TAC 42 Index
The measurement error article by Chris Crocker(1) in TAC36 fascinated me. Having spent so long worrying about whether hills should be Munros or merely Tops, I realised that I had overlooked the more fundamental question: "Which hills could be high enough to qualify even though not listed?" I now know that there are still two "unlikely, but possible" Munros that I have not yet visited. At least I can relax in the knowledge that I have included the "Subs" in my collections of Donalds and Hewitts, and it seems that my use of a 600m limit makes it reasonably certain that I have visited all the summits above 2000 feet (610m).
It seemed an obvious extension to consider the Marilyns. If the heights of summits can be in error, how much more so might drops be? Drops are, after all, the difference between two heights each of which has its own error built in. It quickly became clear that there were a large number of possible sources of error. I have identified the following, referred to below as E1 to E8.
|E 1||Random measurement|
|E 5||Summit spot height or trig point (TP) not at highest point|
|E 6||Col spot height not exactly at the col|
|E 7||Inaccuracies from use of the Newlyn datum|
|E 8||Hills not yet identified|
E1 is the "normal" error described by Chris Crocker, which arises because no measurement can be absolutely precise, and E2 comes from rounding the value as measured by the surveyors (now to the nearer metre). E3 would occur if, for some improbable reason, the whole of a large area had been measured too high or too low. This is the only one that we can ignore, as when we subtract a col height from a summit height it will cancel out. (See Alan Blanco's comments on Skye in TAC36, p4 - Ed.) Of course this assumes that summit and col are fairly close. A few cols are far from their summits, but then the drops are nearly always much greater than 150m.
E4 could arise if, say, the OS determined a height as 976m but accidentally printed it as 967m. There are a number of genuine cases: for example, the first version of Landranger Sheet 25 had all the contours on the western side of Beinn Dronaig 50m too high. In practice there is usually other information on a map which enables this type of mistake to be spotted, and I shall not consider it further. However it is worth adding that using secondary sources such as published lists (even TACit ones!) increases the risk of error; for example, in the new SMC Munro's Tables the height of Millfore is different in the Donald and Graham Tables. (But not in TACit Tables! - Millfore has always been 657m in Graham/Donald booklets - Ed.)
E5 and E6 are the most critical. Taking summit heights first, the evidence from hills that have been studied "on the ground" suggests that, at least for "less important" hills (a category in which, so far as the OS are concerned, most Marilyns belong), it is not unusual to find that there is a higher spot than the point where the TP stands or which the spot height marks. My letter in TAC38 about Swyre Head, subsequently promoted, gives just one example. Col heights (E6) are a different matter:
E7 is not, in practice, of great importance. But consider the five SubMarilyns which are "Island Tops", that is, they are the highest point of their island - Soay, Tiree, Seil, Colonsay and Lundy. (A SubMarilyn, or "Sub" in what follows, is a hill with a drop between 140m and 149m.) For these five hills, by definition between 140m and 149m high, E7 could perhaps arise. Clearly there is no worry about the col height - it is zero. But if local Mean Sea Level were significantly below the assumed MSL derived from Newlyn, then the summit could be higher than its mapped value. I have no idea of the likely size of any such difference, but suspect that it is very small. (In practice the OS could hardly do otherwise. If an area were flooded it would not be too helpful to say, "Ah, but you were well above the Newlyn level"!) As all but one of these Island Tops have TPs, and all but one of their summit heights are below 144m, it seems very unlikely that any are actually 150m above even their local MSL.
The final error (E8) is, one hopes, now theoretical but cannot be overlooked. It is easy to find all possible hills which could be of Munro height since they must either be adjacent to existing Munros, or be current high Corbetts. Barring a geological upheaval of catastrophic proportions, no-one is going to find a hitherto unknown part of Scotland that is 3000ft high or very nearly so! Finding hills with a drop of at least 150m is more awkward. However there is now a list of just over 200 Subs, and recent changes have all resulted from reassessment of hills on that list rather than the discovery of a completely new hill. But it would be rather splendid to find one! (If you want to do so, see later.)
It is important to note that two of these errors are "biased"; that is, they are more likely to work in one direction than another. This is obvious in the case of E5; the true summit height can be larger than that of the TP or spot height but (ignoring the other errors) it cannot possibly be smaller. There is a more subtle bias operating with E6. For the lower hills especially, there is often a road that passes through the relevant col. When this occurs the spot height is usually by the road rather than at the true col. Unfortunately for our purposes, the highest point on the road is nearly always slightly above the true col, for very good engineering reasons to do with dampness, drainage and so on. Thus, when the spot height is at the road summit, once again we find that the true drop may be larger, but cannot be smaller, than that derived from the spot height. (But it is still true that for most hills the col has no such artificial feature, and then E6 can be dealt with as described below.)
Sometimes the spot height is not at the actual road summit; for a good and relevant example, consider Lovely Seat (Section 35A) and the 526m point on OL19 at SD869957. This is below the summit of the road at 868955, and probably below the nearby col too. In fact Lovely Seat, despite its name, is a rather doubtful Marilyn, but as large-scale maps do not give a summit height either, it is difficult to prove that it does not qualify.
Unfortunately there is no sensible way of allowing for these biased errors statistically; it could only be done by inspecting the summits and cols of the Subs to see if any further promotions are justified. This is in practice the only way in which TACers can now "discover" a hill that should be recorded as a Marilyn. If you want to find a completely new (ie unlisted) hill then you have first to find one with a "map" drop slightly less than 140m (or of course one that has escaped Alan Blanco's notice for all these years), and then show that E5 or E6, or both, apply to increase the drop to 150m-plus. A daunting task indeed!
Before proceeding to the final analysis there is one other piece of information that is relevant. About one-quarter of the known Subs have TPs at or very close to the summit and three-quarters do not.
There are many possible errors, as we have seen. Fortunately it is only necessary to consider, in a statistical sense, E1, E2 and E6. But even for some of these it is difficult to give sensible estimates of their size. I shall spare readers the detail, but offer results on three different sets of assumptions. In Case A, I use a maximum error of 0.6m for TPs and 3.3m for all other points including cols. For Case B, I increase the value for a TP to 1.0m, leave that at other summits at 3.3m, and increase the maximum error at a col to 5.4m. And in Case C, I increase the summit spot height value to 4.5m and the col value to 9.0m.
The significance of the three sets of parameters is as follows. Case A uses the values quoted in Harley's authoritative 1975 book(2)2 about the Ordnance Survey. Case B adjusts the maximum TP error to allow for the rounding to the nearer metre (E2), and allows for E6 by assuming rather arbitrarily that the maximum error at a col is the same as that for a point on a contour line, which has been quoted as 5.4m. Case C also makes some allowance for E5, for summits without spot heights, and for the need to estimate the col height by interpolation. (In all cases my values are for the "maximum error" which is three "standard deviations". Over two-thirds of all measurements will be within one standard deviation of the true value. The "maximum error" is not a theoretical "upper limit", but fewer than three values in every 1000 will be in error by more than the quoted maximum.) In practice the assumed increase in the TP error has almost no effect as it is swamped by the error at the col; the conclusions are:
To be at least 99% sure of climbing all hills which in fact qualify as Marilyns, you need to visit Subs with TPs and drops of 147m or more (12 hills), and Subs without TPs and drops of 145m or more (73 hills).
To be similarly 99% sure, you need to visit Subs with TPs and drops of 144m or more (33 hills), and Subs without TPs and drops of 143m or more (104 hills).
In this case you need to visit all 206 Subs - except that you could omit the 2 hills with TPs and drops of exactly 140m, but it is hardly worth it! (Indeed you really need hills without TPs and drops of 139m also, but you might decide, in the absence of a suitable list, to overlook this.)
Which basis should we use? It is clear that the first is optimistic, as there are good reasons why cols could be in error by more than one-third of the contour interval. In my view the last is pessimistic, and could be thought to be "double-counting" errors, though it does derive from a valid analysis of the various factors. Had I visited the hills as determined in Case B, my conscience would be entirely clear! The conclusions may appear rather depressing. Chris Crocker showed that a probabilistic approach to lists such as Munros and Corbetts required the ascent of only a few more hills. But with the Marilyns, many more hills (of the order of 10% of the number in the basic list) need to be visited to be reasonably sure of a full haul. And yet this does not mean that there are a lot more Marilyns. Only a few of these extra hills actually qualify, but even after you have visited them you will not know which they are! Even worse, a similarly small number of current Marilyns with drops thought to be 150m or just over probably do not qualify, so that overall the total may still be very close to 1551. (Actually it is likely to be rather larger as E5 and the "biased" part of E6 operate to increase the numbers.)
The approach that I have used is capable of further refinement. The most important would be to distinguish between Subs where there is a col height on a map (albeit that E6 still applies) and those where estimates from contours have been made. Another would be to note those Subs in regions of the country contoured at 5m intervals, which would reduce their maximum errors somewhat. But I suspect that such extra investigation would hardly change the conclusion: the advice to Marilyn-baggers seems to be, I fear, "visit all the Subs too while you are about it"!
I should like to acknowledge the help that Chris Crocker has given in correspondence since his original article was published. However the responsibility for this article, and especially any statistical errors, is mine alone.
1- "Allowing for measurement error - a principled approach to peak bagging", Chris Crocker, TAC36 (1998), p14.
2 - Ordnance Survey Maps; a descriptive manual, J B Harley (Ordnance Survey, Southampton 1975) - see Chapter 11 in particular.
Ed. - Whernside on 3/6/99 saw David complete 700 Marilyns, all English Marilyns and all English Hewitts. A tidy man. Also given the context, note that in addition to Ann Bowker having completed Grahams (Trallval 12/5/99) and mainland Marilyns (Druim na Cluain-airighe 19/5/99), Ann and Rowland both completed English SubMarilyns with Beacon Hill on Lundy, 9/10/98.
TAC 42 Index