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

The control carries the error of every network above it followed a local survey’s error up a chain of three orders and found that the survey reports eleven millimetres of it and carries fifty-nine. Most of the difference is error the local lines cannot see, because it moves all three control stations together, and the variance factor the adjustment prints rose from 1.000 to 1.036 under the whole of it.

That survey met its control in one way: the three regional stations were held at their published coordinates and the local lines were bent to fit them. There is a second way, older than least squares on this scale, which keeps the two things apart. The local network is adjusted on its own lines with nothing held but the minimum, so that its shape is its own, and afterwards the finished network is picked up and set down on the control by a transformation fitted to the three tied stations. Where the control disagrees with the network, the disagreement is left at the control, as a misfit a surveyor can read, rather than spread through the lines.

That sounds like it should be more honest, and in one precise sense it is. It is also, done the obvious way, eight times worse at locating the stations, and done with a scale it becomes blind. All three statements come out of the same stated survey, and none of them depends on a seed.

The same survey, adjusted on its own lines first #

Nothing about the survey changes. Five local stations about a kilometre apart, every line among them observed and a line from each to three regional stations ten kilometres away, twenty-five distances in all, each good to three millimetres plus two parts per million. The three regional stations carry their full published error, which the three-order propagation already computed: 26 millimetres from the regional order at each and 52 from the national, most of it shared.

What changes is the order of operations. The twenty-five lines are adjusted with all eight stations free — five local and three control — and one station and one bearing held to remove the three numbers a network of distances cannot fix, a position and a rotation. That is a minimal datum, and a coordinate is the output of a solve is the demonstration that the choice of one changes every coordinate and no residual. The result is a network whose shape is exactly what its own lines say, including a shape for the control triangle.

Then comes the fit. The free network’s coordinates for the three control stations are compared with the published ones, and a transformation is chosen to bring them as close as possible: a rigid one — two shifts and a rotation, three numbers — or a similarity, which adds a scale and has four. The transformation is applied to the whole network, which carries the five local stations onto the sheet. Three stations give six coordinates, so a rigid fit leaves three numbers of misfit and a similarity two.

In the vocabulary of a second pin is a measurement of the places, the free network is the shape and the control is a set of pins, and the fit is the registration that puts the one on the other. The fit itself is the one Helmert wrote down for joining a local triangulation to a national one in the 1880s, and it is what a surveyor’s software usually offers when a local job has to be put onto published coordinates. Two parameter sets, one transformation met its seven-parameter form between datums. The plane form with four parameters is its smallest version, and it is fitted here the way it usually is: every control coordinate weighted alike, about the control’s centroid.

Every number below is linear propagation about the stated truth, so each is exact for this survey, and — as the network’s answer is decided before it is measured showed for every quantity of this kind — each is fixed by the geometry and the weights before a line is observed. None is a sample.

Ten-kilometre ties locate the control along themselves only #

The first thing the free adjustment reports is how well the local lines know where the control is, and the answer decides almost everything that follows.

Each control station is reached by five lines from a cluster a kilometre wide. Ten kilometres out, those five lines are nearly parallel, so they fix the station’s distance from the cluster to within a centimetre or two and its position across the lines hardly at all. The ellipses at the control are needles. Along each tie the local lines know the station to a few centimetres; across it, to 179 millimetres at the worst of the three.

An equal-weight fit does not know that. It asks for the transformation that brings all six control coordinates closest to their published values, and it counts a coordinate across a tie — which the lines know to eighteen centimetres — as heavily as one along it, known to two. So the fitted shift and rotation are pulled around by the free network’s worst numbers, and the local stations go wherever those numbers send them. A local station’s own error, the part owed to the local lines and nothing inherited, is 11.2 millimetres when the control is held and 98.9 after the fit. Added to the 52 and 26 the control carries in, the station is uncertain by 115 millimetres against the national origin, where holding left it at 59.

The price rises steeply with the distance to the control. At two kilometres the ties fan out widely enough that the control is known in both directions and the fit costs half as much again. At ten it costs nearly nine times. At twenty-five, twenty-three times, and a local station that holding would place to 27 millimetres is placed by the fit to 62 centimetres — from its own lines’ error alone. The needles lengthen as the square of the distance, because the angular spread of the ties falls as the distance grows while the lateral error at the far end grows with it.

Holding is the fit that weights the control correctly #

The comparison suggests there is a right way to weight the fit, and there is: weight each control coordinate by what the local lines actually know about it, correlations included. The fit weighted that way is not a new procedure. It is exactly the held adjustment.

The identity is worth stating carefully because it removes the apparent choice. Take the free adjustment’s normal equations, add to each control coordinate an overwhelming weight that ties it to its published value, and solve. The local stations’ covariance that comes out equals the held-fixed adjustment’s report, element by element, to seven parts in 10¹³ square metres. Holding the control is the free adjustment conditioned on the control’s coordinates, and conditioning uses every correlation the lines produced.

So on position the held adjustment is not one option among three. It is the best linear use of the same lines and the same published coordinates — the same statement the weights are a guess the solve believes makes about observations, that a fit is only as good as the weights that tell it what each number is worth — and the equal-weight fit is a worse use of them: it throws away the free network’s knowledge of which of its own control coordinates to trust. Anything the fit is going to offer has to come from somewhere other than the coordinates.

Only the shape of the control triangle can be checked #

It comes from the misfit. The fitted transformation cannot bring three free-network stations exactly onto three published ones unless the two triangles are congruent, and they are not, because both carry errors. What is left over is a statement about the control that the surveyor can see.

How much of the control’s error can appear in it is fixed by a count. A rigid transformation can move the free network’s control triangle anywhere and turn it any way, so every error the published triangle shares as a whole — a common shift of all three stations, a common rotation — is absorbed into the fit and never reaches the misfit. What survives is the part a rigid motion cannot produce: the triangle’s shape, which is three numbers, its three sides. The local lines can check the control’s shape and nothing else.

The same count governed the held adjustment all along, and that is why its variance factor rose so little. Holding the control forces the local lines to meet a published triangle whose shape disagrees with theirs, and the residuals absorb that disagreement. The expected excess in their weighted sum of squares is 1.092 — of which 1.077 comes from the regional order and 0.015 from the national, whose error is almost entirely shared by all three stations. The rigid fit’s misfit carries exactly the same excess, 1.092, split exactly the same way. It has to: both are the local lines’ disagreement with the control’s shape, and there is only one such disagreement.

The difference is the denominator. The held adjustment reports its excess as part of the variance factor over all fifteen of its degrees of freedom, twelve of which are the local lines disagreeing with each other and know nothing about the control. The rigid fit reports it over the three degrees of freedom that exist at the control. The same 1.092 is a factor of 1.073 in one report and 1.364 in the other.

Counted where it lives, the control’s error is found more often #

A variance factor is read against its sampling scatter, so a larger one is only worth something if the test built on it fires more often when the control is bad and no more often when it is good.

With perfect control each test fires at its stated rate, which is the check that the statistics are what they claim. With the control carrying its published error, the held adjustment’s global test fires 8.1 per cent of the time and the rigid fit’s misfit test 12.8 — both low, because the published error is mostly shared and the shape moves little, but the second is half as sensitive again. The gap persists as the control gets worse: at twice the published error the rigid fit flags it a third of the time and the held report a fifth; at five times, 68 per cent against 57.

Nothing here is a new observation. The same lines, the same control, the same excess; the rigid fit simply refuses to average the one informative part of the residuals with twelve uninformative ones. That is also why the gain can be had without giving up holding. A surveyor who runs the minimally constrained adjustment first and the held one second can read the difference between their two sums of squares on three degrees of freedom — it is the same statistic — and keep the held coordinates, which are the better ones. Network specifications that ask for a minimally constrained adjustment before a constrained one are asking for this comparison, whether or not they count its degrees of freedom.

A disturbed monument is seen along its ties and not across them #

Published error is a spread. The failure a surveyor actually worries about is a single station that is wrong: a mark knocked by a vehicle, rebuilt on a new foundation, or moved by the ground since it was last observed.

Move one control station five centimetres along its tie and both procedures see the same thing, a non-centrality of 10.6 in their test statistic. Read on three degrees of freedom that is found 79 times in a hundred; read on fifteen, 49. Move it the same five centimetres across its tie and neither sees anything — a non-centrality of 0.06, found at the rate a perfect station is. The ties measure distance, and a station ten kilometres out moved sideways changes its distance to the middle of the cluster by nothing to first order and its distances to the five local stations by at most two or three millimetres each, nearly alike, against lines good to twenty.

So the direction a monument is disturbed in decides whether the local survey can ever know, and the choice of procedure decides only how often it notices when it can. The blunder the network cannot see found the same structure inside a network: an error is visible in proportion to how much the other observations constrain the thing it moved, and a sideways move of a distant station is constrained by nothing. What the held adjustment does with the invisible part is the reassuring half of the result. It moves every local station together by up to 33 millimetres and changes no local length by more than 0.08. A bad station ten kilometres away shifts the local survey without bending it, which is the same blindness what a closed figure cannot see found for a traverse: an error that moves everything together agrees perfectly with itself.

A fitted scale is where the control’s error hides #

The similarity fit looks like the more complete of the two. A real control network’s scale is uncertain, a local survey’s tape or instrument has a scale error of its own, and a fit that estimates the scale seems to deal with both. On this survey it does something else.

Of the excess the local lines can see — the disagreement between the control triangle’s shape and their own — 98 per cent lies in its size. That is not because the control’s error is mostly a change of size. Of the published error that falls in the triangle’s shape at all, only a fifth is a common expansion or contraction; the rest distorts it. It is because size is the one part of the shape the local lines know well. Each control station’s distance from the cluster is what the ties measure, so the lines know the triangle’s size to about nine millimetres, and its other two shape numbers depend on where the stations sit across their ties, which the lines know to the length of the needles in the first figure — about seventeen centimetres. A test reads every disagreement against what the lines know, so a two-centimetre distortion of the triangle is invisible to it and a two-centimetre change of size is not.

A similarity has a parameter for exactly the part that can be seen. It takes the disagreement into its scale, leaves 0.025 of the 1.092 in the misfit, and prints a variance factor of 1.012. The test that the rigid fit ran at 12.8 per cent fires under the similarity at barely above its false-alarm rate, and at five times the published error only 10 per cent of the time. The disturbed monument fares worse: a move along its tie is a change in the triangle’s size, and the similarity flags it at no bearing more often than it would flag a perfect station.

The error the similarity removes from the misfit does not disappear. It goes into the scale, and the scale multiplies every length in the local network. A kilometre line that owes 3.10 millimetres to its own observations and a hundredth of a millimetre to the control when the control is held owes 0.54 millimetres to the control after a similarity fit. That is the one quantity the three-order propagation found a local survey owned outright, and the similarity is the only one of the three procedures that gives some of it away. The rigid fit keeps the local lengths exactly as measured, to rounding.

The direction of a local line is the same in all three, 10.6 millimetres across a kilometre, of which 10.4 is the local network’s own orientation, and that equality is also exact rather than approximate. A network of distances takes its orientation from the angle its ties subtend at the control; holding and fitting read that angle from the same three stations with the same lines.

The spacing makes the contrast plainer. Holding the control lends a local length almost nothing at any distance except the closest, and even at two kilometres only 0.42 millimetres, because a control error moves a small cluster nearly rigidly. The similarity lends a local length the control’s relative error at the scale of the control triangle — 1.37 millimetres a kilometre at two kilometres, where the triangle is small and its relative error large, and still 0.46 at twenty-five. The disturbed monument is the same effect as a single number: five centimetres along one tie becomes a scale error of about two parts per million, and 2.69 millimetres across the longest local line, which the fit reports as a perfect fit.

A surveyor who fits a scale because the instrument might have one has made the scale estimable from the control and nothing else, and the control’s scale is exactly the thing the local lines are best placed to check. Nor does it stay local: a published coordinate is a result is about how numbers computed on a job become the control of the next one. The satellite receiver’s site calibration, which fits a local survey onto a handful of published marks with a scale among its parameters, is this procedure. Its small residuals at the marks are the part of the marks’ disagreement a scale could not absorb, and the part it could has already gone into every distance measured afterwards.

How the numbers were checked #

The fit may not depend on the free adjustment’s datum. Two minimal datums — one at a local station, one at two control stations — give fitted covariances that agree to 8 × 10⁻¹⁶ square metres and identical misfit statistics. Two minimal datums differ by a rigid motion, the fit contains every rigid motion, and the numbers must be the same.

A rigid fit may put no control error into a length. The inherited part of a local line’s length error under the rigid fit is zero to rounding.

Tying the free network to its control must reproduce holding. The free normal equations with the control’s coordinates held down by an overwhelming weight give the held adjustment’s reported covariance to 7 × 10⁻¹³ square metres.

The rigid misfit and the held residuals must carry one excess. Computed independently — one from the misfit at the control projected onto the triangle’s shape, the other from the held adjustment’s residual operator — they agree to 10⁻⁹.

Every test must fire at its stated rate when the control is perfect. In 20,000 seeded draws with no control error the three tests fired 4.99, 5.20 and 5.21 per cent of the time at a stated 5, which is within what 20,000 draws resolve. The power figures for the disturbed monument are computed from the non-central chi-square distribution rather than by sampling, and at zero displacement it returns the test size to twelve decimal places.

Where the stated survey stops #

Equal weights are the usual fit, not the only one. A fit weighted by the free network’s own covariance at the control would give up nothing on position, because that fit is the held adjustment; the price measured above is the price of the fit a surveyor usually runs, and the fitted misfit’s test is computed in the metric the free network supplies whichever weights were used to fit.

Three control stations. Six coordinates against three or four parameters leaves a check of three or two numbers, and with three numbers a disturbed station cannot be told from its neighbours: moving any one of them changes the same three sides. Four stations leave five numbers for a rigid fit, and whether that is enough to name the station rather than only to notice it is the question below.

Distances only. Every local observation is a distance, so the local network owns no orientation and can check none of the control’s. A local order with an azimuth of its own could check a fourth number, the control’s rotation, and would make the rigid fit a transformation with something to test in every parameter.

One epoch, and no scale error in the instrument. The local lines are taken as unbiased. An instrument with a real scale error would make the rigid fit’s misfit carry that as well, indistinguishable from a control triangle that is the wrong size, and would be the case in which estimating a scale is right. Whether the control or the instrument is the likelier to be the wrong size is a question about the survey rather than about the geometry.

Still open: how many control stations it takes to name the bad one #

Three control stations leave three numbers to check, and a disturbed one shows as a triangle of the wrong shape with no way to say which corner moved. Every added control station adds two coordinates and two numbers to the check, and from four stations upward a single disturbed one leaves a pattern in the misfit that the others do not share.

The measurement that would settle how many are enough is the one the blunder the network cannot see made inside a network, moved out to the control: for each control station, the smallest displacement the misfit test would find with a stated probability, and whether the station it points at is the one that moved. The ties’ geometry already says the answer will depend on direction, since no number of distant stations will see a move across its own tie. What the count of stations buys along the ties — whether a fifth control station is worth more to a local survey than five more local lines — is a question three stations cannot ask.