Chronologiesfrom radiocarbon dates to age-depth models

Richard Telford

Bio 351 Quantitative Palaeoecology

Lecture PlanCalibration of single dates

14C years cal yearsBayesian statistics

Calibration of multiple datesin a seriesat the same event

Age-depth models

14C 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

Age yr

%14 C

rem

ain

ing

5730 11460 17190 22920 28650 34380

12

.52

55

01

00

Radioactive Decay14C→14N+

Random process

Atom has 50% chance of decaying in 5730 yrs

Exponential decay

Ndt

dN

Ndt

dN

dtN

dN

tN

N

dtN

dN

00

tN

N

0

ln

e t

N

N 0

e tNN 0

e tAA 0

e tAA

2100

2

e t 21

2

1

t 212

1ln

t 21

2ln

Radioactive Decay equations

What is λ?

Express measured 14C as %modern

A=Ainitiale-ln(2)*age/halflife

ln(A/Ainitial)=ln(2)*age/halflife

Use Libby halflife 5568

age= -8033 ln(A/Ainitial)

assume Ainitial = Amodern

age= -8033 ln(A/Amodern)

Using Radioactive Decay equations

Assumes atmospheric 14C constant

Causes of Non-Constant Atmospheric 14C

1) Changes in production

- Variations in solar activity

solar minimum weak magnetic shieldmaximum 14C production

solar maximumstrong magnetic shieldminimum 14C production

2) Changes in distribution

- rate of ocean turnover- global vegetation changes

- Variations in earth magnetic field strength

Dendrochronological Evidence

Find 14C date of tree rings of known age

INTCAL04

0 2000 4000 6000 8000 10000 12000

02

00

04

00

06

00

08

00

01

00

00

Cal BP

14C

yr

BP

14C Calibration Curves

5000 5200 5400 5600 5800 6000

44

00

46

00

48

00

50

00

52

00

54

00

56

00

Cal BP

14C

yr

BP

Atmospheric

Marine

Calibration: from 14C 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

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

The Intercept Method: Multiple Intercepts

4800 5000 5200 5400 5600Calibrated years BP

420

043

00

440

045

00

460

047

00

480

0

Rad

ioca

rbo

n y

ears

BP

4540±50

5295

4530±50

The Intercept Method: Missing Probabilities

Classical Calibration

Unknown calendar date

() is the true radiocarbon agebut 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).

Classical CalibrationNormal Distribution

ey

Yp

2

))((2

2

2

1)(

22 CS

The probability distribution p(Y) of the 14C ages Y around the 14C date y with a total standard deviation is:

Total standard deviation is, where s and c are the standard deviations of the 14C date and calibration curve respectively:

The calibration curve can be defined as:

Replacing Y with (), p(Y) is:

eyY

Yp

2

)(2

2

2

1)(

To obtain P(), () is determined for each calendar year and the corresponding probability is transferred to the axis.

Y = ()

Classical 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

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.

The Bayesian Paradigm

)()|()|( parametersPparametersdataPdataparametersP

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

The Likelihood

Unknown calendar date

() is the true radiocarbon agebut 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±

ey

yp

2

))((2

2

)|(

This is the likelihood.

The PriorFor 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

Pprior()=

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

Pprior(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

Computation of the PosteriorAnalytically 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

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

theta[1] chains 1:2

iteration

1 2000 4000

0.0

50.0

100.0

150.0

Convergence

Easier to diagnose that it hasn’t converged, than prove that it has.

theta[4] chains 1:2

iteration

1 2000 4000

175.0

200.0

225.0

250.0

275.0

300.0

theta[1] chains 1:2

iteration

1 2000 4000

0.0

50.0

100.0

150.0

ReproducibilityMCMC 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

Outlier DetectionOutliers 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

Dates in Stratigraphic Order

2.0

1.5

1.0

0.5

0.0

0 100 200 300 400 500 600D

ep

th (

cm)

Age (cal yr BP)

2.0

1.5

1.0

0.5

0.0

0 100 200 300 400 500 600D

ep

th (

cm)

Age (cal yr BP)

Wiggle Matching

In material with annual increments (tree-rings & varves)

Time between two dates precisely known

20 years1 2

This additional information can be used in the prior

Wiggle Matching 2

5600 5800 6000 6200 6400

46

00

48

00

50

00

52

00

54

00

Calibrated years BP

Ra

dio

carb

on

ye

ars

BP

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

Wiggle Matching in Unlaminated Sediments

x12

1 3

If the sedimentation rate is assumed to be constant:

(1-2)/(2-3)= x12/x23

This information can be used in the prior

2x23

0.5 1.0 1.5 2.0

01

00

20

03

00

40

05

00

Depth mR

ad

ioca

rbo

n y

ea

rs B

P

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

0 100 200 300 400 500

01

00

20

03

00

40

05

00

Calibrated years BP

Ra

dio

carb

on

ye

ars

BP

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

SoftwareOxcal

• Download from http://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!

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

Linear Interpolation

What assumptions does this make?

Lake Tilo

0 500 1000 1500 2000

02

00

04

00

06

00

08

00

01

00

00

Depth cm

Ca

l BP

Linear Interpolation – Join the Dots

Which dots?

10

50

-50 100 200 300 400 500 600

De

pth

(cm

)Age (cal yr BP)

0 100 200 300 400 500 600

Age (cal yr BP)

285 BP

0 500 1000 1500 2000

02

00

04

00

06

00

08

00

01

00

00

Depth cm

Ca

l BP

123

Linear regression modelsLake Tilo

What assumptions does this make?

Also weighted-least squares

Assess by 2

Polynomial order

Is Sedimentation a Polynomial Function?

2 4 6 8

02

00

06

00

01

00

00

Depth.m.

Ag

e y

r B

P

Holzmaar varve sequence

Conclusions

Bayesian calibration of 14C 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