The network's answer is decided before it is measured

Nine rungs of this ladder are about what an adjustment does with observations: what a tape measures, how a traverse closes, where a misclosure goes, what a solve holds fixed, what the weights believe, and which blunder nobody sees.

There is a quantity underneath all of them that contains no observation at all.

Where the observations are not #

A least-squares adjustment solves N x = AᵀP l, with N = AᵀPA. The covariance of the answer is σ₀²N⁻¹, and every derived quantity a specification talks about comes out of N⁻¹.

A holds the partial derivative of each observation with respect to each coordinate. For a distance between two stations that is a unit vector along the line: pure geometry.

P holds the weights, which are the instrument’s stated precisions: a catalogue.

The observed values are in l, and l appears nowhere in N.

So the cofactor matrix, every error ellipse, every redundancy number and every minimal detectable bias are fixed the day the stations are chosen and the instrument is picked. Not estimated in advance, not predicted — fixed, in the same sense that the area of a triangle is fixed by its vertices.

The measurement #

The claim is testable and it is worth testing, because a claim about what a formula does not contain is exactly the kind that is true on paper and false in code.

Three solutions of one braced quadrilateral, with the stations moved by metres and the pseudo-random noise reseeded, so that every one of the ten observed distances differs by metres between the runs:

Four orders of magnitude separate the two groups, and the separation is the point.

The residue is not noise, and it has a cause. The design matrix holds derivatives evaluated at the current coordinate estimates, and an iterated adjustment updates those. So the independence is exact in the linear model and approximate in the iterated one. Linearising once at a common point makes it exact: the redundancy numbers, the biases and the ellipses come back identical to the last bit across the three observation sets, while the residuals still spread by half.

That is the honest version of the claim, and it is a stronger statement than a bare zero would have been.

What the layout is worth #

If the answer’s quality is decided by the geometry, then choosing the geometry is choosing the answer’s quality, and the size of that choice is measurable.

Every row costs the same. Five stations to occupy, ten distances to observe, the same instrument, the same number of days. The braced quadrilateral is 20.9 times more precise in its worst coordinate than the chain.

And the two columns do not rank the same way. The ring is slightly worse than the braced quadrilateral in precision and considerably better in redundancy — its weakest observation carries 0.255 against 0.144, so a blunder in it is nearly twice as visible. A design that is optimal for precision is not optimal for reliability, which is the trade-off is forced arriving in a place with no projection in it.

Which observation to make next #

The practical form of the whole rung. Given a network that is not good enough, which observation should be added?

The answer is a table, and the table is computable before anybody leaves. Adding the line C–D takes the worst semi-axis from 16.6 millimetres to 11.8; adding C–E takes it to 16.6, which is the same number. One of them is worth a day and the other is worth nothing — the same shape of finding what another common point buys reports for a datum transformation, and the difference is a factor of 1,400 in what the same effort buys.

The best line is neither the longest nor the shortest nor the one to the weakest station. It is the one whose direction is least represented in the existing observations, which is a statement about the null space of A and is not visible by looking at the map.

What a specification is actually saying #

A survey specification states a tolerance on the finished coordinates — twenty millimetres at ninety-five per cent, say — and is checked against the adjustment when it comes back. Everything above says that check is late.

The tolerance is a statement about the semi-major axes of the error ellipses. Those are σ₀ times a quantity that depends only on the layout and the weights, so a specification is met or not met by the design, up to the single scalar σ₀ that the observations supply. A layout whose worst semi-axis is 231 millimetres for σ₀ = 1 cannot meet a twenty-millimetre tolerance with any instrument that has the stated precision, and this is knowable on the morning the marks are chosen.

The distinction matters because the two ways of missing a specification have completely different remedies. If σ₀ comes back too large the instrument or the procedure was worse than stated, and the fix is to re-observe. If the geometric factor is too large the layout could never have met it, and re-observing the same lines with the same instrument produces the same failure however carefully it is done.

An adjustment reports both numbers and a specification check usually reads only their product.

The redundancy number, read forwards #

The redundancy number of an observation is the share of a degree of freedom it carries — how much of its own error the adjustment can push into its residual rather than into the coordinates. A blunder splits between the residuals and the coordinates in the ratio the redundancy sets, which is the previous rung’s finding, and the redundancy numbers sum to the degrees of freedom exactly.

Read forwards, the same list is a design tool with an uncomfortable reading. An observation with a redundancy of 0.055 — the chain layout’s weakest — puts 94.5 per cent of any error it carries straight into the coordinates and 5.5 per cent into its own residual. It is very nearly an unchecked measurement, and it looks exactly like the other nine on the schedule.

The rule of thumb in the reliability literature is that a redundancy under about 0.3 leaves an observation effectively unchecked. Of the four layouts above, only the ring clears it, and it clears it by a hair.

Where the model stops #

These are distances only. A real network mixes distances, directions and height differences, each with its own weight and its own row in A, and adding an angle changes the design in a way a distance cannot. Nothing above changes; the catalogue gets larger.

The weights are stated. They come from the instrument’s specification, and the weights are a guess the solve believes is the rung about what happens when the guess is wrong. If the weights are wrong then the design is wrong too, and it is wrong in advance rather than in the data.

Second-order design is not done here. Choosing where the stations go is first-order design; choosing the weights with the stations fixed is second-order and is a different optimisation with a different answer. Both are classical and only the first is measured.

And a design is not a plan. Terrain, access, sight lines, ownership and weather decide most real layouts, and the best design in this sense is frequently unreachable. What the calculation supplies is the price of each compromise, which is exactly what an argument with a client needs: not a claim that the good layout is unreachable, but a figure in millimetres for what the reachable one will deliver.

Why this is not obvious from the previous rung #

The blunder the network cannot see measures the minimal detectable bias and finds observations whose errors no test can reject. Reading that rung, the natural conclusion is that some observations are unlucky.

They are not unlucky. They were chosen to be undetectable, before any of them was made, by whoever decided where the marks would go — and the calculation that says which ones is the same calculation, run in advance. The redundancy numbers in the table above are the same redundancy numbers that rung reports; what is new is that they are available on the morning of the design rather than the evening of the adjustment.

That inverts what the earlier rung’s finding is about. It is not a limitation of testing. It is a consequence of a decision, and the decision is documented nowhere because nobody records having made it.

What the calculation costs #

The whole of this rung is the adjustment code that already exists, run once with the observations replaced by their own noiseless values — or with nothing at all, since they do not enter.

Building A requires approximate coordinates, which a design has by definition: the marks are on a plan. Building P requires the instrument’s specification, which is on its data sheet. Inverting N for a five-station network is microseconds and for a national one is the same computation the adjustment itself does.

So the answer to should the geometry have been checked first is not that it is expensive. Every table in this essay is a fraction of a second, and the marginal-observation table — which line to add next, over every candidate — is three inversions.

The reason it is not routinely done is more ordinary and worth naming: an adjustment is written to consume observations, so its natural moment is after the fieldwork. Running it before requires believing that a program whose input is measurements will produce something useful with no measurements in it, which is exactly the fact this rung is about and is not obvious from the outside.

There is a version of this that applies before any station is chosen. The number of unknowns is twice the free stations, the number of equations is the observations, and the difference is the degrees of freedom — so a network with as many observations as unknowns has no redundancy anywhere, every observation carries a redundancy of zero, and no blunder in any of them can be detected at all. That is arithmetic on two integers and it rules out a design before a map is opened.

The generalisation #

The quality of an answer is often decided by the question’s shape rather than by the data’s quality, and the shape is settled earlier and by somebody less careful.

The collection has the same structure elsewhere. The best grid a country could have had computes what a national grid’s scale error would have been under a better choice of projection and parameters, and the answer depends on the country’s shape rather than on any measurement. Designing a grid for one region is the same calculation done deliberately. How many sheets an atlas needs prices a layout before any sheet is drawn.

The common shape is that a derivative — of a distance with respect to a coordinate, of a scale factor with respect to a parameter — is a property of the configuration, and every error budget is built out of derivatives. So an error budget can be built before there are any errors.

The one number the observations do supply #

It would be a misreading to conclude that the measurements do not matter, and the arithmetic says exactly what they contribute.

The covariance is σ₀²N⁻¹. The geometry and the weights fix N⁻¹ completely; the observations fix σ₀, the variance of unit weight, which is a single scalar for the whole network. So the observations decide one number, and that number multiplies every error ellipse equally.

That has a consequence worth stating. Good fieldwork scales the whole picture down and cannot change its shape. A station that is weak because of where it sits stays the weakest station however carefully it is observed, and the ratio between the best and worst coordinate in a network is a property of the layout that no amount of care will move.

It also explains the earlier table’s split. σ₀ and the residuals moved by eighty-eight per cent across three observation sets because they are the part the observations own; the ellipse shapes did not move at all because they are not.

Who found it, and when #

Network design is a named field with a standard vocabulary: Grafarend’s classification of the four orders — zero for the datum, first for the configuration, second for the weights, third for densification — dates from the 1970s, and Baarda’s reliability theory from the 1960s supplies the redundancy numbers and the detectable biases the tables above use.

None of it is obscure and all of it is taught. What is unusual is how rarely the calculation is run: a specification is typically written as a tolerance on the finished coordinates and checked afterwards, and the design that would meet it is arrived at by experience. The arithmetic that would say in advance whether a layout can possibly meet a stated tolerance is a few lines on top of the adjustment code every office already has.

Why a calculation this cheap is not run #

Every ingredient is taught, the code is a few lines on top of the adjustment every office already owns, and the calculation is nevertheless rare. The explanation is not ignorance, and it is worth setting out because it points at the remedy.

The calculation has to happen before the work exists. A design analysis is run on a layout — stations, sight lines, an observation plan — that nobody has yet been paid to produce. At the moment it would be most valuable, the job is a tender, the layout is provisional, and the effort is unbillable.

And its output is a reason not to proceed. A design computation can say that a stated tolerance is unreachable with the proposed layout, which is worth a great deal to the client and is an awkward thing for the party proposing the layout to discover and disclose.

Meanwhile the contract is written the other way round. A specification names a tolerance on the finished coordinates, the work is done, and the result is checked against it. That ordering makes the assessment a delivery test rather than a design tool, and it is exactly backwards for a quantity fixed by the configuration: by the time the test is run, the number it measures was determined weeks earlier and nothing about the fieldwork could have changed it.

Which gives a specific and cheap remedy. Require the design computation as a deliverable, before observation, alongside the observation plan. It costs an afternoon, it is checkable, and it moves the conversation about achievability to the one moment at which the answer can still be acted on — by adding a leg, a tie or a station.

There is also nobody whose job it is. The design belongs to whoever plans the observation, the assessment belongs to whoever adjusts it, and those are different people working weeks apart — so the calculation falls between two roles rather than being refused by either.

The alternative is what happens now: experience substitutes for the calculation, which works well for layouts resembling ones the surveyor has done before and fails silently for anything unusual. That is a reasonable heuristic and it is not a measurement, and the difference shows up on exactly the jobs where it matters.

Where the ladder goes next #

Ten rungs price the measurement, its reduction, its adjustment, its weights, its blunders and now its design. What none of them prices is the datum’s own arrival: every network above is adjusted against held coordinates that came from a larger network, which was adjusted against a larger one still, and the uncertainty of that chain is not in any of the error ellipses drawn here. A local network can be internally excellent and sit ten centimetres from where the country thinks it is.