What’s in a bell curve?

Geographers like statistics; they allow us to test for significant differences between samples and to look for empirical relationships between different variables.  Most statistics rely on probability and a definition of what ‘normal looks like’.  We exclude the outliers – data that appears to be anomalous or outside the bounds of probability (Fig. 1).  As such the focus is always on comparing normal with normal.  Was the Boxing Day Tsunami of 2004 normal?  Was it predictable?  If so why did so many people die?  The answer to this lies largely on the time-scale with which one views Earth processes.  In an influential book The Black Swan Nassim Taleb describes such events as outliers and goes on to argue that they are more significant than conventional statistics and science gives them credit.  In geographical terms this speaks to an early debate between those that favour uniformitarianism and those that believed in catastrophism. 

fig1bell

Figure 1: Normal probability curve showing confidence limits and potential outliers.

Early Geographical-Geological Ideas

Early geological thought was plagued by the idea of trying to reconcile religious beliefs (i.e. the account in genesis) with emerging ideas on the antiquity of the Earth.  Many early scientists were profoundly religious yet found their faith was challenged by the observations and inferences they made.

This was particularly true of James Hutton (1726-1797) who many would describe as the founder of geology.  He gave us the concept of uniformitarianism – the key to the past is the present.  Basically that the processes that we observe today however slow, must have always operated.  Since the Earth process he could observe on his estate in the Southern Uplands of Scotland were slow; the earth must be much older than suggested in the bible.  He challenged head on the idea that forces in the past must have been different from the present.  Catastrophism invoked exactly that – processes must have been more active, powerful or different in the past than in the present.  The latter requires much less time than the former and is easier to reconcile with religious estimates of the age of the Earth and such events as the biblical deluge.

Charles Lyell (1797-1875) a friend and contemporary of Charles Darwin codified uniformitarianism in the late nineteenth century.  He envisaged both the uniformity of process but also the uniformity of rate.  The fundamental physics of our universe have not changed through earth history; gravity, light, heat transfer and motion are guided by fundamental physical laws.  The processes by which land erodes and sediment is transported and deposited should not have changed therefore.  The uniformity of process is therefore built on a sound premise, but what about the uniformity of rate?

Let us image rain falling on a slope devoid of vegetation because it had yet to evolve.  Vegetation is good at binding soil and sediment together and resisting erosion.  So erosion rates may have been faster before terrestrial vegetation evolved?  Is this not an example of varying rate potential?  The past may not have been quite the same as the present.  What about catastrophic events like the occasional tsunami?

Humans tend to see things on human rather than geological timescales and set the temporal window accordingly.  Tsunamis occur quite frequently on geological timescales hundreds of thousands to millions of years, but not on human scales of one or two generations (living memory).  So a Black Swan event is only one if we restrict the timescales over which we view things.  We call this the return period for events.  For example, an observed flood event can be described as the one in 20 year event.  It means that in the normal course of events an event of this magnitude occurs once in 20 years; once in a generation or with a probability of one in five each year.  A larger event may have a return period of one in a hundred years; this does not mean that it will happen next year if it has not happened in the last 99 years.  It is just a probability statement.  The point here is that the longer the time period sampled the larger the events that may occur within it.  So in the case of the Boxing Day Tsunami the fact that such an event has a relatively high frequency when sampled over a 1000 years – may four or five – your average person works on ‘living memory’.  Since most humans can’t conceive a 1000 years we tend to work on generational time; the largest event in living memory.  It blinds us in some ways to the hazard that has not happened in living memory but is waiting; that is a Black Swan event.  The importance of temporal sampling is shown in Figure 2.  Only time sample B or E would capture the anomalously large event.  It is one of the reasons why good historical and geological records of event frequency are so important in assessing true risk.

bwell2Figure 2: The effect on temporal sampling on magnitude assessments.

Magnitude and Frequency

This leads us to the question of the relative impact of catastrophic versus non-catastrophic geomorphological events.  Does a large infrequent event have more geomorphological impact than a small but frequent event?

Image a large pile of sand and two students.  One has a t-spoon but no smart phone and diligently moves a spoonful once every minute.  The other student has a spade and a smart phone.  While the spade moves more per shovel full than the spoon, the student is too busy texting and looking for Pokémon Go creatures to move more than one shovelful an hour.  Who will shift most sand in 24 hours?  The chances are the student with the spoon will win out; each spoonful is a low magnitude event, but it is occurring with a high frequency.

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Figure 3: The concept of magnitude and frequency in geomorphology.

This is the concept of Magnitude and Frequency in geomorphology discussed by Wolman and Millar (1960).  Low magnitude, but high frequency processes have the potential to do more geomorphological work than high magnitude but low frequency events.  Essentially there is a trade-off between frequency and magnitude as illustrated in Figure 3.  The classic example is soil creep – the down slope movement of individual soil particles as the ground expands and contracts with temperature and moisture changes.  Because soil creep occurs continuously some researchers have argued that its impact on slope morphology is greater than large mass movements.  There are a number of good illustrations of this type of balance to be had.

The concept is also nicely illustrated by coastal erosion rates.  The classic example is one from a coastal town called Walton on the Naze in Essex, just north of London.  Figure 4 shows the erosion rates along a stretch of coast below the Naze Tower.  The rates of recession are more variable in the south and accelerate toward the north as the cliff height falls.  This feels a little counter intuitive; surely bigger cliffs should erode faster?  The answer to this lies with the magnitude and frequency of the events doing the erosion.  In the south the cliffs are taller and fail periodically by large rotational mass movements founded in the London Clay at the base of the cliff.  The debris acts as a natural sea defence and must be eroded before the cliff can be undercut and steepened again by the sea, ready to fail again.  So the events are large 20 to 30 metres of cliff being lost each time, but the events only occur once every 30 to 40 years so on average the recession rate is modest (Fig. 5).  Contrast this with the cliffs in the north (Fig. 4).  Here the cliffs are only a metre or so high.  The sea is able to erode the base of these cliffs almost continually and when they fail the mass movements are small with little debris involved.  These low magnitude events occur frequently (in fact almost continuously during a high sea) and consequently erosion is fast.  Figure 6 shows this conceptually and is a model that holds for lots of coastal areas around the UK and elsewhere.    There are some good papers based on this case study the original is by Murray Gray and published in the Proceedings of the Geologists’ Association.

walton1

Figure 4: Coastal recession at Walton on the Naze.  Based on Bennett and Doyle (1997).

Figure 5: Periodic coastal erosion via mass movements.

walton2

Figure 6: The concept of magnitude and frequency applied to coastal erosion rates in Britain.  Based on Bennett and Doyle (1997) and Cosgrove et al. (1998).

To this we must add the concept of landscape relaxation.  That is not kicking-back in a nice landscape, but the time it takes to erase the evidence of a geomorphological event.

The last glacial cycle in Britain was a high impact event lasting about 100,000 years.  Its impact on the British landscape was profound and we can still see the evidence of its impact in the landscape ten thousand or more years later.  (As an aside the impact of the Anglian Glaciation (antepenultimate glaciation) was so profound it is still visible today in the form of the north-south UK river pattern and the existence of the Wash.)  The relaxation time from this high magnitude event is considerable.  Now let us consider the beach cusps on Bournemouth beach which are often present.  They form over a single tidal cycle, and can be destroyed over another.  As such they are a lower magnitude event, with a correspondingly lower relaxation time.

So while low magnitude, high frequency events, might do more geomorphological work the legacy of such events may be much more transitory.  Catastrophic events may occur infrequently, but their impact on the landscape may be correspondingly greater.

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