Lichenometry · Beschel, 1950 · a model you can search

The Longer You Look

Lichenometry dates bare rock by the largest map lichen growing on it, read against a curve built from surfaces of known age. But the largest of anything depends on how many you looked at. Search a model moraine boulder by boulder and watch its date grow older: read through a curve calibrated on one square metre, twenty square metres of 150-year-old rock comes out at a median of 192 years, and each tenfold increase in the area searched adds about 30 more. With two real tables, Benedict's thirteen cairns of 1910 and a published calibration, and the field's own rule for the problem.

A glacier leaves bare rock behind as it shrinks. On that rock, after a few years, small yellow-green crusts start to appear. This is Rhizocarpon geographicum, the map lichen: a patch of chartreuse set in a black rim, widening by a fraction of a millimetre a year for centuries. Knut Fægri suggested in 1934 that lichen size could put surfaces in order of age, and from 1950 the botanist Roland Beschel developed the idea into a clock. Find the largest lichen on the surface, measure it, and read its age off a curve built from surfaces whose age you already know, such as gravestones, dated walls and moraines from glacier fronts that someone drew on a map. Geologists have used this method, lichenometry, to date moraines, rockfalls, mudflows and the rubble of earthquakes.

The method measures the largest lichen, and the reason is sound. The largest one is presumably the one that settled earliest, so it tracks the age of the rock most closely. Beschel, as Tom Bradwell summarises him, held that the largest lichen “growing under optimal environmental conditions” gives the closest estimate. But the largest of anything depends on how many you looked at. The biggest lichen on one boulder is not the biggest lichen on twenty boulders, even when the twenty were all bared in the same year.

This moraine is a model, not a photograph, and it is drawn to scale. Every boulder was bared 150 years ago. Lichens start to settle after 10 years, then arrive at random, half a new one per square metre per year. Each one widens at its own steady pace. The median pace is 0.6 mm a year, and about two lichens in three grow within a factor of 1.28 of it, faster or slower (a lognormal spread of 0.25). The calibration curve it is read against was built the same way, on surfaces of known age, each searched over one square metre, which is the area Naveau and colleagues used for their dated rocky slopes in Bolivia in 2007. A single square metre, searched on its own, reads between 131 and 177 years in four draws out of five, centred on 150. Now search the whole moraine, about 20 m² of rock. The median reading is 192 years, and four moraines in five read between 175 and 216. For the whole moraine to read no older than 150, each of its twenty square metres would have to come in at or below the calibration's median, and the chance of that is one half multiplied by itself twenty times, about one in a million.

Nothing is wrong with any single measurement. The error comes from putting a maximum found in a small search next to a maximum found in a large one.

How far it drifts

The moraine is one random draw. The same model also gives the exact distribution of the largest lichen for any area searched, because the number of lichens wider than x is a Poisson count whose mean can be written down (the formula is in engine.mjs, and the check rebuilds it two other ways). This is the median over every rock the model could make, with the band running from the 10th to the 90th percentile:

For a surface 150 years old, searched over the one square metre the curve was calibrated on, the median largest lichen is 117 mm and the curve reads it, correctly, as 150 years. Searched over 100 m², the median largest lichen is 177 mm, and the same curve reads that as 213 years. Over 10,000 m², a large moraine, the reading is 273 years. Each tenfold increase in the area searched adds about 30 years, and it keeps doing so. Searched over a tenth of a square metre, the same surface reads 112 years, too young.

Now move the spread slider to zero, so that every lichen grows at exactly the same pace. The drift nearly vanishes. The largest lichen is then simply the first to arrive, and on a big enough rock the first arrival comes almost as soon as the rock allows. No lichen can be wider than 0.6 mm × (150 − 10) = 84 mm, however much rock you search, and 10,000 m² reads as 151 years. The drift comes from the spread in growth rates. Where lichens differ in pace, a bigger search finds a faster grower, and the fastest one in a large search has no fixed upper limit in this model.

One thing does not change the ages on this page at all. If every lichen grows slower or faster by the same factor, as between sites in Washington and Iceland, every diameter rescales and the curve rescales with them. The check confirms this. The settling rate enters only multiplied by the area, so a rock where lichens settle twice as thickly behaves like a rock twice the size. That does change the ages: at the same ratio of areas, a thicker settling makes both searches effectively bigger, and the drift is a little smaller (at five new lichens per square metre per year instead of half of one, 100 m² reads 200 years rather than 213). So the ages depend on the spread, the lag and the settling rate, all three our choices, and on the two areas searched. The numbers here are an illustration and not a measurement of any real survey. What the model shows is the direction of the effect, and its size under these assumptions: a search that differs in area from the calibration pulls every date the same way.

The field has a rule for this

Lichenometrists know the problem. In a 2009 review of the method's techniques, Bradwell writes:

“Some workers have chosen to use the LL or 5LL technique within a representative sample area (from 25-500 m2), when a whole-surface search is not practical. However, dating curves constructed using this fixed-area approach cannot be directly compared to those constructed using the LL on an entire surface, owing to the different sizes of the search areas (Innes 1983b, 1984).”

T. Bradwell, “Lichenometric dating: a commentary, in the light of some recent statistical studies”, Geografiska Annaler 91A (2009), author's manuscript p. 4. LL is the single largest lichen; 5LL the mean of the five largest.

The next sentence gives the rule: searching only part of a surface can be justified “as long as the same technique is used in the construction of the dating curve and for dating purposes”. The second instrument above shows why the rule matters and roughly how much is lost when it is broken. One consequence is not in Bradwell's sentence, and it is ours: a curve built by searching whole surfaces also mixes areas, because a gravestone is small and a moraine is large.

A second school took the maximum itself as the thing to model. Naveau, Jomelli, Cooley, Grancher and Rabatel wrote in 2007 that “the measurements are not averages but maxima; only the largest lichen diameters provide information about the surface ages”, and they fitted the generalized extreme value distribution, the distribution that maxima of large samples follow, to their lichens. Bradwell's commentary answers that “The largest lichen in any population is by definition an extreme”, but that an empirical curve does not need that statistical machinery. We take no side in that argument. What this page's model adds is small and exact: when the counts are Poisson, the distribution of the maximum can be written down, and so can the effect of the area searched.

Thirteen cairns built in 1910

In the Indian Peaks of the Colorado Front Range, thirteen stone cairns were built in 1910 from stones gathered nearby. A trimline on each stone shows which faces were buried when it was stacked and which were exposed and already lichen-covered, so James Benedict measured only faces that had been bare in 1910. In the summer of 1966, 56 years later, he found this:

“R. geographicum thalli are absent on seven of the 13 cairns and in all cases they are very small, reaching maximum diameters of only 2 mm.” Benedict explained this by the setting: exposed, windswept summits, moisture only from precipitation and condensation, and abrasion by blowing snow. In the model's terms, a surface small enough or harsh enough may hold no lichen at all, and then the largest lichen has no size to read.

A real calibration

Here is a published curve with every control point shown. Larocque and Smith (2004) built it for the Mount Waddington area of British Columbia from 18 surfaces of known age: nine gravestones in the Hagensborg graveyard (from Smith and Desloges, 2000), glacier margins dated by photogrammetry, and moraines dated by the oldest tree growing on them, by cross-dated fossil wood or by radiocarbon. They left out five further points, shown hollow here, and gave their reasons for each.

The curve is y = 13.4098 ln x − 29.515 up to 100 years, then a straight line, y = 0.0909x + 23.2065, to 680 years (y the diameter in mm, x the age in years). The authors state its uncertainty plainly: for a 25-year-old surface the dating error is “+6 and –5 yr”, and for a 150-year-old surface it is “+45 and –30 yr”. They write that “Wherever possible, a minimum sample of 30 largest thalli was measured at each sampling point over a search area of ca. 100 m2”, and that because some control points were one boulder or a small group of boulders, “smaller search areas and fewer thalli (n = 7) were also included”. So this curve, like most, mixes search areas, and says so. The table does not record the area for each row, and we have not tried to estimate whether the mixing moved the curve. The excluded points show the spread on their own. A cairn built during a geological survey in 1958 at Razor Creek Glacier, 43 years old, carried a largest lichen of 9.4 mm, which the authors thought would push the curve's ages too old; they suggest that lichens on a raised structure grow more slowly. At Cathedral Glacier, a partly buried tree bole cross-dated to 309 years sat beside lichens up to 94.9 mm, larger than anything on their 680-year surface; the authors concluded the tree was probably killed by a snow avalanche in about 1691 and did not date the moraine, and left the point out.

What the model leaves out

Real map lichens do not grow at a steady pace. Bradwell and Armstrong marked and photographed 41 thalli on a moraine in southern Iceland in 2001 and again in 2005; 38 showed measurable growth. They found diametral growth that “varies from 0.23 mm to 1.39 mm yr-1; the mean DGR in the 38 thalli being 0.65 mm yr-1”, with a standard deviation of 0.24. Growth was fastest between about 10 and 40 mm across and slower in the smallest and the largest thalli. About 44% of the variation between thalli went with their size (a cubic fit, r² = 0.44). That is where this page's spread comes from. The raw spread is 0.24/0.65 = 0.37 of the mean; removing the part that goes with size leaves roughly 0.37 × √(1 − 0.44) = 0.28, and we use 0.25, a little lower again, on the guess that a rate averaged over a lifetime varies less than one measured over four years. That guess is ours. Try the slider at 0.37 and at 0.10. The drift shrinks at 0.10 but does not go away.

The model also leaves out death. Lichens die, flake off and merge into one another, and the largest lichen on an old surface may simply be the oldest survivor. In 2015 Osborn, McCarthy, LaBrie and Burke wrote that “A major source of error is the assumption that the largest lichen(s) colonized soon after deposition and will survive indefinitely”, and concluded that “it is folly to assign numerical ages to a deposit on the basis of lichen sizes”. M. A. O'Neal answered them in the same journal in 2016, and the authors' reply to him says: “We are cautiously optimistic about the future of lichenometric dating. But at this point in time we continue to invoke Bob Dylan”. We have read the abstract of the 2015 paper and the two letters of 2016, not the full paper, which is behind a paywall.

What is growing

The clock is not a single organism. In 1867 Simon Schwendener proposed that a lichen is a fungus living with an alga, and in 1869 he set out the case in detail, casting the algae as “Nahrung bereitende Diener”, food-preparing servants, and the fungi as their masters. The botanical code still names a lichen after only one partner: “For nomenclatural purposes, names given to lichens apply to their fungal component” (International Code of Nomenclature, Art. F.1.1). So Rhizocarpon geographicum is formally the name of a fungus. In 2016 Spribille and colleagues reported that “many common lichens are composed of the known ascomycete, the photosynthesizing partner, and, unexpectedly, specific basidiomycete yeasts”. Later work found that the yeasts are not everywhere: Lendemer and colleagues (2019) failed to detect them in 330 (97.3%) of the 339 species they sampled, and Tagirdzhanova and colleagues (2021) wrote that their role in the symbiosis “remains unknown”. The lichen that is widening on a boulder is a partnership, and the partnership's growth rate is the thing lichenometry uses as a clock.

Sources

1. T. Bradwell and R. A. Armstrong, “Growth rates of Rhizocarpon geographicum lichens: a review with new data from Iceland”, Journal of Quaternary Science 22 (2007) 311 to 320. Author's manuscript: nora.nerc.ac.uk/id/eprint/2361. (Its abstract gives the mean as 0.64 mm/yr, its results 0.65.)
2. R. A. Armstrong, “Radial growth of Rhizocarpon section Rhizocarpon lichen thalli over six years at Snoqualmie Pass in the Cascade Range, Washington State”, Arctic, Antarctic, and Alpine Research 37 (2005) 411 to 415, doi:10.1657/1523-0430(2005)037[0411:RGORSR]2.0.CO;2: a mean radial growth of 0.07 mm a year, the slow end.
3. T. Bradwell, “Lichenometric dating: a commentary, in the light of some recent statistical studies”, Geografiska Annaler 91A (2009) 61 to 69. Author's manuscript: nora.nerc.ac.uk/id/eprint/7739. The source for Fægri (1934) and for Beschel's “optimal environmental conditions” (Beschel 1961, as Bradwell quotes him).
4. R. Beschel, “Flechten als Altersmaßstab rezenter Moränen”, Zeitschrift für Gletscherkunde und Glazialgeologie 1 (1950) 152 to 161; English translation by W. Barr, “Lichens as a measure of the age of recent moraines”, Arctic and Alpine Research 5 (1973) 303 to 309, doi:10.1080/00040851.1973.12003739. Not read by us; cited for the date.
5. P. Naveau, V. Jomelli, D. Cooley, D. Grancher and A. Rabatel, “Modeling uncertainties in lichenometry studies”, Arctic, Antarctic, and Alpine Research 39 (2007) 277 to 285, doi:10.1657/1523-0430(2007)39[277:MUILS]2.0.CO;2.
6. J. B. Benedict, “Recent glacial history of an alpine area in the Colorado Front Range, U.S.A. I. Establishing a lichen-growth curve”, Journal of Glaciology 6 (1967) 817 to 832, Table I, p. 823. Cambridge Core.
7. S. J. Larocque and D. J. Smith, “Calibrated Rhizocarpon spp. growth curve for the Mount Waddington area, British Columbia Coast Mountains, Canada”, Arctic, Antarctic, and Alpine Research 36 (2004) 407 to 418, Table 2 and equations 1 and 2, p. 410, doi:10.1657/1523-0430(2004)036[0407:CRSGCF]2.0.CO;2.
8. G. Osborn, D. McCarthy, A. LaBrie and R. Burke, “Lichenometric dating: science or pseudo-science?”, Quaternary Research 83 (2015) 1 to 12, doi:10.1016/j.yqres.2014.09.006 (abstract read); M. A. O'Neal, comment, Quaternary Research 86 (2016) 242 to 243, doi:10.1016/j.yqres.2016.05.007; G. Osborn, D. McCarthy, A. Walintschek and R. Burke, reply, 86 (2016) 244 to 245, doi:10.1016/j.yqres.2016.07.004.
9. S. Schwendener, Die Algentypen der Flechtengonidien, Basel, 1869: archive.org/details/diealgentypender00schw. The 1867 proposal (Verhandlungen der Schweizerischen Naturforschenden Gesellschaft 51) we know only at second hand.
10. International Code of Nomenclature for algae, fungi, and plants (Shenzhen Code, 2018), Art. F.1.1: iapt-taxon.org.
11. T. Spribille et al., “Basidiomycete yeasts in the cortex of ascomycete macrolichens”, Science 353 (2016) 488 to 492, doi:10.1126/science.aaf8287; J. C. Lendemer et al., American Journal of Botany 106 (2019) 1090 to 1095, doi:10.1002/ajb2.1339; G. Tagirdzhanova et al., Genome Biology and Evolution 13 (2021) evab047, doi:10.1093/gbe/evab047.