Concept

Error ellipse — where it appears

The two-dimensional confidence region of a position, whose axes are the square roots of the eigenvalues of its covariance matrix. It is the same object as Tissot's indicatrix, and a projection changes the shape of a reported uncertainty by exactly its own anisotropy.

Named by 9 essays across 4 fields — each of them below, with the objects they name alongside it.

One instrument, one covariance, drawn where Mollweide puts it. The same measurement is made at every point: 5 m east by 5 m north, uncorrelated, which on the ground is a circle. Each ellipse is that covariance pushed through the projection's own Jacobian and drawn 26,000 times life size. On Mollweide the axis ratio reaches 4.15, so an instrument that is equally good in every direction is drawn as though it were not.

An error ellipse is an indicatrix

A positional covariance pushed through a projection is the same matrix sandwich that produces Tissot's indicatrix, so the error ellipse drawn on a map and the distortion ellipse drawn beside it are the same ellipse. A five-metre circular accuracy is drawn at an axis ratio of 3.04 on one common projection — and the same projection draws a genuinely lopsided 304 by 100 metre error as a perfect circle.

distortion · Precision

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

A chain of stations, its positions' uncertainty and its baselines'. Nine stations held at the left-hand one, with every neighbour, second neighbour and third neighbour observed. The filled ellipses are each station's own uncertainty, which grows without limit as the chain runs away from the point it is held at — 7.5 mm at the near end and 574 mm at the far. The open ellipses above each leg are the uncertainty of the baseline, drawn at the same scale: they hardly grow at all, because almost everything that is wrong with one end is wrong with the other in the same direction.

The difference of two coordinates

Three essays give a single coordinate a width. Every practical use of one is a difference of two — a distance, a bearing, a movement, an area — and the width of a difference is not the two widths combined, because the errors are not independent. Far from its datum a one-leg baseline is six times more certain than the positions it joins.

distortion · Precision

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

An adjustment is usually judged by what it 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

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

The triangle and the ellipse are two pictures of the same three sights. Three bodies at azimuths of 20°, 140°, 250° with independent errors of 2 km, drawn twice: as the cocked hat their lines bound, dashed, and as the 95 per cent confidence ellipse of the least-squares point, outlined. The ellipse is 4.25 by 3.78 km and covers 50.5 km²; the average hat from the same sights covers 7.06. The triangle is seven times the smaller and holds the ship a quarter of the time; the ellipse holds it 95 per cent of the time, which is what it was constructed to do. The hat is a picture of how much the sights disagree, and the ellipse is a picture of where the ship is.

A confidence ellipse is honest only where the sheet keeps angles

The cocked hat covers seven square kilometres and holds the ship a quarter of the time; the confidence ellipse from the same three sights covers fifty and holds it ninety-five per cent, which is what it was built to do. On a plotting sheet whose longitude is not corrected for latitude that ellipse fails in two unrelated ways — its centre is displaced, which costs coverage and depends on the navigator's own assumed position, and its shape is misread, which costs nothing but meaning and depends only on the latitude.

paths · Position fix

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 $Weighting the low sight turns the region the fix is in. Four bodies, three of them between 44° and 58° up on one side of the ship and one 5° up behind, with each sight's error made of a sextant part of 0.5′ and a tenth of its refraction correction: 0.94, 0.95, 0.93 km for the high bodies and 2.05 km for the low one. Solid: the 95 per cent ellipse of the fit that weights each sight by its variance, 2.30 by 1.55 km. Dashed: the ellipse a navigator draws who gives every sight the same variance, 2.63 by 2.02 km, its long axis turned 20.1° from the weighted one. Dotted: where that unweighted fix actually scatters, 2.98 by 1.57 km, turned 27.7° from the ellipse drawn for it. All three are centred on one point.$

A low sight is worth keeping only if it is weighted

A body five degrees above the horizon is seen through so much air that its refraction correction is ten times a high body's, and a tenth of that correction uncertain doubles the sight's error. Given the same weight as the others, a sight below seven degrees makes the fix worse than throwing it away; weighted by its variance it never does. The ellipse drawn without weights is turned twenty degrees from the right one, and a navigator who must guess the weights loses far less by guessing high.

paths · Position fix

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

Named alongside it

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

Covariance Least-squares Precision Datum Redundancy Adjustment Degrees of freedom Network Propagation Verification Blunder Conformality

All concepts