  ## The use of altimeters in height measurement

Graham Jackson and Chris Crocker

Altimeters have been used by mountaineers for many years, but the recent development of electronic wristwatch 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 recently altimeter measurements have led to the suggestion that Leathad an Taobhain, a Corbett in Glen Feshie (ref. 1), may be over 3000 feet. These claims have also helped to bring these instruments to the attention of hillgoers. However, just how accurately do altimeters measure height? The makers of one of the popular models claims 'On a typical day, minor atmospheric pressure changes may cause the displayed altitude to vary from the actual altitude by 20 metres. With the arrival or departure of a weather front, displayed altitude can change 20 to 50 metres, and a storm can cause a change of more than 50 metres'. That the authors have experienced errors of over 50m on the summits of hills on days when atmospheric pressure changes were equivalent to changes in height reading of only 10 to 20m prompted this study, in which the accuracy of the instrument was tested. This article will show that both barometric drift and temperature may cause large errors to altimeter measurements and corrections must be made for these.

#### The principle of the altimeter and sources of error

The altimeter works on the principle that the pressure within a column of air varies in a known way with height. The mathematical relationship that relates them 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 measured in 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.

The derivation of this relationship is given in Appendix 1.

Electronic altimeters have this relationship programmed into the chip while for hand-held instruments the graduated height scale is calculated from it. 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 on you. That is atmospheric pressure. As you climb to the top of the mountain the column above you is now shorter, but also the air around you is thinner, or less dense. Consequently, at the top of the mountain, the total amount of air pressing down on you, the pressure, is less than at the bottom of the mountain. The equation is merely expressing this change in a quantifiable way.

The problems arise because the detector responds only to changes in pressure. To convert this change to height, the other terms in the equation are assumed constant. Unfortunately for the manufacturers of altimeters and for those of us 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 tested the theory on the hill. Note that what follows does not deal with sources of error associated with the construction of the instrument and which are therefore specific to that instrument or 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.

##### Acceleration due to gravity, g

Firstly, it is assumed that the acceleration due to gravity, g, is constant with latitude and with height. In fact g changes with both but only by very little. Between the equator and the poles of the earth g varies by only 1% and, if it is assumed that instrument manufacturers have used the value at a latitude of 45 degrees, the error would only be 0.1% if the instrument were to be used at a latitude of 52 degrees. The correction for height is even less, 0.03%. Thus, for most purposes errors associated with changes in g can be ignored.

##### The composition of air

Secondly, it is assumed that the composition of the air is constant and therefore that its apparent molecular weight is constant. However, the composition of air varies, the major variant being water vapour. If it is assumed that the altimeter has been calibrated for moist air at about 50% relative humidity then the correction to the altimeter reading for compositional changes is less than 0.3% provided the average air temperature between the starting point and mountain summit is not greater than 13°C (see Appendix 2). This would give a maximum error of 4m for an ascent of Ben Nevis from Fort William. At temperatures greater than 13°C the air is capable of absorbing much larger quantities of water vapour and the resulting correction may become significant. Fortunately, this is rarely the case in the UK.

##### Temperature

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 in cold weather. Suppose the temperature used for the calibration was 10°C or 283K. 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 say 20°C (293K) then the effect will be to change the height reading by 35m per 1000m of ascent. Thus someone climbing Sgorr Dhearg on Beinn a'Bheithir, setting the altimeter at Ballachulish, would measure its height as 1069 or 989m respectively rather than 1024m, a very significant error. 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.

The temperature effect can be understood as follows. Imagine the atmosphere to be very cold, well below the temperature for which the altimeter has been calibrated. The molecules of the air have lost energy and therefore gravity is able pull them closer to the earth. Under these conditions the density and pressure of the air 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. Now the molecules have gained energy and can counter the force of gravity and 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. Now, 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.

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 use the equation to correct for the difference between the calibrated temperature and the actual temperature at the time of the walk. However, as walkers know well, temperature also varies with height. This change, or lapse rate, may be as much as 10°C per 1000m of ascent for dry air, but much less in wet or humid conditions where water vapour is condensing in the atmosphere (see Appendix 3). The usually quoted figure is 6.5°C 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 mountain summit is measured and the temperature at the starting point is calculated on the 6.5°C/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 associated with the assumption that the lapse rate is 6.5°C per 1000m is explored in Appendix 4. 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.

##### Barometric drift

While the walker is in the hills, sea level barometric pressure may be rising or falling. During calm stable weather, this drift may only be the equivalent of 10-20m in one day, but on occasions it may be many tens of metres over a period of a few hours. For example Figure 1 shows the weather pattern for 2 November 1996 for a low centred to the north of Scotland and Figure 2 (circles) is a plot of the resulting height drift recorded over that day, which is typical for many days during the year. 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. Consider a drift profile where a starting height, a finishing height and two intermediate heights are measured for this purpose. How far from the true height reading could the interpolated reading be? A specimen four-point profile can be constructed for the 2 November example (Figure 2). The error resulting from the assumption of a linear drift between the four measured points, shown as squares, would be given by the difference between the actual height readings and the line. In order to estimate this error we conducted a number of similar experiments where barometric drift is followed during an 8-hour period and height readings taken every half-hour. The results are presented in Figure 3. Of the 156 recorded values, 98% are within ±6m of the interpolated reading. ##### Map errors

The height for the start of a walk will usually be obtained from a height contour on a map or more rarely a spot height or trig point. Using reliable sources (refs. 2,3) the maximum errors on OS maps can be taken as ±5.4m for height contours, ±3.3m for spot heights from aerial survey and ±2ft for trig points (±1m 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 one's interpolation. The value for 'Map Error' in table 1 includes an allowance for this.

##### Total error associated with a height determination

The magnitudes of all the sources of error are shown in table 1.

Table 1
all values in metres

 Height diff. Error: g & ρ Error: temperature Error: drift Error: map Total error 500 2 2 8 8 12 1,000 3 6 8 8 14 1,400 4 12 8 8 20

The total error is derived from statistical theory and may be regarded as an interval within which the vast majority of results - over 99% - can be expected to lie, as are the values for the individual components. 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 1324 and 1364m even after corrections for both barometric drift and temperature have been made. For the previous example of Sgorr Dhearg ascended from Ballachulish the measurement will lie in the range 1010 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 601 and 625m.

Details of the calculation are given in Appendix 5.

#### Testing the theory

Having examined the theory, we looked at just how good altimeter measurements are in practice once corrections for temperature and barometric drift have been applied. The results were then compared with the predictions of table 1.

We have measured the heights of over sixty hills using the method described above using an 'Avocet' altimeter calibrated in accordance with the manufacturer's instructions. These hills were not specially chosen for the study, but the ascents ranged from 185 to 866m and thus spanned a wide range; relatively few ascents in the UK are greater than 1000m. On the instrument used for this study, pressing any of the functions buttons tends to alter the current height reading by 5 to 10m temporarily (15 minutes) and, while the altimeter is temperature compensated, it nevertheless gives slightly unstable readings, fluctuating by about 10m, when first put on the wrist. Care was therefore taken when setting the instrument prior to each walk.

As an example, consider the ascent of Mount Battock which the authors made on 27 April 1997. 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. The method of calculating the temperature correction is the same as that described below for the summit height.

Table 2
values in metres

 Grid ref. Map height Altimeter height Corrected height Time Height difference Start NO540790 140 140 140 0 0 Stream (out) NO543827 380 390 386 1 -6 Stream (return) NO543827 380 405 401 2.5 -21 Finish NO540790 140 160 160 3.33 -20

Figure 4 shows the calibration chart constructed from these data. The summit of Mount Battock was reached after 1 hour 50 minutes, when we can see that the barometric drift was –14m. The temperature on the summit of Mount Battock was 4°C and the height difference between the summit and the starting point was 638m which gives an average temperature for the air column of 6.1°C and thereby a temperature correction of –15m. We estimate that the altimeter has been calibrated for an air temperature of 12.6°C (see Appendix 5). The altimeter reading for the summit height 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 measurement for the hill (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. Most of the differences between the OS map height and the corrected altimeter reading lie in the range ±10m. A list of the hills is given in Appendix 6.

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 over 99% within ±15m. In fact two hills were in error by more than 10m: Millfire on the Rhinns of Kells near Corserine (-13m) and Meall Coire nan Saobhaidh a former Corbett 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. If this hill is excluded, the standard deviation of the errors of the remaining 60 hills is 5.0m which is close to the predicted value of 4.5m. Actually we would expect the total error in table 1 to be a slight underestimate as we have not taken into account errors related to instrument design. Apart from one hill, therefore, 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 ca. 50m 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 908 to 920m. Regrettably, that is not good enough to distinguish whether Knight's Peak qualifies as a Munro top or whether it is 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 a correspondent 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 ±10m based on the information provided. Our view is that it would be unwise to get too excited about a single measurement - a statistical outlier could arise from a variety of sources such as instrument malfunction, wind eddies, etc. 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, early mountaineering books and journals show that mountaineers in the early years of this century used altimeters extensively and were familiar with their limitations (refs. 4,5). For a more detailed description of altimeters, the physical principles and the physics of meteorology, ref. 6 is an invaluable source. A general guide to the treatment of measurement errors is provided by ref. 7. Readers wishing to make measurements of their own may find Appendix 7 useful. It is a table of temperature corrections for an altimeter calibrated for an air column of 12.5°C. Corrections for other calibration temperatures can be easily produced in a spreadsheet.

Graham Jackson and Chris Crocker
February 1999

#### References

1. The Great Outdoors, August 1998
2. Ordnance Survey Maps a descriptive manual, J B Harley, Chapter 11
3. Land-Form PROFILE User Guide, Ordnance Survey, 1997
4. J Gall Inglis, SMC Journal, 1907, IX, 243-247; J Rooke Corbett, SMC Journal, 1932, XIX, 324-332; both articles reproduced in The Munroist's Companion, ed. Robin N Campbell, SMC, 1997
5. Scottish Mountaineering Club General Guide Book, 1933
6. Dictionary of Applied Physics Volume 3, published by Peter Smith, New York, 1950
7. Guide to the Expression of Uncertainty in Measurement, International Organisation for Standardization, 1993

#### Appendix 1

##### The barometric equation

Pressure may be defined as the height of a vertical column of fluid of unit cross-section with its base at the point considered. Let p be the pressure at a height z in a column of gas. Then:

dp = –ρg dz   ... equation 1
where
dp is the pressure change for a height change of dz
ρ is the density of the fluid at a height z
g
is the acceleration due to gravity.
ρ is a function of z. The equation can be integrated by expressing ρ in terms of the variables p and/or z.

From the gas equation:

pv = nRT   ... equation 2
where
p is pressure
v is volume
T is absolute temperature
R is the gas constant
n is the number of moles of fluid in the volume v.
Since ρ = Mn/v, where M is the molecular weight of the fluid, equation 2 may be rewritten to give:
p = ρRT/M
Substituting for ρ in equation 1:
dp = –(pgM/RT).dz
Re-arranging
dp/p = –(gM/RT).dz
Upon integration this gives
ln p = –(gM/RT).z + constant
If p is po when z is zero then the equation becomes:
z = –(RT/gM)(ln po – ln p)
or in the general case
z1z2 = –(RT/gM)(ln p1 – ln p2)

#### Appendix 2

##### The effect of the composition of air on altimeter measurements

Consider a unit mass of dry air of volume v and therefore of density 1/v.
Now consider the same volume of moist air of absolute humidity k.

Volume of moist air = v
Mass of water in moist air = kv

Let the density of water vapour be ρw
Then the volume occupied by the water vapour = kvw

Therefore
volume of air = vkvw
= v(1 – kw)
mass of air = v(1 – kw)/v
Therefore
total mass of moist air = (1 – kw) + kv
Now v = 1/ρa. Therefore
total mass of moist air = (1 – kw) + ka
= 1 – k(1/ρw – 1/ρa)
or rearranging
total mass of moist air = 1 – {(kw)(1 – ρwa)}
Now ρwa is equal to 0.622 and so the expression reduces to
total mass of moist air = 1 – 0.378kw
The height reading of the altimeter will be affected by an amount equal to the ratio of the mass of dry air to the mass of the moist air, that is by the ratio:
f = 1/(1 – 0.378kw)
Note that this ratio is greater than unity for moist air, which means that the altimeter will read low if calibrated for dry air.

Now let us assume that the altimeter has in fact been calibrated for air at 50% RH and 12°C. The absolute humidity of air under these conditions is 5.67g/m³ and f is 1.000235. A table may now be constructed for saturated air at different temperature:

Table A1

 Temperature (K) ρw (g/m3) k (g/m3) f Error (%) 260 844 1.67 1.00075 -0.19 266 825 2.77 1.0013 -0.14 270 813 3.83 1.0018 -0.09 273 804 4.85 1.0023 0 278 790 6.8 1.0033 0.06 282 778 8.82 1.0043 0.16 286 767 11.35 1.0056 0.29 290 757 14.48 1.0073 0.46 294 747 18.34 1.0094 0.67

From the table it may be seen that the error becomes significant above 286K or 13°C. For an air column at this average temperature and saturated with water vapour the height reading would be in error by 0.3%. In a tropical climate the error would clearly become much more significant.

#### Appendix 3

##### Lapse rate for dry air

p1v1γ = constant
where
p is pressure
v is volume
γ is the ratio of specific heat at constant pressure to specific heat at constant volume
Therefore if gas at pressure p1 and volume v1 expands to volume v2 at pressure p2 adiabatically:
p1v1γ = p2v2γ
but
pv = nRT
therefore
p1/p2 = (nRT2/p2)γ/(nRT1/p1)γ
rearranging
p1/p2 = (T2/p2)γ.(p1/T1)γ
(p1/p1)γ.(p2γ/p2) = (T2/T1)γ
(p1/p2)1-γ= (T2/T1)γ   ... equation 1
At sea level let p1 be 1014mb and the temperature T1 be 283K. Then for 1000m of ascent, p2 is 897mb. Substituting these values in equation 1 gives a value of 273K for T2 , i.e. the lapse rate is 10K per 1000m of ascent.
##### Lapse rates for saturated air at various temperatures

When air is saturated and begins to rise condensation occurs as the temperature falls. This condensation releases latent heat which warms the air and partially compensates for the adiabatic cooling. The equation which describes this is:

dT/dz = g/(Cp + L.dwS/dT)
where:
wS is the ratio of the mass of saturated water to the mass of air
L is the latent heat of vaporisation of water
Cp is the specific heat of water at constant pressure
The term dwS/dT is readily derived from the gradient of a plot of water content against temperature for saturated air data which is given in table A2.

Table A2
Water content of saturated air

 Temperature (K) Water content (g/m³) Temperature (K) Water content (g/m³) 252 0.82 282 8.82 254 0.98 284 10.02 256 1.17 286 11.35 258 1.4 288 12.83 260 1.67 290 14.48 262 1.98 292 16.32 264 2.35 294 18.34 266 2.77 296 20.58 268 3.26 298 23.06 270 3.83 300 25.78 272 4.48 302 28.77 274 5.19 304 32.06 276 5.95 306 35.67 278 6.8 308 39.61 280 7.75 310 43.93

A plot of lapse rate against temperature is given in Figure A1. From the plot it may be seen that for conditions usually prevailing in the UK, it would be unusual to experience lapse rates below 4K/km. NOTE: Subsequent to this work, information from Richard Cooper suggests that the adiabatic cooling model will apply only about 65% of the time in the UK, typically in overcast conditions. In warm sunny weather cooling will be superadiabatic, resulting in larger temperature gradients, while on clear nights the reverse will apply, giving temperature inversions in extreme cases. It is also possible for the temperature gradient to be non-linear. This clearly increases the scope for error in the walker's calculation of average temperature from the assumed lapse rate. However Richard mentions that very little data is available for mountain conditions and he would expect a greater preponderance of adiabatic lapse rates to apply in this case.

#### Appendix 4

##### Error in temperature correction due to the assumption of lapse rate

The barometric equation may be re-cast by combining all constants into one term:

z = KTΔ(ln p)
Now suppose the average temperature for the column of air is calculated using two different lapse rates, ΔT1 and ΔT2, and found to be T1 and T2 respectively. Let the difference between T1 and T2 be t. Then the height calculated for the two cases is:
z1 = KT1Δ(ln p)
z2 = KT2Δ(ln p)
Then
z1/z2 = T1/T2
But T1 = T2 + t so the expression becomes:
z1/z2 = (T2 + t)/T2
z1 = z2 + t.z2/T2   ... equation 1
Now T is the average of the temperature at the summit of the mountain, Tm and at the starting point, in this case sea level. The latter is calculated from the lapse rate and the height difference between the starting point and the summit. Thus
T1 = {Tm + (Tm + ΔT1.z/1000)}/2   ... equation 2
where z is the height difference between the summit and the starting point. The factor of 1000 appears because lapse rates are quoted in degrees per kilometre. Similarly,
T2 = {Tm + (Tm + ΔT2.z/1000)}/2   ... equation 3
Subtracting equation 3 from equation 2 gives:
T1T2 =T1 – ΔT2).z/2000
And since t = T1T2
t = (ΔT1 – ΔT2).z/2000
Substituting for t in equation 1 then gives:
z1 z2 = (ΔT1 – ΔT2).z2.z/2000T2
Since z, the true height, z1 and z2 are nearly equal this becomes:
z1 z2 = (ΔT1 – ΔT2).z2/2000T2
If ΔT1 is 10K per 1000m, ΔT2 is 6.5K per 1000m and T2 is 283K then the error z1z2 is given by the values in the following table:

 Height difference (m) Error (m) 500 1.5 1000 6 1400 12

#### Appendix 5

##### Statistical estimation of errors

The measure of overall accuracy used by the Ordnance Survey is the root mean square error (rms error), defined by where x1, x2, ... xn are the errors of n heights measured by the same technique.

The rms error is made up of a random component and a systematic component. The generally accepted measure of the random component is the standard deviation. The OS call this the standard error and define it as the root mean square deviation from the mean error: More generally, when the standard deviation of an infinite population is estimated from a sample, the factor n is replaced by n-1 and the statistic is then known variously as the sample standard deviation, the experimental standard deviation or the standard error. Of course, the two statistics converge when n becomes large.

The systematic error, or bias, is estimated by: These statistics are related by the equation: The standard deviation of a measurement subject to multiple sources of variation having component standard deviations σ1, σ2, ...σn, where the errors act additively and independently of each other, is given by ... equation 1.
The total bias is the sum of the individual biases.

For practical purposes the maximum error can be taken as three times the standard deviation plus the bias. For a normal distribution this range will include 99.7% of measurements, on average. Provided a single error source does not dominate, the frequency distribution of the total error in equation 1 will tend to be approximately normal even when the individual errors are not. However even when this approximation does not hold (when n is small and the error distributions are far from normal) it is unlikely that the range ±3σ will contain appreciably less than 99% of measurements. This assumes that the standard deviation is known; if it has been estimated from a sample the percentages will be somewhat less.

Random errors are inherent in any measurement system but it is a goal of metrology to reduce systematic error to a negligible proportion by proper attention to instrument calibration. This is just as well because biases can be difficult to estimate. The problem of estimating the total error in an altimeter reading therefore amounts to one of estimating the standard deviations of all the component errors. The general approach is described in detail in ref. 7.

Map errors

For contours with a 10m vertical spacing, the OS quote an accuracy of ±5m and an rms error of 1.8m (ref. 3). Harley's book (ref. 2) gives some results of accuracy tests on contours which, if typical, would imply that bias can be ignored for practical purposes.

There is less information on spot heights or trig points as only estimates of the maximum error are available. Harley gives these as ±3.3m and ±2ft respectively (ref. 2), although the latter figure is subject to the method of measurement and no general figure can be quoted for trig points. (This error is quoted in feet as the OS network of levelled heights was established in the days of imperial measurements.) If bias is negligible for contour heights it seems reasonable to suppose that this will also be the case here. To estimate the standard deviations it is necessary to make an assumption about the distribution of the population of errors within the quoted accuracy ranges. The maximum expected error in any test accepted by the field accuracy testing group of the OS is defined as three times the standard error plus the systematic error. The OS only systematically test their maps for planimetric errors and contour heights so there is no assurance that this yardstick is applicable to the quoted errors for spot heights and trig points; however if we assume this to be the case, and negligible bias, the standard deviation of a spot height would be estimated as 1.1m and a trig point, 0.2m. The question of the additional error in trig points introduced by metrication is not mentioned by Harley. The error due to rounding to the nearest metre will have a uniform distribution on (-0.5, 0.5m) with standard deviation 0.5/√3. This would increase the standard deviation of a trig point to 0.35m.

When a starting height is estimated by interpolation between two 10m contours, an estimate of the standard deviation of interpolation is needed. If the walker had no idea of his vertical position between the contours then the standard deviation of the interpolation error would be 2.9m. In practice he will have some knowledge and we have subjectively estimated the standard deviation as 2.1m. Using equation 1, the standard deviation of a starting height based on contour interpolation is √(1.8²+2.1²) = 2.8m. This assumes that the errors in the adjacent contour lines are highly correlated, for which we have no proof but which seems intuitively reasonable. Multiplying by 3 and rounding gives the maximum error of 8m quoted in table 1 of the main text.

Error in temperature correction

From Appendix 4, the maximum lapse rate is 10°C/km and is unlikely to fall below 4°C/km in the UK. We have no data on the distribution of actual lapse rates in UK hill conditions. Assuming a most probable lapse rate of 6.5°C/km and a probability approaching zero at the two extremes we have conservatively estimated the standard deviation of the lapse rate as 1.4°C. This may be converted to height using the final equation in Appendix 4. In this case the maximum error, 3.5°C, is smaller than 3 times the standard deviation.

Error in correction for barometric drift

Here we have data from the experiment described in the text. The standard deviation of the 156 values of observed minus interpolated height is 2.8m. There is no correlation between the value and the position on the time axis and, as would be expected, there is no significant bias. The maximum error can therefore be taken as 8.4m. This is consistent with the largest value of 7m recorded in the experiment.

Error in air density and g

Estimation of the error in ρ is subject to the same problem as the error in the lapse rate, viz. our lack of meteorological data on the frequency distribution of relative humidity at different temperatures. A conservative estimate of the standard deviation due to error in ρ would be 0.11% of the height difference. An error in g would introduce a bias rather than a random error. Its magnitude will depend on the instrument's calibration, but as explained in the text will probably be 1m or less for most heights measured in the UK.

Total error

If we assume the errors due to map, temperature, barometric drift and ρ are independent of each other, substitution of their estimated standard deviations in equation 1 gives ... equation 2
which yields values of 3.9, 4.7 and 6.4m for z = 500, 1000 and 1400m respectively. These values, when multiplied by 3, yield the values for maximum total error in table 1. If the error distribution is normal, as is the case with our data, it also follows that the range ±2σ will contain the true height 95% of the time, on average.

Note that the root mean square of the component values in table 1 does not give quite the correct value for total error partly because of rounding and partly because the maximum error for the temperature correction is less than three times the standard deviation of this error component, as explained above.

Other sources of error

The main sources of error not taken into consideration concern the altimeter itself, and will vary with the design and possibly with the individual instrument. Obviously the linearity of the pressure sensor is critical, as is the accuracy of the temperature compensation mechanism. We have no knowledge of these. Other instrument related errors may arise from error in the value assumed for the instrument's calibration temperature, and rounding error from digital readouts. These are considered below. Finally, there is the error in the temperature measurement. For a correctly calibrated thermometer capable of reading to 0.5°C the effect on total error is negligible (and a systematic error in temperature will be unimportant if the instrument's calibration temperature is estimated as described below and the same thermometer is used for all the measurements).

Determination of calibration temperature

To determine the temperature correction, we need to know the temperature at which the altimeter was calibrated. In the absence of a reliable manufacturer's figure, the calibration temperature can be estimated statistically from experimental data. The corrected altimeter height is calculated using the equation where h is the observed altimeter reading, T is the corresponding temperature in °C, l is the lapse rate in degrees per 1000m, d is the difference between the measured height and the height at which the altimeter was set, b is the correction for barometric drift and β is the calibration temperature in °C. This expression is obtained directly from the barometric equation given at the beginning of the article. The only unknown parameter in the equation is β. An estimate of β can be made by finding the value that minimises the sum of the squared errors Σ(hmap-hcorr)2 where hmap is the map height. Because hmap, l, b, d and T are all subject to error, the correct procedure is an orthogonal distance regression, a type of errors-in-variables regression, using estimates of the errors in all the above parameters. Applied to our field data, this gave an estimate for β of 12.6°C with a standard error of 0.25°C. An approximate 95% confidence interval for β would be given by 12.1 to 13.1°C. In fact the method of regression was not critical and an ordinary least squares regression, which can be performed using a spreadsheet solver, gave an almost identical value. Note that the relatively small uncertainty in β was due to our large sample size, viz. 60 hills (we excluded Meall Coire nan Saobhaidh from the original 61). A larger error in β would have a non-negligible effect on the corrected altimeter height and would need to be allowed for by introducing an additional term into equation 2.

Altimeter rounding error

The Avocet instrument reads in multiples of 5m. The standard deviation due to rounding is 2.5/√3 = 1.4m. Adding this term to equation 2 slightly increases the values for total error in table 1 of the main text, by around 0.7m. The maximum errors become 13, 15 and 20m for a height difference of 500, 1000 and 1400m respectively.

Error in the difference between a corrected altimeter reading and an OS map height

If the standard deviation of the error in the corrected altimeter reading is σalt and the standard deviation of the error in map height is σmap, the standard deviation of the difference is given by Typically the estimate of σalt exceeds σmap sufficiently for σdiff to be little greater than σalt.

Error in the average of repeat determinations

Suppose the height of a hill is measured on n separate occasions. If the error standard deviation σ is the same on each occasion, the standard error of the mean of the n determinations is given by σ/√n.

If the standard deviations are different then the expression becomes where σi is the standard deviation of the error in the ith determination.

Note that these expressions require that the measurements are carried out on different days. If they were measured too closely together in time then the errors of some of the components in equation 2 would be correlated and the above expressions will not hold (in effect, you would achieve less averaging).

##### Maximum achievable altimeter accuracy

Our approach has concentrated on obtaining the error which can be expected in typical field conditions, after careful correction for temperature and barometric drift. It is useful to consider what might be achievable under the most favourable conditions, i.e. when the altimeter is set on a known height at a similar altitude to the measured height, in stable weather conditions.

Consider the case of Knight's Peak described previously. The altimeter is set on the summit of Sgurr nan Gillean (spot height 964m), to which the climber returns to make the correction for barometric drift. In the fixed location experiment described in the main text, using only the data for days when barometric pressure was steady, the standard deviation of the error from interpolating between set points was 2.1m. An instrument reading in 1m rather than 5m intervals (a 'precision' altimeter perhaps!) might be expected to do a little better than this, but not by a great deal unless conditions were exceptionally stable. We will optimistically assume a value of 1.6m. The error in air density for a 50m height difference is negligible and the error in lapse rate irrelevant. Using 1.1m for the standard deviation of a spot height, the standard deviation of the total error works out as 1.9m. The maximum error would be three times this figure, or 6m.

On rare occasions the user may be able to set the altimeter on a trig point. Now the standard deviation will be 1.6m and the maximum error 5m. In this case the error is being dominated by barometric drift. And finally, if the OS height were taken as exact (illogical though this would be) you would be left with just the error due to barometric drift, 1.6m and again a maximum error of 5m.

A standard deviation of 1.6m for barometric error may well be unduly optimistic, and in addition we have not taken into account errors associated with the instrument itself. So a maximum error of 5m may be difficult to achieve in practice.

#### Appendix 6

Table of hills

 Name of Hill Grid Reference OS Height (metres) Corrected Altimeter Height 1 GEAL-CHARN MOR NH837123 824 825 2 TROISGEACH NN290194 734 734 3 MEALL AN FHUDAIR NN271192 764 763 4 CREAG MAC RANAICH NN545255 809 815 5 MEALL AN T-SEALLAIDH NN542234 852 844 6 MEALL NA MEOIG NN448642 868 870 7 MEALL NA H-EILDE NN185946 838 840 8 MEALL COIRE NAN SAOBHAIDH NN175951 820-829 845 9 GEAL CHARN NN156942 804 799 10 SGORR CRAOBH A'CHAORAINN NM895758 775 779 11 SGURR GHIUBHSACHAIN NM875751 849 855 12 SGURR BHEALACH AN SGRIODAIN NM874727 770 776 13 STOB COIRE A'CHEARCAILL NN017727 771 772 14 Y GARN SN776852 684 684 15 PUMLUMON FAWR SN790870 752 751 16 PUMLUMON FACH SN787874 664 661 17 PEN PUMLUMON LLYGAD-BYCHAN SN799871 727 719 18 PEN PUMLUMON ARWYSTLI SN815878 741 738 19 CAIRNSMORE OF CARSPHAIRN NX594979 797 793 20 BENINNER NX605971 710 713 21 SHALLOCH ON MINNOCH N TOP NX400920 659 655 22 SHALLOCH ON MINNOCH (Trig Point) NX404907 768 761 23 TARFESSOCK NX409891 697 692 24 TARFESSOCK S TOP NX413886 620 612 25 BENYELLARY NX414839 719 713 26 MERICK NX427855 843 843 27 KIRRIEREOCH HILL NX420870 786 777 28 CORSERINE NX497870 814 810 29 MILLFIRE NX508848 716 729 30 MILLDOWN NX511839 738 738 31 MEIKLE MILLYEA NX518829 746 743 32 BROWN COW HILL NJ221044 829 830 33 CULARDOCH NO194988 900 906 34 CARN LIATH NO165977 862 856 35 BEINN A'BHUIRD NJ092006 1,197 1,189 36 A'CHIOCH NO097987 1,145 1,144 37 CARN NA DROCHAIDE NO127938 818 823 38 MOUNT BATTOCK NO549844 778 786 39 MOEL SIABOD SH705546 872 870 40 SPIDEAN COINICH (QUINAG) NC206277 764 768 41 SAIL GORM (QUINAG) NC198304 776 776 42 SAIL GHARBH (QUINAG) NC209292 808 814 43 BEINN SPIONNAIDH NC362573 772 774 44 CRANSTACKIE NC350556 800 805 45 GANU MOR (FOINAVEN) NC315507 914 909 46 MEALL HORN NC352449 777 785 47 GLAS BHEINN NC254264 776 779 48 CARN BAN NH338875 842 837 49 BEINN A'CHAISTEAL NH370801 787 794 50 CUL BEAG NC140088 769 775 51 CUL MOR NC162119 849 848 52 SAIL MHOR NH033887 767 765 53 AN DUN NN716805 827 826 54 MAOL CREAG AN LOCH NN735807 875 881 55 CARN NA SAOBHAIDHE NH600145 811 809 56 CARN NA LARAICHE MAOILE NH582112 809 809 57 PEN YR OLE WEN SH655619 978 975 58 CARNEDD DAFYDD SH663630 1,044 1,043 59 YR ELEN SH674651 962 952 60 CARNEDD LLEWELYN SH684645 1,064 1,060 61 PEN YR HELGI DU SH698630 833 830