Bayesian palaeoclimate reconstruction

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Bayesian palaeoclimate reconstruction
John Haslett Simon Wilson Michael
Salter-Townshend Andrew Parnell Trinity College
Dublin Alan Gelfand Duke University Brian
Huntley, Judy Allen University of Durham
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Outline

• Context Use of proxies to reconstruct ancient
  climates
• Purpose provide climate reconstructions to
  challenge GCM ? global warming etc.
• 20 minute talk ? a flavour of models and issues
  involved
• SW the science, overview of model, reconstruct
  climate at a single location at a single time
• AP temporal smoothness priors and temporal
  uncertainty
• What we wont look at spatio-temporal
  reconstruction, issues surrounding MCMC, other
  sources of uncertainty that we should be
  modelling, other climate variables

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Glendalough now
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and then
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The goal climate reconstruction over last 12k
years at Glendalough, Ireland
Now
Past

• Time scale reversed
• GDD5 is one aspect of multi-dimensional climate

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The science fossil pollen data

• Pollen deposited in lake bed sediment
• Sediment core extracted and horizontal slices
  taken
• Count pollen grains of different taxa in each
  slice
• Count unknown (data are proportions) but 400
• Slices taken at regular depths ( ? equal time
  intervals)
• 14C dating gives Age Before Present at some
  slices
• Here, time is measured as 14C date Years BP

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The science climate affects pollen proportions

• Pollen proportions vs 14C years BP at Glendalough
• 13 pollen taxa at 150 slices
• Solve the inverse problem climate from pollen

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Modern Training Data
Glendalough

• Data exist on modern pollen compositions at 7815
  sites in Europe and N. America
• Climate known at these locations
• Hence we can learn about relationship between
  climate and pollen

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Likelihood construction

• Simple case 2 pollen taxa A and B, 1 climate
  variable
• Model response (intensity) of pollen taxa to
  climate
• Multi-modal climate likelihoods are natural

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Bayesian Hierarchical Model

• Modern data
• each pim is a 13 vector of pollen
  proportions
• each cim is a 2 vector of climate
• Fossil data
• each pif is a 13 vector of pollen
  proportions
• each cif is a 2 vector of climate
• Inference goal compute

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Bayesian Hierarchical Model (cont)

• Counts , where
• are the multinomial taxon probabilities
• Assume count is 400!
• Smooth response surfaces for each taxon
• better climate for taxa j ? large
• For now assume xi independent a priori in fossil
  climates
• Assume that 14C dates calendar dates
• Multinomial OK? Zero-inflation in data (37 zero)

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Model for
Glendalough (an extreme climate?)
Unknown or impossible climates here?

• Discretise climate space to 778 point discrete
  grid CG
• Map 7815 known climates onto this grid
• Hence
• 
  (13 x 778 10114 parameters)

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Gaussian Process on Climate Space

• Kernel k on climate space grid to interpolate
  between grid points
• Gaussian process prior with constraint for the
  qjg (parameterised by a with flat prior)

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2 Stage MCMC

• Assume that fossil pollen on its own has little
  information on response surfaces
• This allows us to independently sample the
  learning stage and the reconstruction stage
• Learning stage sample by MCMC from

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2 Stage MCMC (cont)

• Reconstruction stage
• given a sample , MCMC sample any past
  climate cif from by sampling from
• Climates assumed conditionally independent this
  leads to rough reconstructions

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Independent reconstructions
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summarised
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Consistent reconstructions
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summarised
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Temporal uncertainty

• Some levels of the fossil core are radiocarbon
  dated
• Radiocarbon dating samples introduces temporal
  uncertainty
• We use a method by Blaauw and Christen (2004) to
  produce date distributions at every required depth

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Temporal uncertainty 2
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Climate smoothness

• Modelled via a long-tailed random walk
• Define ?j over a regular grid every 20 years

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Why t8 ? Why every 20 years?
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Concerns

• t8 inconsistencies
• Chronologies need to be approximated
• Age-depth models need improvement

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General framework

• The masterplan reconstruct climate consistently
  over thousands of years across Europe
• Many more taxa, extra climate variables
• Need to define a coherent prior for climate
  change in space and time
• Useful properties
• Long-tailed in time
• Aggregates consistently
• Includes spatial correlation

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Possible next steps

• Mixtures of Normals
• NIG model
• Stable distributions
• Defined only in terms of characteristic function

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Summary

• We can reconstruct climate via response surfaces
  with a coherent prior for climate change
• The creation of response surfaces is a complex
  computational challenge
• The t8 distribution provides a starting point for
  a coherent prior structure