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

Propagation — where it appears

Carrying an uncertainty through a calculation to find the uncertainty of its result. The standard sandwich AΣAᵀ is a first-order operation, so it is exact for a linear map and approximate for every projection, with an error that grows as the square of the input accuracy.

Named by 4 essays across 3 fields — each of them below, with the objects they name alongside it.

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 A local station's error ellipse is the national network's, arriving. Three orders of a stated survey, each a network of distances adjusted with the stations of the order above held fixed: a national network of seven stations 60 km apart, a regional one of ten stations inside one of its triangles, and a local one of five stations about a kilometre apart tied to three regional stations 10 km away. Every error ellipse is drawn at one scale in all three panels. Dashed: the ellipse each adjustment reports from its own observations. Solid: the full ellipse, with the error of every order above carried in. The national network's centre station is its origin and its eastern station fixes its bearing, so the one has no ellipse and the other's is a line; it reports its full error because nothing is above it. The local stations report 11.2 mm and carry 59.4 mm: their solid ellipses are the size of the national network's at that place.

The control carries the error of every network above it

A local survey is adjusted with its control stations held fixed, and the error ellipses it reports come from its own observations alone. On a stated three-order network a local station reports 11 millimetres and sits 59 from where the national origin puts it: 52 of them from the national order and 26 from the regional. Its own residuals can see almost none of that. What it does own outright is its lengths, and nothing of those is inherited.

practice · Reduction A 900 km circular accuracy at 55° north, projected. Six thousand ground positions drawn from a circular error of 900 kilometres about one place, each projected in Mercator and plotted as a displacement from the projected place. The curve is the nominal 95 per cent ellipse, computed the standard way — the ground covariance sandwiched between the projection's own derivatives. It holds 93.83 per cent of the points, the cloud is measurably longer than it along its own long axis by 5.52 per cent, and it is not symmetric: the third moment along the page's second axis is 0.727 rather than zero.

The error ellipse is not an ellipse

A positional covariance pushed through a projection is the same matrix sandwich that draws an indicatrix. That is a first-order operation on a map with a second derivative, so the propagated distribution is not the ellipse the sandwich draws — and a nominal 95 per cent ellipse holds 93.06 per cent on one projection and 95.63 on another, in opposite directions, from the same input.

distortion · Precision Seven published numbers, twenty-eight actual ones. The correlation matrix of a seven-parameter fit to 64 common points over a region 9° across. Only the diagonal is ever published — seven standard deviations — and the twenty-one off-diagonal entries are not small: the strongest is ty against rx at 0.940. A translation and the rotation that mimics it over a small patch are very nearly the same parameter, so the fit cannot tell them apart and its errors in the two are locked together.

The parameters are not independent

Rung seven gives the seven parameters their own uncertainty and stops at seven numbers. There are twenty-eight, and the twenty-one nobody publishes are not small: a translation and the rotation that mimics it correlate at 0.94, the normal matrix has a condition number of 4 × 10¹⁶, and propagating from the diagonal alone overstates the transformation's uncertainty by up to a factor of thirty-six.

datums · Datum

Named alongside it

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

CovariancePrecisionDatumError ellipseLeast-squaresHelmert transformationAdjustmentConditioningContainmentControl pointsCorrelationDegeneracy

All concepts