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

Standard error — where it appears

The formal uncertainty an adjustment reports for a coordinate, computed from the observations' assumed precision and the network's geometry. It describes what happens if every observation is merely noisy, and says nothing about what happens if one of them is wrong.

Named by 6 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 How large a blunder has to be before the test notices. A blunder of increasing size put into the least-checked observation of a braced quadrilateral, with the standardised residual it produces. The horizontal line is the critical value the test uses, and the vertical one is the minimal detectable bias — δ₀σ/√r, which is 87 millimetres for this observation and is computed from the network's DESIGN, before any observation is made. Below it nothing is flagged; above it everything is. The observation's redundancy number is 0.144, so it is checked by a seventh of an observation and hides six-sevenths of whatever is wrong with it.

The blunder the network cannot see

A least-squares adjustment has no concept of a mistake. The smallest blunder its test will find in the least-checked leg of a braced quadrilateral is 87 millimetres, and by the time it fires a station has moved by nearly ten times the accuracy the same adjustment reports for it.

practice · Reduction A transformation is more certain in some places than others. The horizontal position uncertainty a Helmert transformation carries, from the stated widths of its own seven parameters, along three meridians. It runs from 40 mm to 68 mm — a factor of 1.71 — and it falls towards the poles, because the rotation terms act on the distance from the Earth's axis. A single figure quoted for "the accuracy of the transformation" is the value at some latitude nobody wrote down.

The seven parameters have their own uncertainty

Nine essays on this ladder print a datum transformation as seven exact numbers. Every published set is the output of a least-squares fit and arrives with standard errors as much a part of the result as the parameters — and pushing those widths through to the ground gives an ellipse, not a number, that is 68 mm across at the equator and 43 mm at 70°.

datums · Datum Waiting longer buys almost nothing under the spectrum that is actually there. The scatter of block means against the block length, for the three spectra. White noise falls with a slope of -0.484 — the inverse square root everybody assumes. Flicker falls with a slope of -0.111, so a hundredfold longer average buys a factor of 1.67 rather than ten. A random walk is very nearly flat.

The error that does not average down

Five rungs give a coordinate a width, and all five assume the observations it was averaged from are independent. They are not. Under the noise spectrum every published analysis of a position time series reports, the standard error falls as the eleventh root rather than the square root — so a factor of ten costs a hundred observations if the noise is white and a thousand million if it is not.

distortion · Precision The bias is σ² over the length, over two decades. The amount by which a measured baseline is longer than the true one, against its length, for a twenty-millimetre error on each end. The line is the second-order prediction σ²/d. The measured bias times the length is constant to a factor of 1.0014 across the whole range, and it sits 8.2 per cent below the prediction — which is the fourth-order term the expansion drops. The estimate is antithetic, so the first-order scatter cancels exactly and a bias of thirty-seven microns is measured at a t-statistic of 349.

A length measured from noisy points is too long

A distance is a square root, a square root is concave, and the average of the distances is not the distance between the averages. The gap is a bias with one sign: 37 microns on a ten-metre baseline with twenty-millimetre marks, following σ² over the length across two decades, and it adds rather than cancelling — so the same boundary is 1.5 parts per million longer when it is measured in more pieces.

distortion · Precision 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 budgetPrecisionVerificationAveragingSurveyBiasClosed formLeast-squaresMeasurementNetworkNoiseTolerance

All concepts