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

Adjustment — where it appears

The least-squares solve that turns a set of redundant observations into one consistent set of coordinates. It contains two decisions that are not measurements — what to hold fixed, and what weight each observation carries — and the residuals it reports can test the second only where the network was built to let them.

Named by 11 essays across 2 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 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

Choosing what to hold fixed is a decision inside the least-squares problem that no residual can see. 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

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 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 Ties ten kilometres long locate the control along themselves and hardly at all across. The local order of the stated survey: five stations about a kilometre apart, and the twenty-five lines from them to one another and to three control stations 10 km away. Adjusted on its own lines, the network knows where each control station lies along its ties to within a few centimetres and across them only to 179.0 mm, because five lines leaving a cluster a kilometre wide are nearly parallel ten kilometres out. The long ellipses at the control stations are that knowledge, drawn at one scale. At the local cluster, drawn at the same scale: dashed, the error the held-fixed adjustment reports for each station, 8.0 mm on its long axis at the first station; solid, the same station's own error after an equal-weight rigid fit of the free network onto the three control stations, 70.2 mm. The equal-weight fit trusts each control coordinate as much as every other, and the across-tie ones are worth almost nothing.

Three numbers are all a local survey can check of its control

Adjust a local network on its own lines, then fit it onto its three control stations, and the misfit at the control is the only place the control's error can show. It is exactly the excess the held adjustment's variance factor already carried, counted on three degrees of freedom instead of fifteen, and it catches a disturbed monument 79 times in a hundred where the held report catches it 49. Give the fit a scale and it hides 98 per cent of that excess, and lends the control's error to every length.

practice · Reduction Moved five centimetres, one station of three looks like all three, and one of five looks like itself. The local cluster at the centre, tied to three control stations (left) and to five (right) ten kilometres away, drawn on their true bearings with the distance shortened. The northern station, filled, has been moved 50 mm along its tie and nothing else is wrong. The bar at each station is the expected value of its own test for a displacement along its tie, noise-free. With three stations the three tests read 3.25, 3.22, 3.22: the data say only that the triangle is too large, and any of its corners could have made it so. With five, the moved station reads 4.31 and no other more than 2.44.

Naming the control station that moved takes a fifth

A local survey tied to distant control can tell that a monument has moved, and with three control stations it cannot say which. Each station gives the local lines one number they know well — its distance — and a shift of the whole control uses up two of them, so three stations leave one and four leave two. Five evenly spaced name a station moved five centimetres 83 times in a hundred; three name it 20. Better lines cannot stand in for the missing stations, and every station added also exposes more of the control's own published error.

practice · Reduction Told the control has an error, the test must see a larger move before it names a station. The local cluster at the centre and its 5 control stations ten kilometres away, drawn on their true bearings with the distance shortened. The shaded ellipse at each station is its published error, 53 by 31 mm on average. Outward along each tie, at the same scale, the smallest move of that station its test finds four times in five at a size of 0.001: the thick bar for a test that carries the published error, 74, 79, 78, 78, 78 mm; the thin dashed bar for a test that holds the control exact, 48, 48, 49, 48, 48 mm.

Honest about its control, the test needs eight centimetres where it needed five

A test that holds the control exact reads the control's published error as movement, and at five stations flags a good survey sixteen times in a hundred. Carry the published covariance into the adjustment and the false alarms fall to half a per cent — and the smallest move the test can find grows from 48 millimetres to 77. A five-centimetre move is then named 29 times in a hundred instead of 63, the naming rate of perfect control takes twenty-four stations, and above ten centimetres the two tests agree. What the honest test buys is a flag that means what it says.

practice · Reduction From one survey an optimistic report and a moved mark look alike. The scale on the regional network's published covariance, estimated from the residuals of one survey of five control stations, over 3,000 seeded surveys each: honest report, nothing moved, mean 0.92 and median 0.53; report twice too optimistic, mean 1.91 and median 1.43; honest report, one mark moved 5 cm, mean 4.84 and median 4.48. One survey's estimate has a standard deviation of 3.80, and a five-centimetre move along a tie adds between 3.7 and 4.2 to it: about one standard deviation, and four times what a report twice too optimistic adds.

One survey cannot tell an optimistic report from a moved mark

A local survey that carries its control's published covariance can estimate, from its own residuals, how far that covariance understates the truth. With five control stations the estimate has a standard deviation of 3.8 times the published scale, and a single monument moved five centimetres adds about as much again — four times what a report twice too optimistic would add. Run at the scale it estimated for itself, the honest test stops flooding good surveys with alarms, and names a moved mark seven times in a hundred instead of twenty-five: the move inflates the estimate and the estimate absorbs the move. The report can be checked only across surveys, and telling one twice too optimistic from an honest one takes about two hundred of them with five stations each, or fifty with twelve.

practice · Reduction Adjusted to close on the second base, a sphere's drift of hundreds of ppm becomes a bow of under one. A strip of 40 thirty-kilometre equilateral triangles north from 45°, computed on the conformal sphere, between a base at each end. Unadjusted (dashed), the side error climbs to 315.0 ppm at the second base. Adjusted to close there, each triangle's share of the misclosure equal, what is left (solid) is the sphere's stretch less a straight line between the bases: 0.63 ppm at its largest, at triangle 23. Shaded: three standard deviations of what one-second angles leave in the adjusted strip, 37.6 ppm at the middle and nothing at either base.

An adjustment hides the sphere only in a strip of equal triangles

A strip computed on a substituted sphere drifts by the sphere's own stretch, 315 parts per million over forty triangles north from 45°, and an adjustment that closes it on a second base takes almost all of that away: in a strip of equal triangles what is left is the stretch less a straight line, 0.63 parts per million, a third-order bow sixty times below the noise of the old angles. A real chain is not made of equal triangles. The adjustment spreads the misclosure by how uncertain each triangle's angles are, the sphere put it in by how far north each triangle reaches, and the difference is a first-order residual that never clears the noise at any one side and that the old network's own geometry predicts exactly — so a re-observation matched against it reads the sphere at seven standard deviations.

wrong · Ellipsoid Eight hundred surveys, each a poor estimate of the scale, find the patch where the published covariance is four times too optimistic. The middle one of nine agencies, by how much of the patch its map flags: a region covered by 800 local surveys at random, each tied to five control stations of a regional network whose published covariance is honest everywhere except a disc covering 15% of the region (dashed), where the control's true errors are 4 times its published variance; one survey in ten has a monument moved five centimetres. Filled dots: the 357 surveys whose own residuals put the scale above 2. Shaded: the 18 of 441 grid cells where the count of such surveys within 0.2 of the region's width exceeds what an honest region exceeds there one time in twenty, over the whole map; 18 of them lie in the disc.

An agency can map where its published covariance is optimistic

The local surveys tied to a regional network each estimate, badly, how far the network's published covariance understates the truth, and the agency that receives them can pool what they say. When the understatement is confined to one part of the region, pooling it into one number dilutes it: a patch covering 15 per cent of the region with errors four times their published size reads as 1.30 for the whole. Counted place by place, the same surveys find the patch — two agencies in three find it with 400 surveys and nearly every one with 800, flagging almost nothing outside it. Fed back as one number the scale removes little of the false alarms inside the patch and costs nearly a third of the test's power outside; fed back as a map it cuts the false alarms inside from 20 per cent to 3 and costs little outside.

practice · Reduction

Named alongside it

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

Least-squaresNetworkCovarianceRedundancyVerificationControl pointsReliabilityBlunderDegrees of freedomHierarchyMinimal detectable biasResidual

All concepts