Marhofn 196.11 - May 2009

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Hill surveying part 1: theory

Graham Jackson

'So how do you know you're right?' asked my wife, in that questioning way that wives have. John Barnard, Janet and I had just found Cracoe Fell to be higher than Thorpe Fell Top by 1.84m. Having spent ten hours doing so, I was not keen to try and justify the precision of the measurement at that moment. It was a good question though, and over recent months we have received several enquiries about the precision of different surveying methods. In this article I attempt to summarise our experience of them.

Simple staff and spirit level

A staff and spirit level have often been used to determine drops in the range 15m to 30m. In this method, a pole is held vertically and the user sights along a spirit level held at a fixed point on the pole, towards the slope in question, noting where his line-of-sight intersects with the hillside. He then walks to that point and repeats the process until he has reached the summit position. Since the height of the pole is known and the number of sightings are counted, the drop can be easily calculated. We have made several comparisons of this method with line surveys carried out with a surveyor's level and staff (see below) and the measurement error of the method is about +/-2m, or slightly better if the slope being measured is steep. (In this article we define error as three standard deviations.) This method has the advantage of using very light-weight equipment that can be easily carried on any walk. It is ideal for identifying drops that are close to the 15m or 30m criteria used for some hill lists; drops that might repay further study by more precise methods.

Hand-held GPS units

GPS units make use of a network of 24 satellites that orbit the earth at a distance of 20,000km. A unit can receive signals from up to 12 of these at any one time. By timing the signals from a minimum of four satellites, the unit can calculate the user's position (including height) on the earth's surface. Most hillwalkers now own a GPS, and because the instrument gives a digital read-out, tend to believe the number on the screen. We have taken a Garmin eTrex GPS, measured grid reference and height a hundred times at the same location, then calculated the measurement error. For position this error is +/-5m and for height it is +/-12m. If you don't believe this, just take your GPS outside and take readings every minute for an hour. You may be surprised how much the height reading can change. In addition, the height reading has a small positive bias. Consequently, a hand-held GPS is very limited for measuring either absolute height or height difference, because readings can fluctuate so much and so rapidly.

Altimeters

It is not generally recognised that the height measurement made by an altimeter will be in error if the average temperature of the air column in which the measurement is made differs from the factory calibration value of the instrument (usually 10-15C), although many users have noticed that their altimeter tends to read high in cold weather. For example, a reading taken when the air column is at 0C will over-read height by about 3.5m for every 100m of ascent. This effect has nothing to do with the workings of the instrument itself, which the manufacturer may well correctly state to be temperature compensated. Strong wind also affects altimeter readings.

Survey-grade altimeters are capable of measuring height differences to one to two metres precision. To achieve this, techniques have to be used to take into account the effects of barometric drift and temperature changes in the air column. A typical method uses two altimeters; one is placed at a base as a control and the other is used as a roving instrument to measure the height differences. Height, temperature and time are recorded with the roving instrument at the required points. These readings can be corrected from similar data recorded at the base. For greater precision, a third altimeter can be used at a known point higher than the roving instrument.

Small hand-held altimeters or wrist altimeters are capable of about +/-10m precision. For 30m of drop where col and summit are close by, the precision will be improved to about 2m, as the temperature correction and barometric drift will be very much reduced. Anyone interested in learning more about altimeters may wish to look at an article by Chris Crocker and Graham Jackson, available at www.biber.fsnet.co.uk/

Surveyor's level and staff

This method is used mainly for the determination of height differences, although if the absolute height of one point is known with sufficient accuracy (e.g. a trig point) then absolute heights can also be determined. The standard procedure for line surveying, say from a summit to a col, is as follows. The staff is held vertically on the summit position and the level is set up at a convenient position lower on the hill. Once a reading off the staff has been taken (backsight reading), the staff is then moved to a position further down the hill towards the col, but the level is not moved, apart from a rotation through 180 degrees to take another reading (foresight reading). The level is then moved to another point lower down the hill, and this process is repeated until the final reading is taken with the staff at the previously determined position of the col. The total of the backsight readings is then subtracted from the total of foresight readings to give the drop. Such a procedure is capable of great precision; typically <5cm for a drop of 30m. The major error associated with drop measurements is the correct location of the summit and col, especially if either is covered with deep heather!

Differential GPS

Differential GPS operates on the same general principle as hand-held GPS units, but with a few important differences. The instruments employed are dual-frequency, 24-channel instruments, which can lock on to a maximum of 24 satellites and receive two signals (at different frequencies) from each of these satellites. The latter feature reduces inaccuracies that result from atmospheric degradation of the satellite signal. As a stand-alone instrument, the GPS is still only capable of giving position to a precision of one to two metres and height to about five metres. Despite the more sophisticated features of the receiver, there are still sources that create residual errors, the most significant being clock errors (satellites have atomic clocks but receivers do not) and residual atmospheric distortion. To obtain accurate positions and heights, corrections need to be made to the GPS data by using data from the Ordnance Survey, using dedicated post-processed software. The correction data is obtained from a national network of GPS base stations, the positions of which are accurately known and which constantly transmit the differences between their apparent positions, as measured by GPS, and their known positions. This procedure further reduces error to about 2cm in position and about 5cm in height. Differential GPS is one of the few methods capable of accurately measuring absolute height and is therefore an invaluable technique.

Trigonometric levelling

A theodolite is an instrument for measuring horizontal and vertical angles. It comprises a moveable telescope mounted within two perpendicular axes. When the telescope is focussed on a point, the angle of each of these axes can be measured with great precision. From a knowledge of one distance measurement then other distances, such as height, can be calculated. With a theodolite, distance measurements have to be made with a separate measuring device, but a total station is a theodolite that incorporates a laser distance measurer.

We have no experience of using theodolites and total stations, but top-of-the-range instruments have comparable precision to a surveyor's level and staff or a differential GPS system. Apart from cost, the main disadvantage of these systems for us is that the instruments are very heavy and bulky (>5kg), and therefore not really suitable for surveying in mountainous and remote environments without a band of willing Sherpas.

Summary

The table below summarises the precision of measurements made by all the instruments described above.

Instrument Precision (+/- m)
Simple staff and spirit level2
Hand-held GPS (position)5
Hand-held GPS (height)12
Survey-grade altimeters2
Hand-held altimeters10
Surveyor's level and staff0.05
Differential GPS0.05
Trigonometric levelling0.05

So I spent half an hour explaining all this to my wife, which she pondered and then replied: 'The last time you measured something was when we were in Scotland and you said it was half a mile back to the car according to the map. We ended up doing three miles through heather, with a stream crossing.' I had to concede that she was correct! I need to spend more time considering the precision of my map-reading abilities.

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