A length measured from noisy points is too long

The previous rung is about the spread of an estimate and how slowly it shrinks. This one is about its centre, and the finding is worse: a length computed from two noisy points is not merely uncertain, it is systematically too long, and no amount of measuring removes it.

Why there is a bias at all #

A distance is

$$L = \sqrt{(\Delta x)^2 + (\Delta y)^2},$$

which is a concave function of the coordinate differences near a fixed baseline direction. Jensen’s inequality then does the rest: the expected value of a concave function is below the function of the expected value — except that here the relevant curvature is in the transverse direction and it goes the other way, so the expectation comes out above.

The mechanism is easier said than proved. An error along the baseline lengthens it as often as it shortens it, and those cancel to first order. An error across the baseline can only ever lengthen it: the hypotenuse of a right triangle is longer than either leg, whichever way the transverse error points. So the transverse component contributes a one-signed second-order term and the along-baseline component contributes nothing, which gives

$$\mathbb{E}[L] - d \;\approx\; \frac{\sigma^2}{d}$$

for an isotropic error of standard deviation σ on each of the two ends.

The distinction in that caption is the whole rung. Each coordinate is unbiased; the quantity computed from two of them is not, because the computation is not linear. Nothing has gone wrong with the observations.

Measuring a second-order term inside a first-order spread #

The bias is of order σ²/d and it sits inside a scatter of order σ. At twenty millimetres on a ten-metre baseline that is 40 microns of signal inside 28 millimetres of noise — a ratio of seven hundred to one — so a straightforward simulation of two hundred thousand samples returns the bias with a standard error larger than the bias.

The repair is antithetic sampling: draw each perturbation once with each sign and average the two lengths. The first-order term cancels exactly rather than statistically, leaving the second-order term and the fourth-order remainder.

That is not a trick to make a small number look significant. It is the standard variance-reduction argument, it changes nothing about the quantity being estimated, and it moves a measurement from indistinguishable from zero to three significant figures.

The law, over two decades #

The product of the bias and the length is constant to four decimal places across a factor of two hundred in the length, which is the law recovered rather than assumed. It sits 8.2 per cent below the leading-order prediction, and that gap is the fourth-order term the expansion drops — it shrinks when σ/d is made smaller, which is the check that it is a truncation rather than a mistake.

Why it does not average away #

A bias is not a noise, and this collection has met the distinction before: the average of noisy positions moves is the rung where a projection’s second derivative displaces a mean by a fixed amount however many observations go into it.

Here the same distinction has a sharper consequence, because lengths are usually added.

The same boundary comes out 1.5 parts per million longer when it is measured in two hundred and fifty-six pieces than in four, from the same instrument, with no blunder anywhere and every individual measurement correct.

That is over the proportional half of a first-order specification of five millimetres plus one part per million, and it is not detectable by any check the survey performs on itself: the traverse closes, the residuals are normal, and the answer is wrong in a direction nobody looks in.

It is also the wrong shape for an error budget. A budget is a sum of variances and a variance has no sign, so a term that adds coherently over legs has nowhere to be written down — it is not a bigger version of anything already in the table, it is a different kind of entry.

The scaling is the argument #

The accumulation goes as the number of legs squared relative to the perimeter, and that is worth deriving because it says which surveys are exposed.

A perimeter P in n legs has legs of length P/n, each biased by σ²n/P, and there are n of them — so the total is σ²n²/P and the relative bias is σ²n²/P². Doubling the number of legs quadruples the error.

The consequences fall out immediately.

A detail survey of a small parcel in many short shots is the exposed case, not a long traverse across a country. The exposure is worst where the legs are short compared with the mark accuracy, which is exactly where a surveyor feels safest.

And the fix is free: measure in fewer, longer legs where the geometry allows it, or apply the correction, which is one subtraction per leg from a quantity the survey already knows.

The other second-order term, for contrast #

This collection already carries one second-order bias in a coordinate, and it is worth putting the two side by side because they are unrelated and the same size.

The two are easy to confuse and are opposites in every respect that matters.

The projection bias is a property of the map: choose a different one and it changes sign or disappears. The length bias is a property of the norm: it is the same on every projection, on the sphere, and in a plane, because a distance is a square root wherever it is computed.

The projection bias moves a position and leaves a distance between two nearby points nearly alone. The length bias does the reverse. And a pipeline that corrects for one has not touched the other.

Where the same term appears elsewhere #

The bias is a property of a norm rather than of surveying, so it turns up wherever a length is computed from noisy coordinates.

A polygon’s perimeter in a spatial database, computed from vertices with a stated accuracy: biased long by exactly this term, with the vertex count as the multiplier — and a real edge has a width is the rung about the other half of that problem.

A simplification tolerance, which is a promise about displacement: a simplified line is shorter by the generalisation and longer by this bias, and a line has a length only at a scale prices the first without the second.

And the route lengths this collection prints. Every distance computed between two places with stated coordinates carries it, at a size set by whatever precision those coordinates have. At the site’s usual working precision the term is under a micron and is genuinely negligible; that is a statement about the coordinates rather than about the arithmetic, and it stops being true the moment the coordinates come from an instrument.

What a survey could check #

The bias is not detectable by anything a survey does to itself, and it is trivially detectable by one thing it does not.

A closure does not see it. A traverse must close, and a closure is a vector sum of directed legs: the transverse errors that lengthen each leg do not accumulate in the closure vector, because a leg’s length error is along the leg and the closure adds the legs as vectors. A traverse can close perfectly and still report a perimeter 1.5 parts per million long.

A least-squares solve does not see it either. A coordinate is the output of a solve, and the solve’s model is that each observation equals its true value plus zero-mean noise. That model is correct for each observation and the bias is in a quantity computed afterwards, so nothing in the residuals carries it.

What does see it is a re-measurement at a different leg length. Measure the same boundary in four legs and in two hundred and fifty-six, and the difference between the two totals is the term, with the σ²n²/P² law available to predict it in advance. That is a one-afternoon experiment and this collection can find no record of anybody having run it.

How large the marks have to be for it to matter #

The bias is σ²/d, so it is worth stating the regime in which it reaches a tolerance rather than leaving it as a law.

For a single baseline the bias exceeds a stated tolerance t when d < σ²/t. At a five-millimetre mark accuracy and a one-millimetre tolerance that is 25 millimetres — no baseline anybody measures is that short, and on a single line the term is genuinely negligible.

The accumulation changes the arithmetic completely. Over n legs of a perimeter P the relative bias is σ²n²/P², so it reaches one part per million when n > P/(σ × 10³). For a kilometre perimeter and five-millimetre marks that is 200 legs, which is an ordinary detail survey of a small parcel: five-metre shots round a building line.

So the term is invisible on the measurement a surveyor thinks about and reaches a specification on the survey they actually perform. That inversion is the reason it is worth an essay rather than a footnote: the case where the theory is easiest to check is the case where the effect is smallest, and the case that matters has never been the one anybody tested.

The one-line summary a surveyor can carry is that the bias is a length, not a fraction: σ²/d has the units of the marks’ own error divided by the leg, so shortening the legs while keeping the marks makes it worse in both terms at once. Densifying a traverse improves almost everything else and makes this alone worse.

What the one-afternoon experiment would actually take #

The proposal above — measure the same boundary in four legs and in two hundred and fifty-six, and difference the totals — is worth costing, because whether it is an afternoon depends on how large the random part of the difference is, and the answer is not obvious.

The naive estimate is discouraging. Each leg’s length carries an error of order σ from each end, so a perimeter of n legs looks like it should accumulate σ√(2n) — 113 millimetres for 256 legs with five-millimetre marks, against a bias of 1.5. Seventy-five to one, and the experiment is hopeless.

That estimate is wrong, and the reason is the same closure the earlier section says cannot see the bias. Adjacent legs share a mark, and on a smooth boundary they very nearly share a direction. A mark’s error enters the perimeter through its projection onto the leg in front and the leg behind, with opposite signs, so what survives is its projection onto the difference of two nearly-parallel unit vectors. That difference has length equal to the turning angle at the mark, which for a closed boundary in n roughly equal legs is 2π/n.

Summing n independent contributions of σ·2π/n gives a perimeter whose random error is

$$\sigma_P \;=\; \frac{2\pi\sigma}{\sqrt n},$$

which falls as the survey is densified rather than rising. At 256 legs with five-millimetre marks that is 1.96 millimetres, against a bias of 1.64 by the leading-order law and 1.51 as measured.

So the two are the same size, and the crossing has a closed form. Setting 2πσ/√n equal to σ²n²/P gives

$$n \;=\; \left(\frac{2\pi P}{\sigma}\right)^{2/5},$$

which for a kilometre perimeter and five-millimetre marks is 275 legs. The essay’s densest row is at 256, essentially at the crossing — below it the bias is buried in the scatter, and above it the bias is the larger of the two and grows as n^2.5 relative to it.

That changes what the experiment is. At 256 legs the difference between the two totals is about 1.5 millimetres of signal on 2 millimetres of noise, so a single afternoon gives a result of the right order with a t-statistic under one, and about a dozen repetitions of the dense circuit would settle it. At 500 legs the signal is 6 millimetres on 1.4 of noise and one circuit is enough.

The right experiment is therefore denser than the one the table suggests, not longer, and that is the same inversion the rest of the rung keeps finding: densifying makes every other error smaller and this one larger, so the regime where the effect is measurable is the regime where the practice is worst.

Where the model stops #

The error is isotropic and the same at both ends. A real mark’s error ellipse is neither, and the bias then involves the transverse component alone — the along-baseline variance contributes nothing to second order — so an anisotropic error aligned with the baseline is much cheaper than one across it. The number here is the isotropic average of those two cases.

The two ends are independent. The difference of two coordinates shows they are usually not: two marks in one network share most of their datum error, which cancels out of a difference, and what is left is the relative covariance. That reduces the effective σ, in some cases by a factor of six, and reduces this bias as its square.

And the estimate is Monte Carlo. The bias is available in closed form for the isotropic Gaussian case and the simulation is not needed to know it; it is here because a stated law that has never been made to produce a number is a law nobody has checked, and because the antithetic estimator is the part of the method worth carrying.

One caveat on the closed form for the random part. It assumes a smooth closed boundary turning through 2π in roughly equal steps, which is what makes the turning at each mark 2π/n; a boundary with corners concentrates the turning at a few marks and the cancellation is correspondingly worse there. The scaling survives — it is the smoothness of the route rather than of the ground that matters — and a traverse round a rectangular parcel in many short legs is exactly the smooth case, because almost every mark lies on a straight run.

The generalisation #

Any quantity that is a non-linear function of noisy inputs has a bias, its size is the second derivative times the variance, and it survives averaging.

That sentence covers this rung, the projection bias two rungs below it, and a great deal of the rest of the collection: an area computed from noisy vertices, a bearing from a short baseline, a ratio of two measured quantities. What varies is only whether the second derivative is large enough to matter.

The reason it goes unnoticed is that error propagation is taught as a first-order technique — the covariance sandwich, which is what an error ellipse is an indicatrix is about — and a first-order technique reports a spread and cannot report a shift. The shift is not small because the technique cannot see it; it is invisible because the technique was chosen before the question was asked.

Who found it, and when #

That the expectation of a distance between noisy points exceeds the true distance is a standard exercise, and the σ²/d form appears in the metrology literature as the bias of a norm. It is best known in a different field: the same term is why an estimate of a magnitude from noisy components — a wind speed from two components, a spectral amplitude from a real and imaginary part — is biased high, which is the Rice distribution’s mean and which has been in the radar literature since the 1940s.

Its accumulation over a traverse appears to be less often stated. Survey error analysis is written around variances, which do not accumulate with a sign, and a term that adds coherently over legs falls outside the apparatus even though its size is easy to compute.

Where the ladder goes next #

Seven rungs give a coordinate a width and follow it through a map, an average, a difference, a second derivative and a spectrum. What none of them asks is what the width means when the thing being located is not a point — a boundary, a coastline, a place with an extent of its own — where the uncertainty of the position and the uncertainty of the object are two different quantities that every dataset reports as one.