Error propagation
Error in a taped distance measured in n tape lengths, against a single EDM shot
The same 648 ft line measured two ways. Series propagation over seven tape settings against one EDM observation with a constant-plus-ppm specification, and what the comparison says about when taping is still adequate.
Given
- A line measures 648.55 ft horizontally.
- Method A — taping. The line is taped with a 100 ft steel tape in six full tape lengths plus a partial length of 48.55 ft, so seven independent settings of the tape are made. Systematic effects (tape standardisation, temperature, tension, sag) have already been corrected. The residual random error of each setting — marking, plumbing and reading — is ±0.008 ft.
- Method B — EDM. One observation with an instrument specified at ±(0.010 ft + 2 ppm), the constant and proportional parts to be combined in quadrature as the manufacturer intends.
- The two methods are to be compared as they stand, one pass each.
Required
- The standard error of the taped distance.
- The standard error of the EDM observation.
- The relative precision each method delivers, and how many tape settings the taped method could tolerate before it falls behind.
Work it through yourself before reading on — the solution below shows every step, so there is no way to skim it without giving the answer away.
Worked solution
Series propagation for the taped distance
The taped distance is a sum of seven independently measured pieces. The error of a sum of independent quantities is the square root of the sum of their squared errors, and when the terms are all equal that collapses to the single-setting error times the square root of the number of settings.
The consequence is the whole point of the problem: seven settings at ±0.008 ft give ±0.021 ft, not ±0.056 ft. Random errors of independent settings partially cancel, because they are as likely to be positive as negative and only rarely conspire. If they did accumulate linearly they would not be random errors — they would be systematic, and the answer would be to correct them, not to propagate them.
Count the settings, not the tape lengths. The partial length of 48.55 ft is marked and read with the same care as a full one, so it contributes the same ±0.008 ft. Seven, not six.
E_series = E · √n = 0.008 × √7 = 0.008 × 2.6458 = 0.0212 ft
equivalently √( 7 × 0.008² ) = √0.000448 = 0.0212 ft
the wrong linear answer would be 7 × 0.008 = 0.056 ft, nearly three times too largeThe EDM specification
An EDM specification is written as a constant plus a proportional term. The constant covers the instrument and reflector offsets and the internal phase measurement; the proportional term covers the uncertainty in the modulation frequency and in the atmospheric correction, and grows with the length of the line.
At 648.55 ft the proportional part is tiny — 0.0013 ft — so the whole specification is essentially its constant. That is the general shape of EDM error on short lines: it barely depends on length, which is exactly the opposite of the taped case, where error grows as the square root of length.
Combine the two parts in quadrature rather than adding them. They arise from independent mechanisms, and adding them would overstate the error by about ten per cent here and by more on longer lines.
proportional part = 2 × 10⁻⁶ × 648.55 = 0.0013 ft
σ_EDM = √( 0.010² + 0.0013² ) = √( 0.000100 + 0.00000169 ) = √0.00010169 = 0.0101 ftCompare the two as relative precisions
Divide each standard error into the distance to express it the way a specification is written. Taping delivers about 1:30,600 and the single EDM shot about 1:64,300 — a factor of 2.1 in favour of the instrument.
That factor is smaller than most people expect, and it is worth being honest about. Careful taping over seven settings is not a crude method; on this line it comfortably satisfies a 1:10,000 boundary requirement. The EDM's advantage on a line this short is real but modest, and it comes overwhelmingly from the constant term of the specification, not from anything to do with distance.
The picture reverses with length. Taping error grows as √n and therefore roughly as the square root of the distance, while EDM error grows only through its 2 ppm term. At a mile the taped error would be 0.008 × √53 = 0.058 ft and the EDM √(0.010² + (2 ppm × 5280)²) = 0.015 ft — a factor of nearly four.
| Method | Error model | σ (ft) | Relative precision |
|---|---|---|---|
| Taping, 7 settings | 0.008 × √7 | 0.0212 | 1 : 30,600 |
| EDM, one shot | √(0.010² + (2 ppm × 648.55)²) | 0.0101 | 1 : 64,300 |
| Taping, mean of two passes | 0.0212 / √2 | 0.0150 | 1 : 43,200 |
| EDM, mean of two shots | 0.0101 / √2 | 0.0071 | 1 : 90,900 |
How short a line would taping have to be to match the instrument
Set the series formula equal to the EDM standard error and solve for the number of settings. The answer is 1.59, so a taped line would have to be shorter than about a tape and a half — roughly 159 ft — before one taped measurement matched one EDM shot.
Read that as a statement about the constant term, not about tapes. The instrument's ±0.010 ft floor is worth about one and a half tape settings at ±0.008 ft each; below that length the EDM's constant dominates and the tape is competitive.
The practical form of the same arithmetic: to match the single EDM shot on the actual 648.55 ft line, the taped measurement would have to be repeated (0.0212 / 0.0101)² = 4.4 times, i.e. five independent passes. That is a real cost, and it is the calculation that decides whether the instrument is worth carrying to a particular job.
0.008 √n = 0.0101 → √n = 1.2625 → n = 1.59 settings
so a taped line up to about 1.59 × 100 = 159 ft matches one EDM shot
on the given line, passes needed = ( 0.0212 / 0.0101 )² = 4.4 → 5 passesAnswer
- Taped distance: σ = 0.008 × √7 = 0.0212 ft, so 648.55 ± 0.021 ft, a relative precision of 1 : 30,600.
- EDM: the 2 ppm term contributes only 0.0013 ft, so σ = √(0.010² + 0.0013²) = 0.0101 ft, giving 648.55 ± 0.010 ft, a relative precision of 1 : 64,300.
- The EDM shot is 2.1 times the better observation on this line — a real advantage, but a smaller one than the difference in method suggests.
- Taping would match one EDM shot only on a line shorter than about 159 ft, or by repeating the taping five times on this line.
- The seven settings must not be added linearly: 7 × 0.008 = 0.056 ft is nearly three times the correct 0.021 ft and describes a systematic error, not a random one.
Check
Work the taped error the wrong way deliberately and measure the difference. Seven settings at ±0.008 ft accumulating linearly would give 0.056 ft, a relative precision of 1:11,600 — barely better than the 1:12,500 of a single setting. The correct 0.0212 ft is smaller by a factor of exactly √7 = 2.6458, which is the cancellation the series formula predicts and the reason a taped traverse is better than its worst tape length.
Check the series formula against the general sum formula: √(0.008² + 0.008² + 0.008² + 0.008² + 0.008² + 0.008² + 0.008²) = √(7 × 0.000064) = √0.000448 = 0.0212 ft. Same answer, arrived at without the shortcut.
Check the quadrature combination in the EDM specification against the linear one: 0.010 + 0.0013 = 0.0113 ft linear, against 0.0101 ft in quadrature. Quadrature is 11 per cent smaller, which is the right size of difference when one term is about eight times the other. If the two terms had been equal, quadrature would have been 29 per cent smaller.
Reasonableness: a good crew taping 650 ft should expect a couple of hundredths of a foot, and a modern EDM about a hundredth. Both computed figures land where field experience puts them.
More error propagation
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