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Palaeoclimate Reconstruction: Modelling Temporal Uncertainty

Palaeoclimate Reconstruction: Modelling Temporal Uncertainty. Many collaborators: Stats : Bhattacharya, Gelfand, Salter-Townshend, Parnell, Whiley, Wilson, others Botany : Allen, Huntley, Mitchell, Support: SFI and previously by Enterprise Ireland and PRTLI. Glendalough Co Wicklow.

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Palaeoclimate Reconstruction: Modelling Temporal Uncertainty

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  1. Palaeoclimate Reconstruction:Modelling Temporal Uncertainty Many collaborators: Stats: Bhattacharya, Gelfand, Salter-Townshend, Parnell, Whiley, Wilson, others Botany: Allen, Huntley, Mitchell, Support: SFI and previously by Enterprise Ireland and PRTLI Glendalough Co Wicklow

  2. Sunday Times “Gobsmacking” says Haslett “Not what I said” says Haslett

  3. Courtesy of Sunday Times graphics dept

  4. Reconstruction of GDD5

  5. samples mult. counts by taxa core

  6. Pollen composition changes  Climate changes Recent Shallow Ancient Deep Changing pollen composition Observed pollen proportions vs 14C y BP Use Matts pollen diagram 10,000 BP

  7. Science Changing pollen composition in carefully selected sites • Reflects changing vegetation, which • Reflects changing climate • GDD5 Growing Deg Days > 5o • MTCO Mean Temp Coldest Monthwhence • Can reconstruct climate quantitatively • Can reduce uncertainty about past climate GDD5 eg Avg temp on four successive days 5 8 4 10 Excess over 5 0 3 0 5 Thus GDD5 = 8

  8. Data • Pollen Data – multivariate counts • 14 distinguishable taxa • 115 samples at Sluggan Moss, Lough Neagh • 115 depths of which 32 radiocarbon dated • Climate unknown (2D GDD5, MTCO) • Modern data • 7815 modern sites • Counts known – surface pollen (depth = 0) • Climate known

  9. Statistical Tasks Aspects • Pollen response to climate • Use modern data • Transfer functions/response surfaces • Climate one sample at a time • Climate smoothness in time • Climate history • Greenland ice cores • Dating uncertainties

  10. Statistical Tasks Given • Modern data (pm,cm) (7815 records); and • Fossil data (pf,?) at one depth seek post dist π(cf| pm,cm ,pf) at that depth Additionally, given • ‘Climate smoothness’; and • Fossil data (pf,?, d) at 115 depths • Radiocarbon dating info at 32 depths seek post dist π(cf| pm,cm ,pf) entire climate history

  11. Modern Training Data Glendalough Sluggan Moss • Data on modern pollen compositions pm 7815 sites in Eur/ N. America • Modern climate cm known. • Hence relationship π(pm | cm) • Adopt for fossil data π(pf | cf)

  12. Glendalough An extreme climate? Impossible climates? Unknown climates? Physical and (2D) climate spaces 7815 data locations 7815 data locations - grey points. Computational grid - black points Climate space grid Mean Temp Coldest Month Growing Deg Day > 5o

  13. Pollen response to (2D) climate • π(p | c) pdf • p 14 dim comp vect • c 2 dim climate (here) • Latent Gaussian proc • mixture of multinomials • zero inflation • MCMC 7815 data locations - grey points. Computational grid - black points Two stage implementation 1 MCMC creates/stores many realisations of p(c) 2(a) Draw one at random 2(b) MCMC Climate recon 2(c) Repeat (a,b) MTCO GDD5 Small change in climate c  Small change in vegetation  p = p(c) smooth multivariate function

  14. Reconstruction of GDD5 One sample at a time Here depth to radiocarbon age presumed known Later address dating uncertainty Note dates in Radio-carbon YBP

  15. Post Dist cf given pf Given vector of counts at given depth, whence pf Find π(GDD5f,MTCOf | pf) by MCMC for each depth Here concentrate on GDD5 eg depth 10 m

  16. Post prob C given A high Differential Response to 1D Climate Prop of A high 0 500 1000 1500 2000 2500 A Post prob C given A low B Prop of A low 0 500 1000 1500 2000 2500 C being 1D Climate 0 500 1000 1500 2000 2500 Inverse relationship • Model taxon productivity response to ‘climate’ • Multi-modal climate posteriors natural • Toy example; two taxa one climate dimension

  17. Implied Climate Histories Pointwise realisations of climate Climate apparently volatile

  18. Implied Climate Histories Pointwise realisations of climate Climate apparently volatile

  19. Implied Climate Histories Pointwise realisations of climate Climate apparently volatile

  20. Implied Climate Histories Pointwise realisations of climate Climate apparently volatile

  21. Implied Climate Histories Pointwise realisations of climate Climate apparently volatile

  22. Climate Smoothness • Climate changes δi = c(timei) - c(timei-1) • Mostly small/sometimes large “smooth” • Depends on increments |timei - timei-1| • Reject (most) volatile climates • Issues • How smooth? • Greenland ice cores • Uncertainty in 14C dating? • Random chronology

  23. Temporal uncertainty • 115 samples at Sluggan Moss For 115 : core depths di For 32 : reported 14C agesyi± σi • Seek θi true calendar age θi all di “chronology model” • r(θ) 14C calibration curve • yi ~ N( r(θi), σi2) (outliers, so long tails) • r(θ)~ N( μ(θ), σ2(θ)) prior • Piecewise constant sedimentation rate • Gaussian random walk

  24. Chronology Given complete knowledge of sedimentation history, age may be determined from depth d = depth of accumulated sediment But Only know 14C age at some depths Seek realisations of sediment history, conditional on data Prior: Gaussian random walk with drift constrained to be monotone Piecewise const iid sedimentation rate Calendar age θ

  25. Temporal uncertainty:single dated sample Schematic of Bayesian 14C calibration curve Buck Lab report 3180 ± 30 Implied post dist

  26. Temporal uncertainty:all dated samples Prior: Discrete time (20 year intervals) Random Walk with drift (monotone) • Draw random θi | yiσieach of 32 di • Order constraint θi > θk if di > dk • Stoch. interpolation to undated samples • Sample θm (undated)| θi (dated), all depths

  27. x x x x Draw set of random dates for 14C dated samples Depth d drift Realisations of order constrained radio-carbon dates drift Calendar age θ

  28. x x x x Complete random chronology Depth d Realisations of order constrained stochastic chronology, conditional on radio-carbon derived dates Monotone random walk with drift Given set of depths Calendar age θ Realisation of a set of calendar dates

  29. Climate Smoothness • Climate changes δi = c(timei) - c(timei-1) • Mostly small/sometimes large • Depends on increments |timei - timei-1| • Prior for smoothness  rejection of histories with large |δi |  implicit smoothing / borrowing strength • Issues • Prior for δi long tail random walk

  30. Climate over 100,000 yearsGreenland Ice Core Greenland Ice Core Data 10,000 year intervals Ice Core data time series Oxygen isotope – proxy for Greenland temp Irish study period Temporal structure for climate (20 yr. resolution) Frequent small changes, occasional large changes

  31. Normal prob plot First diffs Climate over 100,000 yearsGreenland Ice Core

  32. Climate Smoothness Greenland Ice Core Data 10,000 year intervals Ice Core data time series Long-tailed Random Walk Prior Model δ = c(t) - c(t-20) as iid NIG Normal Inverse Gamma Random Walk

  33. Sampling Climate Histories Given • Realisation of pollen response surfaces • Sample pollen at each of 115 depths • Realisation of complete chronology • 115 dates given 14C dates for 32 samples • Model for climate smoothness Sample realisations of climate at 115 dates Sample climate history every 20 years

  34. Modelled Climate Histories Climate Smooth mostly

  35. Modelled Climate Histories Climate Smooth mostly

  36. Modelled Climate Histories Climate Smooth mostly

  37. Modelled Climate Histories Climate Smooth mostly

  38. Modelled Climate Histories Climate Smooth mostly

  39. Better Reconstruction of GDD5

  40. Reconstruction of GDD5 Note dates in Radio-carbon YBP

  41. Modern data Climate and pollen Resp Surface Random Climate History length 115 Random set of surfaces Random point-wise histories Point wise Recon- struction Temporal Stochastic Smoothness Fossil Pollen Stochastic Interpolation Dating Random Climate History 12,600y by 20y step Random set of 115 dates Depths and radiocarbon dates Summaries Monte Carlo Modules

  42. Next Stages • Multiple sites • Joint reconstruction of two sites • Probable synchronicity of climate change • Borrow more strength • for dates, for climate smoothness • Joint reconstruction of many sites in space • More climate dimensions and taxa • Many high dim response surfaces • Other proxies, covariates • Confront General Circ. Models

  43. Methodological Issues • MCMC - the way forward? • Speed and convergence • Approximations esp for response surfaces • Model checking and model choice • Technical issues • Zero inflation • Fast high-dim non-parametric smoothing • Long tailed space-time prior for climate • Latent (mixtures of) Gaussian processes

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