Archaeology and Palaeoecology |
The 14Chrono Centre
Data-handling Methods for Quaternary microfossils
An accurate and precise chronology is an essential prerequisite for any
palaeoecological investigation. Chronologies give time-scales for events, and
hence rates for processes. Thay make it possible to compare the temporal
course of events in different sequences. Palaeoecology without chronology is
history without dates.
Dates may be expressed in a number of forms, and it is important to be sure
which is being used by any investigation.
- For most palaeoecological applications, the units of time are
years (yr), thousands of years, etc. Different units are sometimes
used to indicate absolute ages (5ka: 5000 years ago) or
durations (5kyr: an interval lasting 5000 years).
- `Years' may be calendar years, or some other unit closely related
to, but not exactly the same as, calendar years. The most common
non-calendar year unit is `radiocarbon years'. The length of a
radiocarbon year differs from a calendar by a variable amount,
resulting in the need for calibration: see below.
- Year zero: ages may be expressed as years AD or BC, or they be
expressed relative to a zero year. By convention, most
palaeoecological ages are given as `years BP', where `BP' is
defined as AD 1950 (chosen because it predates distortion of
14C abundances in the upper atmosphere by nuclear weapons testing).
Radiocarbon dating of known-age materials, notably tree-rings, has shown that
radiocarbon years and calendar years are not quite the same: on average,
radiocarbon years are slightly longer then calendar years, but by a variable
amount. Cross-matching of tree-rings has made possible calibration for the
whole of Holocene time, and an extension due to comparison of U-Th and
14C ages of corals has provided a calibration back to about 20,000
years ago (Stuiver et al 1998). The difference between radiocarbon
years and calendar years is tolerably even for much of the Holocene, but there
are periods of time (up to 500 calendar years long) near the beginning of the
Holocene and during the late-glacial which give the same radiocarbon age:
so-called radiocarbon `plateaux'. Radiocarbon ages during the late-glacial are
especially difficult to interpret (Pilcher 1991).
- Conventional (bulk) radiocarbon ages. This method of
determining the age of a material works by counting decays from
14C atoms. The number of decays over a known period of time
enables the 14C content of the material to be determined. The 12C
content is determined by other means, and the departure of the
12C:14C ratio from the equilibrium value enables calculation
of the age of the material.
- Accelerator (AMS) radiocarbon ages. This method works by counting
the number of 12C and 14C atoms directly in a mass spectrometer,
and then obtaining the age of the material from the ratio, as with
the conventional method.
- Errors. There is no difference, from a data-handling point of
view, in the errors obtained by AMS and conventional methods. Errors given
with radiocarbon ages are obtained by methods that vary between laboratories.
A conventional radiocarbon age is based on counts of decaying 14C atoms. Some
laboratories assume that the decays are distributed as Poisson processes. In
this case, var(N) = N, where N is the number of observed decays. Thus, the
coefficient of variation is smaller for larger samples, longer counting times,
and younger samples. Other laboratories count the sample in a series of short
time periods (e.g. 100 minutes), and calculate a standard error of the mean
from this series. A standard error of the mean is equal to s/(sqrt[n]),
where s is the sample standard deviation, and n is the number of
individual counts in the series. This statistic cannot be interpreted without
knowing the value of n, but n is not usually reported.
Radiocarbon ages obtained by accelerator mass spectrometry (AMS) depend upon
counts of 14C atoms arriving at a detector, which may be assumed to
be a Poisson process, as for the detection of decaying 14C atoms in
the conventional process. However, the errors on AMS ages depend also upon
complex laboratory factors. I shall assume that the quoted standard deviation
for any radiocarbon age can be treated as a sample standard deviation. This is
obtained from measurements, of course, and is an estimate of the true,
population, standard deviation.
One, albeit crude, approach to this problem is take the mid-point between
the pair of calendar ages that enclose the 95% confidence interval as an
estimate of the calendar age of the sample, and half the distance between this
age and either of the ages marking the confidence interval as the standard
deviation of that estimate.
Two approaches to radiocarbon calibration have been developed. Classically,
radiocarbon ages are calibrated by comparison of the age determination with
the current accepted calibration curve (of which the latest version
is INTCAL98 [Stuiver et al. 1998]). Alternatively, ages may be
calibrated by Bayesian statistics, which means incorporating other
information, appropriately weighted, to constrain the calibration. For example,
if a sample is known to younger than a tephra layer with a known exact
age, this information can, and should, be used. BCal,
an online calibration system, has been developed to do exactly this.
This section is abstracted in large part from Bennett (1994).
Definition of terms:
Three basic types of age-depth model are in use: linear interpolation, spline
interpolation, and polynomial line-fitting. For all types, we proceed with
depth as the independent variable (plotted on the x-axis of a graph), and age
as the dependent variable (plotted on the y-axis). In the real world, the
relationship is the other way round because accumulated sediment thickness is
a function of the passage of time. Statistically, depth is treated as the
independent variable because that is the variable that is controlled, and can
be sampled. Where radiocarbon ages are available for certain depths, we
usually need to find age estimates for other depths. Thus, in line-fitting
models, regress age onto depth, and the resulting gradients are estimates of
DT (rather than SR) in order to reflect the relationship of the variables in
the regression. Different results will be obtained if the
regression is done with the variables reversed. Alternatively, line-fitting
might be carried out using major-axis regression (equivalent to a principal
components analysis on two variables), making no assumption of independent and
- Deposition time (DT)
- time elapsed during
accumulation of unit sediment thickness (units: (radiocarbon years) cm-1)
- Microfossil deposition rate (MDR)
- microfossils incorporated
into the sediment on a unit area of lake bottom each radiocarbon year
(grains cm-2 [radiocarbon year]-1)
- Sedimentation rate (SR)
- inverse of deposition time
- Linear interpolation This is the most frequently used
age-depth model, and the most obvious and basic way to start. Reported
radiocarbon ages are plotted against depth with the points connected by
straight lines (often necessitating extrapolation to the base of the
sequence). Estimates of DT are found from the gradients between adjacent pairs
of points, and interpolated ages read off (or calculated) for intermediate
depths. It is a superficially crude approach, but does provide reasonable
estimates for both ages and gradients. However, it takes no account of the
errors on the radiocarbon ages, and it turns out to be inadequate when
confidence intervals on ages and slopes are obtained. Note also
that the gradient will normally change at every radiocarbon age, which is far from necessarily being a reasonable reflection of what
really happens as basins infill.
- Spline interpolation A spline is a polynomial (see below)
fitted between each pair of points, but whose coefficients are determined
slightly nonlocally: some information is used from other points than the pair
under immediate consideration. This nonlocality is intended to make the fitted
curve smooth overall, and not change gradient abruptly at each data point. The
usual polynomial fitted between pairs of points is a cubic (4-term)
polynomial, producing a cubic spline. This method also takes no account of the
errors on the radiocarbon ages, and can produce 'ruffle-like' bends that
include sections with negative DT.
- Polynomial line-fitting Polynomials with the following form
are fitted to the data:
y = a + bx + cx2 + dx3 etc
x = depth (independent variable),
y = age (dependent variable),
a, b, c, d, etc are
coefficients that must be estimated.
Polynomials may be considered by the
number of terms they include:
y = a + bx has 2 terms, and is a straight line
y = a + bx + cx2 has 3 terms and is a quadratic curve
y = a + bx + cx2 + dx3
has 4 terms and is a cubic curve etc.
The gradients of these curves for any depth x can be
differentiated to obtain dy/dx, the rate of change of y at x.
If y = a + bx, then dy/dx = b (constant gradient for all x
If y = a + bx + cx2, then dy/dx = b + 2cx
If y = a + bx + cx2 + dx3 then dy = b + 2cx + 3dx2
Thus, a straight line regression can be seen as a polynomial
that has just 2 terms.
The idea of fitting a curve is to find a line that is a reasonable model of
the data points. The curve does not necessarily have to pass through all the
points because the points are only statistical estimates of the 'true'
(unknown) radiocarbon age of the sample. For y = a + bx, we need to find
values for a and b such that values of y calculated from
the line at each x are as close as possible to the observed values of
y. "As close as possible" can be defined in many ways, of
which the most usual is 'least-squares'. This means minimising the sum of the
squared distances for the dependent variable. The errors on the radiocarbon
ages are incorporated as weighting on the dependent variable. It will normally
be appropriate to include an age and error estimate for the top sample of a
sequence (use -50+/-50). The procedure for polynomials with
more terms is conceptually identical, but the arithmetic for finding a,
b, c, etc becomes more complex.
The coefficients obtained enable a curve to be plotted and gradients to be
calculated by differentiation. Curves become more 'flexible' with more terms.
We want to use a polynomial that is as simple as possible (few terms), but is
still a 'reasonable' fit.
Goodness-of-fit may be assessed from Chi-squared. The squared distances
from the dependent variable to the fitted curve are weighted by the squared
errors on each age, and summed. This approach assumes that the quoted errors
on the radiocarbon ages are the population values. In practice, they are
sample values from one measurement exercise, and will tend to be slightly too
small as estimators of the population value. The Chi-squared obtained is zero
for a perfect fit (i.e. the fitted curve passes through all the given data
points), and this will always occur when the number of terms is equal to the
number of data points. The Chi-squared value may be assessed from tables or
analytically, for its size is a function of the number of ages, the standard
deviations of the ages, and the number of terms in the polynomial, to provide
a measure of 'goodness-of-fit'. This measure is the probability that the
observed difference between the fitted curve and the data points could have
been obtained by chance if the fitted curve was the 'correct' solution. Thus,
ideally, the goodness-of-fit should exceed 0.05, but values as low as 0.001
may, with caution, be acceptable. Neither 2 nor the 'goodness-of-fit' measure
can make any judgement about the course of the fitted curve between or beyond
the given points: assessment of this remains a matter for the analyst to
The goodness-of-fit will be unacceptably low if one or more of
the following conditions holds:
Condition (1) will often hold for polynomials where the number of terms is
much less than the number of radiocarbon ages. Condition (2) will hold to at
least some extent because of the use of sample values for the errors rather
than population values, and may be substantial depending on the extent to
which laboratory errors have been exhaustively assessed in measuring
radiocarbon ages. Where the goodness-of-fit is low for all numbers of terms,
it may be worth increasing the quoted errors by a laboratory multiplier, as
for calibration of radiocarbon ages. An International Study Group (1982) found
that the quoted errors on radiocarbon ages needed to multiplied by a factor of
between 2 and 3 if these errors are to cover the true variability of
radiocarbon age measurements when compared with known-age material. Condition
(3) is a major problem for using the approach described here with calibrated
- the model is wrong (the polynomial is a poor statement of the
way that sediment has accumulated over time);
- the errors on the radiocarbon ages are too small;
- the errors on the radiocarbon ages are not normally
The calculation of age estimates and deposition times (gradients) through
any of these age-depth models is straightforward, and the results are taken as
means of a distribution to be found by simulation. Ages and gradients are
obtained for each depth of interest, usually the location of each pollen
sample in the sequence. The simulation is then carried out by drawing random
numbers to simulate the radiocarbon ages, plus a value for the surface
samples. Random numbers are drawn from normal distributions of zero mean and
unit variance, then each is multiplied by the error of the age being
simulated, and added to the reported value for the age. One random number is
drawn for each age, then the age-model is fitted and estimates obtained for
the age and gradient at each depth of interest. These values are then
accumulated through a series of simulations (100, for example), and the sample
standard deviation for each age and gradient can then be found. This is a
simple process to implement in any program that is calculating age-depth
relationships (whether the models outlined above, or others), since it
involves only the drawing of random numbers for a given distribution, and
looping through the modelling part of the program while accumulating results.
For 100 simulations, the standard error of the mean is 1/(sqrt) = 0.1 of
the sample standard deviation, and the standard error of the sample standard
deviation is 0.5(sqrt)/(sqrt) = 0.071 of the sample standard
deviation. It is possible to check that the mean and standard deviation of
each simulated distribution is close to the observed mean and standard
deviation and consistent with these error estimates. Simulation results are
used to calculate standard deviations of gradients and interpolated ages, but
the gradients and ages themselves are derived from the observed radiocarbon
Microfossil sequences record changes with time: how rapidly do these changes
take place? We have chronologies to give us a measure of the passage of time:
we also need a measure of change, or difference. Mathematically, this is
usually term `dissimilarity', and there are many ways of measuring it (see
Prentice 1980 for one list). One approach is to measure the chord
distance dissimilarity between any pair of samples (Bennett & Humphry 1995):
pik = proportion
of pollen type k in sample i
pik = proportion of pollen type k in sample j
dij/time = rate of change (e.g., chord distance
per radiocarbon century).
In this approach, we take the time as being the interval between the
age of any pair of samples for which we have a dissimilarity measure.
Another approach is to smooth the sequence and interpolate to constant
time intervals, then to calculate the dissimilarity measures and divide
by the time interval.
Bennett, K.D. 1994. Confidence intervals for age estimates and deposition
times in late-Quaternary sediment sequences. The Holocene, 4,
- Calibration programs
- CALIB 4.12 available
for DOS, Windows 95/98/NT, Linux, and Apple Macintosh.
- Oxcal 3.3 for Windows 95/98/NT
(an earlier version for Windows 3.1 is still available).
- BCal: on-line Bayesian radiocarbon
- Depth-age modelling
- psimpoll offers
depth-age modelling using linear interpolation, splines, and polynomial line
fitting. Confidence intervals can be calculated on age determinations and
sediment accumulation rates. Rates of change may be calculated and
samples can be interpolated to constant intervals. Exact calculation of
- Tilia offers depth-age
modelling using linear interpolation, spline interpolation, and
polynomials. Rates of change and interpolation are possible. Deposition times
are calculated as tangents to fitted curves, rather than exactly.
- POLSTA offers
depth-age modelling using linear interpolation or power curves.
Built in algebra routines enable more complex models.
- DEP-AGE, by L.J. Maher,
Jr, provides depth-age modelling using linear interpolation, cubic splines,
exponential functions, power functions, and best-fit polynomials. Details in
INQUA Data-handling Newsletter.
Bennett, K.D. & Humphry, R.W. 1995. Analysis of late-glacial and
Holocene rates of vegetational change at two sites in the British Isles.
Review of Palaeobotany and Palynology, 85, 263-287, 1995.
Pilcher, J.R. 1991. Radiocarbon dating for the Quaternary scientist.
Quaternary Proceedings 1, 27-33.
Prentice, I.C. 1980. Multidimensional scaling as a research tool in
Quaternary palynology: a review of theory and methods. Review of
Palaeobotany and Palynology, 31, 71-104.
Stuiver, M., Reimer, P.J., Bard, E., Beck, J.W., Burr, G.S., Hughen, K.A.,
Kromer, B., McCormac, F.G., v. d. Plicht, J., and Spurk, M., 1998. INTCAL98
Radiocarbon age calibration 24,000 - 0 cal BP. Radiocarbon 40,
Copyright © 1999 K.D. Bennett
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