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Bio 351 Quantitative Palaeoecology. Chronologies from radiocarbon dates to age-depth models. 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. Richard Telford. - PowerPoint PPT Presentation
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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