QUB | Archaeology and Palaeoecology | The 14Chrono Centre

Data-handling Methods for Quaternary microfossils

K.D. Bennett

November 1999

Depth-age modelling

Introduction

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.

Radiocarbon ages

Terminology
Dates may be expressed in a number of forms, and it is important to be sure which is being used by any investigation.
Radiocarbon dating

Calibration

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).

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.

Depth-age modelling

This section is abstracted in large part from Bennett (1994).

Definition of terms:

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
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 dependent variables.

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 values)
If y = a + bx + cx2, then dy/dx = b + 2cx
If y = a + bx + cx2 + dx3 then dy = b + 2cx + 3dx2
etc.

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 explore. The goodness-of-fit will be unacceptably low if one or more of the following conditions holds:

  1. the model is wrong (the polynomial is a poor statement of the way that sediment has accumulated over time);
  2. the errors on the radiocarbon ages are too small;
  3. the errors on the radiocarbon ages are not normally distributed.
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 radiocarbon ages.

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[100]) = 0.1 of the sample standard deviation, and the standard error of the sample standard deviation is 0.5(sqrt[2])/(sqrt[100]) = 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 ages.

Rate of change

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):

where:
pik = proportion of pollen type k in sample i
pik = proportion of pollen type k in sample j
so
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.

Implementation and links

References

Bennett, K.D. 1994. Confidence intervals for age estimates and deposition times in late-Quaternary sediment sequences. The Holocene, 4, 337-348.

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, 1041-1083.

Copyright © 1999 K.D. Bennett


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