1. Chronologies from radiocarbon dates to age-depth models Richard Telford Bio 351 Quantitative Palaeoecology Lecture Plan Calibration of single dates 14 C years  cal years Bayesian statistics Calibration of multiple dates in a series at the same event Age-depth models

2. 14 C half-life is 5730 years Suitable for organic material and carbonates Useful for sediments 200 - 50 000 years old The most widely used dating tool for late- Quaternary studies Unique amongst absolute-dating methods in not giving a date in calendar years Radiocarbon Dating

3. Radioactive Decay 14 C→ 14 N+  Random process Atom has 50% chance of decaying in 5730 yrs Exponential decay

4. Radioactive Decay equations What is λ?

5. Express measured 14 C as %modern A=A initial e -ln(2)*age/halflife ln(A/A initial )=ln(2)*age/halflife Use Libby halflife 5568 age= -8033 ln(A/A initial ) assume A initial = A modern age= -8033 ln(A/A modern ) Using Radioactive Decay equations Assumes atmospheric 14 C constant

6. Causes of Non-Constant Atmospheric 14 C 1) Changes in production - Variations in solar activity solar minimum weak magnetic shield maximum 14 C production solar maximum strong magnetic shield minimum 14 C production 2) Changes in distribution - rate of ocean turnover - global vegetation changes - Variations in earth magnetic field strength

7. Dendrochronological Evidence Find 14 C date of tree rings of known age

8. INTCAL04

9. 14 C Calibration Curves Atmospheric Marine

10. Calibration: from 14 C Age to Calibrated Age The intercept method –quick, easy and entirely inappropriate Classical calibration (CALIB) –fast and simple Bayesian calibration –allows use of prior information

11. Calibration of marine dates Use either classical or Bayesian calibration Use the marine calibration curve Set ΔR – the local reservoir affect offset Set σΔR – the uncertainty Do not subtract R

12. The Intercept Method: Multiple Intercepts 48005000520054005600 Calibrated years BP 4200 4300 4400 4500 4600 4700 4800 Radiocarbon years BP 4540±50 5295 4530±50

13. The Intercept Method: Missing Probabilities

14. C lassical Calibration  Unknown calendar date  (  ) is the true radiocarbon age  but cannot be measured precisely Radiocarbon date y is a realisation of Y =  (  ) + noise Noise is assumed to have a Normal distribution with mean 0, and standard deviation  Thus Y~N(  (  ),  2 ).

15. C lassical Calibration Normal Distribution The probability distribution p(Y) of the 14 C ages Y around the 14 C date y with a total standard deviation  is: Total standard deviation  is, where  s and  c are the standard deviations of the 14 C date and calibration curve respectively: The calibration curve can be defined as: Replacing Y with  (  ), p(Y) is: To obtain P(  ),  (  ) is determined for each calendar year and the corresponding probability is transferred to the  axis. Y =  (  )

16. C lassical Calibration Quick and simple Fine if we just have one date But difficult to include any a priori knowledge e.g. dates in a sequence To do this we need to use the Bayesian paradigm

17. The Bayesian Paradigm (1702-1761) Bayes, T.R. (1763) An essay towards solving a problem in the Doctrine of Chances. Philosophical Transactions of the Royal Society, 53: 370-418. Can utilise information outside of the data. This prior information and its related uncertainty must be encoded into probabilities. Then it can be combined with data to assess the total value of the combined information. Bayes' Theorem provides a structure for doing this. Simple in theory, but computationally difficult.

18. The Bayesian Paradigm The Likelihood - “How likely are the values of the data observed, given some specific values of the unknown parameters?” The Prior – “How much belief do I attach to possible values of the unknown parameters before observing the data?” The Posterior - “How much belief do I attach to possible values of the unknown parameters after observing the data?” The Posterior The Likelihood The Prior

19. The Likelihood  Unknown calendar date  (  ) is the true radiocarbon age  but cannot be measured precisely Radiocarbon date y is a realisation of Y =  (  ) + noise Noise is assumed to have a Normal distribution with mean 0, and standard deviation  Thus Y~N(  (  ),  2 ). With the calibration curve, we have an estimate of  (  ), and can formalise the relationship between  and y±  This is the likelihood.

20. The Prior For a single date with weak (or no) a priori information we can use an non- informative prior e.g. for a date  known to be post-glacial P prior (  )= Often we know more than this. Perhaps there is stratigraphic information: e.g. dates  1,  2 &  3 are taken from a sediment core and are in chronological order P prior (  1 <  2 <  3 )= The Bayesian paradigm offers the greatest advantage over classical methods when there is a strong prior and overlapping data. constant for -50<  <14000 0 otherwise constant for  1 <  2 <  3 <  0 otherwise

21. Computation of the Posterior Analytically calculation is impossible for all but the simplest cases So instead Produce many simulations from the posterior and use as estimate Markov Chain Monte Carlo does this to give approximate solution Markov Chain? - each simulation depends only on the previous one - selected from range of possible values - the state space Areas with higher probability will be sampled more frequently

22. Markov Chain Continued 1.Start with an initial guess 2.Select the next sample 3.Repeat step 2 until convergence is reached Gibbs sampler - one of the simplest MCMC methods

23. theta[1] chains 1:2 iteration 120004000 0.0 50.0 100.0 150.0 Convergence Easier to diagnose that it hasn’t converged, than prove that it has.

24. Reproducibility MCMC does not yield an exact answer It is the outcome of random process Repeated runs can give different results Calibrate multiple times & verify results are similar Report just one run Acknowledge level of variability

25. Outlier Detection Outliers can have a large impact on the age estimates Extreme but “correct” dates Contamination Erroneous assumptions? Need a method to detect them and reduce their influence Outliers can only be defined based on calibrated dates Christen (1994) Radiocarbon determinations dating the same event should come from N(  (  ),  2 ) An outlier is a determination that needs a shift  j Given the a priori probability that a date is an outlier, posteriori probabilities can be calculated Calibration and outlier detection done together Automatic down-weighting of outliers

26. Dates in Stratigraphic Order

27. Wiggle Matching In material with annual increments (tree-rings & varves) Time between two dates precisely known 20 years 11 22 This additional information can be used in the prior

28. Wiggle Matching 2 Buck et al. (1996) Bayesian approach to interpreting archaeological data. Wiley: Chichester. p232-238

29. Wiggle Matching in Unlaminated Sediments  x 12 11 33 If the sedimentation rate is assumed to be constant: (  1-  2)/(  2-  3)  =  x 12 /  x 23 This information can be used in the prior 22  x 23

30. Wiggle Matching in Unlaminated Sediments Wiggle matching has greatest impact when the calibration curve is very wiggly there is a high density of dates But may be sensitive to the assumption of linear sedimentation Christen et al. (1995) Radiocarbon 37 431-442

31. Sensitivity Tests Bayesian radiocarbon calibration is very flexible and sensitive Apparently small changes in prior information can have a large effect on the results Need to carefully consider the specific representations you choose And investigate what happens when you vary them Report the findings

32. Software Oxcal Download fromhttp://www.rlaha.ox.ac.uk/orau/oxcal.html Fast & easy for simple models BCAL Online at http://bcal.shef.ac.uk Automatic outlier detection WinBugs If you want to implement a novel model Remember to enter your samples oldest first!

33. From Dates to Chronologies Not every level dated –too expensive –insufficient material Fit age-depth to find undated levels –Linear interpolation –Linear regression models –Splines –Mixed-effect models (Heegaard et al. (2005)) Age-depth models based on uncalibrated dates are meaningless

34. Linear Interpolation What assumptions does this make? Lake Tilo

35. Linear Interpolation – Join the Dots Which dots? 285 BP

36. Linear regression models Lake Tilo What assumptions does this make? Also weighted- least squares Assess by  2 Polynomial order

37. Is Sedimentation a Polynomial Function? Holzmaar varve sequence

38. Conclusions Bayesian calibration of 14 C dates - allows inclusion of prior knowledge - produces more precise calibrations - but, if the priors are invalid, lower accuracy Age-depth modelling - lots of different methods - some are worse than others - no currently implemented method properly incorporates the full uncertainties