ellipsoid together with its position and orientation relative to the body, which is what a latitude and longitude are numbers against. Two of them differ by seven parameters, each of which is the output of a fit and each of which has a width the published table does not carry.

Named by 8 essays across 3 fields — each of them below, with the objects they name alongside it.

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. The numbers are identical; only what they refer to differs.

What a coordinate refers to

A latitude and a longitude are not a place. They are a claim about a model, and the model has a shape, a position, a set of marked stones and a date — change any one of the four and the ground under the numbers moves by hundreds of metres.

datums · Datum

Why there are two latitudes, at 45°. A meridian section of an ellipsoid with the flattening exaggerated 12× — at the true value of 1/298 this outline would be indistinguishable from a circle. Geodetic latitude is the angle the surface normal makes with the equatorial plane; geocentric latitude is the angle the radius makes with it. The two lines meet the plane at different points and the angles differ by 11.55 arcminutes on the real WGS84 ellipsoid, which is about 21 km of ground distance.

The Earth is a sphere, and when it is not

Every essay before this one treated the Earth as a ball, and said so. The flattening is one part in three hundred, which is nothing for a distance, everything for a latitude, and exactly enough to make the most-used projection in the world fail the property in its own name.

wrong · Ellipsoid

Two parameter sets 100 metres apart, over the region they were fitted to. Each marker is drawn at a size proportional to how far the two transformations put it apart. The second set differs from the first by 100 metres of translation along the direction this network can least see, with the rotations and the scale re-fitted to absorb it — which is what a second agency's adjustment does when it chooses a different constraint. The worst disagreement anywhere in the region is 5.64 metres and the mean is 3.61. Applied at south-eastern Australia the same two sets differ by 193 metres, because the rotation that absorbed the translation here is a rotation of the whole Earth.

Two parameter sets, one transformation

Agencies publish seven-parameter datum transformations that differ by hundreds of metres in translation, and the usual reading is that one of them is better. Over the region either was fitted to they are the same transformation: a hundred metres of translation, re-absorbed by the rotations and the scale, moves a British coordinate by 5.6 metres and an Australian one by 193.

datums · Datum

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

OSGB36 then DHDN, done four ways. How far each route lands from the truth — applying the two transformations one after the other — at 54.5° north, 2.0° west. Composing them into one affine map is exact, because two affine maps compose into an affine map and nothing is dropped. Describing that composition with seven parameters again leaves 0.19 millimetres, because two linearised rotations compose into something with a symmetric part that seven numbers cannot hold. Adding the fourteen published numbers in pairs — which is what a chain is usually done with — leaves 7.4 millimetres. Logarithmic from a micrometre.

A chain of transformations does not close

Two datum transformations applied one after the other are not the sum of their fourteen published numbers. The gap is seven millimetres in Britain, doing the two in the other order moves the answer twenty, and the rotation matrix everybody prints is not a rotation.

datums · Datum

One meridian, two datums, and the ground between them. "the meridian line of the 141st degree of west longitude" — Anglo-Russian Convention, 1825. The line whose longitude is exactly 141° west, drawn twice: once on nad27 and once on WGS84, with the east–west separation exaggerated 3,000 times. The two are 129.0 metres apart, and remarkably constant — the shift changes by five centimetres over nine degrees of latitude — so the strip between them is a ribbon 129 metres wide and 1039 kilometres long, which is 134.0 square kilometres. The sentence has not changed. The surface the number refers to has.

A meridian boundary moves when its datum does

The 141st meridian is the one boundary description in this collection with no geometric ambiguity in it: every reading of it is the same curve, exactly. It has a different one. A longitude refers to a datum, the 1825 convention named none, and the line of longitude exactly 141° west sits 129.0 metres apart on NAD27 and WGS84 — a ribbon 1,039 kilometres long and 134 square kilometres in area.

practice · Boundary

What a seven-parameter fit's residual is made of. A published transformation accuracy is the root-mean-square residual at the common points, and here it is 1.51 metres. Almost all of it — 1.51 — is the network's own distortion, which a rigid motion and a scale cannot follow and which is present at every point of the country whether it was used in the fit or not. The transformation's own error, measured as the disagreement between the fitted parameters and the true ones over a clean grid, is 0.091 metres: 6 per cent of the quoted figure. The last bar is the control — the same fit with the distortion switched off, at 2.9e-4 metres.

What another common point buys

Rung three finds that a seven-parameter datum fit leaves a pattern rather than noise. Six per cent of the residual it reports is the transformation's own error and the other ninety-four is distortion no seven parameters can follow — so adding common points improves a term that was already small and cannot touch the one that is quoted.

datums · Datum

Millimetres of horizontal move per kilometre of assumed height. How far the transformed latitude and longitude move when the height fed into a datum transformation changes by one kilometre, for four published transformations at five places. It runs from 7.7 to 121 millimetres per kilometre. A latitude and a longitude do not carry a height, so a two-dimensional coordinate cannot be transformed until somebody supplies one that is not in it.

The height a coordinate does not carry

Ten rungs treat a datum shift as a map from one pair of angles to another. It is not one: the transformation runs through Cartesian coordinates, so the answer depends on the height — by 7.7 to 121 millimetres per kilometre depending on the datum and the place, which puts Lhasa 432 millimetres from where a height of zero would have said.

datums · Datum

Named alongside it

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

Helmert transformation Datum Convention Ellipsoid Flattening Realisation Verification Reference frame Tolerance Closed form Conditioning Coordinate semantics

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