The control carries the error of every network above it

The network’s answer is decided before it is measured showed that every quantity a survey specification is written about — the error ellipses, the redundancy numbers, the smallest detectable blunder — comes out of the geometry and the weights and contains no observed value. That made it possible to compare designs on paper, and it left one thing out of every design it compared: the stations each network was held to.

Every adjustment on this side of the collection holds something fixed. A coordinate is the output of a solve found that the choice of what to hold moves every coordinate while moving no residual. What none of those networks asked is where the held coordinates came from. A local survey is held to regional control, the regional control was adjusted against national stations, and the national stations were adjusted against an origin. Each of those adjustments had its own errors, and the held-fixed report of the one below contains none of them.

Three orders, stated #

The arrangement is the one national surveys were built on for two centuries, the principle of working from the whole to the part: a sparse network of long lines observed with the best instruments, a denser one inside each of its figures, and a local one inside that. The retriangulation of Great Britain that began in 1936 was built that way, with a primary network of long sights and secondary and tertiary networks fitted inside it, each held to the one above. Modern networks are built of satellite baselines instead of angles and distances, and the hierarchy of control is the same.

The survey here is stated rather than taken from any real network, because the question is about the propagation and not about any particular country’s numbers.

The national order is seven stations: a centre and six round it at sixty kilometres, with every line to a neighbour observed, twelve lines in all, each good to twenty millimetres plus one part per million. It is adjusted at a minimal datum — its centre is the national origin and the bearing to one neighbour is held — so its errors are errors against that origin, and they run from 61 millimetres at the nearest station to 130 at the far side.

The regional order is ten stations inside one national triangle, twelve kilometres apart, with seventy-two lines among themselves and to the triangle’s three corners, each good to eight millimetres plus two parts per million. The three corners are held at their national coordinates.

The local order is five stations about a kilometre apart, with every line among them and a line from each to three regional stations ten kilometres away, twenty-five lines in all, each good to three millimetres plus two parts per million. The three regional stations are held.

Each order’s reported covariance is the inverse of its own normal matrix. Its full covariance adds what its control carried in: the matrix that turns an error in the held coordinates into an error in the new ones, applied to the control’s own full covariance, $G\,\Sigma_c\,G^{\mathsf T}$ with $G = -N^{-1}A^{\mathsf T}PA_c$. Since the control’s covariance already carries the order above it, the national error reaches the local stations through two applications of the same formula, and every part can be kept apart by the order it came from.

Eleven millimetres reported, fifty-nine carried #

The local adjustment reports its stations to 11.2 millimetres. Against the national origin they are uncertain by 59.4. Of the variance, three quarters came from the national order, nineteen per cent from the regional, and four per cent from the local survey itself. The report is not wrong about the four per cent. It says nothing about the rest, because the rest was held fixed.

The split is not the same for every quantity, and the difference is the most useful thing the measurement finds. The position is inherited almost entirely. The length of a local line is not inherited at all: 3.1 millimetres on a line a kilometre long, every tenth of a millimetre of it the local survey’s own. A regional network wrong by a few centimetres moves all five local stations together, and a common displacement changes no distance between them. The direction sits between the two. The local network is a network of distances and takes its orientation from its ties to control, so most of its directional error is its own weak orientation, 10.4 millimetres across a kilometre, and a further 1.9 is the regional network’s own orientation error at that place.

The middle order sits in the middle. The regional adjustment reports its stations to between 29 and 37 millimetres, and against the national origin they are uncertain by 60 to 66. Its report accounts for between a quarter and a third of its variance, where the local report accounts for less than a twenty-fifth. Each step down the chain adds a report of its own and inherits every report above it, so the share a report can see falls with every order: all of it at the top, where the national network has nothing above it and its report is its full error, and less at each order below.

That is the practical meaning of a survey being internally excellent. A local network measured to three millimetres in its lengths is excellent in its lengths whatever the control is doing. It sits five or six centimetres from where the country thinks it is, and it points where the control points, and neither of those is a statement about the local survey’s quality at all.

The report cannot see what it inherited #

A held-fixed adjustment is not wholly blind to bad control. If the held coordinates disagree with one another by more than the new observations allow, the observations cannot all be satisfied, the residuals grow, and the variance factor the report prints — the root-mean-square standardised residual, which should be about one — rises above one. Surveyors watch that number for exactly this reason.

It rises by almost nothing. The local survey’s expected variance factor, computed from the design, is 1.036 on fifteen degrees of freedom. To make its reported ellipses honest it would have to be 5.32. The regional survey’s is 1.033 where 2.07 would be needed. Neither number is distinguishable from one by an adjustment with that many degrees of freedom; the sampling scatter of a variance factor on fifteen degrees of freedom is about eighteen per cent.

The reason is geometric. A residual can only report an error the observations disagree with, and the new observations disagree with the control only where the control disagrees with itself: two control stations whose held separation is not the separation the new lines measure. The part of the control’s error that all three stations share — a common displacement, most of a common rotation — is something the new network can absorb by moving along with it, at no cost to any residual. The national order’s 52 millimetres at the local site are almost entirely shared by three regional stations ten kilometres apart, and they contribute 1.4 per cent of the tiny excess the variance factor does show.

The numbers under the variance factor make the point exactly. Of the excess the local adjustment’s residuals are expected to show — 1.09 on top of fifteen degrees of freedom — 1.08 comes from the regional order and 0.015 from the national. The regional control stations disagree with one another by their own relative error over seventeen kilometres, and the local lines see some of that. The national order’s error is 52 millimetres at the site and almost identical at all three control stations, so the local lines see almost none of it. The regional adjustment is in the same position one level up: its excess is 3.5 on fifty-two degrees of freedom, all of it the national triangle’s corners disagreeing with each other, and none of it the error they share against the origin.

This is the same blindness what a closed figure cannot see found for a traverse and a traverse must close found for a scale error: a check made from the internal agreement of observations is a check on their agreement, and an error that moves everything together agrees perfectly with itself. The blunder the network cannot see priced what a network misses in its own observations. Here it misses its control’s error, and misses nearly all of it.

Nearer control sharpens the report and imports more of the order above #

A local surveyor chooses how far away to tie. The natural instinct is that nearer control is better, and for the report it is.

The reported directional error falls steadily as the control comes in: 24.9 millimetres across a kilometre with control twenty-five kilometres away, 10.4 at ten, 3.7 at two. A network of distances takes its direction from the angle its ties subtend, and nearer control subtends a wider angle at every local station.

The inherited part does the opposite. With control ten kilometres away it is 1.9 millimetres across a kilometre, and with control two kilometres away 4.5. Three regional stations two kilometres from the site stand three and a half kilometres from one another, and they carry the regional network’s error at that scale; a regional network’s relative error over three and a half kilometres is a larger share of the distance than its relative error over seventeen is of seventeen. Tying to them hands the local network that share as a rotation.

So the report and the truth part company as the control closes in. At ten kilometres the full directional error is 2 per cent above what the report claims. At two it is 60 per cent above. The best-looking local adjustment on the chart is the one whose ellipses understate its direction most. The lengths are indifferent to all of it: between 2.8 and 3.1 millimetres at every spacing, the report and the truth agreeing to within three hundredths of a millimetre.

Whichever order is worst at its own reach decides the position #

Every number so far used one set of precisions. The national order’s is the one that decides a local station’s position, and it is the one that has changed most in practice: a national network of angles and taped bases was good to a few parts per million, and one of continuously operating satellite receivers is good to far better.

At two parts per million in the national distances, a local station’s position error is 104.3 millimetres and all but four of them are national. At one part per million, 59.4. At a tenth of a part per million, the national part falls to 5.2 millimetres and the full error only to 28.7, because the regional order’s 26 millimetres are now the largest term. The two orders contribute equally when the national distances are good to about half a part per million.

So improving the national order has a floor, and the floor is the next order down. That is not a remark about this stated survey but about the structure: a position error is inherited from whichever order is worst in its precision multiplied by its own reach, and the local survey’s own contribution, 11.2 millimetres, is below both of them at every setting drawn. The chain the satellite does not have is what happens when a coordinate can be observed without a chain of control at all; the three orders here are the chain it replaced, and the numbers say which link of it was the long one.

The alternative is to let the control move #

A held-fixed adjustment has an alternative. The control’s coordinates can be treated as observations too, with their full covariance as their weights, and adjusted along with the new stations. The new stations’ ellipses are then honest by construction, since the control’s uncertainty enters the solution.

On the stated survey it changes the local numbers by almost nothing: the local station’s full position error is 59.35 millimetres adjusted that way against 59.38 held. The local lines carry very little information about stations ten kilometres away that the regional network did not already have. What it does change is that the control moves. The local observations shift the three regional stations by between 2.6 and 5.2 millimetres, root-mean-square, and every survey tied to them afterwards would inherit that shift.

That is why control is held. A published coordinate that moved whenever somebody surveyed near it would be a coordinate nobody could quote, and a published coordinate is a result is about exactly that tension: a coordinate is the output of a definition as well as of a measurement, and a definition that changes with every new observation is not doing its job. Holding the control buys stability, and the price is a report that omits most of the error.

How the numbers were checked #

The parts must add. The full covariance of the local stations must equal the sum of the three parts attributed to the three orders, element by element. They agree to 10⁻¹².

The report may not depend on the control. Making the national network ten times worse must leave the local adjustment’s reported covariance unchanged, and does exactly — which is the statement that a held-fixed report cannot see its control.

Perfect control must make the report the truth. With the national and regional distances made perfect, the local station’s full error must equal its reported error, and does to a part in a thousand.

A relative error may not exceed the absolute errors it is made of. The error between two local stations must be no larger than the sum of their position errors, and is far smaller.

Real adjustments must scatter as the propagation says. Four hundred seeded chains of three genuine adjustments — noisy distances at every order, the national network solved at its minimal datum, the regional one against the adjusted national coordinates, the local one against the adjusted regional ones, each iterated to convergence — put the local station 56.8 millimetres from the truth, root-mean-square, against the propagated 59.4, and the two local stations 10.6 millimetres from their true separation against 11.0. Both are within two standard errors of what four hundred trials can resolve. The scatter is also more than three times what the local adjustment reports, which is the result stated as an observation rather than as an equation.

Where the stated survey stops #

Distances only. Every order observes distances and nothing else, which is why the local network takes its direction from its control. A local survey with an azimuth of its own — astronomic, or from a satellite baseline — would own more of its direction and inherit less, and the spacing measurement above is the case without one.

Three orders, one layout each. Real hierarchies have four or five orders and control that is not symmetrically placed. More orders add more inherited terms; the structure of the result, that the position is inherited from the worst order at its reach and the lengths not at all, does not depend on the count.

No time. Every order is observed at one epoch. A national network adjusted decades before its local densification carries the ground’s motion in the interval as well, which the epoch is part of the coordinate measures separately and which would add to the inherited part here.

The origin is a station. “Against the national origin” means against a point the national network holds at zero error. A datum is fitted to a region is about how that origin relates to the Earth, which is another inherited term, and a larger one, above the top of this chain.

Still open: a network fitted to its control afterwards #

There is a third way to meet control, between holding it and adjusting it. The local network can be adjusted on its own at a minimal datum, so that its shape comes only from its own observations, and then fitted onto the control by a similarity transformation — a shift, a rotation and a scale chosen to bring its tied stations as close as possible to their published coordinates.

That keeps the local network’s internal geometry intact and keeps the control stable, and it changes where the control’s error goes. A similarity transformation has four parameters, and three control stations with disagreeing coordinates overdetermine it: the misfit is left at the control rather than distributed into the local lines. Whether that leaves the local lengths untouched where holding the control distorts them, how much of the control’s error the fitted scale absorbs, and whether a fitted network reports its inherited error any more honestly than a held one — the misfit at the control is at least a number the surveyor can see — are questions a network held fixed from the start cannot ask.