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

Network — where it appears

A set of stations joined by redundant observations and solved together, so that no coordinate is the output of one measurement. Its geometry decides everything an adjustment can check: which blunders are visible, which weights are testable, and how much of each observation's own error shows in its residual.

Named by 6 essays across 3 fields — each of them below, with the objects they name alongside it.

Five stations, ten distances, three spare. A braced quadrilateral with a centre point. Every distance between the corners and every distance to the centre is observed, 10 in all, each with a standard deviation of 8 mm. Holding one station and one bearing leaves 8 unknown coordinates, so the network has three degrees of freedom: three independent statements the observations make that could be contradicted. Everything the adjustment can tell anybody about the quality of the work comes out of those three.

A coordinate is the output of a solve

Six essays measure a tape, close a traverse, spread a misclosure and reduce a chain. The coordinate that comes out of the far end is the solution of a least-squares problem, and the problem has a decision in it that is not a measurement: what to hold fixed. Change it and every coordinate moves by centimetres while not one residual moves at all.

practice · Reduction 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 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 Four layouts, the same five stations, the same ten distances. Every panel has five stations, all ten distances between them, the same instrument precision and the same three degrees of freedom. The ellipses are the error ellipses of the adjusted coordinates, drawn at one common exaggeration, and they are computed from the geometry and the weights alone — no observation value enters any of them. The worst semi-axis runs from 11.1 millimetres to 232, a factor of 20.9, and the difference is entirely where the marks were put.

The network's answer is decided before it is measured

Nine rungs measure what an adjustment does with observations. Every quantity a specification is written about — the error ellipses, the redundancy numbers, the smallest detectable blunder — is a function of the geometry and the weights alone, and does not contain an observed value anywhere. Four layouts of five stations with the same ten distances differ by a factor of 20.9 in their worst coordinate.

practice · Reduction 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.

AdjustmentBlunderLeast-squaresRedundancyResidualSurveyCovarianceDatumDegrees of freedomDesignError budgetError ellipse

All concepts