The two ways to spread a misclosure

A traverse closes to 350 millimetres over a 2,920-metre perimeter — one part in 8,344, which most specifications accept. The misclosure has to be got rid of, because coordinates that do not close are unusable.

There are two classical rules for getting rid of it. They give different answers, both are legal, and neither of them finds the leg that is actually wrong.

Bowditch’s rule #

Distribute the misclosure in proportion to each leg’s length.

$$\delta_i = -\,\Delta \cdot \frac{\ell_i}{\sum_j \ell_j}$$

The reasoning is that a long leg has had more opportunity to go wrong than a short one, so it should absorb more of the correction. That is a statement about how errors accumulate — a model — and it is the rule almost every jurisdiction’s specification names.

The Transit rule #

Distribute in proportion to each leg’s component along the axis being corrected: the easting misclosure shared out by each leg’s easting component, the northing misclosure by each leg’s northing component.

$$\delta_{E,i} = -\,\Delta_E \cdot \frac{|\Delta E_i|}{\sum_j |\Delta E_j|}$$

The reasoning is different: that the error is in the lengths rather than in the angles, so a leg running east contributes to an easting misclosure and a leg running north does not. A different model of the same figure, equally plausible from the figure alone.

Both close it exactly #

Applied to the blundered traverse, both rules produce a figure that closes to below a micrometre. That is not a coincidence and it is not a property of these particular numbers: both rules are constructed by subtracting shares of the misclosure that sum to the whole of it, so the corrected figure closes by arithmetic.

So the closure test — which is what got the traverse this far — cannot distinguish them. It is satisfied by both, exactly, and would be satisfied by any number of other distributions somebody cared to invent.

And they disagree #

The corrected leg lengths differ between the two rules by up to 14 millimetres on this figure.

Fourteen millimetres is not nothing. It is inside most specifications and outside some, it is a real difference in the published coordinates of every station in the traverse, and two crews applying the two rules to identical field observations would deliver different answers and both be correct.

The assertion here is the pair. Both must close exactly and they must disagree with each other by more than a millimetre. Requiring only closure would pass on two rules that had become the same rule; requiring only disagreement would pass on a rule that had stopped closing the figure.

What the underdetermination looks like #

The counting is worth doing explicitly, because it is the whole argument and it takes two lines.

A closed traverse of $n$ legs has $n$ unknown length errors and $n$ unknown bearing errors — $2n$ unknowns. The closure supplies two equations, one per axis. So $2n - 2$ degrees of freedom are unconstrained, which for a four-leg traverse is six.

Both rules resolve all six by assumption. Bowditch assumes every bearing is correct and every length error is proportional to its length, which fixes five of the six and leaves the sixth to the closure. Transit assumes every bearing is correct and every length error is proportional to that leg’s component along the axis being fixed. Neither assumption is checked against anything, because there is nothing to check it against.

The only way to reduce the underdetermination is to add observations — more legs closing more loops, a measured baseline, a tie to a second known point — and every one of those brings information from outside the figure. That is the same conclusion a traverse must close reaches from the other direction, and it is not a coincidence: both essays are about what a closed figure does not contain.

Neither finds the blunder #

Leg 3 of this traverse is 350 millimetres too long, planted deliberately. That is the entire cause of the misclosure.

Bowditch puts 98 millimetres of correction on leg 3 — and 98 on leg 1, and 77 on legs 2 and 4. Transit puts 104 on leg 3 and 104 on leg 1. Neither rule puts more than 30 per cent of the blunder on the leg carrying it, and neither singles it out in any way whatever.

This is the finding, and it is asserted rather than described: the correction applied to the guilty leg must be less than 60 per cent of the blunder, under both rules.

Why a convention cannot be an estimator #

The rules cannot find the blunder because they have no information that would let them.

A closed traverse of four legs provides two equations — the easting misclosure and the northing misclosure — and there are four unknown leg errors. The system is underdetermined by two, and no amount of cleverness produces information that is not there. Bowditch and Transit are two different ways of picking one solution out of a two-parameter family, and the pick is made by assumption rather than by evidence.

That is what makes them conventions rather than estimators. A convention produces consistent numbers from insufficient data by declaring a rule in advance. An estimator produces a best guess from evidence about the errors’ distribution, and there is no such evidence in a closed figure.

The distinction is exactly why this collection can discuss traverses at all. numerical-linear-algebra.com owns least squares as a numerical procedure and normaldistribution.xyz owns it as an estimator, and a survey library reaches for network adjustment almost immediately. The misclosure and its distribution rules contain no estimator, which is what keeps them here.

What happens to the blunder #

If neither rule finds the blunder, where does it go?

It gets spread across all four legs in amounts of the same order — around a hundred millimetres each on this figure — so a single 350-millimetre error in one place becomes four errors of about a hundred millimetres in four places. Every station’s coordinates move. None of them moves by the right amount.

That is worse than leaving the misclosure alone in one specific sense: before adjustment, the figure carried a visible symptom of one large mistake; after adjustment, it carries no symptom at all and four small mistakes. The adjustment did not remove the error, it redistributed and concealed it.

Which is exactly why specifications set a rejection threshold. A misclosure inside tolerance is assumed to be accumulated small errors, for which spreading is the right treatment; a misclosure outside tolerance is assumed to contain a blunder, for which spreading is the wrong treatment and re-observation is the only one. The threshold is where the assumption changes, and it is a judgement about which cause is more likely rather than a measurement of the cause.

What the rules are actually for #

Nothing above is an argument against using them. It is an argument against reading their output as a statement about what happened — the same misreading a published coordinate is a result identifies in a control mark’s coordinates, which are also the output of a procedure and are also read as observations.

The purpose of distributing a misclosure is to produce a consistent set of coordinates — a figure that closes, so that every subsequent computation on it is well defined and every user gets the same answer. That is a real requirement and the rules meet it exactly.

What they do not do is tell anybody where the error was. A misclosure inside tolerance is accepted and distributed; a misclosure outside tolerance is investigated in the field, by re-observing, which is the only procedure that brings in new information. The rules are for the first case, and applying them to the second is how a blunder gets buried in four small corrections instead of found.

The same shape, one field along #

The pattern here — a rule that produces a definite answer from data that does not determine one — appears twice more in this collection, and setting the three beside each other is worth doing.

A datum’s seven parameters are fitted to a set of common points, and two authorities fitting the same datum pair over different point sets publish different parameters. Both are right about their own fit, and where a fit leaves residuals shows the residual is a field rather than noise — so the disagreement is structure the seven parameters cannot express, not error.

A projection’s scale factor is chosen by an optimisation whose objective nobody publishes, so two authorities optimising the same country produce different constants. That is the scale factor was chosen, and the answer depends on a judgement about the region.

A misclosure’s distribution is chosen by a rule whose assumption nobody checks, so two crews produce different coordinates.

In all three, the visible output is a definite number and the invisible input is a choice. The collection’s standing response is the same each time: state the choice beside the number, because a number whose provenance is unstated cannot be compared with another number of the same name.

Choosing between them #

If no measurement can settle it, on what basis does a jurisdiction pick?

On what its instruments do. Bowditch’s assumption — error proportional to length — fits distance measurement whose error grows with the distance measured, which is what tape and chain surveying does and what electronic distance measurement does to a lesser degree. Transit’s assumption — error in the lengths rather than the angles — fits work where angles are observed far more precisely than distances, which was the ordinary situation with a good theodolite and a poor tape.

So the choice encodes the technology of the era that made it. Bowditch’s rule has won almost everywhere, and the reason is that modern distance measurement really does have a length-proportional component, which makes its assumption the better-founded of the two.

But the honest version of the claim is comparative rather than absolute. Bowditch is a better model of a modern instrument than Transit is. Neither is a measurement, and a specification that says “adjust by Bowditch’s rule” is specifying a convention for producing agreement, not a method for finding truth.

What was computed, and how #

A closed quadrilateral whose true figure closes exactly, with a 350-millimetre blunder added to one named leg.

Both rules applied, each rebuilding the figure from its corrected legs and recomputing the closure.

Three assertions. Both rules must close the corrected figure to below a micrometre. They must disagree with each other by more than a millimetre. And neither may put more than 60 per cent of the blunder onto the leg that carries it.

The third is the one that could have failed and is the reason the section exists. A rule that did locate the blunder would be an estimator wearing a convention’s clothes, and would mean this collection had taken ground the claim registry gives to another site.

Where the model stops #

Four legs is the smallest interesting case. With more legs the underdetermination is worse — $n$ unknowns against two equations — so the argument strengthens rather than weakening, and the two rules’ answers spread further apart.

Only two rules are computed. Others exist: the Crandall rule, which holds the angles fixed and adjusts only the distances by least squares, and the compass rule’s various local variants. Crandall is genuinely an estimator and is deliberately absent for that reason.

The traverse is computed in the plane, which is what a traverse computation is and is an approximation whose limits how small is flat enough puts a number on. Over a three-kilometre figure it contributes nothing at the millimetre level.

And the blunder is in a length. A blunder in an angle produces a misclosure with a different signature — the figure is not merely the wrong size but the wrong shape — and neither rule finds that either, for the same reason. Only the length case is computed, because it is the one where a reader most expects the rules to work.

The generalisation #

A rule that turns insufficient data into a definite answer is a convention, and its output must not be read as evidence. That sentence covers a great deal of ground outside surveying, and the diagnostic is always the same: count the equations, count the unknowns, and ask what supplied the difference.

Where the difference was supplied by an assumption, the answer is as good as the assumption and no better — and, crucially, the answer looks identical whether the assumption was right or wrong. That is why the demonstration here uses a planted blunder: the only way to show that a convention is not diagnosing anything is to know the truth in advance and watch it be missed.

Who found it, and when #

Nathaniel Bowditch published the rule in The New American Practical Navigator in 1802, as a method for adjusting a dead-reckoning track — not a traverse. The transfer to land surveying came later and the name stuck, which is why it is also called the compass rule.

The Transit rule is older in spirit and has no single author; it belongs to the era when a transit theodolite gave far better angles than any available chain gave distances, and its assumption is a direct statement of that asymmetry.

Both survived the arrival of the instruments that invalidated their premises, which is unremarkable — a convention’s job is to be agreed on, and changing an agreed convention costs more than the improvement is usually worth. What is remarkable is how often the surviving rule is described in modern texts as correcting a traverse, a word that credits it with a diagnosis it has never been able to make.

What a convention is entitled to claim #

The complaint about the word correcting is the essay’s sharpest point and it is worth separating from the arithmetic, because the two rules are not being accused of being wrong.

Both close the traverse exactly, which is what they were built to do. A closed traverse with a misclosure is a set of coordinates that do not join up, and a drawing, a title plan or a set of setting-out marks needs them to join up. Distributing the misclosure produces a consistent figure, and both rules do that perfectly.

Neither claims to locate the error, and neither can. A misclosure is one vector; the observations are many; and no rule distributing one number over many unknowns is recovering information that was not there. The distribution is a choice about presentation, made on an assumption about where error accumulates, and both assumptions are statements about instruments rather than about this traverse.

So the honest verb is distributing rather than correcting. Correcting implies an error has been identified and removed; what has happened is that an inconsistency has been spread according to a convention, and the underlying observations are exactly as wrong as they were.

And the distinction has a consequence rather than being pedantry. A surveyor who believes the traverse has been corrected has no reason to look for the blunder, and the blunder is still in the data — redistributed, invisible, and now spread across every station rather than concentrated at one. That is worse than leaving it visible, and it is what the word encourages.

None of which argues for abandoning either rule, and a surveyor who ran that check first is entitled to use whichever one their office has always used.

The remedy is not a better rule. It is to check the misclosure against what the instruments should have produced before distributing anything: a misclosure much larger than the expected accumulation is evidence of a blunder, and that comparison is the diagnosis neither rule performs.

Where this goes next #

The corrections so far have all been multiplications, which commute. The ones that turn a grid bearing into something to observe do not: setting out runs the chain backwards.