TAC 43 Index
ALTIMETERS have been used by mountaineers for many years, but the recent development of electronic wrist-watch type instruments has made the altimeter easier to use and more popular with walkers and climbers. The inclusion of Knight's Peak in Munro's Tables was justified on height measurements made with an altimeter and last year altimeter measurements led to a suggestion that Leathad an Taobhain, a Corbett in Glen Feshie, may be over 914m. (But see TAC36, pp4-5 re Knight's Peak and TGO Oct 1998 p10 re Leathad an Taobhain - Ed.) So just how accurately do altimeters measure height? Avocet, makers of one of the popular models, claim 'On a typical day, minor atmospheric pressure changes may cause the displayed altitude to vary from the actual altitude by 20m. With the arrival or departure of a weather front, displayed altitude can change 20 to 50m, and a storm can cause a change of more than 50m'. This article aims to show that both barometric drift and temperature may cause large errors in altimeter measurements and corrections must be made for these.
The altimeter works on the principle that the pressure within a column of air decreases in a known way with height. For the scientifically inclined, the relationship is
z = (RT/gM).loge(po/p)
where z is the height difference between the starting height and the measurement height, R is the gas constant, T is temperature of the air in degrees Kelvin, g is the acceleration due to gravity, M is the molar mass of the gas (in this case air), po is the atmospheric pressure at the starting height and p is the atmospheric pressure at the measurement height.
Electronic altimeters have this relationship programmed into the chip, while for hand-held instruments it is used to calculate the graduated height scale. So what is this equation saying? Imagine yourself standing at the foot of a mountain at sea level. Above you is a column of air several miles thick, pressing down. That is atmospheric pressure. As you climb to the top of the mountain the column is now shorter, but the air around you is thinner, or less dense. Consequently, at the top of the mountain, the total amount of air pressing down, the pressure, is less than at the bottom of the mountain. The equation is merely expressing this change in a quantifiable way.
Problems arise because the detector responds only to changes in pressure. To convert this change to height, the other terms are assumed constant. Unfortunately for the manufacturers of altimeters and for those using them, these terms are not constant and so may affect height measurements. We have made a statistical assessment of the errors achievable after making appropriate corrections to the height reading and after having tested the results on the hill. Note that what follows does not deal with sources of error associated with the construction of the instrument and specific to that particular design. Instead it looks at the assumptions made in the application of the equation - and therefore these errors are general and applicable to all instruments.
It is not generally recognised that the height measurement will be in error if the air temperature differs from the value used for the factory calibration of the instrument, although many users have noticed that their altimeter tends to read high at low temperatures. Suppose the temperature used for the calibration was 10°C. For every degree that the air temperature differs from this value the height reading will change by 1/283 or 3.5m per 1000m of ascent. This does not appear to be very much, but if the air temperature is at freezing point or 20°C, then the effect will be to change the height reading by 35m per 1000m of ascent. Thus someone climbing 1024m Sgorr Dhearg on Beinn a'Bheithir, setting the altimeter at Ballachulish, would measure its height as either 1059m or 989m, quite a difference. Temperature has a demon- strably significant effect on altimeter readings and must be corrected for. Note that this has nothing to do with the workings of the instrument itself, which the manufacturer may well correctly state to be 'temperature compensated'. This means that the instrument will give the same reading at whatever temperature it happens to be, not that it can compensate for the effect described above.
To picture what is happening, imagine the atmosphere to be very cold, well below the temperature for which the altimeter has been calibrated. The air molecules have lost energy and therefore gravity is able pull them closer to the Earth. Under these conditions the air's density and pressure fall more rapidly with height. Now imagine the atmosphere to be warmed to a very high temperature, well above that for which the altimeter has been calibrated. The molecules have gained energy and can counter the force of gravity, so the change in density and pressure with height is less. If the altimeter is used under the two conditions it will experience, for the same true height change, a greater pressure change under the cold conditions than it will under the hot conditions. Because it converts pressure change into height change, the altimeter will register a greater height under the cold conditions than it will under the hot conditions even though the actual height ascended is the same.
The error in g can generally be ignored, while the error in M has only a small effect in UK conditions. If the altimeter has been calibrated for moist air at about 50% relative humidity, then on an ascent of Ben Nevis from Fort William an error of up to 4m could result from changes in the moisture content of the air. At temperatures greater than 13ºC the air is capable of absorbing much larger quantities of water vapour and the error could become much more significant.
If it were only this simple then it would be a trivial matter to measure the temperature prior to or during a climb and then to correct for the difference between the calibrated temperature and the actual temperature at the time of the walk. However, temperature also varies with height and this change, or 'lapse rate', may be as much as 10ºC per 1000m of ascent for dry air (although in wet conditions it may be much less than this). The usually quoted figure is 6.5ºC per 1000m of ascent. It is accepted practice that the average temperature between the starting height and the finishing height may be used in calculating the correction for the effect of temperature. Since the walker cannot be in two places at once, the temperature at the summit is measured and the temperature at the starting point is calculated on the 6.5ºC/1000m assumption. Note that it takes most mortals an hour or two to ascend a hill and in this time the starting temperature may vary considerably, so it is not usually acceptable to measure it before you start out on a walk. The error in the assumption that the lapse rate is 6.5ºC per 1000m is summarised in Table 1. It cannot be stressed too much that altimeter measurements must be corrected for temperature in order to be meaningful - but, even then, uncertainty in the determination of the average temperature still leads to a significant source of error.
While the walker is in the hills, sea level barometric pressure may be rising or falling. During stable weather this drift may only be the equivalent of 10-20m a day, but on occasions it may be many tens of metres over a period of a few hours. Figure 2 is a plot of height drift recorded over a typical day. In this case the altimeter was in a fixed location, so the height changes represent pressure changes that have occurred in that location. Of course this drift cannot be continuously monitored during a walk; the best a walker can achieve is the determination of a few points at convenient places where a feature can be accurately identified with a map height. We have estimated the error from performing linear inter- polations between four points measured during the day, shown as squares in Figure 2.
From the results on fifteen experiments we deduce that the error will be within ±8 metres 99% of the time.
The height at the start of a walk will be obtained from a height contour or more rarely a spot height or trig point. Using reliable sources, the maximum errors on OS maps can be taken as ±5.4 metres for height contours, ±3.3 metres for spot heights from aerial survey and ±2 feet for trig points (±1 metre for trigs converted to metric units). In the most common situation the height will be taken between two contours and an additional error arises from the accuracy of interpolation. The value for 'Map Error' in Table 1 includes an allowance for this.
The magnitude of all the sources of error described above are summarised in Table 1 (all measurements in metres).
All values in metres
|Height diff.||Error: g & r||Error: temperature||Error: drift||Error: map||Total error|
The total error is derived from statistical theory and may be regarded as an interval within which the vast majority of results - well over 99% - can be expected to lie. If we again consider our ascent of Ben Nevis, then the best we can do with a perfectly functioning altimeter is to measure the height of the mountain to be between 1324m and 1364m even after correc- tions for both barometric drift and temperature. For the previous example of Sgorr Dhearg ascended from Ballachulish the measurement will lie in the range 1010m to 1038m and even for the ascent of Stac Pollaidh from Loch Lurgainn, which involves just over 500m of ascent, we will only be able to measure the height of the mountain to be between 601m and 625m.
Having examined the theory, we looked at just how good altimeter measurements are in practice once corrections for temperature and barometer drift have been applied. We have measured the heights of over 60 hills by the method described above. These hills were not specially chosen for the study, but the ascents ranged from 185m to 866m and thus spanned a wide range; relatively few ascents in the UK are greater than 1000m. The instrument used was an Avocet altimeter calibrated in accordance with the manufacturer's instructions.
As an example, consider the ascent of Mount Battock which the authors made on 27/4/97. Table 2 shows the barometric drift during the walk. Column 5 shows the height after a temperature correction has been applied to the altimeter readings. In this case a correction of -4m was necessary.
Values in metres
|Grid ref.||Map height||Altimeter height||Corrected height||Time||Height difference|
Figure 4 shows the calibration chart constructed from these data. The summit of Mount Battock was reached after 1hr 50min, when the barometric drift was -14m. The temperature on the summit was 4°C and the height difference between the summit and the starting point was 638m. This gives an average temperature for the air column of 6.1°C and therefore a temperature correction of -15m. We estimate that the altimeter has been calibrated for an air temperature of 12.6°C. The altimeter reading on the summit was 815m, which then gives a height of 786m when the two corrections are added. The OS trig height for Mount Battock is 778m. The results for all the hills are recorded in Figure 5 in which the bars show the corrections that have been made for temperature and barometric drift and the points show how much the resulting answer differs from the OS height (positive numbers mean the OS measurement is higher). It should be noted that some of the temperature corrections are over 20m, in one or two cases 35m, and some of the barometric drift corrections are even greater, up to 65m. Total corrections to the altimeter reading are as high as 85m and so may be very significant indeed.
How does the level of agreement compare with the error estimates in Table 1? To make the comparison, the table values should be increased to take into account the error in the OS summit height. Most of these are spot heights or trigs, however, and the effect is to boost the total error by less than half a metre. From Table 1 and the height ascended for each hill we would predict that, on average, 95% of the corrected altimeter measurements would lie within +-10m of the true height and well over 99% within +-15m. In fact two hills were in error by more than 10m: Millfire on the Rhinns of Kells (-13m) and Meall Coire nan Saobhaidh near Loch Arkaig (-21m). The latter, where the summit height has to be estimated from a ring contour, is well outside the predicted range and on statistical grounds sufficiently far from the other results to merit suspicion. Apart from this one hill, the results are in excellent agreement with theory.
Lastly, consider the controversial case of Knight's Peak. How accurate a measurement can an altimeter be expected to make in the best case? Suppose that the altimeter is set on the summit of Sgurr nan Gillean (964m). A temperature correction is still needed, of course, but the height difference of c50m is so small that the lapse rate can be ignored. A competent climber will reach Knight's Peak in less than an hour so in favourable conditions the barometric drift should not exceed a few metres. We will suppose that he or she returns to Sgurr nan Gillean and makes the recommended correction for barometric drift. We estimate that the maximum error would be 6m. So if the true height of Knight's Peak were 914m we would expect a perfectly functioning altimeter to give values in the range 908m to 920m. Regrettably, that is not good enough to distinguish whether Knight's Peak qualifies as a Munro Top, a Corbett Top or as just another excellent spire on Pinnacle Ridge.
What should we do when we find a measurement outside the expected error range of our altimeter? In the August 1998 issue of TGO Andy Moffat reported a reading of 930m on the trig point of Leathad an Taobhain (912m). As his altimeter read correctly on the 847m spot height to the north both before and afterwards, the result is apparently well outside the expected error range which we would estimate to be +-10 metres based on the information provided. Our view is that it would be unwise to get too excited about a single measurement in case it is a statistical outlier resulting from an instrument quirk, say. In such cases the only way to obtain sufficiently strong evidence to query the mapping is for more measurements to be made. That is a challenge readers may wish to follow up.
For those interested in further reading, old mountaineering books and journals show that mountaineers in the early years of this century used altimeters extensively and were familiar with their limitations. References to the underlying theory, plus the complete set of data and calculations from which this article was written, may be obtained from the authors on request.
TAC 43 Index