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

Noise — where it appears

The part of a measured value that is not the quantity, stated as a standard deviation and carried through every calculation that uses it. Differencing multiplies it by the reciprocal of the spacing, so a derivative taken from noisy samples gets worse as the grid gets finer.

Named by 5 essays across 4 fields — each of them below, with the objects they name alongside it.

Bigger triangles, and how much bigger depends on what is fixed. How well a survey can resolve Gaussian curvature, against the side of its triangles, under three things being held fixed. One triangle: the accuracy improves as the inverse SQUARE of the side, fitted exponent -2.000. A chain of fixed length, which is what every great arc was: bigger triangles mean fewer of them, and the exponent is -1.503 — exactly three halves. A network covering a fixed area: -0.999, exactly one. The trade depends on what a survey is short of.

How many triangles it takes

Rung eleven priced one triangle and recorded that a survey observes hundreds. Averaging n of them divides the noise by √n, and n is not free: a chain of fixed length holds fewer big triangles than small ones, so the accuracy improves as the side to the power three halves rather than two. Struve's 141 triangles of forty kilometres are worth exactly Gauss's seven of eighty-five.

impossibility · Curvature Believing a group takes away the redundancy that would test it. The thirteen degrees of freedom in this network, divided between the two groups, against the weight ratio the adjustment was told. At a ratio of a quarter the edges carry 12.1 of them and at sixteen they carry 0.07. A group's own variance can only be estimated from its own share, so a surveyor who is confident about an instrument has taken away the arithmetic that would have caught the confidence. The two curves sum to thirteen at every ratio, which is the identity that makes this a redistribution rather than a loss.

The weights are a guess the solve believes

Rung seven finds a decision inside the least-squares problem no residual can see: what to hold fixed. There is a second, made more often and thought about less. Every observation enters with a weight nobody measured, the weights move the coordinates by a factor of 1.8, and the standard check on them can be made to pass by a scaling that moves nothing at all.

practice · Reduction A finer grid makes a measured slope worse. The error in the direction of steepest ascent, against the spacing the field was sampled at, for four noise levels. With exact values the curve falls at a fitted slope of 2.00 — second order, which is what a central difference is. Add noise and the same curve turns over: a finite difference divides the noise by the spacing, so halving the grid doubles the noise in the slope while quartering an error that was already negligible. The minimum is where the two meet, and it is not at the fine end.

The slope of a field that was measured

Three rungs differentiate a formula, which is what makes the projection the only thing under test. A real field is a grid of numbers with an error on each of them, and differencing such a thing divides the noise by the spacing — so a finer grid gives a worse slope, there is a best spacing, and it is the cube root of the noise.

distortion · Gradient 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 Two places, one written coordinate. The lattice a coordinate written to 5 decimal places lives on, at 45° of latitude, where a cell is 1.11 metres north to south. Two real places a fifth of a cell apart are drawn as open marks and the single value they both round to as a filled one. The distance between them, computed from what was written, is exactly zero — an error a noise model cannot produce, since independent noise never puts two different points in the same place.

Rounding is not noise

Eight rungs treat a coordinate's error as noise that averages down. A published coordinate has a second error that does not: it is deterministic, it is shared between every point in the same cell, and at five centimetres apart ninety-three per cent of pairs come out as the same place — which no amount of independent noise can produce.

distortion · Precision

Named alongside it

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

Error budgetVerificationAveragingClosed formConditioningLeast-squaresMeasurementNetworkPrecisionResidualSamplingStandard error

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