Cambridge Underground 1988 pp 16-18

Treatment of Errors in Surveying

Patrick Warren

The errors that occur in surveying are no different to those that occur when making measurements in science or technology. They can be divided into two types: systematic errors which occur because of faulty calibration of instruments, or bad technique using them, and random errors which occur because all instruments are limited in precision and no two readings can be exactly repeated. A further type of error often considered is the gross error, when the surveyor makes a silly mistake. These can be avoided by taking care at all stages of producing a survey, but can usually be recognised when plotting the survey by, for example, unusually large misclosures, reversed legs, the centre line drifts above ground, etc.

I won't cover surveying methods and reduction of survey data here; the best overall guide is "Surveying Caves" (BCRA, Ed. B. Ellis, 1976) which also includes a chapter on errors in surveying, but doesn't cover random errors.

1 Systematic Errors

These can and should be taken care of by calibrating instruments and knowing how to use them properly. We consider only the classical surveying instruments compass, clino and tape.

Compass Correction - this can be found by taking two known points on a map and sighting the compass between them on the ground. Comparison between the compass reading and the bearing found from the map gives the amount to be added to or taken from a compass reading to get a correct bearing. The process should be repeated more than once in different directions to obtain a mean value. It is possible to go directly from a compass reading to a bearing with respect to True North by including the magnetic deviation in the correction. The effect on a survey of a systematic compass error is to rotate the whole survey by the error. No misclosure errors will be introduced into networks by missing out the compass correction, which is really only needed when marking North on the survey. If more than one compass is used however, the effect is not so simple and corrections for both should be found before plotting the survey.

Clino Correction - this can be found more simply than the compass correction by picking two points on the ground and measuring the vertical angle from one to the other and vice versa. These two readings should be of equal magnitude and opposite sign. The clino correction is simply the amount that has to be added to or taken from the two readings to make them satisfy this requirement. Again the calibration should be repeated more than once with different elevations to find a mean value.

For the same reason that clino calibration is easier than compass calibration, its effect on a survey is more drastic. A systematic clino error causes the vertical elevation to become completely unpredictable - it depends on which legs were surveyed forwards and which backwards. In network caves such an error can lead to large vertical misclosures which a computer program will almost certainly not distribute correctly. For this reason it is much more important to have a good idea of the clino correction than the compass correction.

Tape correction - just make sure lengths are measured from the end of the tape! Some tapes (non metal ones) can sustain damaged ends and it may be easier to cut the end off and measure from further along. On old tapes, check that nobody has done this already!

2 Random Errors

By their nature these can only be treated statistically, and estimates of their size will only be needed for surveys including closed loops. If none of your surveys are like that, don't bother confusing yourself over this part! For surveys with loops, or networks, some estimate of the relative size of the random errors which cause misclosures is needed. Once these have been found the misclosure can be distributed back through the network. The only sensible way to do this is by least squares - see the author's detailed article in the previous Journal (Cambridge Underground 1986-7, also a forthcoming paper in Cave Science). The analysis shows that what we need to know are the standard deviations (std devs) for the Easting, Northing and Height changes (from now on I call these x,y,z respectively). Misclosure errors are distributed weighted with the square of this standard deviation, so a leg with twice the std. dev. as another gets four times the error. The network can be treated independently in the x, y and z directions (under some simplifying assumptions). A typical leg has Length (L), Clino (C) and Bearing (T) readings. When these have been corrected for systematic errors, we can work out the Easting, Northing and Height changes (x,y,z):

x = L cos(C) sin(T)
y = L cos(C) cos(T)
z = L sin(C)

The sources of random error in these are in the measurement of L, T and C. We take the approximate magnitudes of these errors to be dL, dT and dC respectively. These can be viewed as being the precision with which the measurement is made. A further source of error is in the station position. This we take as about dp in any direction, distributed equally over each of the three directions x,y,z.

We can work out approximate standard deviations for the Easting, Northing and Height changes. Calling them sx, sy and sz respectively we get:

sx² = dp²/3 + [x dL / L]² + [z sin(T) dC]² + [y dT]²
sy² = dp²/3 + [y dL / L]² + [z cos(T) dC]² + [x dT]²
sz² = dp²/3 + [z dL / L]² + [L cos(C) dC]²

Note that dC and dT must be in radians eq:- dT(radians) = dT(degrees)/57.30

The formulae are ambiguous for vertical legs, where there is no compass bearing. We should have instead:

sx² = sy² = dp²/3 + ½ [L dC]²
sz² = dp²/3 + dL²

The effect can be achieved by using the previous formulae with a value of 45° for the bearing. This doesn't of course have any affect on x,y,z.

What are the values of dp, dL, dC, dT? The BCRA survey centre line gradings (Surveying Caves) suggest for Grades 3 and 5 we use the values in table 1. In table 2 there are some examples of standard deviations worked out for a range of leg lengths, for both approximately horizontal, and vertical legs. As can be seen there is some variation within one Grade, but Grade 3 errors are all about five times greater than Grade 5 errors. This leads us to conclude that one should either take all legs within a grade to have about the same error, or work each error out exactly - there aren't really any half-way options. It seems sensible only to include exact errors in a computer program, which can work them out easily enough. Such a program could also distribute the errors back according to least squares. The factor of five between Grades 3 and 5 should be acknowledged however when distributing misclosures back - legs surveyed to Grade 3 should receive twenty-five times the error appropriated to legs surveyed to Grade 5.

Further Considerations

Legs surveyed to standards other than Grade 3 or 5 can have their errors taken into account in the same way. Since errors arising from measurement of L, T, C and station position should be roughly of the same magnitude, a guide is that the standard deviation is about the same as any one of these errors. The above calculations shed light on the practice of distributing errors back directly proportional to L, the slope length. This isn't right because it ignores station position error, and doesn't take into account the variations with compass bearing (T) and clino reading (C), which are significant. The size of these variations spoils any proportionality to slope length. Comparing standard deviations with slope lengths in table 2 illustrates this. Be careful of using these formulae too freely - a computer which distributed a vertical misclosure along a canal passage would give a slope to the water's surface! The answer here is to set to zero (or make very small) the vertical standard deviations for legs surveyed along the canal. Other methods must be applied to other techniques of surveying such as levelling or using a theodolite. Radio location is typical of these - not much has been done to estimate the errors which can occur. Generally depth measurement is less accurate than location of the null point; in good conditions the errors are quoted as being 2m-3m for location, and 5m-10m for depth, for a total depth of about 100m [Surveying Caves]. It is quite important to know the errors since most radio locations are part of loops, surveyed back on the surface to the cave entrance, and we need to know how much mislosure can be attributed to the radio location. Too little would distort the survey and too much would remove the point of doing a radio location in the first place.

Source of Random ErrorSymbolGrade 3Grade 5
Station position errordP0.5m0.1m
Length errordL0.5m
Compass errordT2.5° (0.044 rad)1.0° (0.018 rad)
Clino errordC2.5° (0.044 rad)1.0° (0.018 rad)
Table 1 - Suggested values for random errors.
A2.18+05010 0.382.140.19
B6.09+03297 -5.422.760.32
C10.83-07135 7.60-7.60-1.32
D3.47-90(045) 0.000.00-3.47
E6.70-90(045) 0.000.00-6.70
F11.35-90(045) 0.000.00-11.35
sxsysz sxsysz
Grade 3 Grade 5
Table 2 - Examples of legs, and their random error estimates.


'Surveying Caves', BCRA, ed. B.Ellis 1976

'Distributing Misclosure Errors in Surveying', P.B.Warren, CUCC Journal 1986-87, pp 19-31.

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