Concept

Reliability — where it appears

How large an error in one observation must be before a network's own test is likely to find it. It is computed from the design alone, and it says both what a check can notice and how far the coordinates will already have moved by the time it does.

Named by 3 essays across one field — each of them below, with the objects they name alongside it.

How large a blunder has to be before the test notices. A blunder of increasing size put into the least-checked observation of a braced quadrilateral, with the standardised residual it produces. The horizontal line is the critical value the test uses, and the vertical one is the minimal detectable bias — δ₀σ/√r, which is 87 millimetres for this observation and is computed from the network's DESIGN, before any observation is made. Below it nothing is flagged; above it everything is. The observation's redundancy number is 0.144, so it is checked by a seventh of an observation and hides six-sevenths of whatever is wrong with it.

The blunder the network cannot see

A least-squares adjustment has no concept of a mistake. The smallest blunder its test will find in the least-checked leg of a braced quadrilateral is 87 millimetres, and by the time it fires a station has moved by nearly ten times the accuracy the same adjustment reports for it.

practice · Reduction

Ties ten kilometres long locate the control along themselves and hardly at all across. The local order of the stated survey: five stations about a kilometre apart, and the twenty-five lines from them to one another and to three control stations 10 km away. Adjusted on its own lines, the network knows where each control station lies along its ties to within a few centimetres and across them only to 179.0 mm, because five lines leaving a cluster a kilometre wide are nearly parallel ten kilometres out. The long ellipses at the control stations are that knowledge, drawn at one scale. At the local cluster, drawn at the same scale: dashed, the error the held-fixed adjustment reports for each station, 8.0 mm on its long axis at the first station; solid, the same station's own error after an equal-weight rigid fit of the free network onto the three control stations, 70.2 mm. The equal-weight fit trusts each control coordinate as much as every other, and the across-tie ones are worth almost nothing.

Three numbers are all a local survey can check of its control

Adjust a local network on its own lines, then fit it onto its three control stations, and the misfit at the control is the only place the control's error can show. It is exactly the excess the held adjustment's variance factor already carried, counted on three degrees of freedom instead of fifteen, and it catches a disturbed monument 79 times in a hundred where the held report catches it 49. Give the fit a scale and it hides 98 per cent of that excess, and lends the control's error to every length.

practice · Reduction

Moved five centimetres, one station of three looks like all three, and one of five looks like itself. The local cluster at the centre, tied to three control stations (left) and to five (right) ten kilometres away, drawn on their true bearings with the distance shortened. The northern station, filled, has been moved 50 mm along its tie and nothing else is wrong. The bar at each station is the expected value of its own test for a displacement along its tie, noise-free. With three stations the three tests read 3.25, 3.22, 3.22: the data say only that the triangle is too large, and any of its corners could have made it so. With five, the moved station reads 4.31 and no other more than 2.44.

Naming the control station that moved takes a fifth

A local survey tied to distant control can tell that a monument has moved, and with three control stations it cannot say which. Each station gives the local lines one number they know well — its distance — and a shift of the whole control uses up two of them, so three stations leave one and four leave two. Five evenly spaced name a station moved five centimetres 83 times in a hundred; three name it 20. Better lines cannot stand in for the missing stations, and every station added also exposes more of the control's own published error.

practice · Reduction

Named alongside it

The objects these essays reach for when they reach for this one.

Adjustment Least-squares Network Control points Covariance Degrees of freedom Hierarchy Minimal detectable bias Redundancy Verification Baarda Blunder

All concepts