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

Reduction — the series

19 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. Four steps between a tape and a drawing. A slope distance of 76.895 km measured at 3.2° on ground 220 m above sea level, reduced to the British National Grid, with every step drawn on a logarithmic scale in millimetres. The slope reduction is the largest by a wide margin at 119.90 m and is the one everybody applies. The grid reduction is 30.23 m. The fourth bar is not a step at all: it is what using the levelled height where the height above the ellipsoid is wanted costs, with a geoid separation of 48.5 m — 583 mm, which is 1.9% of the grid reduction it sits beside and 29 times the tolerance the job closes to. Reading a correction's importance off its share of the chain is how it gets dropped.

    What a tape measures

    Four steps stand between an instrument reading and a coordinate, and their sizes are not in the order anybody expects. On a 77-kilometre line the slope reduction is 120 metres, the grid reduction 30, and a step nobody names — using the height above sea level where the height above the ellipsoid is wanted — is 583 millimetres.

    part 1 · practice
  2. A closed traverse cannot see a scale error. The same closed figure with two different errors in it. On the left every leg is 400 parts per million too long, which is roughly what forgetting the grid scale factor costs — and the figure closes to 2.3e-13 m, which is the last bit of a double rather than a measurement. Scaling every leg of a closed figure by the same factor produces a similar figure and a similar figure is still closed, so the check every specification leans on is blind to it. Every dimension in that figure is wrong: the perimeter is out by 1.17 m. On the right one angle is 20 seconds out — a far smaller disturbance in its own units — and the figure fails to close by 0.062 m. Closure tests the shape and says nothing about the size.

    A traverse must close

    The check every survey specification leans on cannot see the error this whole field is about. Scale every leg of a closed figure by four hundred parts per million and it closes to the last bit of a double, while every dimension in it is wrong.

    part 2 · practice
  3. Two conventions, and neither of them finds the blunder. A blunder of 350 mm added to leg 3 of a closed traverse, producing a misclosure of 350 mm — one part in 8344, which most specifications would accept. Bowditch's rule shares the misclosure out in proportion to leg length; the Transit rule shares it in proportion to each leg's component along the axis being corrected. Both close the figure exactly, so both are valid; they disagree with each other by up to 14 mm; and neither puts more than 104 mm of correction on the leg that is actually wrong. A rule for distributing a misclosure is a convention for producing consistent numbers, and it is not an instrument for finding errors.

    The two ways to spread a misclosure

    Bowditch's rule and the Transit rule take the same closed figure and the same misclosure and disagree about which legs were wrong. Both close it exactly, neither puts the correction on the leg that actually carries the blunder, and no measurement can settle which is right.

    part 3 · practice
  4. The order the corrections go in, and what it costs. Setting out a 51.8 km line on the UTM zone 31N means turning a grid bearing into an azimuth to observe, and there are two corrections: the meridian convergence, 0.8669° here, and the arc-to-chord correction, 6.242″. Each bar carries the lateral offset it produces at the far end of the line, because a bearing error is a number nobody can picture and a sideways miss is the thing that misses. The exact arc-to-chord and the classical formula every manual gives differ by 0.0044″, which is 1.10 mm at the far end — small, real, and the reason the corrections have an order rather than a sum.

    Setting out runs the chain backwards

    Turning a design coordinate into something to observe on the ground means undoing the reduction chain, and undoing a chain reverses the order as well as the operations. Two of the corrections do not commute, and getting them the wrong way round misses by a millimetre at fifty kilometres.

    part 3 · practice
  5. Only one of a grid's five parameters changes the map. The same ground point written on the British National Grid and on UTM zone 31N, with the difference between the two coordinate pairs taken apart. The largest term by four orders of magnitude is where the two grids put zero — 5544 km, being a different central meridian, a different true origin and different false constants, all of which move every coordinate and no distance. The datum, which is the term everybody names, moves the ground 106 m. And the whole geometric difference between two transverse Mercators is their scale factors — 0.9996012717 against 0.9996 — which at this point is 50 cm. Four of a grid's five parameters are bookkeeping; the fifth is the map.

    Two grids over the same ground

    One point in the English Midlands is 406,788 east on the British grid and 167,478 east on UTM. Separating the difference properly leaves a surprise — only one of a grid's five parameters changes the map, and it is not any of the four that dominate the number.

    part 4 · practice
  6. Which corrections a 10 mm job may leave out. Each correction inverted: the line length — or, for the last row, the patch radius — at which that correction alone reaches 10 mm, for a job 150 m above the ellipsoid on 2° of ground, 0.0° from the British National Grid's central meridian. The tightest is the slope reduction at 16 m. Three of the four rows are linear in the tolerance, so this ranking is the same at a millimetre and at a decimetre; what does change it is the site — move the job up a mountain or onto steeper ground and the order is different, which is why a specification's list is not transferable.

    The tolerance decides the model

    Every correction inverts to a distance — the line length at which it alone exceeds a stated tolerance. The ranking those distances produce is the same at a millimetre and at a decimetre, and it is different for two jobs at the same tolerance on different ground.

    part 4 · practice
  7. A published coordinate is a definition being quoted. Re-observing a control point perfectly, with an instrument good to 20 mm, and comparing the answer to what is published for it. The published coordinate was fixed on a datum realisation that has since been superseded, which accounts for 106 m, and the ground it marks has moved 750 mm in 30 years at 25 mm a year. Both numbers are larger than the observation and neither is an error in it: subtracting a published coordinate from an observed one measures the interval between two definitions.

    A published coordinate is a result

    Re-observe a control mark perfectly, with an instrument good to twenty millimetres, and the answer disagrees with the published value by a hundred metres. Neither number is wrong. The difference measures the interval between two definitions.

    part 5 · practice
  8. What a 20 mm closure tolerance lets each station hide. A closed traverse of seven stations in plan, each labelled with the angle blunder that would leave the closure inside a 20 millimetre tolerance. An angle error at a station rotates everything downstream of it about that station, so the closing point moves by the distance from the station to the close — and the last station before the close stands 72 metres from it and can hide 57 arcseconds, against 2.1 at the worst-placed station. The check is not insensitive; it is unevenly sensitive, and nothing in the specification says so.

    What a closed figure cannot see

    A closed traverse imposes exactly two conditions on its observations, so everything else is free — and the freedom is not spread evenly. At a twenty-millimetre tolerance the worst-placed station in a seven-station loop hides an angle blunder of 2.1 arcseconds and the station standing seventy-two metres from the close hides 57, because a rotation about a point near the finish moves the finish hardly at all.

    part 5 · practice
  9. Two routes to one grid distance, 76.7 kilometres apart. The first four bars are the tape's chain: the chord between two marks, the height difference taken off it, the reduction to the ellipsoid, and the projection's own scale applied. The last is the satellite route — project both marks and subtract — which needs none of those steps because the observation already contains the positions the chain exists to supply. They agree to 0.17 millimetres, and that agreement is what makes the chain checkable at all: two routes to one number, sharing no arithmetic.

    The chain the satellite does not have

    The four-step reduction from a tape reading to a grid coordinate exists because a tape does not know where it is. A satellite observation does, so the same distance can be got by projecting both ends and subtracting — and the two answers agreeing to a fifth of a millimetre is the first check the chain has ever had.

    part 6 · practice
  10. 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.

    part 7 · practice
  11. 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.

    part 8 · practice
  12. 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.

    part 9 · practice
  13. 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.

    part 10 · practice
  14. 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.

    part 11 · practice
  15. 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.

    part 12 · practice
  16. 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.

    part 13 · practice
  17. 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.

    part 14 · practice
  18. 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.

    part 15 · practice
  19. 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.

    part 16 · practice

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