Transcript Chronologies

Bio 351 Quantitative Palaeoecology
Chronologies
from radiocarbon dates to age-depth models
Lecture Plan
Richard Telford
Calibration of single dates
14C years  cal years
Bayesian statistics
Calibration of multiple dates
in a series
at the same event
Age-depth models
Radiocarbon Dating
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 lateQuaternary studies
Unique amongst absolute-dating methods in not
giving a date in calendar years
Radioactive Decay
100
14C→14N+b
Random process
Atom has 50% chance of decaying in 5730 yrs
50
dN
 N
dt
12.5
25
%
14
C remaining
Exponential decay
5730
11460
17190
22920
Age yr
28650
34380
Radioactive Decay equations
dN
 N
dt
dN
 dt
N
N
t
dN
 N  0  dt
N0
N 
ln    t
 N0 
N
 e  t
N0
N  N 0 e  t
A  A0 e t
What is λ?
A0 

A0 e t1 2
2
1
 e   t1 2
2
ln
1
  t 1 2
2

ln 2
t
12
Using Radioactive Decay equations
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)
Assumes atmospheric 14C constant
Causes of Non-Constant Atmospheric 14C
1) Changes in production
- Variations in solar activity
solar minimum
weak magnetic shield
maximum 14C production
- Variations in earth magnetic field strength
2) Changes in distribution
- rate of ocean turnover
- global vegetation changes
solar maximum
strong magnetic shield
minimum 14C production
Dendrochronological Evidence
Find 14C date of tree rings of known age
INTCAL04
14C
Calibration Curves
10000
Atmospheric
2000
4000
14
14
C yr BP
6000
8000
C yr BP
4400 4600 4800 5000 5200 5400 5600
Marine
5200
0
5000
0
2000
4000
6000
Cal BP
8000
10000
5600
5400
Cal BP
12000
5800
6000
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
4700
4600
4500
4540±50
4300
4400
4530±50
4200
Radiocarbon years BP
4800
The Intercept Method: Multiple Intercepts
4800
5000
5200
5295
Calibrated years BP
5400
5600
The Intercept Method: Missing Probabilities
Classical Calibration
 Unknown calendar date
m() is the true radiocarbon age, but cannot be measured
precisely
Radiocarbon date y is a realisation of Y = m() + noise
Noise is assumed to have a Normal distribution with mean 0,
and standard deviation s.
Thus Y~N(m(), s2).
Classical Calibration
Normal Distribution
The probability distribution p(Y) of the
14C ages Y around the 14C date y with a
total standard deviation s is:
Total standard deviation s is, where ss
and sc are the standard deviations of the
14C date and calibration curve
respectively:
The calibration curve can be defined as:
Replacing Y with m(), p(Y) is:
1
p(Y ) 
s 2
s s
 ( Y  y )2 


2 
 2s 
e
s C
2
S
2
Y = m()
1
p(Y ) 
s 2
 ( m ( )  y )2 


2

2s


e
To obtain P(), m() is determined for each calendar year and the correspondi
probability is transferred to the  axis.
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
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.
(1702-1761)
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
P( parameters | data)  P(data | parameters)  P( parameters)
The Likelihood
 Unknown calendar date
m() is the true radiocarbon age, but cannot be measured precisely
Radiocarbon date y is a realisation of Y = m() + noise
Noise is assumed to have a Normal distribution with mean 0, and
standard deviation s. Thus Y~N(m(), s2).
With the calibration curve, we have an estimate of m(), and can
formalise the relationship between  and y±s
 ( m ( )  y )2 


2

2s


p( | y)  e
This is the likelihood.
The Prior
For a single date with weak (or no) a priori information we can use an noninformative prior
e.g. for a date  known to be post-glacial
Pprior()=
constant for -50<<14000
0 otherwise
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)=
constant for a<1<2<3<b
0 otherwise
The Bayesian paradigm offers the greatest advantage over classical
methods when there is a strong prior and overlapping data.
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
Markov Chain Continued
1.
2.
3.
Start with an initial guess
Select the next sample
Repeat step 2 until convergence is reached
Gibbs sampler - one of the simplest MCMC methods
Convergence
theta[4] chains 1:2
theta[4]
300.0
300.0
275.0
275.0
250.0
250.0
225.0
225.0
200.0
200.0
175.0
175.0
1
1
2000
2000
4000
4000
iteration
iteration
theta[1] chains 1:2
theta[1] chains 1:2
150.0
150.0
100.0
100.0
50.0
0.0
50.0
0.0
1
1
2000
2000
4000
4000
iteration
iteration
Easier to diagnose that it hasn’t converged, than prove that it has.
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
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(m(), s2)
An outlier is a determination that needs a shift dj
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
1.0
1.5
2.0
Depth (cm)
0.5
0.0
0
100
Age (cal yr BP)
200
300
400
500
600
Wiggle Matching
In material with annual increments (tree-rings & varves)
Time between two dates precisely known
20 years
1
2
This additional information can be used in the prior
5200
5000
4800
4600
Radiocarbon years BP
5400
Wiggle Matching 2
5600
5800
6000
6200
6400
Calibrated years BP
Buck et al. (1996) Bayesian approach to interpreting archaeological data.
Wiley: Chichester. p232-238
Wiggle Matching in Unlaminated Sediments
1
Dx12
2
Dx23
3
If the sedimentation rate is assumed to be
constant:
(1-2)/(2-3) = Dx12/Dx23
This information can be used in the prior
500
200
300
400
Christen et al. (1995) Radiocarbon 37
431-442
0
100
Radiocarbon years BP
400
300
200
100
0
Radiocarbon years BP
500
Wiggle Matching in Unlaminated Sediments
0
100
200
300
400
500
Calibrated years BP
0.5
1.0
Depth m
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
1.5
2.0
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
Software
Oxcal
• 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
10000
Linear Interpolation
6000
4000
2000
0
Cal BP
8000
Lake Tilo
0
500
1000
1500
2000
Depth cm
What assumptions does this make?
Linear Interpolation – Join the Dots
Which dots?
100
200
0
100
200
400
500
600
400
500
600
-5
0
Age (cal yr BP)
300
Depth (cm)
0
285 BP
300
5
Age (cal yr BP)
10000
Linear regression models
Lake Tilo
Also weightedleast squares
6000
1
2
3
2000
4000
Assess by c2
0
Cal BP
8000
Polynomial order
0
500
1000
1500
2000
Depth cm
What assumptions does this make?
2000
6000
Holzmaar varve sequence
0
Age yr BP
10000
Is Sedimentation a Polynomial Function?
2
4
Depth.m.
6
8
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