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

Naming the control station that moved takes a fifth found the count that lets a local survey say which of its distant control stations has moved: each station gives the local lines one number they know well, its distance, and a shift of the whole control uses up two, so five stations are the first count whose tests stop being copies of one another. With perfect control, five stations name a monument moved five centimetres along its tie 83 times in a hundred.

The control is not perfect, and the same essay found what that does. A published coordinate is a result, with an error of its own, and each station carries the regional network’s published error, and a test that treats the control as exact reads that error as movement: with five stations and nothing moved it flags a good survey 16 times in a hundred, and with eight, 20. Every station added lets the local lines see one more number about the control, and the published error has a component in each.

The remedy is standard and was named at the end of that essay: tell the test the control has an error, so that the published error is expected rather than flagged. What was not known is what the remedy costs. A test that expects the control to disagree by its published error must see a move clear that disagreement before it fires, so it will find moved monuments less readily, and the question worth answering is how much less and whether it is worth it.

Control as an observation with a weight #

The survey is unchanged. Five local stations a kilometre apart, twenty-five lines between them each good to three millimetres plus two parts per million, and control stations ten kilometres out spaced evenly round the cluster, three of them first and then four, five, six, eight, twelve, sixteen and twenty-four. Every control station’s published coordinates carry the error the three-order propagation gives them: about sixty millimetres at each station, in an ellipse roughly 53 by 31 millimetres, most of it shared between stations and thirty to forty millimetres of it surviving in the difference between any two.

The exact test holds those coordinates fixed and adjusts the local stations to them. The honest test does what the control carries the error of every network above it called weighting the control instead of holding it. The control’s coordinates become observations in their own right, alongside the twenty-five lines, and their weight is the inverse of their published covariance. The adjustment then solves for the local stations and the control together, moving the control by as much as the local lines have evidence to move it and no more.

The test for a moved station is the same question as before, asked of the new observations. Suppose control station j has moved a distance $\nabla$ along its tie. In the honest adjustment that is a blunder in two of the observations — the control station’s own published coordinates are wrong by $\nabla$ along one direction $h$. The residuals $v$ carry some of it, and the statistic is

$$w_j = \frac{h^{\mathsf T} P\, v}{\sqrt{h^{\mathsf T} P\, Q_v P\, h}},$$

where $P$ now holds the lines’ weights and the inverse of the control’s published covariance in one matrix, and $Q_v$ is the residuals’ covariance under the same model. It is the statistic the blunder the network cannot see used for each distance of a braced quadrilateral, and it is standard normal when nothing has moved and the control carries exactly its published error, which is the whole point: the null hypothesis now includes the error the exact test was mistaking for movement. The critical value is the same 3.29 for a test of size one in a thousand, and the procedure is the same data snooping — test every station, take the largest, name it if it passes.

The alarms go back to their stated rate #

The first thing to check is that the honest test does the one job it was built for. With the control carrying its published error and nothing moved, it flags a survey 0.2 per cent of the time with three stations, 0.5 with five, 0.6 with eight and 1.8 with sixteen. Every one of those sits on the dotted line of k tests each of size one in a thousand. The exact test on the same trials flags 2, 16, 20 and 23 per cent.

That difference is not a tuning. The exact test’s false alarms were never a property of the local lines or of the moved-station hypothesis; they were a mismatch between the distribution the test assumed and the one the residuals had. The published covariance is the missing term in that distribution, and with it in place the statistic is what the textbook says it is. At sixteen stations the honest test’s rate creeps a little above its promise, 1.8 per cent against 1.6, and four thousand trials cannot tell that from the promise.

The price is thirty millimetres of reach #

What a test can find is summarised by its reliability — the smallest move it catches four times in five at its stated size, which is $\delta_0 / \sqrt{h^{\mathsf T} P Q_v P h}$ with $\delta_0 = 4.13$. The exact test at five stations finds 48 millimetres at every station. The honest test finds 74 at the northern station and 78 or 79 at the other four.

The published ellipses in the figure are drawn at the same scale, and they are not what the honest test has to clear. At one standard deviation they reach 30 to 52 millimetres along the ties, and a test that had to clear all of that at 4.13 standard deviations would need moves of 120 to 210 millimetres before it fired. It needs 74 to 79, because most of each station’s published error is shared with its neighbours, and a shared error moves the whole control — which the local lines cannot see, so the honest test has nothing to clear there. What it must clear is the part of the published error that differs from one station to the next along the ties. Expressed as the standard deviation of the test’s own estimate of a move, that is 18.7 millimetres for the honest test against 11.6 for the exact one: the local lines’ uncertainty about where a station sits along its tie, with the control’s differential error added to it.

More stations shrink both reaches and never close the gap. The honest test goes from 92 millimetres at three stations to 77 at five, 65 at eight and 50 at twenty-four; the exact test from 64 to 48, 43 and 39. The difference is 28 millimetres at three stations, 29 at five, and narrows only slowly beyond — 11 millimetres at twenty-four. The honest curve also has a flat step between six and eight stations, 65 millimetres at both, where the exact curve keeps falling: the reach of a station depends on how its own published ellipse lies against its tie and against its neighbours’, and the two added stations happen to buy nothing on average.

So the honest test’s reach improves faster than the exact test’s as stations are added, and the gap closes; it does not close at any count a survey would build. The exact test never had to learn anything about the control’s error, because it assumed the answer was zero, and every millimetre of the gap is the price of not assuming it.

At five centimetres, half the naming and a quarter of the blame #

Five centimetres is the move the earlier measurement was built round, and it sits in the worst place for the honest test: just above the exact test’s reach of 48 millimetres and well below its own of 77. The rates show it. With five stations carrying their published error, the exact test names the moved station 63 times in a hundred and names a different station 17 times. The honest test names it 29 times and a different station 4.

Both halves of that matter, and they point in opposite directions. The honest test finds the move half as often. It also blames the wrong monument a quarter as often, and the wrong monument is the expensive outcome: a surveyor who drops a good control station and refits has made the survey worse — a coordinate is the output of a solve, and removing a good constraint changes every one of them — and has recorded, in the adjustment report, a reason to distrust a mark that is fine.

The last row is the one that makes the trade unavoidable. When the control happens to be perfect — its published error is a statement about what it might be, and on a given day it can be much better — the honest test still names the move only 23 times in a hundred. The honest test cannot know that the control is good today. It has been told how good the control usually is, and it tests against that. The reach is lost whether or not the published error is present, because it is spent on the possibility.

Past ten centimetres the two tests agree #

Sweeping the size of the move shows where the disagreement lives. At a hundred millimetres both tests name the moved station nearly every time, 98 and 99 per cent, and neither names another. At seventy-five the honest test names it 78 per cent of the time against 92; at sixty, 50 against 79; at forty, 14 against 45. Everything the two tests disagree about lies between about two and nine centimetres, and that is the band where a move is a few times the test’s own standard deviation — the band where a moved monument and the control’s ordinary differential error are hardest to tell apart.

The thin lines show the other half of the same range. The exact test names the wrong station most often for moves of two to five centimetres, in about a fifth of all trials — 20 per cent at 25 millimetres, 19 at 40, 17 at 50; the honest test’s wrong-station rate never passes five per cent. At no size is the exact test more reliable per flag. It is more sensitive, and all of its extra sensitivity is in the range where it is also mistaken most.

Twenty-four stations to buy back what the covariance cost #

The earlier measurement’s headline was that five stations name a five-centimetre move 83 times in a hundred. That number belonged to perfect control. The honest test reaches it at twenty-four stations: 29 per cent at five, 44 at six, 55 at eight, 69 at twelve, 76 at sixteen and 85 at twenty-four.

Twenty-four control stations round one kilometre of local survey is not a design anybody would build, and that is the practical content of the figure. At the five-centimetre scale a local survey cannot buy its way back to what the exact test appeared to promise. The appearance was partly real — the exact test does name more moves — and partly an artefact of a null hypothesis that did not hold, since at five stations one flag in every five from the exact test fell on a station that had not moved. A survey that needs to name five-centimetre moves reliably needs control whose differential error is well under five centimetres, which is a statement about the regional network rather than about the local one.

The count still decides which stations can be told apart #

The honest test changes how far each station’s test reaches. It does not change which stations can be separated, and the reason is that separability was never about the control’s error. Three stations leave one well-known number once a shift is removed and four leave two — the count three numbers are all a local survey can check of its control began; that count comes from the geometry of long ties, and carrying a covariance on the control’s coordinates does not give the local lines any new direction to look in. Three stations’ honest tests correlate at 0.984, four stations’ at 0.988 — still one test written several times.

From five stations upwards the honest tests correlate slightly more than the exact ones — 0.613 against 0.567 at five, 0.394 against 0.332 at eight. The published error is mostly shared, and a shared error ties every station’s test to the others’. Carrying it makes each test a little less a statement about its own station and a little more a statement about the control as a whole. So the step between four and five that the earlier essay found is still there, and five stations are still the first count that can name a station at all; what the covariance changes is the size of move it takes to do so.

What a flag is worth depends on how often marks move #

The earlier essay’s question was whether a survey is better served by a test that is honest about its control or one that is sensitive to it. The trials above supply every rate except one, and the missing one is not something any adjustment can estimate: how often a control monument has really moved. It depends on traffic, on foundations and on the ground itself, which in some places carries every mark tens of millimetres a year, as the epoch is part of the coordinate measures. So it goes on the axis.

A flag is useful to the extent that it names the monument that moved. If one survey in twenty contains a moved station, 68 per cent of the honest test’s flags name it correctly; the exact test’s flags do so 17 per cent of the time, because its 16 per cent of false alarms on the nineteen good surveys outnumber its hits on the one bad one. Half the exact test’s flags are right only once a moved station turns up in more than a quarter of all surveys. Even at the extreme where every survey contains one, the honest test’s flags are more often right — 87 per cent against 79 — because its wrong-station rate is lower.

The dotted lines are the exact test’s side of the case. Across all surveys, it finds more moves: at a base rate of one in two, it names the moved station in 32 surveys of a hundred where the honest test names it in 14. A survey that treats every flag as a prompt to re-observe, and can afford the re-observation, gets more moves found that way. A survey that treats a flag as a reason to drop a mark and refit — which is what naming a station is for — should use the test whose flags mean what they say.

Where the covariance approach came from #

Treating control coordinates as weighted observations rather than fixed values is the standard alternative to holding them, usually described as an adjustment with weighted constraints or with the control entered as pseudo-observations. Baarda’s testing procedure, published by the Netherlands Geodetic Commission in 1968, is written for any observation with a known weight, so it applies to the control’s coordinates without change once they are observations, and the reliability it defines — the smallest error found with stated power at stated size — is the reach measured above. What the formulation does not supply is an answer to whether the published covariance is the right weight. A regional network’s reported covariance is the covariance of its own adjustment under its own assumptions, and the weights are a guess the solve believes is the general warning about taking such a number at its word.

What the trials leave out #

The published covariance is taken as the truth. Every trial draws the control’s error from exactly the covariance the honest test carries, so the test is being run under the one condition in which it is guaranteed to hold its size. If the regional network understates its error, the honest test’s false alarms return in proportion; if it overstates, the honest test loses more reach than it needs to. Neither is measured here.

One station moves at a time, by a stated amount, along its tie. A move across a tie is still invisible, for the reason what a closed figure cannot see gave for a traverse: nothing observed depends on it, and two simultaneous moves are not tried. The base rate in the last figure is the rate of exactly that event.

Distances only. A local survey with directions or satellite baselines would locate each control station across its tie as well, and the honest test would then have two numbers per station to test rather than one.

The regional network’s covariance comes from a stated design. Its size, and in particular how much of it is shared between stations, decides every reach in this essay. A regional network with more independent error between stations would push the honest test’s reach further out; one whose error is almost entirely shared would bring it back close to the exact test’s.

Still open: whether the control’s own report can be checked #

Every number above rests on the regional network’s published covariance being right, and a local survey has a way to check it that the regional network does not. With many control stations and no moved monuments, the honest adjustment’s residuals at the control are a sample from the published error’s differential part, and their size is a test of whether the published covariance is honest in its turn.

The difficulty is the same one this essay has spent its length on, turned round. A residual larger than expected could be a moved monument or an understated covariance, and a test for one has to assume the other. Whether a local survey tied to enough stations can estimate a scale factor on the regional network’s covariance while still testing for moved monuments — a variance component for the control, in the language of the weights are a guess the solve believes — and how many stations and how many surveys it would take to tell an optimistic regional report from a disturbed mark, are questions a test that takes the published covariance on trust cannot ask.